def EPS_get_spectrum(
EPS: SLEPc.EPS, mpc: MultiPointConstraint
) -> Tuple[List[complex], List[PETSc.Vec], List[PETSc.Vec]]:
""" Retrieve eigenvalues and eigenfunctions from SLEPc EPS object.
Parameters
----------
EPS
The SLEPc solver
mpc
The multipoint constraint
Returns
-------
Tuple consisting of: List of complex converted eigenvalues,
lists of converted eigenvectors (real part) and (imaginary part)
"""
# Get results in lists
eigval = [EPS.getEigenvalue(i) for i in range(EPS.getConverged())]
eigvec_r = list()
eigvec_i = list()
V = mpc.function_space
for i in range(EPS.getConverged()):
vr = fem.Function(V)
vi = fem.Function(V)
EPS.getEigenvector(i, vr.vector, vi.vector)
eigvec_r.append(vr)
eigvec_i.append(vi) # Sort by increasing real parts
idx = np.argsort(np.real(np.array(eigval)), axis=0)
eigval = [eigval[i] for i in idx]
eigvec_r = [eigvec_r[i] for i in idx]
eigvec_i = [eigvec_i[i] for i in idx]
return (eigval, eigvec_r, eigvec_i)
def test_nonlinear_pde_snes():
"""Test Newton solver for a simple nonlinear PDE"""
# Create mesh and function space
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 12, 15)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
u = fem.Function(V)
v = TestFunction(V)
F = inner(5.0, v) * dx - ufl.sqrt(u * u) * inner(
grad(u), grad(v)) * dx - inner(u, v) * dx
def boundary(x):
"""Define Dirichlet boundary (x = 0 or x = 1)."""
return np.logical_or(x[0] < 1.0e-8, x[0] > 1.0 - 1.0e-8)
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(1.0)
bc = fem.DirichletBC(u_bc, fem.locate_dofs_geometrical(V, boundary))
# Create nonlinear problem
problem = NonlinearPDE_SNESProblem(F, u, bc)
with u.vector.localForm() as u_local:
u_local.set(0.9)
b = dolfinx.cpp.la.create_vector(V.dofmap.index_map, V.dofmap.index_map_bs)
J = dolfinx.cpp.fem.create_matrix(problem.a_comp._cpp_object)
# Create Newton solver and solve
snes = PETSc.SNES().create()
snes.setFunction(problem.F, b)
snes.setJacobian(problem.J, J)
snes.setTolerances(rtol=1.0e-9, max_it=10)
snes.getKSP().setType("preonly")
snes.getKSP().setTolerances(rtol=1.0e-9)
snes.getKSP().getPC().setType("lu")
snes.getKSP().getPC().setFactorSolverType("superlu_dist")
snes.solve(None, u.vector)
assert snes.getConvergedReason() > 0
assert snes.getIterationNumber() < 6
# Modify boundary condition and solve again
with u_bc.vector.localForm() as u_local:
u_local.set(0.5)
snes.solve(None, u.vector)
assert snes.getConvergedReason() > 0
assert snes.getIterationNumber() < 6
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_system(N):
fenics_mesh = dolfinx.UnitCubeMesh(fenicsx_comm, N, N, N)
fenics_space = dolfinx.FunctionSpace(fenics_mesh, ("CG", 1))
u = ufl.TrialFunction(fenics_space)
v = ufl.TestFunction(fenics_space)
k = 2
# print(u*v*ufl.ds)
form = (ufl.inner(ufl.grad(u), ufl.grad(v)) -
k**2 * ufl.inner(u, v)) * ufl.dx
# locate facets on the cube boundary
facets = locate_entities_boundary(
fenics_mesh, 2, lambda x: np.logical_or(
np.logical_or(
np.logical_or(np.isclose(x[2], 0.0), np.isclose(x[2], 1.0)),
np.logical_or(np.isclose(x[1], 0.0), np.isclose(x[1], 1.0))),
np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0))))
facets.sort()
# alternative - more general approach
boundary = entities_to_geometry(
fenics_mesh,
fenics_mesh.topology.dim - 1,
exterior_facet_indices(fenics_mesh),
True,
)
# print(len(facets)
assert len(facets) == len(exterior_facet_indices(fenics_mesh))
u0 = fem.Function(fenics_space)
with u0.vector.localForm() as u0_loc:
u0_loc.set(0)
# solution vector
bc = DirichletBC(u0, locate_dofs_topological(fenics_space, 2, facets))
A = 1 + 1j
f = Function(fenics_space)
f.interpolate(lambda x: A * k**2 * np.cos(k * x[0]) * np.cos(k * x[1]))
L = ufl.inner(f, v) * ufl.dx
u0.name = "u"
problem = fem.LinearProblem(form,
L,
u=u0,
petsc_options={
"ksp_type": "preonly",
"pc_type": "lu"
})
# problem = fem.LinearProblem(form, L, bcs=[bc], u=u0, petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
start_time = time.time()
soln = problem.solve()
if world_rank == 0:
print("--- fenics solve done in %s seconds ---" %
(time.time() - start_time))
def rigid_motions_nullspace(V: _fem.FunctionSpace):
"""
Function to build nullspace for 2D/3D elasticity.
Parameters:
===========
V
The function space
"""
_x = _fem.Function(V)
# Get geometric dim
gdim = V.mesh.geometry.dim
assert gdim == 2 or gdim == 3
# Set dimension of nullspace
dim = 3 if gdim == 2 else 6
# Create list of vectors for null space
nullspace_basis = [_x.vector.copy() for _ in range(dim)]
with ExitStack() as stack:
vec_local = [stack.enter_context(x.localForm()) for x in nullspace_basis]
basis = [np.asarray(x) for x in vec_local]
dofs = [V.sub(i).dofmap.list.array for i in range(gdim)]
# Build translational null space basis
for i in range(gdim):
basis[i][dofs[i]] = 1.0
# Build rotational null space basis
x = V.tabulate_dof_coordinates()
dofs_block = V.dofmap.list.array
x0, x1, x2 = x[dofs_block, 0], x[dofs_block, 1], x[dofs_block, 2]
if gdim == 2:
basis[2][dofs[0]] = -x1
basis[2][dofs[1]] = x0
elif gdim == 3:
basis[3][dofs[0]] = -x1
basis[3][dofs[1]] = x0
basis[4][dofs[0]] = x2
basis[4][dofs[2]] = -x0
basis[5][dofs[2]] = x1
basis[5][dofs[1]] = -x2
_la.orthonormalize(nullspace_basis)
assert _la.is_orthonormal(nullspace_basis)
return PETSc.NullSpace().create(vectors=nullspace_basis)
def test_linear_pde():
"""Test Newton solver for a linear PDE"""
# Create mesh and function space
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 12, 12)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
u = fem.Function(V)
v = TestFunction(V)
F = inner(10.0, v) * dx - inner(grad(u), grad(v)) * dx
def boundary(x):
"""Define Dirichlet boundary (x = 0 or x = 1)."""
return np.logical_or(x[0] < 1.0e-8, x[0] > 1.0 - 1.0e-8)
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(1.0)
bc = fem.DirichletBC(u_bc, fem.locate_dofs_geometrical(V, boundary))
# Create nonlinear problem
problem = NonlinearPDEProblem(F, u, bc)
# Create Newton solver and solve
solver = dolfinx.cpp.nls.NewtonSolver(MPI.COMM_WORLD)
solver.setF(problem.F, problem.vector())
solver.setJ(problem.J, problem.matrix())
solver.set_form(problem.form)
n, converged = solver.solve(u.vector)
assert converged
assert n == 1
# Increment boundary condition and solve again
with u_bc.vector.localForm() as u_local:
u_local.set(2.0)
n, converged = solver.solve(u.vector)
assert converged
assert n == 1
def create_normal_approximation(V: _fem.FunctionSpace, mt: _cpp.mesh.MeshTags_int32, value: int):
"""
Creates a normal approximation for the dofs in the closure of the attached entities.
