Tarea: Introducción a Elementos Finitos - Newton-Cotes

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Introducción a elementos finitos
Tarea 6 I-2016
Aplicando el método de Newton-Cotes construir la tabla de pesos y puntos de muestreo hasta n = 4
n=1
k =1−1=0
Calculando ri
+1
Z
P (r) r0 dr = 0
−1
El polinomio es
P (r) = r − r1
Reemplazando
Z
+1
r − r1 dr = 0
−1
Integrando
Z
+1
r − r1 dr =
1
2
−1
r2 − r1 r
+1
= −2r1
−1
Despejando
r1 = 0
Calculando wi
Z
+1
w1 =
dr = 2
−1
n=2
k =2−1=1
Calculando ri
Z
+1
−1
Z +1
P (r) r0 dr = 0
P (r) r1 dr = 0
−1
El polinomio es
P (r) = (r − r1 )(r − r2 )
Reemplazando
1
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Z
+1
(r − r1 )(r − r2 ) dr = 0
Z
−1
+1
(r − r1 )(r − r2 )r dr = 0
−1
Integrando
r1 + r2 2
r + r1 r2 r
3
2
1
r1 + r2 3 r1 r2 2 r4 −
r +
r
4
3
2
1
+1
r3 −
−1
+1
−1
1
= 2 r1 r2 +
3
2
= − (r1 + r2 )
3
Formando el sistema de ecuaciones
r1 r2 = −
1
3
r1 + r2 = 0
Resolviendo
r
1
r1 = −
3
r
1
r2 =
3
Calculando wi
Z
+1
w1 =
−1
+1
Z
w2 =
−1
r − r2
dr
r1 − r2
r − r1
dr
r2 − r1
Reemplazando
Z
+1
w1 =
−1
Z
+1
w2 =
−1
q
r − 13
q
q dr
− 13 − 13
q
r + 13
q
q dr
1
1
+
3
3
Integrando
+1
√3
1 w1 = −
r2 + r
=1
4
2
−1
+1
√3
1 r2 + r
=1
w2 =
4
2
−1
2
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n=3
k =3−1=2
Calculando ri
Z
+1
P (r) r0 dr = 0
−1
Z +1
−1
Z +1
P (r) r1 dr = 0
P (r) r2 dr = 0
−1
El polinomio es
P (r) = (r − r1 )(r − r2 )(r − r3 )
Reemplazando
Z
+1
(r − r1 )(r − r2 )(r − r3 ) dr = 0
Z
−1
+1
(r − r1 )(r − r2 )(r − r3 )r dr = 0
Z
−1
+1
(r − r1 )(r − r2 )(r − r3 )r2 dr = 0
−1
Integrando
i
r1 + r2 + r3 3 r1 r2 + r1 r3 + r2 r3 2
r +
r − r1 r2 r3 r
4
3
2
h1
r1 + r2 + r3 4 r1 r2 + r1 r3 + r2 r3 3 r1 r2 r3 2 i
r5 −
r +
r −
r
5
4
3
2
h1
r1 + r2 + r3 5 r1 r2 + r1 r3 + r2 r3 4 r1 r2 r3 3 i
r6 −
r +
r −
r
6
5
4
3
h1
+1
r4 −
−1
+1
2
= − (3r1 r2 r3 + r1 + r2 + r3 )
3
=
−1
+1
=−
−1
Formando el sistema de ecuaciones
3r1 r2 r3 + r1 + r2 + r3 = 0
5r1 r2 + 5r1 r3 + 5r2 r3 = −3
5r1 r2 r3 + 3r1 + 3r2 + 3r3 = 0
Resolviendo
r
r1 = −
r2 = 0
r
r3 =
3
3
5
3
5
2
(5r1 r2 + 5r1 r3 + 5r2 r3 + 3)
15
2
(5r1 r2 r3 + 3r1 + 3r2 + 3r3 )
15
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Calculando wi
