Tutorial 2016 Fields Course 01 to 04

An introduction to scientific computing using
free software FreeFem++
Ionut Danaila1 and Frédéric Hecht2
1
Université de Rouen Normandie, France
2
Université Pierre et Marie Curie, Paris
August 8, 2021
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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Scientific Computing = branch of Applied Maths
Physics
obs/equations
Numerical meth.
PDE/num analysis.
Implementation Results
algorithm/code physical detail
Stages in Scientific Computing
models –> mathematical theory of PDEs,
numerical analysis –> mathematically sound methods,
algorithms –> program (software) .
Utility of a free easy-to-use software
check mathematical theories,
perform numerical experiments,
initiate collaborations with physics and industry.
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Stages in Scientific Computing: an example
Num method.
• Bose-Einstein • Gross-Pitaevskii
(Schrödinger) eq.
condensate
• finite elements
• experiment
Physics
Implementation Results
agreement
• num method
• qualitative
• algorithm
• quantitative
• software:
FreeFem++
www.freefem.org
JILA, Colorado
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Why using FreeFem++
FreeFem++ (www.freefem.org)
Free Generic PDE solver using finite elements (2D and 3D)
syntax close to the mathematical formulation,
powerful mesh generator,
mesh interpolation and adaptivity,
easy to implement weak formulations,
use combined P1, P2 and P4 elements,
complex matrices available, etc.
You are welcome to participate in the:
FreeFem++ Days, Paris, December, every year.
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FreeFem++ syntax: close to mathematical
formulation
• create a mesh and a finite element space
border circle(t=0,2*pi)
{label=1;x=Rmax*cos(t);y=Rmax*sin(t);};
mesh Th=buildmesh(circle(nbseg));
fespace Vh(Th,P1);
fespace Vh4(Th,P4);
• solve −∆u = f in Ω and u = 0 on ∂Ω
Vh u,v ;
problem Lap(u,v) =
int2d(Th)(u*v + dx(u)*dx(v)+dy(u)*dy(v))
- int2d(Th)(f*v)
+ on(1,u=0);
Lap; plot(u);
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Mesh adaptivity
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Mesh isotropy
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Computation of Bose-Einstein condensates
Developers: G. Vergez (PhD student), I. Danaila, F. Hecht.
submitted to CCP (to freely distribute scripts)!
• stationary Gross-Pitaevskii (Schrödinger) equation
3D anisotropic mesh adaptation, flexibility for boundary conditions,
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Computation of fluids with phase change
• Navier-Stokes-Boussinesq equations + phase change,
• I. Danaila, R. Moglan, F. Hecht, S. le Masson, JCP, 2014.
ice formation
Kowalewski & Rebow, Int. J. of Comput.
Fluid Dynamics, 1999.
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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How to install FreeFem++
FreeFem++: www.freefem.org
pre-compiled versions for Windows and MacOS,
compilation needed for Linux,
to write programs/scripts: use your preferred Editor (Emacs).
Explore www.freefem.org
instructions for compilation,
full documentation, slides from FreeFem++ days, etc