Where a dof is attached to entities facets, an average is computed
Parameters
----------
V
The function space
mt
The meshtag containing the indices
value
Value for the entities in the mesh tag to compute normal on
Returns
-------
nh
The normal vector
"""
nh = _fem.Function(V)
n_cpp = dolfinx_mpc.cpp.mpc.create_normal_approximation(V._cpp_object, mt.dim, mt.find(value))
nh._cpp_object = n_cpp
return nh
def facet_normal_approximation(V, mt: _cpp.mesh.MeshTags_int32, mt_id: int, tangent=False, jit_params: dict = {},
form_compiler_params: dict = {}):
"""
Approximate the facet normal by projecting it into the function space for a set of facets
Parameters
----------
V
The function space to project into
mt
The `dolfinx.mesh.MeshTagsMetaClass` containing facet markers
mt_id
The id for the facets in `mt` we want to represent the normal at
tangent
To approximate the tangent to the facet set this flag to `True`
jit_params
Parameters used in CFFI JIT compilation of C code generated by FFCx.
See `DOLFINx-documentation <https://github.com/FEniCS/dolfinx/blob/main/python/dolfinx/jit.py#L22-L37>`
for all available parameters. Takes priority over all other parameter values.
form_compiler_params
Parameters used in FFCx compilation of this form. Run `ffcx - -help` at
the commandline to see all available options. Takes priority over all
other parameter values, except for `scalar_type` which is determined by
DOLFINx.
"""
timer = _common.Timer("~MPC: Facet normal projection")
comm = V.mesh.comm
n = ufl.FacetNormal(V.mesh)
nh = _fem.Function(V)
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
ds = ufl.ds(domain=V.mesh, subdomain_data=mt, subdomain_id=mt_id)
if tangent:
if V.mesh.geometry.dim == 1:
raise ValueError("Tangent not defined for 1D problem")
elif V.mesh.geometry.dim == 2:
a = ufl.inner(u, v) * ds
L = ufl.inner(ufl.as_vector([-n[1], n[0]]), v) * ds
else:
def tangential_proj(u, n):
"""
See for instance:
https://link.springer.com/content/pdf/10.1023/A:1022235512626.pdf
"""
return (ufl.Identity(u.ufl_shape[0]) - ufl.outer(n, n)) * u
c = _fem.Constant(V.mesh, [1, 1, 1])
a = ufl.inner(u, v) * ds
L = ufl.inner(tangential_proj(c, n), v) * ds
else:
a = (ufl.inner(u, v) * ds)
L = ufl.inner(n, v) * ds
# Find all dofs that are not boundary dofs
imap = V.dofmap.index_map
all_blocks = np.arange(imap.size_local, dtype=np.int32)
top_blocks = _fem.locate_dofs_topological(V, V.mesh.topology.dim - 1, mt.find(mt_id))
deac_blocks = all_blocks[np.isin(all_blocks, top_blocks, invert=True)]
# Note there should be a better way to do this
# Create sparsity pattern only for constraint + bc
bilinear_form = _fem.form(a, jit_params=jit_params,
form_compiler_params=form_compiler_params)
pattern = _fem.create_sparsity_pattern(bilinear_form)
pattern.insert_diagonal(deac_blocks)
pattern.assemble()
u_0 = _fem.Function(V)
u_0.vector.set(0)
bc_deac = _fem.dirichletbc(u_0, deac_blocks)
A = _cpp.la.petsc.create_matrix(comm, pattern)
A.zeroEntries()
# Assemble the matrix with all entries
form_coeffs = _cpp.fem.pack_coefficients(bilinear_form)
form_consts = _cpp.fem.pack_constants(bilinear_form)
_cpp.fem.petsc.assemble_matrix(A, bilinear_form, form_consts, form_coeffs, [bc_deac])
if bilinear_form.function_spaces[0] is bilinear_form.function_spaces[1]:
A.assemblyBegin(PETSc.Mat.AssemblyType.FLUSH)
A.assemblyEnd(PETSc.Mat.AssemblyType.FLUSH)
_cpp.fem.petsc.insert_diagonal(A, bilinear_form.function_spaces[0], [bc_deac], 1.0)
A.assemble()
linear_form = _fem.form(L, jit_params=jit_params,
form_compiler_params=form_compiler_params)
b = _fem.petsc.assemble_vector(linear_form)
_fem.petsc.apply_lifting(b, [bilinear_form], [[bc_deac]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
_fem.petsc.set_bc(b, [bc_deac])
# Solve Linear problem
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType("cg")
solver.rtol = 1e-8
solver.setOperators(A)
solver.solve(b, nh.vector)
nh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
timer.stop()
return nh
def test_pipeline(master_point, get_assemblers): # noqa: F811
assemble_matrix, assemble_vector = get_assemblers
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 3, 5)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
# Solve Problem without MPC for reference
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
d = fem.Constant(mesh, PETSc.ScalarType(1.5))
c = fem.Constant(mesh, PETSc.ScalarType(2))
x = ufl.SpatialCoordinate(mesh)
f = c * ufl.sin(2 * ufl.pi * x[0]) * ufl.sin(ufl.pi * x[1])
g = fem.Function(V)
g.interpolate(lambda x: np.sin(x[0]) * x[1])
h = fem.Function(V)
h.interpolate(lambda x: 2 + x[1] * x[0])
a = d * g * ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
rhs = h * ufl.inner(f, v) * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Generate reference matrices
A_org = fem.petsc.assemble_matrix(bilinear_form)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
# Create multipoint constraint
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {
l2b([1, 0]): {
l2b([0, 1]): 0.43,
l2b([1, 1]): 0.11
},
l2b([0, 0]): {
l2b(master_point): 0.69
}
}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c)
mpc.finalize()
A = assemble_matrix(bilinear_form, mpc)
b = assemble_vector(linear_form, mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
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 = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, 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)
list_timings(comm, [TimingType.wall])
outfile.write_function(uh)
print("----Verification----")
# --------------------VERIFICATION-------------------------
bilinear_form = fem.form(a)
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
linear_form = fem.form(rhs)
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
solver.setOperators(A_org)
u_ = fem.Function(V)
solver.solve(L_org, u_.vector)
it = solver.getIterationNumber()
print("Unconstrained solver iterations {0:d}".format(it))
u_.x.scatter_forward()
u_.name = "u_unconstrained"
outfile.write_function(u_)
root = 0
comm = mesh.comm
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem._A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem._b, mpc, root=root)
# Gather LHS, RHS and solution on one process
def demo_elasticity():
mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
# Generate Dirichlet BC on lower boundary (Fixed)
def boundaries(x):
return np.isclose(x[0], np.finfo(float).eps)
facets = locate_entities_boundary(mesh, 1, boundaries)
topological_dofs = fem.locate_dofs_topological(V, 1, facets)
bc = fem.dirichletbc(np.array([0, 0], dtype=PETSc.ScalarType),
topological_dofs, V)
bcs = [bc]
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
# Elasticity parameters
E = PETSc.ScalarType(1.0e4)
nu = 0.0
mu = fem.Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = fem.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)))
x = SpatialCoordinate(mesh)
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(sigma(u), grad(v)) * dx
rhs = inner(as_vector((0, (x[0] - 0.5) * 10**4 * x[1])), v) * dx
# Create MPC
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {l2b([1, 0]): {l2b([1, 1]): 0.9}}
mpc = MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, 1, 1)
mpc.finalize()
