def test_assemble_derivatives():
"""This test checks the original_coefficient_positions, which may change
under differentiation (some coefficients and constants are
eliminated)"""
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12)
Q = FunctionSpace(mesh, ("Lagrange", 1))
u = Function(Q)
v = ufl.TestFunction(Q)
du = ufl.TrialFunction(Q)
b = Function(Q)
c1 = Constant(mesh, np.array([[1.0, 0.0], [3.0, 4.0]], PETSc.ScalarType))
c2 = Constant(mesh, PETSc.ScalarType(2.0))
b.x.array[:] = 2.0
# derivative eliminates 'u' and 'c1'
L = ufl.inner(c1, c1) * v * dx + c2 * b * inner(u, v) * dx
a = form(derivative(L, u, du))
A1 = assemble_matrix(a)
A1.assemble()
a = form(c2 * b * inner(du, v) * dx)
A2 = assemble_matrix(a)
A2.assemble()
assert (A1 - A2).norm() == pytest.approx(0.0, rel=1e-12, abs=1e-12)
def test_basic_assembly_constant(mode):
"""Tests assembly with Constant
The following test should be sensitive to order of flattening the
matrix-valued constant.
"""
mesh = create_unit_square(MPI.COMM_WORLD, 5, 5, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
c = Constant(mesh, np.array([[1.0, 2.0], [5.0, 3.0]], PETSc.ScalarType))
a = inner(c[1, 0] * u, v) * dx + inner(c[1, 0] * u, v) * ds
L = inner(c[1, 0], v) * dx + inner(c[1, 0], v) * ds
a, L = form(a), form(L)
# Initial assembly
A1 = assemble_matrix(a)
A1.assemble()
b1 = assemble_vector(L)
b1.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
c.value = [[1.0, 2.0], [3.0, 4.0]]
A2 = assemble_matrix(a)
A2.assemble()
b2 = assemble_vector(L)
b2.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
assert (A1 * 3.0 - A2 * 5.0).norm() == pytest.approx(0.0)
assert (b1 * 3.0 - b2 * 5.0).norm() == pytest.approx(0.0)
def J_mono(self, snes, x, J, P):
J.zeroEntries()
assemble_matrix(J, self.a, bcs=self.bcs, diagonal=1.0)
J.assemble()
if self.a_precon is not None:
P.zeroEntries()
assemble_matrix(P, self.a_precon, bcs=self.bcs, diagonal=1.0)
P.assemble()
def test_assembly_dx_domains(mode, meshtags_factory):
mesh = create_unit_square(MPI.COMM_WORLD, 10, 10, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
# Prepare a marking structures
# indices cover all cells
# values are [1, 2, 3, 3, ...]
cell_map = mesh.topology.index_map(mesh.topology.dim)
num_cells = cell_map.size_local + cell_map.num_ghosts
indices = np.arange(0, num_cells)
values = np.full(indices.shape, 3, dtype=np.intc)
values[0] = 1
values[1] = 2
marker = meshtags_factory(mesh, mesh.topology.dim, indices, values)
dx = ufl.Measure('dx', subdomain_data=marker, domain=mesh)
w = Function(V)
w.x.array[:] = 0.5
# Assemble matrix
a = form(w * ufl.inner(u, v) * (dx(1) + dx(2) + dx(3)))
A = assemble_matrix(a)
A.assemble()
a2 = form(w * ufl.inner(u, v) * dx)
A2 = assemble_matrix(a2)
A2.assemble()
assert (A - A2).norm() < 1.0e-12
bc = dirichletbc(Function(V), range(30))
# Assemble vector
L = form(ufl.inner(w, v) * (dx(1) + dx(2) + dx(3)))
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
L2 = form(ufl.inner(w, v) * dx)
b2 = assemble_vector(L2)
apply_lifting(b2, [a], [[bc]])
b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b2, [bc])
assert (b - b2).norm() < 1.0e-12
# Assemble scalar
L = form(w * (dx(1) + dx(2) + dx(3)))
s = assemble_scalar(L)
s = mesh.comm.allreduce(s, op=MPI.SUM)
assert s == pytest.approx(0.5, 1.0e-12)
L2 = form(w * dx)
s2 = assemble_scalar(L2)
s2 = mesh.comm.allreduce(s2, op=MPI.SUM)
assert s == pytest.approx(s2, 1.0e-12)
def test_refine_create_form():
"""Check that forms can be assembled on refined mesh"""
mesh = create_unit_cube(MPI.COMM_WORLD, 3, 3, 3)
mesh.topology.create_entities(1)
mesh = refine(mesh, redistribute=True)
V = FunctionSpace(mesh, ("Lagrange", 1))
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = form(ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx)
assemble_matrix(a)
def test_complex_assembly():
"""Test assembly of complex matrices and vectors"""
mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)
P2 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 2)
V = FunctionSpace(mesh, P2)
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
g = -2 + 3.0j
j = 1.0j
a_real = form(inner(u, v) * dx)
L1 = form(inner(g, v) * dx)
b = assemble_vector(L1)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
bnorm = b.norm(PETSc.NormType.N1)
b_norm_ref = abs(-2 + 3.0j)
assert bnorm == pytest.approx(b_norm_ref)
A = assemble_matrix(a_real)
A.assemble()
A0_norm = A.norm(PETSc.NormType.FROBENIUS)
x = ufl.SpatialCoordinate(mesh)
a_imag = form(j * inner(u, v) * dx)
f = 1j * ufl.sin(2 * np.pi * x[0])
L0 = form(inner(f, v) * dx)
A = assemble_matrix(a_imag)
A.assemble()
A1_norm = A.norm(PETSc.NormType.FROBENIUS)
assert A0_norm == pytest.approx(A1_norm)