lots of examples (.edp scripts).
FreeFem++-cs: http://www.ann.jussieu.fr/∼lehyaric/ffcs/
pre-compiled versions for Windows, MacOS and Ubuntu,
IDE integrated development environment,
different graphical interface.
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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Building a mesh with FreeFem++ (v01)
Any computation starts with a mesh
label=1
1
mesh circle v1.edp
0.5
y
/* Mesh of a circle */
0
// Parameters
-0.5
int nbseg=100;
real R=1, xc=0, yc=0;
-1
-1
-0.5
0
x
0.5
1
// border
border circle(t=0,2*pi){label=1;
x=xc+R*cos(t);
y=yc+R*sin(t);}
plot(circle(nbseg),cmm="border");
// FE mesh
mesh Th = buildmesh(circle(nbseg));
plot(Th, cmm="mesh of a circle");
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Building a mesh with FreeFem++ (v02)
A mesh with a sub-domain:: + circle2(nbseg*2*pi*R2)
mesh circle v2.edp
1
label=1
0.5
label=2
y
/* Mesh of a circle with a subdomain */
// Parameters
int nbseg=10;real R=1, xc=0, yc=0, R2=R/2;
// borders
border circle1(t=0,2*pi){label=1;
x=xc+R*cos(t);
y=yc+R*sin(t);}
border circle2(t=2*pi,0){label=2;
x=xc+R2*cos(t);
y=yc+R2*sin(t);}
plot(circle1(nbseg*2*pi*R)+circle2(-nbseg*2*pi*R2
),cmm="border");
0
-0.5
-1
-1
-0.5
0
x
0.5
1
// FE mesh
mesh Th = buildmesh(circle1(nbseg*2*pi*R)
+circle2(nbseg*2*pi*R2));
plot(Th, cmm="mesh of a circle with subdomain");
// Identify subdomains
cout <<"inner region:: number ="<<
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Th(xc,yc).region <<endl;
cout <<"inner
region::
number
="<<
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Building a mesh with FreeFem++ (v03)
A mesh with a hole inside:: + circle2(-nbseg*2*pi*R2)
1
label=1
0.5
label=2
/* Mesh of a circle with a hole inside */
// Parameters
int nbseg=10;
real R=1, xc=0, yc=0, R2=R/2;
// border
border circle1(t=0,2*pi){label=1;
x=xc+R*cos(t);
y=yc+R*sin(t);}
border circle2(t=0,2*pi){label=2;
x=xc+R2*cos(t);
y=yc+R2*sin(t);}
plot(circle1(nbseg*2*pi*R)+circle2(nbseg*2*pi*R2)
,cmm="border");
// FE mesh
mesh Th = buildmesh(circle1(nbseg*2*pi*R)
+circle2(-nbseg*2*pi*R2));
plot(Th, cmm="mesh of a circle with subdomain");
y
mesh circle v3.edp
0
-0.5
-1
-1
-0.5
0
x
0.5
1
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Building a mesh with FreeFem++ (v04)
A mesh with a hole inside:: using macros to avoid bugs
be carreful with the syntax of EndOfMacro and inside comments
mesh circle v4.edp
/* Mesh of a circle with a hole inside
using macros
*/
macro Bcircle(bname,Rm,xm,ym,labelm)
/* circle border */
border bname(t=0,2*pi)
{label=labelm; x=xm+Rm*cos(t);y=ym+Rm*sin(t);}//EOM
// Parameters
int nbseg=10;
real R=1, xc=0, yc=0, R2=R/2;
// borders
Bcircle(circle1,R ,xc,yc,1);
Bcircle(circle2,R2,xc,yc,2);
plot(circle1(nbseg*2*pi*R)+circle2(nbseg*2*pi*R2),cmm="border");
// FE mesh
mesh Th = buildmesh(circle1(nbseg*2*pi*R)
+circle2(-nbseg*2*pi*R2));
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plot(Th, cmm="mesh of a circle with subdomain");
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Building a mesh with FreeFem++ (v05)
Mesh for a half-disk::
check intersection points
oriented borders (counter clockwise)
2.5
mesh circle v5.edp
y
2
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1
