u, v = ufl.TrialFunction(U), ufl.TestFunction(U)
def free_end(x):
"""Marks the leftmost points of the cantilever"""
return numpy.isclose(x[0], 48.0)
def left(x):
"""Marks left part of boundary, where cantilever is attached to wall"""
return numpy.isclose(x[0], 0.0)
# Locate all facets at the free end and assign them value 1
free_end_facets = locate_entities_geometrical(mesh,
1,
free_end,
boundary_only=True)
mt = dolfinx.mesh.MeshTags(mesh, 1, free_end_facets, 1)
ds = ufl.Measure("ds", subdomain_data=mt)
# Homogeneous boundary condition in displacement
u_bc = dolfinx.Function(U)
with u_bc.vector.localForm() as loc:
loc.set(0.0)
# Displacement BC is applied to the left side
left_facets = locate_entities_geometrical(mesh, 1, left, boundary_only=True)
bdofs = locate_dofs_topological(U, 1, left_facets)
bc = dolfinx.fem.DirichletBC(u_bc, bdofs)
def test_assembly_ds_domains(mesh):
V = dolfinx.FunctionSpace(mesh, ("CG", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
def bottom(x):
return numpy.isclose(x[1], 0.0)
def top(x):
return numpy.isclose(x[1], 1.0)
def left(x):
return numpy.isclose(x[0], 0.0)
def right(x):
return numpy.isclose(x[0], 1.0)
bottom_facets = locate_entities_geometrical(mesh,
mesh.topology.dim - 1,
bottom,
boundary_only=True)
bottom_vals = numpy.full(bottom_facets.shape, 1, numpy.intc)
top_facets = locate_entities_geometrical(mesh,
mesh.topology.dim - 1,
top,
boundary_only=True)
top_vals = numpy.full(top_facets.shape, 2, numpy.intc)
left_facets = locate_entities_geometrical(mesh,
mesh.topology.dim - 1,
left,
boundary_only=True)
left_vals = numpy.full(left_facets.shape, 3, numpy.intc)
right_facets = locate_entities_geometrical(mesh,
mesh.topology.dim - 1,
right,
boundary_only=True)
right_vals = numpy.full(right_facets.shape, 6, numpy.intc)
indices = numpy.hstack(
(bottom_facets, top_facets, left_facets, right_facets))
values = numpy.hstack((bottom_vals, top_vals, left_vals, right_vals))
indices, pos = numpy.unique(indices, return_index=True)
marker = dolfinx.mesh.MeshTags(mesh, mesh.topology.dim - 1, indices,
values[pos])
ds = ufl.Measure('ds', subdomain_data=marker, domain=mesh)
w = dolfinx.Function(V)
with w.vector.localForm() as w_local:
w_local.set(0.5)
# Assemble matrix
a = w * ufl.inner(u, v) * (ds(1) + ds(2) + ds(3) + ds(6))
A = dolfinx.fem.assemble_matrix(a)
A.assemble()
norm1 = A.norm()
a2 = w * ufl.inner(u, v) * ds
A2 = dolfinx.fem.assemble_matrix(a2)
A2.assemble()
norm2 = A2.norm()
assert norm1 == pytest.approx(norm2, 1.0e-12)
# Assemble vector
L = ufl.inner(w, v) * (ds(1) + ds(2) + ds(3) + ds(6))
b = dolfinx.fem.assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
L2 = ufl.inner(w, v) * ds
b2 = dolfinx.fem.assemble_vector(L2)
b2.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
assert b.norm() == pytest.approx(b2.norm(), 1.0e-12)
# Assemble scalar
L = w * (ds(1) + ds(2) + ds(3) + ds(6))
s = dolfinx.fem.assemble_scalar(L)
s = mesh.mpi_comm().allreduce(s, op=MPI.SUM)
L2 = w * ds
s2 = dolfinx.fem.assemble_scalar(L2)
s2 = mesh.mpi_comm().allreduce(s2, op=MPI.SUM)
assert (s == pytest.approx(s2, 1.0e-12)
and 2.0 == pytest.approx(s, 1.0e-12))