b = assemble_vector(L0)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
b1_norm = b.norm(PETSc.NormType.N2)
a_complex = form((1 + j) * inner(u, v) * dx)
f = ufl.sin(2 * np.pi * x[0])
L2 = form(inner(f, v) * dx)
A = assemble_matrix(a_complex)
A.assemble()
A2_norm = A.norm(PETSc.NormType.FROBENIUS)
assert A1_norm == pytest.approx(A2_norm / np.sqrt(2))
b = assemble_vector(L2)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
b2_norm = b.norm(PETSc.NormType.N2)
assert b2_norm == pytest.approx(b1_norm)
def test_assemble_manifold():
"""Test assembly of poisson problem on a mesh with topological
dimension 1 but embedded in 2D (gdim=2)"""
points = np.array([[0.0, 0.0], [0.2, 0.0], [0.4, 0.0], [0.6, 0.0],
[0.8, 0.0], [1.0, 0.0]],
dtype=np.float64)
cells = np.array([[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]], dtype=np.int32)
cell = ufl.Cell("interval", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 1))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
assert mesh.geometry.dim == 2
assert mesh.topology.dim == 1
U = FunctionSpace(mesh, ("P", 1))
u, v = ufl.TrialFunction(U), ufl.TestFunction(U)
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx(mesh)
L = ufl.inner(1.0, v) * ufl.dx(mesh)
a, L = form(a), form(L)
bcdofs = locate_dofs_geometrical(U, lambda x: np.isclose(x[0], 0.0))
bcs = [dirichletbc(PETSc.ScalarType(0), bcdofs, U)]
A = assemble_matrix(a, bcs=bcs)
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[bcs])
set_bc(b, bcs)
assert np.isclose(b.norm(), 0.41231)
assert np.isclose(A.norm(), 25.0199)
def test_assembly_bcs(mode):
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = inner(u, v) * dx + inner(u, v) * ds
L = inner(1.0, v) * dx
a, L = form(a), form(L)
bdofsV = locate_dofs_geometrical(
V,
lambda x: np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0)))
bc = dirichletbc(PETSc.ScalarType(1), bdofsV, V)
# Assemble and apply 'global' lifting of bcs
A = assemble_matrix(a)
A.assemble()
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
g = b.duplicate()
with g.localForm() as g_local:
g_local.set(0.0)
set_bc(g, [bc])
f = b - A * g
set_bc(f, [bc])
# Assemble vector and apply lifting of bcs during assembly
b_bc = assemble_vector(L)
apply_lifting(b_bc, [a], [[bc]])
b_bc.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b_bc, [bc])
assert (f - b_bc).norm() == pytest.approx(0.0, rel=1e-12, abs=1e-12)
def test_vector_assemble_matrix_interior():
mesh = create_unit_square(MPI.COMM_WORLD, 3, 3)
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(ufl.inner(ufl.jump(u), ufl.jump(v)) * ufl.dS)
A = assemble_matrix(a)
A.assemble()
def test_nullspace_check(mesh, degree):
V = VectorFunctionSpace(mesh, ('Lagrange', degree))
u, v = TrialFunction(V), TestFunction(V)
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 = form(inner(sigma(u, mesh.geometry.dim), grad(v)) * dx)
# Assemble matrix and create compatible vector
A = assemble_matrix(a)
A.assemble()
# Create null space basis and test
nullspace = build_elastic_nullspace(V)
la.orthonormalize(nullspace)
ns = PETSc.NullSpace().create(vectors=nullspace)
assert ns.test(A)
# Create incorrect null space basis and test
nullspace = build_broken_elastic_nullspace(V)
la.orthonormalize(nullspace)
ns = PETSc.NullSpace().create(vectors=nullspace)
assert not ns.test(A)
def test_krylov_solver_lu():
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = TrialFunction(V), TestFunction(V)
a = form(inner(u, v) * dx)
L = form(inner(1.0, v) * dx)
A = assemble_matrix(a)
A.assemble()
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
norm = 13.0
solver = PETSc.KSP().create(mesh.comm)
solver.setOptionsPrefix("test_lu_")
opts = PETSc.Options("test_lu_")
opts["ksp_type"] = "preonly"
opts["pc_type"] = "lu"
solver.setFromOptions()
x = A.createVecRight()
solver.setOperators(A)
solver.solve(b, x)
# *Tight* tolerance for LU solves
assert x.norm(PETSc.NormType.N2) == pytest.approx(norm, abs=1.0e-12)
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 = create_unit_square(MPI.COMM_WORLD, N, N)
V = VectorFunctionSpace(mesh, 'Lagrange', 1)
u = TrialFunction(V)
v = TestFunction(V)
facetdim = mesh.topology.dim - 1
bndry_facets = locate_entities_boundary(
mesh, facetdim, lambda x: np.full(x.shape[1], True))
bdofs = locate_dofs_topological(V.sub(0), V, facetdim, bndry_facets)
bc = dirichletbc(PETSc.ScalarType(0), bdofs, V.sub(0))
# 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 = PETSc.KSP().create(mesh.comm)
solver.setType("cg")
# Set matrix operator
solver.setOperators(A)
# Compute solution and return number of iterations
return solver.solve(b, u.vector)
def run_scalar_test(mesh, V, degree):