0.5
0
0.5
1
1.5
2
1.5
2
x
2.5
2
y
/* Mesh of a half-disk */
// Parameters
int nbseg=10;
real R=1, xc=1, yc=1;
// border
border Dcircle(t=0, pi)
{label=1;
x=xc+R*cos(t);
y=yc+R*sin(t);}
border Daxis (t=xc-R,xc+R){label=2;
x=t;
y=yc;}
plot(Dcircle(nbseg*pi*R)+Daxis(nbseg*2*R),cmm="
border");
// FE mesh
mesh Th = buildmesh(Dcircle(nbseg*pi*R)+Daxis(
nbseg*2*R));
plot(Th, cmm="mesh of a half-disk");
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1
0.5
0
0.5
1
x
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Building a mesh with FreeFem++ (v06)
Intermission: Mesh of a smiley
4
y
3
2
1
0
-1
0
1
x
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3
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Building a smiley with FreeFem++ (v06)
mesh circle v6.edp
/* Mesh of a smiley using macros
*/
macro Bellipse(bname,Rmx,Rmy,xm,ym,labelm)
border bname(t=0,2*pi)
{label=labelm; x=xm+Rmx*cos(t);y=ym+Rmy*sin(t);}//EOM
// Parameters
int nbseg=10;
//head
real Rh=2, xh=1, yh=2, Lh=2*pi*Rh;
Bellipse(bs1,Rh,Rh,xh,yh,1);
//eyes
real xy1=xh+Rh/2*cos(pi/4), yy=yh+Rh/2*sin(pi/4), Ry=Rh/4, Ly=2*pi*Ry;
Bellipse(bs2,Ry,Ry,xy1,yy,2);
real xy2=xh-Rh/2*cos(pi/4);
Bellipse(bs3,Ry,Ry,xy2,yy,3);
//mouth
real a=Rh/2, b=Rh/4, Lm=pi*sqrt(2*(aˆ2+bˆ2));
Bellipse(bs4,a,b,xh+0,yh-Rh/2,4);
plot(bs1(nbseg*Lh)+bs2(nbseg*Ly)+bs3(nbseg*Ly)+bs4(nbseg*Lm));
// FE mesh
mesh Th = buildmesh(bs1(nbseg*Lh)+bs2(-nbseg*Ly)+bs3(-nbseg*Ly)+bs4(nbseg*Lm));
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plot(Th, cmm="mesh of a smiley");
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Building the mesh of a rectangle (using ”square”)
Mesh a rectangle::
using the built-in function ”square”
1
0.8
mesh rectangle v02.edp
0.6
0.4
/* Mesh of a rectangle using square
function */
// Parameters
int nbseg=10;
real L=2,H=1;
real xc1=0,
yc1=0;
// FE mesh
mesh Th = square(nbseg*L,
nbseg*H,
[xc1+x*L,yc1+y*H]);
plot(Th, cmm="mesh of a rectangle");
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0
0
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1
1.5
2
0
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2
1
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0.4
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Building the mesh of a rectangle (using macros)
mesh rectangle v01.edp
/* Mesh of a rectangle
*/
macro Bsegment(bname,xP1,yP1,xP2,yP2,Ls,labelm)
real Ls=sqrt((xP1-xP2)ˆ2+(yP1-yP2)ˆ2);
border bname(t=0,Ls)
{label=labelm; x=xP1+t*(xP2-xP1)/Ls;
y=yP1+t*(yP2-yP1)/Ls;}//EOM
// Parameters
int nbseg=10;
real L=2,H=1;
real xc1=0,
yc1=0,
xc2=xc1+L,yc2=yc1,
xc3=xc2, yc3=yc2+H,
xc4=xc1, yc4=yc3+L;
//borders
Bsegment(bs1,xc1,yc1,xc2,yc2,Ls1,1);
Bsegment(bs2,xc2,yc2,xc3,yc3,Ls2,2);
Bsegment(bs3,xc3,yc3,xc4,yc4,Ls3,3);
Bsegment(bs4,xc4,yc4,xc1,yc1,Ls4,4);
plot(bs1(nbseg*Ls1)+bs2(nbseg*Ls2)+bs3(nbseg*Ls3)+bs4(nbseg*Ls4));
// FE mesh
mesh Th = buildmesh(bs1(nbseg*Ls1)+bs2(nbseg*Ls2)+bs3(nbseg*Ls3)+bs4(
nbseg*Ls4));
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plot(Th, cmm="mesh of a rectangle");
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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Finite element representation
• Functional spaces
∂v ∂v
