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fem.py
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fem.py
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from sympy import *
x,y,z = symbols('xyz')
class ReferenceSimplex:
def __init__(self, nsd):
self.nsd = nsd
coords = []
if nsd <= 3:
coords = symbols('xyz')[:nsd]
else:
coords = []
for d in range(0,nsd):
coords.append(Symbol("x_%d" % d))
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 = zeronm(len(equations), 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, r = div(e, c)
A[i,j] = d
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