Jean-Michel Hervouet-Hydrodynamics of Free Surface Flows Modelling with the Finite Element Method-Wiley (2007)

H3A>3H=0<82B >5 A44 (DA5024 ;>FB
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
H3A>3H=0<82B >5 A44 (DA5024 ;>FB
*I>?FFCHA QCNB NB? VHCN? ?F?G?HN G?NBI>
40="8274; 4AE>D4C
(&(0 &" &"
c 2007
Copyright John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
West Sussex PO19 8SQ, England
Telephone (+44) 1243 779777
Email (for orders and customer service enquiries): [email protected]
Visit our Home Page on www.wiley.com
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning
or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the
terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London
W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should
be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate,
Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44)
1243 770620.
Designations used by companies to distinguish their products are often claimed as trademarks. All
brand names and product names used in this book are trade names, service marks, trademarks or
registered trademarks of their respective owners. The Publisher is not associated with any product
or vendor mentioned in this book.
This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering
professional services. If professional advice or other expert assistance is required, the services of a
competent professional should be sought.
Other Wiley Editorial Offices
John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA
Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA
Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany
John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia
John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809
John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3, Canada
Wiley also publishes its books in a variety of electronic formats. Some content that appears in
print may not be available in electronic books.
Anniversary Logo Design: Richard J. Pacifico
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 978-0-470-03558-0 (HB)
Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which
at least two trees are planted for each one used for paper production.
>=C4=CB
!8BC >5 J6DA4B
G8
!8BC >5 C01;4B
G888
!8BC >5 ?;0C4B
GE
2:=>F;4364<4=CB
=CA>3D2C8>=
1Q?HNS S?;LM I@ >?P?FIJG?HN ;N "!#
0IG? MGIINB?L J?<<F?M
0;CHN 3?H;HN ?KO;NCIHM
+;PC?L\0NIE?M ?KO;NCIHM
#CHCN? ?F?G?HN N?=BHCKO?M ;H> IJNCGCT;NCIH
@D0C8>=B >5 5A44 BDA5024 7H3A>3H=0<82B
+IN;NCIHM ;H> =IH=?JNM CH A?IG?NLS
#L?? MOL@;=? +;PC?L\0NIE?M ?KO;NCIHM
+IH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM
IOH>;LS =IH>CNCIHM
%S>LIMN;NC= JL?MMOL? ;H> NB? IOMMCH?MK ;JJLIRCG;NCIH
0IOL=? N?LGM ;H> <I>S @IL=?M
+;PC?L\0NIE?M ?KO;NCIHM QCNB MCAG; NL;HM@ILG
1L;=?L ?KO;NCIHM CH >CG?HMCIHM
0;CHN 3?H;HN ?KO;NCIHM
-L?M?HN;NCIH ;H> <LC?@ L?PC?Q
%SJINB?M?M ;JJLIRCG;NCIHM ;H> =;F=OF;NCIH LOF?M
!?JNB ;P?L;ACHA +;PC?L\0NIE?M ?KO;NCIHM
!C_?L?HN @ILGM I@ ?KO;NCIHM
1B? =B;L;=N?LCMNC= =OLP?M
+INCIHM IH BS>L;OFC= DOGJM
0;CHN 3?H;HN ?KO;NCIHM CH *?L=;NIL JLID?=NCIH
0;CHN 3?H;HN ?KO;NCIHM QCNB JILIMCNS
IOMMCH?MK ?KO;NCIHM
0?LL? ?KO;NCIHM
0IOL=? N?LGM ;H> <I>S @IL=?M CH NQI >CG?HMCIHM
GE88
PC
,+1"+10
IOH>;LS =IH>CNCIHM CH !
1L;=?L ?KO;NCIH CH NQI >CG?HMCIHM
*I>?FFCHA I@ NOL<OF?H=? ;H> >CMJ?LMCIH
/?SHIF>M MNL?MM
7?LI ?KO;NCIH GI>?FM
1OL<OF?H=? MNL?MM IH NB? Q;FFM
"KO;NCIHM I@ NB? A\ GI>?F
,NB?L GI>?FM
%A8=28?;4B >5 C74 J=8C4 4;4<4=C <4C7>3
&HNLI>O=NCIH
&HN?LJIF;NCIH CH VHCN? ?F?G?HNM
3;LC;NCIH;F JLCH=CJF?
'4B>;DC8>= >5 C74 (08=C+4=0=C 4@D0C8>=B
AF;H=? ;N NB? ?RCMNCHA G?NBI>M
*;CH JLIJ?LNC?M I@ ; AII> HOG?LC=;F M=B?G?
#CHCN? >C_?L?H=? M=B?G?M
#CHCN? PIFOG? M=B?G?M @IL BSJ?L<IFC= ?KO;NCIHM
(CH?NC= M=B?G?M
#CHCN? ?F?G?HN M=B?G?M
,P?L;FF PC?Q I@ NB? 1?F?G;= ! ;FAILCNBG
#L;=NCIH;F MN?JM G?NBI>
>P?=NCIH MN;A? OMCHA NB? G?NBI> I@ =B;L;=N?LCMNC=M
-LIJ;A;NCIH >C_OMCIH MIOL=? N?LGM
1CG? >CM=L?NCT;NCIH
0J;=? >CM=L?NCT;NCIH
3;LC;NCIH;F @ILGOF;NCIH
+;NOL;F <IOH>;LS =IH>CNCIHM
0IOL=?M ;H> MCHEM
*;NLCR @ILG I@ NB? MSMN?G
/;>C;NCIH =IH>CNCIHM
/?MIFONCIH I@ NB? IOMMCH?MK ?KO;NCIHM
/?MIFONCIH I@ A\ GI>?F ?KO;NCIHM CH !
>P?=NCIH MN?J
-LI>O=NCIH >C_OMCIH MIOL=? N?LGM
0IFPCHA NB? NL;=?L ?KO;NCIH CH !
);QM I@ =IHM?LP;NCIH CH !
*;MM =IHM?LP;NCIH I@ NB? WOC>
IHM?LP;NCIH I@ NB? NL;=?L
%?;> ;H> GIG?HNOG =IHM?LP;NCIH
IHM?LP;NCIH I@ ?H?LAS
1B? NL?;NG?HN I@ OH=IP?L?> <?>M
,JNCIH =ILL?=NCIH I@ NB? @L?? MOL@;=? AL;>C?HN
,JNCIH G;MECHA I@ ?RJIM?> ?F?G?HNM
-M?O>I Q;P? ?KO;NCIH
0IG? P;FC>;NCIH N?MN =;M?M
,+1"+10 PCC
1?MN I@ ; F;E? ;N L?MN
/;JC> WIQ IP?L ; Q?CL QCNB ; BS>L;OFC= DOGJ >IQHMNL?;G
4;P? JLIJ;A;NCIH
/CNN?L ;H> 0NIE?L MIFONCIHM
'4B>;DC8>= >5 C74 #0E84AO(C>:4B 4@D0C8>=B
&HNLI>O=NCIH
"KO;NCIHM QCNB BS>LIMN;NC= BSJINB?MCM
#L;=NCIH;F MN?JM G?NBI>
0J;NC;F >CM=L?NCT;NCIH
OCF>CHA NB? NBL?? >CG?HMCIH;F G?MB
0CAG; NL;HM@ILG ;H> A?H?L;FCT?> MCAG; NL;HM@ILG
>P?=NCIH MN?J
!C_OMCIH MN?J
-L?MMOL?\=IHNCHOCNS MN?J
+IH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM
%S>LIMN;NC= MN?J
IHNCHOCNS MN?J
;F=OF;NCIH I@ @L?? MOL@;=?
-M?O>I Q;P? ?KO;NCIH ; P;LC;HN I@ NB? BS>LIMN;NC= MN?J
4CNB ?RJFC=CN >C_OMCIH
4CNB CGJFC=CN >C_OMCIH
1L;=?L ?KO;NCIH CH !
1B? =IHM?LP;NCIH I@ G;MM CH !
IHM?LP;NCIH I@ G;MM I@ Q;N?L
IHM?LP;NCIH I@ NL;=?L
0IOL=?M ;H> MCHEM CH NB? +;PC?L\0NIE?M ?KO;NCIHM
0IOL=?M I@ WOC>
0IOL=?M I@ NL;=?L QCNB ; >CMNLC<ONCP? M=B?G?
!CLC=BF?N <IOH>;LS =IH>CNCIHM QCNB ; >CMNLC<ONCP? M=B?G?
0IOL=?M I@ NL;=?L QCNB NB? 02-$ ;>P?=NCIH M=B?G?
!CLC=BF?N <IOH>;LS =IH>CNCIHM QCNB ; 02-$ ;>P?=NCIH M=B?G?
1L?;NG?HN I@ OH=IP?L?> <?>M CH !
,JNCIH =ILL?=NCIH I@ @L?? MOL@;=? AL;>C?HNM
,JNCIH G;MECHA I@ NB? OH=IP?L?> ?F?G?HNM
(>;E8=6 CA0=B?>AC 4@D0C8>=B
*?NBI> I@ =B;L;=N?LCMNC=M
1L;D?=NILS =;F=OF;NCIH
&HN?LJIF;NCIH
*?NBI> I@ =B;L;=N?LCMNC=M CH Q?;E @ILGOF;NCIH
0NL?;GFCH? 2JQCH> -?NLIP\$;F?LECH G?NBI>
IHM?LP;NCP? M=B?G? @IL NB? =IHNCHOCNS ?KO;NCIH
!CMNLC<ONCP? M=B?G?M
+ -0& M=B?G?M CH NQI >CG?HMCIHM
*2/! M=B?G? CH NBL?? >CG?HMCIHM
PCCC
,+1"+10
">34A= C427=8@D4B 8= J=8C4 4;4<4=CB
!?M=LCJNCIH I@ VHCN? ?F?G?HNM
1LC;HAF? QCNB FCH?;L CHN?LJIF;NCIH
.O;MC <O<<F? NLC;HAF?
1?NL;B?>LIH
-LCMG
*?MB >?M=LCJNCIH
1QI >CG?HMCIH;F G?MB
1BL?? >CG?HMCIH;F G?MB
*;NLCR MNIL;A?
!C;AIH;F MNIL;A?
IGJL?MM?> 0J;LM? /IQ
"" MNIL;A?
">A? <;M?> MNIL;A?
,J?L;NCIHM IH G;NLC=?M
OCF>CHA NB? G;NLC=?M
MM?G<FCHA ; G;NLCR @IL ?>A? <;M?> MNIL;A?
MM?G<FS I@ ;H ?F?G?HN;LS P?=NIL
-LI>O=N I@ ; HIH MSGG?NLC= "" G;NLCR ;H> ; P?=NIL
-LI>O=N I@ ; MSGG?NLC= "" G;NLCR ;H> ; P?=NIL
-LI>O=N I@ ; NL;HMJIM?> "" G;NLCR ;H> ; P?=NIL
-LI=?MMCHA I@ NB? !CLC=BF?N <IOH>;LS =IH>CNCIHM
-LI>O=NM <?NQ??H ; >C;AIH;F G;NLCR ;H> ; G;NLCR
*;NLCR\P?=NIL JLI>O=N QCNB ?>A? <;M?> MNIL;A?
*;NLCR\P?=NIL @LIHN;F JLI>O=N
)CH?;L MSMN?G L?MIFONCIH
*?NBI>M I@ >?M=?HN ;H> JL?=IH>CNCIHCHA
IHDOA;N? AL;>C?HN G?NBI>
IHDOA;N? L?MC>O;F G?NBI>
IHDOA;N? AL;>C?HN IH NB? HILG;F ?KO;NCIH
*CHCGOG ?LLIL G?NBI>
$*/"0 G?NBI>
-L?=IH>CNCIHCHA
!C;AIH;F JL?=IH>CNCIHCHA
9\K JL?=IH>CNCIHCHA @IL NB? 0;CHN 3?H;HN ?KO;NCIHM
FI=E >C;AIH;F JL?=IH>CNCIHCHA
-L?=IH>CNCIHCHA QCNB (1 @;=NILCT;NCIH
LION JL?=IH>CNCIHCHA
$;OMM\0?C>?F "" JL?=IH>CNCIHCHA
LION ;H> $;OMM\0?C>?F JL?=IH>CNCIHCHA @IL ?>A? <;M?> MNIL;A?
%0A0;;4;8B<
&HNLI>O=NCIH
-;LNCNCIH I@ NB? >IG;CH
/?KO?MN?> >;N;
IGGOHC=;NCIH <?NQ??H JLI=?MMILM
-ICHN NI JICHN =IGGOHC=;NCIH
,+1"+10 CR
IFF?=NCP? =IGGOHC=;NCIH
-;L;FF?FCMG FC<L;LS
>;JN;NCIH I@ NB? ;FAILCNBGM
!;N; MNLO=NOL? MJ?=CV= NI J;L;FF?FCMG
0=;F;L JLI>O=N
*;NLCR\P?=NIL JLI>O=N
0J?=CV= =;M?M
/?MOFNM
FOMN?L I@ QILEMN;NCIHM %-
/6 1"
,LCACH <CN &/&5 G
%0A0<4C4A 4BC8<0C8>=
&HNLI>O=NCIH
-LCH=CJF? I@ NB? ;>DICHN G?NBI>
1L?;NG?HN I@ !CLC=BF?N =IH>CNCIHM
JJFC=;NCIH NI 1?F?G;= ! ;FAILCNBG
#LC=NCIH =I?^=C?HN ?MNCG;NCIH CH 1?F?G;= !
;F=OF;NCIH I@ LCABN B;H> MC>?M I@ ;>DICHN MSMN?GM
!?M=?HN G?NBI>M @IL GCHCGCTCHA NB? =IMN @OH=NCIH
IHDOA;N? AL;>C?HN
3;FC>;NCIH ;H> L?MOFNM
??;820C8>=B
*;FJ;MM?N >;G <L?;E Q;P?
&HNLI>O=NCIH
1B? *;FJ;MM?N >;G ;==C>?HN
P;CF;<F? >;N;
1QI >CG?HMCIH;F HOG?LC=;F GI>?FFCHA
/?MOFNM
-?L@ILG;H=?M IH P;LCIOM G;=BCH?M
*IL? L?=?HN L?MOFNM
IH=FOMCIH
1C>?M CH NB? "HAFCMB B;HH?F ;H> NB? NF;HNC= ,=?;H
"MNCG;NCIH QCNB ;>DICHN G?NBI>
"MNCG;NCIH QCNB MOLLIA;N? GI>?FM H?OL;F H?NQILEM ;H> A?H?NC=
;FAILCNBGM
0IFCN;LS Q;P?
#FIQ I@ NQI WOC>M I@ >C_?L?HN >?HMCNC?M
Appendix A )83464=4A0C8=6 5>A24
,=?;HC= MN;NC= NC>?
-IN?HNC;F >?J?H>CHA IH NB? A?I=?HNLC= T?HCNB >CMN;H=?
-IN?HNC;F >?J?H>CHA IH NB? F;NCNO>? >?=FCH;NCIH ;H> BIOL ;HAF? !?=IGJIMCNCIH I@ BIOL ;HAF? MC>?L?;F NCG?
*IP?G?HN I@ NB? GIIH
*;MM I@ NB? GIIH
!CMN;H=? NI NB? ";LNB FIHACNO>? I@ NB? GIIH CH CNM IL<CN
R
,+1"+10
!?=FCH;NCIH ;H> LCABN ;M=?HMCIH
*IP?G?HN I@ NB? MOH
*;MM I@ NB? MOH
!CMN;H=? NI NB? ";LNB FIHACNO>? I@ NB? MOH
!?=FCH;NCIH ;H> LCABN ;M=?HMCIH I@ NB? MOH
Appendix B 8QDB8>= <0CA8G F8C7 C4CA0743A0
Appendix C #>C0C8>=B
);NCH ;FJB;<?N
$L??E ;FJB;<?N
*;NB?G;NC=;F MCAHM
81;8>6A0?7H
=34G
!8BC >5 86DA4B
=;F=OF;NCIH >IG;CH
MN?;>S BS>L;OFC= DOGJ
JLIJ;A;NCIH I@ ; DOGJ IH ; Q?N <INNIG
FIHACNO>? ;H> F;NCNO>?
VLMN JB;M? I@ NB? *?L=;NIL JLID?=NCIH
>;GJCHA @OH=NCIHM
B;F@ B?CABN I@ ; D?N
>C_?L?HN MN;A?M I@ ?RJ;HMCIH I@ ; D?N
FCH?;L <;MCM CH IH? >CG?HMCIH
?RN?HN I@ ; <;MCM @OH=NCIH IH ; NLC;HAOF;L G?MB
;H CHPCMC<F? @OH=NCIH @IL NB? AL;>C?HN IJ?L;NIL
J;L;MCNC= @L?? MOL@;=? IH ; MBIL?FCH?
>;G <L?;E IH ; Q?N <INNIG 0NIE?LYM MIFONCIH
G?MB <OCFN QCNB ; =IHMN;HN JLIJILNCIH;FCNS =I?^=C?HN J?L JF;H?
G?MB G;>? I@ BILCTIHN;F JF;H?M
>?=IGJIMCNCIH I@ ; JLCMG CHNI NBL?? N?NL;B?>L;
?R;GJF? I@ ; @IL=?> JLI<F?G
NB? =ILL?MJIH>CHA P?LNC=;F P?FI=CNS
; 02-$ N?MN @OH=NCIH
=;M? QCNB NQI >IQHMNL?;G JICHNM
>?VHCNCIH I@ <7<
L?@?L?H=? NLC;HAF?
KO;MC <O<<F? ?F?G?HN
N?NL;B?>LIH ?F?G?HN
JLCMG;NC= ?F?G?HN
L?@?L?H=? JLCMG
AFI<;F HOG<?LCHA ;H> <IOH>;LS JICHN HOG<?LCHA
CMIJ;L;G?NLC= NL;HM@ILG;NCIH @IL ; KO;>LCF;N?L;F
>IG;CH I@ >?VHCNCIH I@ NB? KO;MC <O<<F? @OH=NCIH MB;>?>
GCHCGOG I@ NB? KO;>L;NC= @OH=NCIH;F
RCC )&01 ,# #&$2/"0
0NLC=EF?L =I?^=C?HN @IL ?;=B TIH?
=FOMN?L I@ MIFONCIHM @IL =I?^=C?HN I@ ;L?; =ILL?F;NCIH <?NQ??H NQI @LC=NCIH =I?^=C?HNM
-;L?NI @LIHN CH ; >O;F =LCN?LCIH JLI<F?G
T?HCNB >CMN;H=?
BILCTIHNB;F ;H> P?LNC=;F @IL=?M
>?=FCH;NCIH
BIOL ;HAF?
MJB?LC=;F NLC;HAF?
LCABN ;M=?HMCIH ;H> BIOL ;HAF?
GIP?G?HN I@ NB? *IIH
G?;H FIHACNO>? I@ NB? GIIH L?;F FIHACNO>?
>?=FCH;NCIH ;H> LCABN ;M=?HMCIH I@ NB? GIIH
!8BC >5 )01;4B
=IHMN;HNM I@ NB? A\ GI>?F
=IHM?LP;NCP? ;H> HIH =IHM?LP;NCP? =IHDOA;N? L?F;NCIHM CH ; BS>L;OFC=
DOGJ
=IHDOA;N? L?F;NCIHM <?NQ??H #LIO>? HOG<?LM CH ; BS>L;OFC= DOGJ
P;LC;NCIHM I@ NB? JBSMC=;F KO;HNCNC?M CH ; BS>L;OFC= DOGJ
G?GILS MJ;=? >?J?H>CHA IH MNIL;A? ;H> ?F?G?HN
P;FO?M I@ NB? FCH?;L <;MCM @OH=NCIHM IH NB? MO< NLC;HAF?M
L?MOFN I@ 9\K JL?=IH>CNCIHCHA
J;L;FF?FCMG IH ; =FOMN?L I@ MN;NCIHM %-
J;L;FF?FCMG IH /6 1"
J;L;FF?FCMG IH ,LCACH G?;MOL?G?HN I@ NB? G;RCGOG Q;N?L F?P?F *;R F?P?F IH NB? JBSMC=;F
GI>?F ;H> ! ;H> ! HOG?LC=;F L?MOFNM
CHWO?H=? I@ >C_OMCIH =I?^=C?HN QCNB ; 0NLC=EF?L =I?^=C?HN I@ CHWO?H=? I@ 0NLC=EF?L =I?^=C?HN
CHWO?H=? I@ 0NLC=EF?L =I?^=C?HN QCNB ;H CHCNC;F >CM=B;LA? I@ =O<C=
G?NL?M
G;RCGOG Q;N?L F?P?F ;N NB? G?;MOL?G?HN JICHNM
-2 NCG? CH M=;F;L GI>?
-2 NCG? CH J;L;FF?F GI>?
L?=?HN L?MOFNM IH ; *%T IH? JLI=?MMIL G;=BCH?
G;AHCNO>? I@ Q;P? *
JB;M? I@ Q;P? * CH >?AL??M
@LC=NCIH =I?^=C?HNM CH >C_?L?HN ;L?;M
@LC=NCIH =I?^=C?HNM I<N;CH?> QCNB MOLLIA;N? GI>?FM
G;AHCNO>? I@ Q;P? * I<N;CH?> QCNB MOLLIA;N? GI>?FM
JB;M? I@ Q;P? *
=IGJON;NCIH NCG? CH NB? =;M? I@ ; MIFCN;LS Q;P?
!8BC >5 %;0C4B
JILIMCNS CH ; L?=N;HAOF;L =B;HH?F ?_?=N IH NB? @L?? MOL@;=? ;H> P?FI=CNS
WIQ CH ; =;H;F QCNB <INNIG @LC=NCIH ;H> >CG?HMCIH;F MCGOF;NCIH
<LC>A? JC?L CH ; =;H;F MCGOF;NCIH QCNB JILIMCNS ;H> >L;A @IL=?
?_?=N I@ G;LCH? NOL<CH?M CH QBCN? L?=N;HAF? IH NC>;F =OLL?HNM
)SH=B N?MN =;M? NC>? IH ; KO;>L;NC= MFIJ?
0;CHN 3?H;HN ?KO;NCIHM PIH (;LG;H ?>>C?M <?BCH> <LC>A? JC?LM
0;CHN 3?H;HN ?KO;NCIHM CHCNC;F @L?? MOL@;=? CH NB? @ILG I@ ; $;OMMC;H BCFF
JLIJ;A;NCIH CH ; MKO;L? <;MCH
NL;HM=LCNC=;F WIQ IP?L ; <OGJ CH ; =B;HH?F QCNB ; BS>L;OFC= DOGJ
?<< NC>? IH ; <?;=B
0;CHN 3?H;HN ?KO;NCIHM ;H> PIH (;LG;H ?>>C?M <?BCH> <LC>A? JC?LM
NL;HM=LCNC=;F WIQ IP?L ; <OGJ CH ; =;H;F QCNB ; BS>L;OFC= DOGJ
JLIJ;A;NCIH I@ ; MIFCN;LS Q;P? CH ; =B;HH?F
0;CHN 3?H;HN ?KO;NCIHM CHCNC;F @L?? MOL@;=? CH NB? @ILG I@ ; $;OMMC;H BCFF
JLIJ;A;NCIH CH ; MKO;L? <;MCH
>;G <L?;E CH ; LCP?L 0NIE?LYM MIFONCIH
*;FJ;MM?N MCN? =IHNIOL I@ =IGJON;NCIH;F >IG;CH JIMCNCIH I@ G?;MOL?G?HN
JLI<?M NI ;H> ?F?=NLC= NL;HM@ILG?LM *;FJ;MM?N MCN? <;NBSG?NLS ;H> >?N;CFM I@ NB? G?MB
*;FJ;MM?N >;G <L?;E P?FI=CNS V?F> ;@N?L M
*;FJ;MM?N >;G <L?;E >?JNB ;H> P?FI=CNS ;@N?L GCHON?M
*;FJ;MM?N >;G <L?;E NB? WII> Q;P? L?;=B?M NB? M?; ;@N?L GCHON?M
*;FJ;MM?N >;G <L?;E WII> ;@N?L M
M?GC >COLH;F NC>? CH NB?
B;HH?F NC>;F L;HA? ;H> JB;M?
M?GC >COLH;F NC>? CH NB?
B;HH?F <;NBSG?NLS G?MB ;H> @LC=NCIH =I?^=C?HNM
GCRCHA I@ NQI WOC>M QCNB >C_?L?HN >?HMCNS
;ONIG;NC= ?MNCG;NCIH I@ NB? @LC=NCIH =I?^=C?HN CH ; =;H;F
2:=>F;4364<4=CB
OCF>CHA ; BS>LI CH@ILG;NC=M MSMN?G CM ; FIHA N?LG GOFNC >CM=CJFCH;LS QILE ;H> NB?
@LOCN I@ NB? =IFF;<IL;NCIH I@ CHN?LH;NCIH;F N?;GM ?@IL? OHL;P?FFCHA CNM GSMN?LC?M
& QIOF> FCE? NI J;S GS NLC<ON?M NI ;H> H;G? ;FF GS =IFF?;AO?M <INB #L?H=B ;H>
@IL?CAH QBI B;P? ;==IGJ;HC?> G? @IL GIL? NB;H NQ?HNS S?;LM IH NBCM ;>P?HNOL? =;FF?>
@L?? MOL@;=? BS>LI>SH;GC=M =IFF?;AO?M QBIM? QILE ;H> C>?;M B;P? F?@N ; G;LE QBC=B
IH? >CM=IP?LM <?NQ??H NB? FCH?M I@ NB? JLIAL;G 0SFP;CH OH;S "FCM;<?NB ;LLIM
-;OF ;N?M *C=B?F ?HICN 1BC?LLS IOF?N 3CH=?HN IS?L
fFCH? B?PC?N '?;H
/I<?LN IOLCP;O> "LC= !;PC> MNLC> !?=I?H? 1BC?LLS !?HIN LH;O> !?MCNN?L
-;NLC=C; "L=IFC HNICH? $;L;JIH *OMN;@; $O?MGC;
F;O>? $OCF<;O> /?CHB;L>
%CHE?FG;HH '?;H *;L= ';HCH ';=?E ';HEIQMEC /?<?EE; (IJG;HH -C?LL? );HA
!IGCHCKO? );OL?H=? ;NB?LCH? )? +ILG;HN HA?FI )?IJ;L>C #L;HdICM )?J?CHNL?
*;LCFSH? )O=E f=CF? *;=B?N H>L?;M *;F=B?L?E #Lf>fLC= *;L=IM -BCFCJJ? -f=BIH
"LC= -?FNC?L "GCF? /;T;VH>L;EINI '?;H -BCFCJJ? /?H;O> !C>C?L /IOAf -;NLC=E
0;OP;A?N ;NB?LCH? 3CFF;L?N !;GC?H 3CIF?;O F;CH 4;NLCH %IFA?L 4?CF<??L
& ;G ;FMI CH>?<N?> NI NB? JLII@ L?;>?LM @IL B;PCHA G?NC=OFIOMFS L?PCM?> NB? G;HO
M=LCJN CH J;LNC=OF;L ;H> MIG?NCG?M IH=? ;A;CH fFCH? B?PC?N !IGCHCKO? );OL?H=?
B;LF?M *IOFCH?= "GCF? /;T;VH>L;EINI
;NB?LCH? 3CFF;L?N !;GC?H 3CIF?;O ;H>
BOH 4;HA
& ?RJL?MM GS JLI@IOH> AL;NCNO>? NI -LI@?MMIL (CG >;H +AOS?H I@ NB? 2HCP?LMCNS
I@ ;?H ; FIHA MN;H>CHA @LC?H> QBI Q;M NB? L?;F CHCNC;NIL I@ NBCM QILE ILCACH;FFS ;
>CMM?LN;NCIH @IL NB? ;==L?>CN;NCIH I@ ; NB?MCM ;H> QBI J;NC?HNFS AOC>?> G? NBLIOAB
NB? JLI=?>OL?M ;H> @ILG;FCNC?M ;MMI=C;N?> QCNB CN
& >OFS NB;HE Z" NL;>O=NCIH[ @IL NB? NL;HMF;NCIH I@ NB? #L?H=B G;HOM=LCJN J;L
NC=OF;LFS "FCM;<?NB ?H 0F;G; %?G; -;LNB;M;L;NS /?HOE; $?ILA? GLCN; /;ABOH
;H>;H ;H> 0BI<;H; 3;LG;
& QIOF> ;FMI FCE? NI NB;HE GS QC@? ;H> NBL?? =BCF>L?H QBI JON OJ QCNB GS MNO>CIOM
MCF?H=? @IL MI G;HS GIHNBM ;H> F?@N G? OH>CMNOL<?> NI QLCN? NBCM <IIE Z;M BOA? ;M
%;LLS -INN?L &3[ ;LIFCH? ;A?> Plate 1: porosity in a rectangular channel: effect on free surface and velocity
flow direction
discharge: 100 m3 /s
depth: 5 m
20
10
0
-10
-20
-150
-100
-50
0
50
100
150
area with porosity
(see the porosity profile below)
5.01
5
4.99
4.98
4.97
4.96
4.95
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
free surface (m)
0
50
100
150
200
250
300
100
150
200
250
300
porosity
velocity (m/s)
0
50
Plate 2: flow in a canal with bottom friction:
2- and 3-dimensional simulation with a Manning coefficient of 0.02
100
90
80
70
60
50
40
30
20
10
0
Velocity field and coloured surfaces of u
velocity (m/s)
1
0.98
0.96
0.94
0.92
0.9
0
100
200
100
90
80
70
60
50
40
30
20
10
0
velocity (m/s)
1.000000
0.8
0.75
300
400
500
0.88
0.86
0.84
0.82
0.8
0.78
0.76
0
100
200
300
400
500
mesh
0.7
0.74
0.72
0.7
0.68
0.66
0.65
free surface (m) with 2D simulation
0.6
free surface (m) with 3D simulation
0.55
0.5
0.45
0
100
200
300
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
400
J.-M. Hervouet
500
Plate 3: bridge pier in a canal: simulation with porosity and drag force
free surface (m)
free surface with pier
20
4.012
10
4.011
0
4.010
-10
4.009
4.008
-20
-100
-80
-60
-40
-20
0
20
40
60
80
100
4.007
4.006
free surface with porosity and drag force
20
4.005
10
4.004
0
4.003
-10
4.002
4.001
-20
-100
-80
-60
-40
-20
0
20
40
60
80
100
4.000
4.0125
4.01
head loss
due to the pier
4.0075
profile of free surface
in the longitudinal axis of the channel
4.005
with pier
4.0025
friction alone, without pier
with porosity and drag force
0
50
100
150
200
Plate 4: effect of 32 marine turbines (in white rectangle) on tidal currents:
drag forces modelled in shallow water equations
velocity norm (m/s)
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
14000
13000
12000
11000
10000
9000
velocity vectors
1.000000
8000
7000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
Plate 5: Lynch test case (tide on a quadratic slope):
linearized shallow water equations without advection
150000
125000
100000
75000
50000
150000.0
125000.0
75000.0
50000.0
0.0
25000.0
0
100000.0
25000
free surface (m)
2.40
2.30
150000
150000
2.20
2.10
125000
125000
2.00
1.90
100000
100000
1.80
75000
1.70
1.60
50000
1.50
1.40
25000
0
1.30
1.20
0
25000 50000 75000 100000 125000 150000
quasi-bubble element
1.10
1.00
75000
50000
25000
0
0
25000 50000 75000 100000 125000 150000
linear element
Plate 6: Saint-Venant equations: von Karman eddies behind bridge piers
10
5
0
-5
-10
-10
velocity
1.000000
-5
0
5
10
-2.5
0
-1
-2
-3
-4
-5
-6
-7
-8
-10
-7.5
velocity
1.000000
-5
Plate 7: Saint-Venant equations: propagation of a wave, effect of boundary conditions
after 2 s, one iso-line every 1/6 m
20
20
15
15
n
10
10
5
5
0
0
0
5
10
with reflection
on the domain
boundaries
15
20
0
5
10
with Thompson
radiation
conditions
15
20
2
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.5
1
1.5
0
5
5
flow
free surface with primitive equations
"exact" free surface
free surface with pseudo-wave equation
bottom
imposed
discharge
mesh
10
10
20
exact free surface with "non-conservative" equations
15
20
constant
free surface
vertical cross-section in the axis y=1
15
Plate 8: transcritical flow over a bump in a channel with a hydraulic jump
Plate 9: ebb tide on a beach: free surface at different times
with imposed sinusoidal elevation on the left
0.1
Saint-Venant equations
t=0 s
0
t=100 s
-0.1
-0.2
t=170 s
-0.3
-0.4
-0.5
-0.6
t=300 s
0
0.1
10
20
-0.1
40
hydrostatic Navier-Stokes
t=0s
0
30
t = 100 s
-0.2
t = 170 s
-0.3
-0.4
-0.5
-0.6
t = 300 s
0
10
20
30
40
0.1
non-hydrostatic Navier-Stokes
t=0s
0
-0.1
t = 100 s
-0.2
-0.3
t = 170 s
-0.4
-0.5
-0.6
t = 300 s
0
10
20
30
40
0.1
water at rest
on a slope
(all equations above)
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0
10
20
30
40
-10
-5
0
5
10
-10
-5
0
5
10
-5
0
5
10
-10
-5
0
5
10
CPU time: 11 s
wave equation, linear discretisation
-10
CPU time: 21.7 s
primitive equations, linear discretisation
-0.42
-0.39
-0.36
-0.33
-0.30
-0.27
-0.24
-0.21
-0.18
-0.15
-0.12
-0.09
-0.06
-0.03
0.00
0.03
0.06
0.09
0.12
0.15
0.18
free surface (m)
-10
-5
0
5
10
-10
-5
0
5
10
-5
0
5
10
-10
-5
0
5
10
CPU time: 20.8 s
wave equation, quasi-bubble velocity
-10
CPU time: 54.7 s
primitive equations, quasi-bubble velocity
Plate 10: Saint-Venant equations and von Karman eddies behind bridge piers: effect of discretization on the free surface
Plate 11: transcritical flow over a bump in a canal with a hydraulic jump:
non-hydrostatic Navier-Stokes equations with five planes vertically in the mesh
2
1.5
1
0.5
0
5
10
mesh seen from above
15
20
vertical cross-section in the longitudinal axis y = 1
0.7
0.6
0.5
0.4
0.3
exact free surface with Saint-Venant equations
0.2
bottom (m)
free surface with Navier-Stokes equations
0.1
0
-0.1
-0.2
0
5
10
velocity (m/s)
10
0
15
20
velocity field in a vertical cross-section
5
10
15
20
dynamic pressure in a vertical cross-section
pressure in Pa divided by 100000
0.008
0.004
0.000
-0.004
-0.008
-0.012
0
5
10
15
20
Plate 12: propagation of a solitary wave in a channel
depth 10 m, wave amplitude 2 m
Saint-Venant equations
2.5
t=0 s
2
t=10 s
t=20 s
t=30 s
1.5
t=40 s
1
0.5
0
-0.5
0
100
200
2.5
300
400
500
600
400
500
600
500
600
Boussinesq equations
2
1.5
1
0.5
0
-0.5
0
100
2.5
200
300
non-hydrostatic Navier-Stokes equations
2
1.5
1
0.5
0
-0.5
0
100
200
300
400
Plate 13: Saint-Venant equations: propagation of a circular wave
initial free surface
20
5
one iso-line every 25 cm
4
15
3
10
2
free surface profile (m)
5
1
0
0
5
10
15
20
0
cross-section
at y = 10.05 m
0
5
10
15
20
free surface after 1 s
20
5
4
15
3
10
2
free surface profile (m)
5
1
0
0
5
10
15
20
0
cross-section
at y = 10.05 m
0
5
10
15
20
Plate 14: dam break in a river, Stoker’s solution
6
5
4
3
computed depth at t = 1200 s
exact depth
initial depth
2
1
free surface at t = 0 and t = 1200 s
0
0
5000
10000
15000
20000
velocity computed at t = 1200 s
3
exact velocity
2
1
0
velocity at t = 1200 s
0
5000
10000
15000
20000
Plate 15: Malpasset site: contour of computational domain, position of measurement
probes (6 to 14) and electric transformers (A,B,C)
6000
5000
A
6
4000
7 8
9
3000
position of dam
10
B
12
11
C
13
14
2000
1000
0
-1000
-2000
2500
5000
7500
10000
12500
15000
17500
Plate 16: Malpasset site: bathymetry and details of the mesh
bottom (m)
100
90
80
70
60
dam
50
40
30
20
10
0
-10
-20
Plate 17: Malpasset dam break: velocity field after 100 s
bottom (m)
100
90
80
70
mesh of 26000 elements
60
50
40
velocity (m/s)
10
30
20
10
0
-10
-20
mesh of 104000 elements
(1 vector every 4 is shown)
Plate 18: Malpasset dam break: depth and velocity after 20 minutes
depth (m)
36
34
32
30
28
26
24
22
20
18
16
14
12
10
velocity (m/s)
10
8
6
4
2
0
Plate 19: Malpasset dam break: the flood wave reaches the sea after 36 minutes
depth (m)
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
mesh of 104000 elements
5
4
3
2
1
0
10000
12500
15000
17500
12
14
16
-2000
-1000
0
1000
2000
3000
4000
2500
5000
7500
10000
Navier-Stokes 6 planes
12500
15000
17500
0
2
4
6
8
7500
10
5000
18
5000
2500
Shallow Water Equations
20
6000
-2000
-1000
0
1000
22
3000
3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
-2000
-1000
0
1000
2000
4000
4000
depth (m)
24
5000
5000
2000
6000
6000
5000
7500
10000
2500
5000
7500
10000
Navier-Stokes 10 planes
2500
Navier-Stokes 2 planes
Plate 20: Malpasset dam break: flood after 2200 s
12500
12500
15000
15000
17500
17500
Plate 21: semi-diurnal tide in the Channel: tidal range and phase
tidal range (m)
2.0
800000
1.9
tidal range
1.8
1.7
1.6
600000
1.5
1.4
1.3
400000
1.2
1.1
1.0
200000
0.9
0.8
0.7
0
0.6
0.5
0.4
-200000
0.3
-600000
-400000
-200000
0
200000
400000
0.2
0.1
0.0
phase
800000
phase (degrees)
360.0
330.0
300.0
600000
270.0
240.0
210.0
400000
180.0
150.0
120.0
200000
90.0
60.0
30.0
0
0.0
-200000
-600000
-400000
-200000
0
200000
400000
Plate 22: semi-diurnal tide in the Channel: bathymetry, mesh and friction coefficients
bathymetry (m)
0
800000
bathymetry and mesh
-250
-500
-750
600000
-1000
-1250
-1500
400000
-1750
-2000
200000
-2250
-2500
0
-2750
-3000
-3250
-200000
-3500
-3750
-400000
-4000
-4250
-600000
-4500
-4750
-1.25e+06 -1e+06 -750000 -500000 -250000
0
800000
250000
500000
-5000
5
600000
Chézy coefficient
100.0
400000
1
200000
0
95.3
4
76.5
62.2
54.3
3
47.2
2
-200000
zonation and
associated friction coefficients
-400000
-600000
-1.25e+06 -1e+06 -750000 -500000 -250000
0
250000
500000
Plate 23: mixing of two fluids with different density:
vertical cross-section in the longitudinal axis of the canal
heavy fluid
0
light fluid
-1
salinity
-2
1.00
-3
0.95
-4
0.90
0
5
10
15
20
25
30
initial conditions
0.85
0.80
0.75
0.70
0
0.65
-1
0.60
-2
0.55
0.50
-3
-4
0.45
0
5
10
15
20
25
30
Navier-Stokes equations with hydrostatic assumption
0.40
0.35
0.30
0.25
0.20
0
0.15
-1
0.10
-2
0.05
-3
0.00
-4
0
5
10
15
20
25
non-hydrostatic Navier-Stokes equations
Plate 24: automatic estimation of the friction coefficient in a canal
(data simulated with a Strickler coefficient of 35, initial guess 5)
0.9
0.8
0.7
0.6
0.5
bottom (m)
0.4
free surface (m)
0.3
0.2
0.1
0
-0.1
100
90
80
70
60
50
40
30
20
10
0
0
0
mesh
100
200
300
100
200
400
300
500
400
500
35
friction (Strickler coefficient)
30
25
20
15
10
5
1.25
iterations
0
1
2
3
4
5
6
7
error on depth
1
0.75
0.5
0.25
0
iterations
0
1
2
3
4
5
6
7
30
70?C4A
=CA>3D2C8>=
)F4=CH H40AB >5 34E4;>?<4=C 0C 0CH=? NB? S?;L CH QBC=B NB? 1?F?G;= JLID?=N Q;M F;OH=B?> NB? /?M?;L=B ;H>
!?P?FIJG?HN !?J;LNG?HN I@ "F?=NLC=CNf >? #L;H=? "!# ?G<;LE?> OJIH ; MOMN;CH?>
?_ILN NI >?P?FIJ ; BS>LI CH@ILG;NC=M MSMN?G >?>C=;N?> NI @L?? MOL@;=? WIQM 1B? MO<
D?=N CM I@ =IHMC>?L;<F? MNL;N?AC= CHN?L?MN !;GM NB?LG;F ;H> HO=F?;L JIQ?L MN;NCIHM IH
M?; MBIL?M IL LCP?L <;HEM ;L? ;FF MNLO=NOL?M QBC=B CHN?L;=N QCNB ;H ;KO;NC= ?HPCLIHG?HN
NB;N B;M NI <? JLIN?=N?> ;H> @LIG QBC=B Q? H??> NI JLIN?=N IOLM?FP?M 1B? KO;FCNS
I@ Q;N?L NB? =;F=OF;NCIH I@ NC>?M ;H> MQ?FFM >OLCHA MNILGM >;G <L?;E WII> Q;P?M
WII>M NB? CGJ;=N ;H> MN;<CFCNS I@ >?P?FIJG?HN ;H> M?>CG?HNIFIAS ;L? ;FF =LO=C;F
KO?MNCIHM @IL "!# ;H> @IL ;FF NB?M? >CM=CJFCH?M ; AII> =IGG;H> I@ NB? G?=B;HC=M I@
@L?? MOL@;=? WIQM CM CH>CMJ?HM;<F?
%CMNILC=;FFS JBSMC=;F GI>?FM VLMN ?HMOL?> NBCM =IGG;H> <ON NB?M? B;P? JLIAL?MM
CP?FS <??H L?JF;=?> <S HOG?LC=;F GI>?FFCHA 1I>;S NB? =IGJON?L B;M <?=IG? ;H
OH;PIC>;<F? CHP?MNCA;NCP? NIIF ;H> JBSMC=;F GI>?FM ;FNBIOAB MNCFF CH>CMJ?HM;<F? CH
=?LN;CH E?S >IG;CHM FCE? MJCFFQ;SM ;H> >SE? MN;<CFCNS B;P? N?H>?> NI <?=IG? P;FC>;
NCIH NIIFM CH INB?L MJB?L?M
&H NB? F;MN >?=;>? HOG?LC=;F MCGOF;NCIH B;M MNII> ION ;M ; PCN;F NIIF @IL ?HPCLIH
G?HN;F BS>L;OFC=M NBLIOAB NB? =IH=?LN?> ;=NCIH I@ NQI CGJILN;HN NL?H>M
1?=BHC=;F JLIAL?MM QCNB IH NB? IH? B;H> ?RJIH?HNC;F JLIAL?MM CH NB? MJ??>
;H> G?GILS =;J;=CNS I@ =IGJON?LM ;H> IH NB? INB?L B;H> NB? ;>P?HN I@ H?Q
;FAILCNBGM ;H> HOG?LC=;F G?NBI>M
1B? LCM? CH L?KOCL?G?HNM <LIOABN ;<ION IH NB? IH? B;H> <S ; L?CH@IL=?G?HN
I@ F?ACMF;NCIH MNI=ECHA I@ BS>LI=;L<IHM ?G?LA?H=S JF;HM >?=L??M IH CHN;E?M
;H> >CM=B;LA? =IGJ?HM;NCIH Q;N?L ?N= ;H> IH NB? INB?L B;H> <S ; M?LC?M I@
MI =;FF?> H;NOL;F =;N;MNLIJB?M %IQ?P?L NB?CL ;HNBLIJIA?HC= ILCACH AL??HBIOM?
?_?=N L?ACIH;F >?P?FIJG?HN =;HHIN <? IP?LFIIE?> ;H> L?KOCL?M L? ?R;GCH;NCIH
I@ NB? BSJINB?M?M IH NB? MCTCHA I@ MNLO=NOL?M
1BOM ?P?LS N?=BHIFIAC=;F ;>P;H=? QB?NB?L CN ;LCM?M @LIG ; H?Q G;=BCH? ;L=BCN?=
NOL? IL ; H?Q GI>?FFCHA N?=BHCKO? B;M <??H OM?> CGG?>C;N?FS ;H> ;>P;HN;A?IOMFS NI
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
MIFP? GIL? ;H> GIL? =IGJF?R JLI<F?GM 1BCM MCNO;NCIH L?G;CHM OH=B;HA?> ;H> MJ?
=CV= NB?IL?NC=;F =B;FF?HA?M FCE? NOL<OF?H=? IL NB? =IGJF?R JBSMC=;F =IH>CNCIHM ;LCMCHA
CH M?>CG?HNIFIAS JIM? JLI<F?GM QBC=B F?;P? OM @;L @LIG IOL AI;F #LIG MCGJF? MI@N
Q;L? MIFPCHA ; MCHAF? NSJ? I@ ?KO;NCIH Q? B;P? AL;>O;FFS JLIAL?MM?> NI NB? HINCIH I@
BS>LI CH@ILG;NC=M ;H> M?NM I@ =I>?M >?;FCHA QCNB =IGJON?LCT?> >;N; ;H> =;J;<F? I@
=IGGOHC=;NCHA @IL NB? JLI=?MMCHA I@ =IGJF?R JLI<F?GM
1B? MN;E?M ;L? BCAB 1B? CHP?MNG?HN @IL ; BS>LI CH@ILG;NC=M MSMN?G CM I@ NB? IL>?L
I@ NI GCFFCIH ?OLIM ; S?;L <ON NB? ;>P;HN;A?M NB;N =;H <? I<N;CH?> @LIG CN
;L? GO=B BCAB?L 1BOM HOG?LC=;F MCGOF;NCIH I@ NB? CGJ;=N I@ =CPCF ?HACH??LCHA QILE
>OLCHA ; WII> L?=?HNFS JL?P?HN?> ;H ?RJ?HM? I@ GCFFCIH ?OLIM HINB?L IH? <S
B?FJCHA NI JL?>C=N CH ; =IHPCH=CHA G;HH?L NB? ?PIFONCIH I@ ; NB?LG;F J;N=B B?FJ?>
NI L?>O=? NB? ?RJ?HM? I@ ; G?;MOL?G?HN =IFF?=NCIH =;GJ;CAH %IQ?P?L NBCM =IH=?LHM
IHFS GIH?S G;NN?LM
1I >?P?FIJ CNM BS>LI CH@ILG;NC=M 1?F?G;= MSMN?G NB? +;NCIH;F );<IL;NILS I@ %S
>L;OFC=M ;H> "HPCLIHG?HN I@ "!# L?FC?> IH NB? VHCN? ?F?G?HN G?NBI> CH QBC=B CN
B;> ;=KOCL?> ?RJ?LC?H=? CH JL?MMOL? WIQ CH J;LNC=OF;L QCNB NB? +0 =I>? ,LCACH
;FFS "!# B;> ;N >CMJIM;F ; <;=EALIOH> CH VHCN? >C_?L?H=?M IH =OLPCFCH?;L G?MBCHA
QBC=B F;LA?FS JL?>?N?LGCH?> NB? ?RINC= ;L=BCN?=NOL? I@ NB? VLMN 1?F?G;= ! =I>?
; NB?IL?NC=;F @L;G?QILE ;H> @ILG;FCMG CH VHCN? ?F?G?HNM <ON QCNB L?AOF;L =OLPCFCH
?;L G?MBCHA 1B? CGG?>C;N? ?HNBOMC;MG I@ NB? VLMN OM?LM @IL L?;F HIH MNLO=NOL?>
G?MBCHA Q;M KOC=EFS @IFFIQ?> <S ; =IHVLG;NCIH I@ NB?CL CHN?L?MN ;H> NB? GI>?FFCHA
;JJLI;=B B;M NB?L?@IL? <??H G;CHN;CH?> %IQ?P?L CN MBIOF> <? HIN?> NB;N CH NB? NBL?? G;CH HOG?LC=;F N?=BHCKO?M L?FSCHA IH G?MBCHA H;G?FS VHCN? >C_?L?H=?M
VHCN? ?F?G?HNM ;H> VHCN? PIFOG?M ;L? ;FF MNCFF P?LS QC>?FS OM?> #CHCN? >C_?L?H=?M
>I CH>??> I_?L NB;HEM NI NB?CL MCGJFC=CNS NB? JIMMC<CFCNS I@ >?P?FIJCHA BCAB IL>?L
HOG?LC=;F M=B?G?M ;N ; FIQ =IMN QBCF? VHCN? ?F?G?HNM I_?L ;N NB? M;G? NCG? ; MNLC=N
NB?IL?NC=;F @L;G?QILE ;H> ; W?RC<CFCNS NI >CM=L?NCT? NB? >IG;CH I@ =IGJON;NCIH QBCF?
VHCN? PIFOG?M B?FJ NI MIFP? MCGJFS ;FF NB? ?KO;NCIHM QLCNN?H CH =IHM?LP;NCP? @ILG
";=B I@ NB?M? NBL?? G;CH @;GCFC?M I@ N?=BHCKO?M CM MO<>CPC>?> CHNI ; F;LA? HOG<?L I@
P;LC;HNM ;H> NI>;S CN CM KOCN? >C^=OFN NI >C_?L?HNC;N? <?NQ??H NB? =OLL?HN ;H> NB?
ION>;N?> IH?M 1B? NL?H> ?G?LACHA =OLL?HNFS QIOF> ;JJ?;L NI <? ; AL;>O;F L?D?=NCIH
I@ VHCN? >C_?L?H=?M ;H> ; =IHP?LA?H=? IH VHCN? ?F?G?HNM ;H> VHCN? PIFOG?M CHPIFPCHA
HIH MNLO=NOL?> G?MBCHA +?Q N?=BHCKO?M FCE? >CMNLC<ONCP? M=B?G?M ;L? J?LB;JM JL?
=OLMILM I@ NBCM ?PIFONCIH ,NB?L N?=BHCKO?M EHIQH ;M Z);AL;HAC;H[ FCE? NB? 0-%
0GIINB?> -;LNC=F? %S>LI>SH;GC=M G?NBI> QBC=B B;P? ;FL?;>S JLI>O=?> JLIG
CMCHA L?MOFNM CH =F;MMC=;FFS >C^=OFN >IG;CHM FCE? GCR?> WIQM =IOF> JF;S ;H CGJILN;HN
LIF? CH NB? @ONOL? 4CNBION JL?>C=NCHA NB? @ILG NB;N @ONOL? G?NBI>M N;E? Q? MB;FF
DOMN MN;N? NB;N NB? VHCN? ?F?G?HN G?NBI> CM =OLL?HNFS NB? GIMN QC>?FS OM?> CH #L;H=?
<S L?M?;L=B >?J;LNG?HNM ;H> ILA;HCT;NCIHM CH ?HPCLIHG?HN;F BS>L;OFC=M &H NBCM QILE
QBCF? L?MNLC=NCHA IOLM?FP?M NI BS>LI>SH;GC=M Q? CHN?H> NI JON @ILQ;L> =?LN;CH N?=B
HC=;F ;>P;H=?M ;=BC?P?> QBCF? <OCF>CHA NB? 1?F?G;= MSMN?G 4? MB;FF ;=BC?P? NBCM CH
; =IB?L?HN @L;G?QILE @LIG NB? ?MN;<FCMBG?HN I@ NB? <;MC= ?KO;NCIHM NI NB?CL HOG?L
C=;F CGJF?G?HN;NCIH &H B;JN?L Q? ?MN;<FCMB M?P?L;F P;LC;HNM I@ NB? +;PC?L\0NIE?M
;H> NB? 0;CHN 3?H;HN ?KO;NCIHM NB? NL;=?L ?KO;NCIH ;H> ; @?Q IONFCH?M I@ NOL<OF?H=?
?KO;NCIHM @N?L ; MBILN B;JN?L IH NB? VHCN? ?F?G?HN G?NBI> B;JN?L ?F;<IL
;N?M NB? HOG?LC=;F L?MIFONCIH I@ NB? 0;CHN 3?H;HN ?KO;NCIHM FMI JLIJIM?> CH NBCM
=B;JN?L CM ; HOG?LC=;F M=B?G? @IL NB? IOMMCH?MK ?KO;NCIHM
B;JN?L >?;FM QCNB
%-1"/ &+1/,!2 1&,+ NB? L?MIFONCIH I@ NB? +;PC?L\0NIE?M ?KO;NCIHM VLMN QCNB ;H> NB?H QCNBION ; BS>LI
MN;NC= BSJINB?MCM
B;JN?L ACP?M ; AFCGJM? I@ MIG? G?NBI>M ;>;JN?> NI ;==IOHN
@IL NB? BSJ?L<IFC= H;NOL? I@ NB? NL;HMJILN ?KO;NCIHM
B;JN?L ACP?M CH >?N;CF NB?
=IGJON?LCT?> CGJF?G?HN;NCIH I@ NB? VHCN? ?F?G?HN G?NBI> ONCFCTCHA GI>?LH @ILGM
I@ G;NLCR MNIL;A? ?F?G?HN <S ?F?G?HN MNIL;A? ;H> ?>A? <;M?> MNIL;A? 1B? IJNCG
CT;NCIH I@ ;FAILCNBGM CM >?;FN QCNB G;CHFS @LIG NB? JICHN I@ PC?Q I@ =IGJON?LM QCNB
P?=NIL ;L=BCN?=NOL? -;L;FF?FCMG QCNB >CMNLC<ON?> G?GILS =IGJON?LM CM >?;FN QCNB
F;N?L CH B;JN?L &H B;JN?L >?N;CFM ;L? ACP?H I@ NB? N?=BHCKO?M I@ J;L;G?N?L
?MNCG;NCIH CH NB? 0;CHN 3?H;HN ?KO;NCIHM OMCHA ;H ;>DICHN G?NBI> #CH;FFS B;JN?L
JL?M?HNM MIG? ;JJFC=;NCIHM QBC=B CH=FO>? NB? =;F=OF;NCIH I@ NB? >;G <L?;E Q;P? I@
NB? *;FJ;MM?N >;G
(><4 B<>>C74A ?411;4B
#IL ; EHIQF?>A?;<F? L?;>?L @;GCFC;L QCNB NB? =F;MMC=;F N?=BHCKO?M ;H> NB? G;CH HO
G?LC=;F M=B?G?M OM?> CH @L?? MOL@;=? WIQ G?=B;HC=M ;H> QBI MCGJFS Q;HNM GIL?
CH@ILG;NCIH ;<ION NB? H?Q HOG?LC=;F M=B?G?M ;H> NB? IJNCGCT;NCIH N?=BHCKO?M JLI
JIM?> CH NBCM <IIE B?L? CM ; MBILN LfMOGf
(08=C+4=0=C 4@D0C8>=B
1B? ;FAILCNBG F?;>CHA NI ; FCH?;L MSMN?G =IOJFCHA Q;N?L >?JNB ;H> P?FI=CNS >;N?M
@LIG ;H> B;M MCH=? <??H =IHNCHOIOMFS ?GJFIS?> <S BOH>L?>M I@ OM?LM &N B;M
MO==?MM@OFFS L?JF;=?> NB? 2T;Q; ;FAILCNBG OM?> ?;LFC?L &NM MNL?HANB AI?M <?SIH>
NB? NB?IL?NC=;F FCGCNM I@ NB? ?KO;NCIHM CH=FO>CHA CNM ?RN?HMCIH @IL MIFONCIH I@ NB?
IOMMCH?MK ?KO;NCIHM GIHA NB? ?MM?HNC;F JICHNM I@ NBCM ;FAILCNBG Q? G;S =CN? NB?
=BIC=? I@ ; HIH =IHM?LP;NCP? @ILG NB? Q?;EFS @ILGOF;N?> CGJ?LG?;<CFCNS =IH>CNCIHM ;N
MIFC> <IOH>;LC?M ;H> JL?=IH>CNCIHCHA =;FF?> Z=\O[ <;M?> IH ; =?F?LCNS\P?FI=CNS @ILG
I@ NB? 0;CHN 3?H;HN ?KO;NCIHM QB?L? NB? >?JNB CM L?JF;=?> <S NB? =?F?LCNS I@ MB;FFIQ
Q;N?L Q;P?M
1B? MJ?=C;F JLI=?MMCHA I@ ?RJIM?> <?>M ;H> >LS ;L?;M QBC=B >I?M HIN L?FS IH
;HS HINCIH I@ GCHCGOG Q;N?L >?JNB ;H> ;MMOL?M ;H ?R;=N =IHM?LP;NCIH I@ G;MM CM
OH>IO<N?>FS NB? ;>P;H=? QBC=B B;M =IHNLC<ON?> GIMN NI NB? MO==?MM I@ NB? 1?F?G;=
! MI@NQ;L? IH ;==IOHN I@ NB? L;HA? I@ ;JJFC=;NCIH >IG;CHM NB;N CN ;FFIQM ;H> NB?
OM?L M;NCM@;=NCIH CN ;=BC?P?M
#IL NB? L?MIFONCIH I@ ?KO;NCIHM CH *?L=;NIL JLID?=NCIH NB? OM? I@ FI=;F =IIL>CH;N?M
@IL ?;=B ?F?G?HN CM ;FMI ; HIP?F ;JJLI;=B QBC=B ;PIC>M =B;HACHA NB? ?KO;NCIHM QCNBION
ACPCHA LCM? NI ; NIJIFIAC=;F JLI<F?G
0CGOFN;H?IOM OM? I@ JILIMCNS ;H> >L;A @IL=? NI MCGOF;N? NB? JL?M?H=? I@ P?LNC=;F
I<MN;=F?M FCE? <LC>A? JC?LM IL NL??M Q;M JL?M?HN?> @IL NB? VLMN NCG? CH NB? S?;L #0E84AO(C>:4B 4@D0C8>=B
1B? JLIJIM?> BS>LIMN;NC= IJNCIH QCNB ; @L;=NCIH;F MN?J G?NBI> ;H> ; MCAG; NL;HM
@ILG CM L?F;NCP?FS =F;MMC=;F 1B? MIFONCIH JL?M?HN?> @IL NB? =IHNCHOCNS ?KO;NCIH CM H?Q
;H> Q? IQ? CN NI '?;H *;L= ';HCH NB? J?LMIH CH =B;LA? ;H> NLOMNQILNBS AO;L>C;H
@IL M?P?L;F S?;LM I@ NB? 1?F?G;= ! MI@NQ;L? #LIG ; MN;AA?L?> P?LNC=;F P?FI=CNS B?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
>L;QM NB;HEM NI NB? >CMNLC<ONCP? M=B?G?M QBC=B B? ;JJFC?M NI JLCMGM ;H ;MNIHCMBCHAFS
=IHM?LP;NCP? ;H> GIHINIHC= GI>?F @IL NB? ;>P?=NCIH\>C_OMCIH ?KO;NCIH I@ ; NL;=?L
#IL HIH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM Q? ;NNLC<ON? NB? JCIH??LCHA L?M?;L=B
QILE NI ';=?E ';HEIQMEC QBI CH CGJF?G?HN?> NB? VLMN P?LMCIH I@ NBCM IJNCIH
CH 1?F?G;= ! &H NB? OM? I@ NB? 0;CHN 3?H;HN ?KO;NCIHM @IL NB? ZBS>LIMN;NC=[
MN?J CH=L?;M?> NB? LI<OMNH?MM NL?G?H>IOMFS NI NB? ?RN?HN I@ I<N;CHCHA NB? =;F=OF;NCIH
I@ NB? MIFCN;LS Q;P? QCNB IHFS IH? F;S?L I@ JLCMGM CH NB? NBL?? >CG?HMCIH;F G?MBCHA
; =;F=OF;NCIH QBC=B CM OH>IO<N?>FS P?LS H?Q
&H NB? M;G? S?;L NB? ;FAILCNBG @IL JLI=?MMCHA ?RJIM?> <?>M CHCNC;FFS >?P?FIJ?> CH
NQI >CG?HMCIHM Q;M ?RN?H>?> NI NBL?? >CG?HMCIHM 1BCM JLIAL?MM F?> NI NB? =;F=OF;NCIH
I@ ;H ?RJIM?> <?;=B QCNB ; HIH BS>LIMN;NC= IJNCIH ; =IGJON;NCIH CH=FO>CHA <INB >LS
;L?;M ;H> NILL?HNC;F WIQ
&H NB? C>?; I@ NB? Q;P? ?KO;NCIH QBC=B Q;M ;FL?;>S OM?> CH NB? 0;CHN 3?H;HN
?KO;NCIHM Q;M ?RN?H>?> NI NBL?? >CG?HMCIHM
8=8C4 4;4<4=C C427=8@D4B 0=3 >?C8<8I0C8>=
1B? >CM=IP?LS I@ NB? ZHIH ;MM?G<F?> G;NLC=?M[ G?NBI> CH ;N "!# A;P? ;H
CGJ?NOM NI NB? 1?F?G;= JLID?=N N NB? M;G? NCG? NB? ?G?LA?H=? I@ MSG<IFC= =IG
JON;NCIH MI@NQ;L? ?H;<F?> NB? L?D?=NCIH I@ $;OMM KO;>L;NOL? &N Q;M IHFS ; FCNNF? F;N?L
NB;N NB? =IHH?=NCIH Q;M G;>? QCNB NB? G?NBI> EHIQH ;M NB? G?LC=;H "" "F?
G?HN S "F?G?HN QBC=B Q;M IF>?L <S @IOL S?;LM /?;>CHA NB? JO<FC=;NCIHM IH NBCM
N?=BHCKO? NB?H M?LP?> NI OM ZIH ; JF;N?[ NB? LION JL?=IH>CNCIHCHA CH "" MNIL;A?
,OL P?=NILCT;NCIH ;FAILCNBG @IL ;MM?G<FCHA P?=NILM @IOH>?> IH ; MJ?=CV= HOG<?LCHA
I@ ?F?G?HNM Q;M =?LN;CHFS ILCACH;F ;M CM ;FMI NB? JLI=?MMCHA I@ !CLC=BF?N <IOH>;LS =IH
>CNCIHM C ? CGJIM?> P;FO?M L?;FCT?> QCNB ; G;NLCR\P?=NIL JLI>O=N &HN?LJL?NCHA NB?
G;CH L?KOCL?> IJ?L;NCIHM CH VHCN? ?F?G?HNM ;M G;NLCR\P?=NIL JLI>O=NM ? A >C;AIH;F
JL?=IH>CNCIHCHA B;M NB?H AL?;NFS @;=CFCN;N?> NB? J;MM;A? NI J;L;FF?FCMG
'?;H *;L= ';HCH =IH=?CP?> NB? C>?; I@ ;MM?G<FCHA G;NLC=?M ;H> MB;LCHA NB? N?LGM
<?NQ??H NB? ?F?G?HNM <S ;H ?R=B;HA? I@ ?F?G?HN;LS N?LGM NBLIOAB M?AG?HNM 1BCM
Q;S I@ MNILCHA QBC=B ;FL?;>S ?H;<F?> ; @LIHN;F G;NLCR\P?=NIL JLI>O=N Q;M =IGJF?N?FS
MOJJF;HN?> <S ; P?LS MCGCF;L ?>A? <;M?> MNIL;A? NBCM C>?; <?CHA JC=E?> OJ >OLCHA ;
=IHAL?MM &H NB? ;<M?H=? I@ JL?=CM? L?@?L?H=?M ;FF ;FAILCNBGM <OCFN IH ?>A? <;M?>
MNIL;A? ;H> ;<ION QBC=B ;H C>?; CM ACP?H CH 0?=NCIH B;P? <??H L?=L?;N?> ;H>
;>;JN?> NI J;L;FF?FCMG
70?C4A
@D0C8>=B >5 5A44 BDA5024
7H3A>3H=0<82B
1B? +;PC?L\0NIE?M ?KO;NCIHM NB? =ILH?LMNIH? I@ ;FF WOC> G?=B;HC=M ;L? ;@N?L ;FF
IHFS ; @ILG ;>;JN?> NI NB? WOC>M I@ NB? @OH>;G?HN;F L?F;NCIH I@ >SH;GC=M "MN;<FCMB
CHA NB?M? ?KO;NCIHM B;M BIQ?P?L <??H ; FIHA JLI=?MM MCH=? ;FGIMN NQI =?HNOLC?M B;P?
J;MM?> <?NQ??H NB? JO<FC=;NCIH I@ +?QNIHYM QILEM CH ;H> 0CL $;<LC?F 0NIE?MY
QLCNCHA I@ NB? @OH>;G?HN;F ?KO;NCIHM I@ BS>LI>SH;GC=M 89 QBC=B BIHIOL BCM H;G?
;H> NB;N I@ NB? #L?H=B ?HACH??L F;O>? )IOCM +;PC?L 89 1B? NB?ILS I@ ?F;MNC=CNS
;H> NB? N?HMIL =;F=OFOM Q?L? ;FIHA QCNB INB?LM NB? JL?L?KOCMCN?M NI ; =ILL?=N QLCN
CHA I@ NB? +;PC?L\0NIE?M ?KO;NCIHM 1B? P;MN FCN?L;NOL? JO<FCMB?> MCH=? MBIQM ;N NB?
M;G? NCG? NB? CGJILN;H=? I@ NB?M? ?KO;NCIHM CH HOG?LIOM M=C?HNCV= ;L?;M ;H> NB?CL
>C^=OFNS >O? CH J;LNC=OF;L NI NB? JB?HIG?HIH I@ NOL<OF?H=? N NB? <?ACHHCHA I@
NB? NBCL> GCFF?HHCOG Q? MNCFF >I HIN EHIQ ?R=?JN CH MIG? MJ?=C;F =;M?M C@ NB?M?
?KO;NCIHM ;>GCN L?AOF;L MIFONCIHM &H J;LNC=OF;L M?? NB? ;LNC=F? <S B;LF?M ) #?_?L
G;H Z"RCMN?H=? ;H> MGIINBH?MM I@ NB? +;PC?L\0NIE?M ?KO;NCIHM[ IH NB? &HN?LH?N MCN?
BNNJQQQ =F;SG;NB ILA 4? ;L? AICHA NI ?MN;<FCMB <?FIQ NB? >C_?L?HN @ILGM I@
NB?M? +;PC?L\0NIE?M ?KO;NCIHM QBC=B QCFF <? >?;FN QCNB CH NB? @IFFIQCHA =B;JN?LM ;H>
Q? MB;FF NB?H ;P?L;A? NB?G IP?L NB? P?LNC=;F NI I<N;CH NB? 0;CHN 3?H;HN ?KO;NCIHM
#>C0C8>=B 0=3 2>=24?CB 8= 64><4CAH
1B? >IG;CH I@ MNO>S CH NB? L?@?L?HNC;F . + N O P +P ;FIHA NB? P?LNC=;F CM
FCGCN?> <?FIQ <S NB? <INNIG >?M=LC<?> <S NB? ?KO;NCIH P 64 N O J IL CH GIMN
MCNO;NCIHM IHFS P 64 N O ;H> ;<IP? <S NB? @L?? MOL@;=? >?M=LC<?> <S NB? ?KO;NCIH
P 6A N O J
&N CM FCGCN?> F;N?L;FFS <S P?LNC=;F FCH?M &NM <IOH>;LS CM 1B?
JLID?=NCIH I@ IH NB? BILCTIHN;F JF;H? + N O NB;N QCFF <? F;N?L IOL NQI >CG?HMCIH;F
>IG;CH CM >?HIN?> B?L? ! M?? #CAOL? 1B? >IG;CH I@ =;F=OF;NCIH CM NBOM
>?VH?> <S
N O P . MO=B NB;N N O ! ;H> 64 N O J P 6A N O J
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
y
z
free surface z=Zs(x,y,t)
Γ
Ω
bottom z=Zf(x,y)
Ω2D
x
#CAOL? =;F=OF;NCIH >IG;CH
$L;PCNS CM >CL?=N?> NIQ;L>M >?=L?;MCHA P
1B? B?CABN I@ NB? Q;N?L =IFOGH IL Q;N?L >?JNB QCFF <? >?HIN?> ;M > ;H> CM ?KO;F
NI 6A 64
1B? <INNIG ?F?P;NCIH CM ACP?H >;N; 1B? Q;N?L >?JNB CM A?H?L;FFS ;H OHEHIQH
KO;HNCNS
1CG? QCFF <? >?HIN?> ;M J
)?N OM HIQ >?VH? NB? HILG;F P?=NILM NI NB? <INNIG ;H> NI NB? @L?? MOL@;=?
1B? @L?? MOL@;=? CM ; OHCP;F?HN @OH=NCIH I@ =IIL>CH;N?M N ;H> O QBC=B ?R=FO>?M
=;F=OF;NCIH I@ Q;P?M IH NB? P?LA? I@ <L?;ECHA ;H> P;LC?M QCNB NCG? &NM ?KO;NCIH CM CH
NB? @ILG I@ P 6A N O J &N =;H ;FMI <? QLCNN?H ;M N O P J QCNB
N O P J P 6A N O J
1B? HILG;F NI NB? @L?? MOL@;=? >CL?=N?> NIQ;L>M CH=L?;MCHA P B?H=? ?RN?LH;F CH
L?F;NCIH NI NB? PIFOG? =IHN;CHCHA Q;N?L CM NB?H NB? P?=NIL HIN HILG;FCT?>
QDA =H7: 1BCM P?=NIL B;M NB? @IFFIQCHA =IGJIH?HNM
QDA
6A
6A
N
O
1B? HILG;F NI NB? <INNIG QIOF> <? ?RJL?MM?> CH NB? M;G? Q;S <ON <S L?JF;=CHA
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 6A <S 64 ;H> QCNB ; GCHOM MCAH NI G;E? CN ?RN?LH;F NI NB? PIFOG? I@ Q;N?L 1B;N CM
QD4
64 64
N O
1B? <INNIG ?F?P;NCIH ;FMI CM ; OHCPI=;F @OH=NCIH I@ =IIL>CH;N?M N ;H> O P?LNC=;F
<INNIG =;H NBOM IHFS <? L?JL?M?HN?> <S ; MN??J MFIJ?
A44 BDA5024 #0E84AO(C>:4B 4@D0C8>=B
#>=7H3A>BC0C82 #0E84AO(C>:4B 4@D0C8>=B
4? MB;FF ?MN;<FCMB VLMN NB? G;MM =IHM?LP;NCIH ?KO;NCIH IL Z=IHNCHOCNS ?KO;NCIH[
QBC=B ?RJL?MM?M NB? =IHM?LP;NCIH I@ WOC> G;MM ;H> NB?H NB? ZGIG?HNOG ?KO;NCIH[
QBC=B CM CH @;=N >?LCP?> @LIG NB? @OH>;G?HN;F L?F;NCIH I@ >SH;GC=M 1B? =IHM?LP;NCIH
I@ ?H?LAS ?KO;NCIH QBC=B QIOF> ?RJL?MM NB? P;LC;NCIHM CH N?GJ?L;NOL? J;LNC=OF;LFS
N;ECHA CHNI ;==IOHN >CMMCJ;NCIH <S @LC=NCIH QCFF <? JON ;MC>? ;H> N?GJ?L;NOL? QCFF
F;N?L ;JJ?;L ;M ; NL;=?L MIF?FS L?MJIHMC<F? @IL NB? ?_?=NM I@ <OIS;H=S
)?N <? NB? >?HMCNS I@ NB? WOC> ;H> 1 NB? P?FI=CNS P?=NIL QBIM? =IGJIH?HNM
;L? 1 2 3
IHM?LP;NCIH I@ NB? WOC> G;MM =IHN;CH?> CH ; >IG;CH CM ?RJL?MM?> ;M
:
:J
.
:
1B? )?C<HCNT NB?IL?G <L?;EM >IQH NBCM N?LG CHNI P;LC;NCIHM I<M?LP?> QCNBCH NB?
>IG;CH ;H> NB? WOR NBLIOAB NB? <IOH>;LC?M
:
:J
.
:
.
:
J
.
Q QD :
1
+
Q QD : CM NB?H L?>O=?> NI ; PIFOG? CHN?AL;F
1B? WOR NBLIOAB NB? <IOH>;LC?M 1
Q : <S NB? $;OMM NB?IL?G ;H> <S ;==?JNCHA NB;N NB? =IHM?LP;NCIH I@ G;MM
:?L 1
MBIOF> <? ?HMOL?> @IL ;HS >IG;CH QB;NMI?P?L Q? I<N;CH ; FI=;F ?RJL?MMCIH I@ NB?
G;MM =IHM?LP;NCIH ?KO;NCIH
+
J
Q :?L 1
&HCNC;FFS NBCM Z=IGJL?MMC<F?[ @ILG QCFF <? L?N;CH?> 4B?H NB? >?HMCNS CM =IHMC>?L?>
;M =IHMN;HN CN QCFF <? MCGJFCV?> CHNI
Q :?L 1
+IQ Q? H??> NI @ILGOF;N? @IL ; WOC> QB;N CM QLCNN?H ;M <Q C 22B) @IL ; LCAC> <I>S
1BCM @ILG QBC=B CM <?MN EHIQH ;M +?QNIHYM M?=IH> F;Q MBIOF> <? GIL? MNLC=NFS
2
Q <?=;OM? @IL ; WOC> NB? >?HMCNS CM CH JLCH=CJF? P;LC;<F? 1B?
QLCNN?H ;M <Q 2B
C1
+
2
2
Q
Q : C ? NB? >?LCP;NCP? QCNB L?MJ?=N
?KOCP;F?HN I@ 2B C1 QCFF <? QLCNN?H 2B
1
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
NI NCG? I@ NB? GIG?HNOG =IHN;CH?> CH NB? >IG;CH M CH NB? =;M? I@ G;MM NBCM
KO;HNCNS CM ;FMI QLCNN?H ;M
.
IL ;M
.
.
Q
1
:
J
Q
1
:
J
Q1
Q QD :
1
Q 1
Q :
1
.
>?MCAH;N?M NB? ?RN?LH;F JLI>O=N <?NQ??H NQI N?HMILM I@ NB? IL>?L ;H> CM NB?
N?HMIL IJ?L;NIL ZH;<F;[
#IL=? <Q CM NB? MOG I@ ?RN?LH;F @IL=?M QBC=B ;L? ;JJFC?> >CL?=NFS CH NB? <I>S ;H>
NB? @IL=?M EHIQH ;M =IHN;=N @IL=?M QBC=B ;L? ;JJFC?> IH CNM MOL@;=?
"RN?LH;F @IL=?M QCFF <? >?HIN?> ;M Q=
"Q <S >CMNCHAOCMBCHA NB? ;==?F?L;NCIH I@
AL;PCNS Q= ;H> NB? INB?L @IL=?M ILCIFCM ;==?F?L;NCIH ?N= QBC=B QCFF <? ?F;<IL;N?>
F;N?L
&@ : CM ; MOL@;=? ?F?G?HN I@ NB? >IG;CH QBIM? ?RN?LH;F HILG;F CM QD NB? =IHN;=N
@IL=? ;JJFC?> NI : CM :"Q QD : QB?L? CM NB? MNL?MM N?HMIL
#IL=?M ;JJFC?> NI IOL >IG;CH ;L? =IHM?KO?HNFS
.
.
Q=
"Q :
QD :
IL ?P?H
.
Q=
"Q :
0I VH;FFS Q? I<N;CH
.
:?L :
Q
1
Q 1
Q :?L Q= "Q
1
J
4? B;P? DOMN ?MN;<FCMB?> NB? +;PC?L\0NIE?M ?KO;NCIHM CH NB? @ILG EHIQH ;M Z=IH
M?LP;NCP?[ QBC=B ;L? =F;MMC=;FFS MOGG?> OJ ;M
IHM?LP;NCIH I@ G;MM
J
Q :?L 1
*IG?HNOG ?KO;NCIH
Q
1
Q 1
Q :?L Q= "Q
1
J
2M? I@ NB? G;MM =IHM?LP;NCIH ?KO;NCIH ;FMI B?FJM OM NI QLCN? NB? MI =;FF?> ZHIH
=IHM?LP;NCP?[ @ILG I@ NB? GIG?HNOG ?KO;NCIH
Q
1
Q 1
Q :?L Q= "Q
1
J
1B? NQI @ILGM ;L? HIN ;<MIFON?FS ?KOCP;F?HN CH NB? =;M? I@ =IGJL?MMC<F? WOC>M ;H>
Q? MB;FF ;FMI HINC=? NBCM >C_?L?H=? QCNB NB? 0;CHN 3?H;HN ?KO;NCIHM CH NB? JL?M?H=?
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 I@ >CM=IHNCHOCNC?M M?? CH J;LNC=OF;L 0?=NCIH )?N OM HIQ =IHMC>?L +?QNIHC;H
WOC>M QB?L? @IL CH=IGJL?MMC<F? WOC>M NB? MNL?MM N?HMIL CM ?RJL?MM?> ;M
F QB?L? F CM NB? JL?MMOL? ?RJL?MM?> CH J;M=;FM CM NB? C>?HNCNS N?HMIL
I@ >SH;GC= PCM=IMCNS ;H>
NB? MNL;CH L;N? N?HMIL
NB? =I?^=C?HN
Q
Q B 1
1
CM ; =IHMN;HN QBC=B >?J?H>M IH NB? WOC> QBC=B =;H ;FMI <? QLCNN?H ;M QB?L?
CM NB? =I?^=C?HN I@ ECH?G;NC= PCM=IMCNS CH G M
3;LC;NCIHM CH >?HMCNS ;L? >O? ?CNB?L NI NB? N?GJ?L;NOL? IL NI NB? JL?M?H=? I@
MO<MN;H=?M CH NB? Q;N?L M;FN IL M?>CG?HNM &H ;HS =;M? Q? MB;FF B?H=?@ILNB =IHMC>?L
NB;N NB?M? P;LC;NCIHM ;L? MO^=C?HNFS GCHIL @IL OMCHA NB? G;MM =IHM?LP;NCIH ?KO;NCIH
CH ;H CH=IGJL?MMC<F? @ILG 4? VH;FFS ?H> OJ QLCNCHA NB? ?KO;NCIHM CH ;LN?MC;H
=IIL>CH;N?M
IHM?LP;NCIH I@ G;MM
1
N
2
O
3
P
IHM?LP;NCIH I@ GIG?HNOG
3
J
1
J
1
1
N
2
1
O
3
1
F
P
N
1
"F
2
J
1
2
N
2
2
O
3
2
F
P
O
2
"G
1
3
N
2
3
O
3
3
F
=
P
P
3
"H
4? OM? NB? =IHM?LP;NCIH I@ G;MM ?KO;NCIH ;H> NB? @;=N NB;N CM =IHMN;HN CH MJ;=?
Q
@IL MCGJFC@SCHA :?L CH 1
)?N OM HIN? NB;N Q? B;P? @IOL OHEHIQHM F 1 2 ;H> 3 ;H> @IOL ?KO;NCIHM M
Q? B;P? HIN S?N @ILG?> ;HS BSJINB?MCM IH JL?MMOL? NB?M? +;PC?L\0NIE?M ?KO;NCIHM
QCFF <? =;FF?> ZHIH BS>LIMN;NC=[
>D=30AH 2>=38C8>=B
1B? >IG;CH MI @;L EHIQH ;M N;E?M IH GIL? =IGJF?R @ILGM @IL @L?? MOL@;=? WIQM &N
CM FCGCN?>
S NB? <INNIG A?H?L;FFS CGJ?LG?;<F?
S NB? MOL@;=? I@ NB? Q;N?L QBC=B CM HIN =LIMM?> <S NB? WOC> <ON QBC=B ?PIFP?M
QCNB NCG?
S <IOH>;LC?M QBC=B =;H <? P?LNC=;F CGJ?LG?;<F? MNLO=NOL?M Q;FF KO;S <ON
;FMI >SE?M LCP?L <;HEM IL <?;=B?M QBC=B QCFF F?N NB? WOC> >IG;CH ?PIFP? GIL?
IL F?MM @L??FS ;H> QCFF JIM? NB? JLI<F?G I@ =IP?LCHA ;H> ?RJIMCHA >LS ;L?;M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&H NB? =;M? I@ HOG?LC=;F GI>?FFCHA <S NB? <IOH>;LC?M QBC=B =IOF> ;FMI <? JOL?FS
CG;ACH;LS <IOH>;LC?M CH NB? IJ?H M?; OJMNL?;G ;H> >IQHMNL?;G FCGCNM I@ ;
LCP?L ;H> HIN =ILL?MJIH> NI ;HS JBSMC=;F JLIJ?LNS 1BCM QCFF ;>> IH? GIL?
>C^=OFNS
1I MN;LN QCNB NB? <IOH>;LS =IH>CNCIHM QCFF <? ?MN;<FCMB?> ;N NB? <INNIG ;H> ;N
NB? @L?? MOL@;=? ;H> MO<M?KO?HNFS ;N NB? Q;FFM ;H> IJ?H <IOH>;LC?M
A44 BDA5024 0=3 1>CC><
&@ IH? @IFFIQM NB? GIP?G?HN I@ ; JICHN IH NB? @L?? MOL@;=? NB? KO;HNCNS M?? 0?=NCIH
N O P J P 6A N O J
L?G;CHM T?LI ;H> NB? >?LCP;NCP? 2
2B CM NBOM T?LI <S NB? );AL;HAC;H BSJINB?MCM QBC=B
JBSMC=;FFS G?;HM NB;N ; J;LNC=F? I@ Q;N?L IH NB? @L?? MOL@;=? L?G;CHM ;NN;=B?> NI
NB? MOL@;=? ;M NCG? AI?M IH &@ NB? P?FI=CNS I@ ; JICHN VR?> IH NB? MOL@;=? B;M NB?
=IGJIH?HNM 1 A 2 A ;H> 3 A NB? ?KO;NCIH 2
CM QLCNN?H QCNB "OF?L P;LC;<F?M
2B 3A 6A
6A
6A
1A
2A
J
N
O
C?
6A
Q A QDA 1
J
N NB? <INNIG IH? I<N;CHM CH NB? M;G? Q;S
34 64
64
64
14
24
J
N
O
IL
64
J
Q 4 QD4 1
QCNB QD4 JICHNCHA >IQHQ;L>M
,0;;B 0=3 >?4= 1>D=30A84B
0NLC=NFS MJ?;ECHA NB? <IOH>;LS =IH>CNCIH @IL P?FI=CNS IH ; MIFC> Q;FF CM ; HI MFCJ
Q =IH>CNCIH 1
%IQ?P?L >O? NI NB? NOL<OF?H=? ;H> NB? ?RCMN?H=? I@ ; <IOH>;LS
F;S?L NB? P?FI=CNS H?;L ; Q;FF KOC=EFS <?=IG?M HIH T?LI ;H> NB? HI MFCJ =IH>CNCIH CM
I@N?H L?JF;=?> <S NQI Q?;E?L =IH>CNCIHM CGJ?LG?;<CFCNS ;H> NB? MJ?=CV=;NCIH I@ ;
N;HA?HNC;F MNL?MM >O? NI NB? @LC=NCIH IH NB? Q;FF
,H ; <IOH>;LS NB? CGJ?LG?;<CFCNS =IH>CNCIH CM NB?H QLCNN?H ;M
Q QD 1
)
<
1B? N;HA?HNC;F MNL?MM CM
&H NB?ILS CN =;H IHFS <? EHIQH C@ NB? WIQ CM
EHIQH ;H> B?H=? MBIOF> HIN <? >;N; &N =IOF> BIQ?P?L <? JLIPC>?> OH>?L =?LN;CH
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 =IH>CNCIHM <S NOL<OF?H=? GI>?FM 4B?H NB? MNL?MM CM T?LI NB?L? CM ; MFCJ =IH>CNCIH
IH NB? Q;FF
,H NB? IJ?H <IOH>;LC?M ;>>CNCIH;F CH@ILG;NCIH CM L?KOCL?> IH NB? JL?MMOL? NB?
Q;N?L >?JNB NB? P?FI=CNS NB? >CM=B;LA? ?N= 1B?M? <IOH>;LC?M QBC=B ;L? JOL?FS
;LNCV=C;F QCFF =;OM? >C^=OFNC?M CH HOG?LC=;F GI>?FFCHA ;M Q? MB;FF M?? F;N?L &H J;L
NC=OF;L =?LN;CH =BIC=?M I@ <IOH>;LS =IH>CNCIHM QCFF HIN H?=?MM;LCFS F?;> NI Q?FF >?VH?>
JLI<F?GM &H @;=N NB;N >?J?H>M IH NB? H;NOL? I@ NB? WIQ ;H> CNM >CL?=NCIH ;M Q? MB;FF
M?? GIL? JL?=CM?FS QCNB NB? NB?ILS I@ =B;L;=N?LCMNC=M @IL NB? 0;CHN 3?H;HN ?KO;NCIHM
0?=NCIH H3A>BC0C82 ?A4BBDA4 0=3 C74 >DBB8=4B@ 0??A>G8<0C8>=
1B? BS>LIMN;NC= JL?MMOL? BSJINB?MCM QBC=B QCFF ;FMI <? G;>? @IL NB? 0;CHN 3?H;HN
?KO;NCIHM =IHMCMNM I@ MCGJFC@SCHA NB? ?KO;NCIH @IL P?LNC=;F P?FI=CNS 3 QBCF? H?AF?=NCHA
>C_OMCIH NB? MIOL=? N?LGM ;H> ;==?F?L;NCIH NI I<N;CH
F
=
P
IL ?P?H <S QLCNCHA ;M =
=
F
= = P
@LIG QBC=B IH? I<N;CHM NB? ?RJL?MMCIH @IL JL?MMOL? ;N ?F?P;NCIH P
F F/B;
= = 6A P
= =
.
.
H
:P
=
1B? JL?MMOL? ;N IH? JICHN IHFS >?J?H>M IH NB? ;NGIMJB?LC= JL?MMOL? IH NB? MOL@;=?
;H> IH NB? Q?CABN I@ NB? =IFOGH I@ Q;N?L ;<IP? CN
!8=40A8I0C8>= O >DBB8=4B@ 0??A>G8<0C8>=
1B? >C_?L?H=? CH >?HMCNS CH L?F;NCIH NI ; L?@?L?H=? P;FO? = CM MOJJIM?>FS MG;FF
&N CM ACP?H <S ;H ?KO;NCIH I@ MN;N? QBC=B =IHH?=NM >?HMCNS I@ NB? WOC> NI NB? =IH
=?HNL;NCIHM I@ NL;=?LM =;FF?> Z;=NCP? NL;=?LM[ NL;HMJILN?> CH NB? G;MM I@ Q;N?L ;H>
QBC=B ;=N IH NB? BS>LI>SH;GC= ?KO;NCIHM 1BCM ?KO;NCIH I@ MN;N? B;M NB? @ILG
< 0 07 0<
=
QB?L? 07 CM NB? ?NB ;=NCP? NL;=?L NL;HMJILN?> 1B? CH=L?;M? CH >?HMCNS CM JLIJILNCIH;F
NI NB? >C_?L?H=?M CH =IH=?HNL;NCIH <S NB? CHN?LG?>C;LS I@ PIFOG?NLC= >CF;N;NCIH =I?@
V=C?HNM 7 , 7 07 07 7
=
7
QB?L? 7 =;H <? JIMCNCP? N?GJ?L;NOL? IL H?A;NCP? M;FCHCNS M?>CG?HN CH MOMJ?HMCIH
07 CM NB? L?@?L?H=? P;FO? I@ NB? NL;=?L ?
12 HYDRODYNAMICS OF FREE SURFACE FLOWS
The pressure gradient term in the first two momentum equations is then developed
to first order:
1
1 ∂p
∆ρ ∂p
≈−
(2.30)
−
1−
ρ ∂x
ρo
ρo ∂x
1 ∂p
∆ρ ∂p
1
−
1−
≈−
(2.31)
ρ ∂y
ρo
ρo ∂y
which gives by using Equation 2.27:
1
1 ∂p
≈−
−
ρ ∂x
ρo
Zs
∆ρ ∂Zs
∂
∆ρ ∆ρ ∂patm
−g 1−
−g
dz
(2.32)
1−
ρo
∂x
ρo
∂x
∂x
ρo
z
1
1 ∂p
≈−
−
ρ ∂y
ρo
Zs
∆ρ ∂Zs
∂
∆ρ ∆ρ ∂patm
−g 1−
−g
dz
(2.33)
1−
ρo
∂y
ρo
∂y
∂y
ρo
z
These two gradients are broken down into a slope effect of the free surface − g ∇(Zs )
and into two source terms due to buoyancy, the first one called barotropic and the second
baroclinic, which can be written along x:
Zs
∆ρ ∂
∆ρ ∂Zs
−g
dz
(2.34)
g
ρo ∂x
∂x
ρo
z
and along y:
∂
∆ρ ∂Zs
−g
g
ρo ∂y
∂y
z
Zs
∆ρ dz
ρo
(2.35)
and to which is added the effect of atmospheric pressure. They are subsequently
integrated in the source terms Fx and Fy .
In view of these hypotheses and conventions, the Navier–Stokes equations with
hydrostatic hypothesis are as follows:
Mass conservation:
∂V
∂W
∂U
+
+
=0
∂x
∂y
∂z
(2.36)
∂U
∂U
∂U
∂U
∂Zs
+U
+V
+W
= −g
+ ν∆(U ) + Fx
∂t
∂x
∂y
∂z
∂x
(2.37)
∂V
∂V
∂V
∂Zs
∂V
+U
+V
+W
= −g
+ ν∆(V ) + Fy
∂t
∂x
∂y
∂z
∂y
(2.38)
Momentum equation:
Pressure:
p = patm + ρo g (Zs − z) + ρo g
z
Zs
∆ρ dz
ρo
(2.39)
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 (>DA24 C4A<B 0=3 1>3H 5>A24B
A82C8>= 0C C74 1>CC><
1B? <INNIG MB?;L MNL?MM ;=NCHA IH NB? WOC> CM IJJIM?> NI NB? P?FI=CNS IHFS BILCTIHN;F
P?FI=CNS QCFF <? =IHMC>?L?> ;H> CM ?KOCP;F?HN NI
1
D
1BCM MNL?MM CM IJJIM?> NI NB? MNL?MM ;JJFC?> <S NB? WOC> NI NB? <INNIG QBC=B CM
CH NB? >CL?=NCIH I@ NB? =OLL?HN M MN;N?> ;<IP? EHIQF?>A? I@ NBCM MNL?MM L?KOCL?M
EHIQF?>A? I@ NB? WIQ 1B? NOL<OF?H=? GI>?FM QCFF JLIPC>? NB? GI>?FFCHA <;M?>
IH EHIQF?>A? I@ NB? =OLL?HN CH NB? PC=CHCNS I@ NB? <INNIG &H IH? >CG?HMCIH CN CM
;=EHIQF?>A?> NB;N
1 !CG?HMCIH;F ;H;FSMCM ;FMI MBIQM NB;N QBC=B CM ; >?VHCNCIH I@ MB?;L P?FI=CNS 1 B;M NB? @ILG
5
2 1
4 1
1B? OHCNM ;L? EA GM ;M 4 CM ; >CG?HMCIHF?MM @LC=NCIH =I?^=C?HN 1BCM A?H?L;F
@ILGOF; @IL MB?;L @IL=?M MOJJIM?M NB;N 1 CM MO^=C?HNFS @;L @LIG NB? Q;FF &N =;H ;FMI
<? QLCNN?H ;M
1
5
2 1
4 1
D
)
1B? MJ?=CV=;NCIH I@ NBCM MNL?MM <
QCFF H;NOL;FFS ;JJ?;L CH NB? P;LC;NCIH;F @IL
GOF;NCIH I@ NB? >C_OMCIH N?LGM CH VHCN? ?F?G?HNM
1B? <INNIG MB?;L MNL?MM =IH>CNCIH QCFF NBOM <? ?HMOL?>
)
"CNB?L <S ; NOL<OF?H=? GI>?F QBC=B QCFF JLIPC>? ; @ILGOF; @IL NB? MNL?MM <
<;M?> IH NB? ?RJL?MMCIH I@ NB? LIOABH?MM ;N NB? <INNIG ;H> WIQ CH NB? PC=CHCNS
I@ NB? <INNIG *IMN I@N?H NB? NOL<OF?H=? GI>?F ACP?M NB? MB?;L P?FI=CNS 2
IL NB? @LC=NCIH =I?^=C?HN 4
,L <S EHIQF?>A? I@ ; @LC=NCIH =I?^=C?HN 4 ;H> CNM ;MMI=C;N?> P?FI=CNS 1 QBC=B
>O? NI NB? F;QM I@ <INNIG MB?;L MNL?MM CH NQI >CG?HMCIHM QCFF <? NB? ;P?L;A?
P?FI=CNS IH NB? P?LNC=;F NB? @LC=NCIH =I?^=C?HNM IH NB? <INNIG QCFF NBOM <? NB?
M;G? CH NQI ;H> NBL?? >CG?HMCIHM 1BCM CM NB? =;M? I@ NB? BfTS 0NLC=EF?L ;H>
*;HHCHA @ILGOF;? ;H> ; @?Q INB?LM ?F;<IL;N?> CH 0?=NCIH BfTS @ILGOF; CM NB?
BfTS =I?^=C?HN
4 =
0NLC=EF?L @ILGOF; / CM NB? 0NLC=EF?L =I?^=C?HN
4 >
=
/
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
*;HHCHA @ILGOF; C CM NB? *;HHCHA =I?^=C?HN ?KO;F NI NB? CHP?LM? I@ NB?
0NLC=EF?L =I?^=C?HN
4 = C
> ,8=3 5>A28=6
4CH> @IL=CHA ;JJ?;LM CH NB? ?KO;NCIHM NBLIOAB NB? NQI >CG?HMCIH;F =IH>CNCIH ;N NB?
MOL@;=?
K
Q M
Q
/7@ 7E7<2 M
D
QCNB /7@ EAG
K CM NB? BILCTIHN;F P?FI=CNS ;N NB? MOL@;=? ;H> M
Q NB? QCH> P?FI=CNS G ;<IP?
NB? Q;N?L 1B? =I?^=C?HN 7E7<2 >CG?HMCIHF?MM CM ACP?H <S #F;NB?L 89
7E7<2 7E7<2 7E7<2 C@ M
Q GM
M
Q C@ GM M
Q GM
C@ M
Q
GM
1B? =I?^=C?HN 7E7<2 BC>?M =IGJF?R JB?HIG?H; ;H> NB? ;<IP? P;FO?M ;L? IHFS
; AOC>?FCH? &H @;=N NB? CHWO?H=? I@ NB? QCH> >?J?H>M IH NB? LIOABH?MM I@ NB? @L??
MOL@;=? QBC=B CM CNM?F@ >?J?H>?HN IH NB? QCH> ;H> NB? >CMN;H=? IP?L QBC=B CN CM ;JJFC?>
=;FF?> Z@?N=B[
>A8>;8B 5>A24 0=3 24=CA85D60; 5>A24
1B?M? NQI @IL=?M ;L? >O? NI NB? LIN;NCIH I@ NB? ";LNB IH CNM IQH ;RCM 4? ?MN;<FCMB
CNM ?RJL?MMCIH <?FIQ
)?N 7 <? ; VR?> L?@?L?HNC;F QCNB CNM ILCACH ;N NB? =?HNL? I@ NB? ";LNB ;H> 8 ;
L?@?L?HNC;F QBC=B NOLHM QCNB NB? ";LNB 1B?S QCFF <? >?HIN?> H;< 7 ;H> H;< 8 CH NB?
@ILGOF;? 1B? ;R?M +N ;H> +O ;L? JF;=?> IH NB? JF;H? I@ NB? ?KO;NIL ;H> NB? ;RCM
+P ;FIHA NB? LIN;NCIH ;RCM 1B? L?F;NCIH <?NQ??H NB? P?FI=CNC?M ?RJL?MM?> CH NB? NQI
L?@?L?HNC;FM CM QLCNN?H ;M
2/20 H/
QBC=B MCAHCV?M
:
H/
:J
@34 /
:
H0
:J
@34 0
H/
1BCM L?F;NCIH CM ; LOF? @IL NCG? >?LCP;NCP?M P;FC> @IL ?P?LS P?=NIL QCNB =IGJIH?HNM
CH L?@?L?HNC;FM 7 ;H> 8 &H @;=N @IL ?P?LS P?=NIL ? I@ NB? MJCHHCHA L?@?L?HNC;F 8 Q?
B;P?
:?
?
:J
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 S >?LCPCHA 2 / QCNB L?MJ?=N NI NCG? Q? A?N
H/
: :J
@34 /
: H0
:J
H/
: :J
@34 /
@34 0
2
@
2B
IL <S OMCHA NB? P;FO? I@
: H0
:J
@34 /
@34 0
:
H0
:J
@34 0
:
H/
:J
@34 /
ACP?H <S NB? VLMN F;Q I@ =IGJIMCNCIH
:
H0
:J
@34 0
H/
NB;N CM <S >?HINCHA NB? ;==?F?L;NCIHM / 2 0
H/
0
QB?L? NB?
ILCIFCM @IL=? CM
20
;H> NB? =?HNLC@OA;F @IL=? CM
H/
,H ; JICHN I@ F;NCNO>? F?N OM HIQ =BIIM? ; L?@?L?HNC;F QCNB +N >CL?=N?> ?;MNQ;L>M
+O HILNBQ;L>M ;H> +P NIQ;L>M NB? P?LNC=;F I@ NB? JF;=? 1B? LIN;NCIH I@ NB? ";LNB
P?=NIL CM CH NBCM L?@?L?HNC;F
F
&
% G & % "$
H
$!
&H NB? GIL? A?H?L;F =;M? QB?L? HILNB G;E?M ;H ;HAF? * QCNB NB? ;RCM +O =IOHN?>
JIMCNCP?FS IH AICHA @LIG +O NI NB? HILNB IH? B;M
F
"$
$! *
"$ * &
% G & % "$
H
$!
;H> NB?
IL ?P?H
ILCIFCM @IL=? CM
G 3 H 2
2 % F 3 H 1 &
F 2 G 1
"$
"$ * 3 $! 2
&
"$
$! * 3 $! 1
2 %
"$
$! * 2 "$
"$ * 1
CM ?KO;F NI NB? P?FI=CNS I@ NB? ";LNBYM LIN;NCIH CH L;>C;HM J?L M?=IH>
CM NB? >OL;NCIH CH M?=IH>M I@ NB? MC>?L?;F >;S
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
S >?VHCNCIH NB? ILCIFCM =I?^=C?HN CM ?KO;F NI $!
1B? CHWO?H=? I@ NB? ILCIFCM @IL=? CH NB? +;PC?L\0NIE?M ?KO;NCIHM CM AIP?LH?> <S
NB? /IMM<S HOG<?L
. L
$!
B
QB?L? L ;H> B ;L? NB? =B;L;=N?LCMNC= P?FI=CNS ;H> F?HANB I@ WIQ 1B? ILCIFCM @IL=?
CM CGJILN;HN IHFS C@ .
NB;N CM @IL F;LA? I=?;HC= IL ;NGIMJB?LC= ;L?;M &N CM
H?AFCAC<F? CH LCP?LM ;H> F;E?M ;H> ?P?H GIL? MI CH Q;MB<;MCHM
1B? P?LNC=;F =IGJIH?HN I@ NB? ILCIFCM @IL=? ;H> NB? N?LGM >O? NI NB? P?LNC=;F
P?FI=CNS ;L? I@N?H CAHIL?> BIQ?P?L NB?CL CGJILN;H=? Q;M BCABFCABN?> CH I=?;HC= WIQM
IH NB? G?MIM=;F? 89
#IL ; A?IJBSMC=;F WIQ NB? =?HNLC@OA;F @IL=? CM HIN ;>>?> NI NB? +;PC?L\0NIE?M
?KO;NCIHM &H @;=N IH? =IHMC>?LM NB;N NB? @L?? MOL@;=? I@ ; WOC> ;N L?MN =ICH=C>?M QCNB
;H ?KOCJIN?HNC;F MOL@;=? I@ ;JJ;L?HN AL;PCNS QBC=B ;FL?;>S CHN?AL;N?M NB? ?_?=N I@
=?HNLC@OA;F @IL=?
40C 4G270=64B
1B? NB?LG;F JIQ?L FC<?L;N?> CHNI NB? ;NGIMJB?L? J?L MOL@;=? OHCN >?HIN?> ;M , CM
MOJJIM?> NI <? JLIJILNCIH;F NI 0 0/7@ QB?L? 0 CM NB? N?GJ?L;NOL? I@ Q;N?L IH NB?
MOL@;=? ;H> 0/7@ NB;N I@ ;CL IH? A?NM , 0 0/7@ QB?L? CM NB? ?R=B;HA?
=I?^=C?HN CH 4G %?;N WOR F?;PCHA NB? FCKOC> >IG;CH CM QLCNN?H
$
(
=H7: 0 D $
(
0
P
QCNB $ 'EA
( CM NB? =I?^=C?HN I@ B?;N >C_OMCIH CH Q;N?L
S ?KO;NCHA NB? NQI @ILGOF;NCIHM IH? >?>O=?M NB? <IOH>;LS =IH>CNCIHM
(
0
0 0/7@
P
>
1B? =I?^=C?HN MBIOF> CH=FO>? JB?HIG?H; FCE? L;>C;NCIH =IHP?=NCIH I@ ;CL CH
=IHN;=N QCNB Q;N?L ;H> F;N?HN B?;N JLI>O=?> <S ?P;JIL;NCIH I@ Q;N?L 0Q??LM 89
?RJL?MM?M NB? =I?^=C?HN CH 4G ;==IL>CHA NI NB? N?GJ?L;NOL? I@ Q;N?L 0
;H> NB? QCH> P?FI=CNS M G?;MOL?> ;N NB? JICHN OH>?L =IHMC>?L;NCIH CH GM
0
8 M 0
0
1B? J;L;G?N?L 8 P;LC?M >?J?H>CHA IH FI=;NCIH &NM ;P?L;A? P;FO? IH NB? MBIL?M I@
NB? B;HH?F ;H> I@ NB? NF;HNC= CM &N CM BCAB?L CH NB? *?>CN?LL;H?;H QB?L?
CN L?;=B?M ;JJLIRCG;N?FS ".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 D>H0=2H C4A<B 0=3 0C<>B?74A82 ?A4BBDA4
M G?HNCIH?> ?;LFC?L NB? ;NGIMJB?LC= JL?MMOL? AL;>C?HN ;H> NB? <OIS;H=S
;LCMCHA @LIG JL?MMOL? AL;>C?HNM B;P? <??H CHN?AL;N?> QCNB NB? MIOL=? N?LGM
;L? L?MJ?=NCP?FS ;FIHA +N ;H> +O
.
.
F/B;
6A
:P
=
=
=
=
N
=
N
N
=
H
.
.
F/B;
6A
:P
=
=
=
=
O
=
O
O
=
H
N?LGM
1B?S
0;FCHCNS / ?RJL?MM?> CH EAG ;H> N?GJ?L;NOL? 0 ;L? MJ?=C;F NL;=?LM QBC=B ?H;<F?
=;F=OF;NCIH I@ P;LC;NCIHM CH >?HMCNS @IL NB? <OIS;H=S N?LGM #CLMN Q? MB;FF ACP? NB?
L?@?L?H=? ?MN;<FCMB?> <S 2+"0 , 89 NB?H MIG? MCGJFCV?> @ILGOF;? NB;N IH? =;H
>?>O=? @LIG CN
@D0C8>=B >5 BC0C4 4BC01;8B743 1H *#($ NB? @ILGOF;? <?FIQ ACP? ; JL?=C
MCIH I@ @IL N?GJ?L;NOL? <?NQ??H ;H> JL?MMOL? <?NQ??H ;H> <;LM ;H> M;FCHCNS <?NQ??H ;H> EAG 0;FCHCNS CM CH @;=N >?VH?> GIL? JL?=CM?FS
<S 2+"0 , <;M?> IH NB? ?F?=NLC= =IH>O=NCPCNS I@ Q;N?L =IGJ;L?> NI NB;N I@ ; MIFO
NCIH I@ JIN;MMCOG =BFILC>? 1B? NL;HMJIMCNCIH NI EAG CM NBOM CH NB?ILS IHFS P;FC>
@IL NBCM M;FN
1B? >?HMCNS I@ JOL? Q;N?L CM VLMN ACP?H ;M ; @OH=NCIH I@ N?GJ?L;NOL?
>C@3 E/B3@ 7
QCNB
7
7
7
7
7
7
7 0
7 0
7 0 7 0 7 0 ,H NB? <;MCM I@ NBCM P;FO? NB? >?HMCNS >?J?H>CHA IH NB? N?GJ?L;NOL? ;H> NB?
M;FCHCNS ;N ;NGIMJB?LC= JL?MMOL? CM ACP?H <S
0 / >C@3 E/B3@
9
8
8
8
8
8
9
8
9 0
8 0
8 0
9 0 / 8 0 : /
8 0 /
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
9 9 : 4=B8CH 5D=2C8>= >5 C4<?4A0CDA4 0=3 B0;8=8CH NB?M? P;LC;NCIHM ;L? ACP?H <S NB?
@IFFIQCHA @ILGOF;
@34 0 0@34
/ 0 ;H> AF / AF
0@34 CM NB? L?@?L?H=? N?GJ?L;NOL? I@ ;H> @34 NB? L?@?L?H=? >?HMCNS ;N NBCM
N?GJ?L;NOL? QB?H NB? M;FCHCNS CM T?LI NB;N CM @34 EAG
4=B8CH 5D=2C8>= >5 B0;8=8CH <S IP?LFIIECHA P;LC;NCIHM CH N?GJ?L;NOL? =BIIMCHA
NB? ;P?L;A? N?GJ?L;NOL? 0 <?=IG?M H?=?MM;LS @IL =;F=OF;NCIHM 1B? P;FO? I@ CH NB?
<;LINLIJC= ;H> <;LI=FCHC= JL?MMOL? AL;>C?HNM CM NB?H
@IL
@34 0 0@34 1B? N?GJ?L;NOL? 0 QBC=B MBIOF> <? ACP?H QCFF CH NBCM =;M? <? =IHMC>?L?> ;M =IH
MN;HN
,H? MCGJFS B;M
/
&H VHCN? ?F?G?HNM QCFF NB?H B;P? @ILG;FFS NB? M;G? L;HA? I@ ;JJLIRCG;NCIH ;M /
!8=40A 5>A<D;0 ; FCH?;L @ILGOF; CM MIG?NCG?M H?=?MM;LS @IL HOG?LC=;F L?;MIHM CN
=;H NB?H <? QLCNN?H ;M
' / /
( 0 0 QB?L? / ;H> 0 ;L? NB? L?@?L?H=? P;FO?M ;H> ' ;H> ( ;L? =;FF?> NB? =I?^=C?HN
I@ M;FCH? =IHNL;=NCIH ;H> =I?^=C?HN I@ B?;N ?RJ;HMCIH L?MJ?=NCP?FS
#0E84AO(C>:4B 4@D0C8>=B F8C7 B86<0 CA0=B5>A<
(86<0 CA0=B5>A<
,H? I@ NB? >C^=OFNC?M QCNB NB? @L?? MOL@;=? +;PC?L\0NIE?M ?KO;NCIHM FC?M CH NB? @;=N
NB;N NB? >IG;CH I@ =;F=OF;NCIH P;LC?M QCNB NCG? =F;MMC=;F ?R;GJF? NI L?G?>S
NBCM JLI<F?G CM NI =;LLS ION ; =B;HA? I@ P?LNC=;F =IIL>CH;N?M MO=B NB;N NB? <INNIG
?F?P;NCIH <?=IG?M ;H> NB? @L?? MOL@;=? ?F?P;NCIH <?=IG?M #IL NBCM JOLJIM? NB?
@IFFIQCHA =B;HA? I@ P;LC;<F?M CM OM?>
P P 64
P 64
6A 64
>
1BCM CM ;H ;^H? NL;HM@ILG;NCIH ACPCHA ; NL;HM@ILG?> >IG;CH 1B? ?F?P;NCIHM
;L? MBC@N?> ;H> GOFNCJFC?> <S ; =I?^=C?HN > #IL ; @OH=NCIH < NB? PIFOG? CHN?AL;FM
;L? NBOM GI>CV?> CH NB? @IFFIQCHA G;HH?L
.
.
.
P
< : <
:
> < :
P
F G B
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 H
#IL GIL? A?H?L;F NL;HM@ILG;NCIHM NB? ';=I<C;H H
#LIG
G;S HIN <? NB? >?JNB
H
HIQ IH Q? MB;FF >?HIN? CN H P M ; G;NN?L I@ @;=N CN QCFF ;JJ?;L CH 0?=NCIH QB?H Q? >?;F QCNB NB? A?H?L;FCT?> MCAG; NL;HM@ILG IH ! G?MB?M NB;N NB? ';=I<C;H
CM NB? FI=;F B?CABN I@ JLCMGM NB? >?JNB > <?CHA NB?H NB? MOG I@ NB?M? B?CABNM P?LNC=;FFS
1B? INB?L =IIL>CH;N?M ;H> NCG? L?G;CH OH=B;HA?> &H INB?L QIL>M J J N N
;H> O O
;L? MBIOF> <? N;E?H HIN NI @IIFCMBFS L?JF;=? N ;H> O ?P?LSQB?L? <S
N ;H> O <?=;OM? ; J;LNC;F >?LCP;NCP? QCNB L?MJ?=N NI N CGJFC?M NB;N IH? >?LCP?M QCNB
4
4
=IHMN;HN O P ;H> J F
;H> F >I HIN B;P? NB? M;G? MCAHCV=;H=? ;H> Q? MB;FF
OM? NB? @IFFIQCHA <;MC= L?F;NCIHM QB?L? NB? CH>C=?M MBIQ QB;N L?G;CHM =IHMN;HN
<
N
<
O
GHB
FHB
<
P
<
J
<
O
GH B
<
P FH B
<
P <
P FGB
FGH
<
N
<
J
FGB
P N
GHB
FGB
P O
FHB
P P
FGB
<
P FGH FGB
FGB
P J
FGH
S N;ECHA < P ;H> OMCHA IOL HIN;NCIH @IL NB? ';=I<C;H NBCM CGG?>C;N?FS ACP?M
JLIJ?LNC?M NB;N QCFF <? OM?> CH NB? H?RN J;L;AL;JB
P N
P O
P J
GHB
P
P
N
FHB
P
P
O
FGH
P
P
J
GH B
FH B
FGH *B45D; 5>A<D;04 F8C7 C74 B86<0 CA0=B5>A<
1BCM H?Q MSMN?G I@ =IIL>CH;N?M CM AICHA NI GI>C@S =?LN;CH N?LGM CH IOL ?KO;NCIHM MI
Q? CHNLI>O=? B?L?;@N?L ; @?Q >?VHCNCIHM ;H> JL?M?HN MIG? ?R;GJF?M NB;N QCFF <? OM?@OF
F;N?L
+4;>28CH >5 C74 <4B7
NB? P?FI=CNS I@ JICHNM FCHE?> NI NB? G?MB CM
3;3A6 P
J
F G H 3 ;3A6 QCFF <? NB? P?=NIL QCNB =IGJIH?HNM 3;3A6 4? MB;FF ;FMI ?H=IOHN?L CH
NB? >?LCP;NCIHM ; KO;HNCNS QBC=B =;H <? CHN?LJL?N?> ;M NB? P?FI=CNS I@ NB? NL;HM@ILG?>
G?MB
3;3A6
P J
FGH
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
"KO;NCIH CM NBOM ;FMI
P3;3A6
3;3A6
;H> NB? @;=N NB;N Q? B;P?
P
P J
P P
J
F?;>M NI
P
P
3;3A6
:?L 3 ;3A6 J
P J
IL
P
J
&H NB? M;G? Q;S Q? VH>
:?L P 3 ;3A6 3 ;3A6
:?L
P
H
J
3E42C8>= C4A<B NB? ;>P?=NCIH N?LGM GOMN <? =;L?@OFFS ?R;GCH?> CH ; GIPCHA
ALC> <?=;OM? NB? ;>P?=N?> @OH=NCIHM P;LS HIN IHFS OH>?L NB? ?_?=N I@ NB? WIQ <ON
;FMI <?=;OM? NB?S ;L? >?VH?> IH GIPCHA JICHNM 1B? >?LCP;NCP? 24
2B QLCNN?H ;M
:<
:J
QB?L? 1 2F
2B 2 @ILGOF; :<
:J
<
J
2G
2B
1
FGH
<
J
1
FGH
<
N
2
;H> 3 2H
2B
<
N
P <
N P 2
<
O <
O 1
P N
1
<
O
3
<
P
;L? NB? P?FI=CNS =IGJIH?HNM HIQ <?=IG?M OMCHA
2
P <
O P 3
< P P P
IL
:<
:J
<
J
1
FGH
<
N
2
2
P O 3
P P
<
P ,H NB? INB?L B;H> NB? P?FI=CNC?M CH NB? NL;HM@ILG?> G?MB ;L? >?VH?> ;M
1 :N
:N
1
:J
:J
2 :O :O
2
:J
:J
:P :J
3 ".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 1BCM P?LNC=;F P?FI=CNS CH NB? NL;HM@ILG?> G?MB CM ;FMI ?KO;F NI
P J
3 1
FGH
P N
2
P O
3
P P
4? NBOM B;P?
:<
:J
<
J
1
FGH
<
N
2
<
O 3 3;3A6
<
P QB?L? NB? P?LNC=;F P?FI=CNS CM L?F;NCP? NI NB? G?MB %IQ?P?L QB?H NB? G?MB CM GIPCHA
< < ;H> < < =ILL?MJIH> NI JICHNM B;PCHA NB? M;G? N O ;H> P <ON HIN NB? M;G? P
;H> Q?
1B? J;LNC;F >?LCP;NCP?M CH NCG? CH IOL ?KO;NCIHM ;L? NBOM CH @;=N 4
B
F G H
B;P?
<
J
<
J
FGH
P J
<
P F G H <
J
FGH
F G H < 3
P ;3A6
"KO;NCIH CGG?>C;N?FS SC?F>M
:<
:J
<
J
1
F G H <
N
2
<
O 3
<
P >P?=NCIH ?KO;NCIHM CH NB? NL;HM@ILG?> G?MB ;H> N;ECHA CHNI ;==IOHN NB? GIP?
G?HN I@ NB? L?;F G?MB QCFF NB?H B;P? NB? MCGJF? @ILG
<
J
1
F G H <
N
2
<
O 3
<
P IGJ;LCHA "KO;NCIHM ;H> ;FMI MBIQM NB;N
1
<
N
2
<
O
3
<
<
1 P
N
2
<
O 3 3;3A6
<
P <
P QBCF? "KO;NCIH ;FMI <?=IG?M
1
<
N
2
<
O
3 3;3A6
<
<
1 P
N
2
<
O 3
8E4A64=24 Q? HIQ Q;HN NI QLCN? :?L 1 CH NB? NL;HM@ILG?> G?MB 2MCHA @ILGOF;
Q? B;P?
:?L 1 2
O
1
N
1
P GHB
FH B
3
P 2
P FGB
P N
FGB
GHB
FGB
P P
P
N
FHB
FGB
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
H
S GOFNCJFSCHA NB? JL?PCIOM ?KO;NCIH <S H
P QBC=B >?J?H>M IHFS IH N ;H>
O <ON HIN IH P IL P ;H> OMCHA NB? @;=N NB;N
1
N
P
P1
N
GH B
GH B
1
P
N
Q? NB?H A?N
P :?L 1 P1
N
P2
O
GH B
FH B
2
1
P
N
P
O
P1
P P2
P P N
FGB
GBH
P
O
FGB
FBH
P3
P FGB
P
P
FGB
1BCM ?KO;NCIH CM MCGJFCV?> NB;HEM NI ; =;F=OF;NCIH I@ NB? N?LG
H+ H FGB
;H>
OMCHA "KO;NCIH Q? VH>
P3 P P
P FGB
P1
P P
J
P FGB
P
P N
P1
P J
P1
P2
P P P
N
P N
FGB
P2
P2
P O
P3
P O
P3
P P P
O
FGB
P3
P P
P P
P P P
"KO;NCIHM NI ACP?
P P J
P P N
P P O
P
P J
FGH
P
P N
GHB
P
P O
FHB
4? HIQ ;MMOG? NB;N NB? ';=I<C;H P CM =IHMN;HN P?LNC=;FFS QBC=B QCFF <? IOL
=;M? IH ?P?LS VHCN? ?F?G?HN ;H> Q? NBOM B;P?
P P P
FGB
FF NBCM F?;>M NI
P3 P FGB
P1
P FGB
P N
P2
P FGB
P O
P3
P FGB
P P
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 P
P
P
1
2
J
N
O
4? >?>O=? @LIG NBCM NB;N
P3
P FGB
P P
P3 P P1
P FGB
FGB
P
N
P
J
P2
P 1
FGB
P
N
2
P
O
P O
S =;LLSCHA IP?L NBCM P;FO? NI "KO;NCIH Q? A?N
P
P :?L 1 J
P1
N
P2
O
GH B
FH B
P3 P FGB
,NB?L N?LGM CH NB? +;PC?L\0NIE?M ?KO;NCIHM MO=B ;M >C_OMCIH N?LGM ;L? GI>CV?>
CH ;H ?P?H GIL? =OG<?LMIG? G;HH?L 1B? HOG?LC=;F L?MIFONCIH I@ ?KO;NCIHM QCNB
NB? B?FJ I@ NBCM MCAG; NL;HM@ILG CM QC>?FS OM?> <ON MIG?NCG?M F?;>M NI ;QEQ;L>
HOG?LC=;F ;LN?@;=NM =L??JCHA >C_OMCIH ?N= 4? MB;FF NLS NI ;PIC> NBCM QB?L?P?L
JIMMC<F? CH J;LNC=OF;L >OLCHA NB? >C_OMCIH MN;A?
)A024A 4@D0C8>=B 8=
38<4=B8>=B
1B? NL;=?L L?JL?M?HNM ; N?GJ?L;NOL? IL ; JBSMC=;F KO;HNCNS =IHN;CH?> CH Q;N?L =IF
IOLCHA ;A?HN M?>CG?HN &@ NB? NL;=?L CHN?L;=NM QCNB NB? BS>LI>SH;GC=M I@ NB? WIQ
CN CM =;FF?> ;=NCP? C@ HIN CN CM J;MMCP? =NCP? NL;=?LM ;L? GIMNFS N?GJ?L;NOL? ;H>
M;FCHCNS MIG?NCG?M M?>CG?HN -;MMCP? NL;=?LM =;H M?LP? @IL Q;N?L KO;FCNS MNO>C?M
-BSMC=;FFS NB? ?PIFONCIH CH NCG? I@ NB? NL;=?L >?J?H>M
,H NB? =OLL?HN QBC=B =;LLC?M CN ;>P?=NCIH
,H >C_OMCIH QBC=B CM ;FGIMN ;FQ;SM NOL<OF?HN
,H MIOL=? IL MCHE N?LGM
&N CM OHH?=?MM;LS NI MJ?=C@S B?L? CH QBC=B OHCN NB? NL;=?L CM ?RJL?MM?>
)A0=B?>ACO38QDB8>= 4@D0C8>=
&H NBL?? >CG?HMCIHM NB? NL;=?L ?KO;NCIH CH ; =IHM?LP;NCP? @ILG CM ;M @IFFIQM
Q QG "A=C@13
0
:?L 0 1
J
QB?L? "A=C@13 CM NB? L;N? I@ =L?;NCIH ;H> QG CM NB? WOR >O? NI NB? GIF?=OF;L >C_OMCIH
;H> F;N?L CH NB? <IIE NOL<OF?H=? ;M
QG (
=H7: 0
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
( CM NB? >C_OMCIH =I?^=C?HN GIF?=OF;L IL NOL<OF?HN CN CM HIN MJ?=CV?> B?L? I@
NB? NL;=?L
&H NB? HIH =IHM?LP;NCP? @ILG <S OMCHA NB? G;MM =IHM?LP;NCIH ?KO;NCIH IH? I<N;CHM
NB? ?KO;NCIH EHIQH ;M NB? NL;HMJILN\>C_OMCIH ?KO;NCIH
0
1 =H7: 0 :?L ( =H7: 0
J
QB?L? - CM ; MIOL=? IL MCHE N?LG
1BCM ?KO;NCIH CM QLCNN?H ;A;CH CH NB? >?P?FIJ?> @ILG
0
J
1
0
0
2
N
O
3
0
P
N
(
0
N
O
(
0
O
-
P
(
0
P
- >D=30AH 2>=38C8>=B
1B? <IOH>;LS =IH>CNCIHM I@ NB? NL;=?LM IH NB? Q;FFM ;L?
+I WOR <IOH>;LS =IH>CNCIH
0
D
QB?L? NB? HIN;NCIH (
< L?JL?M?HNM =H7: 0 D QB?L? D CM NB? J?LJ?H>C=OF;L JICHNCHA
IONQ;L>M @LIG NB? <IOH>;LS
#FOR <IOH>;LS =IH>CNCIH
0
70
D
8
(08=C+4=0=C 4@D0C8>=B
%A4B4=C0C8>= 0=3 1A845 A4E84F
,H OAOMN NB? #L?H=B "=IF? +;NCIH;F? >?M -IHNM ?N B;OMMf?M =?F?<L;N?>
NB? <C=?HN?H;LS I@ NB? <CLNB I@ '?;H F;O>? >BfG;L ;LLf IOHN I@ 0;CHN 3?H;HN
N NB? ;A? I@ B? DICH?> NB? "=IF? -IFSN?=BHCKO? ;H> F;N?L NB? "=IF? >?M -IHNM
?N B;OMMf?M QBC=B B? VLMN ;NN?H>?> CH &H B? JL?M?HN?> NQI J;J?LM
NI NB? =;>fGC? >?M 0=C?H=?M IH NB?IL?NC=;F G?=B;HC=M ;H> NB? >SH;GC=M I@ WOC>M
2JIH L?NCL?G?HN ;@N?L ; =;L??L CH NB? 1?=BHC=;F !?J;LNG?HN I@ NB? CNS I@ -;LCM
B? >?PIN?> BCM NCG? NI M=C?HNCV= L?M?;L=B ;H> ?HN?L?> NB? =;>fGC? >?M 0=C?H=?M CH
==IL>CHA NI (;LF -?;LMIH CH NBCM ZGI>?F M=BIF;L[ =IHP?LN?> BCM BIOM?
CHNI ; BIMJCN;F I@ NB? /?> LIMM @IOH>?> M?P?H S?;LM ?;LFC?L 1BCM G;H QBI OM?> NI
M;S Z>?MCL? BOG<F? J?L@?=NCIH ;H> >I HIN MNLCP? NI G;E? CN HINC=?>[ >C?> CH ';HO;LS
ZNB? MIOF VFF?> QCNB CGGILN;F BIJ? H?P?L N;LHCMB?> <S >IO<N[ ;M QLCNN?H <S BCM
MNO>?HN IOMMCH?MK
1B? ?KO;NCIHM NB?H EHIQH ;M NB? Z;LLf >? 0;CHN 3?H;HN ?KO;NCIHM[ JO<FCMB?>
CH 89 ;L? MNCFF NI>;S I@ ?RNL?G? CGJILN;H=? CH G;LCNCG? IL WOPC;F BS>L;OFC=M
;H> AIP?LH @L?? MOL@;=? WIQM CH MB;FFIQ Q;N?LM B?H=? NB?CL "HAFCMB H;G? ZMB;FFIQ
Q;N?L ?KO;NCIHM[ 1B?S ;L? I<N;CH?> @LIG NB? +;PC?L\0NIE?M ?KO;NCIHM QCNB NB?
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 B?FJ I@ MCGJFC@SCHA BSJINB?M?M ;H> ;L? OM?> CH MJB?L?M ;M P;LC?> ;M JLIN?=NCIH I@
NB? ?HPCLIHG?HN =;F=OF;NCIH I@ NC>?M ;H> MNILG Q;P?M NB? CGJ;=N ;H> MN;<CFCNS I@
MNLO=NOL?M M?>CG?HNIFIAS MCGOF;NCIH I@ >;G <L?;E WII> Q;P?M MNO>S I@ WII>M ?N=
,H? I@ NB? L?MNLC=NCP? BSJINB?M?M I@ NB?M? ?KO;NCIHM CM NB;N NB? Q;P?F?HANB MBIOF>
<? GO=B AL?;N?L NB;H NB? >?JNB
IHM?KO?HNFS =;F=OF;NCHA ; MQ?FF >I?M HIN =IG?
OH>?L NB?CL >IG;CH I@ P;FC>CNS
&H NBCM =B;JN?L NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? JL?M?HN?> QCNB NB?CL >C_?L?HN @IL
GOF;NCIHM CH IL>?L NI BCABFCABN NB? GI>?FF?> JBSMC=;F JB?HIG?H; ;H> NB?CL CGJFC?>
JL?=;ONCIHM CH HOG?LC=;F ;H;FSMCM
1B? JLCH=CJ;F IJ?L;NCIH NI A?N NB? 0;CHN 3?H;HN @LIG NB? +;PC?L\0NIE?M ?KO;NCIHM
CM ;H ;P?L;A? IP?L NB? P?LNC=;F %IQ?P?L NB? ?RCMN?H=? I@ HIH FCH?;L N?LGM H?=?MMCN;N?M
NB? OM? I@ MIG? BSJINB?M?M ;H> ;JJLIRCG;NCIHM 1B?M? BSJINB?M?M QBC=B =IH>CNCIH
NB? M=IJ? I@ P;FC>CNS I@ NB? ?KO;NCIHM MBIOF> <? L?G?G<?L?>
H?>C74B4B 0??A>G8<0C8>=B 0=3 20;2D;0C8>= AD;4B
H3A>BC0C82 ?A4BBDA4
1BCM ;JJLIRCG;NCIH QBC=B CM L?MOG?> B?L? Q;M JL?M?HN?> ?;LFC?L QCNB NB? +;PC?L\
0NIE?M ?KO;NCIHM ==IL>CHA NI CN ;==?F?L;NCIH ;LCMCHA @LIG JL?MMOL? =IOHN?L<;F;H=?M
AL;PCNS &H MO=B ; =;M? NB? JL?MMOL? CM BS>LIMN;NC= 1BCM CM =IOJF?> QCNB MG;FF
GIP?G?HNM CH NB? P?LNC=;F >CL?=NCIH
&H @;=N NBCM BSJINB?MCM IH JL?MMOL? CM HIN MNLC=NFS H?=?MM;LS @IL NB? >?P?FIJG?HN
I@ NB? 0;CHN 3?H;HN ?KO;NCIHM CN CM IHFS H?=?MM;LS NI NL;HMF;N? NB? JL?MMOL? CH N?LGM
I@ NB? B?CABN I@ Q;N?L
1B? JL?MMOL? F N O P J <?CHA >O? IHFS NI NB? Q?CABN I@ NB? =IFOGH I@ Q;N?L ;<IP?
NB? =IIL>CH;N?M N O P Q? A?N
F
= P
NB;N CM
F N O P =P
9IJ
1B? =IHMN;HN 9IJ CM =BIM?H CH MO=B ; Q;S NB;N F N O 6A F/B; QB?L? 6A CM NB?
?F?P;NCIH I@ NB? @L?? MOL@;=? ;H> F/B; NB? ;NGIMJB?LC= JL?MMOL? &@ NB? AL;>C?HN I@ NB?
;NGIMJB?LC= JL?MMOL? CM HIN T?LI CN CM =IHMC>?L?> ;M ; MIOL=? N?LG 1I MCGJFC@S Q?
MB;FF QLCN?
F N O P J = 6A P
&H J;LNC=OF;L ;N NB? <INNIG F = 6A 64 =>
#46;8681;4 E4AC820; E4;>28CH
3?LNC=;F P?FI=CNS QCFF <? H?AFCAC<F? CH NB? 0;CHN 3?H;HN ?KO;NCIHM ;H> QCFF HIN B;P?
;H ?KO;NCIH 1BCM ;JJLIRCG;NCIH AI?M NIA?NB?L QCNB NB? BSJINB?MCM I@ BS>LIMN;NC=M
;==IL>CHA NI QBC=B P?LNC=;F ;==?F?L;NCIHM MBIOF> <? H?AFCAC<F?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
<?4A<4018;8CH >5 C74 BDA5024 0=3 C74 1>CC><
&N CM ;MMOG?> NB;N NB?L? CM HI NL;HM@?L I@ Q;N?L G;MM NBLIOAB NB? <INNIG ;H> NB?
MOL@;=? ;H> NB;N ; J;LNC=F? I@ Q;N?L MCNO;N?> IH IH? I@ NB?M? NQI CHN?L@;=?M QCFF
L?G;CH NB?L? ==IL>CHA NI NB? ?KO;NCIHM ?MN;<FCMB?> CH 0?=NCIH Q? QLCN?
6A
J
1A
6A
N
2A
6A
3A O
64
J
14
64
N
24
64
34 O
;H>
+IN? F;N?L Q? MB;FF MNO>S NB? MIOL=? N?LGM >O? NI NB? ;LLCP;F I@ ; >CM=B;LA?
NBLIOAB NB? <INNIG IL NB? MOL@;=? &H NBCM =;M? NB?L? QCFF <? HI GIL? CGJ?LG?;<CFCNS
)74 !481=8CI AD;4
1B? >?LCP;NCP? I@ ;H CHN?AL;F QCNB P;LC;<F? <IOH>;LC?M ACP?M LCM? NI ; >?LCP;NCP? QCNBCH
NB? CHN?AL;F ;H> WOR N?LGM ;==IL>CHA NI NB? @ILGOF;
. .
. .
6A
64
"
" :P :P
" N O 6A
" N O 64
N .
N
N
N
.
&@ CN CM MN;N?> NB;N
< >
.
.
" :P
.
QB?L? < L?JL?M?HNM NB? ;P?L;A? I@ " IP?L NB? P?LNC=;F <S ;JJFSCHA NB? ;<IP? @ILGOF;
NI NB? >?LCP;NCP?M CH L?F;NCIH NI N O ;H> P Q? A?N
=H7: >< .
.
.
=H7: " :P " N O 6A QDA " N O 64 QD4
QB?L? QDA ;H> QD4 ;L? P?=NILM QBC=B ;L? HIN OHCN P?=NILM >CL?=N?> OJQ;L>M CH NB? VLMN
=;M? ;H> >IQHQ;L>M CH NB? M?=IH>
0CGCF;LFS @IL ; P?=NIL
:?L ><Q .
.
.
:?L "Q :P "Q N O 6A QDA "Q N O 64 QD4
#IL ; >?LCP;NCP? CH NCG? ;H> <?;LCHA CH GCH> NB;N
6A
Q A QDA 1
J
;H>
64
J
Q 4 QD4 1
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 Q? I<N;CH
><
J
.
.
.
"
:P
J
Q A QDA
" N O 6A 1
Q 4 QD4
" N O 64 1
4?C70E4A068=6 #0E84AO(C>:4B 4@D0C8>=B
%A8=28?;4
1B? +;PC?L\0NIE?M ?KO;NCIHM QCNB =IHMN;HN >?HMCNS ;H> BS>LIMN;NC= JL?MMOL? ;L? HIQ
AICHA NI <? ;P?L;A?> P?LNC=;FFS <S CHN?AL;NCIH @LIG NB? <INNIG NI NB? MOL@;=? J;LN
@LIG NB? ;FL?;>S >?VH?> B?CABN I@ NB? Q;N?L NQI H?Q P;LC;<F?M ;L? AICHA NI ;JJ?;L
K >
.
.
.
1 :P ;H> L >
.
.
2 :P
.
1B?M? ;P?L;A?M I@ BILCTIHN;F =IGJIH?HNM I@ NB? NBL?? >CG?HMCIH;F P?FI=CNS P?=NIL
IP?L NB? P?LNC=;F QCFF NB?GM?FP?M <? =;FF?> P?FI=CNC?M IL Z=IGJIH?HNM I@ NB? P?FI=CNS
P?=NIL[ 1B? P?=NIL QCNB =IGJIH?HNM K ;H> L QCFF <? >?HIN?> QK
E4A064 >5 C74 2>=C8=D8CH 4@D0C8>=
1B? =IHNCHOCNS ?KO;NCIH <?=IG?M
.
.
.
1
N
2
O
3
P
:P 4CNB NB? )?C<HCNT LOF? CN ;FMI <?=IG?M
.
.
1
N
.
N
O
.
3
:P 3 6A 3 64
P
.
.
.
2
O
1 :P 1 N O 6A
6A
N
1 N O 64
64
N
2 :P 2 N O 6A
6A
N
2 N O 64
64
O
.
.
1B;N CM N;ECHA CHNI ;==IOHN NB? CGJ?LG?;<CFCNS =IH>CNCIHM
>K
N
>L
O
6A
64
J
J
IL
>
:?L >QK J
%?L? NB? >CP?LA?H=? CM =;F=OF;N?> CH NQI >CG?HMCIHM
4? CGG?>C;N?FS HINC=? NB? ;H;FIAS QCNB NB? G;MM =IHM?LP;NCIH ?KO;NCIH I@ =IG
JL?MMC<F? WOC>M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
:?L QK J
&H NB? 0;CHN 3?H;HN ?KO;NCIHM NB? Q;N?L >?JNB JF;SM NB? LIF? I@ >?HMCNS
E4A064 >5 C74 <><4=CD< 4@D0C8>=
4? ;P?L;A? NB? GIG?HNOG ?KO;NCIH @IOH> CH 0?=NCIH Q
1
J
Q 1
Q 1
=H7: F
:?L Q=
"Q
;H> ;JJFS @LIG NBCM JICHN NB? BSJINB?MCM I@ BS>LIMN;NC= JL?MMOL? "KO;NCIH )?N OM FIIE ;N NB? ;P?L;A? I@ ?;=B I@ NB? N?LGM I@ NBCM ?KO;NCIH OMCHA NB? ?R;GJF? I@
NB? =IGJIH?HN 1
4A8E0C8E4 8= C8<4
. .
.
1
6A
>K
:P 1 6A
J
J
J
1 64
64
J
8QDB8>= C4A<B @IL NB? >C_OMCIH N?LG :?L Q? MOJJIM? NB;N NB? >?HMCNS >I?M
HIN P;LS ;FIHA NB? P?LNC=;F CN =;H NB?H <? L?GIP?> @LIG NB? CHN?AL;FM ;H> NB? )?C<HCNT
@ILGOF; ACP?M
.
.
.
.
:P
4 QD4
AQDA
:?L :P :?L
.
.
HILG;F P?=NILM ;N NB? <INNIG ;H> ;N NB? MOL@;=? ;L? MNCFF HIN HILG;FCT?>
1B? F;MN NQI N?LGM I@ "KO;NCIH L?JL?M?HN NB? MNL?MM ;N NB? MOL@;=? ;H> NB?
<INNIG >O? @IL ?R;GJF? NI NB? QCH> ;H> NI NB? @LC=NCIH 1B?S QCFF <? ?F;<IL;N?>
F;N?L QCNB ;FF NB? MIOL=? N?LGM
+.
1B? NQI =IGJIH?HNM I@ :?L . :P ;L?
.
.
=H7: 1 :P
:?L
.
;H>
:?L
.
.
.
=H7: 2 :P
L?MJ?=NCP?FS
?@IL? ;LLCPCHA ;N NB? VH;F @ILG I@ >C_OMCIH N?LGM Q? MBIOF> =IHMC>?L NB;N
. .
=H7: 1 :P
.
CM B;L>FS >C_?L?HN @LIG
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 > =H7: K
:?L
+ .
.
:P CM VH;FFS N;E?H ;M ?KO;F NI NB? P?=NIL I@ =IGJIH?HNM
:?L > =H7: K
;H>
:?L > =H7: L
IL ?P?H <S =IHMC>?LCHA NB;N NB? >?HMCNS CM =IHMN;HN @IL =;F=OF;NCHA NBCM N?LG
:?L > =H7: K
3E42C8>=B C4A<B
;H> :?L > =H7: L
NB?M? ;L? @IL NB? ?KO;NCIH I@ 1 1
1
N
2
1
O
1
P
3
QBC=B =;H ;FMI <? MN;N?> ;M
11
N
12
O
13
P
NB;HEM NI NB? =IHNCHOCNS ?KO;NCIH )?N OM =;F=OF;N? NB? ;P?L;A? I@ ?;=B I@ NB?M? NBL??
N?LGM IP?L NB? P?LNC=;F
1B? NBCL> N?LG CM NB? MCGJF?MN
.
.
.
1 N O P 3 N O P
P
:P 1 N O 6A 3 N O 6A 1 N O 64 3 N O 64
4? NB?H B;P?
.
.
.
.
.
.
1
:P N
N
12
:P O
O
.
.
.
.
1 :P 1 6A
6A
N
.
.
1 64
6A
O
1 2 :P 1 6A 2 6A
64
N
1 64 2 64
64
O
1B?M? N?LGM ACP? LCM? NI HIH FCH?;LCNS )?N OM >?P?FIJ @IL ?R;GJF? NB? ?RJL?MMCIH
=IHN;CHCHA 1 2 O
.
.
1 2 :P
.
O
.
.
.
>KL
O
K
1 K L
O
.
2 L :P
.
.
1 K 2 L :P
+ .
1 K 2 L :P CM HIN T?LI QB?H NB?L? ;L? B?N?LIA?H?CNC?M I@
1B? N?LG G
.
P?FI=CNS P?LNC=;FFS 1B?M? N?LGM =;FF?> >CMJ?LMCIH N?LGM ;L? A?H?L;FFS CHN?LJL?N?> ;M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;>>CNCIH;F >C_OMCIH ;MC=;FFS NB?S N;E? CHNI ;==IOHN WO=NO;NCIHM ;LIOH> ;H ;P?L;A?
B?L? MJ;NC;F ;H> ;L? IH NB?M? ALIOH>M @ILG;FFS MCGCF;L NI NB? /?SHIF>M MNL?MM &H
L?;FCNS NB?M? N?LGM ;LCM? @LIG NB? @;=N NB;N NL;HMJILN I@ JBSMC=;F KO;HNCNC?M CM =;LLC?>
ION <S L?;F =OLL?HN ;H> HIN <S NB? =OLL?HN ;P?L;A?> IP?L NB? P?LNC=;F !?J?H>CHA
IH NB? >?JNB ;FF J;LNC=F?M ;L? HIN GIP?> ;N NB? M;G? P?FI=CNS &H NB? 0;CHN 3?H;HN
?KO;NCIHM IH NB? =IHNL;LS ; >?JNB ;P?L;A?> =OLL?HN CM =BIM?H %?L? NB? @ILG I@
>C_OMCIH CM ;>IJN?> L?MOFNCHA CH ; N?HMIL H;NOL;FFS <?CHA ;>>?> NI NB? /?SHIF>M
N?HMIL &H @;=N C@ K 1 K ;H> L 2 L NB?M? >CMJ?LMCIH N?LGM ;L? L?MJ?=NCP?FS
;FIHA N ;H> ;FIHA O
> N
.
> N
.
.
KK:P
.
.
KL:P
.
> O
.
> O
.
.
KL:P
LL:P
.
.
.
&H NBCM @ILG CN =;H <? M??H NB;N NB? N?HMIL K7 K8 CM H;NOL;FFS ;>>?> NI NB? MNL?MM
N?HMIL ,H? G;HH?L I@ N;ECHA CN CHNI =IHMC>?L;NCIH QIOF> <? NI GI>C@S NB? P;FO?
I@ PCM=IMCNS
IL NOL<OF?HN WIQ PCM=IMCNS B ;H> L?JF;=? CN <S ;H ?_?=NCP? PCM=IMCNS
CHN?AL;NCHA NB? >CMJ?LMCIH JB?HIG?HIH
1B? NOL<OF?H=? GI>?FM MBIOF> NBOM N;E? >CMJ?LMCIH CHNI ;==IOHN CH NQI >CG?HMCIHM
1BCM QCFF <? M??H GIL? MJ?=CV=;FFS QCNB NB? A\ GI>?F
%A4BBDA4 6A0384=C
@IL JL?MMOL? AL;>C?HNM CH N ;H> O Q? QLCN?
.
.
.
6A
= 6A P :P >=
N
N
6A
= 6A P :P >=
O
O
;H>
.
.
.
1BCM CM NLO? IHFS C@ NB? >?HMCNS >I?M HIN P;LS ;FIHA NB? BILCTIHN;F 1B? P;LC;NCIHM
I@ >?HMCNS QCFF <? N;E?H CHNI ;==IOHN QCNB NB? <OIS;H=S MIOL=? N?LGM
(>DA24 C4A<B 0=3 1>3H 5>A24B @IL =IHMN;HN MIOL=? N?LGM ;FIHA NB? P?LNC=;F NB?
CHN?AL;NCIH ;FIHA P CM ; MCGJF? GOFNCJFC=;NCIH <S NB? Q;N?L >?JNB > ;H> "F ACP?M >"F
2HNCF NB?CL GIL? JL?=CM? MNO>S CH 0?=NCIH @LC=NCIH @IL=?M ;H> NB? N?LGM
QDA
A
QD4
4
QCFF <? CH=FO>?> CH NB?M? <I>S @IL=?M ;H> QCFF NB?L?@IL? <? IGCNN?> B?L?;@N?L
EQUATIONS OF FREE SURFACE HYDRODYNAMICS
31
Recapitulation: the sum of all these terms, a number of which disappear because
of the impermeability of the bottom and open surface, gives the following equation:
−−→
∂
∂
∂Zs
∂(hu)
+
(huu) +
(huv) = −gh
+ hFx + div(hν e grad(u))
∂t
∂x
∂y
∂x
(2.128)
In the same manner, the equation for V gives:
−−→
∂(hv)
∂
∂
∂Zs
+
(huv) +
(hvv) = −gh
+ hFy + div(hν e grad(v))
∂t
∂x
∂y
∂y
(2.129)
ν e is an effective diffusion which will model turbulent viscosity and dispersion.
The W equation has not been retained (it has already been used to obtain the
hydrostatic pressure hypothesis).
With the mass conservation equation, these equations constitute the two-dimensional set of Barré de Saint-Venant equations, in what is called the “conservative” form.
2.3.4
Different forms of equations
Conservative form
Equations 2.119, 2.128 and 2.129 are rewritten by stating:
Qx = hu and Qy = hv
is the vector of the components Qx and Qy . The “discharge” Q is in fact per
Q
unit length, in m2 /s:
Continuity:
∂h
=0
+ div(Q)
∂t
(2.130)
−−→
∂Qx
∂Zs
+ div(Qx u) = −gh
+ hFx + div(hν e grad(u))
∂t
∂x
(2.131)
−−→
∂Qy
∂Zs
+ div(Qy u) = −gh
+ hFy + div(hν e grad(v))
∂t
∂y
(2.132)
Momentum:
Source terms in the continuity equation:
In the presence of water intake or release at the free surface, inside the domain
or at the bottom, or if one takes into account rain or infiltrations in the ground, the
right-hand side of the continuity equation is not zero but equal to a term that we
shall denote as “Sce” for “Source” (it can also be a sink). Sce is expressed in m/s
and is positive for an injection of water into the domain. It is for example the rainfall,
expressed in metres per second. We should multiply Sce by a surface to obtain a real
discharge.