,
∈ L2 (Ω)}
∂x ∂y
V (Ω) = {v ∈ H 1 (Ω) : v |ΓD = 0}.
H 1 (Ω) = {v ∈ L2 (Ω) :
• Approximation spaces
Th ::= triangulation,
t
Tk , (nt is the number of triangles).
Ωh = ∪nk=1
Hh = {v ∈ C 0 (Ωh ) : ∀Tk ∈ Th , v |Tk ∈ P 1 (Tk )},
Vh = {v ∈ Hh : v |ΓD = 0}.
h
• Basis functions
w i ∈ Hh , w i (q j ) = δij (1 if i = j, 0 otherwise).
∇w i |Tk = const,
dim(Hh ) = nv (nv is the number of vertices),
fh ∈ Hh ::= array of nv values.
q2
q0
Tk
q
1
ωi
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Definition of finite element spaces, integrals
FEspace is a type (as real or int) in FreeFem++
fespace v01.edp
/* P1 representation */
// Parameters
int nbseg=100;
real R=1, xc=0, yc=0;
// border
border circle(t=0,2*pi){label=1;x=xc+R*cos(t);
y=yc+R*sin(t);}
// FE mesh
mesh Th = buildmesh(circle(nbseg));plot(Th);
// FE space
fespace Vh(Th, P1);
Vh u=x*x+y*y; plot(u, dim=3,fill=1,wait=1);
// 2d integral
cout.precision(12);
real int2dU = int2d(Th)(u);
cout <<"int2dU="<<int2dU<<endl;
cout <<" exact="<<pi*Rˆ4/2<<endl;
real int1dU = int1d(Th,1)(u);
cout <<"int1dU="<<int1dU<<endl;
cout <<" exact="<<2*pi*Rˆ3<<endl;
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Functions:
x, y , z are reserved names for space variables
(part of) fespace v02.edp
func fu=x*x+y*y;
cout.precision(12);
real int2dU = int2d(Th)(fu);
cout <<"int2dU="<<int2dU<<endl;
cout <<" exact="<<pi*Rˆ4/2<<endl;
real int1dU = int1d(Th,1)(fu);
cout <<"int1dU="<<int1dU<<endl;
cout <<" exact="<<2*pi*Rˆ3<<endl;
// other definition of functions
func string infoMesh(mesh & Th)
{
string message="Mesh size:: nv="+ Th.nv+" nt="+Th.nt;
return message;
}
cout << infoMesh(Th)<<endl;
func real Meshsize(mesh & Th)
{
fespace Ph(Th,P0);
Ph h=hTriangle;
return h[].max;
}
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cout << Meshsize(Th)<<endl;
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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The Poisson equation (1)
• Deformation
of a circular membrane
−∆u = f
for (x, y) ∈ D
u =0
for (x, y) ∈ ∂D = C
• Variational (weak) formulation:
– multiply by a test function
v ∈ V (D) = {v ∈ H 1 (D) : v |C = 0}
C
0
x
y
f
u
– use Green’s formula (integration by parts)
Z
Z
Z
∂u
v dγ,
[−v ∆u] dxdy =
∇v ∇u −
D
D
C ∂n
– to obtain (notice that v |C = 0)
Z
Z
∇v ∇u −
fv = 0,
D
D
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The Poisson equation (2)
Z
Z
∇v ∇u −
D
Z
fv = 0 ⇔
D
D
∂u ∂v
∂u ∂v
+
∂x ∂x
∂y ∂y
Z
−
fv = 0
D
• derive the discrete weak formulation: FreeFem++ will take care!
(part of) lap v01b.edp
// Data of the problem
func fs=4; // RHS (source) function
// FE space
fespace Vh(Th, P1);
// Variational (weak formulation)
Vh u,v;
// u=unknown, v=test function
problem Poisson(u,v) =
int2d(Th)(dx(u)*dx(v)+dy(u)*dy(v))
- int2d(Th)(fs*v)
+ on(1,u=0); // Dirichlet boundary condition
// Solve the problem, plot the solution
Poisson; plot(u,dim=2,fill=1);
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The Poisson equation (3)
• using a macro
for
i the gradient = column array of two componets
h
∂u ∂u
grad(u) = ∂x , ∂y and grad(u)0 is the transposed gradient (row array)
(part of) lap v01.edp
Vh u,v;
// u=unknown, v=test function
macro grad(u) [dx(u), dy(u)]//EOM
