def test_raises(meshes_p1):
mesh1, mesh2 = meshes_p1[:2]
# Wrong FE family
V = VectorFunctionSpace(mesh2, "Discontinuous Lagrange", 1)
c = Function(V)
with pytest.raises(RuntimeError):
get_coordinates(c, mesh2.geometry())
with pytest.raises(RuntimeError):
set_coordinates(mesh2.geometry(), c)
# Wrong value rank
V = FunctionSpace(mesh2, "Lagrange", 1)
c = Function(V)
with pytest.raises(RuntimeError):
get_coordinates(c, mesh2.geometry())
with pytest.raises(RuntimeError):
set_coordinates(mesh2.geometry(), c)
# Wrong value shape
V = VectorFunctionSpace(mesh2, "Lagrange", mesh2.geometry().degree(),
dim=mesh2.geometry().dim() - 1)
c = Function(V)
with pytest.raises(RuntimeError):
get_coordinates(c, mesh2.geometry())
with pytest.raises(RuntimeError):
set_coordinates(mesh2.geometry(), c)
# Non-matching degree
V = VectorFunctionSpace(mesh2, "Lagrange", mesh2.geometry().degree() + 1)
c = Function(V)
with pytest.raises(RuntimeError):
get_coordinates(c, mesh2.geometry())
with pytest.raises(RuntimeError):
set_coordinates(mesh2.geometry(), c)
def amg_solve(N, method):
# Elasticity parameters
E = 1.0e9
nu = 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
# Stress computation
def sigma(v):
return 2.0 * mu * sym(grad(v)) + lmbda * tr(sym(
grad(v))) * Identity(2)
# Define problem
mesh = UnitSquareMesh(MPI.comm_world, N, N)
V = VectorFunctionSpace(mesh, 'Lagrange', 1)
bc0 = Function(V)
with bc0.vector().localForm() as bc_local:
bc_local.set(0.0)
def boundary(x, only_boundary):
return [only_boundary] * x.shape(0)
bc = DirichletBC(V.sub(0), bc0, boundary)
u = TrialFunction(V)
v = TestFunction(V)
# Forms
a, L = inner(sigma(u), grad(v)) * dx, dot(ufl.as_vector(
(1.0, 1.0)), v) * dx
# Assemble linear algebra objects
A = assemble_matrix(a, [bc])
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
# Create solution function
u = Function(V)
# Create near null space basis and orthonormalize
null_space = build_nullspace(V, u.vector())
# Attached near-null space to matrix
A.set_near_nullspace(null_space)
# Test that basis is orthonormal
assert null_space.is_orthonormal()
# Create PETSC smoothed aggregation AMG preconditioner, and
# create CG solver
solver = PETScKrylovSolver("cg", method)
# Set matrix operator
solver.set_operator(A)
# Compute solution and return number of iterations
return solver.solve(u.vector(), b)
def test_vector_p1_3d():
meshc = UnitCubeMesh(MPI.comm_world, 2, 3, 4)
meshf = UnitCubeMesh(MPI.comm_world, 3, 4, 5)
Vc = VectorFunctionSpace(meshc, ("CG", 1))
Vf = VectorFunctionSpace(meshf, ("CG", 1))
def u(x):
values0 = x[:, 0] + 2.0 * x[:, 1]
values1 = 4.0 * x[:, 0]
values2 = 3.0 * x[:, 2] + x[:, 0]
return np.stack([values0, values1, values2], axis=1)
uc, uf = Function(Vc), Function(Vf)
uc.interpolate(u)
uf.interpolate(u)
mat = PETScDMCollection.create_transfer_matrix(Vc._cpp_object,
Vf._cpp_object)
Vuc = Function(Vf)
mat.mult(uc.vector, Vuc.vector)
diff = Vuc.vector
diff.axpy(-1, uf.vector)
assert diff.norm() < 1.0e-12
def test_pointsource_vector_fs(mesh, point):
"""Tests point source when given constructor PointSource(V, point,
mag) with a vector for a vector function space that isn't placed
at a node for 1D, 2D and 3D. Global points given to constructor
from rank 0 processor.
"""
rank = MPI.rank(mesh.mpi_comm())
V = VectorFunctionSpace(mesh, "CG", 1)
v = TestFunction(V)
b = assemble(dot(Constant([0.0] * mesh.geometry().dim()), v) * dx)
if rank == 0:
ps = PointSource(V, point, 10.0)
else:
ps = PointSource(V, [])
ps.apply(b)
# Checks array sums to correct value
b_sum = b.sum()
assert round(b_sum - 10.0 * V.num_sub_spaces()) == 0
# Checks point source is added to correct part of the array
v2d = vertex_to_dof_map(V)
for v in vertices(mesh):
if near(v.midpoint().distance(point), 0.0):
for spc_idx in range(V.num_sub_spaces()):
ind = v2d[v.index() * V.num_sub_spaces() + spc_idx]
if ind < len(b.get_local()):
assert np.round(b.get_local()[ind] - 10.0) == 0
def test_pointsource_vector_fs(mesh, point):
"""Tests point source when given constructor PointSource(V, point,
mag) with a vector for a vector function space that isn't placed
at a node for 1D, 2D and 3D. Global points given to constructor
from rank 0 processor.
"""
rank = MPI.rank(mesh.mpi_comm())
V = VectorFunctionSpace(mesh, "CG", 1)
v = TestFunction(V)
b = assemble(dot(Constant([0.0]*mesh.geometry().dim()), v)*dx)
if rank == 0:
ps = PointSource(V, point, 10.0)
else:
ps = PointSource(V, [])
ps.apply(b)
# Checks array sums to correct value
b_sum = b.sum()
assert round(b_sum - 10.0*V.num_sub_spaces()) == 0
# Checks point source is added to correct part of the array
v2d = vertex_to_dof_map(V)
for v in vertices(mesh):
if near(v.midpoint().distance(point), 0.0):
for spc_idx in range(V.num_sub_spaces()):
ind = v2d[v.index()*V.num_sub_spaces() + spc_idx]
if ind < len(b.get_local()):
assert np.round(b.get_local()[ind] - 10.0) == 0
def get_spaces(mesh):
"""
Return an object of dolfin FunctionSpace, to
be used in the optimization pipeline
:param mesh: The mesh
:type mesh: :py:class:`dolfin.Mesh`
:returns: An object of functionspaces
:rtype: object
"""
from dolfin import FunctionSpace, VectorFunctionSpace, TensorFunctionSpace
# Make a dummy object
spaces = Object()
# A real space with scalars used for dolfin adjoint
spaces.r_space = FunctionSpace(mesh, "R", 0)
# A space for the strain fields
spaces.strainfieldspace = VectorFunctionSpace(mesh, "CG", 1, dim=3)
# A space used for scalar strains
spaces.strainspace = VectorFunctionSpace(mesh, "R", 0, dim=3)
# Spaces for the strain weights
spaces.strain_weight_space = TensorFunctionSpace(mesh, "R", 0)
return spaces
def cg1_cr_interpolation_matrix(mesh, constrained_domain=None):
'''
Compute matrix that allows fast interpolation of CG1 function to
Couzeix-Raviart space.