# Solve Linear problem
petsc_options = {"ksp_type": "preonly", "pc_type": "lu"}
problem = LinearProblem(a, rhs, mpc, bcs=bcs, petsc_options=petsc_options)
u_h = problem.solve()
u_h.name = "u_mpc"
with XDMFFile(MPI.COMM_WORLD, "results/demo_elasticity.xdmf",
"w") as outfile:
outfile.write_mesh(mesh)
outfile.write_function(u_h)
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
bilinear_form = fem.form(a)
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
linear_form = fem.form(rhs)
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A_org)
u_ = fem.Function(V)
solver.solve(L_org, u_.vector)
u_.x.scatter_forward()
u_.name = "u_unconstrained"
with XDMFFile(MPI.COMM_WORLD, "results/demo_elasticity.xdmf",
"a") as outfile:
outfile.write_function(u_)
outfile.close()
root = 0
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem.A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem.b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.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)
# Print out master-slave connectivity for the first slave
master_owner = None
master_data = None
slave_owner = None
if mpc.num_local_slaves > 0:
slave_owner = MPI.COMM_WORLD.rank
bs = mpc.function_space.dofmap.index_map_bs
slave = mpc.slaves[0]
print("Constrained: {0:.5e}\n Unconstrained: {1:.5e}".format(
u_h.x.array[slave], u_.vector.array[slave]))
master_owner = mpc._cpp_object.owners.links(slave)[0]
_masters = mpc.masters
master = _masters.links(slave)[0]
glob_master = mpc.function_space.dofmap.index_map.local_to_global(
[master // bs])[0]
coeffs, offs = mpc.coefficients()
master_data = [
glob_master * bs + master % bs,
coeffs[offs[slave]:offs[slave + 1]][0]
]
# If master not on proc send info to this processor
if MPI.COMM_WORLD.rank != master_owner:
MPI.COMM_WORLD.send(master_data, dest=master_owner, tag=1)
else:
print("Master*Coeff: {0:.5e}".format(
coeffs[offs[slave]:offs[slave + 1]][0] *
u_h.x.array[_masters.links(slave)[0]]))
# As a processor with a master is not aware that it has a master,
# Determine this so that it can receive the global dof and coefficient
master_recv = MPI.COMM_WORLD.allgather(master_owner)
for master in master_recv:
if master is not None:
master_owner = master
break
if slave_owner != master_owner and MPI.COMM_WORLD.rank == master_owner:
dofmap = mpc.function_space.dofmap
bs = dofmap.index_map_bs
in_data = MPI.COMM_WORLD.recv(source=MPI.ANY_SOURCE, tag=1)
num_local = dofmap.index_map.size_local + dofmap.index_map.num_ghosts
l2g = dofmap.index_map.local_to_global(
np.arange(num_local, dtype=np.int32))
l_index = np.flatnonzero(l2g == in_data[0] // bs)[0]
print("Master*Coeff (on other proc): {0:.5e}".format(
u_h.x.array[l_index * bs + in_data[0] % bs] * in_data[1]))
# parse command line options
EPS.setFromOptions()
# Display all options (including those of ST object)
# EPS.view()
EPS.solve()
EPS_print_results(EPS)
return EPS
# Create mesh and finite element
N = 50
mesh = create_unit_square(MPI.COMM_WORLD, N, N)
V = fem.FunctionSpace(mesh, ("CG", 1))
# Create Dirichlet boundary condition
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
def dirichletboundary(x):
return np.logical_or(np.isclose(x[1], 0), np.isclose(x[1], 1))
fdim = mesh.topology.dim - 1
facets = locate_entities_boundary(mesh, fdim, dirichletboundary)
topological_dofs = fem.locate_dofs_topological(V, fdim, facets)
bc = fem.dirichletbc(u_bc, topological_dofs)
bcs = [bc]
def __init__(self,
a: ufl.Form,
L: ufl.Form,
mpc: MultiPointConstraint,
bcs: typing.List[_fem.DirichletBCMetaClass] = None,
u: _fem.Function = None,
petsc_options: dict = None,
form_compiler_params: dict = None,
jit_params: dict = None):
"""Initialize solver for a linear variational problem.
Parameters
----------
a
A bilinear UFL form, the left hand side of the variational problem.
L
A linear UFL form, the right hand side of the variational problem.
mpc
The multi point constraint.
bcs
A list of Dirichlet boundary conditions.
u
The solution function. It will be created if not provided. The function has
to be based on the functionspace in the mpc, i.e.
.. code-block:: python
u = dolfinx.fem.Function(mpc.function_space)
petsc_options
Parameters that is passed to the linear algebra backend PETSc.
For available choices for the 'petsc_options' kwarg, see the
`PETSc-documentation <https://www.mcs.anl.gov/petsc/documentation/index.html>`.
form_compiler_params
Parameters used in FFCx compilation of this form. Run `ffcx --help` at
the commandline to see all available options. Takes priority over all
other parameter values, except for `scalar_type` which is determined by
DOLFINx.
jit_params
Parameters used in CFFI JIT compilation of C code generated by FFCx.
See `DOLFINx-documentation <https://github.com/FEniCS/dolfinx/blob/main/python/dolfinx/jit.py#L22-L37>`
for all available parameters. Takes priority over all other parameter values.
.. code-block:: python
problem = LinearProblem(a, L, mpc, [bc0, bc1], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
"""
# Compile forms
form_compiler_params = {} if form_compiler_params is None else form_compiler_params
jit_params = {} if jit_params is None else jit_params
self._a = _fem.form(a,
jit_params=jit_params,
form_compiler_params=form_compiler_params)
self._L = _fem.form(L,
jit_params=jit_params,
form_compiler_params=form_compiler_params)
if not mpc.finalized:
raise RuntimeError(
"The multi point constraint has to be finalized before calling initializer"
)
self._mpc = mpc
# Create function containing solution vector
if u is None:
self.u = _fem.Function(self._mpc.function_space)
else:
if u.function_space is self._mpc.function_space:
self.u = u
else:
raise ValueError(
"The input function has to be in the function space in the multi-point constraint",
"i.e. u = dolfinx.fem.Function(mpc.function_space)")
# Create MPC matrix
pattern = create_sparsity_pattern(self._a, self._mpc)
pattern.assemble()
self._A = _cpp.la.petsc.create_matrix(
self._mpc.function_space.mesh.comm, pattern)
self._b = _cpp.la.petsc.create_vector(
self._mpc.function_space.dofmap.index_map,
self._mpc.function_space.dofmap.index_map_bs)
self.bcs = [] if bcs is None else bcs
self._solver = PETSc.KSP().create(self.u.function_space.mesh.comm)
self._solver.setOperators(self._A)
# Give PETSc solver options a unique prefix
solver_prefix = "dolfinx_mpc_solve_{}".format(id(self))
self._solver.setOptionsPrefix(solver_prefix)
# Set PETSc options
opts = PETSc.Options()
opts.prefixPush(solver_prefix)
if petsc_options is not None:
for k, v in petsc_options.items():
opts[k] = v
opts.prefixPop()
self._solver.setFromOptions()
A = 1
# Test and trial function space
V = FunctionSpace(mesh, ("Lagrange", deg))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Function(V)
f.interpolate(lambda x: A * k0**2 * np.cos(k0 * x[0]) * np.cos(k0 * x[1]))
a = inner(grad(u), grad(v)) * dx - k0**2 * inner(u, v) * dx
L = inner(f, v) * dx
# Compute solution
uh = fem.Function(V)
uh.name = "u"
problem = fem.LinearProblem(a, L, u=uh, petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
problem.solve()
# Save solution in XDMF format (to be viewed in Paraview, for example)
with XDMFFile(MPI.COMM_WORLD, "plane_wave.xdmf", "w", encoding=XDMFFile.Encoding.HDF5) as file:
file.write_mesh(mesh)
file.write_function(uh)
