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#!/usr/bin/env python
"""FEM library
Demonstrates some simple finite element definitions, and computes a mass
matrix
$ python fem.py
[ 1/60, 0, -1/360, 0, -1/90, -1/360]
[ 0, 4/45, 0, 2/45, 2/45, -1/90]
[-1/360, 0, 1/60, -1/90, 0, -1/360]
[ 0, 2/45, -1/90, 4/45, 2/45, 0]
[ -1/90, 2/45, 0, 2/45, 4/45, 0]
[-1/360, -1/90, -1/360, 0, 0, 1/60]
"""
from sympy import symbols, Symbol, factorial, Rational, zeros, div, eye, \
integrate, diff, pprint, reduced
x, y, z = symbols('x,y,z')
class ReferenceSimplex:
def __init__(self, nsd):
self.nsd = nsd
if nsd <= 3:
coords = symbols('x,y,z')[:nsd]
else:
coords = [Symbol("x_%d" % d) for d in range(nsd)]
self.coords = coords
def integrate(self, f):
coords = self.coords
nsd = self.nsd
limit = 1
for p in coords:
limit -= p
intf = f
for d in range(0, nsd):
p = coords[d]
limit += p
intf = integrate(intf, (p, 0, limit))
return intf
def bernstein_space(order, nsd):
if nsd > 3:
raise RuntimeError("Bernstein only implemented in 1D, 2D, and 3D")
sum = 0
basis = []
coeff = []
if nsd == 1:
b1, b2 = x, 1 - x
for o1 in range(0, order + 1):
for o2 in range(0, order + 1):
if o1 + o2 == order:
aij = Symbol("a_%d_%d" % (o1, o2))
sum += aij*binomial(order, o1)*pow(b1, o1)*pow(b2, o2)
basis.append(binomial(order, o1)*pow(b1, o1)*pow(b2, o2))
coeff.append(aij)
if nsd == 2:
b1, b2, b3 = x, y, 1 - x - y
for o1 in range(0, order + 1):
for o2 in range(0, order + 1):
for o3 in range(0, order + 1):
if o1 + o2 + o3 == order:
aij = Symbol("a_%d_%d_%d" % (o1, o2, o3))
fac = factorial(order) / (factorial(o1)*factorial(o2)*factorial(o3))
sum += aij*fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)
basis.append(fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3))
coeff.append(aij)
if nsd == 3:
b1, b2, b3, b4 = x, y, z, 1 - x - y - z
for o1 in range(0, order + 1):
for o2 in range(0, order + 1):
for o3 in range(0, order + 1):
for o4 in range(0, order + 1):
if o1 + o2 + o3 + o4 == order:
aij = Symbol("a_%d_%d_%d_%d" % (o1, o2, o3, o4))
fac = factorial(order)/ (factorial(o1)*factorial(o2)*factorial(o3)*factorial(o4))
sum += aij*fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)*pow(b4, o4)
basis.append(fac*pow(b1, o1)*pow(b2, o2)*pow(b3, o3)*pow(b4, o4))
coeff.append(aij)
return sum, coeff, basis
def create_point_set(order, nsd):
h = Rational(1, order)
set = []
if nsd == 1:
for i in range(0, order + 1):
x = i*h
if x <= 1:
set.append((x, y))
if nsd == 2:
for i in range(0, order + 1):
x = i*h
for j in range(0, order + 1):
y = j*h
if x + y <= 1:
set.append((x, y))
if nsd == 3:
for i in range(0, order + 1):
x = i*h
for j in range(0, order + 1):
y = j*h
for k in range(0, order + 1):
z = j*h
if x + y + z <= 1:
set.append((x, y, z))
return set
def create_matrix(equations, coeffs):
A = zeros(len(equations))
i = 0
j = 0
for j in range(0, len(coeffs)):
c = coeffs[j]
for i in range(0, len(equations)):
e = equations[i]
d, _ = reduced(e, [c])
A[i, j] = d[0]
return A
class Lagrange:
def __init__(self, nsd, order):
self.nsd = nsd
self.order = order
self.compute_basis()
def nbf(self):
return len(self.N)
def compute_basis(self):
order = self.order
nsd = self.nsd
N = []
pol, coeffs, basis = bernstein_space(order, nsd)
points = create_point_set(order, nsd)
equations = []
for p in points:
ex = pol.subs(x, p[0])
if nsd > 1:
ex = ex.subs(y, p[1])
if nsd > 2:
ex = ex.subs(z, p[2])
equations.append(ex )
A = create_matrix(equations, coeffs)
Ainv = A.inv()
b = eye(len(equations))
xx = Ainv*b
for i in range(0, len(equations)):
Ni = pol
for j in range(0, len(coeffs)):
Ni = Ni.subs(coeffs[j], xx[j, i])
N.append(Ni)
self.N = N
def main():
t = ReferenceSimplex(2)
fe = Lagrange(2, 2)
u = 0
#compute u = sum_i u_i N_i
us = []
for i in range(0, fe.nbf()):
ui = Symbol("u_%d" % i)
us.append(ui)
u += ui*fe.N[i]
J = zeros(fe.nbf())
for i in range(0, fe.nbf()):
Fi = u*fe.N[i]
print(Fi)
for j in range(0, fe.nbf()):
uj = us[j]
integrands = diff(Fi, uj)
print(integrands)
J[j, i] = t.integrate(integrands)
pprint(J)
if __name__ == "__main__":
main()
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