""" Manufactured Poisson problem, solving u = x[1]**p, where p is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
# Get quadrature degree for bilinear form integrand (ignores effect of non-affine map)
a = inner(grad(u), grad(v)) * dx(metadata={"quadrature_degree": -1})
a.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(a)
a = form(a)
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
f = -div(grad(u_exact))
# Set quadrature degree for linear form integrand (ignores effect of non-affine map)
L = inner(f, v) * dx(metadata={"quadrature_degree": -1})
L.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(L)
L = form(L)
u_bc = Function(V)
u_bc.interpolate(lambda x: x[1]**degree)
# Create Dirichlet boundary condition
facetdim = mesh.topology.dim - 1
mesh.topology.create_connectivity(facetdim, mesh.topology.dim)
bndry_facets = np.where(
np.array(compute_boundary_facets(mesh.topology)) == 1)[0]
bdofs = locate_dofs_topological(V, facetdim, bndry_facets)
bc = dirichletbc(u_bc, bdofs)
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
a = form(a)
A = assemble_matrix(a, bcs=[bc])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
uh = Function(V)
solver.solve(b, uh.vector)
uh.x.scatter_forward()
M = (u_exact - uh)**2 * dx
M = form(M)
error = mesh.comm.allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def test_basic_assembly(mode):
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
f = Function(V)
f.x.array[:] = 10.0
a = inner(f * u, v) * dx + inner(u, v) * ds
L = inner(f, v) * dx + inner(2.0, v) * ds
a, L = form(a), form(L)
# Initial assembly
A = assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
assert isinstance(b, PETSc.Vec)
# Second assembly
normA = A.norm()
A.zeroEntries()
A = assemble_matrix(A, a)
A.assemble()
assert isinstance(A, PETSc.Mat)
assert normA == pytest.approx(A.norm())
normb = b.norm()
with b.localForm() as b_local:
b_local.set(0.0)
b = assemble_vector(b, L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
assert isinstance(b, PETSc.Vec)
assert normb == pytest.approx(b.norm())
# Vector re-assembly - no zeroing (but need to zero ghost entries)
with b.localForm() as b_local:
b_local.array[b.local_size:] = 0.0
assemble_vector(b, L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
assert 2.0 * normb == pytest.approx(b.norm())
# Matrix re-assembly (no zeroing)
assemble_matrix(A, a)
A.assemble()
assert 2.0 * normA == pytest.approx(A.norm())
def assemble_div_matrix(k, offset):
mesh = create_quad_mesh(offset)
V = FunctionSpace(mesh, ("DQ", k))
W = FunctionSpace(mesh, ("RTCF", k + 1))
u, w = ufl.TrialFunction(V), ufl.TestFunction(W)
a = form(ufl.inner(u, ufl.div(w)) * ufl.dx)
A = assemble_matrix(a)
A.assemble()
return A[:, :]
def test_overlapping_bcs():
"""Test that, when boundaries condition overlap, the last provided
boundary condition is applied"""
n = 23
mesh = create_unit_square(MPI.COMM_WORLD, n, n)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(u, v) * dx)
L = form(inner(1, v) * dx)
dofs_left = locate_dofs_geometrical(V, lambda x: x[0] < 1.0 / (2.0 * n))
dofs_top = locate_dofs_geometrical(V, lambda x: x[1] > 1.0 - 1.0 /
(2.0 * n))
dof_corner = np.array(list(set(dofs_left).intersection(set(dofs_top))),
dtype=np.int64)
# Check only one dof pair is found globally
assert len(set(np.concatenate(MPI.COMM_WORLD.allgather(dof_corner)))) == 1
bcs = [
dirichletbc(PETSc.ScalarType(0), dofs_left, V),
dirichletbc(PETSc.ScalarType(123.456), dofs_top, V)
]
A, b = create_matrix(a), create_vector(L)
assemble_matrix(A, a, bcs=bcs)
A.assemble()
# Check the diagonal (only on the rank that owns the row)
d = A.getDiagonal()
if len(dof_corner) > 0 and dof_corner[0] < V.dofmap.index_map.size_local:
assert np.isclose(d.array_r[dof_corner[0]], 1.0)
with b.localForm() as b_loc:
b_loc.set(0)
assemble_vector(b, L)
apply_lifting(b, [a], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
b.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
if len(dof_corner) > 0:
with b.localForm() as b_loc:
assert b_loc[dof_corner[0]] == 123.456
def test_assemble_empty_rank_mesh():
"""Assembly on mesh where some ranks are empty"""
comm = MPI.COMM_WORLD
cell_type = CellType.triangle
domain = ufl.Mesh(
ufl.VectorElement("Lagrange", ufl.Cell(cell_type.name), 1))
def partitioner(comm, nparts, local_graph, num_ghost_nodes, ghosting):
"""Leave cells on the curent rank"""
dest = np.full(len(cells), comm.rank, dtype=np.int32)
return graph.create_adjacencylist(dest)
if comm.rank == 0:
# Put cells on rank 0
cells = np.array([[0, 1, 2], [0, 2, 3]], dtype=np.int64)
cells = graph.create_adjacencylist(cells)
x = np.array([[0., 0.], [1., 0.], [1., 1.], [0., 1.]])
