# Create function space
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
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))
# Explicitly compile a UFL Form into dolfinx Form
form = Form(a,
jit_parameters={
"cffi_extra_compile_args": "-Ofast -march=native",
"cffi_verbose": True
})
# Assemble system, applying boundary conditions and preserving symmetry
A = assemble_matrix(form, [bc])
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
# Global time loop
for k in range(len(dtimes)):
t.value = times[k]
dt.value = dtimes[k]
if rank == 0:
logger.info(79 * "#")
logger.info("Time: {:2.2} days.".format(t.value))
logger.info("Step: {}".format(k))
logger.info(79 * "#")
if t.value in loading_times:
bcs_displ = [
DirichletBC(displ_bc_bottom, bottom_left_dofs),
DirichletBC(displ_bc_bottom2, bottom_right_dofs,
w0["displ"].function_space),
DirichletBC(displ_bc_top2, top_load_dofs,
w0["displ"].function_space)
]
displ_bc_top2.interpolate(loading_bc)
else:
bcs_displ = [
DirichletBC(displ_bc_bottom, bottom_left_dofs),
DirichletBC(displ_bc_bottom2, bottom_right_dofs,
w0["displ"].function_space)
]
scale = fecoda.main.solve_displ_system(J[0], F[0], intern_var0,
intern_var1, expr_compiled, w0, w1,
# 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)
with u0.vector.localForm() as u0_loc:
u0_loc.set(0)
facets = locate_entities_boundary(
mesh, 1,
lambda x: np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0)))
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.fem.TrialFunction>` and a
# :py:class:`TestFunction <dolfinx.functions.fem.TrialFunction>` on the
# previously defined :py:class:`FunctionSpace
# <dolfinx.fem.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.
#
# With these ingredients, we can write down the bilinear form ``a`` and
# the linear form ``L`` (using UFL operators). In summary, this reads ::
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)
assert (len(bdofs) < V.dim)
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
# It is a stable, standard element pair for the Stokes
# equations. Now we can define boundary conditions::
# Extract subdomain facet arrays
mf = sub_domains.values
mf0 = np.where(mf == 0)[0]
mf1 = np.where(mf == 1)[0]
# No-slip boundary condition for velocity
# x1 = 0, x1 = 1 and around the dolphin
noslip = Function(W.sub(0).collapse())
noslip.interpolate(lambda x: np.zeros_like(x[:mesh.geometry.dim]))
bdofs = locate_dofs_topological((W.sub(0), noslip.function_space),
sub_domains.dim, mf0)
bc0 = DirichletBC(noslip, bdofs, W.sub(0))
# Inflow boundary condition for velocity
# x0 = 1
def inflow_eval(x):
values = np.zeros((2, x.shape[1]))
values[0] = -np.sin(x[1] * np.pi)
return values
inflow = Function(W.sub(0).collapse())
inflow.interpolate(inflow_eval)
bdofs1 = locate_dofs_topological((W.sub(0), inflow.function_space),
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),
"r",
encoding=XDMFFile.Encoding.ASCII) as xdmf:
mesh = xdmf.read_mesh(name="Grid")
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.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)
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 = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
t1 = time.time()
print("Error assembly time:", t1 - t0)
assert np.absolute(error) < 1.0e-14
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_boundary(mesh, 1, noslip_boundary)
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_boundary(mesh, 1, lid)
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::
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_boundary(mesh, facetdim, boundary)
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_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)
# 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)
bdofs = locate_dofs_geometrical(V, lambda x: np.full(x.shape[1], True))
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
# Create function space
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
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(V, u0, 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)
bcs_co2 = []
if times[k] == t1:
if rank == 0:
logger.info(79 * "!")
logger.info("Replacement")
with displ_bc.vector.localForm() as localbc:
with w0["displ"].vector.localForm() as local:
localbc.array[
replaced_part_interface_dofs_W0] = -local.array[
replaced_part_interface_dofs_W0]
bcs_displ.append(
DirichletBC(displ_bc, replaced_part_interface_dofs_W0))
else:
with displ_bc.vector.localForm() as local:
local.set(0.0)
bcs_displ.append(DirichletBC(displ_bc, leftW0))
bcs_displ.append(DirichletBC(displ_bc, rightW0sub1))
bcs_displ.append(DirichletBC(displ_bc, right_side_dofs_W0sub))
if times[k] == t1:
_scale = fecoda.main.solve_displ_system(Joff[0], Foff[0],
intern_var0, intern_var1,
expr_compiledoff, w0, w1,
bcs_displ, dforce, dt, t,
k)
fecoda.main.post_update(expr_compiledoff, intern_var0, intern_var1,