'''
CG1 = VectorFunctionSpace(mesh,
'CG',
1,
constrained_domain=constrained_domain)
CR = VectorFunctionSpace(mesh,
'CR',
1,
constrained_domain=constrained_domain)
# Get the matrix with approximate sparsity as the interpolation matrix
u = TrialFunction(CG1)
v = TestFunction(CR)
I = PETScMatrix()
assemble(dot(u, v) * dx, tensor=I)
# Fill the interpolation matrix
d = mesh.geometry().dim()
compiled_i_module.compute_cg1_cr_interpolation_matrix(I, d)
return I
def test_vector_p1_3d():
meshc = UnitCubeMesh(MPI.comm_world, 2, 3, 4)
meshf = UnitCubeMesh(MPI.comm_world, 3, 4, 5)
Vc = VectorFunctionSpace(meshc, ("CG", 1))
Vf = VectorFunctionSpace(meshf, ("CG", 1))
@function.expression.numba_eval
def expr_eval(values, x, cell_idx):
values[:, 0] = x[:, 0] + 2.0 * x[:, 1]
values[:, 1] = 4.0 * x[:, 0]
values[:, 2] = 3.0 * x[:, 2] + x[:, 0]
u = Expression(expr_eval, shape=(3, ))
uc = interpolate(u, Vc)
uf = interpolate(u, Vf)
mat = PETScDMCollection.create_transfer_matrix(Vc._cpp_object,
Vf._cpp_object)
Vuc = Function(Vf)
mat.mult(uc.vector(), Vuc.vector())
diff = Vuc.vector()
diff.axpy(-1, uf.vector())
assert diff.norm() < 1.0e-12
def gauss_divergence(u, mesh=None):
'''
This function uses Gauss divergence theorem to compute divergence of u
inside the cell by integrating normal fluxes across the cell boundary.
If u is a vector or tensor field the result of computation is diverence
of u in the cell center = DG0 scalar/vector function. For scalar fields,
the result is grad(u) = DG0 vector function. The fluxes are computed by
midpoint rule and as such the computed divergence is exact for linear
fields.
'''
# Require u to be GenericFunction
assert isinstance(u, GenericFunction)
# Require u to be scalar/vector/rank 2 tensor
rank = u.value_rank()
assert rank in [0, 1, 2]
# For now, there is no support for manifolds
if mesh is None:
_mesh = u.function_space().mesh()
else:
_mesh = mesh
tdim = _mesh.topology().dim()
gdim = _mesh.geometry().dim()
assert tdim == gdim
for i in range(rank):
assert u.value_dimension(i) == gdim
# Based on rank choose the type of CR1 space where u should be interpolated
# to to get the midpoint values + choose the type of DG0 space for
# divergence
if rank == 1:
DG = FunctionSpace(_mesh, 'DG', 0)
CR = VectorFunctionSpace(_mesh, 'CR', 1)
else:
DG = VectorFunctionSpace(_mesh, 'DG', 0)
if rank == 0:
CR = FunctionSpace(_mesh, 'CR', 1)
else:
CR = TensorFunctionSpace(_mesh, 'CR', 1)
divu = Function(DG)
_u = interpolate(u, CR)
# Use Gauss theorem cell by cell to get the divergence. The implementation
# is based on divergence(vector) = scalar and so the spaces for these
# two need to be provided
if rank == 1:
pass # CR, DG are correct already
else:
DG = FunctionSpace(_mesh, 'DG', 0)
CR = VectorFunctionSpace(_mesh, 'CR', 1)
compiled_cr_module.cr_divergence(divu, _u, DG, CR)
return divu
def __init__(self, u, name="Assigned Vector Function"):
self.u = u
assert isinstance(u, ListTensor)
V = u[0].function_space()
mesh = V.mesh()
family = V.ufl_element().family()
degree = V.ufl_element().degree()
constrained_domain = V.dofmap().constrained_domain
Vv = VectorFunctionSpace(mesh, family, degree, constrained_domain=constrained_domain)
Function.__init__(self, Vv, name=name)
self.fa = [FunctionAssigner(Vv.sub(i), V) for i, _u in enumerate(u)]
def get_function_spaces_1():
return (
lambda mesh: FunctionSpace(mesh, "Lagrange", 1),
pytest.mark.slow(lambda mesh: FunctionSpace(mesh, "Lagrange", 2)),
lambda mesh: VectorFunctionSpace(mesh, "Lagrange", 1),
pytest.mark.slow(lambda mesh: VectorFunctionSpace(mesh, "Lagrange", 2)),
pytest.mark.slow(lambda mesh: TensorFunctionSpace(mesh, "Lagrange", 1)),
pytest.mark.slow(lambda mesh: TensorFunctionSpace(mesh, "Lagrange", 2)),
lambda mesh: StokesFunctionSpace(mesh, "Lagrange", 1),
pytest.mark.slow(lambda mesh: StokesFunctionSpace(mesh, "Lagrange", 2)),
lambda mesh: FunctionSpace(mesh, "Real", 0),
pytest.mark.slow(lambda mesh: VectorFunctionSpace(mesh, "Real", 0)),
pytest.mark.slow(lambda mesh: FunctionAndRealSpace(mesh, "Lagrange", 1)),
pytest.mark.slow(lambda mesh: FunctionAndRealSpace(mesh, "Lagrange", 2))
)
def __init__(self, V, subdomains, shape_parametrization_expression):
# Store dolfin data structure related to the geometrical parametrization
self.mesh = subdomains.mesh()
self.subdomains = subdomains
self.reference_coordinates = self.mesh.coordinates().copy()
self.deformation_V = VectorFunctionSpace(self.mesh, "Lagrange", 1)
self.subdomain_id_to_deformation_dofs = dict() # from int to list
for cell in cells(self.mesh):
subdomain_id = int(
self.subdomains[cell]
) - 1 # tuple start from 0, while subdomains from 1
if subdomain_id not in self.subdomain_id_to_deformation_dofs:
self.subdomain_id_to_deformation_dofs[subdomain_id] = list()
dofs = self.deformation_V.dofmap().cell_dofs(cell.index())
for dof in dofs:
global_dof = self.deformation_V.dofmap().local_to_global_index(
dof)
if (self.deformation_V.dofmap().ownership_range()[0] <=
global_dof and global_dof <
self.deformation_V.dofmap().ownership_range()[1]):
self.subdomain_id_to_deformation_dofs[subdomain_id].append(
dof)
# In parallel some subdomains may not be present on all processors. Fill in
# the dict with empty lists if that is the case
mpi_comm = self.mesh.mpi_comm()
if not has_pybind11():
mpi_comm = mpi_comm.tompi4py()
min_subdomain_id = mpi_comm.allreduce(min(
self.subdomain_id_to_deformation_dofs.keys()),
op=MIN)
max_subdomain_id = mpi_comm.allreduce(max(
self.subdomain_id_to_deformation_dofs.keys()),
op=MAX)
for subdomain_id in range(min_subdomain_id, max_subdomain_id + 1):
if subdomain_id not in self.subdomain_id_to_deformation_dofs:
self.subdomain_id_to_deformation_dofs[subdomain_id] = list()
# Subdomain numbering is contiguous
assert min(self.subdomain_id_to_deformation_dofs.keys()) == 0