# Calculate L2 and H1 errors of FEM solution and best approximation.
# This demonstrates the error bounds given in Ihlenburg. Pollution errors
# are evident for high wavenumbers. ::
# Function space for exact solution - need it to be higher than deg
def test_cube_contact(generate_hex_boxes, nonslip,
get_assemblers): # noqa: F811
assemble_matrix, assemble_vector = get_assemblers
comm = MPI.COMM_WORLD
root = 0
# Generate mesh
mesh_data = generate_hex_boxes
mesh, mt = mesh_data
fdim = mesh.topology.dim - 1
# Create functionspaces
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
# Helper for orienting traction
# Bottom boundary is fixed in all directions
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
bottom_dofs = fem.locate_dofs_topological(V, fdim, mt.find(5))
bc_bottom = fem.dirichletbc(u_bc, bottom_dofs)
g_vec = [0, 0, -4.25e-1]
if not nonslip:
# Helper for orienting traction
r_matrix = dolfinx_mpc.utils.rotation_matrix(
[1 / np.sqrt(2), 1 / np.sqrt(2), 0], -theta)
# Top boundary has a given deformation normal to the interface
g_vec = np.dot(r_matrix, [0, 0, -4.25e-1])
# Top boundary has a given deformation normal to the interface
def top_v(x):
values = np.empty((3, x.shape[1]))
values[0] = g_vec[0]
values[1] = g_vec[1]
values[2] = g_vec[2]
return values
u_top = fem.Function(V)
u_top.interpolate(top_v)
top_dofs = fem.locate_dofs_topological(V, fdim, mt.find(3))
bc_top = fem.dirichletbc(u_top, top_dofs)
bcs = [bc_bottom, bc_top]
# Elasticity parameters
E = PETSc.ScalarType(1.0e3)
nu = 0
mu = fem.Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = fem.Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
# Stress computation
def sigma(v):
return (2.0 * mu * ufl.sym(ufl.grad(v)) +
lmbda * ufl.tr(ufl.sym(ufl.grad(v))) * ufl.Identity(len(v)))
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = ufl.inner(sigma(u), ufl.grad(v)) * ufl.dx
rhs = ufl.inner(fem.Constant(mesh, PETSc.ScalarType(
(0, 0, 0))), v) * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Create LU solver
solver = PETSc.KSP().create(comm)
solver.setType("preonly")
solver.setTolerances(rtol=1.0e-14)
solver.getPC().setType("lu")
# Create MPC contact condition and assemble matrices
mpc = dolfinx_mpc.MultiPointConstraint(V)
if nonslip:
with Timer("~Contact: Create non-elastic constraint"):
mpc.create_contact_inelastic_condition(mt, 4, 9)
else:
with Timer("~Contact: Create contact constraint"):
nh = dolfinx_mpc.utils.create_normal_approximation(V, mt, 4)
mpc.create_contact_slip_condition(mt, 4, 9, nh)
mpc.finalize()
with Timer("~TEST: Assemble bilinear form"):
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
with Timer("~TEST: Assemble vector"):
b = assemble_vector(linear_form, mpc)
dolfinx_mpc.apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(b, bcs)
with Timer("~MPC: Solve"):
solver.setOperators(A)
uh = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
# Write solution to file
# u_h = fem.Function(mpc.function_space)
# u_h.vector.setArray(uh.array)
# u_h.x.scatter_forward()
# u_h.name = "u_{0:.2f}".format(theta)
# import dolfinx.io as io
# with io.XDMFFile(comm, "output/rotated_cube3D.xdmf", "w") as outfile:
# 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
dolfinx_mpc.utils.log_info(
"Solving reference problem with global matrix (using numpy)")
with Timer("~TEST: Assemble bilinear form (unconstrained)"):
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, 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)
list_timings(comm, [TimingType.wall])
def test_lifting(get_assemblers): # noqa: F811
"""
Test MPC lifting operation on a single cell
"""
assemble_matrix, assemble_vector = get_assemblers
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 1, 1, CellType.quadrilateral)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
# Solve Problem without MPC for reference
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
x = ufl.SpatialCoordinate(mesh)
f = x[1] * ufl.sin(2 * ufl.pi * x[0])
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
rhs = ufl.inner(f, v) * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Create Dirichlet boundary condition
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(2.3)
def dirichletboundary(x):
return np.isclose(x[0], 1)
mesh.topology.create_connectivity(2, 1)
geometrical_dofs = fem.locate_dofs_geometrical(V, dirichletboundary)
bc = fem.dirichletbc(u_bc, geometrical_dofs)
bcs = [bc]
# Generate reference matrices
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs=bcs)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
# Create multipoint constraint
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {l2b([0, 0]): {l2b([0, 1]): 1}}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c)
mpc.finalize()
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
b = assemble_vector(linear_form, mpc)
dolfinx_mpc.apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(b, bcs)
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 = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
V_mpc = mpc.function_space
u_out = fem.Function(V_mpc)
u_out.vector.array[:] = uh.array
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, root=root)
# constants = dolfinx_mpc.utils.gather_contants(mpc, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ (L_np) # - constants)
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vecto
uh_numpy = K @ (d) # + constants)
assert np.allclose(uh_numpy, u_mpc)
list_timings(comm, [TimingType.wall])
def demo_stacked_cubes(outfile: XDMFFile, theta: float, gmsh: bool = False, ct: CellType = CellType.tetrahedron,
compare: bool = True, res: float = 0.1, noslip: bool = False):
celltype = "hexahedron" if ct == CellType.hexahedron else "tetrahedron"
type_ext = "no_slip" if noslip else "slip"
mesh_ext = "_gmsh_" if gmsh else "_"
log_info(f"Run theta:{theta:.2f}, Cell: {celltype}, GMSH {gmsh}, Noslip: {noslip}")
# Read in mesh
if gmsh:
mesh, mt = gmsh_3D_stacked(celltype, theta, res)
tdim = mesh.topology.dim
fdim = tdim - 1
mesh.topology.create_connectivity(tdim, tdim)
mesh.topology.create_connectivity(fdim, tdim)
else:
mesh_3D_dolfin(theta, ct, celltype, res)
MPI.COMM_WORLD.barrier()
with XDMFFile(MPI.COMM_WORLD, f"meshes/mesh_{celltype}_{theta:.2f}.xdmf", "r") as xdmf:
mesh = xdmf.read_mesh(name="mesh")
tdim = mesh.topology.dim
fdim = tdim - 1
mesh.topology.create_connectivity(tdim, tdim)
mesh.topology.create_connectivity(fdim, tdim)
mt = xdmf.read_meshtags(mesh, "facet_tags")
mesh.name = f"mesh_{celltype}_{theta:.2f}{type_ext}{mesh_ext}"
# Create functionspaces
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
# Define boundary conditions
# Bottom boundary is fixed in all directions
bottom_dofs = fem.locate_dofs_topological(V, fdim, mt.find(5)) # type: ignore
u_bc = np.array((0, ) * mesh.geometry.dim, dtype=PETSc.ScalarType)
bc_bottom = fem.dirichletbc(u_bc, bottom_dofs, V)
g_vec = np.array([0, 0, -4.25e-1], dtype=PETSc.ScalarType)
if not noslip:
# Helper for orienting traction
r_matrix = rotation_matrix([1 / np.sqrt(2), 1 / np.sqrt(2), 0], -theta)