else:
# No cells onm other ranks
cells = graph.create_adjacencylist(np.empty((0, 3), dtype=np.int64))
x = np.empty((0, 2), dtype=np.float64)
mesh = create_mesh(comm, cells, x, domain, GhostMode.none, partitioner)
V = FunctionSpace(mesh, ("Lagrange", 2))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
f, k, zero = Function(V), Function(V), Function(V)
f.x.array[:] = 10.0
k.x.array[:] = 1.0
zero.x.array[:] = 0.0
a = form(inner(k * u, v) * dx + inner(zero * u, v) * ds)
L = form(inner(f, v) * dx + inner(zero, v) * ds)
M = form(2 * k * dx + k * ds)
sum = comm.allreduce(assemble_scalar(M), op=MPI.SUM)
assert sum == pytest.approx(6.0)
# Assemble
A = assemble_matrix(a)
A.assemble()
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
# Solve
ksp = PETSc.KSP()
ksp.create(mesh.comm)
ksp.setOperators(A)
ksp.setTolerances(rtol=1.0e-9, max_it=50)
ksp.setFromOptions()
x = b.copy()
ksp.solve(b, x)
assert np.allclose(x.array, 10.0)
def test_plus_minus_matrix(cell_type, pm1, pm2):
"""Test that ('+') and ('-') match up with the correct DOFs for DG functions"""
results = []
spaces = []
orders = []
for count in range(3):
for agree in [True, False]:
# Two cell mesh with randomly numbered points
mesh, order = two_unit_cells(cell_type, agree, return_order=True)
V = FunctionSpace(mesh, ("DG", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
# Assemble matrices with combinations of + and - for a few
# different numberings
a = form(ufl.inner(u(pm1), v(pm2)) * ufl.dS)
result = assemble_matrix(a, [])
result.assemble()
spaces.append(V)
results.append(result)
orders.append(order)
# Check that the above matrices all have the same values, but
# permuted due to differently ordered dofs
dofmap0 = spaces[0].mesh.geometry.dofmap
for result, space in zip(results[1:], spaces[1:]):
# Get the data relating to two results
dofmap1 = space.mesh.geometry.dofmap
dof_order = []
# For each cell
for cell in range(2):
# For each point in cell 0 in the first mesh
for dof0, point0 in zip(spaces[0].dofmap.cell_dofs(cell),
dofmap0.links(cell)):
# Find the point in the cell 0 in the second mesh
for dof1, point1 in zip(space.dofmap.cell_dofs(cell),
dofmap1.links(cell)):
if np.allclose(spaces[0].mesh.geometry.x[point0],
space.mesh.geometry.x[point1]):
break
else:
# If no matching point found, fail
assert False
dof_order.append((dof0, dof1))
# For all dof pairs, check that entries in the matrix agree
for a, b in dof_order:
for c, d in dof_order:
assert np.isclose(results[0][a, c], result[b, d])
def test_custom_mesh_loop_cffi_rank2(set_vals):
"""Test numba assembler for bilinear form"""
mesh = create_unit_square(MPI.COMM_WORLD, 64, 64)
V = FunctionSpace(mesh, ("Lagrange", 1))
# Test against generated code and general assembler
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(u, v) * dx)
A0 = assemble_matrix(a)
A0.assemble()
A0.zeroEntries()
start = time.time()
assemble_matrix(A0, a)
end = time.time()
print("Time (C++, pass 2):", end - start)
A0.assemble()
# Unpack mesh and dofmap data
num_owned_cells = mesh.topology.index_map(mesh.topology.dim).size_local
num_cells = num_owned_cells + mesh.topology.index_map(
mesh.topology.dim).num_ghosts
x_dofs = mesh.geometry.dofmap.array.reshape(num_cells, 3)
x = mesh.geometry.x
dofmap = V.dofmap.list.array.reshape(num_cells,
3).astype(np.dtype(PETSc.IntType))
A1 = A0.copy()
for i in range(2):
A1.zeroEntries()
start = time.time()
assemble_matrix_cffi(A1.handle, (x_dofs, x), dofmap, num_owned_cells,
set_vals, PETSc.InsertMode.ADD_VALUES)
end = time.time()
print("Time (Numba, pass {}): {}".format(i, end - start))
A1.assemble()
assert (A1 - A0).norm() == pytest.approx(0.0)
def test_complex_assembly_solve():
"""Solve a positive definite helmholtz problem and verify solution
with the method of manufactured solutions"""
degree = 3
mesh = create_unit_square(MPI.COMM_WORLD, 20, 20)
P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), degree)
V = FunctionSpace(mesh, P)
x = ufl.SpatialCoordinate(mesh)
# Define source term
A = 1.0 + 2.0 * (2.0 * np.pi)**2
f = (1. + 1j) * A * ufl.cos(2 * np.pi * x[0]) * ufl.cos(2 * np.pi * x[1])
# Variational problem
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
C = 1.0 + 1.0j
a = form(C * inner(grad(u), grad(v)) * dx + C * inner(u, v) * dx)
L = form(inner(f, v) * dx)
# Assemble
A = assemble_matrix(a)
A.assemble()
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
# Create solver
solver = PETSc.KSP().create(mesh.comm)
solver.setOptionsPrefix("test_lu_")
opts = PETSc.Options("test_lu_")
opts["ksp_type"] = "preonly"