assert len(self.subdomain_id_to_deformation_dofs.keys()) == max(
self.subdomain_id_to_deformation_dofs.keys()) + 1
# Store the shape parametrization expression
self.shape_parametrization_expression = shape_parametrization_expression
assert len(self.shape_parametrization_expression) == len(
self.subdomain_id_to_deformation_dofs.keys())
# Prepare storage for displacement expression, computed by init()
self.displacement_expression = list()
def amg_solve(N, method):
# Elasticity parameters
E = 1.0e9
nu = 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
# Stress computation
def sigma(v):
return 2.0 * mu * sym(grad(v)) + lmbda * tr(sym(
grad(v))) * Identity(2)
# Define problem
mesh = UnitSquareMesh(MPI.comm_world, N, N)
V = VectorFunctionSpace(mesh, 'Lagrange', 1)
bc0 = Function(V)
bc = DirichletBC(V.sub(0), bc0, lambda x, on_boundary: on_boundary)
u = TrialFunction(V)
v = TestFunction(V)
# Forms
a, L = inner(sigma(u), grad(v)) * dx, dot(ufl.as_vector(
(1.0, 1.0)), v) * dx
# Assemble linear algebra objects
A, b = assemble_system(a, L, bc)
# Create solution function
u = Function(V)
# Create near null space basis and orthonormalize
null_space = build_nullspace(V, u.vector())
# Attached near-null space to matrix
A.set_near_nullspace(null_space)
# Test that basis is orthonormal
assert null_space.is_orthonormal()
# Create PETSC smoothed aggregation AMG preconditioner, and
# create CG solver
solver = PETScKrylovSolver("cg", method)
# Set matrix operator
solver.set_operator(A)
# Compute solution and return number of iterations
return solver.solve(u.vector(), b)
def test_read_write_p2_function(tempdir):
mesh = cpp.generation.UnitDiscMesh.create(MPI.comm_world, 3,
cpp.mesh.GhostMode.none)
cmap = fem.create_coordinate_map(mesh.ufl_domain())
mesh.geometry.coord_mapping = cmap
Q = FunctionSpace(mesh, ("Lagrange", 2))
F = Function(Q)
if has_petsc_complex:
F.interpolate(Expression("x[0] + j*x[0]", degree=1))
else:
F.interpolate(Expression("x[0]", degree=1))
filename = os.path.join(tempdir, "tri6_function.xdmf")
with XDMFFile(mesh.mpi_comm(), filename,
encoding=XDMFFile.Encoding.HDF5) as xdmf:
xdmf.write(F)
Q = VectorFunctionSpace(mesh, ("Lagrange", 1))
F = Function(Q)
if has_petsc_complex:
F.interpolate(Expression(("x[0] + j*x[0]", "x[1] + j*x[1]"), degree=1))
else:
F.interpolate(Expression(("x[0]", "x[1]"), degree=1))
filename = os.path.join(tempdir, "tri6_vector_function.xdmf")
with XDMFFile(mesh.mpi_comm(), filename,
encoding=XDMFFile.Encoding.HDF5) as xdmf:
xdmf.write(F)
def _generate_space(expression: Operator):
# Extract mesh from expression (from dolfin/fem/projection.py, _extract_function_space function)
meshes = set(
[ufl_domain.ufl_cargo() for ufl_domain in extract_domains(expression)])
for t in traverse_unique_terminals(
expression): # from ufl/domain.py, extract_domains
if hasattr(t, "_mesh"):
meshes.add(t._mesh)
assert len(meshes) == 1
mesh = meshes.pop()
# The EIM algorithm will evaluate the expression at vertices. However, since the Operator expression may
# contain e.g. a gradient of a solution defined in a C^0 space, we resort to DG1 spaces.
shape = expression.ufl_shape
assert len(shape) in (0, 1, 2)
if len(shape) == 0:
space = FunctionSpace(mesh, "Discontinuous Lagrange", 1)
elif len(shape) == 1:
space = VectorFunctionSpace(mesh,
"Discontinuous Lagrange",
1,
dim=shape[0])
elif len(shape) == 2:
space = TensorFunctionSpace(mesh,
"Discontinuous Lagrange",
1,
shape=shape)
else:
raise ValueError(
"Invalid expression in ParametrizedExpressionFactory.__init__().")
return space
def __init__(self, U_m, mesh):
"""Function spaces and BCs"""
V = VectorFunctionSpace(mesh, 'P', 2)
Q = FunctionSpace(mesh, 'P', 1)
self.mesh = mesh
self.vu, self.vp = TestFunction(V), TestFunction(Q) # for integration
self.u_, self.p_ = Function(V), Function(Q) # for the solution
self.u_1, self.p_1 = Function(V), Function(Q) # for the prev. solution
self.u_k, self.p_k = Function(V), Function(Q) # for the prev. solution
self.u, self.p = TrialFunction(V), TrialFunction(Q) # unknown!
U0_str = "4.*U_m*x[1]*(.41-x[1])/(.41*.41)"
x = [0,
.41 / 2] # evaluate the Expression at the center of the channel
self.U_mean = np.mean(2 / 3 * eval(U0_str))
U0 = Expression((U0_str, "0"), U_m=U_m, degree=2)
bc0 = DirichletBC(V, Constant((0, 0)), cylinderwall)
bc1 = DirichletBC(V, Constant((0, 0)), topandbottom)
bc2 = DirichletBC(V, U0, inlet)
bc3 = DirichletBC(Q, Constant(0), outlet)
self.bcu = [bc0, bc1, bc2]
self.bcp = [bc3]
# ds is needed to compute drag and lift.
ASD1 = AutoSubDomain(topandbottom)
ASD2 = AutoSubDomain(cylinderwall)
mf = MeshFunction("size_t", mesh, 1)
mf.set_all(0)
ASD1.mark(mf, 1)
ASD2.mark(mf, 2)
self.ds_ = ds(subdomain_data=mf, domain=mesh)
return
def test_l2projection(polynomial_order, in_expression):
# Test l2 projection for scalar and vector valued expression
interpolate_expression = Expression(in_expression, degree=3)
xmin, xmax = 0., 1.
ymin, ymax = 0., 1.
property_idx = 5
mesh = RectangleMesh(Point(xmin, ymin), Point(xmax, ymax), 40, 40)
if len(interpolate_expression.ufl_shape) == 0:
V = FunctionSpace(mesh, "DG", polynomial_order)
elif len(interpolate_expression.ufl_shape) == 1:
V = VectorFunctionSpace(mesh, "DG", polynomial_order)
v_exact = Function(V)
v_exact.interpolate(interpolate_expression)
x = RandomRectangle(Point(xmin, ymin), Point(xmax,
ymax)).generate([500, 500])
s = assign_particle_values(x, interpolate_expression)
# Just make a complicated particle, possibly with scalars and vectors mixed
p = particles(x, [x, s, x, x, s], mesh)
vh = Function(V)
lstsq_rho = l2projection(p, V, property_idx)
lstsq_rho.project(vh)
error_sq = abs(assemble(dot(v_exact - vh, v_exact - vh) * dx))
assert error_sq < 1e-15
def initial_conditions(self):
"Return initial conditions for v and s as a dolfin.GenericFunction."