The mass conservation equation is thus written as:
∂h
= Sce
+ div(Q)
∂t
(2.133)
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
0CGCF;LFS CH NB? GIG?HNOG ?KO;NCIHM Q? MBIOF> N;E? CHNI ;==IOHN NB? MOJJFS
@LIG NB? MIOL=? ; P?=NIL I@ =IGJIH?HNM ) J N ;H> ) J O ) J @IL Z*IG?HNOG[
CH G M 1B?M? N?LGM ;L? ?KO;F NI KA13 /9; ;H> LA13 /9; QB?L? KA13 ;H> LA13 ;L? NB?
=IGJIH?HNM I@ NB? P?FI=CNS ;N MIOL=?
#>=2>=B4AE0C8E4 5>A<
&H NB?ILS IHFS ; MNLC=N JLI=?MMCHA I@ =IHM?LP;NCP? ?KO;NCIHM ;FFIQM IH? NI L?;FFS N;E?
CHNI ;==IOHN >CM=IHNCHOCNC?M QBC=B =;H ;JJ?;L CH NB? MIFONCIHM I@ MB;FFIQ Q;N?L ?KO;
NCIHM BS>L;OFC= DOGJ ,NB?L L?;MIHM MN;<CFCNS I@ M=B?G?M OM? I@ NB? G?NBI> I@
=B;L;=N?LCMNC=M B;P? F?> NI NB? ;>IJNCIH I@ ; >C_?L?HN @ILGOF;NCIH QCNB OHEHIQHM
> K ;H> L =;FF?> NB? Z>?JNB\P?FI=CNS[ @ILGOF;NCIH 1BCM @ILGOF;NCIH CM M;C> NI <?
ZHIH =IHM?LP;NCP?[ )?N OM CGG?>C;N?FS G?HNCIH NB;N NBCM H;G? >I?M HIN JL?P?HN OM
@LIG =IHMNLO=NCHA HOG?LC=;F M=B?G?M QBC=B =IHM?LP? NB? G;MM I@ Q;N?L ;M QCFF <?
MBIQH MO<M?KO?HNFS
,H NB? <;MCM I@ NB? =IHM?LP;NCP? @ILG I@ ?KO;NCIHM NB? >?LCP;NCP?M I@ @OH=NCIH
JLI>O=NM ;L? >?P?FIJ?> CH NB? GIG?HNOG ;H> =IHNCHOCNS ?KO;NCIHM
1B? =IHNCHOCNS ?KO;NCIH <?=IG?M
>
QK =H7: >
> :?L QK /9;
J
;H> NB? VLMN =IGJIH?HN I@ NB? GIG?HNOG ?KO;NCIH
>
K
J
K
>
J
K
>K
N
>K
K
N
K
>L
O
6A
>"F :?L > 3 =H7: K
N
NB;N CM <S OMCHA NB? =IHNCHOCNS ?KO;NCIH
=>
>
K
J
>K
K
N
>L
K
6A
=>
O
N
>"F
>L
K
O
KA13 /9;
:?L > 3 =H7: K
KA13 K /9; 1B? NQI G?G<?LM I@ NB? ?KO;NCIH ;L? >CPC>?> <S > S ;JJFSCHA NB? M;G? IJ?L;
NCIH NI L Q? VH;FFS I<N;CH
K
J
K
K
N
L
K
6A
=
O
N
"F
:?L > 3 =H7: K
>
KA13 K
/9;
>
L
J
K
L
N
L
L
6A
=
O
O
"G
:?L > 3 =H7: L
>
LA13 L
/9;
>
1B? =IHM?LP;NCP? ;H> HIH =IHM?LP;NCP? @ILGM ;L? HIN ?KOCP;F?HN CH NB? JL?M?H=? I@
>CM=IHNCHOCNC?M
'13
/?G;LE C@ NB?L? CM ; MIOL=? ;N NB? <INNIG NB? N?LGM KA13 K '13
6 ;H> LA13 L 6
;L? ;>>?> NI NB? LCABN B;H> MC>?M 1B?M? N?LGM ;L? T?LI C@ NB? P?FI=CNS ;N NB? MIOL=?
CM NB? M;G? ;M NB;N I@ NB? =OLL?HN 1B?S QCFF <? CH=FO>?> CH NB? ?RJL?MMCIHM "F ;H>
"G ;H> QCFF NB?L?@IL? HIN ;JJ?;L B?L?;@N?L
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 4;4A8CH >5 F0E4B F8C7 (08=C+4=0=C 4@D0C8>=B
1B? NQI G;CH JB?HIG?H; GI>?FF?> <S NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? ;>P?=NCIH IL
NL;HMJILN <S NB? =OLL?HN ;H> JLIJ;A;NCIH I@ FIHA Q;P?M 1B? JLIJ;A;NCIH CM ?;MCFS
BCABFCABN?> @IL MG;FF >CMNOL<;H=?M @IL QBC=B NB? FCH?;LCT?> ?KO;NCIHM ;L? P;FC> 1B?
;>P?=NCIH N?LGM ;FMI ;L? CAHIL?> ;H> NB? BSJINB?MCM I@ ; BILCTIHN;F <INNIG CM @ILG?>
QBC=B ACP?M CH IH? >CG?HMCIH
K
J
=
>
N
>
J
>
K
N
;H>
QB?L? > CM NB? ;P?L;A? >?JNB
1B? =IG<CH;NCIH
>
N
K
J
=
>
N
J
>
J
>
K
N
ACP?M
>
>
=>
J
N
;H> MCGCF;LFS Q? =;H A?N
K
K
=>
J
N
1B?M? NQI ?KO;NCIHM ?RJL?MM NB;N K ;H> > JLIJ;A;N? ;N NB? P?FI=CNS 9 IL 9 QCNB
9 => #LIG "OF?L Q? EHIQ CH @;=N NB;N NB? MIFONCIH I@ ;H ?KO;NCIH I@ NB? NSJ?
<
<
9
J
N
CM
< N J N 9J
N
9J
NB? @ILG I@ NB? ;H> @OH=NCIHM <?CHA >?N?LGCH?> <S NB? CHCNC;F =IH>CNCIHM ;H> NB?
<IOH>;LS =IH>CNCIHM &H NB? ;<M?H=?
I@ ; =OLL?HN NB? =?F?LCNS I@ Q;P?M CH NB? 0;CHN
3?H;HN ?KO;NCIHM CM B?H=? 9 => 1B? =?F?LCNS I@ FIHA Q;P?M QCNB FIQ ;GJFCNO>? CH
FCH?;L NB?ILS L?;JJ?;LM B?L? #IL ; >?JNB I@ G NB? =?F?LCNS CM NBOM EGB ;
P;FO? QBC=B CM ?_?=NCP?FS ;NN;CH?> <S NC>;F Q;P?M ;H> NMOH;GCM
4;4A8CHOE4;>28CH 5>A<D;0C8>=
,H NB? <;MCM I@ NB? HIH =IHM?LP;NCP? @ILGOF;NCIH QBC=B B;M DOMN <??H I<N;CH?> ;H
INB?L @ILGOF;NCIH CM <LIOABN ION ,H NB? IH? B;H> CN MBIQM NB? ;H;FIAS I@ NB?
0;CHN 3?H;HN ?KO;NCIHM QCNB NB? "OF?L ?KO;NCIHM ;H> IH NB? INB?L B;H> CN =IHMNC
NON?M NB? GIMN H;NOL;F @ILGOF;NCIH @IL JL?M?HNCHA NB? =B;L;=N?LCMNC= =OLP?M #CH;FFS
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
CN QCFF JLIPC>? ;N ; F;N?L MN;A? NB? C>?; I@ ; JL?=IH>CNCIHCHA @IL NB? CN?L;NCP? G?NBI>M
L?MIFPCHA NB? FCH?;L MSMN?G ;LCMCHA @LIG NB? 0;CHN 3?H;HN ?KO;NCIHM
1BCM @ILGOF;NCIH CM I<N;CH?> <S L?JF;=CHA NB? Q;N?L >?JNB > <S 9 = CH NB? ?KO;
NCIHM NI A?N
*;MM =IHM?LP;NCIH
9
9
J
9K
9
N
K
N
9L
9
O
9
L
O
L
9
O
9
:?L QK NB;N CM ;@N?L >CPCMCIH <S 9
9
J
K
9
N
*IG?HNOG
K
J
K
K
N
L
K
O
9
9
64
=
N
N
"F
:?L > 3 =H7: K
>
L
J
K
L
N
L
L
O
9
9
64
=
O
O
"G
:?L > 3 =H7: L
>
1BCM @ILG I@ ?KO;NCIH CM I@N?H MCGJF?L @IL <OCF>CHA ?R;=N MIFONCIHM ? A IH?
>CG?HMCIH;F >;G <L?;E ;H> CM GIL? @OH>;G?HN;F CH J;LNC=OF;L AL;PCN;NCIH;F ;==?F
?L;NCIH HI FIHA?L JF;SM ; J;LN #CH;FFS ;FF NB? OHEHIQHM K L ;H> 9 B;P? NB? M;G?
>CG?HMCIH
=0;>6H F8C7 D;4A 4@D0C8>=B
1B? "OF?L ?KO;NCIHM AIP?LH NB? WIQ I@ =IGJL?MMC<F? CHPCM=C> WOC>M QCNB =IHMN;HN
?HNLIJS NB?S ;L? QLCNN?H CH NQI >CG?HMCIHM
9
J
K
9
N
L
9
O
9
:?L QK K
J
K
K
N
L
K
O
9
9
N
L
J
K
L
N
L
L
O
9
9
O
CM NB? B?;N =;J;=CNS L;NCI NB? L;NCI I@ NB? MJ?=CV= B?;N ;N =IHMN;HN JL?MMOL? NI
NB? MJ?=CV=
6 B?;N ;N =IHMN;HN PIFOG? %?L? 9 CM NB? =?F?LCNS I@ NB? Q;P?M CH NB? WOC>
?KO;F NI >
F <?CHA NB? JL?MMOL? ;H> NB? >?HMCNS
1B?L? CM ; J?L@?=N ;H;FIAS <?NQ??H NB? "OF?L ;H> 0;CHN 3?H;HN ?KO;NCIHM @IL NB?
F;NN?L QIOF> <? ?KO;F NI ;H> @IL ;CL 1BCM ;H;FIAS OH>?LFCH?M NB? @;=N
NB;N NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? @ILG;FFS =IGJL?MMC<F? WOC> ?KO;NCIHM NB? LIF? I@
P;LC;<F? >?HMCNS <?CHA JF;S?> <S NB? Q;N?L >?JNB %IQ?P?L NB? ;H;FIAS ?H>M QCNB
NB? LCABN B;H> MC>?M I@ NB? ?KO;NCIHM NB? MIOL=? N?LGM I@ NB? 0;CHN 3?H;HN ?KO;NCIHM
JIMCHA MJ?=CV= >C^=OFNC?M QBC=B L;JC>FS G;E? NB? L?MJ?=NCP? ;JJLI;=B?M >CP?LA?HN
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 )74 270A02C4A8BC82 2DAE4B
1B? G?NBI> I@ =B;L;=N?LCMNC=M ;LCM?M @LIG NB? A?IG?NLC= NB?ILS I@ >C_?L?HNC;F KO;MC
FCH?;L ?KO;NCIHM &N ACP?M ; JBSMC=;F CHN?LJL?N;NCIH I@ =?LN;CH JLI<F?GM CH WOC> G?=B
;HC=M ;H> MIG?NCG?M ;FFIQM ;H ;H;FSNC=;F MIFONCIH NI <? <OCFN @LIG CN *IL?IP?L CN
CH>C=;N?M NB? HOG<?L ;H> H;NOL? I@ NB? <IOH>;LS =IH>CNCIHM L?KOCL?> ;H> =;H >?
N?=N CFF JIM?> JLI<F?GM
IH=?JNM I@ NB? =B;L;=N?LCMNC=M ;H> NB? /C?G;HH CHP;LC;HNM
CH NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? NBOM @OH>;G?HN;F 4? MB;FF =BIIM? NB? FCH?;LCT?>
=?F?LCNS\P?FI=CNS @ILGOF;NCIH CH IH? >CG?HMCIH QCNBION >C_OMCIH IL MIOL=? N?LGM ;H>
CH ; W;N <INNIG?> =;H;F
K
J
K
K
N
9
9
9
K
J
N
1B? MOG I@ NB? NQI ?KO;NCIHM ACP?M
K
9
K
N
9
K 9
N
K9
K 9
N
K
J
9
N
9
;H> NB? >C_?L?H=?
K 9
J
1B?M? ?KO;NCIHM B;P? NB? @ILG
J
:N :J N
QCNB K 9 IL K 9 QBC=B ;FMI MCAHCV?M 2
2B <S NB? ?KO;NCIH 2F
K
9
IL
K
9
4?
NBOM
B;P?
2B
:K
IH NB? =OLP?M H;G?>
9
:J
L?JL?M?HN?> <S NB? ?KO;NCIH
:N
K
:J
9
;H>
IH NB? =OLP?M H;G?>
,H NB? =OLP?M
,H NB? =OLP?M
: K 9
:J
L?JL?M?HN?> <S NB? ?KO;NCIH
:N
K9
:J
NB? KO;HNCNS K 9 CM CHP;LC;HN
NB? KO;HNCNS K 9 CM CHP;LC;HN
;FIHA NB? =OLP?M L?JL?M?HN?>
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? =OLP?M
;H> ;L? =;FF?> =B;L;=N?LCMNC= =OLP?M 1B? KO;HNCNC?M K 9 ;H>
K 9 ;L? =;FF?> NB? /C?G;HH CHP;LC;HNM 1B? ?RCMN?H=? I@ L?;F =B;L;=N?LCMNC= =OLP?M
JF;=?M NB? 0;CHN 3?H;HN ?KO;NCIHM CH NB? @;GCFS I@ BSJ?L<IFC= >C_?L?HNC;F ?KO;NCIHM
1QI =;M?M NB?H ;JJ?;L ;==IL>CHA NI NB? MCAH I@ K 9 K <?CHA N;E?H ;M JIMCNCP?
&@ K 9 NB? WIQ CM M;C> NI <? MO<=LCNC=;F 1B? Q;P?M ;L? @;MN?L NB;H NB? =OLL?HN
1B? WIQ ;N ;HS JICHN >?J?H>M IH NB? <IOH>;LS =IH>CNCIHM OJMNL?;G ;H> >IQHMNL?;G
&@ K 9 NB? WIQ CM M;C> NI <? MOJ?L=LCNC=;F 1B? Q;P?M ;L? MFIQ?L NB;H NB? =OLL?HN
1B? WIQ ;N ;HS JICHN >?J?H>M IHFS IH NB? OJMNL?;G <IOH>;LS =IH>CNCIHM CN IHFS B;M
;H CHWO?H=? >IQHMNL?;G
&@ K 9 NB? WIQ CM M;C> NI <? =LCNC=;F
1BOM NB? <IOH>;LS =IH>CNCIHM =B;HA? ;==IL>CHA NI NB? H;NOL? I@ WIQ 1BCM
JB?HIG?HIH CM P?LS CGJILN;HN CH HOG?LC=;F GI>?FFCHA ;H> F?;>M NI NB? @;CFOL? I@
KOCN? ; @?Q G?NBI>M
1B? MO<=LCNC=;F IL MOJ?L=LCNC=;F H;NOL? I@ ; WIQ CM ACP?H <S NB? #LIO>? HOG<?L
" K
9
CH IH? >CG?HMCIH IL
" K
L
9
CH NQI >CG?HMCIHM
&@ " NB? WIQ CM MO<=LCNC=;F IL WOPC;F
&@ " NB? WIQ CM =LCNC=;F
&@ " NB? WIQ CM MOJ?L=LCNC=;F
#>C8>=B >= 7H3A0D;82 9D<?B
1B? 0;CHN 3?H;HN ?KO;NCIHM >O? NI NB?CL =IGJL?MMC<F? H;NOL? ;==?JN >CM=IHNCHOIOM
MIFONCIHM QBC=B ;L? =;FF?> BS>L;OFC= DOGJM 4? MB;FF >CMNCHAOCMB <?NQ??H MN?;>S ;H>
OHMN?;>S BS>L;OFC= DOGJM
(C403H 7H3A0D;82 9D<?B 2>=9D60C4 34?C7B A4;0C8>=
)?N OM =IHMC>?L CH IH? >CG?HMCIH J?LG;H?HN WIQ CH ; =;H;F QCNB =IHMN;HN QC>NB ;H> ;
W;N <INNIG &H NB? ;<M?H=? I@ ; MIOL=? N?LG >C_OMCIH ;H> @LC=NCIH NB? 0;CHN 3?H;HN
?KO;NCIHM ?RJL?MM
IHM?LP;NCIH I@ NB? WIQ L;N? J?L OHCN I@ QC>NB - >K
1B? GIG?HNOG <;F;H=? =IHM?LP;NCIH I@ >K
= 6 >?>O=N?> @LIG NB? =IH
M?LP;NCP? GIG?HNOG ?KO;NCIH ;H> QBC=B =;H ;FMI <? >?LCP?> @LIG NB? "OF?L
NB?IL?G
MMOGCHA NB? ?RCMN?H=? I@ ; >CM=IHNCHOCNS QCNB KO;HNCNC?M > ;H> K OJMNL?;G
;H> > ;H> K >IQHMNL?;G M?? #CAOL? CN =;H <? P?LCV?> NB;N
> K > K -
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 > K
=
>
> K
=
>
H ?PC>?HN MIFONCIH QIOF> =?LN;CHFS <? > > ;H> K K
<ON CN CM HIN NB? IHFS
IH?
S L?JF;=CHA K <S -> ;H> K <S -> CH NB? M?=IH> ?KO;NCIH Q? A?N
> > >
>
=
1BCM ?KO;NCIH CM NB? =IHDOA;N? >?JNBM L?F;NCIH IL BS>L;OFC= DOGJ L?F;NCIH QBC=B
>?J?H>M IH NB? WIQ L;N? &N =;H ;FMI <? QLCNN?H ;M
> > >
>1
>
QB?L? >1 CM NB? =LCNC=;F >?JNB MO=B NB;N
>1 =
1B?L? ;FMI ?RCMNM ; L?F;NCIH @IL NB? =IHDOA;N? P?FI=CNC?M
K K
K
K
=K
1
QB?L? K1 CM NB? P?FI=CNS I@ NB? =LCNC=;F WIQ
K1 5
=>1
1B? L?F;NCIH @IL NB? =IHDOA;N? #LIO>? HOG<?LM CM GIL? CHN?L?MNCHA ;M CN CM CH>?
J?H>?HN I@ ;HS INB?L J;L;G?N?L &H @;=N <S HINC=CHA NB;N => %
;H> => %
NB? >?JNBM L?F;NCIH ACP?M
"
"
" "
IL
" 5
"
" " ;H> " JF;S ; MSGG?NLC=;F LIF? CH NB?M? @ILGOF;? 4B?H " " ;H> PC=? P?LM; 1BCM MCAHCV?M NB;N ; MN?;>S BS>L;OFC= DOGJ CM ; NL;HMCNCIH <?NQ??H
MO<=LCNC=;F WIQ ;H> MOJ?L=LCNC=;F WIQ
B?;> FIMM ;FQ;SM N;E?M JF;=? ;=LIMM ; BS>L;OFC= DOGJ NB? >C_?L?H=? CH B?;>
<?NQ??H NB? OJMNL?;G ;H> >IQHMNL?;G ?H>M I@ NB? DOGJ CH OHCNM I@ B?CABN CM
$ $ $ >
K
K
> =
=
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
u2
u1
h1
h2
#CAOL? MN?;>S BS>L;OFC= DOGJ
W
u1
u2
h1
h2
#CAOL? JLIJ;A;NCIH I@ ; DOGJ IH ; Q?N <INNIG
QBC=B =;H <? ?RJL?MM?> ;==IL>CHA NI NB? Q;N?L >?JNBM IH ?CNB?L MC>? I@ NB? DOGJ ;M
$ > >
> >
1BCM P;FO? CM H?=?MM;LCFS JIMCNCP? ;M CN =ILL?MJIH>M NI ; B?;> FIMM 1B?L?@IL? ;
MOJ?L=LCNC=;F WIQ JL?P;CFM OJMNL?;G QBCF? ; MO<=LCNC=;F WIQ JL?P;CFM >IQHMNL?;G 4?
MB;FF L?NOLH NI NB? NB?IL?NC=;F MNO>S I@ BS>L;OFC= DOGJM QB?H Q? ?R;GCH? NB? =IHM?L
P;NCIH I@ KO;HNCNC?M FCE? ?H?LAS B?;> ;H> GIG?HNOG ;N NB? >CM=L?N? F?P?F 0?=NCIHM
;H> ">E8=6 7H3A0D;82 9D<?
)?N OM ;MMOG? NB;N NB? DOGJ GIP?M QCNB ; P?FI=CNS 3 M?? #CAOL? 4? =BIIM?
; L?@?L?HNC;F QBC=B CM ;NN;=B?> NI NB? BS>L;OFC= DOGJ &H NBCM L?@?L?HNC;F NB? WIQ CM
MN?;>S ;H> NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? MNCFF P;FC> 1B? =IHM?LP;NCIH I@ G;MM ;H>
GIG?HNOG IH ?CNB?L MC>? I@ NB? DOGJ ;L? NB?H QLCNN?H ;M
> K 3 > K 3
> K 3
=
>
> K 3
=
>
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 #LIG NB? VLMN ?KO;NCIH Q? >?LCP?
3 > K > K
> >
QBC=B ACP?M NB? P?FI=CNS I@ NB? DOGJ ;==IL>CHA NI NB? OJMNL?;G ;H> >IQHMNL?;G
=B;L;=N?LCMNC=M I@ NB? WIQ &N CM CHN?L?MNCHA NI HINC=? NB;N C@ NB? P?FI=CNS >IQHMNL?;G
K CM T?LI CN =;H <? MBIQH NB;N
3 => >
>
>
+IQ C@ NB? B?CABN I@ NB? BS>L;OFC= DOGJ N?H>M NI T?LI MI NB;N Q? =;H N;E? > > > Q? VH> NB;N 3 => 3 CM NBOM MCGJFS NB? =?F?LCNS I@ FIHA Q;P?M &@ ; Q;P?
CM =IHMC>?L?> ;M ; M?LC?M I@ CHVHCN?FS MG;FF BS>L;OFC= DOGJM Q? VH> CNM =?F?LCNS B?L?
QCNB MCGJF? =IHMC>?L;NCIHM IH NB? =IHM?LP;NCIH I@ G;MM ;H> GIG?HNOG
+IN? 3 >I?M HIN ;FQ;SM B;P? NB? M;G? >CL?=NCIH ;M K ;H> K NB? =;M? I@ ;
BS>L;OFC= <IL?
(08=C+4=0=C 4@D0C8>=B 8= "4A20C>A ?A>942C8>=
0NILG MOLA?M ;H> MNILG CH>O=?> =OLL?HNM >C_?L?H=?M <?NQ??H NB? ;=NO;F F?P?FM I@ NB?
M?; ;H> =OLL?HNM ;H> NB?CL P;FO?M C@ NB? NC>?M ;FIH? Q?L? =IHMC>?L?> ;L? >O? NI NB?
P;LC;NCIHM I@ ;NGIMJB?LC= JL?MMOL? ;H> NB? ?_?=N I@ NB? QCH> IH F;LA? MNL?N=B?M I@
NB? I=?;H 1B? JBSMC=;F =IH>CNCIHM I@ NB?M? JB?HIG?H; ?HMOL? NB;N P?LNC=;F ;==?F?L
;NCIH =;H <? CAHIL?> ;H> NB;N WIQ CM ;FGIMN BILCTIHN;F ;F=OF;NCIH I@ MNILG MOLA?M
;H> MNILG CH>O=?> =OLL?HNM A?H?L;FFS N;E?M JF;=? <S MIFPCHA NB? 0;CHN 3?H;HN ?KO;
NCIHM 1I =;LLS ION HOG?LC=;F MCGOF;NCIHM P;MN G;LCNCG? ;L?;M ;L? >CM=L?NCT?> IH NB?
<;MCM I@ G;JM QBC=B ;L? *?L=;NIL JLID?=NCIHM I@ JILNCIHM I@ NB? ";LNBYM MJB?L? &N
CM NB?H H?=?MM;LS NI VH> ION BIQ NBCM *?L=;NIL JLID?=NCIH GI>CV?M NB? 0;CHN 3?H;HN
?KO;NCIHM
"4A20C>A ?A>942C8>=
1B? *?L=;NIL JLID?=NCIH <?FIHAM NI NB? @;GCFS I@ =IH@ILG;F JLID?=NCIHM QBC=B ?HMOL?
NB? FI=;F =IHM?LP;NCIH I@ ;HAF?M <ON HIN >CMN;H=?M &H INB?L QIL>M NB? M=;F? I@ NB?
G;J CM HIN NB? M;G? ;N ?P?LS JICHN <ON ;N ; ACP?H JICHN NB? M=;F? CM NB? M;G? CH
;FF >CL?=NCIHM 1B? >IG;CH NI <? L?JL?M?HN?> CH JF;H? @ILG CM IH NB? MOL@;=? I@ ;
MJB?L? QBIM? JICHNM ;L? C>?HNCV?> <S NB?CL F;NCNO>?
;H> NB?CL FIHACNO>? M??
#CAOL? %?H=?@ILNB CN CM ;MMOG?> NB;N NB? F;NCNO>? ;H> FIHACNO>? ;L? ;HAF?M
?RJL?MM?> CH L;>C;HM =IOHN?> JIMCNCP?FS CH NB? HILNB?LH B?GCMJB?L? ;H> NI NB? ?;MN
I@ NB? $L??HQC=B *?LC>C;H ;H> H?A;NCP?FS INB?LQCM?
,H NB? <;MCM I@ ; MJB?L? Q? ;LLCP? ;N ; JF;H? =;FF?> NB? *?L=;NIL JF;H? QCNB NB?
B?FJ I@ NB? JLID?=NCIH QBC=B NI ;HS JICHN ) I@ NB? MJB?L? ;MMI=C;N?M ; JICHN )> I@
; =SFCH>?L N;HA?HN NI NB? ";LNB ;N NB? ?KO;NIL M?? #CAOL? 1B? =IIL>CH;N?M N> ;H> O> IH NBCM JF;H? ;L? >?>O=?> @LIG NB? A?IAL;JBC=;F =I
IL>CH;N?M <S
N> . %6!/,!6+*& 0 ,# #/"" 02/# " #),40
Greenw ich meridian
M
O
ϕ
λ
Equator
#CAOL? FIHACNO>? ;H> F;NCNO>?
O> . %!
%IQ?P?L NB? =IIL>CH;N?M QLCNN?H CH NBCM L?@?L?HNC;F >I HIN =ILL?MJIH> NI ; =IH
@ILG;F JLID?=NCIH <?=;OM?
@IL ;H ?;MN\Q?MN >CMJF;=?G?HN :N IH NB? MOL@;=? I@ NB? ";LNB Q? I<N;CH ;
>CMJF;=?G?HN :N> CH NBCM L?@?L?HNC;F MO=B ;M
:N "$
:N>
QBCF? @IL ; HILNB\MIONB >CMJF;=?G?HN :O Q? I<N;CH ; >CMJF;=?G?HN :O> MO=B
;M
:O "$
:O>
1BCM >CMNILNCIH QBC=B CM AL?;N?L CH NB? HILNB\MIONB >CL?=NCIH NB;H CH NB? ?;MN\Q?MN
>CL?=NCIH G;E?M NB? JLID?=NCIH HIH =IH@ILG;F 4? A?N ; =IH@ILG;F JLID?=NCIH <S ;
MCGJF? =B;HA? I@ M=;F? CH NB? O >CL?=NCIH 1B? H?Q IL>CH;N? 5 CM NB?H >?>O=?> @LIG
O> NB?H @LIG
<S
:5 "$
:O> .
"$
:
*IL?IP?L CN CM JIMMC<F? NI =;LLS ION ; =B;HA? CH NB? ILCACH ;N IH? JICHN FIHA NB? BILCTIHN;F ;RCM Q? CGG?>C;N?FS I<N;CH
4 . FIHA NB? P?LNC=;F ;RCM Q? I<N;CH <S CHN?AL;NCIH
! %!
5 . ! %!
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 Mp
M
yp
O
Cylinder tangent
to Earth
at the equator
xp
#CAOL? VLMN JB;M? I@ NB? *?L=;NIL JLID?=NCIH
1B?M? =IIL>CH;N?M 4 ;H> 5 ;L? =IIL>CH;N?M N;E?H IH G;JM ;H> CHJON NI NB?
MI@NQ;L? @IL HOG?LC=;F MCGOF;NCIH %?H=? IH? MBIOF> EHIQ BIQ NI ?RJL?MM NB? 0;CHN
3?H;HN ?KO;NCIHM CH NB?M? =IIL>CH;N?M 4 ;H> 5 <?CHA @OH=NCIHM I@ MJB?LC=;F =IIL>CH
;N?M CN CM NB?L?@IL? JL;=NC=;F NI ?RJL?MM NB? 0;CHN 3?H;HN ?KO;NCIHM CH NB?M? MJB?LC=;F
=IIL>CH;N?M ;N NB? P?LS <?ACHHCHA #IL NBCM Q? OM? NB? >CP?LA?H=? ;H> AL;>C?HN IJ
?L;NILM ?RJL?MM?> CH NB?M? =IIL>CH;N?M
:?L "Q . "$
=H7: < "
. "$
. "$
<
Q;
" "$
<
Q;
.
$CP?H NB? H;NOL? I@ NB? JBSMC=;F KO;HNCNC?M ;H> I@ NB? JLID?=NCIH NB? M=;F;L ;H>
P?=NILC;F KO;HNCNC?M >?JNB P?FI=CNS ;L? OH=B;HA?> 4? B;P? @IL ?R;GJF? " "F
;H> " "G
1B? >?LCP;NCP?M I@ NB? @OH=NCIHM QCNB L?MJ?=N NI ;H> ;L? NB?H L?JF;=?> <S NB?
>?LCP;NCP?M ;FIHA 4 ;H> 5 NB? *?L=;NIL =IIL>CH;N?M
<
< 4
4 <
< 4
4 < 5
<
.
5 4
< 5
.
5 "$
<
5
#LIG NBCM H?Q ?RJL?MMCIHM I@ >CP?LA?H=? ;H> AL;>C?HN ;L? >?>O=?>
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
:?L "Q
"$
"$
"F
4
"$
"F "$
4
=H7: < "$
"G
%!
"G
5
.
"G "$
"$
5
4
,
4
-
4CNB NB? *?L=;NIL =IIL>CH;N?M NB? G;MM =IHM?LP;NCIH ?KO;NCIH QBC=B B;M NI <?
MIFP?> CH ; HIH =IHM?LP;NCP? @ILG CM NBOM
>
J
QK
"$
=H7: >
>
"$
:?L QK >L
%!
.
1B?M? ?KO;NCIHM =IOF> <? MIFP?> CH NBCM @ILG <ON CH VHCN? ?F?G?HNM NBCM QIOF>
JIM? ; JLI<F?G QCNB NB? ;JJLIRCG;NCIH MJ;=? M ; G;NN?L I@ @;=N C@ CM ; JIFSHIGC;F
@OH=NCIH QB;N <?=IG?M I@ "$ &@ "$
;H> $!
;L? JIFSHIGC;FM QB;N CM
%! 1B?M? KO?MNCIHM ;L? @OH>;G?HN;F QB?H G;MM =IHM?LP;NCIH B;M NI <? JLIP?>
+IQ NB? G;MM =IHM?LP;NCIH ?KO;NCIH =;H ;FMI <? JON ;N NB? =IHNCHOOG F?P?F ;M
> "$
:?L "$
> QK J
QB?L? :?L CM HIQ NB? ;LN?MC;H >CP?LA?H=? IJ?L;NIL
4? NB?H I<M?LP? NB;N CHN?AL;NCIH CHNI NB? *?L=;NIL >IG;CH I@ NBCM ?KO;NCIH M??
0?=NCIH IH P;LC;NCIH;F @ILGOF;NCIH ACP?M NB? N?LGM
.
.
> "$
:4:5 ;H>
:?L > QK "$
:4:5
J NBCM F;MN N?LG <?CHA ?KO;F NI
.
> QK QD "$
:
&H NB?M? N?LGM ?;=B F?HANB :4 :5 IL : CM GOFNCJFC?> <S "$
QBC=B L?MNIL?M ;
L?;F FI=;F M=;F? 1B? PIFOG?M IL WOR =;F=OF;N?> ;L? NBOM L?;F KO;HNCNC?M &H NB? M;G?
Q;S NB? AL;>C?HN N?LGM QIOF> ACP? L?;F MFIJ?M
M CH NB? =;M? I@ ;FF IJ?L;NILM NB? =IIL>CH;N?M NB?L?@IL? ;JJ?;L GOFNCJFC?> <S
"$
1B? C>?; CM NI =;LLS ION NBCM IJ?L;NCIH IH=? ;H> @IL ;FF <?@IL? NB? MN;LN I@
=;F=OF;NCIHM 1BCM =;H <? @ILG;FCT?> <S =BIIMCHA ; =IHMN;HN F;NCNO>? J?L ?F?G?HN
1B? ;>P;HN;A? CM NB;N NB? ?KO;NCIHM NI <? QILE?> ION ;L? NB?H MNLC=NFS C>?HNC=;F NI NB?
;LN?MC;H ?KO;NCIHM NBCM CM ;=NO;FFS >O? NI NB? FI=;F =B;L;=N?L I@ NB? IJ?L;NILM ;H>
CM HIN ; A?H?L;F JLIJ?LNS 1BCM =BIC=? I@ =IHMN;HN F;NCNO>? J?L ?F?G?HN CM H;NOL;FFS
;H ;JJLIRCG;NCIH <ON CN ?H;<F?M OM NI L?N;CH MNLC=N VHCN? ?F?G?HN @ILG;FCMG NB?
;JJLIRCG;NCIH <?CHA MIG?BIQ NL;HM@?LL?> NI NB? =IIL>CH;N?M
!OLCHA ; =IGJON;NCIH OMCHA MJB?LC=;F =IIL>CH;N?M QCNB ; G?MBCHA CH NB? *?L=;NIL
JLID?=NCIH Q? MB;FF <OCF> OJ ; M?N I@ =IIL>CH;N?M GOFNCJFC?> <S "$
&@ Q? OM? NB?
=IIL>CH;N?M GI>CV?> CH NBCM G;HH?L NB? *?L=;NIL JLID?=NCIH CM H;NOL;FFS N;E?H CHNI
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 =IHMC>?L;NCIH ;H> HI INB?L GI>CV=;NCIH ;JJ?;LM ;A;CH CH NB? 0;CHN 3?H;HN ?KO;NCIHM
FF NB? G;MM ;H> WIQ =;F=OF;NCIHM ;L? =IB?L?HN ;H> JLIPC>? L?;F P;FO?M
1B? ?LLIL CH N;ECHA ; =IHMN;HN F;NCNO>? J?L ?F?G?HN =;H <? ?P;FO;N?> <S VH>CHA
NB? @ILG QBC=B NB? 0;CHN 3?H;HN ?KO;NCIHM MBIOF> N;E? QCNB NB? =IIL>CH;N?M 4 ;H>
5 MO=B ;M 4 4 "$
;H> 5 5 "$
1B? >C_?L?H=? ;N NB? =IHNCHOOG
F?P?F <?NQ??H NB? ?KO;NCIHM G;E?M NB? CAHIL?> N?LGM QBC=B ;L? I@ NB? IL>?L I@ .
;JJ?;L 1B?S N;E? CHNI =IHMC>?L;NCIH NB? P;LC;NCIHM CH F;NCNO>? QCNBCH NB? ?F?G?HN
&GJILN;HN HIN?M
1B? GOFNCJFC=;NCIH I@ =IIL>CH;N?M I@ NB? JICHNM <S NB? =IMCH? I@ ; =IHMN;HN
F;NCNO>? J?L ?F?G?HN F?;>M NI ; J;L;>IRC=;F MCNO;NCIH ; JICHN QBC=B <?FIHAM NI
M?P?L;F ?F?G?HNM B;M ;M G;HS M?NM I@ >C_?L?HN =IIL>CH;N?M QBC=B MBIOF> <? MNIL?>
?F?G?HN <S ?F?G?HN &H @;=N NBCM >I?M HIN JIM? ; N?=BHC=;F JLI<F?G ;H> >I?M
HIN F?;> NI ;HS =IHNL;>C=NCIH FF =;F=OF;NCIHM I@ P?=NILM IL G;NLC=?M CH VHCN?
?F?G?HNM MN;LN ;N NB? ?F?G?HN;LS F?P?F QB?L? =IB?L?H=? CM L?KOCL?> IHFS <?NQ??H
NB? =IIL>CH;N?M I@ NB? JICHNM I@ NB? ?F?G?HN &N MBIOF> ?P?H <? JIMMC<F? QCNBION
;HS NIJIFIAC=;F JLI<F?G NI <OCF> ; ZNQI >CG?HMCIH;F[ G?MBCHA I@ NB? ";LNB
@IL =;F=OF;NCHA QILF> NC>?M ?;=B ?F?G?HN B;PCHA =IIL>CH;N?M =ILL?MJIH>CHA NI ;
FI=;F JLID?=NCIH JIF?M MBIOF> <? ?R=FO>?> BIQ?P?L
1B? 0;CHN 3?H;HN ?KO;NCIHM CH MJB?LC=;F =IIL>CH;N?M B;P? <??H I<N;CH?> B?L?
<S MCGJFS =B;HACHA NB? >C_?L?HNC;F IJ?L;NILM GIL? =ILL?=N JLI=?>OL? B;M
<??H OM?> <S $L;S ;H> INB?LM 89 &N =IHMCMNM I@ ;P?L;ACHA IP?L NB? P?LNC=;F NB?
+;PC?L\0NIE?M ?KO;NCIHM QLCNN?H CH MJB?LC=;F =IIL>CH;N?M
?LN;CH >C_?L?H=?M
NB?H ;JJ?;L CH J;LNC=OF;L CH N?LGM >?J?H>CHA IH P?LNC=;F P?FI=CNS QBC=B FIAC=
;FFS ;L? IP?LFIIE?> CH NB? 0;CHN 3?H;HN ?KO;NCIHM <ON ;FMI >CMJ?LMCIH N?LGM
;LCMCHA @LIG =OLP;NOL? N?LGM QBC=B GI>C@S $&#& NB? @ILG I@ NB? IJ?L;NIL
I@ >C_OMCIH 1B?M? ;ONBILM BIQ?P?L CH>C=;N? NB;N NBCM >C_?L?H=? CM CGJILN;HN
IHFS @IL >C_OMCIH =I?^=C?HNM AL?;N?L NB;H G M ; P;FO? QBC=B ?R=??>M NB?
G;AHCNO>? =OLL?HNFS ;==?JN?> @IL >C_OMCIH ?P?H C@ IH? CH=FO>?M NB? >CMJ?LMCIH
JB?HIG?HIH
(08=C+4=0=C 4@D0C8>=B F8C7 ?>A>B8CH
FI=;F JILIMCNS =;H <? OM?> NI L?JL?M?HN NB? >?HMCNS I@ <OCF>CHAM CH ;H OL<;H TIH? IL
NL??M CH ; @IL?MN IL ;HS I<MN;=F?M QB?L? Q;N?L =;HHIN WIQ 1BCM =IH=?LHM ; N?=BHCKO?
EHIQH ;M BIGIA?HCT;NCIH QB?L? ; JBSMC=;F JB?HIG?HIH CM HIN >?M=LC<?> =IGJF?N?FS
<ON CM L?JF;=?> <S ; =IHNCHOIOM JLIJ?LNS I@ NB? ;L?; 1B? CHWO?H=? I@ I<MN;=F?M MG;FF?L
NB;H NB? MCT? I@ NB? G?MB =;H NBOM <? L?JL?M?HN?> -ILIMCNS =;H ;FMI <? OM?> NI >?;F
QCNB J;LNC;FFS Q?N ?F?G?HNM &H MO=B ; =;M? CN CM ?KO;F NI NB? J?L=?HN;A? I@ NB? MOL@;=?
=IP?L?> <S Q;N?L ,H? I@ NB? VLMN OM?M I@ JILIMCNS CH NB? 0;CHN 3?H;HN ?KO;NCIHM B;M
<??H JO<FCMB?> <S !?VH; ( 89 %IQ?P?L NB? JLIJIM?> @ILGOF;NCIH Q;M HIN
M;NCM@;=NILS @IL NB? =IHM?LP;NCIH I@ G;MM #IL ; J?L@?=N =IHM?LP;NCIH I@ G;MM Q? ;L?
JLIGJN?> NI JLIJIM? ; G;MM =IHM?LP;NCIH ?KO;NCIH MO=B ;M
>
:?L >
K J
QB?L? CM+ NB? FI=;F JILIMCNS 1B? NIN;F KO;HNCNS I@ Q;N?L CH NB? >IG;CH CM HIQ
QLCNN?H ;M > : &@ NB? JILIMCNS CM NB? NIN;F KO;HNCNS I@ Q;N?L CH NB? >IG;CH
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
CM I@ QB;N CN QIOF> <? QCNB =F;MMC=;F ?KO;NCIHM QCNBION I<MN;=F?M #IL ?R;GJF?
NBCM JILIMCNS N;E?M CHNI ;==IOHN <OCF>CHAM NB;N I==OJS I@ NB? MJ;=? IH NB? ALIOH>
1B?H NB?L? CM NB? KO?MNCIH I@ JILIMCNS CH NB? GIG?HNOG ?KO;NCIH ?;LCHA CH
GCH> NB? MCAHCV=;H=? I@ JILIMCNS CH IOL =;M? ; @;=NIL FCGCNCHA NB? ;P;CF;<F? MJ;=? IH
NB? ALIOH> NB? =IHM?LP;NCP? GIG?HNOG ?KO;NCIH CM QLCNN?H ;M ;H ?R;GJF? I@ NB?
=IGJIH?HN ;FIHA N
>K
J
:?L >K
K =>
6A
N
>"F
:?L >
B
=H7: K
QB?L? IH? ?H>M OJ L?JF;=CHA >?JNB > ?P?LSQB?L? <S > &@ NB? =IHNCHOCNS ?KO;NCIH CM
OM?> QCNB JILIMCNS NI A?N ; HIH =IHM?LP;NCP? @ILG Q? B;P?
K
J
6A
K =H7: K =
N
"F
:?L >
>
B
=H7: K
%?H=? IHFS NB? >C_OMCIH N?LG B;M =B;HA?>
-F;N? CFFOMNL;N?M NB? ?_?=N I@ JILIMCNS CH ; L?=N;HAOF;L =;H;F ;H> ?RJF;CHM CNM
A?IG?NLS &H @;=N CN =IH=?LHM ; MCHAF? >CG?HMCIH =;M? ;H> NB? =OLP?M MBIQH ;L?
FIHACNO>CH;F M?=NCIHM CH NB? ;RCM I@ NB? =;H;F #FIQ @LIG NB? F?@N NI NB? LCABN N;E?M
JF;=? ;N ; =IHMN;HN >CM=B;LA? I@ G M ;H> NB? >?JNB ;N NB? ?RCN CM G MO=B
NB;N NB? P?FI=CNS CM ;JJLIRCG;N?FS GM CH NB? =;H;F @IL GIL? L?;FCMG ; *;HHCHA
@LC=NCIH =I?^=C?HN I@ B;M <??H ;>>?> ;H> NB? NOL<OF?H=? CM GI>?FF?> QCNB NB?
A\ GI>?F 1B? JILIMCNS CM ?P?LSQB?L? ?R=?JN CH NB? TIH? <?NQ??H N ;H>
N G QB?L? CN P;LC?M FCH?;LFS @LIG NI OJ NI N NB?H @LIG NI ;M CM M??H CH NB? <INNIG >C;AL;G 1B? G;CH ?_?=N I@ JILIMCNS CM NB;N NB? P?FI=CNS
CH=L?;M?M @LIG GM NI GM FI=;FFS NI =IHM?LP? ; =IHMN;HN >CM=B;LA? HINB?L
?_?=N CM NB;N NB? Q;N?L >?JNB >?=L?;M?M NI ?HMOL? NB? =IHM?LP;NCIH I@ B?;> 1B? ?_?=N
I@ =IHMNLC=NCIH <S I<MN;=F?M MO=B ;M <LC>A? JC?LM B;M <??H L?JLI>O=?> B?L? %IQ?P?L
NBCM JILIMCNS >I?M HIN GI>?F NB? ?_?=NM I@ B?;> FIMM >O? NI NB? >L;A @IL=? I@ NB?M?
I<MN;=F?M #IL NBCM CN QIOF> <? H?=?MM;LS NI ;>> ;FMI ; >L;A @IL=? NI NB? ?KO;NCIHM
M?? 0?=NCIH >DBB8=4B@ 4@D0C8>=B
'IM?JB IOMMCH?MK \ Q;M ; MNO>?HN I@ 0;CHN 3?H;HN ;N NB? "=IF? +;NCIH;F?
>?M -IHNM ?N B;OMMf?M &H B? JLIJIM?> ;H ?RN?HMCIH I@ NB? 0;CHN 3?H;HN ?KO;
NCIHM NI N;E? CHNI ;==IOHN NB? ;==?F?L;NCIH N?LGM ;FIHA NB? P?LNC=;F &N CM HI FIHA?L
;MMOG?> NB;N JL?MMOL? CM BS>LIMN;NC= HIL CM NB? P?LNC=;F P?FI=CNS NIN;FFS CAHIL?> QB?H
>?;FCHA QCNB GIL? =OLP?> Q;P?M
6 NB;H QCNB 0;CHN 3?H;HN ?KO;NCIHM &H @;=N NB? =?F?LCNS
I@ Q;P?M CM HIN => <ON => 96
QB?L? A CM NB? Q;P? HOG<?L ;H> NB? 0;CHN
96
3?H;HN ?KO;NCIHM =ILL?=NFS JLI=?MM IHFS NB? FCGCN =;M? I@ FIHA Q;P?M 1B? >C_?L?HN
NB?ILC?M CGJLIPCHA IH NB? 0;CHN 3?H;HN ?KO;NCIHM H;G?FS <S IOMMCH?MK 0?LL? IL
(ILN?Q?A ;H> !? 3LC?M ;NN?GJN NI ;JJLI;=B NB? L?;F L?F;NCIH I@ >CMJ?LMCIH
9
7
=>
%! A>
A>
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 &H NB? IOMMCH?MK NB?ILS Q? =IHMC>?L NB;N NB? P?LNC=;F P?FI=CNS CM ; KO;HNCNS >?
6
J?H>CHA IH NB? VLMN IL>?L I@ 6 ;H> IH NB? M?=IH> IL>?L I@ ! QB?L? CM NB? ;GJFCNO>?
I@ NB? Q;P? ;H> ( CNM F?HANB %IQ?P?L NBCM ;JJLIRCG;NCIH I@ NB? P?LNC=;F P?FI=CNS
FCGCNM NI ; AL?;N ?RN?HN NB? ;JJFC=;NCIH >IG;CH ;H> MO<M?KO?HNFS G;HS ;>> IHM IL
BCAB?L ;JJLIRCG;NCIH IL>?LM B;P? <??H JLIJIM?> M?? CH J;LNC=OF;L 89 ;H> 89
%IQ?P?L NB? @ONOL? I@ NB? IOMMCH?MK ?KO;NCIHM M??GM NI <? CH D?IJ;L>S >O? NI =IG
J?NCNCIH @LIG NB? @L?? MOL@;=? HIH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM %?H=? Q?
MB;FF JL?M?HN B?L? IHFS NB? MN;H>;L> IOMMCH?MK ?KO;NCIHM @IL NB? JOLJIM? I@ =IG
J;LCMIH QCNB ; =IGJF?N? NBL?? >CG?HMCIH;F L?MIFONCIH &H ;FF =;M?M NB? 0;CHN 3?H;HN
=IHNCHOCNS ?KO;NCIH L?G;CHM OH=B;HA?> 3?LNC=;F P?FI=CNS CM VH;FFS ?FCGCH;N?> ;H> NBCM
L?MOFNM IHFS CH ; GI>CV=;NCIH I@ NB? GIG?HNOG ?KO;NCIH &H NB? MN;H>;L> IOMMCH?MK
?KO;NCIHM NBCM H?Q N?LG ;>>?> NI NB? LCABN B;H> MC>? CM
$ $ K
K
=H7: :?L
=H7: :?L $
J
J
QB?L? $ CM ; L?@?L?H=? Q;N?L >?JNB 4? MB;FF M?? BIQ NI QILE ION NBCM MOJJF?G?HN;LS
N?LG CH 0?=NCIH <ON ;N NBCM JICHN Q? =;H G;E? NB? @IFFIQCHA L?G;LEM
1B? L?@?L?H=? >?JNB $ ;H> NB? M=IJ? I@ NB? IOMMCH?MK ?KO;NCIHM >I HIN J?LGCN
; L?JL?M?HN;NCIH I@ OH=IP?L?> <?>M &@ NB? L?@?L?H=? >?JNB CM T?LI NB? =;M? I@
;H ;L?; QBC=B CM CHCNC;FFS >LS Q? @;FF <;=E IH NB? 0;CHN 3?H;HN ?KO;NCIHM
1B? MOJJF?G?HN;LS N?LG CM ; >?LCP;NCP? CH NCG? T?LI CH ; MN?;>S WIQ 4?
GOMN MJ?=C@S NB;N CN CM NB?L?@IL? OM?F?MM NI MIFP? NB? IOMMCH?MK ?KO;NCIHM @IL
J?LG;H?HN WIQM
(4AA4 4@D0C8>=B
&H (ILN?Q?A ;H> !? 3LC?M GI>CV?> NB? IOMMCH?MK G?NBI> ;H> <LIOABN ION
; H?Q NB?ILS I@ =OLP?> FIHA Q;P?M NB;HEM NI ; J?LCI>C=CNS BSJINB?MCM F?;>CHA NI
; MIFONCIH =;FF?> NB? Z=HIC>;F[ Q;P? QBC=B CH=FO>?M NB? MIFCN;LS Q;P? &H #LC?>LC=BM JLIJIM?> ; >CMNOL<;H=? G?NBI> <;M?> IH NB? L;NCI I@ NB? ;P?L;A? >?JNB
NI NB? Q;P?F?HANB %? @IOH> NB? 0;CHN 3?H;HN MIFONCIH CM I@ T?LI IL>?L &H NB? M;G?
S?;L (?FF?L >CM=IP?L?> NB? (ILN?Q?A ;H> !? 3LC?M MIFONCIH NI <? I@ VLMN IL>?L 1B?
A?H?L;F NB?ILS I@ #LC?>LC=BM NB?L?@IL? L?JL?M?HNM NB? EHIQH MIFONCIHM I@ FIHA Q;P?M
GIL? ;==OL;N?FS ,H? J;LNC=OF;L =;M? I@ NB? M?=IH> IL>?L CM );CNIH?YM MIFONCIH @IL
NB? MIFCN;LS Q;P? QBC=B QCFF <? >CM=OMM?> CH 0?=NCIH &H ;H ?HACH??L @LIG
-IHNM ?N B;OMMf?M #L;HdICM 0?LL? A;P? ;H ILCACH;F MIFONCIH NI ;==IOHN @IL NB? MFIJ?
I@ NB? <INNIG 1I IOL EHIQF?>A? NB?L? CM HI MI@NQ;L? CH VHCN? ?F?G?HNM NI MIFP? NB?
0?LL? ?KO;NCIHM NBIOAB ; VHCN? >C_?L?H=? JLIAL;G Q;M OM?> ;N );<IL;NICL? +;NCIH;F
>Y%S>L;OFCKO? CH #L;H=? CH NB? M &N CM H?P?LNB?F?MM CGJILN;HN NI BCABFCABN NBCM
QILE 1B? 0?LL? ?KO;NCIHM FCE? NBIM? I@ IOMMCH?MK ;>> GIL? N?LGM NI NB? 0;CHN
3?H;HN GIG?HNOG ?KO;NCIHM ON NB? ILCACH;F BSJINB?MCM NB? FCH?;LCNS I@ P?LNC=;F
P?FI=CNS <?NQ??H NB? <INNIG ;H> NB? @L?? MOL@;=? M??GM GIL? ?F?A;HN NB;H NB? FCGCN?>
>?P?FIJG?HN I@ NB? IOMMCH?MK ?KO;NCIHM 4? MB;FF ?F;<IL;N? NB?M? ;>>CNCIH;F N?LGM
<?FIQ 1BCM >?GIHMNL;NCIH =;H M?LP? ;M ; <;MCM @IL @ONOL? L?M?;L=B QBC=B =IOF> B?FJ
OM NI ?MN;<FCMB ; L?F;NCIH <?NQ??H NB? 0?LL? ;H> NB? +;PC?L\0NIE?M HIH BS>LIMN;NC=
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
?KO;NCIHM QCNB NQI G?MBCHA JF;H?M M?? NB? ;JJFC=;NCIH NI NB? MIFCN;LS Q;P? CH 0?=NCIH
0I Q? >I HIN H?AF?=N P?LNC=;F P?FI=CNS <ON Q? =IHMC>?L CN FCH?;L 1B;N CM
P 64
3
>
3 P 3
2.
QB?L? 3 CM 2B NB? P?FI=CNS ;N NB? <INNIG ;H> 3 3 NB? P?FI=CNS ;N NB? MOL@;=?
2.
3 CM NBOM ;FMI ?KO;F NI 26
2B
2B )?N OM ;FMI I<M?LP? NB;N
K =H7: > > :?L K
>
:>
:J
J
&N @IFFIQM NB;N
3
:?L K
>
3 <?CHA FCH?;L NB? >C_OMCIH N?LG CM =;H=?FF?> ;H> NB? 3 ?KO;NCIH ACP?M NB?
@IFFIQCHA JL?MMOL? AL;>C?HN
F
= P
3
J
1
3
N
2
3
O
3
3
P
&HN?AL;NCHA NBCM ?KO;NCIH <?NQ??H P ;H> NB? @L?? MOL@;=? 6A ACP?M
FP
= 6A P
.
H
.
3
J
1
3
N
2
3
O
3
P
4? HIQ OM? NB? P;FO? I@ 3 @LIG ?KO;NCIH 1B?H
6A P
3
J
3
>
6A P J
6 P
J
+ .
H
64
:P
+
B
:P ACP?M
3
>
S =IHMC>?LCHA NB;N 1 ;H> 2 ;L? =IHMN;HN ;FIHA NB? P?LNC=;F ;H> ?KO;F NI K ;H>
+.
L NB? N?LGM H 1 +
2 +
F
G :P ACP?
6A P K
3
N
6A P
K
N
3
>
K 6A P
N
64
3
>
6A P L
3
O
6A P
L
O
3
>
L 6A P
O
64
3
>
;H>
1B? N?LG
.
H
.
3
:P
P
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 ACP?M
3
3
6A P
>
3
>
6A P
6A P 64
3
>
&H NB? ?RJL?MMCIH F P QBC=B L?MOFNM @LIG NB? F;MN NBL?? ?KO;NCIHM Q? =;H @ILG
NIN;F >?LCP;NCP?M QCNB L?MJ?=N NI NCG? <S NB? @;=N NB;N @IL ?P?LS @OH=NCIH < :
:<
<> > :J
:J
< :>
:<
> :J
:J
3
>
6A P :
3
> :J
<
&N @IFFIQM NB;N
6A P :
>3 64 3
>
:J
FP
= 6A P
4? HIQ FIIE ;N
F P
6A
=
N
N
>H
F
6A P >
N
6A :
>3 64 3
> N :J
6A P > N
:
3
:J
:
3
:J
6A 6A
> N
:
>3 64 3
:J
> :
6A P
>3 64 3
>
N :J
6A P > :
3
>
N :J
1BCM =;H <? L?;LL;HA?> ;M
IL
:
:
6A
F P
=>
6A
>3 64 3
3
N
> N
:J
:J
:
> :
6A P
>3 64 3 >3 64 3
>
N :J
> N :J
:
6A P
> :
3 3
>
N :J
> N :J
.
H
.
.
H
1B? N?LG
.
6A
> 64
:P >
P
P
P
H
6A 64
>
6A P
:P 6A >
>
.
.
6A P
>
:P >
>64
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
.
.
H
CM NB?L?@IL? ?KO;F NI
F P
:P
N
:
6A
:
=>
> 64
3
>3 64 3
N
:J
:J
:
> :
>
64 :
>3 64 3 3 3
N
:J
:J
:J
> N
:
>3 64 3
:J
> N
:
3
:J
:
3
:J
>64 N
1B? N?LG
:
3
:J
>64 N
CM L?JF;=?> <S
> N
64
:
3
:J
> :
64
3
:J
N
,H NB? INB?L B;H> Q? HINC=? NB;N
:
>3 64 3
:J
4? NBOM B;P?
.
H
.
64
:
:
3 >3
:J
:J
F P
>
=>
:P N
N
64
=>
N
>
:
3
:J
>
> :3
:J
:3
:J
3
> :
3
:J
> :
64
3
:J
N
:
64
:J
> N
>
N
:
3
:J
> :3
:J
;H> NI =IH=FO>?
.
H
.
F P
6A
:P =>
N
N
>
2.
64
N
>
N
:3
:J
:
3
:J
:
3
:J
:3
:J
NB? ;>>CNCIH;F N?LGM CH NB? HIH
S L?;JJFSCHA 3 2B ;H> 3 26
2B
=IHM?LP;NCP? 0;CHN 3?H;HN GIG?HNOG ?KO;NCIHM ;L?
: > : 64
64 : 64
: >
>
> N
:J
:J
N
:J
:J
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 2MO;FFS Q? MN;N? NB;N 2 6
2B
;H> 2 .
2B
;H> Q? QLCN? NB?M? N?LGM ;M
>
N
6A
N
>
N
>
O
6A
O
>
O
;FIHA N ;H>
;FIHA O
1B? G;CH >C^=OFNS QCNB NB?M? N?LGM @IL HOG?LC=;F L?MIFONCIH CM NB? JL?M?H=? I@
NB? >?LCP;NCP?M
: 64
: >
;H>
:J
:J
+;gG 89 B;M OM?> NB? 0?LL? ?KO;NCIHM CH IH? >CG?HMCIH @IL MNO>SCHA ; F;H>MFC>?
CGJ;=N 1B? ;>>CNCIH;F N?LGM ;L? QLCNN?H CH NB? @ILG 2
;H> NB? ;FAILCNBG
2B
>?M=LC<?> CH 0?=NCIH CM OM?> QCNB ;H CHN?LG?>C;N? P;LC;<F? K 4CNB ;H CHPCM=C>
WOC> ;H> ; W;N <INNIG NB? 0?LL? NB?ILS ACP?M ; =HIC>;F MIFONCIH >L;Q<;=E I@
NB?M? ?KO;NCIHM CM NB;N NB? 0=BQ;LT =IH>CNCIH ?RJL?MMCHA =IHM?LP;NCIH I@ =OLF ;FIHA ;
NL;D?=NILS CM HIN P?LCV?>
(>DA24 C4A<B 0=3 1>3H 5>A24B 8= CF> 38<4=B8>=B
%?L? Q? MB;FF ?F;<IL;N? NB? N?LGM =;FF?> "F ;H> "G MI @;L =IHMNCNONCHA <I>S @IL=?M "
INB?L NB;H JL?MMOL? ;H> AL;PCNS 0IG? I@ NB?M? @IL=?M FCE? NB? CHWO?H=? I@ NB? QCH>
;L? CH @;=N MOL@;=? @IL=?M CH NBL?? >CG?HMCIHM <ON ;JJ?;L CH NB? 0;CHN 3?H;HN ?KO;NCIHM
;M MIOL=? N?LGM ;JJFC?> NI NB? QBIF? Q;N?L <I>S MCH=? NB? ?KO;NCIHM L?JL?M?HN ;H
;P?L;A? IP?L NB? P?LNC=;F
4? MB;FF MO==?MMCP?FS >CM=OMM
INNIG @LC=NCIH " 4
!L;A @IL=? ;H> B?;> FIMM " B
1B?
ILCIFCM @IL=? " 1
1B? CHWO?H=? I@ NB? QCH> " D
NGIMJB?LC= JL?MMOL? " /
1B? MIOL=?M I@ GIG?HNOG " ?
0J;NC;F P;LC;NCIHM I@ >?HMCNS " 2
1C>? A?H?L;NCHA N?LGM " ;
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
>CC>< 5A82C8>=
#LC=NCIH IH NB? <INNIG CM ; G;DIL OHEHIQH CH ?HPCLIHG?HN;F MNO>C?M ;H> NB? @LC=NCIH
F;QM NB?GM?FP?M ;L? GIMNFS ?GJCLC=;F 1BCM CM QBS =;FC<L;NCHA HOG?LC=;F GI>?FM CM MI
CGJILN;HN
;FC<L;NCIH =;H <? >IH? QCNB ;ONIG;NC= N?=BHCKO?M MO=B ;M J;L;G?N?L
?MNCG;NCIH M?? B;JN?L #;CLFS L?=?HN JO<FC=;NCIHM 89 ;H> 89 MBIQ NB? VLMN
;NN?GJNM ;N ; @LC=NCIH J;L;G?N?LCT;NCIH CH LCP?L WII> GI>?FM OMCHA P?A?N;NCIH B?CABNM
>?LCP?> @LIG ;CL<ILH? M=;HHCHA F;M?L ;FNCG?NLS
?@IL? =IHMC>?LCHA NB? L?;F @LC=NCIH F;QM F?N OM VLMN =CN? NB? FCH?;L F;Q
!8=40A ;0F NBCM F;Q CM L;L?FS OM?> ;M CN >I?M HIN L?JL?M?HN JBSMC=M @LC=NCIH @IL=?M
;L? ; KO;>L;NC= @OH=NCIH I@ P?FI=CNS %IQ?P?L Q? =;H OM? NBCM F;Q @IL GI>?FFCHA INB?L
JB?HIG?H; &NM @ILG CM
"F4 8 K
"G4 8 L
QB?L? 8 CM NB? @LC=NCIH =I?^=C?HN CH M
'40; 5A82C8>= ;0FB CH IH? >CG?HMCIH @LC=NCIH MNL?MM ?R?LN?> <S ; JF;H? BILCTIHN;F
Q;FF IH ; WIQ QBC=B CM J;L;FF?F NI CN CM ?RJL?MM?> ;==IL>CHA NI NB? @ILGOF;
K
4 K
D
4? ;FMI QLCN?
FH FH CM ?RJL?MM?> CH EA GM
K
4 K
D
QCNB
>?HMCNS I@ NB? FCKOC> EAG 4 @LC=NCIH =I?^=C?HN >CG?HMCIHF?MM
K WIQ P?FI=CNS GM
0NL?MM ;=NM CH NB? >CL?=NCIH I@ NB? =OLL?HN ;H> CH NQI >CG?HMCIHM B;M NB? NQI
=IGJIH?HNM
5
K
FH 4 K
L
;H>
5
L
L
GH 4 K
1B? =I?^=C?HN 4 CM B;L>FS OM?> CH @L?? MOL@;=? BS>L;OFC=M ;H> CM OMO;FFS L?JF;=?>
<S INB?LM 1B? <?MN EHIQH
CM NB? BfTS =I?^=C?HN >?HIN?> ;H> 4 ;L? <IOH>
6
<S NB? @ILGOF; 5 >CG?HMCIH G M #IL BfTS NB? P?FI=CNS OM?> CH NB?
@LC=NCIH F;Q CM NB? WIQ P?FI=CNS CH ; =;H;F QBC=B CM ?KO;F NI NB? BILCTIHN;F P?FI=CNS
IHFS C@ NB? P?LNC=;F P?FI=CNS CM H?AFCAC<F?
4? B;P? M??H NB;N CH NB? +;PC?L\0NIE?M ?KO;NCIHM NB? @LC=NCIH ;N NB? <INNIG CM
?KO;F NI 4 QD4 1B? ;P?L;ACHA QBC=B F?;>M NI NB? HIH =IHM?LP;NCP? @ILG I@ NB?
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 +;PC?L\0NIE?M ?KO;NCIHM MCGJFS ACP?M IH? N?LG 6 4 QD4
HILG;F NI NB? <INNIG CM
8
64
64
N
O
1B? HILG I@ NB? P?=NIL
QBC=B CM NB? =IMCH? L?=CJLI=;F I@ NB? MN??J?MN MFIJ? ;N IH? JICHN ;H> QCFF <? >?HIN?>
;M "$ 4? A?N NB? @IFFIQCHA BfTS @ILGOF; @IL NB? MB?;L @IL=? ;N NB? <INNIG NI
<? ;>>?> NI NB? HIH =IHM?LP;NCP? @ILG I@ GIG?HNOG ?KO;NCIH
"Q 4 QK
= 5
K
"$ >
L
QB?L? CM NB? BfTS =I?^=C?HN
BfTSYM @LC=NCIH F;Q Q;M ?MN;<FCMB?> @IL ; OHC@ILG WIQ L?ACG? <ON CM ?RN?H>?> NI
;FF NSJ?M I@ WIQ #CH;FFS CN CM ?RJL?MM?> CH IOL ?KO;NCIHM ;M @IFFIQM
7RIHLB ;0F
K
= 5
K
L
"$ >
L
= 5
"G4 K
L
"$ >
%IQ?P?L NB? BfTS =I?^=C?HN CM HIN L?;FFS ; =IHMN;HN >CG?HMCIH;F ;H;FSMCM ;H>
?RJ?LCG?HNM MBIQ NB;N CN MBIOF> >?J?H> IH NB? >?JNB ;H> IH NB? <INNIG LIOABH?MM
1B? ?GJCLC=;F F;Q I@ *;HHCHA ;H> 0NLC=EF?L =ILL?=NM NBCM JLI<F?G ;H> MJ?=CV?M NB?
P;FO? I@ NB? BfTS =I?^=C?HN &N Q;M MJ?=CV=;FFS >?P?FIJ?> NI I<N;CH ; @LC=NCIH =I?@
V=C?HN CH>?J?H>?HN I@ NB? >?JNB ;H> B?H=? IHFS ; @OH=NCIH I@ NB? <INNIG LIOABH?MM
C ? '> QB?L? ' G M CM NB? 0NLC=EF?L =I?^=C?HN &H IH? >CG?HMCIH NB?
>?JNB > GOMN <? L?JF;=?> <S NB? BS>L;OFC= L;>COM . 1B? ;JJLIRCG;NCIH . >
CM CH JLCH=CJF? NLO? IHFS @IL P?LS F;LA? =;H;FM ' P;LC?M <?NQ??H ;<ION ;H> &NM P;FO? CM NI @IL ; MNL?;G ;H> NI @IL ; =IH=L?N? =;H;F
BfTSYM F;Q QCNB '> F?;>M NI ; H?Q F;Q EHIQH ;M 0NLC=EF?LYM F;Q
"F4 (CA82:;4ALB ;0F
5
K
=
K
L
"$ > '
5
L
=
"G4 K
L
"$ > '
!O? NI NB? >C_?L?H=? <?NQ??H BS>L;OFC= L;>COM ;H> >?JNB =IGJ;LCMIH QCNB =I>?M
CH IH? >CG?HMCIH CM >C^=OFN
;FC<L;NCIH CH IH? >CG?HMCIH QCFF HIN ;FQ;SM <? P;FC>
@IL NQI >CG?HMCIHM
1B? 0NLC=EF?L =I?^=C?HN =;H <? FCHE?> NI NB? <INNIG LIOABH?MM AA <S NB? /;G?NN?
@ILGOF;
=
' AA
1B? *;HHCHA @ILGOF; GIL? =IGGIHFS OM?> CH HAFI 0;RIH =IOHNLC?M CM ; MCGJF?
P;LC;HN I@ NB? 0NLC=EF?L @ILGOF; QCNB ; =I?^=C?HN C NB? CHP?LM? I@ ' 0?? 89 QBC=B
ACP?M ;>PC=? IH =BIC=? I@ =I?^=C?HNM
"F4 %6!/,!6+*& 0 ,# #/"" 02/# " #),40
"0==8=6LB ;0F
"F4 K =C 5
K
"$ >
L
L =C 5
K
L
"$ >
+IN? <?=;OM? CN CM OM?> CH HAFI 0;RIH =IOHNLC?M NB? *;HHCHA =I?^=C?HN CM
MIG?NCG?M ACP?H CH CGJ?LC;F OHCNM
"G4 #8:DA03B4LB ;0F NBCM F;Q MN?GM @LIG ; >?JNB ;P?L;A?> P;FO? I@ NB? FIA;LCNBGC=
NOL<OF?HN P?FI=CNS JLIVF? M?? "KO;NCIH CH 0?=NCIH 1B? =ILL?MJIH>CHA
BfTS =I?^=C?HN CM I<N;CH?> <S NB? @ILGOF;
! >
AA
QB?L? > CM NB? Q;N?L >?JNB ;H> AA NB? LIOABH?MM @;=NIL ;N NB? <INNIG 1BCM F;Q CM
@;PIOL?> <S (F;;MM?H ;H> 3;H >?L 7Q;;L> 89 @IL WII>?> JF;CHM QCNB P?A?N;NCIH
@D8E0;4=24 >5 5A82C8>= 2>4P284=CB 8= 0=3 ; MI@NQ;L? OM?L ?RJ?=NM
NB? @LC=NCIH =I?^=C?HN NI <? CH>?J?H>?HN I@ HOG?LC=;F GI>?FFCHA %?H=? =;FC<L;NCIH
=;LLC?> ION CH ! MBIOF> ;FMI <? P;FC> CH ! %IQ?P?L Q? GOMN ?HMOL? NB;N B?;> FIMM
>O? NI @LC=NCIH CM C>?HNC=;F CH ! ;H> ! 1B? B?;> FIMM?M ;L? ; @OH=NCIH I@ NB? MKO;L?
I@ NB? P?FI=CNS MI Q? MBIOF> <? =;L?@OF NI L?N;CH NB? M;G? P?FI=CNS NB;N CM ;M Q? B;P?
IHFS NBCM IH? CH ! NB? >?JNB ;P?L;A?> P?FI=CNS
"RJL?MM?> CH >CG?HMCIHF?MM =I?^=C?HN N?LGM 4 @LC=NCIH ;JJ?;LM CH NB? 0;CHN
3?H;HN ?KO;NCIHM ;M
5
K
L
INB?L N?LGM 4 K K
J
>
5
L
L
INB?L N?LGM 4 L K
J
>
&@ IH? =IHMC>?LM ; L?=N;HAOF;L L?;=B B;PCHA >?JNB > F?HANB N ;H> QC>NB ( NB?
MB?;L @IL=? ;=NCHA IH NB? WOC> CM NB? JLI>O=N I@ G;MM <S ;==?F?L;NCIH H?A;NCP? >O?
NI @LC=NCIH 1B? =IGJIH?HNM I@ NBCM @IL=? ;L?
4 5
L N (
K K
4 5
L N (
<G L K
&H ! Q? L?;MIH CH N?LGM I@ <INNIG MB?;L MNL?MM QBIM? =IGJIH?HNM ;L?
<F ;H>
1
4 5
1
D
2 1
2
4 5
1
D
2 2
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 #IL NB? M;G? L?;=B MB?;L @IL=? HIQ B;M NB? =IGJIH?HNM
<F <G 4 5
1 1
4 5
2 1
2 N (
2 N (
4? M?? NB;N NI B;P? ;H ?KOCP;F?HN ?_?=N K 1 ;H> L 2 MBIOF> <? P?LCV?> NB;N
CM NB? P?FI=CNS ;N NB? <INNIG CM ?KO;F NI ;P?L;A? P?FI=CNS IP?L NB? P?LNC=;F 1BCM CM
JL?=CM?FS HIN MI QB?H NB?L? CM @LC=NCIH #IL ?KOCP;F?H=? I@ @LC=NCIH <?NQ??H ! ;H> !
CH <INB =;M?M NB? ;P?L;A? P?FI=CNS IP?L NB? P?LNC=;F MBIOF> <? L?N;CH?> @IL =;F=OF;NCHA
NB? @LC=NCIH N?LGM *IL?IP?L C@ @IL HOG?LC=;F L?;MIHM MNL?MM CM G;>? CGJFC=CN ;M
1
1
D
K
L NB?H CH JLCH=CJF? =I?^=C?HN MBIOF> <? =ILL?=N?> <S ;
QCNB @;=NIL K1 1 ?KO;NCIH ;H> L2 2 ?KO;NCIH &H JL;=NC=? NB? M;G? =I?^=C?HN
C D QCFF <? OM?> @IL <INB ?KO;NCIHM 1BCM L?;MIHCHA CGJFC?M NB;N =I?^=C?HN 4 CM
) * NB? M;G? CH ! ;H> ! ;H> NB;N CN CM =;F=OF;N?> QCNB NB? P?FI=CNC?M ;P?L;A?> IP?L NB?
P?LNC=;F -F;N? MBIQM NB? @L?? MOL@;=?M I<N;CH?> CH ; L?=N;HAOF;L =LIMM M?=NCIH =;H;F
QCNB NB? 0;CHN 3?H;HN ;H> NB? +;PC?L\0NIE?M BS>LIMN;NC= ?KO;NCIHM 1B? <INNIG CM
W;N ;N ?F?P;NCIH 1B? >?JNB ;N NB? >IQHMNL?;G ?H> CM G 1B? >CM=B;LA? CM G M ;H> NB? QC>NB I@ NB? =;H;F CM G MO=B NB;N NB? P?FI=CNS ;N NB? >IQHMNL?;G
?H> CM GM 1B? ?F?G?HN MCT? CH NB? G?MB CM ;JJLIRCG;N?FS G 1B? =LIMM M?=NCIH
;N NB? <INNIG I@ NB? JF;N? FIHACNO>CH;F M?=NCIH ;FIHA NB? ;RCM O G =IHVLGM
NB;N NB? @L?? MOL@;=?M ;L? NB? M;G? QCNB ;H C>?HNC=;F 0NLC=EF?L =I?^=C?HN I@ CH NB?
NQI =;F=OF;NCIHM
22>D=C8=6 5>A 3A06 5>A24B 0=3 7403 ;>BB4B
&H NB? 0;CHN 3?H;HN ?KO;NCIHM @IL?MNM ;H> OL<;H ;L?;M ;L? NL;>CNCIH;FFS L?JL?M?HN?>
<S ; @LC=NCIH =I?^=C?HN A?H?L;FFS ; FIQ 0NLC=EF?L =I?^=C?HN C ? L?JL?M?HNCHA ; BCAB
@LC=NCIH %IQ?P?L NBCM ;JJLI;=B B;M ; >?@?=N MCH=? NB? MOL@;=? @LC=NCIH I@ P?LNC=;F
MNLO=NOL?M CH=L?;M?M QCNB Q;N?L >?JNB ;M IJJIM?> NI @LC=NCIH ;N NB? <INNIG %?H=?
NB? BfTS ;H> 0NLC=EF?L F;QM ;L? HIN ;JJFC?> =ILL?=NFS &H NBCM J;L;AL;JB ;H ILCACH;F
;JJLI;=B CM JL?M?HN?> NI N;E? CHNI ;==IOHN CH NB? ?KO;NCIHM NB? ?_?=N I@ P?LNC=;F
MNLO=NOL?M FCE? NL??M IL <LC>A? JC?LM 1B? ;CG CM NI ;PIC> JL?=CM?FS L?JL?M?HNCHA NB?M?
I<MN;=F?M CH ; >IG;CH I@ =;F=OF;NCIHM <ON NI L?JL?M?HN NB?G FI=;FFS ;M ; @IL=? =IG
J;L;<F? NI ; MB?;L @IL=? 1BCM QCFF B;P? NB? M;G? ?_?=N ;M NB? >L;A @IL=? I@ I<MN;=F?M
==IL>CHA NI NB? ;=NCIH\L?;=NCIH JLCH=CJF? NB? MNL?MM I@ NB? WOC> IH ; MNLO=NOL? CM NB?
IJJIMCN? I@ NB? MNLO=NOL?YM MNL?MM IH NB? WOC> )?N OM ;MMOG? NB;N Q? B;P? D P?LNC=;F
=SFCH>?LM QCNB ; >C;G?N?L
>CMNLC<ON?> CH ; >IG;CH QCNB MOL@;=? ;L?; ;H> F?N
OM ;FMI ;MMOG? NB;N NB? P?LNC=;F =SFCH>?LM ;L? MO^=C?HNFS N;FF NI ?G?LA? =IHNCHOIOMFS
#IL ?;=B =SFCH>?L NB? >L;A @IL=? CM
"B 1
>
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
QB?L? CM NB? >L;A =I?^=C?HN I@ NB? =SFCH>?L ;M CH NB? N;<F?M QB?L? CM ;
@OH=NCIH I@ NB? /?SHIF>M HOG<?L ;H> QBIM? P;FO? L;HA?M <?NQ??H ;H> CH GIMN
;JJFC=;NCIHM 1B? MNL?MM CH>O=?> <S NB? D MNLO=NOL?M IH NB? MOL@;=? CM
D"
&@ Q? QCMB NI ;>> NB? ?_?=N I@ NBCM MNL?MM CH NB? HIH =IHM?LP;NCP? 0;CHN 3?H;HN
?KO;NCIHM Q? MBIOF> ;>> ; N?LG 6
CH NB? LCABN B;H> G?G<?LM I@ NB? GIG?HNOG
?KO;NCIH 1B? ?_?=N I@ NB? MNLO=NOL?M =ILL?MJIH>M NI ;H ;>>CNCIH;F MIOL=? N?LG
"FB QBC=B Q? =;H =IGJ;L? NI
D
1
BfTSYM F;Q
=
1
>
/?MIFPCHA >L;A @IL=? CH N?LGM I@ NB? BfTS =I?^=C?HN QIOF> ACP? ; P;FO? NB;N
CM >?J?H>?HN IH NB? >?JNB QBC=B MIG? HOG?LC=;F GI>?FM @;=?> QCNB NBCM JLI<F?G
;=NO;FFS >I 4? JL?@?L NI >CL?=NFS ;>> NB? >L;A @IL=? ;M
"FB D
1
1B? >;N; L?KOCL?> @IL ; =;F=OF;NCIH ;L? ;M @IFFIQM
"FB 1B? HOG<?L I@ MNLO=NOL?M J?L MOL@;=? OHCN D
1B? >C;G?N?L I@ MNLO=NOL?M
1B? >L;A =I?^=C?HN FF NB?M? >;N; =;H <? ?;MCFS =IOHN?> IL G?;MOL?> ;M IJJIM?> NI ; 0NLC=EF?L IL
BfTS =I?^=C?HN QBC=B =;H IHFS <? ?MNCG;N?>
&@ NB? MNLO=NOL?M B;P? >C_?L?HN >C;G?N?LM D QCFF MCGJFS <? L?JF;=?> <S NB? MOG
I@ ;FF NB? >C;G?N?LM
&H N?LGM I@ MCHAOF;L B?;> FIMM CN CM
$ A
1
=
<
QCNB A 9
&H N?LGM I@ FCH?;L B?;> FIMM Q? B;P? ; =I?^=C?HN F
QBC=B QB?H ;JJFC?> NI ;
<
F?HANB N CH>O=?M ; B?;> FIMM ?KO;F NI N
-F;N? MBIQM NB? MCGOF;NCIH I@ WIQ ;LIOH> ; G >C;G?N?L <LC>A? JC?L CH ;
=;H;F QCNBION L?JL?M?HNCHA NB? JC?L CH NB? VHCN? ?F?G?HN G?MB 1B? >L;A @IL=? CM
>CMNLC<ON?> IP?L ;H ;L?; <?NQ??H N G ;H> N G &H NB? M;G? ;L?; ;
JILIMCNS ?KO;F NI CM N;E?H CHNI ;==IOHN NI =L?;N? ;H ?_?=N I@ =IHMNLC=NCIH 1B?
>?JNB CM G NB? >CM=B;LA? G M ;H> NB? P?FI=CNS ;JJLIRCG;N?FS GM >L;A =I?^=C?HN ?KO;F NI L?MNIL?M ; B?;> FIMM C>?HNC=;F NI ; JC?L CH=FO>?> CH NB?
G?MBCHA ,ONMC>? NB? ;L?; <?NQ??H N G ;H> N G C ? >C;G?N?LM
CN =;HHIN <? =IHVLG?> NB;N NB? JC?L B;M <??H MCGOF;N?> ,HFS NB? V?F> H?;L NB?
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 JC?L CM >C_?L?HN 1B? MCGOF;NCIH CM LOH QCNB ; A\ NOL<OF?H=? GI>?F 1B? >L;A
=I?^=C?HN CHN?AL;N?M ; =ILL?=NCIH @;=NIL >O? NI NB? @;=N NB;N NB? P?FI=CNS CH NB? ;RCM
I@ NB? =;H;F QB?L? NB? JC?L CM FI=;N?> CM AL?;N?L NB;H NB? ;P?L;A? P?FI=CNS &@ CN CM
;==?JN?> NB;N NB? L;NCI 1" 1/D3@/53 CM ?KO;F NI ;; QCNB C @IL ; /?SHIF>M
HOG<?L I@ ;<ION M?? &>?FY CE 89 NB?H NB? =ILL?=NCIH NI <? ;JJFC?> NI NB?
>L;A @IL=? CM ;;
QBC=B CM CH NBCM =;M? 1B? L?;F =I?^=C?HN QIOF> CH @;=N
<? QBC=B
=ILL?MJIH>M NI ; P;FO? I@ /?SHIF>M HOG<?L <?NQ??H ;H> 1B? P;FO?M ;L? CH NB? =ILL?=N L;HA? NB? G;CH OH=?LN;CHNS ;LCMCHA JLI<;<FS @LIG
HOG?LC=;F @;=NILM ? A NB? P?FI=CNS JLIVF? ;N NB? ?HNL;H=? I@ NB? =;H;F CM ACP?H ;M
=IHMN;HN ;H> CM HIN S?N Q?FF ?MN;<FCMB?> CH NB? PC=CHCNS I@ NB? JC?L #IL GIL? >?N;CFM
IH NBCM =;M? L?@?L?H=? 89 G;S <? =IHMOFN?>
1B? NB?ILS =;H <? ?RN?H>?> NI MO<G?LA?> MNLO=NOL?M IH NB? =IH>CNCIH NB;N IH?
;>>M NI NB? @IL=? ; GOFNCJFC?L =I?^=C?HN 62
QB?L? : CM NB? B?CABN I@ NB? P?LNC=;F
6
MNLO=NOL?M -F;N? CM ; L?;F ;JJFC=;NCIH QB?L? NB? ?_?=N I@ G;LCH? NOL<CH?M IH NC>;F
=OLL?HNM Q;M GI>?FF?> QCNB >L;A @IL=?M ;M >?M=LC<?> B?L? 1B? =IFIOL?> MOL@;=?M ;L?
NB? P?FI=CNS CHN?HMCNS CH GM 1B? G?MB <?CHA B?;PCFS L?VH?> IHFS ; @?Q P?FI=CNS
P?=NILM ;L? MBIQH
>A8>;8B 5>A24
#IL NB? M;E? I@ =IGJ;NC<CFCNS QCNB NB? H?AFCAC<F? P?LNC=;F P?FI=CNS BSJINB?MCM IHFS NB;N
J;LN I@ @IL=? >O? NI BILCTIHN;F P?FI=CNS CM L?N;CH?> 1B? ! ILCIFCM @IL=? CM
"$
"$ * 3 $! 2
&
"$
$! * 3 $! 1
2 %
"$
$! * 2 "$
"$ * 1
S H?AF?=NCHA 3 ;H> NB? P?LNC=;F =IGJIH?HN Q? A?N
"F1 $!
"G1 $!
L <L
K < K
QB?L? Q? M?N < $! =LIMM MG;FF ;L?;M NBCM =I?^=C?HN =;H <? =IHMC>?L?> ;M ;
=IHMN;HN =LIMM F;LA? ;L?;M QB?L? NB? *?L=;NIL JLID?=NCIH CM OM?> NBCM J;L;G?N?L
P;LC?M CH MJ;=?
1B? ;HAOF;L P?FI=CNS I@ NB? ";LNB CM L>M C ? L;>C;HM J?L
MC>?L?;F >;S ?KO;F NI I@ ; BIOL >;S NB;N CM NI M;S M?=IH>M #IL ;
F;NCNO>? I@ Q? A?N < +GM
=KD4=24 >5 F8=3
&H ; MCGCF;L Q;S ;M @IL @LC=NCIH ;N NB? <INNIG QCH> @IL=CHA N;E?M NB? @ILG @L?? MOL@;=?
MFIJ? B;M <??H CAHIL?>
6
/7@
"FD 7E7<2 1E7<2 1E7<2 2E7<2
> E/B3@
6
/7@
7E7<2 2E7<2 1E7<2 2E7<2
"GD > E/B3@
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
QB?L? 7E7<2 CM NB? QCH> @IL=CHA =I?^=C?HN ;FL?;>S >CM=OMM?> CH 0?=NCIH 1E7<2
;H> 2E7<2 NB? =IGJIH?HNM I@ QCH> P?FI=CNS CH GM MNLC=NFS MJ?;ECHA NBCM CM P?FI=CNS
L?F;NCP? NI NB? WOC> ;H> CM NB? L;NCI I@ NB? >?HMCNC?M I@ ;CL ;H> Q;N?L +OG?LIOM
?GJCLC=;F @ILGOF;? ACP? NB? P;FO? I@ 7E7<2 2H>IO<N?>FS CNM P;FO? >?J?H>M IH NB?
MCN? <?CHA MNO>C?>
C<>B?74A82 ?A4BBDA4
4B?H ;NGIMJB?LC= JL?MMOL? ,/B; CM =IHMC>?L?> >CG?HMCIH J;M=;F EA G M NB? N?LG =H7: ,/B; CM MCGJFS ;>>?> NI NB? LCABN B;H> MC>? I@ NB? GIG?HNOG
?KO;NCIH 1B;N CM
"F/ ,/B;
N
"G/ ,/B;
O
1B? JL?MMOL? N?LG CM I@N?H ;>>?> NI NB? @L?? MOL@;=? QBC=B ;FMI ;JJ?;LM NBLIOAB CNM
AL;>C?HN ON NBCM CM HIN >IH? ;M =;L? MBIOF> <? N;E?H NI L?N;CH ; L?;F @L?? MOL@;=? @IL
VH>CHA OH=IP?L?> ;L?;M 1B? ;NGIMJB?LC= JL?MMOL? AL;>C?HN MBIOF> HIN <? =IHMC>?L?>
CH NBIM? OH=IP?L?> ;L?;M QB?L? CN QIOF> <? ; J;L;MCNC= >LCPCHA N?LG
(>DA24B >5 <><4=CD<
M M??H CH 0?=NCIH MIOL=?M I@ >CM=B;LA? CH>O=? MIOL=?M I@ GIG?HNOG ?KO;F NI
"F? KA13 K /9;
>
"G? LA13 L /9;
>
K L KA13 LA13 ;H> /9; ;L? ?RJL?MM?> CH GM
&GJILN;HN HIN? QB?H NB?M? N?LGM ;L? HIN N;E?H CHNI ;==IOHN CN CM CGJFC?> NB;N
NB? CHN;E? IL IONJON P?FI=CNS I@ NB? MIOL=? CM ?KO;F NI NB;N I@ NB? =OLL?HN
(?0C80; E0A80C8>=B >5 34=B8CH
M NB? 0;CHN 3?H;HN ?KO;NCIHM L?MOFN @LIG ;H ;P?L;A? IP?L NB? P?LNC=;F NB?S =;HHIN
L?JLI>O=? MNL;NCV=;NCIHM ;LCMCHA @LIG P?LNC=;F P;LC;NCIHM I@ >?HMCNS ; @OH=NCIH I@ M;FCH
CNS IL N?GJ?L;NOL? %IQ?P?L CN CM JIMMC<F? NI N;E? CHNI ;==IOHN BILCTIHN;F P;LC;NCIHM
I@ >?HMCNS &@ M;FCHCNS CM >?HIN?> ;M / ;H> N?GJ?L;NOL? ;M 0 CM ; @OH=NCIH / 0
QB?L? / ;H> 0 ;L? @OH=NCIHM I@ N O ;H> J 1B?M? P;LC;NCIHM B;P? NQI ?_?=NM
!CF;N;NCIH I@ Q;N?L
!C_?L?HNC;F ?_?=NM CH AL;PCNS
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 !CF;N;NCIH I@ Q;N?L CM ; M?=IH>;LS ?_?=N QBC=B QCFF <? CAHIL?> ;M CH ! ,H
NB? INB?L B;H> NB? >C_?L?HNC;F ?_?=NM I@ AL;PCNS >O? NI P;LC;NCIHM CH M;FCHCNS =;H <?
BCABFCABN?> CH ?MNO;LC?M ;H> MIG?NCG?M NB?S B;P? NI <? =IHMC>?L?> ? A CH NB? +ILNB
0?; 1B? IOMMCH?MK ;JJLIRCG;NCIH CM OM?> @IL NBCM JOLJIM? OIS;H=S N?LGM CH NB?
0;CHN 3?H;HN ?KO;NCIHM ;L? DOMN NB? ;P?L;A? IP?L NB? P?LNC=;F I@ NB? <OIS;H=S N?LGM
I@ NB? +;PC?L\0NIE?M ?KO;NCIHM QBC=B ACP?
"F2 >
"G2 >
.
.
.
.
.
.
=
=
=
=
F/B;
N
.A
.
6A
% &
:P
=
=
:P = N
N
=
H
F/B;
O
.A
.
6A
% &
:P
=
=
:P = O
O
=
H
1B? ;NGIMJB?LC= JL?MMOL? AL;>C?HN CGG?>C;N?FS ACP?M NB? N?LGM
=
=
F/B;
;H>
N
=
=
F/B;
O
CH QBC=B CM NB? ;P?L;A? I@ IP?L NB? P?LNC=;F
#IL NB? INB?L NQI N?LGM MBIOF> <? =B;HA?> NI &H @;=N ; ! >?HMCNS CM HIN
;==?MMC<F? CH ! 1B? IHFS CH@ILG;NCIH NB;N =;H <? >?>O=?> @LIG ;P?L;A? N?GJ?L;NOL?
;H> M;FCHCNS IP?L NB? P?LNC=;F CM OIS;H=S N?LGM ;L? NB?H CHN?AL;N?> CH
"F2 => N
=> O
1BCM CM ; <;LI=FCHC= JL?MMOL? AL;>C?HN 1B? <;LINLIJC= JL?MMOL? AL;>C?HN JL?M?HN
CH ! >CM;JJ?;LM
"G2 )83464=4A0C8=6 5>A24
;F=OF;NCIHM ;H> ?RJF;H;NCIHM L?F;N?> NI NB? NC>? A?H?L;NCHA @IL=? QCFF <? =IHMC>?L?>
CH JJ?H>CR #IL ; MG;FF ;L?; NBCM @IL=? =;H <? CAHIL?> ;M NB? NC>? CM ?MM?HNC;FFS
ILCACH;NCHA @LIG NB? <IOH>;LS =IH>CNCIHM #IL ;H ;L?; =IP?LCHA NB? B;HH?F ;H>
NB? #L?H=B NF;HNC= =I;MN CN Q;M MBIQH 89 NB;N NB? * Q;P? M?GC >COLH;F FOH;L
;JJFC?> QCNBCH NB? ;L?; Q;M L?MJIHMC<F? @IL ; NC>;F L;HA? I@ ;JJLIRCG;N?FS =G 1BCM
CM ?R;=NFS NB? G;AHCNO>? I@ NB? JL?=CMCIH F?P?F NB;N IH? =;H L?;=B CH ; =;F=OF;NCIH I@
NC>?M 1?MNM CH NB? M;G? >IG;CH <ON QCNBION CH=FO>CHA NC>?M ;N NB? <IOH>;LC?M B;P?
ACP?H ; NC>;F L;HA? I@ M?P?L;F N?HM I@ =?HNCG?NL?M 1BCM MOLJLCMCHA L?MOFN CM ?RJF;CH?>
<S NB? HIH FCH?;LCNS I@ NB? @LC=NCIH N?LG 1BCM N?LG CHN?L@?L?M QCNB NB? ?_?=N I@ NB?
FOH;L @IL=? CH NB? GIG?HNOG ?KO;NCIH 4CNB ;MNL;F JIN?HNC;F ;FIH? P?FI=CNC?M L?G;CH
FIQ ;H> NB? @LC=NCIH N?LG CM I@ NB? M;G? G;AHCNO>? ;M NB? FOH;L @IL=? ,H NB? INB?L
B;H> <S ;JJFSCHA ;H ;P?L;A? * NC>? ;N NB? <IOH>;LC?M I@ NB? >IG;CH NB? P?FI=CNC?M
;H> B?H=? NB? @LC=NCIH N?LG QBC=B P;LC?M ;M NB? MKO;L? I@ P?FI=CNS ;L? AL?;N?L 0I NB?
@LC=NCIH N?LG ;FGIMN =IGJF?N?FS G;MEM NB? ?_?=N I@ NB? FOH;L @IL=?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
>D=30AH 2>=38C8>=B 8= >CMNCH=NCIH MBIOF> <? G;>? <?NQ??H NB? JBSMC=;F ;H> G;NB?G;NC=;F <IOH>;LS =IH
>CNCIHM 1B? F;NN?L ;L? NB? NL;HMF;NCIH I@ NB? @ILG?L CH ;FAILCNBGC= N?LGM &N CM NB?
JBSMC=;F <IOH>;LS =IH>CNCIHM NB;N QCFF <? L?PC?Q?> B?L?
&H N?LGM I@ JBSMC=M ; >CMNCH=NCIH MBIOF> <? G;>? <?NQ??H NQI NSJ?M I@ <IOH>;LC?M
CH NB? =;F=OF;NCIH >IG;CH FCKOC> <IOH>;LC?M ;H> LCAC> <IOH>;LC?M
'8683 1>D=30A84B
1BCM NSJ? I@ <IOH>;LS CM CGJ?LG?;<F? HI WIQ MBIOF> =LIMM ; LCAC> <IOH>;LS 1B?L? CM
@LC=NCIH ;FMI 1B? @LC=NCIH =I?^=C?HN =;H <? JLIPC>?> <S NB? OM?L IL >?N?LGCH?> <S NB?
C
NOL<OF?H=? GI>?F $?H?L;FFS B C
< CM ACP?H CH NB? CGJFC=CN @ILG B < KB QB?L?
KB CM NB? $&#& OHEHIQH =IGJIH?HN I@ NB? P?FI=CNS N;HA?HN NI NB? Q;FF #IL NB?
C
M;E? I@ HOG?LC=;F M=B?G?M IH? CM I<FCA?> NI ;JJFS ; MNLIHA?L =IH>CNCIH B <
K
D
;H> B < L NBCM G?;HM ;JJFSCHA ;H ;LNCV=C;F =IH>CNCIH IH NB? HILG;F =IGJIH?HN
QBC=B CM =;H=?FF?> ;HSQ;S <S NB? CGJ?LG?;<CFCNS =IH>CNCIH
+I =IH>CNCIH IH NB? Q;N?L >?JNB CM ?RJ?=N?> @IL ; LCAC> Q;FF
!8@D83 1>D=30A84B
1BCM NSJ? I@ <IOH>;LS L?KOCL?M NB? GIMN M?HMCNCP? NL?;NG?HN ;M CN MOJJIM?M NB? ?R
CMN?H=? I@ ; WOC> >IG;CH NB;N >I?M HIN <?FIHA NI NB? >IG;CH I@ =;F=OF;NCIH <ON =;H
CHWO?H=? CN 1BCM CHWO?H=? MBIOF> <? NL;HMF;N?> CHNI =IH>CNCIHM ;N NB? <IOH>;LC?M
1B;HEM NI ;H ?R;GCH;NCIH I@ NB? 0;CHN 3?H;HN ?KO;NCIHM QCNBION >C_OMCIH N?LGM
;H> CH F! ;H> QCNB NB? B?FJ I@ NB? NB?ILS I@ =B;L;=N?LCMNC=M Q? =;H >CMNCHAOCMB @IOL
NSJ?M I@ FCKOC> <IOH>;LC?M
&HJON QCNB MOJ?L=LCNC=;F WIQ
&HJON QCNB MO<=LCNC=;F WIQ
,ONJON QCNB MOJ?L=LCNC=;F WIQ
,ONJON QCNB MO<=LCNC=;F WIQ
&HJON ;H> IONJON ;L? >C_?L?HNC;N?> <S NB? MCAH I@ NB? >IN JLI>O=N I@ NB? P?FI
=CNS P?=NIL ;H> NB? HILG;F P?=NIL JICHNCHA IONQ;L>M @LIG NB? Q;FF CHJON H?A;NCP?
JLI>O=N IONJON JIMCNCP? JLI>O=N 0OJ?L=LCNC=;F ;H> MO<=LCNC=;F WIQM ;L? >C_?L?H
NC;N?> QCNB NB? #LIO>? HOG<?L " C56 M M??H ?;LFC?L NBCM HOG<?L ?RJL?MM?M NB?
L;NCI <?NQ??H NB? P?FI=CNS I@ NB? =OLL?HN ;H> NB? =?F?LCNS I@ FIHA Q;P?M 9
&@ " NB? WIQ CM MOJ?L=LCNC=;F C@ " CN CM MO<=LCNC=;F
==IL>CHA NI NB? NB?ILS I@ =B;L;=N?LCMNC=M NB? <IOH>;LS =IH>CNCIHM H?=?MM;LS @IL
NB? <IOH>;LC?M I@ NB? >IG;CH ;L? CH !
1QI =IH>CNCIHM @IL ; MOJ?L=LCNC=;F CHJON
1QI =IH>CNCIHM @IL ; MO<=LCNC=;F CHJON
+I =IH>CNCIH @IL ; MOJ?L=LCNC=;F IONJON
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 ,H? =IH>CNCIH @IL ; MO<=LCNC=;F IONJON
1BCM NB?ILS =;HHIN <? @;CNB@OFFS @IFFIQ?> CH ! VLMN <?=;OM? NB? P?FI=CNS CM ; P?=
NIL 1B?H NB? M?J;L;NCIH I@ ?KO;NCIHM CHNI NQI @L;=NCIH;F MN?JM >OLCHA NB? L?MIFONCIH
<LCHAM ;<ION CHN?LG?>C;N? <IOH>;LS =IH>CNCIHM QBC=B >I HIN ;FQ;SM @IFFIQ JBSMC=M
#IL ?R;GJF? @IL ; FCKOC> <IOH>;LS QCNB P?FI=CNS JICHNCHA CHQ;L>M NB? ;>P?=NCIH MN?J
L?KOCL?M >;N; @IL ;FF KO;HNCNC?M NL;HMJILN?> CH NB? Q;N?L <I>S H ;>>CNCIH;F >C^=OFNS
CM NB;N NB? ZCHJON[ IL ZIONJON[ H;NOL? I@ ; FCKOC> <IOH>;LS =;H P;LS CH NCG? ? A NB?
=;M? I@ ; <IOH>;LS CH NB? IJ?H M?; QCNB NC>?
&H JL;=NC=? >C_?L?HN NSJ?M I@ <IOH>;LS =IH>CNCIHM ;L? =L?;N?> NI =ILL?MJIH> NI
NB? @IOL NSJ?M I@ <IOH>;LC?M G?HNCIH?> ?;LFC?L
&GJIM?> P?FI=CNS ;H> CGJIM?> Q;N?L >?JNB CHJON QCNB MOJ?L=LCNC=;F WIQ
&GJIM?> P?FI=CNS ;H> @L?? Q;N?L >?JNB CHJON QCNB MO<=LCNC=;F WIQ
#L?? P?FI=CNS ;H> @L?? Q;N?L >?JNB IONJON QCNB MOJ?L=LCNC=;F WIQ
#L?? P?FI=CNS ;H> CGJIM?> Q;N?L >?JNB IONJON QCNB MO<=LCNC=;F WIQ
1I NB?M? <;MC= =IH>CNCIHM ;JJFC?> NI NB? OHEHIQHM I@ IOL ?KO;NCIHM MIG? INB?LM
GIL? =IGJF?R IH?M H??> NI <? ;>>?> ;M @IFFIQM
<?>B43 K>F 2>=38C8>=B @IL ?R;GJF? P?LS I@N?H NB? ?R;=N P?FI=CNS JLIVF? IH
; M?=NCIH I@ ; LCP?L CM $&#& HIN EHIQH 1BCM CH@ILG;NCIH QIOF> BIQ?P?L <? H?
=?MM;LS ;N NB? OJMNL?;G ?H> I@ ; >IG;CH I@ =;F=OF;NCIH 1B? IHFS EHIQH P;LC;<F? CM
GIMN I@N?H NB? >CM=B;LA? 0I ; P?FI=CNS JLIVF? CM ;MMOG?> =IHMN;HN IL FIA;LCNBGC=
1B? JL?=CM? HOG?LC=;F P;FO?M I@ P?FI=CNC?M ;L? NB?H I<N;CH?> @LIG ; MCGJF? LOF? I@
JLIJILNCIH MI ;M NI L?NLC?P? NB? L?;F >CM=B;LA? 1BCM JLI=?>OL? CM MNCFF P;FC> C@ >C_?L
?HN >CM=B;LA?M B;P? NI <? ?HN?L?> ;N >C_?L?HN CHJON JICHNM ;H> ;FMI C@ NB? CGJIM?>
>CM=B;LA? CM ; @OH=NCIH I@ NB? Q;N?L >?JNB
=2834=C F0E4 2>=38C8>= NBCM =IH>CNCIH CM G?;HN @IL NB? ?HNLS ;H> ?RCN I@ ;
JLIAL?MMCP? MCHOMIC>;F Q;P? NBLIOAB IJ?H <IOH>;LC?M I@ ; >IG;CH &N CM QLCNN?H @IL ;
Q;P? QBIM? @L?KO?H=S CM ;M
9
K D =6A = A D /0
QB?L? 9 CM NB? Q;P? =?F?LCNS D NB? HILG;F OHCN P?=NIL JICHNCHA IONMC>? I@ NB? <IOH>;LS
I@ NB? >IG;CH ;H> 6A NB? ?F?P;NCIH I@ NB? @L?? MOL@;=? A CM ; Q;P? OHCN P?=NIL I@ NB?
CH=C>?HN MCAH;F QBIM? HILG CM ?KO;F NI 9
/0 ) "$ J ) CM NB? MCHOMIC>;F CH=C>?HN MCAH;F BIGIA?H?IOM NI ;
F?HANB 1B? ILCACH I@ @ILGOF; CM NB? NB?ILS I@ =B;L;=N?LCMNC= =OLP?M ;H> NB?
/C?G;HH CHP;LC;HNM &H MO<=LCNC=;F WIQ Q? QLCN? NB;N NB? /C?G;HH CHP;LC;HN =IGCHA
@LIG IONMC>? NB? >IG;CH =ILL?MJIH>M NI NB? CH=C>?HN Q;P? &H NB? ;<M?H=? I@ ;H
CH=C>?HN Q;P? ;H> CH ! NBCM =IH>CNCIH G?;HM NB;N NB? CHP;LC;HN K 9 CM ?KO;F NI
9 QB?L? 9 CM NB? ?RCMNCHA =?F?LCNS <?SIH> NB? >IG;CH <IOH>;LS
1BCM <IOH>;LS =IH>CNCIH GCHCGCT?M L?W?=NCIH @IL Q;P?M F?;PCHA NB? >IG;CH CH ;
>CL?=NCIH HILG;F NI NB? <IOH>;LS %IQ?P?L NB? L?MOFN >?J?H>M IH NB? ILC?HN;NCIH I@
NB? <IOH>;LS
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
,H NB? INB?L B;H> QB?H CN =IH=?LHM NB? ?HNLS I@ ; MCHOMIC>;F Q;P? NB? OM? I@
NBCM <IOH>;LS =IH>CNCIH L?KOCL?M J?L@?=N EHIQF?>A? I@ NB? CHJON MCAH;F ;GJFCNO>?
JB;M? @L?KO?H=S Q;P? P?=NIL QBC=B CM HIN NB? MCAH;F G?;MOL?> >O? NI NB? L?W?=NCIH
I@ Q;P?M
&H =;M? IHFS NIN;F MCAH;F CH@ILG;NCIH CM ;P;CF;<F? CGJIM?> P;FO?M ;L? JL?@?LL?>
)A024A 4@D0C8>= 8= CF> 38<4=B8>=B
&H ! NB? HOG?LC=;F P;FO? I@ NB? NL;=?L IHFS L?JL?M?HNM ;H ;P?L;A? IP?L NB? P?LNC=;F
1BCM ?R=FO>?M NB? JIMMC<CFCNS I@ L?JL?M?HNCHA N?GJ?L;NOL? IL >?HMCNS MNL;NCV=;NCIHM
==IOHNCHA @IL NB?M? JB?HIG?H; L?KOCL?M NB? ! +;PC?L\0NIE?M ?KO;NCIHM JL?@?L;<FS
NB? HIH BS>LIMN;NC= IH?M
>=B4AE0C8E4 5>A<
P?L;ACHA IP?L NB? P?LNC=;F >IH? ?;LFC?L NI I<N;CH NB? 0;CHN 3?H;HN ?KO;NCIHM ACP?M
NB? =IHM?LP;NCP? @ILG I@ NB? NL;=?L ?KO;NCIH CH !
>0
J
:?L >0 K :?L >
( =H7:
0
0A13 /9;
0A13 CM NB? MIOL=? P;FO? I@ NB? NL;=?L ;H> /9; NB? WIQ MIOL=? CH GM =IHH?=N?> NI
NB? >CM=B;LA? ;N NB? MIOL=? -A13 CM ?RJL?MM?> CH G M
%?L? ( ;FMI N;E?M >CMJ?LMCIH CHNI ;==IOHN
1BCM ?KO;NCIH QCNB OHEHIQH >0 =IOF> <? >?;FN QCNB >CL?=NFS CH NBCM @ILG QBC=B
QIOF> <? GIL? ?^=C?HN NI ?HMOL? =IHM?LP;NCIH I@ NB? NL;=?L &N B;M NB? @IFFIQCHA
>L;Q<;=EM BIQ?P?L
1B? P;LC;<F? 0 =;H IHFS <? I<N;CH?> QCNB ; >CPCMCIH <S NB? >?JNB 1BCM JIM?M
; JLI<F?G QCNB NB? GIHINIHC=CNS I@ 0 HIN NI G?HNCIH OH=IP?L?> <;HEM QB?L?
Q;N?L >?JNB > 4B?H 0 CM =IHMN;HN IP?L NB? QBIF? >IG;CH NB? MCNO;NCIH CM HIN MN;<F? ;M CN
>I?M HIN =ILL?MJIH> NI ;H I<PCIOM MIFONCIH I@ NB? ?KO;NCIH CH >0 M ; G;NN?L
I@ @;=N NB? P;LC;<F? >0 CM HIN =IHMN;HN C@ NB? >?JNB P;LC?M
#>=2>=B4AE0C8E4 5>A<
#IL ;FF NB? L?;MIHM ACP?H CH NB? JL?=?>CHA J;L;AL;JB NB? HIH =IHM?LP;NCP? @ILG I@
NB? NL;=?L ?KO;NCIH CM JL?@?LL?> >?MJCN? CNM >L;Q<;=EM &N CM I<N;CH?> CH NB? G;HH?L
?F;<IL;N?> <?FIQ &N B;M CNM CGJILN;H=? ;M Q? MBIOF> <? =;J;<F? I@ >ICHA CN <;=EQ;L>M
;N NB? >CM=L?N? F?P?F @IL JLIPCHA NB? =IHM?LP;NCIH I@ NB? NL;=?L CH VHCN? ?F?G?HNM
#CLMN NB? >?LCP;NCP?M I@ NB? JLI>O=NM ;L? >?P?FIJ?>
>
0
J
0
>
J
>
K =H7: 0
0 :?L >
K :?L >
( =H7:
0
0A13 /9;
1B? N?LG 0 6
:?L >
K MBIOF> <? HIN?> &N CM L?JF;=?> QCNB NB? B?FJ I@ NB?
B
=IHNCHOCNS ?KO;NCIH <S 0 /9; QB?L? /9; CM NB? MIOL=? N?LG
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 1B? L?G;CHCHA ?KO;NCIH CM NB?H >CPC>?> <S > NI VH;FFS I<N;CH
0
J
K =H7: 0 :?L >
>
( =H7:
0
0A13 0 /9;
>
1B? LCABN B;H> MC>? I@ NBCM ?KO;NCIH CM T?LI C@ NB? P;FO? I@ NB? MIOL=? NL;=?L CM
?KO;F NI NB;N I@ NB? >IG;CH QB?L? NB? MIOL=? ;LLCP?M 1BCM CM A?H?L;FFS HIN NB? =;M?
@IL ;H CHD?=NCIH CHNI NB? >IG;CH #IL ;H IONF?N @LIG NB? >IG;CH Q? B;P? 0A13 0
>D=30AH 2>=38C8>=B
1B? <IOH>;LS =IH>CNCIHM @IL ; NL;=?L ;L?
&GJIM?> P;FO?
#L?? P;FO? ;N ;H IONF?N @IL ?R;GJF?
#FOR =IH>CNCIH
1B? WOR =IH>CNCIH CM ;M CH !
0
70
D
8
QB?L? 7 ;H> 8 ;L? JLIPC>?> <S NB? OM?L
&H NB? =;M? I@ @L?? IONF?N =IH>CNCIH ;H> C@ NB? G?NBI> I@ =B;L;=N?LCMNC=M CM OM?> NB?
L?MOFN QCFF >?J?H> IH NB? ;>P?=NCIH MN?J IHFS 1B? ?_?=N I@ >C_OMCIH ;H> MIOL=? N?LGM
CM NB?L?<S CAHIL?> &H @;=N NB? >C_OMCIH MN?J QIOF> L?KOCL? ; WOR =IH>CNCIH QBC=B
QIOF> OFNCG;N?FS <? ;H ;LNCV=C;F <IOH>;LS =IH>CNCIH HI WOR <IOH>;LS =IH>CNCIH
CM HIN ?KOCP;F?HN NI ; @L?? IONF?N =IH>CNCIH
">34;;8=6 >5 CDA1D;4=24 0=3 38B?4AB8>=
'4H=>;3B BCA4BB
0I @;L IHFS F;GCH;L WIQ B;M <??H =IHMC>?L?> 4B?H NB? /?SHIF>M HOG<?L ?KO;F NI
)!
QB?L? 1 ;H> ( ;L? ; P?FI=CNS ;H> ; F?HANB =B;L;=N?LCMNC= I@ WIQ <?=IG?M AL?;N?L
NB;H ;LIOH> JBSMC=;F KO;HNCNC?M FCE? P?FI=CNS ;H> JL?MMOL? MBIQ L;JC> P;LC;NCIHM
CH NCG? ;H> MJ;=? 1B? P;FO? CM CH @;=N ;H IL>?L I@ G;AHCNO>? P;FC> @IL =;H;FM
;H> P;LC?M AL?;NFS ;==IL>CHA NI WIQM &N ;FMI >?J?H>M IH NB? =BIC=? I@ =B;L;=N?LCMNC=
KO;HNCNC?M 1 ;H> ( #IL F;LA? /?SHIF>M HOG<?LM C ? @IL ; G;DILCNS I@ ;JJFC=;
NCIHM L?AOF;L IL F;GCH;L MIFONCIHM ;L? HI FIHA?L MN;<F? ;H> ;L? L?JF;=?> <S =B;INC=
MIFONCIHM QBIM? P;LC;NCIHM ;L? OHJL?>C=N;<F? 1B?M? WO=NO;NCIHM ;L? HIN >?M=LC<?>
IHFS ;P?L;A? P;FO?M ;L? I@ CHN?L?MN 1B?M? ;P?L;A?M =;H <? ;P?L;A?M CH NCG? NB? =;M?
I@ ; MN?;>S WIQ ;P?L;A?M I@ NB? M;G? ?RJ?LCG?HN =IH>O=N?> ; =?LN;CH HOG<?L I@
NCG?M NB? =;M? I@ OHMN?;>S WIQ IL ;P?L;A?M IP?L MJ;=? NB? =;M? I@ BIGIA?H?IOM
WIQ 2H@ILNOH;N?FS >O? NI NB? HIH FCH?;LCNS I@ ;>P?=NCIH N?LGM NB? ;P?L;A?> WIQ
CM HI FIHA?L NB? MIFONCIH @IL NB? +;PC?L\0NIE?M ?KO;NCIHM ;H> NB? ?KO;NCIHM I@ NBCM
;P?L;A?> WIQ ;L? =;FF?> Z/?SHIF>M ?KO;NCIHM[ &@ NB? =IGJIH?HNM I@ P?FI=CNS 17 ;L?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
<LIE?H >IQH ;M 17 K7 ;P?L;A?
WO=NO;NCIH NB? /?SHIF>M GIG?HNOG ?KO;NCIH
CM QLCNN?H ;M CH NB? =IHM?LP;NCP? @ILG
17
J
F
17 18
N8
N7
78
N8
.78
"7
=7
1B? =IHNCHOCNS ?KO;NCIH CM OH=B;HA?> ;P?L;A?> P;FO?M ;H> WO=NO;NCIHM ;FF <?CHA
T?LI
1B? CH>C=?M ? ;H> @ P;LS @LIG NI 1B? Z"CHMN?CH[ N?HMIL HIN;NCIH B;M <??H
OM?> ;H> NB? L?J?;N?> CH>C=?M =ILL?MJIH> CGJFC=CNFS NI ; MOGG;NCIH "7 L?JL?M?HNM
NB? <I>S @IL=?M INB?L NB;H JL?MMOL? ;H> AL;PCNS =7 L?JL?M?HNM AL;PCNS = = = =
1B? N?LG
78 18
N7
17
N8
CM NB? PCM=IOM J;LN I@ NB? MNL?MM N?HMIL
1B? =IGJIH?HNM I@ N?HMIL .78 ;L? NB? =ILL?F;NCIHM I@ P?FI=CNS WO=NO;NCIHM
.78 K7 K8
&N CM NBCM N?HMIL OH@ILNOH;N?FS OHEHIQH QBC=B G;E?M NB? >C_?L?H=? QCNB NB?
+;PC?L\0NIE?M ?KO;NCIHM
&N CM JIMMC<F? NI @ILG ?KO;NCIHM @IL NB? /?SHIF>M MNL?MM?M .78 <ON NB?S QIOF>
ACP? LCM? NI H?Q OHEHIQHM NLCJF? =ILL?F;NCIHM CH AL?;N?L HOG<?LM 1BCM H?P?L ?H>CHA
BC?L;L=BS I@ ?KO;NCIHM =IHMNCNON?M ;H IJ?H MSMN?G
1I MIFP? NB? /?SHIF>M ?KO;NCIHM NB? JLI<F?G MBIOF> <? =FIM?> 1B;N CM NB? OH
EHIQH N?HMIL .78 MBIOF> <? ?RJL?MM?> IH NB? <;MCM I@ ;P?L;A?> JBSMC=;F KO;HNCNC?M
QCNB NB? B?FJ I@ ; NOL<OF?H=? GI>?F QBC=B =;H <? ;H ;FA?<L;C= L?F;NCIH T?LI ?KO;NCIH
GI>?F ; =IG<CH;NCIH I@ ;H ;FA?<L;C= L?F;NCIH ;H> ;H ?KO;NCIH MCHAF? ?KO;NCIH GI>?F
IL NQI NL;HMJILN\>C_OMCIH ?KO;NCIHM NQI ?KO;NCIH GI>?F IL ?P?H GIL? /?SHIF>M
MNL?MM GI>?F &H ;FF =;M?M NBCM QCFF <? IHFS ; GI>?F <;M?> IH >CG?HMCIH;F ;H;
FSMCM ?GJCLC=CMG IL IH CHN?LJL?N;NCIH I@ JBSMC=;F JB?HIG?H; ;H> NB?CL L?MJ?=NCP?
CGJILN;H=?
1OL<OF?H=? GI>?FM QCNB T?LI IH? IL NQI ?KO;NCIHM ;L? ;FF <;M?> IH ; BSJINB?MCM I@
CMINLIJC= NOL<OF?H=? %IQ?P?L CH =?LN;CH ;JJFC=;NCIHM ; N?HMIL I@ CMINLIJC= NOL<OF?HN
PCM=IMCNS CM HIN ;>;JN?> #IL ?R;GJF? QB?H NB? G?MB CM BOH>L?>M I@ G?NL?M CH NB?
BILCTIHN;F >CL?=NCIH <ON I@ NB? IL>?L I@ G?N?L CH NB? P?LNC=;F >CL?=NCIH IL QB?H
>CMJ?LMCIH ;FIHA NB? P?LNC=;F CM N;E?H CHNI ;==IOHN
&H GIMN NOL<OF?H=? GI>?FM NB? N?LGM
78
N8
.78
;L? GI>?FF?> ;==IL>CHA NI NB? IOMMCH?MK GI>?F 89 QBC=B MN;N?M NB;N
78
N8
.78
N8
B
17
N8
18
N7
A 78
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 QB?L? B CM NB? NOL<OF?HN PCM=IMCNS B CM P;LC;<F? CH NCG? ;H> MJ;=? ;H> CM JLIPC>?> <S
NB? NOL<OF?H=? GI>?F 78 CM NB? (LIH?=E?L N?HMIL ;H> A NB? NOL<OF?HN ECH?NC= ?H?LAS
?KO;F NI K7 K7 QBC=B CM ;FMI JL?M?HN CH NB? A\ GI>?F
#IL ; NL;=?L NBCM IOMMCH?MK BSJINB?MCM CM ?RJL?MM?> <S NB? @ILGOF;
K7 0 0
, HJ N7
B
QB?L? , HJ CM NB? -L;H>NF HOG<?L C@ 0 CM NB? N?GJ?L;NOL? QCNB M;FCHCNS CN QIOF>
<? NB? 0=BGC>N HOG<?L ( 1B? -L;H>NF ;H> 0=BGC>N HOG<?LM L;HA? A?H?L;FFS
<?NQ??H ;H> >?J?H>CHA IH NB? WIQ <ON INB?L P;FO?M ;L? MIG?NCG?M @;PIOL?>
J;LNC=OF;LFS CH NB? JL?M?H=? I@ M?>CG?HNM
%?H=?@ILNB NB? IOMMCH?MK BSJINB?MCM QCFF <? ;==?JN?> 1B? >C_?L?HN GI>?FM
JL?M?HN?> F;N?L QCFF >C_?L <S NB? P;FO? B NB;N NB?S JLIPC>? &H MIG? =;M?M CH IL>?L
NI N;E? CHNI ;==IOHN NB? ?_?=N I@ >CMJ?LMCIH NOL<OF?HN PCM=IMCNS QCFF <? =IHMC>?L?> ;M
;H ;HCMINLIJC= KO;HNCNS QCNB >C_?L?HN P;FO?M ;FIHA ?;=B >CL?=NCIH 1B? =I?^=C?HN B
CM >CG?HMCIH;FFS NB? JLI>O=N I@ ; P?FI=CNS ;H> ; F?HANB &N CM =IHMC>?L?> LIOABFS NI
L?JL?M?HN NB? JLI>O=N I@ P?FI=CNS I@ ;H ?>>S QCNB NB? >CMN;H=? ;FIHA QBC=B CN GCR?M
GIG?HNOG 1BCM CHN?LJL?N;NCIH >?GIHMNL;N?M NB? ;HCMINLIJS I@ NB? JLI=?MM ,H NB?
INB?L B;H> CH NB? IOMMCH?MK GI>?F NB? /?SHIF>M N?HMIL CM ;FQ;SM ;H;FIAIOM NI NB?
>?@ILG;NCIH N?HMIL QCNB ;H CMINLIJC= <?B;PCIOL ;M J?L ?F;MNC= G;N?LC;F
4? MB;FF HIQ ?R;GCH? CH CH=L?;MCHA =IGJF?RCNS NB? >C_?L?HN NOL<OF?HN GI>?FM
OM?> CH @L?? MOL@;=? BS>LI>SH;GC=M
@N?L JL?M?HNCHA ZT?LI ?KO;NCIH[ GI>?FM Q? MB;FF <? CH ; JIMCNCIH NI MJ?=C@S NB?
P;FO?M I@ NOL<OF?HN MNL?MM H?;L Q;FFM >O? NI NOL<OF?H=? ;H> NB?H CH >?N;CF NB? GIMN
=IGGIHFS OM?> NQI ?KO;NCIH GI>?F NB? A\ GI>?F
/4A>4@D0C8>= <>34;B
1B?M? P?LS MCGJF? GI>?FM ;MMOG? NB;N NOL<OF?HN PCM=IMCNS CM =IHMN;HN IL >CL?=NFS
>?J?H> IH EHIQH IL ?;MCFS =;F=OF;<F? J;L;G?N?LM
>=BC0=C E8B2>B8CH
IHMN;HN PCM=IMCNS CM MO^=C?HN QB?H WIQ CM AIP?LH?> <S NB? JL?MMOL? AL;>C?HN ;H> <S
;>P?=NCIH CH NB? NC>? WIQ L?ACG? @IL ?R;GJF? ;H> ?MJ?=C;FFS @IL GI>?FFCHA I=?;HC=
=CL=OF;NCIH IH ; F;LA? M=;F? &H ! NBCM =IHMN;HN PCM=IMCNS MBIOF> CH=FO>? >CMJ?LMCIH
1B? P;FO?M I@ B JLIPC>?> CH NB? FCN?L;NOL? M?? 89 ?RN?H> @LIG G M NL;HMP?LM;F
>C_OMCIH CH NB? *;=E?HTC? /CP?L NI G M FIHACNO>CH;F >C_OMCIH CH NB? *CMMIOLC
/CP?L
;34A <>34;
1B? "F>?L GI>?F @IL NB? 0;CHN 3?H;HN ?KO;NCIHM N;E?M CHNI ;==IOHN NB? LIF? I@ >CMJ?L
MCIH ;H> L?=IGG?H>M NQI >C_?L?HN =I?^=C?HNM IH? ': ;FIHA NB? WIQ FIHACNO>CH;F
>C_OMCIH ;H> NB? INB?L 'B NL;HMP?LM? NI NB? WIQ NL;HMP?LM;F >C_OMCIH
"F>?LYM I<M?LP;NCIHM ;H> =;F=OF;NCIHM <;M?> IH NB? P?FI=CNS JLIVF? CH NB? FIA;L
CNBGC= F;S?L M?? 0?=NCIH <?FIQ F?> BCG NI NB? P;FO?M ': K > ;H> 'B
K > QB?L? K CM NB? MB?;L P?FI=CNS ;H> > NB? Q;N?L >?JNB
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&@ CM NB? ;HAF? +N K I@ NB? P?FI=CNS P?=NIL QCNB L?MJ?=N NI NB? N ;RCM NB?
"F>?L GI>?F L?KOCL?M NB? ;>IJNCIH I@ ; PCM=IMCNS N?HMIL
'FF
'FG
'
'FG
'GG
K ': MO=B NB;N ' K ;H> ' J 'B J QB?L? J CM ; P?=NIL NL;HMP?LM;F NI K
C>?HNCV=;NCIH
'FF ': "$ S
'B $! 'FG ': 'B $! "$ 'GG ': $! 'B "$ #4ID 0=3 #0:060F0 <>34;
1BCM GI>?F JO<FCMB?> CH A?NM CNM CHMJCL;NCIH @LIG NB? ?RJL?MMCIH B K P CH
NB? <IOH>;LS F;S?L ;<IP? NB? <INNIG ON CN ;FMI N;E?M CHNI ;==IOHN NB;N NOL<OF?HN
PCM=IMCNS >?=L?;M?M NI T?LI NIQ;L>M NB? @L?? MOL@;=? ;MMOGCHA NB? ;<M?H=? I@ QCH>
+?TO ;H> +;E;A;Q; 89 NB?H JLIJIM?
B
K P P
>
1BCM GI>?F CM P?LS OM?@OF @IL L?JL?M?HNCHA >C_OMCIH ;FIHA NB? P?LNC=;F
"8G8=6 ;4=6C7 <>34;B
(C0=30A3 <8G8=6 ;4=6C7 <>34;B NBCM GI>?F JLIJIM?> <S -L;H>NF 89 CH ACP?M NB? P;FO? I@ NB? PCM=IMCNS =I?^=C?HN ;M
B
QB?L?
(;
5
78
78
CM NB? MNL;CH L;N? N?HMIL I@ ;P?L;A? GINCIH QCNB
78
17
N8
18
N7
(; CM NB? ZGCRCHA F?HANB[ J;L;G?N?L ?KO;F NI O ;N ; >CMN;H=? O @LIG NB? Q;FF
QCNB PIH (;LG;H =IHMN;HN NCFF NB? MCT? I@ ?>>C?M CM HI FIHA?L CHWO?H=?>
<S NB? Q;FF ;H> L?G;CHM =IHMN;HN CH NB? =;M? I@ WIQ <?NQ??H NQI Q;FFM NB? FCGCN CM
MCNO;N?> ;N NB I@ NB? MJ;=? <?NQ??H NB? Q;FFM 1BCM GI>?F CM OM?> NI >?M=LC<? NB?
P?FI=CNS JLIVF? CH NB? PC=CHCNS I@ ; Q;FF &N CM I<N;CH?> <S ; BSJINB?MCM I@ ?KOCFC<LCOG
NOL<OF?H=? @IL >?N;CFM M?? 0?=NCIH ".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 "8G8=6 ;4=6C7 <>34;B 0;>=6 C74 E4AC820; MNCFF ;==IL>CHA NI NB? -L;H>NF GI>?F
;JJFC?> NBCM NCG? ;FIHA NB? P?LNC=;F NB? GI>?F JLIPC>?M P?LNC=;F PCM=IMCNC?M NI <? OM?>
@IL P?FI=CNC?M IL NL;=?LM
8
1
2
H < .? (;
P
P
;H>
H(
<( .? (;
8
1
P
2
P
QB?L?
.? =
)
H
H
* H
CM NB? /C=B;L>MIH HOG<?L < ;H> <( ;L? NB? >;GJCHA @OH=NCIHM I@ NB? P?FI=CNS ;H>
NL;=?L QBC=B B?FJ NI L?VH? NB? GI>?F ;H> ;>;JN CN NI MJ?=C;F =;M?M J;LNC=OF;LFS @IL
MNL;NCV=;NCIHM &@ MNL;NCV=;NCIH CM MN;<F? QBC=B =ILL?MJIH>M NI ; JIMCNCP? /C=B;L>MIH
HOG<?L < CM H?;L T?LI &@ MNL;NCV=;NCIH CM OHMN;<F? QBC=B =ILL?MJIH>M NI ; H?A;NCP?
/C=B;L>MIH HOG<?L < N;E?M ; P;FO? <?NQ??H ;H> >?J?H>CHA IH NB? ;ONBILM
1B?L? ;L? G;HS ?RJL?MMCIHM @IL >;GJCHA @OH=NCIHM QBC=B ;FQ;SM MBIQ ; N?H>?H=S
NI >?=L?;M? QCNB .? ,H? Q;S I@ I<N;CHCHA ;H ?RJL?MMCIH CM NI MN;LN QCNB NB? /?SHIF>M
N?HMIL ?KO;NCIHM ;H> ;MMOG? ; FI=;F ?KOCFC<LCOG I@ NOL<OF?H=? MI =?LN;CH N?LGM =;H
<? CAHIL?> 1B? L?G;CHCHA N?LGM ;L? NB?H GI>?FF?> 4? A?N ;H ;FA?<L;C= MSMN?G
>?J?H>?HN IH NB? /C=B;L>MIH HOG<?L S =BIIMCHA NB? GI>?FM >?M=LC<?> CH L?@?L?H=?
89 Q? ;LLCP? ;N NB? AL;JB CH #CAOL? @IL >;GJCHA @OH=NCIHM I@ P?FI=CNS ;H>
N?GJ?L;NOL? 1B? @OH=NCIH >?VH?> @IL N?GJ?L;NOL? >I?M HIN ?R=??> @IL .7 <ON CN >I?M @IL P?FI=CNS 0I NB?L? CM ; NOL<OF?HN -L;H>NF HOG<?L >C_?L?HN @LIG CH
; H?ONL;F MCNO;NCIH .? &H NB? =;M? I@ ; NL;=?L >C_?L?HN @LIG ; N?GJ?L;NOL?
INB?L >;GJCHA @OH=NCIHM =;H <? OM?> #IL ?R;GJF?
< .? 7 .?
0
7 ;H> 8 <?CHA NQI =IHMN;HNM >?J?H>CHA IH NB? H;NOL? I@ NB? NL;=?L 1BCM CM ; NIN;FFS
?GJCLC=;F F;Q <ON CN CM MIG?NCG?M OM?> CH NB? JL?M?H=? I@ M?>CG?HN
1B? GCRCHA F?HANB (; =;H <? GI>?FF?> CH >C_?L?HN Q;SM
;0BB820; %A0=3C; <>34; M?? 89
(; P C@ P > B?L? P CM NB? >CMN;H=? NI NB? <INNIG ;H> > NB? Q;N?L >?JNB
>
(; > C@ P
#4ID 0=3 #0:060F0 <>34; M?? 89
7
P
(; P >
&H NBCM =;M? NB? GCRCHA F?HANB P;HCMB?M ;N NB? @L?? MOL@;=? 1BCM GI>?F =;H ACP? ;
FIA;LCNBGC= P?FI=CNS JLIVF? @LIG <INNIG NI @L?? MOL@;=?
&D4C8= <>34; M?? 89
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
(; H
2
QB?L? : CM NB? >CMN;H=? NI NB? @L?? MOL@;=?
)B0=8B <>34; M?? 89
(; P C@ P > B?L? P CM NB? >CMN;H=? NI NB? <INNIG ;H> > NB? Q;N?L >?JNB
(; > C@ P
> ;H> P >
(; : C@ P
> : CM NB? >CMN;H=? NI NB? @L?? MOL@;=?
&H NB? JLI>O=NCIH I@ NOL<OF?H=? ?RJL?MMCIH NB? BILCTIHN;F AL;>C?HNM I@ 1 ;H> 2 ;M
Q?FF ;M NB? AL;>C?HNM I@ 3 ;L? ;MMOG?> H?AFCAC<F? =IGJ;L?> NI NB? P?LNC=;F AL;>C?HNM
1B? N?LG
8
1
2
P
P
5
CM NBOM IHFS ;H ;JJLIRCG;NCIH I@ 78 78 #IL ; =;F=OF;NCIH CH NB? PC=CHCNS I@ ;
MNLO=NOL? KO;S MJCE?M ?N= NB? AL;>C?HNM IP?LFIIE?> B?L? =;H <?=IG? CGJILN;HN
%IQ?P?L NB?CL JL?M?H=? CH NB? P?LNC=;F PCM=IMCNS ?RJL?MMCIH QIOF> HIN <? I@ GO=B
B?FJ NI J?L@ILG ;H ?^=C?HN =;F=OF;NCIH ;LIOH> ; MNLO=NOL? GCRCHA F?HANB GI>?F
@IL BILCTIHN;F PCM=IMCNS QIOF> ;FMI <?=IG? H?=?MM;LS %?H=? ; JL?=CM? =IGJON;NCIH I@
WIQ ;LIOH> MNLO=NOL?M CM HIN JIMMC<F? QCNB NBCM NOL<OF?H=? GI>?FFCHA
p g
y
p
2
1
Velocity
Temperature
0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
Richardson Number
#CAOL? >;GJCHA @OH=NCIHM
"8G8=6 ;4=6C7 <>34;B 5>A 1D>H0=C 94CB NB? OM? I@ GCRCHA F?HANB GI>?FM G?H
NCIH?> ?;LFC?L F?;>M NI ;<HILG;F L?MOFNM CH NB? =;M? I@ ;H IONWIQ &H @;=N >CMMCJ;NCIH
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 δ
#CAOL? B;F@ B?CABN I@ ; D?N
CM IP?L?MNCG;N?> CH NB? PC=CHCNS I@ ;H IONWIQ MCH=? NB? GCRCHA F?HANB (; CM VR?> ;N
> QBC=B CM ;JJLIRCG;N?FS I@ > #IL BCABFS MB?;L WIQM FCE? @L?? D?NM CH ; MNCFF
?HPCLIHG?HN /I>C 89 JLIJIM?M ; GCRCHA F?HANB (; I@ ;<ION <ON I@ NB?
B;F@ B?CABN : @IL NB? D?N
(; : @IL ; JF;H? D?N
(; : @IL ; =CL=OF;L D?N
1B? B;F@ B?CABN CM >?VH?> ;M NB? >CMN;H=? <?NQ??H NB? ?F?P;NCIH I@ NB? D?N ;RCM ;H>
NB? IH? QB?L? P?FI=CNS >I?M HIN ?R=??> I@ CNM P;FO? IH NB? ;RCM M?? #CAOL? 1BCM B;F@ B?CABN I@ NB? D?N CM H?=?MM;LS >;N; CH NBCM =;M? &@ NB? D?N CM HIN MSGG?NLC=
; >CMNCH=NCIH MBIOF> <? G;>? <?NQ??H NB? OJJ?L B;F@ B?CABN ;H> FIQ?L B;F@ B?CABN
&H JL;=NC=? @IL ?;=B P?LNC=;F NB? ?F?P;NCIH I@ NB? G;RCGOG P?FI=CNS P?=NIL CM
>?N?LGCH?> VLMN ?F?P;NCIH I@ NB? D?N ;RCM NB?H NB? OJJ?L ;H> FIQ?L B;F@ B?CABNM 7<4
;H> AC> 1B? GCRCHA F?HANB CH IH? JICHN QCFF <? JLIJILNCIH;F NI NB? FI=;F B;F@ B?CABN
1B? =I?^=C?HN I@ JLIJILNCIH;FCNS CM ;FQ;SM N;E?H ;M C ? @IL NQI L?;MIHM
&N CM N;E?H <?NQ??H NB? =IHMN;HNM @IL JF;H? ;H> =CL=OF;L D?NM ACP?H <S /I>C
&N ;FFIQM ; =IHNCHOCNS I@ NB? GCRCHA F?HANB P;FO? QB?H NB? D?N B;M L?;=B?> NB?
MOL@;=? (; >
N NB? <INNIG NB? FCH?;L >?=L?;M? I@ NB? GCRCHA F?HANB L?G;CHM OH=B;HA?> #CAOL?
CM ; P?LNC=;F JLIVF? MBIQCHA NB? >C_?L?HN MN;A?M I@ >?P?FIJG?HN I@ ; <OIS;HN D?N
#;L @LIG NB? MIOL=? 7<4 CM ?KO;F NI NB? >?JNB
&H NB? =;M? I@ ; =LIMM =OLL?HN >?N?LGCHCHA NB? ;RCM I@ NB? D?N CM F?MM MCGJF? #CLMN
NB? P?LNC=;F JF;H? QCNB G;RCGOG P?FI=CNS B;M NI <? >?N?LGCH?> 1B?H NB? INB?L
P?FI=CNC?M ;L? JLID?=N?> IHNI NBCM IH? 1B? ?F?P;NCIH I@ NB? D?N ;RCM :7<4 ;H> :AC> CM
=;F=OF;N?> CH NBCM JF;H?
&N MBIOF> <? HIN?> NB;N NBCM GI>?F Q;M ?R=FOMCP?FS >?P?FIJ?> @IL <OIS;HN D?NM &N
CM MCGJF? ;H> Q?FF ;>;JN?> NI NBCM NSJ? I@ ;JJFC=;NCIH &N B?FJM NI GI>?F ; JFOG? CH
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
Z
Up p er limit of the jet
dsup
dsup
dinf
Jet axis
dinf
Lower limit of the jet
X
#CAOL? >C_?L?HN MN;A?M I@ ?RJ;HMCIH I@ ; D?N
;H ;FGIMN M;NCM@;=NILS G;HH?L
(<06>A8=B:8 <>34;
1B? 0G;AILCHMEC GI>?F M?? 89 B;M ; G?;HCHA IHFS CH HOG?LC=;F GI>?FFCHA ;H> CM
<;M?> IH NB? GCRCHA F?HANB GI>?F &N <?FIHAM NI NB? =;N?AILS I@ MO< ALC> NOL<OF?H=?
GI>?FM 1B? JLCH=CJF? CM ;M @IFFIQM NOL<OF?H=? CM NB? MIFONCIH I@ NB? +;PC?L\0NIE?M
?KO;NCIHM &N QIOF> H;NOL;FFS ;JJ?;L CH NB? HOG?LC=;F MIFONCIHM C@ NB? MCT? I@ VHCN?
?F?G?HNM ;FFIQ?> NB? L?JLI>O=NCIH I@ ;FF G?=B;HCMGM CH=FO>CHA NB? PCM=IOM >CMMCJ;NCIH
I@ P?LS MG;FF PILNC=?M ,HFS CH NB? @ILG;NCIH I@ MG;FF?L PILNC=?M QB?L? NOL<OF?H=? CM
CHBC<CN?> <S NB? G?MB >I?M GI>?FFCHA N;E? JF;=? CH ; HOG?LC=;F MCGOF;NCIH 0G;AIL
CHMECYM C>?; CM NI ;>> NI NB? GIF?=OF;L PCM=IMCNS ; NOL<OF?HN PCM=IMCNS >?>O=?> @LIG
; GCRCHA F?HANB GI>?F 1BCM GCRCHA F?HANB =ILL?MJIH>M NI NB? MCT? I@ NB? PILNC=?M
MG;FF?L NB;H NB;N I@ NB? G?MB MCT? 4? MCGJFS ;LLCP? ;N NB? @IFFIQCHA @ILGOF;NCIH
B
A 5
78
78
QB?L? A CM ; >CG?HMCIHF?MM =I?^=C?HN NI <? =;FC<L;N?> ;H> NB? G?MB MCT? >?LCP?>
CH ! IL ! @LIG NB? MOL@;=? IL @LIG NB? PIFOG? I@ NB? ?F?G?HNM 1B? P;FO?M I@ A
L;HA? @LIG WIQ CH ; =;H;F NI CMINLIJC= NOL<OF?H=? 1BCM GI>?F CM <?MN
OH>?LMNII> CH ! &H ! CN >I?M HIN N;E? >CMJ?LMCIH CHNI ;==IOHN
)DA1D;4=24 BCA4BB >= C74 F0;;B
@N?L >?M=LC<CHA GCRCHA F?HANB GI>?FM Q? ;L? CH ; JIMCNCIH NI >?M=LC<? NOL<OF?H=?
MNL?MM IH Q;FFM &N >?J?H>M IH NB? P?FI=CNS JLIVF? ;<IP? NB? <INNIG IL CH NB? PC=CHCNS
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 I@ Q;FFM CH QB;N CM =;FF?> NB? <IOH>;LS F;S?L ; F;S?L QBC=B CM CHWO?H=?> <S NB? Q;FF
1B? CGJILN;HN J;L;G?N?LM ;L? MB?;L P?FI=CNS >?VH?> <S 1 ;M M??H ?;LFC?L
;H> NB? >CG?HMCIHF?MM >CMN;H=? NI NB? Q;FF O G) QB?L? O CM NB? >CMN;H=? NI NB?
Q;FF 1L;>CNCIH;FFS NB?L? CM ; >CMNCH=NCIH <?NQ??H
PCM=IOM MO< F;S?L QB?L? NOL<OF?H=? CM H?AFCAC<F? ;H> QB?L? NB? P?FI=CNS JLIVF?
CM FCH?;L ACP?H ;M
1
O1 1
1B? NBC=EH?MM I@ NBCM PCM=IOM MO< F;S?L CM FCGCN?> <S O ;JJLIRCG;N?FS &@
NB? =B;L;=N?LCMNC= LIOABH?MM MCT? I@ NB? Q;FF CM AL?;N?L NB;H NB? NBC=EH?MM I@ NB?
PCM=IOM MO< F;S?L NBCM PCM=IOM MO< F;S?L =;HHIN >?P?FIJ ;H> NB? WIQ CM EHIQH
NI <? BS>L;OFC=;FFS LIOAB &@ NB?L? CM ; PCM=IOM MO< F;S?L NB? WIQ CM EHIQH NI
<? BS>L;OFC=;FFS MGIINB
H CHN?LG?>C;N? F;S?L <?NQ??H O ;H> O 1B?L? CM HI MCGJF?
?RJL?MMCIH I@ P?FI=CNS JLIVF? CH NBCM TIH? <ON IHFS @ILGOF;? =IHH?=NCHA NB?
PCM=IOM MO< F;S?L NI NB? FIA;LCNBGC= F;S?L
FIA;LCNBGC= F;S?L MN;LNCHA ;N O ;H> OHNCF NB? MCT? I@ ?>>C?M CM HI FIHA?L
CHWO?H=?> <S NB? Q;FF ;MMOGCHA NB;N (; O 1B? NOL<OF?HN PCM=IMCNS CM NB?H
QLCNN?H ;M B 1 O &H NBCM F;S?L NB? P?FI=CNS JLIVF? N;E?M NB? @IFFIQCHA @ILG
#IL BS>L;OFC=;FFS MGIINB WIQ
1
!
1
O1 #IL BS>L;OFC=;FFS LIOAB WIQ
1
!
1
O
AA
!
O
AA
QB?L? AA CM NB? LIOABH?MM MCT?
H ?KOCP;F?HN Q;S I@ QLCNCHA NB? P?FI=CNS JLIVF? @IL NB? BS>L;OFC=;FFS LIOAB WIQ
CM
1
!
1
O
O=
9
QB?L? O=
#IL BS>L;OFC=;FFS MGIINB WIQ NB? FCH?;L ;H> FIA;LCNBGC= F;QM ;L? OHCV?> <S NB?
/?C=B;L> F;Q
1
! 1
O
; O ;
G /?@?L?H=? 89 =IHN;CHM NB?IL?NC=;F DOMNCV=;NCIHM I@ NB?M? @ILGOF;?
N NBCM JICHN CN CM JIMMC<F? NI MJ?=C@S NB? F;QM NI <? ;>IJN?> CH HOG?LC=;F MCGOF;
NCIHM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
H3A0D;820;;H B<>>C7 K>F
1B? JLI<F?G CM VH>CHA NB? MB?;L P?FI=CNS 1 &N CM H?=?MM;LS NI B;P? >;N; IH NB?
P?FI=CNS JLIVF? CH NB? PC=CHCNS I@ NB? Q;FF 1B?M? >;N; ;L? I<N;CH?> @LIG NB? @IFFIQCHA
L?;MIHCHA ;MMOG? NB;N NB? P?FI=CNS ;N NB? Q;FF IH ; G?MB CM HIN L?;FFS N;E?H IH NB?