Z
+1
w1 =
−1
+1
Z
w2 =
−1
+1
Z
w3 =
−1
r − r2 r − r3
·
dr
r1 − r2 r1 − r3
r − r1 r − r3
·
dr
r2 − r1 r2 − r3
r − r2 r − r1
·
dr
r3 − r2 r3 − r1
Reemplazando
Z
r−
q
+1
−1
+1
w2 =
−1
Z
3
r−0
5
q
q dr
·
3
3
− 5 − 0 − 5 − 35
q
q
r + 35 r − 35
q ·
q dr
0 + 35 0 − 35
q
r + 35
r−0
q
q dr
·q
3
3
3
−
0
+
5
5
5
w1 =
Z
q
+1
w3 =
−1
Integrando
√
+1
5
15 2 5
3
w1 =
r −
r
=
18
12
9
−1
5
+1
8
w2 = − r3 + r
=
9
9
−1
√
+1
5
15 2 5
3
w3 =
r +
r
=
18
12
9
−1
n=4
k =4−1=3
Calculando ri
Z
+1
−1
Z +1
−1
Z +1
−1
Z +1
P (r) r0 dr = 0
P (r) r1 dr = 0
P (r) r2 dr = 0
P (r) r3 dr = 0
−1
El polinomio es
4
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P (r) = (r − r1 )(r − r2 )(r − r3 )(r − r4 )
Reemplazando
Z
+1
(r − r1 )(r − r2 )(r − r3 )(r − r4 ) dr = 0
Z
−1
+1
(r − r1 )(r − r2 )(r − r3 )(r − r4 )r dr = 0
Z
−1
+1
−1
Z +1
(r − r1 )(r − r2 )(r − r3 )(r − r4 )r2 dr = 0
(r − r1 )(r − r2 )(r − r3 )(r − r4 )r3 dr = 0
−1
Integrando
h1
r1 + r2 + r3 + r4 4 r1 r2 + r1 r3 + r1 r4 + r2 r3 + r2 r4 + r3 r4 3 r1 r2 r3 + r1 r2 r4 + r1 r3 r4 + r2 r3 r4 2
r5 −
r +
r −
r
5
4
3
2
i +1
2
+ r1 r2 r3 r4 r
=
(15r1 r2 r3 r4 + 5r1 r2 + 5r1 r3 + 5r1 r4 + 5r2 r3 + 5r2 r4 + 5r3 r4 + 3)
15
−1
h1
r1 + r2 + r3 + r4 5 r1 r2 + r1 r3 + r1 r4 + r2 r3 + r2 r4 + r3 r4 4 r1 r2 r3 + r1 r2 r4 + r1 r3 r4 + r2 r3 r4 3
r6 −
r +
r −
r
6
5
4
3
+1
r1 r2 r3 r4 2 i
2
r
+
= − (5r1 r2 r3 + 5r1 r2 r4 + 5r1 r3 r4 + 5r2 r3 r4 + 3r1 + 3r2 + 3r3 + 3r4 )
2
15
−1
h1
r1 + r2 + r3 + r4 6 r1 r2 + r1 r3 + r1 r4 + r2 r3 + r2 r4 + r3 r4 5 r1 r2 r3 + r1 r2 r4 + r1 r3 r4 + r2 r3 r4 4
7
r −
r +
r −
r
7
6
5
4
+1
r1 r2 r3 r4 3 i
2
+
r
=
(35r1 r2 r3 r4 + 21r1 r2 + 21r1 r3 + 21r1 r4 + 21r2 r3 + 21r2 r4 + 21r3 r4 + 15)
3
105
−1
h1
r1 + r2 + r3 + r4 7 r1 r2 + r1 r3 + r1 r4 + r2 r3 + r2 r4 + r3 r4 6 r1 r2 r3 + r1 r2 r4 + r1 r3 r4 + r2 r3 r4 5
r8 −
r +
r −
r
8
7
6
5
+1
r1 r2 r3 r4 4 i
2
+
r
= − (7r1 r2 r3 + 7r1 r2 r4 + 7r1 r3 r4 + 7r2 r3 r4 + 5r1 + 5r2 + 5r3 + 5r4 )
4
35
−1
Formando el sistema de ecuaciones
15r1 r2 r3 r4 + 5r1 r2 + 5r1 r3 + 5r1 r4 + 5r2 r3 + 5r2 r4 + 5r3 r4 = −3
5r1 r2 r3 + 5r1 r2 r4 + 5r1 r3 r4 + 5r2 r3 r4 + 3r1 + 3r2 + 3r3 + 3r4 = 0