n = self.num_states() # (Maximal) Number of states in MultiCellModel
VS = VectorFunctionSpace(self.mesh(), "DG", 0, n+1)
vs = Function(VS)
markers = self.markers()
u = TrialFunction(VS)
v = TestFunction(VS)
dy = Measure("dx", domain=self.mesh(), subdomain_data=markers)
# Define projection into multiverse
a = inner(u, v)*dy()
Ls = list()
for (k, model) in enumerate(self.models()):
ic = model.initial_conditions() # Extract initial conditions
n_k = model.num_states() # Extract number of local states
i_k = self.keys()[k] # Extract domain index of cell model k
L_k = sum(ic[j]*v[j]*dy(i_k) for j in range(n_k))
Ls.append(L_k)
L = sum(Ls)
solve(a == L, vs)
return vs
def __init__(self, mesh, mf, bc_dict, move_dict):
"""
Inititalize Stokes solver with a mesh, its corresponding facet function,
a dictionary describing boundary conditions and
a dictionary describing which boundaries are fixed in the shape optimization setting
"""
self.mesh = mesh
self.backup = mesh.coordinates().copy()
self.mf = mf
V2 = VectorElement("CG", mesh.ufl_cell(), 2)
S1 = FiniteElement("CG", mesh.ufl_cell(), 1)
TH = V2 * S1
self.VQ = FunctionSpace(self.mesh, TH)
self.w = Function(self.VQ)
self.f = Constant([0.]*mesh.geometric_dimension())
self.S = VectorFunctionSpace(self.mesh, "CG", 1)
self.move_dict = move_dict
self.J = 0
self._init_bcs(bc_dict)
self.solve()
self.vfac = 1e4
self.bfac = 1e2
self._init_geometric_functions()
self.eval_current_J()
self.eval_current_dJ()
self.outfile = File("output/u_singlemesh.pvd")
self.outfile << self.u
self.gradient_scale = 1
self.iteration_counter = 1
self.create_mapping_for_moving_boundary()
def test_p4_scalar_vector():
perms = itertools.permutations([1, 2, 3, 4])
for p in perms:
cells = numpy.array([[0, 1, 2, 3], p], dtype=numpy.int64)
points = numpy.array(
[[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0],
[0.0, 0.0, 1.0], [1.0, 1.0, 1.0]],
dtype=numpy.float64)
mesh = Mesh(MPI.comm_world, CellType.Type.tetrahedron, points, cells,
[], GhostMode.none)
mesh.geometry.coord_mapping = fem.create_coordinate_map(mesh)
Q = FunctionSpace(mesh, ("CG", 4))
F0 = interpolate(Expression("x[0]", degree=4), Q)
F1 = interpolate(Expression("x[1]", degree=4), Q)
F2 = interpolate(Expression("x[2]", degree=4), Q)
pts = numpy.array([[0.4, 0.4, 0.1], [0.4, 0.1, 0.4], [0.1, 0.4, 0.4]])
for pt in pts:
assert numpy.isclose(pt[0], F0(pt)[0])
assert numpy.isclose(pt[1], F1(pt)[0])
assert numpy.isclose(pt[2], F2(pt)[0])
V = VectorFunctionSpace(mesh, ("CG", 4))
F = interpolate(Expression(("x[0]", "x[1]", "0.0"), degree=4), V)
for pt in pts:
result = F(pt)
assert numpy.isclose(pt[0], result[0])
assert numpy.isclose(pt[1], result[1])
assert numpy.isclose(0.0, result[2])
def test_save_3d_vector(tempdir, encoding):
filename = os.path.join(tempdir, "u_3Dv.xdmf")
mesh = UnitCubeMesh(MPI.comm_world, 2, 2, 2)
u = Function(VectorFunctionSpace(mesh, ("Lagrange", 1)))
u.vector.set(1.0 + (1j if has_petsc_complex else 0))
with XDMFFile(mesh.mpi_comm(), filename, encoding=encoding) as file:
file.write(u)
def increase_order(V):
"""
For a given function space, return the same space, but with a
higher polynomial degree
"""
n = V.num_sub_spaces()
if n > 0:
spaces = []
for i in range(n):
V_i = V.sub(i)
element = V_i.ufl_element()
# Handle VectorFunctionSpaces specially
if isinstance(element, ufl.VectorElement):
spaces += [
VectorFunctionSpace(V_i.mesh(),
element.family(),
element.degree() + 1,
dim=element.num_sub_elements())
]
# Handle all else as MixedFunctionSpaces
else:
spaces += [increase_order(V_i)]
return MixedFunctionSpace(spaces)
if V.ufl_element().family() == "Real":
return FunctionSpace(V.mesh(), "Real", 0)
return FunctionSpace(V.mesh(),
V.ufl_element().family(),
V.ufl_element().degree() + 1)
def divergence_matrix(mesh):
CR = VectorFunctionSpace(mesh, 'CR', 1)
DG = FunctionSpace(mesh, 'DG', 0)
A = cg1_cr_interpolation_matrix(mesh)
M = assemble(dot(div(TrialFunction(CR)), TestFunction(DG)) * dx())
C = compiled_cr_module.cr_divergence_matrix(M, A, DG, CR)
return C
def __init__(self, u, name="Assigned Vector Function"):
self.u = u
assert isinstance(u, ListTensor)
V = u[0].function_space()
mesh = V.mesh()
family = V.ufl_element().family()
degree = V.ufl_element().degree()
constrained_domain = V.dofmap().constrained_domain
Vv = VectorFunctionSpace(mesh,
family,
degree,
constrained_domain=constrained_domain)
Function.__init__(self, Vv, name=name)
self.fa = [FunctionAssigner(Vv.sub(i), V) for i, _u in enumerate(u)]
def test_closed_boundary(advection_scheme):
# FIXME: rk3 scheme does not bounces off the wall properly
xmin, xmax = 0., 1.
ymin, ymax = 0., 1.
mesh = RectangleMesh(Point(xmin, ymin), Point(xmax, ymax), 10, 10)
# Particle
x = np.array([[0.975, 0.475]])
# Given velocity field:
vexpr = Constant((1., 0.))
# Given time do_step:
dt = 0.05
# Then bounced position is
x_bounced = np.array([[0.975, 0.475]])
p = particles(x, [x, x], mesh)
V = VectorFunctionSpace(mesh, "CG", 1)
v = Function(V)
v.assign(vexpr)
# Different boundary parts
bound_left = UnitSquareLeft()
bound_right = UnitSquareRight()
bound_top = UnitSquareTop()
bound_bottom = UnitSquareBottom()
# Mark all facets
facet_marker = MeshFunction('size_t', mesh, mesh.topology().dim() - 1)
facet_marker.set_all(0)
# Mark as closed
bound_right.mark(facet_marker, 1)
# Mark other boundaries as open
bound_left.mark(facet_marker, 2)
bound_top.mark(facet_marker, 2)
bound_bottom.mark(facet_marker, 2)
if advection_scheme == 'euler':
ap = advect_particles(p, V, v, facet_marker)
elif advection_scheme == 'rk2':
ap = advect_rk2(p, V, v, facet_marker)
elif advection_scheme == 'rk3':
ap = advect_rk3(p, V, v, facet_marker)
else:
assert False
# Do one timestep, particle must bounce from wall of
ap.do_step(dt)
xpE = p.positions()
# Check if particle correctly bounced off from closed wall
xpE_root = comm.gather(xpE, root=0)
if comm.rank == 0:
xpE_root = np.float64(np.vstack(xpE_root))
error = np.linalg.norm(x_bounced - xpE_root)
assert(error < 1e-10)
def test_multi_ps_matrix(mesh):
"""Tests point source PointSource(V, source) for mulitple point
sources applied to a matrix for 1D, 2D and 3D. Global points given
to constructor from rank 0 processor.