Q;FF INB?LQCM? CN QIOF> <? T?LI <ON CM N;E?H CH NB? <IOH>;LS F;S?L ;N ; >CMN;H=? O
O CM =BIM?H ;L<CNL;LCFS IL >?J?H>CHA IH NB? FI=;F G?MB MCT? ? A NB? ?>A? JICHN CM ;N
; >CMN;H=? @LIG NB? L?;F Q;FF ?KO;F NI NB I@ NB? MCT? I@ NB? G?MB S EHIQCHA 1
;H> O NB? /?C=B;L> F;Q "KO;NCIH ACP?M 1 @LIG NB? @ILGOF;
1
1
! ; O
G G ;
1BCM F;Q CM CGJFC=CN ;M O >?J?H>M IH 1 ;H> 1 CM @IOH> <S MO==?MMCP? CN?L;NCIHM
4? NB?H QLCN?
1
1 D
H3A0D;820;;H A>D67 K>F
#IL BS>L;OFC=;FFS LIOAB WIQ P;LCIOM @ILGOF;? ACP? NB? @LC=NCIH =I?^=C?HN 4 >?
J?H>CHA IH NB? LIOABH?MM ;N NB? <INNIG &H 0?=NCIH Q? B;P? ?R;GCH?> NB?
BfTS 0NLC=EF?L *;HHCHA ;H> +CEOL;>M? @ILGOF;? #IL +CEOL;>M? @ILGOF; Q? =;H
HIQ OH>?LMN;H> BIQ CN MN?GM @LIG NB? BfTS F;Q ;H> ;H CHN?AL;NCIH P?LNC=;FFS I@ NB?
FIA;LCNBGC= JLIVF? ACP?H CH "KO;NCIH 4B?H NB? LIOABH?MM MCT? AA CM EHIQH NB? @IFFIQCHA F;QM =;H ;FMI <? OM?>
00;0=3 5>A<D;0
4 !
6 ) * 9
6
'0<4CC4 5>A<D;0 ;==IL>CHA NI /;G?NN? 89 NB? LIOABH?MM MCT? AA =;H ;FMI <?
=IHH?=N?> NI NB? BfTS =I?^=C?HN <S NB? ?RJL?MMCIH
AA
>
> AA
1BCM F;MN ?GJCLC=;F @ILGOF; CM HIN >CG?HMCIHF?MM ;H> L?KOCL?M ; MNLC=N L?MJ?=N I@ 0&
OHCNM
@D0C8>=B >5 C74 O <>34;
AO <>34; 8= 1BCM GI>?F CM <;M?> IH NB? =;F=OF;NCIH I@ JBSMC=;F KO;HNCNC?M L?JL?M?HNCHA NOL<OF?H=?
CH NB? WIQ A ;H> L?MJ?=NCP?FS >?MCAH;N? NOL<OF?HN ECH?NC= ?H?LAS ;H> CNM >CMMCJ;NCIH
A
KK
7 7
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 K7 K7
N8 N8
QB?L? K7 CM NB? NOL<OF?HN WO=NO;NCIH I@ P?FI=CNS ;H> QB?L? NB? <;L L?JL?M?HNM ;P?L
;ACHA
A 0=3 4@D0C8>=B CH ! NB? KO;HNCNC?M A ;H> ;L? I<N;CH?> <S MIFPCHA NB?
@IFFIQCHA ?KO;NCIHM M?? 89 ;H> 89
A
J
J
17
N7
N7
17
A
N7
N7
B N7
B A
9 N7
,
A
, #
# A
QB?L?
, CM NB? N?LG @IL NB? JLI>O=NCIH I@ NOL<OF?HN ?H?LAS
, B
17
N8
17
N8
18
N7
B
78
78
# CM ; MIOL=? N?LG >O? NI NB? @IL=?M I@ AL;PCNS CH NB? =;M? I@ N?GJ?L;NOL? AL;>C?HNM
#
0
B
=
, HJ P
, HJ <?CHA NB? -L;H>NF HOG<?L
QCNB (
(HIQCHA A ;H> B CM >?>O=?> @LIG NB? @ILGOF;
A
QBC=B ?MM?HNC;FFS =IG?M @LIG >CG?HMCIH;F ;H;FSMCM 4? ;FMI A?N
B
B(
(
A
&N CM NB? ?KOCFC<LCOG BSJINB?MCM <?NQ??H NB? =L?;NCIH I@ NOL<OF?H=? ;H> >CMMCJ;NCIH
9
QBC=B F?;>M NI NB? GCRCHA F?HANB GI>?F =NO;FFS <S ;MMOGCHA , ;H> ;M B
9
Q? CGG?>C;N?FS VH> B
FMI
A(
QB?L?
(
CM
NB?
=B;L;=N?LCMNC=
MCT?
I@
B
B
B
$
PILNC=?M 4? A?N
A
B
78 78
;H> B A (B S >CPC>CHA NB? M?=IH> ?KO;NCIH <S NB? VLMN IH? G?G<?L <S G?G<?L
Q? VH;FFS A?N
B
(B
5
78
78
1B? =IHMN;HNM I@ NB? A\ GI>?F B;P? <??H >?N?LGCH?> <S =IGJ;LCMIH QCNB MCGJF?
MCNO;NCIHM 1B? =IHMN;HNM ;H> ;L? I<N;CH?> @LIG >;N; @IL NOL<OF?HN WIQ
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1;<F? =IHMN;HNM I@ NB? A\ GI>?F
(
, HJ
C@ # ;H> C@ # 9
H?;L ; LCAC> Q;FF 1B? @L?? >?=;S I@ ALC> NOL<OF?H=? B?FJM OM NI VH> ; P;FO? @IL IH NB? <;MCM I@ Q?FF >I=OG?HN?> ?RJ?LCG?HN;F >;N; 1B? =IHMN;HNM 9 ;H> B;P?
<??H ZIJNCGCT?>[ <;M?> IH NB? J?L@ILG;H=? I@ NB? GI>?F CH <INB NB? N?MN =;M?M DOMN
G?HNCIH?> ,<N;CHCHA CM ?F;<IL;N?> CH L?@?L?H=?M 89 ;H> 89 4? L?N;CH
NB;N CM ?KO;F NI @IL ; MN;<F? MCNO;NCIH C ? C@ # CM H?A;NCP? ;H> ?KO;F NI @IL
OHMN;<F? MNL;NCV=;NCIHM 9 CM MIG?NCG?M =;FF?> Z-L;H>NFYM NOL<OF?HN HOG<?L @IL NB?
ECH?NC= ?H?LAS I@ NOL<OF?H=?[
FF NB?M? =IHMN;HNM ;L? MOGG;LCT?> CH 1;<F? &H CHN?LH;F WIQ GI>?FFCHA NB? N?LG A 78 CM I@N?H IGCNN?> <?=;OM? CN JF;SM NB?
M;G? LIF? ;M JL?MMOL? &@ Q? Q;HN NI B;P? NB? NLO? JL?MMOL? CN CM HIN JIMMC<F? NI
CHN?AL;N? NBCM N?LG QBC=B MBIOF> NBOM ;JJ?;L CH HOG?LC=;F L?MIFONCIH
>D=30AH 2>=38C8>=B >5 C74 AO <>34;
'8683 1>D=30A84B NI >?VH? LCAC> <IOH>;LS =IH>CNCIHM CN CM =IHMC>?L?> NB;N NOL
<OF?H=? CM CH FI=;F ?KOCFC<LCOG ;N NB? Q;FF MO=B NB;N NB? JLI>O=NCIH I@ NOL<OF?H=? >O?
NI MB?;L ;N NB? Q;FF ;H> ;N NB? <INNIG CM ?KO;F NI >CMMCJ;NCIH ;H> NB? P?FI=CNS JLIVF?
CM FI=;FFS FIA;LCNBGC= 1BCM C>?; Q;M ?RJFICN?> <S *?FFIL ;H> 6;G;>; NI ?MN;<FCMB ;
T?LI ?KO;NCIH NOL<OF?H=? GI>?F M?? 89
4CNB NB? ?KOCFC<LCOG BSJINB?MCM ;N ; >CMN;H=? @LIG NB? Q;FF NB? A\ GI>?F
?KO;NCIHM ;L?
K
A
9
K
A
1B? >CMN;H=? CM =BIM?H ?KO;F NI NB I@ NB? FI=;F G?MB MCT? G?;MOL?> ;FIHA NB?
HILG;F >CL?=NCIH NI NB? Q;FF &H NB? JL?M?H=? I@ NOL<OF?H=? NB? <IOH>;LS =IH>CNCIH
@IL P?FI=CNS CM HIQ QLCNN?H ;M
K
K
B
D
<?CHA L?JF;=?> <S
B
!8@D83 1>D=30A84B QCNBION ;>>CNCIH;F CH@ILG;NCIH IH NB? P;FO? I@ A ;H> ;N ;
9
FCKOC> <IOH>;LS ; ZQ?;E[ =IH>CNCIH CM ;JJFC?> C ? B <
;H> B <
0O=B ;
=IH>CNCIH =;H BIQ?P?L ACP? LCM? NI HOG?LC=;F JLI<F?GM ;N NB? ?HNL;H=? I@ ; >IG;CH
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 AO <>34; 0E4A0643 >E4A C74 E4AC820;
1B? CHN?AL;NCIH I@ NB? /?SHIF>M ?KO;NCIH IP?L NB? P?LNC=;F <LCHAM ;<ION >CMJ?LMCIH
N?LGM >O? NI NB? HIH OHC@ILGCNS I@ P?LNC=;F P?FI=CNS JLIVF?M ,H ;==IOHN I@ NB?M?
>CMJ?LMCIH N?LGM /;MNIAC ;H> /I>C 89 JLIJIM?> ;H ?RN?HMCIH I@ NB? =F;MMC=;F A\
GI>?F ;>;JN?> NI NB? 0;CHN 3?H;HN ?KO;NCIHM
A ;H> ;L? ;P?L;A?> P;FO?M IP?L NB? P?LNC=;F >?VH?> <S
A
>
>
.
.
.
.
.
.
K K :P
7 8
K7 K7
:P
N8 N8
1B? A ;H> ?KO;NCIHM N;E? NB? @ILG
A
J
K7
A
B :?L > =H7: A
N7
>
9
, ,9D
B :?L > =H7: K7
, ,D
J
N7
>
A
-LI>O=NCIH N?LGM ;FQ;SM JIMCNCP? ;L? HIQ =;F=OF;N?> QCNB BILCTIHN;F P?FI=CNS
AL;>C?HNM
K7
K8 K7
N8
N7 N8
;L? >O? NI NB? MB?;L @IL=? I@ WIQ ;FIHA NB? P?LNC=;F
, ,9D ;H> ,D
B
,9D 9
K
K
;H> ,D >
>
&H NB?M? @ILGOF;?
5
9 5
;H> 4
4
4 CM NB? >CG?HMCIHF?MM @LC=NCIH =I?^=C?HN ;H> K6CM NB? MB?;L P?FI=CNS IL @LC=NCIH
P?FI=CNS IH NB? <INNIG S >?VHCNCIH CN CM ?KO;F NI QB?L? CM NB? @LC=NCIH MNL?MM
IH NB? <INNIG 4CNB
4 5
K
NB? MB?;L P?FI=CNS CM ;FMI ?KO;F NI
L K K
7
K
K
L
4
L
K
1B? H?Q J;L;G?N?LM 9 ;H> B;P? <??H >?N?LGCH?> IH NB? <;MCM I@ HILG;F WIQ
CH NB? =?HNL? I@ ; L?=NCFCH?;L WOPC;F =;H;F
#CH;FFS NB? @LC=NCIH =I?^=C?HN 4 CM >?>O=?> @LIG NB? =BIM?H @LC=NCIH F;Q ;M
;FL?;>S M??H CH 0?=NCIH K %6!/,!6+*& 0 ,# #/"" 02/# " #),40
>D=30AH 2>=38C8>=B >= A8683 1>D=30A84B Q? =IHMC>?L NB? WIQ ;N ; >CMN;H=? @LIG NB? Q;FF ;H> =IHNCHO? NI ;MMOG? ; FI=;F ?KOCFC<LCOG I@ JLI>O=NCIH ;H> >CMMCJ;NCIH
I@ NOL<OF?H=? %IQ?P?L CH ! JLI>O=NCIH I@ NOL<OF?H=? >O? NI NB? <INNIG MBIOF>
;FMI <? =IHMC>?L?> 1B? P;FO? ;N ; >CMN;H=? @LIG NB? Q;FF @IL ; Q;N?L >?JNB > CM
HIQ ACP?H <S
K
1
K
5
4 >
QB?L? K
1 CM NB? MB?;L P?FI=CNS IH NB? <INNIG M @IL NB? JLI>O=NCIH I@ NOL<OF?HN
ECH?NC= ?H?LAS CN CM ;MMOG?> NB;N CN =IG?M @LIG NQI =IHNLC<ONCIHM NB? Q;FF ;H> NB?
<INNIG 1B? @IFFIQCHA ?RJL?MMCIH CM >?>O=?>
K
A 5 K
1
4 1B? MB?;L P?FI=CNS ;N NB? Q;FF CM =;F=OF;N?> ;==IL>CHA NI NB? =BIM?H @LC=NCIH F;Q ;H>
NB? H;NOL? I@ WIQ BS>L;OFC=;FFS MGIINB IL LIOAB ;M M??H CH 0?=NCIH #IL NB? M;E? I@ HOG?LC=;F MN;<CFCNS NB? MNL?MM IH NB? Q;FF CM ACP?H NB? @ILG
KB
7 KB
D
==IL>CHA NI NB? M;G? ?KOCFC<LCOG BSJINB?MCM <?NQ??H JLI>O=NCIH ;H> >CMMCJ;NCIH
NB? FIA;LCNBGC= L?F;NCIH <?NQ??H MB?;L P?FI=CNS ;H> N;HA?HNC;F P?FI=CNS ;N NB? Q;FF KB
ACP?M NB? @IFFIQCHA L?F;NCIHM
B
KB
K
D
O
QB?L? O CM NB? >CMN;H=? NI NB? Q;FF ;H>
B
OK
QBC=B VH;FFS ACP?M
7
K
KB
>D=30AH 2>=38C8>=B >= ;8@D83 1>D=30A84B BIGIA?H?IOM +?OG;HH =IH>CNCIHM
;L? =BIM?H @IL ;H IONF?N ;H> MFCJ =IH>CNCIHM @IL ; Q;FF
A
;H>
D
D
M @IL ?HNL;H=? =IH>CNCIHM NB?L? CM ;H ;<M?H=? I@ CH@ILG;NCIH NB;N =;H B;L>FS <?
;ME?> I@ ; OM?L "KOCFC<LCOG CM ;MMOG?> <?NQ??H ECH?NC= ?H?LAS JLI>O=?> <S @LC=NCIH
IH NB? <INNIG ;H> CNM >CMMCJ;NCIH L;N? 1B? A ;H> M?N I@ ?KO;NCIHM NB?H ACP? NB?
@IFFIQCHA =IH>CNCIHM @IL ;H IJ?H <IOH>;LS I@ NB? >IG;CH
,9D ;H> A ,9D
,D
".21&,+0 ,# #/"" 02/# " %6!/,!6+*& 0 1B?M? P;FO?M I@ A ;H> =;H ;FMI <? OM?> QCNBCH NB? >IG;CH NI CHCNC;FCT? =;F=OF;
NCIHM
$C74A <>34;B
1BIOAB MO^=C?HNFS >C^=OFN NI CGJF?G?HN HOG?LC=;FFS NB? A\ GI>?F B;M I<PCIOM
FCGCN;NCIHM J;LNC=OF;LFS @IL ?>>C?M <?BCH> I<MN;=F?M ;H> CH NB? JL?M?H=? I@ ; BCAB L;N?
I@ >?@ILG;NCIH N?HMIL 78 H?;L ; MN;AH;NCIH JICHN @IL ?R;GJF? 1B? /?SHIF>M MNL?MM
GI>?F CM ; BCAB?L IL>?L GI>?F 1B? MCR /?SHIF>M MNL?MM ?KO;NCIHM ;L? NB?H >CL?=NFS
MIFP?> ;H> NB? ?KO;NCIH NIA?NB?L QCNB NB? ?KO;NCIHM I@ M=;F;L NOL<OF?HN WOR I@
N?GJ?L;NOL? ;H> M;FCHCNS ,HFS NBCM NSJ? I@ GI>?F CM =;J;<F? I@ L?;FFS N;ECHA CHNI
;==IOHN NOL<OF?H=? ;HCMINLIJS <S L?HIOH=CHA NB? IOMMCH?MK BSJINB?MCM %IQ?P?L NB?
NLCJF? =ILL?F;NCIHM I@ NB? ?KO;NCIHM ;L? >C^=OFN NI GI>?F ;H> NB? <IOH>;LS =IH>CNCIHM
>C^=OFN NI DOMNC@S 1I IOL EHIQF?>A? NBCM CM NB? L?;MIH QBS NB?L? CM MNCFF HI CH>OMNLS
ILC?HN?> ;JJFC=;NCIH I@ NBCM GI>?F CH @L?? MOL@;=? BS>LI>SH;GC=M
70?C4A
%A8=28?;4B >5 C74 J=8C4
4;4<4=C <4C7>3
%)(#"' # "#( ,$ # ?LNL;H> /OMM?FF \
=CA>3D2C8>=
MCGJF? >?M=LCJNCIH I@ NB? G?NBI> CM ME?N=B?> ION <?FIQ &N GCABN L?KOCL? @OLNB?L
>?P?FIJG?HN <ON ; FCMN I@ L?@?L?H=? QILEM CM ACP?H C@ GIL? >?N;CFM ;L? H??>?>
1B? VHCN? ?F?G?HN G?NBI> CM <;M?> IH NQI JLCH=CJ;F C>?;M ;H CHN?LJIF;NCIH G?NBI>
;H> ; P;LC;NCIH;F G?NBI> EHIQH ;M ZQ?CABN?> L?MC>O;FM[ ";=B I@ NB?M? G?NBI>M B;M
; MCGCF;L ;CG NI J;MM @LIG ; =IHNCHOIOM JLI<F?G NI ; >CM=L?N? JLI<F?G &@ ; @OH=NCIH
K CM >?VH?> <S NB? ?KO;NCIH ( K IH ; =IHNCHOIOM >IG;CH NB? CHN?LJIF;NCIH
G?NBI> QCFF JLIPC>? ;H ;JJLIRCG;N? @OH=NCIH I@ K =IHMNLO=N?> @LIG ; VHCN? HOG<?L I@
L?;F HOG<?LM 1B? P;LC;NCIH;F G?NBI> QCFF L?JF;=? NB? =IHNCHOIOM ?KO;NCIH ( K P;FC> ;N ?P?LS JICHN I@ <S ; VHCN? HOG<?L I@ ?KO;NCIHM &H MBILN CHN?LJIF;NCIH
>CM=L?NCT?M NB? OHEHIQH ;H> NB? P;LC;NCIH;F JLCH=CJF? >CM=L?NCT?M NB? ?KO;NCIH 1BCM
MBIOF> <? MJ?=CV?> CH ; MNLC=N G;NB?G;NC=;F @L;G?QILE QBC=B MIG?NCG?M B?FJM NI
MBIQ NB? ?RCMN?H=? ;H> OHCKO?H?MM I@ NB? MIFONCIH 2H@ILNOH;N?FS CH @L?? MOL@;=? WOC>
G?=B;HC=M NB?M? >?GIHMNL;NCIHM IHFS ?RCMN CH MCGJF? =;M?M
1B? >IG;CH I@ =IGJON;NCIH CM ;H IJ?H ;H> <IOH>?> >IG;CH CH # <?CHA
NB? M?N I@ L?;F HOG<?LM * CM NB? MJ;=? >CG?HMCIH ?KO;F NI IL CM NB?
<IOH>;LS L?AOF;L ; HILG;F P?=NIL =;H <? >?VH?> IH CN I@ N NB? =IHNCHOOG
F?P?F JBSMC=;F KO;HNCNC?M JL?MMOL? N?GJ?L;NOL? =IGJIH?HNM I@ NB? P?FI=CNS P?=NIL
?N= ;L? @OH=NCIHM I@ CH *IL?IP?L Q? L?KOCL? NB;N NB?M? @OH=NCIHM ;L? MKO;L?
CHN?AL;<F? ;H> ( >?HIN?M NB? M?N I@ MKO;L? CHN?AL;<F? @OH=NCIHM >IN JLI>O=N
=;H <? >?VH?> IH NBCM M?N ;M
.
K L KL :
MO<M?N I@ ( CM =IGJIM?> I@ @OH=NCIHM QBIM? >?LCP;NCP?M ;FIHA NB? * >CL?=
NCIHM I@ MJ;=? ;L? ;FMI MKO;L? CHN?AL;<F? 1BCM AO;L;HN??M NB;N ; @OH=NCIH AL;>C?HN
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
QIOF> ;FMI <? MKO;L? CHN?AL;<F? 1BCM MO<M?N CM >?HIN?> $ CM >?VH?> CH $ ;M
.
K L KL =H7: K =H7: L :
H?Q >IN JLI>O=N
1B? MJ;=?M ( ;H> $ QCNB NB?CL >IN JLI>O=N ;L? =;FF?> %CF<?LN MJ;=?M
%CF<?LN MJ;=?M ;L? P?LS MCGCF;L NI NB? "O=FC>?;H P?=NIL MJ;=?M ,H <INB I@ NB?G ;H>
QCNB NB? B?FJ I@ NB? >IN JLI>O=N Q? =;H >?VH?
; HILG
5
K K K
; >CMN;H=?
: K L K L
K L
K L
;H> ;H ;HAF?
"$ <ON NB?CL >CG?HMCIH CM CHVHCN? $ B;M <??H >?M=LC<?> QCNB >?LCP;NCP?M I@ NB? VLMN
IL>?L $ 9 =;H <? >?M=LC<?> CH NB? M;G? Q;S QCNB >?LCP;NCP?M I@ NB? ANB IL>?L
FCE? NB? M?N I@ @OH=NCIHM QBIM? >?LCP;NCP?M OJ NI ANB IL>?L ;L? ;FF MKO;L? CHN?AL;<F?
1B? $ 9 MJ;=?M ;L? EHIQH ;M 0I<IF?P MJ;=?M
M @IL NB? ?KO;NCIHM <?CHA QILE?> ION IHFS FCH?;L ?KO;NCIHM QCNB J;LNC;F >?
LCP;NCP?M ;L? =IHMC>?L?> WOC> G?=B;HC=M ?KO;NCIHM ;L? A?H?L;FFS HIH FCH?;L <ON NB?
L?MIFONCIH G?NBI>M OM?> @IL MIFPCHA NB?G QCFF A?H?L;FFS <? QLCNN?H CH ; FCH?;L @ILG
%?L? ;L? MIG? I@ NB? FCH?;L ?KO;NCIHM NB;N Q? MB;FF =IG? ;=LIMM F;N?L
%?FGBIFNT ?KO;NCIH
K
N
K
O
K
-ICMMIH ?KO;NCIH
K
N
K
<
O
&H NBCM =B;JN?L Q? =IHMC>?L NB;N IOL ?KO;NCIH CM CH NB? @ILG
( K <
&HN?LJIF;NCIH ;H> NB? P;LC;NCIH;F G?NBI> QCFF HIQ <? >?M=LC<?> GIL? JL?=CM?FS
=C4A?>;0C8>= 8= J=8C4 4;4<4=CB
1B? A?H?L;F C>?; CM NI L?JF;=? ;H OHEHIQH @OH=NCIH K QBC=B <?FIHAM NI ;H CHVHCN?
>CG?HMCIH;F MJ;=? <S ;H ;JJLIRCG;NCIH K6 >?VH?> IH ; VHCN? >CG?HMCIH;F MJ;=? *
1BCM CM ;H ;JJLIRCG;NCIH ;H> Q? =;HHIN ;FQ;SM ?HMOL? NB;N ( K6 < 4? MCGJFS
NLS NI GCHCGCT? ( K6 < ;H> NBCM QCFF <? NB? AI;F I@ NB? P;LC;NCIH;F G?NBI>
-/&+ &-)"0 ,# 1%" #&+&1" ")"*"+1 *"1%,! y
1
x
0
i-1
i
i+1
#CAOL? FCH?;L <;MCM CH IH? >CG?HMCIH
&H VHCN? ?F?G?HNM ; @OH=NCIH K CM L?JL?M?HN?> <S D L?;F HOG<?LM QBC=B ;L? NB?
?R;=N P;FO?M I@ K ;N MJ?=C;F JICHNM I@ NB? >IG;CH EHIQH ;M >?AL??M I@ @L??>IG #IL NB?
INB?L JICHNM ;H CHN?LJIF;NCIH CM MO^=C?HN 1I >?VH? ; OHCKO? CHN?LJIF;NCIH @OH=NCIH
? A ; JIFSHIGC;F IH? QCNB ?R;=N P;FO?M I@ K @IL NB? D >?AL??M I@ @L??>IG QIOF>
<? P?LS =IGJF?R IL QIOF> F?;> ;N F?;MN NI ; BCAB IL>?L @OH=NCIH 0I VHCN? ?F?G?HNM
ACP? OJ NB? OHCPI=;F H;NOL? I@ NB? CHN?LJIF;NCIH @OH=NCIH ;H> =BIIM? ; FI=;FFS MCGJF?
CHN?LJIF;NCIH @OH=NCIH ? A =IHMN;HN FCH?;L KO;>L;NC= N?MM?FF;NCIH I@ MJ;=? CHNI
M?AG?HNM NLC;HAF?M KO;>LCF;N?L;FM N?NL;B?>L; JLCMGM ?N= CM ?R?=ON?> 1B? >?AL??M
I@ @L??>IG ;L? ;MMCAH?> NI NB?M? ?F?G?HNM IH NB? P?LNC=?M ;N NB? =?HNL? I@ AL;PCNS CH
NB? GC>>F? I@ NB? MC>?M ?N= @IL ?R;GJF? 1BOM ?;=B ?F?G?HN CM >?M=LC<?> <INB <S
NB? =IIL>CH;N?M I@ CNM A?IG?NLC= HI>?M ;H> <S NBIM? I@ CNM CHN?LJIF;NCIH HI>?M 1BCM
CM @IFFIQ?> <S ; MCGJF? >?VHCNCIH I@ CHN?LJIF;NCIH QCNBCH ?;=B ?F?G?HN #IL ?R;GJF?
CH IH? >CG?HMCIH N;ECHA NB? P?LNC=?M I@ M?AG?HNM ;M >?AL??M I@ @L??>IG ; FCH?;L
CHN?LJIF;NCIH QCFF <? =BIM?H 1B? @OH=NCIH K6 ;JJLIRCG;NCHA K =;H NB?H <? QLCNN?H
;M
K6 <
,
K 7 7
7
QB?L? 7 CM ; <;MCM @OH=NCIH >?VH?> >C_?L?HNFS <ON CH ; FCH?;L G;HH?L IH ?;=B
M?AG?HN ?H>CHA ;N NB? JICHN ? ,ONMC>? NB?M? M?AG?HNM 7 CM T?LI #CAOL? MBIQM
NBCM <;MCM @OH=NCIH CH IH? >CG?HMCIH IH ; HIH L?AOF;L G?MB 1B? P;FO? I@ 7 CM ;N
JICHN C ;H> IH )
;FF NB? INB?L >?AL??M I@ @L??>IG
<
1B? JLIJ?LNS 7 7 CM P?LS CGJILN;HN ;H> QCFF <? NB? E?S NI M?P?L;F >?GIH
MNL;NCIHM &H NQI >CG?HMCIHM QCNB NLC;HAF?M ;H> ; FCH?;L CHN?LJIF;NCIH ?;=B <;MCM
7 N O QCFF <? QLCNN?H ;M 7N 8O 9 7 8 ;H> 9 >?J?H>CHA IH NB? NLC;HAF? MI ;M NI
I<N;CH NB? P;FO? ;N JICHN ? ;H> ;N NB? INB?L IH?M #CAOL? MBIQM NB? ?RN?HN
;H> NB? HI>;F P;FO?M I@ ; FCH?;L <;MCM IH ; VHCN? ?F?G?HN G?MB
1B?M? <;MCM @OH=NCIHM ;L? MIG?NCG?M =;FF?> ZMB;J? @OH=NCIHM[ 4CNB ; FCH?;L CH
N?LJIF;NCIH ; FCH?;L @OH=NCIH K QCFF <? L?JL?M?HN?> ?R;=NFS &N CM M;C> NI <? CH NB?
;JJLIRCG;NCIH MJ;=? 1B? CHN?LJIF;NCIH =;H ;FMI <? KO;>L;NC= IL I@ ;HS INB?L IL>?L
QBC=B ?H;<F?M L?JL?M?HN;NCIH I@ NB? @OH=NCIH IH NB? >?AL??M I@ @L??>IG <ON ;FMI MIG?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
0
0
0
0
1
0
0
#CAOL? ?RN?HN I@ ; <;MCM @OH=NCIH IH ; NLC;HAOF;L G?MB
I@ CNM >?LCP;NCP?M
0IG?NCG?M ;H> CN CM MI @IL KO;>LCF;N?L;FM NB? ?F?G?HN CM NII =IGJF?R NI A?N ;
MCGJF? >?VHCNCIH I@ NB? CHN?LJIF;NCIH 0I Q? AI <;=E NI ; GIL? L?AOF;L L?@?L?H=?
?F?G?HN ; MKO;L? CH NB? =;M? I@ ; KO;>LCF;N?L;F CH ; ZL?@?L?H=?[ MJ;=? ;H> >?VH?
; A?IG?NLC= NL;HM@ILG;NCIH <?NQ??H NB? NQI MJ;=?M &HN?LJIF;NCIH CM MCGJF? IHFS CH
NB? L?@?L?H=? MJ;=? ?R=?JN CH NB? =;M? I@ MCGJFC=C;F ?F?G?HNM M?AG?HNM NLC;HAF?M ;H>
N?NL;B?>L; QCNB FCH?;L CHN?LJIF;NCIH 1B? OM? I@ ; L?@?L?H=? MJ;=? QCFF <? ?R?GJFCV?>
CH 0?=NCIH @IL <OCF>CHA VHCN? ?F?G?HN G;NLC=?M 1B? VHCN? ?F?G?HNM NI <? OM?>
QCFF <? JL?M?HN?> CH >?N;CF CH B;JN?L JL?=CM? >?M=LCJNCIH I@ ; F;LA? HOG<?L I@
INB?L ?F?G?HNM =;H ;FMI <? @IOH> CH L?@?L?H=? 89
+0A80C8>=0; ?A8=28?;4
%;PCHA L?MNLC=N?> IOL @OH=NCIH K NI ; L?JL?M?HN;NCIH K6 <;M?> IH ; >CM=L?N? HOG<?L
I@ OHEHIQHM K7 Q? HIQ
+ QCMB NI GCHCGCT? ( K6 < 1B? P;LC;NCIH;F G?NBI> MN;N?M
NB;N NB? >IN JLI>O=N ( K6 < 7 : CM T?LI @IL MIG? @OH=NCIHM 7 =;FF?> N?MN
@OH=NCIHM 1B? =BIC=? I@ NB?M? @OH=NCIHM >?VH?M NB? P;LC;HNM I@ NB? VHCN? ?F?G?HN
G?NBI> 1B? GIMN =F;MMC= N?=BHCKO? NB? $;F?LECH N?=BHCKO? =IHMCMNM I@ =BIIMCHA N?MN
@OH=NCIHM
?KO;F NI NB? <;MCM @OH=NCIHM 7 7 &H NBCM =;M? C@ K6 CM >?=IGJIM?> ;M
)<
K
NB? P;LC;NCIH;F G?NBI> QCFF =F?;LFS F?;> NI ; FCH?;L MSMN?G I@ OHEHIQHM
7
7
7
K7
*IL?IP?L NI ?;=B FCH?;L IJ?L;NIL ( CM ;MMI=C;N?> ; <CFCH?;L @ILG 7 MO=B ;M
.
7 K L ( K < L :
1B? VHCN? ?F?G?HN NB?ILS ;CGM ;N >?N?LGCHCHA NB? =IH>CNCIHM IH ( ;H> IH 7 @IL
QBC=B NB? JLI<F?G QIOF> B;P? ; OHCKO? MIFONCIH H CGJILN;HN =IH>CNCIH CM NB;N @IL
-/&+ &-)"0 ,# 1%" #&+&1" ")"*"+1 *"1%,! u
1
x
0
-1
#CAOL? ;H CHPCMC<F? @OH=NCIH @IL NB? AL;>C?HN IJ?L;NIL
?P?LS HIH T?LI @OH=NCIH K NB?L? ?RCMNM ; L?;F JIMCNCP? MO=B NB;N ( K &@ NB? HILG ( CM >?VH?> <S
( $&#
C
K
( K K
NBCM =IH>CNCIH MCAHCV?M NB;N ( MBIOF> B;P? ; MNLC=NFS JIMCNCP? HILG ,NB?LQCM? C@ ;
HIH T?LI @OH=NCIH K> P?LCV?M ( K> NB? MIFONCIH QCFF HIN <? OHCKO? <?=;OM? C@ K CM
; MIFONCIH I@ NB? ?KO;NCIH ( K < K K> QCFF ;FMI <? IH? 4CNBCH NB? P;LC;NCIH;F
G?NBI> @L;G? NB? =IH>CNCIH CM QLCNN?H ;M
7 K L
! $&#
C
K L
D
1BCM CM EHIQH ;M NB? ZCH@\MOJ =IH>CNCIH[ &N JL?P?HNM NB? I==OLL?H=? I@ J;L;MCNC=
MIFONCIHM QBC=B =IOF> ;LCM? @LIG NB? <CFCH?;L @ILG #IL ?R;GJF? @IL ; IH? >CG?HMCIH;F
>IG;CH QCNB L?AOF;LFS MJ;=?> >?AL??M I@ @L??>IG C@ ( CM NB? AL;>C?HN IJ?L;NIL ;H>
C@ NB? <;MCM @OH=NCIHM 7 ;L? FCH?;L ;H> ?KO;F NI NB? N?MN @OH=NCIHM 7 ; @OH=NCIH K
QBIM?
P;FO?M ;N NB? HI>?M ;L? ; MO==?MMCIH I@ ;H> M?? #CAOL? QIOF> F?;> NI
+
(
K
7 : @IL ?P?LS N?MN @OH=NCIH 7 ;H> NB? CH@\MOJ =IH>CNCIH QIOF> HIN <?
P?LCV?>
1BCM MCNO;NCIH ;LCM?M QB?H NB? <IOH>;LS =IH>CNCIHM ;L? HIN =IHMC>?L?> <ON CN
=;H N;E? JF;=? ;FMI @;L @LIG NB? <IOH>;LC?M I@ ; =;F=OF;NCIH >IG;CH 1B? <CFCH?;L
@ILGM L?MOFNCHA @LIG NB? P;LC;NCIH;F @ILGOF;NCIHM I@ NB? +;PC?L\0NIE?M ?KO;NCIHM >I
HIN ;FQ;SM M;NCM@S NB? CH@\MOJ =IH>CNCIH Q?FF EHIQH =IH>CNCIH @IL CHN?LH;F WIQM
I@ CH=IGJL?MMC<F? WOC>M =IHMCMNM I@ =BIIMCHA >C_?L?HN ?F?G?HNM NI L?JL?M?HN JL?MMOL?
;H> P?FI=CNS @IL ?R;GJF? ; KO;>L;NC= P?FI=CNS ;H> ; FCH?;L JL?MMOL? &H JLCH=CJF?
NBCM =BIC=? CM HIN G;H>;NILS @IL =IGJL?MMC<F? WOC>M ;H> <S ?RN?HMCIH @IL NB? 0;CHN
3?H;HN ?KO;NCIHM %IQ?P?L NB? =BIC=? I@ >C_?L?HN ?F?G?HNM @IL P?FI=CNS ;H> Q;N?L
>?JNB ACP?M GIL? L?AOF;L L?MOFNM 1BCM CM ?R?GJFCV?> CH -F;N? QB?L? ; NC>? CM
MCGOF;N?> IH ; KO;LN?L =CL=F? <?;=B QCNB ; KO;>L;NC= <INNIG ?F?P;NCIH >P?=NCIH
;H> ;FF >CMMCJ;NCP? N?LGM ;L? L?GIP?> @LIG ?KO;NCIHM &N CM =F?;L NB;N CH NBCM ;=;>?GC=
MCNO;NCIH MJOLCIOM IM=CFF;NCIHM I==OL QCNB ; FCH?;L >CM=L?NCT;NCIH <ON NB?S >CM;JJ?;L
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
C@ NB? P?FI=CNS CM NL?;N?> QCNB ; KO;MC <O<<F? >CM=L?NCT;NCIH M?? 89 @IL GIL? >?N;CFM
HINB?L MIFONCIH ?^=C?HN QCNB L?AOF;L G?MB ;H> NB? VHCN? >C_?L?H=? G?NBI> CM NI
B;P? NQI >C_?L?HN ;H> MN;AA?L?> G?MB?M @IL JL?MMOL? ;H> P?FI=CNS #CH;FFS @IL ?FFCJNC=
?KO;NCIHM Q? =;H OM? ; >CM=L?NCT;NCIH I@ NB? );JF;=C;H EHIQH ;M ZHIH =IGJ;NC<F?[
@IL QBC=B NB? );JF;=C;H CM HIN ;N NB? >CM=L?N? F?P?F ?KO;F NI NB? >CP?LA?H=? I@ NB?
AL;>C?HN 1B? L?MOFN I@ NBCM G?NBI> CM >?N;CF?> CH 0?=NCIH QB?L? NB? 0;CHN 3?H;HN
?KO;NCIHM ;L? QLCNN?H CH NB? @ILG I@ ; Q;P? ?KO;NCIH
GIL? NBILIOAB ?R;GCH;NCIH I@ NB?M? KO?MNCIHM ?MJ?=C;FFS @IL NB? +;PC?L\0NIE?M
?KO;NCIHM =;H <? @IOH> CH L?@?L?H=?M 89 ;H> 89 ;H> CH NB? GIL? L?=?HN QILE <S
-CLIHH?;O 89 IH NB? G?NBI>M I@ VHCN? ?F?G?HNM @IL WOC>M
70?C4A
'4B>;DC8>= >5 C74
(08=C+4=0=C 4@D0C8>=B
*&-(" ' *) (# & -#) # "#( & ' ( -#) * (& (# ! ( $&'
?LNL;H> /OMM?FF
6;0=24 0C C74 4G8BC8=6 <4C7>3B
-L?M?HNCHA ;FF NB? G?NBI>M OM?> NI MIFP? NB? 0;CHN 3?H;HN ?KO;NCIHM L;CM?M ?P?H GIL?
=B;FF?HA?M &H @;=N ; =IG<CH;NCIH I@ P;LC;HNM I@ NB? ?KO;NCIHM ;H> I@ L?MIFONCIH
G?NBI>M F?;>M NI BOH>L?>M I@ HOG?LC=;F M=B?G?M
IH=?LHCHA NB? ?KO;NCIHM NB? @IFFIQCHA MBIOF> <? ;>>?> NI NB? =IHM?LP;NCP? ;H>
HIH =IHM?LP;NCP? @ILGM I@ NB? JLCGCNCP? ?KO;NCIHM M??H ?;LFC?L
1B? ZJM?O>I Q;P? ?KO;NCIH @ILG[ CH QBC=B P?FI=CNS CM QCNB>L;QH @LIG NB?
=IHNCHOCNS ?KO;NCIH NB;HEM NI NB? GIG?HNOG ?KO;NCIH M?? 89
1B? ZQ;P? ?KO;NCIH[ @ILG I<N;CH?> <S >?LCPCHA NB? ?KO;NCIHM QCNB L?MJ?=N NI
MJ;=? ;H> NCG? ;M B;M <??H >IH? CH 0?=NCIH 1B? IFNTG;HH ?KO;NCIH NB? MN;LNCHA JICHN @IL ZECH?NC=[ M=B?G?M QBC=B QCFF
<? <LC?WS >CM=OMM?> @OLNB?L
IH=?LHCHA NB? L?MIFONCIH G?NBI>M Q? =;H FCMN NB? @IFFIQCHA
#CHCN? >C_?L?H=?M
#CHCN? PIFOG?M
#CHCN? ?F?G?HNM
);AL;HAC;H IL M?GC );AL;HAC;H G?NBI>M
0J?=NL;F G?NBI>M
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
HOG?LC=;F M=B?G? QCFF =IG<CH? ; =BIC=? I@ ILCACH;F ?KO;NCIHM ;H> ; =BIC=? I@
L?MIFONCIH G?NBI>M &@ ; @L;=NCIH;F MN?JM G?NBI> CM OM?> M?? 0?=NCIH <?FIQ CN
CM JIMMC<F? NI =IG<CH? >C_?L?HN G?NBI>M <ON NB;N =IGJFC=;N?M ?P?H GIL? N;RIHIGS
;H> =IGJL?B?HMCIH 1BOM NB? ;FAILCNBG JL?M?HN?> CH ;H> ;@N?L 0?=NCIH CM @IOH>?>
G;CHFS IH NB? VHCN? ?F?G?HN G?NBI> ;H> ;N NB? M;G? NCG? <ILLIQM @LIG NB? M?GC
);AL;HAC;H G?NBI>M @IL ;>P?=NCIH N?LGM <S OMCHA NB? G?NBI> I@ =B;L;=N?LCMNC=M IL
@LIG VHCN? PIFOG?M <S OMCHA WO=NO;NCIH MJFCNNCHA M=B?G?M B;JN?L 1B? =BIC=?
I@ NB? =IGJON;NCIH >IG;CH CM ;HINB?L MIOL=? I@ >CP?LMCNS &H @;=N VHCN? >C_?L?H=?M
=IOF> <? =IHMC>?L?> ;M ; @ILG I@ VHCN? ?F?G?HNM QB?L? =?LN;CH G;NLC=?M ;L? L?JF;=?> <S
>C;AIH;FM <ON NB?S ;L? CGJF?G?HN?> IH L?AOF;L G?MB?M QB?L? CN CM ?;MC?L NI =;F=OF;N?
BCAB IL>?L >?LCP;NCP?M ACPCHA ;==?MM NI ; QC>?L =BIC=? I@ HOG?LC=;F M=B?G?M
?@IL? >?N;CFCHA IOL VHCN? ?F?G?HN M=B?G? Q? MB;FF ?RJF;CH QB;N A?H?L;F JLIJ?LNC?M
;L? ?RJ?=N?> @LIG ;H ?^=C?HN HOG?LC=;F M=B?G? ;H> ;FMI <LC?WS >?M=LC<? MIG? INB?L
G?NBI>M
"08= ?A>?4AC84B >5 0 6>>3 =D<4A820; B274<4
4? =IHMC>?L MSMN?GM I@ J;LNC;F >C_?L?HNC;F ?KO;NCIHM CH=FO>CHA NCG? ;H> MJ;=? >?LCP
;NCP?M ;>P?=NCIH>C_OMCIH WOR ? A >C_OMCP? WOR ;H> MIOL=? N?LGM +IH FCH?;LCNS
I@ ?KO;NCIHM L?MOFNM CH <INB NOL<OF?H=? ;H> >CM=IHNCHOIOM MIFONCIHM ?P?H NBIOAB NB?
CHCNC;F =IH>CNCIHM ;L? =IHNCHOIOM "KO;NCIHM ;L? BSJ?L<IFC= ;H> NB? >CM=L?NCT;NCIH I@
MIG? I@ NB?CL N?LGM H??>M NI N;E? CHNI ;==IOHN NB? >CL?=NCIH I@ NB? WIQ &H NB?M?
=IH>CNCIHM NB? HOG?LC=;F JLIJ?LNC?M ?RJ?=N?> @IL ; L?MIFONCIH M=B?G? ;L? ;M @IFFIQM
%A428B8>=
+OG?LC=;F MIFONCIHM MBIOF> <? ;M =FIM? ;M JIMMC<F? NI NB? ?R;=N MIFONCIHM QCNBION
>CP?LACHA #IL J?LCI>C= MIFONCIHM IL CH ; GIL? A?H?L;F G;HH?L @IL Q;P?M ; >CMNCH=NCIH
CM G;>? <?NQ??H JL?=CMCIH IH NB? ;GJFCNO>? ;H> JL?=CMCIH IH NB? JB;M?
>=B8BC4=2H
1BCM =IH>CNCIH L?KOCL?M NB;N ?KO;NCIHM ;L? Q?FF >CM=L?NCT?> C ? C@ NB? NCG? MN?J ;H>
G?MB MCT? N?H> NI T?LI Q? ;L? <;=E NI =IHNCHOIOM ?KO;NCIHM
(C018;8CH
0IFONCIHM MBIOF> HIN >CP?LA? 4? =;H MBIQ NB;N =IHMCMN?H=S ;H> MN;<CFCNS CGJFS =IH
P?LA?H=? NI NB? =ILL?=N MIFONCIH @IL ; FCH?;L JLI<F?G Q?FF JIM?> QCNBION <IOH>;LS
=IH>CNCIHM ;H> QCNB CHCNC;F P;FO?M NBCM CM NB? );R NB?IL?G 89
*=8@D4=4BB
1BCM JLIJ?LNS CM A?H?L;FFS M;NCMV?> <S ; =IH>CNCIH EHIQH ;M Z?HNLIJS[ %IQ?P?L
G;HS L?;F JLI<F?GM >I HIN B;P? DOMN IH? MIFONCIH ?G?LA?H=? I@ PIH (;LG;H PILNC=?M
<S NB? LOJNOL? I@ MSGG?NLS NOL<OF?H=? I;H>; ?_?=N ?N= &H NB? 0;CHN 3?H;HN
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 ?KO;NCIHM ?HNLIJS CM NB? CHN?AL;F I@ NB? B?;> ;FIHA NB? Q;N?L >?JNB &N CM ;FMI NB?
NIN;F ?H?LAS I@ ; =IFOGH I@ Q;N?L QCNB ; B?CABN > &N CM QLCNN?H ;M
!
>K
=
>
=>64
;H> NB? ?HNLIJS =IH>CNCIH CM
!
J
>
=
:?L K !
1BCM ?HNLIJS =IH>CNCIH <?=IG?M ?KO;FCNS CH NB? ;<M?H=? I@ >CM=IHNCHOCNC?M
">=>C>=828CH
H ;>P?=NCIH ;H> >C_OMCIH M=B?G? MBIOF> HIN =L?;N? GCHCG; IL G;RCG; INB?L NB;H
NBIM? I@ NB? CHCNC;F =IH>CNCIHM 0=B?G?M OMCHA FIQ CHN?LJIF;NCIH IL>?LM QCFF <? GIHI
NIHC= GIL? ?;MCFS <ON ;FMI F?MM JL?=CM? ;H> GIL? >C_OMCP? Q;P? ;GJFCNO>? QCFF <?
>?=L?;M?> ;LNCV=C;FFS %CAB?L IL>?L M=B?G?M QCFF <? GIL? JL?=CM? <ON MO<D?=N NI
MJOLCIOM IM=CFF;NCIHM
>=B4AE0C8>=
HOG?LC=;F M=B?G? MBIOF> ?HMOL? =IHM?LP;NCIH I@ JBSMC=;F KO;HNCNC?M FCE? G;MM GI
G?HNOG B?;N ?N=
470E8>DA 8= C74 ?A4B4=24 >5 B7>2:B
"CNB?L NB? M=B?G? CM =;J;<F? I@ L?JLI>O=CHA >CM=IHNCHOCNC?M QCNBION ; MJ?=CV= NL?;N
G?HN NBCM N?=BHCKO? CM =;FF?> ZMBI=E =;JNOLCHA[ IL MBI=EM MBIOF> <? @IFFIQ?> ;H>
NB?S L?=?CP? ; MJ?=C;F NL?;NG?HN ZMBI=E VNNCHA[
8=8C4 38Q4A4=24 B274<4B
#CHCN? >C_?L?H=? M=B?G?M QBIM? OM? CM FCGCN?> NI L?AOF;L ALC>M ;L? JLIAL?MMCP?FS L?
JF;=?> <S N?=BHCKO?M ;JJFC=;<F? NI HIH MNLO=NOL?> G?MB?M %?L? Q? ;L? DOMN JL?M?HN
CHA ; G?NBI> QC>?FS OM?> CH NB? J;MN NB? !& FN?LH;NCHA !CL?=NCIH &GJFC=CN
G?NBI> &N OM?M NB? L?AOF;LCNS I@ NB? G?MB @IL NB? NL?;NG?HN I@ NB? >?LCP;NCP?M ;FIHA O
VLMN NB?H ;FIHA N 1I G;E? CN =F?;L Q? ;JJFS NB? G?NBI> NI NB? @IFFIQCHA FCH?;LCT?>
;H> MCGJFCV?> ?KO;NCIHM
>
J
K
> :?L K =H7: > K
J
K
K
N
L
K
O
=
>
N
L
J
K
L
N
L
L
O
=
>
O
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
0I @LC=NCIH ;H> >C_OMCIH ;L? CAHIL?> ;H> CN CM ;MMOG?> NB;N > K ;H> L L?G;CH
=FIM? NI > K ;H> L 1B? NBL?? ?KO;NCIHM =;H <? QLCNN?H CH ; P?=NIL @ILG
"
J
"
N
"
O
QCNB
>
K
" % K & % =
L
>
K
& ;H> %
K
>
L
L
=
L
&
CM >CM=L?NCT?> ;M B 0J;NC;F >CM=L?NCT;NCIH I@ NB? IJ?L;NIL J F
ACP?M
; G;NLCR , 0J;NC;F >CM=L?NCT;NCIH I@ NB? IJ?L;NIL J G ACP?M ; G;NLCR - #IL MJ;=?
>?LCP;NCP?M ; L;HE\+C=BIFMIH M=B?G? CM OM?> 1B;N CM
B
"
" <
N
N
QCNB
" <
N
@IL NB? M;E? I@ MN;<CFCNS "KO;NCIH CM NB?H QLCNN?H ;M
%
- "<
,
,H NB? <;MCM I@ " " <
% , - "<
" < ?;G ;H> 4;LGCHA 89 QLCN? NB? ?KO;NCIH ;M
%
,
- "<
- " ,
;H> @;=NILCT? NB? F?@N B;H> MC>? ;M
%
, %
- "
1B? ;JJLIRCG;NCIH L?MJ?=NM NCG? IL>?L ;==OL;=S
#CLMN Q? L?MIFP?
%
- "<
, 4 ,
QB?L? 4 CM ;H CHN?LG?>C;N? OHEHIQH 1B?H
%
- " 4
1B? FCH?;L ?KO;NCIHM I<N;CH?> @IL ?;=B L?MIFONCIH ;JJFC=;<F? IHFS NI IH? >CG?H
MCIH ;L? NLC >C;AIH;F ;M CH ; IH? >CG?HMCIH;F JLI<F?G -?;=?G;H ;H> /;=B@IL> 89
<L?;E >IQH NB? JLI<F?G >C_?L?HNFS <S CHNLI>O=CHA ;H CHN?LG?>C;N? G;NLCR " < %
%
,
"<
"<
%
%
,
"<
"<
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 8=8C4 E>;D<4 B274<4B 5>A 7H?4A1>;82 4@D0C8>=B
1I MCGJFC@S G;NN?LM ?RJF;H;NCIHM ;L? CH IH? >CG?HMCIH #CHCN? PIFOG? M=B?G?M >?;F
QCNB ?KO;NCIHM CH NB? @IFFIQCHA =IHM?LP;NCP? @ILG
"
J
# "
/ "
N
QB?L?
>
>K
" >K
#" >K
=6
;H> / " CM NB? LCABN B;H> MC>? =IHN;CHCHA NB? >C_OMCIH N?LGM NB? MIOL=? N?LGM ;H>
.
@IL NB? M?=IH> =IGJIH?HN NB? <INNIG AL;>C?HN = F 1B? ;H;FSNC=;F MIFONCIH I@
NBCM MSMN?G CHPIFP?M NB? ';=I<C;H G;NLCR I@ NB? MSMN?G
#
"
QBIM? ?CA?HP;FO?M ;L? K
=> K K
=> ;H> K => ;H> ?CA?HP?=NILM ;L?
K
;H>
=>
K
=>
1B?M? ?CA?HP;FO?M G;E? NB? 0;CHN 3?H;HN ?KO;NCIHM BSJ?L<IFC= 1B? VHCN? PIFOG?
G?NBI> FIIEM @IL =IHMN;HN KO;HNCNC?M IP?L ?;=B PIFOG? IL =?FF 1B?M? G;S ?CNB?L
=ICH=C>? QCNB NB? ?F?G?HNM I@ ; HIH MNLO=NOL?> G?MB IL <? =?HNL?> IH NB? HI>?M I@
; HIH MNLO=NOL?> G?MB ;MC=;FFS C@ Q? >CM=;L> / " "KO;NCIH CM MOGG?> IP?L
?;=B PIFOG? 7 QBC=B ACP?M
LEB 7
"
J
.
# " D : QB?L? LEB 7 >?HIN?M NB? F?HANB NB? MOL@;=? IL NB? PIFOG? I@ NB? ?F?G?HN >?J?H>CHA
IH NB? >CG?HMCIH ;H> D NB? HILG;F P?=NIL %?H=? CN MO^=?M NI EHIQ # " ;N NB?
<IOH>;LC?M I@ NB? =IHNLIF =?FF NI VH> NB? ?PIFONCIH CH NCG? I@ " M " CM =IHMN;HN
IH ?;=B =?FF CN CM >CM=IHNCHOIOM ;N NB? =?FF <IOH>;LC?M #CH>CHA # " G?;HM VH>CHA
NB? MIFONCIH I@ NB? /C?G;HH JLI<F?G QCNB =IHMN;HN P?FI=CNC?M ;H> Q;N?L >?JNB IH
?CNB?L MC>? I@ ; >CM=IHNCHOCNS 1BCM JLI<F?G MCGCF;L NI ; >;G <L?;E IH ; Q?N <INNIG
M?? 0?=NCIH CM L?MIFP?> <S NB? NB?ILS I@ NB? =B;L;=N?LCMNC= =OLP?M ;H> NB?
/C?G;HH CHP;LC;HNM /I? 89 JLIJIM?> ;H ;JJLIRCG;NCIH I@ NB? /C?G;HH JLI<F?G
<;M?> IH ; FCH?;LCT?> ';=I<C;H G;NLCR QBC=B CM NB? MN;LNCHA JICHN I@ G;HS M=B?G?M
%IQ?P?L NB? /I? MIFONCIH MBIOF> <? GI>CV?> CH NB? JL?M?H=? I@ >LS TIH?M ;M CN ACP?M
LCM? NI HIH JBSMC=;F BS>L;OFC= DOGJM QBC=B MBIOF> <? L?JF;=?> <S ?RJ;HMCIH Q;P?M
1B? =;F=OF;NCIH I@ WOR MBIOF> N;E? CHNI ;==IOHN NB? >CL?=NCIH I@ NB? WIQ 1BCM CM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
I<N;CH?> QCNB NB? B?FJ I@ NB? ';=I<C;H G;NLCR &H NQI >CG?HMCIHM "KO;NCIH CM
QLCNN?H ;M
"
J
#F "
N
#G "
/ "
O
1B? JLCH=CJF? I@ L?MIFONCIH CM NB? M;G? S MIFPCHA NB? /C?G;HH JLI<F?G NB?
VHCN? PIFOG? G?NBI>M I<N;CH ?R=?FF?HN L?MOFNM @IL ;FF MSMN?GM I@ ?KO;NCIHM QCNB MBI=EM
1B?S P?LC@S NB? =IHM?LP;NCIH I@ G;MM ;H> GIG?HNOG JLIJ?LNC?M ,H NB? INB?L B;H>
NB? >C_?L?HN MIOL=? N?LGM =IHN;CH?> CH / " G;CHFS NB? <INNIG AL;>C?HN ;H> NB?
@LC=NCIH N?LG ;L? ; >C^=OFNS #LC=NCIH ;N NB? <INNIG CM >?;FN QCNB CH ;H ;>>CNCIH;F
@L;=NCIH;F MN?J A?H?L;FFS CGJFC=CN 1BCM ;_?=NM NB? <;F;H=? ?MN;<FCMB?> CH NB? VLMN
MN?J +?Q M=B?G?M FCE? NB;N I@ ?LGO>?T ( 89 @IL ?R;GJF? JLIJIM? ;H OJQCH>
=IGJON;NCIH I@ NB? <INNIG AL;>C?HN ;M Q?FF ;M NB? WOR NI @OFVF NB? JLIJ?LNC?M I@
=IHM?LP;NCIH 1B?S ;FMI JLIJIM? ;H CGJFC=CN P;LC;HN I@ NB?CL G?NBI> QBCF? NB? =F;MMC=
M=B?G?M ;L? ?RJFC=CN QCNB NB? P;FO?M I@ NB? IOL;HN HOG<?L CB
F FCGCN?> NI IL 8=4C82 B274<4B
(CH?NC= M=B?G?M CH J;LNC=OF;L M?? 89 ;L? VHCN? PIFOG? M=B?G?M <;M?> IH ;H
ILCACH;F C>?; MCH=? NB?S ;L? MOJJILN?> <S NB? IFNTG;HH ECH?NC= ?KO;NCIHM @IL A;M?M
&H @;=N NB? ECH?NC= ;JJLI;=B =IHMCMNM I@ ;MMI=C;NCHA ; JICHN QBIM? Q;N?L >?JNB CM >
;H> P?FI=CNS K ;H> ; >?HMCNS ) I@ J;LNC=F?M QBIM? P?FI=CNC?M P;LS ;LIOH> ;H ;P?L;A?
K 1BCM >?HMCNS CM >?VH?> NBOM
>
K
) > 6 % 6 &
56
56
QB?L? NB? @OH=NCIH CM ; E?LH?F QCNB NB? @IFFIQCHA JLIJ?LNC?M
.
: &
;H>
.
: &
1B? IFNTG;HH ?KO;NCIH I<N;CH?> @IL ; A;M I@ * J;LNC=F?M QB?H * N?H>M NI
CHVHCNS CM
)
J
)
64 )
=
- ) )
N
N +
+
2FNCG;N?FS
- CM ; Z=IFFCMCIH[ N?LG QBC=B P?LCV?M & -: ;H> & -: CN QCFF HIN <? H?=?MM;LS NI =;F=OF;N? NBCM OHEHIQH N?LG 1B? G;CH JICHN CM NB;N
NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? =IHN;CH?> CH "KO;NCIH =;FF?> NB? ZGC=LIM=IJC=
?KO;NCIH[ 1BCM CM CFFOMNL;N?> <S NB? @IFFIQCHA ?KO;NCIHM
.
>
) > K :
&
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 >K .
&
>K
=>
) > K :
.
&
) > K :
1B? OHEHIQHM I@ NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? B?H=? L?JF;=?> <S NB? MCHAF?
@OH=NCIH ) ;H> ;L? @IOH> IH NB? <;MCM I@ CNM GIG?HNM *IL?IP?L NB? ?KO;NCIH I@
) CM FCH?;L <ON ) CM ; HIH FCH?;L @OH=NCIH I@ > ;H> K 1B? =BIC=? I@ NB? @OH=NCIH >CMNCHAOCMB?M >C_?L?HN P;LC;HNM I@ NB? G?NBI> -?LNB;G? ;H> 0CG?IHC 89 IJN @IL ;
@OH=NCIH MO=B NB;N NB? @OH=NCIH ) GCHCGCT?M NB? ?H?LAS JLIPC>?> NB;N CNM T?LI ;H>
VLMN IL>?L GIG?HNM L?MNIL? > ;H> >K 1B? ?HNLIJS =IH>CNCIH CM NBOM M;NCMV?> ,H
; W;N <INNIG NB? 0;CHN 3?H;HN MSMN?G CM L?>O=?> NI NB? L?MIFONCIH I@ ;H ;>P?=NCIH
?KO;NCIH QCNB MIOL=? N?LG "KO;NCIH 1BCM ?KO;NCIH CM L?MIFP?> CH VHCN? PIFOG?M
1B? >CL?=NCIH I@ NB? =OLL?HN CM N;E?H CHNI ;==IOHN CH NB? =;F=OF;NCIH I@ WOR N?LGM ;H>
NBCM CM ;>;JN?> NI ?;=B P;FO? I@ P?FI=CNS 1B? G;=LIM=IJC= P;FO? I@ WOR?M CM NB?H
@IOH> <S MOGGCHA IP?L 1B? M=B?G?M NBOM =IH=?CP?> ?HMOL? =IHM?LP;NCIH I@ G;MM
;H> JIMCNCPCNS I@ NB? Q;N?L >?JNB 1BCM JLIJ?LNS CM ?RN?H>?> NI NB? NL;=?L ?KO;NCIHM
1B? =IH>CNCIH I@ MN;<CFCNS B;M NB? @ILG
' K => J
N
8=8C4 4;4<4=C B274<4B
1B? =F;MMC= $;F?LECH G?NBI> OMCHA =?HNL?> >?LCP;NCP?M ?P?H @IL NL;HMJILN N?LGM IJ
JIM?M NB? BSJ?L<IFC= H;NOL? I@ ?KO;NCIHM ;M CN >I?M HIN >CMNCHAOCMB <?NQ??H NB? OJ
MNL?;G ;H> >IQHMNL?;G >CL?=NCIHM I@ NB? WIQ ,NB?L MJ?=CV= N?=BHCKO?M Q?L? >?MCAH?>
NI =IJ? QCNB NBCM >C^=OFNS 1B? !$ B;L;=N?LCMNC=\!CMMCJ;NCP?\$;F?LECH M?? 89
G?NBI> CM <;M?> IH NB? 02-$ G?NBI> QBC=B QCFF <? OM?> ;H> >?M=LC<?> CH 0?=NCIH
1B? 1;SFIL\$;F?LECH 89 G?NBI> CM ;H ?RJFC=CN @ILGOF;NCIH C>?HNC=;F NI NB? );R\
4?H>LI_ @;GCFS I@ VHCN? >C_?L?H=? M=B?G?M 1B? OHEHIQHM ;L? >?P?FIJ?> CH M?LC?M
CH L?F;NCIH NI NCG? &N =;H <? MBIQH NB;N NB? G?NBI> CHNLI>O=?M ;H ;LNCV=C;F >C_OMCIH
NI ?KO;NCIHM >?J?H>?HN IH NB? NCG? MN?J M?? 89
1B? F?;MN MKO;L?M G?NBI> 89 CM ;H CGJFC=CN M=B?G? QB?L? NB? L?MC>O;F CM GCHCGCT?>
<S F?;MN MKO;L?M
%C=EM ;H> 0N?`?L 89 ACP? ; #IOLC?L ;H;FSMCM I@ NB? NBL?? G?NBI>M DOMN G?HNCIH?>
;M Q?FF ;M ; =IGJ;LCMIH IH NB? N?MN =;M? I@ NB? 0NIE?L MIFONCIH M?? 0?=NCIH 1B? ;LNC=F? ?MN;<FCMB?M NB? MOJ?LCILCNS I@ NB? !$ G?NBI> QBIM? HOG?LC=;F >C_OMCIH
CM F?MM NB;H 1;SFIL\$;F?LECH 1B? F?;MN MKO;L?M G?NBI> CM =IHMC>?L?> OH;==?JN;<F?
@IL OHMN?;>S WIQM ;H> ACP?M LCM? NI >C^=OFNC?M @IL MOJ?L=LCNC=;F WIQ
$E4A0;; E84F >5 C74 )4;4<02 0;6>A8C7<
H IP?LPC?Q I@ NB? G?NBI>M OM?> QCFF B?FJ NB? L?;>?L NI A?N ; A?H?L;F C>?; I@ NB?
L?MIFONCIH M=B?G?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? CHCNC;F M?N I@ ?KO;NCIHM CM NB? HIH =IHM?LP;NCP? 0;CHN 3?H;HN ?KO;NCIHM CH NB?
>?JNB\P?FI=CNS @ILG &@ NB? G?NBI> I@ =B;L;=N?LCMNC=M CM =BIM?H NB?M? ?KO;NCIHM
;L? MIFP?> CH NQI MN;A?M OMCHA NB? @L;=NCIH;F MN?JM G?NBI>
&H NB? VLMN MN?J ;>P?=NCIH N?LGM QBC=B =ILL?MJIH> NI NB? NL;HMJILN I@ JBSMC=;F
KO;HNCNC?M K L ;H> QB?L? ;JJFC=;<F? > A ;H> 0 1B? BSJ?L<IFC= H;NOL? I@
?KO;NCIHM DOMNCV?M NB? OM? I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M L?F?P;HN NI CN
&H NB? M?=IH> MN?J NB? L?G;CHCHA ?KO;NCIHM ;L? =IHMC>?L?> JLIJ;A;NCIH >C_O
MCIH ;H> MIOL=? N?LGM JFOM ;>P?=NCIH N?LGM QB?H NB? G?NBI> I@ =B;L;=N?LCMNC=M
=;HHIN <? OM?> 1BCM MN;A? CM L?MIFP?> <S NB? VHCN? ?F?G?HN N?=BHCKO? NCG?
>CM=L?NCT;NCIH ;FFIQM NB? ?FCGCH;NCIH I@ HIH FCH?;L N?LGM @LIG NB? ?KO;NCIHM
3;LC;NCIH;F @ILGOF;NCIH ;H> >CM=L?NCT;NCIH CH MJ;=? ;H> NCG? NB?H NL;HM@ILG
=IHNCHOIOM ?KO;NCIHM CHNI ; >CM=L?N? FCH?;L MSMN?G QBIM? OHEHIQHM ;L? NB? P;F
O?M I@ NB? JBSMC=;F KO;HNCNC?M > K L ;N NB? HI>?M I@ NB? G?MB 1BCM MSMN?G CM
NB?H L?MIFP?> <S ;H CN?L;NCP? G?NBI> I@ NB? =IHDOA;N? AL;>C?HN NSJ?
A02C8>=0; BC4?B <4C7>3
1BCM G?NBI> CM IHFS ;JJFC?> QB?H NB? G?NBI> I@ =B;L;=N?LCMNC=M CM OM?>
M @IL NB? =IGJON;NCIH >IG;CH NCG? CM >CM=L?NCT?> ;H> NB? OHEHIQHM > K L ?N=
;L? IHFS >?N?LGCH?> ;N CHMN;HNM J
0 J
0 J
0 ;H> MI IH !?HINCHA
J< J
D 0 NB? >?LCP;NCP? CH NCG? I@ ; @OH=NCIH < CM >CM=L?NCT?> ;M
<
<< <<
J
J
QB?L? < < CM NB? @OH=NCIH < ;N NCG? J< H?Q NCG? MN?J ;H> < < CM NB? @OH=NCIH <
;N NCG? J< IF> NCG? MN?J 1BCM CM ; VLMN IL>?L ;JJLIRCG;NCIH ?R=?JN C@ IH? =IHMC>?LM
NB;N NB? >?LCP;NCP? CM ?RJL?MM?> ;N NCG? J< B CH QBC=B =;M? Q? I<N;CH M?=IH> IL>?L
;==OL;=S #IL NBCM L?;MIH Q? NLS NI >CM=L?NCT? NB? INB?L N?LGM I@ NB? ?KO;NCIHM ;N
NB? M;G? NCG? 1BCM CM HIN ;FQ;SM JIMMC<F? J;LNC=OF;LFS @IL ;>P?=NCIH IL CH =;M? I@
HIH FCH?;LCNS
+IN? NBCM JL?M?HN;NCIH I@ >?LCP;NCP?M CH NCG? CM ; ZVHCN? >C_?L?H=?[ ;JJLI;=B &H
VHCN? ?F?G?HNM NB? >C_?L?H=? < < < < =IOF> ;FMI <? CHN?LJL?N?> ;M NB? L?MOFN I@
+ B
NB? CHN?AL;NCIH <S J;LNM I@ ; N?LG B 4
B :J QBC=B QIOF> ;LCM? @LIG ; P;LC;NCIH;F
@ILGOF;NCIH CH=FO>CHA NCG?
1B? A?H?L;F JLCH=CJF? I@ L?MIFONCIH CM ;M @IFFIQM
#LIG ;H CHCNC;F MIFONCIH < ACP?H ;N NCG? J NB? MIFONCIH < < CM <OCFN QCNB MO==?MMCP?
CN?L;NCIHM IL NCG? MN?JM VLMN <S VH>CHA < NB?H < ?N=
4CNBCH ; NCG? MN?J ;H> CH IL>?L NI VH> < < @LIG < < NB? @L;=NCIH;F MN?JM G?NBI>
=IGJON?M CHN?LG?>C;N? P;FO?M <S =IHMC>?LCHA IHFS ; J;LN I@ NB? N?LGM I@ NB? ?KO;NCIHM
&H IOL =;M? Q? MB;FF MIFP?
<1 < <
J
;>P?=NCIH N?LGM /"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 NB?H
<1
INB?L N?LGM J
QB?L? < L?JL?M?HNM > K L A IL 0
1B? MOG I@ NB?M? NQI ?KO;NCIHM L?MNIL?M NB? CHCNC;F M?N BIQ?P?L ;JJLIRCG;NCIH CH
NCG? QCFF <? I@ NB? VLMN IL>?L ;M MIG? N?LGM ;>P?=NCIH ;L? HIN >CM=L?NCT?> ;N NCG?
J< B
1B? CH=IHP?HC?H=? I@ NCG? IL>?L CM =IGJ?HM;N?> <S NB? @;=N NB;N ?;=B I@ NB?M?
N?LGM =;H <? NL?;N?> QCNB NB? N?=BHCKO? GIMN ;JJLIJLC;N? NI CN
/?G;LE CH @;=N HINBCHA JL?P?HNM OM @LIG A?NNCHA M?=IH> IL>?L CH NCG? QCNB NB?
G?NBI> I@ =B;L;=N?LCMNC=M @IL ?R;GJF? <S =IGJONCHA NQI MN;A?M I@ ;>P?=NCIH QCNB NQI
>C_?L?HN NCG? MN?JM J ;H> J ;H> ;H ;>P?=NCIH IJ?L;NIL ?KO;F NI K<
K <
1B? L?MOFNM I@ NB? NQI ;>P?=NCIHM ;L? >?HIN?> <1 ;H> <1 1B? M?=IH> IL>?L CM I<N;CH?>
<S L?JF;=CHA 4 B4 <S
<<
<1
J
%IQ?P?L NBCM MIFONCIH CM KOCN? ;QEQ;L>
<<
<1
3E42C8>= BC064 DB8=6 C74 <4C7>3 >5 270A02C4A
8BC82B
1B? QBIF? IL J;LN I@ NB? @IFFIQCHA ?KO;NCIHM CM MIFP?> >?J?H>CHA IH NB? =BIC=? I@
P;LC;<F?M NI <? NL?;N?> QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M <?NQ??H NCG? J< ;H> NCG?
J< 1
> ><
J
K
1 K<
J
L1 L <
J
01 0 <
J
1
A A<
J
K =H7: > K =H7: K K =H7: L K =H7: 0 K =H7: A 1
< K =H7: J
";=B I@ NB?M? ?KO;NCIHM MCGJFS G?;HM NB;N NB? NIN;F >?LCP;NCP? I@ @OH=NCIHM > K
L ?N= CM T?LI ;FIHA MIG? NL;D?=NILC?M =;FF?> =B;L;=N?LCMNC=M 1B? KO;HNCNC?M NL;HMJIL
N?> L?G;CH OH=B;HA?> ;FIHA NB?M? =B;L;=N?LCMNC=M 1B? =;F=OF;NCIH I@ =B;L;=N?LCMNC=
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
=OLP?M CM >?N;CF?> CH B;JN?L 0J?=C;F ;NN?HNCIH MBIOF> <? ACP?H NI NB? VH;F L?G;LE
L?A;L>CHA NB? NL?;NG?HN I@ NB? N?LGM QBC=B HILG;FFS ;JJ?;L IH NB? LCABN B;H> MC>?
I@ ?KO;NCIHM
1B? JL?M?H=? I@ > CH NB? ;<IP? FCMN G;S ;JJ?;L MOLJLCMCHA 1BCM CM <?=;OM? Q? =;H
MIFP? NB? =IHNCHOCNS ?KO;NCIH CH NQI @L;=NCIH;F MN?JM <S MJFCNNCHA NB? N?LG :?L >
K
CHNI K =H7: > > :?L K QB?L? NB? VLMN N?LG =;H <? CHN?LJL?N?> ;M ; NL;HMJILN I@
>
%A>?060C8>= 38QDB8>= B>DA24 C4A<B
4B?H NB? G?NBI> I@ =B;L;=N?LCMNC=M CM HIN OM?> NBCM MN?J L?MIFP?M ;FF NB? ?KO;NCIHM
;H> B?H=? ;FMI N;E?M NL;HMJILN N?LGM CHNI ;==IOHN
)8<4 38B2A4C8I0C8>=
M ?RJF;CH?> ?;LFC?L @IL NB? @L;=NCIH;F MN?JM G?NBI> NB? >?LCP;NCP? CH NCG? I@ ; @OH=NCIH
< N;E?M NB? @ILG
<
<< <<
J
J
0B4 >5 0 ?A8>A 03E42C8>= BC064
&H NBCM =;M? Q? GOMN MIFP?
><
K<
L<
1
>
J
K
1
6A
=
J
N
L1
6A
=
J
O
> :?L K /9;
"F
:?L > 3 =H7: K
>
"G
:?L > 3 =H7: L
>
QB?L? 1
> K
1 ;H> L1 ;L? NB? L?MOFN I@ ;>P?=NCIH I@ > K ;H> L
0NLC=NFS MJ?;ECHA NB? N?LGM IH NB? LCABN B;H> MC>? MBIOF> =IG? @LIG CHN?AL;NCIH
CH NCG? ;FIHA NB? =B;L;=N?LCMNC= =OLP?M M?? NB? L?G;LE CH 0?=NCIH 0B4 F8C7>DC ?A8>A 03E42C8>= BC064
><
><
J
K =H7: >
> :?L K /9;
K<
K<
J
6A
K =H7: K =
N
"F
:?L > 3 =H7: K
>
L<
L<
J
6A
K =H7: L =
O
"G
:?L > 3 =H7: L
>
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 ==IL>CHA NI NB? =BIC=? I@ HOG?LC=;F M=B?G?M IH NB? >C_?L?HN P;LC;<F?M Q? B;P?
CH @;=N ;H ;G;FA;G I@ NB? NQI ?;LFC?L =;M?M N NBCM JICHN CN CM HIN S?N MJ?=CV?>
C@ NB? N?LGM > K L MO<MCMNCHA CH NB?M? ?KO;NCIHM ;L? P;LC;<F?M ;N NCG? J< IL J<
0NLC=NFS MJ?;ECHA C@ Q? B;P? CH GCH> NB? M?=IH> IL>?L ;==OL;=S CH NCG? IH? MIFONCIH
QIOF> <? NI QLCN? > >< >< ;H> ;H;FIAIOM @ILGOF;? @IL K ;H> L 1BCM NCG?
>CM=L?NCT;NCIH =;FF?> M?GC CGJFC=CN;NCIH MBIOF> <? ;JJFC?> =;L?@OFFS =NO;FFS #IOLC?L
;H;FSMCM I@ NB? M?GC CGJFC=CN M=B?G? QIOF> ACP? ;H ;GJFCV=;NCIH =I?^=C?HN ?KO;F NI
0O=B M=B?G?M CH JLCH=CJF? J?L@?=N N?H> NI >CP?LA? CH JL;=NC=? 1BCM CM <?=;OM?
NB? #IOLC?L ;H;FSMCM >I?M HIN =IHMC>?L ?CNB?L HIH FCH?;LCNS IL <IOH>;LS =IH>CNCIHM M;@?NS G;LACH CM JLIPC>?> <S QLCNCHA @IL @OH=NCIH < < <<
<<
QCNB <ON P?LS =FIM? NI 1B? H;G? ZM?GC CGJFC=CN[ CM L?N;CH?> ?P?H C@
CM AL?;N?L NB;H (D18C4A0C8>=B
1BCM M?GC CGJFC=CN;NCIH >I?M HIN MIFP? ;FF JLI<F?GM =NO;FFS NB? HIH FCH?;L N?LG
> :?L K <LCHAM ION ; JLI>O=N I@ OHEHIQHM ><
;H> K<
4CNBION ;HS INB?L
NL?;NG?HN Q? QIOF> A?N ;N NB? ?H> I@ NB? =;F=OF;NCIH ; HIH FCH?;L MSMN?G NB;N Q?
QIOF> HIN EHIQ BIQ NI MIFP? 0I NB? @IFFIQCHA @ILGOF; CM ;>IJN?>
> :?L K >>@=> :?L
C
K<
C
K<
QB?L? C CM =;FF?> NB? ZL?F;R;NCIH =I?^=C?HN @IL P?FI=CNS[
>>@=> CM =;FF?> NB? ZB?CABN I@ JLIJ;A;NCIH[ NBCM QIOF> <? OM?> NI =;F=OF;N? NB?
=?F?LCNS I@ JLIJ;A;NCIH I@ FIHA Q;P?M 1I A?N ; FCH?;L JLI<F?G CN MBIOF> <? ?KO;F NI
>< <ON Q? =;H N;E? ; P;FO? =FIM?L NI >< >< <S MIFPCHA CH MO==?MMCIH NB? M;G?
NCG? MN?J NB?M? NCG? MN?JM ;L? =;FF?> MO< CN?L;NCIHM QCNB NB? @ILGOF;
>>@=> 6>
<
6
><
6 CM NB? L?F;R;NCIH =I?^=C?HN @IL Q;N?L >?JNB
> < CM NB? =OLL?HN ?MNCG;NCIH I@ ><
&N CM ?KO;F NI >< QCNBION MO< CN?L;NCIH
<
<ON QCNB MO< CN?L;NCIHM CN CM ?KO;F NI >
NB? P;FO? L?MOFNCHA @LIG NB? JL?=?>CHA
MO< CN?L;NCIH
0CGCF;LFS QB?H ;>P?=NCIH N?LGM ;L? HIN JLI=?MM?> <S NB? G?NBI> I@ =B;L;=N?LCMNC=M
Q? QLCN? K =H7: < ;M
K =H7: < K 1=<D =H7:
4<
<
4
<<
C K < ;H> K < CM CNM?F@
K <
QB?L? K 1=<D CM NB? ;>P?=NCIH V?F> ?KO;F NI C ;H ;JJLIRCG;NCIH I@ K < L?MOFNCHA @LIG MO< CN?L;NCIHM
1B? L?F;R;NCIH =I?^=C?HNM ;L? CH JLCH=CJF? ;FQ;SM AL?;N?L NB;H A?H?L;FFS IL ACP? AII> L?MOFNM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
8QDB8>= C4A<
0?GC CGJFC=CN;NCIH I@ NB? >C_OMCIH N?LGM ACP?M NB? ?KO;NCIH ?R;GJF? I@ K
:?L > 3 =H7:
>
2 <
CK
2
C
K<
1I A;CH GIL? W?RC<CFCNS CH NB? CGJF?G?HN;NCIH I@ NB? >C_?L?HN HOG?LC=;F M=B?G?M
; MJ?=CV= L?F;R;NCIH =I?^=C?HN 2C CM OM?> B?L? @IL >C_OMCIH QBC=B =;H <? >C_?L?HN
@LIG C ? A CH NB? =;M? I@ NB? JM?O>I Q;P? ?KO;NCIH
=2A4<4=C 5>A<D;0C8>= 5>A F0C4A 34?C7
1B? @OH=NCIH 6A @L?? MOL@;=? MBIOF> <? ?RJL?MM?> ;M ; @OH=NCIH I@ NB? <INNIG ?F?P
;NCIH ;H> Q;N?L >?JNB > 1B? M?GC CGJFC=CN;NCIH I@ > ACP?M NB? @IFFIQCHA ?KO;NCIH
= =H7: 6A = =H7:
6>
<
><
64
>< = =H7: 6A<
6
NB;N CM NI M;S ;FMI
= =H7: 6A =
6
=H7: ><
QB?L? 6A< >< 64
IHMC>?LCHA NBCM ?RJL?MMCIH ;H> NB? =IHNCHOCNS ?KO;NCIH CN CM GIL? ?F?A;HN NI
L?JF;=? NB? OHEHIQH ><
<S NB? CH=L?G?HN >< >< &H @;=N CH NB? =;M? I@ ;
F;E? ;N L?MN QCNB ;H CLL?AOF;L <INNIG NB? N?LG = =H7: 6A< QBC=B L?;JJ?;LM IH
NB? LCABN B;H> MC>? I@ NB? VH;F FCH?;L MSMN?G QCFF <? T?LI ;M Q?FF ;M NB? CH=L?G?HN
>< >< QBC=B CM ;H OHEHIQH I@ NB? MSMN?G #L?? MOL@;=? ?F?P;NCIH CH NB? =;M? I@
; F;E? ;N L?MN QCNB ;H CLL?AOF;L <INNIG CM NBOM H;NOL;FFS =IHMN;HN 1BCM QIOF> HIN <?
NB? =;M? C@ NB? <INNIG AL;>C?HN ;H> NB? Q;N?L >?JNB AL;>C?HN Q?L? NL?;N?> M?J;L;N?FS
0O<M?KO?HNFS NB? CH=L?G?HN I@ > QCFF <? >?HIN?> ;M > (HIQCHA > CN CM NB?H P?LS
?;MS NI VH> ><
&N QIOF> <? F?MM H;NOL;F NI CHNLI>O=? NB? CH=L?G?HNM I@ K ;H> L CH J;LNC=OF;L NB?S
QIOF> CH=L?;M? NB? LCABN B;H> MC>? I@ NB? =IHNCHOCNS ?KO;NCIH CHMN?;> I@ >?=L?;MCHA
CN
0J?=CV= NL?;NG?HN I@ NB? @L?? MOL@;=? AL;>C?HN CH NB? JL?M?H=? I@ OH=IP?L?> <?>M
CM >?N;CF?> CH 0?=NCIH +IN? QB?H JILIMCNS CM ;>>?> NI NB? 0;CHN 3?H;HN ?KO;NCIHM NB? N?LG 6
QCFF
B
6 6
1BCM N?LG CM NB?H QLCNN?H ;M
;JJ?;L QB?H >CM=L?NCTCHA NB? >?LCP;NCP?
B
<
>< ><
J
<
>< < ><
J
QB?L? NB? OHEHIQH >< >< L?;JJ?;LM CH NB? VLMN N?LG < CM ;MMOG?> NI <?
EHIQH @IL MIFPCHA NB? ?KO;NCIHM <?NQ??H J< ;H> J< 1B? M?=IH> N?LG CM ?HNCL?FS
EHIQH ;H> CM ;>>?> NI NB? LCABN B;H> MC>? I@ NB? ?KO;NCIH
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 )8<4 38B2A4C8I0C8>= >5 5A82C8>= C4A<B
#LC=NCIH N?LGM ;L? HIH FCH?;L ?R=?JN I@ =IOLM? QCNB NB? FCH?;L F;Q 1B?S B;P? NI <?
NL?;N?> CH ;H CGJFC=CN Q;S C@ Q? Q;HN NI ;PIC> OH>?MCL;<F? HOG?LC=;F ?_?=NM FCE? ;
NIN;FFS ;LNCV=C;F CHP?LMCIH I@ P?FI=CNC?M
OH>?L NB? ?_?=N I@ @LC=NCIH &H IL>?L NI =IHM?LP?
; FCH?;L @ILG K K
L ;H> L K
L ;L? >CM=L?NCT?> ;M @IFFIQM
K
L
5
K
5
K
L
K<
L
L<
6
6
K<
L<
K<
L<
4? NBOM I<N;CH ;H ?HNCL?FS CGJFC=CN NL?;NG?HN I@ @LC=NCIH N?LGM =ILL?MJIH>CHA NI
; MIG?QB;N M?GC CGJFC=CN;NCIH CH NB? @ILG K K< K< H ?PC>?HN CGJ?L@?=NCIH
<ON HIN P?LS >CMNOL<CHA CH JL;=NC=? CM NB;N ; NCG? MN?J QCNB CHCNC;F =IH>CNCIHM I@ T?LI
P?FI=CNS CM HIN MO<D?=N NI @LC=NCIH
(?024 38B2A4C8I0C8>=
4? FIIE @IL @OH=NCIH P;FO?M ;N NB? >CM=L?NCT;NCIH JICHNM <S JLID?=NCHA NB?M? @OH=NCIHM
IHNI NB? <;M?M
<
<
,
< 7 7
7
D B?L? CM NB? HOG<?L I@ >CM=L?NCT;NCIH JICHNM < 7 CM NB? P;FO? I@ NB? @OH=NCIH < ;N JICHN
? ;H> 7 NB? <;MCM @OH=NCIH ;MMI=C;N?> NI NB? M;G? JICHN ";=B I@ NB? <;M?M CM NBOM
;MMI=C;N?> NI ; >?AL?? I@ @L??>IG M Q? B;P? M??H CH B;JN?L 7 CM ; @OH=NCIH
QBIM? P;FO? CM ;N NBCM JICHN ;H> ;N ;FF NB? INB?LM
4CNBCH ; ACP?H ?F?G?HN ; @OH=NCIH CM CHN?LJIF;N?> @LIG NB? P;FO?M ;N NB? >CM=L?N
CT;NCIH HI>?M
&HN?LJIF;NCIH @OH=NCIHM ;L? MCGJF? IHFS IH ; L?@?L?H=? ?F?G?HN ";=B ;=NO;F ?F?
G?HN CM NB? CG;A? I@ NBCM L?@?L?H=? ?F?G?HN <S MCGJF? A?IG?NLC= NL;HM@ILG;NCIH NB?L?
CM IH? @IL ?;=B ?F?G?HN 0?? 0?=NCIH @IL ; >?M=LCJNCIH I@ <;MCM @OH=NCIHM
&GJILN;HN HIN? NI ;PIC> ; JIMMC<F? L?MOLA?H=? I@ NB? CH@\MOJ =IH>CNCIH M??
B;JN?L MJ;NC;F >CM=L?NCT;NCIH CM HIN ;FQ;SM NB? M;G? @IL P;LC;<F?M > K ;H> L
1QI M?NM I@ <;MCM @OH=NCIHM ;L? NB?L?@IL? H?=?MM;LS CH P;LC;NCIH;F @ILGOF;NCIH JL?M?H
N?> H?RN
+0A80C8>=0; 5>A<D;0C8>=
P;LC;NCIH;F @ILGOF;NCIH Q;M ?MN;<FCMB?> CH $ION;FYM NB?MCM 89 QCNB NB? P;LC;<F?M
K ;H> > 1B? Q;N?L >?JNB ;H> NB? P?FI=CNS MBIOF> <?FIHA NI $ ;H> ( L?MJ?=NCP?FS ( CM NB? M?N I@ MKO;L? CHN?AL;<F? @OH=NCIHM >?VH?> IH NB? >IG;CH ;H>
$ CM NB? MO<M?N I@ ( =IHMCMNCHA I@ NB? @OH=NCIHM QBIM? >?LCP;NCP?M ;L? MKO;L?
CHN?AL;<F? ;H> T?LI ;FIHA NB? <IOH>;LS I@ !?GIHMNL;NCIHM ?MN;<FCMB?> CH L?@?L?H=?
89 QCFF HIN <? L?J?;N?> B?L? 1B?S MOJJIM? !CLC=BF?N NSJ? <IOH>;LS =IH>CNCIHM @IL
P?FI=CNS K 1B?S MBIOF> <? ?RN?H>?> NI INB?L NSJ?M I@ <IOH>;LS =IH>CNCIHM MO=B
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;M NB? !CLC=BF?N =IH>CNCIH @IL Q;N?L >?JNB > ;H> ;FMI NI NB? =;M? I@ CGJ?LG?;<CFCNS
<IOH>;LS =IH>CNCIH CH ; Q?;E @ILG QBC=B QCFF <? ?R;GCH?> MBILNFS
IHP?LA?H=? I@ IOL ;FAILCNBG QCNB FIQ /?SHIF>M HOG<?L B;M ;FMI <??H ?MN;<FCMB?>
<S ?LH;L>C ;H> -CLIHH?;O 89
&H JL;=NC=? NB? P;LC;NCIH;F @ILGOF;NCIH =IHMCMNM I@ =BIIMCHA ; M?N I@ @OH=NCIHM
=;FF?> ZN?MN @OH=NCIHM[ >?HIN?> 8 ;H> ?;=B ;MMI=C;N?> NI ; JICHN I@ >CM=L?NCT;NCIH
>?AL??
I@ @L??>IG I@ HOG<?L @ NB?H L?JF;=CHA ;H ?KO;NCIH QCNB NB? @ILG ! <S
+
!
:
8
&H NBCM =B;JN?L Q? ;MMOG? NB;N NB? N?MN @OH=NCIHM ;L? >C_?L?HN @LIG NB? <;MCM
@OH=NCIHM ;H> NB;N >CM=L?NCT;NCIH I@ NB? Q;N?L >?JNB =;H <? >C_?L?HN @LIG NB;N I@
P?FI=CNS %?H=? Q? >?HIN?
67 <;MCM ;N JICHN ? NI ?RJL?MM NB? Q;N?L >?JNB
67 N?MN @OH=NCIH ;N JICHN ? @IL NB? =IHNCHOCNS ?KO;NCIH
C7 <;MCM ;N JICHN ? NI ?RJL?MM NB? =IGJIH?HNM I@ P?FI=CNS
C7 N?MN @OH=NCIH ;N JICHN ? @IL NB? NQI GIG?HNOG ?KO;NCIHM
1B? HOG<?L I@ <;M?M L?F;N?> NI Q;N?L >?JNB CM >?HIN?> ;M DF>
1B? HOG<?L I@ <;M?M L?F;N?> NI P?FI=CNS CM >?HIN?> ;M DFK
DFK ;H> DF> ;L? HIN H?=?MM;LCFS ?KO;F NB? =;M? I@ ; FCH?;L Q;N?L >?JNB ;H>
; KO;>L;NC= P?FI=CNS @IL ?R;GJF? M NB? ?RCMN?H=? I@ ; JLCIL MN?J I@ ;>P?=NCIH
IHFS MOJJL?MM?M N?LGM Q? MB;FF >?;F QCNB NB? A?H?L;F =;M? QB?L? ;>P?=NCIH N?LGM
MNCFF ;JJ?;L -ON MCGJFS NB?M? F;MN N?LGM ;L? NL?;N?> FCE? NB? INB?LM QBC=B QIOF>
=ILL?MJIH> NI =?HNL?> M=B?G?M I@ JIIL KO;FCNS 4B;N CM L?;FFS >IH? CM ?RJF;CH?> CH
>?N;CF CH B;JN?L %?H=? NB? CHCNC;F ?KO;NCIHM ;@N?L NCG? >CM=L?NCT;NCIH ;L?
IHNCHOCNS
><
><
J
K 1=<D =H7: >
>>@=> :?L K /9;
*IG?HNOG
K<
K<
J
><
6=
L<
L<
J
><
6=
K 1=<D =H7: K
><
6 <
= A
N
N
"F
:?L > 3 =H7: K
>
"G
:?L > 3 =H7: L
>
K 1=<D =H7: L
><
6 <
= A
O
O
4B?H > K IL L ;L? G?HNCIH?> QCNBION ; NCG? CH>?R D IL D NB?M? ;L? M?GC
CGJFC=CN P;FO?M QBC=B B;P? HIN <??H >?P?FIJ?> @IL NB? M;E? I@ =IH=CM?H?MM #IL NB?
>?VHCNCIH I@ K 1=<D ;H> >>@=> M?? 0?=NCIH ,H NB? <;MCM I@ NB?M? ?KO;NCIHM NB? IJ?L;NCIHM ;L? M?J;L;N?> CHNI NQI MN;A?M
QLCNCHA NB? CHN?AL;F IH NB? >IG;CH QCNB GOFNCJFC=;NCIH <S ?;=B N?MN @OH=NCIH NB?H
JLID?=NCIH I@ NB? OHEHIQH @OH=NCIHM IHNI NB? <;M?M
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 =C46A0C8>= >= C74 3><08= 0=3 <D;C8?;820C8>= 1H C74 C4BC 5D=2C8>=B
*OFNCJFC=;NCIH <S NB? N?MN @OH=NCIHM ;H> CHN?AL;NCIH IH NB? >IG;CH ACP? NB? @IFFIQCHA
?KO;NCIHM
.
><
><
J
.
67 :
K 1=<D =H7: > 67 :
.
.
>>@=> :?L K 67 :
/9;67 :
@IL ?P?LS >?AL?? I@ @L??>IG I@ Q;N?L >?JNB >?HIN?> ? <?NQ??H ;H> DF> ;H>
.
K<
.
K<
J
C7 :
.
K 1=<D =H7: K C7 :
.
6 <
>< C
= A C7 :
7 : N
N
.
.
"F C7 :
:?L > 3 =H7: K C7 :
>
.
.
L< L<
C
7 :
K 1=<D =H7: L C7 :
J
.
.
>< > < C
6 <
= A C7 :
7 : 6=
O
O
.
.
"G C7 :
:?L > 3 =H7: L C7 :
>
><
6=
@IL ?P?LS ? >?AL?? I@ @L??>IG I@ P?FI=CNS <?NQ??H ;H> DFK
&H IL>?L NI G;E? NB? CGJ?LG?;<CFCNS <IOH>;LS =IH>CNCIH ;JJ?;L CH ZQ?;E @ILG[
NB? N?LG
.
>>@=> :?L K 67 :
I@ NB? =IHNCHOCNS ?KO;NCIH CM CHN?AL;N?> <S J;LNM NI ACP?
.
>>@=> :?L K 67 : .
>>@=> K D 67 : .
K =H7: >>@=> 67 :
4CNB NB? =F;MMC= L?MIFONCIH G?NBI>M I@ NB? 0;CHN 3?H;HN IL +;PC?L\0NIE?M ?KO;
NCIHM IL CH NB? @ILGOF;NCIH <S $ION;F M?? 89 CGJ?LG?;<CFCNS CM A?H?L;FFS NL?;N?> CH
; ZMNLIHA @ILG[ >O? NI !CLC=BF?N =IH>CNCIHM CH NB? =IHM?LP;NCIH I@ GIG?HNOG ?KO;
NCIH &H NBCM =;M? NB? =IHM?LP;NCIH I@ GIG?HNOG ?KO;NCIH CM HI FIHA?L NL?;N?> ;FIHA
NB? <IOH>;LC?M
M ; =IHNCHO;NCIH I@ P;LC;NCIH;F @ILGOF;NCIH NB? >C_OMCIH N?LGM FCE?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
.
:?L > 3 =H7: K C7 :
>
;L? MCGJFCV?> ;H> NL?;N?> ;M
.
:?L
3 =H7:
K C7 :
NBCM N?LG ;FMI MBIOF> <? CHN?AL;N?> <S J;LNM ;M CN =IHN;CHM M?=IH> >?LCP;NCP?M HIN
L?F?P;HN NI NB? MJ;=?M =BIM?H @IL IOL @OH=NCIHM IOH>;LS N?LGM ;FMI ;JJ?;L
.
:?L 3 =H7: K C7 :
.
C7 3 =H7:
K D : .
3 =H7:
K =H7: C7 :
1B? <IOH>;LS N?LGM QCFF JLIPC>? NB? <IOH>;LS =IH>CNCIHM @IL @LC=NCIH IH NB? Q;FFM
=NO;FFS 3 =H7: K D CM ; MB?;L MNL?MM
+IN? ; MNLC=N?L NL?;NG?HN I@ >C_OMCIH QCNB NB? N?LG > ACP?M LCM? NI NB? JLI<F?G
I@ L?JL?M?HNCHA > CH NB? MJ;=? I@ ;JJLIRCG;NCIH ,H? MIFONCIH =IHMCMNM I@ QLCNCHA
.
.
:?L > 3 =H7: K C7 :
:?L > 3 =H7: K C7 :
>
>
7 QBC=B ;FFIQM ; M;NCM@;=NILS >?GIHMNL;NCIH I@ =IHM?LP;NCIH I@ GIG?HNOG M?? 0?=NCIH
@IL ; MCGCF;L >?GIHMNL;NCIH @IL NL;=?L G;MM 1BCM MIFONCIH ;FMI <LCHAM ION
JLI<F?GM I@ >CPCMCIH <S T?LI CH NB? JL?M?H=? I@ OH=IP?L?> <?>M
%A>942C8>= >5 5D=2C8>=B >=C> C74 10B4B
M CH>C=;N?> ?;LFC?L Q? HIQ QLCN?
>
<>6
,
>8 68 K
8
<>C
,
K8 C8 L
8
<>C
,
L8 C8 ?N=
8
@N?L NBCM JLID?=NCIH NB? ?KO;NCIHM <?=IG?
IHNCHOCNS
<>6
,
8
><8
><8
J
.
.
68
67
:
>8
>>@=> K D 67 : <>6
,
8
<>C
,
K8
8
/9;8
.
.
8
*IG?HNOG
<>6
,
.
K 1=<D =H7: 68 67 :
C8 =H7: >>@=> 67 :
68 67 :
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 <>C
,
8
K<8
K<8
J
.
"F C7
:
><8
8
.
8
L8<
"G C7
<>6
,
><8
8
.
:
.
6=
C8 C7 :
C7 3 =H7:
<>C
,
K8
.
K 1=<D =H7: C8 C7 :
<>C
,
L8
.
<>C
,
=
3 =H7:
.
6A< C
:
N 7
C8 =H7: C7 : K 1=<D =H7: C8 C7 :
68 C
: 6=
O 7
L D : .
8
8
><8
.
68 C
: N 7
K D : .
><8
K8
8
C7 3 =H7:
L8<
J
<>C
,
C8 C7 :
<>C
,
.
<>6
,
.
L8
8
.
.
=
3 =H7:
6A< C
:
O 7
C8 =H7: C7 : ?@IL? =B;HACHA NB?M? ?KO;NCIHM NI ; FCH?;L MSMN?G NB? <IOH>;LS CHN?AL;FM I<
N;CH?> ;L? >?M=LC<?> GIL? JL?=CM?FS
#0CDA0; 1>D=30AH 2>=38C8>=B
1B? H;NOL;F <IOH>;LS =IH>CNCIHM ;JJ?;L CH NB? JL?=?>CHA ?KO;NCIHM ;M <IOH>;LS
CHN?AL;FM
"0BB 2>=B4AE0C8>=
&H NB? G;MM =IHM?LP;NCIH ?KO;NCIH NB?M? <IOH>;LS CHN?AL;FM ;L?
.
>>@=> K D 67 :
C
.
>>@=> K<
D 67 :
C
.
>>@=> D 67 :
K < 1B?M? N?LGM ;L? ?RJL?MMCIHM I@ WOR NBLIOAB Q;FFM 1QI =;M?M ;JJ?;L
&H NB? =;M? I@ ; LCAC> Q;FF NB?S ;L? MCGJFS =;H=?FF?> 1BCM G?NBI> CM L?@?LL?>
NI ;M ZCGJ?LG?;<CFCNS <IOH>;LS =IH>CNCIH CH Q?;E @ILG[
&H NB? =;M? I@ ; FCKOC> <IOH>;LS NB? CHN?AL;FM B;P? NI <? =;F=OF;N?> JLI<F?G
;JJ?;LM @IL NB? N?LG
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
C
.
>>@=> K<
D 67 :
QBC=B =IHN;CHM NB? OHEHIQH K<
1?MNM I@ NL?;NG?HN I@ NBCM N?LG QCNB K < <?CHA =IHMC>?L?> ;M ;H OHEHIQH ;L?
MNCFF HIN =IH=FOMCP? NB? =;M? I@ @L?? IONJON QB?L? J;L;MCNC= L?W?=NCIHM I==OL K<
CM NBOM L?JF;=?> <S K < ?R=?JN C@ NB? =IGJIH?HNM I@ P?FI=CNS ;L? ACP?H ;N NCG? J<
JICHN QCNB !CLC=BF?N NSJ? =IH>CNCIH @IL P?FI=CNC?M
1B? CGJ?LG?;<CFCNS <IOH>;LS =IH>CNCIH CH Q?;E @ILG Q;M >?M=LC<?> @IL NB? VLMN
NCG? CH CH L?@?L??H=? 89 4?MN?LCHE ( 89 ;H> 89 OM?> CN NI MIFP?
NB? 0;CHN 3?H;HN ?KO;NCIHM CH ; Q;P? ?KO;NCIH @ILG &H NBCM @L;G?QILE NB?S MBIQ?>
NB;N NBCM =IH>CNCIH MOJJL?MM?M J;L;MCNC= GI>?M I<M?LP?> ?;LFC?L -F;N? L?JL?M?HNM
PIH (;LG;H ?>>C?M H;G?FS PILN?R MNL??NM <?BCH> <LC>A? JC?LM !?N;CFM I@ NB? P?FI=CNS
V?F> ;LIOH> NB? OJJ?L JC?L ;L? ACP?H ;N NB? <INNIG I@ NB? JF;N? &N L?MOFNM @LIG ;
=IG<CH;NCIH I@ ; MFCJ =IH>CNCIH ;H> ;H CGJ?LG?;<CFCNS =IH>CNCIH CH Q?;E @ILG 1B?
=IH>CNCIH K D CM @OFVFF?> QCNBION ;HS MO<M?KO?HN =ILL?=NCIH
"><4=CD<
&H ?;=B GIG?HNOG ?KO;NCIH ; MCHAF? <IOH>;LS N?LG ;JJ?;LM
.
C7 3 =H7: K D :
IL
.
C7
3
=H7: L D :
+IQ <S >?VHCNCIH
<
=H7: < D D
0I NB? <IOH>;LS =IH>CNCIHM IH @LC=NCIH ;L? OM?> ? A NB? FIA;LCNBGC= F;Q NI
=;F=OF;N? NB? <IOH>;LS N?LGM I@ NB? =IHM?LP;NCIH I@ GIG?HNOG ?KO;NCIH IH NB?
<;MCM I@ ; P;FO? I@ NB? MB?;L MNL?MM 3 C
#IL ; =F;MMC= =IH>CNCIH I@ @LC=NCIH Q? A?N
<
@IL ?R;GJF?
KB
K
D
%IQ?P?L CN CM JL?@?L;<F? NI OM? ;H CGJFC=CN NL?;NG?HN @IL @LC=NCIH MO=B ;M
3
KB
7KB
D
QBC=B I_?LM GIL? MN;<CFCNS 1BCM =;H <? I<N;CH?> HOG?LC=;FFS <S =BIIMCHA
3
7
K
K<B
QB?L? K<B CM NB? N;HA?HNC;F P?FI=CNS I@ NB? JL?PCIOM NCG? MN?J
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 1B? VH;F @ILG I@ NB? <IOH>;LS N?LG CM B?H=? QLCNN?H ;M
.
.
C7 3 =H7: K C7 7 K :
D : M IJJIM?> NI NB? =IHNCHOCNS ?KO;NCIH NBCM N?LG CM NL?;N?> CGJFC=CNFS MI CN QCFF
;JJ?;L CH NB? G;NLC=?M ;H> HIN CH NB? LCABN B;H> N?LGM M?? NB? @IFFIQCHA MO<M?=NCIH
4B?H MO=B ; @LC=NCIH =IH>CNCIH CM ;JJFC?> NB? P?FI=CNS IH NB? MIFC> <IOH>;LC?M
CM HIN T?LI ;H> K >?J?H>M IH NB? FI=;F WIQ ;H> CM $&#& ;H OHEHIQH &H @;=N
Q? B;P? NI ;MMOG? NB;N NB? P?FI=CNS CM HIN L?;FFS N;E?H ;N NB? <IOH>;LS <ON ;N ;
MG;FF >CMN;H=? @LIG NB? <IOH>;LS ;H> NB;N Q? B;P? ; FIA;LCNBGC= JLIVF? ;M ACP?H
CH @ILGOF;? ;H> CH 0?=NCIH #IL ?R;GJF? CH ; BS>L;OFC=;FFS LIOAB
WIQ Q? B;P?
K
!
K !
K
AA
QBC=B ;FFIQM OM NI ?P;FO;N? K K
9
&@ K CM N;E?H ;N NB? <IOH>;LS CM ?CNB?L ;H ;L<CNL;LCFS MG;FF >CMN;H=? IL ; @L;=NCIH
I@ NB? FI=;F G?MB MCT? <ON CH <INB =;M?M CN CM ; H?Q J;L;G?N?L NB;N =;H CHWO?H=?
NB? L?MOFNM HINB?L MIFONCIH NI ?P;FO;N? K =IHMCMNM I@ =BIIMCHA @IL K NB? H?;L?MN
JICHN CHMC>? NB? >IG;CH ;H> @IL NB? >CMN;H=? <?NQ??H NBCM JICHN ;H> NB? <IOH>;LS
1B? ;MMOGJNCIH CM NBOM NB;N NBCM CHH?L JICHN CM MNCFF CH NB? FIA;LCNBGC= JLIVF? 4CNB
; BS>L;OFC=;FFS MGIINB WIQ ;H ?RNL; >C^=OFNS CM NB;N @ILGOF; ACPCHA K CM
CGJFC=CN MI NB;N ;H CN?L;NCP? JLI=?MM CM H?=?MM;LS NI A?N NB? @LC=NCIH P?FI=CNS
(>DA24B 0=3 B8=:B
&N B;M <??H M??H NB;N CH NBCM =;M? NB? =IHNCHOCNS ?KO;NCIH CM QLCNN?H ;M
>
Q /9;
:?L J
QCNBION MJ?=C@SCHA NB? P;FO? I@ NB? MIOL=? N?LG /9; S MN;NCHA NB;N NB? @OH=NCIH
/9; MBIOF> IHFS L?@?L NI NB? MIOL=? JICHN QBIM? HOG<?L QCFF <? ?I9; Q? QLCN? /9; 47A13 7A13 <?CHA NB? <;MCM I@ NB? JICHN ?I9; ;H> 4 ;H OHEHIQH =I?^=C?HN 4?
;CG ;N CGJIMCHA ;H CHWIQ -A13 ;N NB? MIOL=? JICHN &H NB? ;<M?H=? I@ WOR ;N NB?
<IOH>;LC?M ;H> <S CHN?AL;NCHA NB? ?KO;NCIH IH NB? ?HNCL? =;F=OF;NCIH >IG;CH Q? A?N
QCNB NB? OMO;F >CM=L?NCT;NCIH I@ NB? NCG? >?LCP;NCP?
.
.
>< ><
47A13 :
: J
ON NB?H Q? MBIOF> B;P?
.
QBC=B NB?H CGJIM?M
><
><
: -A13
J
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
H> Q? VH;FFS A?N
4+
-A13
:
7A13
-A13 7A13
/9; +
:
7A13
0I NBCM =IH=?JN I@ MIOL=?M ;H> MCHEM CHNLI>O=?M ;N NB? =IHNCHOOG F?P?F N?LGM
QBC=B >?J?H> IH NB? >CM=L?NCT;NCIH I@ NB? >IG;CH 1BCM CM >O? NI NB? CHCNC;F BSJINB?MCM
/9; 47A13 QBC=B MOJJIM?M NB;N NB? MIOL=? CM HIN L?>O=?> NI ; MCHAF? JICHN <ON
MJL?;> ION IH NB? >IG;CH I@ >?VHCNCIH I@ NB? <;M? 7A13 HINB?L MIFONCIH =IHMCMNM
I@ QLCNCHA NB;N /9; -A13 ?I9; QB?L? ?I9; CM NB? !CL;= @OH=NCIH L?F;NCP? NI NB?
<;MCM 7A13 QBC=B CM ?KOCP;F?HN NI >CL?=NFS ;>>CHA NB? N?LG -A13
+ NI NB? LCABN B;H>
MC>? I@ NB? =IHNCHOCNS ?KO;NCIH ;N NB? JICHN ?I9; ;>GCNNCHA NB;N /9;7 : CM ?KO;F
NI -A13 C@ ? ?I9; ;H> INB?LQCM?
"0CA8G 5>A< >5 C74 BHBC4<
1B? M?N I@ ?KO;NCIHM ?F;<IL;N?> CH 0?=NCIH IHFS =IHN;CHM FCH?;L =IG<CH;NCIHM I@
OHEHIQHM %?H=? NB?S =IHMNCNON? ; FCH?;L MSMN?G NB;N =;H B;P? NB? @ILG 4 &H NBCM MSMN?G NB? JICHNM QCNB NB? !CLC=BF?N =IH>CNCIH C ? CGJIM?> P;FO?M ;L? E?JN
@ILG;FFS ;M OHEHIQHM M?? 0?=NCIH 4 CM NB? P?=NIL I@ OHEHIQHM =IGJIM?> I@
NBL?? P?=NILM QCNB L?MJ?=NCP? >CG?HMCIHM DF> DFK ;H> DFK ;H> =IHN;CHCHA NB? P;FO?M
> K ;H> L 1B?M? P?=NILM QCFF <? =;FF?> $ 1 ;H> 2 $
4% 1 &
2
CM ;FMI =IGJIM?> I@ NBL?? P?=NILM
2 % 2 &
2 CM ; G;NLCR CHCNC;FFS =IGJIM?> I@ HCH? MKO;L? IL L?=N;HAOF;L G;NLC=?M GIHA
NB?M? HCH? G;NLC=?M MIG? I@ NB?G ;L? T?LI >O? NI NB? ;<M?H=? I@ =IOJFCHA <?NQ??H
NB? NQI =IGJIH?HNM I@ P?FI=CNS 1BCM CM NB? =IHM?KO?H=? I@ NL?;NCHA CH Q?;E @ILG NB?
CGJ?LG?;<CFCNS =IH>CNCIH ;N NB? Q;FFM CM QLCNN?H ;M
) ) ) &
% ) ) ) ) ?@IL? ACPCHA NB? ?RJL?MMCIH I@ G;NLC=?M =IGJIMCHA ; @?Q INB?LM QBC=B ;JJ?;L
CH NB? FCH?;L MSMN?G I@ 0?=NCIH QCFF <? >?VH?>
1B? G;MM G;NLC=?M
.
6
)78 68 67
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 .
C
)78
C8 C7
1B? ;>P?=NCIH G;NLC=?M
0F6 78
.
K1=<D K1=<D
68 6
:
N 7
L1=<D
68 6
:
O 7
K1=<D
C8 C
:
N 7
L1=<D
C8 C
:
O 7
C8 =H7: C7 :
.
0G6 78 L1=<D .
0FC 78 K1=<D .
0GC 78 L1=<D 1B? >C_OMCIH G;NLCR
C
78
.
3 =H7:
1B? ZAL;>C?HN[ IL Z>CP?LA?H=?[ NSJ? G;NLC=?M ;L? ;M @IFFIQM
&H NB? =IHNCHOCNS ?KO;NCIH
.
C6
C8
>>@=> 67 :
F 78 >>@=> N
.
C8
>>@=> 67 :
GC678 >>@=> O
&H @;=N NB? NQI JL?=?>CHA G;NLC=?M ;L? HIN NLOFS =F;MMC= AL;>C?HN IL >CP?LA?H=?
G;NLC=?M >O? NI NB? JL?M?H=? I@ NB? N?LG >>@=> CH NB? CHN?AL;FM
4? ;FMI B;P? CH NB? GIG?HNOG ?KO;NCIH
FC678 GC678
.
68 C
:
N 7
.
68 C
:
O 7
C7
3
1B? <IOH>;LS G;NLCR
#C78 .
7 C7 C8 :
1B? G;NLC=?M I@ @LC=NCIH ;L? ;M @IFFIQM
4CNB NB? NCG? >CM=L?NCT;NCIH =BIM?H NB? P;FO? I@ @LC=NCIH N?LGM CM ?R;GJF? I@
=IGJIH?HN K
.
4 < 5 <
K
K
L < C7 :
<
"$ >
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B?M? N?LGM ;L? NL?;N?> CH NB? @IFFIQCHA MCGJFCV?> @ILG
.
6
4
<
<
<
K7
L7
C7 :
K7
"$ ><7
QBC=B ACP?M
"77C 4
"$ ><7
6
K<7
.
L7<
C7 :
1BCM ;JJLIRCG;NCIH QBC=B CM ; G;MM FOGJCHA B;M NB? ;>P;HN;A? I@ ACPCHA >C;AIH;F
G;NLC=?M H ?HNCL?FS MNLC=N NL?;NG?HN CM HIN JIMMC<F? IH ;==IOHN I@ MKO;L? LIINM
@IL ?R;GJF? %IQ?P?L GIL? =IGJF?N? G;NLC=?M =IOF> <? E?JN K< MBIOF> HIN <?
L?GIP?> @LIG NB? CHN?AL;F ;M K<7 <ON N?MNM MBIQ NB;N NB? L?MOFNM >I HIN =B;HA?
MCAHCV=;HNFS ,H NB? INB?L B;H> NB? =IH>CNCIHCHA I@ NB? MSMN?G >?AL;>?M MFCABNFS
4? HINC=? NB;N NB? ?KO;NCIHM CH K ;H> CH L ACP? NB? M;G? G;NLCR &@ 4 CM P;LC;<F? CH
MJ;=? NI =;F=OF;N? "77C NB? P;FO? I@ 4 CM N;E?H ;N JICHN ?
FI=EM I@ G;NLC=?M CH G;NLCR 1B? G;NLC=?M =IHN;CH?> CH ;L? NB?H
) )6
J
6
6 0F 78
6
6 0G 78
K1=<D
L1=<D
1B? G;NLC=?M 0 6 >CM;JJ?;L C@ IH? ?R?=ON?M ; JLCIL @L;=NCIH;F MN?J QCNB NB? G?NBI>
I@ =B;L;=N?LCMNC=M
1B? G;MM G;NLCR =;H <? ZFOGJ?>[ C@ H?=?MM;LS 1BCM MCAHCV?M NB;N CN CM L?JF;=?>
<S ; >C;AIH;F
QB?L? ?;=B =I?^=C?HN 7 CM ?KO;F NI NB? MOG I@ NB? N?LGM
)@ I@ NB?
=ILL?MJIH>CHA FCH? I@ ) &@ H CM NB? L;HE I@ NB? G;NLCR Q? NBOM A?N 7 8 )78
)@
*;MM FOGJCHA =;H <? J;LNC;F CH QBC=B =;M? IHFS ; J?L=?HN;A? I@ 8 )78 CM =;LLC?>
IP?L NI NB? >C;AIH;F I@ ) 1B? ?_?=N I@ G;MM FOGJCHA CM NI L?CH@IL=? NB? >C;AIH;F
I@ ) ;H> B?H=? CGJLIP? NB? =IH>CNCIHCHA I@ NB? VH;F FCH?;L MSMN?G ,H NB? INB?L
B;H> NB?L? CM ; MFCABN @;FF CH JL?=CMCIH Q? N?H> NIQ;L>M ; @ILGOF;NCIH MCGCF;L NI VHCN?
>C_?L?H=?M ;FNBIOAB ;FF NB? JLIJ?LNC?M I@ G;MM =IHM?LP;NCIH L?G;CH OH=B;HA?> 4?
AI IH QCNB NB? G;NLC=?M CH ) ) )C
J
C
C 0F
K1=<D
C
C 0G
C
L1=<D
" C #C
QB?L? NB? G;NLC=?M 0 C >CM;JJ?;L C@ IH? ?R?=ON?M ; JLCIL @L;=NCIH;F MN?J QCNB NB?
G?NBI> I@ =B;L;=N?LCMNC=M
) C6
C F
>>@=>
) =
C6
6 F
) ) =
C6
C G
>>@=>
C6
6 G
1B? LCABN B;H> MC>?M CH P?=NIL ;L?
2 6
0F6
0G6 $ <
C
FC6 1 <
GC6 2 <
) 6 /!
0 /"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 QB?L? 0 L?JL?M?HNM NB? N?LG 2 )C <
1
J
C
+
>>@=> K D 67 :
0FC
0GC 1 < =FC6 6A<
) C "F
)C <
C
0GC 2 < =GC6 6A< ) C "G
2
C 0F
J
+IN? QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M Q? MBIOF> L?JF;=?
<
1
CH NB? ?RJL?MMCIH @IL 2 6 0F6 0G6 $ < <S "
B $ $
<
C
1
0GC 1 < <S "
CH NB? ?RJL?MMCIH @IL 2 "
C 0F
B 1
B 1
" <
" 1
C
C
<
CH NB? ?RJL?MMCIH @IL 2 B 2
0G 2 <S B 2
C 0F
/?G;LE NB? G;DILCNS I@ NB? G;NLC=?M I@ NB? MSMN?G ;L? >?J?H>?HN IH NCG?
,H=? NB? G;NLC=?M ;H> NB? LCABN B;H> MC>?M I@ NB? MSMN?G ;L? <OCFN NB? !CLC=BF?N
NSJ? <IOH>;LS =IH>CNCIHM CGJIM?> P;FO?M B;P? NI <? N;E?H CHNI ;==IOHN ;H> NB?
FCH?;L MSMN?G B;M NI <? L?MIFP?> 1B?M? IJ?L;NCIHM ;L? CH>?J?H>?HN I@ NB? NSJ? I@
?KO;NCIH NI <? MIFP?> ;H> ;L? >?M=LC<?> CH 0?=NCIHM ;H> 2 '0380C8>= 2>=38C8>=B
1B? ?RCMN?H=? I@ ;LNCV=C;F <IOH>;LC?M CH NB? =;F=OF;NCIH >IG;CHM ? A CH NB? IJ?H
M?; I@N?H L;CM?M NB? KO?MNCIH I@ <IOH>;LS =IH>CNCIHM ;H> F?;>M NI CFF JIM?> JLI<F?GM
&H NB? IJ?H M?; A?H?L;FFS IHFS NB? @L?? MOL@;=? ?F?P;NCIH CM EHIQH &N CM N?GJNCHA
NI ;>IJN ; =IH>CNCIH QCNB CGJIM?> ?F?P;NCIH ;H> @L?? P?FI=CNS 4B?H NB? =OLL?HN CM
>CL?=N?> CHQ;L>M IH NB?M? <IOH>;LC?M NB? JLI<F?G CM CFF JIM?> ;H> NB? MIFONCIH =IOF>
>CP?LA? 1B? NB?ILS I@ =B;L;=N?LCMNC=M QBC=B QIOF> B?FJ NI ?GJB;MCT? NB? L?KOCL?>
=IH>CNCIHM =;HHIN <? ;JJFC?> MNLC=NFS CH NQI >CG?HMCIHM 1B? =B;L;=N?LCMNC= =OLP?M
BCABFCABN?> CH IH? >CG?HMCIH ;L? NL;HM@ILG?> CHNI =IH?M M?? 89 1B? ZL;>C;NCIH[
<IOH>;LS =IH>CNCIHM OM? NB? NB?ILS I@ =B;L;=N?LCMNC=M QCNB ; @?Q ;MMOGJNCIHM 1BCM CM
J;LNC=OF;LFS MI @IL NB? 1BIGJMIH G?NBI> 89 &H JL;=NC=? NB? @IFFIQCHA ;JJLI;=B
CM OM?>
FF NB? >;N; ;P;CF;<F? ;N NB? IJ?H <IOH>;LC?M ?F?P;NCIH ;H> P?FI=CNC?M ;L?
JLIPC>?> ;H> >?=F;L?> ;M CGJIM?> P;FO?M
1B? FI=;F =IH>CNCIHM ;N NB? <IOH>;LC?M ;L? MNO>C?> CH NB? FCABN I@ NB? NB?ILS
I@ =B;L;=N?LCMNC=M ;H> NB? /C?G;HH CHP;LC;HNM NI >?N?LGCH? QBC=B CH@ILG;NCIH
CM L?;FFS H?=?MM;LS ;H> QBC=B MBIOF> <? E?JN ;MC>? 1B? H?Q ?F?P;NCIHM ;H>
P?FI=CNC?M NI <? CGJIM?> ;L? >?>O=?> 1B?M? ;L? HIN H?=?MM;LCFS ACP?H <S NB?
OM?L 1B? 1BIGJMIH G?NBI> ;>;JN?> NI NB? 0;CHN 3?H;HN ?KO;NCIHM CM ACP?H
<?FIQ 1B? OHCKO?H?MM I@ NB? MIFONCIHM NBOM I<N;CH?> B;M <??H MNO>C?> <S )C=B
;H> *O;N 89
)?N OM MN;LN QCNB NB? =IHM?LP;NCP? @ILG I@ NB? 0;CHN 3?H;HN ?KO;NCIHM OMCHA NB?
HIN;NCIHM I@ 0?=NCIH ;H> "KO;NCIH L?=;FF?> B?L?
"
J
#F "
N
#G "
/ "
O
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1;ECHA CHNI ;==IOHN ;FF NB? MIOL=? N?LGM ;H> NB? JIMMC<CFCNS I@ >CM=B;LA? MIOL=?M
QCNBCH NB? >IG;CH NB? LCABN B;H> MC>? CM
/9;
>"F &
/ " % => .
F
.
=> G
>"G
4? =IHMC>?L ;H IJ?H <IOH>;LS QCNB ; FI=;F HILG;F P?=NIL D ;H> ; FI=;F N;HA?HN
P?=NIL J S ; LIN;NCIH Q? CHNLI>O=? >?LCP;NCP?M QCNB L?MJ?=N NI ;H> QB?L? CM NB? HILG;F ;H> NB? N;HA?HN ;N NB? <IOH>;LS
# "
# "
"
/ "
J
1B? >?N?LGCH;HN I@ NB? ';=I<C;H G;NLCR
F
F
&
G
G
CM ?KO;F NI 1BOM Q? A?N
#F
#G ;H> # #F
N
O
N
1B? MSMN?G CM NB?H QLCNN?H ;M
# "
J
"
#G
O
"
/ "
QCNB
'
%
F
G
9 K
9 K
G KL
F KL
F
K F
L G
L F
K F
G
K G
L G
(
&
IL ?P?H <S >?HINCHA K ;M NB? HILG;F =IGJIH?HN I@ P?FI=CNS ;H> K NB? N;HA?HNC;F
=IGJIH?HN
'
%
F 9
G 9
F
KK
LK
K F
K
L F
G
(
K &
G
K L G
0O<M?KO?HNFS Q? CAHIL? NB? P;LC;NCIHM ;FIHA NB? N;HA?HN ;N NB? <IOH>;LS ;H> NLS
NI MIFP? NB? MSMN?G
"
J
"
/ "
QB?L?
/9;
.
'
/ " % => F .
=> G >"F (
&
>"G
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 CM >C;AIH;FCT?> ;M ( ( QCNB
K
G
'
( % 9 K
F
9 K F
F
(
&
G
G
;H>
%
K
K
9
K 9
&
S MN;NCHA NB;N :3 (:" Q? NB?H A?N <;=E NI NB? >C;AIH;FCT?> MSMN?G
3
J
3
(/
?;=B I@ QBIM? FCH?M CM ; MCGJF? NL;HMJILN ?KO;NCIH QCNB MIOL=? N?LG 1BIGJMIH
JLIJIM?M NI =IHMC>?L NB;N ( CM =IHMN;HN CH NB? PC=CHCNS I@ ; <IOH>;LS JICHN ;H> NI
QLCN? 3 (" QB?L?
K
F
G
'
(
( % 9 K
F
G &
9 K F
G
NB? IP?L<;L P;FO?M <?CHA =IHMC>?L?> ;M =IHMN;HN 1B? /C?G;HH CHP;LC;HNM I@ NB? P?=NIL
3 ;L? NBOM
3 > K K
3 > 9
3 > 9
;>P?=NCIH QCNB P?FI=CNS K K K
;>P?=NCIH QCNB P?FI=CNS K
K K
9
;>P?=NCIH QCNB P?FI=CNS K 9
;H> NI QBC=B =;H <? ;>>?> C@ ; NL;=?L 0 ;FMI B;M NI <? =IHMC>?L?>
3 > 0 0
;>P?=NCIH QCNB P?FI=CNS K -OL? ;>P?=NCIH CM NL?;N?> QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M 0IOL=? N?LGM C@
;HS ;L? =IHMC>?L?> CH ; @L;=NCIH;F MN?J ,H=? NB? /C?G;HH CHP;LC;HNM ;L? EHIQH NB?
JLCGCNCP? P;LC;<F?M ;L? L?MNIL?> <S NB? @IFFIQCHA @ILGOF;?
>
K
L
G 3
3 3
9
F 3
>
G 3
F 3
>
>9 F
K
>9 G
L
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
3
0
>
1B? CFF JIM?> H;NOL? I@ ; JLI<F?G ;N NB? <IOH>;LS ;JJ?;LM QCNB NBCM G?NBI> QB?H
GIPCHA ;FIHA NB? =B;L;=N?LCMNC= =OLP?M CH NB? OJMNL?;G >CL?=NCIH C@ ; =B;L;=N?LCMNC=
F?;P?M NB? >IG;CH QCNBION NB? =ILL?MJIH>CHA CH@ILG;NCIH <?CHA ACP?H <S NB? OM?L
,H NB? INB?L B;H> ; L?>OH>;HN =IH>CNCIH QCFF MCGJFS <? JON ;MC>? &H JLCH=CJF?
NB? G?NBI> CM HIN NCG? =IHMOGCHA ;M GIPCHA ;FIHA NB? =B;L;=N?LCMNC= CH NB? OJMNL?;G
>CL?=NCIH CM L?KOCL?> IHFS @IL JICHNM MCNO;N?> IH ;H IJ?H <IOH>;LS %IQ?P?L NBL?? IL
@IOL QCNB ; NL;=?L =B;L;=N?LCMNC=M J?L JICHN B;P? NI <? =;F=OF;N?> ;H> ?;=B NCG?
NB? =ILL?MJIH>CHA ;>P?=NCIH V?F> JL?J;L?> 1BCM CM >IH? @IL NB? ?HNCL? =;F=OF;NCIH
>IG;CH &H @;=N =?LN;CH NC>? MCGOF;NCIHM ;L? ?R?=ON?> QCNB IOL;HN HOG<?L P;FO?M
1B
F ;LIOH> 1BCM JLIBC<CNM L?MNLC=NCIH NI ; =;F=OF;NCIH I@ NB? FI=;F ;>P?=NCIH V?F>
;N NB? <IOH>;LS
-F;N? MBIQM ;H ;JJFC=;NCIH I@ NB? 1BIGJMIH L;>C;NCIH =IH>CNCIHM @IL F?NNCHA ;
=CL=OF;L Q;P? JLIJ;A;NCHA CH ; MKO;L? L?M?LPICL F?;P? NB? >IG;CH M?? ;FMI 0?=NCIH
&H NBCM =;M? HI CH@ILG;NCIH Q;M ACP?H ;N NB? <IOH>;LC?M %IQ?P?L Q? I<M?LP?
NB;N NB? Q;P? AI?M IONQ;L>M QCNB IHFS ; MFCABN L?W?=NCIH ;N NB? =ILH?LM >O? NI NB?
@;=N NB;N N;HA?HN N?LGM ;N NB? <IOH>;LS Q?L? H?AF?=N?> CH IOL ;JJLI;=B
0 '4B>;DC8>= >5 C74 >DBB8=4B@ 4@D0C8>=B
3?LS @?Q GI>?FM B;P? <??H JLIJIM?> @IL MIFPCHA NB? IOMMCH?MK ?KO;NCIHM CH VHCN?
?F?G?HNM 1B? G;CH >C^=OFNS CM NB? BCAB?L IL>?L I@ >?LCP;NCP?M CH NB? N?LG
$ $ K
K
=H7: :?L
=H7: :?L $
J
J
QBC=B CM ;>>?> NI NB? 0;CHN 3?H;HN GIG?HNOG ?KO;NCIH !I ;LGI ;H> 0?;<L;
0;HNIM 89 B;P? JLIJIM?> IH? I@ NB? VLMN GI>?FM 4CNB ;H ;MMOGJNCIH I@ =OLF @L??
WIQ NB?M? ;ONBILM MBIQ NB;N NB? MOJJF?G?HN;LS N?LG F?;>M NI NB? @IFFIQCHA LCABN
B;H> N?LGM CH NB? =IHM?LP;NCIH I@ GIG?HNOG ?KO;NCIHM
&H NB? K ?KO;NCIH
$
K
N J
K
O J
"C
J
$
L
N J
L
O J
"D
J
&H NB? L ?KO;NCIH
QCNB
"C $
K
$
N
$ L
N O
$ L
O N
L
$
NO
$ K
N N
"D $
L
$
O
$ K
N O
$ K
O N
K
$
NO
$ L
O O
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 1B? C>?; I@ NB? ;FAILCNBG CM NB?H NI ;>> NQI ;ORCFC;LS P;LC;<F?M H ;H> I MO=B ;M
H K
$
K
N
K
O
"C
IL
$
L
N
L
O
"D
1B? 0;CHN 3?H;HN GIG?HNOG ?KO;NCIHM N;E? CHNI ;==IOHN NB? IOMMCH?MK N?LG C@
D
@
A
NB? >?LCP;NCP?M C
1B? H?Q ?KO;NCIHM =;H <?
B ;H> B ;L? =B;HA?> CHNI B ;H> B
MIFP?> CH NBL?? MN?JM
;F=OF;NCIH I@ > QCNB NB? =IHNCHOCNS ?KO;NCIH
;F=OF;NCIH I@ H ;H> I QCNB NB? GIG?HNOG ?KO;NCIH
!?>O=NCIH I@ K ;H> L @LIG H ;H> I
1B? GIMN NCG? =IHMOGCHA MN;A? CM NB? =;F=OF;NCIH I@ K ;H> L IH NB? <;MCM I@ H ;H>
I 1B? G;NLCR =IH>CNCIHCHA ;@N?L >CM=L?NCT;NCIH I@ NB? ?KO;NCIH
K
K
N
$
K
O
"C H
CM KOCN? JIIL 0O=B =IH>CNCIHCHA CM =IGGIH NI ;FF NB? L?MIFONCIH G?NBI>M I@ NB?
IOMMCH?MK ?KO;NCIHM
1B? ?;LFC?L ;FAILCNBG CM M;NCM@;=NILS @IL L?JLI>O=CHA NB? JLIJ;A;NCIH I@ Q;P?M CH
; L?=NCFCH?;L =;H;F %IQ?P?L NB? ;MMOGJNCIH I@ =OLF @L?? WIQ JLIP?> NI <? NB? =;OM?
I@ CHMN;<CFCNC?M QCNB =IGJF?R WIQM GIL? LI<OMN ;FAILCNBG CM I<N;CH?> <S >CL?=NFS
;>>CHA NB? MOJJF?G?HN;LS IOMMCH?MK N?LG NI NB? ;FAILCNBG ?F;<IL;N?> @IL NB? 0;CHN
3?H;HN ?KO;NCIHM
1B? >CM=L?NCT;NCIH I@ NB? N?LG =IHN;CHM ;H CGJFC=CN N?LG
$ $ K<
=H7: :?L K<
=H7: :?L $ J J CH NB? F?@N B;H> MC>? I@ NB? GIG?HNOG ?KO;NCIH ;H> ;H ?RJFC=CN N?LG
$ $ K<
=H7: :?L K< =H7: :?L $ J J CH NB? LCABN B;H> MC>?
1BCM M?=IH> N?LG CM ?;MCFS >?>O=?> @LIG NB? G;NLC=?M NI <? <OCFN @IL NB? F?@N B;H>
N?LG &H NB? P;LC;NCIH;F @ILGOF;NCIH NBCM N?LG <?=IG?M
.
$ =H7: :?L K<
J 7 : .
$ K<
=H7: :?L $ J 7 :
IL ?P?H <S CHN?AL;NCHA <S J;LNM ;H> CAHILCHA NB? <IOH>;LS N?LGM
J
.
:?L K<
=H7:
$
7
:
J
.
:?L $ K<
=H7:
$
7
:
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
QBC=B VH;FFS ACP?M
J
.
=H7: 7 :
$ :?L K<
.
$ =H7: $ :?L K < 7 :
J .
$ K < =H7: $ =H7: 7 :
J .
K < =H7: $ 7 =H7: $ :
J 1BCM N?LG CM ; P?=NIL QCNB NQI =IGJIH?HNM &@ Q? JLID?=N K < IHNI NB? <;M?M
8 ;H> =IHMC>?L NB? =IGJIH?HN ;FIHA N ;H> NB? =IGJIH?HN ;FIHA O I@ NB? P?=NIL Q?
A?N ;FIHA N ; N?LG
) ) 1 <
) * 2 <
* ) 1 <
* * 2 <
;H> ;FIHA O ; N?LG
QB?L? ) ) CM @IL ?R;GJF? NB? G;NLCR
.
.
8 7
$ 8
$ $
$
) ) :
7 :
J N N
J N N
.
.
$
7
$ $
$
8
:
8 7 :
J N
N
J N N
0I ; QBIF? =F;MM I@ G;NLC=?M ;JJ?;LM ;M
.
< $ < $ < 8 < 7 :
QB?L? NB? @OH=NCIHM < NI < =;H <? NB? C>?HNCNS NB? >?LCP;NCP? QCNB L?MJ?=N NI N IL
NB? >?LCP;NCP? QCNB L?MJ?=N NI O N?LGCHIFIAS CM HIQ JLIJIM?> QB?L? C>?HNCNS CM
L?JL?M?HN?> <S Z[ NB? >?LCP;NCP? QCNB L?MJ?=N NI N <S Z4[ ;H> NB? >?LCP;NCP? QCNB
L?MJ?=N NI O <S Z5 [ 0I Q? >?HIN? ) , - ;M NB? G;NLCR
.
$
7
$
8
:
N
O
)?N OM JICHN ION CHN?L?MNCHA JLIJ?LNC?M I@ NB?M? G;NLC=?M MO=B ;M ) , - ), ;H> ) ,- B ) , 4? =;H HIQ QLCN? ) ) ) * * ) ;H> * * QCNB ; FCGCN?> HOG<?L I@ H?Q
G;NLC=?M NI <? <OCFN
) ) J
)
,,
),,
B
)
,,
)
,,
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 ) * J
)
* ) J
)
* * J
)
B
),
, -
)
),
- ,
,-
B
)- B
,-
)
--
)
)
, -
B
)
- ,
)
--
--
1B? G;CH ?RNL; =IMN CM ; =IOJFCHA I@ K< ;H> L <
1B? G;NLCR
) ) ) &
% ) ) ) ) <?=IG?M
) % ) ) ) ) * )
) ) * &
) QCNB ) ) ) ) ;H> ) ) * *
,H NB? LCABN B;H> MC>? I@ NB? ?KO;NCIH @IL K<
) ) 1 < ) * 2 < MBIOF> <?
;>>?>
,H NB? LCABN B;H> MC>? I@ NB? ?KO;NCIH @IL L <
* ) 1 < * * 2 < MBIOF> <?
;>>?>
1B? H?Q MSMN?G NI <? MIFP?> B;M NB? M;G? JIIL =IH>CNCIHCHA ;M NB? %?FGBIFNT
?KO;NCIHM @IOH> CH NB? VLMN G?NBI> <ON NB? ;FAILCNBG CM GO=B GIL? LI<OMN
'4B>;DC8>= >5 O <>34; 4@D0C8>=B 8= 3E42C8>= BC4?
#IL NB? L?MIFONCIH I@ A\ GI>?F ?KO;NCIHM NB? @L;=NCIH;F MN?JM G?NBI> CM ;FMI OM?>
IHM?KO?HNFS NB? ;>P?=NCIH MN;A? I@ NOL<OF?HN KO;HNCNC?M CM NL?;N?> M?J;L;N?FS ;H>
CM MIFP?> ;N NB? M;G? NCG? ;M BS>L;OFC= KO;HNCNC?M C@ NB? G?NBI> I@ =B;L;=N?LCMNC=M CM
L?KOCL?>
%A>3D2C8>= 38QDB8>= B>DA24 C4A<B
)8<4 38B2A4C8I0C8>=
M CH NB? JLIJ;A;NCIH MN;A? ;H> EHIQCHA 1
A ;H> 1
NB? L?MOFNM I@ NB? ;>P?=NCIH MN?J
Q? H??> NI MIFP?
A<
<
1
A
:?L
J
1
:?L
J
B
B
9
=H7: A
=H7: #
,9D #
A
,D A
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
)CH?;LCT;NCIH CM ?R?=ON?> ;==IL>CHA NI NB? JLCH=CJF? I@ M?GC CGJFC=CN;NCIH ?RJF;CH?>
?;LFC?L <ON B?L? QCNB NB? =BIC=? I@ =I?^=C?HNM QBC=B M?J;L;N? NB? P;LC;<F?M A ;H> 1BCM F?;>M NI NB? MSMN?G
A<
:?L
J
<
:?L
J
B
B
9
< <
A
A<
=H7: A <
=H7: <
< <
A<
1
A
J
1
J
#<
<
,9D
< <
#
A<
<
,D
)?N OM JICHN ION NB? MJ?=CV= NL?;NG?HN I@ NB? N?LG IH NB? LCABN B;H> MC>? I@ NB?
?KO;NCIH @IL A QLCNN?H ;M
< <
A
A<
1BCM MO<MNCNONCIH CM CH>CMJ?HM;<F? NI M?J;L;N? NB? P;LC;<F?M ;H> @IL NB? HOG?LC=;F
MN;<CFCNS I@ NB? GI>?F < A < CM NB? CHP?LM? I@ ; NCG? H ?R;GCH;NCIH I@ "KO;NCIHM
;H> MBIQM NB;N NB? =BIC=? I@ ; P;LC;<F? A CH JF;=? I@ QIOF> J?LB;JM
<? GIL? ;JJLIJLC;N?
(?0C80; 38B2A4C8I0C8>=
3;LC;NCIH;F @ILGOF;NCIH ;M Q?FF ;M MJ;NC;F >CM=L?NCT;NCIH ;L? ?R?=ON?> ?R;=NFS ;M CH NB?
BS>LI>SH;GC= MSMN?G ;@N?L CHN?AL;NCIH <S J;LNM I@ >C_OMCIH N?LGM @IL >CM=L?NCT;NCIH
CH MJ;=? 1B? <IOH>;LS N?LGM I<N;CH?> >OLCHA NBCM CHN?AL;NCIH <S J;LNM ;L? ;MMOG?>
NI <? T?LI 1BCM >I?M HIN ;_?=N NB? ?HNLS JICHNM I@ NB? >IG;CH IL NB? Q;FFM MCH=?
A ;H> ;L? CGJIM?> 1BCM =IH@ILGM NI ;H IL>CH;LS P;LC;NCIH;F @ILGOF;NCIH QB?L?
NB? MJ;=? I@ N?MN @OH=NCIHM CM FCGCN?> NI T?LI @OH=NCIHM IH NB? J;LNM I@ NB? <IOH>;LS
QB?L? NB? OHEHIQH CM ACP?H !CLC=BF?N =IH>CNCIH #IL ; FCKOC> ?RCN NBCM CM NB? M;G?
;M ;MMOGCHA NB;N NB? AL;>C?HNM I@ A ;H> ;FIHA NB? HILG;F ;N NB? ?RCN ;L? T?LI
&@ NB? M;G? HIN;NCIHM ;L? E?JN ;M <?@IL? @IL NB? G;NLC=?M G;MM >C_OMCIH NB?
VH;F MSMN?G =;H N;E? NB? @ILG
'
<
A<
)
QCNB
J
9
<
< )
J
A
9
1
'
<
)
#< ,9D
J
1
<
<
< #< ,9D
J
A
0CH=? NB? ?KO;NCIHM @IL A ;H> ;L? OH=IOJF?> NB?S ;L? MIFP?> M?J;L;N?FS
)
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 (>;E8=6 C74 CA024A 4@D0C8>= 8= -;L;>IRC=;FFS NBCM ?KO;NCIH CM GIL? =IGJF?R NI MIFP? CH ! NB;H CH ! <?=;OM? NB?
P;FO? I@ NB? NL;=?L CM ;H ;P?L;A? IP?L NB? P?LNC=;F
1B? ?KO;NCIH L?MIFP?> @IL NB? NL;=?L ;N ; >?AL?? I@ @L??>IG ? CM ;M @IFFIQM
.
0<
0<
7 :
J
.
K =H7: 0 7 : >
7
>7
.
.
:?L > B =H7: 0 <
7 :
0A13 0 /9;7 :
&H NBCM ?KO;NCIH
0A13 CM NB? P;FO? I@ NB? NL;=?L ;N NB? MIOL=?
/9; CM NB? MIOL=? I@ NB? NL;=?L CH GM &N CM =IHH?=N?> NI NB? >CM=B;LA? I@ NB?
MIOL=?M -A13 CH G M
0 (0<
( 0 < QB?L? ( CM NB? L?F;R;NCIH =I?^=C?HN @IL NB? NL;=?L
<
> CM NB? Q;N?L >?JNB ;H> CH NB?ILS MBIOF> <? ?KO;F NI ( ><
( > M??
0?=NCIH <?FIQ
>7 CM NB? Q;N?L >?JNB > >?VH?> ;<IP? ;H> N;E?H ;N JICHN ?
&N CM CGJIMMC<F? NI QILE ION NB? N?LGM CH > CH NB? P;LC;NCIH;F @ILGOF;NCIH MCH=?
NB? @OH=NCIH > CM HIN CH NB? ;JJLIRCG;NCIH MJ;=? 1B? N?LG > B;M <??H L?GIP?>
@LIG NB? CHN?AL;F IH ;H> L?JF;=?> <S >7 1BCM =BIC=? B;M ;A;CH <??H >C=N;N?> <S
=IHMC>?L;NCIHM IH NB? =IHM?LP;NCIH I@ NL;=?L KO;HNCNS M?? 0?=NCIH *IL?IP?L >C_OMCIH N?LGM ;L? ?HNCL?FS CGJFC=CN
1B? >C_?L?HN IJ?L;NCIHM @IL NB? L?MIFONCIH I@ NB? NL;=?L ?KO;NCIH ;L? C>?HNC=;F NI
NBIM? >?M=LC<?> @IL NB? A\ GI>?F 1B? IHFS >C_?L?H=? =IG?M @LIG NB? <IOH>;LS
N?LG ;LCMCHA @LIG NB? CHN?AL;NCIH I@ NB? >C_OMCIH N?LG QBC=B CM L?N;CH?> ;N NBCM
JICHN %IQ?P?L CNM NL?;NG?HN CM MCGCF;L NI CNM =IOHN?LJ;LNM CH NB? BS>LI>SH;GC=
MSMN?G =NO;FFS ;N NB? Q;FFM NB? <IOH>;LS =IH>CNCIH @IL NB? NL;=?L CM NB? >;N; I@
>CMJ?LMCP? WOR CH NB? @ILG
B
0
70
D
8
;M CH !
#IL ; @L?? ?RCN ;H> QB?H NB? G?NBI> I@ =B;L;=N?LCMNC=M CM OM?> @IL ;>P?=NCIH NB?