problem Poisson(u,v)=int2d(Th)(grad(u)’*grad(v))
-int2d(Th)(fs*v)
+on(1,u=0); // Dirichlet bc
// Solve the problem, plot the solution
Poisson; plot(u,dim=2,fill=1);
// Compare with the exact solution
Vh uexact=Rˆ2-xˆ2-yˆ2, error=abs(u-uexact);
plot(error,dim=3,fill=1);
cout.precision(12);
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FreeFem++ program for the Poisson equation
lap v01.edp
/* Solving the Posson equation in a disk */
int nbseg=100;real R=1, xc=0, yc=0;
border circle(t=0,2*pi){label=1;x=xc+R*cos(t);
y=yc+R*sin(t);}
mesh Th = buildmesh(circle(nbseg));plot(Th);
// Data of the problem
func fs=4; // RHS (source) function
// FE space
fespace Vh(Th, P1);
// Variational (weak formulation)
Vh u,v;
// u=unknown, v=test function
macro grad(u) [dx(u), dy(u)]//EOM
problem Poisson(u,v)=int2d(Th)(grad(u)’*grad(v))
-int2d(Th)(fs*v)
+on(1,u=0); // Dirichlet bc
// Solve the problem, plot the solution
Poisson; plot(u,dim=2,fill=1);
// Compare with the exact solution
Vh uexact=Rˆ2-xˆ2-yˆ2, error=abs(u-uexact);
plot(error,dim=3,fill=1);
cout.precision(12);
cout<<"Maximum error ="<<error[].linfty<<endl;
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cout<<"Maximum error ="<<error[].max<<endl;
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Convergence of the FE method
1.000000
numerical result
theoretical h2
0.100000
ku − uex k2 =
(u − uex )2
Ω
ku − uex k2 ≤ Ch2
0.010000
L2error
sZ
0.001000
0.000100
0.000010
0.01
0.1
1
h
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Convergence of the FE method (script part1)
(part 1 of) lap v02.edp
/* Convergence of the P1 FEM */
int nbseg=5;real R=1, xc=0, yc=0;
border circle(t=0,2*pi){label=1;x=xc+R*cos(t);
y=yc+R*sin(t);}
mesh Th = buildmesh(circle(nbseg));
fespace Vh(Th, P1);
func fs =4; // RHS (source) function
func fex=Rˆ2-xˆ2-yˆ2;
func real Meshsize(mesh & Th)
{
fespace Ph(Th,P0);
Ph h=hTriangle;
return h[].max;
}
Vh u,v,uexact,error;
// u=unknown, v=test function
macro grad(u) [dx(u), dy(u)]//EOM
problem Poisson(u,v) =
int2d(Th)(grad(u)’*grad(v))
- int2d(Th)(fs*v)+ on(1,u=0); // Dirichlet bc
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Convergence of the FE method (script part 2)
(part 2 of) lap v02.edp
{verbosity=0; // lowest verbosity mode (no internal cout)
ofstream ferror("lap_v02_error.dat");
real eL2, h, eL2old=1,hold=1;
for(int irun=0;irun<8;irun++)
{
nbseg *= 2;
Th=buildmesh(circle(nbseg));
Poisson;
eL2 = sqrt(int2d(Th)(square(u-fex)));
h
= Meshsize(Th);
real order =log(eL2/eL2old)/log(h/hold);
eL2old=eL2;
hold =h;
cout<<"nbseg= "<<nbseg<<" Mesh size= "<<h <<"
nv= "<<Th.nv
<<"
L2error= "<<eL2<<" order= "<<order<<endl;
ferror<<h <<" "<<eL2<<" "<< order<<endl;
}
}
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Convergence of the FE method
1.000000
numerical result
theoretical h2
0.100000
ku − uex k2 =
(u − uex )2
Ω
ku − uex k2 ≤ Ch2
0.010000
L2error
sZ
0.001000
0.000100
0.000010
0.01
0.1
h
1
Gnuplot commands
set logscale xy
plot "lap_v02_error.dat" w lp pt 7 t "num",x*x/6 w l t "
theory"
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Boundary conditions (1)
• Consider the Poisson (heat) equation:
−∆u = f ,
in
Ω = [0, L] × [0, H]
• with boundary conditions:
H/2
Γ