"""
c_ids = [0, 1, 2]
rank = MPI.rank(mesh.mpi_comm())
V = VectorFunctionSpace(mesh, "CG", 1, dim=2)
u, v = TrialFunction(V), TestFunction(V)
A = assemble(Constant(0.0) * dot(u, v) * dx)
source = []
if rank == 0:
for c_id in c_ids:
cell = Cell(mesh, c_id)
point = cell.midpoint()
source.append((point, 10.0))
ps = PointSource(V, source)
ps.apply(A)
# Checks b sums to correct value
a_sum = MPI.sum(mesh.mpi_comm(), np.sum(A.array()))
assert round(a_sum - 2 * len(c_ids) * 10) == 0
def get_custom_space(self, family, degree, shape, boundary=False):
"""Get a custom function space. """
if boundary:
mesh = self.BoundaryMesh
key = (family, degree, shape, boundary)
else:
mesh = self.mesh
key = (family, degree, shape)
space = self._spaces.get(key)
if space is None:
rank = len(shape)
if rank == 0:
space = FunctionSpace(mesh, family, degree)
elif rank == 1:
space = VectorFunctionSpace(mesh, family, degree, shape[0])
else:
space = TensorFunctionSpace(mesh,
family,
degree,
shape,
symmetry={})
self._spaces[key] = space
return space
def test_save_1d_vector(tempfile, file_options):
mesh = UnitIntervalMesh(MPI.comm_world, 32)
u = Function(VectorFunctionSpace(mesh, "Lagrange", 2))
u.vector()[:] = 1.0
VTKFile(tempfile + "u.pvd", "ascii").write(u)
for file_option in file_options:
VTKFile(tempfile + "u.pvd", file_option).write(u)
def test_nullspace_check(mesh, degree):
V = VectorFunctionSpace(mesh, ('Lagrange', degree))
u, v = TrialFunction(V), TestFunction(V)
mesh.geometry.coord_mapping = fem.create_coordinate_map(mesh)
E, nu = 2.0e2, 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
def sigma(w, gdim):
return 2.0 * mu * ufl.sym(grad(w)) + lmbda * ufl.tr(
grad(w)) * ufl.Identity(gdim)
a = inner(sigma(u, mesh.geometry.dim), grad(v)) * dx
zero = Function(V)
L = inner(zero, v) * dx
# Assemble matrix and create compatible vector
A, L = assembling.assemble_system(a, L, [])
# Create null space basis and test
null_space = build_elastic_nullspace(V)
assert null_space.in_nullspace(A, tol=1.0e-8)
null_space.orthonormalize()
assert null_space.in_nullspace(A, tol=1.0e-8)
# Create incorrect null space basis and test
null_space = build_broken_elastic_nullspace(V)
assert not null_space.in_nullspace(A, tol=1.0e-8)
null_space.orthonormalize()
assert not null_space.in_nullspace(A, tol=1.0e-8)
def test_advect_periodic(advection_scheme):
xmin, ymin, zmin = 0., 0., 0.
xmax, ymax, zmax = 1., 1., 1.
pres = 10
mesh = UnitCubeMesh(10, 10, 10)
lims = np.array([[xmin, xmin, ymin, ymax, zmin, zmax],
[xmax, xmax, ymin, ymax, zmin, zmax],
[xmin, xmax, ymin, ymin, zmin, zmax],
[xmin, xmax, ymax, ymax, zmin, zmax],
[xmin, xmax, ymin, ymax, zmin, zmin],
[xmin, xmax, ymin, ymax, zmax, zmax]])
vexpr = Constant((1., 1., 1.))
V = VectorFunctionSpace(mesh, "CG", 1)
v = Function(V)
v.assign(vexpr)
x = RandomBox(Point(0., 0., 0.), Point(1., 1.,
1.)).generate([pres, pres, pres])
x = comm.bcast(x, root=0)
dt = 0.05
p = particles(x, [x * 0, x**2], mesh)
if advection_scheme == 'euler':
ap = advect_particles(p, V, v, 'periodic', lims.flatten())
elif advection_scheme == 'rk2':
ap = advect_rk2(p, V, v, 'periodic', lims.flatten())
elif advection_scheme == 'rk3':
ap = advect_rk3(p, V, v, 'periodic', lims.flatten())
else:
assert False
xp0 = p.positions()
t = 0.
while t < 1. - 1e-12:
ap.do_step(dt)
t += dt
xpE = p.positions()
xp0_root = comm.gather(xp0, root=0)
xpE_root = comm.gather(xpE, root=0)
assert len(xp0) == len(xpE)
num_particles = p.number_of_particles()
if comm.Get_rank() == 0:
xp0_root = np.float32(np.vstack(xp0_root))
xpE_root = np.float32(np.vstack(xpE_root))
# Sort on x positions
xp0_root = xp0_root[xp0_root[:, 0].argsort(), :]
xpE_root = xpE_root[xpE_root[:, 0].argsort(), :]
error = np.linalg.norm(xp0_root - xpE_root)
assert error < 1e-10
assert num_particles - pres**3 == 0
def test_advect_periodic(advection_scheme):
# FIXME: this unit test is sensitive to the ordering of the particle
# array, i.e. xp0_root and xpE_root may contain exactly the same entries
# but only in a different order. This will return an error right now
xmin, xmax = 0., 1.
ymin, ymax = 0., 1.
pres = 3
mesh = RectangleMesh(Point(xmin, ymin), Point(xmax, ymax), 10, 10)
lims = np.array([[xmin, xmin, ymin, ymax], [xmax, xmax, ymin, ymax],
[xmin, xmax, ymin, ymin], [xmin, xmax, ymax, ymax]])
vexpr = Constant((1., 1.))
V = VectorFunctionSpace(mesh, "CG", 1)
x = RandomRectangle(Point(0.05, 0.05), Point(0.15, 0.15)).generate([pres, pres])
x = comm.bcast(x, root=0)
dt = 0.05
v = Function(V)
v.assign(vexpr)
p = particles(x, [x*0, x**2], mesh)
if advection_scheme == 'euler':
ap = advect_particles(p, V, v, 'periodic', lims.flatten())
elif advection_scheme == 'rk2':
ap = advect_rk2(p, V, v, 'periodic', lims.flatten())
elif advection_scheme == 'rk3':
ap = advect_rk3(p, V, v, 'periodic', lims.flatten())
else:
assert False
xp0 = p.positions()
t = 0.
while t < 1.-1e-12:
ap.do_step(dt)
t += dt
xpE = p.positions()
# Check if position correct
xp0_root = comm.gather(xp0, root=0)
xpE_root = comm.gather(xpE, root=0)
num_particles = p.number_of_particles()
if comm.Get_rank() == 0:
xp0_root = np.float32(np.vstack(xp0_root))
xpE_root = np.float32(np.vstack(xpE_root))
# Sort on x positions
xp0_root = xp0_root[xp0_root[:, 0].argsort(), :]
xpE_root = xpE_root[xpE_root[:, 0].argsort(), :]
error = np.linalg.norm(xp0_root - xpE_root)
assert error < 1e-10
assert num_particles - pres**2 == 0
def make_function_spaces(mesh, element):
'Create mixed function space on the mesh.'
assert hasattr(element, 'V') and hasattr(element, 'Q')
# Create velocity space
n = len(element.V)
assert n > 0
V = VectorFunctionSpace(mesh, *element.V[0])
for i in range(1, n):
V += VectorFunctionSpace(mesh, *element.V[i])
# Create pressure space
assert len(element.Q) == 1
Q = FunctionSpace(mesh, *element.Q[0])
M = MixedFunctionSpace([V, Q])
return V, Q, M
class SNDiscretization(Discretization):
def __init__(self, problem, SN_order, verbosity=0):
super(SNDiscretization, self).__init__(problem, verbosity)
if self.verb > 1: print0("Obtaining angular discretization data")
t_ordinates = Timer("Angular discretization data")
from transport_data import ordinates_ext_module, quadrature_file
from common import check_sn_order
self.angular_quad = ordinates_ext_module.OrdinatesData(SN_order, self.mesh.topology().dim(), quadrature_file)
self.M = self.angular_quad.get_M()
self.N = check_sn_order(SN_order)
ord_mat = [self.angular_quad.get_xi().tolist(), self.angular_quad.get_eta().tolist()]
if self.angular_quad.get_D() == 3: ord_mat.append(self.angular_quad.get_mu().tolist())
self.ordinates_matrix = as_matrix(ord_mat)
if self.verb > 2:
self.angular_quad.print_info()
def init_solution_spaces(self, group_GS=False):
if self.verb > 1: print0("Defining function spaces" )
self.t_spaces.start()
self.Vpsi1 = VectorFunctionSpace(self.mesh, "CG", self.parameters["p"], self.M)
if not group_GS:
self.V = MixedFunctionSpace([self.Vpsi1]*self.G)
else:
# Alias the solution space as Vpsi1
self.V = self.Vpsi1
self.Vphi1 = FunctionSpace(self.mesh, "CG", self.parameters["p"])
self.ndof1 = self.Vpsi1.dim()
self.ndof = self.V.dim()
if self.verb > 1:
print0(" {0} direction(s), {1} group(s), {2}".format(self.M, self.G, "group GS" if group_GS else "full system"))
print0(" NDOF (solution): {0}".format(self.ndof) )
print0(" NDOF (1gr): {0}".format(self.ndof1) )
print0(" NDOF (1dir, 1gr): {0}".format(self.Vpsi1.sub(0).dim()) )
print0(" NDOF (XS): {0}".format(self.ndof0) )
def test_multi_ps_matrix_node_vector_fs(mesh):
"""Tests point source applied to a matrix with given constructor
PointSource(V, source) and a vector function space when points
placed at 3 vertices for 1D, 2D and 3D. Global points given to
constructor from rank 0 processor.