opts["pc_type"] = "lu"
solver.setFromOptions()
x = A.createVecRight()
solver.setOperators(A)
solver.solve(b, x)
# Reference Solution
def ref_eval(x):
return np.cos(2 * np.pi * x[0]) * np.cos(2 * np.pi * x[1])
u_ref = Function(V)
u_ref.interpolate(ref_eval)
diff = (x - u_ref.vector).norm(PETSc.NormType.N2)
assert diff == pytest.approx(0.0, abs=1e-1)
def test_custom_mesh_loop_ctypes_rank2():
"""Test numba assembler for bilinear form"""
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 64, 64)
V = FunctionSpace(mesh, ("Lagrange", 1))
# Extract mesh and dofmap data
num_owned_cells = mesh.topology.index_map(mesh.topology.dim).size_local
num_cells = num_owned_cells + mesh.topology.index_map(
mesh.topology.dim).num_ghosts
x_dofs = mesh.geometry.dofmap.array.reshape(num_cells, 3)
x = mesh.geometry.x
dofmap = V.dofmap.list.array.reshape(num_cells,
3).astype(np.dtype(PETSc.IntType))
# Generated case with general assembler
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(u, v) * dx)
A0 = assemble_matrix(a)
A0.assemble()
A0.zeroEntries()
start = time.time()
dolfinx.fem.petsc.assemble_matrix(A0, a)
end = time.time()
print("Time (C++, pass 2):", end - start)
A0.assemble()
# Custom case
A1 = A0.copy()
for i in range(2):
A1.zeroEntries()
mat = A1.handle
start = time.time()
assemble_matrix_ctypes(mat, (x_dofs, x), dofmap, num_owned_cells,
MatSetValues_ctypes,
PETSc.InsertMode.ADD_VALUES)
end = time.time()
print("Time (numba, pass {}): {}".format(i, end - start))
A1.assemble()
assert (A0 - A1).norm() == pytest.approx(0.0, abs=1.0e-9)
def test_basic_interior_facet_assembly():
mesh = create_rectangle(
MPI.COMM_WORLD,
[np.array([0.0, 0.0]), np.array([1.0, 1.0])], [5, 5],
cell_type=CellType.triangle,
ghost_mode=GhostMode.shared_facet)
V = FunctionSpace(mesh, ("DG", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = ufl.inner(ufl.avg(u), ufl.avg(v)) * ufl.dS
a = form(a)
A = assemble_matrix(a)
A.assemble()
assert isinstance(A, PETSc.Mat)
L = ufl.conj(ufl.avg(v)) * ufl.dS
L = form(L)
b = assemble_vector(L)
b.assemble()
assert isinstance(b, PETSc.Vec)
def run_vector_test(mesh, V, degree):
"""Projection into H(div/curl) spaces"""
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(u, v) * dx)
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[0]**degree
L = form(inner(u_exact, v[0]) * dx)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a)
A.assemble()
# Create LU linear solver (Note: need to use a solver that
# re-orders to handle pivots, e.g. not the PETSc built-in LU solver)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType("preonly")
solver.getPC().setType('lu')
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.x.scatter_forward()
# Calculate error
M = (u_exact - uh[0])**2 * dx
for i in range(1, mesh.topology.dim):
M += uh[i]**2 * dx
M = form(M)
error = mesh.comm.allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def test_matrix_assembly_block_nl():
"""Test assembly of block matrices and vectors into (a) monolithic
blocked structures, PETSc Nest structures, and monolithic structures
in the nonlinear setting
"""
mesh = create_unit_square(MPI.COMM_WORLD, 4, 8)
p0, p1 = 1, 2
P0 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), p0)
P1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), p1)
V0 = FunctionSpace(mesh, P0)
V1 = FunctionSpace(mesh, P1)
def initial_guess_u(x):
return np.sin(x[0]) * np.sin(x[1])
def initial_guess_p(x):
return -x[0]**2 - x[1]**3
def bc_value(x):
return np.cos(x[0]) * np.cos(x[1])
facetdim = mesh.topology.dim - 1
bndry_facets = locate_entities_boundary(
mesh, facetdim,
lambda x: np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0)))
u_bc = Function(V1)
u_bc.interpolate(bc_value)
bdofs = locate_dofs_topological(V1, facetdim, bndry_facets)
bc = dirichletbc(u_bc, bdofs)
# Define variational problem
du, dp = ufl.TrialFunction(V0), ufl.TrialFunction(V1)
u, p = Function(V0), Function(V1)
v, q = ufl.TestFunction(V0), ufl.TestFunction(V1)
u.interpolate(initial_guess_u)
p.interpolate(initial_guess_p)
f = 1.0
g = -3.0
F0 = inner(u, v) * dx + inner(p, v) * dx - inner(f, v) * dx
F1 = inner(u, q) * dx + inner(p, q) * dx - inner(g, q) * dx
a_block = form([[derivative(F0, u, du),
derivative(F0, p, dp)],
[derivative(F1, u, du),
derivative(F1, p, dp)]])
L_block = form([F0, F1])
# Monolithic blocked
x0 = create_vector_block(L_block)
scatter_local_vectors(x0, [u.vector.array_r, p.vector.array_r],
[(u.function_space.dofmap.index_map,
u.function_space.dofmap.index_map_bs),
(p.function_space.dofmap.index_map,