on
Γ
u
=
u
Dirichlet
BC
4
hot




on Γ2 u = ucold
Dirichlet BC



-H/2
∂u
=0
Neumann BC
on Γ1
∂n




∂u


+ αu = 0 Fourier BC(α > 0)
 on Γ3
∂n
• Weak formulation:
Z
Z
Z
∂u
[−v ∆u] =
∇v ∇u −
v,
Ω
Ω
Γ ∂n
Z
Z
Z
∂u
fv −
v = 0.
∇v ∇u −
Ω
Ω
Γ ∂n
Γ3
Γ2
4
L
Γ1
Ω
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Boundary conditions
Z
Z(2)
Ω
4 Z
X
∂u
v = 0.
∇v ∇u −
fv −
Ω
Γi ∂n
i=1
Vh u,v;
// u=unknown, v=test function
problem Poisson(u,v) =int2d(Th)(dx(u)*dx(v)+dy(u)*dy(v))
-int2d(Th)(fs*v)



on Γ4 u = uhot
Dirichlet
v =0






+on(4, u = uhot)






on Γ2 u = ucold
Dirichlet
v =0





+on(2, u = ucold);
∂u


on Γ1 ∂u
Neumann
=0

∂n = 0

∂n




nothing to be done




∂u


= −αu
on Γ3 ∂u

∂n + αu = 0 Fourier


∂n


+int1d(Th, 3)(alpha ∗ u ∗ v );
Z
Γ4
Z
Γ2
Z
Γ1
Z
Γ3
∂u
v =0
∂n
∂u
v =0
∂n
∂u
v =0
∂n
∂u
v=
∂n
Z
(−αuv )
Γ3
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Exercises for the Poisson equation (1)
In the previous problem (see lap v01.edp): we chose f = 4
corresponding to uex (x, y ) = R 2 − x 2 − y 2 which satisfies the BC.
1
Solve the same problem on the half disk. What BC should be
applied at the axis (y = 0)? How this new BC appears in the
variational formulation?
2
Solve the following PDE
−∆u + u = f
in R,
and
u|∂R = g,
where R = [0, 1] × [0, 1] is a square domain. Find the expression
for f (x, y ) and g(x, y ) such as that the exact solution should be
uex (x, y) = 1 − x 2 − y 2 .
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Exercises for the Poisson equation (2)
3
Potential flow around a cylinder (solution script: lap ex3.edp).
For each point (x, y) the flow velocity is derived
from the potential ϕ