"""
point = [0.0, 0.5, 1.0]
rank = MPI.rank(mesh.mpi_comm())
V = VectorFunctionSpace(mesh, "CG", 1, dim=2)
u, v = TrialFunction(V), TestFunction(V)
w = Function(V)
A = assemble(Constant(0.0)*dot(u, v)*dx)
dim = mesh.geometry().dim()
source = []
point_coords = np.zeros(dim)
for p in point:
for i in range(dim):
point_coords[i - 1] = p
if rank == 0:
source.append((Point(point_coords), 10.0))
ps = PointSource(V, source)
ps.apply(A)
# Checks array sums to correct value
A.get_diagonal(w.vector())
a_sum = MPI.sum(mesh.mpi_comm(), np.sum(A.array()))
assert round(a_sum - 2*len(point)*10) == 0
# Check if coordinates are in portion of mesh and if so check that
# diagonal components sum to the correct value.
mesh_coords = V.tabulate_dof_coordinates()
for p in point:
for i in range(dim):
point_coords[i] = p
j = 0
for i in range(len(mesh_coords)//(dim)):
mesh_coords_check = mesh_coords[j:j+dim-1]
if np.array_equal(point_coords, mesh_coords_check) is True:
assert np.round(w.vector()[j//(dim)] - 10.0) == 0.0
j += dim
def save_tstep_solution_h5(tstep, q_, u_, newfolder, tstepfiles, constrained_domain,
output_timeseries_as_vector, u_components,
scalar_components, NS_parameters):
"""Store solution on current timestep to XDMF file."""
timefolder = path.join(newfolder, 'Timeseries')
if output_timeseries_as_vector:
# project or store velocity to vector function space
for comp, tstepfile in tstepfiles.iteritems():
if comp == "u":
if not hasattr(tstepfile, 'uv'): # First time around only
V = q_['u0'].function_space()
Vv = VectorFunctionSpace(V.mesh(), V.ufl_element().family(), V.ufl_element().degree(),
constrained_domain=constrained_domain)
tstepfile.uv = Function(Vv)
tstepfile.d = dict((ui, Vv.sub(i).dofmap().collapse(Vv.mesh())[1])
for i, ui in enumerate(u_components))
# The short but timeconsuming way:
#tstepfile.uv.assign(project(u_, Vv))
# Or the faster, but more comprehensive way:
for ui in u_components:
q_[ui].update()
vals = tstepfile.d[ui].values()
keys = tstepfile.d[ui].keys()
tstepfile.uv.vector()[vals] = q_[ui].vector()[keys]
tstepfile << (tstepfile.uv, float(tstep))
elif comp in q_:
tstepfile << (q_[comp], float(tstep))
else:
tstepfile << (tstepfile.function, float(tstep))
else:
for comp, tstepfile in tstepfiles.iteritems():
tstepfile << (q_[comp], float(tstep))
if MPI.rank(mpi_comm_world()) == 0:
if not path.exists(path.join(timefolder, "params.dat")):
f = open(path.join(timefolder, 'params.dat'), 'w')
cPickle.dump(NS_parameters, f)
def pot():
# Define mesh and boundaries.
mesh = RectangleMesh(0.0, 0.0, 0.4, 0.1, 120, 30, "left/right")
V = VectorFunctionSpace(mesh, "CG", 2)
Q = FunctionSpace(mesh, "CG", 1)
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] < GMSH_EPS
left_boundary = LeftBoundary()
class RightBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] > 0.4 - GMSH_EPS
right_boundary = RightBoundary()
class LowerBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] < GMSH_EPS
lower_boundary = LowerBoundary()
class UpperBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] > 0.1 - GMSH_EPS
upper_boundary = UpperBoundary()
# Boundary conditions for the velocity.
u_bcs = [
DirichletBC(V, (0.0, 0.0), lower_boundary),
DirichletBC(V.sub(1), 0.0, upper_boundary),
DirichletBC(V.sub(0), 0.0, right_boundary),
DirichletBC(V.sub(0), 0.0, left_boundary),
]
p_bcs = []
return mesh, V, Q, u_bcs, p_bcs, [lower_boundary], [right_boundary, left_boundary]
def save_tstep_solution_h5(tstep, q_, u_, newfolder, tstepfiles, constrained_domain,
output_timeseries_as_vector, u_components,
scalar_components, NS_parameters):
"""Store solution on current timestep to XDMF file."""
timefolder = path.join(newfolder, 'Timeseries')
if output_timeseries_as_vector:
# project or store velocity to vector function space
for comp, tstepfile in tstepfiles.iteritems():
if comp == "u":
V = q_['u0'].function_space()
# First time around create vector function and assigners
if not hasattr(tstepfile, 'uv'):
Vv = VectorFunctionSpace(V.mesh(), V.ufl_element().family(), V.ufl_element().degree(),
constrained_domain=constrained_domain)
tstepfile.uv = Function(Vv)
tstepfile.fa = [FunctionAssigner(Vv.sub(i), V) for i, ui in enumerate(u_components)]
# Assign solution to vector
for i, ui in enumerate(u_components):
tstepfile.fa[i].assign(tstepfile.uv.sub(i), q_[ui])
# Store solution vector
tstepfile << (tstepfile.uv, float(tstep))
elif comp in q_:
tstepfile << (q_[comp], float(tstep))
else:
tstepfile << (tstepfile.function, float(tstep))
else:
for comp, tstepfile in tstepfiles.iteritems():
tstepfile << (q_[comp], float(tstep))
if MPI.rank(mpi_comm_world()) == 0:
if not path.exists(path.join(timefolder, "params.dat")):
f = open(path.join(timefolder, 'params.dat'), 'w')
cPickle.dump(NS_parameters, f)
def init_solution_spaces(self, group_GS=False):
if self.verb > 1: print0("Defining function spaces" )
self.t_spaces.start()
self.Vpsi1 = VectorFunctionSpace(self.mesh, "CG", self.parameters["p"], self.M)
if not group_GS:
self.V = MixedFunctionSpace([self.Vpsi1]*self.G)
else:
# Alias the solution space as Vpsi1
self.V = self.Vpsi1
self.Vphi1 = FunctionSpace(self.mesh, "CG", self.parameters["p"])
self.ndof1 = self.Vpsi1.dim()
self.ndof = self.V.dim()
if self.verb > 1:
print0(" {0} direction(s), {1} group(s), {2}".format(self.M, self.G, "group GS" if group_GS else "full system"))
print0(" NDOF (solution): {0}".format(self.ndof) )
print0(" NDOF (1gr): {0}".format(self.ndof1) )
print0(" NDOF (1dir, 1gr): {0}".format(self.Vpsi1.sub(0).dim()) )
print0(" NDOF (XS): {0}".format(self.ndof0) )
def _evaluateLocalEstimator(cls, mu, w, coeff_field, pde, f, quadrature_degree, epsilon=1e-5):
"""Evaluation of patch local equilibrated estimator."""
