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_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 solve(dtype=np.float32):
"""Solve the variational problem"""
# Process forms. This will compile the forms for the requested type.
a0 = fem.form(a, dtype=dtype)
if np.issubdtype(dtype, np.complexfloating):
L0 = fem.form(L, dtype=dtype)
else:
L0 = fem.form(ufl.replace(L, {fc: 0, gc: 0}), dtype=dtype)
# Create a Dirichlet boundary condition
bc = fem.dirichletbc(value=dtype(0), dofs=dofs, V=V)
# Assemble forms
A = fem.assemble_matrix(a0, [bc])
A.finalize()
b = fem.assemble_vector(L0)
fem.apply_lifting(b.array, [a0], bcs=[[bc]])
b.scatter_reverse(common.ScatterMode.add)
fem.set_bc(b.array, [bc])
# Create a Scipy sparse matrix that shares data with A
As = scipy.sparse.csr_matrix((A.data, A.indices, A.indptr))
# Solve the variational problem and return the solution
uh = fem.Function(V, dtype=dtype)
uh.x.array[:] = scipy.sparse.linalg.spsolve(As, b.array)
return uh
def solve(self) -> fem.Function:
"""Solve the problem."""
# Assemble lhs
self._A.zeroEntries()
fem.assemble_matrix(self._A, self._a, bcs=self.bcs)
self._A.assemble()
# Assemble rhs
with self._b.localForm() as b_loc:
b_loc.set(0)
fem.assemble_vector(self._b, self._L)
# Apply boundary conditions to the rhs
fem.apply_lifting(self._b, [self._a], [self.bcs])
self._b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
fem.set_bc(self._b, self.bcs)
# Solve linear system and update ghost values in the solution
self._solver.solve(self._b, self.u.vector)
self.u.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
return self.u
def F_nest(self, snes, x, F):
assert x.getType() == "nest" and F.getType() == "nest"
# Update solution
x = x.getNestSubVecs()
for x_sub, var_sub in zip(x, self.soln_vars):
x_sub.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
with x_sub.localForm() as _x:
var_sub.x.array[:] = _x.array_r
# Assemble
bcs1 = bcs_by_block(extract_function_spaces(self.a, 1), self.bcs)
for L, F_sub, a in zip(self.L, F.getNestSubVecs(), self.a):
with F_sub.localForm() as F_sub_local:
F_sub_local.set(0.0)
assemble_vector(F_sub, L)
apply_lifting(F_sub, a, bcs=bcs1, x0=x, scale=-1.0)
F_sub.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
# Set bc value in RHS
bcs0 = bcs_by_block(extract_function_spaces(self.L), self.bcs)
for F_sub, bc, x_sub in zip(F.getNestSubVecs(), bcs0, x):
set_bc(F_sub, bc, x_sub, -1.0)
# Must assemble F here in the case of nest matrices
F.assemble()
def F(self, snes: PETSc.SNES, x: PETSc.Vec, b: PETSc.Vec):
"""Assemble the residual F into the vector b.
Parameters
==========
snes: the snes object
x: Vector containing the latest solution.
b: Vector to assemble the residual into.