# and the part of the boundary on which the condition applies.
# This boundary part is identified with degrees of
# freedom in the function space to which we apply the boundary conditions.
# A method ``locate_dofs_geometrical`` is provided to extract the boundary
# degrees of freedom using a geometrical criterium.
# In our example, the function space is ``V``,
# the value of the boundary condition (0.0) can represented using a
# :py:class:`Function <dolfinx.functions.Function>` and the Dirichlet
# boundary is defined immediately above. The definition of the Dirichlet
# boundary condition then looks as follows: ::
# Define boundary condition on x = 0 or x = 1
u0 = Function(V)
u0.vector.set(0.0)
facets = locate_entities_geometrical(mesh, 1,
lambda x: np.logical_or(x[0] < np.finfo(float).eps,
x[0] > 1.0 - np.finfo(float).eps),
boundary_only=True)
bc = DirichletBC(u0, locate_dofs_topological(V, 1, facets))
# Next, we want to express the variational problem. First, we need to
# specify the trial function :math:`u` and the test function :math:`v`,
# both living in the function space :math:`V`. We do this by defining a
# :py:class:`TrialFunction <dolfinx.functions.function.TrialFunction>`
# and a :py:class:`TestFunction
# <dolfinx.functions.function.TrialFunction>` on the previously defined
# :py:class:`FunctionSpace <dolfinx.functions.FunctionSpace>` ``V``.
#
# Further, the source :math:`f` and the boundary normal derivative
# :math:`g` are involved in the variational forms, and hence we must
# specify these.
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):
return np.full(x.shape[1], True)
facetdim = mesh.topology.dim - 1
bndry_facets = locate_entities_geometrical(mesh, facetdim, boundary, boundary_only=True)
bdofs = locate_dofs_topological(V.sub(0), V, facetdim, bndry_facets)
bc = DirichletBC(bc0, bdofs, V.sub(0))
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 = PETSc.KSP().create(mesh.mpi_comm)
solver.setType("cg")
# Set matrix operator
solver.setOperators(A)
# Compute solution and return number of iterations
return solver.solve(b, u.vector)
def test_3d(tempdir, cell_type, encoding):
filename = os.path.join(tempdir, "meshtags_3d.xdmf")
comm = MPI.COMM_WORLD
mesh = UnitCubeMesh(comm, 4, 4, 4, cell_type)
bottom_facets = locate_entities_geometrical(
mesh, 2, lambda x: np.isclose(x[1], 0.0))
bottom_values = np.full(bottom_facets.shape, 1, dtype=np.intc)
left_facets = locate_entities_geometrical(mesh, 2,
lambda x: np.isclose(x[0], 0.0))
left_values = np.full(left_facets.shape, 2, dtype=np.intc)
indices, pos = np.unique(np.hstack((bottom_facets, left_facets)),
return_index=True)
mt = MeshTags(mesh, 2, indices,
np.hstack((bottom_values, left_values))[pos])
mt.name = "facets"
top_lines = locate_entities_geometrical(mesh, 1,
lambda x: np.isclose(x[2], 1.0))
top_values = np.full(top_lines.shape, 3, dtype=np.intc)
right_lines = locate_entities_geometrical(mesh, 1,
lambda x: np.isclose(x[0], 1.0))
right_values = np.full(right_lines.shape, 4, dtype=np.intc)
indices, pos = np.unique(np.hstack((top_lines, right_lines)),
return_index=True)
mt_lines = MeshTags(mesh, 1, indices,
np.hstack((top_values, right_values))[pos])
mt_lines.name = "lines"
with XDMFFile(comm, filename, "w", encoding=encoding) as file:
mesh.topology.create_connectivity_all()
file.write_mesh(mesh)
file.write_meshtags(mt)
file.write_meshtags(mt_lines)
with XDMFFile(comm, filename, "r", encoding=encoding) as file:
mesh_in = file.read_mesh()
mesh_in.topology.create_connectivity_all()
mt_in = file.read_meshtags(mesh_in, "facets")
mt_lines_in = file.read_meshtags(mesh_in, "lines")
assert mt_in.name == "facets"
assert mt_lines_in.name == "lines"
with XDMFFile(comm,
os.path.join(tempdir, "meshtags_3d_out.xdmf"),
"w",
encoding=encoding) as file:
file.write_mesh(mesh_in)
file.write_meshtags(mt_lines_in)
file.write_meshtags(mt_in)
# Check number of owned and marked entities
lines_local = comm.allreduce(
(mt_lines.indices < mesh.topology.index_map(1).size_local).sum(),
op=MPI.SUM)
lines_local_in = comm.allreduce(
(mt_lines_in.indices < mesh_in.topology.index_map(1).size_local).sum(),
op=MPI.SUM)
assert lines_local == lines_local_in
def test_assembly_solve_taylor_hood(mesh):
"""Assemble Stokes problem with Taylor-Hood elements and solve."""