L?MOFN I@ NB? ;>P?=NCIH MN?J CM CGJIM?> CH NB? >C_OMCIH MN?J 1B? ?_?=N I@ >C_OMCIH CM
NBOM IP?LFIIE?> CH NBCM =;M? 1BCM JLI=?MM CM JL?@?LL?> NI CGJIMCHA ;H ;LNCV=C;F Q;FF
<IOH>;LS =IH>CNCIH IH ; FCKOC> <IOH>;LS MIG?NBCHA NB;N QIOF> B;JJ?H C@ ; JICHN CH
; FCKOC> ?RCN Q;M NL?;N?> ;M ; >?AL?? I@ @L??>IG CH NB? >C_OMCIH MN?J
1B? VH;F FCH?;L MSMN?G CM L?MIFP?> <S CN?L;NCP? G?NBI>M >?M=LC<?> CH 0?=NCIH !0FB >5 2>=B4AE0C8>= 8= (& #" ' (&" 6OH 0IH !I \ (IL?;
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
IHNLIFFCHA NB? G;MM CH HOG?LC=;F MCGOF;NCIHM ?GJB;MCT?M ; =?LN;CH HOG<?L I@ >C^
=OFNC?M ;H> >?GIHMNL;NCIHM I@ G;MM =IHM?LP;NCIH ;L? ; AOC>? @IL <OCF>CHA ;FAILCNBGM
NB?S MIG?NCG?M KO?MNCIH NB?G M?? @IL ?R;GJF? NB? NL?;NG?HN I@ NB? *?L=;NIL JLID?=
NCIH 1B? JLII@ I@ G;MM =IHM?LP;NCIH I@ WOC> ;H> NB? JLII@ I@ G;MM =IHM?LP;NCIH I@
; NL;=?L QCFF <? ?R;GCH?> MO==?MMCP?FS ;H> MO<M?KO?HNFS NB? B?;> GIG?HNOG ;H>
?H?LAS =IHM?LP;NCIH
"0BB 2>=B4AE0C8>= >5 C74 KD83
)?N OM HIQ =IHMC>?L NB? =;M? I@ NB? =IHNCHOCNS ?KO;NCIH QCNBION ; MIOL=? N?LG
>
J
:?L >
K QBC=B =;H <? QLCNN?H ;M
>
J
K :?L >>@=> ;M <?@IL?
@N?L NCG? >CM=L?NCT;NCIH P;LC;NCIH;F @ILGOF;NCIH CHN?AL;NCIH <S J;LNM ;H> L?MIF
ONCIH I@ NB? VH;F FCH?;L MSMN?G Q? MB;FF L?MIFP? @IL ?;=B JICHN ? QB?L? NB? =IHNCHOCNS
?KO;NCIH CM NL?;N?>
.
><
><
7 :
J
.
7 >>@=> K D : .
>>@=> K =H7: 7 :
1CG? >CM=L?NCT;NCIH I@ K CM HIN MJ?=CV?> 1BCM >I?M HIN =B;HA? NB? >?GIHMNL;NCIH
CH ;HS Q;S
1I MBIQ IP?L;FF G;MM =IHM?LP;NCIH Q? ;>> NB? ?KO;NCIHM I@ ;FF NB? JICHNM 1;ECHA
)<>=7< )<>=7<
1B? MOG I@ ;FF NB?
CHNI ;==IOHN NB;N 7 7 Q? A?N 7 =H7: 7 ?KO;NCIHM NB?H ACP?M
.
><
>< : J
.
>>@=> K D :
1BCM ?KO;NCIH ?RJL?MM?M NB;N NB? ?PIFONCIH I@ G;MM CH NB? >IG;CH CM MCGJFS >O?
NI NB? WOR NBLIOAB NB? <IOH>;LC?M I@ 1BCM CM NB? JLII@ I@ G;MM =IHM?LP;NCIH
1BCM @ILGOF; ;FMI MBIQM BIQ NI =;F=OF;N? WOR?M ;N NB? <IOH>;LC?M <S MNLC=NFS
@IFFIQCHA NB? VHCN? ?F?G?HN @ILGOF;NCIH 4? M?? CH J;LNC=OF;L NB;N NB? =BIC=? I@
>>@=> L?G;CHM @L??
1B? >?GIHMNL;NCIH @;CFM QB?H
*;MM =IHM?LP;NCIH B;M HIN <??H L?MIFP?> @IL ;FF NB? JICHNM 2H@ILNOH;N?FS NBCM
B;JJ?HM QB?H Q;N?L >?JNB =IH>CNCIHM ;L? CGJIM?> GIL?IP?L CH NBCM F;MN =;M?
NB? IONWIQ CM ?P;FO;N?> DOMN ;N NBIM? JICHNM QB?L? G;MM =IHM?LP;NCIH B;M HIN
<??H L?MIFP?> 1BCM ?RJF;CHM NB? J?LMCMN?H=? I@ ; L?MC>O;F ?LLIL ;N FCKOC> <IOH>
;LC?M ?P?H QB?H L?;FFS =IHM?LP;NCP? M=B?G?M ;L? OM?>
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 !CM=L?NCT?> ?KO;NCIHM =;HHIN <? L?>O=?> NI ; =IHM?LP;NCP? @ILG 1BCM CM MI C@
;>P?=NCIH I@ > CM >IH? QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M IL QB?H >CM=L?NCT;NCIH
CH ; HIH =IHM?LP;NCP? @ILG I@ K =H7: > ;H> > :?L K CH NB? =IHNCHOCNS ?KO;NCIH
CM HIN =IGJ;NC<F? NB;N CM NB?S =;HHIN <? L?>O=?> NI ; >CP?LA?H=? IJ?L;NIL
1B? FIMM I@ =IGJ;NC<CFCNS CM A?H?L;FFS >O? NI NCG? >CM=L?NCT;NCIH ;H> FCH?;LCT;NCIH
QBC=B ?FCGCH;N?M NB? JLI>O=NM I@ OHEHIQH @OH=NCIHM 1BCM CM QBS NB? OM? I@ MO<
CN?L;NCIHM QB?L? HIH FCH?;L N?LGM ;L? ;=NO;FCT?> L?MNIL?M G;MM =IHM?LP;NCIH
>=B4AE0C8>= >5 C74 CA024A
,H NB? <;MCM I@ QB;N Q;M G?HNCIH?> @IL NB? WOC> ?P?LSNBCHA <?=IG?M =IGJFC=;N?> NI
MO=B ; JICHN NB;N <OCF>CHA HIH =IHM?LP;NCP? M=B?G?M @IL ; HIH =IHM?LP;NCP? ?KO;NCIH
I@ NB? NL;=?L L?G;CHM ; J;LNC;FFS OHL?MIFP?> JLI<F?G
)?N OM ;MMOG? NB;N NB? ?KO;NCIH L?MIFP?> @IL NB? NL;=?L CM ;M @IFFIQM
.
0<
>7
.
.
0<
7 :
J
K =H7: 0 7 : >
7
.
0A13 /9;7 : >
7
.
0
><
:?L > B =H7: 0 <
7 :
><
J
:?L >
K
7 :
1BCM ?KO;NCIH B;M <??H M??H CH 0?=NCIH QCNB NB? M;G? HIN;NCIHM <ON <S
L?JF;=CHA NB? N?LG
.
0 /9;7 :
>7 <S
>7
.
><
0
><
J
:?L >
K
7 :
NB;N CM <S L?JF;=CHA NB? LCABN B;H> MC>? I@ NB? =IHNCHOCNS ?KO;NCIH <S CNM F?@N B;H>
MC>? 4LCNCHA NBCM CM CH>CMJ?HM;<F? @IL NB? >?GIHMNL;NCIH ;H> QCFF <LCHA ION ;H ;>>C
NCIH;F =IHMNL;CHN !?GIHMNL;NCIH I@ NB? NL;=?L =IHM?LP;NCIH OM?M NB? >?GIHMNL;NCIH
I@ NB? G;MM I@ WOC> =IHM?LP;NCIH 1B?L? MBIOF> <? MNLC=N =IGJ;NC<CFCNS <?NQ??H NB?
=IHNCHOCNS ?KO;NCIH ;H> NB? NL;=?L ?KO;NCIH 1BCM =IGJ;NC<CFCNS =;HHIN <? ;MMOL?>
C@ NB? NL;=?L CM NL?;N?> QCNB ; >C_?L?HN NCG? MN?J 1BCM CM >O? NI NB? JLI=?MMCHA I@
HIH FCH?;L N?LGM ; G;MM =IHM?LP;NCIH ?KO;NCIH L?MIFP?> CH NB? >CM=L?NCT?> @ILG @LIG
NB? CHMN;HN J< NI J<
NB?H @LIG J< NI J<
>I?M HIN ACP? NB? M;G? L?MOFN ;M ;
L?MIFONCIH CH ; MCHAF? MN?J @LIG J< NI J<
1B? >?GIHMNL;NCIH @IL =IHM?LP;NCIH I@ NB? NL;=?L @IFFIQM
";=B NL;=?L ?KO;NCIH ;N JICHN ? CM GOFNCJFC?> <S >7 Q? HIQ OH>?LMN;H> QBS 6
B;M <??H JON IONMC>? NB? CHN?AL;F I@ NB? >C_OMCIH N?LG NB?H ;FF NB? ?KO;NCIHM ;L?
;>>?> S OMCHA NB? >?=IGJIMCNCIH I@ > IH NB? N?MN @OH=NCIHM ;MMOGCHA NB;N NB? N?MN
@OH=NCIHM I@ NB? NL;=?L ;L? NB? <;MCM @OH=NCIHM I@ NB? Q;N?L >?JNB Q? NB?H A?N
.
0<
0<
> :
J
.
>
K =H7: 0 : .
:?L >
(
=H7: 0 <
:
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
.
.
0A13 /9; : ><
0
><
J
:?L >
K
:
&H NB? JL?=?>CHA ?KO;NCIH Q? ;MMOG?> NB;N
0 (
0<
(0
<
;H> > ><
6
6>
<
N NBCM JICHN Q? L?;FCT? NB;N NB? =IH>CNCIH 6
( CM H?=?MM;LS MO=B NB;N NB?
N?LGM =IGCHA @LIG NCG? >?LCP;NCP?M =IG<CH? NI ACP? NB? @IFFIQCHA NCG? >?LCP;NCP?
><
0<
J
>< 0 <
,H NBCM =IH>CNCIH Q? VH;FFS A?N
.
><
0<
J
>< 0 <
:
.
:?L >0 K : .
.
:?L >
(
=H7: 0 <
:
0A13 /9; :
1BCM ?KO;NCIH ?RJL?MM?M NB;N NB? P;LC;NCIH I@ NB? NL;=?L KO;HNCNS CH NB? >IG;CH CM
>O? NI NB? WOR ;N NB? <IOH>;LC?M <S CH=FO>CHA ; >C_OMCP? WOR ;H> ;N NB? L;N? I@
=L?;NCIH
&N ;FMI MBIQM NB;N NB? NL;=?L KO;HNCNS CH NB? >IG;CH MBIOF> <? =;F=OF;N?>
+
;M >0 : QBC=B =;H <? CHN?LJL?N?> ;M NB? >IN JLI>O=N ) $ 0 QB?L? ) CM NB?
G;MM G;NLCR ;H> $ ;H> 0 NB? P?=NILM =IHN;CHCHA NB? Q;N?L >?JNB ;H> NL;=?L P;FO?M
1B? =IGJ;NC<F? @ILG I@ WOR?M QCNB NB?CL MJ?=CV= >CM=L?NCT;NCIH I@ > CM ;FMI ACP?H
403 0=3 <><4=CD< 2>=B4AE0C8>=
,OL CHCNC;F =BIC=? I@ NB? HIH =IHM?LP;NCP? 0;CHN 3?H;HN ?KO;NCIHM ;H> NB? ZMBI=E =;J
NOLCHA[ ;JJLI;=B CH QBC=B NB?L? CM HI L?;F NL?;NG?HN I@ MBI=EM L;CM?M ; @OH>;G?HN;F
KO?MNCIH QB;N QIOF> <? NB? HOG?LC=;F MIFONCIH QB?H NB? L?;F NB?IL?NC=;F MIFONCIH
B;M >CM=IHNCHOCNC?M 1B? L?;MIHCHA QCFF N;E? JF;=? CH NQI MN;A?M #CLMN Q? ?R;GCH?
NB? H;NOL? I@ >CM=IHNCHOCNC?M QBC=B =IOF> ;JJ?;L CH MIFONCIHM I@ NB? HIH =IHM?LP;NCP?
?KO;NCIHM 1BCM VLMN ?R;GCH;NCIH QCFF N?H> NI JLIP? NB;N NB? HIH =IHM?LP;NCP? ?KO;
NCIHM F?;> NI CH=ILL?=N MIFONCIHM &H NB? M?=IH> MN;A? Q? ;==?JN NB;N NBCM ;JJLI;=B CM
HIN P;FC> C@ NB? HOG?LC=;F G?NBI> OM?> >I?M HIN ;FFIQ >CM=IHNCHOCNC?M ;M CM NB? =;M?
QCNB VHCN? ?F?G?HNM QCNB FCH?;L CHN?LJIF;NCIH
8B2>=C8=D8C84B F8C7 C74 =>=2>=B4AE0C8E4 4@D0C8>=B
)?N OM L?=;FF NB? =IH>CNCIHM I@ 0?=NCIH IH BS>L;OFC= DOGJM ! MIOL=? N?LGM
>C_OMCIH ;H> @LC=NCIH <?CHA H?AF?=N?> W;N <INNIG ;H> MN?;>S WIQ 1B? GIG?HNOG
?KO;NCIH CH =IHM?LP;NCP? @ILG CM L?>O=?> NI
N
>K
=
>
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 QBCF? NB? M;G? ?KO;NCIH CH HIH =IHM?LP;NCP? @ILG ACP?M
K
=> N &H NB? HIH =IHM?LP;NCP? @ILG I@ NB? 0;CHN 3?H;HN GIG?HNOG ?KO;NCIH NB? JBSMC=;F
KO;HNCNS =IHM?LP?> IH ?CNB?L MC>? I@ NB? BS>L;OFC= DOGJ CM HI FIHA?L
>K
=
>
<ON NB? B?;>
K
=
0I Q? VH> >C_?L?HN =IHDOA;N? >?JNBM ;H> P?FI=CNC?M L?F;NCIHM
>
> >
>1
>
>
K K K
K
=-
;H>
"
"
"
"
QBC=B ;L? =IGJ;L?> QCNB NB? L?;F IH?M CH 1;<F?M ;H> 1B? =OLP?M " < "
I<N;CH?> QCNB NB? =IHM?LP;NCIH I@ GIG?HNOG IH NB? IH? B;H> ;H> NB? =IHM?LP;NCIH
I@ B?;> IH NB? INB?L B;H> ;L? P?LS =FIM? CH NB? PC=CHCNS I@ " 1B?S >I HIN
>C_?L <S GIL? NB;H OJ NI ; #LIO>? HOG<?L I@ ;H> HIN GIL? NB;H OJ
NI 1B?M? ZHIH =IHM?LP;NCP?[ L?F;NCIHM ;L? ?_?=NCP?FS I<M?LP?> IH <INB MC>?M
I@ ; NL;HM=LCNC=;F WIQ IP?L ; <OGJ ; =;M? QB?L? NB? B?;> CM =IHM?LP?> QBCF? NB?
GIG?HNOG CM HIN
1;<F? =IHM?LP;NCP? ;H> HIH =IHM?LP;NCP? =IHDOA;N? L?F;NCIHM CH ; BS>L;OFC= DOGJ
IHDOA;N? >?JNBM
IHM?LP;NCP?
@ILG I@
GIG?HNOG ?KO;NCIH
+IH =IHM?LP;NCP?
@ILG I@
GIG?HNOG ?KO;NCIH
> > >
6 6
6 6
>
%
5
%
5
IHDOA;N? P?FI=CNC?M
C C
C C
K K K
K
5%
=-
1I CFFOMNL;N? NBCM CH 1;<F? NB?L? CM ; =IGJ;LCMIH I@ NB? P;LC;NCIHM CH B?;> GI
G?HNOG ;H> NIN;F ?H?LAS CH ; BS>L;OFC= DOGJ QCNB =IHM?LP;NCP? ;H> HIH =IHM?LP;NCP?
@ILGM 1B? N;<F? ACP?M NB? >C_?L?H=? <?NQ??H NB? OJMNL?;G ;H> >IQHMNL?;G P;FO?M
1BOM Q? I<M?LP? NB;N QCNB <INB @ILGOF;NCIHM NB? NIN;F ?H?LAS I@ ; =IFOGH I@ Q;N?L
CH=L?;M?M >OLCHA ; BS>L;OFC= DOGJ ;M >I?M ?HNLIJS CH ; MBI=E %IQ?P?L NB?L? CM
>?VHCN?FS ; FIMM I@ ?H?LAS CH <INB =;M?M MCH=? CNM WOR >?=L?;M?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1;<F? =IHDOA;N? L?F;NCIHM <?NQ??H #LIO>? HOG<?LM CH ; BS>L;OFC= DOGJ
IHDOA;N? #LIO>? HOG<?LM
IHM?LP;NCP?
@ILG I@
GIG?HNOG ?KO;NCIH
+IH =IHM?LP;NCP?
@ILG I@
GIG?HNOG ?KO;NCIH
"
"
"
"
1;<F? P;LC;NCIHM I@ NB? JBSMC=;F KO;HNCNC?M CH ; BS>L;OFC= DOGJ
%?;>
IHM?LP;NCP?
@ILG I@
GIG?HNOG ?KO;NCIH
+IH =IHM?LP;NCP?
@ILG I@
GIG?HNOG ?KO;NCIH
6 6
6 6
*IG?HNOG
56 6 6 6
1IN;F ?H?LAS
5
> >
56 6 6 6
6 6
?0A03>G
1B? KO?MNCIH ;N NBCM JICHN CM NI EHIQ QB?NB?L NB? OM? I@ ; HIH =IHM?LP;NCP? ?KO;NCIH
QCFF F?;> NB? HOG?LC=;F MCGOF;NCIHM NI NB? ?LLIH?IOM L?F;NCIHM I@ NB? BS>L;OFC= DOGJ
Q? B;P? DOMN M??H =IHMCMN?HN M=B?G? NIA?NB?L QCNB NB? NCG? ;H> MJ;=? MN?JM
N?H>CHA NI NB? T?LI FCGCN MBIOF> ACP? NB? Z?R;=N[ MIFONCIH =ILL?MJIH>CHA NI NB? CHCNC;F
?KO;NCIHM &H @;=N ?P?LS ZVHCN? ?F?G?HN[ MIFONCIH QCNBION MJ?=C;F NL?;NG?HN I@ MBI=EM
CM =IHNCHOIOM %?H=? NB? NQI @ILGOF;NCIHM ;L? ?KOCP;F?HN ;H> MBIOF> F?;> NI NB?
M;G? MIFONCIH C@ =IHMCMN?HN HOG?LC=;F M=B?G?M ;L? OM?> 1BCM MIFONCIH QCFF =IHM?LP?
GIG?HNOG ;H> B?;> ;H> QCFF B?H=? <? @;FM? CH <INB =;M?M &H JL;=NC=? Q? I<M?LP?
NB;N NBCM MIFONCIH CM ; AII> ;JJLIRCG;NCIH I@ NB? =IHM?LP;NCP? >CM=IHNCHOIOM MIFONCIH
;M CM MBIQH CH NB? L?MOFN I@ -F;N? M?? 0?=NCIH WIQ IP?L ; Q?CL QCNB ; BS>L;OFC=
DOGJ ?MC>?M Q? =;HHIN LOF? ION NB;N FI=;F HOG?LC=;F >C_OMCIH DOMN ;JJ?;LM ;H>
OM?M OJ ;FF IL ; J;LN I@ NB? ?H?LAS QBIM? >CM;JJ?;L;H=? CM @IL?M??H <S NB? NB?ILS
>=B4AE0C8>= >5 4=4A6H
&H NB? ;<M?H=? I@ @LC=NCIH >C_OMCIH MIOL=? N?LGM ;H> >CM=IHNCHOCNC?M NB? NIN;F ?H?LAS
I@ ; WIQ MBIOF> <? =IHM?LP?> 4? ;FMI ;MMOG? ; W;N <INNIG NI MCGJFC@S G;NN?LM
1B? HIH =IHM?LP;NCP? 0;CHN 3?H;HN ?KO;NCIHM ;L? NB?H QLCNN?H ;M
>
J
K
J
:?L >
K K =H7: K
=
>
N
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 K =H7: L
L
J
=
>
O
S GOFNCJFSCHA "KO;NCIH <S => C D "KO;NCIH <S >K ;H> "KO;NCIH
<S >L NB? MOG I@ NB? NBL?? L?MOFN;HN ?KO;NCIHM NB?H ACP?M
L
L
=>
K
K
:?L =>
K >
>
J
QBC=B ?RJL?MM?M NB? =IHM?LP;NCIH I@ NB? @IFFIQCHA KO;HNCNS
=>
>
L
K
&HN?AL;N?> IH NB? ! >IG;CH NBCM QCFF JLIPC>? NB? NIN;F ?H?LAS ?R=?JN @IL IH?
@;=NIL &H @;=N NBCM KO;HNCNS CM NB? MOG I@ JIN?HNC;F ?H?LAS ;H> ECH?NC= ?H?LAS I@
; =IFOGH I@ Q;N?L QBIM? B?CABN CM > P?FI=CNS K ;H> =LIMM M?=NCIH ?KO;F NI OHCNS 4?
;A;CH VH> NB? =IH>CNCIH I@ ?HNLIJS I@ "KO;NCIH CH NB? ;<M?H=? I@ >CM=IHNCHOCNS
/?G;LE NB? ;<M?H=? I@ ; @;=NIL CH @LIHN I@ NB? N?LG => QCNBCH NB? >CP?LA?H=?
=;H FIIE FCE? ;H ?LLIL ;H> GCABN MOAA?MN NB;N IHFS B;F@ NB? ?H?LAS ?HN?LCHA ;N NB?
<IOH>;LS CM ;P;CF;<F? CHMC>? NB? >IG;CH &H @;=N 56 CM HIN JIN?HNC;F ?H?LAS <ON
NB? CHN?AL;F I@ JIN?HNC;F ?H?LAS ;FIHA ; =IFOGH I@ Q;N?L QBC=B CM HIN ;H ?RN?HMCP?
KO;HNCNS S GOFNCJFSCHA "KO;NCIH <S > Q? MBIQ NB;N
> J
K K =H7: > :?L > &@ Q? =IHMC>?L NB? MCGJFCV?> =;M? I@ NB? MFIQ VFFCHA I@ ; L?M?LPICL QCNB ; >?JNB
QBC=B L?G;CHM =IHMN;HN CH MJ;=? MI ;M NI =;H=?F NB? N?LG K =H7: > Q? I<M?LP?
NB;N NB? WOR > ;N NB? <IOH>;LC?M =;OM?M ; P;LC;NCIH I@ > CH NB? >IG;CH
)?N OM ?R;GCH? NB? =IHM?LP;NCIH I@ ?H?LAS ;N NB? >CM=L?N? F?P?F 4? IOABN NI EHIQ
QB?NB?L CN CM JIMMC<F? NI I<N;CH ; =IHM?LP;NCIH I@ ?H?LAS ?KO;NCIH ;M
56
>C
D <
J
56
>C
D <
:?L
>
=>
L
K
K
,OL >?AL??M I@ @L??>IG ;L?
BIC=? I@ >CM=L?NCT;NCIH I@ KO;HNCNC?M =>
;H> ;L? GOFNCJFC?>
C D >K ;H> >L <S QBC=B "KO;NCIHM
BIC=? I@ >CM=L?NCT;NCIH I@ NB? M;G? NBL?? ?KO;NCIHM
&H IL>?L NI @ILG NB? >?LCP;NCP?M CH NCG? ;N NB? >CM=L?N? F?P?F CN CM H?=?MM;LS
@IL => C D NI <? QLCNN?H ;M
=
><
><
K<
K<
L<
L<
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
@IL >K NI <? QLCNN?H ;M
>K ><
><
K<
K<
><
><
L<
L<
@IL >L NI <? QLCNN?H ;M
>L 4CNB NBCM =BIC=? Q? =;H P?LC@S NB;N => 6
B CM ;=NO;FFS >CM=L?NCT?> ;M
>
=>
J
C 6
B
&H NB? M;G? Q;S >K C
B
K
>K
J
&H IL>?L NB;N NB? N?LG
5 6
J
56
<?=IG?M
K >
J
6 C
J
6 C
>
K =H7: =>
=> :?L >
K
;=NO;FFS ACP?M :?L => K
CN NB?H <?=IG?M H?=?MM;LS NB;N :?L >
K CM >CM=L?NCT?> ;M
<
K< ><
K<
>
:?L >
K :?L
;H> =H7: => ;M
=H7: => =H7: =
><
><
#CH;FFS NB? >CM=L?NCT;NCIH I@
K
L
>K
K =H7: K
:?L >
K
>L K =H7: L
K NB? IHFS >?AL??M I@ @L??>IG F?@N <?CHA
MBIOF> ACP? ; >CM=L?NCT;NCIH I@ :?L > C D NB? >CM=L?NCT;NCIH I@ K =H7: K ;H> K =H7: L 1BCM <LCHAM OM <;=E NI CGJIMCHA
NB;N NB? N?LG
K<
><
K<
><
<?=IG?M ?KO;F NI
K<
L<
K<
L<
K =H7: K
:?L
><
><
><
><
K<
K<
K<
K<
K =H7: L
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 :?L
><
><
K<
K<
L<
L<
K<
K<
1BCM CM JIMMC<F? IHFS <S >CM=L?NCTCHA
K < =H7: K<
K =H7: K ;M
K<
=H7: K<
K < =H7: L <
K =H7: L ;M
K<
=H7: L <
&H =IH=FOMCIH Q? VH> NB;N
#IL ;H OHMN?;>S WIQ IHFS ; HOG?LC=;F M=B?G? J?L@?=NFS =?HNL?> CH NCG? ;H>
B?H=? HIH FCH?;L CH NBCM =;M? =;H =IHM?LP? ?H?LAS
1B? CHCNC;F =BIC=? I@ HIH =IHM?LP;NCP? ?KO;NCIHM >I?M HIN ;JJ?;L ;M ; B;H>C=;J
@IL =IHM?LP;NCIH I@ ?H?LAS
M ;FF NB? =IH>CNCIHM @IOH> IHFS =IH=?LH NCG? >CM=L?NCT;NCIH NB? =IHM?LP;NCIH
I@ ?H?LAS CM ;MMOL?> @IL MN?;>S WIQ
4B;N B;M DOMN <??H >IH? CM BIQ?P?L P;FC> IHFS QCNBCH NB? @L;G?QILE I@ ; MNLC=N
=IH@ILGCNS QCNB NB? $;F?LECH G?NBI> 1B? OM? I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M IL NB?
02-$ G?NBI> @IL ?R;GJF? ;>>M N?LGM QBC=B =;HHIN <? JON NIA?NB?L CH ; >CP?LA?H=?
;H> B?H=? LOCHM NB? >?GIHMNL;NCIH
)74 CA40C<4=C >5 D=2>E4A43 143B
#+ +) ' &*& +(#)( +(& M;SCHA @LIG ;H=C?HN "ASJN
1B? MIFONCIH OMO;FFS ;>IJN?> @IL >?;FCHA QCNB OH=IP?L?> <?>M IL >LS TIH?M CM NI FCGCN
NB? P;FO? I@ NB? Q;N?L >?JNB NI ; NBL?MBIF> 1BCM MIFONCIH B;M M?P?L;F >CM;>P;HN;A?M
VLMN ;HS ;L<CNL;LS CHN?LP?HNCIH IH NB? Q;N?L >?JNB LOCHM NB? =IHM?LP;NCIH I@ G;MM
NB?H NB? =IHM?LP;NCIH I@ GIG?HNOG ?KO;NCIH ;JJFC?> NI ; VH? VFG I@ Q;N?L =L?;N?M ;
F;GCH;L WIQ QBIM? P?FI=CNC?M ;L? HIN H?AFCAC<F? ;H> ;=N ;M J;L;MCNC= P;FO?M QBC=B ;L?
NL;HMGCNN?> NI TIH?M OH>?L Q;N?L 1QI INB?L MIFONCIHM B;P? <??H ?F;<IL;N?> ;H> VH?
NOH?> 1B? VLMN IH? =IHMCMNM I@ NL?;NCHA NB? ?KO;NCIHM CH NB? QBIF? >IG;CH ;H> CH
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
free surface = bottom elevation + water depth
free surface
mesh nodes
bottom elevation
#CAOL? J;L;MCNC= @L?? MOL@;=? IH ; MBIL?FCH?
NB?CL NIN;FCNS MCGJFS <S =ILL?=NCHA NB? N?LGM NB;N ;L? ?PC>?HNFS L?H>?L?> @;FM? <S NB?
;<M?H=? I@ Q;N?L FCE? NB? @L?? MOL@;=? AL;>C?HN &H @;=N CH ;H OH=IP?L?> ;L?; NB? @L??
MOL@;=? AL;>C?HN <?=IG?M NB? <INNIG AL;>C?HN ;H> =L?;N?M J;L;MCNC= >LCPCHA N?LGM ;M
CM MBIQH CH ! CH #CAOL? IHMC>?LCHA NB? ?HNCL? @L?? MOL@;=? AL;>C?HN @IL ; F;E? ;N L?MN QCNB <;HEM J;L
NC;FFS =IP?L?> QCNB Q;N?L BCABFCABNM NB? JL?M?H=? I@ J;L;MCNC= >LCPCHA N?LGM >O? NI NB?
?LLIH?IOM CHNLI>O=NCIH CH NB? ?KO;NCIHM I@ >LS <INNIG AL;>C?HN N?LGM
1B? M?=IH> IJNCIH =IHMCMNM I@ L?GIPCHA @LIG NB? =;F=OF;NCIH ;FF ?F?G?HNM QBC=B
;L? HIN =IGJF?N?FS Q?N
$?C8>= 2>AA42C8>= >5 C74 5A44 BDA5024 6A0384=C
1B? B;F@ >LS B;F@ Q?N ?F?G?HNM ;L? =B;L;=N?LCT?> <S ; >CM=IHNCHOCNS CH NB? @L?? MOL@;=?
AL;>C?HN QBC=B J;MM?M @LIG ; P;FO? =FIM? NI QB?L? NB?L? CM Q;N?L NI ; P;FO? =FIM?
NI =H7: 64 QB?L? NB?L? CM HIH? 0I FCH?;L CHN?LJIF;NCIH CM CFF ;>;JN?> ;H> ; MJ?=CV=
NL?;NG?HN I@ NB? N?LG = =H7: 6A< MBIOF> N;E? JF;=?
M?=IH> @OH>;G?HN;F C>?; CM NI ;<;H>IH NB? GCHCGOG NBL?MBIF> =LCN?LCIH NI >?
N?LGCH? C@ ;H ?F?G?HN CM >LS IL Q?N 4? >?=C>? NB;N ?F?G?HNM L?KOCLCHA ; MJ?=C;F =;F=O
F;NCIH ;L? NBIM? QBIM? <INNIG ?F?P;NCIH I@ IH? HI>? L?@?LL?> NI ;M ZBCAB[ CM BCAB?L
NB;H NB? ?F?P;NCIH I@ NB? @L?? MOL@;=? I@ ;HINB?L IH? L?@?LL?> NI ;M ZFIQ[ #CH;FFS
CH NB?M? =IHN?HNCIOM ?F?G?HNM NB? @L?? MOL@;=? AL;>C?HN CM HIN <LON;FFS =;H=?FF?> ,H?
MCGJFS L?>O=?M NB? @L?? MOL@;=? ?F?P;NCIH I@ NB? ZBCAB[ HI>? <S NB? >C_?L?H=? <?NQ??H
CNM <INNIG ?F?P;NCIH ;H> NB? @L?? MOL@;=? ?F?P;NCIH I@ NB? ZFIQ[ HI>? NBCM CM >IH? IHFS
NI =;F=OF;N? NB? @L?? MOL@;=? AL;>C?HN 1B? JLI<F?G I@ ; F;E? ;N L?MN CM MIFP?> <S NBCM
G?NBI> QBC=B GIL?IP?L =IHM?LP?M L?;F @L?? MOL@;=? AL;>C?HNM ? A QB?H ; NLC;HAF?
=IHN;CHM NQI Q?N HI>?M ;H> IH? >LS HI>?
1B? IHFS FCGCN;NCIH NI NB? G?NBI> =IG?M @LIG NB? =LCN?LCIH >?=C>CHA QB?NB?L ;H
?F?G?HN CM >LS IL Q?N &H =?LN;CH =;M?M F;LA? ?F?G?HNM ;FIHA NB? MFIJ? ;H> FIQ Q;N?L
>?JNB NBCM =LCN?LCIH CM @;FM? ;H> >?=F;L?M ;M >LS ?F?G?HNM QBC=B ;L? ?HNCL?FS Q?N 1BCM
FCGCN;NCIH MBIOF> <? N;E?H CHNI ;==IOHN QBCF? <OCF>CHA NB? G?MB &N MBIOF> <? ?HMOL?>
NB;N NB? >C_?L?H=? CH F?P?F IH ;H ?F?G?HN CM F?MM NB;H NB? >?JNB
-F;N? CFFOMNL;N?M NB? =;J;=CNS I@ NBCM ;JJLI;=B NI MCGOF;N? >LSCHA IH ; <?;=B NI
T?LI >?JNB &H NBCM N?MN =;M? NB? ?F?P;NCIH I@ NB? @L?? MOL@;=? ;N NB? F?@N <IOH>;LS
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 I@ NB? =;F=OF;NCIH >IG;CH ; G FIHA =;H;F QBIM? =LIMM M?=NCIH CM MBIQH CH NB?
JF;N? CM FIQ?L?> ;==IL>CHA NI ; MCHOMIC>;F F;Q ;M >OLCHA ; NC>? 1B? <?;=B ?GJNC?M
;H> Q;N?L L?N;CH?> CH ; JIIF @ILGM ; JO>>F? QBC=B AL;>O;FFS AI?M <;=E NI L?MN 1B?
JLI<F?G I@ ; ZF;E? ;N L?MN[ CM ;FMI <LIOABN ION CH NBCM =IHVAOL;NCIH ;N NB? <INNIG I@
NB? JF;N? 1B? MCNO;NCIH MBIQH CM MN;<F? NB? JIMN JLI=?MMIL <S FCH?;L CHN?LJIF;NCIH
>L;QM ; MFIJCHA @L?? MOL@;=? ;N NB? JICHN QB?L? NB? @L?? MOL@;=? L?;=B?M NB? <?;=B
1B? ;FAILCNBG >?;FCHA QCNB ?RJIM?> <?>M >I?M HIN =IHMC>?L NBCM J;L;MCNC= AL;>C?HN
;H> >I?M HIN A?H?L;N? ;HS GIP?G?HN
$?C8>= <0B:8=6 >5 4G?>B43 4;4<4=CB
&H NBCM IJNCIH NB? =LCN?LCIH OM?> NI >?=C>? QB?NB?L ;H ?F?G?HN CM J;LNC;FFS >LS >?
J?H>M IH ; =IGJ;LCMIH <?NQ??H ;H ;P?L;A? P;FO? I@ NB? <INNIG J?L ?F?G?HN ;H> ;H
;P?L;A? P;FO? I@ NB? @L?? MOL@;=? *IL?IP?L ;FF ?F?G?HNM NB;N ;L? HIN ?HNCL?FS Q?N ;L?
?FCGCH;N?> @LIG NB? G?MB #IL NBCM JOLJIM? ?;=B ?F?G?HN CM ;MMI=C;N?> QCNB ; G;ME
?KO;F NI @IL ; Q?N ?F?G?HN ;H> ?KO;F NI @IL ;H ?RJIM?> ?F?G?HN ?P?H J;LNC;FFS
?RJIM?> FF =IHNLC<ONCIHM I@ NB? ?F?G?HNM CH NB? =;F=OF;NCIH I@ P?=NILM ;H> G;NLC=?M
;L? GOFNCJFC?> <S NBCM G;ME QBC=B F?;>M OM NI ; GI>CV=;NCIH I@ NB? G?MB <S NB?
L?GIP;F I@ ?F?G?HNM
1BCM N?=BHCKO? CM HIN ;M AII> ;M NB? =ILL?=NCIH I@ NB? @L?? MOL@;=? AL;>C?HN ;H> CH
J;LNC=OF;L JIM?M JLI<F?GM CH NB? >SH;GC=M I@ Q?NNCHA ;H> >LSCHA ;M Q?FF ;M CH G;MM
=IHM?LP;NCIH 1B?M? JLI<F?GM ;H> ; @?Q MIFONCIHM QCFF <? ACP?H CH >?N;CF NIA?NB?L
QCNB NB? NL?;NG?HN I@ ?RJIM?> <?>M CH ! @IL QBC=B NB? G?NBI> B;M <??H GIL? QC>?FS
;JJFC?>
%B4D3> F0E4 4@D0C8>=
1B? JM?O>I Q;P? ?KO;NCIH CM I<N;CH?> @LIG NB? 0;CHN 3?H;HN ?KO;NCIHM <S ?FCGCH;N
CHA NB? P?FI=CNS @LIG NB? =IHNCHOCNS ?KO;NCIH OMCHA ; P;FO? I@ NBCM P?FI=CNS ACP?H <S
NB? GIG?HNOG ?KO;NCIH 1I I<N;CH NB? Q;P? ?KO;NCIH P?FI=CNS CM ?FCGCH;N?> CH NB?
M;G? Q;S <ON <S =IHMC>?LCHA NB? J;LNC;F NCG? >?LCP;NCP? I@ NB? =IHNCHOCNS ?KO;NCIH
1B? HOG?LC=;F JLI<F?GM A?H?L;FFS ?H=IOHN?L?> CH NB? Q;P? ?KO;NCIH B;P? F?> NI ;
P;LC;HN NB? A?H?L;FCT?> Q;P? ?KO;NCIH QB?L? ; J?H;FNS CM ;>>?> NI ?HMOL? AII> G;MM
=IHM?LP;NCIH %IQ?P?L NB? J?H;FNS =I?^=C?HN B;M NI <? ;>DOMN?> ;M NB? =;M? G;S <?
NI ?FCGCH;N? MJOLCIOM IM=CFF;NCIHM IL ?HMOL? NB? =IHNCHOCNS ?KO;NCIH ,H NB?CL MC>?
MOJJILN?LM I@ NB? A?H?L;FCT?> Q;P? ?KO;NCIH ;^LG NB;N NB? MIFONCIHM I@ JLCGCNCP?
?KO;NCIHM NB?GM?FP?M MO_?L @LIG J;L;MCNC= IM=CFF;NCIHM /?@?L?H=? 89 ; =IGGOHC=
;NCIH JL?M?HN?> ;N NB? IHAL?MM IGJON;NCIH;F *?NBI>M CH 4;N?L /?MIOL=?M MBIQM
NB;N NB?M? IM=CFF;NCIHM QBC=B M??G NI <? >O? NI ; L?MOLA?H=? I@ NB? CH@\MOJ =IH>CNCIH
=;H <? ?FCGCH;N?> ?CNB?L <S MO^=C?HN JL?=CMCIH IL <S NB? ;>IJNCIH I@ ; >C_?L?HN
>CM=L?NCT;NCIH @IL NB? Q;N?L >?JNB ;H> @IL P?FI=CNS <S OMCHA @IL ?R;GJF? NB? ?F?G?HN
EHIQH ;M NB? ZKO;MC <O<<F?[ @IL NB? P?FI=CNS
S ?MN;<FCMBCHA ; <LC>A? <?NQ??H JLCGCNCP? ?KO;NCIHM ;H> NB? JM?O>I Q;P? ?KO;
NCIH Q? MB;FF MBIQ B?L? NB;N NB?L? CM HI L?;F >C_?L?H=? <?NQ??H <INB ;JJLI;=B?M
?R=?JN NB? OM? CH NB? F;NN?L =;M? I@ ; ZHIH =IGJ;NC<F?[ );JF;=C;H NI >CM=L?NCT? NB?
>CP?LA?H=? I@ ; AL;>C?HN CH VHCN? ?F?G?HNM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? C>?; =IG?M @LIG ;H ;NN?GJN NI MCGJFC@S NB? FCH?;L MSMN?G =IG<CHCHA NB?
Q;N?L >?JNB ;H> P?FI=CNS I@ 0?=NCIH <S ?FCGCH;NCHA P?FI=CNS @LIG NB? =IHNCHOCNS
?KO;NCIH ;N NB? >CM=L?N? F?P?F 1I MCGJFS >L;Q ;H ?RJL?MMCIH I@ P?FI=CNS @LIG NB?
GIG?HNOG ?KO;NCIH Q? FIIE @IL ; M?N I@ IJNCIHM @IL QBC=B NB? G;NLC=?M ) ;H>
) ;L? >C;AIH;F 1B? @ILG I@ IOL FCH?;L MSMN?G CM
) ) ) $
2 % ) ) & % 1<
& % 2 &
<
) ) 2
2 ) ;H> ) =IHN;CH G;NLC=?M ;LCMCHA @LIG NCG? >?LCP;NCP?M I@ >C_OMCIH @LC=NCIH
;H> CH NB? =;M? I@ NB? 02-$ G?NBI> I@ ;>P?=NCIH 1B? @LC=NCIH G;NLCR CM >C;AIH;F
NB? G;MM G;NLCR L?MOFNCHA @LIG NB? NCG? >?LCP;NCP? =;H <? G;>? >C;AIH;F <S G;MM
FOGJCHA ;H> >C_OMCIH =;H <? =;LLC?> IP?L NI NB? LCABN B;H> MC>? C@ CN CM QLCNN?H CH ;H
?RJFC=CN Q;S #CH;FFS NB? OM? @IL NL;HMJILN N?LGM I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M
IL I@ ;HS INB?L @ILG;FFS ?RJFC=CN G?NBI> -0& >CMNLC<ONCP? M=B?G? @IL ?R;GJF? ;FMI
?H;<F?M OM NI MBC@N NB?G NI NB? LCABN B;H> MC>? -ON NIA?NB?L NB?M? =IH>CNCIHM ACP?
>C;AIH;F G;NLC=?M ) ;H> ) ;H> Q? A?N NB? @IFFIQCHA ?RJL?MMCIHM @IL 1 ;H> 2 1 ) 2 ) ) $
2 ) 2 ) ) $
S =;LLSCHA IP?L NB?M? P;FO?M NI NB? >CM=L?NCT?> =IHNCHOCNS ?KO;NCIH Q? A?N
) ) ) ) ) ) ) $
2 ) ) 2 ) ) 2 QB?L? Q? =;H L?JF;=?
) ) ) ) ) ) <S NB? HIN;NCIH #
1BOM Q? =;H QLCN? NB;N NB? H?Q ?KO;NCIH MSMN?G CM
) #
$
%
& % 1<
&
) <
) 2
2 ) ) 2 ) ) 2 &
2 ) $
%
2 ) $
&H NBCM MSMN?G NB? NBL?? ?KO;NCIHM ;L? OH=IOJF?> MI $ =;H <? =IGJON?> VLMN
!?JNB ;H> P?FI=CNS G;S B;P? >C_?L?HN >CM=L?NCT;NCIH MI Q?
NB?H 1 < ;H> 2 <
QCFF <? ;<F? NI =IOJF? NB? JM?O>I Q;P? ?KO;NCIH ;H> KO;MC <O<<F? >CM=L?NCT;NCIH I@
NB? P?FI=CNS
1BCM MSMN?G CM L?MIFP?> ;M MO=B <S ;MOFFC 89 CH VHCN? >C_?L?H=?M &H @;=N Q? =;H
<OCF> QCNBION ;JJLIRCG;NCIH CH VHCN? >C_?L?H=?M ; G;NLCR FCE? ) ) ) /"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 &H ! @IL ?R;GJF? ) QIOF> <? NLC >C;AIH;F ;H> ) ) ) J?HN; >C;AIH;F
M Q? >I HIN QCMB NI >?;F QCNB =IGJF?R G;NLC=?M CH VHCN? ?F?G?HNM NB? G;NLCR #
QCFF <? ;JJLIRCG;N?> &H IL>?L NI VH> ION NB? H;NOL? I@ # Q? L?MOG? ?FCGCH;NCIH I@
P?FI=CNS ;N NB? =IHNCHOOG F?P?F IH NB? <;MCM I@ ?KO;NCIHM >CM=L?NCT?> IHFS CH NCG? ;M
@IFFIQM
IHNCHOCNS
>
J
C :?L
/?=;FF NB;N > ><
*IG?HNOG
K<
K<
J
>< K<
C
:?L >< K<
><
K<
:?L > 3 =H7: >
K<
K K = =H7: 6A < H?9 5 C <S >?HINCHA @IL NB? M;E? I@ MCGJFCV=;NCIH < H?9 6 NB? ?R;GJF? I@ 0NLC=EF?LYM
@LC=NCIH @ILGOF;
4CNB NB? M;G? ;MMOGJNCIHM ;>IJN?> ;N NB? >CM=L?N? F?P?F NBCM F;MN ?KO;NCIH <?
=IG?M
K<
K
1
=
J
6 =H7:
K<
> = =H7: 6A< < H?9 K<
:?L > 3 =H7: >
4? >?LCP?
K<
C
B
= =H7: 6A<
> 3 =H7: K<
< H?9
B
6 :?L
S =;LLSCHA IP?L NBCM P;FO? K<
Q;P? ?KO;NCIH
>
J
C :?L
B
:?L
6 C =>
B
><
< H?9
6 =H7:
>
NI NB? =IHNCHOCNS ?KO;NCIH Q? VH> NB? JM?O>I
< < H?9
=
=H7: >
K
1
= =H7: 6A<
J
C
:?L >< K<
K<
:?L > 3 =H7: >
";=B I@ NB?M? >C_?L?HN N?LGM =;H <? =IGJ;L?> QCNB CNM >CM=L?N? ?KOCP;F?HN
>
<?=IG?M ) $
J
C
K < <?=IG?M 2 :?L >< %6!/,!6+*& 0 ,# #/"" 02/# " #),40
C :?L
B
><
< H?9
<?=IG?M
K
1
= =H7: 6A<
J
K<
:?L > 3 =H7: >
) ) 2 ) ) 2 ;H>
:?L
6 C =>
< < H?9
B
=H7: >
<?=IG?M
) ) ) ) ) ) #
4? L?=IAHCT? CH
:?L
6 C =>
< < H?9
B
=H7: >
; >C_OMCIH N?LG QCNB ; >C_OMCIH =I?^=C?HN ?KO;F NI
6 C =>
<
< H?9
B
QBIM? >CG?HMCIH CM G M 1BCM N?LG =;H <? QLCNN?H CH ; =F;MMC= G;HH?L <S CHN?AL;NCIH
<S J;LNM ACPCHA ; >C_OMCIH G;NLCR QBC=B QCFF <? ;H ;JJLIRCG;NCIH I@ # #IL NBCM
Q? QLCN?
.
6 C =>< 67 :?L
=H7: > :
< H?9
B
.
67
B
6 C =>
< < H?9
=H7: > D :
.
6 C =>
<
< H?9
B
=H7: > =H7: 67 : H ;JJLIRCG;NCIH I@ # CM NBOM NB? G;NLCR ACP?H <S NB? N?LG
.
<
6 C => =H7: 68 =H7: 67 :
<
H?9
B
1BCM P;FO? I@ # CM >?FC<?L;N?FS E?JN CH NBCM @ILG C@ NB? P?FI=CNS CM KO;MC <O<<F?
&H ! IOL ?KO;NCIH =;H B;P? NQI CHN?LJL?N;NCIHM ?CNB?L >C_OMCIH QCNB =I?^=C?HN
6 C =>
B
<
< H?9
IL <S >CPC>CHA CN <S J Q;P? JLIJ;A;NCIH QCNB =?F?LCNS
8
<
6 C =>
9
J < H?9
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 1I A?N ; JBSMC=;F =?F?LCNS CH NB? ;<M?H=? I@ @LC=NCIH CN CM H?=?MM;LS NB;N 6 C 1B? ;JJLIRCG;N? =;F=OF;NCIH I@ # CM CH @;=N ; );JF;=C;H =;F=OF;NCIH =;FF?> ZHIH
=IGJ;NC<F?[ 1B;N CM Q? >I HIN P?LC@S ;N NB? >CM=L?N? F?P?F NB;N
:?L =H7: <
<
1B? OM? I@ NB? HIH =IGJ;NC<F? );JF;=C;H CM IH? I@ NB? Q?FF EHIQH MIFONCIHM @IL
L?GIPCHA MJOLCIOM IM=CFF;NCIHM ;LCMCHA @LIG ; PCIF;NCIH I@ NB? CH@\MOJ =IH>CNCIH ;H>
L?MIFONCIH QCNB NB? JM?O>I Q;P? ?KO;NCIH G;S <?H?VN @LIG NBCM ?_?=N
4? B;P? DOMN MBIQH NB;N ; MJ?=CV= =IHVAOL;NCIH CH IOL L?MIFONCIH I@ JLCGCNCP?
?KO;NCIHM MO=B ;M
=BIC=? I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M @IL NB? ;>P?=NCIH I@ P?FI=CNC?M
G;MM FOGJCHA I@ NB? G;MM G;NLCR I@ NB? GIG?HNOG ?KO;NCIH
?RJFC=CN >C_OMCIH
OM? I@ ; HIH =IGJ;NC<F? );JF;=C;H
=IOF> <? CHN?LJL?N?> ;M ; L?MIFONCIH I@ ; JM?O>I Q;P? ?KO;NCIH 4? =;H =IGJ;L?
JLCGCNCP? ?KO;NCIHM ;H> JM?O>I Q;P? ?KO;NCIHM @IL ;H OHMN?;>S =;M? NB?H @IL ; MN?;>S
=;M? -F;N? MBIQM NB? L?MOFNM I@ @IOL >C_?L?HN L?MIFONCIHM @IL WIQ ;LIOH> ; <LC>A?
JC?L
-LCGCNCP? ?KO;NCIHM QCNB FCH?;L >CM=L?NCT;NCIH
-LCGCNCP? ?KO;NCIHM QCNB ZKO;MC <O<<F?[ P?FI=CNS
-M?O>I Q;P? ?KO;NCIH QCNB FCH?;L >CM=L?NCT;NCIH
-M?O>I Q;P? ?KO;NCIH QCNB ZKO;MC <O<<F?[ P?FI=CNS
#IL ;FF NB? =;F=OF;NCIHM NB? G;MM G;NLCR I@ P?FI=CNS CM ZFOGJ?>[ ;H> >C_OMCIH CM
G;>? ?RJFC=CN ; =IHVAOL;NCIH QBC=B CH @;=N CM HIN NB? <?MN @IL JLCGCNCP? ?KO;NCIHM
&N CM =F?;L NB;N JLCGCNCP? ?KO;NCIHM ;H> JM?O>I Q;P? ?KO;NCIH ;L? P?LS =FIM? 1B?
G;CH >C_?L?H=? CH NB? @IOL L?MOFNM ;JJ?;LM NI <? >O? NI NB? >CM=L?NCT;NCIH I@ P?FI=CNS
4CNB FCH?;L CHN?LJIF;NCIH MIG? ZNL?G<FCHA[ I@ NB? @L?? MOL@;=? CM PCMC<F? NB?M? MJOLCIOM
IM=CFF;NCIHM ;L? L?GIP?> <S ; KO;MC <O<<F? P?FI=CNS 1B? -2 NCG?M I<N;CH?> QCNB ;
*%T %?QF?NN -;=E;L> ;L? ;FMI ACP?H CH -F;N? IOL H?Q ZJM?O>I Q;P?
?KO;NCIH[ ;FAILCNBG CM ;FQ;SM @;MN?L <S ; @;=NIL I@ IL ?P?H GIL? CN CM A?H?L;FFS
GIL? LI<OMN ;H> M??GM GIL? ;JJLIJLC;N? @IL MN?;>S WIQ &H @;=N CH NBCM =;M? G;MM
FOGJCHA I@ NB? NCG? >?LCP;NCP? ;H> ?RJFC=CN;NCIH I@ >C_OMCIH B;P? HI ?_?=N IH NB?
VH;F MIFONCIH NB? IHFS >C_?L?H=? CM NB?H NB? NL?;NG?HN I@ NB? );JF;=C;H @IL QLCNCHA
:?L =H7: > 1BCM >C_?L?H=? ;FNBIOAB J?LB;JM CGJILN;HN @IL MN;<CFCNS CM GCHCG;F CH
NB? L?MOFNM ;M CM MBIQH <S NB? MOJ?LJIMCNCIH I@ =OLP?M CH -F;N? L?F;NCP? NI NB? WIQ
IP?L ; Q?CL QCNB ; BS>L;OFC= DOGJ M?? 0?=NCIH <?FIQ
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
(><4 E0;830C8>= C4BC 20B4B
4? B;P? G?HNCIH?> >OLCHA NB? JL?M?HN;NCIH I@ NB? ;FAILCNBGM MIG? N?MNM <S MBIQCHA
NB? G;CH J;LNC=OF;LM %?L? CM ; L?=;JCNOF;NCIH I@ NB? G;CH JICHNM @IL NB? P;FC>;NCIH I@
; =I>? L?MIFPCHA NB? 0;CHN 3?H;HN ?KO;NCIHM ;M Q?FF ;M ; @?Q H?Q =;M?M I@ P;FC>;NCIH
)4BC >5 0 ;0:4 0C A4BC
&H ; F;E? ;N L?MN QCNB ; P;LC;<F? <INNIG NB? HOG?LC=;F MIFONCIH MBIOF> HIN A?H?L;N?
;HS GIP?G?HN 1BCM ?PC>?H=? CM HIN AO;L;HN??> C@ NB? <INNIG AL;>C?HN ;H> NB?
>?JNB AL;>C?HN ;L? HIN NL?;N?> CH ; NIN;FFS =IGJ;NC<F? G;HH?L B?H=? NB? H?=?MMCNS I@
NB? M;G? OJQCH>CHA CH VHCN? PIFOG?M &H IOL ;FAILCNBG NBCM =IH>CNCIH CM I<N;CH?>
<?=;OM? NB? OHEHIQH ZQ;N?L >?JNB[ CM CH @;=N NB? CH=L?G?HN >< >< 1B? >LCPCHA
N?LG CH NB? =IHM?LP;NCIH I@ GIG?HNOG ?KO;NCIH CM =H7: 6A< QBC=B CM ;=NO;FFS T?LI
C@ IH? MN;LNM @LIG ; W;N @L?? MOL@;=? 1B? N?MN <?=IG?M =IGJFC=;N?> C@ >LS TIH?M ;L?
;>>?> CH NB? =;F=OF;NCIH >IG;CH M?? NB? NL?;NG?HN I@ ?RJIM?> <;HEM 0?=NCIH ;H> -F;N? '0?83 K>F >E4A 0 F48A F8C7 0 7H3A0D;82 9D<? 3>F=
BCA40<
1BCM N?MN =;M? L?@?LM NI ; WIQ CH ; =;H;F QBC=B CM G QC>? ;<ION G FIHA QCNB ;
W;N <INNIG QBIM? ?F?P;NCIH CM G ?R=?JN <?NQ??H NB? ;<M=CMM;? N G ;H>
N G QB?L? NB?L? CM ; J;L;<IFC= Q?CL =OFGCH;NCHA ;N N G ;N ?F?P;NCIH ;==IL>CHA NI NB? @ILGOF;
64 N ' N 1B? G?MB CM G;>? OJ I@ MKO;L?M QBIM? MC>?M G?;MOL? =G =ON CHNI NLC;HAF?M =IHMN;HN >CM=B;LA? I@ - G M ?HN?LM NB? =;H;F >?JNB I@ =G CM G;CHN;CH?>
>IQHMNL?;G 0NLC=EF?LYM @LC=NCIH F;Q CM OM?> QCNB ; =I?^=C?HN I@ 3CM=IMCNS CM
T?LI MMOGCHA ; J?LG;H?HN WIQ ;H> ; =IHMN;HN QC>NB @IL NB? =;H;F NB? GIG?HNOG
?KO;NCIH <?=IG?M
K
6A
K
=
=
N
N
'>
;H> QLCNCHA NB;N 6A > 64 Q? VH>
K
S L?JF;=CHA K <S
%
6!
=( >
1B? =LCNC=;F Q;N?L >?JNB CM
>
64
N
N
( ' >
G
=(
&H NB? ;<M?H=? I@ @LC=NCIH NBCM =LCNC=;F Q;N?L >?JNB CM I<M?LP?> ;N NB? NIJ I@ NB? Q?CL
%?L? CN CM ;NN;CH?> C@
>1 64
N
( ' >1
=
' >1
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 1B? @ILG I@ NB? Q?CL Q;M =BIM?H MO=B NB;N NB? =LCNC=;F >?JNB CM ;NN;CH?> IH? ?F?G?HN
=G ;@N?L NB? NIJ I@ NB? Q?CL NB? MFIJ? CM NB?H ?KO;F NI 1BCM B?FJM OM
NI <OCF> ;H Z?R;=N[ MIFONCIH <S G;ECHA OM? I@ NB? >C_?L?HNC;F ?KO;NCIH
>
N
.
F
%
! 6
%
5! 6
QCNB ; /OHA?\(ONN; G?NBI> NI M?=IH> IL>?L ;==OL;=S 1B? MIFONCIH OJMNL?;G I@ NB?
=LCNC=;F M?=NCIH CM =;F=OF;N?> QCNB H?A;NCP? MJ;=? MN?JM MN;LNCHA @LIG NBCM M?=NCIH
4? =IGJON? >IQHMNL?;G MNCFF QCNB MN?JM MN;LNCHA @LIG NB? M;G? M?=NCIH 1BCM
ACP?M ; VLMN MIFONCIH QBC=B CM MOJ?L=LCNC=;F >IQHMNL?;G MO<=LCNC=;F MIFONCIH CM
NB?H =;F=OF;N?> <?ACHHCHA @LIG NB? LCABN I@ NB? =;H;F ;H> MN;LNCHA @LIG NB? <IOH>;LS
=IH>CNCIH 1B? J;MM;A? @LIG NB? MOJ?L=LCNC=;F MIFONCIH NI NB? MO<=LCNC=;F MIFONCIH CM
;=BC?P?> <S ; BS>L;OFC= DOGJ QB?H NB? DOGJ =IH>CNCIHM ;L? @OFVFF?> NB;N CM QB?H
> > >
> >1 ;M MBIQH CH 0?=NCIH &H -F;N? NB? Z?R;=N[ MIFONCIH CM
=IGJ;L?> QCNB NB? MIFONCIHM I<N;CH?> QCNB HIH =IHM?LP;NCP? JLCGCNCP? ?KO;NCIHM ;H>
QCNB NB? JM?O>I Q;P? ?KO;NCIH 1B? OM? I@ ; HIH =IHM?LP;NCP? @ILG JLIP?M NI <? HIN
P?LS >?NLCG?HN;F ;H> HI J;L;MCNC= IM=CFF;NCIH CM I<M?LP?> IH ?CNB?L MC>? I@ NB? BS>L;OFC=
DOGJ 1B? @;FM? Z?R;=N[ MIFONCIH I<N;CH?> QCNB ; HIH =IHM?LP;NCP? @ILG >;MB?> FCH?
CH -F;N? >C_?LM <S NB? JIMCNCIH I@ NB? BS>L;OFC= DOGJ &H @;=N NB? =LCNC=;F Q?CL
OJMNL?;G ;H> NB? Q;N?L >?JNB CGJIM?> >IQHMNL?;G >?=C>? NB? MIFONCIH <ON NB?
JF;=? QB?L? NB? BS>L;OFC= DOGJ L?F;NCIH NLCAA?LM NB? NL;HMCNCIH @LIG NB? MOJ?L=LCNC=;F
WIQ CGJIM?> <S NB? OJMNL?;G =IH>CNCIH ;H> NB? MO<=LCNC=;F WIQ CGJIM?> ;N NB?
>IQHMNL?;G ?H> CM MN;AA?L?> 4? I<M?LP? ;M CH>C=;N?> CH 0?=NCIH NB;N NB?
HOG?LC=;F =;F=OF;NCIH ACP?M ; P?LS AII> ;JJLIRCG;NCIH I@ NB? ZL?;F[ =IHM?LP;NCP?
MIFONCIH GIL? ;>P;H=?> NB?IL?NC=;F MNO>S QIOF> <? H?=?MM;LS NI OH>?LMN;H> NBCM
MOLJLCMCHA L?MOFN <?NN?L
1BCM =;M? =;H ;FMI <? MCGOF;N?> QCNB NB? HIH BS>LIMN;NC= ! +;PC?L\0NIE?M ?KO;
NCIHM M?? -F;N? 1B? HOG?LC=;F J;L;G?N?LM ;L? NB? M;G? ;M NBIM? I@ NB? 0;CHN
3?H;HN ?KO;NCIHM BIQ?P?L ; PCM=IMCNS I@ G M ;FIHA NB? P?LNC=;F CM OM?> ; T?LI
>C_OMCIH QIOF> HIN NL;HMGCN NB? MB?;L MNL?MM I@ @LC=NCIH IH NB? <INNIG 1B? >SH;GC=
JL?MMOL? V?F> ;N NB? <INNIG I@ -F;N? <;MC=;FFS NBCM CM NB? P;LC;<F? >B
L?P?;FM
NB;N NB? WIQ CM BS>LIMN;NC= ?P?LSQB?L? ?R=?JN ;N NB? FI=;NCIH I@ NB? BS>L;OFC= DOGJ
1B? BS>L;OFC= DOGJ CM F?MM G;LE?> ;H> CM @IFFIQ?> <S ;H OH>OF;NCIH QBC=B CM GIL?
CH =IH@ILGCNS QCNB NB? I<M?LP;NCIHM
,0E4 ?A>?060C8>=
1QI JICHNM B;P? NI <? P?LCV?>
-LIJ;A;NCIH I@ Q;P?M QCNB NB? =ILL?=N =?F?LCNS 1BCM A?H?L;FFS CM >IH? CH ;
=;H;F QCNB ; L?=N;HAOF;L M?=NCIH ;H> =IHMN;HN >?JNB &N =;H <? P?LCV?> QCNB
NB? IOMMCH?MK ?KO;NCIHM CH -F;N? NB;N
5 NB? Q;P? =LIMM?M G CH M
=ILL?MJIH>CHA NI NB? NB?IL?NC=;F =?F?LCNS = > QCNB = GM >
G ;H> G
1B? ;<M?H=? I@ JIF;LCT;NCIH C ? ; JLIJ;A;NCIH I@ C>?HNC=;F Q;P?M CH ;FF >CL
?=NCIHM 1BCM CM NB? ;CG I@ NB? N?MN =;M? JL?M?HN?> CH -F;N? QBIM? CHC
NC;F =IH>CNCIH CM ; $;OMMC;H MB;J?> @L?? MOL@;=? QCNB ;H ?F?P;NCIH ?KO;F NI
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;F G 4? =;H =F?;LFS M?? NB;N ; =CL=OF;L Q;P? CM JLIJ;A;N?>
;H> NB?H L?W?=N?> <S NB? Q;FFM I@ NB? MKO;L? >IG;CH 0BIQH ;N NB? NIJ I@ -F;N?
CM NB? L?MOFN I@ NB? L?W?=NCIH <S NB? Q;FFM
'8CC4A 0=3 (C>:4A B>;DC8>=B
1B?M? ! MIFONCIHM L?JLI>O=? ; >;G <L?;E QCNB ; L?M?LPICL ;N =IHMN;HN >?JNB >5 CH ; LCP?L QCNB ; =IHMN;HN L?=N;HAOF;L =LIMM M?=NCIH &H NB? =;M? I@ /CNN?L 89 NB?
<INNIG IH NB? >IQHMNL?;G MC>? CM >LS &H NB? =;M? I@ 0NIE?L 89 NB? >?JNB IH NB?
>IQHMNL?;G MC>? CM =IHMN;HN ;H> P?FI=CNS CM T?LI &@ NB? LCP?L WIQM ;FIHA NB? N ;RCM
;H> NB? >;G CM ;N N NB? @L??5MOL@;=? CM ; J;L;<IF; CH NB? /CNN?L MIFONCIH 5
FCGCN?>
OJMNL?;G <S NB? ;<M=CMM; N =>5 J ;H> >IQHMNL?;G <S NB? ;<M=CMM; N =>5 J
S =IHMC>?LCHA5NB? /C?G;HH CHP;LC;HNM ;H> NB? NB?ILS I@ =B;L;=N?LCMNC=M ;H> <S
>?HINCHA 95 =>5 Q? VH> NB;N
9
=> K
F
B
95 F
B
95
&H NB? 0NIE?L
MIFONCIH QCNB ; >?JNB I@ >2 >IQHMNL?;G NI QBC=B CM ;MMI=C;N?> ;
=?F?LCNS 92 =>2 NB? /CNN?L ?RJ;HMCIH Q;P? CM FCGCN?> IH NB? >IQHMNL?;G MC>? <S ;
TIH? QCNB =IHMN;HN >?JNB >; ;H> =IHMN;HN P?FI=CNS K; CNM?F@ =IHH?=N?> NI NB? LCP?L
;N L?MN <S ; BS>L;OFC= DOGJ QBC=B ;>P;H=?M ;N ; MJ??> I@
3 >; K ;
>; >2
; @ILGOF; ?MN;<FCMB?> CH 0?=NCIH QCNB K2 B?L? &H NB? =;M? I@ 0NIE?L NB?L?
;L? @IOL >CMNCH=N TIH?M I@ MIFONCIHM M?J;L;N?> <S NBL?? JICHNM
!;H> " M?? #CAOL?
1B? ;<M=CMM;? I@ NB? JICHNM
! ;H> " ;L? >?HINCHA 9; =>; N 95 J
N K; 9; J
N 3 J
1B? @IOL TIH?M NB?H B;P? NB? @IFFIQCHA MIFONCIHM
?NQ??H NB? OJMNL?;G JIMCNCIH ;H> NB? JICHN
?NQ??H
L?M?LPICL ;N L?MN
;H> ! NB? /CNN?L MIFONCIH
?NQ??H ! ;H> " > >; ;H> K K;
!IQHMNL?;G I@ " LCP?L ;N L?MN QCNB ; >?JNB I@ >2
/"0,)21&,+ ,# 1%" 0&+1 3"++1 ".21&,+0 C
D
E
W
hg
h
hm
hd
#CAOL? >;G <L?;E IH ; Q?N <INNIG 0NIE?LYM MIFONCIH
1B? P?FI=CNS K; =;H <? @IOH> ;N JICHN ! QBC=B <?FIHAM NI NB? ?RJ;HMCIH Q;P? @IL
QBC=B K 9 95
0I K; 95 9; 1B? F;MN OHEHIQH 9; CM @IOH> <S QLCNCHA NB? =IHM?LP;NCIH I@
GIG?HNOG ;=LIMM NB? BS>L;OFC= DOGJ CH ; L?F;NCP? L?@?L?HNC;F ;NN;=B?> NI NB? DOGJ
>;
>
= 2
>2 3
1BCM CM ?RJL?MM?> <S ;H ?KO;NCIH I@ NB? MCRNB >?AL?? L?MIFP?> <S CN?L;NCIHM
>; K ; 3
9; 92 9;
=
95 92 9; 92 92
95 9;
92 1B? =;M? 92 L?MNIL?M NB? /CNN?L MIFONCIH &H 4O ( 89 JO<FCMB?> ;
H?Q MIFONCIH NI NBCM JLI<F?G =IH=?LHCHA ; LCP?L QCNB ; NL;J?TIC>;F M?=NCIH
-F;N? MBIQM ; HOG?LC=;F MCGOF;NCIH I@ NB? 0NIE?L MIFONCIH IH ; EG FIHA L?;=B
QCNB ; G?MB G;>? I@ G FIHA ?F?G?HNM %?L? >5 G ;H> >2 G 1B? NCG? MN?J CM
M ;H> Q? >CMJF;S NB? MIFONCIH ;N J M &H NBCM =;F=OF;NCIH NB? P?FI=CNS CM ?HNCL?FS
CGJFC=CN ;H> >CM=L?NCT?> OMCHA KO;MC <O<<F? ?F?G?HNM 1B? ;>P?=NCIH I@ P?FI=CNS CM >IH?
QCNB NB? >CMNLC<ONCP? -0& M=B?G? 0?=NCIH 1B? =IHNCHOCNS ?KO;NCIH CM L?MIFP?>
CH NB? =IHM?LP;NCP? @ILG QCNB ; 02-$ OJQCH>CHA 0?=NCIH 1B? BS>L;OFC= DOGJ
CM >?F;S?> <S IH? ?F?G?HN ;H> NB? NL;HMCNCIH N;E?M JF;=? CH IH? ?F?G?HN QCNB MFCABN
IM=CFF;NCIHM OJMNL?;G ;H> >IQHMNL?;G 1BCM MBIQM NB? KO;FCNS I@ NB? M=B?G? QBC=B
>I?M HIN B;P? ;HS MJ?=CV= NL?;NG?HN @IL MBI=EM
70?C4A '4B>;DC8>= >5 C74
#0E84AO(C>:4B 4@D0C8>=B
* (& ' "# $ #'#$ &# " " +( $'(!# # '"$#'('
# + & " )" " ." #)& +- - (& " &&#& ) " #)& &# "
)' ' + $&# *;R ILH \
=CA>3D2C8>=
1B? VLMN HOG?LC=;F NBL?? >CG?HMCIH;F GI>?FM @IL @L?? MOL@;=? WIQ >;N? @LIG NB? M
89 Q?FF ?MN;<FCMB?> JL;=NC=? =IHMCMNM I@ ;MMOGCHA BS>LIMN;NC= JL?MMOL? 89 F?;>
CHA NI QB;N CM EHIQH ;M NB? NBL?? >CG?HMCIH;F 0;CHN 3?H;HN ?KO;NCIHM 0?P?L;F ?R
;GJF?M I@ NB?M? GI>?FM ?RCMN CH VHCN? >C_?L?H=?M
1B? -LCH=?NIH ,=?;H *I>?F -,* M?? 89 &H NBCM GI>?F NB? AL;>C?HNM I@
<;LI=FCHC= ;H> <;LINLIJC= JL?MMOL?M ;L? M?J;L;N?> CHNI NQI >CMNCH=N GI>?M
1B? !;PC?M ;H> );QL?H=? GI>?F 89 ;JJFC?> NI =;F=OF;NCIHM I@ NC>?M CH NB? &LCMB
0?;
1B? 1LCG! GI>?F M?? 89 89 ;H> 89 QBC=B =IG<CH?M ; M?GC CGJFC=CN
@ILGOF;NCIH ;H> ; );AL;HAC;H\"OF?LC;H GI>?F @IL NL;HMJILN ?KO;NCIHM KOCN?
=FIM? NI QB;N Q? MB;FF <? >ICHA CH VHCN? ?F?G?HNM 1B? JL?MMOL? ;H> P?FI=CNS
HI>?M ;L? MN;AA?L?> 4CNB ; G?MB G;>? I@ IH? F;S?L NB? BS>LIMN;NC= JL?MMOL?
BSJINB?MCM L?MNIL?M NB? 0;CHN 3?H;HN ?KO;NCIHM I@ NB? 1LCG! MI@NQ;L? 1BCM
GI>?F B;M <??H ;JJFC?> NI =;F=OF;NCIHM I@ M;FCHCNS CH NB? <;S ;L?; I@ 0;H #L;H=CM=I
89
1B? !?F@N! #FIQ GI>?F 89 @OH=NCIHM QCNB ; =OLPCFCH?;L G?MB @ILG?> QCNB
B?R;B?>L; QBC=B B;M NB? ;>P;HN;A? I@ <?CHA ;<F? NI @IFFIQ NB? <IOH>;LS FCH?M
NI ; =?LN;CH ?RN?HN 1B? CH=IHP?HC?H=? CH L?F;NCIH NI VHCN? ?F?G?HNM CM NB;N FI=;F
L?VHCHA JLIJ;A;N?M ;FIHA NB? G?MB FCH?M FI=;F A?IG?NLC= NL;HM@ILG;NCIH CM
;JJFC?> NI >I NB? =;F=OF;NCIHM CH ; L?AOF;L G?MB 1BCM CHNLI>O=?M ; G?NLC=
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
QBC=B =IGJFC=;N?M NB? MJ;NC;F >?LCP;NCP?M 1BCM G?NLC= =;H <? MCGJFCV?> C@ NB?
NQI @;GCFC?M I@ NB? G?MB FCH?M ;L? J?LJ?H>C=OF;L
FF NB? GI>?FM G?HNCIH?> MI @;L OM? ; MCAG; NL;HM@ILG HINB?L MIFONCIH JLIJIM?>
CH VHCN? >C_?L?H=?M ;H> =IGJ;L?> NI NB? MCAG; NL;HM@ILG 89 CM =;FF?> ZA?IJIN?HNC;F
T =IIL>CH;N?[ &N =IHMCMNM MCGJFS I@ QILECHA QCNB VR?> MCT? ?F?G?HNM <ON P;LSCHA CH
HOG<?L ;FIHA NB? P?LNC=;F NI @IFFIQ NB? ?PIFONCIH I@ NB? MOL@;=? 1B? <INNIG ;H> NB?
@L?? MOL@;=? B;P? ; MN;CL=;M? MB;J? ;H> NBCM F?;>M NI ?LLILM CH NB? ?P;FO;NCIH I@ MNL?MM
IL NL;HMJILN N?LGM M?? 89
&H #L?>?LCEM?H ;H> 4;NNM 89 JLIJIM?> ; MJ;=? ;H> NCG? VHCN? ?F?G?HN NI
;==IOHN @IL P;LC;NCIH I@ NB? @L?? MOL@;=?
FMI CH VHCN? ?F?G?HNM )O?NNC=B ;H> =IFF?;AO?M 89 =IGJCF?> NB? MI@NQ;L? !
&/ !P;H=?> &/ OF;NCIH GI>?F @IL I=?;HC= =I;MN;F ;H> ?MNO;LCH? Q;N?LM >?>
C=;N?> NI I=?;HC= =CL=OF;NCIH ;H> ?MNO;LC?M ;H> OM?> <S NB? 2 0 LGS ILJM I@
"HACH??LM 1BCM MI@NQ;L? ;FMI MIFP?M NB? 0;CHN 3?H;HN ?KO;NCIHM ;H> B;M <??H OM?>
IH J;L;FF?F =IGJON?L ;L=BCN?=NOL? QCNB NB? *-& FC<L;LS M?? B;JN?L #IL GIL?
CH@ILG;NCIH L?@?L NI NB? &HN?LH?N Q?<MCN? BNNJQQQ G;LCH? OH= ?>O
1B? HIH BS>LIMN;NC= GI>?FM ;L? HIN MI =IGGIH 1B? C>?; I@ >?=IGJIMCHA NB? JL?M
MOL? CHNI BS>LIMN;NC= ;H> >SH;GC= J;LNM QBC=B QCFF <? >CM=OMM?> F;N?L Q;M @ILGOF;N?>
CH <S ;MOFFC ;H> 0N?FFCHA 89 &H *;B;>?P;H ( 89 CHNLI>O=?> ;H
I=?;HC= HIH BS>LIMN;NC= GI>?F NI =IHP?HC?HNFS L?JLI>O=? OJQ?FFCHA ;H> >IQHQ?FFCHA
JB?HIG?H; 1B?S ;FMI MBIQ?> NB;N NB? HIH BS>LIMN;NC= JLI<F?G Q;M Q?FF JIM?>
,H? I@ NB? G;CH >C^=OFNC?M I@ NBL?? >CG?HMCIH;F GI>?FM CM NB? L?JLI>O=NCIH I@
NB? @L?? MOL@;=? 0?P?L;F H?Q N?=BHCKO?M B;P? <??H >?P?FIJ?> NI NLS NI IP?L=IG? NBCM
JLI<F?G
1B? )" G?NBI> L<CNL;LS );AL;HAC;H\"OF?LC;H 89 1B? C>?; CM NI ;MMCAH
; MJ?=CV= P?FI=CNS NI NB? G?MB CH IL>?L NI @IFFIQ NB? GIP?G?HNM I@ NB? >IG;CH
1B? ILCACH;FCNS I@ NB? G?NBI> CM NB;N NBCM P?FI=CNS CM HIN NB? ;=NO;F P?FI=CNS I@
J;LNC=F?M I@ Q;N?L
1B? * G?NBI> *;LE?L H> ?FF 89 OM?M V=NCNCIOM J;LNC=F?M QCNBION G;MM
NI G;LE NB? ?F?G?HN @OFF I@ Q;N?L 1B? JLCH=CJ;F ;>P;HN;A? I@ NBCM G?NBI> CM
NB;N CN B?FJM NI >?;F QCNB <L?;ECHA Q;P?M IL QCNB ;HS WIQ QBIM? @L?? MOL@;=?
CM HIN ; OHCPI=;F @OH=NCIH 1B? CH=IHP?HC?H=? CM NB? NCG? =IMN @IL ;>>CNCIH;F
=;F=OF;NCIH NI @IFFIQ NB? J;LNC=F?M CH ;>>CNCIH NI NB? L?MIFONCIH I@ ?KO;NCIHM
AIP?LHCHA NB? GIP?G?HN I@ NB? WOC>
1B? 3,# G?NBI> 3IFOG? ,@ #FOC>M 89 L?MILNM NI ; KO;HNCNS 9 NB? PIFOG?
I==OJ;NCIH L;NCI QBC=B CH>C=;N?M C@ ; PIFOG? CM @OFF I@ FCKOC> 9 IL ?GJNS
9 1B? ?PIFONCIH I@ 9 CM AIP?LH?> <S ; =IHM?LP;NCIH ?KO;NCIH
9
:?L 9
K J
1B? L?MIFONCIH I@ NBCM ?KO;NCIH L?KOCL?M J?L@?=NFS =IHM?LP;NCP? ;H> GIHINIHC= HO
G?LC=;F M=B?G?M &H CNM ILCACH;F P?LMCIH NB? 3,# G?NBI> IHFS MIFP?M ?KO;NCIHM I@ NB?
FCKOC> JB;M? JI=E?N I@ A;M ?H=FIM?> <S ; <L?;ECHA Q;P? QCFF HIN <? MO<D?=N NI ;HS
?_?=N I@ <OIS;H=S 1B? I@N?H MJ?=N;=OF;L L?MOFNM I@ NBCM G?NBI> F;=E L?;FCMG BIQ?P?L
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
&H GIL? L?=?HN P?LMCIHM L?@?L @IL ?R;GJF? NI 89 NBCM ?LLIL CM =ILL?=N?> <S MIFPCHA
NB? ?KO;NCIHM I@ NB? NQI JB;M?M QBC=B ;L? M?J;L;N?> <S NB? CHN?L@;=? QBIM? ?KO;NCIH
CM 9 0I CN CM H?=?MM;LS NI JF;H ; MJ?=CV= JLI=?MMCHA I@ PIFOG?M =IHN;CHCHA NB?
CHN?L@;=? ?;=B I@ NB?CL HI>?M <?CHA ;NNLC<ON?> JBSMC=;F KO;HNCNC?M @IL ?;=B I@ NB? NQI
JB;M?M C@ H?=?MM;LS <S ; MCGJF? ?RNL;JIF;NCIH NI ;PIC> <;MCHA NB? HOG?LC=;F M=B?G?
IH HIH >?VH?> P;FO?M HINB?L MIFONCIH JO<FCMB?> <S );@;OLC? ( 89 CH ;H> N;E?H OJ GIL? L?=?HNFS <S *OT;@?LCD; ;H> -?LC= 89 =IHMCMNM I@ =IHMC>?LCHA IHFS
IH? WOC> NB? =B;HA? I@ JB;M? <?CHA IHFS ; >CM=IHNCHOCNS I@ >?HMCNS
HINB?L G?NBI> 0-% 0GIINB?> -;LNC=F?M %S>LI>SH;GC=M Q;M VLMN CHNLI>O=?>
CH ;MNLIJBSMC=M 89 NB?H ;JJFC?> NI @L?? MOL@;=? WIQM 89 1BCM N?=BHCKO? JLIP?M NI
<? GIL? JL?=CM? ;H> GIL? JLIGCMCHA NB;H NB? * ;H> 3,# G?NBI>M ;H> >?M?LP?M
GIL? ?RJF;H;NCIHM &N CM ; =IGJF?N?FS );AL;HAC;H ;JJLI;=B QCNBION ; G?MB QCNB
J;LNC=F?M GIPCHA >O? NI NB? CHWO?H=? I@ JL?MMOL? @IL=?M PCM=IMCNS ;H> AL;PCNS ";=B
J;LNC=F? JIMM?MM?M >?HMCNS JL?MMOL? ;H> P?FI=CNS 1B? ILCACH;FCNS I@ 0-% FC?M CH NB?
Q;S CH QBC=B ; =IHNCHOIOM >?VHCNCIH I@ NB? V?F>M CM ACP?H IH NB? <;MCM I@ HI>;F P;FO?M
I@ J;LNC=F?M &HN?LJIF;NCIH CM CHMJCL?> <S NB? ?KO;FCNS
.
H H H H :
QB?L? CM NB? !CL;= >CMNLC<ONCIH 4CNB 0-% Q? QLCN?
.
H M6 H H :
H + >
QB?L? M6 CM ;H ;JJLIRCG;NCIH I@ ; MJB?LC=;F E?LH?F B;PCHA ; =IGJ;=N MOJJILN QCNB
; >C;G?N?L > > <?CHA =;FF?> NB? ;NN?HO;NCIH F?HANB 1B? E?LH?F >?J?H>M IHFS IH
NB? >CMN;H=? H NI CNM =?HNL? %?L? CM ;H ?R;GJF? <;M?> IH ; MJFCH? I@ IL>?L CH NQI
>CG?HMCIHM
4? >?HIN? G H>
G
G C@ G #
"
M6 G !
>
G
C@
G
,H=? NB? E?LH?F M6 CM >?VH?> "KO;NCIH CM L?QLCNN?H ;M ; MOG IP?L ;FF NB?
J;LNC=F?M
H , C0
0
0
0 M6 H H0
QB?L? C0 CM NB? G;MM I@ NB? J;LNC=F? 8 0 CNM >?HMCNS C0 0 CM NB? PIFOG? I@ NB?
J;LNC=F? ;H> L?JF;=?M NB? PIFOG? ?F?G?HN : CH @ILGOF; 1B? CHN?L?MN CH NBCM @ILG
I@ CHN?LJIF;NCIH CM NB;N NB? =;F=OF;NCIH I@ >?LCP;NCP?M CM MCGJF? 1BOM NB? AL;>C?HN I@
; V?F> CM MCGJFS
H , C0
0
0
0 M6 H H0
MMOGCHA NB? WIQ NI <? =IGJL?MMC<F? NB? ?KO;NCIHM >?M=LC<CHA NB? GIP?G?HN I@
J;LNC=F?M ;L?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
G;MM =IHM?LP;NCIH ?KO;NCIH QLCNN?H ;M
: 1 =H7: :?L 1
:J
GIG?HNOG ?KO;NCIH ;M
:1
=H7: F
:J
1
" 3FB
H ?KO;NCIH I@ MN;N?
F
9
9 CM NB? MJ??> I@ MIOH> CH NB? WOC> ;H> NB? L;NCI I@ MJ?=CV= B?;NM ;N =IHMN;HN
JL?MMOL? ;H> =IHMN;HN PIFOG? 1B? G?NBI> CM MNLC=NFS ;JJFC?> NI ; =IGJL?MMC<F? WOC>
*IL?IP?L NB? G;MM =IHM?LP;NCIH ?KO;NCIH IHFS =IHM?LP?M G;MM HIN PIFOG?
1B? ?KO;NCIHM ;L? L?MIFP?> CH ;H ?RJFC=CN G;HH?L QCNB ; FCGCN;NCIH I@ NB? NCG? MN?J
<S NB? P;FO?
7
>
>
J ! "0
9
QB?L? "0 CM NB? G;RCGOG @IL=? ;JJFC?> NI ; J;LNC=F? &H JL;=NC=? CH IL>?L NI A;CH
MN;<CFCNS NB? P;FO?M ;H> 9 ;>IJN?> ;L? >C_?L?HN @LIG NB? L?;F P;FO?M @IL Q;N?L
ACPCHA ; MFCABNFS GIL? =IGJL?MMC<F? WOC>
HINB?L ILCACH;F @?;NOL? I@ NB? 0-% G?NBI> CM NB? NL?;NG?HN I@ <IOH>;LS =IH>C
NCIHM NB? Q;FFM NB?GM?FP?M <?CHA GI>?FF?> <S J;LNC=F?M NI QBC=B NB? Z)?HH;L> 'IH?M[
NSJ? I@ L?JOFMCIH @IL=?M ;L? ;MMI=C;N?>
&H NB? @ONOL? 0-% =IOF> <?=IG? NB? G?NBI> QBC=B N;E?M IP?L VHCN? ?F?G?HNM @IL
=IGJF?R =;M?M J;LNC=OF;LFS @IL GCR?> WIQ Q;P? <L?;ECHA ;H> ;FF NB? =;M?M QB?L? @L??
MOL@;=? CM HIN OHCPI=;F
@N?L NBCM L;JC> IP?LPC?Q Q? MB;FF MNO>S CH >?N;CF NB? L?MIFONCIH I@ NB? +;PC?L\
0NIE?M ?KO;NCIHM CH VHCN? ?F?G?HNM VLMN QCNB NB? BS>LIMN;NC= BSJINB?MCM <S OMCHA
NB? @L;=NCIH;F MN?JM G?NBI> NB?H QCNB ;H ;FAILCNBG ;>;JN?> NI NB? HIH BS>LIMN;NC=
?KO;NCIHM
@D0C8>=B F8C7 7H3A>BC0C82 7H?>C74B8B
A02C8>=0; BC4?B <4C7>3
1B? <;MCM ;FAILCNBG =IHMCMNM I@ NBL?? @L;=NCIH;F MN?JM 4? QLCN?
< < < < < < < <
<
J
J
1BCM F?;>M NI ; MO==?MMCP? L?MIFONCIH I@ NBL?? MN?JM
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
H ;>P?=NCIH MN?J
< < <
J
>C_OMCIH MN?J
< <
J
;>P?=NCIH N?LGM < 1 2 ' IL 0
>C_OMCIH N?LGM MIOL=? N?LGM < 1 2 ' IL 0
#IL ; NL;=?L 0 NB? MIOL=? N?LGM HIN IHFS ;L? >O? NI MIOL=?M IL MCHEM <ON G;S ;FMI
=IHN;CH ; JLI>O=NCIH IL >?MNLO=NCIH N?LG
JL?MMOL? =IHNCHOCNS MN?J
<<
<
J
JL?MMOL? N?LGM 1BCM F;MN MN?J CH=FO>?M ;FMI NB? L?MIFONCIH I@ NB? =IHNCHOCNS ?KO;NCIH :?L 1 QBC=B B?FJM NI >?>O=? NB? Q;N?L >?JNB > ;H> NB? P?LNC=;F =IGJIH?HN I@ P?FI=CNS 3
";=B I@ NB?M? MN?JM QCFF <? L?MIFP?> CH >?N;CF IH? ;@N?L NB? INB?L -LCIL NI NBCM
BIQ?P?L >CM=L?NCT;NCIH CH MJ;=? ;H> <OCF>CHA OJ NB? G?MB B;P? NI <? MJ?=CV?>
(?0C80; 38B2A4C8I0C8>=
1B? MJ;NC;F >CM=L?NCT;NCIH =BIM?H CM ; VHCN? ?F?G?HN >CM=L?NCT;NCIH OMCHA JLCMGM M??
0?=NCIH -LCMGM B;P? MCR HI>?M 1B?CL KO;>L;HAOF;L MC>?M ;L? P?LNC=;F 1B?
JLCMG B;M <??H L?N;CH?> <?=;OM? CNM NQI >CG?HMCIH;F BILCTIHN;F JLID?=NCIH =IHMNCNON?M
IH? I@ NB? VHCN? ?F?G?HNM NLC;HAF? QCNB FCH?;L CHN?LJIF;NCIH OM?> NI L?MIFP? NB? 0;CHN
3?H;HN ?KO;NCIHM CH NQI >CG?HMCIHM &N CM NBOM JIMMC<F? NI <OCF> ; NBL?? >CG?HMCIH;F
G?MB @LIG ; NQI >CG?HMCIH;F G?MB &N IHFS L?KOCL?M NI G?MB NB? NQI >CG?HMCIH;F
>IG;CH QCNB NLC;HAF?M NB?H NI >OJFC=;N? NBCM G?MB P?LNC=;FFS ;M ME?N=B?> CH #CAOL?
;F=OF;NCIH CH NQI ;H> NBL?? >CG?HMCIHM QCNB NB? M;G? G?MB CM NBOM JIMMC<F?
D8;38=6 C74 C7A4438<4=B8>=0; <4B7
1I <OCF> NB? NBL?? >CG?HMCIH;F >IG;CH QCNB JLCMGM CN MO^=?M NI <OCF> NB? NLC;HAO
F;NCIH I@ NB? NQI >CG?HMCIH;F >IG;CH NB?H NI L?J?;N CN ;FIHA NB? P?LNC=;F CH MOJ?L
CGJIM?> F;S?LM =;FF?> ZJF;H?M[ 1B?M? ZJF;H?M[ ;L? CH @;=N MOL@;=?M QCNB P;LC;<F?
?F?P;NCIHM 1B? IHFS =IH>CNCIH CM NB;N NB? ?F?P;NCIH I@ NB? JICHNM <?FIHACHA NI ; P?L
NC=;F MBIOF> CH=L?;M? @LIG NB? <INNIG NI NB? @L?? MOL@;=? 0?P?L;F MCGJF? @ILGOF;?
AO;L;HN?? NBCM =IH>CNCIH ;M CH NB? @IFFIQCHA @IOL G?NBI>M
"4C7>3 ?;0=4B 4E4=;H B?0243 0;>=6 C74 E4AC820;
1BCM VLMN IJNCIH NB? GIMN =F;MMC= =IHMCMNM I@ MN;LNCHA QCNB NB? =IIL>CH;N? HI>?M
N O I@ NB? ! G?MB NB?H >?VHCHA NB? ?F?P;NCIH @IL ?;=B JICHN I@ NB? ! G?MB QCNB
=IIL>CH;N?M N O P ;==IL>CHA NI NB? @ILGOF;
P N O ?F 64 N O
?F 6A N O 64 N O
DF %6!/,!6+*& 0 ,# #/"" 02/# " #),40
64 ;FQ;SM <?CHA NB? <INNIG ?F?P;NCIH 6A NB? @L?? MOL@;=? ?F?P;NCIH ;H> ?F CM NB? L;HE
I@ NB? JF;H? OH>?L =IHMC>?L;NCIH JF;H?M L;HA? @LIG NI DF C@ Q? AI @LIG NB? <INNIG
NI NB? @L?? MOL@;=? 1BCM CM NB? MCGJF?MN G?NBI> <;M?> IH NB? MCAG; NL;HM@ILG &H
NBCM =;M? NB? P?LNC=;F =IIL>CH;N? P I@ ?;=B JF;H? CH NB? NL;HM@ILG?> G?MB CM MCGJFS
7>
<> "4C7>3 30C0 >5 0 5D=2C8>=
2>=BC0=C 5>A 4027 ?;0=4
7>
<ON ; ACP?H
1B? =IIL>CH;N? P I@ JF;H? ?F CH NB? NL;HM@ILG?> G?MB CM HI FIHA?L <>
HOG<?L 7> MO=B NB;N 7> ;H> @IL CHN?LG?>C;LS JF;H?M 7> 7> 7>
4? HIQ B;P?
P N O ?F 64 N O
7>
6A N O 64 N O
#CAOL? MBIQM ;H ?R;GJF? I@ ; ! G?MB <OCFN QCNB NB? @IFFIQCHA P;FO?M
1B? =BIC=? I@ P;FO?M I@ 7> CM @L?? NB? IHFS
=IHMNL;CHNM <?CHA ; P;FO? I@ @IL NB? <INNIG ?F ; P;FO? I@ @IL NB? @L?? MOL@;=?
?F DF ;H> ; M?LC?M I@ CHN?LG?>C;N? P;FO?M CH ; MNLC=NFS CH=L?;MCHA IL>?L
"4C7>3 3>D1;4 B86<0 CA0=B5>A<
%?L? Q? JL?M=LC<? ; =IHMN;HN ?F?P;NCIH @IL IH? JF;H? NB?H NB? JF;H?M <?NQ??H NB?
<INNIG ;H> NBCM JF;H? ;L? ?P?HFS MJ;=?> ;H> NB? M;G? CM >IH? <?NQ??H NBCM JF;H?
;H> NB? @L?? MOL@;=?
Free surface
0
-2
-4
-5
-6
Bottom
#CAOL? G?MB <OCFN QCNB ; =IHMN;HN JLIJILNCIH;FCNS =I?^=C?HN J?L JF;H?
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
"4C7>3 7>A8I>=C0; ?;0=4B
&H IL>?L NI L?JL?M?HN NB? N?GJ?L;NOL? MNL;NCV=;NCIH TIH?M NB?LGI=FCH?M <?NN?L CN
CM MIG?NCG?M QILNB CGJIMCHA NB;N ; G;RCGOG HOG<?L I@ CHN?LG?>C;N? ZJF;H?M[ <?
BILCTIHN;F &H NBCM =;M? CN CM JIMMC<F? NI CH>C=;N? NB? >?MCL?> ?F?P;NCIH @IL ?;=B JF;H?
QBC=B QCFF <? >?HIN?> ;M P7> 1BCM ?F?P;NCIH CM HIN ;FQ;SM ;==?MMC<F? <?=;OM? CN =IOF>
<? MCNO;N?> <?FIQ NB? <INNIG IL ;<IP? NB? @L?? MOL@;=? &H NBCM =;M? NB? ?F?P;NCIH
I@ NB? JICHN I@ NB? JF;H? ?F QCFF <? G;CHN;CH?> MFCABNFS ;<IP? NB? <INNIG IL MFCABNFS
<?FIQ NB? MOL@;=? 1BCM =;H <? ?R?=ON?> QCNB NB? B?FJ I@ ; Z*CH*I>[ FCGCN?L <S NB?
@ILGOF;
P
!
' 64 N O
?F DF ?F
: P7> 6A N O J :
DF DF QB?L? :;7< CM ;H CGJIM?> GCHCGOG >CMN;H=? 4? CGJIM? NB;N JF;H? NB? <INNIG
CM ;N ; GCHCGOG >CMN;H=? : @LIG NB? @L?? MOL@;=? ;H> NB;N NB? JF;H? DF NB? @L??
MOL@;=? CM ;N ; GCHCGOG >CMN;H=? :;7< @LIG NB? <INNIG
#CAOL? MBIQM ;H ?R;GJF? I@ ; ! G?MB <OCFN QCNB NBCM G?NBI> ;H> NB? @IFFIQCHA
?F?P;NCIHM P7> P P P P P Free surface
0
-2
-4
-5
-6
Bottom
#CAOL? G?MB G;>? I@ BILCTIHN;F JF;H?M
&H =IH@ILGCNS QCNB NB? ?;LFC?L @ILGOF; QB?H NB? <INNIG LCM?M NI NB? F?P?F I@ JF;H?
HOG<?L HILG;FFS MCNO;N?> ;N ?F?P;NCIH G NBCM JF;H? HOG<?L NB?H ;>IJNM NB?
@ILG I@ NB? <INNIG ;N ; >CMN;H=? :;7< 1B? ;>P;HN;A? I@ NBCM >CMNLC<ONCIH CM NB? ?FCGCH;NCIH I@ NLOH=;NCIH ?LLILM CH NB?
=;F=OF;NCIH I@ <OIS;H=S N?LGM 1B?M? ?LLILM =;H A?H?L;N? OH;==?JN;<F? CHMN;<CFCNS
QB?H NB?M? N?LGM ;L? >IGCH;HN =NO;FFS QB?H NB? JF;H?M ;L? HIN BILCTIHN;F CN
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
CM CGJIMMC<F? NI L?JL?M?HN =ILL?=NFS ; @OH=NCIH < NB;N IHFS >?J?H>M IH P +IQ NB?
>?HMCNS I@ NB? WOC> MBIOF> N?H> NI NBCM NSJ? I@ @OH=NCIH =ILL?MJIH>CHA NI ; JIMCNCIH
I@ ?KOCFC<LCOG QCNBION BILCTIHN;F JL?MMOL? AL;>C?HNM
4CNB NBCM G?NBI> HIH BILCTIHN;F G?MB JF;H?M ;L? CH?PCN;<F? <ON IHFS ;N ; >CM
N;H=? F?MM NB;H :;7< @LIG NB? <INNIG IL @LIG NB? MOL@;=? &H NB?M? TIH?M QCNB
KO;MC T?LI PIFOG? CN CM MOAA?MN?> NB;N NB? @IL=?M ;LCMCHA @LIG NB? J;L;MCNC= BILCTIHN;F
JL?MMOL? AL;>C?HNM =;FF?> BS>LIMN;NC= CH=IHMCMN?H=C?M <? =;H=?FF?> <S NB? CHN?LG?
>C;LS I@ ; VFN?L QBC=B ;=NM CH JF;=?M QB?L? NB? JLCMGM ;L? MO=B NB;N ; HI>? I@ NB?
FIQ?L <;M? HOG<?L IL B;M ;H ?F?P;NCIH BCAB?L NB;H ; HI>? I@ NB? OJJ?L <;M?
HOG<?L IL 1BCM VFN?L ;FMI =;H=?FM NB? >C_?L?HN >C_OMCIH =I?^=C?HNM IH NB?M?
?F?G?HNM
(86<0 CA0=B5>A< 0=3 64=4A0;8I43 B86<0 CA0=B5>A<
M NB? @L?? MOL@;=? ?PIFP?M QCNB NCG? NB? ?F?P;NCIHM P I@ NB? ! G?MB P;LS @LIG
IH? NCG? MN?J NI ;HINB?L 1B? P;LC;<F?M < <
< < ;H> < < < K L IL 0 ;L?
?;=B >?VH?> IH >C_?L?HN G?MB?M %IQ?P?L CN CM MNCFF JIMMC<F? NI ?R?=ON? ; =B;HA? I@
P;LC;<F?M MI ;M NI VR NB? G?MB >OLCHA ; NCG? MN?J M Q? B;P? ;FL?;>S M??H CH 0?=NCIH
NBCM =;H <? J;LNC=OF;LFS CHN?L?MNCHA CH NB? ;>P?=NCIH MN?J QBC=B B;M ; >?VHCN?
CHWO?H=? IH NB? ?PIFONCIH I@ NB? @L?? MOL@;=? 1B? =B;HA? I@ P;LC;<F?M =OLL?HNFS
;>IJN?> CM NB? MCAG; NL;HM@ILG QBC=B =IHMCMNM I@ J;MMCHA @LIG NB? =IIL>CH;N? P NI NB?
=IIL>CH;N? P CH>?J?H>?HN I@ NCG?
#IL NB? =F;MMC=;F MCAG; NL;HM@ILG Q? L?=;FF NB? @ILGOF; OM?>
P P 64
6A 64
4B?H ; JF;H? B;M ; JL?M=LC<?> =IHMN;HN ?F?P;NCIH ; >IO<F? MCAG; NL;HM@ILG CM
JL?@?LL?> QCNB ; @OH=NCIH P P;LSCHA <?NQ??H ;H> &@ 6>@3A1@7032 CM NB? L?KOCL?>
=IHMN;HN ?F?P;NCIH Q? B;P? <?FIQ NB? VR?> JF;H?
P P 6>@3A1@7032
6>@3A1@7032 64
P P 6>@3A1@7032
6A 6>@3A1@7032
;H> ;<IP?
#IL ; A?H?L;FCT?> MCAG; NL;HM@ILG CN CM =IHP?HC?HN NI OM? NB? @IFFIQCHA @OH=NCIH
P <?NQ??H JF;H?M ?F ;H> ?F P ?F P P7>
P7> P7>
QB?L? P7> CM NB? ?F?P;NCIH I@ JF;H? ?F ;H> P7> NB? ?F?P;NCIH I@ JF;H? ?F 1BCM
@OH=NCIH P HIQ P;LC?M <?NQ??H ;H> DF 1B?M? @ILGM I@ NL;HM@ILG;NCIH ;L? H?=?MM;LS QB?H >?;FCHA QCNB NB? IP?L;FF G?MB
? A >OLCHA ;H ;>P?=NCIH MN?J QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M 1B? >?VHCNCIH I@ P /"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
CM NBOM OHCPI=;F IP?L NB? ?HNCL? P?LNC=;F #IL FI=;F =IGJON;NCIHM QCNBCH ; MCHAF? VHCN?
?F?G?HN @IL ?R;GJF? ; FI=;F >?VHCNCIH QCFF <? MCGJF?L ;H> Q? G;S QLCN? NB;N
P P P7>
P7> P7>
C ? Q? B;P? ; MCGJF? MCAG; NL;HM@ILG <?NQ??H NQI JF;H?M
&N CM CHN?L?MNCHA NI HINC=? NB;N NB? @IOL G?NBI>M ACP?H ;<IP? MJFCN CHNI NQI >C_?L?HN
H
@;GCFC?M 1B? VLMN NQI ;L? MO=B NB;N H
> NB?S NBOM L?FS IH NB? MCAG; NL;HM@ILG
#IL NB? F;MN NQI QBC=B =ILL?MJIH> NI A?H?L;FCT?> MCAG; NL;HM@ILGM
P
P7>
P P7> P
C ? NB? QC>NB <?NQ??H NQI JF;H?M QBC=B Q? >?HIN? P &N CM IHFS JC?=?QCM? =IHMN;HN
P?LNC=;FFS
3E42C8>= BC4?
1B? L?MIFONCIH I@ NB? ;>P?=NCIH ?KO;NCIHM CM >?M=LC<?> CH B;JN?L 1BCM MN?J CM
?R?=ON?> CH ; VR?> NL;HM@ILG?> G?MB <S NB? MCAG; NL;HM@ILG QBC=B G;E?M CN H?
=?MM;LS NI =;F=OF;N? ; P?LNC=;F P?FI=CNS ;N ?;=B NCG? MN?J
3 :P P :J
J
1
P N
2
P O
3
P P
IH NB? <;MCM I@ L?MOFNM I@ NB? ?;LFC?L NCG? MN?J
2M? I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M IH NB? VR?> G?MB ;LCMCHA @LIG NB? MCAG;
NL;HM@ILG CM HIN ; JLI<F?G <ON CN G;S <? QCNB INB?L N?=BHCKO?M &H IL>?L NI MBIQ NB?
=IGJFC=;NCIHM ;LCMCHA @LIG NB? MCAG; NL;HM@ILG B?L? CM ;H ?R;GJF? I@ ;>P?=NCIH QCNB
NB? 02-$ G?NBI> >?M=LC<?> CH B;JN?L 1BCM G?NBI> =IHMCMNM I@ OJQCH>CHA NB?
<;MCM @OH=NCIHM CH NB? >CL?=NCIH I@ NB? WIQ
7 7
J Q 1 =H7: 7
0I NB? H?Q <;MCM @OH=NCIHM 7 L?G;CH FCH?;L <ON <?=IG? >CM=IHNCHOIOM
P;LC;NCIH;F @ILGOF;NCIH CM ;JJFC?> NI NB? @IFFIQCHA ?KO;NCIH
<
1 =H7:
< <<
J
<< 1B?
QB?L? CM NB? L?F;R;NCIH =I?^=C?HN 1 NB? P?FI=CNS CH NB? VR?> G?MB ;H> =H7:
NB? AL;>C?HN IJ?L;NIL CH NB? M;G? G?MB 1B? ?;LFC?L ?KO;NCIH CM GOFNCJFC?> <S NB?
>?@ILG?> <;MCM @OH=NCIH
7 7
J 1 =H7: 7
;H> <S OMCHA NB? "CHMN?CH HIN;NCIH < <8 8 Q? A?N
<8 8 7
J
7 1 =H7: <8 8
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
<8< 8 7
7 1 =H7: <8< 8
J
0NLC=NFS MJ?;ECHA NB? >?@ILG?> <;MCM @OH=NCIH MBIOF> <? OM?> @IL ;FF NB? N?LGM CH
NB? ?KO;NCIH &H JL;=NC=? IHFS NB? ;>P?=NCIH N?LG CM NL;HM@ILG?> QBC=B >I?M HIN
;FN?L NB? =IHM?LP;NCIH JLIJ?LNS @N?L >CPC>CHA <S Q? A?N
<8
<8
<8<
8 7
J
7 1 =H7: 8
J Q 1 =H7: 7 1 =H7: 8
7 1 =H7: 8
<8<
8 7
J
J Q 1 =H7: 7 1 =H7: 8
S GOFNCJFSCHA <S > ;H> <S CHN?AL;NCHA IH NB? NIN;F VR?> >IG;CH Q? VH;FFS A?N
<8
C78
J
L=H78
K=K=78 <8<
C78
J
L=H78
K=K=78
QCNB
C78 .
>7 8 : .
7 8 :
) QBIM? A?H?L;F N?LG CM C78 CM NB? G;MM G;NLCR &N G;S <? ?MNCG;N?> ?CNB?L CH
NB? L?;F G?MB IL CH NB? VR?> G?MB
.
L=H78 >7 QK =H7: 8 :
2 #. QBIM? A?H?L;F N?LG CM L=H78 CM NB? G;NLCR I<N;CH?> @IL =F;MMC= ;>P?=NCIH <S
VHCN? ?F?G?HNM ?RJL?MM?> CH VR?> G?MB
.
J
Q K=K=78 >1
=H7: 7 1 =H7: 8 :
1 #1 # QBIM? A?H?L;F N?LG CM K=K=78 CM NB? ;>>CNCIH;F G;NLCR @LIG 02-$ 4?
MBIQ 89 NB;N K=K=78 CM ?KOCP;F?HN NI ;H ;LNCV=C;F >C_OMCIH QBC=B CH ; >CG?HMCIH
AL?;N?L NB;H IH? CM IHFS JLI>O=?> CH NB? >CL?=NCIH I@ NB? =OLL?HN &H ! NB? =;F=O
F;NCIH I@ NB? ?;LFC?L G;NLC=?M CM L?F;NCP?FS MBILN @IL NLC;HAOF;L ?F?G?HNM #IL JLCMGM
QCNB ; NLC;HAOF;L <;M? NB? HOG<?L I@ IJ?L;NCIHM H?=?MM;LS NI @ILG NB? G;NLC=?M NB?H
=F?;LFS CH=L?;M?M &H IL>?L NI MCGJFC@S NB? =;F=OF;NCIH I@ NB?CL =I?^=C?HNM ;H> I<
Q
M?LPCHA NB;N CH NB?ILS CN >I?M HIN B;LG NB? =IHM?LP;NCIH JLIJ?LNS Q? =;H FCGCN 1
CH @ILGOF; NI NB? BILCTIHN;F =IGJIH?HNM I@ P?FI=CNS &H NB? =;M? I@ ?MM?HNC;FFS
BILCTIHN;F WIQ NBCM ;JJLIRCG;NCIH CM HIN ; JLI<F?G ,HFS NB? A?H?LC= =I?^=C?HN I@
NB? 1 #1 # G;NLCR CM GI>CV?> CH NBCM Q;S &N <?=IG?M
.
J
K=K=78 > 1 6 =H7: 7 1 =H7: 8 :
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
QCNB 1 6 1 2
4B?H NBCM ;JJLIRCG;NCIH CM G;>? NB? G;NLCR 1 #1 # CM HI
FIHA?L MSGG?NLC=
4B?H ;>P?=NCIH I@ ; P;LC;<F? CM >IH? QCNB NB? B?FJ I@ NBCM G?NBI> NB? MN?JM I@
;>P?=NCIH ;H> >C_OMCIH ;L? CH @;=N NL?;N?> ;N NB? M;G? NCG?
8QDB8>= BC4?
#IL NB? =IH>CNCIHM I@ ?RCMN?H=? ;H> OHCKO?H?MM I@ NB? MIFONCIH @IL NBCM NSJ? I@ J;L;<IFC=
JLI<F?G NB? L?;>?L =;H L?@?L NI /;PC;LN ;H> 1BIG;M 89 IL NI NB? <IIE <S -CLIHH?;O
89 CH #L?H=B
$CP?H NB? =IGJF?RCNS I@ ; >C_OMCIH MN;A? QCNB NB? MCAG; NL;HM@ILG ;H> NB? HO
G?LC=;F ;LN?@;=NM QBC=B ;L? CNM =IHM?KO?H=? NBCM MN;A? CM ?R?=ON?> CH NB? L?;F G?MB
%IQ?P?L NB? ';=I<C;H I@ NB? CMIJ;L;G?NLC= NL;HM@ILG;NCIH @IL ;HS JLCMG CM HIN ; =IH
MN;HN +IQ NBCM ';=I<C;H ;JJ?;LM CH NB? >?HIGCH;NIL CH NB? =;F=OF;NCIH I@ ; >C_OMCIH
G;NLCR NB?L?@IL? Q? =;HHIN A?N ;H ?R;=N JIFSHIGC;F ?RJL?MMCIH 1B? H?=?MM;LS ;J
JLIRCG;NCIHM NB?GM?FP?M CH>O=? >CMNOL<CHA HOG?LC=;F ?LLILM 1BCM JLI<F?G >I?M HIN
?RCMN QCNB N?NL;B?>L; ;H> ; FCH?;L CHN?LJIF;NCIH NB? ';=I<C;H NB?H <?CHA ; =IHMN;HN
+IQ ; JLCMG =;H <? <LIE?H >IQH CHNI NBL?? N?NL;B?>L; M?? #CAOL? QB?L? NB?
N?NL;B?>L; ;L? @ILG?> <S NB? M?NM I@ @IOL JICHNM ;H> &N
MO^=?M NI =IHMC>?L NB;N CHN?LJIF;NCIH IH NB? JLCMG CM FCH?;L <S JC?=?M ?R;=NFS ;M CH
NB? KO;MC <O<<F? NLC;HAF? 4? =BIIM? NI ?R?=ON? NB? >C_OMCIH MN;A? QCNB ; H?Q NSJ? I@
?F?G?HN M HI >?AL?? I@ @L??>IG CM ;>>?> NB? NIJIFIAS I@ ; JLCMG;NC= ?F?G?HN G;NLCR
=;H <? @ILG;FFS L?N;CH?> @N?L =;F=OF;NCHA NB? ?F?G?HN G;NLC=?M IH NB? N?NL;B?>L; ;
JL? ;MM?G<FS L?MNIL?M NB? ?F?G?HN G;NLC=?M JLCMG <S JLCMG 4LCNCHA NBCM JL? ;MM?G<FS
MBIQM NB;N ;FF NB? JLIJ?LNC?M I@ ; >C_OMCIH G;NLCR MOG I@ NB? N?LGM I@ ?;=B FCH? ?KO;F
NI T?LI MOG I@ NB? N?LGM I@ ?;=B =IFOGH ?KO;F NI T?LI ;L? ;=NO;FFS NL;HM@?LL?> @LIG
NB? N?NL;B?>L; NI NB? JLCMGM HINB?L JIMMC<CFCNS CM NI VH> ;H ;JJLIRCG;NCIH I@ NB?
>C_OMCIH G;NLCR CH JLCMGM ? A <S =IHMC>?LCHA ; =IHMN;HN ;P?L;A? ';=I<C;H IH ?P?LS
JLCMG NI A?N ; JIFSHIGC;F ?RJL?MMCIH M?? 0?=NCIH D>H0=2H C4A<B
1B? MIOL=? N?LGM I@ <OIS;H=S ;L? IH? I@ NB? MNL;N?AC= JICHNM I@ ! ?KO;NCIHM 1B?CL
=ILL?=N NL?;NG?HN =IH>CNCIHM NB? L?JL?M?HN;NCIH I@ MNL;NCV=;NCIHM ; =OLL?HN MNOG<FCHA
<FI=E I@ GI>?FFCHA 1B?M? MIOL=? N?LGM ;L? C@ Q? >I HIN =IHMC>?L ;NGIMJB?LC= JL?MMOL?
M?? 0?=NCIH .
.
6A
"F =
=
:P
=
N GB
N
=
H
GHB
"G =
=
6A
O
FB
=
O
.
H
.
:P
=
FHB
1B? N?LGM "F ;H> "G ;L? =;F=OF;N?> IH NB? G?MB ;N NCG? J< IH NB? <;MCM I@
>C_?L?H=?M CH >?HMCNS 1B?M? >C_?L?H=?M CH >?HMCNS NB?GM?FP?M B;P? <??H ?P;FO;N?>
?;LFC?L QCNB NB? B?FJ I@ ; MN;N? @OH=NCIH CH QBC=B NB? >C_?L?HN =IH=?HNL;NCIHM I@ ;=NCP?
NL;=?LM ;JJ?;L
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
6
4
5
3
1
2
#CAOL? >?=IGJIMCNCIH I@ ; JLCMG CHNI NBL?? N?NL;B?>L;
CDP?F>M 89 =IGJ;L?M NB? NL?;NG?HN I@ NB?M? N?LGM CH MCAG; NL;HM@ILG @ILGOF;
NCIH CH >C_?L?HN MI@NQ;L? $CP?H <?FIQ ;L? NQI GI>?M I@ =;F=OF;NCIH FCGCN?> NI NB?
N?LG "F NB? VLMN CH NB? L?;F G?MB ;H> NB? M?=IH> CH NB? NL;HM@ILG?> G?MB
0;2D;0C8>= 8= A40; <4B7 NB? M?=IH> N?LG I@ "F =;HHIN <? >CL?=NFS ?RJL?MM?>
CH NB? @ILG I@ P?=NILM CH VHCN? ?F?G?HNM &N B;M NI <? NL;HM@ILG?> QCNB NB? )?C<HCNT
@ILGOF; NI I<N;CH
=
N
.
.
H
:P
=
GHB
=
.
.
H
N
:P
=
=
:P
6I 6A
=
N
4? VH;FFS A?N
"F =
.
.
H
N
=
6A
N
P
6I
=
=
1BCM @ILGOF; CM MCGJFS =;F=OF;N?> <S OMCHA P?LNC=;F FCH?M I@ NB? G?MB @IL NB? CHN?AL;F
;FIHA P ;H> NI VH> NB? P;FO? I@ 6I ;<IP? IH? JICHN I@ ?F?P;NCIH P
0;2D;0C8>= 8= CA0=B5>A<43 <4B7
"F =
=
6A
N
GB
=
N
.
H
CH NB? NL;HM@ILG?> G?MB Q? B;P?
.
:P
=
GH B
P
N
GH B
=
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
1B? P;LC;<F? PI P 6A CM CHNLI>O=?> "F CM NB?H MCGJFCV?> ;M
.
PI
:P
"F =
N
N GH B =
HA =
GH B
N NBCM MN;A? I@ L?;MIHCHA CN CM QILNB JOLMOCHA NB? =;F=OF;NCIH I@ "F ;N NB? >CM=L?N?
F?P?F <?=;OM? NB? ?RJL?MMCIH @ILGOF;N?> ;<IP? =;H <? @OLNB?L MCGJFCV?> 1B? P;LC;NCIH
I@ NBCM N?LG QCFF <? =;F=OF;N?> <?NQ??H NQI HI>?M I@ NB? G?MB >?HIN?> ;M ZCH@[ ;H>
ZMOJ[ IH? FI=;N?> DOMN ;<IP? NB? INB?L QCNB NB? EHIQF?>A? NB;N IH NB? MOL@;=? "F
CM T?LI
.
HA :P
"F "F! = N
=
HA
GH B
=
PI
N
GH B
=
PI!
N
GH B
!
=
4CNB FCH?;L @OH=NCIHM ;FIHA P NBCM ACP?M
!
!
"F "F! =
PI PI
N =
=
!
PI
PI!
=
N
N
=
GH B =
GH B
;H> Q? VH;FFS A?N
"F "F! =
!
=
=
= !
PI PI
N
!
PI
N
!
=
=
PI
GH B
GH B
1B? NQI >?LCP;NCP?M ;FIHA N ;L? >CM=IHNCHOIOM @OH=NCIHM ;N NB? HI>?M I@ NB? G?MB
;H> ;L? =;F=OF;N?> ;M @IFFIQM
#IL ?P?LS HI>? ) Q? =;H >?VH? NB? M?N !(!) ) I@ ?F?G?HNM I@ NB? G?MB
B;PCHA NB? HI>? ) ;GIHA NB?CL P?LNC=?M
#IL ?;=B I@ NB? ?F?G?HNM ? I@ !(!) ) Q? =;H =;F=OF;N? NB?
I@ < ;N ) IH NB? MC>? I@ NB? ?F?G?HN ?
4
F
7
>?LCP;NCP?
1B? >?LCP;NCP? I@ < ;N ) CM NB?H ?MNCG;N?> <S
<
N
) )
+
7 !" " +
4
F
" :
QB?L? " CM NB? <;MCM @OH=NCIH ;MMI=C;N?> NI HI>? )
7
" :
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
A82C8>= C4A<B
IGJ;NC<CFCNS <?NQ??H @LC=NCIH N?LGM CH ! ;H> ! B;M ;FL?;>S <??H >CM=OMM?> CH
0?=NCIH BIQ?P?L NBCM CM IHFS P;FC> @IL @ILGOF;? L?FSCHA IH ; >?JNB CHN?AL;N?>
P?FI=CNS MO=B ;M NBIM? I@ 0NLC=EF?L IL BfTS &@ Q? Q;HN NI OM? ; FI=;F LIOABH?MM @;=NIL
;H> ;MMOG? ; FIA;LCNBGC= JLIVF? H?;L NB? <INNIG Q? ;L? L;NB?L CH NB? =;M? I@ 0?=NCIH
;H> NB? @LC=NCIH P?FI=CNS 1 B;M NI <? ?P;FO;N?> 4? =;H OM? NB? @;=N NB;N NB?
G?MB I@ JLCMGM CM G;>? I@ JCF?> OJ G?MB?M I@ NLC;HAF?M ;H> =IHMC>?L NB? VLMN JF;H?
;<IP? NB? <INNIG &@ Q? ;MMOG? NB;N NBCM VLMN JF;H? CM MNCFF CH NB? FIA;LCNBGC= JLIVF?
Q? =;H QLCN? NB? ?R;GJF? I@ ; BS>L;OFC=;FFS LIOAB WIQ
1 1>:/<3
! H
9
QB?L? 1>:/<3 CM NB? P?FI=CNS ;N NB? VLMN JICHN ;<IP? NB? <INNIG ;H> P NB? ;FNCNO>?
I@ NBCM JICHN ;<IP? NB? <INNIG
$C74A B>DA24 C4A<B
1B? INB?L MIOL=? N?LGM CH=FO>?> CH NB? N?LGM "F ;H> "G ;L? ;FMI NL?;N?> CH NB?
>C_OMCIH MN;A?
%A4BBDA4O2>=C8=D8CH BC4?
MMI=C;N?> QCNB NB? BSJINB?MCM I@ ; BS>LIMN;NC= JL?MMOL? CHN?AL;NCIH ;FIHA NB? P?LNC=;F
I@ NB? =IHNCHOCNS ;H> GIG?HNOG ?KO;NCIHM @IL 1 ;H> 2 B?FJM NI I<N;CH ?KO;NCIHM
MCGCF;L NI NB? 0;CHN 3?H;HN ?KO;NCIHM &HN?AL;NCIH I@ NB? GIG?HNOG ?KO;NCIHM =IH
=?LHCHA 1 ;H> 2 ;M NB?S ;L? QLCNN?H N;E?M JF;=? ;@N?L NL?;NCHA ;>P?=NCIH ;H> >C_OMCIH
0I ;>P?=NCIH IL >C_OMCIH IL ?P?H @LC=NCIH N?LGM ;N NB? <INNIG >I HIN ;JJ?;L CH NB?
CHN?AL;N?> ?KO;NCIHM S L?MIFPCHA NB?M? ?KO;NCIHM <?NQ??H NB? CHMN;HNM J< ;H> J<
Q? I<N;CH NB? @L?? MOL@;=? ;H> NB? ;P?L;A? BILCTIHN;F P?FI=CNC?M ;N NB? CHMN;HN J<
#LIG NBCM Q? >?>O=? NB? G?MB ;N J<
NB? P;LC;<F?M 1 ;H> 2 NB?H 3 <S L?MIFPCHA
NB? =IHNCHOCNS ?KO;NCIH
'4B>;DC8>= >5 4@D0C8>=B 0E4A0643 >E4A C74 E4AC820;
@N?L NB? ;>P?=NCIH ;H> >C_OMCIH MN?JM NB? =IHNCHOCNS ;H> GIG?HNOG ?KO;NCIHM
L?F;N?> NI 1 ;H> 2 ;L? QLCNN?H ;M
1
N
2
O
3
P
1<
1
F6
J
N
2<
2
F6
J
O
QB?L? F6 CM NB? BS>LIMN;NC= JL?MMOL?
S CHN?AL;NCIH ;FIHA NB? P?LNC=;F
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
< N O J >
.
.
< N O P J :P
.
Q? I<N;CH ; MSMN?G =ILL?MJIH>CHA NI NB? 0;CHN 3?H;HN ?KO;NCIHM QCNBION ;>P?=NCIH
IL >C_OMCIH
C6
D6
6
$
$
B
F
G $
$
$
"
C C
= .
B
F
$
$
$
$
$
D D = .
B
G
1B? N?LGM ;P?L;A?> IP?L NB? P?LNC=;F ;L? =;F=OF;N?> IH NB? <;MCM I@ NB? ! V?F> <
<S NB? NL;J?TIC>;F LOF? 1B;N CM
<
<>
,
6A 64 7>
<7>
<7>
P7>
P7>
QB?L? DF CM NB? HOG<?L I@ JF;H?M &H NB? =;M? I@ ; =F;MMC=;F MCAG; NL;HM@ILG NBCM =;H
<? MCGJFCV?> NI
<
<>
,
7>
<7>
<7>
P7>
P7>
1B? 0;CHN 3?H;HN NSJ? I@ ?KO;NCIHM QBC=B B;P? DOMN <??H I<N;CH?> ;L? L?MIFP?>
<S NB? G?NBI>M ;FL?;>S ACP?H CH B;JN?L @N?L NB? L?MIFONCIH Q? B;P?
><
K<
;H> L<
1B? H?Q G?MB ;N NCG? J< CM <;M?> IH NB? H?Q >?JNB QBC=B ACP?M NB? H?Q @L??
MOL@;=? 4B;N L?G;CHM CM NB? ?P;FO;NCIH I@ NB? =IGJIH?HNM I@ ! P?FI=CNS H;G?FS 1
2 ;H> 3 ;N CHMN;HN J<
0;2D;0C8>= >5 1 0=3 2
1B? L?G;CHCHA J;LN I@ NB? +;PC?L\0NIE?M ?KO;NCIHM NI <? MIFP?> @IL NB? =;F=OF;NCIH I@
1 ;H> 2 CM
)
)
$
= .
"
B
F
$ * *
.
= G
B
1BCM MSMN?G =IG<CH?> QCNB NB? MSMN?G CHN?AL;N?> ;FIHA NB? P?LNC=;F <?=IG?M
) $
" ) B) C B
$ * *
D * B
B
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? =;F=OF;NCIH I@ 1 <
I@ P?=NILM
/?G;LEM
;H> 2 <
B?H=?@ILNB CM L?>O=?> NI ; MCGJF? =IG<CH;NCIH
1B? CGJ?LG?;<CFCNS =IH>CNCIH IH NB? MIFC> Q;FFM CM N;E?H CHNI =IHMC>?L;NCIH CH
NB? L?MIFONCIH I@ NB? 0;CHN 3?H;HN ?KO;NCIHM &N CM ;A;CH N;E?H CHNI ;==IOHN
+. CH NB? =;F=OF;NCIH I@ P?FI=CNC?M 1 < ;H> 2 < QB?L? 6 . 1 QD :P QKQD CM
CGJIM?> NB;N CM NB? =IH>CNCIH CGJIM?> IH ! P?FI=CNC?M CM =IHMCMN?HN QCNB NB?
IH? CGJIM?> IH ! P?FI=CNC?M 4? ;MMOG? NB;N NB? CGJ?LG?;<CFCNS =IH>CNCIH CM
;FL?;>S P?LCV?> @IL QKQD ;H> Q? FCGCN IOLM?FP?M NI JLID?=N 1 QK IHNI NB? N;HA?HN
;N NB? Q;FF
1B? MSMN?G =;H <? L?MIFP?> <S MN;LNCHA HIN @LIG NB? L?MOFN I@ >C_OMCIH <ON
@LIG NB? P;FO?M I@ NCG? N< ;P?L;A?> ;FIHA NB? P?LNC=;F 1B? >C_OMCIH ;>P?=NCIH
;H> MIOL=? N?LGM MBIOF> NB?H <? ;>>?> ;FF ;P?L;A?> ;FIHA NB? P?LNC=;F 4?
I<N;CH NB? =IGJF?N? M?N I@ 0;CHN 3?H;HN ?KO;NCIHM B?H=? GIL? JBSMC=;F &H
HILG;F WIQ QB?L? NB? ?_?=N I@ NB? @L?? MOL@;=? MFIJ? CM <;F;H=?> <S @LC=NCIH
NB?M? NQI N?LGM ;L? <;F;H=?> CN CM NBOM CGJILN;HN NI NL?;N NB?G CH NB? M;G?
@L;=NCIH;F MN?J
><?DC8=6 C74 E4AC820; E4;>28CH
1B? P?LNC=;F P?FI=CNS 3 CM =;F=OF;N?> IH NB? <;MCM I@ NB? G;MM =IHM?LP;NCIH ?KO;NCIH
M JL?MMOL? CM ;MMOG?> BS>LIMN;NC= 3 CM L?KOCL?> IHFS @IL NB? ;>P?=NCIH MN?J ;N NB?
H?RN NCG? MN?J M NBCM ;>P?=NCIH CM ?R?=ON?> CH NB? VR?> NL;HM@ILG?> G?MB CN CM
3 NB;N Q? IOABN NI EHIQ &N CM GIL? DO>C=CIOM NI MIFP? NB? ! G;MM =IHM?LP;NCIH
?KO;NCIH CH ; VR?> G?MB 1BOM Q? ;PIC> NB? NI ;H> @LI GIP?G?HN <?NQ??H NB? L?;F
;H> VR?> G?MB QBC=B G;E?M NB? =;F=OF;NCIH OHQC?F>S
&H @;=N ?R=?JN CH NB? =;M? I@ ;>P?=NCIH <S NB? G?NBI> I@ =B;L;=N?LCMNC=M QB?L? 3 CM L?;FFS OM?> CN CM NB? P3 ALIOJCHA QBC=B ;JJ?;LM I@N?H ;H> QBC=B CM BIGIA?H?IOM
NI ; P?FI=CNS Q? L?=;FF NB;N P CM NB? >?JNB <?NQ??H NQI JF;H?M M?? 0?=NCIH 1B? @ILGOF; @IL J;MMCHA @LIG 3 NI 3 C ? 3 P3 INB?L N?LGM M?? "KO;NCIH
<?FIQ Q? NL?;N B?L? NB? A?H?L;FCT?> MCAG; NL;HM@ILG =;M? MBIQM NB;N P3 MBIOF> B;P? NB? M;G? >CM=L?NCT;NCIH ;M 3 &N ;JJ?;LM NB;N CH ; VR?> G?MB NB?
L?;F OHEHIQH MBIOF> <? P3 ;H> HIN 3 4? MB;FF HIQ ?R;GCH? ; G?NBI> @IL
=;F=OF;NCHA P3 NB?H BIQ 3 CM @IOH> IH NB? <;MCM I@ P3 4? VLMN B;P? NI
?MN;<FCMB NB? @ILG I@ NB? =IHNCHOCNS ?KO;NCIH CH NB? NL;HM@ILG?> G?MB
>=C8=D8CH 4@D0C8>= 8= C74 CA0=B5>A<43 <4B7 Q? L?=;FF B?L? NB? =IHNCHOCNS
?KO;NCIH :?L 1 QLCNN?H CH NB? NL;HM@ILG?> G?MB "KO;NCIH P
P
J
P1
N
GH B
P2
O
FH B
P3 P FGB
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
#IL ?;=B >?AL?? I@ @L??>IG ? Q? MBIOF> NBOM MIFP?
.
P
J
P1
N
P2
O
GH B
FH B
P3 P FGB
7 : 4CNB ; MCGJF? MCAG; NL;HM@ILG Q? QIOF> CH @;=N A?N
.
>
J
>1
N
FG
>2
O
GH B
FH B
>3 P FGB
7 : QB?L? Q? M?? NB? OH>?LFSCHA >?JNB CHN?AL;N?> =IHNCHOCNS ?KO;NCIH
&N CM ;=NO;FFS NBCM @ILG I@ NB? G;MM =IHM?LP;NCIH ?KO;NCIH NB;N CM @IOH> QBCF?
>?GIHMNL;NCHA NB? G;MM =IHM?LP;NCIH I@ NB? NL;=?L M?? 0?=NCIH !CM=L?NCT?> ;M
MO=B NBCM ?KO;NCIH QBIM? OHEHIQH CM 3 IL L;NB?L P3 F?;>M NI ; MSMN?G NB;N CM
P?LS CFF =IH>CNCIH?> IL ?P?H NI ; @IL=?> MSMN?G 1B? <IOH>;LS =IH>CNCIHM ;L? 3 ;N NB? <INNIG ;H> NB? @L?? MOL@;=? C@ NB?S ;L? CGJ?LG?;<F? 1B? N?LG
.
P3 7 :
P FGB
QCNB NB? OHEHIQH P3 F?;>M NI ; G;NLCR QCNB A?H?L;F N?LG
.
8 78 7 :
P
,ONMC>? NB? <IOH>;LC?M NB? >C;AIH;F N?LGM I@ NBCM G;NLCR ;L? T?LI #LIG NB?
<IOH>;LS =IH>CNCIHM ;N NB? <INNIG ;H> NB? @L?? MOL@;=? Q? ;LLCP? ;N >CM=IHH?=N?>
MIFONCIHM <?NQ??H NQI MO==?MMCP? JF;H?M J;L;MCNC= IM=CFF;NCIHM IL ;H CGJIMMC<CFCNS
1BCM CM <?=;OM? NB? IHFS OHEHIQH L?N;CH?> @IL MIFPCHA G;MM =IHM?LP;NCIH CM NB? P?LNC=;F
P?FI=CNS #CAOL? MBIQM NB? =;M? I@ ; G?MB L?>O=?> NI NQI JF;H?M QB?L? HI >?AL??
I@ @L??>IG CM ;P;CF;<F? IH NB? P?LNC=;F ? NI VH> ; P?LNC=;F P?FI=CNS @IL L? ?MN;<FCMBCHA ;
>CP?LA?H=? @L?? WIQ
P?LNC=;F P?FI=CNS >?VH?> IH ; MN;AA?L?> G?MB QIOF> <? H?=?MM;LS ;M CH #CAOL?
1I =CL=OGP?HN NBCM JLI<F?G ; MIFONCIH =IHMCMNM I@ =B;HACHA NB? OHEHIQH IL NI <?
GIL? JL?=CM? I@ ACPCHA ; MJ?=CV= >?VHCNCIH I@ NB? OHEHIQH P3 1BCM >?VHCNCIH
QBC=B QCFF <? ACP?H CH NB? H?RN J;L;AL;JB B;M <??H CHMJCL?> <S NB? H?=?MMCNS NI A?N
;H ?R;=N G;MM =IHM?LP;NCIH I@ NL;=?LM QCNB >CMNLC<ONCP? M=B?G?M M ; G;NN?L I@ @;=N
CN QCFF CH >O? =IOLM? ;JJ?;L NB;N NBCM >?VHCNCIH I@ P3 CM ?HNCL?FS =IGJ;NC<F? QCNB
NB?M? >CMNLC<ONCP? M=B?G?M
0;2D;0C8>= >5 P3 .
P
<
P
<
B?L? Q? MNLC=NFS MIFP? @IL ?P?LS JICHN ? I@ NB? G?MB
7 :
J
.
P 1 =H7: 7 : J
.
P 1 QD 7 :
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
i
i-1
i+1
#CAOL? ?R;GJF? I@ ; @IL=?> JLI<F?G
+IN? IH NB? LCABN B;H> MC>? >CM=L?NCT;NCIH I@ P CM HIN MJ?=CV?> =NO;FFS CN
GOMN <? =IHMCMN?HN QCNB NB? ! =IHNCHOCNS ?KO;NCIH ;M QCFF <? ?RJF;CH?> F;N?L QBC=B
A?H?L;FFS CGJIM?M NB;N P CM N;E?H ;N NCG? J<
4? MB;FF HIQ MBIQ NB;N NB? JLI<F?G CM Q?FF JIM?> C@ Q? =IHMC>?L NB;N NB? OHEHIQHM
;L? NB? ;P?L;A? I@ P3 ;FIHA NB? P?LNC=;F I@ ?;=B JLCMG
&@ NB? BILCTIHN;F ;H> P?LNC=;F AL;>C?HNM ;L? >C_?L?HNC;N?> NB? =IHNCHOCNS ?KO;NCIH
@IL ; JICHN ? =;H <? QLCNN?H ;M
.
.
<
P3 7 : P < 7 :
P
P
J .
.
P 1 =H7: 7 :
P 1 QD 7 :
&H NBCM @ILG NB? LCABN B;H> MC>? =IHN;CHM IHFS EHIQH N?LGM ;MMOGCHA NB;N NB?
WOR <IOH>;LS =IH>CNCIHM ;L? ACP?H ;N NB? <IOH>;LC?M 0I Q? =IH=?HNL;N? IH NB?
F?@N B;H> MC>? QBC=B =IHN;CHM OHEHIQHM #IL NBCM CN MBIOF> <? =IHMC>?L?> NB;N Q?
QILE QCNB JLCMGM CH NB? NL;HM@ILG?> VR?> G?MB B?H=? QCNB BILCTIHN;F <;M? ;H> NIJ
&H NB?M? =IH>CNCIHM JLCMGM ;L? BIGINB?NC= NI NB? L?@?L?H=? JLCMG ;H> NB? ! <;M?M
;L? JLI>O=NM I@ IH? @OH=NCIH I@ N ;H> O ;H> IH? @OH=NCIH I@ P F?N OM L?=;FF NB;N @IL
?R;GJF? 1B? @OH=NCIH I@ N ;H> O CM NB? M;G? ;M NB? <;MCM
@OH=NCIHM I@ NLC;HAF?M QCNB FCH?;L CHN?LJIF;NCIH 1BCM ;FFIQM OM NI <L?;E >IQH ?;=B
<;M? ;M 7 67 D7 67 >?J?H>M IHFS IH N ;H> O QBCF? D7 >?J?H>M IHFS IH P 1BOM
D
7
6 7
7
P P 1B? @OH=NCIH P3 =;H <? >?VH?> IH NB? QBIF? >IG;CH ;M @IFFIQM
P3 <>=7<
,
8
P3 P 8 67 N O
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
i
i-1
i+1
#CAOL? NB? =ILL?MJIH>CHA P?LNC=;F P?FI=CNS
QB?L? DFE?D CM NB? HOG<?L I@ >?AL??M I@ @L??>IG CH NB? ! G?MB P3 P CM ;
@OH=NCIH >?VH?> <?NQ??H ;H> @IL ?;=B P?LNC=;F FCH? I@ HI>?M I@ NB? NL;HM@ILG?>
G?MB
1B? F?@N B;H> MC>? I@ "KO;NCIH NB?H <?=IG?M
<>=7<
,
8
.
67 68 :
.
P3 P 8
1B? G;MM G;NLCR I@ NB? G?MB I@ NLC;HAF?M ;JJ?;LM CH
+
D7 :P
P H 67 68 :
CH ; JLCMG CM ?KO;F NI
==IL>CHA NI NB? JIMCNCIH I@ NB? JICHN ? NB? P;FO? I@
H QB?L? P CM NB? B?CABN I@ NB? JLCMG NB?M? <;M?M ;L? FCH?;L NB?CL P;FO? CM ;N NB? NIJ ;H> ;N NB? <INNIG IL PC=? P?LM;
#IL ; <;MCM MCNO;N?> IH NB? JF;H? ?F ; JF;H? >C_?L?HN @LIG NB? <INNIG ;H> NB?
@L?? MOL@;=? Q? A?N
.
D7 P3 P 8
:P P P7> P7>
P7>
P7>
.
H H .
H H P3 P 8 :P P3 P 8 :P 1B? KO;HNCNC?M L?NLC?P?> B?L? ;L? NB? IH?M QBC=B B?FJ NI =;F=OF;N? NB? =I?^=C?HNM
I@ >CMNLC<ONCP? M=B?G?M M?? 0?=NCIH )?N OM >?HIN?
PM
8
7> P7> P7>
.
H H P3 P 8 :P %6!/,!6+*& 0 ,# #/"" 02/# " #),40
4? VH;FFS A?N
.
P3 P 8
D7 :P PM
P 8
7> PM
8
7>
#IL ; JICHN ;N NB? <INNIG Q? B;P?
.
P3 P 8
D7 :P P .
P3 P 8
D7 :P PM
P PM
8
;H> ;N NB? MOL@;=?
8
<> 4? ;LLCP? ;N ; M?LC?M I@ FCH?;L MSMN?GM IH? J?L JF;H? QBIM? @ILG CM
<>=7<
,
.
8
J
67 68 : .
P
<
P
.
<
3
7 :
8
7>
PM
.
PM
8
7> P 1 =H7:
P 1 QD7 :
4
7 :
1 =H7: G?;HM NB;N IHFS NB? VLMN NQI =IGJIH?HNM I@ NB? ! P?FI=CNS ;L? N;E?H
8
CHNI ;==IOHN &H NBCM @ILG NB? H?Q OHEHIQHM PM 7> =;H <? @IOH> JF;H? <S
JF;H? MN;LNCHA @LIG NB? <INNIG IL CH NB? M;G? Q;S @LIG NB? MOL@;=?
1B? MSMN?GM G;NLCR CM ;FQ;SM NB? M;G? G;MM G;NLCR ;H> CM ?;MCFS CHP?LNC<F? ,H NB?
INB?L B;H> NB?L? ;L? MNCFF NII G;HS ?KO;NCIHM ;H> CN CM FCE?FS NB;N C@ Q? MN;LN @LIG NB?
<INNIG NB? <IOH>;LS =IH>CNCIH ;N NB? @L?? MOL@;=? QCFF HIN <? L?NLC?P?> 1BCM QIOF>
HIN B;JJ?H C@ NB? ! =IHNCHOCNS ?KO;NCIH Q;M L?MIFP?> CH ; =IGJ;NC<F? Q;S NI =;F=OF;N?
NB? @L?? MOL@;=? &H @;=N NB? MOG I@ ;FF ?KO;NCIHM JF;H? ;@N?L JF;H? ?FCGCH;N?M
8
;FF NB? OHEHIQHM PM 7> ;H> L?MNIL?M NB? G;MM =IHM?LP;NCIH ?RJL?MMCIH I@ NB?
0;CHN 3?H;HN ?KO;NCIHM @IL GIL? >?N;CFM M?? H?RN J;L;AL;JB &@ NB? F;NN?L B;M <??H
=ILL?=NFS L?MIFP?> NB? FCH?;L MSMN?G I@ NB? F;MN JF;H? CM >?>O=?> @LIG NB? INB?LM ;H>
B?H=? CN =;HHIN ACP? LCM? NI =IHNL;>C=NCIHM
&GJILN;HN HIN? "KO;NCIH =;H <? MCGJFCV?> <S G;MM FOGJCHA &H NBCM =;M?
NB? F?@N B;H> MC>? CM
.
3
4
7
7
67 : PM 7> PM 7> +
;H> NB? MSMN?G CM MIFP?> <S ; G?L? >CPCMCIH I@ NB? LCABN B;H> MC>? <S 67 : 1B? M;G? G;MM FOGJCHA GOMN ;FMI <? >IH? QB?H =IGJONCHA NB? =ILL?MJIH>CHA N?LGM
CH NB? NL;=?L ?KO;NCIH M?? 0?=NCIH ;H> NB? =IGJON;NCIH I@ NB? N?LGM 87 /"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
><?0C818;8CH 14CF44= 0=3 2>=C8=D8CH 4@D0C8>=B
&H "KO;NCIH NB? N?LG
J
.
P <
CM CH @;=N NL?;N?> CH NB? L?;F >IG;CH ;M
.
7 :<
J P < 7 :
.
7 :<
QB?L? NB? L?;F >IG;CH CM CH>?R?> QCNB D ;H> D ;M CN =B;HA?M QCNB NCG? 4B?H
MOGG?> IP?L NB? P?LNC=;F NB?M? N?LGM H;NOL;FFS SC?F> NB? F?@N B;H> MC>? I@ NB? !
=IHNCHOCNS ?KO;NCIH C ? .
>< >< 7 : J QB?L? 7 CM NB? ! <;MCM CH NB? ! >IG;CH %IQ?P?L NB? ! =IHNCHOCNS ?KO;
NCIH CM MIG?NCG?M =IGJON?> QCNB G;MM FOGJCHA QCNB ; F?@N B;H> MC>? ;JJLIRCG;N?>
CH NB? @ILG
.
.
><7 ><7
>< >< 7 : 7 : J J
+
+
NB?H NB? N?LGM 7 :< ;H> 7 :< GOMN <? GI>CV?> ;==IL>CHAFS ?P?LS
QB?L? NB?S ;L? OM?> M ; G;NN?L I@ @;=N Q? GOMN QLCN?
.
.
.
7 : >7 :
>7
7 :
NI <? =IGJ;NC<F? QCNB NB? G;MM FOGJCHA CH !
+
HINB?L JLI<F?G CM NB? N?LG P 1 =H7: 7 : QBC=B ;@N?L MOGG;NCIH
+
IP?L NB? P?LNC=;F GOMN ACP? > K =H7: 7 : &@ ; A?H?L;FCT?> MCAG; NL;HM
@ILG CM OM?> NBCM CM I<N;CH?> IHFS C@ Q? QLCN?
.
P 1 =H7:
7
.
P 1 : >
=H7:
>
7 :
;H> =IHMC>?L > IH NB? IH? B;H> ;H> H6 ) IH NB? INB?L B;H> ;M FCH?;L @OH=NCIHM S
MOGGCHA IP?L NB? P?LNC=;F Q? NB?H M?? NB;N Q? A?N NB? =ILL?=N ! N?LG C@ K CM ;
>?JNB ;P?L;A?> P;FO? I@ 1 I<N;CH?> QCNB NB? NL;J?TIC>;F LOF?
#>30; E0;D4B >5 3 ,H? QIOF> HIQ ?RJ?=N NB;N Q? >?>O=? 3 @LIG P3 <S >CPC>CHA <S P 1BCM CM HIN
; AII> C>?; M ; G;NN?L I@ @;=N Q? H??> HI>;F P;FO?M I@ 3 ? A @IL NB? ;>P?=NCIH
CH NB? NL;HM@ILG?> G?MB QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M <ON C@ Q? QLCN? 3 CH NB?
NL;HM@ILG?> G?MB Q? A?N
3 :P
P
:J
J
1
P
N
2
P
O
3
P
P %6!/,!6+*& 0 ,# #/"" 02/# " #),40
;H> MI Q? B;P?
P3 P
P
P
1
2
J
N
O
3
4? =;H >?>O=? @LIG NBCM F;NN?L ?KO;NCIH NB;N C@ 3 CM ; FCH?;L @OH=NCIH NB?H P3 CM ;FMI ; FCH?;L @OH=NCIH 1I A?N NBCM FCH?;L @OH=NCIH Q? H??> NI B;P? HI>;F P;FO?M
I@ P3 QBC=B =;H <? I<N;CH?> <S NB? @IFFIQCHA @ILGOF; OMCHA NB? F;S?L ;P?L;A?>
P;FO?M I@ NB? JL?PCIOM J;L;AL;JB
PM
8
7>
8
7> PM
PM
8
7>
N NB? <IOH>;LC?M C ? ;N NB? <INNIG ;H> ;N NB? @L?? MOL@;=? Q? EHIQ NB;N NB?