p.function_space.dofmap.index_map_bs)])
x0.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Ghosts are updated inside assemble_vector_block
A0 = assemble_matrix_block(a_block, bcs=[bc])
b0 = assemble_vector_block(L_block, a_block, bcs=[bc], x0=x0, scale=-1.0)
A0.assemble()
assert A0.getType() != "nest"
Anorm0 = A0.norm()
bnorm0 = b0.norm()
# Nested (MatNest)
x1 = create_vector_nest(L_block)
for x1_soln_pair in zip(x1.getNestSubVecs(), (u, p)):
x1_sub, soln_sub = x1_soln_pair
soln_sub.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
soln_sub.vector.copy(result=x1_sub)
x1_sub.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
A1 = assemble_matrix_nest(a_block, bcs=[bc])
b1 = assemble_vector_nest(L_block)
apply_lifting_nest(b1, a_block, bcs=[bc], x0=x1, scale=-1.0)
for b_sub in b1.getNestSubVecs():
b_sub.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
bcs0 = bcs_by_block([L.function_spaces[0] for L in L_block], [bc])
set_bc_nest(b1, bcs0, x1, scale=-1.0)
A1.assemble()
assert A1.getType() == "nest"
assert nest_matrix_norm(A1) == pytest.approx(Anorm0, 1.0e-12)
assert b1.norm() == pytest.approx(bnorm0, 1.0e-12)
# Monolithic version
E = P0 * P1
W = FunctionSpace(mesh, E)
dU = ufl.TrialFunction(W)
U = Function(W)
u0, u1 = ufl.split(U)
v0, v1 = ufl.TestFunctions(W)
U.sub(0).interpolate(initial_guess_u)
U.sub(1).interpolate(initial_guess_p)
F = inner(u0, v0) * dx + inner(u1, v0) * dx + inner(u0, v1) * dx + inner(u1, v1) * dx \
- inner(f, v0) * ufl.dx - inner(g, v1) * dx
J = derivative(F, U, dU)
F, J = form(F), form(J)
bdofsW_V1 = locate_dofs_topological((W.sub(1), V1), facetdim, bndry_facets)
bc = dirichletbc(u_bc, bdofsW_V1, W.sub(1))
A2 = assemble_matrix(J, bcs=[bc])
A2.assemble()
b2 = assemble_vector(F)
apply_lifting(b2, [J], bcs=[[bc]], x0=[U.vector], scale=-1.0)
b2.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b2, [bc], x0=U.vector, scale=-1.0)
assert A2.getType() != "nest"
assert A2.norm() == pytest.approx(Anorm0, 1.0e-12)
assert b2.norm() == pytest.approx(bnorm0, 1.0e-12)
def J(self, x, A):
"""Assemble Jacobian matrix."""
A.zeroEntries()
assemble_matrix(A, self.a, bcs=[self.bc])
A.assemble()
def test_eigen_assembly(tempdir): # noqa: F811
"""Compare assembly into scipy.CSR matrix with PETSc assembly"""
def compile_eigen_csr_assembler_module():
dolfinx_pc = dolfinx.pkgconfig.parse("dolfinx")
eigen_dir = dolfinx.pkgconfig.parse("eigen3")["include_dirs"]
cpp_code_header = f"""
<%
setup_pybind11(cfg)
cfg['include_dirs'] = {dolfinx_pc["include_dirs"] + [petsc4py.get_include()]
+ [pybind11.get_include()] + [str(pybind_inc())] + eigen_dir}
cfg['compiler_args'] = ["-std=c++17", "-Wno-comment"]
cfg['libraries'] = {dolfinx_pc["libraries"]}
cfg['library_dirs'] = {dolfinx_pc["library_dirs"]}
%>
"""
cpp_code = """
#include <pybind11/pybind11.h>
#include <pybind11/eigen.h>
#include <pybind11/stl.h>
#include <vector>
#include <Eigen/Sparse>
#include <petscsys.h>
#include <dolfinx/fem/assembler.h>
#include <dolfinx/fem/DirichletBC.h>
#include <dolfinx/fem/Form.h>
template<typename T>
Eigen::SparseMatrix<T, Eigen::RowMajor>
assemble_csr(const dolfinx::fem::Form<T>& a,
const std::vector<std::shared_ptr<const dolfinx::fem::DirichletBC<T>>>& bcs)
{
std::vector<Eigen::Triplet<T>> triplets;
auto mat_add
= [&triplets](const xtl::span<const std::int32_t>& rows,
const xtl::span<const std::int32_t>& cols,
const xtl::span<const T>& v)
{
for (std::size_t i = 0; i < rows.size(); ++i)
for (std::size_t j = 0; j < cols.size(); ++j)
triplets.emplace_back(rows[i], cols[j], v[i * cols.size() + j]);
return 0;
};
dolfinx::fem::assemble_matrix(mat_add, a, bcs);
auto map0 = a.function_spaces().at(0)->dofmap()->index_map;
int bs0 = a.function_spaces().at(0)->dofmap()->index_map_bs();
auto map1 = a.function_spaces().at(1)->dofmap()->index_map;
int bs1 = a.function_spaces().at(1)->dofmap()->index_map_bs();
Eigen::SparseMatrix<T, Eigen::RowMajor> mat(
bs0 * (map0->size_local() + map0->num_ghosts()),
bs1 * (map1->size_local() + map1->num_ghosts()));
mat.setFromTriplets(triplets.begin(), triplets.end());
return mat;
}
PYBIND11_MODULE(eigen_csr, m)
{
m.def("assemble_matrix", &assemble_csr<PetscScalar>);
}
"""
path = pathlib.Path(tempdir)
open(pathlib.Path(tempdir, "eigen_csr.cpp"),
"w").write(cpp_code + cpp_code_header)
rel_path = path.relative_to(pathlib.Path(__file__).parent)
p = str(rel_path).replace("/", ".") + ".eigen_csr"
return cppimport.imp(p)
def assemble_csr_matrix(a, bcs):
"""Assemble bilinear form into an SciPy CSR matrix, in serial."""