"""
# We need to assign the vector to the function
x.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
x.copy(self.u.vector)
self.u.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Zero the residual vector
# import pdb; pdb.set_trace()
with b.localForm() as b_local:
b_local.set(0.0)
assemble_vector(b, self.F_form)
# Apply boundary conditions
apply_lifting(b, [self.J_form], [self.bcs], [x], -1.0)
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, self.bcs, x, -1.0)
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 monolithic_solve():
"""Monolithic (interleaved) solver"""
P2_el = ufl.VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1_el = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
TH = P2_el * P1_el
W = FunctionSpace(mesh, TH)
(u, p) = ufl.TrialFunctions(W)
(v, q) = ufl.TestFunctions(W)
a00 = ufl.inner(ufl.grad(u), ufl.grad(v)) * dx
a01 = ufl.inner(p, ufl.div(v)) * dx
a10 = ufl.inner(ufl.div(u), q) * dx
a = a00 + a01 + a10
p00 = ufl.inner(ufl.grad(u), ufl.grad(v)) * dx
p11 = ufl.inner(p, q) * dx
p_form = p00 + p11
f = Function(W.sub(0).collapse()[0])
p_zero = Function(W.sub(1).collapse()[0])
L0 = inner(f, v) * dx
L1 = inner(p_zero, q) * dx
L = L0 + L1
a, p_form, L = form(a), form(p_form), form(L)
bdofsW0_P2_0 = locate_dofs_topological(W.sub(0), facetdim, bndry_facets0)
bdofsW0_P2_1 = locate_dofs_topological(W.sub(0), facetdim, bndry_facets1)
bc0 = dirichletbc(bc_value, bdofsW0_P2_0, W.sub(0))
bc1 = dirichletbc(bc_value, bdofsW0_P2_1, W.sub(0))
A = assemble_matrix(a, bcs=[bc0, bc1])
A.assemble()
P = assemble_matrix(p_form, bcs=[bc0, bc1])
P.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[[bc0, bc1]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc0, bc1])
ksp = PETSc.KSP()
ksp.create(mesh.comm)
ksp.setOperators(A, P)
ksp.setType("minres")
pc = ksp.getPC()
pc.setType('lu')
def monitor(ksp, its, rnorm):
# print("Num it, rnorm:", its, rnorm)
pass
ksp.setTolerances(rtol=1.0e-8, max_it=50)
ksp.setMonitor(monitor)
ksp.setFromOptions()
x = A.createVecRight()
ksp.solve(b, x)
assert ksp.getConvergedReason() > 0
return b.norm(), x.norm(), A.norm(), P.norm()
def F(self, x, b):
"""Assemble residual vector."""
with b.localForm() as b_local:
b_local.set(0.0)
assemble_vector(b, self.L)
apply_lifting(b, [self.a], bcs=[[self.bc]], x0=[x], scale=-1.0)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [self.bc], x, -1.0)
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 test_assembly_dx_domains(mode):
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(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 F(self, snes, x, F):
"""Assemble residual vector."""
x.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
x.copy(self.u.vector)
self.u.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
with F.localForm() as f_local:
f_local.set(0.0)
fem.assemble_vector(F, self.L)
fem.apply_lifting(F, [self.a], [[self.bc]], [x], -1.0)
F.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
fem.set_bc(F, [self.bc], x, -1.0)
def F_mono(self, snes, x, F):
x.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
with x.localForm() as _x:
self.soln_vars.x.array[:] = _x.array_r
with F.localForm() as f_local:
f_local.set(0.0)
assemble_vector(F, self.L)
apply_lifting(F, [self.a], bcs=[self.bcs], x0=[x], scale=-1.0)
F.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(F, self.bcs, x, -1.0)
def F(self, x: PETSc.Vec, b: PETSc.Vec):
"""Assemble the residual F into the vector b.
Parameters
----------
x
The vector containing the latest solution
b
Vector to assemble the residual into
"""
# Reset the residual vector
with b.localForm() as b_local:
b_local.set(0.0)
fem.assemble_vector(b, self._L)
# Apply boundary condition
fem.apply_lifting(b, [self._a], [self.bcs], [x], -1.0)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
fem.set_bc(b, self.bcs, x, -1.0)
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 project(v, target_func, bcs=[]):
# Ensure we have a mesh and attach to measure
V = target_func.function_space
dx = ufl.dx(V.mesh)
# Define variational problem for projection
w = ufl.TestFunction(V)
Pv = ufl.TrialFunction(V)
a = ufl.inner(Pv, w) * dx
L = ufl.inner(v, w) * dx
# Assemble linear system
A = assemble_matrix(a, bcs)
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
solve(A, target_func.vector, b)
def _solve_varproblem(*args, **kwargs):
"Solve variational problem a == L or F == 0"