# Top boundary has a given deformation normal to the interface
g_vec = np.dot(r_matrix, g_vec)
top_dofs = fem.locate_dofs_topological(V, fdim, mt.find(3)) # type: ignore
bc_top = fem.dirichletbc(g_vec, top_dofs, V)
bcs = [bc_bottom, bc_top]
# Elasticity parameters
E = PETSc.ScalarType(1.0e3)
nu = 0
mu = fem.Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = fem.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
# NOTE: Traction deactivated until we have a way of fixing nullspace
# g = fem.Constant(mesh, PETSc.ScalarType(g_vec))
# ds = Measure("ds", domain=mesh, subdomain_data=mt, subdomain_id=3)
rhs = inner(fem.Constant(mesh, PETSc.ScalarType((0, 0, 0))), v) * dx
# + inner(g, v) * ds
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
mpc = MultiPointConstraint(V)
if noslip:
with Timer("~~Contact: Create non-elastic constraint"):
mpc.create_contact_inelastic_condition(mt, 4, 9)
else:
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 data and finialize MPC"):
mpc.finalize()
# Create null-space
null_space = rigid_motions_nullspace(mpc.function_space)
num_dofs = V.dofmap.index_map.size_global * V.dofmap.index_map_bs
with Timer(f"~~Contact: Assemble matrix ({num_dofs})"):
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
with Timer(f"~~Contact: Assemble vector ({num_dofs})"):
b = assemble_vector(linear_form, mpc)
apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(b, bcs)
# Solve Linear problem
opts = PETSc.Options()
opts["ksp_rtol"] = 1.0e-8
opts["pc_type"] = "gamg"
opts["pc_gamg_type"] = "agg"
opts["pc_gamg_coarse_eq_limit"] = 1000
opts["pc_gamg_sym_graph"] = True
opts["mg_levels_ksp_type"] = "chebyshev"
opts["mg_levels_pc_type"] = "jacobi"
opts["mg_levels_esteig_ksp_type"] = "cg"
opts["matptap_via"] = "scalable"
# opts["pc_gamg_square_graph"] = 2
# opts["pc_gamg_threshold"] = 1e-2
# opts["help"] = None # List all available options
# opts["ksp_view"] = None # List progress of solver
# Create functionspace and build near nullspace
A.setNearNullSpace(null_space)
solver = PETSc.KSP().create(mesh.comm)
solver.setOperators(A)
solver.setFromOptions()
u_h = fem.Function(mpc.function_space)
with Timer("~~Contact: Solve"):
solver.solve(b, u_h.vector)
u_h.x.scatter_forward()
with Timer("~~Contact: Backsubstitution"):
mpc.backsubstitution(u_h.vector)
it = solver.getIterationNumber()
unorm = u_h.vector.norm()
num_slaves = MPI.COMM_WORLD.allreduce(mpc.num_local_slaves, op=MPI.SUM)
if mesh.comm.rank == 0:
num_dofs = V.dofmap.index_map.size_global * V.dofmap.index_map_bs
print(f"Number of dofs: {num_dofs}")
print(f"Number of slaves: {num_slaves}")
print(f"Number of iterations: {it}")
print(f"Norm of u {unorm:.5e}")
# Write solution to file
u_h.name = f"u_{celltype}_{theta:.2f}{mesh_ext}{type_ext}".format(celltype, theta, type_ext, mesh_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 scipy)")
with Timer("~~Contact: Reference problem"):
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
root = 0
with Timer("~~Contact: Compare LHS, RHS and solution"):
compare_mpc_lhs(A_org, A, mpc, root=root)
compare_mpc_rhs(L_org, 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)
list_timings(mesh.comm, [TimingType.wall])
def test_pipeline(u_from_mpc):
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 5, 5)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
# Solve Problem without MPC for reference
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
d = fem.Constant(mesh, PETSc.ScalarType(0.01))
x = ufl.SpatialCoordinate(mesh)
f = ufl.sin(2 * ufl.pi * x[0]) * ufl.sin(ufl.pi * x[1])
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx - d * ufl.inner(u, v) * ufl.dx
rhs = ufl.inner(f, v) * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Generate reference matrices
A_org = fem.petsc.assemble_matrix(bilinear_form)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
# Create multipoint constraint
def periodic_relation(x):
out_x = np.copy(x)
out_x[0] = 1 - x[0]
return out_x
def PeriodicBoundary(x):
return np.isclose(x[0], 1)
facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, PeriodicBoundary)
arg_sort = np.argsort(facets)
mt = meshtags(mesh, mesh.topology.dim - 1, facets[arg_sort], np.full(len(facets), 2, dtype=np.int32))
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V, mt, 2, periodic_relation, [], 1)
mpc.finalize()
if u_from_mpc:
uh = fem.Function(mpc.function_space)
problem = dolfinx_mpc.LinearProblem(bilinear_form, linear_form, mpc, bcs=[], u=uh,
petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
problem.solve()
root = 0
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem.A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem.b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh.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)
else:
uh = fem.Function(V)
with pytest.raises(ValueError):
problem = dolfinx_mpc.LinearProblem(bilinear_form, linear_form, mpc, bcs=[], u=uh,
petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
problem.solve()
def test_vector_possion(Nx, Ny, slave_space, master_space,
get_assemblers): # noqa: F811
assemble_matrix, assemble_vector = get_assemblers
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, Nx, Ny)
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
def boundary(x):
return np.isclose(x.T, [0, 0, 0]).all(axis=1)
# Define boundary conditions (HAS TO BE NON-MASTER NODES)
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
bdofsV = fem.locate_dofs_geometrical(V, boundary)
bc = fem.dirichletbc(u_bc, bdofsV)
bcs = [bc]
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
x = ufl.SpatialCoordinate(mesh)
f = ufl.as_vector((-5 * x[1], 7 * x[0]))
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
rhs = ufl.inner(f, v) * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Setup LU solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
# Create multipoint constraint
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {l2b([1, 0]): {l2b([1, 1]): 0.1, l2b([0.5, 1]): 0.3}}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, slave_space, master_space)
mpc.finalize()
with Timer("~TEST: Assemble matrix"):
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
with Timer("~TEST: Assemble vector"):
b = dolfinx_mpc.assemble_vector(linear_form, mpc)
dolfinx_mpc.apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(b, bcs)
solver.setOperators(A)
uh = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
# Generate reference matrices for unconstrained problem
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, 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)
list_timings(comm, [TimingType.wall])
P2 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
TH = P2 * P1
W = fem.FunctionSpace(mesh, TH)
V, V_to_W = W.sub(0).collapse()
Q, _ = W.sub(1).collapse()
def inlet_velocity_expression(x: NDArray[np.float64]) -> NDArray[np.bool_]:
return np.stack((np.sin(np.pi * np.sqrt(x[0]**2 + x[1]**2)),
5 * x[1] * np.sin(np.pi * np.sqrt(x[0]**2 + x[1]**2))))