gdim = mesh.geometry.dim
P2 = dolfinx.function.VectorFunctionSpace(mesh, ("Lagrange", 2))
P1 = dolfinx.function.FunctionSpace(mesh, ("Lagrange", 1))
def boundary0(x):
"""Define boundary x = 0"""
return x[0] < 10 * numpy.finfo(float).eps
def boundary1(x):
"""Define boundary x = 1"""
return x[0] > (1.0 - 10 * numpy.finfo(float).eps)
def initial_guess_u(x):
u_init = numpy.row_stack((numpy.sin(x[0]) * numpy.sin(x[1]),
numpy.cos(x[0]) * numpy.cos(x[1])))
if gdim == 3:
u_init = numpy.row_stack((u_init, numpy.cos(x[2])))
return u_init
def initial_guess_p(x):
return -x[0]**2 - x[1]**3
u_bc_0 = dolfinx.Function(P2)
u_bc_0.interpolate(lambda x: numpy.row_stack(tuple(x[j] + float(j) for j in range(gdim))))
u_bc_1 = dolfinx.Function(P2)
u_bc_1.interpolate(lambda x: numpy.row_stack(tuple(numpy.sin(x[j]) for j in range(gdim))))
facetdim = mesh.topology.dim - 1
bndry_facets0 = locate_entities_geometrical(mesh, facetdim, boundary0, boundary_only=True)
bndry_facets1 = locate_entities_geometrical(mesh, facetdim, boundary1, boundary_only=True)
bdofs0 = dolfinx.fem.locate_dofs_topological(P2, facetdim, bndry_facets0)
bdofs1 = dolfinx.fem.locate_dofs_topological(P2, facetdim, bndry_facets1)
bcs = [dolfinx.DirichletBC(u_bc_0, bdofs0),
dolfinx.DirichletBC(u_bc_1, bdofs1)]
u, p = dolfinx.Function(P2), dolfinx.Function(P1)
du, dp = ufl.TrialFunction(P2), ufl.TrialFunction(P1)
v, q = ufl.TestFunction(P2), ufl.TestFunction(P1)
F = [inner(ufl.grad(u), ufl.grad(v)) * dx + inner(p, ufl.div(v)) * dx,
inner(ufl.div(u), q) * dx]
J = [[derivative(F[0], u, du), derivative(F[0], p, dp)],
[derivative(F[1], u, du), derivative(F[1], p, dp)]]
P = [[J[0][0], None],
[None, inner(dp, q) * dx]]
# -- Blocked and monolithic
Jmat0 = dolfinx.fem.create_matrix_block(J)
Pmat0 = dolfinx.fem.create_matrix_block(P)
Fvec0 = dolfinx.fem.create_vector_block(F)
snes = PETSc.SNES().create(MPI.COMM_WORLD)
snes.setTolerances(rtol=1.0e-15, max_it=10)
snes.getKSP().setType("minres")
snes.getKSP().getPC().setType("lu")
snes.getKSP().getPC().setFactorSolverType("superlu_dist")
problem = NonlinearPDE_SNESProblem(F, J, [u, p], bcs, P=P)
snes.setFunction(problem.F_block, Fvec0)
snes.setJacobian(problem.J_block, J=Jmat0, P=Pmat0)
u.interpolate(initial_guess_u)
p.interpolate(initial_guess_p)
x0 = dolfinx.fem.create_vector_block(F)
with u.vector.localForm() as _u, p.vector.localForm() as _p:
dolfinx.cpp.la.scatter_local_vectors(
x0, [_u.array_r, _p.array_r],
[u.function_space.dofmap.index_map, p.function_space.dofmap.index_map])
x0.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
snes.solve(None, x0)
assert snes.getConvergedReason() > 0
# -- Blocked and nested
Jmat1 = dolfinx.fem.create_matrix_nest(J)
Pmat1 = dolfinx.fem.create_matrix_nest(P)
Fvec1 = dolfinx.fem.create_vector_nest(F)
snes = PETSc.SNES().create(MPI.COMM_WORLD)
snes.setTolerances(rtol=1.0e-15, max_it=10)
nested_IS = Jmat1.getNestISs()
snes.getKSP().setType("minres")