# Extract arguments
eq, u, bcs, J, tol, M, form_compiler_parameters, petsc_options \
= _extract_args(*args, **kwargs)
# Solve linear variational problem
if isinstance(eq.lhs, ufl.Form) and isinstance(eq.rhs, ufl.Form):
a = fem.Form(eq.lhs, form_compiler_parameters=form_compiler_parameters)
L = fem.Form(eq.rhs, form_compiler_parameters=form_compiler_parameters)
b = fem.assemble_vector(L._cpp_object)
fem.apply_lifting(b, [a._cpp_object], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
fem.set_bc(b, bcs)
A = fem.assemble_matrix(a._cpp_object, bcs)
A.assemble()
comm = L._cpp_object.mesh().mpi_comm()
ksp = PETSc.KSP().create(comm)
ksp.setOperators(A)
ksp.setOptionsPrefix("dolfin_solve_")
opts = PETSc.Options()
opts.prefixPush("dolfin_solve_")
for k, v in petsc_options.items():
opts[k] = v
opts.prefixPop()
ksp.setFromOptions()
ksp.solve(b, u.vector)
# Solve nonlinear variational problem
else:
raise RuntimeError("Not implemented")
def project(v, target_func, degree, bcs=[]):
# Ensure we have a mesh and attach to measure
V = target_func.function_space
dx = ufl.dx(V.mesh, degree=degree)
# Define variational problem for projection
w = ufl.TestFunction(V)
Pv = ufl.TrialFunction(V)
a = fem.form(ufl.inner(Pv, w) * dx)
L = fem.form(ufl.inner(v, w) * dx)
# Assemble linear system
A = fem.assemble_matrix(a, bcs)
A.assemble()
b = fem.assemble_vector(L)
fem.apply_lifting(b, [a], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
fem.set_bc(b, bcs)
# Solve linear system
solver = PETSc.KSP().create(A.getComm())
solver.setOperators(A)
solver.solve(b, target_func.vector)
np.array([0, 0, 0], dtype=PETSc.ScalarType),
locate_dofs_geometrical(
V,
lambda x: np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[1], 1.0))),
V)
# Assembly and solve
# ------------------
# ::
# Assemble system, applying boundary conditions
A = assemble_matrix(a, bcs=[bc])
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[[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 (required for smoothed aggregation AMG)
null_space = build_nullspace(V)
# Attach near nullspace to matrix
A.setNearNullSpace(null_space)
# Set solver options
opts = PETSc.Options()
opts["ksp_type"] = "cg"
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 test_manufactured_poisson(degree, filename, datadir):
""" Manufactured Poisson problem, solving u = x[1]**p, where p is the
degree of the Lagrange function space.
"""
with XDMFFile(MPI.comm_world, os.path.join(datadir, filename)) as xdmf:
mesh = xdmf.read_mesh(GhostMode.none)
V = FunctionSpace(mesh, ("Lagrange", degree))
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)
# 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)
t0 = time.time()
L = fem.Form(L)
t1 = time.time()
print("Linear form compile time:", t1 - t0)
u_bc = Function(V)
u_bc.interpolate(lambda x: x[1]**degree)
# Create Dirichlet boundary condition
mesh.create_connectivity_all()
facetdim = mesh.topology.dim - 1
bndry_facets = np.where(
np.array(mesh.topology.on_boundary(facetdim)) == 1)[0]
bdofs = locate_dofs_topological(V, facetdim, bndry_facets)
assert (len(bdofs) < V.dim())
bc = DirichletBC(u_bc, bdofs)
t0 = time.time()
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
t1 = time.time()
print("Vector assembly time:", t1 - t0)
t0 = time.time()
a = fem.Form(a)
t1 = time.time()
print("Bilinear form compile time:", t1 - t0)
t0 = time.time()
A = assemble_matrix(a, [bc])
A.assemble()
t1 = time.time()
print("Matrix assembly time:", t1 - t0)
# 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
t0 = time.time()
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
t1 = time.time()
print("Linear solver time:", t1 - t0)
M = (u_exact - uh)**2 * dx
t0 = time.time()
M = fem.Form(M)
t1 = time.time()
print("Error functional compile time:", t1 - t0)
t0 = time.time()
error = assemble_scalar(M)
error = MPI.sum(mesh.mpi_comm(), error)
t1 = time.time()
print("Error assembly time:", t1 - t0)
assert np.absolute(error) < 1.0e-14
a = inner(sigma(u), grad(v)) * dx
L = inner(f, v) * dx
u0 = Function(V)
with u0.vector.localForm() as bc_local:
bc_local.set(0.0)
# Set up boundary condition on inner surface
bc = DirichletBC(u0, locate_dofs_geometrical(V, boundary))
# Assemble system, applying boundary conditions and preserving symmetry)
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 (required for smoothed aggregation AMG).