# ----------------------Defining boundary conditions----------------------
# Inlet velocity Dirichlet BC
inlet_velocity = fem.Function(V)
inlet_velocity.interpolate(inlet_velocity_expression)
inlet_velocity.x.scatter_forward()
W0 = W.sub(0)
dofs = fem.locate_dofs_topological((W0, V), 1, mt.find(3))
bc1 = fem.dirichletbc(inlet_velocity, dofs, W0)
# Collect Dirichlet boundary conditions
bcs = [bc1]
# Slip conditions for walls
n = dolfinx_mpc.utils.create_normal_approximation(V, mt, 1)
with Timer("~Stokes: Create slip constraint"):
mpc = MultiPointConstraint(W)
mpc.create_slip_constraint(W.sub(0), (mt, 1), n, bcs=bcs)
mpc.finalize()
def demo_periodic3D(celltype: CellType):
# Create mesh and finite element
if celltype == CellType.tetrahedron:
# Tet setup
N = 10
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)
V = fem.VectorFunctionSpace(mesh, ("CG", 1))
else:
# Hex setup
N = 10
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
V = fem.VectorFunctionSpace(mesh, ("CG", 2))
def dirichletboundary(x: NDArray[np.float64]) -> NDArray[np.bool_]:
return np.logical_or(
np.logical_or(np.isclose(x[1], 0), np.isclose(x[1], 1)),
np.logical_or(np.isclose(x[2], 0), np.isclose(x[2], 1)))
# Create Dirichlet boundary condition
zero = PETSc.ScalarType([0, 0, 0])
geometrical_dofs = fem.locate_dofs_geometrical(V, dirichletboundary)
bc = fem.dirichletbc(zero, geometrical_dofs, V)
bcs = [bc]
def PeriodicBoundary(x):
return np.isclose(x[0], 1)
facets = locate_entities_boundary(mesh, mesh.topology.dim - 1,
PeriodicBoundary)
arg_sort = np.argsort(facets)
mt = meshtags(mesh, mesh.topology.dim - 1, facets[arg_sort],
np.full(len(facets), 2, dtype=np.int32))
def periodic_relation(x):
out_x = np.zeros(x.shape)
out_x[0] = 1 - x[0]
out_x[1] = x[1]
out_x[2] = x[2]
return out_x
with Timer("~~Periodic: Compute mpc condition"):
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V.sub(0), mt, 2,
periodic_relation, bcs, 1)
mpc.finalize()
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx
x = SpatialCoordinate(mesh)
dx_ = x[0] - 0.9
dy_ = x[1] - 0.5
dz_ = x[2] - 0.1
f = as_vector((x[0] * sin(5.0 * pi * x[1]) +
1.0 * exp(-(dx_ * dx_ + dy_ * dy_ + dz_ * dz_) / 0.02),
0.1 * dx_ * dz_, 0.1 * dx_ * dy_))
rhs = inner(f, v) * dx
petsc_options: Dict[str, Union[str, float, int]]
if complex_mode:
rtol = 1e-16
petsc_options = {"ksp_type": "preonly", "pc_type": "lu"}
else:
rtol = 1e-8
petsc_options = {
"ksp_type": "cg",
"ksp_rtol": rtol,
"pc_type": "hypre",
"pc_hypre_typ": "boomeramg",
"pc_hypre_boomeramg_max_iter": 1,
"pc_hypre_boomeramg_cycle_type": "v",
"pc_hypre_boomeramg_print_statistics": 1
}
problem = LinearProblem(a, rhs, mpc, bcs, petsc_options=petsc_options)
u_h = problem.solve()
# --------------------VERIFICATION-------------------------
print("----Verification----")
u_ = fem.Function(V)
u_.x.array[:] = 0
org_problem = fem.petsc.LinearProblem(a,
rhs,
u=u_,
bcs=bcs,
petsc_options=petsc_options)
with Timer("~Periodic: Unconstrained solve"):
org_problem.solve()
it = org_problem.solver.getIterationNumber()
print(f"Unconstrained solver iterations: {it}")
# Write solutions to file
ext = "tet" if celltype == CellType.tetrahedron else "hex"
u_.name = "u_" + ext + "_unconstrained"
# NOTE: Workaround as tabulate dof coordinates does not like extra ghosts
u_out = fem.Function(V)
old_local = u_out.x.map.size_local * u_out.x.bs
old_ghosts = u_out.x.map.num_ghosts * u_out.x.bs
mpc_local = u_h.x.map.size_local * u_h.x.bs
assert (old_local == mpc_local)
u_out.x.array[:old_local + old_ghosts] = u_h.x.array[:mpc_local +
old_ghosts]
u_out.name = "u_" + ext
fname = f"results/demo_periodic3d_{ext}.bp"
out_periodic = VTXWriter(MPI.COMM_WORLD, fname, u_out)
out_periodic.write(0)
out_periodic.close()
root = 0
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(org_problem.A,
problem.A,
mpc,
root=root)
dolfinx_mpc.utils.compare_mpc_rhs(org_problem.b,
problem.b,
mpc,
root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(org_problem.A, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(org_problem.b, root=root)
u_mpc = dolfinx_mpc.utils.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_surface_integrals(get_assemblers): # noqa: F811
assemble_matrix, assemble_vector = get_assemblers
N = 4
mesh = create_unit_square(MPI.COMM_WORLD, N, N)
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
# Fixed Dirichlet BC on the left wall
def left_wall(x):
return np.isclose(x[0], np.finfo(float).eps)
fdim = mesh.topology.dim - 1
left_facets = locate_entities_boundary(mesh, fdim, left_wall)
bc_dofs = fem.locate_dofs_topological(V, 1, left_facets)
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
bc = fem.dirichletbc(u_bc, bc_dofs)
bcs = [bc]
# Traction on top of domain
def top(x):
return np.isclose(x[1], 1)
top_facets = locate_entities_boundary(mesh, 1, top)
arg_sort = np.argsort(top_facets)
mt = meshtags(mesh, fdim, top_facets[arg_sort], np.full(len(top_facets), 3, dtype=np.int32))
ds = ufl.Measure("ds", domain=mesh, subdomain_data=mt, subdomain_id=3)
g = fem.Constant(mesh, PETSc.ScalarType((0, -9.81e2)))
# Elasticity parameters
E = PETSc.ScalarType(1.0e4)
nu = 0.0
mu = fem.Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = fem.Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
# Stress computation
def sigma(v):
return (2.0 * mu * ufl.sym(ufl.grad(v))
+ lmbda * ufl.tr(ufl.sym(ufl.grad(v))) * ufl.Identity(len(v)))
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = ufl.inner(sigma(u), ufl.grad(v)) * ufl.dx
rhs = ufl.inner(fem.Constant(mesh, PETSc.ScalarType((0, 0))), v) * ufl.dx\
+ ufl.inner(g, v) * ds
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Setup LU solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
# Setup multipointconstraint
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {}
for i in range(1, N):
s_m_c[l2b([1, i / N])] = {l2b([1, 1]): 0.8}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, 1, 1)
mpc.finalize()
with Timer("~TEST: Assemble matrix old"):
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
with Timer("~TEST: Assemble vector"):
b = assemble_vector(linear_form, mpc)
dolfinx_mpc.apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(b, bcs)
solver.setOperators(A)
uh = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
# Write solution to file
# u_h = dolfinx.Function(mpc.function_space)
# u_h.vector.setArray(uh.array)
# u_h.name = "u_mpc"
# outfile = dolfinx.io.XDMFFile(MPI.COMM_WORLD, "output/uh.xdmf", "w")
# outfile.write_mesh(mesh)
# outfile.write_function(u_h)
# outfile.close()
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
# Generate reference matrices and unconstrained solution
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, 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)
list_timings(comm, [TimingType.wall])
def __init__(self,
a: ufl.Form,
L: ufl.Form,
bcs: typing.List[fem.DirichletBC] = [],
u: fem.Function = None,
petsc_options={},
form_compiler_parameters={},
jit_parameters={}):
"""Initialize solver for a linear variational problem.