snes.getKSP().setTolerances(rtol=1e-12)
snes.getKSP().getPC().setType("fieldsplit")
snes.getKSP().getPC().setFieldSplitIS(["u", nested_IS[0][0]], ["p", nested_IS[1][1]])
ksp_u, ksp_p = snes.getKSP().getPC().getFieldSplitSubKSP()
ksp_u.setType("preonly")
ksp_u.getPC().setType('lu')
ksp_u.getPC().setFactorSolverType('superlu_dist')
ksp_p.setType("preonly")
ksp_p.getPC().setType('lu')
ksp_p.getPC().setFactorSolverType('superlu_dist')
problem = NonlinearPDE_SNESProblem(F, J, [u, p], bcs, P=P)
snes.setFunction(problem.F_nest, Fvec1)
snes.setJacobian(problem.J_nest, J=Jmat1, P=Pmat1)
u.interpolate(initial_guess_u)
p.interpolate(initial_guess_p)
x1 = dolfinx.fem.create_vector_nest(F)
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)
x1.set(0.0)
snes.solve(None, x1)
assert snes.getConvergedReason() > 0
assert nest_matrix_norm(Jmat1) == pytest.approx(Jmat0.norm(), 1.0e-12)
assert Fvec1.norm() == pytest.approx(Fvec0.norm(), 1.0e-12)
assert x1.norm() == pytest.approx(x0.norm(), 1.0e-12)
# -- Monolithic
P2_el = ufl.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1_el = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
TH = P2_el * P1_el
W = dolfinx.FunctionSpace(mesh, TH)
U = dolfinx.Function(W)
dU = ufl.TrialFunction(W)
u, p = ufl.split(U)
du, dp = ufl.split(dU)
v, q = ufl.TestFunctions(W)
F = inner(ufl.grad(u), ufl.grad(v)) * dx + inner(p, ufl.div(v)) * dx \
+ inner(ufl.div(u), q) * dx
J = derivative(F, U, dU)
P = inner(ufl.grad(du), ufl.grad(v)) * dx + inner(dp, q) * dx
bdofsW0_P2_0 = dolfinx.fem.locate_dofs_topological((W.sub(0), P2), facetdim, bndry_facets0)
bdofsW0_P2_1 = dolfinx.fem.locate_dofs_topological((W.sub(0), P2), facetdim, bndry_facets1)
bcs = [dolfinx.DirichletBC(u_bc_0, bdofsW0_P2_0, W.sub(0)),
dolfinx.DirichletBC(u_bc_1, bdofsW0_P2_1, W.sub(0))]
Jmat2 = dolfinx.fem.create_matrix(J)
Pmat2 = dolfinx.fem.create_matrix(P)
Fvec2 = dolfinx.fem.create_vector(F)
snes = PETSc.SNES().create(MPI.COMM_WORLD)
snes.setTolerances(rtol=1.0e-15, max_it=10)
snes.getKSP().setType("minres")
snes.getKSP().getPC().setType("lu")
snes.getKSP().getPC().setFactorSolverType("superlu_dist")
problem = NonlinearPDE_SNESProblem(F, J, U, bcs, P=P)
snes.setFunction(problem.F_mono, Fvec2)
snes.setJacobian(problem.J_mono, J=Jmat2, P=Pmat2)
U.interpolate(lambda x: numpy.row_stack((initial_guess_u(x), initial_guess_p(x))))
x2 = dolfinx.fem.create_vector(F)
x2.array = U.vector.array_r
snes.solve(None, x2)
assert snes.getConvergedReason() > 0
assert Jmat2.norm() == pytest.approx(Jmat0.norm(), 1.0e-12)
assert Fvec2.norm() == pytest.approx(Fvec0.norm(), 1.0e-12)
assert x2.norm() == pytest.approx(x0.norm(), 1.0e-12)
def test_matrix_assembly_block():
"""Test assembly of block matrices and vectors into (a) monolithic
blocked structures, PETSc Nest structures, and monolithic structures
in the nonlinear setting
"""