null_space = build_nullspace(V)
# Attach near nullspace to matrix
A.setNearNullSpace(null_space)
# Set solver options
opts = PETSc.Options()
opts["ksp_type"] = "cg"
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)
# 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)
with common.Timer("Linear form compile"):
L = fem.Form(L)
with common.Timer("Function interpolation"):
u_bc = Function(V)
u_bc.interpolate(lambda x: x[1]**degree)
# Create Dirichlet boundary condition
mesh.topology.create_connectivity_all()
facetdim = mesh.topology.dim - 1
bndry_facets = np.where(
np.array(cpp.mesh.compute_boundary_facets(mesh.topology)) == 1)[0]
bdofs = locate_dofs_topological(V, facetdim, bndry_facets)
bc = DirichletBC(u_bc, bdofs)
with common.Timer("Vector assembly"):
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
with common.Timer("Bilinear form compile"):
a = fem.Form(a)
with common.Timer("Matrix assembly"):
A = assemble_matrix(a, [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)
with common.Timer("Solve"):
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
with common.Timer("Error functional compile"):
M = (u_exact - uh)**2 * dx
M = fem.Form(M)
with common.Timer("Error assembly"):
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
common.list_timings(MPI.COMM_WORLD, [common.TimingType.wall])
assert np.absolute(error) < 1.0e-14
L = form(ufl.inner(f_exact, v) * dx)
zero_u = Function(V_1)
zero_u.x.array[:] = 0.0
boundary_facets = dolfinx.mesh.locate_entities_boundary(
mesh, mesh.topology.dim - 1, lambda x: np.full(x.shape[1], True, dtype=bool))
boundary_dofs = dolfinx.fem.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()
sigma_h = S(ufl.as_tensor([[x_h[0], x_h[1]], [x_h[2], x_h[3]]]))
# A = - A10 * A00^{-1} * A01
A[:, :] = - A10 @ np.linalg.solve(A00, A01)
# Prepare a Form with a condensed tabulation kernel
Form = Form_float64 if PETSc.ScalarType == np.float64 else Form_complex128
integrals = {IntegralType.cell: ([(-1, tabulate_condensed_tensor_A.address)], None)}
a_cond = Form([U._cpp_object, U._cpp_object], integrals, [], [], False, None)
A_cond = assemble_matrix(a_cond, bcs=[bc])
A_cond.assemble()
b = assemble_vector(b1)
apply_lifting(b, [a_cond], bcs=[[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
uc = Function(U)
solver = PETSc.KSP().create(A_cond.getComm())
solver.setOperators(A_cond)
solver.solve(b, uc.vector)
# Pure displacement based formulation
a = form(- ufl.inner(sigma_u(u), ufl.grad(v)) * ufl.dx)
A = assemble_matrix(a, bcs=[bc])
A.assemble()
# Create bounding box for function evaluation
bb_tree = geometry.BoundingBoxTree(mesh, 2)
def solve_displ_system(J,
F,
intern_var0,
intern_var1,
expr_compiled,
w0,
w1,
bcs,
df,
dt,
t,
k,
io_callback=None,
refinement_callback=None,
rtol=1.e-6,
atol=1.e-16,
max_its=20):
"""Solve system for displacement"""
rank = MPI.COMM_WORLD.rank
t0 = time()
A = create_matrix(J)
if rank == 0:
logger.info(f"[Timer] Preallocation matrix time {time() - t0:.3f}")
t0 = time()
t0 = time()
b = create_vector(F)
if rank == 0:
logger.info(f"[Timer] Preallocation vector time {time() - t0:.3f}")
with b.localForm() as local:
local.set(0.0)
assemble_vector(b, F)
apply_lifting(b, [J], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
resid_norm0 = b.norm()
if rank == 0:
print(f"Initial DISPL residual: {resid_norm0:.2e}")
iter = 0
# Total number of NR iterations including refinement attempts
iter_total = 0
converged = False
refine = 0
scale = 1.0
rel_resid_norm = 1.0
bcs_init = []
for bc in bcs:
bcs_init += [bc.value.vector.duplicate()]
with bcs_init[-1].localForm() as loc0, bc.value.vector.localForm(
) as loc1:
loc1.copy(loc0)
df0 = np.copy(df.value)
while converged is False:
if iter > max_its or rel_resid_norm > 1.e+2:
refine += 1
iter = 0
if rank == 0:
logger.info(79 * "!")
logger.info(
f"Restarting NR with rel. stepsize: {1.0 / (2 ** refine)}")
with w0["displ"].vector.localForm(
) as w_local, w1["displ"].vector.localForm() as w1_local:
w_local.copy(w1_local)
df.value = df0
scale = 1.0 / 2**refine
if refine > 10:
raise ConvergenceError(
"Inner adaptivity reqiures > 10 refinements.")