∂ϕ
 ∂x 
→
−
v = ∇ϕ =  ∂ϕ
 , −∆ϕ = 0.
∂y
Ω = [0, L] × [−H, H].
The cylinder of radius r is centred in the domain.
on Γ1 ∪Γ2 ∪Γ3 ∪Γ4 :
~v = v0 ~ex ,
on the cylinder Γ5 :
H
Γ3
Γ4
r
Γ2
L
0
-H
Ω
Γ1
where ~ex is the unit vetor alog x-axis.
~v · ~n = 0,
zero normal velocity.
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Outline
1
Scientific Computing and FreeFem++
Purpose of the Course
Why using FreeFem++
Examples of complex problems computed with FreeFem++
2
How to install FreeFem++
3
Building a mesh with FreeFem++
4
Finite-element spaces, 2d and 1d integrals
5
Solving the Poisson equation in 10 lines of code
6
From steady to time-dependent problems
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Solving the heat equation with various BC
−∆θ = f
in Ω,
θ=
T − Tcold
Thot − Tcold
smiley heat A.edp
include "./mesh_circle_v6.edp";
// data
real Tcold=20, Thot=300, dT=Thot-Tcold;
real ThetaCold = 0;//(Tcold-Tcold)/dT
real ThetaHot = 1;//(Thot-Tcold)/dT;
real alpha=100;
// FE space
fespace Vh(Th, P1);
// Variational (weak formulation)
Vh u,v;
// u=unknown, v=test function
macro grad(u) [dx(u), dy(u)]//EOM
problem Smiley(u,v)=int2d(Th)(grad(u)’*grad(v))
+int1d(Th,4)(alpha*u*v) //Fourier bc
+on(2, u=ThetaCold) // Dirichlet bc
+on(1, u=ThetaHot); // Dirichlet bc
// Solve the problem, plot the solution
Smiley; plot(u,dim=2,fill=1,cmm="Non-dim temp");
Vh T = u*dT+Tcold;plot(T,cmm="Temp dist");
cout <<"T min = "<< T[].min <<endl;
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cout <<"T max = "<< T[].max <<endl;
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Solving the heat equation in complex domains
smiley heat B.edp
include "./mesh_circle_v6.edp";
// New FE mesh
Th = buildmesh(bs1(nbseg*Lh)+bs2(-nbseg*Ly)+bs3(-nbseg*Ly)
+bs4(nbseg*Lm));plot(Th, cmm="New mesh of a smiley");
smiley heat B.edp
fespace Vh(Th, P1);
fespace Ph(Th, P0);
// Identification of the mouth:: characteristic
function
int regMouth = Th(xh+0,yh-Rh/2).region;
Ph
IMouth = (region == regMouth)?1:0;
plot(IMouth,cmm="Mouth",dim=3,wait=1);
// Variational (weak formulation)
Vh u,v;
// u=unknown, v=test function
macro grad(u) [dx(u), dy(u)]//EOM
problem Smiley(u,v)=int2d(Th)(grad(u)’*grad(v))
-int2d(Th)(q*IMouth*v) //Fourier bc
+on(2, u=ThetaCold) // Dirichlet bc
+on(1, u=ThetaHot); // Dirichlet bc
// Solve the problem, plot the solution I. Danaila & F. Hecht: Graduate Course on FreeFem++, Fields Institute, March 2016.
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Solving the time-dependent heat equation (1)
∂θ
− ∆θ = 0, for (x, y ) ∈ Ω, 0 ≤ t ≤ tmax
∂t
+Boundary Conditions(in space) + Initial Condition(t = 0).
• Discretisation in time (FD finite-difference type)
[0, tmax ] =
N−2
[
[tn , tn +δt], tn = nδt, n = 0, 1, . . . , N −1, δt = T /(N −1).
n=0
Notation θn (x) = θ(x, tn ).
θn+1 (x) − θn (x)
− ∆θn+1 (x) = 0
δt
θn+1 (x) − θn (x)
− ∆θn (x) = 0
δt
(implicit scheme)
(explicit scheme)
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Solving the time-dependent heat equation (2)
• Discretisation in space (FE finite-element type): implicit scheme
Z
Ω
Z
Ω
θn+1
v−
δt
θn+1
v−
δt
Z
Ω
Z
Ω
θn
v+
δt
θn
v+
δt
Z h
i
−v ∆θn+1 = 0
Z
∇θ
Ω
n+1
Z
∇v −
Ω
Γ
∂θn+1
v =0
∂n
• Weak formulation ready to use with FreeFem++: impose BC on θn+1
as for the steady problem.