# prepare numerical flux and f
sigma_mu, f_mu = evaluate_numerical_flux(w, mu, coeff_field, f)
# ###################
# ## MIXED PROBLEM ##
# ###################
# get setup data for mixed problem
V = w[mu]._fefunc.function_space()
mesh = V.mesh()
mesh.init()
degree = element_degree(w[mu]._fefunc)
# data for nodal bases
V_dm = V.dofmap()
V_dofs = dict([(i, V_dm.cell_dofs(i)) for i in range(mesh.num_cells())])
V1 = FunctionSpace(mesh, 'CG', 1) # V1 is to define nodal base functions
phi_z = Function(V1)
phi_coeffs = np.ndarray(V1.dim())
vertex_dof_map = V1.dofmap().vertex_to_dof_map(mesh)
# vertex_dof_map = vertex_to_dof_map(V1)
dof_list = vertex_dof_map.tolist()
# DG0 localisation
DG0 = FunctionSpace(mesh, 'DG', 0)
DG0_dofs = dict([(c.index(),DG0.dofmap().cell_dofs(c.index())[0]) for c in cells(mesh)])
dg0 = TestFunction(DG0)
# characteristic function of patch
xi_z = Function(DG0)
xi_coeffs = np.ndarray(DG0.dim())
# mesh data
h = CellSize(mesh)
n = FacetNormal(mesh)
cf = CellFunction('size_t', mesh)
# setup error estimator vector
eq_est = np.zeros(DG0.dim())
# setup global equilibrated flux vector
DG = VectorFunctionSpace(mesh, "DG", degree)
DG_dofmap = DG.dofmap()
# define form functions
tau = TrialFunction(DG)
v = TestFunction(DG)
# define global tau
tau_global = Function(DG)
tau_global.vector()[:] = 0.0
# iterate vertices
for vertex in vertices(mesh):
# get patch cell indices
vid = vertex.index()
patch_cid, FF_inner, FF_boundary = get_vertex_patch(vid, mesh, layers=1)
# set nodal base function
phi_coeffs[:] = 0
phi_coeffs[dof_list.index(vid)] = 1
phi_z.vector()[:] = phi_coeffs
# set characteristic function and mark patch
cf.set_all(0)
xi_coeffs[:] = 0
for cid in patch_cid:
xi_coeffs[DG0_dofs[int(cid)]] = 1
cf[int(cid)] = 1
xi_z.vector()[:] = xi_coeffs
# determine local dofs
lDG_cell_dofs = dict([(cid, DG_dofmap.cell_dofs(cid)) for cid in patch_cid])
lDG_dofs = [cd.tolist() for cd in lDG_cell_dofs.values()]
lDG_dofs = list(iter.chain(*lDG_dofs))
# print "\nlocal DG subspace has dimension", len(lDG_dofs), "degree", degree, "cells", len(patch_cid), patch_cid
# print "local DG_cell_dofs", lDG_cell_dofs
# print "local DG_dofs", lDG_dofs
# create patch measures
dx = Measure('dx')[cf]
dS = Measure('dS')[FF_inner]
# define forms
alpha = Constant(1 / epsilon) / h
a = inner(tau,v) * phi_z * dx(1) + alpha * div(tau) * div(v) * dx(1) + avg(alpha) * jump(tau,n) * jump(v,n) * dS(1)\
+ avg(alpha) * jump(xi_z * tau,n) * jump(v,n) * dS(2)
L = -alpha * (div(sigma_mu) + f) * div(v) * phi_z * dx(1)\
- avg(alpha) * jump(sigma_mu,n) * jump(v,n) * avg(phi_z)*dS(1)
# print "L2 f + div(sigma)", assemble((f + div(sigma)) * (f + div(sigma)) * dx(0))
# assemble forms
lhs = assemble(a, form_compiler_parameters={'quadrature_degree': quadrature_degree})
rhs = assemble(L, form_compiler_parameters={'quadrature_degree': quadrature_degree})
# convert DOLFIN representation to scipy sparse arrays
rows, cols, values = lhs.data()
lhsA = sps.csr_matrix((values, cols, rows)).tocoo()
# slice sparse matrix and solve linear problem
lhsA = coo_submatrix_pull(lhsA, lDG_dofs, lDG_dofs)
lx = spsolve(lhsA, rhs.array()[lDG_dofs])
# print ">>> local solution lx", type(lx), lx
local_tau = Function(DG)
local_tau.vector()[lDG_dofs] = lx
# print "div(tau)", assemble(inner(div(local_tau),div(local_tau))*dx(1))
# add up local fluxes
tau_global.vector()[lDG_dofs] += lx
# evaluate estimator
# maybe TODO: re-define measure dx
eq_est = assemble( inner(tau_global, tau_global) * dg0 * (dx(0)+dx(1)),\
form_compiler_parameters={'quadrature_degree': quadrature_degree})
# reorder according to cell ids
eq_est = eq_est[DG0_dofs.values()].array()
global_est = np.sqrt(np.sum(eq_est))
# eq_est_global = assemble( inner(tau_global, tau_global) * (dx(0)+dx(1)), form_compiler_parameters={'quadrature_degree': quadrature_degree} )
# global_est2 = np.sqrt(np.sum(eq_est_global))
return global_est, FlatVector(np.sqrt(eq_est))#, tau_global
def karman():
#mesh = Mesh('../meshes/2d/karman.xml')
mesh = Mesh('../meshes/2d/karman-coarse2.xml')
V = VectorFunctionSpace(mesh, 'CG', 2)
Q = FunctionSpace(mesh, 'CG', 1)
# Define mesh and boundaries.
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] < DOLFIN_EPS
left_boundary = LeftBoundary()
class RightBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[0] > 2.5 - DOLFIN_EPS
class LowerBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] < DOLFIN_EPS
lower_boundary = LowerBoundary()
class UpperBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] > 0.4-DOLFIN_EPS
upper_boundary = UpperBoundary()
class ObstacleBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary \
and x[0] > DOLFIN_EPS and x[0] < 2.5-DOLFIN_EPS \
and x[1] > DOLFIN_EPS and x[1] < 0.4-DOLFIN_EPS
obstacle_boundary = ObstacleBoundary()