# Reset and scale boundary condition
for i, bc in enumerate(bcs_init):
with bcs[i].value.vector.localForm(
) as bcsi_local, bc.localForm() as bc_local:
bc_local.copy(bcsi_local)
bcsi_local.scale(scale)
df.value *= scale
dt.value *= scale
if refinement_callback is not None:
refinement_callback(scale)
if rank == 0:
logger.info("Newton iteration for displ {}".format(iter))
if iter > 0:
for bc in bcs:
with bc.value.vector.localForm() as locvec:
locvec.set(0.0)
A.zeroEntries()
with b.localForm() as local:
local.set(0.0)
size = A.getSize()[0]
local_size = A.getLocalSize()[0]
t0 = time()
assemble_matrix(A, J, bcs)
A.assemble()
_time = time() - t0
Anorm = A.norm()
if rank == 0:
logger.info(f"[Timer] A0 size: {size}, local size: {local_size}")
logger.info(f"[Timer] A0 assembly: {_time:.4f}")
logger.info(f"[Timer] A0 assembly dofs/s: {size / _time:.1f}")
logger.info(f"A0 norm: {Anorm:.4f}")
t0 = time()
assemble_vector(b, F)
apply_lifting(b, [J], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
bnorm = b.norm()
if rank == 0:
logger.info(f"[Timer] b0 assembly {time() - t0:.4f}")
logger.info(f"b norm: {bnorm:.4f}")
nsp = build_nullspace(w1["displ"].function_space)
ksp = PETSc.KSP()
ksp.create(MPI.COMM_WORLD)
ksp.setOptionsPrefix("disp")
opts = PETSc.Options()
A.setNearNullSpace(nsp)
A.setBlockSize(3)
ksp.setOperators(A)
x = A.createVecRight()
ksp.setFromOptions()
t0 = time()
ksp.solve(b, x)
t1 = time() - t0
opts.view()
xnorm = x.norm()
if rank == 0:
its = ksp.its
t1 = time() - t0
dofsps = int(size / t1)
logger.info(f"[Timer] A0 converged in: {its}")
logger.info(f"[Timer] A0 solve: {t1:.4f}")
logger.info(f"[Timer] A0 solve dofs/s: {dofsps:.1f}")
logger.info(f"Increment norm: {xnorm}")
# TODO: Local axpy segfaults, could ghostUpdate be avoided?
w1["displ"].vector.axpy(1.0, x)
w1["displ"].vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
#
# Evaluate true residual and check
#
with b.localForm() as local:
local.set(0.0)
assemble_vector(b, F)
apply_lifting(b, [J], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
norm = b.norm()
rel_resid_norm = norm / resid_norm0
rel_dx_norm = x.norm() / w1["displ"].vector.norm()
if rank == 0:
logger.info("---")
logger.info(f"Abs. resid norm: {norm:.2e}")
logger.info(f"Rel. dx norm: {rel_dx_norm:.2e}")
logger.info(f"Rel. resid norm: {rel_resid_norm:.2e}")
logger.info("---")
iter += 1
iter_total += 1
if rel_resid_norm < rtol or norm < atol:
if rank == 0:
logger.info(
f"Newton converged in: {iter}, total: {iter_total}")
if io_callback is not None:
io_callback(intern_var1, w1, t.value + dt.value)
return scale
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 = RectangleMesh(MPI.COMM_WORLD, [np.array([0.0, 0.0, 0.0]),
np.array([1.0, 1.0, 0.0])], [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 = inner(sigma_S, tau_S) * dx - b(tau_S, u) + b(sigma_S, v)
L = inner(f_exact, v) * dx
V_1 = V.sub(1).collapse()
zero_u = Function(V_1)
with zero_u.vector.localForm() as zero_u_local:
zero_u_local.set(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])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
# 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.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Recall that x_h has flattened indices.