35r1 r2 r3 r4 + 21r1 r2 + 21r1 r3 + 21r1 r4 + 21r2 r3 + 21r2 r4 + 21r3 r4 = −15
7r1 r2 r3 + 7r1 r2 r4 + 7r1 r3 r4 + 7r2 r3 r4 + 5r1 + 5r2 + 5r3 + 5r4 = 0
Resolviendo
5
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s
3 2
+
7 7
r1 = −
r
6
5
s
r
3 2 6
−
r2 = −
7 7 5
s
r
3 2 6
r3 =
−
7 7 5
s
r
3 2 6
r4 =
+
7 7 5
Calculando wi
Z
+1
w1 =
−1
+1
Z
w2 =
−1
+1
Z
w3 =
−1
+1
Z
w4 =
−1
r − r2 r − r3 r − r4
6r2 r3 r4 + 2r2 + 2r3 + 2r4
·
·
dr = −
r1 − r2 r1 − r3 r1 − r4
3(r1 − r2 )(r1 − r3 )(r1 − r4 )
r − r1 r − r3 r − r4
6r1 r3 r4 + 2r1 + 2r3 + 2r4
·
·
dr =
r2 − r1 r2 − r3 r2 − r4
3(r1 − r2 )(r2 − r3 )(r2 − r4 )
r − r2 r − r1 r − r4
6r1 r2 r4 + 2r1 + 2r2 + 2r4
·
·
dr = −
r3 − r2 r3 − r1 r3 − r4
3(r2 − r3 )(r1 − r3 )(r3 − r4 )
r − r3 r − r2 r − r1
6r1 r2 r3 + 2r1 + 2r2 + 2r3
·
·
dr =
r4 − r3 r4 − r2 r4 − r1
3(r3 − r4 )(r2 − r4 )(r1 − r4 )
Reemplazando
r
q r
q r
r
3 −
3
7
+
r
2
7
2
7
6
5
6
5
+
3
7
3
7
q r
−
2
7
2
7
6
5
6
5
q r
3
7
−
2
7
3
7
q
6
5
−
2
7
6
5
−
r
2
7
3
7
−
6
5
2
7
2
7
6
5
3
7
2
7
6
5
3
7
2
7
6
5
3
7
Simplificando
6
2
7
q
3
7
6
5
−
r
2
7
r
3
7
q
6
5
−
2
7
6
5
−
r
3
7
+
2
7
6
5
q
q
6
+
−
+
−2
+
− 2 37 − 72 65 + 2 37 + 72 65
r
r
r
w3 = − r
q
q r
q
q r
q
q 3
2
6
3
2
6
3
2
6
3
2
6
3
2
6
3
2
6
3 − 7−7 5− 7−7 5
− 7+7 5− 7−7 5
−
−
+
7
7
5
7
7
5
r
r
r
r
r
r
q
q
q
q
q
q
6 37 + 27 65 37 − 27 65 37 − 72 65 − 2 37 + 27 65 − 2 37 − 27 65 + 2 37 − 27 65
r
r
r
w 4 = r
q
q r
q
q r
q
q 3
2
6
3
2
6
3
2
6
3
2
6
3
2
6
3
2
6
3
−
−
+
−
−
−
+
−
+
−
+
7
7
5
7
7
5
7
7
5
7
7
5
7
7
5
7
7
5
3
7
q
3
7
q
r
q
6
−
−
+
+2
−
+2
−
+ 2 37 + 72 65
r
r
r
w1 = − r
q
q r
q
q r
q
q 3
6
3
6
3
6
3
6
3
6
3
2
2
2
2
2
2
3 − 7+7 5+ 7−7 5
− 7+7 5− 7−7 5
− 7 + 7 5 − 7 + 7 65
r
r
r
r
q r
q r
q
q
q
q
3
6
3
6
3
6
3
6
3
6
2
2
2
2
2
−6 7 + 7 5 7 − 7 5 7 + 7 5 − 2 7 + 7 5 + 2 7 − 7 5 + 2 37 + 27 65
r
r
r
w2 = r
q
q r
q
q r
q
q 3
7
6
5
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√
18 − 30
36√
18 + 30
36√
18 + 30
36√
18 − 30
36
w1 =
w2 =
w3 =
w4 =
Tabla resumen
n
r
w
1
0
q
− 13
q
2
q
− 35
5
9
2
1
1
3
3
0
q
8
9
3
5
5
9
r
−
3
7
+
2
7
q
6
5
√
18− 30
36
3
7
−
2
7
q
6
5
√
18+ 30
36
r
4
−
r
1
3
7
−
2
7
q
6
5
√
18+ 30
36
3
7
+
2
7
q
6
5
√
18− 30
36
r
7