def test_surf_approx(n, f, true=None):
'''Surface integral (for benchmark problem)'''
omega = UnitCubeMesh(n, n, 2 * n)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
gamma = StraightLineMesh(A, B, 2 * n)
V3 = FunctionSpace(omega, 'CG', 1)
V1 = FunctionSpace(gamma, 'CG', 1)
f3 = interpolate(f, V3)
shape = SquareRim(P=lambda x: np.array([0.25, 0.25, x[2]]), degree=10)
Pi = average_matrix(V3, V1, shape=shape)
x = Pi * f3.vector()
f1 = Function(V1, x)
if true is None:
true = f
error = sqrt(abs(assemble(inner(true - f1, true - f1) * dx)))
return gamma.hmin(), error, f1
def test_id(n, f, align=True):
'''Surface integral (for benchmark problem)'''
omega = UnitCubeMesh(n, n, 2 * n)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
lmbda = StraightLineMesh(A, B, 2 * n if align else n) # The curve
# Coupling surface
walls = [
'near((x[0]-0.25)*(x[0]-0.75), 0) && ((0.25-tol) < x[1]) && ((0.75+tol) > x[1])',
'near((x[1]-0.25)*(x[1]-0.75), 0) && ((0.25-tol) < x[0]) && ((0.75+tol) > x[0])'
]
walls = ['( %s )' % w for w in walls]
chi = CompiledSubDomain(' || '.join(walls), tol=1E-10)
surfaces = MeshFunction('size_t', omega, 2, 0)
chi.mark(surfaces, 1)
gamma = EmbeddedMesh(surfaces, 1)
V3 = FunctionSpace(omega, 'CG', 1)
V2 = FunctionSpace(gamma, 'CG', 1)
V1 = FunctionSpace(lmbda, 'CG', 1)
# What we want to extend
f1 = interpolate(f, V1)
# To 2d
E = uniform_extension_matrix(V1, V2)
x = E * f1.vector()
f2 = Function(V2, x)
# Extend to 3d by harmonic extension
u, v = TrialFunction(V3), TestFunction(V3)
a, L = inner(grad(u), grad(v)) * dx, inner(Constant(0), v) * dx
bcs = [DirichletBC(V3, f, 'on_boundary'), DirichletBC(V3, f, surfaces, 1)]
A, b = assemble_system(a, L, bcs)
f3 = Function(V3)
solver = KrylovSolver('cg', 'hypre_amg')
solver.parameters['relative_tolerance'] = 1E-13
solver.solve(A, f3.vector(), b)
# Come back
shape = SquareRim(P=lambda x: np.array([0.25, 0.25, x[2]]), degree=10)
Pi = average_matrix(V3, V1, shape=shape)
x = Pi * f3.vector()
f10 = Function(V1, x)
error = sqrt(abs(assemble(inner(f10 - f, f10 - f) * dx)))
return gamma.hmin(), error, f10
def poisson_1d(n, u_true, f):
'''1d part of benchmark'''
A, B = np.array([0.5, 0.5, 0]), np.array([0.5, 0.5, 1])
mesh = StraightLineMesh(A, B, n)
V = FunctionSpace(mesh, 'CG', 1)
bc = DirichletBC(V, u_true, 'on_boundary')
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
L = inner(f, v) * dx
A, b = assemble_system(a, L, bc)
solver = KrylovSolver('cg', 'amg')
solver.parameters['relative_tolerance'] = 1E-13
uh = Function(V)
solver.solve(A, uh.vector(), b)
return mesh.hmin(), errornorm(u_true, uh, 'H1'), uh
def test_Pi(n, f3, f1, Pi_f3):
'''(Pi u_3d, u_1d)_Gamma.'''
# True here is the reduced one
omega = UnitCubeMesh(n, n, 2*n)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
gamma = StraightLineMesh(A, B, 2*n)
V3 = FunctionSpace(omega, 'CG', 1)
V1 = FunctionSpace(gamma, 'CG', 1)
u3 = TrialFunction(V3)
v1 = TestFunction(V1)
shape = SquareRim(P=lambda x: np.array([0.25, 0.25, x[2]]),
degree=10)
dx_ = Measure('dx', domain=gamma)
a = inner(Average(u3, gamma, shape=shape), v1)*dx_
A = ii_assemble(a)
# Now check action
f3h = interpolate(f3, V3)
x = A*f3h.vector()
f1h = interpolate(f1, V1)
Pi_f = Function(V1, x)
result = f1h.vector().inner(x)
true = assemble(inner(Pi_f3, f1)*dx_)
error = abs(true - result)
return gamma.hmin(), error, Pi_f
def test_ET(n, f3, f1):
'''(E u_1d, T u_3d)_Gamma.'''
