from dolfin import *
from fenicstools import divergence_matrix
from numpy import cos, pi
mesh = UnitSquareMesh(10, 10)
x = mesh.coordinates()
x[:] = (x - 0.5) * 2
x[:] = 0.5*(cos(pi*(x-1.) / 2.) + 1.)
CG = VectorFunctionSpace(mesh, 'CG', 1)
u = interpolate(Expression(("sin(2*pi*x[0])", "cos(3*pi*x[1])")), CG)
C = divergence_matrix(mesh)
DG = FunctionSpace(mesh, 'DG', 0)
cc = Function(DG)
cc.vector()[:] = C * u.vector()
plot(cc, title='Gauss div')
plot(div(u), title="Projection")
divu_dg = project(div(u), DG)
plot(divu_dg - cc, title="Difference between FV and FEM")
def setup(low_memory_version, u_components, u, v, p, q, velocity_degree,
bcs, scalar_components, V, Q, x_, dim, mesh,
constrained_domain, velocity_update_type, **NS_namespace):
"""Preassemble mass and diffusion matrices.
Set up and prepare all equations to be solved. Called once, before
going into time loop.
"""
P = None
Rx = None
if not low_memory_version:
# Constant pressure gradient matrix
P = dict((ui, assemble(v*p.dx(i)*dx)) for i, ui in enumerate(u_components))
# Constant velocity divergence matrix
if V == Q:
Rx = P
else:
Rx = dict((ui, assemble(q*u.dx(i)*dx)) for i, ui in enumerate(u_components))
# Mass matrix
M = assemble(inner(u, v)*dx)
# Stiffness matrix (without viscosity coefficient)
K = assemble(inner(grad(u), grad(v))*dx)
# Pressure Laplacian. Either reuse K or assemble new
if V == Q and bcs['p'] == []:
Ap = K
else:
Bp = assemble(inner(grad(q), grad(p))*dx)
[bc.apply(Bp) for bc in bcs['p']]
Ap = Matrix()
Bp.compressed(Ap)
# Allocate coefficient matrix (needs reassembling)
A = Matrix(M)
# Create dictionary to be returned into global NS namespace
d = dict(P=P, Rx=Rx, A=A, M=M, K=K, Ap=Ap)
# Allocate coefficient matrix and work vectors for scalars. Matrix differs from velocity in boundary conditions only
if len(scalar_components) > 0:
d.update(Ta=Matrix(M))
if len(scalar_components) > 1:
# For more than one scalar we use the same linear algebra solver for all.
# For this to work we need some additional tensors. The extra matrix
# is required since different scalars may have different boundary conditions
Tb = Matrix(M)
bb = Vector(x_[scalar_components[0]])
bx = Vector(x_[scalar_components[0]])
d.update(Tb=Tb, bb=bb, bx=bx)
# Allocate for velocity update
if velocity_update_type.upper() == "GRADIENT_MATRIX":
from fenicstools.WeightedGradient import weighted_gradient_matrix
dP = weighted_gradient_matrix(mesh, range(dim), degree=velocity_degree,
constrained_domain=constrained_domain)
dp = Function(V)
d.update(dP=dP, dp=dp)
from fenicstools import divergence_matrix
C = divergence_matrix(mesh)
MA = assemble(q*TrialFunction(FunctionSpace(mesh, "DG", 0))*dx)
d.update(C=C, MA=MA)
elif velocity_update_type.upper() == "LUMPING":
ones = Function(V)
ones.vector()[:] = 1.
ML = M * ones.vector()
ML.set_local(1. / ML.array())
d.update(ML=ML)
else:
Mu = Matrix(M) if len(scalar_components) > 0 else M # Copy if used by scalars
[bc.apply(Mu) for bc in bcs['u0']]
d.update(Mu=Mu)
# Setup for solving convection
u_ab = as_vector([Function(V) for i in range(len(u_components))])
a_conv = 0.5*inner(v, dot(u_ab, nabla_grad(u)))*dx # Faster version
#a_conv = 0.5*inner(v, dot(U_AB, nabla_grad(u)))*dx
a_scalar = a_conv
d.update(u_ab=u_ab, a_conv=a_conv, a_scalar=a_scalar)
return d