mesh = dolfinx.generation.UnitSquareMesh(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 = dolfinx.function.FunctionSpace(mesh, P0)
V1 = dolfinx.function.FunctionSpace(mesh, P1)
def boundary(x):
return numpy.logical_or(x[0] < 1.0e-6, x[0] > 1.0 - 1.0e-6)
def initial_guess_u(x):
return numpy.sin(x[0]) * numpy.sin(x[1])
def initial_guess_p(x):
return -x[0]**2 - x[1]**3
def bc_value(x):
return numpy.cos(x[0]) * numpy.cos(x[1])
facetdim = mesh.topology.dim - 1
bndry_facets = locate_entities_geometrical(mesh, facetdim, boundary, boundary_only=True)
u_bc = dolfinx.function.Function(V1)
u_bc.interpolate(bc_value)
bdofs = dolfinx.fem.locate_dofs_topological(V1, facetdim, bndry_facets)
bc = dolfinx.fem.dirichletbc.DirichletBC(u_bc, bdofs)
# Define variational problem
du, dp = ufl.TrialFunction(V0), ufl.TrialFunction(V1)
u, p = dolfinx.function.Function(V0), dolfinx.function.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 = [[derivative(F0, u, du), derivative(F0, p, dp)],
[derivative(F1, u, du), derivative(F1, p, dp)]]
L_block = [F0, F1]
# Monolithic blocked
x0 = dolfinx.fem.create_vector_block(L_block)
dolfinx.cpp.la.scatter_local_vectors(
x0, [u.vector.array_r, p.vector.array_r],
[u.function_space.dofmap.index_map, p.function_space.dofmap.index_map])
x0.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
# Ghosts are updated inside assemble_vector_block
A0 = dolfinx.fem.assemble_matrix_block(a_block, [bc])
b0 = dolfinx.fem.assemble_vector_block(L_block, a_block, [bc], x0=x0, scale=-1.0)
A0.assemble()
assert A0.getType() != "nest"
Anorm0 = A0.norm()
bnorm0 = b0.norm()
# Nested (MatNest)
x1 = dolfinx.fem.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 = dolfinx.fem.assemble_matrix_nest(a_block, [bc])
b1 = dolfinx.fem.assemble_vector_nest(L_block)
dolfinx.fem.apply_lifting_nest(b1, a_block, [bc], x1, scale=-1.0)
for b_sub in b1.getNestSubVecs():
b_sub.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
bcs0 = dolfinx.cpp.fem.bcs_rows(dolfinx.fem.assemble._create_cpp_form(L_block), [bc])
dolfinx.fem.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 = dolfinx.function.FunctionSpace(mesh, E)
dU = ufl.TrialFunction(W)
U = dolfinx.function.Function(W)
u0, u1 = ufl.split(U)
v0, v1 = ufl.TestFunctions(W)
U.interpolate(lambda x: numpy.row_stack((initial_guess_u(x), initial_guess_p(x))))
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)
bdofsW_V1 = dolfinx.fem.locate_dofs_topological((W.sub(1), V1), facetdim, bndry_facets)
bc = dolfinx.fem.dirichletbc.DirichletBC(u_bc, bdofsW_V1, W.sub(1))
A2 = dolfinx.fem.assemble_matrix(J, [bc])
A2.assemble()
b2 = dolfinx.fem.assemble_vector(F)
dolfinx.fem.apply_lifting(b2, [J], bcs=[[bc]], x0=[U.vector], scale=-1.0)
b2.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
dolfinx.fem.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 test_assembly_solve_block():
"""Solve a two-field nonlinear diffusion like problem with block matrix
approaches and test that solution is the same.