• In programs, in the ”time loop” we use only two variables:
u = θn+1 and uold = θn .
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Script for the time-dependent heat equation (1)
(part 1 of) smiley heat C.edp
include "./mesh_circle_v6.edp";
// data
real Tcold=20, Thot=300, dT=Thot-Tcold;
real ThetaCold = 0;//(Tcold-Tcold)/dT
real ThetaHot = 1;//(Thot-Tcold)/dT;
real alpha=100;
//Time-evolution data
real tmax=0.5, dt=0.01, idt=1./dt;
// FE space
fespace Vh(Th, P1);
// Variational (weak formulation)
Vh u,v, uold=0;
// u=unknown, v=test function
macro grad(u) [dx(u), dy(u)]//EOM
problem SmileyT(u,v)=int2d(Th)(idt*u*v)-int2d(Th)(idt*uold*v)
+int2d(Th)(grad(u)’*grad(v))
+int1d(Th,4)(alpha*u*v) //Fourier bc
+on(2, u=ThetaCold) // Dirichlet bc
+on(1, u=ThetaHot); // Dirichlet bc
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Script for the time-dependent heat equation (2)
(part 2 of) smiley heat C.edp
//Time loop
real t=0; int iter=0, nplot=2;Vh T;
verbosity=0;
while (t <= tmax)
{
iter++;t+=dt;SmileyT;
if(!(iter%nplot))// visualisation every nplot
{
T = u*dT+Tcold;
plot(T,cmm="Temp t="+t,fill=1);
cout <<"t="<<t<<" T min= "<< T[].min
<<" T max="<< T[].max <<endl;
}
uold=u;
}
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Solving the wave equation (1)
∂2u
− ∆u = 0,
∂t 2
for (x, y) ∈ Ω,
0 ≤ t ≤ tmax
+Boundary Conditions(in space) + Initial Condition for u,
∂u
(t = 0).
∂t
• Discretisation in time (FD finite-difference type)
[0, tmax ] =
N−2
[
[tn , tn +δt], tn = nδt, n = 0, 1, . . . , N −1, δt = T /(N −1).
n=0
Notation u n (x) = u(x, tn ).
u n+1 (x) − 2u n (x) + u n−1 (x)
− ∆u n (x) = 0.
δt 2
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Solving the wave equation (2)
• Discretisation in space (FE finite-element type): implicit scheme
Z
Ω
Z
Ω
u n+1
v−
δt
u n+1
v−
δt
Z
Ω
Z
Ω
2u n
v+
δt
un
v+
δt
Z
Ω
Z
Ω
u n−1
v+
δt
u n−1
v+
δt
Z
Z
−v ∆u n = 0
Ω
n
Z
∇u ∇v −
Ω
Γ
∂u n
v =0
∂n
• Weak formulation ready to use with FreeFem++: impose BC on u n as
for the steady problem.
• In programs, in the ”time loop” we use only three variables:
u = u n+1 , uold = u n , uvold = u n−1 .
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Script for the wave equation (1)
(part 1 of) smiley heat D.edp
include "./mesh_circle_v6.edp";
// New FE mesh (close the eyes and the mouth)
Th = buildmesh(bs1(nbseg*Lh)+bs2(-nbseg*Ly)+bs3(nbseg*Ly)
+bs4(nbseg*Lm));plot(Th, cmm="New mesh of a smiley");
//Time-evolution data
real tmax=5, dt=0.01, idt2=1./(dt*dt);
// FE space
fespace Vh(Th, P1);
// Variational (weak formulation)
Vh u,v, uold=0,uvold=0;
macro grad(u) [dx(u), dy(u)]//EOM
problem SmileyW(u,v)=int2d(Th)(idt2*u*v)-int2d(Th)(2*idt2*uold*v)+int2d
(Th)(idt2*uvold*v)
+int2d(Th)(grad(uold)’*grad(v))
; // only Neumann boundary conditions
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Script for the wave equation (2)
(part 2 of) smiley heat D.edp
//Time loop
real t=0; int iter=0, nplot=2;
// Initial condition (u is Gaussian and du/dt=0)
uvold = exp(-2*((x-xh)ˆ2 + (y-yh)ˆ2));
uold = uvold;
u
= uold;
verbosity=0;
while (t <= tmax)
{
iter++;t+=dt;SmileyW;
if(!(iter%nplot))// visualisation every nplot
{
plot(u,cmm="Wave t="+t,fill=1,dim=3);
cout <<"t="<<t<<" u min= "<< u[].min
<<" u max="<< u[].max <<endl;
}
uvold=uold;
uold =u;
}
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