# Boundary conditions for the velocity.
# Proper inflow and outflow conditions are a matter of voodoo. See for
# example Gresho/Sani, or
#
# Boundary conditions for open boundaries
# for the incompressible Navier-Stokes equation;
# B.C.V. Johansson;
# J. Comp. Phys. 105, 233-251 (1993).
#
# The latter in particularly suggest for the inflow:
#
# u = u0,
# d^r v / dx^r = v_r,
# div(u) = 0,
#
# where u and v are the velocities in normal and tangential directions,
# respectively, and r\in{0,1,2}. The setting r=0 essentially means to set
# (u,v) statically at the left boundary, r=1 means to set u and control
# dv/dn, which is what we do here (namely implicitly by dv/dn=0).
# At the outflow,
#
# d^j u / dx^j = 0,
# d^q v / dx^q = 0,
# p = p0,
#
# is suggested with j=q+1. Choosing q=0, j=1 means setting the tangential
# component of the outflow to 0, and letting the normal component du/dn=0
# (again, this is achieved implicitly by the weak formulation).
#
inflow = Expression('100*x[1]*(0.4-x[1])', degree=2)
u_bcs = [DirichletBC(V, (0.0, 0.0), upper_boundary),
DirichletBC(V, (0.0, 0.0), lower_boundary),
DirichletBC(V, (0.0, 0.0), obstacle_boundary),
DirichletBC(V.sub(0), inflow, left_boundary),
#DirichletBC(V.sub(1), 0.0, right_boundary),
]
dudt_bcs = [DirichletBC(V, (0.0, 0.0), upper_boundary),
DirichletBC(V, (0.0, 0.0), lower_boundary),
DirichletBC(V, (0.0, 0.0), obstacle_boundary),
DirichletBC(V.sub(0), 0.0, left_boundary),
#DirichletBC(V.sub(1), 0.0, right_boundary),
]
# If there is a penetration boundary (i.e., n.u!=0), then the pressure must
# be set at the boundary to make sure that the Navier-Stokes problem
# remains consistent.
# It is not quite clear now where exactly to set the pressure to 0. Inlet,
# outlet, some other place? The PPE system is consistent in all cases.
# TODO find out more about it
#p_bcs = [DirichletBC(Q, 0.0, right_boundary)]
p_bcs = []
return mesh, V, Q, u_bcs, dudt_bcs, p_bcs
def peter():
base = '../meshes/2d/peter'
#base = '../meshes/2d/peter-fine'
mesh = Mesh(base + '.xml')
subdomains = MeshFunction('size_t', mesh, base+'_physical_region.xml')
workpiece_index = 3
subdomain_materials = {1: 'SiC',
2: 'carbon steel',
3: 'GaAs (liquid)'}
submesh_workpiece = SubMesh(mesh, subdomains, workpiece_index)
W = VectorFunctionSpace(submesh_workpiece, 'CG', 2)
Q = FunctionSpace(submesh_workpiece, 'CG', 2)
# Define melt boundaries.
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
# Be especially careful in the corners when defining the r=0 axis:
# For the PPE to be consistent, we need to have n.u=0 *everywhere*
# on the non-(r=0) boundaries.
return on_boundary \
and x[0] < GMSH_EPS \
and 0.005+GMSH_EPS < x[1] and x[1] < 0.050-GMSH_EPS
class SurfaceBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary \
and 0.055-GMSH_EPS < x[1] \
and 0.030-GMSH_EPS < x[0] and x[0] < 0.075+GMSH_EPS
class StampBoundary(SubDomain):
def inside(self, x, on_boundary):
# Well well. If we don't include the endpoints here, the PPE turns
# out inconsistent. This is weird since the endpoints should be
# picked up by RightBoundary and StampBoundary.
return on_boundary \
and 0.050-GMSH_EPS < x[1] and x[1] < 0.055+GMSH_EPS \
and x[0] < 0.030+GMSH_EPS
class LowerBoundary(SubDomain):
def inside(self, x, on_boundary):
# Danger, danger.
# If the rightmost point (0.075, 0.010) is excluded, the PPE will
# end up inconsistent.
# This is actually weird since it should be picked up by
# RightBoundary.
return on_boundary \
and (x[1] < 0.005+GMSH_EPS
or (x[0] > 0.070-GMSH_EPS and x[1] < 0.010+GMSH_EPS)
)
class RightBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary \
and x[0] > 0.075-GMSH_EPS
left_boundary = LeftBoundary()
lower_boundary = LowerBoundary()
right_boundary = RightBoundary()
surface_boundary = SurfaceBoundary()
stamp_boundary = StampBoundary()
# Define workpiece boundaries.
wp_boundaries = FacetFunction('size_t', submesh_workpiece)
wp_boundaries.set_all(0)
left_boundary.mark(wp_boundaries, 1)
lower_boundary.mark(wp_boundaries, 2)
right_boundary.mark(wp_boundaries, 3)
surface_boundary.mark(wp_boundaries, 4)
stamp_boundary.mark(wp_boundaries, 5)
# For local use only:
wp_boundary_indices = {'left': 1,
'lower': 2,
'right': 3,
'surface': 4,
'stamp': 5}
# Boundary conditions for the velocity.
u_bcs = [DirichletBC(W, (0.0, 0.0), stamp_boundary),
DirichletBC(W, (0.0, 0.0), right_boundary),
DirichletBC(W, (0.0, 0.0), lower_boundary),
DirichletBC(W.sub(0), 0.0, left_boundary),
DirichletBC(W.sub(1), 0.0, surface_boundary),
]
p_bcs = []
# Boundary conditions for the heat equation.
# Dirichlet
theta_bcs_d = []
# Neumann
theta_bcs_n = {}
# Robin, i.e.,
#
# -dtheta/dn = alpha (theta - theta0)
#
# (with alpha>0 to preserve coercivity of the scheme).
#
theta_bcs_r = {
wp_boundary_indices['stamp']: (100.0, 1500.0),
wp_boundary_indices['surface']: (100.0, 1550.0),
wp_boundary_indices['right']: (300.0, Expression('200*x[0] + 1600', degree=1)),
wp_boundary_indices['lower']: (300.0, Expression('-1200*x[1] + 1614', degree=1)),
}
return mesh, subdomains, subdomain_materials, workpiece_index, \
wp_boundaries, W, u_bcs, p_bcs, \
Q, theta_bcs_d, theta_bcs_n, theta_bcs_r
def crucible_without_coils():
base = '../meshes/2d/crucible'
mesh = Mesh(base + '.xml')
subdomains = MeshFunction('size_t', mesh, base+'_physical_region.xml')
workpiece_index = 4
subdomain_materials = {1: 'SiC',
2: 'boron trioxide',
3: 'GaAs (solid)',
4: 'GaAs (liquid)'}
submesh_workpiece = SubMesh(mesh, subdomains, workpiece_index)
V = VectorFunctionSpace(submesh_workpiece, 'CG', 2)
P = FunctionSpace(submesh_workpiece, 'CG', 1)
Q = FunctionSpace(mesh, 'CG', 2)
# Define mesh and boundaries.
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
# Be especially careful in the corners when defining the r=0 axis:
# For the PPE to be consistent, we need to have n.u=0 *everywhere*
# on the non-r=0 boundaries.
return on_boundary \
and x[0] < GMSH_EPS and 0.366 + GMSH_EPS < x[1] \
and x[1] < 0.411 - GMSH_EPS
left_boundary = LeftBoundary()
class SurfaceBoundary(SubDomain):
def inside(self, x, on_boundary):
# Make sure to catch the entire surface, so don't be too
# restrictive about x[0].
return on_boundary \
and x[1] > 0.38-GMSH_EPS \
and x[0] > 0.04-GMSH_EPS and x[0] < 0.07+GMSH_EPS
class OtherBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary \
and not left_boundary.inside(x, on_boundary) # \
#and not surface_boundary.inside(x, on_boundary)
other_boundary = OtherBoundary()
# Boundary conditions for the velocity.
u_bcs = [DirichletBC(V, (0.0, 0.0), other_boundary),
DirichletBC(V.sub(0), 0.0, left_boundary),
#DirichletBC(V.sub(1), 0.0, surface_boundary),
]
#u_bcs = [DirichletBC(V, (0.0, 0.0), 'on_boundary')]
p_bcs = []
class HeaterBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary \
and ((x[1] < 0.38 and GMSH_EPS < x[0]
and x[0] < 0.075+GMSH_EPS)
or x[0] < GMSH_EPS and x[1] < 0.365+GMSH_EPS)
heater_boundary = HeaterBoundary()
background_temp = 1400.0
# Heat with 1580K at r=0, 1590K at r=0.075.
heater_bcs = [(heater_boundary, '1580.0 + 133.0 * x[0]')]
return mesh, subdomains, subdomain_materials, workpiece_index, \
V, Q, P, u_bcs, p_bcs, background_temp, heater_bcs