"""
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 12, 11)
p = 1
P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), p)
V0 = dolfinx.function.FunctionSpace(mesh, P)
V1 = V0.clone()
def bc_val_0(x):
return x[0]**2 + x[1]**2
def bc_val_1(x):
return numpy.sin(x[0]) * numpy.cos(x[1])
def initial_guess_u(x):
return numpy.sin(x[0]) * numpy.sin(x[1])
def initial_guess_p(x):
return -x[0]**2 - x[1]**3
def boundary(x):
return numpy.logical_or(x[0] < 1.0e-6, x[0] > 1.0 - 1.0e-6)
facetdim = mesh.topology.dim - 1
bndry_facets = locate_entities_geometrical(mesh, facetdim, boundary, boundary_only=True)
u_bc0 = dolfinx.function.Function(V0)
u_bc0.interpolate(bc_val_0)
u_bc1 = dolfinx.function.Function(V1)
u_bc1.interpolate(bc_val_1)
bdofs0 = dolfinx.fem.locate_dofs_topological(V0, facetdim, bndry_facets)
bdofs1 = dolfinx.fem.locate_dofs_topological(V1, facetdim, bndry_facets)
bcs = [dolfinx.fem.dirichletbc.DirichletBC(u_bc0, bdofs0),
dolfinx.fem.dirichletbc.DirichletBC(u_bc1, bdofs1)]
# Block and Nest variational problem
u, p = dolfinx.function.Function(V0), dolfinx.function.Function(V1)
du, dp = ufl.TrialFunction(V0), ufl.TrialFunction(V1)
v, q = ufl.TestFunction(V0), ufl.TestFunction(V1)
f = 1.0
g = -3.0
F = [inner((u**2 + 1) * ufl.grad(u), ufl.grad(v)) * dx - inner(f, v) * dx,
inner((p**2 + 1) * ufl.grad(p), ufl.grad(q)) * dx - inner(g, q) * dx]
J = [[derivative(F[0], u, du), derivative(F[0], p, dp)],
[derivative(F[1], u, du), derivative(F[1], p, dp)]]
# -- Blocked version
Jmat0 = dolfinx.fem.create_matrix_block(J)
Fvec0 = dolfinx.fem.create_vector_block(F)
snes = PETSc.SNES().create(MPI.COMM_WORLD)
snes.setTolerances(rtol=1.0e-15, max_it=10)
snes.getKSP().setType("preonly")
snes.getKSP().getPC().setType("lu")
snes.getKSP().getPC().setFactorSolverType("superlu_dist")
problem = NonlinearPDE_SNESProblem(F, J, [u, p], bcs)
snes.setFunction(problem.F_block, Fvec0)
snes.setJacobian(problem.J_block, J=Jmat0, P=None)
u.interpolate(initial_guess_u)
p.interpolate(initial_guess_p)
x0 = dolfinx.fem.create_vector_block(F)
dolfinx.cpp.la.scatter_local_vectors(
x0, [u.vector.array_r, p.vector.array_r],
[u.function_space.dofmap.index_map, p.function_space.dofmap.index_map])
x0.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
snes.solve(None, x0)
assert snes.getKSP().getConvergedReason() > 0
assert snes.getConvergedReason() > 0
J0norm = Jmat0.norm()
F0norm = Fvec0.norm()
x0norm = x0.norm()
# -- Nested (MatNest)
Jmat1 = dolfinx.fem.create_matrix_nest(J)
Fvec1 = dolfinx.fem.create_vector_nest(F)
snes = PETSc.SNES().create(MPI.COMM_WORLD)
snes.setTolerances(rtol=1.0e-15, max_it=10)
nested_IS = Jmat1.getNestISs()
snes.getKSP().setType("fgmres")
snes.getKSP().setTolerances(rtol=1e-12)
snes.getKSP().getPC().setType("fieldsplit")
snes.getKSP().getPC().setFieldSplitIS(["u", nested_IS[0][0]], ["p", nested_IS[1][1]])
ksp_u, ksp_p = snes.getKSP().getPC().getFieldSplitSubKSP()
ksp_u.setType("preonly")
ksp_u.getPC().setType('lu')
ksp_u.getPC().setFactorSolverType('superlu_dist')
ksp_p.setType("preonly")
ksp_p.getPC().setType('lu')
ksp_p.getPC().setFactorSolverType('superlu_dist')
problem = NonlinearPDE_SNESProblem(F, J, [u, p], bcs)
snes.setFunction(problem.F_nest, Fvec1)
snes.setJacobian(problem.J_nest, J=Jmat1, P=None)
u.interpolate(initial_guess_u)