Parameters
----------
a
A bilinear UFL form, the left hand side of the variational problem.
L
A linear UFL form, the right hand side of the variational problem.
bcs
A list of Dirichlet boundary conditions.
u
The solution function. It will be created if not provided.
petsc_options
Parameters that is passed to the linear algebra backend PETSc.
For available choices for the 'petsc_options' kwarg, see the
`PETSc-documentation <https://www.mcs.anl.gov/petsc/documentation/index.html>`.
form_compiler_parameters
Parameters used in FFCx compilation of this form. Run `ffcx --help` at
the commandline to see all available options. Takes priority over all
other parameter values, except for `scalar_type` which is determined by
DOLFINx.
jit_parameters
Parameters used in CFFI JIT compilation of C code generated by FFCx.
See `python/dolfinx/jit.py` for all available parameters.
Takes priority over all other parameter values.
.. code-block:: python
problem = LinearProblem(a, L, [bc0, bc1], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
"""
self._a = fem.Form(a,
form_compiler_parameters=form_compiler_parameters,
jit_parameters=jit_parameters)
self._A = fem.create_matrix(self._a)
self._L = fem.Form(L,
form_compiler_parameters=form_compiler_parameters,
jit_parameters=jit_parameters)
self._b = fem.create_vector(self._L)
if u is None:
# Extract function space from TrialFunction (which is at the
# end of the argument list as it is numbered as 1, while the
# Test function is numbered as 0)
self.u = fem.Function(a.arguments()[-1].ufl_function_space())
else:
self.u = u
self.bcs = bcs
self._solver = PETSc.KSP().create(
self.u.function_space.mesh.mpi_comm())
self._solver.setOperators(self._A)
# Give PETSc solver options a unique prefix
solver_prefix = "dolfinx_solve_{}".format(id(self))
self._solver.setOptionsPrefix(solver_prefix)
# Set PETSc options
opts = PETSc.Options()
opts.prefixPush(solver_prefix)
for k, v in petsc_options.items():
opts[k] = v
opts.prefixPop()
self._solver.setFromOptions()
def test_linearproblem(master_point):
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 3, 5)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
# Solve Problem without MPC for reference
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
d = fem.Constant(mesh, PETSc.ScalarType(1.5))
c = fem.Constant(mesh, PETSc.ScalarType(2))
x = ufl.SpatialCoordinate(mesh)
f = c * ufl.sin(2 * ufl.pi * x[0]) * ufl.sin(ufl.pi * x[1])
g = fem.Function(V)
g.interpolate(lambda x: np.sin(x[0]) * x[1])
h = fem.Function(V)
h.interpolate(lambda x: 2 + x[1] * x[0])
a = d * g * ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
rhs = h * ufl.inner(f, v) * ufl.dx
# Generate reference matrices
A_org = fem.petsc.assemble_matrix(fem.form(a))
A_org.assemble()
L_org = fem.petsc.assemble_vector(fem.form(rhs))
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
# Create multipoint constraint
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {
l2b([1, 0]): {
l2b([0, 1]): 0.43,
l2b([1, 1]): 0.11
},
l2b([0, 0]): {
l2b(master_point): 0.69
}
}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c)
mpc.finalize()
problem = dolfinx_mpc.LinearProblem(a,
rhs,
mpc,
bcs=[],
petsc_options={
"ksp_type": "preonly",
"pc_type": "lu"
})
uh = problem.solve()
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem._A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem._b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh.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)
list_timings(comm, [TimingType.wall])
def calculate_norm_tensor_function(C):
norm: float = 0.0
for i in range(3):
for j in range(3):
norm += fem.assemble_scalar(fem.form(C[i, j] * C[i, j] * dx))
return np.sqrt(norm)
# mesh and inputs
comm = MPI.COMM_WORLD
mesh = create_unit_cube(comm, 2, 2, 2, ctype.tetrahedron)
mesh.topology.create_connectivity(2, 3)
V = fem.VectorFunctionSpace(mesh, ("CR", 1))
Vp = fem.VectorFunctionSpace(mesh, ("CG", 1))
u = fem.Function(V, name="u")
Cn = fem.Function(V, name="Cn")
uplot = fem.Function(Vp, name="disp")
v = TestFunction(V)
du = TrialFunction(V)
WCv = TensorElement(
"Quadrature",
mesh.ufl_cell(),
degree=metadata["quadrature_degree"],
quad_scheme="default",
)
Ws = TensorElement(
"Quadrature",
mesh.ufl_cell(),
degree=metadata["quadrature_degree"],
def nitsche_ufl(mesh: dmesh.Mesh, mesh_data: Tuple[_cpp.mesh.MeshTags_int32, int, int],
physical_parameters: dict = {}, nitsche_parameters: Dict[str, float] = {},
plane_loc: float = 0.0, vertical_displacement: float = -0.1,
nitsche_bc: bool = True, quadrature_degree: int = 5, form_compiler_params: Dict = {},
jit_params: Dict = {}, petsc_options: Dict = {}, newton_options: Dict = {}) -> _fem.Function:
"""
Use UFL to compute the one sided contact problem with a mesh coming into contact
with a rigid surface (not meshed).
Parameters
==========
mesh
The input mesh
mesh_data
A triplet with a mesh tag for facets and values v0, v1. v0 should be the value in the mesh tags
for facets to apply a Dirichlet condition on. v1 is the value for facets which should have applied
a contact condition on
physical_parameters
Optional dictionary with information about the linear elasticity problem.
Valid (key, value) tuples are: ('E': float), ('nu', float), ('strain', bool)
nitsche_parameters
Optional dictionary with information about the Nitsche configuration.
Valid (keu, value) tuples are: ('gamma', float), ('theta', float) where theta can be -1, 0 or 1 for
skew-symmetric, penalty like or symmetric enforcement of Nitsche conditions
plane_loc
The location of the plane in y-coordinate (2D) and z-coordinate (3D)
vertical_displacement
The amount of verticial displacment enforced on Dirichlet boundary
nitsche_bc
Use Nitche's method to enforce Dirichlet boundary conditions
quadrature_degree
The quadrature degree to use for the custom contact kernels
form_compiler_params
Parameters used in FFCX compilation of this form. Run `ffcx --help` at
the commandline to see all available options. Takes priority over all
other parameter values, except for `scalar_type` which is determined by
DOLFINX.
jit_params
Parameters used in CFFI JIT compilation of C code generated by FFCX.