u_error_numerator = np.sqrt(mesh.mpi_comm().allreduce(assemble_scalar(
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.mpi_comm().allreduce(assemble_scalar(
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.mpi_comm().allreduce(assemble_scalar(
inner(sigma_exact - sigma_h, sigma_exact - sigma_h) * dx(mesh, metadata={"quadrature_degree": 5})), op=MPI.SUM))
sigma_error_denominator = np.sqrt(mesh.mpi_comm().allreduce(assemble_scalar(
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 demo_stacked_cubes(outfile: XDMFFile,
theta: float,
gmsh: bool = True,
quad: bool = False,
compare: bool = False,
res: float = 0.1):
log_info(
f"Run theta:{theta:.2f}, Quad: {quad}, Gmsh {gmsh}, Res {res:.2e}")
celltype = "quadrilateral" if quad else "triangle"
if gmsh:
mesh, mt = gmsh_2D_stacked(celltype, theta)
mesh.name = f"mesh_{celltype}_{theta:.2f}_gmsh"
else:
mesh_name = "mesh"
filename = f"meshes/mesh_{celltype}_{theta:.2f}.xdmf"
mesh_2D_dolfin(celltype, theta)
with XDMFFile(MPI.COMM_WORLD, filename, "r") as xdmf:
mesh = xdmf.read_mesh(name=mesh_name)
mesh.name = f"mesh_{celltype}_{theta:.2f}"
tdim = mesh.topology.dim
fdim = tdim - 1
mesh.topology.create_connectivity(tdim, tdim)
mesh.topology.create_connectivity(fdim, tdim)
mt = xdmf.read_meshtags(mesh, name="facet_tags")
# Helper until meshtags can be read in from xdmf
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
r_matrix = rotation_matrix([0, 0, 1], theta)
g_vec = np.dot(r_matrix, [0, -1.25e2, 0])
g = Constant(mesh, PETSc.ScalarType(g_vec[:2]))
def bottom_corner(x):
return np.isclose(x, [[0], [0], [0]]).all(axis=0)
# Fix bottom corner
bc_value = np.array((0, ) * mesh.geometry.dim, dtype=PETSc.ScalarType)
bottom_dofs = locate_dofs_geometrical(V, bottom_corner)
bc_bottom = dirichletbc(bc_value, bottom_dofs, V)
bcs = [bc_bottom]
# Elasticity parameters
E = PETSc.ScalarType(1.0e3)
nu = 0
mu = Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = Constant(mesh, 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(len(v)))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(sigma(u), grad(v)) * dx
ds = Measure("ds", domain=mesh, subdomain_data=mt, subdomain_id=3)
rhs = inner(Constant(mesh, PETSc.ScalarType(
(0, 0))), v) * dx + inner(g, v) * ds
def left_corner(x):
return np.isclose(x.T, np.dot(r_matrix, [0, 2, 0])).all(axis=1)
# Create multi point constraint
mpc = MultiPointConstraint(V)
with Timer("~Contact: Create contact constraint"):
nh = create_normal_approximation(V, mt, 4)
mpc.create_contact_slip_condition(mt, 4, 9, nh)
with Timer("~Contact: Add non-slip condition at bottom interface"):
bottom_normal = facet_normal_approximation(V, mt, 5)
mpc.create_slip_constraint(V, (mt, 5), bottom_normal, bcs=bcs)
with Timer("~Contact: Add tangential constraint at one point"):
vertex = locate_entities_boundary(mesh, 0, left_corner)
tangent = facet_normal_approximation(V, mt, 3, tangent=True)
mtv = meshtags(mesh, 0, vertex, np.full(len(vertex), 6,
dtype=np.int32))
mpc.create_slip_constraint(V, (mtv, 6), tangent, bcs=bcs)
mpc.finalize()
rtol = 1e-9
petsc_options = {
"ksp_rtol": 1e-9,
"pc_type": "gamg",
"pc_gamg_type": "agg",
"pc_gamg_square_graph": 2,
"pc_gamg_threshold": 0.02,
"pc_gamg_coarse_eq_limit": 1000,
"pc_gamg_sym_graph": True,
"mg_levels_ksp_type": "chebyshev",
"mg_levels_pc_type": "jacobi",
"mg_levels_esteig_ksp_type": "cg"
# , "help": None, "ksp_view": None
}
# Solve Linear problem
problem = LinearProblem(a, rhs, mpc, bcs=bcs, petsc_options=petsc_options)