4? HIQ B;P? ; FCH?;L @OH=NCIH
CGJ?LG?;<CFCNS =IH>CNCIH CGJIM?M NB;N P3 P3 4B?H =IGJONCHA NB? =B;L;=N?LCMNC=M CH NB? NL;HM@ILG?> G?MB NB? P?LNC=;F
P?FI=CNS CH ; F;S?L QCFF <? NBCM FCH?;L @OH=NCIH P3 >CPC>?> <S P QBC=B CM FI=;FFS ;
@OH=NCIH I@ N ;H> O
%0BB8=6 5A>< P3 C> 3
M NB? ;>P?=NCIH CM >IH? CH NB? NL;HM@ILG?> G?MB 3 CM ;=NO;FFS H?P?L OM?> <ON OM?LM
A?H?L;FFS ?RJ?=N NI M?? CN CH NB? L?MOFNM ;H> MI CN GOMN <? <OCFN
1BCM CM >IH? QCNB "KO;NCIH QBC=B CM >CM=L?NCT?> CH NCG? CH NB? @ILG
P3 3<
P<
<
P<
J
1
P <
N
2
P <
O
1BCM ?KO;NCIH CM CHN?AL;N?> F;S?L <S F;S?L IP?L NB? P?LNC=;F CH NB? NL;HM@ILG?>
G?MB ;H> NB?H GOFNCJFC?> <S ; N?MN @OH=NCIH ;H> CHN?AL;N?> IH NB? ! G?MB &N ACP?M
.
.
H :P 3<
H .
.
H 7 :
P<
H .
P<
J
P <
1
N
.
H P3
<
:P H P <
2
O
:P
7 :
7 :
QB?L? ?F CM NB? HOG<?L I@ NB? FIQ?L JF;H? I@ NB? F;S?L ;H> ? CM NB? AFI<;F HOG<?L I@
; HI>? CH NB? ! G?MB M 3 CM FCH?;L Q? B;P? MCGJFS
.
H H 3<
<
:P 37>
<
37>
P7>
S OMCHA MCGCF;L @ILGOF;? @IL H BH Q? A?N
.
.
<
<
37>
37> 7 : P3 <
7>
7 :
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
J
P7>
.
.
<
P7>
<
P7>
.
H H <
<
P7>
P7>
P <
1
N
P <
2
O
7 :
:P
7 :
4? L?=IAHCT? B?L? ; M?LC?M I@ FCH?;L MSMN?GM IH NB? ! G?MB NB? G;NLCR <?CHA
<
<
NB? G;MM G;NLCR #IL ?P?LS F;S?L CH NB? G?MB Q? MB;FF A?N 37>
37>
&@ Q?
MN;LN @LIG ?F NB? VLMN P;FO? 3 < CM ACP?H <S NB? <IOH>;LS =IH>CNCIH ;N NB?
<INNIG M?? "KO;NCIH ;H> NB?H NB? <IOH>;LS =IH>CNCIH ;N NB? @L?? MOL@;=?
MBIOF> <? H;NOL;FFS ?HMOL?> HINB?L JIMMC<CFCNS CM NI MN;LN @LIG NB? @L?? MOL@;=? QCNB
"KO;NCIH ;H> AICHA >IQH NI NB? <INNIG )?MN NB? NLOH=;NCIH ?LLILM IL ; F;=E I@
=IGJ;NC<CFCNS CH NB? >CM=L?NCT;NCIH MBIOF> ACP? ; QLIHA <IOH>;LS =IH>CNCIH ;@N?L AICHA
C>
2=E<
OJ IL >IQH Q? <OCF> <INB MIFONCIHM =L?;NCHA ; M?LC?M I@ 37>
;H> 37>
QB?L? 3 C>
2=E<
CM I<N;CH?> <S NB? <IOH>;LS =IH>CNCIH ;N NB? <INNIG ;H> 3<> CM I<N;CH?> @LIG NB?
<IOH>;LS =IH>CNCIH ;N NB? @L?? MOL@;=? 1B? VH;F P;FO? I@ NB? P?LNC=;F P?FI=CNS CM NB?H
I<N;CH?> <S NB? @ILGOF;
<
37>
C>
2=E<
?F 37>
DF ?F 37>
DF QB?L? DF CM MNCFF NB? HOG<?L I@ MOJ?LCGJIM?> G?MB?M I@ NLC;HAF?M
#>=7H3A>BC0C82 #0E84AO(C>:4B 4@D0C8>=B
1B? BSJINB?MCM IH BS>LIMN;NC= JL?MMOL? CM JON ;MC>? ;H> Q? HIQ MIFP?
IHNCHOCNS
:?L 1 *IG?HNOG ?KO;NCIH
1
F
J
N
INB?L N?LGM
2
F
J
O
INB?L N?LGM
3
F
J
P
=
INB?L N?LGM
1B? INB?L N?LGM G?HNCIH?> CH ?;=B ?KO;NCIH ;L? ;>P?=NCIH NOL<OF?HN >C_OMCIH
;H> MIOL=? N?LGM <ON HIN <OIS;H=S N?LGM 1B?M? N?LGM B;P? HIN <??H >?N;CF?> ;M
NB?S ;L? NL?;N?> CH NB? M;G? G;HH?L ;M QCNB NB? BS>LIMN;NC= BSJINB?MCM 4? HIQ
B;P? ;H ?KO;NCIH @IL NB? P?LNC=;F =IGJIH?HN I@ P?FI=CNS 3 MCGCF;L NI NB? 1 ;H> 2
?KO;NCIHM QCNB ; AL;PCNS N?LG ;M Q?FF 1B? A?H?L;F C>?; I@ NB? L?MIFONCIH CM ;A;CH NI
OM? NB? @L;=NCIH;F MN?JM G?NBI> <S MCGJFS ;>>CHA ; MN?J ;@N?L NB? L?MIFONCIH I@ NB?
BS>LIMN;NC= ?KO;NCIHM ,HFS NB? Z=IHNCHOCNS[ MN?JM ;L? >C_?L?HN 0I Q? <L?;E >IQH
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
JL?MMOL? F CHNI ; BS>LIMN;NC= N?LG ;FL?;>S >?HIN?> ;M F6 ;H> ; N?LG =;FF?> Z>SH;GC=
JL?MMOL?[ >?HIN?> ;M F2 4? QLCN?
F F6
F2 = 6A P
.
=
.
H
:P
F2
@N?L NB? ;>P?=NCIH ;H> >C_OMCIH MN?JM NB? ?KO;NCIHM ;L? MIFP?> CH NQI INB?L MN?JM
; ZBS>LIMN;NC=[ MN?J
1 1<
1
=H7: F6
J
INB?L N?LGM
; =IHNCHOCNS MN?J
:?L 1 <
<
1
1
1
=H7: F2
J
1B? NQI MN?JM ;L? ACP?H CH >?N;CF <?FIQ
H3A>BC0C82 BC4?
4CNB NB? IOMMCH?MK ;JJLIRCG;NCIH
F6 = = 6A P
= =
.
H
.
:P
=
1 ;H> 21 <?=IG?
;H> NB? NLOH=;N?> M?LC?M ?RJ;HMCIH I@ NB? ?KO;NCIHM I@ 1
1 1<
1
6A
=
J
N
6A
=
=
= N
N
.
21 2 <
6A
=
J
O
6A
=
=
= O
O
.
.
H
.
H
:P
=
INB?L N?LGM
:P
=
INB?L N?LGM
1B?M? ?KO;NCIHM ;L? MCGCF;L IH ;FF JICHNM NI NB? ZBS>LIMN;NC=[ P?LMCIH 1B? <OIS
;H=S N?LGM CH=FO>?> CH NB? >C_OMCIH MN?J ;L? JL?M?HN A;CH @IL NL?;NCHA NB? @L??
MOL@;=? AL;>C?HNM NB? 0;CHN 3?H;HN ?KO;NCIHM ;L? OM?> ;M CH NB? BS>LIMN;NC= IJNCIH
H?Q @L?? MOL@;=? CM =IGJON?>
1B? ?KO;NCIH I@ 3 =IHN;CHM IHFS >C_OMCIH ;H> ;>P?=NCIH N?LGM &H @;=N
<S >?VHCNCIH I@ BS>LIMN;NC= JL?MMOL?
F6
P
=
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
>=C8=D8CH BC4?
1BCM MN?J B;M <??H EHIQH ;M NB? ZJLID?=NCIH MN?J[ @IL NB? +;PC?L\0NIE?M ?KO;NCIHM
MCH=? CNM JO<FC=;NCIH CH 89 NB? AL;>C?HN I@ >SH;GC= JL?MMOL? ;=NCHA ;M ; =ILL?=
NCIH ?HMOLCHA NB? >CP?LA?H=? @L?? P?FI=CNS 4B;N CM F?@N CM
<
1
1B? =IH>CNCIH :?L 1 <
:?L
1
1
=H7: F2
J
CM NBOM ?RJL?MM?> ;M
=H7: F2
1
:?L 1
J
1B? P;LC;NCIH;F @ILGOF;NCIH CM QLCNN?H ;M
.
.
1 7 :
:?L
:?L 1
=H7: F2 7 : J .
1B? F?@N B;H> MC>? I@ NBCM ?KO;NCIH CM CHN?AL;N?> <S J;LNM
7 D : =H7: F2 .
=H7: F2 =H7: 7 : J
.
1 7 :
:?L 1
.
1 7 :
:?L 1
;H> Q? ;LLCP? ;N NB? MSMN?G
.
J =H7: F2 =H7: 7 : .
J7 D : =H7: F2 1 NB?H Q?
&@ Q? ;MMOG? NB;N NB? F;N?L;F <IOH>;LS =IH>CNCIHM ;L? ;FL?;>S M?N IH 1
GOMN B;P? =H7: F2 D IH NB?M? <IOH>;LC?M 1B? <IOH>;LS N?LGM ;L? NBOM HIH
T?LI IHFS ;N NB? <INNIG ;H> NB? @L?? MOL@;=? 1B?S ;L? >?>O=?> @LIG NB? >SH;GC=
<IOH>;LS =IH>CNCIHM "KO;NCIHM ;H> N NB? @L?? MOL@;=?
.
J7 =H7: F2 QD : .
1 A QDA 1
7
QDA J7 =H7: F2 QD : .
1 4 QD4 1
7
QD4 N NB? <INNIG
.
.
B
.
B
:
:
1B? CHN?AL;FM B?L? ;L? =IGJON?> IH NB? ! <IOH>;LC?M <ON C@ NB?S ;L? =IGJON?>
IH NB? ! W;N G?MB NB?S <?=IG? MCGJF?L
N NB? @L?? MOL@;=?
.
.
J7 1 A QDA 6A :
7 1
=H7: F2 QD : J
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
N NB? <INNIG
.
J7 =H7: F2 QD : .
7
1 4 QD4
1
64
J
:
1B? G;NLCR I@ NB? FCH?;L MSMN?G I<N;CH?> CM ; >C_OMCIH NSJ? G;NLCR <OCFN QCNB
CH NB? JF;=? I@ PCM=IMCNS 1B? <IOH>;LS =IH>CNCIH IH NB? @L?? MOL@;=? CM F2 0?NNCHA MO=B ; !CLC=BF?N <IOH>;LS =IH>CNCIH ;N NB? @L?? MOL@;=? QCFF CH @;=N =;H=?F NB?
<IOH>;LS N?LGM I@ "KO;NCIH ,H=? F2 CM EHIQH 1 < CM L?NLC?P?> <S NB? @ILGOF;
<
1
1
1
=H7: F2
J
QBC=B =;H ;FMI <? MO<D?=N?> NI ; P;LC;NCIH;F @ILGOF;NCIH
)C ;H> #F?GCHA 89 OM? MO=B ; JLID?=NCIH MN?J CH VHCN? >C_?L?H=?M <ON NB? @L??
MOL@;=? AL;>C?HN CM ;FMI NL?;N?> 1B? CGJILN;HN JICHN CM NB;N NB? MN?J >?;FCHA QCNB
NB? @L?? MOL@;=? AL;>C?HN MBIOF> >I MI =IHMC>?LCHA NB? =IHMNL;CHN I@ G;MM =IHM?LP;NCIH
QBC=B CM NB? =;M? QB?H Q? OM? NB? 0;CHN 3?H;HN ?KO;NCIHM CH NB? ZBS>LIMN;NC=[ MN?J
0;2D;0C8>= >5 5A44 BDA5024
,H=? NB? VH;F P?FI=CNC?M 1 < ;L? EHIQH NB? @L?? MOL@;=? QBC=B B;M ;FL?;>S <??H
OJ>;N?> <S NB? BS>LIMN;NC= MN?J =IOF> <? =IGJON?> ;A;CH <ON CM HIN CH IL>?L NI
G;CHN;CH ; =IGJ;NC<CFCNS <?NQ??H NB? >?JNB ;H> NB? >?JNB ;P?L;A?> =IHNCHOCNS ?KO;
NCIH 1BCM =IGJ;NC<CFCNS CM H?=?MM;LS @IL NL;=?L =IHM?LP;NCIH *IL?IP?L N?MNM B;P?
MBIQH NB;N =IGJONCHA ; M?=IH> NCG? NB? @L?? MOL@;=? ;N NBCM F?P?F CM F?MM MN;<F? ;H>
;>>M HI L?;F ;==OL;=S NI NB? L?MOFNM 0IFPCHA >SH;GC= <IOH>;LS =IH>CNCIHM @IL NB?
@L?? MOL@;=? QIOF> ;FMI MJICF NB? >?JNB ;P?L;A?> =IHNCHOCNS ?KO;NCIH
%B4D3> F0E4 4@D0C8>= 0 E0A80=C >5 C74 7H3A>
BC0C82 BC4?
,8C7 4G?;828C 38QDB8>=
1BCM CM ; P?LS CHN?L?MNCHA P;LC;HN QBC=B ALIOJM NB? >C_OMCIH MN?J ;H> NB? =IHNCHOCNS
?KO;NCIH I@ 0?=NCIH CH ; MCHAF? MN?J ;H> L?JF;=?M NB? BS>LIMN;NC= MN?J I@ NB?
JL?PCIOM M?=NCIH M B;M <??H >IH? CH ! M?? 0?=NCIH B?L? ;A;CH NB? P?FI=CNS
CM ?FCGCH;N?> @LIG NB? =IHNCHOCNS ?KO;NCIH ;P?L;A?> IP?L NB? P?LNC=;F )?N OM =IHMC>?L
L?MIFONCIH QCNB ; BS>LIMN;NC= BSJINB?MCM S ;MMOGCHA NB;N ;H ;>P?=NCIH MN?J ?RJFC=CN
IL G?NBI> I@ =B;L;=N?LCMNC=M B;M ;FL?;>S ACP?H ; P?FI=CNS V?F> 1 HIQ NB? @IFFIQCHA
?KO;NCIHM B;P? NI <? MIFP?> MCGOFN;H?IOMFS
><
;H>
><
J
:?L >
C
K<
C
K < /9;
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
<
1
1
IK 1 <
J
= =H7: 6A
"
:?L
B =H7:
1
QB?L? " MJ?=CV=;FFS =IHN;CHM NB? <OIS;H=S N?LGM 1B;N CM
" =
=H7: 6A = =H7:
=
.
.
H
:P
=
1B? N?LG IK 1 < L?JL?M?HNM NB? G;HH?L CH QBC=B ;HS JIMMC<F? CGJFC=CN MIOL=?
N?LGM ;L? NL?;N?> IK CM ; @OH=NCIH >CM=L?NCT?> ;M NB? P?FI=CNS
&HCNC;FFS Q? QLCN? NB? P;LC;NCIH;F @ILGOF;NCIH I@ NB? >?JNB ;P?L;A?> =IHNCHOCNS
?KO;NCIH QCNB CHN?AL;NCIH <S J;LNM I@ NB? >CP?LA?H=? N?LG
.
7 ><
QCNB
.
>
C
K<
< BKN: ? .
7 >
;H>
I: ? ><
J
:
< BKN: ?
C
K < =H7: 7 : C
.
K<
C
7 /9; : I: ?
K < D
:
1BCM ?RJL?MMCIH @IL I: ? CM ;=NO;FFS MFCABNFS CH=ILL?=N <ON CM F?@N ;M MO=B B?L?
@IL MCGJFC=CNS 4? MB;FF M?? CH 0?=NCIH NB;N NBCM N?LG B;M NI <? ; MOG IP?L NB?
P?LNC=;F I@ ! MIOL=? N?LGM @IL L?;MIHM I@ =IGJ;NC<CFCNS <?NQ??H ! ;H> ! =IHNCHOCNS
?KO;NCIHM
< BKN: ? =;H <? NL?;N?> ;M CH ! ? A IH MIFC> Q;FFM <S MCGJFS =;H=?FFCHA NB?
WOR?M <ON CH ; ! @ILG
.
3 4
< BKN: ? D : 7 C 1 <
C 1 < NB? <;M? 7 <?CHA ; @OH=NCIH I@ IHFS N ;H> O CH NB? CHN?AL;FM &H @;=N < BKN: ? =;H
<? I<N;CH?> <S ; MOGG;NCIH IP?L NB? P?LNC=;F I@ ? I@ NB? N?LGM
.
3 <4 <
< BKN: @ 1
1 D : C
C
8
1B?H NB? P?FI=CNC?M ;P?L;A?> IP?L NB? P?LNC=;F ;L? L?JF;=?> <S NB?CL >?VHCNCIH
.
7 ><
><
J
:
< BKN: ?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
.
C
.
.
<
1
:P
.
C
.
.
.
<
1 :P =H7: 7 : I: ?
#IL L?;MIHM I@ =IGJ;NC<CFCNS <?NQ??H ! ;H> ! =IHNCHOCNS ?KO;NCIHM WOR?M FCE?
.
.
.
1 =H7: 7 : .
QCFF <? NL?;N?> ;M
,
8 IH NB? P?LNC=;F I@ 7
.
P 1 =H7: : 8
4? NBOM A?N
.
7 .
P
><
,
8 IH NB? P?LNC=;F I@ 7
><
J
<
C1
:
C
< BKN: ?
< =H7: 8
1
: I: ?
&H NB? GIG?HNOG ?KO;NCIH CH=L?G?HNM I@ NB? Q;N?L >?JNB ;L? G;>? NI ;JJ?;L
<S ; M?GC CGJFC=CN;NCIH I@ NB? @L?? MOL@;=? ;M @IL NB? 0;CHN 3?H;HN ?KO;NCIHM
<
1
1
IK 1 <
J
= =H7: 6A<
"
=H7: ><
=
6
:?L
B =H7:
1
><
&>?;FFS Q? QIOF> FCE? NI I<N;CH ; P;LC;NCIH;F @ILGOF;NCIH I@ NBCM ?KO;NCIH QBC=B
JLIPC>?M ;H ?RJL?MMCIH I@ 1 < QBC=B =;H <? CHNLI>O=?> CHNI NB? =IHNCHOCNS ?KO;NCIH
;H> IH=? >< CM EHIQH B?FJM NI L?NLC?P? CH ; =IGJ;NC<F? G;HH?L NB? P;FO? I@ 1 <
CNM?F@
1B? >C_OMCIH N?LG QCFF NBOM <? QLCNN?H ;M
+
:?L B =H7: 1 7 : +
7 : ; @ILG CH QBC=B NB? >CM=L?NCT;NCIH CH NCG? CM MNCFF HIN MJ?=CV?> MCH=? Q? QCMB NI L?N;CH
NB? @LC=NCIH N?LGM CH ;H CGJFC=CN @ILG ;H> NB? INB?LM CH ;H ?RJFC=CN @ILG
1B? AL;>C?HN N?LG = 6 =H7: >< >< MCGJFCV?> CH = 6 =H7: > QCFF <?
E?JN CH NB? =IHNCHOIOM @ILG @IL NB? N?LG CHNLI>O=?> CHNI NB? =IHNCHOCNS ?KO;NCIH ;H>
=;F=OF;N?> DOMN FCE? = =H7: 6A< ;M ; HI>;F P;FO? @IL NB? VH;F =;F=OF;NCIH I@ 1 <
1B;N CM
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
+
=
6
=H7: > = =H7: 6A< 7 : +
7 : )?N OM HIQ ?R;GCH? NB? >C_OMCIH N?LG QBC=B MBIOF> <? CHN?AL;N?> <S J;LNM
.
:?L B =H7: 1 7 : .
7
B =H7: 1 D : .
B
: =H7: 1 =H7: 7
S MJ?=C@SCHA NB? >CM=L?NCT;NCIH Q? QLCN? ;M G?HNCIH?>
.
:?L B =H7: 1
7
: .
7
B
<
=H7: 1
+IQ NI >?;F QCNB
.
B
D : < : =H7: 1 =H7: 7
B
<
=H7: 1
D Q? QLCN? NB;N
B
=H7: 1 <
D 7K8EH 1 <
8K8EH
B
=H7: 2 <
D 7L8EH 2 <
8L8EH
;H> NB;N
QB?L? 7L8EH 7K8EH L?JL?M?HNM NB? MNL?MM >O? NI @LC=NCIH ;H> 8K8EH ;H> 8L8EH ;H
?RJFC=CN MNL?MM >O? @IL ?R;GJF? NI QCH> ;N NB? @L?? MOL@;=? IL ;HS INB?L MNL?MM ;N NB?
<INNIG IL NB?+ F;N?L;F <IOH>;LC?M
<
1B? N?LG D : VH;FFS N;E?M NB? @ILG I@ ; >C;AIH;F
B =H7: 1
7
G;NLCR GOFNCJFC?> <S NB? P?FI=CNS P?=NIL JFOM ;H ?RJFC=CN MNL?MM >?HIN?> 8K8EH QCNB
=IGJIH?HNM 8K8EH ;H> 8L8EH 4? HIN? NB;N
+
<
: 7 7K8EH 1
+
< H?9: 1 <
7 : &H NB? ZG;MM FOGJ?>[ @ILG < H?9: CM MO=B NB;N
+
+
: 7K8EH 7 : 7
+
7K8EH
< H?9: +
"$ : : 7
7
QB?L? CM NB? MFIJ? ;HAF? MO=B NB;N
8
"$ M ; G;NN?L I@ @;=N Q? B;P?
64
N
64
O
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
: 8
64
N
64
O
:
IH NB? <INNIG <?=;OM? NB? ! >IG;CH CM W;N QB?L?;M NB?L? CM ; MFIJ? I@ NB? !
<IOH>;LS I@ NB? ! >IG;CH
4? ;FMI HIN? NB;N
:?< < +
;H> VH;FFS Q? A?N
+
B
+
< : : =H7: 1 =H7: 7
7
+
+
8K8EH
: 7 : 7
:?L
+
B =H7: 1
:
7
7 : < H?9: 1 <
:?< <
1B? ?RJL?MMCIH 1 < =;LLC?> IP?L CH NB? G;MM =IHM?LP;NCIH ?KO;NCIH QCFF <?
3
<
4
1
1 J= 6 =H7: > = =H7: 6A<
" :?< < QCNB
J < H?9:
IK
4? B;P?
J < H?9:
IK
<ON CN CM CHN?HNCIH;FFS E?JN CH NB? @ILG <?=;OM?
H?RN M?=NCIH
1B? GIG?HNOG ?KO;NCIH CM QLCNN?H ;M
<
1
J
+
=
6
QCFF <?=IG? ; G;NLCR CH NB?
=H7: > = =H7: 6A< 7 : +
7 : " :?< <
1
,<M?LPCHA NB;N ;N NB? =IHNCHOOG F?P?F
<
C1
C
<
1 C
3
1
J
4
J " :?< < = =H7: 6A<
=
C
<
1
>
4
J " :?< < = =H7: 6A<
6
C =H7:
;H> HINCHA NB;N
/CF
1
3
1
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
NB? =IHNCHOCNS ?KO;NCIH <?=IG?M
.
7 ><
.
.
IL ?P?H
3
3
J
/CF
C1
=
6
C
.
C =H7:
><
.
3
.
.
><
7 J
:P
.
/CF
C1
><
:
4 =H7: 7 : < 4 1 =H7: 7 : .
><
J
3
C
=
6
><
J
C =H7:
><
:
I: < BKN:
><
4 =H7: 7 : < 4 1 =H7: 7 : I: < BKN:
&H NB? M?=IH> FCH? I@ NBCM ?KO;NCIH NB?L? CM ; >C_OMCIH G;NLCR <OCFN QCNB ; ZPCM=IMCNS[
E/D3 =J
E/D3
6
C
.
.
:P
.
/?G;LE NB? N?LG
.
3
/CF
C1
CM NL?;N?> ;M
.
.
.
.
3
/CF
C1
#CH;FFS NB? =IHNCHOCNS ?KO;NCIH CM
.
><
7 .
E/D3
=H7: ><
.
3
/CF
C1
C
C
< 4 1 =H7: 7 : C
<4
:P=H7: 7 : 1
><
J
:
>< =H7: 7 : < 4 1 =H7: 7 : I: < BKN:
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
H> NB? GIG?HNOG ?KO;NCIH CM
<
1
1 /CF
+
J
=
6
=H7: >< >< 7 : +
7 : ,8C7 8<?;828C 38QDB8>=
1B? @;=N NB;N NB? >C_OMCIH CM NL?;N?> CH ;H ?RJFC=CN Q;S CM ; >L;Q<;=E I@ NB? G?NBI>
QBC=B =;H CGJIM? FCGCN;NCIHM IH NB? NCG? MN?J )?N OM HIQ ;MMOG? NB;N NB? >C_OMCIH
CM CGJFC=CN ;H> NB;N Q? NBOM B;P? :?< < ) 1 <
QB?L? ) CM ; G;NLCR FF
NB;N B;M <??H QLCNN?H CH NB? JL?PCIOM M?=NCIH L?G;CHM P;FC> C@ Q? ?L;M? NB? N?LG
:?< < ;H> =B;HA?
NI
J )
QIOF> NB?H <? ; G;NLCR <ON ?P?H C@ Q? =;H
CHP?LN CN <S G?;HM I@ ;H CN?L;NCP? MSMN?G @IL =IGJONCHA 1 /CF QCNB "KO;NCIH N?LGM FCE? =H7: >< >< CH "KO;NCIH M??G ; HCABNG;L? %IQ?P?L C@ Q?
IHFS B;P? ; P?LNC=;F >C_OMCIH FCHECHA NIA?NB?L NB? P?FI=CNC?M IP?L NB? P?LNC=;FM CH NB?
! G?MB
CM =IGJIM?> I@ ; M?LC?M I@ NLC >C;AIH;F G;NLC=?M IH? @IL ?P?LS P?LNC=;F
;H> =;H <? ?;MCFS CHP?LN?> <S MIFPCHA ; M?LC?M I@ CH>?J?H>?HN NLC >C;AIH;F MSMN?GM
=H7: >< >< =;H <? =IGJON?> CH ;H CGJFC=CN Q;S C ? E??JCHA NB? OHEHIQH
<
>
<?=;OM? =H7: >< >< CM ; ! N?LG QBC=B CM =IHMN;HN IH ?P?LS P?LNC=;F 1BOM
Q? B;P?
>< =H7: ><
=H7: ><
><
L;9JEH KD?JO
QB?L? L;9JEH KD?JO CM ; P?=NIL QCNB ;FF =IGJIH?HNM ?KO;F NI &N CM NBOM JIMMC<F? NI NL?;N J;LN I@ NB? >C_OMCIH N?LGM CH ;H CGJFC=CN Q;S CN MO^=?M
NI L?GIP? @LIG :?< < NB? N?LGM
<
18
+
;H> L?JF;=? NB?G <S
<
18
+
B
=H7: : =H7: 8
+ 7 7 : B
=H7: : =H7: 8
+ 7 7 : C@ Q? =BIIM? >C;AIH;F N?LGM IL ?RNL; >C;AIH;F N?LGM FCHECHA P?FI=CNC?M I@ NB? M;G?
P?LNC=;F &@ NB?L? CM P?LNC=;F >C_OMCIH IHFS CN =;H NB?H ;G;TCHAFS <? NL?;N?> CH ;
@OFFS CGJFC=CN Q;S &@ NB?L? CM ; BILCTIHN;F >C_OMCIH MIG? ?RNL; >C;AIH;F N?LGM QCFF
B;P? NI L?G;CH ?RJFC=CN ;H> NBCM >C_?L?HN NL?;NG?HN QIOF> IHFS <? P;FC> CH MN?;>S
MN;N? =IH>CNCIHM &N CM ;FMI QILNB HINC=CHA NB;N QB?H NB?L? CM ; P?LNC=;F >C_OMCIH IHFS
NB? >?JNB CHN?AL;NCIH I@ NB? >C_OMCIH N?LGM MBIOF> L?MOFN IHFS CH NB? MNL?MM?M ;N NB?
<INNIG ;H> ;N NB? @L?? MOL@;=? MI NB;N NB? N?LG >?HIN?> B?L? <S :?< < MBIOF> B;P?
HI CHWO?H=? IH NB? ! =IHNCHOCNS ?KO;NCIH 1BCM CM BIQ?P?L HIN MI I<PCIOM NI MBIQ
;N NB? >CM=L?N? F?P?F
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
)A024A 4@D0C8>= 8= 1B? =F;MMC=;F NL;=?L ?KO;NCIH
1 =H7: 0 :?L
0
J
(
=H7: 0
-
QBC=B CM CH @;=N CH ; GIPCHA ALC>
4 3 1 3 ;3A6 =H7: 0 :?L
0
J
(
=H7: 0
-
CM L?MIFP?> CH NQI MN?JM ;>P?=NCIH NB?H >C_OMCIH ;H> MIOL=? N?LGM >P?=NCIH N;E?M
JF;=? CH NB? VR?> NL;HM@ILG?> G?MB QCNB NB? G?NBI>M JL?M?HN?> CH B;JN?L IL CM
=IOJF?> QCNB >C_OMCIH NB? =;M? I@ NB? 02-$ M=B?G? 1B? ;>P?=NCIH ?KO;NCIH CM
QLCNN?H ;M
.
P 0
<
0
<
7 :
J
.
P 1 =H7: 0 7 : 4? L?=;FF NB;N NB? P?FI=CNS I@ NB? G?MB CM ;FMI N;E?H CHNI ;==IOHN <S NBCM @ILGOF;
;M ?RJF;CH?> CH 0?=NCIH &H ';HCH M?? 89 @IOH> ; L?G;LE;<F? MIFONCIH @IL <OCF>CHA ; GIHINIHC=
;H> ;N NB? M;G? NCG? =IHM?LP;NCP? M=B?G?
1B? OM? I@ ; >CMNLC<ONCP? M=B?G? =;FF?> NB? *2/! M=B?G? M?? 0?=NCIH QCNB 0 QLCNN?H CH ;H
1B? =BIC=? I@ ;>P?=NCIH CH ; G?MB VR?> CH NCG? J<
?RJFC=CN Q;S CH NB? N?LG =H7: 0 M?? NB? >?GIHMNL;NCIH I@ =IHM?LP;NCIH I@ NB?
NL;=?L CH 0?=NCIH =IGJ;NC<F? L?MIFONCIH I@ NB? G;MM =IHM?LP;NCIH ?KO;NCIH QCNB ; P;LC;<F?
P3 ;P?L;A?> J?L F;S?L ;M M??H CH 0?=NCIH ,HFS ; AII> =IGJL?B?HMCIH I@ NB?M? NBL?? JICHNM ;FFIQM IH? NI ;JJL?=C;N? NB?
CHN?L?MN ;H> ILCACH;FCNS I@ NBCM ;JJLI;=B
!C_OMCIH B;M @?Q?L =IHMNL;CHNM ;H> CM JIMMC<F? CH NB? L?;F G?MB &H G;MM
=IHM?LP;NCIH I@ NB? NL;=?L Q;M ?RN?H>?> NI NB? 02-$ ;>P?=NCIH M=B?G? OMCHA NB?
M;G? C>?;M
)74 2>=B4AE0C8>= >5 <0BB 8= >=B4AE0C8>= >5 <0BB >5 F0C4A
&H 0?=NCIH Q? B;P? M??H NB;N =IHM?LP;NCIH I@ G;MM IH ; >IG;CH CM QLCNN?H ;M
.
:
J
.
1 D : %6!/,!6+*& 0 ,# #/"" 02/# " #),40
$CP?H NB? =SFCH>LC=;F @ILG I@ ;H> NB? >?VHCNCIH I@ > ;@N?L >CM=L?NCT;NCIH CH NCG?
NBCM ?KO;NCIH CM ?KOCP;F?HN NI
.
><
:
.
>< :
J
.
1 QD :
QB?L? CM NB? G?MB I@ NLC;HAF?M ;H> NB? <IOH>;LS I@ NB? ! >IG;CH
&@ Q? ?HMOL? ; =IGJ;NC<F? =;F=OF;NCIH I@ NB? WOR ;N NB? F;N?L;F <IOH>;LC?M
.
1 QD : .
>
K QD :
Q? L?=IAHCT? NB? =IHM?LP;NCIH I@ G;MM QLCNN?H @IL NB? 0;CHN 3?H;HN ?KO;NCIHM 1BOM
NBCM =IH>CNCIH QCFF <? ?HMOL?> CH NB? BS>LIMN;NC= =;M? <S L?MILNCHA NI NB? 0;CHN 3?H;HN
?KO;NCIHM @IL =;F=OF;NCHA NB? @L?? MOL@;=?
&@ NBCM ?KO;FCNS CM HIN P?LCV?> NB? G;MM I@ Q;N?L CM HIN =IHM?LP?> 1B? >C_?L
?H=? <?NQ??H NB? F?@N B;H> MC>? ;H> NB? LCABN B;H> MC>? =IHMNCNON?M NB? A;CH CH G;MM
<?NQ??H NB? CHMN;HNM J< ;H> J<
1BCM ?RJL?MMCIH CM ;FMI ?KOCP;F?HN NI
.
1
N
2
O
3
P
: <ON IP?L;FF =IHM?LP;NCIH I@ G;MM CM JIMMC<F? QCNBION ; FI=;F L?MIFONCIH I@ =IHNCHOCNS
B?H=? QCNBION ; >CP?LA?H=? @L?? ! P?FI=CNS V?F> GIL? FI=;FCT?> =IHM?LP;NCIH QCFF
<? CGJIM?> @IL NB? =IHM?LP;NCIH I@ NB? NL;=?L
1B? =IHM?LP;NCIH I@ G;MM CM NBOM H;NOL;FFS P?LCV?> <S NB? L?MIFONCIH I@ NB? 0;CHN
3?H;HN ?KO;NCIHM CH NB? =;M? I@ ?KO;NCIHM QCNB ; BS>LIMN;NC= BSJINB?MCM 1BCM CM
;JJFC=;<F? ;FMI @IL HIH BS>LIMN;NC= ?KO;NCIHM C@ NB? @L?? MOL@;=? CM =;F=OF;N?> QCNB NB?
=IHNCHOCNS ?KO;NCIH ,H NB? INB?L B;H> NBCM =IHM?LP;NCIH QIOF> <?=IG? JLI<F?G;NC=
QCNB ; >SH;GC= <IOH>;LS =IH>CNCIH ;JJFC?> NI NB? @L?? MOL@;=?
>=B4AE0C8>= >5 CA024A
1B? =IHM?LP;NCIH I@ NB? NL;=?L CM QLCNN?H ;M
.
0<
0 < :
J
.
0 1 D : J
.
0
: J
B
D
.
- : QB?L? - CM ; MIOL=? N?LG &@ !CLC=BF?N NSJ? <IOH>;LS =IH>CNCIHM ;L? CGJIM?> IH
NB? <IOH>;LC?M NBCM MJICFM NB? =IHM?LP;NCIH ;H> ; =ILL?=NCIH CM H??>?> M?? <?FIQ CH
0?=NCIH 1B? HIH =IHM?LP;NCP? ?KO;NCIH L?MIFP?> @IL ?;=B >?AL?? I@ @L??>IG CM ;M @IFFIQM
.
0
J
1 =H7: 0 B 0
- 7 : /"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
%?L?;@N?L Q? MB;FF >LIJ NB? >C_OMCIH N?LG ;H> NB? =L?;NCIH N?LG - QBC=B ;L? HIN
L?;FFS ; JLI<F?G @IL NB? =IHM?LP;NCIH I@ G;MM S MOGGCHA IP?L ;FF NB? N?MN @OH=NCIHM
;H> ;@N?L >CM=L?NCT;NCIH CH NCG? NB? ;>P?=NCIH ?KO;NCIH I@ NB? NL;=?L <?=IG?M
.
.
1 =H7: 0 : 0 < 0 < : J
1I ;LLCP? ;N
.
0<
0 < :
J
.
0 1 D : CN QIOF> <? H?=?MM;LS NI ;>> NI NB? LCABN B;H> MC>? NB? N?LG
.
J
0 :?L 1 :
QBC=B CH JLCH=CJF? CM T?LI 1BCM CM QB?L? ; =IGJ;NC<CFCNS =IH>CNCIH QCNB NB?
=IHNCHOCNS ?KO;NCIH ;JJ?;LM
1BCM @ILGOF;NCIH CH NB? L?;F >IG;CH G;MEM NB? JLI<F?G I@ NB? ?PIFONCIH I@ ;H>
NB? @;=N NB;N NB? ;>P?=NCIH MN?J I@ NB? NL;=?L MBIOF> <? >IH? IH ; =IHMN;HN G?MB S
OMCHA NB? MCAG; NL;HM@ILG NB? =IHM?LP;NCIH I@ NB? NL;=?L CM
.
.
P < 0 < P < 0 < : J
P0 1 D : +IQ NB? MOG I@ ;FF NB? ;>P?=NCIH ?KO;NCIHM I@ NB? NL;=?L QCNB ; VR?> G?MB ;N ;H
CHMN;HN N;E?H <?NQ??H J< ;H> J< ACP?M
.
.
P 0 < 0 < : J
P 1 =H7: 0 : &H NBCM ?KO;NCIH C@ NB? VR?> G?MB CM =BIM?H ;N CHMN;HN J<
MBIOF> <? N;E?H
P 6 P
<
6 J
NB? @IFFIQCHA
P <
6
4? NB?H H??> NI =BIIM? ;H CGJFC=CN ?RJL?MMCIH I@ 0 @IL =;F=OF;NCHA CNM AL;>C?HN &@
Q? >?HIN?
0 (0
<
0<
(
NB? CHN?AL;F ?KO;NCIH ;<IP? <S ;>>CHA NB? @IFFIQCHA HOFF ?RJL?MMCIH NI CN
.
.
J
P0 1 D : J
P0 1 D :
=;H NB?H <? QLCNN?H ;M
.
P < 0 <
.
60
<
P
<
<
P 0
P
<
:
<
:
J
.
(
J
.
P0 1 D :
P 1 =H7: 0 < :
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
( J
J
(
.
.
P0 < 1 D : P 1 =H7: 0 <
.
.
:
6
( J
0<
.
P <
P0 <
P < :
1 D :
&N B;M NI <? JLIP?> NB;N NB? F;MN NQI N?LGM ;L? T?LI NI AO;L;HN?? NB? =IHM?LP;NCIH
I@ NB? NL;=?L S <L?;ECHA >IQH 0 < ;H> 0 < IH NB? <;MCM 7 I@ NB? NL;HM@ILG?>
G?MB NQI =IH>CNCIHM ;L? MO^=C?HN
6
(
;H>
J
.
.
P <
P 1 =H7: 7 :
P < 7 :
J
.
7 P 1 D : N NBCM JICHN NB? P;LC;NCIH;F @ILGOF;NCIH I@ NB? G;MM =IHM?LP;NCIH ?KO;NCIH ;J
J?;LM
>
:?L > 1 J
QCNB ;H CHN?AL;NCIH <S J;LNM I@ NB? >CP?LA?H=? N?LG C>?HNC=;F NI NB? IH? OM?> @IL
L?MIFPCHA NB? 0;CHN 3?H;HN ?KO;NCIHM
4? ;LLCP? ;N CF> 5D=30<4=C0; 2>=2;DB8>=B
1B? =IHM?LP;NCIH I@ NB? G;MM I@ NB? NL;=?L L?KOCL?M ; =IGJ;NC<F? NL?;NG?HN I@
NB? =IHNCHOCNS ?KO;NCIH
1B? =IH>CNCIH 6
( QBC=B MCAHCV?M @IL CHMN;H=? NB;N C@ ;H ?RJFC=CN
+
M=B?G? CM OM?> NI =;F=OF;N? NB? P;FO? I@ P 1 =H7: 0 : NB? ;>P?=NCIH
MN?J MBIOF> <? >IH? J;L;>IRC=;FFS CH NB? G?MB VR?> ;N NCG? J<
GIL? >?
N;CF?> >?GIHMNL;NCIH I@ NL;=?L =IHM?LP;NCIH QCFF <? ACP?H CH 0?=NCIH <?FIQ
QCNB ;H ?R;GJF? I@ NB? >CMNLC<ONCP? M=B?G?
(>DA24B 0=3 B8=:B 8= C74 #0E84AO(C>:4B 4@D0
C8>=B
&H IL>?L NI >?;F QCNB Q;N?L IL NL;=?L CHJONM CHNI ; >IG;CH ? A >L;CH;A? JCJ?M
QBIM? >CG?HMCIHM ;L? F?MM NB;H NB? MCT? I@ VHCN? ?F?G?HNM Q? OM? NB? =IH=?JN I@
MIOL=?M ;H> MCHEM 1B? WOC> WIQ QCNB IL QCNBION NL;=?L CM =IHMC>?L?> ;M ?HN?LCHA
NB? =;F=OF;NCIH >IG;CH ;N ; MCHAF? JICHN 1BCM =IH=?JN ;FL?;>S NL?;N?> @IL NB? 0;CHN
3?H;HN ?KO;NCIHM CM ;LNCV=C;F CH =?LN;CH ;MJ?=NM ;H> @IL NB? +;PC?L\0NIE?M ?KO;NCIHM
L?KOCL?M ; NBILIOAB ?R;GCH;NCIH
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
(>DA24B >5 KD83
&H ! CN CM HIQ NB? =IHNCHOCNS ?KO;NCIH :?L 1 QBC=B CM HI FIHA?L P?LCV?> FI=;FFS
&H NB? NL;HM@ILG?> G?MB MN;LNCHA ;A;CH @LIG "KO;NCIH Q? HIQ B;P? NI MIFP?
.
P
J
P1
N
FG
P2
O
GH B
.
FH B
P3 P FGB
7 :
P :?L 1 7 :
<S =IHMC>?LCHA NB;N IH NB? LCABN B;H> MC>? :?L 1 QCFF HIN <? T?LI CH =?LN;CH JICHNM
+
+
+IQ P :?L 1 7 : :?L 1 7 : 1B? CHN?AL;F :?L 1 ;LIOH> ;
>IG;CH MOLLIOH>CHA ; MIOL=? JICHN QCFF <? NB? >CM=B;LA? I@ NB? MIOL=? -A13 M CH !
<ON N;ECHA CHNI ;==IOHN NB? P?LNC=;F IH? MIFONCIH =IHMCMNM I@ ;>>CHA NI NB? LCABN B;H>
MC>? I@ NB? =IHNCHOCNS ?KO;NCIHM NB? N?LG
+
2
,
-A13 2
7A13 72 :2
+
2
7A13 :2
72 IH NB? P?LNC=;F I@ 7 2
QB?L? ?I9; CM NB? JICHN QCNB NB? MIOL=? 4? B;P? >CMNCHAOCMB?> B?L? @IL =F;LCNS C@ HIN
=IH=CM?H?MM NB? ! ;H> ! >IG;CHM ;H> <;MCM @OH=NCIHM <S CH>?RCHA NB?G QCNB :
;H> : &N CM QILNB HINC=CHA NB;N
,
72 IH NB? P?LNC=;F I@ 7 2
+
2
-A13 2
7A13 72 :2
+
2
7A13 :2
+
2
-A13 7A13
7 2 :
+
2
7A13 : 2
2
1B? >IG;CH CM =IHMC>?L?> ;N ;H CHMN;HN CH =IHMCMN?H=S QCNB NB? INB?L WOR N?LGM
1B? M?LC?M I@ FCH?;L MSMN?GM ACPCHA NB? P?LNC=;F P?FI=CNC?M ;P?L;A?> IP?L ?;=B JF;H? CM
NB?H QLCNN?H @IL ?P?LS JICHN ? CH NB? ! G?MB ;M
<>=7<
,
8
.
J
.
.
P <
P 1 QD7 :
67 68 : 3
P < 7 :
,
8
7>
PM
.
72 IH NB? P?LNC=;F I@ 7 2
PM
8
7> P 1 =H7:
+
4
7 :
- 2 2 ::
2 + A13 7A13 72
2 ::
2 7A13
(>DA24B >5 CA024A F8C7 0 38BCA81DC8E4 B274<4
%;PCHA M??H <INB NB? =IHM?LP;NCIH I@ NL;=?L G;MM ;H> NB? MIOL=?M I@ Q;N?L
HIQ CH ; JIMCNCIH NI OH>?LMN;H> BIQ NI >?;F QCNB MIOL=?M I@ NL;=?LM M ;
I@ @;=N CN CM H?=?MM;LS NI L?FS IH NB? JLII@ I@ =IHM?LP;NCIH I@ NB? NL;=?L G;MM
ION QBC=B MIOL=? N?LGM ;JJFS NI CN 1B? NL?;NG?HN I@ NB? ;>P?=NCIH N?LGM
Q? ;L?
G;NN?L
NI VH>
B;M CNM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
CGJILN;H=? ;H> Q? N;E? B?L? NB? ?R;GJF? I@ NB? >CMNLC<ONCP? M=B?G? QBC=B QCFF <?
JL?M?HN?> CH 0?=NCIH &N CM ;H ?RJFC=CN M=B?G? ;H> CN CM ;<ION NB? IHFS NBCHA NI
L?=;FF CH QB;N @IFFIQM )?N OM >?HIN?
.
" (1 %* 0 ? P 1 =H7: 7 :
" (1 2 !. ? +
.
P3 7
:
P -A13 7A13 7 :
+
?
:
7A13
.
" (1 !40 ? P 1 QD7 :
/+1 .! ? 1B? =IHNCHOCNS ?KO;NCIH CM NB?H QLCNN?H ;M
.
<
P < 7 :
P
J " (1 %* 0 ?
" (1 2 !. ? " (1 !40 ?
/+1 .! ?
1B? ?KO;NCIH @IL NB? NL;=?L 0 CM ;M @IFFIQM
.
<
P <
0
0 < 7 : J .
P 1 =H7: 0 < 7 : /+1 .!:0 . ?
+
QB?L? NB? N?LG P 1 =H7: 0 < 7 : CM ?RJFC=CN ;H> =;F=OF;N?> <S NB? >CMNLC<ONCP?
M=B?G?
/+1 .!:0 . ? CM NB? MIOL=? N?LG Q? FIIE @IL 1B? IP?L;FF =IHM?LP;NCIH I@
NB? NL;=?L G;MM CM MBIQH <S VLMN GOFNCJFSCHA ;FF NB? =IHNCHOCNS ?KO;NCIHM <S 07< NB?H
<S ;>>CHA NB?G ;H> ?P?HNO;FFS <S ;>>CHA NI NB?G ;FF NB? NL;=?L ?KO;NCIHM QBC=B
ACP?M
.
.
<
< <
<
P 0 :
P <
0 < :
P
0
J J ,
,
,
" (1 %* 0 ? 07<
" (1 2 !. ? 07< " (1 !40 ? 07<
7
,
7
/+1 .! ? 07< 7
.
P 1 =H7: 0 < :
+IQ NB? >CMNLC<ONCP? M=B?G? ?HMOL?M NB;N
7
,
7
/+1 .!:0 . ? /"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
,
" (1 %* 0 ?
,
07<
7
" (1 2 !. ?
07<
7
.
P 1 =H7: 0 < : <?=;OM? CN MCGJFS J?L@ILGM ; L?>CMNLC<ONCIH I@ NB? VLMN NQI N?LGM QCNBION =B;HACHA
CNM MOG 4? VH;FFS ;LLCP? ;N
.
<
0 < P < 0 < :
P
J ,
,
" (1 !40 ? 07<
7
/+1 .! ? 07<
7
,
/+1 .!:0 . ? 7
&H NB? =;M? I@ ; MIOL=? QCNB ; NL;=?L P;FO? 0A13 NB? =IHM?LP;NCIH I@ NL;=?L G;MM
CGJIM?M NB;N
,
,
/+1 .! ? 07<
/+1 .!:0 . ? -A13 0A13
7
7
,H? MIFONCIH @IL ; MIOL=? CM NB?H
/+1 .!:0 . ? 0A13 07< /+1 .! ?
&@ NB?L? ;L? M?P?L;F MIOL=?M NB?L? MBIOF> <? ; >CMNCH=NCIH <?NQ??H NB? /+1 .! ?
P?=NILM @IL ?;=B MIOL=?
&@ NB? MIOL=? CM ;=NO;FFS ; MCHE NB?H /+1 .! ? CM H?A;NCP? ;H> ; >C_?L?HN NL?;N
G?HN CM H?=?MM;LS M ; G;NN?L I@ @;=N CH ; MCHE NB? P;FO? I@ NB? NL;=?L HIQ ?RCNCHA
NB? >IG;CH CM HIN 0A13 <ON L;NB?L NB? FI=;F P;FO? 07 MI NB;N /+1 .!:0 . ?
GOMN CH @;=N <? =;H=?FF?> "P?LSQB?L? 0A13 ;JJ?;LM CN GOMN <? L?JF;=?> <S 07 B?L?
07< <?=;OM? Q? B;P? ;H ?RJFC=CN M=B?G? <ON CN =IOF> ;M Q?FF <? 07< QCNB ;H CGJFC=CN
M=B?G? 1I N;E? CHNI ;==IOHN MCHEM ;M Q?FF ;M MIOL=?M Q? GOMN NBOM =IHMC>?L NB;N
/+1 .!:0 . ? 0A13 07< ) 4 /+1 .! ? 8A827;4C 1>D=30AH 2>=38C8>=B F8C7 0 38BCA81DC8E4 B274<4
M NB? JL?PCIOM MO<M?=NCIHM ACP? ; @OFF JLII@ I@ NB? G;MM =IHM?LP;NCIH I@ NB? NL;=?L
Q? ;L? HIQ CH ; JIMCNCIH NI >?;F QCNB NB? JLI<F?G I@ CGJIM?> <IOH>;LS =IH>CNCIHM
1B? >CMNLC<ONCP? M=B?G?M J;S HI ;NN?HNCIH NI JL?M=LC<?> P;FO?M ;N NB? <IOH>;LC?M
NB?S DOMN N;E? NB? NL;=?L ;N NCG? J< ;H> ?HMOL? G;MM =IHM?LP;NCIH ;H> GIHINIHC=CNS
OM?L G;S Q;HN NI JL?M=LC<? ACP?H P;FO?M ;N ;H ?HNL;H=? NI NB? >IG;CH IL ; ACP?H
NL;=?L WOR &@ HINBCHA MJ?=C;F CM >IH? NB? WOR I@ NB? NL;=?L NBLIOAB NB? <IOH>;LC?M
B?L? JIMCNCP? C@ ?HN?LCHA NB? >IG;CH QCFF <?
.
E8I;HL;: < BKN P0 < 1 QD :
,
7
07<
.
,
P 1 QD7 : 07< " (1 !40 ?
7
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;H> NB? CGJIM?> P;FO?M QCFF HIN <? ?HMOL?> 1B? MOGG;NCIH IH ? GOMN <? OH>?LMNII>
;M ? <?CHA IH NB? <IOH>;LS QCNB JL?M=LC<?> P;FO?M 1B? OM?L QIOF> CH @;=N FCE? NI
B;P?
M7DJ;: < BKN .
P0 7;>=A32 1 QD : ;H> 07<
077;>=A32 IH <IOH>;LC?M
INB =IH>CNCIHM ;L? HIN =IGJ;NC<F? LON;FFS M?NNCHA 07< 077;>=A32 IL 07< IH <IOH>;LC?M QCFF ;>> IL L?GIP? NL;=?L G;MM CH NB? >IG;CH <?=;OM? NB?
<;MCM @OH=NCIHM IH NB? <IOH>;LC?M AI CHMC>? NB? >IG;CH NBOM =L?;NCHA ;H ;LNCV=C;F WOR
4? JLIJIM? B?L? ; L?F;R;NCIH I@ NB? !CLC=BF?N =IH>CNCIHM QBC=B =IHMCMNM I@ L?JF;=CHA
07< <S
077;>=A32
07<
077;>=A32 07< 07<
0 <
IH FCKOC> <IOH>;LC?M QCNB CGJIM?> P;FO?M QCNB CM =IGJON?> MI NB;N
NB? MOG I@ NB? $&#& ;>>?> G;MM ;H> NB? I<M?LP?> WOR ACP?M NB? Q;HN?> WOR 1B?
;>>?> G;MM CM
7::;: C7II 077;>=A32 07<
.
P < 7 : 077;>=A32 07<
.
7 :<
NB? H?Q I<M?LP?> WOR CM
3
E8I;HL;: < BKN 07<
M Q? Q;HN
7::;: C7II
077;>=A32 07<
4
" (1 !40 ?
J E8I;HL;: < BKN J M7DJ;: < BKN
Q? B;P? NI =BIIM?
" (1 !40 ?
+
<
" (1 !40 ?
B 7 :
1BCM FI=;F P;FO? I@ CM =IHMC>?L?> IHFS @IL ?HNL;H=?M C ? QB?H " (1 !40 ? CM
H?A;NCP? NB?H CM ;FQ;SM <?NQ??H ;H> @N?L ;JJFSCHA NB? >CMNLC<ONCP? M=B?G? NB? P;FO? I@ NB? NL;=?L CHCNC;FFS M?N ;N NB?
<IOH>;LS 07<
077;>=A32 07< QCFF B;P? <??H MFCABNFS GI>CV?> <S NB? ;FAILCNBG
<ON MNCFF ?HMOL?M GIHINIHC=CNS ;H> NB? =ILL?=N JL?M=LC<?> WOR QCFF <? I<M?LP?> C@ ;
G;MM <;F;H=? CM >IH?
(>DA24B >5 CA024A F8C7 C74 (*% 03E42C8>= B274<4
4? B;P? N;E?H NB? ?R;GJF? I@ ; >CMNLC<ONCP? M=B?G? @IL IOL ?RJF;H;NCIHM CH NB? F;MN
NQI MO<M?=NCIHM 1B? NB?ILS QCNB ; 02-$ ;>P?=NCIH M=B?G? M?? 0?=NCIH QIOF>
<? MCGCF;L <ON QCNB ; @?Q GI>CV=;NCIHM >?N;CF?> B?L?;@N?L >O? NI NB? @;=N NB;N NBCM
M=B?G? CM CGJFC=CN NB? NL;=?L =IHMC>?L?> <?CHA ( 0 <
( 0 < 1B? G?MB OM?>
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
@IL NB? ;>P?=NCIH ?KO;NCIH GOMN <? N;E?H ;N NCG? CH NCG? I@ NB? NL;=?L CM NBOM CH NB? NL;HM@ILG?> G?MB
.
<
<
( P <
0
( P
J (
J<
(
J< 1B? >?LCP;NCP?
0 < 7 :
IHM?KO?HNFS CH NB? JLII@ I@ NL;=?L G;MM =IHM?LP;NCIH "KO;NCIH CM HI FIHA?L
GOFNCJFC?> <S 07< <ON <S ( 07<
( 07< MI NB;N NB? =IG<CH;NCIH ;H> MOG I@
;FF ?KO;NCIHM @ILG IH NB? F?@N B;H> MC>? NB? N?LG
.
P < 0 < P < 0 :
J M QCFF <? M??H F;N?L NB? 02-$ NB?ILS CHNLI>O=?M ;H ?RNL; N?LG CHNI NB? NL;=?L
?KO;NCIH CH NB? @ILG
.
K
P 1 =H7: 0 < A =H7: 7 :
K
IH NB? LCABN B;H> MC>? <ON NBCM N?LG CM =;H=?FF?> QB?H MOGG?> IP?L ;FF <;M?M 7
<?=;OM? NB?CL MOG ?KO;FM M CH IOL JLII@ /+1 .! ? CH NB? =IHNCHOCNS ?KO;NCIH CM HIQ GOFNCJFC?> <S
<
( 07< NB? MIOL=? I@ NB? NL;=?L CM ;==IL>CHAFS GI>CV?> CH NB? NL;=?L
( 07
?KO;NCIH
/+1 .!:0 . ? 0A13 <
( 07
(
07< /+1 .! ?
8A827;4C 1>D=30AH 2>=38C8>=B F8C7 0 (*% 03E42C8>=
B274<4
&H NB? =;M? I@ ;H CGJFC=CN ;>P?=NCIH M=B?G? ;>P?=NCIH ;H> >C_OMCIH ;L? MIFP?> NI
A?NB?L 1B? >C^=OFNS I@ ?HMOLCHA ; =ILL?=N WOR =ILL?MJIH>CHA NI JL?M=LC<?> <IOH>;LS
=IH>CNCIHM CM NB;N NB? WOR =;HHIN <? $&#& EHIQH ;M CN Q;M QCNB ;H ?RJFC=CN M=B?G?
H $#'(&#& =ILL?=NCIH CM NBOM H?=?MM;LS %IQ?P?L Q? =BIIM? NI MN;LN QCNB NB? $&#& =ILL?=NCIH I@ NB? >CMNLC<ONCP? M=B?G? NBOM ;>>CHA NB? CH=L?G?HN 0 < NI 0 < ;M
CH "KO;NCIH 1B?H NB? ;>P?=NCIH\>C_OMCIH ;FAILCNBG CM ;JJFC?> MN;LNCHA @LIG
07< 0 < ;H> SC?F>M ; L?MOFN >?HIN?> 0 "P?HNO;FFS ;H CH=L?G?HN 0 < CM ;>>?>
NI 0 MO=B NB;N
7 FEIJ;H?EH? 7::;: C7II
0 <
.
7 :<
J " (1 !40 ?
07<
0 < 07
M NB? I<M?LP?> WOR CM
E8I;HL;: < BKN J 07<
;H> ;M
0 <
07 07< 0 <
" (1 !40 ?
174 HYDRODYNAMICS OF FREE SURFACE FLOWS
a priori added mass =
∆Tin
Ωn
Ψi dΩn = −∆t FLUEXT(i) Tiimposed − Tin − ∆T n
we can check that we recognize the required property:
a priori added mass + a posteriori added mass + observed f lux
= −Tiimposed ∆t F LU EXT (i)
which is the wanted flux.
5.8
5.8.1
Treatment of uncovered beds in 3D
Option 1: correction of free surface gradients
This option used in 2D has been considered impossible for a long time in 3D, on
account of severely distorted elements which appear when, in the absence of water,
the free surface and the bottom become identical. It appeared later that the work with
these severely distorted elements was manageable on condition that some divisions
by 0 were avoided. For example, to calculate the vertical average of the horizontal
velocity, calculation with the trapezoidal rule should be replaced on the exposed beds
by an arithmetic average. Apart from a few numerical problems, for which a remedy
exists, the correction of the free surface gradients is formally the most elegant because,
when a free surface gradient is treated with a resolution of the Saint-Venant equations,
nothing else is left in the actual 3D part of the resolution for both the hydrostatic
hypothesis and the non-hydrostatic equations. Plate 9, shown earlier, shows that
exposure on a beach can be treated in 3D, in this case with 10 planes along the
vertical, with zero water depth on the right side of the calculation domain. In the
non-hydrostatic case, we notice a trough in the free surface. This trough becomes
more prominent when the flow at the exit of the pool which is emptied, becomes
supercritical. The water trapped in the puddle seems to take longer to return to rest.
This is because the dynamic pressure gradient is not subject to any specific treatment
on the exposed banks.
5.8.2
Option 2: masking of the uncovered elements
As in the case of 2D, the columns of dry or partially dry elements have to be extracted
from the calculation domain. This is done with the help of an array equal to 1 or
0, called a mask. The detailed description of this technique is mainly a catalogue of
dissuasive difficulties.
Untimely masking–unmasking
In situations at the limit between covering and exposing, with fluxes of water changing
signs from one time step to another, the elements balance endlessly between a normal
situation and a situation where they are removed from the mesh. In the presence
/"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0
I@ ; <INNIG MFIJ? NB? WIQ CM AOC>?> <S NB? MFIJ? ;H> NBIM? ?F?G?HNM QBIM? <INNIG
?F?P;NCIH CM NB? BCAB?MN QCFF >LS VLMN ,H NB? INB?L B;H> IH ; W;N <INNIG NB? WIQ CM
GO=B GIL? ?LL;NC= ;H> NB? JLI<F?G CM GIL? ;=ON? &N CM NB?L?@IL? H?=?MM;LS NI G;ME
NB? ?HNCL? W;N TIH? QB?H IH? I@ NB? ?F?G?HNM I@ NBCM TIH? B;M NB? =LCN?LCIH @IL G;MECHA
1BCM =BIC=? ;FMI J;LNC;FFS B?FJM CH MIFPCHA NB? @IFFIQCHA >C^=OFNS
$22DAA4=24 >5 B8=6D;0A ?>8=CB
1B? L?GIP;F I@ ;H ?F?G?HN >I?M HIN ;FQ;SM ACP? ; NIJIFIAC=;FFS =ILL?=N G?MB JICHN =;H @ILG ;H CMNBGOM QCNB T?LI QC>NB QB?L? NQI >C_?L?HN =I;MNM NIO=B <ON
QCNBION ; =B;H=? I@ WOR 1BCM NSJ? I@ JICHN MBIOF> <? ?FCGCH;N?> <S ?RN?H>CHA NB?
G;ME?> TIH? ;LIOH> NB? JICHN ;N IH? I@ NB? NQI ;HAOF;L M?=NILM HIN S?N G;ME?> 1BCM
JLI=?MM MBIOF> <? CN?L;N?> NCFF ; =ILL?=N NIJIFIAS CM I<N;CH?> JLCIL ;H;FSMCM I@
NB? NIJIAL;JBS B?FJM NI ;==?F?L;N? NB? JLI=?MM &H ?RJIM?> TIH?M NB? @L?? MOL@;=? CM
=IHMC>?L?> FI=;FFS KO;MC BILCTIHN;F ;H> <S EHIQCHA CNM ?F?P;NCIH ;N ; ACP?H CHMN;HN
;FF NB? >LS IL J;LNC;FFS >LS ?F?G?HNM =;H <? >?>O=?> @LIG CN &H @;=N C@ ;H ?F?G?HN
CM G;ME?> <?=;OM? NB? Q;N?L >?JNB CM <?FIQ ; ACP?H =LCN?LCIH ;FF NB? H?CAB<IOLCHA
?F?G?HNM QCNB ; AL?;N?L IL ?KO;F <INNIG ?F?P;NCIH MBIOF> ;FMI <? G;ME?> ;H> MI IH
@LIG IH? NI NB? H?RN 2H@ILNOH;N?FS NBCM BSJINB?MCM CM HIN P;FC> CH TIH?M QCNB BCAB
P;LC;NCIHM CH NB? @L?? MOL@;=? <ON CN B?FJM NI ?FCGCH;N? NB? LCME I@ ; MCHAOF;L JICHN C@
@LIG NB? MN;LN =?LN;CH L?AOF;LCNS CM ;MMOL?> CH NB? <;NBSG?NLS &H JL;=NC=? CN B;M NI <?
?HMOL?> NB;N <S FI=;FFS =B;HACHA ; <INNIG ?F?P;NCIH C@ H?=?MM;LS C@ Q? NOLH ;LIOH>
;H CHN?LH;F JICHN CH NB? >IG;CH <S AICHA IP?L NB? ?F?G?HNM NI QBC=B CN <?FIHAM Q?
J;MM IP?L ; MCHAF? GCHCGOG ;H> ; MCHAF? G;RCGOG I@ NB? <INNIG
A8C4A8>= 5>A <0B:8=6
1B? NBCL> >C^=OFNS ?H=IOHN?L?> =IH=?LHM >?N?LGCHCHA NB? =LCN?LCIH @IL G;MECHA 1B?L?
CM HI =LCN?LCIH @IL OHG;MECHA MCH=? CH NB? ;FAILCNBG Q? MN;LN <S OHG;MECHA ;FF NB?
?F?G?HNM 1B? =LCN?LCIH @IL G;MECHA H??>M NB? =IGJON;NCIH I@ NB? Q;N?L >?JNB J?L
?F?G?HN QBC=B =;HHIN <? F?MM NB;H ; ACP?H NBL?MBIF> P;FO? 1BCM =;F=OF;NCIH I@ NB?
B?CABN IHFS L?KOCL?M ;H ?MNCG;NCIH I@ NB? @L?? MOL@;=? ?F?P;NCIH @IL ?;=B ?F?G?HN QBC=B
CM =IGJ;L?> QCNB IH? <INNIG ?F?P;NCIH J?L ?F?G?HN 1B? VLMN C>?; QIOF> <? NI L?N;CH
NB? ?F?P;NCIH I@ NB? FIQ?MN @L?? MOL@;=? I@ NB? NBL?? P?LNC=?M I@ ?;=B NLC;HAF? CH IL>?L
NI <? =?LN;CH I@ G;MECHA ;FF NB? ?F?G?HNM NB;N L?KOCL? G;MECHA %IQ?P?L C@ NB? P?LN?R
CH KO?MNCIH CM MOLLIOH>?> <S G;ME?> ?F?G?HNM NB?L? CM HI =B;H=? NB;N CNM ?F?P;NCIH
QCFF =B;HA? ;H> NB? G;ME?> ?F?G?HNM QCFF L?G;CH NB? M;G? QB;N?P?L NB? ?PIFONCIH I@
NB? MCNO;NCIH ;LIOH> NB?G 1I ;PIC> NBCM IL NI FCGCN NBCM LCME ;M @;L ;M JIMMC<F? NB?
?F?P;NCIH I@ NB? @L?? MOL@;=? J?L ?F?G?HN CM N;E?H ;M NB? ;P?L;A? I@ NB? ?F?P;NCIHM I@
NB? NBL?? P?LNC=?M 1B?L?OJIH ;FGIMN ;FF NB? ?F?G?HNM <?=IG? OHG;ME?> QB?H NB?
F?P?F I@ NB? @L?? MOL@;=? LCM?M ?R=?JN CH MIG? MJ?=CV= JF;=?M QCNB QBC=B Q? B;P? HIN
S?N >?;FN MO=B ;M <;MCHM QCNB ; MCHAF? HI>? I@ G?MB ;N NB? =?HNL? &@ MIG? CMIF;N?>
?F?G?HNM B;P? HIN <??H =ILL?=NFS OHG;ME?> ;N NB? NCG? I@ LCM? CH F?P?F I@ NB? @L??
MOL@;=? Q? MBIOF> L?VH? NB? G?MB FI=;FFS ; P?LS =OG<?LMIG? MIFONCIH @IL ; MNO>S
NB;N CM ;FL?;>S OH>?L Q;S IL L?MILN NI ; MCGJFCV=;NCIH I@ NB? <;NBSG?NLS
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
)A40C<4=C >5 <0B:43 =>34B
#ILG;FFS ;M NB? N?=BHCKO? I@ G;MECHA JL?M?HN?> B?L? =IHMCMNM I@ ?R=FO>CHA @LIG NB?
=;F=OF;NCIH >IG;CH NBIM? ?F?G?HNM NB;N VN NB? =LCN?LCIH G?HNCIH?> ;<IP? NB?L? G;S
<? HI>?M MOLLIOH>?> <S G;ME?> ?F?G?HNM =;FF?> G;ME?> HI>?M @IL QBC=B NB? ?KO;
NCIHM ;L? HI FIHA?L L?MIFP?> 1B? P;LC;<F?M CH NB?M? HI>?M K L 0 A ;L? NB?H
;L<CNL;LCFS GI>CV?> 4B?H OHG;MECHA NB?M? HI>?M NB?M? P;FO?M ;L? IH=? ;A;CH N;E?H
CHNI ;==IOHN CH NB? =;F=OF;NCIH I@ NB? MIFONCIH 1I L?G?>S NBCM MBILN=IGCHA Q? MBIOF>
L?N;CH >OLCHA NB? ?HNCL? >OL;NCIH I@ NB? G;MECHA I@ ; HI>? NB? P;FO?M I@ NB? P;LC;<F?M
=;F=OF;N?> DOMN <?@IL? NBCM G;MECHA 0I CN CM NB? CH=L?G?HNM I@ NB? P;LC;<F?M ;H> HIN
NB? P;LC;<F?M NB?GM?FP?M QBC=B ;L? ;L<CNL;LCFS @IL=?> NI T?LI
70?C4A (>;E8=6 CA0=B?>AC 4@D0C8>=B
1B? NL?;NG?HN I@ NL;HMJILN ?KO;NCIHM CM IH? I@ NB? MNL;N?AC= JICHNM CH NB? HOG?LC=;F
L?MIFONCIH I@ WOC> G?=B;HC=M ?KO;NCIHM 1B? BSJ?L<IFC= H;NOL? I@ NB?M? ?KO;NCIHM
CGJFC?M NB;N NB? MJ;=? >?LCP;NCP?M H??> ;H OJQCH> NL?;NG?HN HOG<?L I@ HOG?LC=;F
G?NBI>M NB;N B;P? <??H >?P?FIJ?> IH MNLO=NOL?> G?MB?M ? A .OC=E?MN 2FNCG;N?
+CLP;H; <S )?IH;L> =;HHIN <? ;JJFC?> NI NB? VHCN? ?F?G?HNM <?=;OM? NB?S ;L? >?
J?H>?HN IH G?MB L?AOF;LCNS 4? MB;FF >?M=LC<? <?FIQ NBL?? G?NBI>M NB;N =;H <?
;JJFC?> NI NB? VHCN? ?F?G?HNM
1B? G?NBI> I@ =B;L;=N?LCMNC=M
1B? 02-$ 0NL?;GFCH? 2JQCH> -?NLIP\$;F?LECH G?NBI> NB;N Q? MB;FF ;FMI
;JJFS NI NB? =IHNCHOCNS ?KO;NCIH
!CMNLC<ONCP? M=B?G?M QBC=B ;L? ; FCHE QCNB N?=BHCKO?M NB;N ;L? OM?> G;CHFS @IL
VHCN? PIFOG?M
"4C7>3 >5 270A02C4A8BC82B
1B? G;CH ;>P;HN;A?M I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M ;L? OH=IH>CNCIH;F MN;<CFCNS ;H>
GIHINIHC=CNS I@ NB? MIFONCIHM &H CNM ZQ?;E[ @ILG G;MM =IHM?LP;NCIH MBIOF> ;FMI <?
;>>?> 1B? <;MC= JLCH=CJF? CM NI N?GJIL;LCFS ;>IJN ; );AL;HAC;H ;JJLI;=B ;H> NI
QLCN? NB;N NB? P;FO? I@ ; @OH=NCIH ;N NCG? J< IH ; HI>? ) I@ NB? G?MB CM ?KO;F NI
NB? P;FO? NB;N NB? M;G? @OH=NCIH B;> ;N NCG? J< ;H> ;N JICHN - C>?HNCV?> <S AICHA
<;=EQ;L>M CH NCG? ;FIHA NB? NL;D?=NILS @LIG JICHN ) ;H> @IL ; F?HANB I@ NCG? J
1BOM /C?G;HHYM CHP;LC;HN CM OM?> CH ;MMI=C;NCIH QCNB NB? =B;L;=N?LCMNC= ?KO;NCIH
:
N
1
:J
4? =;FF NB? JICHN - ZNB? @IIN I@ NB? =B;L;=N?LCMNC=[
#IL ?;=B JICHN ) NB? G?NBI> I@ =B;L;=N?LCMNC=M =IHMCMNM I@ NQI MN?JM
#CH>CHA NB? @IIN I@ NB? =B;L;=N?LCMNC= JICHN - <S =L?;NCHA NB? NL;D?=NILS NB;N
J;MM?M <S ) ;N NB? GIG?HN J<
&HN?LJIF;NCIH QCNBCH NB? ;LLCP;F ?F?G?HN
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;F=OF;NCHA NB? NL;D?=NILS JIM?M NB? JLI<F?G I@ NCG? IL>?L
QBC=B MBIOF> <? OM?> NI =;F=OF;N? NB? NL;D?=NILS CM
1 J 1<
J J< <
1
J
1B? P?FI=CNS V?F>
2H@ILNOH;N?FS NB? ;>P?=NCIH V?F> CM OHEHIQH ;N NCG? J< <?@IL? NB? ?H> I@ NB?
NCG? MN?J 1B?L? ;L? NQI IJNCIHM ?CNB?L IH? =BIIM?M NB? ;>P?=NCIH V?F> ;N NCG? J<
BSJINB?MCM I@ ; VR?> V?F> IL IH? =BIIM?M NI ;JJLI;=B NB? ;>P?=NCIH V?F> ;N NCG?
J<
;H> NB? M;G? NCG? MN?J CM L?J?;N?> M?P?L;F NCG?M 1B? F;NN?L IJNCIH CM NCG?
=IHMOGCHA <?=;OM? NB? HOG<?L I@ CN?L;NCIHM L?KOCL?> GOFNCJFC?M NB? ?R?=ONCIH NCG?
1B? BSJINB?MCM I@ ; VR?> V?F> CM NBOM L?N;CH?> @IL A?H?L;F =;M?M *IL?IP?L QCNBCH
NB? @L;G?QILE I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M NB? HINCIH I@ NCG? IL>?L CM =IGJF?R
<?=;OM? CN CM ;FMI >?J?H>?HN IH NB? =;F=OF;NCIH I@ NB? =B;L;=N?LCMNC=M
1B? =B;L;=N?LCMNC=M NL;D?=NILC?M ;L? =;F=OF;N?> <S ; /OHA?\(ONN; IL>?L G?NBI>
;H> CHN?LJIF;NCIH ;N NB? @IIN I@ NB? =B;L;=N?LCMNC= =IH@ILGM NI NB? NSJ? I@ VHCN? ?F?G?HN
=BIM?H 1B? M?=IH> IL>?L /OHA?\(ONN; G?NBI> B;M ;FL?;>S <??H N?MN?> <ON >I?M HIN
;>> ;HS MCAHCV=;HN CGJLIP?G?HN CN CM ;FMI ?RJ?HMCP? BCAB?L IL>?L CHN?LJIF;NCIH
QIOF> =?LN;CHFS <LCHA ;<ION ; AL?;N?L CGJLIP?G?HN NB;H ; BCAB?L IL>?L /OHA?\(ONN;
G?NBI>
)A0942C>AH 20;2D;0C8>=
%?L? NB? JLCH=CJF? CM ?RJF;CH?> CH NQI >CG?HMCIHM ";=B NCG? MN?J J CM CNM?F@ >C
B
PC>?> CHNI MO< NCG? MN?JM <AC0
>OLCHA QBC=B IH? =IHMC>?LM NB;N NB? P?FI=CNS L?G;CHM
=IHMN;HN DIK8 CM =;F=OF;N?> FI=;FFS MI NB;N CN N;E?M @IL ?R;GJF? NBL?? MO<MN?JM NI
J;MM NBLIOAB ?;=B ?F?G?HN &@ 4AB/@B ;H> 5AB/@B ;L? NB? =IIL>CH;N?M I@ ; JICHN
IH
NB? NL;D?=NILS <?CHA =;F=OF;N?> NB? H?RN JICHN QCNB =IIL>CH;N?M 4/@@7D/: 5/@@7D/:
CM @IOH> <S G?;HM I@ NB? @IFFIQCHA @ILGOF;? NB? GCHOM MCAH CM OM?> <?=;OM? IH? CM
GIPCHA <;=EQ;L>M @LIG NCG? J< NI NCG? J< 4/@@7D/: 4AB/@B KAB/@B J
DIK8
J
DIK8
M NBCM JICHN CM
QB?L? KAB/@B ;H> LAB/@B ;L? NB? P?FI=CNS =IGJIH?HNM I@ JICHN
HIN ;FQ;SM ; >IG;CH HI>? KAB/@B ;H> LAB/@B ;L? I<N;CH?> <S CHN?LJIF;NCIH QCNBCH ;H
?F?G?HN 1B? G;CH JLI<F?G CH VHCN? ?F?G?HNM CM NI JCHJICHN NB? ;LLCP;F JICHN NB;N
CM NI M;S NI FI=;N? QBC=B ?F?G?HN CN <?FIHAM NI &N CM ?MM?HNC;F NI EHIQ NBCM ?F?G?HN
CH IL>?L NI VH> NB? P?FI=CNS ;N NBCM JICHN <S CHN?LJIF;NCIH )I=;FCT;NCIH ;N NB? @IIN
I@ NB? =B;L;=N?LCMNC= - CM ;FMI H?=?MM;LS CH IL>?L NI CHN?LJIF;N? NB? KO;HNCNC?M OH>?L
;>P?=NCIH
4BCF? =;F=OF;NCHA NB? NL;D?=NILS NQI ?P?HNM =;H I==OL <?@IL? NB? ;LLCP;F ;N NCG?
J< 5/@@7D/: 5AB/@B LAB/@B ,H? L?;=B?M ; FCKOC> <IOH>;LS CH NBCM =;M? NB? NL;D?=NILS NB?IL?NC=;FFS =IHNCHO?M
IONMC>? NB? >IG;CH M NB? CH@ILG;NCIH NI =IHNCHO? NB? =;F=OF;NCIH I@ NBCM
NL;D?=NILS CM F;=ECHA NB? JICHN - CM N;E?H ;N NB? JICHN I@ CHN?LM?=NCIH I@ NB?
0,)3&+$ 1/+0-,/1 ".21&,+0 <IOH>;LS ;H> NB? NL;D?=NILS 1B? CHN?LJIF;NCIH MBIOF> <? >IH? ;M JL?=CM?FS ;M
JIMMC<F? ;N NB? GIG?HN I@ F?;PCHA NB? >IG;CH QBC=B CM CHN?LG?>C;LS <?NQ??H
J< ;H> J<
,H? L?;=B?M ; LCAC> <IOH>;LS NBCM NB?IL?NC=;FFS CGJIMMC<F? =;M? =;H I==OL >O?
NI ;JJLIRCG;NCIHM CH NB? /OHA?\(ONN; G?NBI> 1B? JICHN I@ CHN?LM?=NCIH I@ NB?
NL;D?=NILS QCNB NB? <IOH>;LS NB?H L?JF;=?M JICHN QBC=B QIOF> @;FF IONMC>?
NB? >IG;CH 1B? P?FI=CNS V?F> CM NB?H JLID?=N?> IHNI NB? <IOH>;LS CH IL>?L NI
;PIC> ;HINB?L ?RCN I@ NB? >IG;CH ;N NB? H?RN MO< NCG? MN?J
+;PCA;NCIH ;H> FI=;FCT;NCIH CH NB? G?MB >OLCHA NB? =IHMNLO=NCIH I@ NB? =B;L;=
N?LCMNC=M CM NB? =ILH?LMNIH? I@ NB? QBIF? G?NBI> ,H? ?GJFISM ; >;N; MNLO=NOL? NB;N
;FFIQM NB? HOG<?LM I@ ?F?G?HNM FI=;N?> <?BCH> ?;=B MC>? I@ ; ACP?H ?F?G?HN NI <? C>?H
NCV?> M?? NB? &#,/ N;<F? CH 0?=NCIH 1B? ;FAILCNBGC= A?IG?NLS L?KOCL?> ;H>
NB? G;H;A?G?HN I@ NLOH=;NCIH ?LLILM G;E? NB? G?NBI> I@ =B;L;=N?LCMNC=M ; >C^=OFN
N?=BHCKO? NI OM?
=C4A?>;0C8>=
M CH>C=;N?> ;<IP? CHN?LJIF;NCIH CM =IGJ;NC<F? QCNB NB? NSJ? I@ ?F?G?HN =BIM?H CH NB?
M?=IH> MN;A? I@ NB? =;F=OF;NCIH &@ NB?M? CHN?LJIF;NCIHM ;L? @IOH> NI <? NII >C_OMCP?
@IL =?LN;CH =;M?M I@ ;JJFC=;NCIH CN QIOF> NB?H <? H?=?MM;LS NI =IHMC>?L BCAB?L IL>?L
?F?G?HNM #IL GIL? CH@ILG;NCIH ;<ION NB? G?NBI> I@ =B;L;=N?LCMNC=M IH? =;H L?@?L NI
NB? @IFFIQCHA QILEM 89 ;H> 89 >P?=NCIH JLIAL;GG?M @IFFIQCHA NB? G?NBI> I@
=B;L;=N?LCMNC=M B;P? NI <? =;L?@OFFS IJNCGCT?> ;H> =;H <? J;LNC;FFS P?=NILCT?>
&GJILN;HN HIN? NBCM JL?M?HN;NCIH CM =IHMCMN?HN QCNB NB? HINCIH I@ @L;=NCIH;F MN?JM
<ON ACP?M HI CH>C=;NCIH ;<ION NB? NL?;NG?HN I@ INB?L N?LGM QBC=B HILG;FFS VAOL? CH NB?
LCABN B;H> MC>? I@ NB? ?KO;NCIHM 1B?M? =;H CH @;=N <? CH=FO>?> CH NB? =B;L;=N?LCMNC=M
NL?;NG?HN &H NBCM =;M? NB? P;LC;NCIH I@ ; @OH=NCIH < ;FIHA CNM =B;L;=N?LCMNC= =OLP?
>OLCHA NB? NCG? CHN?LP;F N;E?H <?NQ??H J< ;H> J< CM HI FIHA?L T?LI <ON ?KO;F NI
+ B
EJ>;H J;HCI :J QB?L? NB? HIN;NCIH MCAHCV?M NB;N NB?M? INB?L N?LGM ;JJFS
B
NI NB? J;LNC=F? NL;HMJILN?> <S NB? =OLL?HN ;H> NBOM L?@?LM NI NB? =B;L;=N?LCMNC= =OLP?
,H? ;FMI M??M GIL? =F?;LFS NB? H??> @IL ; M?GC CGJFC=CN;NCIH I@ ;FF NB? N?LGM NB;N >I
HIN <?FIHA NI NB? );AL;HAC;H >?LCP;NCP? &@ NB? ?RJL?MMCIH I@ NB?M? N?LGM ;N NCG? J<
JIM?M HI JLI<F?G IH? HINC=?M IH NB? INB?L B;H> NB;N NB? ?RJFC=CN N?LGM N;E?H ;N
NCG? J< MBIOF> <? N;E?H ;N NB? @IIN I@ NB? =B;L;=N?LCMNC=
"4C7>3 >5 270A02C4A8BC82B 8= F40: 5>A<D;0C8>=
1B? NQI G;CH MBILN=IGCHAM I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M ;L? NB? >C_OMCP? H;NOL?
;H> NB? F;=E I@ ?R;=N G;MM =IHM?LP;NCIH 1B?M? NQI JLI<F?GM ;L? =ILL?=N?> <S ;
ZQ?;E[ ;JJLI;=B QB?L? NB? G?NBI> CM ;JJFC?> QCNBCH ; @L;G?QILE I@ VHCN? ?F?G?HNM
?RN?H>?> NI NCG? 1BCM G?NBI> Q;M VLMN JL?M?HN?> CH <S ?HKO? ( 89
GIL? >?N;CF?> >?M=LCJNCIH =;H <? @IOH> CH L?@?L?H=? 89 %?L? Q? MB;FF L?MIFP? ;
=IHM?LP;NCP? NL;HMJILN ?KO;NCIH
<
J
:?L < K %6!/,!6+*& 0 ,# #/"" 02/# " #),40
QBC=B ;@N?L P;LC;NCIH;F @ILGOF;NCIH ?RN?H>?> NI NCG? <?=IG?M
.
B
:J
B
.
<
J
7 : :?L < K
2JIH CHN?AL;NCIH <S J;LNM NBCM ?KO;NCIH F?;>M @ILG;FFS NI
.
<
<
<7
: .
<
<
<7 :
.
.
:J
B
B
:J
B
B
.
.
<
7
J
K =H7: 7
7 < K D :
:
&H NBCM ?KO;NCIH IH? B;M CHN?HNCIH;FFS >CMNCHAOCMB?> NB? N?MN @OH=NCIH 7 ;N NCG?
J< ;H> ;N NCG? J<
&H @;=N NB? ILCACH;FCNS I@ NBCM G?NBI> FC?M CH CNM GI<CF? G?MB
QBC=B @IFFIQM NB? Q;N?L J;LNC=F?M CH NB?CL GIP?G?HN >OLCHA NB? NCG? MN?J J 2H>?L
NB?M? =IH>CNCIHM IH? =;H M;S NB;N NB?L? CM ;H ;>P?=NCIH I@ NB? N?MN @OH=NCIHM ;H> NBOM
NB;N
7 K =H7: 7 J
%?L? IH? CM >?;FCHA QCNB ; HIH =IHM?LP;NCP? @ILG I@ NB? ;>P?=NCIH ?KO;NCIH @IL NB?
N?MN @OH=NCIHM NBIOAB Q? MN;LN?> @LIG ; =IHM?LP;NCP? @ILG "KO;NCIH =;H NB?H <?
MCGJFCV?> NI
.
<
<
<7
: .
<
<
<7 :
.
B
B
:J
.
7 < K D :
1B? ?PIFONCIH I@ NB? N?MN @OH=NCIHM CM >IH? @LIG NCG? J< NI NCG? J< ;H> IH? =IH
MC>?LM NB;N NB? G?MB ;N NCG? J< =ICH=C>?M QCNB NB? =F;MMC= G?MB 1B? N?MN @OH=NCIHM
;N NCG? J< ;L? NBOM >?P?FIJ?> <S AICHA OJ NB? =B;L;=N?LCMNC= =OLP?M ;H> <S =;F=OF;NCHA
NB? JIMCNCIH I@ NB?CL @IIN ;N NCG? J< 1BCM CM >IH? <S NB? =F;MMC= G?NBI> I@ =B;L;=
N?LCMNC=M ZMNLIHA[ @ILG 1B? F?@N B;H> MC>? I@ "KO;NCIH CM ; Q?FF EHIQH N?LG
; G;MM G;NLCR GOFNCJFC?> <S NB? P?=NIL I@ OHEHIQHM <7<
,H NB? LCABN B;H> MC>?
NB?L? CM ; N?LG >?HINCHA ; WOR
NBLIOAB
NB?
<IOH>;LS
QBC=B
CM ACP?H <S NB? <IOH>
+
;LS =IH>CNCIHM ;H> NB? N?LG < < <7 : QBC=B CM JLI<F?G;NC= <?=;OM? CN CM ?KO;F NI
)<>=7< + < <
< 8 <7 : 1B? >?=IGJIMCNCIH I@ < < CM ?_?=NCP?FS EHIQH IH NB? <;MCM
8
8
8 I@ NB? =F;MMC=+ G?MB 1B?L?@IL? IH? B;M NI EHIQ BIQ NI =;F=OF;N? ; G;NLCR QBIM?
A?H?L;F N?LG CM <7 <8 : QB?H NB? <;M?M ;L? MN;AA?L?> <?NQ??H NB? NCG? J< ;H>
NB? NCG? J<
1B? =;F=OF;NCIH >IG;CH CM NBOM ; OHCIH I@ CHN?LM?=NCIHM I@ NQI NLC
;HAF?M ;L<CNL;LCFS F;C> IH NB? G;J /?@?L?H=? 89 JLIJIM?M ; G?;HM I@ =;F=OF;NCHA
NB?M? N?LGM JL?=CM?FS <S L?@?LLCHA NI NB? >CMNLC<ONCIH NB?ILS %IQ?P?L NBCM G?NBI>
CM ?RJ?HMCP? ;H> M?HMCNCP? NI =IGJON?L A?H?L;N?> NLOH=;NCIH ?LLILM &H %;M<;HC
( 89 =;G? OJ QCNB ;H ILCACH;F <ON ;JJLIRCG;N? MIFONCIH NB;N =IHMCMN?> I@ =;F=O
F;NCHA NB? CHN?AL;F QCNB ; $;OMM KO;>L;NOL? 1BCM N?=BHCKO? Q;M ;>IJN?> CH NB? 20
QB?L? NB? G?NBI> I@ =B;L;=N?LCMNC=M CH CNM Q?;E @ILG CM OM?> OH>?L NB? H;G? "))*
"OF?LC;H );AL;HAC;H )I=;FCT?> >DICHN *?NBI>
0,)3&+$ 1/+0-,/1 ".21&,+0 Galerkin test function
SUPG test function
1
x i-1
xi
x i+1
DX
#CAOL? ; 02-$ N?MN @OH=NCIH
(CA40<;8=4 *?F8=3 %4CA>EO0;4A:8= <4C7>3
1B? JLCH=CJF? I@ NB? 02-$ G?NBI> >?P?FIJ?> <S LIIEM ;H> %OAB?M 89 CM NI OM?
N?MN @OH=NCIHM >?@ILG?> <S NB? >CL?=NCIH I@ NB? =OLL?HN 4CNB NB? -?NLIP\$;F?LECH
G?NBI> NB?M? N?MN @OH=NCIHM ;L? I<N;CH?> QCNB KO;>L;NC= @OH=NCIHM NB? ILCACH;FCNS
I@ NB? 02-$ G?NBI> CM NI ?GJFIS FCH?;L <ON >CM=IHNCHOIOM @OH=NCIHM NBOM NB? N?MN
@OH=NCIH 67 CM L?JF;=?> <S
67
K 7 A =H7: 67
K 7
QB?L? K 7 CM NB? P?FI=CNS N;E?H ;N HI>? ? I@ NB? >IG;CH ;H> A ; =I?^=C?HN QBC=B CM
S?N NI <? >?VH?> ,H ; IH? >CG?HMCIH;F G?MB NB? MB;J? I@ NB?M? H?Q N?MN @OH=NCIHM
CM ;M ACP?H CH #CAOL? &H NB?ILS NB?M? N?MN @OH=NCIHM MBIOF> <? ;JJFC?> NI ;FF NB? N?LGM I@ ;H ?KO;NCIH
@IL NB? M;E? I@ =IH@ILGCNS QCNB NB? VHCN? ?F?G?HN JLCH=CJF? 4? B;P? <??H =IGJ?FF?>
NI GI>C@S NBL?? JICHNM I@ NB? G?NBI> <?=;OM? I@ NB? MJ?=CV=CNC?M I@ IOL ?KO;NCIHM
1B? 02-$ N?MN @OH=NCIH =;HHIN <? ;JJFC?> NI ;FF NB? N?LGM I@ NB? =IHNCHOCNS
?KO;NCIH &N M??GM NB;N CNM ;JJFC=;NCIH NI JLIJ;A;NCIH N?LG :?L >
K JLI>O=?M ;<?L
L;HN N?LGM 1B?L? CM ; JLI<F?G S?N NI <? L?MIFP?> JLI<;<FS >O? NI NB? @;=N NB;N NB?
JLIJ;A;NCIH N?LGM MBIOF> HIN <? MO<D?=N NI ;H OJQCH> NL?;NG?HN
1B? P?=NIL K 7 P?=NIL ?KO;F NI NB? HI>;F P;FO? I@ NB? P?FI=CNS P?=NIL ;N JICHN ?
B;> NI <? L?JF;=?> ?CNB?L <S ; P?=NIL K 3: =IHMN;HN <S ?F?G?HN NBCM =;H @IL ?R;GJF?
<? NB? ;P?L;A? P?FI=CNS ;N ?;=B HI>? IL <S NB? L?;F P?=NIL P?FI=CNS K N;E?H ;N NCG? J<
QBC=B CM CNM?F@ ; P;LC;<F? @OH=NCIH CH MJ;=? 1BCM GCHIL GI>CV=;NCIH >I?M HIN =B;HA?
NB? MN;<CFCNS ;H;FSMCM <ON CM ?MM?HNC;F CH IL>?L NI I<N;CH ;H ?R;=N =IHM?LP;NCIH I@ NB?
G;MM 1B? NQI IJNCIHM K ;H> K 3: B;P? <??H JLIAL;GG?> ;H> NB? GIL? ?F?A;HN ;H>
J;L;>IRC=;FFS =B?;J?L =BIC=? I@ K Q;M VH;FFS L?N;CH?> &H NB? 02-$ @ILGOF;NCIH
K 7 QCFF NBOM B?H=?@ILNB <? L?JF;=?> <S K
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? =BIC=? I@ A NB? IH? LIIEM ;H> %OAB?M L?=IGG?H>?> L?MOFNM @LIG NB?
B?;N ?KO;NCIH ;H;FSMCM QBC=B >I?M HIN ;JJFS NI IOL =;M? ,OL =BIC=? I@ A QCFF HIQ <?
?RJF;CH?>
&N CM Q?FF EHIQH NB;N IH? =;H I<N;CH ;H OJQCH> ;>P?=NCIH N?LG I@ M?=IH> IL>?L
;==OL;=S <S ;>>CHA ;H ;LNCV=C;F >C_OMCIH C, NI ; =?HNL?> N?LG )?N OM HIQ FIIE
;N NB? ?_?=N I@ 02-$ <;M?M IH VHCN? ?F?G?HNM QB?H NB?S ;L? ;JJFC?> IHFS NI NB?
;>P?=NCIH N?LG B?L? Q? N;E? NB? ?R;GJF? I@ ;>P?=NCIH I@ ; @OH=NCIH < 1BOM
+ K =H7: < 67 : <?=IG?M
.
K :
K =H7: <
67 A =H7: 67
K
IGJ;L?> QCNB NB? =?HNL?> =;M? Q? NBOM B;P? NI ;>> NB? N?LG
.
K A
K =H7: < =H7: 67 :
K
QBC=B NB?H GI>CV?M NB? G;NLC=?M ;H> NB? LCABN B;H> MC>?M <?=;OM? I@ NB? M?GC
CGJFC=CN;NCIH I@ < 1BCM N?LG CH IH? >CG?HMCIH CM FCE? ; >C_OMCIH =I?^=C?HN AK
1BOM Q? M?? NB;N NB? =I?^=C?HN A QBC=B ACP?M NB? >C_OMCIH C, QBC=B ;>>M ;H OJ
QCH> NL?;NG?HN NI ; =?HNL?> ;>P?=NCIH N?LG CM A , 4? MB;FF B?H=?@ILNB QLCN?
A CH NB? @ILG A , QB?L? CM NB? OJQCH> =I?^=C?HN QBC=B =BIC=? H??>M NI <?
=F;LCV?>
1B? MOJJF?G?HN;LS N?LG L?MOFNCHA @LIG 02-$ =;H <? CHN?LJL?N?> ;M ;H ;LNCV=C;F
>C_OMCIH NB;N MN;<CFCT?M ; =?HNL?> M=B?G? &H NQI >CG?HMCIHM <?=;OM? I@ NB? ?_?=N I@
>IN JLI>O=NM NBCM >C_OMCIH ;JJFC?M IHFS CH NB? >CL?=NCIH I@ NB? WIQ 1BLIOAB #IOLC?L
;H;FSMCM IH? =;H MBIQ NB;N CH IL>?L NI ?HMOL? NB? MN;<CFCNS I@ NB? M=B?G? M?? 89 CN
C B
CM MO^=C?HN @IL NB? =I?^=C?HN NI <? ?KO;F NI NB? IOL;HN HOG<?L F A CM NB?H
C B
?KO;F NI
%;PCHA G;>? NBCM =BIC=? CN CM JIMMC<F? NI MBIQ NB;N NB? 02-$ G?NBI> NBOM
GI>CV?> >I?M HIN ;FN?L NB? G;MM =IHM?LP;NCIH I@ IOL G;MM =IHM?LP;NCIH ?KO;NCIH
1B? MOG I@ GI>CV?> <;M?M L?G;CHM ?KO;F NI <?=;OM?
<3:
,
7
67
<3:
,
7
K 4 =H7: 67 K
<3:
4 K , 6
7 =H7:
K
7
4? NB?H HIN? NB;N IH? =;H NL?;N ?;=B N?LG I@ ; =IHM?LP;NCP? ?KO;NCIH QCNB N?MN
@OH=NCIHM I@ >C_?L?HN H;NOL? QCNBION ;FN?LCHA NB? G;MM =IHM?LP;NCIH ;M FIHA ;M NB?
MOG I@ NB?M? N?MN @OH=NCIHM CM ?KO;F NI IH NB? G?MB #IL NB? 02-$ N?MN @OH=NCIHM
NB? F;NN?L KO;FCNS CM ?HMOL?> <S NB? =BIC=? I@ ; =I?^=C?HN QBC=B CM =IHMN;HN J?L
?F?G?HN ;H> <S NB? =BIC=? I@ K CHMN?;> I@ K7
>=B4AE0C8E4 B274<4 5>A C74 2>=C8=D8CH 4@D0
C8>=
1BCM M=B?G? CM <;M?> IH ; =IHM?LP;NCP? =IHNCHOCNS ?KO;NCIH CH NB? @ILG
>
>
C K< K<
:?L >
K :?L >>@=> C J
J
0,)3&+$ 1/+0-,/1 ".21&,+0 &@ IH? OM?M NB? 02-$ G?NBI> NB? N?MN @OH=NCIHM I@ NBCM G?NBI> =;H <? ;JJFC?>
IHFS NI ;>P?=NCIH N?LGM MI NB;N IH? ;=NO;FFS L?MIFP?M
.
.
> 6
:?L >>@=> 67 :
K<
7 :
C
J
.
K K < =H7: >>@=> A =H7: 67 :
C
K
.
C
:?L >>@=> K < 67 :
C
.
K K < =H7: >>@=> A =H7: 67 :
K
1BCM CM ; =IHM?LP;NCP? M=B?G? QCNBION MO< CN?L;NCIH
8BCA81DC8E4 B274<4B
!CMNLC<ONCP? M=B?G?M Q?L? VLMN JO<FCMB?> <S /I? ;H> 0C>CFEIP?L 89 OH>?L NB? H;G?
I@ ZWO=NO;NCIH MJFCNNCHA M=B?G?M[ 4BCF? NB?S Q?L? VLMN FCGCN?> NI L?AOF;L G?MB?M
NB?S Q?L? F;N?L ?RN?H>?> NI NLC;HAOF;L G?MB?M M?? 89 ;H> 89
&H ;FF NB;N @IFFIQM Q? MB;FF J;S J;LNC=OF;L ;NN?HNCIH NI L?MIFPCHA NB? ?KO;NCIH
< K =H7: < J
<S VLMN FCGCNCHA IOLM?FP?M NI NLC;HAOF;L G?MB?M 4? MN;LN <S ?R;GCHCHA NB? =F;MMC=
=?HNL?> $;F?LECH G?NBI> NI MBIQ NB? >L;Q<;=EM Q? =IOF> L?G?>S <S OMCHA >CM
NLC<ONCP? M=B?G?M
0;4A:8=LB <4C7>3
#IL ?;=B N?MN @OH=NCIH I@ NB? >IG;CH >?HIN?> 7 Q? L?MIFP?
.
< 7
K =H7: < : J
1B? >?LCP;NCP? QCNB L?MJ?=N NI NCG? QCFF NBOM =F;MMC=;FFS JLI>O=? ; N?LG
) <
"
J
"<
;H> " < L?JL?M?HN NB?
QB?L? ) CM NB? G;MM G;NLCR J NB? NCG? MN?J ;H> " <
P?=NILM QBIM? =IGJIH?HNM ;L? NB? <7 N;E?H L?MJ?=NCP?FS ;N NB? CHMN;HNM J< ;H> J<
0CH=? NB? N?LG K =H7: < CM =IHMN;HN J?L ?F?G?HN C@ Q? =IHMC>?L NB;N NB? P?FI=CNS CM
=IHMN;HN IH ; NLC;HAF? QBC=B QCFF <? >IH? QCNB NB? >CMNLC<ONCP? M=B?G?M NBCM ACP?M
; =IHNLC<ONCIH IH ?;=B NLC;HAF? I@
.
M7 K =H7: <
7 :
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
3
&
u
1
2
#CAOL? =;M? QCNB NQI >IQHMNL?;G JICHNM
+
+IQ IH ; NLC;HAF? 7 : >I?M HIN >?J?H> IH NB? =IHMC>?L?> JICHN ;H> CM ?KO;F
NI /( QB?L? /( CM NB? MOL@;=? I@ NB? NLC;HAF? 1B? =IHNLC<ONCIHM I@ ?;=B JICHN ;L?
NBOM ?KO;F CH>?J?H>?HNFS I@ NB?CL OJMNL?;G IL >IQHMNL?;G H;NOL? ;H> ACP? LCM? NI
; >L;Q<;=E 1B?M? =IHNLC<ONCIHM NB?H H??> NI <? ;MM?G<F?> CH IL>?L NI I<N;CH NB?
?RJL?MMCIH I@ K =H7: < ;M JLID?=N?> IHNI ; N?MN @OH=NCIH 7
# %( B274<4B 8= CF> 38<4=B8>=B
#>C0C8>=B 0=3 34J=8C8>=B
#IL ?;=B NLC;HAF? IH? >?VH?M NBL?? HILG;F P?=NILM HIN OHCN P?=NILM ;N NB? P?LNC=?M
D O O
N N
D O O
N N
D O O
N N
) ,H? HIN?M NB;N
7 D 7 #IL L?;MIHM I@ MCGJFC=CNS Q? MB;FF =IHMC>?L NB;N NB? P?FI=CNS K CM =IHMN;HN IH ?;=B
NLC;HAF? 1BCM BSJINB?MCM CM G;>? MSMN?G;NC=;FFS CH NB? FCN?L;NOL? =NO;FFS NBCM CM
HIN H?=?MM;LS ;M QCFF <? MBIQH F;N?L
1B? MOL@;=? I@ NB? NLC;HAF? CM >?HIN?> /( &N CM
/( N O N O
N O N O
N O N O
IL CH @ILG I@ ; >?N?LGCH;HN
N
N
N
/( O
O
O
0,)3&+$ 1/+0-,/1 ".21&,+0 1
fin &
u
&
u
2
3
fin
#CAOL? >?VHCNCIH I@ <7<
#IL ? ;H> Q? >?VH? NB? KO;HNCNC?M A7 K D 7 &@ A7 CM JIMCNCP? NB?
P?FI=CNS K CM ?RCNCHA ;N JICHN ? NB? M?AG?HN IJJIMCN? JICHN ? CM ;H ?HNLS 1B? NIN;F
A
A
A CM T?LI ;H> NB?L? ;L? NBOM NQI JIMMC<F? =;M?M
,H? I@ NB? A7 CM JIMCNCP? ;H> NB? NQI INB?LM ;L? H?A;NCP? NB? =;M? QCNB IH?
?HNLS
,H? I@ NB? A7 CM H?A;NCP? ;H> NB? NQI INB?LM ;L? JIMCNCP? NB? =;M? QCNB NQI
?HNLC?M
&H NB? IH? ?HNLS =;M? NB?L? ;L? NQI OJMNL?;G JICHNM ;H> IH? >IQHMNL?;G JICHN
&H NB? NQI ?HNLS =;M? NB?L? ;L? IH? OJMNL?;G JICHN ;H> NQI >IQHMNL?;G JICHNM
#CAOL? 4? MB;FF HIQ >?VH? NQI H?Q KO;HNCNC?M <7< NB? P;FO? I@ < ;N ?HNLS ;H> <=CB NB?
P;FO? I@ < ;N ?RCN
&@ NB?L? CM IHFS IH? OJMNL?;G JICHN <7< QCFF >?HIN? NB? P;FO? I@ < ;N NBCM JICHN
&@ NB?L? ;L? NQI OJMNL?;G JICHNM <7< QCFF <? >?VH?> ;M NB? P;FO? I@ < ;N NB? JICHN I@
NB? OJMNL?;G MC>? QB?L? NB? P?FI=CNS K NOLHM NIQ;L>M NB? >IQHMNL?;G JICHN <INB
=;M?M ;L? CFFOMNL;N?> CH #CAOL? 0CGJF? A?IG?NLC=;F =;F=OF;NCIHM CH NBCM =;M? ;N ; >IQHMNL?;G JICHN #CAOL? IH
NB? LCABN ACP? NB? @IFFIQCHA P;FO? @IL <7< <7< K D <
K D
K D
K D
<
FF =;M?M =;H CH @;=N <? =IP?L?> <S NB? @IFFIQCHA @ILGOF;
<7< )
7
)
7
! A7 <7
! A7
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&H NB? M;G? Q;S IH? VH>M
<=CB )
' A7 <7
7
)
' A7
7
)?N OM HIQ NLS NI ?P;FO;N? NB? AL;>C?HN I@ < &N =;H <? QLCNN?H ;M
=H7: < , <7 D 7
/( 7
&@ NB? @OH=NCIH < N O CM QLCNN?H CH NB? FCH?;L @ILG 7 8N 9O @IL ?;=B P?LN?R I@
NB? NLC;HAF? IH? MBIOF> B;P? < N7 O7 7 8N7 9O7 1B? =IGJIH?HNM I@ NB? AL;>C?HN
I@ < ;L? 8 ;H> 9 QBC=B ;L? MIFONCIHM I@ ; (L;G?L MSMN?G
< O N < < O N < < O N < 8 ;H> 9 N O N O N O N O N O N O 1B? >?HIGCH;NIL CM /( ;H> NB? NQI HOG?L;NILM ;L? ?_?=NCP?FS NB? =IGJIH?HNM I@
7 <7 D 7 1BCM AL;>C?HN CM =IHMN;HN IH NB? NLC;HAF? 1B? KO;HNCNS K =H7: < =;H
HIQ <? ?RJL?MM?> ;M
, <7 K D 7
K =H7: < /( 7
)
IL QCNB IOL >?VHCNCIH I@ A7 ,
<7 A7
K =H7: < /( 7
1BCM =;H <? L?ILA;HCT?> ;M
K =H7: < A < <
/(
A < < ! A7 <7
)
QBC=B IHFS MBIQM OJ >C_?L?H=?M I@ NB? @OH=NCIH < 1B? KO;HNCNS 7 <7 A7
CM >?HIN?> ( S M?J;L;NCHA NBCM MOG CHNI NQI J;LNM ;==IL>CHA NI NB? MCAH I@ A7 Q?
I<N;CH
( ,
' A7 <7
7
,
7
1BCM <S OMCHA NB? ?RJL?MMCIHM @IL <7< ;H> <=CB ;H> NB? @;=N NB;N
,
7
' A7 ,
7
! A7
0,)3&+$ 1/+0-,/1 ".21&,+0 VH;FFS JLI>O=?
( <=CB <7<
,
' A7
7
1BCM ?RJL?MMCIH MBIQM NB;N NB? =IHNLC<ONCIH ( QCFF <? T?LI C@ < CM CH NB? M;G?
MN;N? ;N NB? ?HNLS ;H> ?RCN 1B? H;G? Z0NL?;GQCM? &HP;LC;HN[ I@ NB? -0& M=B?G?
NB;N Q? MB;FF FIIE ;N <?FIQ =IG?M @LIG NBCM JLIJ?LNS *?F8=3 <D;C838<4=B8>=0; B274<4B
4CNB NB? $;F?LECH G?NBI> Q? M;Q NB;N ?;=B =IHNLC<ONCIH @IL NB? NBL?? P?LNC=?M I@ ;
NLC;HAF? CH NB? ;MM?G<FS I@ NB? N?LG
.
7 K =H7: < :
CM ( 1BCM KO;HNCNS CAHIL?M NB? @;=N NB;N MIG? NLC;HAF? P?LNC=?M ;L? ?HNLC?M ;H>
INB?LM ?RCNM 4? MB;FF >?HIN? M M ;H> M ;M NB?M? =IHNLC<ONCIHM *;MM =IHM?LP;NCIH
CM ?HMOL?> ;M FIHA ;M
M
M (
M
,H? =;H NBOM GI>C@S NB? M7 C@ NB? OJMNL?;G ;H> >IQHMNL?;G JICHNM
) CH NB? NLC;HAF?
;L? N;E?H CHNI =IHMC>?L;NCIH <S QLCNCHA M7 7 ( ;M FIHA ;M 7 7 1B?
=I?^=C?HNM 7 ;L? =;FF?> Z>CMNLC<ONCIH =I?^=C?HNM[
&@ NB?L? CM IHFS IH? >IQHMNL?;G JICHN ;H> C@ IH? =IHMC>?LM NB;N NB? OJMNL?;G JICHNM
CH ; NLC;HAF? MBIOF> HIN >?J?H> IH QB;N B;JJ?HM CH NBCM NLC;HAF? IH? =;H =;H=?F NB?CL
=IHNLC<ONCIHM ;H> NB? NIN;F =IHNLC<ONCIH ( QCFF <? ;_?=N?> NI NB? >IQHMNL?;G JICHN
&H NB? =;M? I@ NQI >IQHMNL?;G JICHNM Q? =;H=?F NB? OJMNL?;G JICHN =IHNLC<ONCIH
<ON IH? MNCFF B;M NI >?=C>? BIQ NI MB;L? ( <?NQ??H NB? NQI >IQHMNL?;G JICHNM
1BCM =BIC=? QCFF >?N?LGCH? >C_?L?HN P;LC;HNM I@ NB? M=B?G?M ;H> CM >IH? <;M?> IH
=IHMC>?L;NCIHM L?A;L>CHA JIMCNCPCNS NB? <?B;PCIOL CH NB? JL?M?H=? I@ FCH?;L @OH=NCIHM
IL G;MM =IHM?LP;NCIH 4? MB;FF HIQ MNO>S NB? >C_?L?HN JLIJ?LNC?M I@ >CMNLC<ONCP?
M=B?G?M
%>B8C8E8CH CH NB? =;M? I@ -0& M=B?G?M NB? JLI<F?G I@ JIMCNCPCNS CM =IHMC>?L?> CH
NB? BSJINB?MCM I@ ; ZFOGJ?>[ G;MM G;NLCR NB;N CM NI M;S ?KO;F NI ; >C;AIH;F G;NLCR
QBIM? A?H?L;F N?LG CM
.
/7 7 :
*IL?IP?L NQI C>?;M GOMN <? >CMNCHAOCMB?> FI=;F ;H> AFI<;F JIMCNCPCNS
)I=;F JIMCNCPCNS >?G;H>M NB;N NB? @OH=NCIH < L?G;CHM JIMCNCP? ;N ; JICHN ? QBCF?
IHFS =IHMC>?LCHA NB? =IHNLC<ONCIH I@ ; MCHAF? NLC;HAF? 4CNB G;MM FOGJCHA ;H> =IH
MC>?LCHA IHFS IH? NLC;HAF? IOL ;>P?=NCIH ?KO;NCIH QCFF <? QLCNN?H ;M
/7 <7<
<7<
(7 (
J
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
IL ?P?H
/7 <7<
QBC=B ACP?M @IL <7<
,
<7<
(
<8 A8
7
J
8
<7<
<7<
J A7 (7
/7
J ( ,
<8 A8
/7 7
8 7
)
1BCM M=B?G? N;E?M NB? @ILG <7< 8 98 <8 &H IL>?L NI B;P? FI=;F JIMCNCPCNS
;FF NB? 98 B;P? NI <? JIMCNCP? &@ ? CM NB? IHFS >IQHMNL?;G JICHN A7 JIMCNCP? ;H> <INB
NB? INB?L A8 H?A;NCP? IH? =BIIM?M (7 FF NB? 98 ;L? NB?H JIMCNCP? C@ J '9 NB? NCG? MN?J CM NBOM FCGCN?> )I=;F JIMCNCPCNS CH NB? =;M? I@ NQI >IQHMNL?;G JICHNM
CM F?MM ?;MCFS P?LCV?>
&H IL>?L NI >CMNCHAOCMB NB? JLIJ?LNS I@ AFI<;F JIMCNCPCNS IH? FIIEM ;N NB? =IHNLC
<ONCIHM I@ ;FF NB? NLC;HAF?M NI QBC=B NB? JICHN <?FIHAM 4? VH> NB? M;G? JIMCNCPCNS
=IHMNL;CHN NB;N ;JJFC?M NI =I?^=C?HN 98 <ON @IL L?;MIHM I@ =IHMCMN?H=S IH? ;FMI B;M NI
)
B;P? 8 98 )I=;F JIMCNCPCNS CM ; MNLIHA?L =IH>CNCIH NB;H AFI<;F JIMCNCPCNS
%A4B4AE0C8>= >5 ;8=40A8CH NBCM JLIJ?LNS >?G;H>M NB;N C@ < CM FCH?;L NB? JLI<F?G
ACP?M ;H ?R;=N MIFONCIH CH MN?;>S WIQ 1BCM JLIJ?LNS CM AO;L;HN??> C@ NB? >CMNLC<ONCIH
=I?^=C?HNM 7 ;L? <IOH>?>
>=C8=D8CH NBCM JLIJ?LNS L?KOCL?M NB;N NB? >CMNLC<ONCIH =I?^=C?HNM P;LS =IHNCHO
IOMFS C@ NB? >CL?=NCIH I@ NB? P?=NIL K CM =B;HA?> &H NB? IJJIMCN? =;M? NB? J?LG;H?HN
=;M? =IOF> =IHP?LA? NIQ;L>M =S=FC=;F MIFONCIHM
# B274<4
)
)
)?N OM AI <;=E NI NB? ?RJL?MMCIH ( 7 <7 A7 S OMCHA NB? @;=N NB;N 7 A7 Q? =;H JLIJIM? >C_?L?HN P;LC;HNM I@ NBCM ?RJL?MMCIH 4? MB;FF JON ( CHNI ; @ILG QB?L?
IHFS NB? >C_?L?H=?M I@ < P;FO?M ;JJ?;L ;NNLC<ON?> QCNB JIMCNCP? =I?^=C?HNM 1B;N CM
NI M;S
( ,
,
7 8
78
<7 <8
QB?L? NB? =I?^=C?HNM 77 ;L? T?LI &N CM ;FQ;SM JIMMC<F? NI QLCN? ( NBCM Q;S #IL
?R;GJF? C@ A ;H> A ;L? JIMCNCP? ;H> A H?A;NCP? IH? B;M
( A < <
A < <
QBC=B ACP?M ;H> JIMCNCP? ;H> NB? INB?LM T?LI &@ A ;H> A Q?L? H?A;NCP? IH?
QIOF> N;E? ;H> JIMCNCP? ;H> NB? INB?LM T?LI 1B?M? NQI MIFONCIHM ;=NO;FFS
=IP?L ;FF JIMMC<F? MCNO;NCIHM
0,)3&+$ 1/+0-,/1 ".21&,+0 1B? @ILGOF; 78 ' ! A7 A8 AO;L;HN??M NB? >?MCL?> L?MOFN Q? =;H
=B?=E CN QCNB NQI ?R;GJF?M
1B? + M=B?G? + @IL H;LLIQ =BIIM?M @IL NB? =IHNLC<ONCIH I@ JICHN ? NB? CHH?L MOG
)
I@ NB? ?RJL?MMCIH I@ ( NB;N CM NI M;S
1BCM ?RJL?MMCIH L?MJ?=NM
78 <7 <8
8
NB? GIHINIHC=CNS ;M ;FF NB? <8 =I?^=C?HNM ;L? H?A;NCP? NB?S QCFF NBOM ;JJ?;L QCNB ;
JIMCNCP? =I?^=C?HN CH NB? @ILGOF; NB;N ACP?M <7 ==IL>CHA NI NB? HIN;NCIHM ;>IJN?>
;<IP? NB? >CMNLC<ONCIH =I?^=C?HNM ;L? NBOM
7 )
8
)
)
78
<7 <8
:8
: 8
<: <8
)
&N CM BIQ?P?L JL?@?L;<F? NI JL?M?LP? NB? @ILG 7 ( 8 78 <7 <8 QBC=B
=;LLC?M HI LCME I@ >CPCMCIH <S T?LI
1BCM M=B?G? >I?M HIN L?MJ?=N FCH?;LCNS ;M NB? >CMNLC<ONCIH =I?^=C?HNM ;L? HIN
<IOH>?> ,H NB? INB?L B;H> CN CM NLOFS GIHINIHIOM M @;L ;M G;MM FOGJCHA CM
=IH=?LH?> NB? M=B?G? SC?F>M ;N NCG? J< <7<
<7< % ,
H?CAB< IOLCHA NLC;HAF?M
J &
78
/7
,
H?CAB< IOLCHA NLC;HAF?M
J
/7
<
78 <8
1B? MOG I@ ;FF NB? =I?^=C?HNM CM ?KO;F NI ;H> JIMCNCPCNS CM AO;L;HN??> C@ NB?
=IH>CNCIH
/
)7
J 78
H?CAB< IOLCHA NLC;HAF?M
CM =B?=E?> 1BCM F;NN?L =IH>CNCIH CM NBOM NB? =LCN?LCIH @IL + M=B?G? MN;<CFCNS
#IL NB? =;M? I@ HIH =IHMN;HN P?FI=CNS IH NB? ?F?G?HN CH IL>?L NI =IHMNLO=N NB? +
M=B?G? Q? B;P? IHFS OM?> NB? @;=N NB;N
( ,
7
<7 A7 ;H>
,
A7 7
+
K =H7: < : ;H> NB? NQI JL?=?>CHA JLIJ?LNC?M ;L? NBOM
+IQ ( B@7/<5:3
;FQ;SM =B?=E?> ?P?H C@ K CM HIN =IHMN;HN IH ?;=B ?F?G?HN %IQ?P?L @IL NB? NLC;HAF?M
NB? =B;HA? NI ; FCH?;L P?FI=CNS >I?M HIN <LCHA ;HS CGJLIP?G?HN ;M NB? @ILGOF;?
I<N;CH?> IHFS MBIQ NB? ;P?L;A? P?FI=CNS IH NB? ?F?G?HN
-?LNB;G? A;P? ; JLII@ I@ =IHP?LA?H=? I@ NB? + M=B?G? CH ) 89
%( B274<4
1B? -IMCNCP? 0NL?;GQCM? &HP;LC;HN -0& M=B?G? CM I<N;CH?> G?L?FS <S FCGCNCHA NB?
>CMNLC<ONCIH =I?^=C?HNM CH NB? + M=B?G? MI NB;N NB?S @;FF <?NQ??H ;H> 1I >I NBCM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
IH? ;JJFC?M NB? @OH=NCIH =;FF?> NB? Z*CH*I> FCGCN?L[ QBC=B CM HINBCHA GIL? NB;H
) ?D) E: N '
! N 1BCM FCGCN;NCIH =;H M??G ;L<CNL;LS <ON CH @;=N CN >I?M HIN =B;HA? NB? MOG I@
=I?^=C?HNM ;H> NB? M=B?G? L?G;CHM =IHM?LP;NCP? ;N F?;MN IH? I@ NB? NBL?? >CMNLC<O
NCIH =I?^=C?HNM CH M=B?G? + CM T?LI ;H> NB? =ILL?=NCIHM ;JJFC?> NI NB? INB?L NQI
=IGJ?HM;N? ?;=B INB?L
4? NB?H B;P?
'
$
7
'
! #
7 4? B;P? M??H HIH?NB?F?MM NB;N CN CM JL?@?L;<F? @IL NB? + M=B?G? NI @ILG NB?
=IHNLC<ONCIHM #
7 ( >CL?=NFS CH IL>?L NI ;PIC> >CPCMCIH 4? >I NB? M;G? CH NBCM =;M?
&@ ( CM JIMCNCP? Q? I<N;CH
'
( $
7
'
! #
7 ( ( &@ ( CM H?A;NCP? Q? I<N;CH
'
( $
7
'
! #
7 ( ( 1B? NQI @ILGOF;? =;H <? ?RJL?MM?> ;M
'
$
( I?=D (
7
'
! I?=D ( #
7 ( I?=D ( ( "*' B274<4 8= C7A44 38<4=B8>=B
1B? *2/! M=B?G? I@ ';HCH 89 QBC=B MCAHCV?M Z*OFNC>CG?HMCIH;F 2JQCH> /?MC>O;F
!CMNLC<ONCIH[ CM ;H ;>;JN;NCIH NI JLCMGM I@ NB? + ;H> -0& M=B?G?M NB;N Q? B;P? DOMN
>?M=LC<?> @IL NB? NLC;HAOF;L G?MB?M 4? MB;FF FIIE MO==?MMCP?FS ;N NB? NQI P;LC;HNM
+ ;H> -0&
#CH?4 "*' B274<4
4? QILE QCNB NB? NL;HM@ILG?> VR?> G?MB ;H> Q? Q;HN NI AO;L;HN?? NB? AFI<;F =IH
M?LP;NCIH I@ NB? @OH=NCIH < <S MBIQCHA NB;N IH? B;M
.
.
<< <<
<
P
P 1 =H7: < < : :
J
<;M?> IH IOL FI=;F ?KO;NCIHM 4? L?=;FF NB;N 1 CM G;>? OJ I@ NB? =IGJIH?HNM
1 2 3 1B? VLMN N?LG CM CHCNC;FFS QLCNN?H ;M
.
P <
<<
4? F;N?L >?HIN?
<>
<
< .
,
<7>
<7>
<<
7> P <
: J
J
7>
:
0,)3&+$ 1/+0-,/1 ".21&,+0 .
7> P <
)<>
4 4 '
B
: /7>
NB? VLMN N?LG I@ NB? ?KO;NCIH I@ ; JICHN ?F
#IL NB? M?=IH> N?LG P 1 =H7: < < : Q? MB;FF =;F=OF;N? CN ?F?G?HN <S
?F?G?HN ;H> @IL ?;=B ?F?G?HN MB;L? NB? =IHNLC<ONCIHM IP?L NB? JICHNM CN CM G;>? OJ
I@ 1B? MOG I@ ;FF IOL ?KO;NCIHM QCFF ;A;CH ACP? OM NB? AFI<;F ?KO;NCIH >?MCL?>
S FCGCNCHA IOLM?FP?M NI ; JLCMG , I@ NB? NL;HM@ILG?> G?MB Q? MB;FF NBOM NLS NI
=;F=OF;N?
.
$ P 1 =H7: < :, NI I<N;CH
7>
+
$
<S NLSCHA NI JON CN CHNI NB? @ILG
$ ,
,
78
7 8
<7< <8<
QCNB NB? =I?^=C?HNM 78 JIMCNCP? IL T?LI ;M Q? >C> @IL NB? + M=B?G? S >?=IGJIMCHA
< Q? ;FL?;>S EHIQ NB;N
$ ,
<7<
7
+IQ NB? N?LGM
.
P 1 =H7: 7 :, $
.
P 1 =H7: 7 :, 87 .
$
=;H <? >?=IGJIM?> CH BILCTIHN;F IL P?LNC=;F AL;>C?HNM CH NB? @ILG 77
.
.
7
7
P1
P2
:,
:, 77 N
O
$
$
87 QCNB
;H>
P3 $
7
:, P M Q? >C> CH 0?=NCIH Q? HIQ >?=IGJIM? NB? <;M?M 7 N O P I@ NB? JLCMG
CHNI NB? JLI>O=N 67 N O D7 P QBC=B ACP?M
.
.
67
67
D
P1 7 :,
P2 D7 :, 77 N $ O $ ;H>
87 D7
P .
$
P3 67 :, 1B? JLIJ?LNC?M I@ NB? <;M?M 67 N O ;H> D7 P NB?H ACP? OM ; ALIOJ I@ JLIJ?LNC?M
@IL =I?^=C?HNM 77 ;H> 87 %6!/,!6+*& 0 ,# #/"" 02/# " #),40
7
7
7 CH@?LL?> <S 6
6
6 7
7
7 CH@?LL?> <S 6
6
6 8
8 CH@?LL?> <S D
D 8
8 CH@?LL?> <S D
D 8
8 CH@?LL?> <S D
D %?L? Q? B;P? ;A;CH OM?> NB? FI=;F HOG<?LCHA I@ NB? JICHNM CH ; JLCMG
;M?> IH NB? + M=B?G? QBC=B CM <OCFN IH NLC;HAF?M ; VLMN M?N I@ =I?^=C?HNM
=IOF> <? NB? @IFFIQCHA
' ! 77 78 @IL NB? JICHNM ? ;H> @ I@ NB? FIQ?L NLC;HAF? I@ NB?
78 JLCMG IL @IL NBIM? I@ NB? OJJ?L NLC;HAF?
77 ' 87 7 7
' 87
78
@IL NB? JICHNM ? I@ NB? FIQ?L NLC;HAF?
@IL NB? JICHNM ? I@ NB? OJJ?L NLC;HAF?
@IL NB? INB?L N?LGM QBC=B ;L? Z=LIMM?>[ N?LGM FCHECHA NB? JICHNM QBC=B
78 ;L? H?CNB?L ;N NB? M;G? F?P?F HIL IH NB? M;G? P?LNC=;F FCH?
4? ;FL?;>S JIMM?MM ;FF NB? AII> JLIJ?LNC?M NI =IHMNLO=N ;H + NSJ? M=B?G? ;FF NB?
=I?^=C?HNM ;L? JIMCNCP? IL T?LI %IQ?P?L IH? =IOF> I<N;CH ; F?MM >C_OMCP? M=B?G?
<S NL;HM@ILGCHA NB? MOGM I@ BILCTIHN;F ;H> P?LNC=;F WOR?M CHNI =LIMM?> WOR?M QBC=B
=;H <? >IH? <S HINC=CHA NB;N @IL ; M?N I@ NBL?? =I?^=C?HNM 78 89 ;H> 79 IH? >I?M
HIN =B;HA? NB? P;FO? I@ $ <S =;LLSCHA ION NB? @IFFIQCHA M?LC?M I@ IJ?L;NCIHM
79
79
78
78
89
89
!
78 89
!
78 89
!
78 89
,H? CH @;=N I<M?LP?M NB;N NB? =I?^=C?HNM I@ <7< <8< ;H> <9< L?G;CH OH=B;HA?>
4? HIQ B;P? ; MOJJF?G?HN;LS JLIJ?LNS QBC=B CM NB;N C@ ; =I?^=C?HN 78 CM MNLC=NFS
JIMCNCP? NB?H ;FF NB? =I?^=C?HNM 89 ;L? T?LI
4? MB;FF =BIIM? ;M CH NQI >CG?HMCIHM NB? >CMNLC<ONCIH =I?^=C?HNM 7 ;M
7 $ ,
8
78
<7< <8<
CH IL>?L NI I<N;CH ;H + NSJ? M=B?G? %?L? ;A;CH Q? VH> ; MN;<CFCNS =LCN?LCIH CH IL>?L
NI ?HMOL? NB? GIHINIHC=CNS I@ NB? M=B?G?
J /
)7
H?CAB< IOLCHA JLCMG M
78
0,)3&+$ 1/+0-,/1 ".21&,+0 %(CH?4 "*' B274<4
;M?> IH IOL H?QFS @IOH> + NSJ? M=B?G? Q? H??> NI =IHMNLO=N NB? -0& NSJ? M=B?G?
NB;N QCFF ;FMI AO;L;HN?? NB;N NB? >CMNLC<ONCIH =I?^=C?HNM @;FF <?NQ??H ;H> 1I >I
NBCM CH NB? =;F=OF;NCIH I@ $ Q? H??> NI M?J;L;N? NB? JIMCNCP? ;H> H?A;NCP? =IHNLC<O
NCIHM
$ $ $
,
'
78
<7<
8
<8< ,
78
8
' <8< <7< S =IGJ;LCHA $ ;H> $ IH? =;H GI>C@S NB? M=B?G? + =I?^=C?HNM
&@ $ $
&@ <8< <7< NB?H 78 CM =;H=?FF?>
INB?LQCM? 78 CM L?JF;=?> <S 78
&@ $ $
&@ <8< <7< NB?H
78
CM L?JF;=?> <S 78
INB?LQCM? 78 CM =;H=?FF?>
1B? @IFFIQCHA CM ;H ?RJF;H;NCIH I@ NB?M? G;HCJOF;NCIHM &H NB? =;M? QB?L? NB?
JIMCNCP? =IHNLC<ONCIHM ;L? NB? GIMN CGJILN;HN Q? GOFNCJFS NB?G <S ;H> Q? =;H=?F
NB? H?A;NCP? =IHNLC<ONCIHM 4B;N Q;M $ CM NBOM =;H=?FF?> QB;N Q;M $ <?=IG?M
$ 1B? VH;F ?RJL?MMCIH @IL $ L?G;CHM OH=B;HA?> 1B? =;M? $ $
$
=;H ?;MCFS <? >?>O=?>
#IFFIQCHA NBCM GI>CV=;NCIH I@ NB? =I?^=C?HNM 78 ;FF NB? N?LGM 78 <7< <8< ;L?
JIMCNCP? ;H> ;FF NB? N?LGM 7 @;FF <?NQ??H ;H> =NO;FFS
7 )
8
78
<7< <8<
$
)
8
)
)
: 8
78
<7< <8<
<
78 <:
<8<
FF NB? N?LGM I@ NB? MOGG;NCIHM ;L? JIMCNCP? ;H> NB? HOG?L;NIL CM MG;FF?L NB;H
NB? >?HIGCH;NIL
'4<0A:B >= <0BB 2>=B4AE0C8>= F8C7 C74 "*' B274<4
1B? *2/! M=B?G? CM J?L@?=NFS =IGJ;NC<F? QCNB NB? JLII@ I@ NL;=?L G;MM =IHM?LP;NCIH
M?N ION CH 0?=NCIH ;H> QCNB NB? =IHNCHOCNS MN?J >?M=LC<?> CH 0?=NCIH 4?
B;P? CH @;=N M;C> CH NBCM M?=NCIH NB;N
P3 <>=7<
,
P3 P 8 67 N O
8
;H> P3 CM H?=?MM;LS B?L? NI =;F=OF;N? NB? =I?^=C?HNM
.
D7
87 P3 67 :, P $ %6!/,!6+*& 0 ,# #/"" 02/# " #),40
4? NBOM VH> NB;N
87 <>=7<
,
8
.
67 68 :
D7
P .
H H P3 8 :P 4? M?? NB;N NB? N?LG CH <L;=E?NM CM CAHILCHA NB? MCAH
P7> P7>
8
.
H H P3 P 8 :P QBC=B Q? B;P? >?HIN?> PM 7> 1B?M? N?LGM ;L? JLI>O=?> NI L?MIFP? NB? =IH
NCHOCNS ?KO;NCIH
8
&@ G;MM FOGJCHA B;M <??H OM?> QB?H =IGJONCHA PM 7> M?? NB? HIN? ;N NB?
?H> I@ 0?=NCIH NB?H NB? =I?^=C?HNM 87 GOMN <? =IGJON?> CH ; =IGJ;NC<F? Q;S
QBC=B CM
.
D . H 7
87 67 : P3 7 :P P H
70?C4A ">34A= C427=8@D4B 8= J=8C4
4;4<4=CB
4? MB;FF >?M=LC<? CH NBCM =B;JN?L NB? VHCN? ?F?G?HNM OM?> CH 1?F?G;= NB? G?MB ;H>
NB? MNIL;A? I@ NB? VHCN? ?F?G?HN G;NLC=?M
4B2A8?C8>= >5 J=8C4 4;4<4=CB
)A80=6;4 F8C7 ;8=40A 8=C4A?>;0C8>=
1B? L?@?L?H=? NLC;HAF? M?? #CAOL? CM G;>? OJ I@ NB? JICHNM QBIM? =IIL>CH;N?M
;L? ;H> ;H> QBIM? HOG<?LCHA CM ;H> L?MJ?=NCP?FS ,H NBCM
L?@?L?H=? ?F?G?HN NB? <;MCM @OH=NCIHM ;L?
, , , &D0B81D11;4 CA80=6;4
1B? KO;MC <O<<F? ?F?G?HN M?? #CAOL? Q;M JO<FCMB?> CH M?? 89 &N CM
>?LCP?> @LIG ; NLC;HAF? <S ;>>CHA NI NB? NBL?? P?LNC=?M - - - ;H ?RNL; JICHN
- 1B? =?HNL? I@ AL;PCNS I@ NB? NLC;HAF? CM ; H;NOL;F =BIC=? @IL NBCM @IOLNB JICHN 1B?
CHCNC;F NLC;HAF? CM NBOM >CPC>?> CHNI NBL?? MO< NLC;HAF?M 1 1 ;H> 1
S =BIIMCHA ; FCH?;L >CM=L?NCT;NCIH NB? <;MCM @OH=NCIHM I@ NB? KO;MC <O<<F? NLC;HAF?
CH NB? VHCN? ?F?G?HN NB?ILS ;L? NB? @IOL JC?=?QCM? FCH?;L @OH=NCIHM 7 @IL ?;=B JICHN
- NI - MJ?=CV?> IH NB? NLC;HAF? ;H> P?LC@SCHA
7 ,8 78
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
β
1
3
01
α
2
1
#CAOL? L?@?L?H=? NLC;HAF?
)4CA0743A>=
1B? L?@?L?H=? ?F?G?HN CM >?VH?> <S NB? @IOL JICHNM QBIM? =IIL>CH;N?M ;L? ;H> M?? #CAOL? 1B? <;MCM @OH=NCIHM ;L?
, , , , %A8B<
1B? JLCMG CM G;>? OJ I@ NBL?? P?LNC=;F L?=N;HAOF;L MC>?M ;H> NQI NLC;HAOF;L MC>?M IH?
OJJ?L IH? FIQ?L HIN H?=?MM;LCFS BILCTIHN;F M?? #CAOL? 1B? >CACNM CH NB? VAOL?
L?@?L NI NB? FI=;F HOG<?LCHA I@ NB? HI>?M
"P?LS @OH=NCIH < CM ;JJLIRCG;N?> <S
<
,
<7 7
7
QB?L? <7 ;H> 7 ;L? L?MJ?=NCP?FS NB? P;FO?M I@ < ;H> NB? <CFCH?;L <;MCM @OH=NCIHM
>?VH?> IH NB? HI>?M ? I@ NB? G?MB I@ >IG;CH 7 ?KO;FM IH NB? HI>? ? ;H> IH
;HS INB?L HI>?
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
P1
T1
T3
P4
T2
P2
P3
#CAOL? KO;MC <O<<F? ?F?G?HN
&H IL>?L NI MCGJFC@S NB? =;F=OF;NCIHM ;H> @IL ?P;FO;NCHA NB? <;MCM @OH=NCIHM 7
IH ?;=B ?F?G?HN I@ NB? NLC;HAOF;LCT;NCIH IH? VLMN IJ?L;N?M IH ; MCGJF?L JLCMG NB;N
=IHMNCNON?M NB? L?@?L?H=? ?F?G?HN M?? #CAOL? 1B? <;MCM @OH=NCIH ,7 ;MMI=C;N?> QCNB NB? HI>?M I@ NB? L?@?L?H=? ?F?G?HN ;L?
, , , , , , 1B? L?MNLC=NCIHM I@ NB? <;MCM @OH=NCIHM 7 IH ;HS JLCMG I@ NB? G?MB I@ >IG;CH ;L?
>?LCP?> @LIG NB? =IGJIMCNCIH I@ NB? @OH=NCIHM ,7 QCNB NB? CMIJ;L;G?NLC= NL;HM@ILG;
NCIH " QBC=B NL;HM@ILGM NB? L?@?L?H=? JLCMG CHNI NB? JLCMG OH>?L =IHMC>?L;NCIH
#IL ;HS JLCMG I@ G?MB QBIM? P?LN?R =IIL>CH;N?M ;L? N7 O7 P7 " ;MMI=C;N?M
; HI>? ) QCNB =IIL>CH;N?M I@ NB? L?@?L?H=? ?F?G?HN QCNB ; HI>? ) QCNB
=IIL>CH;N?M N O P I@ NB? JLCMG OH>?L =IHMC>?L;NCIH IH ;==IOHN I@ NB? L?F;NCIHM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
y
3
4
2
x
1
z
#CAOL? N?NL;B?>LIH ?F?G?HN
)
$
$
N
N7 ,7 $
$
$
7
$
"
)
O7 ,7 O
$
7
$
$
$
$
$ P ) P7 ,7 7
1B? @OH=NCIHM ,7 ;JJ?;LCHA CH NB? >?VHCNCIH I@ " ;L? NB? <;MCM @OH=NCIHM MJ?=CV?>
IH NB? L?@?L?H=? ?F?G?HN ;M NBCM =BIM?H L?@?L?H=? ?F?G?HN CM CMIJ;L;G?NLC= NB? HI>?M
I@ CHN?LJIF;NCIH ;L? NB? A?IG?NLC= HI>?M
1B? >?VHCNCIH I@ " =;H <? MCGJFCV?> <?=;OM?
O O
" N N
N N
O O
N N
O O
4? NBOM A?N NB? @IFFIQCHA L?F;NCIHM @IL " N N
N
N
O O
O
O
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
6
4
5
3
1
2
#CAOL? JLCMG;NC= ?F?G?HN
P P
P
P
P
P P 1B? @OH=NCIHM 7 ;L? >?VH?> <S 7 N O P ,7 " N O P
1B? <CFCH?;LCNS I@
NB? NL;HM@ILG;NCIH CGJFC?M NB;N A?H?L;FFS NB? <;MCM @OH=NCIHM CH NB? ;=NO;F >IG;CH
B;P? ; HIH JIFSHIGC;F >?VHCNCIH +IN? NB;N ; JF;H? QCNB F?P?F P=IHMN;HN CH NB?
;=NO;F G?MB G;S HIN <? ; JF;H? CH NB? L?@?L?H=? MJ;=? 1B;N G;S B;P? @;L L?;=BCHA
=IHM?KO?H=?M @IL NB? NL?;NG?HN I@ MNL;NCV=;NCIHM
"4B7 34B2A8?C8>=
)F>38<4=B8>=0; <4B7
&H NBCM MO<M?=NCIH Q? MB;FF >?VH? NB? HIN;NCIHM ;H> >;N; MNLO=NOL? H?=?MM;LS @IL
H;PCA;NCHA CHMC>? NB? G?MB 1B? P;LC;<F?M CH =;JCN;F F?NN?LM ;L? NB? IH?M NI <? OM?> CH
NB? #,/1/+ JLIAL;G ?R;GJF?M ACP?H F;N?L
G?MB CM G;>? OJ I@ +")"* ?F?G?HNM ;H> +-,&+ P?LNC=?M >?VH?> <S NB?CL
=IIL>CH;N?M 5 6 7 HS ECH> I@ ?F?G?HN NLC;HAF? QCNB FCH?;L CHN?LJIF;NCIH KO;MC
<O<<F? NLC;HAF? JLCMG QCNB =IHMN;HN P;FO? ?N= =;H <? L?@?LL?> NI <S ; =I>? &")*
? A @IL FCH?;L NLC;HAF?M ;H> B;M +!- HI>?M +!- @IL ; FCH?;L NLC;HAF?
1B? HI>?M I@ ?;=B ?F?G?HN ;L? ;MMI=C;N?> NI ; FI=;F HOG<?L L;HACHA @LIG NI +!1B? FCHEM <?NQ??H NBCM HOG<?LCHA MJ?=CV= NI IH? ?F?G?HN FI=;F HOG<?LCHA ;H>
NB? HOG<?LCHA MJ?=CV= NI NB? QBIF? G?MB HI>?M L;HACHA @LIG NI +-,&+ AFI<;F
HOG<?LCHA ;L? MJ?=CV?> <S NB? =IHH?=NCPCNS N;<F? &()" 1B? AFI<;F HOG<?L I@ ;
HI>? QBIM? FI=;F HOG<?L CM &!- CH NB? ?F?G?HN &")"* CM &()"&")"* &!- #IL
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
γ
1
4
6
5
1
0
β
1
α
-1
1
3
2
#CAOL? L?@?L?H=? JLCMG
?;=B NLC;HAF? NB? FCMN I@ FI=;F JICHNM CM JLIPC>?> ;HNC=FI=EQCM? MI ;M NI =IGJON? NB?
NLC;HAF? ;L?; QCNBION GCMN;ECHA NB? MCAH (HIQF?>A? I@ &()" JLIPC>?M ;FMI ;H CGJFC=CN
FI=;F HOG<?LCHA I@ NB? MC>?M I@ ?;=B NLC;HAF? MC>? HOG<?L CM G;>? OJ I@ FI=;F JICHNM
;H> ;H> MI IH
1B? N;<F?M 5 6 7 ;H> &()" ;L? MO^=C?HN @IL ; @OFF >?M=LCJNCIH I@ NB? G?MB %IQ
?P?L CN CM OM?@OF NI >?VH? INB?L N;<F?M QBC=B QCFF B?FJ CH NB? ?F;<IL;NCIH I@ ;FAILCNBGM
G;CHFS QB?H >?;FCHA QCNB <IOH>;LS =IH>CNCIHM
1B? JICHNM IH NB? <IOH>;LS I@ NB? G?MB ;L? HOG<?L?> @LIG NI +-1#/ 1B?
FCHE QCNB NB? AFI<;F HOG<?LCHA CM G;>? <S NB? N;<F? +,/ +,/&-1#/ CM NB?
AFI<;F HOG<?L I@ NB? JICHN &-1#/ I@ NB? <IOH>;LS S =BIIMCHA NBCM >;N; MNLO=
NOL? ;HS JICHN IH NB? <IOH>;LS =;H <? KOC=EFS FI=;N?> &H NQI >CG?HMCIHM NB?
FCMN I@ NB?M? JICHNM JLIPC>?M GIL? CH@ILG;NCIH QB?H @IFFIQCHA NB? CH=L?;MCHA HOG
<?LM IH? GIP?M ;FIHA NB? <IOH>;LS QCNB NB? >IG;CH ;N IH?YM F?@N ;HNC=FI=EQCM?
>CMJF;=?G?HN ?R=?JN @IL CMF;H>M QBIM? HOG<?LCHA CM =FI=EQCM? 1B? JICHN &-1#/ I@ NB? ?>A? @IFFIQM NB? JICHN &-1#/ ?R=?JN QB?H NB? HOG<?LCHA B;M NI MBC@N @LIG
NB? ?RN?LH;F ?>A? NI ;H CMF;H> IL @LIG IH? CMF;H> NI ;HINB?L 4B;N?P?L NB? =;M?
NB? N;<F? (-,/ QCFF <? OM?> NI >?N?LGCH? NB? H?RN JICHN IH NB? <IOH>;LS ?>A?
2MO;FFS (-,/&-1#/&-1#/ <ON CH NB? =;M? I@ ; G?MB QCNBION ;H CM
F;H> (-,/+-1#/ &H NQI >CG?HMCIHM IH NB? <IOH>;LS NB? HOG<?LCHA I@
JICHNM G;S <? OM?> @IL HOG<?LCHA NB? M?AG?HNM 0?AG?HN & CM FCGCN?> <S JICHNM & ;H>
(-,/& 0IG?NCG?M CN G;S <? OM?@OF NI ;>> ; M?=IH> >CG?HMCIH NI (-,/ NI
>?M=LC<? NB? <IOH>;LS CH NB? IJJIMCN? >CL?=NCIH
(-,/&-1#/ CM NB? HI>? DOMN ;@N?L &-1#/ QBCF? (-,/&-1#/ CM
NB? HI>? DOMN <?@IL? CN
1I >?M=LC<? NB? <IOH>;LS INB?L N;<F?M ;L? ;P;CF;<F? &(),/ CM ; =IHH?=NCPCNS
N;<F? @IL NB? <IOH>;LS MC>?M CN ACP?M NB? AFI<;F HOG<?LM I@ NB? JICHNM =IHMNCNONCHA
NB? MC>? +"),/ ;MMI=C;N?M NB? HOG<?LM I@ NB? M?AG?HNM IH NB? <IOH>;LS NI NB?