p.interpolate(initial_guess_p)
x1 = dolfinx.fem.create_vector_nest(F)
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)
snes.solve(None, x1)
assert snes.getKSP().getConvergedReason() > 0
assert snes.getConvergedReason() > 0
assert x1.getType() == "nest"
assert Jmat1.getType() == "nest"
assert Fvec1.getType() == "nest"
J1norm = nest_matrix_norm(Jmat1)
F1norm = Fvec1.norm()
x1norm = x1.norm()
assert J1norm == pytest.approx(J0norm, 1.0e-12)
assert F1norm == pytest.approx(F0norm, 1.0e-12)
assert x1norm == pytest.approx(x0norm, 1.0e-12)
# -- Monolithic version
E = P * P
W = dolfinx.function.FunctionSpace(mesh, E)
U = dolfinx.function.Function(W)
dU = ufl.TrialFunction(W)
u0, u1 = ufl.split(U)
v0, v1 = ufl.TestFunctions(W)
F = inner((u0**2 + 1) * ufl.grad(u0), ufl.grad(v0)) * dx \
+ inner((u1**2 + 1) * ufl.grad(u1), ufl.grad(v1)) * dx \
- inner(f, v0) * ufl.dx - inner(g, v1) * dx
J = derivative(F, U, dU)
u0_bc = dolfinx.function.Function(V0)
u0_bc.interpolate(bc_val_0)
u1_bc = dolfinx.function.Function(V1)
u1_bc.interpolate(bc_val_1)
bdofsW0_V0 = dolfinx.fem.locate_dofs_topological((W.sub(0), V0), facetdim, bndry_facets)
bdofsW1_V1 = dolfinx.fem.locate_dofs_topological((W.sub(1), V1), facetdim, bndry_facets)
bcs = [dolfinx.fem.dirichletbc.DirichletBC(u0_bc, bdofsW0_V0, W.sub(0)),
dolfinx.fem.dirichletbc.DirichletBC(u1_bc, bdofsW1_V1, W.sub(1))]
Jmat2 = dolfinx.fem.create_matrix(J)
Fvec2 = dolfinx.fem.create_vector(F)
snes = PETSc.SNES().create(MPI.COMM_WORLD)
snes.setTolerances(rtol=1.0e-15, max_it=10)
snes.getKSP().setType("preonly")
snes.getKSP().getPC().setType("lu")
snes.getKSP().getPC().setFactorSolverType("superlu_dist")
problem = NonlinearPDE_SNESProblem(F, J, U, bcs)
snes.setFunction(problem.F_mono, Fvec2)
snes.setJacobian(problem.J_mono, J=Jmat2, P=None)
U.interpolate(lambda x: numpy.row_stack((initial_guess_u(x), initial_guess_p(x))))
x2 = dolfinx.fem.create_vector(F)
x2.array = U.vector.array_r
snes.solve(None, x2)
assert snes.getKSP().getConvergedReason() > 0
assert snes.getConvergedReason() > 0
J2norm = Jmat2.norm()
F2norm = Fvec2.norm()
x2norm = x2.norm()
assert J2norm == pytest.approx(J0norm, 1.0e-12)
assert F2norm == pytest.approx(F0norm, 1.0e-12)
assert x2norm == pytest.approx(x0norm, 1.0e-12)
# field::
P2 = ufl.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1 = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
V, Q = FunctionSpace(mesh, P2), FunctionSpace(mesh, P1)
# We can define boundary conditions::
# No-slip boundary condition for velocity field (`V`) on boundaries
# where x = 0, x = 1, and y = 0
noslip = Function(V)
with noslip.vector.localForm() as bc_local:
bc_local.set(0.0)
facets = locate_entities_geometrical(mesh,
1,
noslip_boundary,
boundary_only=True)
bc0 = DirichletBC(noslip, locate_dofs_topological(V, 1, facets))
# Driving velocity condition u = (1, 0) on top boundary (y = 1)
lid_velocity = Function(V)
lid_velocity.interpolate(lid_velocity_expression)
facets = locate_entities_geometrical(mesh, 1, lid, boundary_only=True)
bc1 = DirichletBC(lid_velocity, locate_dofs_topological(V, 1, facets))
# Collect Dirichlet boundary conditions
bcs = [bc0, bc1]
# We now define the bilinear and linear forms corresponding to the weak
# mixed formulation of the Stokes equations in a blocked structure::