# Build near nullspace
null_space = rigid_motions_nullspace(mpc.function_space)
problem.A.setNearNullSpace(null_space)
u_h = problem.solve()
it = problem.solver.getIterationNumber()
if MPI.COMM_WORLD.rank == 0:
print("Number of iterations: {0:d}".format(it))
unorm = u_h.vector.norm()
if MPI.COMM_WORLD.rank == 0:
print(f"Norm of u: {unorm}")
# Write solution to file
ext = "_gmsh" if gmsh else ""
u_h.name = "u_mpc_{0:s}_{1:.2f}{2:s}".format(celltype, theta, ext)
outfile.write_mesh(mesh)
outfile.write_function(u_h, 0.0,
f"Xdmf/Domain/Grid[@Name='{mesh.name}'][1]")
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
if not compare:
return
log_info("Solving reference problem with global matrix (using numpy)")
with Timer("~MPC: Reference problem"):
# Generate reference matrices and unconstrained solution
A_org = assemble_matrix(form(a), bcs)
A_org.assemble()
L_org = assemble_vector(form(rhs))
apply_lifting(L_org, [form(a)], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(L_org, bcs)
root = 0
with Timer("~MPC: Verification"):
compare_mpc_lhs(A_org, problem.A, mpc, root=root)
compare_mpc_rhs(L_org, problem.b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = gather_PETScMatrix(A_org, root=root)
K = gather_transformation_matrix(mpc, root=root)
L_np = gather_PETScVector(L_org, root=root)
u_mpc = gather_PETScVector(u_h.vector, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc, rtol=rtol)
def test_matrix_assembly_block(mode):
"""Test assembly of block matrices and vectors into (a) monolithic
blocked structures, PETSc Nest structures, and monolithic
structures"""
mesh = create_unit_square(MPI.COMM_WORLD, 4, 8, ghost_mode=mode)
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)
# Locate facets on boundary
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)))
bdofsV1 = locate_dofs_topological(V1, facetdim, bndry_facets)
u_bc = PETSc.ScalarType(50.0)
bc = dirichletbc(u_bc, bdofsV1, V1)
# Define variational problem
u, p = ufl.TrialFunction(V0), ufl.TrialFunction(V1)
v, q = ufl.TestFunction(V0), ufl.TestFunction(V1)
f = 1.0
g = -3.0
zero = Function(V0)
a00 = inner(u, v) * dx
a01 = inner(p, v) * dx
a10 = inner(u, q) * dx
a11 = inner(p, q) * dx
L0 = zero * inner(f, v) * dx
L1 = inner(g, q) * dx
a_block = form([[a00, a01], [a10, a11]])
L_block = form([L0, L1])
# Monolithic blocked
A0 = assemble_matrix_block(a_block, bcs=[bc])
A0.assemble()
b0 = assemble_vector_block(L_block, a_block, bcs=[bc])
assert A0.getType() != "nest"
Anorm0 = A0.norm()
bnorm0 = b0.norm()
# Nested (MatNest)
A1 = assemble_matrix_nest(a_block, bcs=[bc], mat_types=[["baij", "aij"], ["aij", ""]])
A1.assemble()
Anorm1 = nest_matrix_norm(A1)
assert Anorm0 == pytest.approx(Anorm1, 1.0e-12)
b1 = assemble_vector_nest(L_block)
apply_lifting_nest(b1, a_block, bcs=[bc])
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)
b1.assemble()
bnorm1 = math.sqrt(sum([x.norm()**2 for x in b1.getNestSubVecs()]))
assert bnorm0 == pytest.approx(bnorm1, 1.0e-12)
# Monolithic version
E = P0 * P1
W = FunctionSpace(mesh, E)
u0, u1 = ufl.TrialFunctions(W)
v0, v1 = ufl.TestFunctions(W)
a = inner(u0, v0) * dx + inner(u1, v1) * dx + inner(u0, v1) * dx + inner(
u1, v0) * dx
L = zero * inner(f, v0) * ufl.dx + inner(g, v1) * dx
a, L = form(a), form(L)
bdofsW_V1 = locate_dofs_topological(W.sub(1), mesh.topology.dim - 1, bndry_facets)
bc = dirichletbc(u_bc, bdofsW_V1, W.sub(1))
A2 = assemble_matrix(a, bcs=[bc])
A2.assemble()
b2 = assemble_vector(L)
apply_lifting(b2, [a], bcs=[[bc]])
b2.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b2, [bc])