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
HOG<?LM I@ NB? ?F?G?HNM NI QBC=B NB?S <?FIHA ;H> +2),+" CM ; N;<F? QBC=B =IHH?=NM
NB? FI=;F HOG<?L I@ NB? <IOH>;LS JICHNM QCNBCH NB? MC>?M I@ QBC=B NB?S ;L? ; J;LN NI
NB? FI=;F HOG<?L I@ NB? M;G? JICHNM QCNBCH NB? ?F?G?HN NI QBC=B NB?S <?FIHA #CAOL?
MBIQM BIQ NI OM? NB?M? N;<F?M @IL ; NLC;HAOF;L ?F?G?HN
)?N OM =IHMC>?L NB? NLC;HAF? - QBIM? HOG<?L CM &")"* <OCFN IH NB? NBL?? HI>?M
QBIM? AFI<;F HOG<?LM ;L? +$ +$ ;H> +$ 1B? MC>? FCGCN?> <S NB? JICHNM +$
;H> +$ J?LN;CHM NI NB? <IOH>;LS ;H> B;M NB? HOG<?L &##/ ?KO;F NI &-1#/
1B? HI>?M +$ ;H> +$ ;L? IH NB? <IOH>;LS 1B?CL HOG<?LM ;M <IOH>;LS JICHNM
;L? &-1#/ ;H> &-1#/ 1B? HI>?M +$ +$ ;H> +$ B;P? FI=;F HOG<?LM ;H> QCNBCH NB? NLC;HAF?
triangle IELEM
NG3
IPTFR2
NG1
side IFAFR
NG2
IPTFR1
#CAOL? AFI<;F HOG<?LCHA ;H> <IOH>;LS JICHN HOG<?LCHA
,H? A?NM
&()"&")"* +$
&()"&")"* +$
&()"&")"* +$
+,/&-1#/ +$
+,/&-1#/ +$
+"),/&##/ &")"*
&(),/&##/ +$
&(),/&##/ +$
+2),+"&##/ +2),+"&##/ (-,/&-1#/ &-1#/
(-,/&-1#/ &-1#/
)7A4438<4=B8>=0; <4B7
*IMN I@ NB? =IH=?JNM ?MN;<FCMB?> CH NQI >CG?HMCIHM QCFF <? OM?@OF MCH=? NB? NBL??
>CG?HMCIH;F G?MB CM ; JCF? I@ G?MB?M G;>? I@ NLC;HAF?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? HOG<?LCHA I@ NB? HI>?M ;H> ?F?G?HNM CH NB? ! G?MB CM >?>O=?> @LIG NB?
HOG<?LCHA I@ NB? ! G?MB 1B? VLMN JF;H? HI>?M ;H> NB? VLMN F;S?L I@ JLCMGM CH NB?
! G?MB B;P? NB? M;G? HOG<?LM ;M NB? HI>?M ;H> NLC;HAF?M CH NB? ! G?MB #IL NB?
JF;H?M ;H> F;S?LM ;<IP? NB? HOG<?LCHA CM MCGJFS MBC@N?> @IL NB? HI>?M <S NB? HOG<?L
I@ HI>?M CH NB? ! G?MB ;H> @IL NB? JLCMGM <S NB? HOG<?L I@ ?F?G?HNM CH NB? !
G?MB
1B? <IOH>;LS MC>?M I@ NB? JLCMGM ;L? I@ NQI NSJ?M 1B?S G;S <? NLC;HAF?M IL
KO;>LCF;N?L;FM >CG?HMCIH GOMN <? ;>>?> NI NB? N;<F?M +"),/ &(),/ ;H>
+2),+" CH IL>?L NI C>?HNC@S NB? ECH> I@ MC>? OH>?L =IHMC>?L;NCIH NLC;HAF? KO;>LC
F;N?L;F ?N= QBC=B G;S <? H?=?MM;LS @IL =;F=OF;NCHA NB? <IOH>;LS G;NLCR 16-")
CM NB?H OM?> @IL NBCM JOLJIM? 16-")&")* JLIPC>?M NB? =I>? QCNB NB? VLMN
NSJ? I@ MC>? I@ NB? &")* NSJ? ?F?G?HN -LCMGM QCNB FCH?;L CHN?LJIF;NCIH L?@?LL?> NI
<S NB? =I>? B;P? NQI NSJ?M I@ MC>?M KO;>LCF;N?L;FM ;H> NLC;HAF?M
IHM?KO?HNFS
16-") =I>? &")* @IL NB? FCH?;L KO;>LCF;N?L;F ;H> 16-") =I>? &")* @IL NB? FCH?;L NLC;HAF?
"0CA8G BC>A064
4CNBCH NB? OMO;F N?=BHCKO?M @IL MNILCHA G;NLC=?M CH VHCN? ?F?G?HNM Q? =;H JICHN ION
NB? >C;AIH;F MNIL;A? QCNB IJNCGCT;NCIH I@ NB? G;NLCR <;H>QC>NB ;H> NB? =IGJ;=N
MNIL;A? NB;N M;P?M G?GILS MJ;=? <ON G;E?M NB? MSGG?NLC= G;NLCR =;F=OF;NCIHM GIL?
>C^=OFN 1BIOAB Q? MB;FF HIN OM? NB?M? NQI NSJ?M I@ MNIL;A? Q? MB;FF >?M=LC<? NB?G
CH IL>?L NI MBIQ NB?CL CGJ?L@?=NCIHM 1B?M? G?NBI>M ;L? MFIQFS <ON MOL?FS >LIJJ?>
@IL NB? <?H?VN I@ GIL? ?^=C?HN IH?M 1B?M? ;L? ?H>IQ?> QCNB ; >;N; MNLO=NOL? QBIM?
L?AOF;LCNS CM P?LS MCGCF;L NI NB? L?AOF;LCNS I@ MNLO=NOL?> G?MB?M 4? MB;FF NB?H >?N;CF
MO==?MMCP?FS "" "F?G?HN S "F?G?HN MNIL;A? QBC=B OM?M P?=NIL =IGJON?L ;L=BC
N?=NOL? J?L@ILG;H=?M CH NB? <?MN JIMMC<F? G;HH?L ;H> ?>A? <;M?> MNIL;A? QBC=B
M??GM FCE? NB? IJNCGOG ;H> VH;F MIFONCIH ;H> CM P?LS =FIM? NI NB? >;N; MNLO=NOL?
OM?> CH VHCN? PIFOG? G?NBI>M
806>=0; BC>A064
1BCM G?NBI> ;MMOG?M ;H> CN CM IH? I@ CNM G;CH >CM;>P;HN;A?M NB;N NB? G;NLCR B;M
; FIQ HOG<?L I@ HIH T?LI >C;AIH;FM >C;AIH;FM ;L? B?L? OH>?LMNII> ;M FCH?M J;L;FF?F
NI NB? G;CH >C;AIH;F 1B;N L?KOCL?M ?R=?JN CH NB? MCGJF? =;M?M I@ L?AOF;L G?MB?M ;
JL?PCIOM HOG<?LCHA I@ NB? HI>?M CH IL>?L NI A?N ; L?>O=?> <;H>QC>NB @IL NB? G;NLCR
&@ H CM NB? IL>?L I@ NB? G;NLCR ;H> E NB? HOG<?L I@ MO=B HIH T?LI >C;AIH;FM NB? G;NLCR
CM G;>? OJ I@
; N;<F? 4H E =IHN;CHCHA NB? ?F?G?HNM I@ NB? HIH T?LI >C;AIH;FM
; N;<F? &6H E MJ?=C@SCHA NB? =IFOGH CH>?R I@ NB? ?F?G?HNM I@ 4
; N;<F? JICHNCHA NI NB? FI=;NCIH I@ NB? HIH T?LI >C;AIH;F
AL?;N HOG<?L I@ T?LI P;FO?M G;S <? MNIL?> <S NBCM G?NBI> QBIM? ?^=C?H=S >?
J?H>M F;LA?FS IH NB? KO;FCNS I@ IJNCGCT;NCIH I@ NB? <;H>QC>NB #OLNB?LGIL? QILECHA
ION NB? G;NLC=?M CM P?LS >C^=OFN
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
><?A4BB43 (?0AB4 '>F
1BCM G?NBI> ;FMI EHIQH <S CNM ;=LIHSG 0/ ;FFIQM IH? NI FCGCN NB? MNIL;A? NI NB?
HIH T?LI ?F?G?HNM I@ NB? G;NLCR &N H??>M
; N;<F? 4 =IHN;CHCHA NB? HIH T?LI ?F?G?HNM MNIL?> LIQ <S LIQ
; N;<F? &6 MJ?=C@SCHA NB? =IFOGH CH>?R I@ NB? ?F?G?HNM I@ 4
; N;<F? JICHNCHA NI NB? FI=;NCIH CH NB? ;LL;S 4 I@ NB? HIH T?LI ?F?G?HNM IH ?;=B
LIQ
1B? G;DIL >CM;>P;HN;A? I@ NB? 0/ G?NBI> CM NB? >C^=OFNS I@ >CL?=NFS ;==?MMCHA
;H ?F?G?HN I@ NB? G;NLCR L?@?LL?> NI <S CH>C=?M C ;H> D 1BCM CM <?=;OM? CH NB? VHCN?
?F?G?HN JLI=?MMCHA NB? HOG<?L I@ H?CAB<IOLM @IL ?;=B JICHN CM P;LC;<F? @LIG JICHN
NI JICHN ;H> B?H=? NB? HOG<?L I@ HIH T?LI ?F?G?HNM QCNBCH ; LIQ CM P;LC;<F? NII
M ; =IHM?KO?H=? ; MSGG?NLC= G;NLCR MBIOF> <? MNIL?> @OFFS 1BCM >CM;>P;HN;A?
P;HCMB?M QCNB NB? ?>A? <;M?> MNIL;A? NBIOAB CN MNIL?M ;FMI NB? HIH T?LI ?F?G?HNM I@
NB? G;NLC=?M IHFS P;LC;HN I@ 0/ =;FF?> *0/ *I>CV?> 0J;LM? /IQ JONM ;MC>?
NB? >C;AIH;F I@ NB? G;NLC=?M 1BCM CM QB;N QCFF <? >IH? CH NB? @IFFIQCHA NQI ECH>M I@
MNIL;A?
BC>A064
4CNBCH VHCN? ?F?G?HN MI@NQ;L? QCNB CN?L;NCP? G?NBI>M I@ =IGJON;NCIH NB? G;NLC=?M
;L? A?H?L;FFS GOFNCJFC?> <S P?=NILM HS INB?L ECH> I@ IJ?L;NCIH QCNB G;NLC=?M CM F?MM
@L?KO?HN ;H> ;M QCFF <? M??H CH 0?=NCIH NB?M? IJ?L;NCIHM =;H <? ?F;<IL;N?> IH NB?
M;G? ;L=BCN?=NOL? ;M NB? G;NLCR\P?=NIL JLI>O=N 1B? "" N?=BHCKO? I@ MNIL;A? B;M
<??H =IH=?JNO;FCT?> NI G;E? NB? G;NLCR\P?=NIL JLI>O=N ;M ?^=C?HN ;M JIMMC<F? 1B?
@OH>;G?HN;F C>?; MN;N?M NB;N CH VHCN? ?F?G?HN NB?ILS CN CM HIN H?=?MM;LS NI ;MM?G<F?
; G;NLCR CH IL>?L NI GOFNCJFS CN <S ; P?=NIL 4CNBCH ; G?MB QCNB +")"* ?F?G?HNM
; G;NLCR ) CM ?RJL?MM?> PC; NB? ?F?G?HN G;NLC=?M )3 IH ?;=B ?F?G?HN
)
<3:3;
,
,3 )3 B ,3
3
QB?L? ,3 CM ; G;NLCR QBC=B G;E?M NB? NL;HMCNCIH <?NQ??H NB? ?F?G?HN ;H> NB? IP?L;FF
G?MB ,3 CM >?LCP?> @LIG NB? =IHH?=NCPCNS N;<F? M ;H ?R;GJF? @IL ; NLC;HAOF;L
?F?G?HN QBIM? HOG<?L CM ; ;H> QBIM? P?LNC=?M B;P? NB? AFI<;F HOG<?LM +$ +$ ;H>
+$ )3 QCFF <? ; G;NLCR I@ >CG?HMCIH ;H> ,3 ; G;NLCR I@ >CG?HMCIH +-,&+ 1B? =I?^=C?HN I@ ,3 FI=;N?> ;N NB? CHN?LM?=NCIH I@ LIQ & ;H> =IFOGH ' ?KO;FM C@
&&()"; ' ;H> INB?LQCM?
(
' (
'
' (
(
'
(
,3 '
(
'
' (
(
'
&
%
LIQ &()"? LIQ &()"? LIQ &()"? %6!/,!6+*& 0 ,# #/"" 02/# " #),40
&@ 4 CM ; P?=NIL NB? JLI>O=N ) 4 CM ?RJL?MM?> ;M
)4
<3:3;
,
3
,3 )3 B ,3 4
1BCM CM ?KOCP;F?HN NI GOFNCJFSCHA )3 <S NB? =IGJIH?HNM I@ 4 ;MMI=C;N?> QCNB NB?
HI>?M I@ ?F?G?HN ; ?F?G?HN;LS JLI>O=NM NB?H MOGGCHA IP?L NB? QBIF? ?F?G?HNM I@
NB? G?MB ;MM?G<FS &N CM H?P?L H?=?MM;LS NI <OCF> ,3 CNM?F@ QBC=B CM DOMN ; MJ;LM?
@ILG I@ NB? =IHH?=NCPCNS N;<F? &()"
G;NLCR ) =;H <? MNIL?> ;M +")"* G;NLC=?M )3 #IL ; G?MB G;>? I@ FCH?;L
NLC;HAF?M NB;N F?;>M NI +")"* =I?^=C?HNM 1BCM HOG<?L =;H <? L?>O=?> <S E??JCHA
IHFS NB? ?RNL; >C;AIH;F ?HNLC?M @IL ?;=B ?F?G?HN G;NLCR ;H> <S ;MM?G<FCHA NB? >C;AIH;F
?HNLC?M ;M >?M=LC<?> B?L?;@N?L
&@ NB? >C;AIH;F ;H> ?RNL; >C;AIH;F J;LN I@ )3 ;L? >?HIN?> 3 ;H> 43 L?MJ?=NCP?FS
)3 3 43 IH? =;H QLCN?
)4 <3:3;
,
3
,3
3
B
,3 4
<3:3;
,
3
,3 43 B ,3 4 4
<3:3;
,
3
,3 43 B ,3 4 QB?L?
CM NB? >C;AIH;F I@ ) L?MOFNCHA @LIG NB? ;MM?G<FS I@ >C;AIH;FM 3
HS G;NLCR =;H <? MNIL?> QCNBCH NQI N;<F?M IH? ;/B =IHN;CHCHA NB? ;MM?G<F?>
>C;AIH;F ;H> NB? INB?L 4;/B =IHN;CHCHA NB? ?RNL; >C;AIH;F ?HNLC?M I@ NB? ?F?G?HN
G;NLC=?M #IL ; G;NLCR ;MMI=C;N?> QCNB ; G?MB I@ NLC;HAOF;L ?F?G?HNM +")"*
+-,&+ =I?^=C?HNM QCFF B;P? NI <? MNIL?> QBC=B G;E?M ;H ?=IHIGS I@ ;<ION
+")"* =I?^=C?HNM CH NB? G?GILS MJ;=? =IGJ;L?> QCNB NB? MNIL;A? I@ NB? QBIF?
?F?G?HN G;NLC=?M #IL ; MSGG?NLC= G;NLCR IHFS B;F@ I@ NB? ?RNL; >C;AIH;F N?LGM QCFF
<? MNIL?> ;H> NB? ?=IHIGS CM I@ +")"* GIL? =I?^=C?HNM
#OLNB?LGIL? NB? OM? I@ ?F?G?HN G;NLC=?M ;FFIQM IH? NI P?=NILCT? NB? G;NLCR\P?=NIL
JLI>O=N IH P?=NIL =IGJON?L ;L=BCN?=NOL?M 1B? FIIJ IH NB? ?F?G?HN;LS JLI>O=NM =;H
<? P?=NILCT?> ;H> NB? ;MM?G<FCHA FIIJ ;M Q?FF ;M FIHA ;M NB? HOG<?LCHA B;M <??H
B;H>F?> QCNB =;L? 1B? >?N;CF?> =IGJON;NCIH I@ G;NLCR\P?=NIL JLI>O=NM ;H> NB?
JLI=?MM I@ ;MM?G<FCHA QCFF <? ?R;GCH?> @OLNB?L CH 0?=NCIH QBC=B >?;FM QCNB G;NLCR
IJ?L;NCIHM
4? MB;FF OM? NB? @IFFIQCHA =IHP?HNCIH @IL NB? MNIL;A? I@ ?RNL; >C;AIH;F ?F?G?HNM
)?N ; <? ;H ?F?G?HN G;>? OJ I@ +!- HI>?M 1B?H ;H ?F?G?HN G;NLCR )3 B;M
+!- +!- ?RNL; >C;AIH;F ?HNLC?M 478 FI=;N?> ;N NB? CHN?LM?=NCIH I@ LIQ C ;H>
=IFOGH D I@ )3
4? MB;FF >?HIN?
4) 0 ; 4
4) 0 ; 4
4) 0 ; * , 4
<2>
4) 0 ; * , * , 4<2>
<2>
@IL NB? ?HNLC?M I@ NB? OJJ?L NLC;HAOF;L J;LN I@ )3
&@ )3 CM MSGG?NLC= 4) 0
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
1;<F? G?GILS MJ;=? >?J?H>CHA IH MNIL;A? ;H> ?F?G?HN
"F?G?HN NSJ?
)CH?;L KO;>LCF;N?L;F
)CH?;L NLC;HAF?
.O;>L;NC= NLC;HAF?
.O;>L;NC= KO;>LCF;N?L;F
)CH?;L <LC=E
)CH?;L JLCMGLCMG
+-,&+
+-,&+
+-,&+
+-,&+
+-,&+
+-,&+
"" MNIL;A?
+")"* +-,&+
+")"* +-,&+
+")"* +-,&+
+")"* +-,&+
+")"* +-,&+
+")"* +-,&+
0/
+-,&+
+-,&+
+-,&+
+-,&+
+-,&+
+-,&+
CM @OFFS >?VH?> ,NB?LQCM? CN CM H?=?MM;LS NI MNIL? NB? FIQ?L NLC;HAOF;L J;LN I@ )3 ;M
@IFFIQM
4) 0 ; * , * , 4) 0 ; * , * , 4
4
4) 0 ; * , * , * , 4<2>
4) 0 ; * , * , 4<2> <2>
&H NB? G;NLCR )3 B?L?;@N?L G;>? OJ I@ ; NLC;HAF? QCNB FCH?;L CHN?LJIF;NCIH NB? >C@
@?L?HN J;LNM I@ NB? N;<F? 5*1 B;P? <??H FI=;N?> ;M NB?S MBIOF> NB? CH NB? N;<F?
L?@?LM NI NB? >C;AIH;F ?HNLC?M NB;N ;L? ;MM?G<F?> ;H> HIN MNIL?> ;N NB? ?F?G?HN;LS
F?P?F
)3 % 4) 0 ; 4) 0 ; 4) 0 ; 4) 0 ; 4) 0 ; 4) 0 ; &
1;<F? MOGG;LCT?M @IL MIG? ECH>M I@ ?F?G?HNM NB? G?GILS MJ;=? H?=?MM;LS
@IL NB? MNIL;A? I@ ; G;NLCR MIG? P;FO?M ;L? JOL?FS MN;NCMNC=;F ;H> >?J?H> IH NB?
ILA;HCT;NCIH I@ NB? ?F?G?HNM ;H> IH NB? HOG<?L I@ H?CAB<IOLM
36410B43 BC>A064
1BCM CM ; H?Q N?=BHCKO? ?H;<FCHA ;H ;MM?G<F?> G;NLCR NI <? MNIL?> CH ; L?AOF;L
;H> IJNCG;F Q;S 1B? =IH=?JN
CM NB;N @IL ; G?MB G;>? I@ NLC;HAF?M QCNB FCH?;L
+
CHN?LJIF;NCIH NB? JLI>O=NM 7 8 : I@ NQI <;MCM @OH=NCIHM 7 ;H> 8 QCNB >CMNCH=N
? ;H> @ QCFF <? HIH T?LI IHFS C@ ? ;H> @ ;L? FCHE?> <S ; M?AG?HN I@ NB? G?MB "P?LS
M?AG?HN CM NB?H NB? C>?;F ZJF;=?[ NI MNIL? NB? ?RNL; >C;AIH;F ?HNLC?M I@ NB? G;NLC=?M
#IL ; HIH MSGG?NLC= G;NLCR ; M?AG?HN L?JL?M?HNM NQI ?HNLC?M 1BCM C>?; CM P?LS
MCGJF? @IL ;FF MCGJFC=C;F ?F?G?HNM FCE? NLC;HAF?M IL N?NL;B?>L; &N =;H <? ?RN?H>?> NI
;FF INB?L ?F?G?HNM <S ;>>CHA ?RNL; CG;ACH;LS M?AG?HNM #IL ; KO;MC <O<<F? NLC;HAF?
NBL?? M?AG?HNM FCHECHA NB? NBL?? P?LNC=?M NI NB? =?HNL;F JICHN QCFF <? H?=?MM;LS
1B? >;N; MNLO=NOL? H?=?MM;LS NI B;H>F? ;H> <OCF> ?>A? <;M?> MNIL?> G;NLC=?M CM ;M
@IFFIQM
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
H ;LL;S H;G?> $),0"$ MCGCF;L NI &()" @IL NB? NLC;HAF?M ACP?M NB? HOG<?LM
I@ NB? NQI JICHNM >?VHCHA NB? M?AG?HN &NM >CG?HMCIH CM +0"$ QB?L? +0"$
CM NB? HOG<?L I@ M?AG?HNM CH NB? =;M? I@ NLC;HAF?M +0"$ # !" # $ ( & 1B? M?=IH> >CG?HMCIH L?@?LM NI NB? NQI JICHNM I@ NB? M?AG?HNM
H ;LL;S H;G?> ")10"$ QBIM? >CG?HMCIH CM +")"* +0 QB?L? +0 CM NB?
HOG<?L I@ M?AG?HNM =IHMNCNONCHA ;H ?F?G?HN NBL?? @IL ; NLC;HAF? #IL ?;=B
?F?G?HN ")10"$ JLIPC>?M NB? HOG<?LCHA I@ CNM ;MMI=C;N?> M?AG?HNM
H ;LL;S H;G?> ,/&0"$ QBIM? >CG?HMCIH CM +")"* +0 ,/&0"$ CM FCHE?>
NI ")10"$ ;H> CH>C=;N?M NB? ILC?HN;NCIH I@ NB? M?AG?HN QCNBCH NB? ?F?G?HN &NM
P;FO? CM C@ IH? GIP?M ;HNC=FI=EQCM? ;LIOH> NB? ?F?G?HN @IFFIQCHA NB? CH=L?;MCHA
FI=;F HOG<?LCHA I@ HI>?M ;H> INB?LQCM?
H ?>A? <;M?> MNIL?> ;MM?G<F?> G;NLCR QCFF <? >?VH?> <S ; >C;AIH;F ;H> NQI
N;<F?M 5 ;H> 5 L;HE +0"$ @IL ; HIH MSGG?NLC= G;NLCR IL IHFS IH? N;<F?
5 @IL ; MSGG?NLC= G;NLCR
5 CH=FO>?M NB? =I?^=C?HN I@ JICHN CH NB? JICHN L?F;N?> ?KO;NCIH
5 CH=FO>?M NB? =I?^=C?HN I@ JICHN CH NB? JICHN L?F;N?> ?KO;NCIH
1B? Q;S I@ ;MM?G<FCHA ; G;NLCR @IL ?>A? <;M?> MNIL;A? ;M Q?FF ;M NB? ?RJL?MMCIH
@IL NB? G;NLCR\P?=NIL JLI>O=N QCFF <? >?M=LC<?> QB?H L?KOCL?>
$?4A0C8>=B >= <0CA824B
1BCM =B;JN?L >?M=LC<?M BIQ NI <OCF> NB? G;NLC=?M ;H> C@ H?=?MM;LS ;M CH NB? =;M? I@
?>A? <;M?> MNIL;A? BIQ NI ;MM?G<F? NB?G &N ;FMI >?M=LC<?M NB? G;CH IJ?L;NCIHM IH
G;NLC=?M J?L@ILG?> CH VHCN? ?F?G?HN MI@NQ;L? CH NB? =;M? I@ "" MNIL;A? 4? MB;FF
>CM=OMM MO==?MMCP?FS
1B? =IHMNLO=NCIH I@ ; HIH ;MM?G<F?> G;NLCR
1B? ;MM?G<FS JLI=?MM @IL ?>A? <;M?> MNIL;A?
1B? JLI>O=N I@ ; HIH MSGG?NLC= G;NLCR ;H> ; P?=NIL
1B? JLI>O=N I@ ; MSGG?NLC= G;NLCR ;H> ; P?=NIL
1B? JLI>O=N I@ ; NL;HMJIM?> G;NLCR ;H> ; P?=NIL
1B? NL?;NG?HN I@ !CLC=BF?N NSJ? <IOH>;LS =IH>CNCIHM
1B? >C;AIH;F JL?=IH>CNCIHCHA
1B? GIMN CGJILN;HN ;FAILCNBG H;G?FS NB? G;NLCR\P?=NIL JLI>O=N ;FAILCNBG
HS INB?L =;H <? <;M?> IH IL L?>O=?> NI NBCM IH?
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
D8;38=6 C74 <0CA824B
2HFCE? NB? N?=BHCKO?M <;M?> IH $;OMM KO;>L;NOL? NB? G;NLC=?M B?L? ;L? =;F=OF;N?> <S
?R;=N ;H;FSNC=;F CHN?AL;NCIH 1B? ?HNLC?M I@ ; VHCN? ?F?G?HN G;NLCR ;L? OMO;FFS JIFS
HIGC;F CHN?AL;FM QBC=B =;H <? ?P;FO;N?> <S MO=B ;H ;H;FSNC=;F CHN?AL;NCIH 4LCNCHA
NB? CHN?AL;NCIH CM ; FIHA =OG<?LMIG? ;H> LCMES N;ME ?P?H NBIOAB MIG? NLC=EM =;H
<? OM?> 1BCM CM NB? L?;MIH QBS Q? B;P? =BIM?H NI OM? @ILG;F =IGJON;NCIH MI@NQ;L?
/"!2 " *-)" 3 QBC=B =;H JLIPC>? NB? ?R;=N L?MOFN I@ ;H CHN?AL;F =;F=OF;NCIH
H ?R;GJF? I@ MO=B ; =IGJON;NCIH CM ACP?H H?RN
G0<?;4 >5 0 <0CA8G 2><?DC0C8>=
)?N OM =;F=OF;N? NB? G;MM G;NLCR @IL ; G?MB G;>? I@ KO;>LCF;N?L;FM QCNB FCH?;L CH
N?LJIF;NCIH 4? PIFOHN;LCFS =BIIM? ; HIH MCGJFC=C;F ?F?G?HN QBIM? ';=I<C;H CM HIN
=IHMN;HN CH IL>?L NI ;PIC> NII MCGJF? ;H CMIJ;L;G?NLC= NL;HM@ILG;NCIH NB;N QIOF>
BC>? NB? >C^=OFNC?M
1B? ?R;GJF? I@ =IGJON;NCIH QCFF <? G;>? IH ;H ;L<CNL;LS KO;>LCF;N?L;F . >?
N?LGCH?> <S CNM P?LNC=?M - - - ;H> - #CAOL? QBIM? =IIL>CH;N?M ;L?
R S R+ S R S R S 1B? ?HNLS )78 I@ NB? G;MM G;NLCR QCFF <? QLCN
N?H )78 7 8 : QB?L? 7 CM NB? <;MCM @OH=NCIH ;MMI=C;N?> NI NB? HI>? ? ?
IL 1B? =IGJON;NCIH I@ NB? CHN?AL;F =;H NB?H <? G;>? IH ; L?@?L?H=? ?F?G?HN NB;HEM
NI ;H CMIJ;L;G?NLC= NL;HM@ILG;NCIH 0 HS HI>? I@ NB? L?@?L?H=? ?F?G?HN . QCNB =I
IL>CH;N?M CM ;MMI=C;N?> QCNB IH? HI>? I@ NB? KO;>LCF;N?L;F . QBIM? =IIL>CH;N?M
;L? N O MO=B NB;N
N J
O J
J J J J J J P1
η
P2
+1
4
1
2
3
+1
-1
1
2
-1
ζ
4
P4
T
3
Y
P3
X
#CAOL? CMIJ;L;G?NLC= NL;HM@ILG;NCIH @IL ; KO;>LCF;N?L;F
1B? CG;A? J?L 0 I@ ?P?LS P?LN?R CH NB? L?@?L?H=? ?F?G?HN CM NBOM ; P?LN?R I@ NB?
KO;>LCF;N?L;F S ; JLI=?MM I@ C>?HNCV=;NCIH IH? =;H >?>O=? NB? =I?^=C?HNM J J
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
?N= I@ NB? NL;HM@ILG;NCIH QCNB L?MJ?=N NI NB? =IIL>CH;N?M N
O I@ NB? P?LNC=?M
J N
N
N
J N
N
N N N
J N N
N N O
O
O
J O
O
O O J O O
J O
O O
N
O
N O N N J N N
J O
N O O O O <;MCM @OH=NCIH CH NB? ;=NO;F ?F?G?HN =ILL?MJIH>M NI ; JIFSHIGC;F , ? MO=B ;M
N O , ? 0 N O
&H NB? ?R;GJF? ;<IP? NB?L? ;L? @IOL JIFSHIGC;FM ;MMI=C;N?> NI NB? @IOL <;M?M I@
NB? ;=NO;F ?F?G?HN
, , , , M @IL NB? @OH=NCIHM NB? P;FO? I@ ?;=B JIFSHIGC;F CM ;N IH? P?LN?R ;H> @IL
NB? INB?LM
&H NB? L?@?L?H=? ?F?G?HN NB? CHN?AL;F CM
.
%
7 8 :- .
.
,7 ,8 & ::
QB?L? & CM NB? ';=I<C;H I@ NB? NL;HM@ILG;NCIH 0 ?KO;F NI NB? >?N?LGCH;HN I@ NB?
';=I<C;H G;NLCR
F
G
F
G
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
0I
& J
J J
J J
J J
J 4? MB;FF ;MMOG? ; JIMCNCP? P;FO? @IL & QBC=B CM ;=BC?P?> QCNB ;H ;HNC=FI=EQCM?
FI=;F HOG<?LCHA I@ NB? HI>?M ;FIHA NB? ?>A? ?=;OM? I@ NB? JIFSHIGC;F H;NOL? I@ &
NB? =IGJON;NCIH CM L?>O=?> NI NB? =;F=OF;NCIH I@ ; JIFSHIGC;F QBIM? BCAB?MN IL>?L
CM ?RJL?MM?> ;M 1B?M? >;N; ;L? MO^=C?HN NI ;=BC?P? ;H ;H;FSNC=;F =IGJON;NCIH QCNB @ILG;F =;F=O
F;NCIH MI@NQ;L? 1B? L?MOFN G;S <? CH=FO>?> CH #,/1/+ MI@NQ;L? #IL ?R;GJF?
NB? @ILG;F =;F=OF;NCIH I@ ; G;MM G;NLCR IH ; KO;>LCF;N?L;F QCFF ACP?
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
) 0
4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 4 5 5 4 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 4 5 5 4 5 4 5 5 4 5 5 4 5 5 4 5 4 5 5 4 5 4 5 4 5 5 4 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 5 4 5 4 5 5 4 5 4 5 5 4 5 5 4 5 5 4 5 4 5 5 4 5 4CNB ; P?=NIL =IGJON?L ;L=BCN?=NOL? NB? JL?PCIOM ?RJL?MMCIH CH=FO>?> CH ; FIIJ
=;H <? P?=NILCT?> +I FCH? >?J?H>M IH NB? L?MOFN I@ ; JL?PCIOM IH? NB?L? CM HI
<;=EQ;L> >?J?H>?H=?
1B? >?GIHMNL;NCIH DOMN MBIQH =;H <? ;JJFC?> NI ;HS G;NLCR NB;N F?;>M NI NB? CH
N?AL;F I@ ; JIFSHIGC;F ?RJL?MMCIH 1BCM QIOF> <? NB? =;M? @IL G;MM G;NLC=?M ;H>
>CP?LA?H=? NSJ? G;NLC=?M &N QIOF> HIN ;FQ;SM <? ;JJLIJLC;N? @IL >C_OMCIH NSJ?
G;NLC=?M QB?L? ; ';=I<C;H ;JJ?;LM ;M NB? >?HIGCH;NIL
#IL HIH MCGJFC=C;F ?F?G?HNM NB? ;H;FSNC=;F =;F=OF;NCIH QCFF HIN <? MOCN;<F? @IL NB?
>C_OMCIH G;NLCR ;H> NB? $;OMM KO;>L;NOL? QCFF DOMN ACP? ;H ;JJLIRCG;N? MIFONCIH
2H@ILNOH;N?FS NB? JLCMG CM I@ NBCM NSJ? ;H> L?KOCL?M ;JJLIRCG;NCIHM
JJ?H>CR MBIQM NB? VF? I@ *-)" 3 CHMNLO=NCIHM @IL NB? ;H;FSNC=;F =;F=OF;NCIH
I@ ; >C_OMCIH G;NLCR CH ; N?NL;B?>LIH QCNB FCH?;L CHN?LJIF;NCIH &H NBCM =;M? NB?
';=I<C;H CM =IHMN;HN IH ?P?LS ?F?G?HN
%A>?4AC84B >5 J=8C4 4;4<4=C <0CA824B
?MC>?M <?CHA ZMJ;LM?[ C ? =IHN;CHCHA ; FIN I@ T?LI ?HNLC?M NB? VHCN? ?F?G?HN G;NLC=?M
B;P? CHN?L?MNCHA JLIJ?LNC?M QBC=B =;H <? OM?> CH IL>?L NI IJNCGCT? -2 NCG? IL NB?
G?GILS MJ;=? IL QBC=B =;H ?RJF;CH ; =?LN;CH <?B;PCIOL I@ NB? HOG?LC=;F M=B?G?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
(H<<4CAH
NBCM CM NB? =;M? I@ NB? G;MM G;NLCR
.
7 8 :
;H> I@ NB? >C_OMCIH G;NLCR
.
=H7: 7 =H7: 8 :
1B;HEM NI NB? ECH>M I@ MNIL;A? OM?> IHFS B;F@ NB? ?RNL; >C;AIH;F N?LGM ;L? MNIL?>
M&D0B8<0682N B@D0A4 ; >C_OMCIH G;NLCR CM ; KO;MC G;AC= MKO;L? ,H ?P?LS LIQ
;M Q?FF ;M IH ?P?LS =IFOGH NB? MOG I@ ;FF NB? N?LGM CM T?LI 1B? =IGJ;LCMIH MNIJM B?L?
<?=;OM? ;>>CHA OJ NB? N?LGM I@ NB? >C;AIH;F
; HIH T?LI NIN;F 1B?M? JLIJ?LNC?M
)ACP?M
<>=7<
=;H <? >?GIHMNL;N?> <S OMCHA NB? ?KO;NCIH 7 7 1B?S B?FJ NI IJNCGCT? NB?
=;F=OF;NCIHM QBCF? L?>O=CHA NLOH=;NCIH ?LLILM NI NB? GCHCGOG 1BOM @IL NB? >C_OMCIH
G;NLCR I@ IH? ?F?G?HN QCNB D >?AL??M I@ @L??>IG << ?F?G?HN;LS N?LGM MBIOF>
#IL ; NLC;HAF? QCNB FCH?;L CHN?LJIF;NCIH NB? ?RNL;
<? =;F=OF;N?> CHMN?;> I@ <<
>C;AIH;F N?LGM I@ NB? >C_OMCIH G;NLCR ;L? @OFFS =;F=OF;N?> @LIG NB? P;FO?M I@ NB?
N?LGM IH NB? >C;AIH;F
"<0CA8G C@ ; G;NLCR =;H <? QLCNN?H ;M NB? MOG I@ ; >C;AIH;F QBIM? ?HNLC?M
;L? MNLC=NFS JIMCNCP? ;H> I@ ?RNL; >C;AIH;F N?LGM 4 ;FF MNLC=NFS H?A;NCP? CM EHIQH
;M ;H * G;NLCR &N =;H <? ?;MCFS JLIP?> NB;N NB? MSMN?G 4 NB?H B;M ; JIMCNCP?
MIFONCIH C@ ;FF NB? ?HNLC?M I@ ;L? JIMCNCP? >C_OMCIH G;NLCR MBIOF> <? ;H * G;NLCR
<ON MIG? J;NBIFIAC=;F =;M?M =;H <? @IOH> @IL QBC=B NBCM JLIJ?LNS CM HIN P?LCV?>
&H MO=B =;M?M NB? >C_OMCIH CM HI FIHA?L GIHINIHIOM #IL ?R;GJF? F?N OM =IHMC>?L
; NLC;HAF? =IGJLCMCHA NBL?? JICHNM ) ) ;H> ) QBIM? =IIL>CH;N?M ;L? N O
N O ;H> N O 1B? FCH?;L <;MCM @OH=NCIHM IH NBCM NLC;HAF? ;L?
N O O O
N
/
N N
O
/
N O N O
/
N O O O
N
/
N N
O
/
N O N O
/
N O N O
/
O O
N N
N
O
/
/
QB?L? / CM NB? ;L?; I@ NB? NLC;HAF? ?KO;F NI
N O N O N O
N O N O
N O N O
4? A?N
.
B@7/<5:3
N N
O O
:
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
O O O O
/
N N
N N ) ) ) )
/
IL
.
B@7/<5:3
N N
O O
: 1BCM KO;HNCNS CM H?A;NCP? IHFS C@ NB? ;HAF? ;N ) CM ;=ON? >C_OMCIH G;NLCR
=;H NBOM FIM? NB? =B;L;=N?LCMNC=M I@ ;H * G;NLCR C@ IH? ;HAF? CH NB? G?MB CM I<NOM?
%IQ?P?L Q? B;P? =IHMC>?L?> IHFS IH? ?F?G?HN <ON CN QIOF> <? H?=?MM;LS NI ?R;GCH?
NB? ?RNL; >C;AIH;F N?LGM ;@N?L ;MM?G<FS
8QDB8>= <0CA8G F8C7 ?A8B<0C82 4;4<4=C
#IL >C_OMCIH G;NLC=?M I@ NB? @ILG
.
)78 =H7: 7 =H7: 8 :
NB? ';=I<C;H & I@ NB? NL;HM@ILG;NCIH 0 M?? 0?=NCIH ;<IP? ;JJ?;LM CH NB?
>?HIGCH;NIL >O? NI NB? NQI AL;>C?HNM QCNBCH NB? CHN?AL;F &H NB;N =;M? C@ & CM HIN
=IHMN;HN CH MJ;=? NB? @OH=NCIH NI <? CHN?AL;N?> CM HIN ; JIFSHIGC;F ;H> NB?L? CM HI
BIJ? I@ VH>CHA ;H ?R;=N CHN?AL;F 1B?L?@IL? ;H ;JJLIRCG;NCIH GOMN <? >IH? ? A
N;ECHA ;H ;P?L;A? =IHMN;HN ';=I<C;H IH NB? JLCMG HINB?L JLI<F?G CM NB? @;=N NB;N
NBCM >C_OMCIH G;NLCR CM @;L @LIG <?CHA ;H * G;NLCR #CLMN NB? M;G? JLI<F?G ;LCM?M ;M
QCNB NLC;HAF?M <ON NB?L? CM ;FMI NB? ?RNL; >C^=OFNS NB;N MIG? I@ NB? ?RNL; >C;AIH;F
N?LGM MN?GGCHA @LIG P?LNC=;F >C_OMCIH ;L? I<PCIOMFS ;FQ;SM JIMCNCP? #IL ?R;GJF?
NB? N?LG
.
:
H
P P
QCNB CH>C=?M ;H> L?@?LLCHA NI IOL FI=;F HOG<?LCHA I@ JICHNM CH ; JLCMG QCFF <?
JIMCNCP? <?=;OM? <INB >?LCP;NCP?M QCNB L?MJ?=N NI P ;L? H?A;NCP? 1I =IJ? QCNB NBCM
JLI<F?G Q? L?G;LE NB;N <?=;OM? IOL JLCMGM ;L? <;M?> IH NBL?? P?LNC=;F FCH?M CN MBIOF>
<? ?;MS NI =IGJON? NB? M?=IH> >?LCP;NCP? QCNB L?MJ?=N NI P I@ ; @OH=NCIH NI OM? JICHNM
IH ; MCHAF? P?LNC=;F &H INB?L QIL>M QB?H Q? =IGJON? N?LGM MO=B ;M
.
< :
H
P P
CN CM MNL;HA? NI OM? NB? P;FO? I@ < ;N JICHN QB?L?;M 4
H =IOF> <? N;E?H GIL? MCGJFS
4 4
;M H QB?L? P CM NB? >CMN;H=? <?NQ??H JICHNM ;H> Q? >?VH? CH NB? M;G?
Q;S P @IL JICHNM ;H> ;H> P @IL JICHNM ;H> #IFFIQCHA NBCM C>?; ;H> C@
Q? >?HIN?
. .
H
P
: . .
H
P
: ;H> . .
H
:
P
Q? A?N NI ; P?LNC=;F J;LN I@ NB? >C_OMCIH G;NLCR CH NB? JLCMG CH NB? @ILG
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
'
'
'
'
'
'
'
%
&
H
&
H
&
H
&
H
&
H
&
H
&
H
&
H
&
H
&
H
&
H
&
H
(
(
(
(
(
(
(
&
4? HIQ B;P? ;H * G;NLCR &N CM ;FMI =F?;L NB;N NB? ZKO;MC G;AC=[ MKO;L? JLIJ?LNC?M
;L? =IHM?LP?> ;H> IHFS JICHNM QBC=B ;L? IH NB? M;G? P?LNC=;F FCH? ;L? FCHE?> 1BCM
F;NN?L JLIJ?LNS F?;>M NI NB? CGJILN;HN @;=N NB;N C@ NB?L? CM IHFS P?LNC=;F >C_OMCIH CN
=;H <? =IGJON?> <S MIFPCHA ; M?LC?M I@ NLC >C;AIH;F MSMN?GM =ILL?MJIH>CHA NI ;M G;HS
P?LNC=;F FCH?M CH NB? G?MB I@ JLCMGM
"0CA824B F8C7 0 @D0B81D11;4 4;4<4=C
1B? G;NLC=?M OH>?L =IHMC>?L;NCIH ;L? >?VH?> ;M @IFFIQM
)78 .
< 7 8 " 1 2 3 :
QB?L? " 1 2 3 ;L? >C_?L?HN @OH=NCIHM M=;F;L IL P?=NILC;F QBC=B NB? G;NLC=?M >?J?H>
IH #IL NB?M? G;NLC=?M NB? N?MN @OH=NCIHM ;H> NB? <;MCM @OH=NCIHM G;S <? I@
>C_?L?HN NSJ?M FCH?;L IL KO;MC <O<<F? DOMN FCE? NB? >CM=L?NCT;NCIH @OH=NCIHM I@ P;LC;<F?M
" 1 2 ?N=
1QI =;M?M G;S I==OL
;H> ;L? I@ ; KO;MC <O<<F? NSJ?
1B? ?F?G?HN G;NLC=?M B;P? ?HNLC?M 1B? =;F=OF;NCIH @IL NB?M? ?HNLC?M CM ?;MS
NB;HEM NI NB? J;LNCNCIH I@ NLC;HAF? 1 CHNI NBL?? MO< NLC;HAF?M 0I Q? A?N C@ (7 8 CM NB?
L?MNLC=NCIH I@ NB? <;MCM @OH=NCIH 7 NI NB? MO< NLC;HAF? 0 @
)78 .
< (7 (8 " 1 2 3 :
.
< (7 (8 " 1 2 3 :
.
< (7 (8 " 1 2 3 :
QBC=B CM ; MOG I@ NBL?? CHN?AL;FM >?HIN?> % % ;H> %
+IQ NB? <;MCM @OH=NCIHM I@ NB? KO;MC <O<<F? ?F?G?HN L?MNLC=N?> NI NB? MO< NLC;HAF?M
;L? NB? P?LS <;MCM @OH=NCIHM - IH NB? M;G? =?FFM 1B? =;F=OF;NCIH I@ NB? NBL?? CHN?AL;FM
CM NB?H MN;H>;L>
1B? H?RN MN?J CM NI ;>> OJ NB?M? CHN?AL;FM #OLNB?LGIL? NB? >IG;CHM I@ NB? KO;MC
<O<<F? <;MCM @OH=NCIHM L;L?FS CHN?LM?=N IH NB? NBL?? MO< NLC;HAF?M <ON GIMN I@N?H IH
IHFS IH? ?RNL; >C;AIH;F N?LGM M ; =IHM?KO?H=? IH? IL NQI I@ NB? CHN?AL;FM % %
;H> % ;L? T?LI
,H? =;H I<M?LP? NB;N IHFS NB? @OH=NCIH CM >?VH?> IH NB? QBIF? NLC;HAF? 1
IHMC>?LCHA NB? N?LG ) IH? A?NM
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
P1
1 2
T1
3 3
3
T3
P4
P2
2
1
1
2
T2
P3
#CAOL? >IG;CH I@ >?VHCNCIH I@ NB? KO;MC <O<<F? @OH=NCIH MB;>?>
)
.
< (7 (8 " 1 2 3 ?N= :
.
< (7 (8 " 1 2 3 ?N= :
%
% C
C
QB?L? C C C ;L? NB? FCH?;L ?F?G?HN G;NLC=?M =IGJON?> IH NB? MO< NLC;HAF?M 0 0 0 M ; G;NN?L I@ @;=N ( CM NB? <;MCM @OH=NCIH ;MMI=C;N?> NI NB? VLMN JICHN
I@ MO< NLC;HAF? 0 ;H> ( CM NB? <;MCM @OH=NCIH ;MMI=C;N?> NI NB? M?=IH> JICHN I@
MO< NLC;HAF? 0 M?? #CAOL? 1B? KO;MC <O<<F? G;NLCR ) CM NB?H ; L?MOFN I@ NB? JL? ;MM?G<FCHA JLI=?MM I@ FCH?;L
MO< G;NLC=?M 1B? FCMNM I@ IJ?L;NCIHM CM ACP?H <?FIQ
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
C
C
C
C C
C
C
C C
C
C
C C
C
C
C
C
C
C
C
C
C C
C
C
C
C
;H> ;L? I@ >C_?L?HN NSJ?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1;<F? P;FO?M I@ NB? FCH?;L <;MCM @OH=NCIHM IH NB? MO< NLC;HAF?M
#OH=NCIH , , , , , , 1B? ?F?G?HN G;NLC=?M ;L? L?=N;HAOF;L ;H> =IHN;CH ?HNLC?M 4? =IHMC>?L B?L? NB?
=;M? QB?L? CM FCH?;L ;H> KO;MC <O<<F? NB? MSGG?NLC=;F MCNO;NCIH CM ;ONIG;NC=;FFS
>?>O=?> @LIG NBCM =;M? 1B? L?F;NCIH )78 % % % CM ;FQ;SM NLO? <ON OHFCE? NB?
JL?PCIOM =;M? NB? @OH=NCIHM L?MNLC=N?> NI NB? MO< NLC;HAF?M HI FIHA?L =ILL?MJIH> NI
NB? <;MCM @OH=NCIHM - IH NB?M? MO< NLC;HAF?M &H @;=N 7 ,8 78 @IL @ ;H>
7 , <?=;OM? - CM NB? =?HNL? I@ AL;PCNS I@ NB? NLC;HAF? 1 4CNBCH NB? CHN?LH;F
HOG<?LCHA I@ NB? MO< NLC;HAF?M 1C - ;FQ;SM =ILL?MJIH>M NI NB? JICHN HOG<?L -
;H> MI NB? <;MCM @OH=NCIHM CH NB? L?@?L?H=? NLC;HAF? ;L? NBIM? ACP?H CH 1;<F? 1B? CMIJ;L;G?NLC= NL;HM@ILG;NCIH M;P?M NB? P;FO?M I@ NB? @OH=NCIHM IH NB? HI>?M
;H> NB? ?RJ?=N?> L?MOFN CM VH;FFS I<N;CH?> QCNBCH NB? ;=NO;F G?MB
1B? G;NLCR =I?^=C?HNM L?F;N?> NI NB? NLC;HAF? 1 ;L? I<N;CH?> <S ;H ;MM?G<FS IH
NB? MO< NLC;HAF?M
)
)
)
)
)
)
)
)
)
)
)
)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C C
C
C
C C
C
C
C C
C
C
4? QIOF> A?N @IL NB? CHP?LM? =;M? I@ ; KO;MC <O<<F? FCH?;L G;NLCR
)
)
)
)
)
)
)
)
)
)
)
C
C
C
C
C
C C C
C
C
C
C
C C C C
C
C
C C
C
C
C
C
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
) C
C
C
BB4<1;8=6 0 <0CA8G 5>A 436410B43 BC>A064
4? >?M=LC<? <?FIQL BIQ NI CGJF?G?HN ; MNIL;A? <S M?AG?HN @LIG ; G;NLCR CHCNC;FFS
MNIL?> CH ; HIH ;MM?G<F?> "" G;HH?L M CH>C=;N?> CH 0?=NCIH NB? ?RNL;
>C;AIH;F N?LGM I@ NB? G;NLCR ;L? MNIL?> CH NQI N;<F?M 5 ;H> 5 NB?GM?FP?M
CH=FO>?> CH NB? >CG?HMCIH N;<F? 5*0 0 ;M CH M?AG?HN CHCNC;FFS M?N NI 1B? N;<F?
5*1 1 ;M CH NLC;HAF? L?@?LM NI NB? "" MNIL;A? 1B? P;FO? I@ NB? N;<F? ,/&0"$
QCNB CH>?R IL CH>C=;N?M C@ CN CM H?=?MM;LS NI ;>> IH? ?F?G?HN I@ NB? G;NLCR NI
5 IL NI 5 1B? P;FO? ,/&0"$ ?KO;FM ;H> L?MJ?=NCP?FS QB?H ,/&0"$
?KO;FM ;H> &N CM OM?> NI MNIL? NB? FIQ?L J;LN I@ NB? G;NLCR
&+&1&)&71&,+
!, &0"$ +0"$
5*0&0"$ !
5*0&0"$ !
"+!!,
00"*)6
!, &")"* +")"*
N?LG 5*0")10"$&")"* ,/&0"$&")"*
5*0")10"$&")"* ,/&0"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* ,/&0"$&")"*
5*0")10"$&")"* ,/&0"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* ,/&0"$&")"*
5*0")10"$&")"* ,/&0"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* ,/&0"$&")"*
5*0")10"$&")"* ,/&0"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* ,/&0"$&")"*
5*0")10"$&")"* ,/&0"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* ,/&0"$&")"*
5*0")10"$&")"* ,/&0"$&")"* 5*1&")"* "+!!,
1BCM ;FAILCNBG CM ?;MCFS MCGJFCV?> @IL ; MSGG?NLC= G;NLCR 4B?H NB? G;NLCR
CM L?=N;HAOF;L ? A C@ CN B;M <??H @ILG?> QCNB FCH?;L <;M?M ;H> KO;MC <O<<F? N?MN
@OH=NCIHM >C_?L?HN HOG<?LM I@ M?AG?HNM B;P? NI <? =IHMC>?L?> M?? +0"$ ;H>
+0"$ B?L?;@N?L ;H> NB? JLIAL;G GOMN <? =IGJF?N?> QCNB
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
.20& 2)" 00"*)6
!, &")"* +")"*
5*0")10"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* 5*1&")"* N?LG 5*0")10"$&")"* 5*1&")"* N?LG "+!!,
M NB? 5*1 N?LGM ;L? ;FL?;>S ;MM?G<F?> NBCM CM CH @;=N DOMN ; =IJS
BB4<1;H >5 0= 4;4<4=C0AH E42C>A
4? HIQ =IG? <;=E NI NB? "" MNIL;A? 1B? JLI>O=N I@ ;H "" MNIL?> G;NLCR
;H> ; P?=NIL QCFF L?MOFN CH ; HIH ;MM?G<F?> P?=NIL ;FMI MNIL?> ?F?G?HN <S ?F?G?HN
?@IL? >?N;CFCHA NB? G;NLCR\P?=NIL JLI>O=N Q? MB;FF MNO>S NB? JLI=?MM I@ ;MM?G<FCHA
;H ?F?G?HN;LS P?=NIL 1BCM JLI=?MM CM ;FMI H?=?MM;LS NI ;MM?G<F? NB? >C;AIH;F I@ ;HS
G;NLCR
4CNB ; P?=NIL <?CHA >?VH?> IH ?;=B ?F?G?HN <S ; P?=NIL 33 >CG?HMCIH +!- ;M
@IL ?R;GJF? NB? >C;AIH;F N?LGM ;H> I@ ; G;NLCR IH ; NLC;HAF? NB?
N;LA?N CM NI =IHMNLO=N NB? AFI<;F P?=NIL . >CG?HMCIH HJICH 4CNB NB? HIN;NCIHM I@
0?=NCIH .
<3:3;
,
,3 33
3
1B? NLC;HAF?M QCNB FCH?;L CHN?LJIF;NCIH QCFF ;A;CH <? OM?> ;M ;H ?R;GJF? 4 4
;H> 4 ;L? NB? =IGJIH?HNM I@ P?=NIL 3 MNIL?> ?F?G?HN <S ?F?G?HN 4&")"* CM
NB? =IGJIH?HN I@ NB? P?=NIL ;N HI>? I@ ?F?G?HN &")"* ?N=
&H NB? #,/1/+ JLIAL;GGCHA F;HAO;A? NB? MN?JM @IL =IHMNLO=NCHA P?=NIL . ;L?
. B;M <??H JL?PCIOMFS M?N NI !, &")"* +")"*
/&()"&")"* /&()"&")"* 4&")"*
/&()"&")"* /&()"&")"* 4&")"*
/&()"&")"* /&()"&")"* 4&")"*
"+!!,
QB?L? &()" CM NB? =IHH?=NCPCNS N;<F? &()"&")"* & CM NB? AFI<;F HOG<?L I@ NB?
&NB FI=;F JICHN I@ ?F?G?HN &")"*
4CNB ; P?=NIL =IGJON?L NBCM FIIJ CM HIN ;ONIG;NC=;FFS P?=NILCT?> <?=;OM? I@ NB?
LCME I@ ; MCGOFN;H?IOM ;MM?G<FS ;N NB? M;G? FI=;NCIH CH P?=NIL . 3?=NILCT;NCIH QCFF
BIQ?P?L <? JIMMC<F? OH>?L ; @?Q =IH>CNCIHM ;M @IFFIQM
1B? ;MM?G<FS FIIJ CM VLMN MJFCN CHNI NBL?? =IHM?=ONCP? FIIJM
!, &")"* +")"*
/&()"&")"* /&()"&")"* 4&")"*
"+!!,
!, &")"* +")"*
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
/&()"&")"* /&()"&")"* 4&")"*
"+!!,
!, &")"* +")"*
/&()"&")"* /&()"&")"* 4&")"*
"+!!,
1B?M? NBL?? FIIJM ;L? MNCFF HIN ;ONIG;NC=;FFS P?=NILCT?> IH ; P?=NIL =IGJON?L &H
@;=N P?=NILCT;NCIH QILEM MCGOFN;H?IOMFS IH ; =?LN;CH HOG<?L I@ CH>C=?M CH NB? FIIJ
1BCM HOG<?L CM EHIQH ;M NB? F?HANB I@ NB? =IGJON?L P?=NIL L?ACMN?L ;H> P;LC?M @LIG
IH? G;=BCH? NI ;HINB?L &N CM @IL ; /6 6*- ;H> @IL ; #2'&102 G;=BCH?
0I IH ; /6 G;=BCH? ; FIIJ L;HACHA @LIG NI +")"* QCFF <? J?L@ILG?> <S ALIOJM
I@ ?F?G?HNM &@ NB? VLMN FIIJ B;> <??H P?=NILCT?> IH ; /6 NB? MN;N?G?HN
/&()"&")"* /&()"&")"* 4&")"*
QIOF> B;P? <??H ?R?=ON?> ;N NB? M;G? NCG? @IL ?F?G?HNM NI NB?H NI ?N=
1B? L?MOFN CM =ILL?=N IHFS C@ QCNBCH ; ALIOJ I@ ?F?G?HNM ?;=B =IGJIH?HN I@ P?=NIL
. CM OM?> IH=? ;H> IHFS IH=? 1QI ?F?G?HNM &")"* ;H> &")"* MO=B NB;N
&()"&")"* &()"&")"* MBIOF> HIN ;JJ?;L CH NB? M;G? ALIOJ I@ ?F?G?HNM
4B?H =IGJCFCHA NB? P?=NIL =IGJON?L QBC=B CM OH;Q;L? I@ NB? =IHN?HN I@ &()"
C>?HNCV?M NBCM =IHWC=N I@ <;=EQ;L> >?J?H>?H=? ;H> >I?M HIN P?=NILCT? NB? FIIJ
&N CM JIMMC<F? NI @IL=? P?=NILCT;NCIH <S ;H IL>?L NI NB? =IGJCF?L <ON CN GOMN <?
AO;L;HN??> NB;N HI <;=EQ;L> >?J?H>?H=? I==OLM QCNBCH NB? G?MB
1B? ;>P;HN;A? I@ MJFCNNCHA NB? CHCNC;F FIIJ ;MM?G<FS CHNI NBL?? FIIJM <?=IG?M =F?;L
4CNBION NBCM L?MJ?=NCHA NB? <;=EQ;L> >?J?H>?H=? =IH>CNCIH QIOF> B;P? <??H GIL?
>C^=OFN NI ;=BC?P? &()" QIOF> B;P? NI P?LC@S
&()"&")"* & &()"&")"* '
@IL & ' ;H> ;H> @IL ;FF >C_?L?HN ?F?G?HNM &")"* ;H> &")"* QCNBCH ;
ALIOJ I@ ?F?G?HNM
4B?H NB? FIIJ CM MJFCN ; LCME I@ <;=EQ;L> >?J?H>?H=? L?G;CHM C@ @IL NQI >C_?L
?HN ?F?G?HNM &")"* ;H> &")"* &")"* &")"* >?HIN?M CHN?A?L
>CPCMCIH ;H> C@ NB?L? CM ;H CH>?R & IL MO=B NB;N
&()"&")"* &&()"&")"* &
1B? C>?; <?BCH> NB? P?=NILCT;NCIH I@ NB? NBL?? FIIJM CM NI VH> ; HOG<?LCHA I@ ?F?
G?HNM NB;N QIOF> ?FCGCH;N? ;HS <;=EQ;L> >?J?H>?H=? I==OLL?H=? 1BCM CM CGJIMMC<F?
@IL G?MB?M G;>? I@ NII @?Q ?F?G?HNM <ON CN CM P?LS ?;MS NI J?L@ILG QB?H NB? HOG<?L
I@ ?F?G?HNM ?R=??>M ; @?Q BOH>L?>M @IL ; P?=NIL L?ACMN?L F?HANB I@ 4? MB;FF >?N;CF =IHM?=ONCP?FS ; Z=IOHN?L ?R;GJF?[ NB?H ; ZB?OLCMNC=[ ;FAILCNBG
?^=C?HN CH NB? =;M? I@ MO^=C?HN HOG<?L I@ ?F?G?HNM
>D=C4A4G0<?;4
&G;ACH? ; G?MB I@ NLC;HAF?M QCNB @?Q?L NB;H ?F?G?HNM CH QBC=B IH? JICHN
<?FIHAM NI @IOL >C_?L?HN NLC;HAF?M 1BCM JICHN QCFF B;P? NB? M;G? FI=;F HOG<?L %6!/,!6+*& 0 ,# #/"" 02/# " #),40
IL CH ;N F?;MN NQI I@ NB?M? NLC;HAF?M 1B;N QCFF L?MOFN CH ; <;=EQ;L> >?J?H>?H=?
QBC=B =;HHIN <? ;PIC>?>
4DA8BC82 0;6>A8C7< 5>A =D<14A8=6 C74 4;4<4=CB
1B? C>?; CM NI VH> ION NB? >?J?H>?H=? MCNO;NCIHM ;H> ?FCGCH;N? NB?G QB?H NB?S
I==OL 4? MN;LN @LIG ;H ?RCMNCHA HOG<?LCHA NB? HOG<?L I@ ?F?G?HNM MBIOF> <? AL?;N?L
NB;H ;H> JLIAL?MM CH NBCM HOG<?LCHA ?R;GCHCHA ;FF NB? ALIOJM I@ ?F?G?HNM
M MIIH ;M ; =LCNC=;F ?F?G?HN ;JJ?;LM CNM HOG<?L CM ?R=B;HA?> QCNB ; BCAB?L L;HE
?F?G?HN NB? =IHWC=NM ;L? NBOM JOMB?> @ILQ;L> ;M NB? JLIAL?MMCIH AI?M IH 1BCM
;FAILCNBG @;CFM C@ NB? BCAB?MN L;HE CM L?;=B?> ;H> >?J?H>?H=?M L?G;CH &H NBCM =;M?
NB? ?R=B;HA?M ;L? G;>? QCNB FIQ?L L;HE ?F?G?HNM QBC=B QCFF CHP;FC>;N? NB? JL?PCIOM
=ILL?=NCIHM 1B? QBIF? ;FAILCNBG CM NI <? >IH? ;A;CH OHNCF CN QILEM
-L;=NC=;FFS NBCM ;FAILCNBG CM P?LS ?^=C?HN ;H> L;L?FS H??>M GIL? NB;H NQI CN?L;
NCIHM NI MO==??> 1BCM CM >O? NI NB? @;=N NB;N ;M MIIH ;M NB? HOG<?L I@ ?F?G?HNM ?R
=??>M ; @?Q BOH>L?>M NB? =IG<CH;NILC=M JLIPC>?M G;HS ;JJLIJLC;N? MIFONCIHM 4B?H
NB? HOG<?LCHA CM >IH? NB? =IGJCF?L B;M NI <? CH@ILG?> NB;N HI GIL? <;=EQ;L> >?
J?H>?H=? L?G;CHM CH NB? FIIJM NB? IL>?L CM !&/&3!"- IH ; /6 G;=BCH? &@
NB? P?=NIL L?ACMN?L F?HANB CM BCAB NB? MIFONCIH G;S HIN ?RCMN @IL G?>COG MCT? G?MB?M
MIG? NBIOM;H>M I@ ?F?G?HNM %IQ?P?L P?=NILCT;NCIH =;H MNCFF <? J?L@ILG?> <S
MJFCNNCHA NB? FIIJM CHNI MO< FIIJM =IHN;CHCHA ALIOJM I@ ?F?G?HNM QCNBION <;=EQ;L>
>?J?H>?H=? 1B? P?=NILCT;NCIH MBIOF> <? @IL=?> @IL NB?M? MO< FIIJM
1B? @IFFIQCHA FIIJ QBC=B =;H <? P?=NILCT?> IH ; /6 G;=BCH? C@ NB?L? CM HI
<;=EQ;L> >?J?H>?H=? CH ;HS ALIOJ I@ ?F?G?HNM
!, &")"* +")"*
/&()"&")"* /&()"&")"* 4&")"*
"+!!,
=;H <? NL;HMJIM?> QCNBION ;HS =B;HA? CH NB? G?MB NI ; #2'&102 G;=BCH? QBIM?
P?=NIL L?ACMN?L F?HANB CM ;M @IFFIQM
* +")"*
+ *,!+")"* !, ( *
!, &")"*( ( /&()"&")"* /&()"&")"* 4&")"*
,+1&+2"
,+1&+2"
!, &")"** * +
/&()"&")"* /&()"&")"* 4&")"*
,+1&+2"
1B? P?=NILCT;NCIH I@ FIIJM ;H> G;S <? @IL=?>
0IG? N?MNM IH NB? /6 6*- L?P?;F NB;N P?=NILCT;NCIH JLIPC>?M =IHMC>?L;<F?
?=IHIGS CH =IGJON;NCIH NCG? @IL NB? P?=NIL ;MM?G<FS
L;NCI @IL KO;>LCF;N?L;FM QCNB <CFCH?;L CHN?LJIF;NCIH @IOL FIIJM
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
L;NCI @IL NLC;HAF?M QCNB FCH?;L CHN?LJIF;NCIH NBL?? FIIJM
M NB? ;MM?G<FS I@ P?=NILM CM CHN?HMCP?FS OM?> CH P;LCIOM ;FAILCNBGM NB? ?=IHIGS
CH NCG? @IL NB? IP?L;FF =IGJON;NCIH CM KOCN? HIN?QILNBS ;H> =;H L?;=B ; L;NCI I@ +IN? NB? B?OLCMNC= ;FAILCNBG G;S FIIE NII =B;FF?HACHA <?=;OM? NB? <;=EQ;L>
>?J?H>?H=?M ;L? ?FCGCH;N?> @IL ;FF NB? ALIOJM I@ ?F?G?HNM NI NI NI
?N=
$&#& CN QIOF> <? ?HIOAB NI =IHMC>?L NB? ALIOJM NI NI ?N=
%IQ?P?L CN CM ; JLIN?=NCIH ;A;CHMN NB? <?B;PCIOL I@ NB? =IGJCF?L QBC=B @IL MIG?
CHN?LH;F ;L=BCN?=NOL;F L?;MIHM =;H >?=C>? NB? HOG<?L I@ ?F?G?HNM NI <? B;H>F?> CH
?;=B ALIOJ ;H> NBOM CHNLI>O=? MIG? ;L<CNL;LS MBC@NM #OLNB?LGIL? >?;FCHA QCNB NB?
ALIOJM NI NI ?N= M??GM NI G;E? NB? =IHP?LA?H=? I@ NB? ;FAILCNBG GIL?
>C^=OFN
%A>3D2C >5 0 =>=BH<<4CA82 <0CA8G 0=3 0 E42C>A
1B? P?=NIL . NI <? =IGJON?> CM NB? JLI>O=N I@ G;NLCR ) ;H> P?=NIL 2 1B? @IFFIQCHA
?KO;FCNS GOMN <? L?G?G<?L?>
. )2 2
<3:3;
,
,3 !3 ,3B 2
3
1B? ?R;GJF? QCFF =IHNCHO? QCNB NLC;HAF?M 1B? G;NLCR ) CM MNIL?> CH NQI N;<F?M
!*+-,&+ ;H> 5*+")"* 1B? #,/1/+ CHMNLO=NCIHM ;L? ;M @IFFIQM
!C;AIH;F CHJON -LI>O=N !3 P?=NILCT;<F? FIIJ
!, & +-,&+
/& !*& 3&
,+1&+2"
&HJON I@ ?RNL; >C;AIH;F N?LGM MNIL?> CH QILEMJ;=? 4 4 ;H> 4 1BCM FIIJ
=;H ;FMI <? P?=NILCT?>
!, &")"* +")"*
$FI<;F HOG<?LCHA I@ NB? HI>?M I@ ?F?G?HN ACP?H <S N;<F? &()"
& &()"&")"* & &()"&")"* & &()"&")"* N NB? ?F?G?HN F?P?F NB? L?MOFNM I@ HI>?M QCNB >C_?L?HN FI=;F HOG<?LM
;L? MNIL?> M?J;L;N?FS @IL HOG<?L 4 ?N= 4&")"* 5*&")"* 3& 5*&")"* 3&
4&")"* 5*&")"* 3& 5*&")"* 3&
4&")"* 5*&")"* 3& 5*&")"* 3&
,+1&+2"
1B? P?=NIL >?VH?> <S 4 4 ;H> 4 CM ;MM?G<F?> M?? 0?=NCIH NB?H ;>>?>
OJ NI . QBC=B ;FL?;>S =IHN;CHM NB? >C;AIH;F CHJON
1B? G?;HCHA I@ NBCM ;FAILCNBG CM P?LS MCGJF? )?N OM =IHMC>?L 5*&")"* QBC=B CM <S =IHP?HNCIH ?KO;F NI *1 @IL NB? ?F?G?HN &")"* &N CM J;LN I@ NB?
=I?^=C?HN I@ JICHN J?LN;CHCHA NI NB? ?F?G?HN &")"* QCNBCH NB? ?KO;NCIH I@ JICHN
J?LN;CHCHA NI NB? M;G? ?F?G?HN 1B? JLI>O=N 5*&")"* 3& GOMN NB?H <?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;>>?> NI NB? L?MOFN /& 1BCM CM >IH? CH NB? M?=IH> FIIJ ;H> >OLCHA NB? ;MM?G<FS
MN;A? PC; NB? QILEMJ;=? 4
M ; MOGG;LS IH? =IOF> M;S NB;N NB? ;MM?G<FS JLI=?MM CM HI FIHA?L J?L@ILG?> IH
NB? G;NLC=?M <ON IH NB? P?=NILM NB?GM?FP?M 4CNB =F;MMC=;F N?=BHCKO?M QILECHA IH ;
=F;MMC=;F =IGJ;=N MNIL;A? NB? P?=NILCT;NCIH I@ NB? G;NLCR\P?=NIL JLI>O=N CM Z<LIE?H
OJ[ <S ;H CHN?LH;F FIIJ IH NB? H?CAB<IOLM I@ ?;=B JICHN ?R=?JN @IL NB? Z@LIHN;F[
JLI>O=N QBC=B QCFF <? OM?> ;FMI @IL NB? ?>A? <;M?> MNIL;A?
%A>3D2C >5 0 BH<<4CA82 <0CA8G 0=3 0 E42C>A
4B?H NB? G;NLCR CM MSGG?NLC= NB? N?LGM
5*&")"* +!- +!- NI 5*&")"* +!- +!;L? HIN MNIL?> <?=;OM? NB?S ;L? L?MJ?=NCP?FS ?KO;F NI
5*&")"* 5*&")"* +!- +!- 1B?S ;L? MO<MNCNON?> CH NB? JL?PCIOM FIIJ QBC=B F?;>M NI
4&")"* 5*&")"* 3& 5*&")"* 4&")"* 5*&")"* 3& 5*&")"* 4&")"* 5*&")"* 3& 5*&")"* 3&
3&
3&
%A>3D2C >5 0 CA0=B?>B43 <0CA8G 0=3 0 E42C>A
1B? ?F?G?HN;LS JLI>O=NM 5*&")"* & QCFF MCGJFS <? ?R=B;HA?> QCNB
5*&")"* & +!- +!- C@ &+!- +!- ;H> QCNB
5*&")"* & +!- +!- C@ &+!- +!- QBC=B QCFF F?;> @IL G;NLC=?M <OCFN IH NLC;HAF?M NI
4&")"* 5*&")"* 3& 5*&")"* 3&
4&")"* 5*&")"* 3& 5*&")"* 3&
4&")"* 5*&")"* 3& 5*&")"* 3&
1B?M? F;MN NQI ?R;GJF?M JLIP? NB;N NB? JLI<F?GM I@ MSGG?NLS ;H> NL;HMJIMCNCIH
CH "" MNIL;A? ;L? P?LS MCGJFS MIFP?>
%A>24BB8=6 >5 C74 8A827;4C 1>D=30AH 2>=38C8>=B
1B? =F;MMC=;F VHCN? ?F?G?HN N?=BHCKO?M ?FCGCH;N? NB? EHIQH ;H> CGJIM?> P;FO?M JICHNM
QCNB !CLC=BF?N =IH>CNCIHM @LIG NB? FCH?;L MSMN?GM QBC=B L?MOFNM CH ; L?HOG<?LCHA I@
NB? OHEHIQH P;LC;<F?M
&H IL>?L NI ;PIC> NBCM L?HOG<?LCHA ;H> CNM =IMN NB? JICHNM QCNB !CLC=BF?N =IH>CNCIHM
;L? HIN L?GIP?> @LIG NB? G;NLC=?M 1B?S ;L? E?JN ;H> ;MMCAH?> QCNB ;H ?KO;NCIH I@
NB? NSJ? N CGJIM?> P;FO? &H NB? INB?L ?KO;NCIHM NB? =I?^=C?HN ;MMI=C;N?> NI ;
!CLC=BF?N =IH>CNCIH HI>? QCFF <? M?N NI ;H> NB? LCABN B;H> MC>?M I@ NB? ?KO;NCIHM
QCFF <? =B;HA?> ;==IL>CHAFS S >ICHA MI NB? MSGG?NLS I@ NB? G;NLCR CM HIN ;_?=N?>
&H INB?L QIL>M @IL ; JICHN QCNB ; !CLC=BF?N =IH>CNCIH NB? P;FO?M IH NB? >C;AIH;F
I@ NB? G;NLCR ;L? M?N NI QB?L?;M NB? ?RNL; >C;AIH;F N?LGM ;L? M?N NI ;@N?L B;PCHA
N;E?H CHNI ;==IOHN NB? EHIQH P;FO?M IH NB? LCABN B;H> MC>?M 1B? JLI<F?G CM NB;N @IL
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
; ACP?H JICHN HI >;N; MNLO=NOL? ?RCMNM NB;N =IOF> B?FJ CH VH>CHA KOC=EFS NB? ?F?G?HNM
NI QBC=B NBCM JICHN <?FIHAM &N NBOM M??GM CGJIMMC<F? NI M?N NI NB? ;MMI=C;N?>
?RNL; >C;AIH;F N?LGM C@ NB?S B;P? <??H MNIL?> ?F?G?HN <S ?F?G?HN %IQ?P?L NBCM
=;H <? >IH? OMCHA NB? @IFFIQCHA FIIJ I@ CHMNLO=NCIHM <;M?> IH NB? =IH=?JN I@ NB?
G;NLCR\P?=NIL JLI>O=N
&H QB;N @IFFIQM ?P?LS =IGJIH?HN I@ P?=NIL 2 ?KO;FM @IL ; MN;H>;L> JICHN ;H> @IL ; !CLC=BF?N =IH>CNCIH JICHN S >ICHA MI ;HS ?F?G?HN I@ NB? G;NLCR QBC=B QIOF>
ZNIO=B[ ; !CLC=BF?N JICHN QB?H =IGJONCHA ; G;NLCR\P?=NIL JLI>O=N QIOF> <? M?N NI
NB? INB?L ?F?G?HNM L?G;CHCHA OH=B;HA?> QBC=B CM NB? ?RJ?=N?> L?MOFN #IL ; G;NLCR
<OCFN IH FCH?;L NLC;HAF?M NB? FIIJ CM ;M @IFFIQM
!, &")"* +")"*
& &()"&")"* & &()"&")"* & &()"&")"* 5*&")"* 5*&")"* 3& 3&
5*&")"* 5*&")"* 3& 3&
5*&")"* 5*&")"* 3& 3&
5*&")"* 5*&")"* 3& 3&
5*&")"* 5*&")"* 3& 3&
5*&")"* 5*&")"* 3& 3&
"+!!,
&H NB? HIH ;MM?G<F?> G;NLC=?M NB? P;FO?M QBC=B QCFF @ILG NB? FCH? ;H> NB? =IFOGH
L?F;N?> NI ; JICHN QCNB ; !CLC=BF?N =IH>CNCIH QCFF <? M?N NI ?R=?JN @IL NB? >C;AIH;F
N?LGM
M ; MJ?=CV= N;<F? ?RCMNM @IL NB? >C;AIH;F CN CM P?LS ?;MS NI M?N NI ;HS N?LG
;MMI=C;N?> QCNB ; !CLC=BF?N JICHN &H NB? M;G? Q;S NB? CGJIM?> P;FO?M GOMN <?
QLCNN?H ;@N?LQ;L>M IH NB? LCABN B;H> MC>? I@ NB? FCH?;L MSMN?G
1B? G?NBI> >?M=LC<?> ;<IP? QCFF <? >?N;CF?> ;A;CH ;H> @ILG;FCT?> CH 0?=NCIH I@ NB? =B;JN?L L?F;N?> NI J;L;G?N?L ?MNCG;NCIH
%A>3D2CB 14CF44= 0 3806>=0; <0CA8G 0=3 0 <0CA8G
1B?M? JLI>O=NM ;JJ?;L >OLCHA NB? >C;AIH;F IL <FI=E >C;AIH;F JL?=IH>CNCIHCHA I@ ;
FCH?;L MSMN?G QB?L? NB? G;NLCR ) CM =B;HA?> CHNI )
<?CHA ; >C;AIH;F G;NLCR
M?? 0?=NCIH &N CM H?=?MM;LS NI EHIQ BIQ NI =IGJON? NB? JLI>O=N ) IH IH? B;H> ;H> NB?
JLI>O=N ) IH NB? INB?L
&H NB? @IFFIQCHA #,/1/+ JLIAL;G ?R;GJF?M
CM >?=F;L?> ;M ; L?;F ;LL;S I@
>CG?HMCIH +-,&+ ;H> !& >?HIN?M NB? &NB ?F?G?HN I@ NB? >C;AIH;F G;NLCR 0CH=? NB?
=IGJON;NCIH I@ NB? >C;AIH;F I@ NB? L?MOFNCHA G;NLCR CM I<PCIOM IHFS NB? ;FAILCNBGM
@IL NB? ?RNL; >C;AIH;F N?LGM ;L? ACP?H <?FIQ CH NB? =;M? I@ FCH?;L NLC;HAF?M
-LI>O=N !*
!, &")"* +")"*
& &()"&")"* & &()"&")"* %6!/,!6+*& 0 ,# #/"" 02/# " #),40
& &()"&")"* 5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
!&
!&
!&
!&
!&
!&
"+!!,
#IL ;MM?G<F?> G;NLC=?M NB? @ILGOF; QIOF> <?
)
;<
;
);<
5*&")"* >?HIN?M @IL CHMN;H=? ; J;LN I@ N?LG *& & ;H> CN QCFF <? FIAC=
;FFS GOFNCJFC?> <S !& M @IL JLI>O=N )
CN QCFF <? GOFNCJFC?> <S !&
-LI>O=N *!
!, &")"* +")"*
& &()"&")"* & &()"&")"* & &()"&")"* 5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
5*&")"*
!&
!&
!&
!&
!&
!&
"+!!,
1B? L?MOFN CNM?F@ N;E?M NB? @ILG I@ ; HIH ;MM?G<F?> G;NLCR ;H> NB? NQI JL?PCIOM
FIIJM ;L? P?=NILCT?>
"0CA8GOE42C>A ?A>3D2C F8C7 436410B43 BC>A064
M ; G;NLCR QCNB ?>A? <;M?> MNIL;A? CM @OFFS ;MM?G<F?> ?F;<IL;NCHA ; G;NLCR\P?=NIL
JLI>O=N CM HI FIHA?L ; JLI<F?G *OFNCJFSCHA <S 5 NI A?N ; P?=NIL 4 VLMN L?KOCL?M
NB;N Q? =IGJON? 4 5 >C;AIH;F =IHNLC<ONCIH NB?H ?R?=ON? NB? @IFFIQCHA FIIJ
!, &0"$ +0"$
5$),0"$&0"$ 5$),0"$&0"$ 5&0"$ 6$),0"$&0"$ 5$),0"$&0"$ 5$),0"$&0"$ 5&0"$ 6$),0"$&0"$ "+!!,
MODERN TECHNIQUES IN FINITE ELEMENTS 223
This loop is a little more complex for rectangular matrices because some values
of X are not initialized by the contribution of the diagonal, and need to be set to 0
first.
However, as the edge-based storage provides assembled matrices, it allows a more
efficient matrix–vector product, the frontal product, detailed in the following subsection.
7.4.10
Matrix–vector frontal product
The idea is to write the matrix–vector product directly in its classical form:
X(i) =
npoin
Aij Y (j)
j=1
As the matrix A is sparse, we do not want really to write loops from 1 to NPOIN,
but we want to use an optimal storage, like the edge-based storage.
For each point, a data structure will be defined containing the number of neighbours and their addresses, and the entry of the matrix related to it. In the classical
versions of this kind of matrix–vector product, the internal loop is a loop on the
number of neighbours for each point, which inhibits any attempt at vectorization.
With the frontal product, the points are numbered according to the number of their
neighbours, in ascending order. This results in long sequences of nodes, with the same
number of neighbours, for which a specific loop is written. The maximum number
of neighbours for one point, denoted MXPTVS, is thus restricted by what has been
specified in the program.
A 2-dimension table, denoted LIMVOI, determines the arrangement of the nodes:
LIMVOI(K,1) is the global number of the first point having K neighbours.
LIMVOI(K,2) is the global number of the last point having K neighbours.
The address of the neighbours of points is denoted IY, with size (NPOIN,MXPTVS).
The address of the elements of the matrix, related to the neighbours, is denoted
IW, which has the same dimension as IY.
The table X is, as usual, initialized by the contribution of the diagonal DA Y .
The loop for those nodes having three neighbours is given below. In this case, the
extra-diagonal terms are stored in the table W:
DO I=LIMVOI(3,1),LIMVOI(3,2)
X(I)=X(I)+W(IW(I,1))*Y(IY(I,1))+W(IW(I,2))*Y(IY(I,2))
+W(IW(I,3))*Y(IY(I,3))
ENDDO
Unfortunately, W changes when the matrix is symmetric, and IW must be written
in two versions. Transposed products, like the computation of X = t A Y , requires
two other versions.
The whole data structure is thus stored in an array named IKLEM1 with four
dimensions: NPOIN, MXPTVS, 4 and 2. The third dimension is coded as follows:
1: IW for a normal product
2: IY for a normal product
3: IW for a transposed product
4: IY for a transposed product
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? @IOLNB >CG?HMCIH C ? CM =I>?> ;M @IFFIQM
HIH MSGG?NLC= G;NLCR
MSGG?NLC= G;NLCR
!8=40A BHBC4< A4B>;DC8>=
4? ACP? B?L? ;H IP?LPC?Q I@ CN?L;NCP? N?=BHCKO?M @IL MIFPCHA FCH?;L MSMN?GM CH=FO>CHA
JL?=IH>CNCIHCHA
4 CM NB? FCH?;L MSMN?G NI <? MIFP?> CM ; G;NLCR L?MOFNCHA @LIG NB?
P;LC;NCIH;F @ILGOF;NCIH CH NB? VHCN? ?F?G?HN G?NBI> CM NBOM ; MJ;LM? G;NLCR QBIM?
MNIL;A? G;S <? >IH? CH >CP?LM? @ILGM "" IL ?>A? <;M?> #OLNB?LGIL? =IHMC>?LCHA
NB? L?MIFONCIH I@ NB? 0;CHN 3?H;HN MSMN?G I@ ?KO;NCIHM IL ;HS INB?L L?MIFONCIH =IOJFCHA
G;HS P;LC;<F?M G;S <? <LIE?H >IQH CHNI <FI=EM I@ M?P?L;F G;NLC=?M ? A @IOL IL
HCH?
1B? >CL?=N L?MIFONCIH G?NBI>M NII =IMNFS ;M MIIH ;M NB? P;LC;<F?M ;L? NII HOG?L
IOM ;L? HIN ;JJLIJLC;N? <?=;OM? NB?S L?KOCL? EHIQF?>A? I@ NB? ;MM?G<F?> P;FO?M I@
NB? G;NLCR ,H NB? =IHNL;LS NB? CN?L;NCP? G?NBI>M <;M?> IH NB? GCHCGCT;NCIH I@ ;
@OH=NCIH;F H??> IHFS G;NLCR\P?=NIL ;H> >IN JLI>O=NM ;H> ;L? J?L@?=NFS L?F?P;HN NI
;HS ECH> I@ MNIL;A? ,H NB? INB?L B;H> NB?M? G?NBI>M ;L? P?LS M?HMCNCP? NI G;NLCR
=IH>CNCIHCHA M?? M?=NCIH <?FIQ 1BCM CM QBS CN QCFF <? H?=?MM;LS NI J?L@ILG ; JL?PC
IOM JL?=IH>CNCIHCHA CH IL>?L NI L?>O=? NB? HOG<?L I@ CN?L;NCIHM L?KOCL?> NI L?;=B ;
ACP?H JL?=CMCIH
1B? JLCH=CJF?M I@ NB? G?NBI>M I@ >?M=?HN ;L? ACP?H H?RN @IFFIQ?> <S MIG? I@ NB?CL
;FAILCNBGM NB?H <S JL?=IH>CNCIHCHA
"4C7>3B >5 34B24=C 0=3 ?A42>=38C8>=8=6
#LIG HIQ IH ?R=?JN C@ MJ?=CV=;FFS G?HNCIH?> QCFF <? ; MSGG?NLC= JIMCNCP? >?VHCN?
G;NLCR 4 CM NB? P?=NIL AL;>C?HN I@ NB? KO;>L;NC= @OH=NCIH;F
4 4 4
QB?L? NB? <L;=E?NM MN;H> @IL NB? >IN JLI>O=N "KO;NCHA NBCM P?=NIL NI G?;HM VH>CHA
;H ?RNL?GOG I@ & 4 NB;N QCFF <? ; GCHCGOG C@ & CM JIMCNCP? B?H=? NB? CHN?L?MN CH
; JIMCNCP? >?VHCN? G;NLCR
)?N 6 <? NB? @OH=NCIH 6 & 4 & N N N N< D <?CHA NB? HOG<?L I@
OHEHIQHM CH NB? MSMN?G #CAOL? 1B? AL;>C?HN I@ & 4 CM HILG;F NI NB? CMI & FCH?M &@ & CM =IHP?R QBC=B CM NB? =;M?
@IL ; KO;>L;NC= JIMCNCP? >?VHCN? @OH=NCIH;F & >?=L?;M?M CH NB? >CL?=NCIH IJJIMCN? NI
CNM AL;>C?HN 1B? C>?; <?BCH> NB? AL;>C?HN G?NBI>M CM NI ;JJLI;=B NB? GCHCGOG I@
& <S JLIAL?MMCHA CH NB? >CL?=NCIH IJJIMCN? NI NB? AL;>C?HN 1B;N F?;>M NI ;H CN?L;NCP?
G?NBI> I@ NB? ECH>
& 4 4;
4 ; 4 ; QB?L? NB? =BIC=? I@ GOMN <? MJ?=CV?>
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
Z=J(X)
minimum Z
x2
solution
x1
#CAOL? GCHCGOG I@ NB? KO;>L;NC= @OH=NCIH;F
1B? =IHP?LA?H=? =LCN?LCIH I@ NBCM G?NBI> <?=IG?M =F?;L QB?H QLCNCHA NB?M? CN?L;
NCIHM ;M @IFFIQM
;
4
%
4;
&H @;=N ;H CN?L;NCP? G?NBI> MO=B ;M
) 4;
* 4;
=IHP?LA?M C@ NB? MJ?=NL;F L;>COM I@ ) * C ? % CM FIQ?L NB;H #OLNB?LGIL?
NB? HILG I@ NB? L?MC>O;F 4 ; CM GOFNCJFC?> ;N ?;=B CN?L;NCIH <S NB? MJ?=NL;F
L;>COM I@ % &@ NB? ?CA?HP;FO?M I@ ;L? >?HIN?> <S 7 NB?S ;L? L?;F ;H> JIMCNCP?
@IL ; MSGG?NLC= JIMCNCP? >?VHCN? G;NLCR NB? ?CA?HP;FO?M I@ % ;L? ?KO;F NI 7
1B? <?MN G?NBI> QCNB =IHMN;HN CM NI =BIIM?
"
"
QB?L? " ;H> ;L? L?MJ?=NCP?FS NB? BCAB?MN ;H> NB? FIQ?MN ?CA?HP;FO? I@ 1B;N BCABFCABNM ; @OH>;G?HN;F J;L;G?N?L @IL NB? L?MIFONCIH I@ FCH?;L MSMN?GM H;G?FS
NB? Z=IH>CNCIHCHA[ ;FMI =;FF?> Z=IH>CNCIH HOG<?L[
"
-L?=IH>CNCIHCHA ;CGM ;N =B;HACHA NB? MSMN?G 4 NI ; MCGJF?L IH? 4 @IL QBC=B NB? =IH>CNCIHCHA I@ QCFF <? <?NN?L =FIM?L NI NB;H @IL 1B? =IGJON;NCIH I@ GOMN <? MCGJF? ?HIOAB NI E??J NB? <?H?VN I<N;CH?> <S
;==?F?L;NCHA NB? L?MIFONCIH
,NB?L AL;>C?HN G?NBI>M ;L? CH @;=N @;L GIL? ?^=C?HN NB;H ; =IHMN;HN IJNCG;F
P;FO? I@ ,JNCGCTCHA ;N ?;=B CN?L;NCIH ;H> GCHCGCTCHA
& 4;
& 4 ; 4 ; %6!/,!6+*& 0 ,# #/"" 02/# " #),40
QBC=B CM ; KO;>L;NC= @OH=NCIH I@ F?;>M NI NB? IJNCG;F P;FO? I@ =; =;
= ; = ;
QB?L? = ; MN;H>M @IL 4 ; 1B? EHIQF?>A? I@ NB? ?CA?HP;FO?M CM HI FIHA?L H?=?M
M;LS
*IMN I@ NB? @IFFIQCHA G?NBI>M ;LCM? @LIG NBCM IH?
1B? >C_?L?HN ;FAILCNBGM >?N;CF?> <?FIQ @ILG;FFS CH=FO>? JL?=IH>CNCIHCHA QCNB ;
G;NLCR QBC=B CM HIN >?VH?> <ON QCFF <? CH 0?=NCIH 1B? MNIJ N?MN CM NB? M;G? QB;N?P?L NB? G?NBI> &N?L;NCIHM AI IH OHNCF NB? JL?=CMCIH
IL NIF?L;H=? ;FI?BED >?J?H>CHA IH NB? N?MN
4 ; IL
4 ;
;FI?BED
;FI?BED
C@ L?F;NCP? JL?=CMCIH
C@ ;<MIFON? JL?=CMCIH
4? =BIIM? NB? "O=FC>?;H HILG ( ;
: <
:,
4 9
N7
7
1B? ( HILG
4 ' N7 QIOF> <? M;@?L ;M CN ?HMOL?M ; =IHP?LA?H=? ;N ;HS JICHN <ON CN MBIQM HI =IB?L?H=?
QCNB NB? HILG OM?> CH NB? >?M=?HN G?NBI>M ;H> =IHM?KO?HNFS QCNB NB? >?GIHMNL;NCIHM
MBIQCHA NB;N NB? L?MC>O;F CM >?=L?;MCHA
>=9D60C4 6A0384=C <4C7>3
1B? Q?;EH?MM I@ NB? IJNCG;F AL;>C?HN ;FAILCNBG QBC=B GCHCGCT?M
& 4;
& 4 ; 4 ; CM NB;N NB? GCHCGCT;NCIH @IFFIQCHA NB? >CL?=NCIH I@ >?M=?HN = ; 4 ; QBC=B CM
;FMI NB? >CL?=NCIH I@ NB? AL;>C?HN CM @ILAINN?H ;N NB? H?RN CN?L;NCIH 1B? C>?; <?BCH>
NB? =IHDOA;N? AL;>C?HN G?NBI> M?? 89 CM NI >CMNCHAOCMB NB? L?MC>O;F = ; ;H> NB?
>CL?=NCIH I@ >?M=?HN :; ;H> AO;L;HN?? NB;N ?P?LS H?Q CN?L;NCIH QCFF F?;> NI ; GCH
CGOG @IL ;HS >CL?=NCIH I@ >?M=?HN JL?PCIOMFS N?MN?> 4CNB L?=OLL?H=? @ILGOF;? NBCM
=;H <? ;=BC?P?> QCNBION MNILCHA ;FF NB?M? >CL?=NCIHM IHP?LA?H=? CM AO;L;HN??> C@ CM ; MSGG?NLC= JIMCNCP? >?VHCN? G;NLCR ,LNBIAIH;FCNS L?F;NCIHM MO=B ;M = 7 = 8 C@ ? @ ;L? OM?> NI JLIP? NB;N NB? ?R;=N MIFONCIH CM L?;=B?> CH D CN?L;NCIHM C@ D CM
NB? HOG<?L I@ OHEHIQHM -L;=NC=;FFS @IL MIFPCHA WOC> G?=B;HC=M MSMN?GM CN N;E?M ;
@?Q CN?L;NCIHM <?NQ??H ;H> NI =IHP?LA? QCNB ; JL?=CMCIH L;HACHA <?NQ??H ;H> *,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
>=9D60C4 6A0384=C 0;6>A8C7<
&HCNC;FCT;NCIH
H 4 L?MIFONCIH I@ = H
: =
H =
: :
4 4 :
&N?L;NCIHM
H; H; ; :;
L?MIFONCIH I@ = ; H;
H; = ;
:;
H; = ;
:; = ;
; 4;
H ; :;
:; :;
4 ; ; : ;
>=9D60C4 A4B83D0; <4C7>3
1B? JLCH=CJF? I@ NBCM G?NBI> CM ?R;=NFS NB? M;G? ;M NB? JLCH=CJF? I@ NB? =IHDOA;N?
AL;>C?HN G?NBI> <ON CN ;CGM ;N GCHCGCTCHA ; >C_?L?HN @OH=NCIH;F H;G?FS
& 4 4 4 IHP?LA?H=? CM L?;=B?> C@ CM ; MSGG?NLC= JIMCNCP? >?VHCNCP? G;NLCR
>=9D60C4 A4B83D0; 0;6>A8C7<
&HCNC;FCT;NCIH
H 4 L?MIFONCIH I@ = H
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
: =
L?MIFONCIH I@ : :
= :
: :
4 4 :
&N?L;NCIHM
H; H; ; :;
= ; = ; ; :;
:; = ; = ; :;
:;
:; :;
= ; :;
:;
:; :;
:; = ; L?MIFONCIH I@ :; :;
; 4;
:; = ;
:; :;
4 ; ; : ;
>=9D60C4 6A0384=C >= C74 =>A<0; 4@D0C8>=
1B? JL?PCIOM G?NBI>M MBIOF> CH JLCH=CJF? <? L?MNLC=N?> NI MSGG?NLC= JIMCNCP? >?VH
CN? G;NLC=?M 1B?S =;H BIQ?P?L <? NL;HMJIM?> ?;MCFS NI HIH MSGG?NLC= G;NLC=?M
&N MO^=?M NI MO<MNCNON? <S B CH ?P?LS ;FAILCNBG NB?H NI GOFNCJFS NB? LCABN B;H>
MC>? <S B NI A?N ;H ?KOCP;F?HN MSMN?G QCNB ; MSGG?NLC= G;NLCR 1B? IHFS L?KOCL?G?HN
CM NB? L?AOF;LCNS I@ 1BCM JLCH=CJF? OM?> NIA?NB?L QCNB NB? =IHDOA;N? AL;>C?HN G?NBI> L?MOFNM CH NB?
HILG;F ?KO;NCIH G?NBI>
%IQ?P?L <S MO<MNCNONCHA <S B NB? AL;>C?HN = 4 QIOF> <? =B;HA?> CH
B
4 &H @;=N NB? >?VHCNCIH I@ = CM G;CHN;CH?> ;H> NB? H?Q ;FAILCNBG CM ;M
@IFFIQM
#>A<0; 4@D0C8>= 0;6>A8C7<
&HCNC;FCT;NCIH
H 4 *,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
L?MIFONCIH I@ = H
L?MIFONCIH I@ B = =
: B =
L?MIFONCIH I@ : :
: :
: : 4 4 :
&N?L;NCIHM
H; H; ; :;
= ; = ; ; :;
L?MIFONCIH I@ B = ; = ;
:; B =
B
= ; B = ;
:;
B = ; B = ;
;
L?MIFONCIH I@ :; :;
;
B
4;
= ; B =
:; :;
;
4 ; ; : ;
"8=8<D< 4AA>A <4C7>3
#IL NB? JOLJIM? I@ ?F;<IL;NCHA ;H CN?L;NCP? G?NBI> @IL ; MCGJFS L?AOF;L G;NLCR NB?
@IFFIQCHA @OH=NCIH;F CM =BIM?H
& 4 ; 4 ; 4 ; "8=8<D< 4AA>A 0;6>A8C7<
&HCNC;FCT;NCIH
H 4 %6!/,!6+*& 0 ,# #/"" 02/# " #),40
L?MIFONCIH I@ = H
L?MIFONCIH I@ B = =
: B = = =
: :
4 4 :
&N?L;NCIHM
H; H; ; :;
L?MIFONCIH I@ = ; H;
L?MIFONCIH I@ B = ; = ;
=; =;
:;
= ; = ;
:; B = ;
; 4;
=; =;
:; : ;
4 ; ; : ;
"'( <4C7>3
1B? $*/"0 G?NBI> $?H?L;FCT?> *CHCGOG /"0C>O;F G?NBI> JO<FCMB?> CH 89 CM ; QILNBS CHHIP;NCIH ?MJ?=C;FFS OM?@OF @IL JIILFS =IH>CNCIH?> MSMN?GM 4? MB;FF
?RJF;CH NB? @OH>;G?HN;F JLCH=CJF?M <?@IL? >?N;CFCHA NB? ;FAILCNBG QBIM? JLIAL;GGCHA
GOMN <? B;H>F?> =;L?@OFFS
1BCM ;FAILCNBG CM CN?L;NCP? ;H> ;N ?;=B CN?L;NCIH ; GCHCGCT;NCIH I@ 4 CM MIOABN ON 4 MBIOF> <? ; P?=NIL I@ NB? Z(LSFIP MO< MJ;=?[ A?H?L;N?> <S NB?
P?=NILM H 4 ; H H 9 H QB?L? A CM NB? >CG?HMCIH I@ NB? (LSFIP
MO< MJ;=? NBCM >CG?HMCIH MBIOF> <? ; J;L;G?N?L I@ NB? G?NBI>
1BOM NB? MIFONCIH B;M NI <? @IOH> CH ; L?MNLC=N?> MJ;=? 1B? C>?; ILCACH;N?M @LIG
);H=TIMY G?NBI>M QBC=B QCFF HIN <? JL?M?HN?> B?L? M?? 89
4? MB;FF >?N;CF <?FIQ NB? N?=BHCKO?M @IL <OCF>CHA NB? (LSFIP MO< MJ;=? ;H> NB?
GCHCGCT;NCIH N?=BHCKO?M IH NBCM MO< MJ;=?
#CLMN ;H ILNBIHILG;F <;M? CH NBCM MO< MJ;=? QCFF <? <OCFN OMCHA NB? GI>CV?>
$L;G\0=BGC>N G?NBI>
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
)?N H <? NB? MN;LNCHA L?MC>O;F ?KO;F NI 4 ; 1B?H K C
C
C
C
@
@
;H> K
2 K
K ;H> K ;L? <INB
=;H NB?H <? >?VH?> 4CNB K
2 K NB?H K OHCN ;H> ILNBIAIH;F P?=NILM
1BCM C>?; I@ MO<NL;=NCHA @LIG ?;=B P?=NIL CNM JLID?=NCIHM IH NB? JL?PCIOM P?=NILM I@
NB? <;M? =IHNCHO?M @IL ;FF NB? P?=NILM I@ NB? <;M? 1B? ;FAILCNBG I@ ILNBIAIH;FCT;NCIH
CM NBOM NB? @IFFIQCHA
@LIG ? NI A
K
17 K7
@LIG @ NI ?
17 K8 ;H> K
17 L?JF;=?> <S K
17 7 8 K8
7 8 K
C
K7 C /?G;LE NB? P?=NIL K9 CM <OCFN 1BIOAB CN CM OM?@OF @IL NB? JL?M?HN;NCIH IHFS
1
K9 CM H??>?> ;H> =;H <? ?;MCFS >?>O=?> @LIG NB? @ILGOF;? K
17
L?JF;=?> <S
K
17 7 8 K8 M NB? <;M? CM ILNBIAIH;F Q? A?N
1
K9
K9 9
,
9
8
8
1BCM ;FAILCNBG I@ ILNBIAIH;FCT;NCIH =;H <? MOGG;LCT?> <S >?VHCHA NBL?? G;NLC=?M
19 K K K9 G;NLCR D A
19
K K K9
G;NLCR D A
;H>
' 1
K
'
'
1
K
'
$9 '
'
'
99
'
%
1
K9 9 9 9 9
1
K9 (
(
(
(
(
(
(
(
&
1B? G;NLCR $9 B;M A LIQM ;H> A =IFOGHM 1B? ;FAILCNBG =;H <? MOGG;LCT?>
<S 19 19 $9 $9 CM ;FGIMN ;H OJJ?L NLC;HAOF;L G;NLCR IHFS NB? >C;AIH;F
=IHN;CHCHA NB? N?LGM 1
K 1
K IP?LF;JM NB? FIQ?L B;F@ 1BCM G;NLCR CM =;FF?> NB?
OJJ?L %?MM?H<?LA G;NLCR
&@ 4 ; ;H> H; 4 ; L?MOFN @LIG NB? CGG?>C;N?FS JL?PCIOM CN?L;NCIH ; <?NN?L
;JJLIRCG;NCIH QCFF <? MIOABN FCE? 4 ; 6 IL <S GCHCGCTCHA
4 ;
6 H;
6
M CH>C=;N?> ?;LFC?L 6 MBIOF> <? MIOABN CH NB? (LSFIP MO< MJ;=? CH NB? @ILG
6
9
,
8
O8 K 8
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&@ 5 CM ; P?=NIL QBIM? =IGJIH?HNM ;L? >?HIN?> O8 6 CM NB? G;NLCR\P?=NIL JLI>O=N
19 5 &@ ! CM NB? P?=NIL QBIM? =IGJIH?HNM ;L? H; >CG?HMCIH A H; CM QLCNN?H ;M 19 ! 4? NB?H B;P?
H;
6 19
!
19 5 19
!
$9 5
M 19 CM G;>? OJ I@ ILNBIAIH;F ;H> OHCN P?=NILM NB? HILG I@ 19 ! $9 5
?KO;FM NB? HILG I@ ! $9 5 4? L?=;FF NB;N NB? HILG I@ ; G;NLCR CM ACP?H <S
' ,
QB?L? 4 ,
1B? F;MN MN?J CM NI VH> 5 NB? P?=NIL QBC=B GCHCGCT?M ! $9 5 1B? N?=BHCKO? OM?> CM ; @;=NILCT;NCIH N?=BHCKO? =;FF?> -. &N NL;HM@ILGM ; G;NLCR
) QBIM? >CG?HMCIHM ;L? C D CHNI ; JLI>O=N -. QB?L? - CM ;H ILNBIAIH;F G;NLCR
QBIM? >CG?HMCIHM ;L? C C ;H> . CM ;H OJJ?L NLC;HAOF;L G;NLCR CNM >CG?HMCIHM
<?CHA C D !C_?L?HN P;LC;HNM G;S <? @IOH> CH L?@?L?H=? 89
#;=NILCT;NCIH QCFF N;E? JF;=? ;M $ .$9 QB?L? $ CM ;H OJJ?L NLC;HAOF;L G;NLCR
I@ >CG?HMCIH A A ;H> . ;H ILNBIAIH;F G;NLCR I@ >CG?HMCIH A A 1B?
JLI=?MM QCFF NB?H AI IH QCNB $ QBIM? CHP?LMCIH CM ?;MC?L NB;H @IL NB? %?MM?H<?LA
G;NLCR $9 ?@IL? >?VHCHA . CN =;H <? HIN?> ;N NBCM JICHN NB;N C@ ! .! Q?
MBIOF> GCHCGCT?
!
$9 5 . !
. $5
!
$5
M $ CM ;H OJJ?L NLC;HAOF;L G;NLCR I@ >CG?HMCIH A A NB? A NB =IGJIH?HN
I@ $5 CM T?LI ;H> MI NB? A NB =IGJIH?HN I@ ! $5 CM ;9
1B? INB?L
=IGJIH?HNM I@ ! $5 =;H <? G;>? T?LI <S ; DO>C=CIOM =BIC=? I@ 5 ;H> <S CHP?LNCHA
$ &N @IFFIQM NB;N NB? GCHCGOG I@ ! $5 I<N;CH?> QCNB NB? <?MN =BIC=? I@ 5 CM
?KO;F NI ;9 4? NBOM VH> NB;N 5 CM NB? MIFONCIH I@ NB? MSMN?G <?FIQ
>
>
>9
;
O
'
'
( '
(
>
(
'
(' O ( ' ; (
% >9 9 >9 9 & % & % &
O9
;9
>9 9
;H> NB? L?MC>O;F QCFF <? ?KO;F NI ;9 )?N OM HIQ MJ?=C@S BIQ NB? G;NLCR . CM <OCFN 4? B;P? NI >?VH? VLMN $CP?HMY
LIN;NCIHM QBIM? MCT? CM A A ;H> QBC=B ;L? ?KO;F NI NB? C>?HNCNS G;NLCR
?R=?JN @IL IH? LIN;NCIH L?F;NCP? NI NQI LIQM ;H> NQI =IFOGHM )?N .8 <? NB? LIN;NCIH
(
'
(
'
(
'
(
'
(
'
98 I8
.8 '
(
(
'
I
9
8
8
(
'
&
%
1B? ?R;GJF? ;<IP? CM ACP?H @IL A ;H> @
I?^=C?HNM 98 ;L? FI=;N?> CH
NB? JIMCNCIH @ @ ;H> @ @ QCNB 98 I8 1B? G;NLCR . CM ?KO;F NI NB?
JLI>O=N I@ A G;NLC=?M =;FF?> $CP?HMY LIN;NCIHM
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
.9 .9 . .
$CP?HMY LIN;NCIHM QCFF =;H=?F NB? JLI<F?G =;OMCHA >C;AIH;F QCNB N?LGM 1
K 1
K ?N= I@ NB? %?MM?H<?LA G;NLCR $9 1I @ILG $ $9 QCFF <? NL;HM@ILG?> =IFOGH <S
=IFOGH #IL ?R;GJF? NB? VLMN =IFOGH I@ $9 CM
>
' > (
(
' 1
'
(
' K (
( >?HIN?> ;M '
(
'
'
(
(
'
% &
% &
1BCM =IFOGH QCFF <? GI>CV?> <S LIN;NCIH .
9 5
>
;H>
QCNB
>
>
>
I 5
>
>
1B? VLMN =IFOGH I@ $9 CM GI>CV?> CHNI
5
>
>
'
'
'
'
%
(
(
(
(
&
&N QCFF HIN <? ;_?=N?> <S NB? @IFFIQCHA LIN;NCIHM . NI .9 QBC=B QCFF ;=N QCNB
L?A;L>M NI CN ;M NB? C>?HNCNS 1B? @NB =IFOGH I@ $9 QCFF NB?H IHFS <? NL;HM@ILG?> <S
NB? JLI>O=N .8 .8
.
1B? JLI>O=N .8 .8
. F?;>M NI ; =IFOGH
QB?L? 0
>88 ;H> 0
>8
;H>
>8
>8
0
>88
'
'
'
'
'
'
' 0
' >8
'
'
% 8
(
(
(
(
(
(
(
(
(
(
&
8
;L? CHN?LG?>C;N? L?MOFNM M @IL NB? VLMN =IFOGH F?N OM =BIIM?
98 6
0
>88
0
>88
0
>8
8
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
0
>8 8
I8 6
0
>8
>88 0
8
1B? @NB =IFOGH I@ $ CM =B;HA?> ;@N?L <?CHA GOFNCJFC?> <S .8 CHNI
'
'
'
'
'
'
'
'
'
'
'
%
>8
>8
6
0
>8
>88 0
8
(
(
(
(
(
(
(
(
(
(
(
&
1BIOAB NBCM NL;HM@ILG;NCIH I@ $9 CHNI $ M??GM =IGJF?R NI QLCN? CN CM ?;MCFS
NL;HMF;N?> CHNI NB? #,/1/+ F;HAO;A? 1B? =B;HA? CM G;>? ZIH NB? MJIN[ QCNBION
OMCHA >C_?L?HN L?ACMN?LM @IL $9 ;H> $ CH NB? G?GILS MJ;=?
1B? $*/"0 G?NBI> CM P?LS ?^=C?HN @IL HIH MSGG?NLC= ;H> JIILFS =IH>CNCIH?>
G;NLC=?M ,H? I@ CNM >CM;>P;HN;A?M CM NB? J;L;G?N?L A NB;N GOMN <? ;>DOMN?> %IQ?P?L
NB? G?NBI> CM HIN P?LS M?HMCNCP? NI CN 3;FO?M L;HACHA @LIG NI IL ?P?H NI =;H
<? @IOH> CH NB? FCN?L;NOL? ,@ =IOLM? NB? BCAB?L NB? P;FO? I@ A NB? AL?;N?L NB? MJ;=?
L?KOCL?> CH NB? G?GILS ,H? MBIOF> HINC=? ;FMI NB;N NB? HOG<?L I@ G;NLCR\P?=NIL
JLI>O=NM J?L CN?L;NCIH CM A 4CNB A NB? $*/"0 G?NBI> QCFF B;P? HI =B;H=? NI
J?L@ILG <?NN?L NB;H ; =IHDOA;N? AL;>C?HN NB;N =IHP?LA?M CH CN?L;NCIHM
(HIQF?>A? I@ NB? $*/"0 G?NBI> B?FJM NI OH>?LMN;H> CNM FCGCNM 1B? G?NBI>
QCFF MOL?FS >?A?H?L;N? C@ NB? G;NLCR I@ NB? MSMN?G NI <? MIFP?> CM >C;AIH;F &H NBCM
=;M? NB? P?=NILM H H H A?H?L;NCHA NB? (LSFIP MO< MJ;=? ;L? ;FF =IFCH?;L
%A42>=38C8>=8=6
806>=0; ?A42>=38C8>=8=6
)?N 4 <? ; FCH?;L MSMN?G NI <? MIFP?> 1B? Z>C;AIH;F JICHN[ JL?=IH>CNCIHCHA
CM ; L?;F IH? @LIG ;H ?NSGIFIAC=;F JICHN I@ PC?Q ;M CN CM ;JJFC?> <?@IL? NB? L?MIFONCIH
4? @ILG NB? >C;AIH;F G;NLCR
MO=B ;M 77 &@ 77 =;H <? H?A;NCP? NB?H 77 =;H ;FMI <? N;E?H ;M ON CH ;HS =;M? 77 GOMN <? HIH T?LI 1B? ILCACH;F MSMN?G CM NB?H MIFP?> <ON
QLCNN?H ;M
1B? @IFFIQCHA N?LGM ;JJ?;L
H?Q G;NLCR H?Q OHEHIQH P?=NIL 4 4
H?Q LCABN B;H> MC>? 4
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
1;<F? L?MOFN I@ 9\K JL?=IH>CNCIHCHA
+ILG;F ?KO;NCIH
$*/"0
4CNBION 9\K JL?=IH>CNCIHCHA
+I =IHP?LA?H=?
M
4CNB 9\K JL?=IH>CNCIHCHA
M
M
,H ;==IOHN I@ NBCM <OCF>CHA ;H> QB?H 77 CM ;FQ;SM JIMCNCP? NB? >C;AIH;F I@ CM IHFS G;>? I@ IH?M NBCM =;H <? OM?@OF @IL NB? JOLJIM? I@ IJNCGCT;NCIH N?=BHCKO?M
1B? L?MOFN I@ JL?=IH>CNCIHCHA B?L? CM NI ACP? ; =IGJ;L;<F? Q?CABN NI ;FF NB? ?KO;NCIHM
,H=? NB? MSMN?G 4 B;M <??H MIFP?> CN CM ?;MS NI VH> 4 NB;N CM ?KO;F NI
4
1BCM MBIQM NB? AL?;N CHN?L?MN CH ;MM?G<FCHA NB? >C;AIH;F I@ NB? G;NLCR QCNB NB?
"" N?=BHCKO? I@ MNIL;A? NBOM ?H;<FCHA NB? OM? I@
O ?A42>=38C8>=8=6 5>A C74 (08=C+4=0=C 4@D0C8>=B
HINB?L JL?=IH>CNCIHCHA L?@?LL?> NI ;M 9\K JL?=IH>CNCIHCHA CM MJ?=CV= NI NB? 0;CHN
3?H;HN ?KO;NCIHM &N CM <;M?> IH NB? @;=N NB;N NB? FCH?;L MSMN?GM >?LCP?> @LIG ;
P;LC;NCIH;F @ILGOF;NCIH MO=B ;M NB? IH? >?M=LC<?> CH 0?=NCIH <ON QCNB NB? =?F?LCNS
;H> P?FI=CNS P;LC;<F?M B;P? ;H ?R=?FF?HN =IH>CNCIHCHA 1B;N =;H <? ?RJF;CH?> <S NB?
@;=N NB;N NB? NBL?? OHEHIQHM I@ NB? FCH?;L MSMN?G ;L? BIGIA?H?IOM NI ; P?FI=CNS ;H>
NBOM ; MSGG?NLS ;JJ?;LM =NO;FFS IH? =;H QLCN? ) B ) ;H> ) B ) B;HACHA NB? MCAH I@ NB? =IHNCHOCNS ?KO;NCIH QIOF> F?;> NI ; MSGG?NLC=
G;NLCR <ON OH@ILNOH;N?FS HIN JIMCNCP? >?VHCN? 1B? JL?=IH>CNCIHCHA B;M NI <? >IH?
;N NB? F?P?F I@ NB? FCH?;L MSMN?G &H NB? =?F?LCNS\P?FI=CNS IJNCIH NB? =B;HA?M CH
NB? P;LC;<F?M Q?L? G;>? ;N NB? =IHNCHOOG F?P?F 1B? C>?; B?L? CM NI MO<MNCNON? NB?
OHEHIQH Q;N?L >?JNB <S NB? Q;P? =?F?LCNS ;N NB? >CM=L?N? F?P?F ;H> NLS NI A?N NB?
MSGG?NLS I@ NB? G;NLCR ;M @;L ;M JIMMC<F?
&HCNC;FFS NB? P;FO? I@ NB? =?F?LCNS 9 IH ?;=B HI>? >?HIN?> <S 9< ? CM =IGJON?>
OMCHA NB? L?MOFNM I@ NB? ?;LFC?L NCG? MN?J IL NB? <IOH>;LS =IH>CNCIHM &H NB? =;M? I@
NC>;F W;N ;L?;M NBCM P;FO? =;H <? E?JN QCNBCH <IOH>M NI ;PIC> T?LI >CPCMCIH ?LLIL 1B?
OHEHIQH >< ? I@ NCG? J<
CM NB?H MO<MNCNON?> CH NB? QBIF? FCH?;L MSMN?G QCNB
>< ? 157 QB?L? = CM NB? AL;PCNS 1B? =IHNCHOCNS ?KO;NCIH CM NB?H GOFNCJFC?> <S
5
1 7
,H=? NB? H?Q MSMN?G CM MIFP?> NB? P;FO? I@ >< ? B;M MCGJFS NI <? ?RJL?MM?>
;==IL>CHA NI NB? H?Q OHEHIQH I@ NB? MSMN?G
1BCM IJ?L;NCIH CM @ILG;FFS ?KOCP;F?HN NI ; >C;AIH;F JL?=IH>CNCIHCHA NB? G;NLCR I@ NB? MSMN?G CM MO<MNCNON?> QCNB QB?L? ;H> ;L? >C;AIH;F G;NLC=?M
@OH=NCIHM I@ 9<
1BCM JL?=IH>CNCIHCHA G?NBI> Q;M JO<FCMB?> CH M?? 89 &N B;M JLIP?> NI
<? ?^=C?HN @IL NB? =IGJON;NCIH I@ NC>;F =OLL?HNM IP?L ;L?;M MNL?N=BCHA <?SIH> NB?
=IHNCH?HN;F MB?F@ NB;N CM P?LS >??J ;L?;M #IL ; GI>?F I@ NB? #L?H=B NF;HNC= =I;MN
;H> NB? B;HH?F M?? 0?=NCIH NB? -2 NCG?M IH ; /6 ;L? ACP?H CH
1;<F? =IGJON;NCIH I@ @IOL NC>?M G?MB G;>? I@ ?F?G?HNM NCG? MN?J M
;H> QCNB ;H ;>>CNCIH;F >C;AIH;F JL?=IH>CNCIHCHA
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
;>2: 3806>=0; ?A42>=38C8>=8=6
1BCM JL?=IH>CNCIHCHA G;E?M M?HM? IHFS C@ NB? G;NLCR ) CM ; <FI=E I@ MKO;L? G;NLC=?M
1B? JLCH=CJF? ;H> >?N;CF?> ?RJF;H;NCIHM QCFF <? ACP?H H?RN @IL ; @IOL G;NLCR <FI=E
5>DA<0CA8G 1;>2:
)?N 4 <? NB? MSMN?G NI MIFP? QB?L?
4
4
4
;H> 1B? JOLJIM? CM NI ;JJFS NB? C>?; I@ >C;AIH;F JL?=IH>CNCIHCHA HIN IHFS NI G;NLC=?M
;H> <ON ;FMI NI NB? ?RNL; >C;AIH;F G;NLC=?M ;H> M ; L?MOFN NB?
@IOL G;NLC=?M I@ NB? <FI=E QIOF> B;P? ; >C;AIH;F G;>? I@ IH?M
;H> QCFF L?MJ?=NCP?FS >?HIN? NB? >C;AIH;FM I@ ;H> 1B? <;MC= C>?; CM NI ;JJLIRCG;N? QCNB NB? G;NLCR
1 CH NB? @ILG (
;H> OM? ; ( 1 @;=NILCT;NCIH I@ 1 1B? CHCNC;F MSMN?G 4 =;H NB?H <? QLCNN?H ;M
1
(
1
1 4 (
1BCM MSMN?G CM ?KO;F NI
( 1 1 4 ( &H NBCM @ILG NB? MSMN?G =;H <? PC?Q?> ;M ; >C;AIH;F JL?=IH>CNCIHCHA I@ NB? MSMN?G
4 QCNB NB? JL?=IH>CNCIHCHA G;NLCR
<?CHA ACP?H ;H> QCNB
( 1 4 1 4
( ,H=? NB? MSMN?G CM MIFP?> NB? OHEHIQH 4 CM =IGJON?> <S NB? @ILGOF;
4 1 4
1B? @IFFIQCHA IJ?L;NCIHM B;P? NI <? >IH? MO==?MMCP?FS
IGJON;NCIH I@ (
;H> 1 <S ; ( 1 @;=NILCT;NCIH I@
1
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
IGJON;NCIH I@ 4 ;H> /?MIFONCIH I@ NB? MSMN?G 4 QCNB ; MCGJF? >C;AIH;F JL?=IH>CNCIHCHA
QB?L? NB? >C;AIH;F
CM ACP?H M?? 0?=NCIH ;<IP?
IGJON;NCIH I@ 4 ;M ; @OH=NCIH I@ 4 1B? >?N;CFM I@ IJ?L;NCIHM ;H> ;L? ;M @IFFIQM
1
502C>A8I0C8>= >5 1
CM >?=IGJIM?> ;M
%
( %
%
1 %
S C>?HNCV=;NCIH IH? =;H VH>
( 1 (
1 &H JL;=NC=? NB? JLIAL;GGCHA QCFF OM? NB? M;G? G?GILS MJ;=? @IL >C;AIH;FM
;H> ;H> ( ;H> 1 ;H> 1BCM L?KOCL?M NB? MO==?MMCP? IJ?L;NCIHM
CM MO<MNCNON?> QCNB CM MO<MNCNON?> QCNB CM MO<MNCNON?> QCNB 1 CM NB?H QLCNN?H CH NB? @ILG
%
%
%
%
1B? >C;AIH;FM ;H> ;L? CHP?LN?> NB?CL MKO;L? LIIN CM =IGJON?> ;H> NB?S
;L? E?JN ;MC>? @IL @OLNB?L >C;AIH;F JL?=IH>CNCIHCHA IJ?L;NCIH 1B?S ;L? HI FIHA?L
H?=?MM;LS @IL IJ?L;NCIHM ;H> ><?DC0C8>= >5 0=3 ,H? OM?M NB? @IFFIQCHA @ILGOF;?
%
%
%
%
;H>
%
1B? JLI>O=N
%
%
%
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
%
%
CM ?KO;F NI
M @IL NB? ( 1 @;=NILCT;NCIH NB? =IGJON;NCIH I@ CM G;>? ZIH NB? MJIN[ NB;N
CM CH NB? P?LS G?GILS MJ;=? QB?L? B;M <??H MNIL?>
,H? MN;LNM QCNB NB? IJ?L;NCIHM
CM MO<MNCNON?> QCNB CM MO<MNCNON?> QCNB *OFNCJFSCHA IH NB? LCABN MC>? <S 1 NB?H L?KOCL?M NB;N
CM MO<MNCNON?> QCNB 1 CM MO<MNCNON?> QCNB 1 #IFFIQCHA NB?M? IJ?L;NCIHM NB? G;NLCR N;E?M NB? G?GILS MJ;=? I@ NB? G;NLCR
1B? =IGJON;NCIH I@ 4 CM ;FMI G;>? ZIH NB? MJIN[ 4 <?CHA MO<MNCNON?> QCNB
4
4 4 OH=B;HA?>
#IL NB? =IGJON;NCIH I@ CM MO<MNCNON?> QCNB OH=B;HA?>
><?DC0C8>= >5 1BCM N;E?M JF;=? <S MO<MNCNONCHA 4 QCNB 4 4 5 OH=B;HA?>
#8=4<0CA8G 1;>2:
)?N 4 <? NB? MSMN?G NI MIFP? QB?L?
4
% & 4 % 4 & ;H> % &
4
CD >?HIN?M NB? >C;AIH;F I@ CD 1B? JL?=IH>CNCIHCHA JLCH=CJF? CM NB? M;G? ;M @IL
; @IOL G;NLCR <FI=E
1B? @IFFIQCHA IJ?L;NCIHM B;P? NI <? >IH?
IGJON;NCIH I@ (
;H> 1 <S ( 1 @;=NILCT;NCIH I@
1 % &
IGJON;NCIH I@ 4 ;H> /?MIFONCIH I@ NB? MSMN?G 4 OMCHA ; MCGJF? >C;AIH;F JL?=IH>CNCIHCHA
QB?L? NB? >C;AIH;F G;NLCR
CM ACP?H
IGJON;NCIH I@ 4 ;M ; @OH=NCIH I@ 4 %?L?;@N?L Q? >?N;CF IJ?L;NCIHM ;H> CH IL>?L NI MBIQ NB? G?=B;HCMGM I@
G;NLCR >?=IGJIMCNCIH CH NB? M;G? G?GILS MJ;=?
1
502C>A8I0C8>= >5 *,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
1 CM <LIE?H >IQH ;M
%
%
& %
%
%
%
%
& %
&
%
1B? >?=IGJIMCNCIH >OLCHA QBC=B ;H> I==OJS ;FN?LH;NCP?FS NB? M;G? G?GILS
MJ;=? N;E?M JF;=? ;M @IFFIQM
L?G;CHM OH=B;HA?>
CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S
CM MO<MNCNON?> <S >CPCMCIH <S CM CHN?HNCIH;FFS IGCNN?>
CM MO<MNCNON?> <S ?KO;FM CH @;=N CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S NI =ILL?=N NB? IGCMMCIH G?HNCIH?> NBL?? FCH?M
;<IP?
1B? >CPCMCIHM ;L? CH @;=N GOFNCJFC=;NCIHM <?=;OM? I@ NB? JL?PCIOM CHP?LMCIH I@ NB?
>C;AIH;FM ;H> QBC=B QCFF <? OM?> CH NBCM @ILG F;N?L #IFFIQCHA NB?
CHP?LMCIH NB? MKO;L? LIIN CM =IGJON?> ;H> E?JN ;MC>? @IL NB? >C;AIH;F JL?=IH>CNCIHCHA
><?DC0C8>= >5 0=3 1B? @IFFIQCHA @ILGOF;? ;L? OM?>
;H>
%
%
%
%
&
%
%
%
%
%
%
& % %
%
%
%
& %
%
%
%
&%
%
%
%
&
%
&
1B?M? NQI >?=IGJIMCNCIHM ;L? OM?> NI =IGJON? E??JCHA CH GCH> NB;N CM
GOFNCJFC?> IH NB? F?@N <S NB? FIQ?L J;LN ;H> GOFNCJFC?> IH NB? LCABN <S NB? OJJ?L J;LN
1B? =B;HA?M ;L? ;FQ;SM G;>? ZIH NB? MJIN[
1B? F?@N GOFNCJFC=;NCIH CM =;LLC?> ION QCNB NB? @IFFIQCHA IJ?L;NCIHM
CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S 1B? LCABN GOFNCJFC=;NCIH CM =;LLC?> ION QCNB NB? @IFFIQCHA IJ?L;NCIHM
CM MO<MNCNON?> <S CM MO<MNCNON?> <S %6!/,!6+*& 0 ,# #/"" 02/# " #),40
CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S CM MO<MNCNON?> <S @N?L NB?M? IJ?L;NCIHM G;NLCR CM CH NB? G?GILS MJ;=? JL?PCIOMFS OM?> <S G;NLCR
1B? =IGJON;NCIH I@ 4 CM G;>? QCNB NB? M;G? =IH>CNCIHM @IFFIQCHA NB? MN?JM
4 4
4
4 4
4
4
4 L?G;CHM OH=B;HA?>
1B? =IGJON;NCIH I@ CM =;LLC?> ION <S @IFFIQCHA NB? MN?JM
L?G;CHM OH=B;HA?> ;H>
><?DC0C8>= >5 1BCM N;E?M JF;=? <S @IFFIQCHA NB? IJ?L;NCIHM
4 L?G;CHM OH=B;HA?> ;H>
4 4 4 4 4
4 4
%A42>=38C8>=8=6 F8C7 502C>A8I0C8>=
1BCM ECH> I@ JL?=IH>CNCIHCHA CM HIN ;JJLIJLC;N? @IL HIH ;MM?G<F?> G;NLC=?M %IQ?P?L
CNM =IGJL?B?HMCIH CM H?=?MM;LS @IL CHNLI>O=CHA NB? LION JL?=IH>CNCIHCHA N?=BHCKO?
)?N ( ;H> 1 L?MJ?=NCP?FS <? ; FIQ?L ;H> ;H OJJ?L G;NLCR QBIM? JLI>O=N (1 CM =FIM?
NI 1B? ?^=C?H=S I@ NB? JL?=IH>CNCIHCHA CM >?N?LGCH?> <S NB? =BIC=? I@ ( ;H> 1
0IG? ?R;GJF?M QCFF <? ACP?H CH NB? @IFFIQCHA MO<M?=NCIHM )?N OM MOJJIM? NB;N ( ;H>
1 ;L? EHIQH &HMN?;> I@ NB? MSMN?G 4 NB? @IFFIQCHA MSMN?G QCFF <? MIFP?>
( 1 1 4 ( M ; L?MOFN H?Q KO;HNCNC?M ;JJ?;L
H?Q G;NLCR ( 1 H?Q OHEHIQH P?=NIL 4 1 4
H?Q LCABN B;H> MC>? ( ,H=? NB? MSMN?G CM MIFP?> 4 CM ACP?H <S NB? @ILGOF;
4 1 4
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
&@ CM ; MSGG?NLC= G;NLCR CN CM L?=IGG?H>?> NI =BIIM? ; (1 @;=NILCT;NCIH QB?L?
1 CM B ( 4B?H (1 ( B ( NB? CN?L;NCP? G?NBI>M MO=B ;M =IHDOA;N? AL;>C?HN =;H <?
;>;JN?> CH IL>?L NI OM? IHFS IH? CHP?LMCIH <S (B ( 0?? 89
A>DC ?A42>=38C8>=8=6
%A8=28?;4
LION JL?=IH>CNCIHCHA M?? 89 OM?M NB? C>?; I@ NB? (1 @;=NILCT;NCIH <ON ;>;JNM CN
NI NB? G;NLC=?M MNIL?> QCNB NB? "" N?=BHCKO? CM ;JJLIRCG;N?> <S ; >?=IGJIMCNCIH
CH NB? @ILG * ( 1 QB?L? ( ;H> 1 ;L? L?MJ?=NCP?FS NB? FIQ?L ;H> OJJ?L G;NLC=?M
QCNB IH?M IH NB? >C;AIH;F
)?N OM ;MMOG? NB;N ;FL?;>S B;M ; OHCN >C;AIH;F C@ HIN ;H> C@ NB? >C;AIH;F G;E?M
CN JIMMC<F? ; >C;AIH;F JL?=IH>CNCIHCHA QIOF> B;P? NI <? J?L@ILG?> <?@IL? LION
JL?=IH>CNCIHCHA
=;H NB?H <? QLCNN?H OMCHA NB? ?RNL; >C;AIH;F ?F?G?HN G;NLC=?M 43 QCNB NB?
HIN;NCIHM I@ 0?=NCIH <3:3;
,
%
,3 43 ,3B
3
1B? C>?; @IL ;JJLIRCG;NCHA CM NI ;JJFS NB? C>?HNCNS
<
,
<
-
7
7
7
7
1B? MG;FF?L NB? 7 NB? NLO?L NBCM C>?HNCNS &H NB? =;M? I@ NBCM ;JJLIRCG;NCIH
F?;>M NI
<3:3;
-
%
,3 43 ,3B
3
23 <? 43 %3 QB?L? %3 CM NB? C>?HNCNS
43 CM ; G;NLCR QCNB T?LI >C;AIH;F )?N 4
G;NLCR ;N NB? ?F?G?HN;LS F?P?F
)?N %13 <? ?KO;F NI NB? C>?HNCNS G;NLCR % IP?L NB? QBIF? G?MB <ON QCNB T?LI
P;FO?M IH NB? >C;AIH;F CH NB? JIMCNCIHM =ILL?MJIH>CHA NI NB? HI>?M I@ ?F?G?HN ;
%13 % ,3 %3 ,3B
23 (3 3 13 QB?L? (3 CM ;
23 C ? 4
1B? LION >?=IGJIMCNCIH CM ;JJFC?> NI 4
FIQ?L NLC;HAOF;L G;NLCR QCNB OHCN >C;AIH;F 3 CM ; >C;AIH;F G;NLCR ;H> 13 CM ; OJJ?L
NLC;HAOF;L G;NLCR QCNB OHCN >C;AIH;F CM NB?H ;JJLIRCG;N?> <S
<3:3;
3
%13
,3 (3
%13
,3
3 13
,3B
"P?HNO;FFS * CM ACP?H <S ; MCGCF;L <ON MSGG?NLC=;F ?RJL?MMCIH
*
<3:3;
3
%13
,3 (3 ,3B
<3:3;
3
B
3 ,3
-
3<3:3;
%13
,3 13 ,3B
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
*<3:3; 1
1B? M?=IH> ALIOJ I@ JLI>O=NM
%3 ,3 3 ,3B CM ; >C;AIH;F G;NLCR >?
3
HIN?>
QBC=B =;H <? ?;MCFS =IGJON?> 1B? =ILL?MJIH>CHA IJ?L;NCIH CM @ILG;FFS
?KOCP;F?HN NI ; P?=NIL ;MM?G<FS <ON QCNB GOFNCJFC=;NCIHM CHMN?;> I@ ;>>CNCIHM B?H=?
NB? H;G? ZGOFNCJFC=;NCP? ;MM?G<FS[ 1B? INB?L G;NLC=?M CH NB? JLI>O=N ;L? ?CNB?L
FIQ?L NLC;HAOF;L IL OJJ?L NLC;HAOF;L 0IFPCHA ; * 4 NSJ? I@ FCH?;L MSMN?G CM
NB?H ; MO==?MMCIH I@ >?M=?H>CHA ;H> ;M=?H>CHA =IGJON;NCIHM VLMN OMCHA NB? OJJ?L
NLC;HAOF;L G;NLC=?M 13 NB?H ;@N?L CHP?LMCIH I@ NB? >C;AIH;F G;NLCR
OMCHA NB? FIQ?L
NLC;HAOF;L G;NLC=?M (3
G0<?;4B DB8=6 CA80=6;4B F8C7 ;8=40A 8=C4A?>;0C8>=
1B? G;NLCR CM >?VH?> <S CNM OHCN >C;AIH;F * , +%* ;H> CNM ?RNL; >C;AIH;F
N?LGM 4 * !(!) CM HIN ;MMOG?> NI <? MSGG?NLC=
( 1 502C>A8I0C8>= NB? ?F?G?HN G;NLC=?M ;L? >?=IGJIM?> ;M ( 1 JLI>O=NM OMCHA
NB? LION ;FAILCNBG 1B? L?MOFN CM MNIL?> CH * * , +%* @IL NB? 3 G;NLC=?M ;@N?L
GOFNCJFC=;NCP? ;MM?G<FS ;H> CH 4* * !(!) @IL G;NLC=?M (3 ;H> 13
1B? LION ;FAILCNBG ;JJFC?> NI NB? G;NLCR QCNB ?F?G?HNM 778 CM ;M @IFFIQM
#IL D NI H ;JJFS
#IL C NI H ;JJFS
7
)
78 778 79 98 C@ C NB? MOG CM T?LI
9
NB?H
#IL CD NI H ;JJFS
8
)
78 778 79 98 C@ D NB? MOG CM T?LI
9
1BCM L?MOFNM CH ; (1 @;=NILCT;NCIH ( =IHMCMNM I@ 78 ;H> 1 I@ 78 1B? ( 1
@;=NILCT;NCIH CM ?P?HNO;FFS I<N;CH?> <S >CPC>CHA ?;=B 78 <S 77
( CM VH;FFS NB? FIQ?L NLC;HAOF;L G;NLCR >?VH?> <S (78 78 @ ? (77 (78 ? @
CM NB? >C;AIH;F G;NLCR >?VH?> <S
77
77
1 CM NB? OJJ?L NLC;HAOF;L G;NLCR
178 78 @ ? 177 178 78
?@
77
&H #,/1/+ ;H> QCNB "" MNIL;A? NBCM <?=IG?M
!, &")"* +")"*
*;NLCR NI <? >?=IGJIM?>
;@N?L >C;AIH;F JL?=IH>CNCIHCHA NB;N CM QCNB ; OHCN >C;AIH;F
/IQ I@ ?F?G?HN G;NLCR
!
5*&")"* *,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
5*&")"* /IQ I@ ?F?G?HN G;NLCR
5*&")"* !
5*&")"* /IQ I@ ?F?G?HN G;NLCR
5*&")"* 5*&")"* !
LION )2 @;=NILCT;NCIH
IFOGH "1 CM IGCNN?>
)# )# IFOGH "1 "1 )# "1
)# )# "1"1
IFOGH "1 "1 )# "1
"1 )# "1 )# "1
)!2 @;=NILCT;NCIH
"RNL; >C;AIH;F N?LGM ;L? MNIL?> CH 5+
!C;AIH;F N?LGM ;L? MNIL?> CH 4 ;H> 4
4 CM HIN H?=?MM;LS <?=;OM? "1
5+&")"* "1
5+&")"* "1
5+&")"* "1"1
5+&")"* )#
5+&")"* )#
5+&")"* )#
4&")"* "1
4&")"* "1
"+!!,
*;NLCR ) CM MNIL?> CH
5+&")"* 5+&")"* ;H> 5+&")"* *;NLCR 2 CM MNIL?> CH 5+&")"* 5+&")"* ;H> 5+&")"* %6!/,!6+*& 0 ,# #/"" 02/# " #),40
LL;SM 4 ;H> 4 ;L? NB?H GOFNCJFC=;NCP?FS ;MM?G<F?>
CH !+ JL?PCIOMFS CHCNC;FCT?> NI ;M @IFFIQM
]]]]]]]]]]]]]]]]]]]]]]]]]
),,- 4&1% &*-,0"! 3" 1,/&71&,+ L;S =IGG;H>
]]]]]]]]]]]]]]]]]]]]]]]]]
!&/ &3!"!, &")"* +")"*
!+&()"&")"* !+&()"&")"* ,+1&+2"
!&/ &3!"!, &")"* +")"*
!+&()"&")"* !+&()"&")"* ,+1&+2"
4&")"*
4&")"*
1B? FIIJ CH NB? GOFNCJFC=;NCP? ;MM?G<FS =;H <? P?=NILCT?> @IL NB? M;G? L?;MIHM ;M
NB? =F;MMC=;F ;MM?G<FS C@ NB? ?F?G?HNM B;P? <??H HOG<?L?> ;JJLIJLC;N?FS
3?=NIL * CM NB?H CHP?LN?> <?=;OM? IHFS NB? N?LG
* CM H?=?MM;LS
!+ CHP?LMCIH
!, & +-,&+
!+& !!+&
,+1&+2"
(HBC4< * 4 AOF;L G;NLC=?M
MIFPCHA NBCM MSMN?G CM ; MO==?MMCIH I@ CHP?LMCIHM I@ FIQ?L NLC;H
&+&1&)&71&,+ 5 LCABN B;H> 0&!"
!, & +-,&+
5& &
"+!!,
&+3"/0&,+0 ,# ),4"/ 1/&+$2)/ *1/& "0
!&/ &3!"!, &")"* +")"*
5&()"&")"* 5&()"&")"* 5+&")"* 5&()"&")"* 5&()"&")"* 5&()"&")"* 5+&")"* 5&()"&")"* 5+&")"* 5&()"&")"* "+!!,
*,!"/+ 1" %+&.2"0 &+ #&+&1" ")"*"+10
*2)1&-)& 1&,+ 6 !+ -/"3&,20)6 &+3"/1"!
!, & +-,&+
5& 5& !+&
"+!!,
&+3"/0&,+0 ,# 2--"/ 1/&+$2)/ *1/& "0
!&/ &3!"!, &")"* +")"* 5&()"&")"* 5&()"&")"* 5+&")"* 5&()"&")"* 5&()"&")"* 5&()"&")"* 5+&")"* 5&()"&")"* 5+&")"* 5&()"&")"* "+!!,
&HP?LMCIH FIIJM =;HHIN <? P?=NILCT?> ?P?H NBIOAB IH? N;E?M NB? M;G? JL?=;ONCIHM
;M @IL P?=NILCTCHA NB? ;MM?G<FS I@ P?=NILM &N QIOF> <? H?=?MM;LS NI P?LC@S
&()"&")"* & &()"&")"* '
@IL & ' ;H> @IL ?P?LS >C_?L?HN ?F?G?HN &")"* ;H> &")"* =BIM?H CH ; <FI=E
I@ ?F?G?HNM CH NB? =;M? I@ ; /6 =IGJON?L 1BCM MN;N?G?HN CM NLO? IHFS C@
&' =IHMC>?LCHA IOL ;FAILCNBG I@ ?F?G?HN HOG<?LCHA M?? 0?=NCIH %IQ?P?L
P?=NILCT;NCIH =;H <? @IL=?> &N L?MOFNM CH ;H ?LLIL G;>? <S NB? =IGJON?L ;H> NB?
L?MOFN QCFF HIN <? ; MIFONCIH I@ NB? MSMN?G * 4 %IQ?P?L NB? ;CG I@ <OCF>CHA
* Q;M NI ;JJLIRCG;N? ) NB;N CM NI ;JJLIRCG;N? NB? MIFONCIH ) 4 0I NB?
L?MOFN ACP?H <S @IL=?> P?=NILCT;NCIH =;H <? OM?> ;M Q?FF CN QCFF DOMN ACP? ; >C_?L?HN
JL?=IH>CNCIHCHA #OLNB?LGIL? ;N NB? ?H> I@ ;HS CN?L;NCP? G?NBI> ; MNIJ N?MN CM
J?L@ILG?> NI AO;L;HN?? NB? ;>?KO;=S I@ NB? MIFONCIH 1B? N?MNM J?L@ILG?> B;P?
JLIP?> NB;N LION JL?=IH>CNCIHCHA QB?H ;JJLIJLC;N? CM ?RNL?G?FS ?^=C?HN ?P?H QCNB
; @IL=?> P?=NILCT;NCIH &H NB? =;M? I@ >C_OMCIH G;NLC=?M NB? =IGJON;NCIH =IMN =;H <?
>?=L?;M?> NI QB;N?P?L NB? NCG? H?=?MM;LS NI <OCF> NB? JL?=IH>CNCIH?L
0DBBO(4834; ?A42>=38C8>=8=6
1B? JLCH=CJF? CM MCGCF;L NI LION JL?=IH>CNCIHCHA MMOGCHA ; OHCN >C;AIH;F @IL )
( CM >?VH?> ;M NB? FIQ?L J;LN I@ ) QCNB OHCN >C;AIH;F QB?L?;M 1 CM >?VH?> ;M NB?
OJJ?L J;LN I@ ) QCNB ; OHCN >C;AIH;F 1BOM ( 1 ) %
1B?H NB? MO==?MMCP? >?M=?H>CHA ;H> ;M=?H>CHA =IGJON;NCIHM @IFFIQ NB? M;G? JLCH
=CJF? ;M @IL LION JL?=IH>CNCIHCHA
A>DC 0=3 0DBBO(4834; ?A42>=38C8>=8=6 5>A 436410B43
BC>A064
FF NB? L?;MIHCHA G;>? ?;LFC?L @IL NB? LION ;H> $;OMM\0?C>?F "" JL?=IH>CNCIHCHA
=;H <? ;JJFC?> QCNB NB? M;G? ?^=C?H=S NI NB? ?>A? <;M?> MNIL;A? M?? @IL ?R;GJF?
89 ?R=?JN @IL NQI >C_?L?H=?M VLMN ?RNL; >C;AIH;F N?LGM ;L? ;MM?G<F?> P;FO?M NB?H
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
NB? G;NLC=?M NI <? >?=IGJIM?> B;P? ; >CG?HMCIH QB;N?P?L NB? VHCN? ?F?G?HN
OH>?L =IHMC>?L;NCIH
MMOGCHA ; JL?PCIOM >C;AIH;F JL?=IH>CNCIHCHA Q? HIQ =IHMC>?L NB? ?>A? <;M?>
>?=IGJIMCNCIH I@ NB? G;NLC=?M
7
7
CH
7
1BCM CM
7
7
7
7
7
LION JL?=IH>CNCIHCHA IL
7
7
7
7
1BCM CM $;OMM\0?C>?F JL?=IH>CNCIHCHA 1B? F;NN?L CM A?H?L;FFS GIL? ?^=C?HN ;M CN
;PIC>M >CPCMCIH <S NB? >C;AIH;FM %IQ?P?L NB? M;PCHA I@ NCG? L?MOFNCHA @LIG MO=B
JL?=IH>CNCIHCHA CM I@N?H KOCN? FIQ 1B? HOG<?L I@ CN?L;NCIHM I@ G?NBI>M MO=B ;M NB?
=IHDOA;N? AL;>C?HN CM >CPC>?> <S <ON NB? =IMN I@ JL?=IH>CNCIHCHA CM CNM?F@ ?KOCP;F?HN
NI IH? CN?L;NCIH
70?C4A %0A0;;4;8B<
=CA>3D2C8>=
-;L;FF?FCMG =IHMCMNM I@ OMCHA MCGOFN;H?IOMFS ; =FOMN?L I@ =IGJON?LM IL ; =FOMN?L I@
JLI=?MMILM CH NB? M;G? =IGJON?L NI MIFP? ; MCHAF? JLI<F?G 2MCHA D JLI=?MMILM
QIOF> C>?;FFS >CPC>? <S D NB? NCG? H?=?MM;LS NI MIFP? NB? M;G? JLI<F?G QCNB IHFS IH?