See https://github.com/FEniCS/dolfinx/blob/main/python/dolfinx/jit.py
for all available parameters. Takes priority over all other parameter values.
petsc_options
Parameters that is passed to the linear algebra backend
PETSc. For available choices for the 'petsc_options' kwarg,
see the `PETSc-documentation
<https://petsc4py.readthedocs.io/en/stable/manual/ksp/>`
newton_options
Dictionary with Newton-solver options. Valid (key, item) tuples are:
("atol", float), ("rtol", float), ("convergence_criterion", "str"),
("max_it", int), ("error_on_nonconvergence", bool), ("relaxation_parameter", float)
"""
# Compute lame parameters
plane_strain = physical_parameters.get("strain", False)
E = physical_parameters.get("E", 1e3)
nu = physical_parameters.get("nu", 0.1)
mu_func, lambda_func = lame_parameters(plane_strain)
mu = mu_func(E, nu)
lmbda = lambda_func(E, nu)
sigma = sigma_func(mu, lmbda)
# Nitche parameters and variables
theta = nitsche_parameters.get("theta", 1)
gamma = nitsche_parameters.get("gamma", 1)
(facet_marker, top_value, bottom_value) = mesh_data
assert(facet_marker.dim == mesh.topology.dim - 1)
# Normal vector pointing into plane (but outward of the body coming into contact)
# Similar to computing the normal by finding the gap vector between two meshes
n_vec = np.zeros(mesh.geometry.dim)
n_vec[mesh.geometry.dim - 1] = -1
n_2 = ufl.as_vector(n_vec) # Normal of plane (projection onto other body)
# Scaled Nitsche parameter
h = ufl.CellDiameter(mesh)
gamma_scaled = gamma * E / h
# Mimicking the plane y=-plane_loc
x = ufl.SpatialCoordinate(mesh)
gap = x[mesh.geometry.dim - 1] + plane_loc
g_vec = [i for i in range(mesh.geometry.dim)]
g_vec[mesh.geometry.dim - 1] = gap
V = _fem.VectorFunctionSpace(mesh, ("CG", 1))
u = _fem.Function(V)
v = ufl.TestFunction(V)
metadata = {"quadrature_degree": quadrature_degree}
dx = ufl.Measure("dx", domain=mesh)
ds = ufl.Measure("ds", domain=mesh, metadata=metadata,
subdomain_data=facet_marker)
a = ufl.inner(sigma(u), epsilon(v)) * dx
zero = np.asarray([0, ] * mesh.geometry.dim, dtype=_PETSc.ScalarType)
L = ufl.inner(_fem.Constant(mesh, zero), v) * dx
# Derivation of one sided Nitsche with gap function
n = ufl.FacetNormal(mesh)
def sigma_n(v):
# NOTE: Different normals, see summary paper
return ufl.dot(sigma(v) * n, n_2)
F = a - theta / gamma_scaled * sigma_n(u) * sigma_n(v) * ds(bottom_value) - L
F += 1 / gamma_scaled * R_minus(sigma_n(u) + gamma_scaled * (gap - ufl.dot(u, n_2))) * \
(theta * sigma_n(v) - gamma_scaled * ufl.dot(v, n_2)) * ds(bottom_value)
# Compute corresponding Jacobian
du = ufl.TrialFunction(V)
q = sigma_n(u) + gamma_scaled * (gap - ufl.dot(u, n_2))
J = ufl.inner(sigma(du), epsilon(v)) * ufl.dx - theta / gamma_scaled * sigma_n(du) * sigma_n(v) * ds(bottom_value)
J += 1 / gamma_scaled * 0.5 * (1 - ufl.sign(q)) * (sigma_n(du) - gamma_scaled * ufl.dot(du, n_2)) * \
(theta * sigma_n(v) - gamma_scaled * ufl.dot(v, n_2)) * ds(bottom_value)
# Nitsche for Dirichlet, another theta-scheme.
# https://doi.org/10.1016/j.cma.2018.05.024
if nitsche_bc:
disp_vec = np.zeros(mesh.geometry.dim)
disp_vec[mesh.geometry.dim - 1] = vertical_displacement
u_D = ufl.as_vector(disp_vec)
F += - ufl.inner(sigma(u) * n, v) * ds(top_value)\
- theta * ufl.inner(sigma(v) * n, u - u_D) * \
ds(top_value) + gamma_scaled / h * ufl.inner(u - u_D, v) * ds(top_value)
bcs = []
J += - ufl.inner(sigma(du) * n, v) * ds(top_value)\
- theta * ufl.inner(sigma(v) * n, du) * \
ds(top_value) + gamma_scaled / h * ufl.inner(du, v) * ds(top_value)
else:
# strong Dirichlet boundary conditions
def _u_D(x):
values = np.zeros((mesh.geometry.dim, x.shape[1]))
values[mesh.geometry.dim - 1] = vertical_displacement
return values
u_D = _fem.Function(V)
u_D.interpolate(_u_D)
u_D.name = "u_D"
u_D.x.scatter_forward()
tdim = mesh.topology.dim
dirichlet_dofs = _fem.locate_dofs_topological(V, tdim - 1, facet_marker.find(top_value))
bc = _fem.dirichletbc(u_D, dirichlet_dofs)
bcs = [bc]
# DEBUG: Write each step of Newton iterations
# Create nonlinear problem and Newton solver
# def form(self, x: _PETSc.Vec):
# x.ghostUpdate(addv=_PETSc.InsertMode.INSERT, mode=_PETSc.ScatterMode.FORWARD)
# self.i += 1
# xdmf.write_function(u, self.i)
# setattr(_fem.petsc.NonlinearProblem, "form", form)
problem = _fem.petsc.NonlinearProblem(F, u, bcs, J=J, jit_params=jit_params,
form_compiler_params=form_compiler_params)
# DEBUG: Write each step of Newton iterations
# problem.i = 0
# xdmf = _io.XDMFFile(mesh.comm, "results/tmp_sol.xdmf", "w")
# xdmf.write_mesh(mesh)
solver = _nls.petsc.NewtonSolver(mesh.comm, problem)
null_space = rigid_motions_nullspace(V)
solver.A.setNearNullSpace(null_space)
# Set Newton solver options
solver.atol = newton_options.get("atol", 1e-9)
solver.rtol = newton_options.get("rtol", 1e-9)
solver.convergence_criterion = newton_options.get("convergence_criterion", "incremental")
solver.max_it = newton_options.get("max_it", 50)
solver.error_on_nonconvergence = newton_options.get("error_on_nonconvergence", True)
solver.relaxation_parameter = newton_options.get("relaxation_parameter", 0.8)
def _u_initial(x):
values = np.zeros((mesh.geometry.dim, x.shape[1]))
values[-1] = -0.01 - plane_loc
return values
# Set initial_condition:
u.interpolate(_u_initial)
# Define solver and options
ksp = solver.krylov_solver
opts = _PETSc.Options()
option_prefix = ksp.getOptionsPrefix()
# Set PETSc options
opts = _PETSc.Options()
opts.prefixPush(option_prefix)
for k, v in petsc_options.items():
opts[k] = v
opts.prefixPop()
ksp.setFromOptions()
# Solve non-linear problem
_log.set_log_level(_log.LogLevel.INFO)
num_dofs_global = V.dofmap.index_map_bs * V.dofmap.index_map.size_global
with _common.Timer(f"{num_dofs_global} Solve Nitsche"):
n, converged = solver.solve(u)
u.x.scatter_forward()
if solver.error_on_nonconvergence:
assert(converged)
print(f"{num_dofs_global}, Number of interations: {n:d}")
return u