assert A2.getType() != "nest"
assert A2.norm() == pytest.approx(Anorm0, 1.0e-9)
assert b2.norm() == pytest.approx(bnorm0, 1.0e-9)
def test_assembly_solve_block(mode):
"""Solve a two-field mass-matrix like problem with block matrix approaches
and test that solution is the same"""
mesh = create_unit_square(MPI.COMM_WORLD, 32, 31, ghost_mode=mode)
P = ufl.FiniteElement("Lagrange", mesh.ufl_cell(), 1)
V0 = FunctionSpace(mesh, P)
V1 = V0.clone()
# Locate facets on boundary
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)))
bdofsV0 = locate_dofs_topological(V0, facetdim, bndry_facets)
bdofsV1 = locate_dofs_topological(V1, facetdim, bndry_facets)
u0_bc = PETSc.ScalarType(50.0)
u1_bc = PETSc.ScalarType(20.0)
bcs = [dirichletbc(u0_bc, bdofsV0, V0), dirichletbc(u1_bc, bdofsV1, V1)]
# Variational problem
u, p = ufl.TrialFunction(V0), ufl.TrialFunction(V1)
v, q = ufl.TestFunction(V0), ufl.TestFunction(V1)
f = 1.0
g = -3.0
zero = Function(V0)
a00 = form(inner(u, v) * dx)
a01 = form(zero * inner(p, v) * dx)
a10 = form(zero * inner(u, q) * dx)
a11 = form(inner(p, q) * dx)
L0 = form(inner(f, v) * dx)
L1 = form(inner(g, q) * dx)
def monitor(ksp, its, rnorm):
pass
# print("Norm:", its, rnorm)
A0 = assemble_matrix_block([[a00, a01], [a10, a11]], bcs=bcs)
b0 = assemble_vector_block([L0, L1], [[a00, a01], [a10, a11]], bcs=bcs)
A0.assemble()
A0norm = A0.norm()
b0norm = b0.norm()
x0 = A0.createVecLeft()
ksp = PETSc.KSP()
ksp.create(mesh.comm)
ksp.setOperators(A0)
ksp.setMonitor(monitor)
ksp.setType('cg')
ksp.setTolerances(rtol=1.0e-14)
ksp.setFromOptions()
ksp.solve(b0, x0)
x0norm = x0.norm()
# Nested (MatNest)
A1 = assemble_matrix_nest([[a00, a01], [a10, a11]], bcs=bcs, diagonal=1.0)
A1.assemble()
b1 = assemble_vector_nest([L0, L1])
apply_lifting_nest(b1, [[a00, a01], [a10, a11]], bcs=bcs)
for b_sub in b1.getNestSubVecs():
b_sub.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
bcs0 = bcs_by_block([L0.function_spaces[0], L1.function_spaces[0]], bcs)
set_bc_nest(b1, bcs0)
b1.assemble()
b1norm = b1.norm()
assert b1norm == pytest.approx(b0norm, 1.0e-12)
A1norm = nest_matrix_norm(A1)
assert A0norm == pytest.approx(A1norm, 1.0e-12)
x1 = b1.copy()
ksp = PETSc.KSP()
ksp.create(mesh.comm)
ksp.setMonitor(monitor)
ksp.setOperators(A1)
ksp.setType('cg')
ksp.setTolerances(rtol=1.0e-12)
ksp.setFromOptions()
ksp.solve(b1, x1)
x1norm = x1.norm()
assert x1norm == pytest.approx(x0norm, rel=1.0e-12)
# Monolithic version
E = P * P
W = FunctionSpace(mesh, E)
u0, u1 = ufl.TrialFunctions(W)
v0, v1 = ufl.TestFunctions(W)
a = inner(u0, v0) * dx + inner(u1, v1) * dx
L = inner(f, v0) * ufl.dx + inner(g, v1) * dx
a, L = form(a), form(L)
bdofsW0_V0 = locate_dofs_topological(W.sub(0), facetdim, bndry_facets)
bdofsW1_V1 = locate_dofs_topological(W.sub(1), facetdim, bndry_facets)
bcs = [dirichletbc(u0_bc, bdofsW0_V0, W.sub(0)), dirichletbc(u1_bc, bdofsW1_V1, W.sub(1))]
A2 = assemble_matrix(a, bcs=bcs)
A2.assemble()
b2 = assemble_vector(L)
apply_lifting(b2, [a], [bcs])
b2.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b2, bcs)
A2norm = A2.norm()
b2norm = b2.norm()
assert A2norm == pytest.approx(A0norm, 1.0e-12)
assert b2norm == pytest.approx(b0norm, 1.0e-12)
x2 = b2.copy()
ksp = PETSc.KSP()
ksp.create(mesh.comm)
ksp.setMonitor(monitor)
ksp.setOperators(A2)
ksp.setType('cg')
ksp.getPC().setType('jacobi')
ksp.setTolerances(rtol=1.0e-12)
ksp.setFromOptions()
ksp.solve(b2, x2)
x2norm = x2.norm()
assert x2norm == pytest.approx(x0norm, 1.0e-10)