module = compile_eigen_csr_assembler_module()
A = module.assemble_matrix(a, bcs)
if a.function_spaces[0] is a.function_spaces[1]:
for bc in bcs:
if a.function_spaces[0].contains(bc.function_space):
bc_dofs, _ = bc.dof_indices()
# See https://github.com/numpy/numpy/issues/14132
# for why we copy bc_dofs as a work-around
dofs = bc_dofs.copy()
A[dofs, dofs] = 1.0
return A
mesh = create_unit_square(MPI.COMM_SELF, 12, 12)
Q = FunctionSpace(mesh, ("Lagrange", 1))
u = ufl.TrialFunction(Q)
v = ufl.TestFunction(Q)
a = form(ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx)
bdofsQ = locate_dofs_geometrical(
Q,
lambda x: np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0)))
bc = dirichletbc(PETSc.ScalarType(1), bdofsQ, Q)
A1 = assemble_matrix(a, [bc])
A1.assemble()
A2 = assemble_csr_matrix(a, [bc])
assert np.isclose(A1.norm(), scipy.sparse.linalg.norm(A2))
def J(self, snes, x, J, P):
"""Assemble Jacobian matrix."""
J.zeroEntries()
assemble_matrix(J, self.a, bcs=[self.bc])
J.assemble()
def run_dg_test(mesh, V, degree):
"""Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
# Exact solution
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
# Coefficient
k = Function(V)
k.x.array[:] = 2.0
# Source term
f = -div(k * grad(u_exact))
# Mesh normals and element size
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2.0
# Penalty parameter
alpha = 32
dx_ = dx(metadata={"quadrature_degree": -1})
ds_ = ds(metadata={"quadrature_degree": -1})
dS_ = dS(metadata={"quadrature_degree": -1})
a = inner(k * grad(u), grad(v)) * dx_ \
- k("+") * inner(avg(grad(u)), jump(v, n)) * dS_ \
- k("+") * inner(jump(u, n), avg(grad(v))) * dS_ \
+ k("+") * (alpha / h_avg) * inner(jump(u, n), jump(v, n)) * dS_ \
- inner(k * grad(u), v * n) * ds_ \
- inner(u * n, k * grad(v)) * ds_ \
+ (alpha / h) * inner(k * u, v) * ds_
L = inner(f, v) * dx_ - inner(k * u_exact * n, grad(v)) * ds_ \
+ (alpha / h) * inner(k * u_exact, v) * ds_
for integral in a.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
a)
for integral in L.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
L)
a, L = form(a), form(L)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a, [])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.x.scatter_forward()
# Calculate error
M = (u_exact - uh)**2 * dx
M = form(M)
error = mesh.comm.allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def test_biharmonic():
"""Manufactured biharmonic problem.
Solved using rotated Regge mixed finite element method. This is equivalent
to the Hellan-Herrmann-Johnson (HHJ) finite element method in
two-dimensions."""
mesh = create_rectangle(
MPI.COMM_WORLD,
[np.array([0.0, 0.0]), np.array([1.0, 1.0])], [32, 32],
CellType.triangle)
element = ufl.MixedElement([
ufl.FiniteElement("Regge", ufl.triangle, 1),
ufl.FiniteElement("Lagrange", ufl.triangle, 2)
])
V = FunctionSpace(mesh, element)
sigma, u = ufl.TrialFunctions(V)
tau, v = ufl.TestFunctions(V)
x = ufl.SpatialCoordinate(mesh)
u_exact = ufl.sin(ufl.pi * x[0]) * ufl.sin(ufl.pi * x[0]) * ufl.sin(
ufl.pi * x[1]) * ufl.sin(ufl.pi * x[1])
f_exact = div(grad(div(grad(u_exact))))
sigma_exact = grad(grad(u_exact))
# sigma and tau are tangential-tangential continuous according to the
# H(curl curl) continuity of the Regge space. However, for the biharmonic
# problem we require normal-normal continuity H (div div). Theorem 4.2 of
# Lizao Li's PhD thesis shows that the latter space can be constructed by
# the former through the action of the operator S:
def S(tau):
return tau - ufl.Identity(2) * ufl.tr(tau)
sigma_S = S(sigma)
tau_S = S(tau)
# Discrete duality inner product eq. 4.5 Lizao Li's PhD thesis
def b(tau_S, v):
n = FacetNormal(mesh)
return inner(tau_S, grad(grad(v))) * dx \
- ufl.dot(ufl.dot(tau_S('+'), n('+')), n('+')) * jump(grad(v), n) * dS \
- ufl.dot(ufl.dot(tau_S, n), n) * ufl.dot(grad(v), n) * ds
# Non-symmetric formulation
a = form(inner(sigma_S, tau_S) * dx - b(tau_S, u) + b(sigma_S, v))
L = form(inner(f_exact, v) * dx)
V_1 = V.sub(1).collapse()[0]
zero_u = Function(V_1)
zero_u.x.array[:] = 0.0
# Strong (Dirichlet) boundary condition
boundary_facets = locate_entities_boundary(
mesh, mesh.topology.dim - 1,
lambda x: np.full(x.shape[1], True, dtype=bool))
boundary_dofs = locate_dofs_topological(
(V.sub(1), V_1), mesh.topology.dim - 1, boundary_facets)
bcs = [dirichletbc(zero_u, boundary_dofs, V.sub(1))]
A = assemble_matrix(a, bcs=bcs)
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
# Solve
solver = PETSc.KSP().create(MPI.COMM_WORLD)
PETSc.Options()["ksp_type"] = "preonly"
PETSc.Options()["pc_type"] = "lu"
# PETSc.Options()["pc_factor_mat_solver_type"] = "mumps"
solver.setFromOptions()
solver.setOperators(A)
x_h = Function(V)
solver.solve(b, x_h.vector)
x_h.x.scatter_forward()
