Question

我有一堆矩阵，我必须用它来计算下面代码中的最终4x4矩阵[Ke]：

import numpy as num
from numpy import *
import scipy as sci
from scipy.integrate import *
from scipy import *
import sympy as sym
from sympy import *

x1 = 3; x2 = 0; x3 = 0; x4 = 3
y1 = 3; y2 = 3; y3 = 0; y4 = 0

K = []

s = symbols('s')
t = symbols('t')
k = symbols('k')

N1 = (s+1)*(t+1)/4
N2 = (1-s)*(t+1)/4
N3 = (1-s)*(1-t)/4
N4 = (s+1)*(1-t)/4

dN1s = diff(N1, s); dN1t = diff(N1, t)
dN2s = diff(N2, s); dN2t = diff(N2, t)
dN3s = diff(N3, s); dN3t = diff(N3, t)
dN4s = diff(N4, s); dN4t = diff(N4, t)

x = (x1*N1)+(x2*N2)+(x3*N3)+(x4*N4)
y = (y1*N1)+(y2*N2)+(y3*N3)+(y4*N4)

dxs = sym.diff(x, s)
dxt = sym.diff(x, t)
dys = sym.diff(y, s)
dyt = sym.diff(y, t)

print dxs, dxt, dys, dyt

J = sym.Matrix([[dxs, dys], [dxt, dyt]])
print J

J_inv = J.inv()
print J_inv

J_det = J.det()
print J_det

ST = sym.Matrix([[dN1s, dN2s, dN3s, dN4s], [dN1t, dN2t, dN3t, dN4t]])
print ST

XT = J_inv * ST
print XT

XT_trans = XT.transpose()
print XT_trans

k_small = sym.Matrix([[5, 0], [0, 5]])
print k_small

Ke = num.matrix(XT_trans*k_small*XT*J_det)
print Ke

for i in range(16):
    K.append([])
    K.append(dblquad(lambda t, s: Ke.item(i), -1, 1, lambda s: -1, lambda s: 1))

print K

下一步是将矩阵[Ke]相对于't'和's'双重积分在-1到1的范围内。我尝试循环[Ke]矩阵，因为scipy.integrate.dblquad没有直接支持矩阵。但是，当我运行代码时，我收到以下错误：

quadpack.error: Supplied function does not return a valid float.

问题在于我尝试整合的行，但我无法弄清问题是什么。任何帮助将不胜感激。谢谢。

PS：[Ke]总是一个4x4矩阵，[k_small]是一个对角线矩阵，可以对角线有任何值。

Answer 1

终于发现我做错了什么。它更多的是我用于集成的模块而不是语法。在我的问题的代码中，我使用scipy进行集成，但它没有工作。原因是，只有当我指定集成在dblquad函数中的等式时，scipy中的积分才有效。由于我一直在尝试迭代矩阵Ke并通过索引调用元素，因此我收到了错误。现在，我已经修改了我的集成以使用来自sympy的integrate函数，因为这可以处理一个需要方程式的变量。修改后的代码如下。

import numpy as num
from scipy import integrate
import sympy as sym

x1 = 3; x2 = 7; x3 = 6; x4 = 3
y1 = 3; y2 = 3; y3 = 4; y4 = 5

K = []

s = sym.symbols('s')
t = sym.symbols('t')
k = sym.symbols('k')

N1 = (s+1)*(t+1)/4
N2 = (1-s)*(t+1)/4
N3 = (1-s)*(1-t)/4
N4 = (s+1)*(1-t)/4

dN1s = sym.diff(N1, s); dN1t = sym.diff(N1, t)
dN2s = sym.diff(N2, s); dN2t = sym.diff(N2, t)
dN3s = sym.diff(N3, s); dN3t = sym.diff(N3, t)
dN4s = sym.diff(N4, s); dN4t = sym.diff(N4, t)

x = (x1*N1)+(x2*N2)+(x3*N3)+(x4*N4)
y = (y1*N1)+(y2*N2)+(y3*N3)+(y4*N4)

dxs = sym.diff(x, s)
dxt = sym.diff(x, t)
dys = sym.diff(y, s)
dyt = sym.diff(y, t)

print dxs, dxt, dys, dyt

J = sym.Matrix([[dxs, dys], [dxt, dyt]])
print J

J_inv = J.inv()
print J_inv

J_det = J.det()
print J_det

ST = sym.Matrix([[dN1s, dN2s, dN3s, dN4s], [dN1t, dN2t, dN3t, dN4t]])
print ST

XT = J_inv * ST
print XT

XT_trans = XT.transpose()
print XT_trans

k_small = sym.Matrix([[k, 0], [0, k]])
print k_small

Ke = num.matrix(XT_trans*k_small*XT*J_det)
print Ke

for i in range(16):
    K.append(integrate(Ke.item(i), (t, -1, 1), (s, -1, 1)))

print K

请注意，我实际上在矩阵k中使用了变量k_small并且它仍然有效（这是我的实际问题，因为，集成应该能够处理变量代替k并返回一个自己变量的答案。）

quadpack.error：当尝试在矩阵上进行双重积分时，提供的函数不会返回有效的float

1 个答案: