Gauss-Legendre integration in Python
''' I = gaussQuad2(f,xc,yc,m).
Gauss-Legendre integration of f(x,y) over a
quadrilateral using integration order m.
{xc},{yc} are the corner coordinates of the quadrilateral.
'''
from gaussNodes import *
from numpy import zeros,dot
def gaussQuad2(f,x,y,m):
def jac(x,y,s,t):
J = zeros((2,2))
J[0,0] = -(1.0 - t)*x[0] + (1.0 - t)*x[1] \
+ (1.0 + t)*x[2] - (1.0 + t)*x[3]
J[0,1] = -(1.0 - t)*y[0] + (1.0 - t)*y[1] \
+ (1.0 + t)*y[2] - (1.0 + t)*y[3]
J[1,0] = -(1.0 - s)*x[0] - (1.0 + s)*x[1] \
+ (1.0 + s)*x[2] + (1.0 - s)*x[3]
J[1,1] = -(1.0 - s)*y[0] - (1.0 + s)*y[1] \
+ (1.0 + s)*y[2] + (1.0 - s)*y[3]
return (J[0,0]*J[1,1] - J[0,1]*J[1,0])/16.0
def map(x,y,s,t):
N = zeros(4)
N[0] = (1.0 - s)*(1.0 - t)/4.0
N[1] = (1.0 + s)*(1.0 - t)/4.0
N[2] = (1.0 + s)*(1.0 + t)/4.0
N[3] = (1.0 - s)*(1.0 + t)/4.0
xCoord = dot(N,x)
yCoord = dot(N,y)
return xCoord,yCoord
s,A = gaussNodes(m)
sum = 0.0
for i in range(m):
for j in range(m):
xCoord,yCoord = map(x,y,s[i],s[j])
sum = sum + A[i]*A[j]*jac(x,y,s[i],s[j]) \
*f(xCoord,yCoord)
return sum
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