# Recall that x_h has flattened indices.
u_error_numerator = np.sqrt(
mesh.comm.allreduce(assemble_scalar(
form(
inner(u_exact - x_h[4], u_exact - x_h[4]) *
dx(mesh, metadata={"quadrature_degree": 5}))),
op=MPI.SUM))
u_error_denominator = np.sqrt(
mesh.comm.allreduce(assemble_scalar(
form(
inner(u_exact, u_exact) *
dx(mesh, metadata={"quadrature_degree": 5}))),
op=MPI.SUM))
assert np.absolute(u_error_numerator / u_error_denominator) < 0.05
# Reconstruct tensor from flattened indices.
# Apply inverse transform. In 2D we have S^{-1} = S.
sigma_h = S(ufl.as_tensor([[x_h[0], x_h[1]], [x_h[2], x_h[3]]]))
sigma_error_numerator = np.sqrt(
mesh.comm.allreduce(assemble_scalar(
form(
inner(sigma_exact - sigma_h, sigma_exact - sigma_h) *
dx(mesh, metadata={"quadrature_degree": 5}))),
op=MPI.SUM))
sigma_error_denominator = np.sqrt(
mesh.comm.allreduce(assemble_scalar(
form(
inner(sigma_exact, sigma_exact) *
dx(mesh, metadata={"quadrature_degree": 5}))),
op=MPI.SUM))
assert np.absolute(sigma_error_numerator / sigma_error_denominator) < 0.005
def test_curl_curl_eigenvalue(family, order):
"""curl curl eigenvalue problem.
Solved using H(curl)-conforming finite element method.
See https://www-users.cse.umn.edu/~arnold/papers/icm2002.pdf for details.
"""
slepc4py = pytest.importorskip("slepc4py") # noqa: F841
from slepc4py import SLEPc
mesh = create_rectangle(
MPI.COMM_WORLD,
[np.array([0.0, 0.0]), np.array([np.pi, np.pi])], [24, 24],
CellType.triangle)
element = ufl.FiniteElement(family, ufl.triangle, order)
V = FunctionSpace(mesh, element)
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = inner(ufl.curl(u), ufl.curl(v)) * dx
b = inner(u, v) * dx
boundary_facets = locate_entities_boundary(
mesh, mesh.topology.dim - 1,
lambda x: np.full(x.shape[1], True, dtype=bool))
boundary_dofs = locate_dofs_topological(V, mesh.topology.dim - 1,
boundary_facets)
zero_u = Function(V)
zero_u.x.array[:] = 0.0
bcs = [dirichletbc(zero_u, boundary_dofs)]
a, b = form(a), form(b)
A = assemble_matrix(a, bcs=bcs)
A.assemble()
B = assemble_matrix(b, bcs=bcs, diagonal=0.01)
B.assemble()
eps = SLEPc.EPS().create()
eps.setOperators(A, B)
PETSc.Options()["eps_type"] = "krylovschur"
PETSc.Options()["eps_gen_hermitian"] = ""
PETSc.Options()["eps_target_magnitude"] = ""
PETSc.Options()["eps_target"] = 5.0
PETSc.Options()["eps_view"] = ""
PETSc.Options()["eps_nev"] = 12
eps.setFromOptions()
eps.solve()
num_converged = eps.getConverged()
eigenvalues_unsorted = np.zeros(num_converged, dtype=np.complex128)
for i in range(0, num_converged):
eigenvalues_unsorted[i] = eps.getEigenvalue(i)
assert np.isclose(np.imag(eigenvalues_unsorted), 0.0).all()
eigenvalues_sorted = np.sort(np.real(eigenvalues_unsorted))[:-1]
eigenvalues_sorted = eigenvalues_sorted[np.logical_not(
eigenvalues_sorted < 1E-8)]
eigenvalues_exact = np.array([1.0, 1.0, 2.0, 4.0, 4.0, 5.0, 5.0, 8.0, 9.0])
assert np.isclose(eigenvalues_sorted[0:eigenvalues_exact.shape[0]],
eigenvalues_exact,
rtol=1E-2).all()