JLI=?MMIL
1B? N;ME QIOF> <? ?;MS C@ NB? JLI<F?G =IOF> <? <LIE?H >IQH CHNI MO< N;MEM
CH>?J?H>?HN I@ ?;=B INB?L &N <?=IG?M GO=B GIL? >C^=OFN QB?H ?;=B JLI=?MMIL
H??>M NB? L?MOFNM @LIG NB? INB?LM 4? MB;FF @I=OM B?L?;@N?L IH J;L;FF?FCMG J?L@ILG?>
QCNB JLI=?MMILM ?;=B B;PCHA CNM IQH G?GILS ;H> =IGGOHC=;NCHA QCNB NB? INB?LM
PC; G?MM;A? NL;HMGCMMCIH ; =FOGMS Q;S QB?H =IGJ;L?> QCNB >CL?=N ;==?MM CHNI NB?
/* 1BCM CM NB? =;M? QCNB H?NQILEM I@ QILEMN;NCIHM IL - M ;H> CM EHIQH ;M ZMB;L?>
G?GILS J;L;FF?FCMG[
1B? ?G?LA?H=? I@ P?=NIL =IGJON?LM ;M Q? B;P? M??H ?;LFC?L F?> NI =B;HA?M CH
MIG? ;FAILCNBGM %IQ?P?L NB? JIMMC<CFCNS I@ P?=NILCT;NCIH CM HIQ >?N?=N?> ;ONIG;N
C=;FFS ;H> NBCM CM NLO? @IL GIL? ;H> GIL? =IGJF?R IJ?L;NCIHM >O? NI NB? JLIAL?MM CH
=IGJON?L N?=BHIFIAS 1B? ;LLCP;F I@ J;L;FF?F =IGJON?LM QCNB =FOMN?LM I@ JLI=?MMILM
CHCNC;FFS <LIOABN NB? M;G? BIJ? %IQ?P?L CH NB? V?F> I@ WOC> G?=B;HC=M MI@NQ;L?
NBCM BIJ? B;M HIN <??H @OFVFF?> *;HS ?RJ?LC?H=?M CH ;ONIG;NC= J;L;FF?FCMG QB?L?
=IGJCF?LM NB?GM?FP?M J?L@ILG NB? N;ME I@ IJNCGCTCHA NB? JLIAL;G MBIQ?> P?LS JIIL
CGJLIP?G?HN CH -2 NCG? #OLNB?LGIL? P?=NILCT;NCIH ;H> J;L;FF?FCMG ;JJ?;L?> NI
<? =IHNL;>C=NILS 3?=NILCT;NCIH L?KOCL?M MCGJF? FIIJM NI J?L@ILG MO< N;MEM QB?L?;M
MB;L?> G?GILS J;L;FF?FCMG B;H>M IP?L =IGJF?R N;MEM NI ?;=B JLI=?MMIL MI ;M NI IJNCG
CT? =IGGOHC=;NCIH NCG? ;H> >;N; NL;HMGCMMCIH NCG? "^=C?HNFS P?=NILCT?> MI@NQ;L?
QIOF> H;NOL;FFS <? JIILFS J;L;FF?FCT?> M ;H ?R;GJF? NB? ;MM?G<FS FIIJM L?MOFN
CHA @LIG ;H "" G;NLCR\P?=NIL JLI>O=N M?? 0?=NCIH =;HHIN <? ;==?F?L;N?> <S
GIL? NB;H ; L;NCI I@ NQI ?P?H C@ Q? IP?LFIIE NB? =IGGOHC=;NCIH NCG? 1BCM CM >O?
NI NB? @;=N NB;N NB? L?MOFNM I@ ?;=B JLI=?MMIL B;P? NI <? ;MM?G<F?> 1B;N F?;>M NI ;
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
248 HYDRODYNAMICS OF FREE SURFACE FLOWS
new cost which limits drastically the overall time to be gained. This is known as a
“granularity” problem, the size of the tasks to be parallelized being too small. It is
unlikely that progress in the algorithm would help in solving this theoretical bottleneck. From that, the idea emerges to break up the problem in another way: not by
isolating “small” tasks, but in a kind of geographical way by decomposing the domain
of computation.
This idea of domain decomposition aims at assigning to each processor one part
of the domain over which it would solve the fluid mechanics problem. The results of
the other processors would help in defining artificial boundary conditions arising from
the partition. The assignments given to each processor are the same, but the data
differ. According to the Flynn classification (see [51]), this corresponds to SMPD
(Single Program Multiple Data) parallelism. The implementation is quite easy in the
case of explicit algorithms: each equation is only a function of the variables related
to the nodes in the immediate vicinity, computed at the earlier time step. Each
processor is then in charge of the equations related to a group of nodes, and of the
data concerning the neighbours of these nodes, resulting from the previous time step.
This is executed merely by a partition of the domain, with an overlap of one element,
and data transmission at the end of each time step.
In the case of our implicit algorithms, and especially with linear systems solved
on the whole domain, the implementation is more complex, though possible. In fact,
we can guarantee that the results of a parallel computation will be the same as a
scalar computation, except for truncation errors because computations will not be
performed in the same order. Parallelism (initially implemented on our algorithms
by Reinhard Hinkelmann at the University of Hanover [101]) can be limited to the
following problems:
• Partition of the domain.
• Communication between processors.
• Implementation of some basic algorithms: vector assembly, dot product, computation of normal vector and so on.
We shall examine these three points hereafter.
8.2
Partition of the domain
In the case of finite element meshes and implicit algorithms, it is better to partition
without overlapping, edge to edge. Though the techniques of partition are beyond the
scope of this book, we shall mention some of them and give the original references.
Then we shall specify the kind of additional data structures necessary to describe the
domain in order to implement parallelization.
The specifications for an efficient domain decomposer would be the following:
• Realizing a partition of meshes of triangles (the sub-domains of three-dimensional meshes of prisms will be made from sub-domains of triangles).
• Partitioning the domain into blocks in order to ensure a balance between the
processors. Because the main algorithms are made of loops over the elements
-/))")&0*
;H> HI>?M NB? J;LNCNCIH QCFF AO;L;HN?? ; <;F;H=? <?NQ??H NB? HOG<?LM I@ ?F?
G?HNM IL MCGCF;LFS <?NQ??H NB? HOG<?L I@ HI>?M QCNBCH ?;=B MO< >IG;CH
*CHCGCTCHA NB? HOG<?L I@ HI>?M MB;L?> <S >C_?L?HN MO< >IG;CHM ;H> @IL QBC=B
=IGGOHC=;NCIHM <?NQ??H JLI=?MMILM QCFF <? H?=?MM;LS
,NB?L =LCN?LC; @IOH> CH NB? FCN?L;NOL? ;CG?> ;N MCGJFC@SCHA NB? L?MIFONCIH I@ ;
FCH?;L MSMN?G IP?L ?;=B MO< >IG;CH ;L? HIN L?F?P;HN NI IOL ;JJLI;=B M?? 89
1B? MCGJF?MN N?=BHCKO? CM EHIQH ;M L?=OLMCP? =IIL>CH;N? <CM?=NCIH 1B? >IG;CH CM
J;LNCNCIH?> CHNI NQI ;L?;M <S ; P?LNC=;F IL BILCTIHN;F FCH? QBIM? JIMCNCIH CM >?N?LG
CH?> CH IL>?L NI <;F;H=? NB? HOG<?L I@ HI>?M IH ?CNB?L MC>? 1B? J;LNCNCIH CM L?J?;N?>
M?P?L;F NCG?M IH ?;=B J;LN L?MOFNCHA CH ; HOG<?L I@ MO< >IG;CHM ?KO;F NI ; JIQ?L I@
"P?HNO;FFS NB? MO< >IG;CHM B;P? M;Q NIINB?> CHN?L@;=?M QBC=B >I HIN GCHCGCT?
NB? HOG<?L I@ HI>?M ;FIHA NB? CHN?L@;=?M &N CM NB?H H?=?MM;LS NI ;JJFS NB? N?=B
HCKO? I@ (?LHCAB;H ;H> )CH 89 QBC=B GIP?M MIG? ?F?G?HNM @LIG IH? MO< >IG;CH
NI ;HINB?L @IL GCHCGCTCHA NB? CHN?L@;=C;F HI>?M QCNBION OJM?NNCHA NB? <;F;H=? CH NB?
HOG<?L I@ ?F?G?HNM <?NQ??H NB? MO< >IG;CHM
FF NB? INB?L N?=BHCKO?M ;L? <;M?> G;CHFS IH AL;JB NB?ILS ;H> OM? CH@ILG;NCIH
L?F;NCP? NI NB? =IHH?=NCPCNS I@ NB? G?MB 0IG? P;LC;HNM ;FMI OM? A?IG?NLC= CH@ILG;NCIH
$L??>S ;FAILCNBGM M?? 89 <OCF> MO< >IG;CHM CHCNC;N?> @LIG ; HI>? ;H> ?RN?H>
CHA IP?L NB? ;L?;M I@ NB? G?MB HIN S?N CH=FO>?> CH ;HINB?L MO< >IG;CH ,NB?L >IG;CH
>?=IGJIM?LM ;LN@OFFS OM? ;FAILCNBGM I@ GCHCGCT;NCIH I@ ; G;NLCR <;H> 1B? L?MOFNCHA
J;LNCNCIH GCHCGCT?M NB? FCHEM <?NQ??H MO< >IG;CHM NB;N CM NB? HOG<?L I@ >CMNCH=N
<IOH>;LC?M I@ ?;=B MO< >IG;CH 2H@ILNOH;N?FS NB? HOG<?L I@ CHN?L@;=C;F HI>?M CM HIN
IJNCGCT?>
,H? I@ NB? GIMN JIJOF;L G?NBI>M CM 0CGIHYM L?=OLMCP? MJ?=NL;F <CM?=NCIH M??
89 1B? >IG;CH CM >CPC>?> CHNI NQI J;LNM ;H> NB? HOG<?L I@ CHN?L@;=C;F M?AG?HNM
CM GCHCGCT?> 1B? G?NBI> FIIEM @IL NB? ?CA?HP;FO?M I@ NB? =IHH?=NCPCNS N;<F? I@ NB?
=ILL?MJIH>CHA AL;JB ;H> JLIPC>?M ; <;F;H=? I@ NB? HOG<?L I@ HI>?M &H IL>?L NI A?N
;H ?KO;F HOG<?L I@ ?F?G?HNM CN CM H?=?MM;LS NI >?;F QCNB ; Z>O;F[ AL;JB <OCFN @LIG
NB? =?HNL;F HI>?M I@ NB? ?F?G?HNM &H NBCM =;M? (?LHCAB;H ;H> )CHYM G?NBI> GOMN <?
;JJFC?> NI GCHCGCT? NB? HOG<?L I@ CHN?L@;=C;F M?AG?HNM
1I>;S Q? B;P? @L?? >?=IGJIM?LM IH NB? MI@NQ;L? G;LE?N QBC=B =;H >CPC>? ;
>IG;CH CHNI ;HS HOG<?L I@ MO< >IG;CHM 4? L?=IGG?H>
*"1&0 89 ;P;CF;<F? IH NB? Q?<MCN?
BNNJQQQ OM?LM =M OGG ?>OUE;LSJCMG?NCM
% , 89
1,-!,*!"
89
1B? Q?<MCN?
BNNJQQQ ?L= GMMN;N? ?>OU<C>?L<;L?M?;L=BINB?L:MCN?M BNGF
JLIPC>?M M?P?L;F ;>>L?MM?M CH NB? V?F> I@ >IG;CH J;LNCNCIHCHA
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
'4@D4BC43 30C0
";=B MO< >IG;CH CM ;MMCAH?> NI ; JLI=?MMIL NIA?NB?L QCNB NB? CH@ILG;NCIH I@ ; MN;H>
;L> >IG;CH =IHH?=NCPCNS N;<F? &()" HI>? =IIL>CH;N?M ;H> MI IH <ON ;FMI QCNB
;>>CNCIH;F CH@ILG;NCIH NI B?FJ CH ;MM?G<FCHA NB? L?MOFNM IP?L NB? QBIF? >IG;CH ;N
NB? ?H> I@ NB? =IGJON;NCIH 1BCM CH@ILG;NCIH CM ;M @IFFIQM
H ;LL;S (+,)$ MNILCHA NB? AFI<;F HOG<?LM I@ NB? HI>?M CH NB? QBIF? G?MB
1B? CHP?LM? ;LL;S (+,$) CM >?VH?> QCNBCH ; FIIJ
!, & +-,&+
(+,$)(+,)$&&
"+!!,
H ;LL;S + % QBIM? >CG?HMCIH CM +-1&/ QB?L? +-1&/ CM NB? HOG<?L
I@ CHN?L@;=C;F HI>?M <?NQ??H NB? MO< >IG;CH ;H> NB? INB?LM 1B?M? CHN?L@;=C;F
HI>?M ;L? HOG<?L?> @LIG NI +-1&/ + % &/ CM NB? AFI<;F HOG<?L I@
NB? CHN?L@;=C;F HI>? &/ 1BCM ;LL;S A;NB?LM NB? CH@ILG;NCIH NI <? NL;HMGCNN?>
NI NB? INB?L JLI=?MMILM 1B? CHN?A?LM + % &/ NI + % &/ ;L? NB?
HOG<?LM I@ NB? INB?L MO< >IG;CHM NI QBC=B NB? HI>?M <?FIHA 1B?S G;S B;P?
; P;FO? C@ NB?M? MO< >IG;CHM ;L? F?MM NB;H @IOL 1B? >CG?HMCIH VP? GOMN <?
CH=L?;M?> C@ ; HI>? <?FIHAM NI GIL? NB;H VP? MO< >IG;CHM
0IG? ;>>CNCIH;F CH@ILG;NCIH B;M NI <? >?VH?> IH NB? ;=NO;F <IOH>;LS HI>?M I@ ;
NQI >CG?HMCIH;F >IG;CH H;G?FS &0"$ 50"$ 60"$ ;H> +2*)&.
&@ &0"$ NB? <IOH>;LS HI>? DOMN ;@N?L <?FIHAM NI ;HINB?L MO< >IG;CH CNM
AFI<;F HOG<?L CM &0"$ ;H> CNM L?;F =IIL>CH;N?M ;L? 50"$ ;H> 60"$
&@ &0"$ NB? <IOH>;LS HI>? DOMN <?@IL? <?FIHAM NI ;HINB?L MO< >IG;CH CNM
AFI<;F HOG<?L CM &0"$ ;H> CNM L?;F =IIL>CH;N?M ;L? 50"$ ;H> 60"$
&@ &0"$ NB? HI>? P?LCV?M NB? NQI =IH>CNCIHM ;<IP? NBCM J;NBIFIAC=;F
=;M? CM HIN =IHMC>?L?> <S NB? MI@NQ;L?
+2*)&. CM ; MJ?=CV= HOG<?LCHA I@ NB? FCKOC> <IOH>;LC?M QBC=B ;FFIQM ;MMI=C;NCIH
I@ <IOH>;LS =IH>CNCIHM NI NB?G ? A ;H CGJIM?> >CM=B;LA? 1B? HOG<?LCHA CM ?;MS
NI >?VH? QB?H NB? QBIF? >IG;CH CM EHIQH JIMMC<F? =IHP?HNCIH CM NI MN;LN NB?
HOG<?LCHA I@ NB?M? FCKOC> <IOH>;LC?M CHWIQ ;H> IONWIQ @LIG NB? ?RNL?G? MIONB
Q?MN HI>? I@ NB? >IG;CH ;H> JLI=??> CH NB? ;HNC=FI=EQCM? >CL?=NCIH ;FIHA NB? ?>A?
&H NB? =;M? I@ ; MO< >IG;CH @IL QBC=B CN CM HIN JIMMC<F? NI AI IP?L NB? QBIF? =IHNIOL
NB? HOG<?LCHA GOMN <? MJ?=CV?> <?=;OM? CN =;HHIN <? MCGJFS L?=IGJON?>
><<D=820C8>= 14CF44= ?A>24BB>AB
-LIAL;GGCHA <S =IGGOHC=;NCP? JLI=?MM L?KOCL?M NB? OM? I@ ; FC<L;LS I@ =IGGOHC=
;NCIH @OH=NCIHM MO=B ;M -3* 89 IL *-& 89 1B?M? FC<L;LC?M J?L@ILG >C_?L?HN
IJ?L;NCIHM
0?NNCHA OJ NB? J;L;FF?F G;=BCH? NB;N CM ILA;HCTCHA NB? =IGGOHC=;NCIH <?NQ??H
P;LCIOM =IGJON?LM IL JLI=?MMILM
-/))")&0*
/OHHCHA JLIAL;GM IH ;FF NB? JLI=?MMILM MJ;QH IL>?L QCNB -3* &H MIG?
=;M?M IH? JLI=?MMIL B;M NB? @OH=NCIH I@ ; G;MN?L NB? INB?LM <?CHA NB? MF;P?M
1B? JLIAL;G CM VLMN LOH IH NB? G;MN?L JLI=?MMIL ;H> CH=FO>?M ; MN;LN IL>?L
@IL NB? MF;P?M ZALIOJ[ H;G? CM ACP?H NI NB? =IGJON;NCIH J;LN @LIG NBCM
CHCNC;FCT;NCIH MN?J NB?L? CM NB?H HI GIL? BC?L;L=BS <?NQ??H NB? JLI=?MMILM
3;LCIOM ECH>M I@ =IGGOHC=;NCIH JICHN NI JICHN IL =IFF?=NCP?
%>8=CC>?>8=C 2><<D=820C8>=
1BCM CM NB? <;MC= =IGGOHC=;NCIH <?NQ??H NQI JLI=?MMILM NB? L?=?JNIL ;H> NB? NL;HM
GCNN?L 1B? ZM?H>[ @OH=NCIH ;H> NB? ZL?=?CP?[ @OH=NCIH =;H >C_?L >?J?H>CHA IH NB?
NSJ?M I@ >;N; CHN?A?LM L?;F HOG<?LM ?N= 1B? @OH=NCIHMY ;LAOG?HNM ;L?
C>?HNCV?L I@ L?=?JNIL IL NL;HMGCNN?L
;>>L?MM I@ NB? >;N; NI NL;HMGCN IL L?=?CP?
; G?MM;A? W;A
1L;HMGCMMCIHM =;H <? G;>? CH NQI Q;SM 0SH=BLIHIOM NL;HMGCMMCIH I==OLM QB?H
Q;CNCHA @IL NB? L?=?JNIL NI <? L?;>S NBCM ;PIC>M ; =IJS CH NB? <O_?L MSH=BLIHIOM
NL;HMGCMMCIH I==OLM QB;N?P?L NB? MN;N? I@ NB? L?=?JNCIH
>;;42C8E4 2><<D=820C8>=
IFF?=NCP? =IGGOHC=;NCIHM OM? MCGOFN;H?IOMFS ;FF NB? JLI=?MMILM 1BL?? NSJ?M G;S
<? >CMNCHAOCMB?>
LI;>=;MN ; JLI=?MMIL A?H?L;FFS NB? G;MN?L M?H>M NB? M;G? >;N; NI ;FF NB?
MF;P?M
0SH=BLIHCT;NCIH L?@?LM NI ; <;LLC?L QBC=B ;FF JLI=?MMILM GOMN L?;=B <?@IL? AICHA
IHNI NB? =IGJON;NCIH
IH>CNCIH;F NL;HMGCMMCIH ?;=B JLI=?MMIL M?H>M >;N; ;H> L?=?CP?M CH L?NOLH NB?
MOG NB? G;RCGOG IL NB? GCHCGOG I@ ;FF NB? >;N; NL;HMGCNN?>
%0A0;;4;8B< ;81A0AH
1I G;E? NB? MIOL=? JLIAL;GM CH>?J?H>?HN I@ NB? =BIC=? I@ NB? =IGGOHC=;NCIH @OH=
NCIHM FC<L;LS CN CM QILNB QLCNCHA ;H CHN?L@;=? FC<L;LS NI >?;F QCNB ;FF NB? =IGGOHC=;
NCIHM <?NQ??H NB? JLI=?MMILM @?Q @OH=NCIHM ;L? H?=?MM;LS
-:!*55 G;RCGOG I@ 5 IP?L ;FF NB? JLI=?MMILM 5 CM ; >IO<F? JL?=CMCIH L?;F
-:!*&+5 GCHCGOG I@ 5 IP?L ;FF NB? JLI=?MMILM 5 CM ; >IO<F? JL?=CMCIH L?;F
-:!02*5 MOG I@ ;FF NB? P;FO?M I@ 5 JLIPC>?> <S ;FF NB? JLI=?MMILM 5 CM ;
>IO<F? JL?=CMCIH L?;F
-:&*5& G;RCGOG I@ & IP?L ;FF NB? JLI=?MMILM & CM ;H CHN?A?L
-:&*&+& GCHCGOG I@ & IP?L ;FF NB? JLI=?MMILM & CM ;H CHN?A?L
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
-:&02*& MOG I@ ;FF NB? P;FO?M I@ & JLIPC>?> <S ;FF NB? JLI=?MMILM & CM ;H
CHN?A?L
-:*&) %&+" + / <LI;>=;MNCHA ; =B;L;=N?L MNLCHA %&+" QCNB ;
F?HANB + / NI ;FF NB? JLI=?MMILM
-:/"!2##"/ +61"0 0,2/ " 16-" L?;>CHA I@ +61"0 <SN?M I@
>;N; QBIM? NSJ? CM 16-" NI <? MNIL?> ;N NB? ;>>L?MM 2##"/ ;H> M?HN <S NB?
JLI=?MMIL 0,2/ "
-:06+ MNIJM NB? JLI=?MMILM NCFF ;FF I@ NB?G =;FF NBCM @OH=NCIH
-:4/&12##"/ +61"0 !"01 16-" QLCNCHA I@ +61"0 <SN?M I@ >;N;
QBIM? NSJ? CM 16-" FI=;N?> ;N NB? ;>>L?MM 2##"/ ;H> CHN?H>?> @IL NB? JLI=?MMIL
!"01
30?C0C8>= >5 C74 0;6>A8C7<B
1B? >;N; MNLO=NOL? MJ?=CV= NI J;L;FF?FCMG QCFF <? >?M=LC<?> H?RN ;M Q?FF ;M NB? GI>C
V=;NCIHM G;>? NI NB? G;CH ;FAILCNBGM
0C0 BCAD2CDA4 B?428J2 C> ?0A0;;4;8B<
#LIG NB? <;MC= CH@ILG;NCIH JLIPC>?> <S NB? >IG;CH >?=IGJIM?L @IL ?P?LS MO< >IG;CH
;M G?HNCIH?> CH 0?=NCIH NB? @IFFIQCHA >;N; GOMN <? >?VH?>
H ;LL;S &+!-2 CHP?LM? I@ + % QBIM? N;ME CM NI B?FJ MNIL? NB? >;N;
L?=?CP?> @LIG NB? INB?L JLI=?MMILM &+!-2+ % && $CP?H NB? HOG<?L
I@ ;H CHN?L@;=C;F HI>? &+!-2 M?H>M <;=E CNM AFI<;F HOG<?L QCNBCH NB? MO<
>IG;CH
H ;LL;S # JLIPC>CHA @IL ?;=B HI>? NB? CHP?LM? I@ NB? HOG<?L I@ MO< >IG;CHM
NI QBC=B CN <?FIHAM M?? NB? ;FAILCNBG @IL >IN JLI>O=N <?FIQ
(20;0A ?A>3D2C
1B? >IN JLI>O=N I@ NQI P?=NILM IP?L NB? QBIF? >IG;CH CGJFC?M FI=;F =IGJONCHA VLMN
NB?H MOGGCHA OJ IP?L ;FF NB? MO< >IG;CHM 4BCF? MOGGCHA NB? CHN?L@;=? HI>?M ;L?
N;E?H CHNI ;==IOHN M?P?L;F NCG?M 1B? ;LL;S # CM NB?H OM?> NI =ILL?=N NBCM ?LLIL
1B? J;L;FF?F >IN JLI>O=N -:!,1 I@ NB? P?=NILM 5 ;H> 6 CM QLCNN?H ;M @IFFIQM
-:!,1 !
!, & +-,&+
-:!,1-:!,1 5& 6& # &
"+!!,
-:!,1-:!02*-:!,1
,H ; P?=NIL =IGJON?L GOFNCJFSCHA <S # & >I?M HIN ;_?=N NB? NCG? I@ =IGJO
N;NCIH ,H NB? =IHNL;LS NB? =;FF NI NB? -:02* LIONCH? =;OM?M ; MSH=BLIHCT;NCIH I@
NB? JLI=?MMILM
-/))")&0*
"0CA8GOE42C>A ?A>3D2C
#CLMN <OCF>CHA NB? G;NLC=?M ;H> =IGJONCHA NB? JLI>O=N ;L? >IH? FI=;FFS CH>?J?H>
?HNFS I@ NB? INB?L MO< >IG;CHM 1B? L?MOFN CM ; P?=NIL QCNBION ;HS =IHNLC<ONCIH @LIG
NB? H?CAB<IOLCHA MO< >IG;CHM IH NB? CHN?L@;=C;F HI>?M 1B? GCMMCHA =IHNLC<ONCIHM
L?MOFN @LIG NB? =IGJON;NCIH I@ NB? P?=NIL IH NB? INB?L MO< >IG;CHM #IL ?R;GJF? C@
NB? HI>? & <?FIHAM NI NQI >C_?L?HN MO< >IG;CHM @IL QBC=B Q? A?N 5& ;H> 5&
L?MJ?=NCP?FS NB? ;=NO;F P;FO? I@ 5& QCFF <? NB? MOG 1BCM =;H <? ;=BC?P?> C@
JLI=?MMIL M?H>M NB? P;FO? NI JLI=?MMIL ;H> JLI=?MMIL M?H>M NB? P;FO? NI
JLI=?MMIL FNBIOAB CN ;JJ?;LM MCGJF? NBCM IJ?L;NCIH CM KOCN? =IGJF?R NI ;=BC?P?
<?=;OM? CN =;H F?;> NI ; @;N;F LCME <INB JLI=?MMILM Q;CNCHA ?H>F?MMFS @IL NB? =IHNLC
<ONCIH I@ NB? INB?L <?@IL? M?H>CHA NB?CL IQH 1B? NL;HMGCMMCIH CM NBOM MJFCN CHNI @IOL
>C_?L?HN N;MEM >?J?H>CHA IH NB? L;HE I@ NB? JLI=?MMILM
NL;HMGCMMCIH I@ NB? >;N; NI NB? BCAB?L L;HE JLI=?MMILM
L?=?JNCIH <S NB? BCAB?L L;HE JLI=?MMILM
NL;HMGCMMCIH I@ NB? >;N; NI NB? FIQ?L L;HE JLI=?MMILM
L?=?JNCIH <S NB? FIQ?L L;HE JLI=?MMILM
+?Q >;N; ;L? L?KOCL?> @IL ?P?LS JLI=?MMIL ";=B JLI=?MMIL JL?J;L?M CNM IQH HOG
<?LCHA I@ BCAB?L L;HE ;H> FIQ?L L;HE JLI=?MMILM ;M Q?FF ;M ; HOG<?LCHA I@ HI>?M
CHN?L@;=CHA QCNB ?;=B I@ NB?M? JLI=?MMILM #IOL H?Q ;LL;SM &(- +%- &(* ;H> +%*
;L? H?=?MM;LS NI H;PCA;N? @LIG IH? HOG<?LCHA NI ;HINB?L
&(- &(-&7% L?@?LM NI NB? JLI=?MMIL QBC=B CM NB? %6$NB FI=;F BCAB?L
L;HE JLI=?MMIL &(-&7% CM NB? HOG<?L I@ CHN?L@;=C;F HI>?M MB;L?> QCNB NBCM
JLI=?MMIL
&(* &(*&7% L?@?LM NI NB? JLI=?MMIL QBC=B CM NB? %6$NB FI=;F FIQ?L
L;HE JLI=?MMIL &(*&7% CM NB? HOG<?L I@ CHN?L@;=C;F HI>?M MB;L?> QCNB NBCM
JLI=?MMIL
+%- +-1&/ +%-&7% &/ CM NB? AFI<;F HOG<?L CH NB? MO< >IG;CH I@ ;
JICHN QBIM? HOG<?L CM &/ CH NB? CHN?L@;=? QCNB NB? %6$NB BCAB?L L;HE JLI=?MMIL
+%* +-1&/ +%*&7% &/ CM NB? AFI<;F HOG<?L CH NB? MO< >IG;CH I@ ;
JICHN QBIM? HOG<?L CM &/ CH NB? CHN?L@;=? QCNB NB? %6$NB FIQ?L L;HE JLI=?MMIL
&)*5 CM NB? G;RCGOG >CMN;H=? CH IL>?L <?NQ??H NB? JLI=?MMIL ;H> CNM H?CAB
<IOLM &N QCFF L?MNLC=N NB? MCT? I@ NB? FIIJM IP?L NB?M? H?CAB<IOLM
IHMC>?LCHA NBCM CH@ILG;NCIH NB? >;N; NL;HMGCMMCIH @IL P?=NIL 3 >?VH?> IP?L NB?
QBIF? >IG;CH ;H> @IL JLI=?MMIL &-&! CM QLCNN?H ;M @IFFIQM
BC>A064 >5 C74 30C0 C> 14 B4=C 8= 0 JABC 1DQ4A
!, & +-1&/
2#&3+ % &
"+!!,
CA0=B<8BB8>= C> 78674A A0=: ?A>24BB>AB 46 ?A>24BB>A !%
MNIL;A? I@ NB? >;N; L?F;NCP? NI NB? JLI=?MMIL CH ; M?=IH> <O_?L
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
!, & &(-&)- "/$2#&2#&+!-2+%-&)- &
"+!!,
M?H>CHA
)) -:4/&1"/$2# &(-&)- &(-&)- 0&(-&)- &-&!
A424?C8>= 5A>< ;>F4A A0=: ?A>24BB>AB 46 ?A>24BB>A !"
L?=?JNCIH
)) -:/"!"/$2# &(*&)* &(*&)* 0&(*&)* &-&!
MNIL;A? CH P?=NIL 3
!, & &(*&)* 3+%*&)* &3+%*&)* & "/$2#&
"+!!,
CA0=B<8BB8>= C> ;>F4A A0=: ?A>24BB>AB 46 ?A>24BB>A !"
MNIL;A? I@ NB? >;N; L?F;NCP? NI NB? JLI=?MMIL CH ; M?=IH> <O_?L
!, & &(*&)* "/$2#&2#&+!-2+%*&)* &
"+!!,
M?H>CHA
)) -:4/&1"/$2# &(*&)* &(*&)* 0&(*&)* &-&!
A424?C8>= 5A>< D??4A A0=: ?A>24BB>AB 46 ?A>24BB>A !%
L?=?JNCIH
)) -:/"!"/$2# &(-&)- &(-&)- 0&(-&)- &-&!
MOGGCHA CHNI P?=NIL 3
!, & &(-&)- 3+%-&)- &3+%-&)- & "/$2#&
"+!!,
1BCM ;FAILCNBG =;H <? G;>? GIL? =IGJF?R NI JLI=?MM M?P?L;F P?=NILM ;N NB? M;G?
NCG? IL @IL N;MEM >C_?L?HN @LIG MOGGCHA
(?428J2 20B4B
&N CM QILNB HINCHA NB;N DOMN ; @?Q ;FN?L;NCIHM I@ NB? M=;F;L ;H> G;NLCR\P?=NIL JLI>O=NM
;L? MO^=C?HN NI MIFP? ; FCH?;L ?KO;NCIH QCNB J;LNC;F >?LCP;NCP?M CH MB;L?> G?GILS
#OLNB?LGIL? NB? =IHNLC<ONCIHM I@ NB? CHN?L@;=C;F N?LGM =;H <? JIMNJIH?> OHNCF NB?
VH;F MIFPCHA 1B?S QIOF> <? H?=?MM;LS <?@IL? IHFS C@ L?;F HI>;F P;FO?M >IN JLI>O=N
G;NLCR\P?=NIL JLI>O=N IL VH;F L?MOFNM ;L? H??>?> #IL ; FCH?;L ?KO;NCIH NB?M? IJ?L;
NCIHM =;H <? =IHVH?> CH NB? L?MIFONCIH I@ NB? FCH?;L MSMN?G ,H=? NB? NIIFM B;P? <??H
M?N OJ NI =IGGOHC=;N? <?NQ??H JLI=?MMILM J;L;FF?FCT;NCIH I@ WOC> G?=B;HC=M MI@NQ;L?
CM KOCN? ?;MS %IQ?P?L MJ?=CV= =;M?M ;LCMCHA G;CHFS @LIG HIH FCH?;LCNS B;P? NI <?
=IHMC>?L?> 1B? <;MC= C>?; CM NI A?N C>?HNC=;F L?MOFNM IH P?=NIL IL J;L;FF?F =IGJON?LM
?R=?JN @IL NLOH=;NCIH ?LLILM 0IG? ?R;GJF?M @IFFIQ
!C;AIH;F JL?=IH>CNCIHCHA NB? >C;AIH;F OM?> @IL JL?=IH>CNCIHCHA B;M NI <? =IG
JF?N?> ;N NB? CHN?L@;=?M
-/))")&0*
IGJON;NCIH I@ HI>;F P;FO?M ;@N?L JLID?=NCIH IHNI ; <;MCM F?N OM =IHMC>?L NB?
AL;>C?HN I@ ; @OH=NCIH < G?;H HI>;F P;FO? I@ =H7: < CM ACP?H <S
=H7: <
7
+ =H7: < 7 :
+
:
7
CMMO?>+@LIG NB? G?;H P;FO? NB?IL?G &H NBCM =;M? NB? =IGJON;NCIH I@ NB? A?H?L;F
N?LG 7 : B;M NI <? =IGJF?N?> ;N NB? CHN?L@;=?M 1B? P?=NIL CH=FO>CHA
.
=H7: < 7 :
CM OH=B;HA?> QB?H ;>>?> NI NB? LCABN B;H> MC>? I@ ; FCH?;L MSMN?G NB? MIFP?L
CNM?F@ QCFF =IGJF?N? NBCM N?LG %IQ?P?L CN GOMN <? =IGJF?N?> QB?H OM?> NI
=IGJON? ;HINB?L N?LG MO=B ;M ; NOL<OF?HN JLI>O=NCIH
HS AFI<;F =IH=?JN G;RCGOG IOL;HN HOG<?L G;RCGOG #LIO>? HOG<?L
?N= B;P? NI <? NL;HMGCNN?> <?NQ??H JLI=?MMILM
1B? G?NBI> I@ =B;L;=N?LCMNC=M @IL BCAB P;FO?M I@ NB? IOL;HN HOG<?L ; =B;L
;=N?LCMNC= =OLP? G;S F?;P? ; MO< >IG;CH ;H> ?HN?L INB?LM -LIAL;GGCHA NBCM
MCNO;NCIH QCNBCH NB? J;L;FF?F ;L=BCN?=NOL? QIOF> L?KOCL? ; BOA? HOG<?L I@ L;H
>IG NL;HMGCMMCIHM >?J?H>CHA IH NB? WIQ &@ NB? IOL;HN HOG<?L CM E?JN OH>?L
NB? NL;HMGCMMCIHM QIOF> =IH=?LH IHFS NB? CHN?L@;=C;F HI>?M 1B? JLIAL;G QCFF
NB?H =IHMC>?L NB? L?MOFN I@ NB? >IG;CH L?;=B?> <S NB? =B;L;=N?LCMNC= =OLP? T?LI L?MOFN =;H ;FMI <? ;MMCAH?> NI NB? JICHNM QBIM? =B;L;=N?LCMNC= =OLP? F?;P?M
NB? MO< >IG;CH &H NBCM =;M? IHFS NB? L?MOFN QCNB BCAB?L ;<MIFON? P;FO? CM
=IHMC>?L?> QBC=B CM ; JLI=?MM P?LS MCGCF;L NI ; =IGJF?G?HN ;N NB? CHN?L@;=?M
%IQ?P?L FCGCNCHA NB? NCG?\MN?J CM HIN @OFFS M;NCM@;=NILS @IL ;H OH=IH>CNCIH;FFS
MN;<F? G?NBI>
'4BD;CB
0IG? ?R;GJF?M I@ J;L;FF?F =IGJON;NCIHM OMCHA NB? JL?PCIOM N?=BHCKO?M ;L? ACP?H <?FIQ
&H ;FF =;M?M NB? J;LNCNCIHCHA I@ NB? >IG;CH B;M <??H ;=BC?P?> QCNB 0CGIHYM L?=OLMCP?
MJ?=NL;F <CM?=NCIH
;DBC4A >5 F>A:BC0C8>=B %
IGGOHC=;NCIH FC<L;LS -3*
4;P? JLIJ;A;NCIH CH ; MKO;L? N;HE OMCHA 1?F?G;= ! MI@NQ;L? M?? 1;<F? 1;<F? J;L;FF?FCMG IH ; =FOMN?L I@ MN;NCIHM %-
-2 NCG?
0J??> OJ @;=NIL
JLI=?MMIL
M
JLI=?MMILM
M
JLI=?MMILM
M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
'. ) IGGOHC=;NCIH FC<L;LS -3*
NCG? MN?JM I@ NB? *;FJ;MM?N >;G <L?;E Q;P? JLIJ;A;NCIH QCNB 1?F?G;= !
MI@NQ;L? M?? 1;<F? 1;<F? J;L;FF?FCMG IH
/6 1"
JLI=
M
JLI=
M
-2 NCG?
0J??> OJ @;=NIL
JLI=
M
JLI=
M
1BCM ;H> NB? JL?PCIOM =;M? IONFCH? NB? CHN?L?MN I@ ; >?>C=;N?> =IGJON?L CH>?J?H>
?HN I@ NB? ?RN?LCIL H?NQILE ;H> CNM FIQ NL;HMGCMMCIH MJ??>
$A868=
18C '- <
*;=BCH? =IHMCMNM I@ JLI=?MMILM I@ *%T
IGGOHC=;NCIH FC<L;LS *-&
0NO>S I@ ; L?;=B I@ NB? /CP?L /BCH? <?NQ??H EG ;H> EG OMCHA 1?F?G;=
! MI@NQ;L? *?MB G;>? I@ ?F?G?HNM =IGJON;NCIH I@ NCG? MN?JM I@ M L?@?L NI 1;<F? QBIM? >;N; B;P? <??H AL;=CIOMFS JLIPC>?> <S ';=?E ';HEIQMEC
@LIG OH>?M;HMN;FN @jL 4;MM?L<;O CH (;LFMLOB?
1;<F? J;L;FF?FCMG IH ,LCACH -2 NCG?
0J??> OJ @;=NIL
JLI=
M
JLI=
M
JLI=
M
JLI=
M
JLI=
M
JLI=
M
1BCM ;G;TCHA L?MOFN NB? MJ??> OJ @;=NIL CM BCAB?L NB;H NB? HOG<?L I@ JLI=?MMILM
H??>M =IGG?HNCHA IH &H NBCM =;M? ; BCAB HOG<?L I@ ?F?G?HNM CH NB? QBIF? G?MB QIOF>
CHBC<CN NB? ?^=C?H=S I@ IH? JLI=?MMIL <?=;OM? I@ NB? G?GILS MCT? I@ NB? <O_?L ;MMCAH?>
NI >;N; *<SN?M 0I ?^=C?H=S CH=L?;M?M QB?H NB? MO< >IG;CH MCT? >?=L?;M?M
,NB?L L?MOFNM ;L? ACP?H CH NB? =B;JN?L L?F;NCP? NI ;JJFC=;NCIHM 0?=NCIH 70?C4A %0A0<4C4A 4BC8<0C8>=
=CA>3D2C8>=
-;L;G?N?L ?MNCG;NCIH ;FMI =;FF?> NB? CHP?LM? JLI<F?G =IHMCMNM I@ L?NLC?PCHA >;N; I@
; JLI<F?G @LIG CNM MIFONCIH #IL @L?? MOL@;=? WIQ BS>L;OFC=M CN CHPIFP?M ;MM?MMCHA
BC>>?H IL >C^=OFN NI ;==?MM J;L;G?N?LM MO=B ;M @LC=NCIH =I?^=C?HNM IH NB? <INNIG
>?J?H>CHA IH NB? I<M?LP;NCIHM ;H> G?;MOL?G?HNM M?LC?M I@ BCMNILC=;F WII>M NC>;F
L;HA? CH JILNM ?N= 1B? QILEM IH J;L;G?N?L ?MNCG;NCIH <?A;H CH NB? 1?F?G;=
MSMN?G CH QCNB ;LLIMYM NB?MCM 89 1B? ;>DICHN G?NBI> >?P?FIJ?> <S )CIHM
;H> B;P?HN M?? 89 ;H> 89 B;M <??H =BIM?H &N CM <;M?> IH NB? =IHMNLO=NCIH
I@ ; );AL;HAC;H NI QBC=B IJNCG;FCNS =IH>CNCIHM ;L? ;JJFC?> C ? T?LI >?LCP;NCP?M QCNB
L?MJ?=N NI J;L;G?N?LM MN;N? P;LC;<F?M ;H> ;>DICHN P;LC;<F?M 4? NLS NI GCHCGCT? ; =IMN
@OH=NCIH & >C_?L?H=?M <?NQ??H G?;MOL?G?HNM ;H> NB? L?MOFN I@ ; HOG?LC=;F MCGOF;NCIH
I@ ;H ?P?HN <S ; AL;>C?HN G?NBI> QBC=B =IHM?KO?HNFS L?KOCL?M NB;N Q? EHIQ BIQ NI
=;F=OF;N? NB? AL;>C?HN I@ NB? @OH=NCIH & QCNB L?MJ?=N NI ;FF NB? J;L;G?N?LM NB;N ;L?
FIIE?> @IL 1B? ;>DICHN MSMN?G ;H> ;>DICHN P;LC;<F?M ;L? OM?> NI A?N NBCM AL;>C?HN
,H? I@ NB? CGJILN;HN =BIC=?M I@ IOL JLI=?>OL? CM NI QILE L;NB?L IH NB? >CM=L?NCT?>
?KO;NCIHM NB;H IH NB? ILCACH;F ?KO;NCIHM QLCNN?H CH NB? =IHNCHOOG QBC=B QCFF JLIPC>?
; P?LS JL?=CM? P;FC>;NCIH #CLMN I@ ;FF Q? MB;FF ?RJF;CH NB? JLCH=CJF? I@ NB? G?NBI>
;JJFC?> NI ; FCH?;L MSMN?G IL ; M?LC?M I@ MSMN?GM CH NB? =;M? I@ OHMN?;>S WIQM NB?H Q?
MB;FF ;JJFS CN NI NB? FCH?;L MSMN?GM I<N;CH?> @LIG NB? 0;CHN 3?H;HN ?KO;NCIHM /?=?HN
N?MNM ;FMI MBIQ?> NB? ?^=C?H=S I@ NIN;FFS >C_?L?HN N?=BHCKO?M <;M?> IH MOLLIA;N?
GI>?FM H?OL;F H?NQILEM ;H> A?H?NC= ;FAILCNBGM ;H ?R;GJF? QCFF <? ACP?H CH NB? H?RN
=B;JN?L CH 0?=NCIH %A8=28?;4 >5 C74 039>8=C <4C7>3
4? >?;F B?L? QCNB NB? JLCGCNCP? MB;FFIQ Q;N?L ?KO;NCIHM MIFP?> ;M ?RJF;CH?> CH
B;JN?L @LIG 0?=NCIH IH ,OL Z>CL?=N[ JLI<F?G CM ; M?LC?M I@ FCH?;L MSMN?GM
I@ NB? @ILG 7B 4 7B 7B QB?L? NB? CHN?A?L ?J P;LC?M <?NQ??H ;H> D?J NB? NIN;F
HOG<?L I@ CN?L;NCIHM IL NCG? MN?JM CM ; G;NLCR 4 CM NB? P?=NIL I@ OHEHIQHM NB?
>?JNB ;H> NB? NQI =IGJIH?HNM I@ NB? P?FI=CNS ;H> NB? P?=NIL I@ LCABN B;H> MC>?M
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
4? MB;FF >?HIN? 4 ;H> ;M NB? ?F?G?HNM I@ NB? P?=NILM 4 ;H> ?F L;HACHA @LIG
NI DF NB? HOG<?L I@ OHEHIQHM QBC=B CM ;FMI NB? L;HE I@ ,H ; G?MB I@ NLC;HAF?M
I@ D HI>?M QCNB ; FCH?;L >CM=L?NCT;NCIH Q? QIOF> NBOM B;P? D OHEHIQH P;LC;<F?M CH
IOL =;M? ;N ?;=B NCG? MN?J
4? B;P? M??H NB;N IOL FCH?;L MSMN?GM >?J?H> IH NB? L?MOFN I@ NB? JL?PCIOM CN?L;NCIH
7B ;H> 7B ;L? NB?L?@IL? $&#& @OH=NCIHM I@ 4 7B 7B ;H> 7B ;FMI >?J?H> IH
J;L;G?N?LM Q;N?L ?F?P;NCIH MIOL=? N?LGM <IOH>;LS =IH>CNCIHM @LC=NCIH =I?^=C?HNM
?N= ;H> Q? MB;FF =IHMC>?L IH? ;GIHA NB?G G;LE?> / ;M ; P?=NIL I@ L;HE DI DI
=;H <? NB? HOG<?L I@ HI>?M I@ NB? G?MB ;M Q?FF ;M NB? HOG<?L I@ <IOH>;LS JICHNM
/ QCFF <? @IL ?R;GJF? NB? P;FO? I@ NB? @LC=NCIH =I?^=C?HN ;N ?;=B >?AL?? I@ @L??>IG
+IN? QB?H OHEHIQH P;LC;<F?M ;L? ACP?H IH ; <IOH>;LS ? A CH NB? =;M? I@ ;
<IOH>;LS =IH>CNCIH I@ !CLC=BF?N NSJ? 7B ;H> 7B ;L? ;FMI @OH=NCIHM I@ =?LN;CH P;FO?M
I@ P?=NIL 4 7B 1BCM @;=N QCFF <? =IHMC>?L?> F;N?L 0?=NCIH ;H> QIOF> IHFS G;E?
NB? JL?M?HN;NCIH N?>CIOM
&H MBILN ;H> @IL HIQ 7B ;H> 7B ;L? @OH=NCIHM I@ / ;H> 4 7B
*IL?IP?L ; P?=NIL I@ G?;MOL?G?HNM G;LE?> ) 7B CM EHIQH %IQ?P?L NB? G?;M
OL?G?HNM I@ ;FF NB? P;LC;<F?M ;L? =?LN;CHFS HIN ;P;CF;<F? CH ;FF NB? JICHNM I@ NB? G?MB
@IL ;FF NB? NCG? MN?JM I@ =;F=OF;NCIH 4B?H G?;MOL?G?HNM ;L? GCMMCHA NBCM QCFF <?
N;E?H CHNI ;==IOHN QCNB NB? B?FJ I@ ; ZQ?CABN[ P?=NIL G;LE?> 3 T?LI Q?CABN QCFF
<? ;MMCAH?> NI GCMMCHA G?;MOL?G?HNM
1B?L?@IL? Q? NLS NI VH> ; P?=NIL / NB;N GCHCGCT?M NB? =IMN @OH=NCIH & >?VH?>
<S
&
<>
<7B ,
,
7B
7>
7B
7B
37> 47>
)7>
NB;N CM NB? MOG IP?L NCG? ;H> MJ;=? I@ NB? MKO;L?M I@ >C_?L?H=?M <?NQ??H NB? L?MOFNM
I@ NB? =;F=OF;NCIH ;H> NB? G?;MOL?G?HNM
VLMN MNL;CABN@ILQ;L> C>?; @IL NB? Q?CABNM 37> QIOF> <? NI N;E? NB?G ?KO;F NI C@ IH? G?;MOL?G?HN CM EHIQH ;H> NI INB?LQCM? %IQ?P?L NB? P;FO?M I@ 37> =;H <?
OM?> @IL ACPCHA GIL? Q?CABN NI =?LN;CH JICHNM ? A >?J?H>CHA IH NB? ?RN?HN I@ NB?CL
<;MCM @OH=NCIH IL GIL? JLIM;C=;FFS ;==IL>CHA NI NB? L?FC;<CFCNS I@ NB? G?;MOL?G?HN
1B? A?H?L;F JLCH=CJF? I@ NB? M?;L=B CM ; AL;>C?HN G?NBI> NB;N QCFF =IHMCMN I@ ; M?LC?M
I@ CN?L;NCIHM I@ NB? @ILG
;
/7A
;
/7A
& ;
;
/7A
MN;LNCHA @LIG ;H CHCNC;F P;FO? / ; M?N I@ J;L;G?N?LM ?P;FO;N?> $&#& ?I P;LC?M @LIG
NI DI ;H> C CM NB? =OLL?HN CN?L;NCIH I@ NB? M?;L=B JLI=?MM 4? >I HIN MJ?=C@S B?L?
BIQ NB? =I?^=C?HN CM =BIM?H NBCM CM ;HINB?L MO<D?=N ;H> M?P?L;F AL;>C?HN G?NBI>M
;L? JIMMC<F? 1B? M?;L=B CM MNIJJ?> QB?H ; FI=;F GCHCGOG I@ & CM @IOH> @IL ; M?N I@
J;L;G?N?LM /
1BCM JLI=?>OL? L?KOCL?M EHIQF?>A? I@ AL;>C?HNM '
QBC=B ;L? DI CH HOG<?L
1BCM =IOF> <? >IH? CH ; P?LS =FOGMS Q;S <S =;LLSCHA ION M?P?L;F =;F=OF;NCIHM QB?L?
Q? QIOF> OM? ?;=B NCG? NB? >?VHCNCIH I@ NB? >?LCP;NCP? <S MFCABNFS P;LSCHA IH? I@ NB?
J;L;G?N?LM 4? MB;FF OM? NBCM C>?; @IL =;FC<L;NCIH JOLJIM?M <ON CN CM NB? ;>DICHN MSMN?G
NB;N CM AICHA NI SC?F> NB? H?=?MM;LS CH@ILG;NCIH GIL? MCGJFS IH NB? & AL;>C?HNM
-/*"1"/ "01&*1&,+ #LIG NB? >?VHCNCIH I@ & Q? VH> NB;N
<>
<7B ,
7B
,
47>
& ;
7B
7B
3
4
)
7>
7>
7>
;
;
/7A
/7A
7B 7>
QBC=B =;H ;FMI <? QLCNN?H CH NB? @ILG I@ >IN JLI>O=NM
<7B
,
& ;
4 7B
7B
7B
3
4
)
;
;
/7A
/7A
7B
)<>
QB?L? 4 5 L?JL?M?HNM 7> 47> 57> 4? MB;FF ;FMI OM? NB? @;=N NB;N
4 5 4B 5
7B
QB?L?
&@ NB? =BIC=? I@ NB? =IMN @OH=NCIH CM GIL? =IGJF?R I@ NB? @ILG & < 47>
& CM ; @OH=NCIH I@ ;FF NB? =IGJIH?HNM I@ NB? P?=NIL 4 @IL ;FF NB? NCG? MN?JM Q? MB;FF
B;P? CH ; GIL? A?H?L;F Q;S
<7B
,
& ;
;
/7A
7B
< 4 7B
;
4 7B /7A
4
1BCM QCFF <? NB?
1B? N?LG 3 4 7B ) 7B CM NB?L?@IL? MCGJFS L?JF;=?> <S ,
=;M? QB?H Q? NLS NI VH> NC>;F L;HA?M G?;MOL?> CH JILNM QBC=B ;L? NB? L?MOFN I@ #IOLC?L
;H;FSMCM I@ P;LC;NCIHM I@ NB? @L?? MOL@;=?
,
&H ;HS =;M? NB? G;CH >C^=OFNS L?G;CHM NB? HOG<?L I@ AL;>C?HNM I@ NB? @ILG ' NB;N B;P? NI <? =;F=OF;N?> MCH=? NB?S ;L? NB? HOG<?L I@ DI DF D?J 1B;HEM NI NB?
N?=BHCKO? JLIJIM?> <S B;P?HN 89 NBCM QCFF HIN <? H?=?MM;LS ;H> CN QCFF MO^=? @IL
OM NI =;F=OF;N? ; FCGCN?> HOG<?L I@ FCH?;L =IG<CH;NCIHM I@ NB?M? AL;>C?HNM 4? MB;FF
HIQ ?RJF;CH NB? A?H?L;F JLCH=CJF? CH );AL;HAC;H N?LGM <?@IL? JL?M?HNCHA =;F=OF;NCIH
>?N;CFM
1B? );AL;HAC;H CM @ILG?> ;M
( 4 / , & 4 /
, 4 QB?L? , CM NB? P?=NIL I@ );AL;HA? GOFNCJFC?LM ;L<CNL;LCFS CHNLI>O=?> P;LC;<F?M NB;N
QCFF <? MJ?=CV?> ;@N?LQ;L>M %?L? CN CM =IHMC>?L?> NB;N 4 / ;H> , ;L? CH>?J?H>?HN
P;LC;<F?M ;H> >?J?H> IH / ;H> 4 4? NLS NI GCHCGCT? & 4 / OH>?L =IHMNL;CHN
NB? =IHMNL;CHN <?CHA NB;N Q? ;FQ;SM GOMN B;P? 4 1B? GCHCGOG FIIE?> @IL
=ILL?MJIH>M NI NB? NLCJF?N 4 / , QBC=B M;NCMV?M NB? IJNCG;FCNS =IH>CNCIHM
QBC=B ACP?M 4 ( 4 / ,
,
C ? NB? >CL?=N MSMN?G
( 4 / ,
4
QBC=B ACP?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&
4
,
4 4
IL C@ >I?M HIN >?J?H> IH 4
&
4
, 4
4
4
QBC=B CM NB? Z;>DICHN[ MSMN?G
( 4 / ,
/
QBC=B ACP?M
&
4 ,
/
/
QBC=B CM NB? ?KO;NCIH I@ NB? & AL;>C?HN ACP?H <S );AL;HA? GOFNCJFC?LM IL Z;>DICHN
P;LC;<F?M[
??;820C8>= C> 0= D=BC403H ?A>1;4<
4? HIQ =IHMC>?L NB;N IOL JLI<F?G CM ; MO==?MMCIH I@ FCH?;L MSMN?GM NB? G;NLC=?M ;H>
LCABN B;H> MC>?M I@ QBC=B >?J?H> IH NB? L?MOFN I@ JL?PCIOMFS L?MIFP?> MSMN?GM
S MN;LNCHA @LIG NB? FCH?;L MSMN?G I@ CN?L;NCIH ?J 7B 4 7B 7B Q? A?N
:
7B 7B
7B
; 4 :/7A
QBC=B ACP?M DI >C_?L?HN ?KO;NCIHM ";=B I@ NB?M? ?KO;NCIHM CM NL;HM@ILG?> CHNI ;H
?KO;FCNS <?NQ??H ; MOG I@ NQI G;NLCR\P?=NIL JLI>O=NM ;H> ; LCABN B;H> MC>?
:7B 7B
; 4
:/7A
M >?J?H>M IH 4 7B
7B
:4 7B
: 7B
;
;
:/7A
:/7A
Q? NB?H QLCN? NB;N
:7B
7B
;
;
:/7A
/7A
<>
,
7>
7B
7B 47>
;
7B
/7A
47>
1BCM CM NB? M;G? @IL 7B
: 7B
;
;
:/7A
/7A
;H> 2,
2' I@ 4 7B , ' <>
7B
,
7B 47>
;
7B
/7A
47>
7>
HI @ILG;F >?J?H>?H=? <?NQ??H NB? =IGJIH?HNM I@ 4 7B ;H> NBIM?
1B?M? ?KO;FCNC?M B;P? ; G?;HCHA ?P?H C@ ?J <ON CH NBCM =;M?
NB? CHCNC;F =IH>CNCIHM >I HIN >?J?H> IH NB? =BIC=? I@ /
, ' <?=;OM?
-/*"1"/ "01&*1&,+ "KO;NCIH CM VH;FFS QLCNN?H ;M
4 7B
7B ;
/7A
%
<>
,
7>
<>
7B
7B
7B 47> & 7B % , 7B 47> &
7B 7B 7B
4
;
;
; / ; 4
7B
7B
/7A
/7A
/7A
47>
47>
7A
7>
+IQ Q? ;L? AICHA NI =IHMNLO=N NB? ;>DICHN P;LC;<F?M G;LE?> , I@ NB? M;G? L;HE
;M 4 IOL I<D?=NCP? <?CHA NB;N NB? AL;>C?HN I@ & =;H VH;FFS <? QLCNN?H ;M
<>
<7B ,
,
& ;
7B
,7>
;
/7A
7B 7>
7B
7B 7B
;
;4
/7A
/7A
7>
1BCM ?KO;NCIH QCFF <? NLO? IHFS C@ NB? ;>DICHN P?=NIL , @OFVFM =?LN;CH =IH>CNCIHM
QBC=B Q? ;L? AICHA NI F;S >IQH ;H> Q? MB;FF M?? NB;N NB?M? =IH>CNCIHM >?J?H> H?CNB?L
IH NB? =BIC=? I@ / HIL IH CNM H;NOL? 4? L?=IAHCT? IH NB? LCABN I@ NB? ?KO;FM MCAH NB?
LCABN B;H> MC>? I@ "KO;NCIH QBC=B Q? ;L? AICHA NI L?JF;=? <S NB? F?@N B;H> MC>?
I@ NB? M;G? ?KO;NCIH CH IL>?L NI A?N
<>
<7B ,
7B
,
& ;
7B 7B 4
,7>
;
;
/7A
/7A
7B 7>
<>
<7B ,
,
7B
7>
7B %
,7>
%
<>
,
8>
<>
,
8>
7B
7B 48> & 7B 4
;
7B
/7A
48>
7>
7B
7B 48> &
;
7B
/7A
48>
7>
,H? I@ NB? NQI CH>C=?M =;FF?> ?F B;> NI <? L?H;G?> ;M @F
1B? H?Q LCABN B;H> MC>? CM AICHA NI <? MJFCN CHNI NBL?? N?LGM
1?LG <>
<7B ,
,
7B
QBC=B CM ;FMI
)<7B
7B
7B
7B
,7>
7>
, 7B 7B ,
' IL ?P?H
<>
<7B ,
,
7B
B
7>
4 7B
;
/7A
)<7B
7B
7B , 7B
B
7>
7B , 7B ,
' IL VH;FFS
7B
47>
7>
;
/7A
1B? INB?L NQI N?LGM ;L?
1?LG <>
<7B ,
,
7B
1?LG 7>
7B %%
,7>
<>
,
8>
7B
7B 48> & 7B &
4
;
7B
/7A
48>
7>
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
<>
<7B ,
,
7B
7B
,7>
7>
%
<>
,
8>
7B
7B 48> &
;
7B
/7A
48>
7>
1B? NL;HM@ILG;NCIH I@ N?LGM ;H> CM MBIQH <?FIQ CH NB? ?R;GJF? I@ N?LG 4LCNN?H CH NB? @ILG I@ >IN JLI>O=NM CN CM ?KO;F NI
<7B
,
7B
%, 7B %
<>
,
8>
7B
7B 48> & 7B &
4
;
7B
/7A
48>
IL ?P?H <S JONNCHA , 7B <;=E CHNI NB? MOG IP?L @F ;H> @;=NILCTCHA NB? N?LGM
NI
<>
<7B ,
,
7B 8>
7B
, 7B 4 7B
7B
48>
7B
48>
;
/7A
B 7B 7B 7B
, 4
7B
48>
7B
48>
;
/7A
,
' ;FMI ?KO;F NI
<>
<7B ,
,
7B 8>
;H> VH;FFS <S L?H;GCHA NB? GON? CH>?R @F ;M ?F NI
<>
<7B ,
,
7B 7>
B 7B 7B 7B
, 4
7B
47>
7B
47>
;
/7A
&H NB? M;G? Q;S Q? VH> @IL N?LG <>
<7B ,
,
7B
7>
7B %
,7>
<>
,
8>
7B
7B 48> &
;
7B
/7A
48>
7>
<>
<7B ,
,
7B 7>
7B
, 7B 7B
47>
7B
47>
;
/7A
"R;GCH;NCIH I@ N?LGM ;H> ACP?M NB? C>?; I@ @;=NILCTCHA
, ' , ' <ON NBCM CM
QBC=B ;JJ?;LM CH N?LGM ;H> )?N OM HIN? NB;N NBCM F;MN N?LG MNCFF B;M ; G?;HCHA C@ ?J <?=;OM?
, ' <7B
0I ;H CN?L;NCIH D?J CM >?VH?> <S M?F?=NCHA HI ;>DICHN P;LC;<F?M ,
#IL N?LGM ;H> Q? =;H NB?L?@IL? QLCN?
1?LG <>
<7B ,
,
7B 7>
B 7B
, 7B
7B
47>
4 7B
7B
47>
;
/7A
-/*"1"/ "01&*1&,+ 1?LG <>
<7B ,
,
,
7B 7>
7B
7B
47>
7B
7B
47>
;
/7A
4? B;P? NBOM MBIQH NB;N IOL KO;HNCNS
<>
<7B ,
,
7B
7B
7B 7B
;
;4
/7A
/7A
7B
,7>
7>
7>
CM ?KO;F NI
<>
<7B ,
,
7B
7>
B
7B
,
7B
7>
4
7B
B 7B
, 7B
7B
47>
,
7B
7B
7B
47>
7B
47>
;
/7A
+IQ Q? L?=;FF NB;N
<>
<7B ,
7B
,
47>
& ;
7B
7B
3
4
)
7>
7>
7>
;
;
/7A
/7A
7B 7>
QBC=B <LCHAM OM NI NB? =IH=FOMCIH NB;N C@ Q? ;L? =;J;<F? I@ VH>CHA ;>DICHN P;LC;<F?M
, MO=B NB;N
B
7B
,
7B
7>
4
7B
B 7B
, 7B
7B
47>
,
7B
7B
7B
47>
7B
7B
37> 47>
)7>
1B?H ?KO;NCIH QCFF <? MIFP?>
%?L? Q? B;P? ; M?N I@ FCH?;L MSMN?GM NB? OHEHIQHM I@ QBC=B ;L? NB? P?=NILM I@
;>DICHN P;LC;<F?M , 7B ;H> NB?M? FCH?;L MSMN?GM =IOJF? NIA?NB?L , 7B ;H> , 7B
4?
=;HHIN VH> , QCNBION B;PCHA @IOH> , <ON , <7B
NB?L?@IL? Q? =;H VH>
, <7B 1BCM <LCHAM OM NI NB? =IH=FOMCIH NB;N NB? FCH?;L MSMN?GM ACPCHA ;>DICHN P;LC;<F?M
QCFF B;P? NI <? L?MIFP?> <S AICHA <;=EQ;L>M CH NCG? FF NB? P;LC;<F?M 4 7B =;H <?
EHIQH C@ ; JL?PCIOM >CL?=N =;F=OF;NCIH CM >IH? NB;N CM ; MCGOF;NCIH N?MN QCNB NB? M?N
I@ J;L;G?N?LM /
1B? ;>DICHN P;LC;<F?M QCFF NB?L?@IL? <? @IOH> <S NB? M?LC?M I@ NB? @IFFIQCHA FCH?;L
MSMN?GM
B
7B
,
7B
7>
7B
7B
37> 47>
)7>
4
B 7B
, 7B
7B
47>
7B
,
7B
7B
7B
47>
IL ?P?H
B
7B
,
7B
7>
7B
37> 47>
7B
)7>
7B
4 7B
7B
47>
7B
, 7B
7B
47>
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? VLMN MSMN?G NI <? L?MIFP?> CM NB? IH? ACPCHA , <7B B
<7B , <7B
7>
<7B
<7B
37> 47>
)7>
/?MIFPCHA NB? M?LC?M I@ ;>DICHN MSMN?GM @ILG;FFS L?JL?M?HNM NB? M;G? ?_ILN ;M NB?
>CL?=N =;F=OF;NCIH M;G? HOG<?L I@ >CL?=N ;H> ;>DICHN MSMN?GM M;G? HOG<?L I@ OH
EHIQHM G;NLC=?M I@ >CL?=N ;H> ;>DICHN MSMN?GM <?CHA NL;HMJIMCNCIHM I@ ?;=B INB?L
1B?M? ;>DICHN MSMN?GM >I HIN >?J?H> IH NB? J;L;G?N?L QBC=B CM NI <? C>?HNCV?>
NB? >?J?H>?H=? ;JJ?;LM IHFS QB?H Q? =;F=OF;N? NB? AL;>C?HN I@ & <S OMCHA NB? ;>DICHN
P;LC;<F?M "KO;NCIH )A40C<4=C >5 8A827;4C 2>=38C8>=B
0I @;L Q? B;P? H?AF?=N?> NB? @;=N NB;N 7B ;H> 7B ;L? ;FMI @OH=NCIHM I@ 4 7B CH NB?
=;M? I@ !CLC=BF?N NSJ? <IOH>;LS =IH>CNCIHM
1B? JL?PCIOM =;F=OF;NCIHM =;H <? >IH? ;A;CH <S ;>>CHA NBCM >?J?H>?H=? ;H> QIOF>
F?;> NI NB? @IFFIQCHA ;>DICHN FCH?;L MSMN?GM
7B
7B
B 7B 7B
7B
7B
, 7> 37> 47>
)7>
4 7B , 7B
7B
7B
47>
47>
7B 7B 7B 7B
4 ,
7B
7B
47>
47>
&H NB? 1?F?G;= MSMN?G !CLC=BF?N =IH>CNCIHM ;L? NBOM JLI=?MM?> M?? 0?=NCIH ;M @IFFIQM
&@ IH? OHEHIQH N B;M ;H CGJIM?> P;FO? N ?KO;NCIH I@ N <?=IG?M N N
N CM ?FCGCH;N?> @LIG ;FF INB?L ?KO;NCIHM OMCHA CNM P;FO? N NB? LCABN B;H> MC>?M
I@ ?KO;NCIHM ;L? =B;HA?>
4CNB NBCM JLCH=CJF? NB? HOG<?LCHA I@ ?KO;NCIHM L?G;CHM OH=B;HA?> L;HE NII
<ON =?LN;CH FCH?M ;H> =IFOGHM I@ ;L? =;H=?FF?> ;J;LN @LIG NB? >C;AIH;F N?LG QB?L?
CM JF;=?>
1B? JLI=?>OL? =;H NBOM <? @ILG;FCT?> ;M @IFFIQM
)?N <? ; >C;AIH;F G;NLCR NB? N?LGM I@ QBC=B ;L? @IL ; HILG;F JICHN ;H> @IL
NB? !CLC=BF?N JICHNM
B;M NB? @IFFIQCHA JLIJ?LNC?M
;H>
%
%
CM ; >C;AIH;F QCFF ;FF N?LGM ?KO;F NI % 4 7B CM ; P?=NIL QBC=B =IHN;CHM IHFS NB? CGJIM?> P;FO?M I@ 4 1;ECHA CHNI
;==IOHN EHIQH N?LGM !CLC=BF?N CH 4 7B G?;HM =B;HACHA CH
% 4 7B
1B? H?Q LCABN B;H> MC>? GOMN NB?L?@IL? <? ?KO;F NI % 4 7B @IL NB?
7B
HILG;F JICHNM ;H> NI % 4 @IL !CLC=BF?N JICHNM 1BCM =;H <? MOGG;LCT?> CH IH?
@ILGOF; CM =B;HA?> NI QCNB
% 4 7B %
4 7B
-/*"1"/ "01&*1&,+ ;H=?FFCHA FCH?M ;H> =IFOGHM I@ !CLC=BF?N JICHNM ;H> JF;=CHA ; IH NB? >C;AIH;F
MCGCF;LFS G?;HM =B;HACHA NI QCNB
%
1B?M? F;MN NQI ?KO;NCIHM MBIQ NB;N I@ NB? CN?L;NCIH ?J >I?M HIN >?J?H> IH 4 7B
IHFS >?J?H>M IH ?J
(HIQCHA NB;N L?MIFP?> >CL?=N MSMN?GM N;E? CHNI ;==IOHN !CLC=BF?N P;LC;<F?M ;H>
;L? CH @;=N I@ NB? @ILG 4 NB? KO?MNCIH HIQ CM NI EHIQ QBC=B ;>DICHN MSMN?G
L?MOFNM @LIG CN ;H> QBC=B =IH>CNCIH GOMN @OFVF !CLC=BF?N ;>DICHN P;LC;<F?M FF NB;N
B;M <??H >IH? QCNB ;H> =;H <? >IH? ;A;CH QCNB ;H> <ON NI =IHMNLO=N Q? GOMN $&#& EHIQ NB? !CLC=BF?N P;FO?M I@ ;>DICHN P;LC;<F?M 1B? G;NLCR I@ ;H
;>DICHN MSMN?G QCFF <? B 1B? ?KO;NCIH I@ ; !CLC=BF?N ;>DICHN P;LC;<F? I@ ?F HOG<?L
7B
QCFF NBOM <? L?>O=?> NI ,7>
;H OHEHIQH LCABN B;H> MC>? 4? ;L? AICHA NI MBIQ
NB;N NB? P;FO? ACP?H NI !CLC=BF?N ;>DICHN P;LC;<F?M >I?M HIN ;_?=N NB? INB?LM ;H> NB;N
CN =;H <? =BIM?H CH ;H ;L<CNL;LS G;HH?L #IL NBCM Q? ;L? AICHA NI MBIQ NB;N NB?
LCABN B;H> MC>? I@ NB? ?KO;NCIH I@ ; HILG;F ;>DICHN P;LC;<F? >I?M HIN >?J?H> IH NB?M?
;L<CNL;LS P;FO?M 1B? N?LG
7B 7B 7B 7B
4 ,
7B
7B
47>
47>
NB;N Q? B;P? ;>>?> NI NB? LCABN B;H> MC>? NI N;E? CHNI ;==IOHN !CLC=BF?N =IH>CNCIHM CM
T?LI @IL ; >?AL?? I@ @L??>IG ?F 1B? N?LG
7B
7B
7B
7B
4
,
7B
7B
47>
47>
CM <LIE?H >IQH CHNI 1B? N?LG
4 7B
, 4 7B
, , 7B
NB? N?LG % , 7B
;H>
, 7B
, <S =B;HACHA NI <?=IG?M
7B
7B
47>
4
7B
,
7B
>I?M HIN >?J?H> IH 4 1BCM N?LG CM ?KO;F NI
7B
7B
47>
4
7B
,
7B
0I CN CM M??H NB;N !CLC=BF?N ;L<CNL;LS P;FO?M I@ , 7B QCFF <? =;H=?FF?> <S NB? JLI>O=N
QCNB
7B
&@ Q? FIIE ;N NB? N?LG ,
CN HIQ <?=IG?M
,
7B 7B % 4 7B
7B
47>
%
4 7B
,
7B
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
IL ?P?H
7B 7B
,
7B
47>
7B 7B % 7B
47>
4 7B
,
7B
<?=;OM?
=NO;FFS
,
,
% 4 7B 7B
,
7B
47>
4 7B
%
7B
47>
,
7B
CM ; T?LI P?=NIL ?R=?JN C@ NB? JICHN HOG<?L ?F CM I@ !CLC=BF?N NSJ?
&H NBCM =;M? NB? >IN JLI>O=N CM ?KO;F NI NB? =IGJIH?HN ?F I@ % , 7B QBC=B G?;HM
7B
,7>
NB? ;>DICHN P;LC;<F? I@ ; !CLC=BF?N P;FO? B?H=? T?LI 4? NB?H QLCN?
7B 7B % 7B
47>
4 7B
,
7B
7B 7B % 7B
47>
4 7B
,
7B
;H> CN CM ;A;CH @IOH> NB;N !CLC=BF?N ;L<CNL;LS P;FO?M I@ , 7B QCFF <? =;H=?FF?> <S NB?
JLI>O=N QCNB
&N L?G;CHM NI =B?=E NB;N CH NB? ?RJL?MMCIH M?F?=N?> ;L<CNL;LS P;FO?M >I HIN
=IG? CH ?CNB?L &@ CH NB? ?RJL?MMCIH Q? =B;HA? NI ;H> NI 7B 7B
7B
;
;4
/7A
/7A
7>
<?=IG?M
7B 7B % 4 7B
;
/7A
%
4 7B
7B
;
/7A
%
4
7B
7>
FF NB? P?=NILM % 4 7B B;P? HIH T?LI N?LGM IHFS @IL !CLC=BF?N P;FO?M QBC=B
>I HIN >?J?H> IH NB? J;L;G?N?L / MI NB?M? N?LGM ;L? ?FCGCH;N?> 4B;N L?G;CHM CM
IHFS
7B 7B % 4 7B
7B
4 7B
;
;
/7A
/7A
7>
CH QBC=B NB?L? CM ;FQ;SM ; JLI>O=N <S
IL>?L NI I<N;CH
<7B
,
& ;
;
/7A
7B
IH NB? F?@N NB;N =;H <? <LIOABN CHNI , 7B CH
7B 7B % , ;
/7A
7B
4 7B
7B
7B
4
;
/7A
CH QBC=B ;A;CH !CLC=BF?N P;FO?M I@ ;>DICHN P;LC;<F?M >I HIN =IG? CH ;HS FIHA?L @IL
, 7B =IH=?LHM IHFS HILG;F P;FO?M
&H =IH=FOMCIH I@ NBCM M?=NCIH CN CM NB?L?@IL? M??H NB;N ;>DICHN MSMN?GM QCFF B;P? NB?
M;G? NL?;NG?HN I@ !CLC=BF?N =IH>CNCIHM ;M >CL?=N MSMN?GM ;H> NB;N !CLC=BF?N ;>DICHN
P;LC;<F?M =;H <? M?F?=N?> ;L<CNL;LCFS
-/*"1"/ "01&*1&,+ ??;820C8>= C> )4;4<02 0;6>A8C7<
A82C8>= 2>4P284=C 4BC8<0C8>= 8= )4;4<02 &H NB? =;M? I@ 1?F?G;= ! @LC=NCIH =IG?M CHNI NB? GIG?HNOG ?KO;NCIH ;H> CM NL?;N?>
CH ;H CGJFC=CN @ILGOF;NCIH 1B? NQI =IGJIH?HNM I@ @LC=NCIH @IL=? ;L? L?MJ?=NCP?FS
5
K
"F 4 K
L
>
5
L
"G 4 K
L
>
CH QBC=B > CM NB? Q;N?L >?JNB 4 ; >CG?HMCIHF?MM @LC=NCIH =I?^=C?HN ;H> K ;H> L NB?
NQI =IGJIH?HNM I@ BILCTIHN;F P?FI=CNS 1B?M? @IL=?M ;L? >CM=L?NCT?> L?MJ?=NCP?FS CH NB?
@ILG I@
"F "G 5
K<
4 K
>
5
L<
4 K
>
L
L
L <?CHA N;E?H ;N NB? NCG? J< 1B?M? @IL=?M
NB? G;AHCNO>? I@ NB? P?FI=CNS K
QLCNN?H CH ;H CGJFC=CN @ILG =IG? NI GI>C@S NB? G;NLCR I@ NB? VH;F FCH?;L MSMN?G
L?G;CHCHA OH=B;HA?> 4? @OLNB?L =BIIM? NI >I ; ZG;MM FOGJCHA[ I@ NB?M? @LC=NCIH
@IL=?M QBC=B G?;HM NB;N @IL NB? J;LN =IH=?LHCHA P?FI=CNC?M CM =B;HA?> NI "
QB?L? " A% % CM NB? C>?HNCNS G;NLCR ;H> A NB? P?=NIL ?KO;F NI
.
5
A
L
7 :
4 K
>
QB?L? 7 CM NB? <;MCM L?F;NCP? NI NB? JICHN QCNB CH>?R ?
0I Q? B;P?
<>
<7B ,
,
& ;
7B
,7>
;
/7A
7B 7>
7B
7B 7B
;
;4
/7A
/7A
7>
<>
<7B ,
,
7B
7B
,7>
7>
7B 7B
;4
/7A
7>
@IL >I?M HIN >?J?H> IH / IL ?P?H
<7B
,
& ;
;
/7A
7B
, 7B 7B 7B
;4
/7A
<7B
,
7B
, 7B A 7B
;4
/7A
&N CM NB?L?@IL? H?=?MM;LS NI EHIQ NB? >?J?H>?H=? I@ A CH N?LGM I@ / NBOM MJ?=C@SCHA
9
NB? F;Q OM?> #IL ; BfTS IL 0NLC=EF?L @ILGOF; A M=;F?M IH / ;H> '
NBOM B;M ;
P;FO? I@
9
'
@IL CH>?R ? ?I ;H> @IL NB? INB?LM 1BOM Q? A?N
<7B
7B
,
& ;
7B A47A
,
7A
;
;
/7A
/7A
7B
268 HYDRODYNAMICS OF FREE SURFACE FLOWS
Note: we have assumed so far that a friction coefficient was estimated per point. If
we look for a limited number of coefficients by dividing the mesh into different zones,
or arranging points class-wise, then the gradient becomes:
nit
∂J m
=
m
∂Sis
it=1
i ∈ classe(is)
Piit
2kXiit
m
Sis
(9.40)
m
is
in which index i covers all the points to which the same friction coefficient Sis
assigned.
9.4.2
Calculation of right-hand sides of adjoint systems
Principle
Let us recall that the right-hand side of the adjoint system has for component ip:
∂Ait+1 it+1 ∂B it+1 it+1
∂Ait it ∂B it it
it
it
2Wip (Xip − Mip ) −
X
−
,P
X −
,P
−
it
it
it
it
∂Xip
∂Xip
∂Xip
∂Xip
(9.41)
it
), let us recall that the term 2Wip
With a cost function of the formulation f (Xip
∂f
it
it
(Xip
− Mip
) is replaced by ∂X
it . A and B originate from a variational formulation,
ip
which gives to each point a weight proportional to the integral of its basis Ω Ψi dΩ.
For the
homogeneity, it would be necessary to take a cost function J which is
of
nitsake
np
it
it 2
− Mip
) but rather:
not it=1 ip=1 Wip (Xip
nit it=1
Ω
it
it 2
Ψi Wip (Xip
− Mip
) dΩ
which complicates the calculation of its gradient. Without changing the calculation
of
the gradient, it can also be considered that Wip is equal not to 1 or 0 but to Ω Ψi dΩ
or 0, which restores the homogeneity of the right-hand side. In any case, it is easy to
it
it
− Mip
).
calculate 2Wip (Xip
In the case of Telemac-2D, the matrix A is composed of blocks and vector X
contains the water depth and the two components of the velocity. The three blocks
of the right-hand sides are marked CV 1, CV 2 and CV 3:






AM 1 BM 1 BM 2
H
CV 1
0  , X =  U  , B =  CV 2 
A =  CM 1 AM 2
CM 2
0
AM 3
V
CV 3
Note: strictly speaking, the unknown H is in fact in Telemac-2D: δH = H n+1 −
H ; we consider here for more clarity and symmetry that the unknown is H n+1 . This
changes slightly the right-hand sides which become (with the notations of Section
4.5.6):
n
CV 1 =
Mh n
H + (θh − 1)(Txh + Tyh )H n + (θu − 1)(Bxuh U n + Byuh V n ) + M h SCE + T B1
∆t
(9.42)
-/*"1"/ "01&*1&,+ QB?L?
"
B
$ < ;JJ?;LM CH 2 2 )C <
1
J
C
0FC
QB?L? =FC6 6A< <?=IG?M = 2 )C <
2
J
C
0GC 1 < = 6
6
FC6 6A<
) C "F
6
GC6 6A<
) C "G
FC6 6A< CH 2 0FC
0GC 2 < = QB?L? =GC6 6A< <?=IG?M = 6 GC6 6A< CH 2 &@ NB? G?NBI> I@ =B;L;=N?LCMNC=M CM OM?> @IL ;>P?=NCIH N?LGM NB?M? NBL?? LCABN B;H>
MC>?M ;L? QLCNN?H ;M
2 )6 1
$
J
FC6 1 <
C
2 GC6 2 <
)C 1
1 = J
FC6 6A<
6
) 6 /!
0 ) C "F
)C 1
2 = 6 GC6 6A< ) C "G
J
#IL NB? ;>DICHN MSMN?G NB? @IFFIQCHA HIN;NCIHM QCFF <? ;>IJN?>
2 B
) B
% B ) B
) B
) B
) B
) ) /28
& % ) /28
B
) /28
) $/28
, % 1/28 &
2/28
/28
1BCM ;>DICHN MSMN?G CM MJFCN CHNI NBL?? <FI=EM
B
) $/28
B
) 1/28
B
) /28
) /28
) /28
) /28
2 /28
% 2 /28 &
2 /28
&
) 2/28 2 /28
B
) $/28
B
) 1/28 2 /28
B
) $/28
B
) 2/28 2 /28
)?N OM L?=;FF NB;N NB? LCABN B;H> MC>? I@ NB? ;>DICHN MSMN?G CM "KO;NCIH B 7B
7B
7B
7B
7B
7B
7B
,
,
37> 47> )7> 4
7B
7B
47>
47>
1BCM ACP?M @IL ?;=B JICHN ?F I@ NB? G?MB
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
2 /28
7B
$/28
2 7B
7B
$7>
$
$
1
7B
7B
2 7B
7>
7B
$/28
7B
1/28
B ) 7B
7B
$7>
7B
$/28
B ) 7B
7B
$7>
7B
$/28
$
7>
$ 7B
7B
1/28
B ) 7B
7B
17>
7B
$/28
B ) 7B
7B
17>
7B
$/28
1
2
7B
2 /28
7B
$/28
2 7B
27>7B
$
7>
7B
$
1
7B
7B
2 7B
B ) 7B
7B
$7>
7B
2/28
B ) 7B
7B
$7>
7B
1/28
B ) 7B
7B
$7>
7B
$/28
$ 7B
1
7B
2
7B
2 7B
27>7B
B ) 7B
27>7B
7B
2/28
7B
2/28
7B
2/28
B ) 7B
7B
17>
7B
1/28
B ) 7B
7B
17>
7B
2/28
7B
$/28
2 7B
7B
17>
B ) 7B
7B
17>
7>
7B
37> 2 7B 2;3A
7B
1/28
7B
2/28
2 7B
7B
$7>
2 7B
7B
17>
B ) 7B
7B
17>
B ) 7B
7B
17>
7B
7B
$/28
7>
7B
37> 1 7B 1;3A
7B
1/28
7B
2 7B
7B
$7>
B ) 7B
7B
$7>
B ) 7B
7B
$7>
2 7B
7B
17>
7B
1/28
7B
2 /28
7B
37> $ 7B $;3A
7>
7B
2/28
2 7B
27>7B
-/*"1"/ "01&*1&,+ $
7B
B ) 7B
27>7B
7B
1/28
1 7B
B ) 7B
27>7B
7B
$/28
B ) 7B
27>7B
7B
$/28
2
7B
1 7B
7B
$
2
7B
B ) 7B
27>7B
7B
2/28
B ) 7B
27>7B
7B
1/28
B ) 7B
27>7B
7B
2/28
?MC>?M NB? NBL?? >C_?L?H=?M @LIG NB? G?;MOL?G?HNM NB?L? ;L? N?LGM >?J?H>CHA
IH NB? G;NLC=?M ;H> LCABN B;H> MC>?M I@ NB? >CL?=N MSMN?G QBC=B Q? B;P? NI =;F=OF;N?
VLMN ;H;FSMCM ;FFIQM OM NI ?FCGCH;N? I@ NB?G CH @;=N
) ;H> ) >?J?H> H?CNB?L IH $ HIL IH 1 ;H> 2 MCR N?LGM
) ;H> ) >?J?H> IHFS IH $ @IOL N?LGM
) >I?M HIN >?J?H> IH $ IH? N?LG
2 >I?M HIN >?J?H> IH 2 IH? N?LG
2 >I?M HIN >?J?H> IH 1 IH? N?LG
4? ;L? F?@N QCNB
2 /28
7>
7B
37> $ 7B $;3A
0 0 0 0 0 0 0 7>
2 /28
7>
7B
37> 1 7B 1;3A
7>
0 0 0 0 0 2 /28
7>
7B
37> 2 7B 2;3A
7>
0 0 0 0 0 <S >?HINCHA
2 7B
7B
$7>
7B
1/28
2 7B
7B
$7>
7B
2/28
2 7B
7B
$7>
0 0 0 0 1
7B
$/28
7B
B ) 7B
7B
$7>
7B
$/28
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
0 1
7B
0 2 7B
0 2
0 0 0 1
0 2
0 0 7B
7B
7B
0 1 7B
0 2
7B
2/28
7B
2 7B
7B
17>
2 7B
7B
17>
B ) 7B
7B
17>
7B
$/28
B ) 7B
7B
17>
7B
1/28
B ) 7B
7B
17>
7B
2/28
7B
2/28
B ) 7B
7B
$7>
7B
$/28
0 $
7B
$/28
7B
1/28
7B
7B
1/28
B ) 7B
7B
$7>
7B
$/28
0 $
7B
B ) 7B
7B
$7>
2 7B
27>7B
2 7B
27>7B
B ) 7B
27>7B
7B
$/28
B ) 7B
27>7B
7B
1/28
B ) 7B
27>7B
7B
2/28
0;2D;0C8>= AD;4B
FF NB?M? N?LGM B;P? NB? @ILG ,
, =;N?AILS IL
4? MB;FF >?;F QCNB NB?M? NQI =;N?AILC?M MO==?MMCP?FS
0C46>AH 7B
, ,
, 4 ,
=;N?AILS -/*"1"/ "01&*1&,+ 0B4 F74A4 8B >5 C74 5>A< ) 4 7B NB? =IGGIH?MN =;M? CM NB;N QB?L?
) CM ; G;NLCR QBC=B >I?M HIN >?J?H> IH 4 7B ? A ; G;MM IL AL;>C?HN G;NLCR 0I Q?
B;P?
, 7B
47>
+IQ
,
,
) 4 7B
, 7B
47>
CM ?KO;F NI ) 4 7B
, 7B
47>
4 7B B
) , 7B
47>
CM ; P?=NIL ;FF =IGJIH?HNM I@ QBC=B ;L? T?LI ?R=?JN =IGJIH?HN ?F QBC=B
,
,
B ) , 7B
B
CM NB?L?@IL? ?KO;F NI
LCABN B;H> MC>? QBIM? =IGJIH?HN ?F CM
, 7B
, 7B
) , 7B
7>
1B? P?=NIL I@ NB?
QCFF NB?L?@IL? <? B ) , 7B
IH=FOMCIH C@ ) >I?M HIN >?J?H> IH 4 ; LCABN B;H> MC>? ) 4 7B I@ NB? >CL?=N
MSMN?G =ILL?MJIH>M NI ; LCABN B;H> MC>? B ) , 7B I@ NB? ;>DICHN MSMN?G
0C46>AH , 4 ,
0B4 F74A4 8B 0 3806>=0; >5 64=4A0; C4A< < 47 CM NB? =;M? I@ @LC=NCIH G;NLC=?M
4 ,
47>
CH NBCM =;M? QBC=B
<
47> ,7>
47>
)0B4 F74A4 8B 0 ;8=40A 5D=2C8>= >5 4 C@ Q? <L?;E >IQH 4 CHNI NB?
<
@ILG 7 47 7 Q? HINC=? NB;N NB? >?LCP;NCP? QCNB L?MJ?=N NI 47> ?FCGCH;N?M ;FF NB?
N?LGM ?R=?JN 47> 7> QBC=B ;@N?L >C_?L?HNC;NCIH QCNB L?MJ?=N NI 47> ACP?M 7> 1B?
G;NLCR ,
CM NB?L?@IL? ; G;NLCR I<N;CH?> @LIG <S L?JF;=CHA 4 <S 7>
0;2D;0C8>= >5 C4A<B 0 C> 0 %?H=?@ILNB Q? =IHMC>?L NB;N =;F=OF;NCIHM ;L? >IH? QCNBION MO< CN?L;NCIH @IL NL?;NCHA
HIH FCH?;L N?LGM ;H> NB?L?@IL? NB;N >>@=> >< K1=<D K< ;H> L1=<D L < )?N OM
HIQ =;F=OF;N? NB? N?LGM 0 NI 0 )4A< 0 0 7>
7B
$/28
2 7B
7B
$7>
1B? >?J?H>?H=?M I@ > CH 2 ;L?
"
<
B $ =;M? QBC=B ACP?M
0 : 6
0F6
0G6
7B
) 6 7B
$
J /28
$ 7B =;M? QBC=B ACP?M
0 : 6
B
0F6
0G6
7B
7B
$/28
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
C
FC6 1 7B
GC6 2 7B =;M? QBC=B ACP?M
0 : FC6/28 1 7B
C
GC6/28 2 7B
<S >?HINCHA FC6/28 ;M NB? G;NLCR FC6 QB?L? NB? Q;N?L >?JNB B;M <??H L?JF;=?> <S
NB? ;>DICHN Q;N?L >?JNB C ? ? @ .
C8
6 >7B
N 7 /28
:
GC6/28 ? @ .
C8
6 7B
>
O 7 /28
:
FC6/28
;H> ;FMI
0 ?KO;F NI ; N?LG
+
+ 7B <
>>@=> K D 7B
7 : ;H> MCGJFCV?> B?L? CH > K D 7 : ACP?M
0 :
0I Q? B;P?
0 C
7>
0 6/28 7> ) 6 7B
$
J /28
6
FC6/28 1 <
)4A<B 0 0=3 0 0 7B
1/28
.
B
<
>7B
/28 K D 7> :
0F6
0G6
GC6/28 2 <
2 7B
7B
$7>
7B
7B
$/28
0 6/28
1B? IHFS >?J?H>?H=? IH > CH 2 CM NB? N?LG = 6 FC6 6A< ?KOCP;F?HN CH
M?HMCNCPCNS NI = 6 FC6 $ < 4? ;L? MNCFF CH NB? =;M? QBC=B CH>C=;N?M NB;N
0 0 = 6
6
= 6
6
6
6
B
7B
FC6 1/28
B
6
7B
) 1/28
6
7B
) /28 1/28
B
7B
GC6 2/28
7B
) /28 2/28
6
B
6
7B
) 2/28
-/*"1"/ "01&*1&,+ )4A<B 0 0=3 0 0 7>
1
B ) 7B
7B
$7>
7B
7B
$/28
) 7B
7B
$7>
1
7B
7B
$/28
4? ;L? CH NB? =;M? I@ =;N?AILS QB?L? EHIQCHA ) Q? B;P?
0 7>
<
< ,
,
0 7 8
C
.
>7>
C8
7B 6
> 7 : K7B
8
N
>7B
/28 7
IL ?P?H
7>
<
< ,
,
C
.
C8
7 8
6 6 : K7B
8
N 7> 7
>7B
/28
7
QBC=B VH;FFS ACP?M
0 7>
C
.
K7B
6 >7B
N 7> /28
:
NB;N CM NI M;S
0 C6
7B
C F /28 1
0 C6
7B
C G /28 2
;H> ;FMI
1B? N?LGM 0 ;H> 0 =;H <? L?ALIOJ?> QCNB 0 : <?=;OM? NB?S L?KOCL? NB?
M;G? G;NLC=?M
)4A<B 0 0=3 0 0 7>
1 7B
B ) 7B
7B
$7>
7B
1/28
1B? >?J?H>?H=?M I@ > CH ) =IG? IHFS @LIG @LC=NCIH ;H> NBOM @LIG NB? G;NLCR
;FL?;>S HIN?>
.
6
4
7B
7B
"77C K
L
C7 :
7
7
"$ >7B
7
4? ;L? CH =;M? QBC=B ACP?M
0 7>
"77C 7B
K/28
>7B
7>
7>
K7B
7>
C
"7> 7> K7B
/28
>7B
7>
7>
K7B
7>
C@ NB?L? CM HI >?J?H>?H=? I@ 4 CH L?F;NCIH NI > QBC=B CM HIN NB? =;M? @IL NB? 0NLC=EF?L
@ILGOF; @IL QBC=B Q? QIOF> NB?H B;P?
0 7>
7B
C
"7>
7> K/28
>7B
7>
7>
K7B
7>
276 HYDRODYNAMICS OF FREE SURFACE FLOWS
So we have:
T 15 = −
1 u it+1 it+1
4
it+1
F Uadj U
(Chézy) or T 15 = −
F u Uadj
U it+1 (Strickler)
H it
3H it
(9.73)
and:
T 17 = −
1 u it+1 it+1
4
it+1
F Vadj V
(Chézy) or T 17 = −
F u Vadj
V it+1 (Strickler)
it
H
3H it
(9.74)
Terms T 21 and T 31:
T 21 =
it+1
Hadj
,
∂CV 1it+1
it
∂Uip
The dependences of CV 1 in relation to the velocity are in the transport term (θh −
1)(Txh + Tyh )it H it , in the term (θu − 1)Bxuh U it and the boundary term T B1. By
x
as the matrix of the general term:
denoting Tadj
∂hit u h
x
(i, j) =
Tadj
Ψi ϕj
Ω ∂x
and:
y
(i, j) =
Tadj
Ω
∂hit u h
Ψ ϕ
∂y i j
we find:
it+1
it+1
x
Hadj
+ (θu − 1)t Bxuh Hadj
+ T B1uadj
T 21 = (θh − 1)Tadj
(9.75)
y
it+1
it+1
Hadj
+ (θu − 1)t Byuh Hadj
+ T B1vadj
T 31 = (θh − 1)Tadj
(9.76)
by denoting:
T B1uadj
ip
=−
Γ
n u
hit+1
adj h Ψip nx dΓ and
Terms T 22 and T 32:
T 22 =
it+1
Uadj
,
T B1vadj
∂CV 2it+1
it
∂Uip
ip
=−
Γ
n u
hit+1
adj h Ψip ny dΓ
The dependence in relation to the velocity u in CV 2 originates from:
M u it
U and (θu − 1)(Txu + Tyu )it U it
∆t
M u it
M u it+1
∆t U gives ∆t Uadj
∂((Txu +Tyu )it U it )
) equal to:
it
∂Uip
where
it+1
and (θu − 1)(Txu + Tyu )it U it gives a term (θu − 1)(Uadj
,
-/*"1"/ "01&*1&,+ C
7B
1/28
+ 7B 7B
C
=H7:
K
:
K
7
7B
17>
,L ?P?H
C
.
7B C 7B
K 7> 1/28 :
N
C
.
L 7B
C7> 7B
:
1
O /28
;H> NB?L?@IL?
) C 7B
0 1
J /28
C
.
7B C 7B
K 7> 1/28 :
N
C
.
L 7B
.
K7B
C7> 7B
:
1
O /28
4? VH> CH NB? M;G? Q;S NB;N
) C 7B
0 2
J /28
C
.
7B C 7B
L 7> 2/28 :
O
C
C7> 7B
:
2
N /28
)4A<B 0 0=3 0 0 $
B ) 7B
7B
17>
7B
7B
$/28
1B? >?J?H>?H=? I@ ) CH L?F;NCIH NI K ILCACH;N?M IHFS @LIG NB? NL;HMJILN N?LG
6
0G6 ;H> ?P?H IHFS 6 0F6 QBC=B ACP?M ; N?LG
6 0F
6
.
C7>
7B
6
6 0F /28 $/28
<S >?HINCHA
+
NB? G;NLCR I@ A?H?L;F N?LG +
NB? G;NLCR I@ A?H?L;F N?LG 4? B;P?
?KO;F NI >7B
>7B
/28 :
N
0F6/28
0G6/28
6 C 6
F 7 8 :
6 C 6
G 7 8 :
0 7B
6
6 0F /28 $/28
0 7B
6
6 0G /28 $/28
;H>
)4A<B 0 0=3 0 0 0=3 0 0 1
7B
B ) 7B
7B
17>
7B
1/28
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? >?J?H>?H=?M I@ ) ;==IL>CHA NI K ;L? NB? NL;HMJILN G;NLCR
@LC=NCIH 4? >?HIN?
+
0FC/28 NB? G;NLCR I@ A?H?L;F N?LG CF C7 C8 :
+
0GC/28 NB? G;NLCR I@ A?H?L;F N?LG CG C7 C8 :
+
0FD /28 NB? G;NLCR I@ A?H?L;F N?LG DF C7 C8 :
+
0GD/28 NB? G;NLCR I@ A?H?L;F N?LG DG C7 C8 :
4? B;P?
0 7B
C
C 0F /28 1/28 0 7B
D
C 0F /28 2/28
0 7B
C
C 0G /28 1/28
0 7B
D
C 0G /28 2/28 1 7B 1 7B
1 7B
1 7B 2 7B
1 7B
2 7B 1 7B
1 7B
2 7B 2 7B
1 7B
7B
1/28
2 7B
7B
2/28
2 7B
7B
1/28
2 7B
7B
2/28
2
7B
C
C 0F
;H> NB?
"C
"C
"C
"C
4B24=C <4C7>3B 5>A <8=8<8I8=6 C74 2>BC 5D=2
C8>=
>=9D60C4 6A0384=C
4? OM? NB? =F;MMC= ;FAILCNBG I@ NB? =IHDOA;N? AL;>C?HN CH QBC=B NB? AL;>C?HN I@ NB?
@OH=NCIH;F Q? Q;HN NI GCHCGCT? CHN?LP?H?M &H NB? =F;MMC= =;M? I@ NB? L?MIFONCIH I@ NB?
FCH?;L MSMN?G 4 NBCM AL;>C?HN CM ?KO;F NI 4 @IL & 4 4 4
%?L? NBCM AL;>C?HN CM ACP?H <S NB? ;>DICHN P;LC;<F?M 4? B;P?
&HCNC;FCT;NCIH
= &
/7>
: =
&
: =
=; &
;
/7>
&N?L;NCIH C
:; =;
=; = ;
:;
=; =;
-/*"1"/ "01&*1&,+ ; /;
&
:; = ;
/ ; ; : ;
+0;830C8>= 0=3 A4BD;CB
&H L?;F =;M?M NB?L? CM ;FQ;SM ; FCGCN?> HOG<?L I@ G?;MOL?G?HNM GO=B F?MM NB;H NB?
HOG<?L I@ OHEHIQHM NB?L?@IL? Q? =;HHIN BIJ? NI C>?HNC@S CH ; OHCKO? Q;S ; @LC=NCIH
=I?^=C?HN J?L >?AL?? I@ @L??>IG 1B;N CM QBS Q? ;L? L?MNLC=NCHA IOLM?FP?M NI ; FCGCN?>
HOG<?L I@ TIH?M ;M G;HS ;M NB? G?;MOL?G?HNM IH QBC=B Q? ;MMOG? NB;N NB? @LC=NCIH
=I?^=C?HN CM =IHMN;HN 1?MN =;M?M QCNB MCGOF;N?> >;N; CFFOMNL;N? NB? JLI=?>OL? -F;N?
L?JILNM ; P?LS MCGJF? =;M? I@ ; MCHAF? 0NLC=EF?L =I?^=C?HN CH ; =;H;F QCNB MN?;>S
MN;N? WIQ 1B? CHCNC;F AO?MM CM QB?L?;M NB? P;FO? OM?> NI =IGJON? NB? MCGOF;N?>
>;N; CM 1B? =ILL?=N >?JNB ;H> =I?^=C?HN ;L? @IOH> ;@N?L M?P?H CN?L;NCIHM GIL?
=IGJF?R =;M? CM NB? WIQ ;LIOH> <LC>A? JC?LM M?? -F;N? QCNB >C_?L?HN 0NLC=EF?L
=I?^=C?HNM CH @IOL L?=N;HAOF;L TIH?M ;M J?L #CAOL? <?FIQ
S=10
S=15
S=5
S=20
#CAOL? 0NLC=EF?L =I?^=C?HN @IL ?;=B TIH?
/?MOFNM I@ ; MCGOF;NCIH QCNB MO=B =I?^=C?HNM ;L? E?JN ;MC>? ;H> =IHMC>?L?> ;@
N?LQ;L>M ;M >;N; 0N;LNCHA @LIG ; =IHMN;HN @LC=NCIH =I?^=C?HN I@ Q? NB?H NLS
NI C>?HNC@S NB? P;FO?M OM?> @IL =IHMNCNONCHA NB? >;N; 1B? @IOL P;FO?M ;L? @IOH> ;N
;LIOH> CH CN?L;NCIHM %IQ?P?L MO=B ; L?MOFN CM HIN ; P;FC>;NCIH I@ ; =ILL?=N
=;F=OF;NCIH I@ NB? AL;>C?HN I@ NB? =IMN @OH=NCIH GCMN;E? CH NB? G;AHCNO>? =IOF>
;FFIQ ; >?=L?;M? I@ & ; >?PC;NCIH CH NB? >CL?=NCIH =IOF> AI OHHINC=?> NB? MIFONCIH
<?CHA @IOH> <S >?M=LC<CHA ; MJCL;F 4? =;H ;JJLIRCG;N? NB? AL;>C?HN <S P;LSCHA NB?
@LC=NCIH =I?^=C?HN CH IH? MCHAF? JICHN ;H> =IGJ;LCHA NB? I<N;CH?> P;FO?M I@ & 4?
QLCN?
&
& /
/ & /
/
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&@ / CM NB? >C_?L?H=? QCNB NB? AL;>C?HN I<N;CH?> <S NB? ;>DICHN MSMN?G CM
;<ION L?;F ;JJFC=;NCIH I@ NB? J;L;G?N?L ?MNCG;NCIH CM JL?M?HN?> CH NB? @IFFIQCHA
=B;JN?L 0?=NCIH 70?C4A
??;820C8>=B
%I@MN;>N?LYM F;Q ( +-' (' #"& (" -#) ,$( *" +" -#) ( "(#
#)"( #'((&/' +
"0;?0BB4C 30<1A40: F0E4
=CA>3D2C8>=
1B? HOG?LC=;F MCGOF;NCIH I@ NB? *;FJ;MM?N >;G <L?;E Q;P? QCNB 1?F?G;= ! MI@N
Q;L? M?LP?> ;M ; N?MN =;M? CH NB? "OLIJ?;H JLID?=N =;FF?> - " ,41"/ -;L;FF?F
IGJONCHA I@ "HPCLIHG?HN ,;MN;F ;H> F;E? MB;FFIQ 41"/ >SH;GC=M CH NB?
@L;G?QILE I@ NB? "0-/&1 JLIAL;GG? <?NQ??H ;H> 1B? J;LNH?LM CH NB?
JLID?=N Q?L? "!# CNM &N;FC;H ?KOCP;F?HN "+") NB? L?M?;L=B =?HNL? /0 CH 0;L>CHC;
/?CN?L /?M?;L=B OL?;O )N> ;H> NB? ?F?=NLC=CNS JLI>O=?L (?GCDIEC ,S CH #CHF;H> 4?
MB;FF <LC?WS L?=;FF NB? =CL=OGMN;H=?M ;H> =IHM?KO?H=?M I@ NB? ;==C>?HN <?@IL? JL?M?HN
CHA NB? HOG?LC=;F MCGOF;NCIH CNM P;FC>;NCIH ;H> J?L@ILG;H=?M I<N;CH?> CH J;L;FF?FCMG
CH NBCM =;M? #IL GIL? >?N;CFM ;H> ; FCMN I@ L?@?L?H=? >I=OG?HNM M?? 89
)74 "0;?0BB4C 30< 022834=C
1B? *;FJ;MM?N ;==C>?HN CM ; L;L? ?R;GJF? I@ NIN;F ;H> CHMN;HN;H?IOM =IFF;JM? I@ ;H
;L=B >;G 1B? >;G QCNB ; =;J;=CNS I@ GCFFCIH =O<C= G?NL?M B;> <??H =IHMNLO=N?>
@IL CLLCA;NCIH ;H> ;M ; L?M?LPICL @IL >LCHECHA Q;N?L &N Q;M MCNO;N?> CH ; H?=E I@
NB? /?SL;H P;FF?S 3;L >?J;LNG?HN CH NB? MIONB ?;MN I@ #L;H=? ;<ION EG @LIG
#LfDOM &NM B?CABN Q;M G =LIQH F?HANB G ;H> NBC=EH?MM P;LC?> @LIG G NI G ,H !?=?G<?L ;N J G ;@N?L ; J?LCI> I@ B?;PS L;CH NB;N
KOC=EFS VFF?> NB? L?M?LPICL NB? L?;MIH @IL QBC=B ; VR?> QB??F A;N? B;> <??H IJ?H?>
NB? >;G A;P? Q;S MO>>?HFS 1B?L? Q?L? HI QCNH?MM?M CH NB? PC=CHCNS ;H> HI ;<HILG;F
CH>C=;NCIH Q;M HINC=?> <S NB? ?HACH??LM QBI B;> PCMCN?> NB? MCN? MIG? BIOLM <?@IL?
NB? ;==C>?HN <ION EG ;Q;S MNLIHA PC<L;NCIHM Q?L? @?FN CH NB? ALIOH> ;H> ;
<LC?@ LOG<FCHA Q;M B?;L> @IFFIQ?> <S ; MNLIHA QCH> 1B?H ; VLMN Q;P? >?M=?H>?> IH
NB? P;FF?S @IFFIQ?> <S ; Q;FF I@ Q;N?L "F?=NLC=CNS Q;M =ON ;H> NB? Q;P? MJL?;> IP?L
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
NB? JIJOF;N?> TIH?M =F;CGCHA PC=NCGM 0IG? EG >IQHMNL?;G IP?L G I@
NB? "MN?L?F BCABQ;S Q;M >;G;A?> ;H> ; <LC>A? Q;MB?> ;Q;S %OA? <IOF>?LM Q?L?
MQ?JN ;Q;S ;H> >?JIMCN?> @OLNB?L >IQHMNL?;G 1B?L? CM ;FGIMN HINBCHA F?@N I@ NB?
;L=B IH NB? MCN? ;H> NB? >IQHMNL?;G GILJBIFIAS B;M <??H G;LE?>FS =B;HA?> 1B?
LCP?L HI FIHA?L WIQM IH CNM <?> <ON IH ;H ;FFOPC;F MO<MNL;NOG
1B? CHKOCLS MBIQ?> NB;N NB? G;CH =;OM? I@ NB? ;==C>?HN Q;M ;H OH>?LALIOH> @;OFN
>IQHMNL?;G I@ NB? >;G 2H>?L NB? Q?CABN I@ Q;N?L NB? ;L=B <?=;G? >?N;=B?> @LIG
CNM @IOH>;NCIH ;H> NOLH?> CH IH? JC?=? IH CNM LCABN MC>? QBCF? NB? F?@N MC>? =IFF;JM?>
NB?H NB? =?HNL;F J;LN 1B? F?A;F JLI=??>CHAM NB;N @IFFIQ?> =IH=FO>?> NB;N NB? =;OM?
I@ NB? ;==C>?HN B;> <??H OHJL?>C=N;<F? BIQ?P?L NBCM QIOF> HIN <? NB? =;M? NI>;S
QCNB NB? A?IN?=BHC=;F MN;H>;L>M MO<M?KO?HNFS CGJIM?>
@N?L NB? ;==C>?HN ; -?LG;H?HN 1?=BHC=;F IGGCNN?? @IL !;GM Z IGCNf 1?=B
HCKO? -?LG;H?HN >?M ;LL;A?M[ 1- Q;M =IHMNCNON?> CH #L;H=? 1B? *;S >?=L?? MNCJOF;N?M NB;N ?;=B >;G ?R=??>CHA G ;H> QCNB ; =;J;=CNS I@ GIL? NB;H
GCFFCIH =O<C= G?NL?M GOMN <? CHMJ?=N?> <S NB? 1- GIHAMN INB?L G?;MOL?M
; JL?>C=NCP? >;G <L?;E Q;P? MCGOF;NCIH ;FIHA QCNB ; G;J I@ WII>?> TIH?M CH =;M?
I@ ;==C>?HN GOMN <? @OLHCMB?> 1B? HOG?LC=;F MCGOF;NCIHM ;L? GIMNFS =;LLC?> ION CH
>CG?HMCIH IH? QCNB 0;CHN 3?H;HN ?KO;NCIHM &H MIG? =;M?M QB?H NB? Q;P? MJL?;>M
IP?L QC>? JF;CHM IL QB?H NB?L? CM ;H IJ?HCHA CHNI NB? M?; ; NQI >CG?HMCIH;F HOG?LC=;F
MCGOF;NCIH CM H?=?MM;LS 1BCM CM NB? L?;MIH QBS NB? 1?F?G;= ! MI@NQ;L? CM OM?> CH
"!# 1B? >;N; =IFF?=N?> ;N NB? NCG? I@ NB? *;FJ;MM?N ;==C>?HN ;M Q?FF ;M ; JBSMC=;F
GI>?F ;FFIQ ; P;FC>;NCIH I@ NBCM ECH> I@ =;F=OF;NCIH
E08;01;4 30C0
1B? >;N; ;P;CF;<F? @IL IOL MCGOF;NCIH ;L? ;M @IFFIQM
-BINIAL;JBM I@ NB? >;G ;H> NB? P;FF?S ;@N?L NB? ;==C>?HN
H IH NB? MJIN CHKOCLS <S NB? FI=;F JIFC=? MBIQCHA ?F?P;NCIHM ;NN;CH?> IH NB?
F?@N ;H> LCABN <;HEM
-LIJ;A;NCIH NCG? I@ NB? <L?;E Q;P? &H @;=N NB? Q;P? >?MNLIS?> NBL?? NL;HM
@ILG?LM IH CNM J;MM;A? QBC=B QCFF <? L?@?LL?> NI ;M ;H>
NB? <L?;E>IQH
NCG? I@ QBC=B B;M <??H L?=IL>?> ?R;=NFS 1L;HM@ILG?L Q;M MNLO=E M ;@N?L
NB? <L?;E <ON NBCM VAOL? G;S <? ?LLIH?IOM <?=;OM? NB? NCG? I@ NB? >;G <L?;E
;LIOH> J G CM HIN EHIQH ?R;=NFS 1B? NL;HMCN NCG? <?NQ??H ;H> CM
GO=B <?NN?L EHIQH M ;H> QCFF M?LP? @IL P;FC>;NCIH 1B? Q;P? L?;=B?> NB?
M?; ;N ;LIOH> J G
1IJIAL;JBS L?=?HN G;JM ;L? HI FIHA?L P;FC> ;H> CN Q;M H?=?MM;LS NI L?MILN NI
NB? &$+ &HMNCNON $fIAL;JBCKO? +;NCIH;F G;J I@ 0;CHN 1LIJ?T HOG<?L
>;N?> 1B? P;FF?S CM P?LS H;LLIQ >IQHMNL?;G I@ NB? >;G QCNB NQI LCABN
;HAF?> <?H>M 1B? P;FF?S NB?H QC>?HM IH >L;CHCHA ; @?Q MG;FF NLC<ON;LC?M ;H>
H;LLIQM >IQH ;A;CH <?@IL? VH;FFS L?;=BCHA NB? =I;MN;F JF;CH
-BSMC=;F GI>?F CH ; JBSMC=;F GI>?F IH NB? M=;F? I@ B;> <??H JL?
J;L?> ;N );<IL;NICL? +;NCIH;F >Y%S>L;OFCKO? +;NCIH;F %S>L;OFC=M );<IL;NILS
NI L?JLI>O=? NB? ;==C>?HN ;H> ; IH? >CG?HMCIH;F MCGOF;NCIH Q;M ;NN?GJN?> 0I
--)& 1&,+0 Q? B;P? G?;MOL?G?HNM NB;N Q?L? =;LLC?> ION IH NBCM GI>?F NB? =;FC<L;NCIH I@
QBC=B F?> NI ?MNCG;N? NB;N NB? 0NLC=EF?L =I?^=C?HN CH NB? P;FF?S Q;M CH NB? L;HA?
I@ \ G?;MOL?G?HN JICHNM Q?L? =IFF?=N?> IH @L?? MOL@;=? ?F?P;NCIH CH NBCM
GI>?F I@ QBC=B Q?L? CH NB? L?M?LPICL CNM?F@ N ?;=B G?;MOL?G?HN JICHN @L??
MOL@;=? G;RCGOG ?F?P;NCIH Q;M HIN?> 1B?M? >;N; ;L? JON NIA?NB?L CH 1;<F?
QCNB IH? ;H> NQI >CG?HMCIH;F L?MOFNM &H NB? =;M? I@ 1?F?G;= ! =IG
JON;NCIH CM >IH? QCNB ; 0NLC=EF?L =I?^=C?HN I@ QCNBION CHCNC;F >CM=B;LA? CHNI
NB? LCP?L 1B? N ;H> O =IIL>CH;N?M CH G?NL?M I@ G?;MOL?G?HN JICHNM ;L? ACP?H
CH <L;=E?NM
,H? >CG?HMCIH;F =;F=OF;NCIHM NB?M? =IG? @LIG NB? /2-12/ MI@NQ;L? 89
1B? =LIMM M?=NCIHM ;L? M?N OJ ?P?LS G 1QI 0NLC=EF?L =I?^=C?HNM B;P? <??H
OM?> OJMNL?;G I@ NB? >;G ;H> ?FM?QB?L? 1B? L?MOFNM ACP? ;H ;P?L;A?
P?FI=CNS I@ NB? Q;P? ;M EGB NI <? =IGJ;L?> QCNB NB? I<M?LP?> EGB
1BCM >C_?L?H=? CM ?RJF;CH?> <S NB? @;=N NB;N IH? >CG?HMCIH;F MCGOF;NCIHM >I HIN
L?JLI>O=? NB? MFIQCHA >IQH ?_?=N I@ <?H>M OHF?MM CN CM CHN?AL;N?> ;LNCV=C;FFS
CH NB? ?KO;NCIHM CH NB? @ILG I@ FI=;F B?;> FIMM H ?LLIL I@ G CM HINC=?>
;N G?;MOL?G?HN JICHN -ICHNM ;H> Q?L? HIN CH=FO>?> CH NB? =;F=OF;NCIH
>IG;CH
1;<F? G?;MOL?G?HN I@ NB? G;RCGOG Q;N?L F?P?F *;R F?P?F IH NB? JBSMC=;F
GI>?F ;H> ! ;H> ! HOG?LC=;F L?MOFNM
*?;MOL?G?HN JICHN
!CMN;H=? NI >;G
G
G
G
G
G
G
G
G
G
INNIG
*;R F?P?F
!
!
)F>38<4=B8>=0; =D<4A820; <>34;;8=6
1B? >IG;CH B;M <??H >CM=L?NCT?> CHNI ?F?G?HNM M?? -F;N?M ;H> 1B?
>CG?HMCIHM ;L? G G 1B? ?F?P;NCIH I@ NB? <INNIG AI?M @LIG G
M?; <?> NI G NB? ?MNCG;N?> P;FO? I@ NB? @L?? MOL@;=? ?F?P;NCIH CH NB? F;E? ;<ION
QBC=B NB?L? CM MIG? OH=?LN;CHNS BIQ?P?L NB? N?MNM MBIQ?> NB;N ;H ?LLIL I@ =G
B;> IHFS ?_?=N IH NB? JLIJ;A;NCIH NCG? I@ NB? Q;P?
1B? >;G CM GI>?FF?> <S ; MNL;CABN FCH? <?NQ??H NB? JICHNM I@ =IIL>CH;N?M G G ;H> G G 1B? JIMCNCIH I@ NB? NBL?? NL;HM@ILG?LM ;H> CM G G G G ;H> G G L?MJ?=NCP?FS
1B? G?MB MCT? P;LC?M @LIG G >IQHMNL?;G I@ NB? >;G NI G M?;
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1B? L?M?LPICL CM CH=FO>?> CH NB? =;F=OF;NCIHM 1B? CHCNC;F =IH>CNCIH =IHMCMNM I@ ;
Q;N?L <I>S G ;<IP? M?; F?P?F OJMNL?;G I@ NB? >;G ;H> Q;N?L ;N L?MN ;N ?F?P;NCIH
CH NB? M?; >?VH?> <S NB? ;L?; QB?L? NB? <INNIG ?F?P;NCIH CM H?A;NCP? N NCG?
J NB?L? CM NBOM ; G BCAB Q;FF I@ Q;N?L ;N NB? FI=;NCIH I@ NB? >;G &H NB? VLMN
=IHVAOL;NCIH NB?L? CM HI Q;N?L CH NB? LCP?L ;N NB? MN;LN I@ =;F=OF;NCIH M?LC?M I@
N?MNM B;M ;FMI <??H >IH? QCNB ; =IHMN;HN >CM=B;LA? I@ G M ;H> ; MFCABN ;==?F?L;NCIH
I@ NB? Q;P? CM HINC=?> 4? EHIQ NB;N ; A;N? CH NB? >;G Q;M IJ?H?> <ON NB? ?R;=N
P;FO? I@ >CM=B;LA? =IGCHA ION I@ CN <?NQ??H ;H> G M CM HIN EHIQH
#D<4A820; ?0A0<4C4AB
1B? =IHNCHOCNS ?KO;NCIH CM >?;FN QCNB CH =IHM?LP;NCP? @ILG ;M CH>C=;N?> CH 0?=
NCIH 1B? FIMM I@ G;MM >OLCHA =;F=OF;NCIH NCG? CM FCNL?M IP?L ; NIN;F
I@ G IL ; L?F;NCP? JL?=CMCIH I@ NI NB? IL>?L I@ JL?=CMCIH I@
G;=BCH?M
1B? P?FI=CNS ;>P?=NCIH MN;A? CM =;LLC?> ION QCNB NB? G?NBI> I@ =B;L;=N?LCMNC=M
@IL M=;F;L G;=BCH?M ;H> CM L?JF;=?> <S NB? >CMNLC<ONCP? M=B?G? + CH J;L;FF?FCMG
!CM=L?NCT;NCIH I@ NB? ?F?P;NCIH CM FCH?;L ;H> NB;N I@ NB? P?FI=CNS CM KO;MC <O<<F?
1B? ?P?HN F;MNM M ;H> CM L?JLI>O=?> <S NCG? MN?JM I@ M 1B?
IOL;HN HOG<?L <;M?> IH NB? P?FI=CNS C FD B CM NI NB? IL>?L I@ NCG?
MN?J I@ M CM ;FMI JIMMC<F? <ON NB? @L?? MOL@;=? NB?H MBIQM MIG? IM=CFF;NCIHM
CGG?>C;N?FS >IQHMNL?;G I@ NB? >;G CH NB? VLMN @?Q M?=IH>M
1B? CN?L;NCP? MIFP?L =BIM?H CM $*/"0 QCNB ; L?F;NCP? JL?=CMCIH I@ MO< MJ;=? >CG?HMCIH I@ ;H> MCGJF? >C;AIH;F JL?=IH>CNCIHCHA
; (LSFIP
%7HB820; ?0A0<4C4AB
1B? NOL<OF?HN >C_OMCIH =I?^=C?HN ;FMI CH=FO>CHA >CMJ?LMCIH B;M <??H N;E?H ;M
=IHMN;HN ;H> ?KO;F NI G M M?HMCNCPCNS ;H;FSMCM MBIQM NB;N CN B;M FCNNF?
CHWO?H=? IH NL;HMCN NCG?M <?NQ??H NB? NL;HM@ILG?LM M?? 1;<F? ,HFS ;
P;FO? I@ G M ?HN;CFM ; HIN;<F? =B;HA?
1B? @LC=NCIH =I?^=C?HN CM GO=B GIL? MCAHCV=;HN ?MJ?=C;FFS @IL JLIJ;A;NCIH NCG?
;M Q? =;H M?? CH 1;<F? 1;<F? CHWO?H=? I@ >C_OMCIH =I?^=C?HN QCNB ; 0NLC=EF?L =I?^=C?HN I@ !C_OMCIH =I?^=C?HN
+;NOL?
G M
G M
G M
G M
G M
LLCP;F ;N M M
M
M
M
M
1CG? @LIG NI M
M
M
M
M
M
1CG? @LIG NI
M
M
M
M
M
M
--)& 1&,+0 1;<F? CHWO?H=? I@ 0NLC=EF?L =I?^=C?HN
0NLC=EF?L =I?^=C?HN
+;NOL?
LLCP;F ;N M M
M
M
M
M
1CG? @LIG NI M
M
M
M
M
M
1CG? @LIG NI
M
M
M
M
M
M
4? I<M?LP? P;LC;NCIHM @LIG \ NI I@ NL;HMCN NCG? <?NQ??H ;H> ,H NB? INB?L B;H> NB? =BIC=? B;M FCNNF? CHWO?H=? IH ;LLCP;F NCG? ;N 1;<F? MBIQM NB? L?MOFNM I<N;CH?> QCNB ;H CHCNC;F >CM=B;LA? I@ G M CH NB? LCP?L 1B?
>C_?L?H=?M ;L? @IL NB? NL;P?F NCG? <?NQ??H ;H>
;H> <?NQ??H ;H> 1BCM CM ?PC>?HNFS >O? NI NB? @;=N NB;N JLIJ;A;NCIH IH ; Q?N <INNIG CM @;MN?L *IL?IP?L
=?LN;CH NLIOABM ;L? ;FL?;>S VFF?> QCNB Q;N?L <?@IL? NB? ;LLCP;F I@ NB? Q;P?
1;<F? CHWO?H=? I@ 0NLC=EF?L =I?^=C?HN QCNB ;H CHCNC;F >CM=B;LA? I@ =O<C= G?NL?M
0NLC=EF?L =I?^=C?HN
+;NOL?
LLCP;F ;N M M
M
1CG? @LIG NI M
M
M
1CG? @LIG NI
M
M
M
0NCFF QCNB ;H CHCNC;F >CM=B;LA? CH NB? LCP?L NB? G;RCGOG ?F?P;NCIHM I@ G?;MOL?G?HN
JICHNM ;L? ;M ACP?H CH 1;<F? 1;<F? G;RCGOG Q;N?L F?P?F ;N NB? G?;MOL?G?HN JICHNM
*?;MOL?G?HN JICHN
0NLC=EF?L 0NLC=EF?L 1B? CHWO?H=? I@ @LC=NCIH IH NB? ?F?P;NCIHM L?G;CHM Q?;E QCNBION G;LE?> N?H>?H=S
CH IH? >CL?=NCIH IL NB? INB?L GIL? JL?=CM? =;FC<L;NCIH QIOF> H?=?MMCN;N? P;LSCHA
NB? 0NLC=EF?L =I?^=C?HN CH MJ;=? &H NB? L?;F JL?>C=NCP? =IGJON;NCIHM I@ >;G <L?;E
Q;P?M NBCM =I?^=C?HN CM CH @;=N VR?> <S NB? 1- ;==IL>CHA NI ; M=;F? QBC=B N;E?M
CHNI ;==IOHN NB? H;NOL? I@ NB? N?LL;CH =LIMM?> NIQHM @IL?MNM ?N= %6!/,!6+*& 0 ,# #/"" 02/# " #),40
'4BD;CB
%?L? Q? JL?M?HN ; =;M? QCNBION CHCNC;F >CM=B;LA? CH NB? LCP?L QCNB ; 0NLC=EF?L =I?^=C?HN
I@ 1B? G?MB B;M ?F?G?HNM <ON ; VH?L G?MB B;M ;FMI <??H OM?> NI MBIQ
NB;N ; MO^=C?HN JL?=CMCIH CH MJ;=? Q;M ;NN;CH?> ";=B NLC;HAF? I@ NB? CHCNC;F G?MB B;M
<??H >CPC>?> CHNI @IOL ;H> NCG? MN?J >CPC>?> <S 1B?H =;F=OF;NCIH N;E?M BIOLM
IH ; /6 1B? ;LLCP;F NCG? I@ NB? Q;P? ;N CM >?=L?;M?> <S NB? NCG?
<?NQ??H ;H> L?G;CHM OH=B;HA?>
-F;N? MBIQM NB? P?FI=CNS V?F> >IQHMNL?;G I@ NB? >;G GCHON? ;@N?L NB? <L?;E
1B? ?_?=N I@ NB? <?H> CM ;FL?;>S HINC=?;<F? QCNB ; MN;LN I@ L?=CL=OF;NCIH IH NB? LCABN
<;HE *;RCGOG P?FI=CNS CM GM
IGJ;L?> QCNB ; NB?IL?NC=;F IH? >CG?HMCIH;F
>;G <L?;E IH ; W;N <INNIG QBC=B QIOF> ACP? GM NBCM MBIQM NB? ;==?F?L;NCIH
?_?=N I@ NB? MFIJ? ;H> NB? H?=E 1B? NIJ I@ -F;N? CM NB? L?MOFN I<N;CH?> QCNB
NB? G?MB I@ ?F?G?HNM NB? <INNIG QCNB ; G?MB G;>? I@ ?F?G?HNM IH?
P?=NIL IHFS ION I@ @IOL CM >L;QH CH NBCM =;M? @IL =F;LCNS 1B?L? CM HI ;JJL?=C;<F?
>C_?L?H=? <ON CH NB? VH?L G?MB MIG? MG;FF ?>>C?M ;L? GIL? PCMC<F? 1B? Q;N?L >?JNB
;H> P?FI=CNC?M ;@N?L GCHON?M ;L? MBIQH CH -F;N? &H NBCM JF;N? NB? G?MB CM PCMC<F?
C@ NB? >?JNB CM F?MM NB;H =G N NB? JF;=? I@ NB? NQI LCABN ;HAF?> <?H>M CN CM I<M?LP?>
NB;N WIQ >I?M HIN I==OJS NB? QBIF? Q?N M?=NCIH #IL ?R;GJF? CN CM =IH=?HNL;N?> IH
NB? LCABN <;HE @IL NB? VLMN <?H> 1BCM =IHVAOL;NCIH CM >O? NI ; MCAHCV=;HN NL;HMP?LM?
MFIJ? G <?NQ??H NQI <;HEM ;FMI HIN?> IH NB? JBSMC=;F GI>?F
-F;N? L?JL?M?HNM NB? Q;N?L >?JNB ;H> P?FI=CNC?M ;N NB? NCG? QB?H NB? Q;P?
L?;=B?M NB? M?; #IL GIL? =F;LCNS IHFS ; MG;FF JLIJILNCIH I@ P?FI=CNS P?=NILM CM MBIQH
1B? GI>?F J?LB;JM >I?M HIN L?JL?M?HN L?;FCNS P?LS Q?FF CH NB? =I;MN;F TIH? @IL NB?
+C=?\*;LM?CFF?M L;CFQ;S NL;=E B;M HIN <??H CH=FO>?> CH NB? GI>?F ;H> OH>IO<N?>FS
JF;S?> ; LIF? CH =IH=?HNL;NCHA NB? Q;P? <?@IL? <?CHA >?MNLIS?>
%4A5>A<0=24B >= E0A8>DB <0278=4B
;F=OF;NCIH QCNB ; G?MB I@ ?F?G?HNM ;H> NCG? MN?JM I@ M B;M <??H
=BIM?H ;M ; N?MN @IL ?P;FO;NCHA J?L@ILG;H=?M I@ P;LCIOM G;=BCH?M VLMN CH M=;F;L GI>?
NB?H CH J;L;FF?F GI>? QCNB >IG;CH >?=IGJIMCNCIH
= B20;0A <>34
1;<F? FCMNM NB? -2 NCG? @IL HCH? =IGJON?LM &N CM M??H NB;N NB?M? NCG?M ;L?
LIOABFS JLIJILNCIH;F NI =FI=E J?LCI> ?R=?JN @IL NB? /6 QBC=B CM ; P?=NIL =IGJON?L
4? M?? B?L? NB? CHN?L?MN CH IJNCGCT;NCIH I@ ;FAILCNBGM QCNB IOL P;LCIOM G;NLCR MNIL;A?
G?NBI>M 1B? 0CFC=IH $L;JBC=M QILEMN;NCIH CM ;HINB?L ?R=?JNCIH <?=;OM? CN CM ; <CN
G;=BCH? QBC=B CM ;<ION NQC=? ;M @;MN ;M ; <CN G;=BCH? I@ NB? M;G? =FI=E J?LCI>
- M B;PCHA 4CH>IQM +1 M??G NI J?L@ILG F?MM Q?FF NBIOAB CN GOMN <? JICHN?> ION
NB;N NB?M? L?MOFNM ;FMI >?J?H> IH NB? =IGJCF?LM OM?>
= ?0A0;;4; <>34
1;<F? =IGJ;L?M NB? LOHM CH J;L;FF?F I@ ;H &* 0- && /6 1" 2+&5 MN;NCIH
H?NQILE ;H> ; - H?NQILE QCNB IL MO< >IG;CHM JLI=?MM?> CH J;L;FF?F 1B?
F;HAO;A? I@ =IGGOHC=;NCIH CM *-& IH ,LCACH ;H> -3* CH ;FF INB?L =;M?M
--)& 1&,+0 1;<F? -2 NCG? CH M=;F;L GI>?
*;=BCH?
P?=NIL =IGJON?L <CNM
/6 1" JLI=?MMIL
&* 0- && JLI=?MMIL
0CFC=IH $L;JBC=M ,=N;H? *%T <CNM
02+ 0J;L= *%T
%?QF?NN -;=E;L> *%T
-"+1&2* && 4CH>IQM +1 *%T
%?QF?NN -;=E;L> *%T
%?QF?NN -;=E;L> *%T
/6
1;<F? *;=BCH?
,LCACH &* 0- &&
/6 1"
%- -?HNCOG *%T
-2 NCG?
B GCH
B GCH
B GCH
B GCH
B GCH
B GCH
B GCH
B GCH
B GCH
-2 NCG? CH J;L;FF?F GI>?
JLI=?MMIL
1CG?
M
M
M
M
M
JLI=?MMILM
0J??> OJ
JLI=?MMILM
0J??> OJ
JLI=?MMILM
0J??> OJ
1B? MJ??> OJ @;=NIL >?=L?;M?M MFCABNFS QCNB NB? HOG<?L I@ JLI=?MMILM ?R=?JN QCNB
NB? ,LCACH ;M Q? B;P? M??H CH 0?=NCIH <?=;OM? NB? HOG<?L I@ =IGGO
HC=;NCIHM CH=L?;M?M CH ; =IH=IGCN;HN G;HH?L &N CM L?G;LE;<F? NB;N ;N F?;MN ?CABN
JLI=?MMILM I@ NB? /6 1" ;L? L?KOCL?> @IL =IGJ?NCHA QCNB NB? /6 P?=NIL
=IGJON?L )?N OM L?=;FF B?L? NB;N NB? NCG? A;CH >O? NI P?=NILCT;NCIH I@ 1?F?G;= ! CM
; @;=NIL I@ "RJ?LCG?HNM IH NB? QILEMN;NCIHMY H?NQILE IL - M ACP? KOCN? L;H>IG
L?MOFNM 1;<F? MBIQM IHFS NB? <?MN J?L@ILG;H=?M &N M??GM >C^=OFN NI J?L@ILG
J;L;FF?FCMG CH ;H ?^=C?HN G;HH?L C@ Q? >I HIN OM? >?>C=;N?> H?NQILEM QBC=B L?GIP?
;FF NB? =IGJ?NCNCIH @LIG NB? INB?L OM?LM %CAB?L A;CHM =;H <? I<N;CH?> C@ NB? >?=L?;M?
CH G?GILS L?KOCL?G?HNM I@ ; MO< >IG;CH CH L?F;NCIH NI NB? ?HNCL? >IG;CH L?GIP?M NB?
H?=?MMCNS I@ L?=IOLM? NI NB? PCLNO;F G?GILS >CPCMCIH <S I@ NB? =;F=OF;NCIH NCG?
=IOF> NBOM <? I<N;CH?> QCNB NQI QILEMN;NCIHM
">A4 A424=C A4BD;CB
0CH=? NB? Q;P? ?KO;NCIH B;M <??H CGJF?G?HN?> CH 1?F?G;= ! ;H> ;FFIQ?> F;LA?L
NCG? MN?JM M?=IH>M CH NBCM =IGJON;NCIH *IL?IP?L CH NB? MCGOF;NCIH =IOF>
<? >IH? CH NBL?? >CG?HMCIHM QCNB HIH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM ?P?H QCNB
NQI JF;H?M IP?L NB? P?LNC=;F QBC=B CM NI M;S QCNB IHFS IH? F;S?L I@ JLCMG;NC= ?F?G?HNM
<?NQ??H NB? <INNIG ;H> NB? @L?? MOL@;=? /?MOFNM ;L? =IGJ;L?> QCNB NB? 0;CHN 3?H;HN
?KO;NCIHM CH -F;N? 1BIOAB NB? WII> CM MFCABNFS >?F;S?> QCNB NQI JF;H?M NB?L? CM ;
L?G;LE;<F? ;AL??G?HN <?NQ??H L?MOFNM CH NQI ;H> NBL?? >CG?HMCIHM IGJON?L NCG?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
@IL P;LCIOM =IHVAOL;NCIHM ;L? ACP?H CH 1;<F? 1;<F? L?=?HN L?MOFNM IH ; *%T IH? JLI=?MMIL G;=BCH?
*;FJ;MM?N >;G <L?;E NCG? MN?JM I@ M
0;CHN 3?H;HN ?KO;NCIHM Q;P? ?KO;NCIH
+;PC?L\0NIE?M ?KO;NCIHM JF;H?M
+;PC?L\0NIE?M ?KO;NCIHM JF;H?M
1CG? IH %- *%T
M
GCHON?M
BIOL GCHON?M
>=2;DB8>=
1BCM MCGOF;NCIH I@ NB? *;FJ;MM?N >;G <L?;E Q;P? MBIQM NB;N NQI >CG?HMCIH;F ;H>
?P?H NBL?? >CG?HMCIH;F =;F=OF;NCIHM I@ >;G <L?;E Q;P?M IL JLIJ;A;NCIH I@ WII>M IP?L
>IG;CHM I@ M?P?L;F ECFIG?NL?M ;L? HIQ =OLL?HN CH>OMNLC;F ;JJFC=;NCIHM 1B? CGJLIP?
G?HN CH L?F;NCIH NI IH? >CG?HMCIH;F MCGOF;NCIHM CM I<PCIOM @LIG NB? GIG?HN NB? A?I
G?NLC?M =IGJLCM? QC>? JF;CHM IL G;LE?> <?H>M 1B? =?F?LCNS I@ WII> Q;P?M IL >;G
<L?;EM CM ;FMI <?NN?L L?MJ?=N?>
)834B 8= C74 =6;8B7 70==4; 0=3 C74 C;0=C82
$240=
BC8<0C8>= F8C7 039>8=C <4C7>3
1BCM F;LA? NC>;F GI>?F ;<ION EG EG CM OM?> NI JLIPC>? <IOH>;LS =IH>C
NCIHM @IL GIL? FI=;F MNO>C?M &N ?RN?H>M @LIG NB? ;S I@ CM=;S ;H> &<?LC;H -?HCHMOF;
OJ NI DOMN MIONB I@ NB? +ILNB 0?; 1B? Q?MN?LH <IL>?L I@ NB? >IG;CH CM MCNO;N?>
<?SIH> NB? =IHNCH?HN;F MB?F@ LIOABFS NIQ;L>M NB? >?AL?? Q?MN FIHACNO>? 1B?
G?MB G;>? I@ ?F?G?HNM ;H> HI>?M CM L?VH?> H?;L NB? =I;MNM QCNB ?F?
G?HNM P;LSCHA <?NQ??H EG ;H> EG 1B? MSMN?G I@ =IIL>CH;N?M CM ; *?L=;NIL
JLID?=NCIH NB?L?@IL? NB? JLI=?>OL? >?M=LC<?> CH 0?=NCIH CM OM?> B?L? #IOL NC>?M
C ? ; >OL;NCIH I@ M ;L? MCGOF;N?> MN;LNCHA @LIG ; MN;N? I@ L?MN ,H NB? IJ?H
<IOH>;LC?M Q;N?L ?F?P;NCIHM I@ NB? * Q;P? M?GC >COLH;F Q;P? >O? NI NB? GIIH ;L?
CGJIM?> 1B?M? Q;N?L ?F?P;NCIHM ;L? I<N;CH?> <S ; #IOLC?L ;H;FSMCM I@ G?;MOL?G?HNM
NI ?RNL;=N NB? * =IGJIH?HN 1B? NCG? MN?J CM M 1B? IOL;HN HOG<?L 1B
F
=;F=OF;N?> QCNB NB? Q;P? =?F?LCNS CM NB?L?@IL? A?H?L;FFS GIL? NB;H 4CNB ; JIIL L?F
;NCP? JL?=CMCIH MIG? IM=CFF;NCIHM I@ NB? @L?? MOL@;=? ;L? I<M?LP?> IH =LIMMCHA NB?
=IHNCH?HN;F MB?F@ 1B?M? IM=CFF;NCIHM ;L? L?GIP?> ?CNB?L <S ; BCAB?L JL?=CMCIH IL <S NB? M?F?=NCIH I@ ; KO;MC <O<<F? ?F?G?HN @IL NB? P?FI=CNS 1B? >C_OMCIH =I?^=C?HN
CM M?N ;N ;H> NB? P?FI=CNS ;>P?=NCIH CM CAHIL?> -F;N? MBIQM NB? G;AHCNO>? ;H>
JB;M? I@ NB? * Q;P? CH =IFIOL?> MOL@;=?M 1B? ?_?=N I@ ;GJFCV=;NCIH I@ NB? *IHN
0;CHN *C=B?F <;S CH #L;H=? ;H> NB? 0?P?LH ?MNO;LS CH "HAF;H> CM KOCN? PCMC<F? 1B?
;GJBC>LIGC= JICHN QCFF <? HINC=?> CH NB? +ILNB 0?; QB?L? ;FF NB? JB;M?M =IHP?LA?
3?LCV=;NCIH I@ NB? GI>?F CM >IH? QCNB G?;MOL?G?HNM I@ NC>;F L;HA? M?? 1;<F? ;H> JB;M? I@ NB? Q;P? M?? 1;<F? CH JILNM IL ;N NB? FI=;NCIH I@ ; H;PCA;NCIH
--)& 1&,+0 1;<F? G;AHCNO>? I@ Q;P? *
*?;MOL?G?HN JICHN
7??<LOAA?
;F;CM
IOFIAH?
!C?JJ?
B?L<IOLA
0;CHN *;FI
OIS ,O?MM;HN
*?;MOL?G?HN
G
G
G
G
G
G
G
G
*;HO;F =;FC<L;NCIH
G
G
G
G
G
G
G
G
"MNCG;NCIH
G
G
G
G
G
G
G
G
1;<F? JB;M? I@ Q;P? * CH >?AL??M
*?;MOL?G?HN JICHN
7??<LOAA?
;F;CM
IOFIAH?
!C?JJ?
B?L<IOLA
0;CHN *;FI
OIS ,O?MM;HN
*?;MOL?G?HN
*;HO;F =;FC<L;NCIH
"MNCG;NCIH
<?;=IH 1B? 7??<LOAA? JILN CM MCNO;N?> CH NB? +ILNB 0?; NB? H;PCA;NCIH <?;=IH CH NB? NF;HNC= I_ NB? LCNN;HS =I;MN
1B? BfTS @ILGOF; CM OM?> @IL MCGOF;NCHA @LC=NCIH ;H> VP? >C_?L?HN =I?^=C?HNM ;L?
OM?> ;H> ;JJFC?> NI ;M G;HS >CMNCH=N TIH?M QBC=B @ILG ; N?MM?FF;NCIH I@ NB? >IG;CH
1B?M? TIH?M ;L? ACP?H CH -F;N? QBC=B ;FMI MBIQM NB? <;NBSG?NLS ;H> ?HNCL? G?MBCHA
1QI MCGOF;NCIHM QBC=B >C_?L IHFS <S @LC=NCIH =I?^=C?HNM ;L? =IGJ;L?> QCNB
G?;MOL?G?HNM NB? VLMN IH? CM NB? L?MOFN I@ ; G;HO;F =;FC<L;NCIH NB? M?=IH> IH? OM?M
; M?N I@ @LC=NCIH =I?^=C?HNM ACP?H <S ;ONIG;NC= J;L;G?N?L ?MNCG;NCIH 1B? NQI M?NM I@
=I?^=C?HNM ;L? FCMN?> CH 1;<F? #IL G;HO;F =;FC<L;NCIH ; MNO>S QBC=B F;MN?> @IL
;<ION MCR GIHNBM NB? G;RCGOG ?LLIL CM =G 0;CHN *;FI JILN IH NB? G;AHCNO>?M
;H> >?AL??M IH NB? JB;M?M CH 7??<LOAA? JILN NB? @OLNB?MN ;Q;S @LIG NB? <IOH>;LS
=IH>CNCIHM 1B? G?;H MKO;L? >?PC;NCIH CM =G @IL NB? G;AHCNO>?M ;H> >?AL??M
@IL NB? JB;M?M 4CNB ;ONIG;NC= =;FC<L;NCIH QBC=B L?KOCL?M ; BOH>L?> =;F=OF;NCIHM
HIQ GO=B F?MM NB;H ; >;S IH ; QILEMN;NCIH NB? G;RCGOG ?LLIL CM =G ;F;CM
@IL NB? G;AHCNO>?M ;H> >?AL??M @IL NB? JB;M?M 1B? G?;H MKO;L? >?PC;NCIH @IL
NB? G;AHCNO>?M CM =G =G A;CH ;H> >?AL??M @IL NB? JB;M?M %IQ?P?L CN
CM H?=?MM;LS NI MJ?=C@S NB;N NB? M?N I@ =I?^=C?HNM I<N;CH?> <S ;ONIG;NC= =;FC<L;NCIH
>?J?H>M ; FIN IH CHCNC;F P;FO?M @IL NB? G?NBI> =;H =IHP?LA? NIQ;L>M FI=;F GCHCG;
%?L? NB? CHCNC;F P;FO? CM N;E?H @LIG G;HO;F =;FC<L;NCIH ;H> Q? VH> NB;N NB? @LC=NCIH
P;FO?M ;L? HIN P?LS @;L @LIG CN <ON NB?S JLIPC>? BIQ?P?L ; =IHMC>?L;<F? CGJLIP?G?HN
@IL G;AHCNO>?M
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
1;<F? @LC=NCIH =I?^=C?HNM CH >C_?L?HN ;L?;M
7IH?
*;HO;F J;L;G?N?L ?MNCG;NCIH
ONIG;NC= J;L;G?N?L ?MNCG;NCIH
AL?;N?L JL?=CMCIH I@ ;<ION =G =;H <? I<N;CH?> QCNB ; VH?L G?MBCHA I@ ?F?G?HNM %IQ?P?L MO=B ; JL?=CM? =;FC<L;NCIH CM DOMN ;H CFFOMCIH ;H> ; HOG<?L I@
@;=NILM =;H <? L?MJIHMC<F? @IL NB? ?LLIL CH G;AHCNO>? P;LSCHA @LIG NI =G M?HMCNCPCNS ;H;FSMCM B;M MBIQH NB;N ; P;LC;NCIH I@ I@ <IOH>;LS =IH>CNCIHM CM ?HIOAB
NI JLI>O=? MO=B ;H ?LLIL &N CM ;FMI NB? IL>?L I@ G;AHCNO>? I@ >C_?L?H=?M I<N;CH?>
QB?H NB? NC>? A?H?L;NCHA @IL=? CM ;>>?> CHMC>? NB? >IG;CH
BC8<0C8>= F8C7 BDAA>60C4 <>34;B =4DA0; =4CF>A:B 0=3
64=4C82 0;6>A8C7<B
1BCM M?=NCIH ;CGM ;N N?MNCHA N?=BHCKO?M NB;N ;L? HIN <;M?> IH ;H ;>DICHN MSMN?G 1B?
G;CH CHN?L?MN CM NB;N NB? >?N;CFM I@ NB? L?MIFONCIH ;H> NB? JLI=?MM F?;>CHA @LIG NB? L?M
OFNM Q;N?L P?FI=CNC?M ;H> >?JNB NI NB? =IMN @OH=NCIH G;S <? =IHMC>?L?> ;M <F;=E <IR?M
$?H?L;FFS NB?M? N?=BHCKO?M L?KOCL? ; F;LA? HOG<?L I@ CN?L;NCIHM ;H> =;HHIN <? =IH
N?GJF;N?> CH =IGJON;NCIH;F WOC> >SH;GC=M %IQ?P?L LOHHCHA NB? GI>?F NBIOM;H>M
I@ NCG?M G;S HIN <? H?=?MM;LS C@ Q? OM? ; MOLLIA;N? GI>?F NB;N QCFF <? MO<MNCNON?> @IL
NB? MCGOF;NCIH GI>?F @OLNB?L ;>P;HN;A? I@ NB?M? N?=BHCKO?M CM NB;N NB? IJNCGCT;
NCIH =;H <? G;>? QCNB L?MJ?=N NI M?P?L;F =LCN?LC; H;G?FS B?L? NB? NC>? ;GJFCNO>? ;H>
JB;M? ;N P;LCIOM G?;MOL?G?HN JICHNM &H L?;F FC@? MNO>C?M NB? >C_?L?HN =LCN?LC; G;S ;FMI
=ILL?MJIH> NI M?P?L;F WII> =IH>CNCIHM 1BCM MNO>S Q;M >IH? QCNBCH NB? @L;G?QILE I@
NB? ) & JLID?=N >?>C=;N?> NI J;L;G?N?L ?MNCG;NCIH QCNB EHIQF?>A? <;M?> G?NB
I>M QBC=B L;H @LIG *;L=B NI +IP?G<?L ;H> Q;M @OH>?> <S NB? #L?H=B
*CHCMNLS I@ &H>OMNLS ;H> /?M?;L=B ;M J;LN I@ NB? JLIAL;GG? Z/fM?;O +;NCIH;F >?M
1?=BHIFIAC?M )IAC=C?FF?M[ ,NB?L J;LNH?LM Q?L? NB? ?HNL? >Y"NO>?M *;LCNCG?M ?N
#FOPC;F?M "1*"# IGGCMM;LC;N b FY"H?LAC? NIGCKO? " !CP?LA?HN @LIG
NB? 2HCP?LMCNS I@ IGJCeAH? 3?LH? >?L ;H> );/&2-'3 @LIG NB? 2HCP?LMCNS I@
GC?HM ;H> !?FN;=;> =IHMOFN;HNM '?;H *;L= *;LNCH?T $CFF?M LH;O> ;H> *C=B?F
!OG;M @LIG " B;P? B?FJ?> QLCNCHA NBCM M?=NCIH
(DAA>60C4 <>34;B
H?OLIH;F MOLLIA;N? GI>?F CM <OCFN JLCIL NI NB? IJNCGCT;NCIH MN;A? 4? MB;FF L?PC?Q
B?L? NB? G;DIL JLCH=CJF?M I@ NB? G?NBI>IFIAS CH QBC=B NB? MOLLIA;N? GI>?F MOJ?L
M?>?M NB? GI>?F CH NB? M?;L=B @IL NB? <?MN M?N I@ =I?^=C?HNM GCHCGCTCHA NB? >?PC;NCIH
<?NQ??H =IGJON;NCIHM ;H> G?;MOL?G?HNM 3;LCIOM IJNCGCT;NCIH N?=BHCKO?M NB;N ;L?
CH=IHMCMN?HN QCNB NB? GI>?F =IGJON?L NCG?M QCFF NBOM <?=IG? MOCN;<F? INB ZL;Q
@IL=?[ C ? ; L;H>IG M?;L=B ;H> A?H?NC= ;FAILCNBGM G;S <? OM?> 1B? MOLLIA;N?
--)& 1&,+0 GI>?F QCFF JLIPC>? @IL ;HS M?N I@ @LC=NCIH =I?^=C?HNM ; AII> ;JJLIRCG;NCIH I@ NB?
MCGOF;NCIH L?MOFNM CH ; P?LS MBILN -2 NCG? H ?RJ?LCG?HN;F JLINI=IF >?N?LGCH?M
NB? M?N I@ JICHNM NI QBC=B NB? MOLLIA;N? GI>?F QCFF <? VNN?> MI NB;N CN QCFF L?JL?M?HN
NB? ILCACH;F GI>?F ;M ;==OL;N?FS ;M JIMMC<F? &H NB? JL?M?HN MNO>S NB? ?RJ?LCG?HN;F
JLINI=IF JLIPC>?M M?NM I@ P;FO?M @IL NB? VP? @LC=NCIH =I?^=C?HNM =ILL?MJIH>CHA NI
NB? VP? ;L?;M CH NB? B;HH?F #IL ?;=B M?N NB? NC>? MCGOF;NCIH JLIPC>?M L?MOFNM
C ? Q;P? ;GJFCNO>? ;H> JB;M? P;FO?M ;M G?;MOL?> CH ?CABN B;L<IOLM H?OLIH;F
MOLLIA;N? GI>?F CM >?LCP?> @LIG NB?M? M?NM I@ L?MOFNM 1BOM ;=NO;F @OH=NCIHM
I@ NB? VP? @LC=NCIH =I?^=C?HNM ;L? OFNCG;N?FS I<N;CH?> ;H> QCFF G;E? CN JIMMC<F? NI
?MNCG;N? NB? ?CABN NC>;F L;HA?M ;H> NB? ?CABN JB;M?M @IL INB?L M?NM I@ @LC=NCIH =I?@
V=C?HNM 1B? MCGOF;NCIH ?R;GJF?M L?KOCL?> @IL N?;=BCHA NB? H?OLIH;F H?NQILEM Q?L?
MJ?=CV?> <S ; );NCH %SJ?L=O<? 0;GJFCHA )%0 C ? CH>?J?H>?HN L;H>IG P;LC;<F?M
?;=B <?CHA ?P?HFS >CMNLC<ON?> QCNBCH NB? 8 9 L;HA? =ILL?MJIH>CHA NI 0NLC=EF?L
@LC=NCIH =I?^=C?HNM 1B? HOG<?L I@ ?RJ?LCG?HNM Q;M M?N NI C ? 1?F?G;= !
=IGJON;NCIHM MI NB;N NB? G?NBI> =;H <? =IGJ;L?> QCNB NB? ;>DICHN G?NBI> NB;N B;>
;FMI CHPIFP?> =IGJON;NCIHM 1B? H?OLIH;F H?NQILEM ;L? I@ NB? *)- *OFNC );S?L
-?L=?JNLIH NSJ? QCNB ; MCHAF? BC>>?H F;S?L 1B? F?;LHCHA IJ?L;NCIHM Q?L? =;LLC?> ION
IH =?HNL?> ;H> L?>O=?> >;N; CHJONM ;H> IONJONM &N MBIOF> <? JICHN?> ION NB;N ;
ZGI>OFI [ JLI=?MMCHA CM ;JJFC?> NI NB? ?CABN JB;M?M <?CHA =IGJON?> <S 1?F?G;= !
;FF NB? =IGJON?> JB;M?M ;L? NB?H QCNBCH NB? 8 9 L;HA?
(8=6;4 >A <D;C8?;42A8C4A8>= ?A>1;4<
1QI IJNCGCT;NCIH JLI<F?GM B;P? <??H MIFP?>
1B? VLMN IH? CM ; MCHAF? =LCN?LCIH JLI<F?G NB? =IMN @OH=NCIH < N CM NB? MOG I@
MKO;L?> ;GJFCNO>? >?PC;NCIHM ;N NB? ?CABN G?;MOLCHA JICHNM
1B? M?=IH> IH? CM ; >O;F =LCN?LCIH JLI<F?G ;>>CNCIH;FFS ;CGCHA ;N GCHCGCTCHA
NB? MOG < N I@ MKO;L?> JB;M? >?PC;NCIHM
1B? VLMN JLI<F?G CM MIFP?> QCNB >C_?L?HN G?NBI>M NB? @ILG?L ZL;Q @IL=?[ CG
JF?G?HNM ; L;H>IG M?;L=B NB;N CM @I=OMM?> IH NB? GIMN JLIGCMCHA ;L?;M CH NB? M?;L=B
>IG;CH ;M NB? CHP?MNCA;NCIH JLI=??>M NB? F;NN?L OM?M A?H?NC= ;FAILCNBGM 0CH=? NB?
A?H?NC= ;FAILCNBGM CH>O=? =B;HA?M CH ; JIJOF;NCIH I@ CH>CPC>O;FM L?JL?M?HNCHA NB? JI
N?HNC;F MIFONCIHM NI NB? JLI<F?G HIN NB? GCHCGOG I@ @OH=NCIH < <ON NB? M?N I@ JICHNM
N G??NCHA < N < CM M?;L=B?> @IL G;S <? CHN?LJL?N?> ;M ;H IL>?L I@
G;AHCNO>? I@ NB? ?LLIL =;OM?> <S NB? MOLLIA;N? GI>?F 1B?L? QCFF HIN <? ; MCHAF?
MIFONCIH <ON ; M?N I@ JICHNM CH NB? PC=CHCNS I@ NB? MIFONCIH #LIG NB? L?MOFNM I@ NB;N
IJNCGCT;NCIH CN QCFF ;JJ?;L NB;N NB? NIJIFIAS I@ NB;N PC=CHCNS QCFF JLIPC>? P?LS OM?@OF
CH@ILG;NCIH ;<ION NB? <?B;PCIOL I@ NB? MIFONCIHM
1B? M?=IH> JLI<F?G CM MIFP?> <S G?;HM I@ A?H?NC= ;FAILCNBGM "R;GCHCHA NB?
-;L?NI @LIHN QCFF <? I@ NB? ONGIMN CGJILN;H=? @IL ;H;FSMCHA NB? JLI<F?G 1B? -;L?NI
@LIHN CM =IGJIM?> I@ MIFONCIHM QBC=B ;JJ?;L NI <? NB? <?MN ;M L?A;L>M IH? I@ NB?
=LCN?LC; ;GIHA M?NM I@ MIFONCIHM QBC=B ACP? NB? M;G? J?L@ILG;H=? IH INB?L =LCN?LC;
#IL NB? MCHAF? =LCN?LCIH =;M? NB? @IFFIQCHA JLI<F?G CM MIFP?>
#CH> NB? JICHNM N G??NCHA
< N < QCNB < ;H>
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
cost
friction of zone 4
#CAOL? =FOMN?L I@ MIFONCIHM @IL =I?^=C?HN I@ ;L?; 1B;N JLI<F?G CM =F;MMC=;FFS @ILGOF;N?> ;M ;H IJNCGCT;NCIH JLI<F?G OH>?L =IH
MNL;CHN 1B? MIFONCIHM NI NB;N JLI<F?G ;L? >CMJF;S?> CH NB? @ILG I@ M=;NN?L JFINM
&H #CAOL? NB? @LC=NCIH =I?^=C?HN I@ TIH? CM NB? ;<M=CMM; ;H> NB? =IMN CM NB?
IL>CH;N? %?L? NB? @LC=NCIH =I?^=C?HN I@ H?;LFS ;FF MIFONCIHM CM =F?;LFS CH NB? L;HA?
8 9 ,NB?L =;M?M MG;FF TIH?M @IL ?R;GJF? MBIQ P?LS M=;NN?L?> L?MOFNM &H S?N
INB?L =;M?M ; <CGI>;F MIFONCIH ;JJ?;LM QCNB NQI JIMMC<F? P;FO?M -FINM QB?L? IH?
=I?^=C?HN CM NB? ;<M=CMM; ;H> ;HINB?L NB? IL>CH;N? MBIQ JIMMC<F? =ILL?F;NCIHM ;M CH
#CAOL? QCNB =I?^=C?HNM I@ ;L?;M ;H> %?L? Q? B;P? ; H?A;NCP? =ILL?F;NCIH
BCAB P;FO?M I@ IH? =I?^=C?HN =ILL?MJIH>CHA NI FIQ P;FO?M I@ NB? INB?L
#IL NB? >O;F =LCN?LCIH JLI<F?G NB? JOLJIM? CM HIQ NI VH> ; M?N I@ @LC=NCIH =I?^
=C?HNM QBC=B GCHCGCT?M NQI @OH=NCIHM < N ;H> < N QB?L? < N >?HIN?M NB? MOG
I@ MKO;L?> ;GJFCNO>? >?PC;NCIHM ;N ;FF ?CABN G?;MOLCHA JICHNM ;H> < N CM NB? MOG I@
MKO;L?> JB;M? >?PC;NCIHM #CAOL? MBIQM NB? QBIF? -;L?NI @LIHN "LLILM IH NC>;F
L;HA? CH G?NL?M ;L? NB? ;<M=CMM; ;H> ?LLILM IH JB;M?M CH >?AL??M ;L? NB? IL>CH;N?
1B? @LIHN CM MJFCN CHNI NQI J;LNM <S ; ECH> I@ >CM=IHNCHOCNS 1B? J;LN IH NB? F?@N
=ILL?MJIH>CHA NI NB? FIQ?L NC>;F L;HA? ?LLILM M??GM NI <? NB? IH? GIMN J?LNCH?HN NI
NB? JLI<F?G 1B? ?LLIL IH < N P;LC?M <S ; @;=NIL 1B? NQI =LCN?LC; ;L? =IHNL;>C=N
ILS ;H> NB? GCHCG; I@ ?;=B @OH=NCIH ;L? I<N;CH?> @IL KOCN? >C_?L?HN M?NM I@ @LC=NCIH
=I?^=C?HNM ,H INB?L ?R;GJF?M CN ;JJ?;LM NB;N NB? @OH=NCIHM ?RBC<CN >CM=IHNCHOCNC?M
;H> ;L? HIN GIHINIHC=
'4BD;CB 0=3 0=0;HB8B
1B? M?=IH> ;H> NBCL> =IFOGHM I@ 1;<F? ACP? NB? 0NLC=EF?L @LC=NCIH =I?^=C?HNM
I<N;CH?> L?MJ?=NCP?FS QCNB ; L;H>IG M?;L=B ;H> QCNB A?H?NC= ;FAILCNBGM #IL NB?
F;NN?L ; JIJOF;NCIH I@ CH>CPC>O;FM ;H> A?H?L;NCIHM B;P? <??H OM?> QBC=B
--)& 1&,+0 friction of zone 5
friction of zone 1
#CAOL? =ILL?F;NCIH <?NQ??H NQI @LC=NCIH =I?^=C?HNM
CGJFC?> ;MM?MMG?HNM
1;<F? @LC=NCIH =I?^=C?HNM I<N;CH?> QCNB MOLLIA;N? GI>?FM
7IH?
/;H>IG M?;L=B
$?H?NC= ;FAILCNBGM
1;<F? ;H> 1;<F? ACP? NB? H?Q P;FO?M I<N;CH?> @IL NB? ;GJFCNO>?M ;H>
JB;M?M
&N CM VLMN I<M?LP?> NB;N NB? M?NM I@ @LC=NCIH =I?^=C?HNM ;=BC?P?> NBLIOAB NB? P;LCIOM
G?NBI>M ;L? L;NB?L >C_?L?HN 1BOM CN ;JJ?;LM NB;N M?P?L;F MIFONCIHM =;H <? @IOH> 4?
B;P? M??H NB;N NB? MN;H>;L> >?PC;NCIHM IP?L ;GJFCNO>?M ;H> JB;M?M QCNB NB? G;HO;F
M?NNCHA Q?L? =G ;H> >?AL??M L?MJ?=NCP?FS ;H> NB;N NB? ;>DICHN G?NBI> QCNB
=IGJON;NCIHM <LIOABN ;H CGJLIP?G?HN MCH=? NB? >?PC;NCIHM MBC@N >IQH NI =G
;H> >?AL??M &H =IGJ;LCMIH NB? MOLLIA;N? GI>?F <;M?> N?=BHCKO?M B;JJ?H NI <?
;NNL;=NCP? M ; G;NN?L I@ @;=N Q? A?N
/?@?L?H=? ?RJ?LN ?MNCG;NCIH ;>DICHN G?NBI> =G ;H> >?AL??M
/;H>IG M?;L=B M?P?L;F GCFFCIH L;H>IG M?F?=NCIHM =G ;H> >?AL??M
$?H?NC= ;FAILCNBGM =G ;H> >?AL??M
FF NB? L?MOFNM ;L? <?NN?L NB;H NB? L?@?L?H=? &N M??GM ;@N?L INB?L N?MNM QCNB ?CABN
;L?;M HIN L?JILN?> B?L? NB;N NB? >C_?L?H=?M ;GIHA NB? P;LCIOM G?NBI>M IHFS ?R?LN ;
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
error on phase
error on tidal range
#CAOL? -;L?NI @LIHN CH ; >O;F =LCN?LCIH JLI<F?G
MG;FF ?_?=N IH NB? ;==OL;=S <?CHA ;=BC?P?> NBIOAB NB?S F?;> NI L;NB?L >C_?L?HN M?NM
I@ =I?^=C?HNM ;H> NBCM IH=? ;A;CH L;CM?M NB? KO?MNCIH I@ FI=;F GCHCG; 0BC@NCHA OJ NI
?CABN ;L?;M IL NI >O;F =LCN?LCIH IJNCGCT;NCIH ;=NO;FFS <LCHAM FCNNF? CGJLIP?G?HN ;H>
L?KOCL?M GIL? ?RJ?LNCM?
>=2;DB8>= 0=3 ?A>B?42CB
1BIM? G?NBI>M <;M?> IH ; GCHCGCT;NCIH @LIG ; L?MJIHM? MOL@;=? L?P?;F B?L? NB;N ;N
NB? M;G? =IGJON;NCIH;F =IMN NB?S =;H JLIPC>? <?NN?L L?MOFNM #CH>CHA M?NM I@ VP?
=I?^=C?HNM NB;N ;L? <?NN?L NB;H ;H IJNCGCT;NCIH CHPIFPCHA ;H ?RJ?LN @IL MCR GIHNBM
;H> ; NB?MCM CM ; P?LS JLIGCMCHA ;=BC?P?G?HN 1B? JLCG;LS ;>P;HN;A? I@ NB? H?Q
G?NBI>M <?CHA N?MN?> B?L? CM NB;N NB?S =IHMC>?L NB? ?P;FO;NCIH I@ NB? =LCN?LC; ;H> NB?
=IGJON;NCIH;F ;FAILCNBGM ;M <F;=E <IR?M QB?L?;M NB? ;>DICHN G?NBI>M MBIOF> @IFFIQ
NB?M? ;FAILCNBGM ;M =FIM?FS ;M JIMMC<F? NBOM B?;PCFS <OL>?HCHA <INB >?P?FIJG?HN ;H>
G;CHN?H;H=? 1B?M? H?Q G?NBI>M BIQ?P?L ;FMI CGJFS ; BOG;H MECFF ?MJ?=C;FFS QCNB
M?P?L;F =LCN?LC; ? A <S >CMJF;SCHA ;H> ;H;FSMCHA NB? -;L?NI @LIHNM 1B? ION=IG?
I@ NB;N ;JJLI;=B QCFF NB?H >?J?H> IH ?CNB?L NB? OM?L @LC?H>FCH?MM IL NB? L?KOCL?>
EHIQF?>A? F?P?F @IL CGJF?G?HNCHA NB?G CH L?;F FC@? =IH>CNCIHM
(>;8C0AH F0E4
0IFCN;LS Q;P?M JLIJ;A;NCHA QCNBION >CMNILNCIH ;L? HINC=?> ;N NCG?M CH NB? M?; IL CH ;
=;H;F ;@N?L NB? MO>>?H =FIMCHA I@ ; FI=E &H );CNIH? 89 JLIJIM?> ;H ;JJLIR
CG;NCIH NI NB? VLMN IL>?L =IGJ;L?> QCNB M?? <?FIQ I@ MO=B ; Q;P? QBC=B CM ;
J;LNC=OF;L FCGCN =;M? I@ ; =HIC>;F Q;P? &H ; =;H;F QCNB ; =IHMN;HN >?JNB ;FIHA ;RCM
APPLICATIONS 295
Table 10.13: magnitude of wave M2 obtained with surrogate models.
Measurement point
Zeebrugge
Calais
Boulogne
Dieppe
Cherbourg
Saint-Malo
Buoy C1
Ouessant
Measurement
1.59 m
2.46 m
2.93 m
3.08 m
1.87 m
3.74 m
1.29 m
2.06 m
Random
1.63
2.39
2.93
3.03
1.94
3.65
1.29
1.98
search
m
m
m
m
m
m
m
m
Genetic algorithm
1.66 m
2.42 m
2.95 m
3.04 m
1.93 m
3.65 m
1.29 m
1.98 m
Table 10.14: phase of wave M2
Measurement point Measurement (degrees) Random search Genetic algorithm
Zeebrugge
14
14.79
14.63
Calais
345
343.78
343.94
Boulogne
331
332.31
332.84
Dieppe
311
313.49
314.300
Cherbourg
228
226.56
226.95
Saint-Malo
178
175.70
178.18
Buoy C1
103
105.41
105.40
Ouessant
111
114.68
114.77
Ox, this solution is as follows:
U (x, t) =
A
gH0
H0 cosh(
W (x, z, t) = 3gH0
A
H0
3/2
1
3A
(x
4H03
− ct))2
3A
tanh(
(x − ct))
4H 3
z
0
H0 cosh( 3A3 (x − ct))2
4H
(10.1)
(10.2)
0
h(x, t) = H0 +
cosh(
A
3A
(x
4H03
− ct))2
(10.3)
where H0 is the reference depth. The magnitude of the wave is A, and its celerity is:
c = g(H0 + A)
Second-order [20] and third-order approximations [180] have also been found since
Laitone’s first publication.
With a depth of 10 m and a magnitude A of 2 m, we are in the application domain
of the Boussinesq equations. On the other hand, the Saint-Venant equations reveal
non-physical behaviour and the solitary wave is deformed and tends to break. A series
of tests has been carried out in a 600 m long canal. The initial condition represents a
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
MIFCN;LS Q;P? QBIM? G;RCGOG CM ;FL?;>S G ;Q;S @LIG NB? F?@N ?HNLS I@ NB? =;H;F
C ? ;N ; >CMN;H=? MO=B NB;N NB? Q;N?L >?JNB ;N NB? ?HNLS CM F?MM NB;H NB I@ CNM
G;AHCNO>? 1B? =;F=OF;NCIH =IHMCMNM I@ CN?L;NCIHM I@ M ;@N?L QBC=B NB? Q;P?
A?NM =FIM? NI NB? IONF?N IH NB? LCABN I@ NB? =;H;F 1B? MCT? I@ ?F?G?HNM CM ;<ION G
QBC=B ACP?M ; IOL;HN HOG<?L =;F=OF;N?> @LIG NB? Q;P? =?F?LCNS I@ NB? IL>?L I@ 1B? @IFFIQCHA B;P? <??H N?MN?>
1B? 0;CHN 3?H;HN ?KO;NCIHM
1B? IOMMCH?MK ?KO;NCIHM
1B? +;PC?L\0NIE?M BS>LIMN;NC= ?KO;NCIHM
1B? +;PC?L\0NIE?M HIH BS>LIMN;NC= ?KO;NCIHM QCNB P;LCIOM HOG<?LM I@ JLCM
G;NC= ?F?G?HN F;S?LM
1B? @;CFOL? I@ NB? BS>LIMN;NC= BSJINB?MCM CM HINC=?> QCNB NB? 0;CHN 3?H;HN ;M Q?FF ;M
QCNB NB? +;PC?L\0NIE?M ?KO;NCIHM 1B? IOMMCH?MK ?KO;NCIHM ;H> NB? HIH BS>LIMN;NC=
+;PC?L\0NIE?M ?KO;NCIHM ACP? =IGJ;L;<F? L?MOFNM 1B? Q;P? CM HIN ?R;=NFS MIFCN;LS ;H>
;N NCG?M F?;P?M CH CNM Q;E? MIG? OH>OF;NCIHM QBC=B GOMN <? >O? NI NB? @;=N NB;N IOL
CHCNC;F =IH>CNCIHM ;L? IHFS ;H ;JJLIRCG;NCIH I@ ; MIFCN;LS Q;P? ;H> ;FMI NI NB? @;=N
NB;N Q? B;P? IHFS ; VLMN IL>?L ;JJLIRCG;NCIH
-F;N? MBIQM MO==?MMCP? JLIVF?M I@ NB? Q;P? ;N >C_?L?HN NCG?M I@ ;H> M?=IH>M QCNB @IOL NSJ?M I@ ?KO;NCIHM 1;<F? =IGJ;L?M NB? =;F=OF;NCIH
NCG?M QCNB ; NQI >CG?HMCIH;F G?MB G;>? I@ JICHNM ;H> ?F?G?HNM QBC=B CM
L?JLI>O=?> ;FIHA NB? P?LNC=;F @IL NBL?? >CG?HMCIH;F ?KO;NCIHM L?G;LE;<F? L?MOFN CM
NB? L;NB?L AII> <?B;PCIOL I@ NB? HIH BS>LIMN;NC= ?KO;NCIHM QCNB IHFS NQI =;F=OF;NCIH
JF;H?M QBC=B ;L? NB? <INNIG ;H> @L?? MOL@;=? &H NBCM =;M? NB? P?LNC=;F P?FI=CNS CM
FCH?;L <?NQ??H NB? <INNIG ;H> @L?? MOL@;=? QBC=B MOAA?MNM ;H ;H;FIAS QCNB NB? 0?LL?
?KO;NCIHM 0?=NCIH 1B?L?@IL? NB? HIH BS>LIMN;NC= ?KO;NCIHM ;JJ?;L NI <? F?MM
NCG? =IHMOGCHA NB;H NB? IOMMCH?MK ?KO;NCIHM $CP?H NB? FCGCN;NCIHM I@ NB? >IG;CH
I@ P;FC>CNS I@ NB? IOMMCH?MK ?KO;NCIHM ;H> ?P?H C@ @;MN?L MIFP?LM =;H OH>IO<N?>FS
<? =IHMNLO=N?> @LIG NBCM CN CM J?LB;JM QCM?L NI =IH=?HNL;N? L?M?;L=B B?H=?@ILNB IH
NB? HIH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM L;NB?L NB;H NI QILE IH NB? IOMMCH?MK
?KO;NCIHM QCNB BCAB?L ;H> BCAB?L IL>?LM QBC=B ;L? >IIG?> NI <?=IG? I<MIF?N? MIIH?L
IL F;N?L >O? NI JLIAL?MM CH G;=BCH?M ;H> HOG?LC=;F ;H;FSMCM )C ;H> #F?GCHA 89
;FL?;>S G?HNCIH?> ;FMI L?;=B?> NB? M;G? =IH=FOMCIHM CH ';HO;LS QCNB ; VHCN?
>C_?L?H=? GI>?F
1;<F? =IGJON;NCIH NCG? CH NB? =;M? I@ ; MIFCN;LS Q;P?
"KO;NCIHM
0;CHN 3?H;HN
IOMMCH?MK
! BS>LIMN;NC= JF;H?M
! HIH BS>LIMN;NC= JF;H?M
! HIH BS>LIMN;NC= JF;H?M
! HIH BS>LIMN;NC= JF;H?M
IGJON?L NCG?
M
M
M
M
M
M
--)& 1&,+0 $OCFFIO ( 89 ;N NB? 2HCP?LMCNS I@ ;?H B;P? ;FMI JL?M?HN?> NBCM =;M? I@ ;
MIFCN;LS Q;P? ;M ;H ;JJFC=;NCIH I@ ; HOG?LC=;F ! GI>?F QCNB VHCN? >C_?L?H=?M OMCHA
NB? MCAG; NL;HM@ILG ;H> BILCH JLID?=NCIH G?NBI> @IL =IHNCHOCNS
;>F >5 CF> KD83B >5 38Q4A4=C 34=B8C84B
L?=N;HAOF;L N;HE G FIHA G QC>? ;H> G >??J =IHN;CHM NQI WOC>M ;N L?MN QCNB
MFCABNFS >C_?L?HN >?HMCNC?M <?=;OM? I@ NB?CL M;FCHCNS AF CH NB? F?@N B;F@ ;H> AF CH NB?
LCABN B;F@ 1B? L?MJ?=NCP? >?HMCNC?M ;L? EAG ;H> EAG
NB? Q;N?L CM ;N >?AL??M ?FMCOM 1B? WIQ CM AIP?LH?> <S >?HMCNS ?_?=NM 1B?
B?;PS WOC> MCHEM <?FIQ NB? FCABN?L IH? <S @ILGCHA ; M;FCH? Q?>A? QBC=B ;==IL>CHA NI
I<M?LP;NCIHM G;E?M ;H ;HAF? I@ ;<ION QCNB NB? <INNIG 1B? FCABN WOC> MJL?;>M
IP?L NB? MOL@;=? 1BCM =;M? MBIQM NB? P;FO? I@ ; HIH BS>LIMN;NC= MCGOF;NCIH QBC=B ACP?M
; GO=B GIL? JBSMC=;F MIFONCIH NB;H NB? @LIHN I@ L?=N;HAOF;L MB;J? I@ NB? BS>LIMN;NC=
MIFONCIH &@ Q? =IHMC>?L NB;N NB? QBIF? JIN?HNC;F ?H?LAS I@ NB? CHCNC;F =IH>CNCIHM CM
NL;HM@ILG?> CHNI ECH?NC= ?H?LAS Q? =;H MBIQ NB;N NB? @LIHN P?FI=CNS CM QB?L?
7
=
>
1B? G?;MOL?G?HNM MBIQ ; MFCABN ;MSGG?NLS ; P?FI=CNS I@ @IL NB? @LIHN ;N
NB? <INNIG ;H> IH NB? MOL@;=? 1B? =LIMM M?=NCIHM I@ -F;N? =ILL?MJIH> NI
NB? NCG? J M C ? ;@N?L NCG? MN?JM I@ M?=IH> 1B? G?MB CM =IGJIM?> I@
BILCTIHN;F JF;H?M ?;=B =IHN;CHCHA NLC;HAF?M CH MO=B ; Q;S NB;N NB? G?MB MCT?
CM G IH NB? BILCTIHN;F ;M Q?FF ;M IH NB? P?LNC=;F %?L? NB? NB?IL?NC=;F P?FI=CNS
CM GM NB? ;P?L;A? P?FI=CNS HOG?LC=;FFS I<N;CH?> CH HIH BS>LIMN;NC= GI>?
<?CHA ;LIOH> GM C ? ;H ?LLIL I@ ;<ION 1BCM OH>?L?MNCG;N?> L?MOFN Q;M
JL?>C=N;<F? <?=;OM? Q? MN;LN @LIG ; HIH ?MN;<FCMB?> MCNO;NCIH QCNB T?LI P?FI=CNS
QBC=B G;E?M NB? =;F=OF;NCIH CH=ILL?=N ,H NB? =IHNL;LS Q? =;H VH> NB? ;HAF? I@ <?NQ??H NB? @LIHN ;H> NB? <INNIG 1B? BS>LIMN;NC= IJNCIH QCNB ;H ;P?L;A? P?FI=CNS
I@ GM ;H> ;H ;HAF? =FIM? NI >?AL??M CM >CMKO;FCV?> @IL NBCM NSJ? I@ WIQ
1B? L?;>?L CHN?L?MN?> CH GCRNOL?M I@ WOC>M I@ >C_?L?HN >?HMCNC?M ;H> CH NB? NB?ILS I@
CHN?L@;=C;F Q;P?M QCFF VH> ; FIN I@ CH@ILG;NCIH CH L?@?L?H=?M 89 89 89 ;H> 89
??4=38G )83464=4A0C8=6 5>A24
()& )"+& # (' '$ "#)& " #&-
#' (# (' $'( +(#)( '& #& !!#&-
" (# & &(#&
' "#(" !#& (" $") )! (( #' ('
" '(&' " ( "."(- # (! " '$
&-(! # +(
#LC?>LC=B PIH 0=BCFF?L
1B? >C_?L?H=? <?NQ??H NB? JOFF ?R?LN?> <S ; B?;P?HFS <I>S C ? NB? GIIH IL NB?
MOH ;N NB? =?HNL? I@ NB? ";LNB ;H> NB? JOFF ;N ; JICHN IH NB? MOL@;=? CM L?MJIHMC<F?
@IL NB? JB?HIG?HIH I@ NC>?M 1B? NC>? A?H?L;NCHA @IL=? CM ?RNL?G?FS Q?;E <ON CM
;JJFC?> IP?L ?HILGIOM >CMN;H=?M 1B? Q;P?M JLI>O=?> ;L? GIL?IP?L >?@ILG?> ;H>
;GJFCV?> <S I=?;HC= <;MCHM NBOM P;LC;NCIHM CH Q;N?L F?P?FM I@ G ;L? I<M?LP?> QBCF?
; MN;NC= NB?ILS QIOF> ACP? P;FO?M HIN ?R=??>CHA G #IL MG;FF >IG;CHM MO=B ;M @IL
?R;GJF? NB? "HAFCMB B;HH?F NB? ?MM?H=? I@ NB? NC>? CM >O? NI NB? Q;P?M =L?;N?> CH
NB? F;LA? I=?;HM 1B?L?@IL? @IL HOG?LC=;F MCGOF;NCIHM CH NB? "HAFCMB B;HH?F Q?
B;P? I@N?H H?AF?=N?> NB? NC>? A?H?L;NCHA @IL=? CHMC>? NB? >IG;CH NI =IHMC>?L IHFS NB?
Q;P?M =IGCHA @LIG NB? NF;HNC= ,=?;H IL @LIG NB? +ILNB 0?; 1B? F;LA?L NB? >IG;CH
NB? F?MM NB? DOMNCV=;NCIH @IL H?AF?=NCHA NB? NC>? A?H?L;NCHA @IL=? CH NB? CHN?LCIL I@ NBCM
>IG;CH 1;ECHA CHNI ;==IOHN NBCM @IL=? CH NB? 0;CHN 3?H;HN ?KO;NCIHM NB?H <?=IG?M
CH>CMJ?HM;<F?
1B? NC>? A?H?L;NCHA @IL=? >?LCP?M @LIG ; JIN?HNC;F NB;N =;H <? M?J;L;N?> CHNI ;
MN;NC= J;LN ;H> ; >SH;GC= J;LN 1B? >SH;GC= J;LN >?J?H>M IH NB? MN;N? I@ NB? NC>?
;H> =;HHIN MCGJFS <? N;E?H CHNI ;==IOHN 1B? MN;NC= J;LN CNM?F@ CM MJFCN CHNI
H I=?;HC= MN;NC= NC>? <ILH >CL?=NFS @LIG NB? CHWO?H=? I@ NB? GIIH ;H> NB?
MOH
N?LL?MNLC;F NC>? @LIG >?@ILG;NCIH I@ NB? ";LNB OH>?L NB? CHWO?H=? I@ NB? NC>?
0N;NC= >CMNOL<;H=?M >O? NI NB? CHWO?H=? I@ NB? N?LL?MNLC;F NC>? IH NB? >CMNLC<ONCIH
I@ Q;N?L <I>C?M
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
z
A
∆
T θ
x
r
M
y
#CAOL? T?HCNB >CMN;H=?
1B? F;MN NQI N?LGM L?MOFN @LIG NB? VLMN IH? CH ; FCH?;L G;HH?L QCNB NB? B?FJ
I@ )IP? HOG<?LM 89 &@ "D CM NB? I=?;HC= MN;NC= NC>? > "D CM NB? @IL=? >O? NI
NB? N?LL?MNLC;F NC>? > 1B? @IL=? >O? NI NB? MN;NC= >CMNOL<;H=?M CM A "D
A 1B? NIN;F MN;NC= @IL=? NB?L?@IL? CM "D
1B? JOLJIM? I@ NBCM ;JJ?H>CR CM NI MBIQ ;FF NB? >;N; @ILGOF;? L?;MIHCHA ;H>
;JJLIRCG;NCIHM NB;N ;FFIQ OM NI =;LLS ION JLIJ?LFS NB? =;F=OF;NCIH I@ NB? I=?;HC=
MN;NC= NC>? #CLMN NB? A?H?L;F ?RJL?MMCIH I@ @IL=?M I@ AL;PCNS CM ?MN;<FCMB?> @IL ;HS
B?;P?HFS <I>S >?J?H>CHA IH NB? G;MM >CMN;H=? @LIG NB? ";LNB >?=FCH;NCIH ;H> LCABN
;M=?HMCIH I@ NB? B?;P?HFS <I>S 1B?M? @IOL J;L;G?N?LM ;L? ACP?H CH >?N;CF F;N?L IH @IL
NB? GIIH ;H> NB? MOH
$240=82 BC0C82 C834
%>C4=C80; 34?4=38=6 >= C74 64>24=CA82 I4=8C7 38BC0=24
G02C 4G?A4BB8>=
)?N OM MOJJIM? NB;N Q? ;L? IH ; JF;H? =IHN;CHCHA NB? =?HNL? I@ NB? ";LNB 1 ; JICHN
IH CNM MOL@;=? ;H> ; B?;P?HFS <I>S * @IL *IIH 1B? ;HAF? 1 1* >?HIN?>
CM NB? T?HCNB >CMN;H=?
4? >?HIN? NB? >CMN;H=? * ;H> H NB? >CMN;H=? 1* M?? #CAOL? 7 CM NB?
L;>COM I@ NB? ";LNB
L?@?L?HNC;F CM M?F?=N?> 1 N O P FCHE?> NI NB? ";LNB QB?L? N ;H> O ;L? IH NB?
?KO;NIL ;H> P IH NB? ;RCM DICHCHA NB? =?HNL? I@ NB? ";LNB ;H> NB? +ILNB -IF? 1B?
=IIL>CH;N?M I@ ;L? N O P 1B? =IIL>CH;N?M I@ * ;L? N" O" P"
==?F?L;NCIH I@ CM >O? NI
==?F?L;NCIH I@ 1
?HNLC@OA;F @IL=?
$L;PCNS I@ NB? ";LNB
$L;PCNS I@ NB? B?;P?HFS <I>S
APPENDIX A 1&!" $"+"/1&+$ #,/ "
1B? NC>? A?H?L;NCHA @IL=? CM <S >?VHCNCIH NB? @IL=? ;=NCHA IH ;H> >?J?H>CHA IH
* C?
#C" *
#C"
" H
*
1*
"
1*
"
QB?L? # CM NB? =IHMN;HN I@ OHCP?LM;F AL;PCN;NCIH ?KO;F NI G M ;H> C"
NB? G;MM I@ NB? B?;P?HFS <I>S
# =;H <? =;F=OF;N?> @LIG NB? N?LL?MNLC;F AL;PCN;NCIH = MCH=? = ;
QB?L? C(
/
CM NB? G;MM I@ NB? ";LNB
1B? @IL=?M " ;H> " ?;=B >?LCP? @LIG ; JIN?HNC;F 2 ;H> 2 CH MO=B ; Q;S NB;N
" =H7: 2
2
$CP?H C C C ;M NB? =IIL>CH;N?M I@ NB? P?=NIL
NB? =IIL>CH;N?M I@ NB?
1 P?=NIL 1 1* 1* $CP?H F F F ;M
-IN?HNC;F 2 ;N JICHN CM
#C"
C N
H
P 7 F MI
2 N O P +IQ N 7 F
O 7 F
#C"
7F C
H
NB? >CMN;H=? CM ?KO;F NI
5
N N"
2 F C
C O
F C O O"
C P
#C"
7 "$
H
P P"
;H> IH? =;H ?;MCFS =IH=FO>? @LIG NBCM NB;N JIN?HNC;F 2 CM ?KO;F NI
NB? JIN?HNC;F I@ NB? A?H?L;NCHA @IL=? CM
7
2 #C"
"$
H
QCNB H
7 7H "$
;
1B?L?@IL?
QBC=B ACP?M
#C" 6
2 H
7
"$
H
/
@
/@
"$
4E4;>?43 4G?A4BB8>=
0CH=? NB? L;>COM I@ NB? ";LNB CM MG;FF CH L?F;NCIH NI NB? >CMN;H=? I@ NB? B?;P?HFS <I>S
NB? >?P?FIJG?HN CM >IH? CH L?F;NCIH NI /@
&@ 7 H Q? =;H QLCN? QCNB )?A?H>L? JIFSHIGC;FM
6
/
@
/@ "$
,
7
H
<
<
,< "$
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
z
Fv
A
∆
Fh
T
r
M
y
x
#CAOL? BILCTIHNB;F ;H> P?LNC=;F @IL=?M
QCNB , N , N N , N 1B? L?=OLMCIH @ILGOF; CM
F ;H> , N F F
D D
,< N ,< N
D D 1BOM Q? VH> MNIJJCHA ;N NB? N?LG CH /@ ,<
2 N #C"
H
#C"
H
#C"
H
7
H
7
H
"$
"$
"$
1B? >?P?FIJ?> ?RJL?MMCIH A?H?L;FFS =BIM?H CM
2 #C"
H
7
H
"$
4? >I HIN N;E? CHNI ;==IOHN NB? N?LG ;@ QBC=B CM CH>?J?H>?HN I@ NB? JIMCNCIH
I@ NB? >?VHCNCIH I@ 2 CH=FO>?M ;H ;L<CNL;LS =IHMN;HN Q? OM? IHFS CNM >?LCP;NCP?
;H> Q? CAHIL? NB? N?LG CH /@ L?MJIHMC<F? @IL ; >COLH;F Q;P? ;H> NBCL> >COLH;F
+4AC820; 0=3 7>A8I>=C0; 2><?>=4=CB >5 C74 5>A24 <06=8CD34
&H NB? 1* JF;H? NB? P?LNC=;F @IL=? "D CM
#CAOL? 1BOM
3?LNC=;F @IL=? ?R;=N ?RJL?MMCIH
#C"
"D H
7
7
"$
H
H
*
/
;H> NB? BILCTIHN;F @IL=? "6 CM / *
"$
7
"$
H
APPENDIX A 1&!" $"+"/1&+$ #,/ "
3?LNC=;F @IL=? >?P?FIJ?> ?RJL?MMCIH
#C" 7
"$
H
H
%ILCTIHN;F @IL=? ?R;=N ?RJL?MMCIH
7
7
#C"
$!
"$
"6 H
H
H
"D %ILCTIHN;F @IL=? >?P?FIJ?> ?RJL?MMCIH
#C" 7
$! H
H
1BCM CM NB? BILCTIHN;F =IGJIH?HN I@ NB? @IL=? NB;N CM NB? =;OM? I@ NB? NC>? 1B?
P?LNC=;F =IGJIH?HN CM CH @;=N =IGJF?N?FS G;ME?> <S AL;PCNS CN CM ;<ION GCFFCIH
NCG?M Q?;E?L NB;H AL;PCNS
"6 %>C4=C80; 34?4=38=6 >= C74 ;0C8CD34 342;8=0C8>= 0=3
7>DA 0=6;4
G?A4BB8>= 5>A C74 64>24=CA82 I4=8C7 38BC0=24
1B? A?I=?HNLC= T?HCNB >CMN;H=? CM NB? MCGJF?MN J;L;G?N?L @IL ?MN;<FCMBCHA JIN?HNC;F
4? GOMN HIQ ?RJL?MM CN >?J?H>CHA IH GIL? ?;MCFS ;==?MMC<F? ;HAF?M
P: pole
π/2-δ
π/2-ϕ
T
m
A
θ
#CAOL? >?=FCH;NCIH
M: heavenly body
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&H #CAOL? NB? JICHNM - ;H> G <?FIHA NI NB? MOL@;=? I@ NB? ";LNB ;H> NB?
MJB?LC=;F NLC;HAF? -G B;M MC>?M
- ;HAF? QB?L? CM NB? F;NCNO>? I@ G- ;HAF? QB?L? CM NB? >?=FCH;NCIH I@ NB? B?;P?HFS <I>S ?KOCP;F?HN NI ;
F;NCNO>?
G ;HAF? NB? A?I=?HNLC= T?HCNB >CMN;H=?
P;LC?M @LIG NI ;JJLIRCG;N?FS @IL NB? MOH ;H> @LIG NI ;JJLIRCG;N?FS @IL NB? GIIH
%IOL ;HAF? % CM >?VH?> IH ; JLID?=NCIH IH NB? ?KO;NIL M?? #CAOL? P lane of t he equat or
M
AH
T
A
#CAOL? BIOL ;HAF?
4? MB;FF M?? CH 0?=NCIH NB;N NBCM BIOL ;HAF? % CM ?KO;F NI J:=1 QB?L?
J:=1 CM NB? FI=;F MC>?L?;F NCG? I@ JF;=? ;H> NB? LCABN ;M=?HMCIH I@ NB? B?;P?HFS
<I>S M?? ;FMI #CAOL? #IL NB? NLC;HAF? -G MJB?LC=;F NLCAIHIG?NLS NBOM ACP?M OM
"$
$! $! "$ "$ "$ $
)?N OM L?=;FF NB;N @IL ; MJB?LC=;F NLC;HAF? QCNB P?LNC=?M QB?L? NB? MC>?M 7
8 ;H> 9 M?? #CAOL? ;L? >?VH?> <S ;HAF?M ? A 9 CM NB? ;HAF? , QB?L? , CM
NB? =?HNL? I@ NB? MJB?L? Q? B;P?
0CH? F;Q
$! 7
$! 8
$! 9
$! $! $! IMCH? F;Q
"$ 7 "$ 8 "$ 9
$! 8 $! 9 "$ "$ "$ "$ $! $! "$ 7
#4F 5>A<D;04 5>A C74 ?>C4=C80; F0E4 "
S L?JF;=CHA "$
<S CNM H?Q ?RJL?MMCIH CH NB? @ILGOF; @IL NB? JIN?HNC;F Q? QIOF> I<
N;CH ; H?Q ?R;=N @ILGOF; QBC=B CM HIN QLCNN?H B?L? IGJF?N? QLCNCHA I@ NB? >?P?FIJ?>
APPENDIX A 1&!" $"+"/1&+$ #,/ "
A
c
b
B
C
a
#CAOL? MJB?LC=;F NLC;HAF?
?RJL?MMCIH IH NB? INB?L B;H> ;FFIQM OM NI MBIQ M?P?L;F NSJ?M I@ NC>;F =IGJIH?HNM
4? VH>
$! #C" 7
$! 2 H
H
#C"
H
7
H
$! $! "$ % #C" 7 "$ "$ "$ %
H
H
1B? N?LGM CH "$ % ;H> "$ % ;L? >COLH;F ;H> M?GC >COLH;F Q;P?M L?MJ?=N
CP?FS 1B? M?GC >COLH;F Q;P? CM =;FF?> NB? * Q;P? ;H> NB? P;FO? I@ CNM JIN?HNC;F
CM
2 #C"
H
7
H
"$ "$ "$ %
><?>=4=CB >5 C74 64=4A0C8=6 5>A24
1B? P?LNC=;F =IGJIH?HN B;M JL?PCIOMFS <??H ACP?H B?L? CM NB? J;LN I@ NBCM @IL=? >O?
NI NB? * Q;P?
#C" 7
"$ "$ "$ %
H
H
,H NB? INB?L B;H> NB? @ILG ACP?H @IL NB? BILCTIHN;F =IGJIH?HN >I?M HIN ;FFIQ OM
NI L?F;N? CN NI ; EHIQH L?@?L?HNC;F QBC=B Q? ;L? AICHA NI >I HIQ =NO;FFS NB? Q?MN\
?;MN ;H> MIONB\HILNB =IGJIH?HNM I@ NB? BILCTIHN;F @IL=? ;L? >?LCP;NCP?M L?MJ?=NCP?FS
I@ NB? JIN?HNC;F CH L?F;NCIH NI NB? J;L;FF?F ;L= ;H> G?LC>C;H
*
4?MN\?;MN =IGJIH?HN "F / GCHOM MCAH @IL % CM >CL?=N?> NIQ;L>M
NB? ?;MN\Q?MN >CL?=NCIH
0IONB\HILNB =IGJIH?HN "G / *
"D %6!/,!6+*& 0 ,# #/"" 02/# " #),40
"R;=N @ILGOF;
#C"
"$ $! % 7KN
H
#C"
"$ $! $! "$ "$ % 7KN
H
"F "G QB?L?
7KN 7
$! $! H
7
H
"$ "$ "$ %
!?P?FIJ?> @ILG
"F #C" 7 $! $! $! %
H
H
"G $! %
"$ "$ #C" 7 $! $! H
H
#C" 7
"$ $! "$ % H
H
* Q;P?
#C" 7 $! "$ H
H
"F "$ %
#C" 7 "$ "$ H
H
"G $! %
#C" 7 $! "$ H
H
"$ %
G0<?;4 >5 0 JG43 J2C8C8>DB <>>= 8= C74 ?;0=4 >5 C74 4@D0C>A
)?N OM =;F=OF;N? CH MO=B ; =;M? NB? NBL?? =IGJIH?HNM I@ NB? @IL=? I@ NB? * Q;P?
4? B;P? ;H> % J = QB?L? CM NB? LIN;NCIH I@ NB? ";LNB ;H> = NB?
BIOL ;HAF? ;N J )?N OM N;E? = Q? A?N
"D " "$ QCNB " ;JJ?;LM
; /
@ @
"$ J
"F " "$ $! J
"G " $! "$ "$ J
QBC=B Q? =;H ;FMI JON CH ; @ILG QB?L? NB? N?LL?MNLC;F AL;PCNS =
" C"
=
C(
7
H
APPENDIX A 1&!" $"+"/1&+$ #,/ "
/
QCNB ;
; ;H> @ @LIG QBC=B "
1B? G;AHCNO>? I@ NB? BILCTIHN;F @IL=? I@ NB? * Q;P? CM
" "$ 5
$! J
$! "$ J
&@ Q? ;L? CHN?L?MN?> CH NIN;F
QBC=B ACP?M ;N J ;H> @IL ; @IL=? ?KO;F NI
@IL=? QBC=B CM =;F=OF;N?> QCNB NB? B?FJ I@ ?R;=N @ILGOF;? Q? VH>
#C"
"F $! J
H
7
"$ "$ J
H
#C"
$! "$ J
"G H
7
H
7
"$ "$ J
H
7
H
QBIM? G;AHCNO>? CM
H
7
7
"$ "$ J
H
7
H
"
5
$! J
$! "$ J
;N J ;H> @IL Q? A?N ; G;AHCNO>? ?KO;F NI " C ? NQC=? NB? @IL=? >O? NI
NB? * Q;P? ;FIH?
42><?>B8C8>= >5 7>DA 0=6;4 B834A40; C8<4
42><?>B8C8>= >5 7>DA 0=6;4
1B? BIOL ;HAF? QBC=B ;JJ?;LM CH NB? @ILGOF;? ACP?H NCFF HIQ >?J?H>M <INB IH NB? JF;=?
;H> IH NB? B?;P?HFS <I>S #CAOL? MBIQM NB? >?=IGJIMCNCIH CHNI FI=;F MC>?L?;F
NCG? ;H> NB? LCABN ;M=?HMCIH I@ NB? B?;P?HFS <I>S
1B? JICHN CM NB? P?LH;F JICHN MCNO;N?> IH NB? FCH? ;N NB? CHN?LM?=NCIH <?NQ??H
NB? ?=FCJNC= JF;H? ;H> ?KO;NIL JF;H?
#CAOL? MBIQM NB;N % J:=1 FI=;F MC>?L?;F NCG? LCABN ;M=?HMCIH
/CABN ;M=?HMCIH >?J?H>M IHFS IH NB? L?F;NCP? GIP?G?HN I@ NB? ";LNB ;H> NB?
B?;P?HFS <I>S &N QCFF <? ACP?H CH 0?=NCIHM ;H> @IL FOH;L ;H> MIF;L GIP?G?HN
+IQ Q? B;P? NI =IHH?=N FI=;F MC>?L?;F NCG? NI =CPCF NCG?
!>20; B834A40; C8<4 O 28E8; C8<4
#LIG ; >;N? ;H> NCG? ;N ; ACP?H JF;=? NB? M?LC?M I@ NL;HM@ILG;NCIHM QBC=B F?;>M NI
FI=;F MC>?L?;F NCG? CM ;M @IFFIQM
-;MM;A? NI OHCP?LM;F NCG? 21 #L?H=B NCG? @IL ?R;GJF? CM 21
;H> 21 CH MOGG?L
CH QCHN?L
;F=OF;NCIH I@ 'OFC;H >;S '! =ILL?MJIH>CHA NI NCG? BIOL I@ NB? M;G? >;N?
M?? @IFFIQCHA MO<M?=NCIH
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
Plane of the equator
γ
α
t loc
AH: hour angle
Heavenly body
Place
#CAOL? LCABN ;M=?HMCIH ;H> BIOL ;HAF?
;F=OF;NCIH CH 'OFC;H =?HNOLC?M I@ 0 NB? NCG? J;MM?> MCH=? !?=?G<?L ;N HIIH NBCM NCG? M?LP?M ;M ; L?@?L?H=? CH GIMN ;MNLIHIGC=;F =;F=OF;NCIHM
0 '! ;F=OF;NCIH I@ MC>?L?;F NCG? CH BIOLM CH $L??HQC=B ;N BIOL 89 J 0
0
1BCM NCG? GOMN <? <LIOABN NI ; P;FO? <?NQ??H ;H> C@ H?=?MM;LS
;F=OF;NCIH I@ MC>?L?;F NCG? ;N $L??HQC=B IH BIOL 21 89 J 21
1B? =I?^=C?HN GOFNCJFSCHA 21 CM NB? L;NCI <?NQ??H BIOLM ;H> NB? >OL;NCIH
I@ ; LIN;NCIH I@ NB? ";LNB ;LIOH> CNM IQH ;RCM
;F=OF;NCIH I@ FI=;F MC>?L?;F NCG? CH L;>C;HM
J(
%?L? ( CM B?L? NB? FIHACNO>? I@ NB? JF;=? CH L;>C;HM
+IQ CN L?G;CHM NI =;F=OF;N? NB? 'OFC;H >;S =ILL?MJIH>CHA NI BIOL I@ ; >;N?
@LIG NB? $L?AILC;H =;F?H>;L
APPENDIX A 1&!" $"+"/1&+$ #,/ "
D;80= 20;4=30A
1B? $L?AILC;H =;F?H>;L QCNB CNM F?;J S?;LM ;H> L?@ILGM CM HIN L?AOF;L ?HIOAB NI <?
ONCFCT?> CH ;MNLIHIGS MNLIHIG?LM OM? NB? HINCIH I@ 'OFC;H >;S ; J?LCI> I@ BIOLM
MN;LNCHA ;N HIIH CH IOL =;F?H>;L 1B? MN;LNCHA JICHN @IL =IOHNCHA 'OFC;H >;SM B;M <??H
VR?> ;N ; >;N? AICHA <;=E GIL? NB;H S?;LM CH MO=B ; Q;S NB;N *;S ;N HIIH =ILL?MJIH>M NI 'OFC;H >;S 1B? >C^=OFNS CH =B;HACHA NI 'OFC;H >;S
=IG?M @LIG NB? CLL?AOF;LCNC?M I@ NB? $L?AILC;H =;F?H>;L 1B? @IFFIQCHA #,/1/+
@OH=NCIH >L;QH @LIG 89 L?NOLHM NB? 'OFC;H >;S '2)!6 NI HIIH I@ NB? >;S &!
I@ NB? GIHNB ** ;H> NB? S?;L &666 I@ NB? $L?AILC;H =;F?H>;L 1I I<N;CH NB? 'OFC;H
>;S '! QBC=B =ILL?MJIH>M NI BIOL Q? GOMN NB?H >?>O=N #2+ 1&,+ '2)!6** &! &666
-/*"1"/ &$/"$ &# &666 ". 01,- 1B?L? CM HI 6?;L 7?LI
&# &666 )1 &666&666 &# ** $1 1%"+
'6&666
'*** ")0"
'6&666 '*** "+!&#
'2)!6&+1 '6 &+1 '* &! &# &! ** &666 $" &$/"$ 1%"+
'&+1 '6
'2)!6'2)!6 ' &+1 '
"+!&#
/"12/+
"+!
0IG? LOF?M I@ NB? $L?AILC;H =;F?H>;L ;>IJN?> IH ,=NI<?L ;L?
+I S?;L Q? AI @LIG NI
6?;LM GOFNCJF?M I@ ;L? F?;J S?;LM ?R=?JN QB?H NB?S =;H <? >CPC>?> <S ;H>
HIN <S %?H=? Q;M HIN ; F?;J S?;L Q;M
">E4<4=C >5 C74 <>>=
&H NB? @ILGOF;? ACPCHA NB? NC>? A?H?L;NCHA @IL=? ; B?;P?HFS <I>S CHN?LP?H?M NBLIOAB
@IOL J;L;G?N?LM
G;MM
>CMN;H=? NI NB? ";LNB
>?=FCH;NCIH
LCABN ;M=?HMCIH
%?L? Q? NLS HIN NI ACP? @OFF ?RJF;H;NCIHM <ON MCGJFS @OLHCMB =;F=OF;NCIH @ILGOF;?
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
"0BB >5 C74 <>>=
1B? L;NCI I@ NB? G;MM I@ NB? GIIH NI NB? G;MM I@ NB? ";LNB CM
C"
C(
8BC0=24 C> C74 0AC7 ;>=68CD34 >5 C74 <>>= 8= 8CB >A18C
1B? GIIH >?M=LC<?M ;H ?FFCJNC=;F IL<CN I@ M?GC G;DIL ;RCM 9 EG ;H> ?==?HN
LC=CNS ; 4B?L?P?L NB? N?LG 7H =IG?M CHNI NB? @ILGOF;? I@ NB? JIN?HNC;F Q? =;H L?JF;=?
CN <S /1 @1 1BCM N?LG 9H M?GC G;DIL ;RCM I@ NB? FOH;L IL<CN >CPC>?> <S NB? >CMN;H=?
<?NQ??H NB? ";LNB ;H> NB? GIIH QCFF <? ACP?H B?L?;@N?L <S >?P?FIJCHA CN ;==IL>CHA NI
NB? JIQ?LM I@ ; NB? ?==?HNLC=CNS 1B? MCNO;NCIH CM =IGJFC=;N?> <?=;OM? CN CM H?=?MM;LS
NI =IHMC>?L M?? #CAOL? NBL?? JF;H?M NB? JF;H? I@ NB? ?KO;NIL JF;H? I@ FOH;L
IL<CN JF;H? I@ NB? ?=FCJNC=
perigee
ecliptic
i
N: ascending node
γ
ω'
vernal point
I
equator
G
orbit
γ'
#CAOL? GIP?G?HN I@ NB? *IIH
-?LCA?? CM NB? JICHN I@ FOH;L IL<CN QB?L? NB? GIIH CM =FIM?MN NI NB? ";LNB
1B? JIMCNCIH I@ NB? GIIH CH CNM IL<CN CM G?;MOL?> @LIG ; V=NCNCIOM JICHN CH MO=B
; Q;S NB;N NB? ;HAF?M * ;H> * ;L? ?KO;F
1B? GIIH @ILGM QCNB NB? JICHN ;H ;HAF? G?;MOL?> @LIG NB? =?HNL? I@ NB?
";LNB =;FF?> I NB? G?;H FIHACNO>? I@ NB? GIIH CH CNM IL<CN 1B? G?;H FIHACNO>? I@
J?LCA?? CM >?HIN?> F I ;H> F CH >?AL??M 89 J ;H> 89 J ;L? ACP?H
;M @OH=NCIHM I@ 0 NB? HOG<?L I@ 'OFC;H =?HNOLC?M J;MM?> MCH=? !?=?G<?L ;N
APPENDIX A 1&!" $"+"/1&+$ #,/ "
HIIH M?? 0?=NCIH I 0 F 0
0 0 0 0 &H @;=N NB? G?;H FIHACNO>? I@ NB? GIIH ;M CNM H;G? CH>C=;N?M CM HIN NB? L?;F
FIHACNO>? >?HIN?> ( #CAOL? MBIQM NB? >C_?L?H=? CH ; P?LS ?R;AA?L;N?> G;HH?L
1B? HINCIH I@ NB? G?;H GIIH CM ; A?IG?NLC= ;LNCV=? M ; VLMN ;JJLIRCG;NCIH CNM
GIP?G?HN CM L?AOF;L ;LIOH> , NB? =?HNL? I@ NB? =CL=F? I@ L;>COM 9 QBCF? NB? GIIH
>?M=LC<?M ;H ?FFCJM? I@ QBC=B ";LNB I==OJC?M IH? I@ NB? @I=C
L: Moon
OL=c
Mean Moon
TL=r
ε
OT=ec
M
v
u
O
T: Earth
P: perigee
#CAOL? G?;H FIHACNO>? I@ NB? GIIH L?;F FIHACNO>?
1B? FIHACNO>? ( CM QLCNN?H ;M F L QB?L? L CM NB? L?;F ;HIG;FS
) CM NB? G?;H ;HIG;FS K CM NB? ?==?HNLC= ;HIG;FS
(?JF?LYM ?KO;NCIH K ; $! K ) ;FFIQM OM NI J;MM @LIG K NI ) 1BCM ?KO;NCIH
CM CH@?LL?> @LIG (?JF?LYM F;QM S MCGJF? A?IG?NLC= =IHMC>?L;NCIHM ;H> <S CAHILCHA
NB? BCAB?L JIQ?LM I@ ; Q? VH>
L)
+IQ ) I F ;H> K )
#CH;FFS Q? VH>
(I
; $! K
; $! K
; $! I F
; $! I F
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
&H NB? NLC;HAF? ,1) I@ #CAOL? Q? B;P?
9 H "$ ;
;H> NI M?=IH> IL>?L "$ ;
;9 "$ K
B?H=? H 9 ; "$ K @LIG QBC=B Q? CH@?L
9
H
; "$ I F
; "$ I F
1I =IHMC>?L NB? CHWO?H=? I@ NB? MOH ?P?=NCIH
NI ( NB? KO;HNCNS
C; $! I >
;H> NI
1
@
F
P;LC;NCIH CN CM ;JJLIJLC;N? NI ;>>
C $! I >
C "$ I >
NB? KO;HNCNS
C; "$ I >
F
QB?L?
C CM NB? L;NCI I@ NB? ;P?L;A? GIP?G?HN I@ NB? MOH NI NB? ;P?L;A? GIP?G?HN I@ NB?
GIIH ?KO;F NI > CM NB? G?;H FIHACNO>? I@ NB? GIIH CH CNM IL<CN 89 J > 0
0
%?H=? Q? GOMN N;E?
(I
; $! I F
C; $! I >
9
; "$ IF
H
QB?L?
1
@
F
; $! I F
C $! I >
C; "$ I> F
; "$ IF
C "$ I>
CM =;FF?> NB? L?;F J;L;FF;R I@ NB? GIIH
9
H
;
; "$ I F
C; "$ I >
F
; "$ I >
C "$ I >
APPENDIX A 1&!" $"+"/1&+$ #,/ "
P
Moon
L
δ
N
l
Equator
Ecliptic
i
Ω
γ
ω'
v
I
L'
x
G
Lunar orbit
γ"
ξ
#CAOL? >?=FCH;NCIH ;H> LCABN ;M=?HMCIH I@ NB? GIIH
42;8=0C8>= 0=3 A867C 0B24=B8>=
&H #CAOL? >?=FCH;NCIH ;H> LCABN ;M=?HMCIH I@ NB? GIIH ;L? NB? ;HAF?M ( ( ;H> ( L?MJ?=NCP?FS
0IG? ?RJF;H;NCIHM ;<ION NB? ;HAF?M
* FIHACNO>? I@ ;M=?H>CHA HI>?
( LCABN ;M=?HMCIH I@ NB? GIIH
LCABN ;M=?HMCIH I@ #
;HAF? # ;HAF? #( B ( N
;HAF? #( N
4? EHIQ ( ?KO;F NI NB? ;HAF? ( M?? JL?PCIOM MO<M?=NCIH
MNLIHIGS ;FMI JLIPC>?M OM QCNB NB? @IFFIQCHA ;HAF?M
CH=FCH;NCIH I@ NB? ?KO;NIL IH NB? ?=FCJNC= NBCM CM NB? ;HAF? QBC=B >?VH?M NB?
NLIJC=M 89 J 0 0
0 CH >?AL??M ? CH=FCH;NCIH I@ NB? FOH;L IL<CN IH NB? ?=FCJNC= ? 89
J FIHACNO>? I@ NB? ;M=?H>CHA HI>? I@ NB? GIIH 89 J %6!/,!6+*& 0 ,# #/"" 02/# " #),40
* 0
0
0 ;F=OF;NCIH I@ NB? >?=FCH;NCIH 1BCM CM >IH? NBLIOAB ;H> %
;F=OF;NCIH I@ &H NB? NLC;HAF? * # MJB?LC=;F NLCAIHIG?NLS ACP?M
$! ? "% "% $! "$ ? "$ ;M * * NB? ;HAF? #* CM CH>??> ?KO;F NI 1BCM @ILGOF; ACP?M
"% $! ? "% "$ ? "$ $! NB?L?@IL? Q? B;P? S >?P?FIJCHA "% NI NB? VLMN IL>?L QCNB L?MJ?=N NI Q? VH> NB? @IFFIQCHA
;JJLIRCG;N? @ILGOF;
? "% $! P;LC?M @LIG NI CH S?;LM
;F=OF;NCIH I@ %
0NCFF CH NB? NLC;HAF? * # Q? B;P?
"$ % "$ "$ ? $! $! ? "$ #CH;FFS Q? I<N;CH <?=;OM? CH NB? NLC;HAF? #(( Q? B;P? NB? ;HAF? #( ( CM ;
LCABN ;HAF?
$! $! B $! % $! ( N $! %
;F=OF;NCIH I@ NB? LCABN ;M=?HMCIH 0NCFF CH NB? NLC;HAF? #(( Q? B;P?
%! N %! B "$ % $! ( N $! %
&H NB? NLC;HAF? * # MJB?LC=;F NLCAIHIG?NLS ;A;CH ACP?M
$! "% ?
"$ "$ $! "%
B?H=?
%!
(HIQCHA
$! $! "% ?
"$ "$ *
;H> N Q? MCGJFS B;P? N
">E4<4=C >5 C74 BD=
M @IL NB? GIIH Q? MB;FF >?N;CF NB? @IOL J;L;G?N?LM ;JJ?;LCHA CH NB? @ILGOF;?
APPENDIX A 1&!" $"+"/1&+$ #,/ "
"0BB >5 C74 BD=
1B? L;NCI I@ NB? MOHYM G;MM C' NI NB? ";LNBYM G;MM CM 89 J C'
C(
8BC0=24 C> C74 0AC7 ;>=68CD34 >5 C74 BD=
$CP?H ; NB? ?==?HNLC=CNS I@ NB? N?LL?MNLC;F IL<CN 89 J ; 0 0
&H NB? =;M? I@ NB? MOH ;H> >?HINCHA ! ;M NB? ?==?HNLC= ;HIG;FS ;H> ) NB? G?;H
;HIG;FS Q? B;P?
!)
; $! ! (?JF?LYM ?KO;NCIH
QCNB
) 0 0
0 CH >?AL??M ! =;H <? >?>O=?> @LIG (?JF?LYM ?KO;NCIH NBLIOAB MO==?MMCP? CN?L;NCIHM C@ H??>?>
C ? !< ) ; $! !< ;H> ! )
1B? ";LNB\MOH >CMN;H=? CM NB?L?@IL?
H ; "$ !
?RJL?MM?> CH ;MNLIHIGC=;F OHCNM ,H? ;MNLIHIGC=;F OHCN ?KO;FM EG IL
GCF?M
*?;H P;FO? /@ 1B? L?;F ;HIG;FS I@ NB? MOH L CM ACP?H <S 89 J 7
L
;
!
%!
%!
;
1B? L?;F FIHACNO>? I@ NB? MOH CM NB?H 89 J ( 0
(
L ) QCNB
0 CH >?AL??M
( <?CHA NB? G?;H FIHACNO>?
42;8=0C8>= 0=3 A867C 0B24=B8>= >5 C74 BD=
1B? F;NCNO>? I@ NB? MOH IH NB? ?=FCJNC= >I?M HIN ?R=??> M?=IH>M I@ ;L= NB?L?@IL?
Q? =;H =IHMC>?L NB;N NB? MOH ?PIFP?M CH NB? ?=FCJNC= 1B? LCABN ;M=?HMCIH ;H> NB?
>?=FCH;NCIH ;L? NBOM ACP?H <S
"$ $!
"$
$! $! $!
%! ??4=38G 8QDB8>= <0CA8G F8C7
C4CA0743A0
*-)" 3 VF? @IL <OCF>CHA NB? ?F?G?HN;LS >C_OMCIH G;NLCR @IL ; N?NL;B?>LIH QCNB
FCH?;L CHN?LJIF;NCIH
QCNBFCH;FA
0?NNCHA NB? P;FO?M I@ ;LL;SM IH NB? N?NL;B?>L; 35 36 3 31 3# 3$ 3%
32 33 ;H> 34
"R;GJF?M 3C< CJ P;FO? I@ <;M? C< ;N CNM FI=;F JICHN CJ 35CJ P;FO? I@ 5
;N CNM FI=;F JICHN CJ
IIL>CH;N?M 5 6 ;H> 7 CH ;LL;SM 35 36 ;H> 37
35;LL;S 85 5 5 59
36;LL;S 86 6 6 69
37;LL;S 87 7 7 79
;MCM @OH=NCIHM 3 N?MN @OH=NCIHM 31 @OH=NCIHM # $ ;H> % ;LL;SM 3# 3$
;H> 3% @IL MNILCHA N?HMILC;F PCM=IMCNS
3;LL;S 88 9 8 9 8 9 8 99
31;LL;S 88 9 8 9 8 9 8 99
3#;LL;S 8# # # #9
3$;LL;S 8$ $ $ $9
3%;LL;S 8% % % %9
!?VHCNCIH I@ FCH?;L <;MCM @OH=NCIHM IH NB? L?@?L?H=? N?NL;B?>LIH =IIL>CH;N?M )
-% "1 $*
-0&;LL;S 8 )-% "1 $* )-% "1 $*9
/?MOFNCHA G;NLCR
*1G;NLCR 0CGJFCV=;NCIHM C@ ILCACH CM N;E?H ;N JICHN
5
6
7
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
!5!)-%3589 3589
!5!"13589 3589
!5!$*3589 3589
!6!)-%3689 3689
!6!"13689 3689
!6!$*3689 3689
!7!)-%3789 3789
!7!"13789 3789
!7!$*3789 3789
';=I<C;H G;NLCR @IL CMIJ;L;G?NLC= NL;HM@ILG;NCIH
*1' G;NLCR 8!5!)-% !5!"1 !5!$* !6!)-%
!6!"1 !6!$* !7!)-% !7!"1 !7!$*9
';=I<C;H I@ CMIJ;L;G?NLC= NL;HM@ILG;NCIH
' >?N*1' &HP?LM? I@ NB? NL;HMJIM? I@ ';=I<C;H G;NLCR
1*1' G;NLCR 8!5!)-% !6!)-% !7!)-%
!5!"1 !6!"1 !7!"1 !5!$* !6!$* !7!$*9
&+3' CHP?LM?1*1' IGJON;NCIH I@ NB? ?F?G?HN;LS N?LGM I@ NB? G;NLCR
@IL C @LIG NI >I
@IL D @LIG NI >I
38D 9 -0&89 38D 9 -0&89 38D 9 -0&89 38D 9 -0&89
$/!GOFNCJFS&+3' P?=NIL8>C_ )-%
>C_ "1 >C_ $*9
5$/!89
6$/!89
7$/!89
1318C 9 -0&89 318C 9 -0&89 318C 9 -0&89 318C 9 -0&89
$/!1GOFNCJFS&+3' P?=NIL8>C_1 )-%
>C_1 "1 >C_1 $*9
15$/!189
16$/!189
17$/!189
#3#89 -0&89 3#89 -0&89 3#89 -0&89 3#89 -0&89
$3$89 -0&89 3$89 -0&89 3$89 -0&89 3$89 -0&89
%3%89 -0&89 3%89 -0&89 3%89 -0&89 3%89 -0&89
#OH=NCIH NI <? CHN?AL;N?> IP?L NB? L?;F ?F?G?HN
#,+ # 5 15 $ 6 16 % 7 17
#,+ MCGJFC@S#,+ ' APPENDIX B !&##20&,+ *1/&5 4&1% 1"1/%"!/ &HN?AL;NCIH IP?L NB? L?@?L?H=? N?NL;B?>LIH
#,+ CHN#,+ )-%
#,+ MO<M)-% "1 $* #,+ MO<M)-% #,+ #,+ CHN#,+ "1
#,+ MCGJFC@SMO<M"1 $* #,+ MO<M"1 #,+ #,+ CHN#,+ $*
*18C D9MCGJFC@SMO<M$* #,+ MO<M$* #,+ I>
I>
#,/1/+ JLCHNION I@ NB? L?MOFN IH ; VF? H;G?> /"02
QLCN?NIX/"02X
@ILNL;H*1
??4=38G #>C0C8>=B
1B? HIN;NCIHM OM?> CH NB? =IOLM? I@ NBCM QILE ;L? >?M=LC<?> B?L?;@N?L
!0C8= 0;?7014C
7 @LC=NCIH =I?^=C?HN ;N NB? Q;FFM
7E7<2 =I?^=C?HN I@ QCH> CHWO?H=?
8 FCH?;L @LC=NCIH =I?^=C?HN
9 =?F?LCNS I@ FIHA Q;P?M
BfTS =I?^=C?HN
0 NCG? MN?J
"F MIOL=? N?LG ;FIHA N CH NB? GIG?HNOG ?KO;NCIH
"G MIOL=? N?LG ;FIHA S CH NB? GIG?HNOG ?KO;NCIH
/
" MIOL=? N?LG ;LCMCHA @LIG ;NGIMJB?LC= JL?MMOL?
1
" ILCIFCM @IL=?
2
" @IL=? ;LCMCHA @LIG MJ;NC;F P;LC;NCIHM I@ >?HMCNS
4
" @LC=NCIH @IL=? IH NB? <INNIG
;
" NC>? >LCPCHA N?LGM
?
" MIOL=? N?LG @IL =IHM?LP;NCIH I@ GIG?HNOG
B
" >L;A @IL=?
D
" CHWO?H=? I@ NB? QCH>
= ;==?F?L;NCIH I@ AL;PCNS
> Q;N?L >?JNB G?;MOL?> <?NQ??H <INNIG ;H> @L?? MOL@;=?
>>@=> B?CABN I@ JLIJ;A;NCIH
A NOL<OF?HN ?H?LAS
' 0NLC=EF?L @LC=NCIH =I?^=C?HN
AA LIOABH?MM MCT? IH NB? <INNIG
C *;HHCHA =I?^=C?HN
D 4 HILG;F P?=NIL ;N NB? <INNIG JICHNCHA IONMC>?
D A HILG;F P?=NIL ;N NB? @L?? MOL@;=? JICHNCHA IONMC>?
F JL?MMOL?
F6 BS>LIMN;NC= JL?MMOL?
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
%6!/,!6+*& 0 ,# #/"" 02/# " #),40
F2 >SH;GC= JL?MMOL?
,/B; ;NGIMJB?LC= JL?MMOL?
-F WIQ J?L OHCN F?HANB CH G M ?KO;F NI >K
-G WIQ J?L OHCN F?HANB CH G M ?KO;F NI >L
-A13 MIOL=? I@ >CM=B;LA? ;N NB? <INNIG CH G M CGJIM?> ;N =?LN;CH JICHNM I@ NB?
G?MB
. ";LNB L;>COM
.78 /?SHIF>M N?HMIL
/ M;FCHCNS
/9; MIOL=? N?LG I@ NB? G;MM =IHM?LP;NCIH ?KO;NCIH ?RJL?MM?> CH GM ;H> >?>O=?>
@LIG -A13
J NCG?
0 NL;=?L IL N?GJ?L;NOL?
K VLMN =IGJIH?HN I@ P?FI=CNS CH NB? 0;CHN 3?H;HN ?KO;NCIHM
1 VLMN =IGJIH?HN I@ P?FI=CNS CH NB? +;PC?L\0NIE?M ?KO;NCIHM
K1=<D VLMN =IGJIH?HN I@ ;>P?=NCIH P?FI=CNS
KA13 VLMN =IGJIH?HN I@ MIOL=?M P?FI=CNS
1E7<2 VLMN =IGJIH?HN I@ QCH> P?FI=CNS
1 1 1 =IGJIH?HNM I@ P?FI=CNS CH NB? +;PC?L\0NIE?M ?KO;NCIHM OM?> CH "CHMN?CH
HIN;NCIH 1 1 1 2 1 3 L M?=IH> =IGJIH?HN I@ P?FI=CNS CH NB? 0;CHN 3?H;HN ?KO;NCIHM
2 M?=IH> =IGJIH?HN I@ P?FI=CNS CH NB? +;PC?L\0NIE?M ?KO;NCIHM
L1=<D M?=IH> =IGJIH?HN I@ ;>P?=NCIH P?FI=CNS
LA13 M?=IH> =IGJIH?HN I@ MIOL=?M P?FI=CNS
2E7<2 M?=IH> =IGJIH?HN I@ QCH> P?FI=CNS
4 *?L=;NIL ;<M=CMM;
5 *?L=;NIL IL>CH;N?
6A @L?? MOL@;=? ?F?P;NCIH
64 <INNIG ?F?P;NCIH
A44: 0;?7014C
;HAF? I@ NB? <INNIG MFIJ? J?L ?F?G?HN
NOL<OF?HN >CMMCJ;NCIH
6 =I?^=C?HN I@ M?GC CGJFC=CN;NCIH @IL NB? Q;N?L >?JNB
C =I?^=C?HN I@ M?GC CGJFC=CN;NCIH @IL NB? P?FI=CNS
( =I?^=C?HN I@ M?GC CGJFC=CN;NCIH @IL NB? NL;=?L
JILIMCNS
PIH (;LG;H =IHMN;HN ?KO;F NI F;NCNO>?
F;NCNO>? I@ NB? ILCACH JICHN
=I?^=C?HN I@ GIF?=OF;L >C_OMCIH CH G M
B =I?^=C?HN I@ NOL<OF?HN >C_OMCIH CH G M
3 >C_OMCIH =I?^=C?HN CH G M CH=FO>CHA >CMJ?LMCIH ;H> NOL<OF?H=?
>?HMCNS I@ Q;N?L
/7@ >?HMCNS I@ ;CL
78 PCM=IOM J;LN I@ MNL?MM N?HMIL
APPENDIX C +,11&,+0 4 PCM=IOM MNL?MM ;N NB? <INNIG
A PCM=IOM MNL?MM ;N NB? MOL@;=?
FIHACNO>?
FIHACNO>? I@ NB? ILCACH JICHN
67 N?MN @OH=NCIH ;N JICHN ? @IL NB? 0;CHN 3?H;HN =IHNCHOCNS ?KO;NCIH
C7 N?MN @OH=NCIH ;N JICHN ? @IL NB? NQI 0;CHN 3?H;HN GIG?HNOG ?KO;NCIHM
67 <;MCM @OH=NCIH ;N JICHN ? @IL NB? 0;CHN 3?H;HN =IHNCHOCNS ?KO;NCIH
C7 <;MCM @OH=NCIH ;N JICHN ? @IL NB? NQI 0;CHN 3?H;HN GIG?HNOG ?KO;NCIHM
P?=NIL LIN;NCIH I@ NB? ";LNB
"0C74<0C820; B86=B
?RN?LH;F JLI>O=N <?NQ??H NQI VLMN IL>?L N?HMILM
N?HMILC;F IJ?L;NIL ZH;<F;[
<?NQ??H NQI P?=NILM >?MCAH;N?M NB? >IN JLI>O=N
NB? M?N I@ L?;F HOG<?LM
N O >IN JLI>O=N I@ N ;H> O
N "O=FC>?;H HILG I@ P?=NIL N
81;8>6A0?7H
89 +!"/0,+ * $ 1"0 - ! %"/3,2"1 ' *
IGJON;NCIH I@ ; WII>
?P?HN OMCHA ; >CG?HMCIH;F VHCN? ?F?G?HN GI>?F ;H> =IGJ;LCMIH NI V?F> >;N;
0J?=C;FNS =IH@?L?H=? IH GI>?FFCHA I@ WII> JLIJ;A;NCIH IP?L CHCNC;FFS >LS ;L?;M
*CF;H &N;FS \ 'OH? 89 1(&+0,+ ' % 4"01"/&+( ' ' %"/3,2"1 ' * 0CGCF;LCNC?M <?NQ??H
NB? KO;MC <O<<F? ;H> NB? A?H?L;FCT?> Q;P? =IHNCHOCNS ?KO;NCIH MIFONCIHM NI NB?
MB;FFIQ Q;N?L ?KO;NCIHM &HN?LH;NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M CH #FOC>M
\ 89 // ! !?HMCG?NLC= ?R=B;HA? WIQ CH L?=N;HAOF;L =B;HH?FM &&& );LA? M=;F?
?RJ?LCG?HNM ); %IOCFF? F;H=B? \ 89 //,0 " "MNCG;NCIH >? J;L;GeNL?M >;HM F?M fKO;NCIHM >? 0;CHN 3?H;HN
1BeM? >? FY2HCP?LMCNf -C?LL? ?N *;LC? OLC? -;LCM 3& 0Jf=C;FCNf *;NB
fG;NCKO?M JJFCKOf?M 89 1"0 - ! %"/3,2"1 ' * H?Q G?NBI> @IL GIPCHA <IOH>;LS BS
>LI>SH;GC= JLI<F?GM CH MB;FFIQ Q;N?L -LI=??>CHAM I@ NB? /IS;F 0I=C?NS \ 89 1"0 - ! %,//&11 * 0 %"/3,2"1 ' * &HP?MNCA;NCHA NQI
>CG?HMCIH;F VHCN? ?F?G?HN JL?>C=NCIHM I@ WII>JF;CH CHOH>;NCIH OMCHA @L;=N;F
A?H?L;N?> NIJIAL;JBS %S>LIFIAC=;F -LI=?MM?M \ 89 1"0 - ! 01"4/1 * ! 0&$$"/0 $ 0*&1% + %"/3,2"1
' * 0"))&+ / % ' &HN?LH;F ;H> ?RN?LH;F P;FC>;NCIH I@ ; NQI >CG?HMCIH;F
VHCN? ?F?G?HN =I>? @IL LCP?L WII> MCGOF;NCIHM -LI=??>CHAM I@ NB? & " 4;N?L
;H> *;LCNCG? "HACH??LCHA \ 89 "* / * 4/*&+$ / # H CGJFC=CN @;=NIL?> M=B?G? @IL NB? =IGJL?MM
C<F? +;PC?L\0NIE?M ?KO;NCIHM & 'IOLH;F \ 89 "+'*&+ 1 $L;PCNS =OLL?HNM ;H> L?F;N?> JB?HIG?H; 'IOLH;F I@ #FOC>
*?=B;HC=M \ 89 "+.2" ' - )!&" $ /,+1 ' 2H? GfNBI>? >YfFfG?HNM VHCM JIOL
F; LfMIFONCIH >?M fKO;NCIHM >? +;PC?L\0NIE?M =IOJFf?M b OH? fKO;NCIH NB?LGCKO?
IGGOHC=;NCIH ;O CeG? 0SGJIMCOG &HN?LH;NCIH;F MOL F?M GfNBI>?M >YfFfG?HNM
VHCM JIOL F?M JLI<FeG?M >? Gf=;HCKO? >?M WOC>?M 1IESI \ 'OFS Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
&)&,$/-%6
89 "+,&01 $ %"/3,2"1 ' * )!&" $ )Y?RJfLC?H=? >Y"!# ?H
G;NCeL? >? =;F=OF >?M IH>?M >? MO<G?LMCIH &HN?LH;NCIH;F G??NCHA IH >;G <L?;E
WII> Q;P? MCGOF;NCIH *IHNLf;F \ JLCF 89 "/*2!"7 !"/3&"25 !"0&!"/& ' 37.2"7 * " 2JQCH>
M=B?G?M @IL NB? NQI >CG?HMCIH;F MB;FFIQ Q;N?L ?KO;NCIHM QCNB P;LC;<F? >?JNB
OMCHA OHMNLO=NOL?> G?MB?M /;JJILN >? L?=B?L=B? >? FY&+/& H !?=?G<?L
89 "/+/!&
IF>M HOG<?L
-&/,++"2 , ,H NB? MB;FFIQ Q;N?L ?KO;NCIHM ;N FIQ /?SH
IGGOHC=;NCIHM CH -;LNC;F !C_?L?HNC;F "KO;NCIHM \
89 &'3")!0 * +OG?LC=;F GI>?FFCHA I@ ?MNO;LCH? WIQ IP?L MN??J NIJIAL;JBS
IGGOHC=;NCIHM IH %S>L;OFC= ;H> $?IN?=BHC=;F "HACH??LCHA 12 !?F@N #;=OFNS
I@ CPCF "HACH??LCHA /?JILN *;L=B 89 )2*"/$ *")),/ $ >?M=LCJNCIH I@ ; NBL?? >CG?HMCIH;F =I;MN;F
I=?;H =CL=OF;NCIH GI>?F 1BL?? >CG?HMCIH;F I;MN;F ,=?;H *I>?FM + %?;JM
"> 3IFOG? I@ I;MN;F ;H> "MNO;LCH? 0=C?H=?M G?LC=;H $?IJBSMC=;F 2HCIH
89 ,2)"1 1 /,2$" ! 1?F?G;= ! 3?LMCIH !IMMC?L >? P;FC>;NCIH "!#
/?JILN %" 89 ,200&+"0. ' "MM;C MOL F; NBfILC? >?M ?;OR =IOL;HN?M *fGICL? JLfM?HNf
J;L >CP?LM M;P;HNM b FY=;>fGC? >?M 0=C?H=?M >? -;LCM \ 89 /,,(0 + %2$2"0 1 ' / 0NL?;GFCH? 2JQCH> -?NLIP $;F?LECH @ILGO
F;NCIHM @IL =IHP?=NCIH >IGCH;N?> WIQM QCNB J;LNC=OF;L ?GJB;MCM IH NB? +;PC?L\
0NIE?M ?KO;NCIHM IGJON?L *?NBI>M CH JJFC?> *?=B;HC=M ;H> "HACH??LCHA
\ 89 2/$"0 &
%"/$2& ' $&/,2 ! /".2"+ 0 *?MM;A? -;MMCHA &H
N?L@;=? &>LCM *-& P?LMCIH 89 611 0*&1% ' $ H CHN?AL;F ?KO;NCIH @IL OHMN?;>S MOL@;=? Q;P?M ;H> ;
=IGG?HN IH NB? IOMMCH?MK ?KO;NCIH 'IOLH;F I@ #FOC> *?=B;HC=M \
89
/"6 $ # '&+$ + MN;<F? F?;MN MKO;L?M VHCN? ?F?G?HN G?NBI> @IL
HIH FCH?;L BSJ?L<IFC= MSMN?GM &HN?LH;NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M CH
#FOC>M \ 89
02))& 3 0?GC CGJFC=CN VHCN? >C_?L?H=? G?NBI>M @IL NB? NQI >CG?HMCIH;F
MB;FFIQ Q;N?L ?KO;NCIHM 'IOLH;F I@ IGJON;NCIH;F -BSMC=M \ 89
02))& 3
11+& " 0N;<CFCNS ;==OL;=S ;H> ?^=C?H=S I@ ; M?GC CGJFC=CN
G?NBI> @IL NBL?? >CG?HMCIH;F MB;FFIQ Q;N?L WIQ IGJON?LM *;NB?G;NC=M
QCNB JJFC=;NCIHM \ &)&,$/-%6
89
02))& 3
%"+$ / 0?GC CGJFC=CN VHCN? >C_?L?H=? G?NBI>M @IL NBL??
>CG?HMCIH;F MB;FFIQ Q;N?L WIQ &HN?LH;NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M
CH #FOC>M \ 89
02))& 3
%"+$ / $/1+"/ ' 1C>;F /?MC>O;F &HN?LNC>;F *O>W;N
1/&* GI>?F ;H> CNM ;JJFC=;NCIHM NI 0;H #L;H=CM=I ;S ;FC@ILHC; "MNO;LCH?
I;MN;F ;H> 0B?F@ 0=C?H=? \ 89
02))& 3 01"))&+$ $ 0CGOF;NCIH I@ NBL?? >CG?HMCIH;F HIH BS>LIMN;NC=
@L?? MOL@;=? WIQM @IL ?MNO;LC?M ;H> =I;MN;F M?;M "MNO;LCH? ;H> =I;MN;F GI>?FFCHA
-LI=??>CHAM I@ NB? NB &HN?LH;NCIH;F IH@?L?H=? 0;H !C?AI 20 * 0J;OF>CHA
;H> / B?HA ">M 0 " \ 89
%3"+1 $ H;FSM? @IH=NCIHH?FF? ?N C>?HNCV=;NCIH >? =I?^=C?HNM LfJ;LNCM
>;HM F?M fKO;NCIHM ;OR >fLCPf?M J;LNC?FF?M 1BeM? >? F; #;=OFNf >?M 0=C?H=?M >?
-;LCM 0Jf=C;FCNf *;NBfG;NCKO?M 89
%,/&+ ' +OG?LC=;F MIFONCIH I@ NB? +;PC?L\0NIE?M ?KO;NCIHM *;NB?G;NB
C=M I@ IGJON;NCIH \ 89
&/)"1 - $ 1B? VHCN? ?F?G?HN G?NBI> @IL ?FFCJNC= JLI<F?GM +ILNB %IFF;H>
89
,6 ! * *0,+ ! %,//&11 * 0 ;H> 1"0 - ! 1QI
>CG?HMCIH;F BS>L;OFC= WII> GI>?FFCHA OMCHA ; VHCN? ?F?G?HN G?MB >?=IGJIM?>
;==IL>CHA NI P?A?N;NCIH ;H> NIJIAL;JBC= @?;NOL?M >?LCP?> @LIG ;CL<ILH? M=;H
HCHA F;M?L ;FNCG?NLS %S>LIFIAC=;F -LI=?MM?M \ 89
,21&+%, ) $ 1/&$ ) */1&+0 * ! )3"0 ' ) !
H ?>A? <;M?> JL?=IH>CNCIH?L @IL HIH MSGG?NLC= VHCN? ?F?G?HN ?KO;NCIHM -LI
=??>CHAM I@ NB &<?LI );NCH G?LC=;H IHAL?MM IH IGJON;NCIH;F *?NBI>M CH
"HACH??LCHA L;MCFC; L;TCF 3IFOG? \ 89 !)0" , 0 $,21) + %"/3,2"1 ' * 32'0&+,3& 6 /?=?HN
;>P;H=?M CH MB;FFIQ Q;N?L =IGJON;NCIHM IGGOHC=;NCIH ;O CeG? =IFFIKO?
CHN?LH;NCIH;F MOL F; MCGOF;NCIH >Yf=IOF?G?HNM J;L fFfG?HNM VHCM HNC<?M 89 !2"/1 $/##" , .O?FKO?M ;MJ?=NM >?M f=IOF?G?HNM JL?MKO? BILC
TIHN;OR b >CG?HMCIHM ?H JF;H ?N HIH J?LG;H?HNM ); %IOCFF? F;H=B? \ 89 !2"/1 , $"01 )"-"&+1/" #
I>? ! b MOL@;=? FC<L? *&1%/&!
1" fKO;NCIHM >? <;M? HIN? >Y;P;H=?G?HN "!# /?JILN )+% %" 89 !3&"0 )4/"+ " ' NBL?? >CG?HMCIH;F GI>?F I@ NB? * NC>? CH NB?
&LCMB 0?; 1B? CGJILN;H=? I@ IJ?H <IOH>;LS =IH>CNCIHM ;H> CHWO?H=? I@ QCH>
'IOLH;F I@ $?IJBSMC=;F /?M?;L=B \ 89 !"/!,## ' HOG?LC=;F MNO>S I@ NBL?? >CG?HMCIH;F NOL<OF?HN =B;HH?F WIQ
;N F;LA? /?SHIF>M HOG<?LM 'IOLH;F I@ #FOC> *?=B;HC=M J;LN \
&)&,$/-%6
89 !" ,+&+ ( % -&))"/" % 01/2&'0 / /," - ) *OFNC>CG?HMCIH;F
OJQCH> M=B?G?M <;M?> IH WO=NO;NCIH MJFCNNCHA @IL MSMN?GM I@ =IHM?LP;NCIH F;QM
IGJON;NCIH;F *?=B;HC=M H \ 89 !"#&+ !Y)-,0 ) *11& %&, H?Q M?N I@ ?KO;NCIHM @IL P?LS
MB;FFIQ Q;N?L ;H> J;LNC;FFS >LS ;L?;M MOCN;<F? NI ! HOG?LC=;F >IG;CHM -LI
=??>CHAM I@ NB? MJ?=C;FCNS =IH@?L?H=? IH GI>?FFCHA I@ WII> JLIJ;A;NCIH IP?L
CHCNC;FFS >LS ;L?;M *CF;H &N;FS \ 'OH? 89 !")#1 %6!/2)& 0 !?F@N! #FIQ G;HO;F !?F@N %S>L;OFC=M /?F?;M? 89 !%11 $ 1,27,1 $ 2H? JLfM?HN;NCIH >? F; GfNBI>? >?M fFfG?HNM VHCM
IFF?=NCIH 2HCP?LMCNf >? IGJCeAH? -L?MM?M >? FY2HCP?LMCNf );P;F 89 !, /*, 0"/ 0+1,0 ' 0 ,H <L?;ECHA Q;P?M ;H> Q;P? =OLL?HN
CHN?L;=NCIH CH MB;FFIQ Q;N?L ; !% VHCN? ?F?G?HN GI>?F &HN?LH;NCIH;F 'IOLH;F
I@ +OG?LC=;F *?NBI>M CH #FOC>M \ 89 !,+" ' L<CNL;LS );AL;HAC;H "OF?LC;H VHCN? ?F?G?HN G?NBI>M
NCIH;F *?NBI>M @IL 1L;HMC?HN H;FSMCM \ IGJON;
89 !,+" ' 1;SFIL $;F?LECH G?NBI> @IL =IHP?=NCP? NL;HMJILN JLI<F?GM &H
N?LH;NCIH;F 'IOLH;F IH +OG?LC=;F *?NBI>M CH "HACH??LCHA \ 89 !,++/0 - %2$2") 4/)27") 2NCFCM;NCIH >YOH =I>?
>Yf=IOF?G?HN NLC>CG?HMCIHH?F JIOL F? =;F=OF >?M fPIFONCIHM >O @IH> "!#
/?JILN )+% %" 89 "0-,0&1, - /fMIFONCIH <C>CG?HMCIHH?FF? >?M fKO;NCIHM >? NL;HMJILN J;L F;
GfNBI>? >?M =;L;=NfLCMNCKO?M "!# /?JILN %" 89 #/%1
MCGJF? ;H> ?^=C?HN ;ONIG;NC= #"* >IG;CH >?=IGJIM?L
JON?LM ;H> 0NLO=NOL?M \ IG
89 #/%1
)+1"/& 0 0&*,+ % ! 1,-!,*!" MI@NQ;L? NIIF @IL
G?MB J;LNCNCIHCHA ;H> J;L;FF?F JLI=?MMCHA IGJONCHA 0SMN?GM CH "HACH??LCHA
\ 89 #/%1
)"0,&++" * ONIG;NC= J;LNCNCIHHCHA I@ OHMNLO=NOL?> G?MB?M
@IL NB? J;L;FF?F MIFONCIH I@ JLI<F?GM CH =IGJON;NCIH;F G?=B;HC=M &HN?LH;NCIH;F
'IOLH;F @IL +OG?LC=;F *?NBI>M CH "HACH??LCHA \ 89 #&0 %"/ % *CRCHA CH CHF;H> ;H> =I;MN;F Q;N?LM =;>?GC= -L?MM 89 #)1%"/ / /?MOFNM @LIG MOLA? JL?>C=NCIH GI>?F I@ NB? +ILNB 4?MN
"OLIJ?;H =IHNCH?HN;F MB?F@ @IL JLCF +IP?G<?L ;H> !?=?G<?L &HMNCNON?
I@ ,=?;HIAL;JBS 2( /?JILN H 89 #)6++ * 3?LS BCAB MJ??> =IGJONCHA -LI=??>CHAM I@ NB? &""" \
&)&,$/-%6
89 #,/12+1, -1&01 * *I>?FFCHA H?;L <INNIG ;>P?=NCP? ;=
=?F?L;NCIH CH MOL@;=? Q;N?L GI>?FM IGJON;NCIH;F *?NBI>M CH 4;N?L /?MIOL=?M
5 \ (FOQ?L =;>?GC= -L?MM 89 #/"!"/&(0"+
4110 #CHCN? ?F?G?HN G?NBI> @IL NCG? >?J?H>?HN
CH=IGJL?MMC<F? @L?? MOL@;=? WIQ 'IOLH;F I@ IGJON;NCIH;F -BSMC=M \
89 $))+! '
$,21) + %"/3,2"1 ' * H?Q HOG?LC=;F GI>?F @IL
MIFPCHA MB;FFIQ Q;N?L ?KO;NCIHM >P;H=?M CH 4;N?L /?MIOL=?M \
89 $"&01 "$2")&+ !,+$// ' '&+$ 4 *+ %"( / 02+
!"/* 3 -3* 2M?LYM AOC>? ;H> /?@?L?H=? *;HO;F ,;E /C>A? +;NCIH;F
);<IL;NILS 89 $&/2)1 3 /3&/1 - #CHCN? ?F?G?HN G?NBI>M @IL NB? +;PC?L\0NIE?M
?KO;NCIHM 0JLCHA?L 3?LF;A 89 $,)2 $ % 3+ ),+
# *;NLCR =IGJON;NCIHM H> ?>CNCIH 'IBHM
%IJECHM 2HCP?LMCNS -L?MM 89 $,21) + /fMIFONCIH >?M fKO;NCIHM >? 0;CHN 3?H;HN ?H LfACG? NL;HM=LCNCKO?
J;L OH? GfNBI>? >YfFfG?HNM VHCM JJFC=;NCIH ;OR <;H=M >f=IOPL;HNM 1BeM? >?
>I=NIL;N >? FY2HCP?LMCNf -;LCM 3& 89 $,21) +
+IN? >? JLCH=CJ? >O =I>? /OJNOL "!# /?JILN %" 89 $/0 / /f@LCAfL;NCIH >?M =?HNL;F?M NB?LGCKO?M =F;MMCKO?M IO HO=Ff;CL?M J;L
JF;H >Y?;O ?N LCPCeL?M "!# /?JILN % H 89 $/6 4 $ (,)/ / ) )2"11& % / 4"01"/&+( ' ' 0B;FFIQ
Q;N?L GI>?FFCHA CH MJB?LC=;F =IIL>CH;N?M ?KO;NCIH @ILGOF;NCIH HOG?LC=;F CG
JF?G?HN;NCIH ;H> ;JJFC=;NCIH 'IOLH;F I@ %S>L;OFC= /?M?;L=B \ 89 $/,00 " 0 (,0"## ' / *,+&0*&1% 0 $ 1BL?? >CG?HMCIH;F M;FCHCNS
MCGOF;NCIHM I@ MIONB 0;H #L;H=CM=I <;S 'IOLH;F I@ %S>L;OFC= "HACH??LCHA
+IP?G<?L \ 89 $OC>? @IL M?F?=NCHA *;HHCHAYM LIOABH?MM =I?^=C?HNM @IL H;NOL;F =B;HH?FM ;H>
WII> JF;CHM 2HCN?> 0N;N?M $?IFIAC=;F 0OLP?S 4;N?L 0OJJFS -;J?L 89 $2&)),2 0 //6 + +$26"+ ( !
;F=OF HOGfLCKO? >?M IH>?M >?
MOL@;=? J;L OH? GfNBI>? >? JLID?=NCIH ;P?= OH G;CFF;A? ?OFfLC?H ;>;JN;NC@
IGJN? /?H>O b FY=;>fGC? >?M 0=C?H=?M >? -;LCM 0fLC? && < \ 89 %/),4 # 4") % ' +OG?LC=;F =;F=OF;NCIH I@ NCG? >?J?H>?HN PCM=IOM
CH=IGJL?MMC<F? WIQ I@ WOC> QCNB @L?? MOL@;=? 1B? -BSMC=M I@ #FOC>M &)&,$/-%6
89 %0+& 6 )&3+" " "/ ,3&"/ * #CHCN? ?F?G?HNM ;H> =B;L;=N?LCMNC=M
;JJFC?> NI ;>P?=NCIH >C_OMCIH ?KO;NCIHM IGJON?LM ;H> #FOC>M \
89 %"+!/& (0,+ )")+! / 1B?
1?=BHC=;F /?JILN 0+! B;=I OM?LYM AOC>? 3?LMCIH 0;H>C;
89 %"/3,2"1 ' * JJFC=;NCIH >? F; GfNBI>? >?M =;L;=NfLCMNCKO?M ?H @ILGO
F;NCIH @;C<F? b F; LfMIFONCIH >?M fKO;NCIHM >Y;>P?=NCIH <C>CG?HMCIHH?FF?M MOL >?M
G;CFF;A?M ALCFF?M "!# /?JILN %" 89 %"/3,2"1 ' *
/ GI>OF? >? =IHP?=NCIH ?H fFfG?HNM VHCM J;L F;
GfNBI>? >?M =;L;=NfLCMNCKO?M "!# /?JILN %" 89 %"/3,2"1 ' * JJFC=;NCIH I@ NB? G?NBI> I@ =B;L;=N?LCMNC=M CH NB?CL Q?;E
@ILGOF;NCIH NI MIFPCHA NQI >CG?HMCIH;F ;>P?=NCIH ?KO;NCIHM IH G?MB ALC>M IG
JON;NCIH;F 1?=BHCKO?M @IL #FOC> #FIQ 3IFOG? CH NB? M?LC?M Z/?=?HN ;>P;H=?M
CH HOG?LC=;F G?NBI>M CH WOC>M[ -CH?LC>A? -L?MM 89 %"/3,2"1 ' *
IGJ;LCMIH I@ ?RJ?LCG?HN;F >;N; ;H> F;M?L G?;MOL?G?HNM
QCNB NB? =IGJON;NCIH;F L?MOFNM I@ 1?F?G;= =I>? MB;FFIQ Q;N?L ?KO;NCIHM -LI
=??>CHAM I@ Z%S>LI=IGJ Y[ !O<LIPHCE 6OAIMF;PC; 'OH? 89 %"/3,2"1 ' * 3?=NILCM;NCIH ?N MCGJFCV=;NCIH >?M ;FAILCNBG?M ?H fFfG?HNM
VHCM "!# OFF?NCH >? F; !CL?=NCIH >?M "NO>?M ?N /?=B?L=B?M 0fLC?
*;NB
fG;NCKO?M &H@ILG;NCKO? H \ 89 %"/3,2"1 ' * 1?F?G;= ; @OFFS P?=NILCM?> VHCN? ?F?G?HN MI@Q;L? @IL MB;FFIQ
Q;N?L ?KO;NCIHM IGJON?L *?NBI>M ;H> 4;N?L /?MIOL=?M /;<;N *ILI==I \
,=NI<?L 89 %"/3,2"1 ' *
B;L;=N?LCMNC=M ;H> G;MM =IHM?LP;NCIH +?Q >?P?FIJG?HNM
CH 1?F?G;= ! "!# /?JILN %" 89 %"/3,2"1 ' * "F?G?HN <S ?F?G?HN G?NBI>M @IL MIFPCHA MB;FFIQ Q;N?L
?KO;NCIHM QCNB # " * &5 &HN?LH;NCIH;F IH@?L?H=? IH IGJON;NCIH;F *?NB
I>M CH 4;N?L /?MIOL=?M !?HP?L IFIL;>I 20 \ 'OH? 89 %"/3,2"1 ' * 0IFPCHA MB;FFIQ Q;N?L ?KO;NCIHM QCNB L;JC> WIQM ;H> NC>;F
W;NM %S>LIMI@N 3;F?H=C; 0J;CH \ 'OFS 89 %"/3,2"1 ' * *fNBI>?M CNfL;NCP?M JIOL F; LfMIFONCIH >?M MSMNeG?M G;NLC
=C?FM "!# /?JILN %" 89 %"/3,2"1 ' * 3;FC>;NCHA NB? HOG?LC=;F MCGOF;NCIH I@ >;G <L?;EM ;H>
WII>M MN &HN?LH;NCIH;F IH@?L?H=? IH %S>LI 0=C?H=? ;H> "HACH??LCHA & %"
4;MBCHANIH ! 20 \ 'OH? 89 %"/3,2"1 ' * #CHCN? "F?G?HN ;FAILCNBGM @IL GI>?FFCHA WII> JLIJ;A;NCIH
-LI=??>CHAM I@ NB? MJ?=C;FCNS =IH@?L?H=? IH GI>?FFCHA I@ WII> JLIJ;A;NCIH IP?L
CHCNC;FFS >LS ;L?;M *CF;H &N;FS \ 'OH? &)&,$/-%6
89 %"/3,2"1 ' *
IGJON;NCIH I@ @L?? MOL@;=? WIQM ; MN;N? I@ NB? ;LN CH
NB? 1?F?G;= MSMN?G #CLMN &HN?LH;NCIH;F IH@?L?H=? IH %S>LICH@ILG;NC=M !?F@N
1B? +?NB?LF;H>M \ 0?JN?G<?L 89 %"/3,2"1 ' * /?=?HN ;>P;H=?M CH NB? 1?F?G;= MSMN?G Z- ,* [
1BCL> MC;H -;=CV= IH@?L?H=? 0?IOF 0IONB (IL?; \ 0?JN?G<?L 89 %"/3,2"1 ' * HOG?LC=;F MCGOF;NCIH I@ NB? *;FJ;MM?N >;G <L?;E QCNB
! 0;CHN 3?H;HN ?KO;NCIHM -LI=??>CHAM I@ NB? 0;CHN 3?H;HN MSGJIMCOG *;LH?
F; 3;FFf? #L;H=? \ OAOMN 89 %"/3,2"1 ' *
IGJ;LCHA HOG?LC=;F MCGOF;NCIHM I@ @L?? MOL@;=? WIQM OM
CHA HIH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM ;H> IOMMCH?MK ?KO;NCIHM 5&& &H
N?LH;NCIH;F IH@?L?H=? IH IGJON;NCIH;F *?NBI>M CH 4;N?L /?MIOL=?M 1B?
B?LMIH?M? (L?N; $L??=? \ 'OH? 89 %"/3,2"1 ' * BCAB L?MIFONCIH ! >;G <L?;E GI>?F OMCHA J;L;FF?FCT;
NCIH %S>LIFIAC=;F -LI=?MM?M \ 89 %"/3,2"1 ' * ,H MJOLCIOM IM=CFF;NCIHM CH JLCGCNCP? MB;FFIQ Q;N?L ?KO;
NCIHM 5&&& &HN?LH;NCIH;F IH@?L?H=? IH IGJON;NCIH;F *?NBI>M CH 4;N?L /?
MIOL=?M ;FA;LS ;H;>; \ 'OH? 89 %"/3,2"1 ' * $2"0*& * 0IFONCIH I@ IOMMCH?MK ?KO;NCIHM CH VHCN?
?F?G?HNM I;MN;F "HACH??LCHA )?GHIM $L??=? \ *;S 89 %"/3,2"1 ' * '+&+ ' * /?VH?> WIQ GI>?FFCHA CH =I;MN;F ;L?;M
I;MN;F ;H=OH *?RC=I \ 0?JN?G<?L 89 %"/3,2"1 ' * '+&+ ' * */ ,0 # +OG?LC=;F GI>?FFCHA I@ @L??
MOL@;=? WIQM !?PCMCHA MNL;N?AC?M @IL J;L;FF?FCMG IGJON?L *I>?FFCHA I@ 0?;M
;H> I;MN;F /?ACIHM ); ILOH; 0J;CH \ 'OH? 89 %"/3,2"1 ' * '+&+ ' * *,2)&+
-L?=IH>CNCIHCHA ;H> MIFPCHA
FCH?;L MSMN?GM @IL NB? =IGJON;NCIH I@ @L?? MOL@;=? WIQM 5 &HN?LH;NCIH;F IH
@?L?H=? IH IGJON;NCIH;F *?NBI>M CH 4;N?L /?MIOL=?M %?C>?F<?LA $?LG;HS
89 %"/3,2"1 ' * '+(,40(& '
IGJ;LCHA HOG?LC=;F MCGOF;NCIHM I@ @L??
MOL@;=? WIQM OMCHA HIH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM ;H> IOMMCH?MK
?KO;NCIHM %S>LICH@ILG;NC=M &IQ; &HMNCNON? I@ %S>L;OFC= /?M?;L=B &IQ; CNS
20 \ 'OFS 89 %"/3,2"1 ' * )"-"&+1/" # -LCM? ?H =IGJN? >?M JIN?HNC?FM ;MNL;OR
>;HM F?M =;F=OFM >? G;Lf? "!# /?JILN %" 89 %"/3,2"1 ' * */ ,0 # -;L;FF?FCMG ;H> =I>? =IOJFCHA CH NB? 1?F?G;=
MSMN?G -LI=??>CHAM I@ NB? %S>LICH@ILG;NC=M IH@?L?H=? IJ?HB;A?H !?H
G;LE \ OAOMN &)&,$/-%6
89 %"/3,2"1 ' * *,2)&+
+IOP?;OR M=BfG;M >? =IHP?=NCIH >;HM
1?F?G;= ! P?LMCIH ;JJILN >? F; GfNBI>? 02-$ "!# /?JILN %"
89 %"/3,2"1 ' * -"1&1'"+ *;FJ;MM?N >;G <L?;E L?PCMCN?> QCNB NQI
>CG?HMCIH;F =IGJON;NCIHM 'IOLH;F I@ %S>L;OFC= /?M?;L=B \ 89 %"/3,2"1 ' * 0*&" / *,/"2 *I>?FFCHA OL<;H ;L?;M CH >;G
<L?;E WII> Q;P? HOG?LC=;F MCGOF;NCIHM &HN?LH;NCIH;F 0?GCH;L ;H> 4ILEMBIJ
IH NB? /?M=>;G -LID?=N 0?CHcDIEC #CHF;H> \ ,=NI<?L 89 %"/3,2"1 ' * 3+ %/"+ )
I>? 1?F?G;= ! 3?LMCIH +IN? >?
-LCH=CJ? "!# /?JILN %" 89 %"/3,2"1 ' * 3+ %/"+ ) /?=?HN ;>P;H=?M CH HOG?LC=;F G?NBI>M
@IL WOC> WIQ &H #FII>JF;CH -LI=?MM?M 'IBH 4CF?S 0IHM )N> 89 %"01"+"0 * / 01&"#") " *?NBI>M I@ =IHDOA;N? AL;>C?HNM @IL MIFPCHA
FCH?;L MSMN?GM 'IOLH;F I@ /?M?;L=B I@ NB? +0 3 89 %& (0 # " 01"##)"/ - * =B;L;=N?LCMNC= >CMMCJ;NCP? $;F?LECH M=B?G?
@IL IJ?H =B;HH?F WIQ 0 " 'IOLH;F I@ %S>L;OFC=M "HACH??LCHA \
89 %& (0 # " 01"##)"/ - *
IGJ;LCMIH I@ VHCN? ?F?G?HN G?NBI>M @IL NB?
0;CHN 3?H;HN ?KO;NCIHM &HN?LH;NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M CH #FOC>M
\ 89 %&+(")*++ /
-;L;FF?FCMC?LOHA ?CH?M );AL;HA? "OF?L 3?L@;BL?HM @jL
0NLhGOHAM OH> 0NI_NL;HMJILNJLIT?MM? CH ,<?LWc=B?HA?QcMM?LH &HMNCNON @jL
0NLhGOHAMG?=B;HCE OH> "F?ENLIHCE /?=BH?H CG ;OQ?M?H >?L 2HCP?LMCNcN
%;HHIP?L ?LC=BN H 89 %&/1
+& %,)0 3IFOG? I@ WOC> 3,# G?NBI> @IL NB? >SH;GC=M I@
@L?? <IOH>;LC?M 'IOLH;F I@ IGJON;NCIH;F -BSMC=M \ 89 &!")Y &( & " *fG?HNI >?M J?LN?M >? =B;LA? ">CNCIHM "SLIFF?M 89 '+&+ ' *
I>? 1?F?G;= ! 3?LMCIH /?JILN )+% %" !?M=LCJNC@ CH@ILG;NCKO? "!#
89 '+&+ ' *
IHM?LP;NCPCNf ?N JIMCNCPCNf >;HM OH GI>OF? >? NL;HMJILN >?
M=;F;CL? f=LCN ?H fFfG?HNM VHCM JJFC=;NCIH b 1?F?G;= ! "!# /?JILN %"
89 '+&+ ' * !3&! " !"+,1 1
I>? 1?F?G;= ! 3?LMCIH *;HO?F
ONCFCM;N?OL "!# /?JILN %" 89 '+&+ ' * !3&! " !"+,1 1
I>? 1?F?G;= ! 3?LMCIH ;NCIH >I=OG?HN "!# /?JILN )+% %" 3;FC>
&)&,$/-%6
89 '+&+ ' * )"-"&+1/" # -" %,+ - 1?F?G;= ! ; VHCN? ?F?G?HN =I>?
NI MIFP? ! @L?? MOL@;=? WIQ JLI<F?GM IGJON?L *I>?FFCHA I@ 0?;M ;H> I;MN;F
/?ACIHM 0IONB;GJNIH 2( JLCF 89 '+(,40(& ' HIH BS>LIMN;NC= GI>?F @IL @L?? MOL@;=? WIQM &HMNCNON
@jL 0NLhGOHAMG?=B;HCE OH> "F?ENLIHCE /?=BH?H CG ;OQ?M?H >?L 2HCP?LMCNcN
%;HHIP?L ?LC=BN H 89 (/6-&0 $ (2*/ 3 *?NCM G?MB J;LNCNCIH?L 2HMNLO=NOL?> AL;JB J;LNC
NCIHCHA ;H> MJ;LM? G;NLCR IL>?LCHA MSMN?G 2HCP?LMCNS I@ *CHH?MIN; 1?=BHC=;F
/?JILN 89 (4*2/ 1 *&61 % 0CGOF;NCIH I@ HIHFCH?;L MBCJ WIQM <S >?HMCNS
@OH=NCIH G?NBI> 'IOLH;F I@ NB? 0I=C?NS I@ +;P;F L=BCN?=NM ';J;H \
89 ("/+&$%+ 4 )&+ 0 H ?^=C?HN B?OLCMNC= JLI=?>OL? @IL J;LNCNCIHCHA
AL;JBM 1B? ?FF 0SMN?G 1?=BHC=;F 'IOLH;F 89 ()00"+ $ ' P;H >?L 74/! ' ' /IOABH?MM =I?^=C?HNM I@ P?A?N;N?>
WII> JF;CHM 'IOLH;F I@ %S>L;OFC= /?M?;L=B H 89 (,)/ / ) $/6 4 $ 4"01"/&+( ' ' )2"11& % / 0B;FFIQ
Q;N?L GI>?FFCHA CH MJB?LC=;F =IIL>CH;N?M ?KO;NCIH @ILGOF;NCIH HOG?LC=;F CG
JF?G?HN;NCIH ;H> ;JJFC=;NCIH 'IOLH;F I@ %S>L;OFC= /?M?;L=B \ 89 (,)/ / ) $/6 4 $ 4"01"/&+( ' ' IOH>;LS =IH>CNCIHM CH MB;FFIQ
Q;N?L GI>?FM ;H ;FN?LH;NCP? CGJF?G?HN;NCIH @IL VHCN? ?F?G?HN =I>?M &HN?LH;
NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M CH #FOC>M \ 89 (,/+ $ (,/+ 1 * *;NB?G;NC=;F B;H><IIE @IL ?HACH??LM ;H> M=C?HNCMNM
*=$L;Q %CFF 89 )!&" $ )0)"&7 - .O?FKO?M GfNBI>?M >? LfMIFONCIH >O JLI<FeG? >?
0NIE?M ?H fFfG?HNM VHCM "!# /?JILN %" 89 )!&" $ -"/,+ 0 %"/3,2"1 ' * 0NO>S I@ B?;> FIMM?M CH ; =IH
WO?H=? I@ LCP?LM QCNB ; M=;F? GI>?F ;H> ; HOG?LC=;F GI>?F MIFPCHA MB;FFIQ
Q;N?L ?KO;NCIHM /?VH?> WIQM GI>?FFCHA 55&3 &%/ IHAL?MM *;>LC> \
0?JN?G<?L 89 )#2/&" +/!,+"
0 /!,3"))& / 7)"0(& 0 7+"11&
$ *I>?FFCHA G?LACHA ;H> @L;AG?HN;NCIH CH GOFNCJB;M? WIQM QCNB 02/#"/
'IOLH;F I@ IGJON;NCIH;F -BSMC=M \ 89 )&1,+" " 1B? M?=IH> ;JJLIRCG;NCIH NI =HIC>;F ;H> MIFCN;LS Q;P?M 'IOLH;F
I@ #FOC> *?=B;HC=M \ 89 )2+!"/ " 0-)!&+$ ! 1B? HOG?LC=;F =IGJON;NCIH I@ NOL<OF?HN
WIQM IGJON?L *?NBI>M CH JJFC?> *?=B;HC=M ;H> "HACH??LCHA &)&,$/-%6
89 )2/"+ " ! 3&,))"1 - )
IOLM >? 0CGOF;NCIH +OGfLCKO? >O )+%
1OL<OF?H=? 1B?LGIBS>L;OFCKO? ?N 0NL;NCV=;NCIHM 1IG? "!# /?JILN !"/
%" 89 )5 - ! 4"+!/,## 0SMN?GM I@ =IHM?LP;NCIH F;QM
-OL? ;H> JJFC?> *;NB?G;NC=M \ IGGOHC=;NCIHM CH
89 )"%#")!1 / ),00 0 ,H NB? HOG?LC=;F GI>?FFCHA I@ MNL;NCV?> WIQM -BSM
C=;F JLI=?MM?M CH ?MNO;LC?M ' !LIE?LM 4 3;H )?OMM?H ?>M 0JLCHA?L
3?LF;A 89 )"-"&+1/" #
IGG?HN LfMIO>L? F?M fKO;NCIHM >? 0;CHN 3?H;HN MJBfLCKO?M
;P?= OH =I>? LfMIFP;HN F?M fKO;NCIHM >? 0;CHN 3?H;HN ?H =IIL>IHHf?M =;LNfMC
?HH?M "!# /?JILN %" ;H> %" 89 )& #)"*&+$
1BL?? >CG?HMCIH;F GI>?F I@ +;PC?L\0NIE?M ?KO;NCIHM
@IL Q;N?L Q;P?M 'IOLH;F I@ 4;N?LQ;S -ILN I;MN;F ;H> ,=?;H "HACH??LCHA
';HO;LS\#?<LO;LS 89 )& % 1 $ *21 ) ( IOH>;LS =IH>CNCIHM @IL NB? NQI >CG?HMCIH;F 0;CHN
3?H;HN ?KO;NCIH MSMN?G JJFC?> *;NB?G;NC=;F *I>?FFCHA \ 89 )&,+0 ' ) 0IG? G?NBI>M CH NB? G;NB?G;NC=;F ;H;FSMCM I@ MSMN?GM ;H> NB?CL
=IHNLIF 0=C?H=? -L?MM 89 ),%+"/ / *,/$+ ( -"/&/" ' ;H> 3%!1& * #CHCN? ?F?G?HN
WOR =ILL?=N?> NL;HMJILN #"*\# 1 @IL NB? "OF?L ;H> +;PC?L\0NIE?M ?KO;
NCIHM &HN?LH;NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M CH #FOC>M \
89 ),3" " %
0IG? JLI<F?GM I@ A?I>SH;GC=M
;G<LC>A? 2HCP?LMCNS -L?MM
89 )2 .2&+ -&/,++"2 , &HNLI>O=NCIH ;O =;F=OF M=C?HNCVKO? *;MMIH
89 )2"11& % / 4"01"/&+( ' ' 0 %"##+"/ + 4 ! &/ H ;>
P;H=?> NBL?? >CG?HMCIH;F =CL=OF;NCIH GI>?F @IL MB?FP?M =I;MNM ;H> ?MNO;LC?M
!?JN I@ NB? LGS 20 LGS ILJM I@ "HACH??LM 89 )2*)"6 ' ) GI>?F @IL =IGJON;NCIH I@ MNL;NCV?> NOL<OF?HN WIQMY -LI=??>
CHAM I@ &HN?LH;NCIH;F 0SGJIMCOG IH 0NL;NCV?> #FIQM +IPIMMC<CLME JJ \
G?LC=;H 0I=C?NS I@ CPCF "HACH??LM 89 *!0"+ - 0,/"+0"+ , / H?Q @ILG I@ NB? IOMMCH?MK ?KO;NCIHM
QCNB CGJLIP?> FCH?;L >CMJ?LMCIH =B;L;=N?LCMNC= -;LN && MFIQFS P;LSCHA <;NBS
G?NLS I;MN;F "HACH??LCHA \ 89 *%!"3+ ,)&$"/ ' 01/""1 / HIH BS>LIMN;NC= G?MI M=;F?
I=?;H GI>?F 'IOLH;F I@ -BSMC=;F ,=?;HIAL;JBS \ 89 *+,% *fNBI>?M MCGJF?M >Y?MNCG;NCIH >? F; >CFONCIH >YOH L?D?N ?H G?L
"!# /?JILN )+% %" &)&,$/-%6
89 */ ,0 # '+&+ ' *
I>? 1?F?G;= !
"!# /?JILN )+% %" 3?LMCIH +IN? NBfILCKO?
89 */ ,0 # '+&+ ' * +IOP?;OR >fP?FIJJ?G?HNM >;HM FYfN;J? >? =IH
P?=NCIH >C_OMCIH >? 1?F?G;= ! "!# /?JILN %" ,=NI<?L 89 *0,+ !
,6 ! * %,//&11 * 0 ;H> 1"0 - ! #FII>JF;CH
@LC=NCIH J;L;G?N?LCT;NCIH CH NQI >CG?HMCIH;F LCP?L WII> GI>?FM OMCHA P?A?N;NCIH
B?CABNM >?LCP?> @LIG ;CL<ILH? M=;HHCHA F;M?L ;FNCG?NLS %S>LIFIAC=;F -LI=?MM?M
\ 89 *""20 ' MNLIHIGC=;F @ILGOF;? @IL =;F=OF;NILM 4CFFG;HH ?FF 89 *")),/ $ ) 6*! 1 BC?L;L=BS I@ NOL<OF?H=? =FIMOL? GI>?FM @IL
JF;H?N;LS <IOH>;LS F;S?LM 'IOLH;F I@ NGIMJB?LC= 0=C?H=?M \
89 *"4&0 - %,)7 ( - KO;MC <O<<F? @OH=NCIH ;JJLI;=B @IL MB;FFIQ Q;N?L
Q;P?M >P;H=?M CH %S>LI 0=C?H=? ;H> "HACH??LCHA \ 89 *,+$%+ ' ' 0GIINB?> J;LNC=F?M BS>LI>SH;GC=M HHO;F /?PC?Q I@ M
NLIJBSMC=M 89 *,+$%+ ' ' 0CGOF;NCHA @L?? MOL@;=? WIQM QCNB 0-% 'IOLH;F I@
N;NCIH;F -BSMC=M \ IGJO
89 *27#"/&' 0 -"/& *
IGJON;NCIH I@ @L?? MOL@;=? WIQM OMCHA
CHN?L@;=? NL;=ECHA ;H> CHN?L@;=? =;JNOLCHA G?NBI>M &H +IHFCH?;L 4;N?L 4;P?
&HN?L;=NCIH , *;BL?HBIFNT * *;LEC?QC=T ">M IGJON;NCIH;F *?=B;HC=M
-O<FC=;NCIHM 89 +&* * *I>fFCM;NCIH HOGfLCKO? >?M ?_?NM BS>LI>SH;GCKO?M JLIPIKOfM J;L
OH AFCMM?G?HN >? N?LL;CH >;HM OH? L?N?HO? 1BeM? >? FY2HCP?LMCNf 'IM?JB #IOLC?L
$L?HI<F? 89 +3&"/
*fGICL? MOL F?M FICM >O GIOP?G?HN >?M WOC>?M *fGICL?M >?
FY=;>fGC? >?M 0=C?H=?M \ 89 +"72 & +($4 % 1OL<OF?H=? CH IJ?H =B;HH?F WIQM &%/ GIHI
AL;JB M?LC?M ;FE?G; 89 +&%,2) ' '*/1 1BL?? >CG?HMCIH;F GI>?FM I@ G;LCH? ;H> ?MNO;LCH?
>SH;GC=M 3IFOG? I@ "FM?PC?L ,=?;HIAL;JBS 0?LC?M "FM?PC?L 89 +4,$2 , FN?LH;NCP? @ILG I@ IOMMCH?MK ?KO;NCIHM @IL H?;LMBIL? Q;P?
JLIJ;A;NCIH 'IOLH;F I@ 4;N?LQ;SM -ILN I;MN;F ;H> ,=?;H "HACH??LCHA \ 89 -&))"/" % *OFNC>CG?HMCIH;F OJQCH> L?MC>O;F >CMNLC<ONCIH M=B?G?M @IL NB?
"OF?L ;H> +;PC?L\0NIE?M ?KO;NCIHM IH OHMNLO=NOL?> ALC>M 1BeM? >? FY2HCP?LMCNf
)C<L? >? LOR?FF?M &)&,$/-%6
89 -" "*+ ! 4 / %#,/! % % 1B? HOG?LC=;F MIFONCIH I@ J;L;<IFC=
;H> ?FFCJNC= >C_?L?HNC;F ?KO;NCIHM 0&* 'IOLH;F \ 89 -" %,+ MOL@;=? FC<L?
I>? *&1%/&!1" JIOL F?M f=IOF?G?HNM NLC>CG?HMCIHH?FM b
;BC?L >?M =B;LA?M "!# /?JILN )+% %" 89 -"/1%*" IHP?LA?H=? I@ + M=B?G?M @IL FCH?;L ;>P?=NCIH ?KO;NCIHM -CN
G;H *IHIAL;JBM 0OLP?SM CH -OL? ;H> JJFC?> *;NB?G;NC=M )IHAG;H
89 -"/1%*" 0&*",+&
ECH?NC= M=B?G? @IL NB? 0;CHN 3?H;HN MSMN?G
QCNB ; MIOL=? N?LG /;JJILN >? /?=B?L=B? !* >O !fJ;LN?G?HN >?
G;NBfG;NCKO?M ?N ;JJFC=;NCIHM "=IF? +ILG;F? 0OJfLC?OL? +/0 2*/ *;S 89 -&/,++"2 , *fNBI>? >?M fFfG?HNM VHCM JIOL F?M WOC>?M *;MMIH 89 -/+!1) ) a<?L >C? ;OMA?<C>?N? 1OL<OF?HT 7?CNM=BLC@N @jL ;OA?Q;H>N?
*;NB?G;NCE OH> *?=B;HCE 89 -/"00 4 % #)++"/6 - 1"2(,)0(6 0 3"11"/)&+$ 4 1
+OG?LC=;F L?=CJ?M NB? ;LN I@ M=C?HNCV= =IGJONCHA #,/1/+ P?LMCIH ;G
<LC>A? 2HCP?LMCNS -L?MM 89 .2"1&+ *I>eF?M G;NBfG;NCKO?M >? =;F=OF >?M f=IOF?G?HNM CH>OCNM J;L F?
P?HN CeG? =IHALeM >? FY&/% ;>?H ;>?H \ IiN 89 /*"11" * $OC>? >YBS>L;OFCKO? WOPC;F? "!# /?JILN )+% %" 89 /01,$& ( /,!& 4 -L?>C=NCIHM I@ B?;N ;H> G;MM NL;HM@?L CH IJ?H
=B;HH?FM 'IOLH;F I@ NB? %S>L;OFC=M !CPCMCIH 0 " H %6 \ 89 /3&/1 - 1%,*0 ' * &HNLI>O=NCIH b FY;H;FSM? HOGfLCKO? >?M fKO;
NCIHM ;OR >fLCPf?M J;LNC?FF?M *;MMIH 89 /6+ $
/"!2 " 0I@NQ;L? @IL ;FA?<L;C= =IGJON;NCIH 0JLCHA?L\3?LF;A
89 /&11"/ !C? #ILNJW;HTOHA >?L 4;MM?LQ?FF?H 7?CNM=BLC@N >?M 3?L?CH?M
>?ONM=B?L &HA?HC?OL? \ 89 /,!& 4 1OL<OF?H=? GI>?FM ;H> NB?CL ;JJFC=;NCIHM CH BS>L;OFC=M MN;N? I@
NB? ;LN L?PC?Q ">CNCIH &/% 89 /," - ) JJLIRCG;N? /C?G;HH MIFP?LM J;L;G?N?L P?=NILM ;H> >C_?L?H=?
M=B?G?M 'IOLH;F I@ IGJON;NCIH;F -BSMC=M \ 89 /," - ) 0&!&)(,3"/ ! ,JNCGOG JIMCNCP? FCH?;L M=B?G?M @IL ;>P?=NCIH CH
NQI ;H> NBL?? >CG?HMCIHM 0&* 'IOLH;F I@ +OG?LC=;F H;FSMCM \
&)&,$/-%6
89 0! 6 0 %2)17 * % $*/"0 ; A?H?L;FCT?> GCHCG;F L?MC>O;F ;FAILCNBG
@IL MIFPCHA HIHMSGG?NLC= FCH?;L MSMN?GM /?M?;L=B /?JILN 6)"2// 89 0&+1 3"++1 '
1BfILC? >O GIOP?G?HN HIH J?LG;H?HN >?M ?;OR
;P?= ;JJFC=;NCIH ;OR =LO?M >?M LCPCeL?M ?N b FYCHNLI>O=NCIH >?M G;Lf?M >;HM F?OL
FCN IGJN? /?H>O b FY=;>fGC? >?M 0=C?H=?M >? -;LCM \ 89 0")*&+ 3 !,+" ' .2/1-"))" ) #CHCN? ?F?G?HN G?NBI>M @IL HIH
FCH?;L ;>P?=NCIH IGJON?L *?NBI>M CH JJFC?> *?=B;HC=M ;H> "HACH??LCHA
\ 89 0"//" #
IHNLC<ONCIH b FYfNO>? >?M f=IOF?G?HNM J?LG;H?HNM ?N P;LC;<F?M >;HM
F?M =;H;OR ); %IOCFF? F;H=B? 'OCH\'OCFF?N 89 0 %2/"*+ - *;HO;F I@ B;LGIHC= ;H;FSMCM ;H> JL?>C=NCIH I@ NC>?M 20
$IP?LHG?HN -LCHNCHA ,^=? 89 0%(& # %2$%"0 1 ' / ',%+ 7 GOFNC ?F?G?HN ALIOJ JL?=IH>C
NCIH?> $*/"0 ;FAILCNBG @IL HIHMSGG?NLC= MSMN?GM ;LCMCHA CH VHCN? ?F?G?HN
;H;FSMCM IGJON?L *?NBI>M CH JJFC?> *?=B;HC=M ;H> "HACH??LCHA \
89 0&*,+ % ! -;LNCNCIHCHA I@ OHMNLO=NOL?> JLI<F?GM @IL J;L;FF?F JLI=?MMCHA
IGJONCHA 0SMN?GM CH "HACH??LCHA \ 89 0*$,/&+0(& ' $?H?L;F =CL=OF;NCIH ?RJ?LCG?HNM QCNB NB? JLCGCNCP? ?KO;
NCIHM & 1B? <;MC= ?RJ?LCG?HN *IHNBFS 4?;NB?L /?PC?Q \ 89 0,)&3 '
*I>eF? NLC>CG?HMCIHH?F >Yf=IOF?G?HNM GfMI GfNfILIFIACKO?M
"NO>? >? F; =IHP?=NCIH >C_OMCIH >YOH JIFFO;HN J;MMC@ b =?NN? f=B?FF? 1BeM?
"+- 89 0-"7&)" $ ,H HIH FCH?;L E ) ;H> E ?JMCFIH GI>?FM I@ NOL<OF?H=? 'IOLH;F
I@ #FOC> *?=B;HC=M \ 89 01,("/ ' ' 4;N?L Q;P?M 4CF?S &HN?LM=C?H=? -O<FCMB?LM \ 89 01,("0 $ ,H NB? NB?ILC?M I@ NB? CHN?LH;F @LC=NCIH I@ WOC>M GINCIH ;H> I@
NB? ?KOCFC<LCOG ;H> GINCIH I@ ?F;MNC= MIFC>M 1L;HM;=NCIHM I@ NB? ;G<LC>A?
-BCFIMIJBC=;F 0I=C?NS \ 89 02
% *&/&" / * ,H B?;> IH =IFFCMCIHM <?NQ??H NQI MIFCN;LS Q;P?M
'IOLH;F I@ #FOC> *?=B;HC=M \ 89 04""/0 % " *IHIAL;GM NI ?MNCG;N? NB? B?;N ?R=B;HA? =I?^=C?HN ;N NB? ;CL
Q;N?L CHN?L@;=? ;M ; @OH=NCIH I@ QCH> MJ??> ;H> N?GJ?L;NOL? ; =LCNC=;F MOLP?S
I@ MIG? FCN?L;NOL? 'IOLH;F I@ %S>LIFIAS H \ 89 1%,*-0,+ ( 4 1CG? >?J?H>?HN <IOH>;LS =IH>CNCIHM @IL BSJ?L<IFC= MSM
N?GM 'IOLH;F I@ IGJON;NCIH;F -BSMC=M \ &)&,$/-%6
89 10+&0 & 0CGOF;NCIH I@ QCH> CH>O=?> Q;N?L =OLL?HNM 'IOLH;F I@ %S>L;OFC=
"HACH??LCHA \ 89 12/+"/ ' OIS;H=S ?_?=NM CH WOC>M
;G<LC>A? 2HCP?LMCNS -L?MM 89 2+"0 , &HN?LH;NCIH;F ,=?;HIAL;JBC= 1;<F?M 3IFOG? 2+"0 , 1?=BHC=;F
J;J?LM CH G;LCH? M=C?H=? H 89 3&,))"1 - ) *I>eF?M >? NOL<OF?H=? b J;LNCL >? AL;H>?OLM MN;NCMNCKO?M ?H OH
JICHN GI>eF? ( ?JMCFIH "!# OFF?NCH >? F; !CL?=NCIH >?M "NO>?M ?N /?=B?L=B?M
0fLC? +O=Ff;CL? %S>L;OFCKO? 1B?LGCKO? \ 89 3&,))"1 - ) ,H NB? HOG?LC=;F GI>?FFCHA I@ MNL;NCV?> WIQM -BSMC=;F JLI
=?MM?M CH ?MNO;LC?M ' !LIE?LM 4 3;H )?OMM?H ?>M 89 3&,))"1 - )
%/! ' - "0-,0&1, - )2/"+ " ! *f=;HCKO?
>?M WOC>?M ;JJFCKOf? -L?MM?M >? FY"=IF? +;NCIH;F? >?M -IHNM ?N B;OMMf?M 89 4"01"/&+( ' '
,++,/ ' ' 01,)7"+ % ( ! JLCGCNCP? JM?O>I
Q;P? ?KO;NCIH @ILGOF;NCIH @IL MIFPCHA NB? B;LGIHC= MB;FFIQ Q;N?L ?KO;NCIHM
>P;H=?M CH 4;N?L /?MIOL=?M 3IFOG? 89 4"01"/&+( ' ' )2"11& % / 'L 42 ' ( (,)/ / ) 1B? CHWO?H=?
I@ HILG;F WIQ <IOH>;LS =IH>CNCIHM IH MJOLCIOM GI>?M CH VHCN? ?F?G?HN MIFONCIHM
NI NB? MB;FFIQ Q;N?L ?KO;NCIHM &HN?LH;NCIH;F 'IOLH;F @IL +OG?LC=;F *?NBI>M CH
#FOC>M \ 89 42
%2+$ $ 7%"+$ 6 1B?IL?NC=;F MIFONCIH I@ >;G <L?;E MBI=E Q;P?
'IOLH;F I@ %S>L;OFC= "HACH??LCHA +IP?G<?L \ 89 6)&+ * 0 *?=B;HC=M I@ M?>CG?HN NL;HMJILN -?LA;GIH -L?MM 89 6&%
0 0NL;NCV?> WIQM =;>?GC= -L?MM Index
ADI (Alternating-Direction Implicit),
85
advection, 20, 91, 141, 177
ALE (Arbitrary Lagrangian–Eulerian),
134
assembly, 215, 216
baroclinic, 12, 18, 57, 133
barotropic, 12, 18, 57, 133
bisection, 249
Boltzman equation, 88
bottom shear stress, 13
boundary conditions
k–ε model, 72
Navier–Stokes, 9
Saint-Venant, 58, 99
tracer, 24, 61
Boussinesq, 24, 44
approximation, 11, 57, 156
equations, 44, 108
model, 62
buoyancy, 12, 17, 57, 143
celerity, 33, 39, 58
centrifugal force, 14
Chézy
coefficient, 50, 54, 70
formula, 13, 51, 53, 146, 267, 276,
289
characteristic curves, 35, 91, 177
cnoidal wave, 45
conditioning, 104, 109, 111, 225
conjugate gradient, 226, 228, 278
conjugate residual, 227
connectivity table, 199, 250
conservation of energy, 118
conservation of mass
Navier–Stokes, 9, 27, 153, 165
Saint-Venant, 27, 42, 43, 96, 98, 114,
153
tracer, 115, 166
conservation of momentum
Navier–Stokes, 9
Saint-Venant, 31, 96, 98, 116
consistency, 84
continuity, see conservation of mass
Coriolis
coefficient, 16
force, 14, 55
cost function, 258
Crank–Nicholson scheme, 86
Crout preconditioning, 241, 245
CSR (Compressed Sparse Row), 203
dam break, 130, 281
damping functions, 65
density of water, 17, 56, 297
diagonal storage, 202
diffusion, 92
dilatation coefficients, 11
Dirac function, 102, 135
Dirichlet boundary conditions
(imposed values), 4, 95, 97, 100,
102, 105, 112, 158, 166, 171, 173,
220, 264
dispersion, 29
dissipation, 70
distributive scheme, 169, 183
divergence, 21, 41
drag force, 53
EBE storage (element-by-element), 203
edge-based storage, 205
eigenvalue, 87, 225
eigenvector, 87
Elder model, 63
entropy, 117
Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method
ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0
J.-M. Hervouet
340 INDEX
Euler, 33
equations, 33
fetch, 14
finite element, 77, 195
finite volume, 87
fractional steps, 90, 136
friction, 13, 50, 103, 146, 267
friction velocity, 73
Froude number, 36, 58
Galerkin method, 89
Gauss–Seidel preconditioning, 245
genetic algorithms, 290
GMRES, 230
Goutal, 95, 97
gradient, 41
Haaland, 70
head loss, 37, 53
heat, 16
heat capacity ratio, 34
Helmholtz equation, 78, 111
Hilbert space, 78
hydraulic jump, 36, 116, 128
impermeability, 26, 148
incident wave, 59
inf–sup condition, 81
isoparametric transformation, 207
Jacobian, 19, 22, 87, 106, 143,
207–209, 211, 318
Janin, 3
Jankowski, 4
jet, 66
Julian calendar, 309
k–ε model, 44, 55, 70, 111
kernel, 88, 135
kinematic viscosity, 9
kinetic schemes, 88
Lagrangian, 10, 83, 133, 135, 177,
257, 259
Laplacian, 123
Leibnitz, 26
logarithmic layer, 69
M-matrix, 210
MAC (Marker And Cell), 134
Malpasset, 281
Manning
coefficient, 44
formula, 14, 52
mass lumping, 104, 124, 127
matrix–vector product, 220, 222, 253
Mercator projection, 39
minimum error, 229
mixing length model, 64, 71
momentum, see conservation of
momentum
monotonicity, 85
MURD scheme, 190
N scheme, 184, 188
nabla, 8
Navier, 5
Navier–Stokes equations, 7, 133, 155
Neumann, 74
Newton, 5
second law, 7
Newtonian fluid, 9
Nezu and Nakagawa model, 64
Nikuradse formula, 52, 70
no-flux boundary condition, 24, 61
parallelism, 247
parameter estimation, 257, 288
partition, 248
Poisson equation, 78
porosity, 43, 94
Prandtl, 64, 71
preconditioning, 34, 224, 234
pressure, 12
atmospheric, 11, 17, 56
hydrostatic, 11, 25
prism, 196
production of turbulent energy, 71
propagation, 33, 92
pseudo wave equation, 83, 94, 123, 158
PSI scheme, 184, 189
quasi-bubble, 123, 126, 195, 212
Quetin model, 65
radiation conditions, 105
Ramette, 70
INDEX 341
Reichard law, 70
relaxation
coefficient for diffusion, 94
coefficient for the tracer, 113
coefficient for velocity, 93
coefficient for water depth, 93
Reynolds
equations, 61
number, 54, 61, 96
stress, 30, 61, 62, 75
tensor, 30, 63, 65
Riemann
invariants, 35, 59, 105
problem, 87
Ritter solution, 130
Roe scheme, 87
Rossby, 16
rough flow, 69, 101, 146
Saint-Venant, 24
equations, 24, 83
salinity, 11, 17, 63
Schmidt number, 63
sediment, 11, 23, 63
Serre equations, 45
shallow water equations,
see Saint-Venant
sigma transform, 18, 140
sinks, 168
Smagorinski model, 68
smooth flow, 69, 70, 101
Sobolev space, 78
solitary wave, 294
source terms, 31, 49, 56, 92
sources, 168
space discretization, 95
SPH (Smoothed Particles
Hydrodynamics), 135
stability, 84
Stoker solution, 130
Stokes, 5
strain rate tensor, 9, 64
stress tensor, 9, 62
Strickler
coefficient, 51, 54, 279
formula, 13, 51, 53, 146, 267, 276
subcritical, 36, 58
supercritical, 36, 38, 58, 89, 129, 174
SUPG (Streamline Upwind Petrov
Galerkin), 141, 172, 181
surrogate model, 290
Telemac, 1, 195
2D, 89, 256, 267, 281
3D, 255
temperature, 11, 17, 23, 63
test function, 80
tetrahedron, 196, 317
Thompson method, 105
time discretization, 90, 92, 95
tracer, 23, 60, 113, 115, 165, 166
triangle, 195
Tsanis model, 66
tsunami, 33
turbulence, 61
turbulent fluctuation, 71
turbulent kinetic energy, 70
uncovered beds
2D, 121
3D, 174
uniqueness, 84, 105
variational formulation, 80, 95, 112, 157
viscous layer, 69
VOF (Volume Of Fluids), 134
von Karman
constant, 64
eddies, 84, 100
weir, 128, 129
wind, 14, 55