# True here is the reduced one
omega = UnitCubeMesh(n, n, 2 * n)
# Coupling surface
walls = [
'near((x[0]-0.25)*(x[0]-0.75), 0) && ((0.25-tol) < x[1]) && ((0.75+tol) > x[1])',
'near((x[1]-0.25)*(x[1]-0.75), 0) && ((0.25-tol) < x[0]) && ((0.75+tol) > x[0])'
]
walls = ['( %s )' % w for w in walls]
chi = CompiledSubDomain(' || '.join(walls), tol=1E-10)
surfaces = MeshFunction('size_t', omega, 2, 0)
chi.mark(surfaces, 1)
gamma = EmbeddedMesh(surfaces, 1)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
lmbda = StraightLineMesh(A, B, 2 * n) # The curve
V3 = FunctionSpace(omega, 'CG', 1)
V2 = FunctionSpace(gamma, 'CG', 2)
V1 = FunctionSpace(lmbda, 'CG', 1)
u1 = TrialFunction(V1)
v3 = TestFunction(V3)
dx_ = Measure('dx', domain=gamma)
a = inner(Extension(u1, gamma, 'uniform'), Trace(v3, gamma)) * dx_
A = ii_assemble(a)
# Now check action
f1h = interpolate(f1, V1)
x = A * f1h.vector()
f3h = interpolate(f3, V3)
result = f3h.vector().inner(x)
true = assemble(inner(f3, f1) * dx_)
error = abs(true - result)
return gamma.hmin(), error, f3h
def test_ext(n, f, true=None):
'''Surface integral (for benchmark problem)'''
omega = UnitCubeMesh(n, n, 2 * n)
A = np.array([0.5, 0.5, 0.0])
B = np.array([0.5, 0.5, 1.0])
lmbda = StraightLineMesh(A, B, 2 * n) # The curve
# Coupling surface
walls = [
'near((x[0]-0.25)*(x[0]-0.75), 0) && ((0.25-tol) < x[1]) && ((0.75+tol) > x[1])',
'near((x[1]-0.25)*(x[1]-0.75), 0) && ((0.25-tol) < x[0]) && ((0.75+tol) > x[0])'
]
walls = ['( %s )' % w for w in walls]
chi = CompiledSubDomain(' || '.join(walls), tol=1E-10)
surfaces = MeshFunction('size_t', omega, 2, 0)
chi.mark(surfaces, 1)
gamma = EmbeddedMesh(surfaces, 1)
V3 = FunctionSpace(omega, 'CG', 1)
V2 = FunctionSpace(gamma, 'CG', 1)
V1 = FunctionSpace(lmbda, 'CG', 1)
# What we want to extend
f1 = interpolate(f, V1)
E = uniform_extension_matrix(V1, V2)
x = E * f1.vector()
f2 = Function(V2, x)
if true is None:
true = f
error = sqrt(abs(assemble(inner(true - f2, true - f2) * dx)))
return gamma.hmin(), error, f2
# I check here assembly of coupling integrals that are needed in Federica's
# LM formulation
from dolfin import *
from weak_bcs.burman.generation import StraightLineMesh
import numpy as np
from xii import *
H, n = 4, 10
# Vasculature
mesh_1d = StraightLineMesh(np.array([0.5, 0.5, 0]), np.array([0.5, 0.5, H]),
3 * n) # Just to make it finer
# This is a model background mesh where all the cells would be intersected
# by a line. FIXME: how to compute this
mesh = BoxMesh(Point(0, 0, 0), Point(1, 1, H), 1, 1, n)
# Now we have a bulk unknown, the 1d unknown and the multiplier
V3 = FunctionSpace(mesh, 'CG', 1) # This
V1 = FunctionSpace(mesh_1d, 'CG', 1)
Q = FunctionSpace(mesh, 'DG', 0) # and this would be different
u3, u1 = list(map(TrialFunction, (V3, V1)))
q = TestFunction(Q)
# 3d->1d
avg_shape = Circle(lambda x: 0.05, degree=10)
Pi_u = Average(u3, mesh_1d, avg_shape)
# To aveluate the multiplier on the line we use trace (Average of None shape)
Tq = Average(q, mesh_1d, None)
# The coupling is still on the line