def test_nonlinear_pde():
"""Test Newton solver for a simple nonlinear PDE"""
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 12, 5)
V = FunctionSpace(mesh, ("Lagrange", 1))
u = 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
bc = dirichletbc(
PETSc.ScalarType(1.0),
locate_dofs_geometrical(
V, lambda x: np.logical_or(np.isclose(x[0], 0.0),
np.isclose(x[0], 1.0))), V)
# Create nonlinear problem
problem = NonlinearPDEProblem(F, u, bc)
# Create Newton solver and solve
u.x.array[:] = 0.9
solver = _cpp.nls.petsc.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 < 6
# Modify boundary condition and solve again
bc.g.value[...] = 0.5
n, converged = solver.solve(u.vector)
assert converged
assert n > 0 and n < 6
def test_assemble_manifold():
"""Test assembly of poisson problem on a mesh with topological
dimension 1 but embedded in 2D (gdim=2)"""
points = np.array([[0.0, 0.0], [0.2, 0.0], [0.4, 0.0],
[0.6, 0.0], [0.8, 0.0], [1.0, 0.0]], dtype=np.float64)
cells = np.array([[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]], dtype=np.int32)
cell = ufl.Cell("interval", geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 1))
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
assert mesh.geometry.dim == 2
assert mesh.topology.dim == 1
U = FunctionSpace(mesh, ("P", 1))
u, v = ufl.TrialFunction(U), ufl.TestFunction(U)
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx(mesh)
L = ufl.inner(1.0, v) * ufl.dx(mesh)
a, L = form(a), form(L)
bcdofs = locate_dofs_geometrical(U, lambda x: np.isclose(x[0], 0.0))
bcs = [dirichletbc(PETSc.ScalarType(0), bcdofs, U)]
A = assemble_matrix(a, bcs=bcs)
A.assemble()
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[bcs])
set_bc(b, bcs)
assert np.isclose(b.norm(), 0.41231)
assert np.isclose(A.norm(), 25.0199)
def test_linear_pde():
"""Test Newton solver for a linear PDE"""
# Create mesh and function space
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 12, 12)
V = function.FunctionSpace(mesh, ("Lagrange", 1))
u = function.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 = function.Function(V)
u_bc.vector.set(1.0)
u_bc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
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)
n, converged = solver.solve(problem, u.vector)
assert converged
assert n == 1
# Increment boundary condition and solve again
u_bc.vector.set(2.0)
u_bc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
n, converged = solver.solve(problem, u.vector)
assert converged
assert n == 1
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_linear_pde():
"""Test Newton solver for a linear PDE"""
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12)
V = FunctionSpace(mesh, ("Lagrange", 1))
u = Function(V)
v = TestFunction(V)
F = inner(10.0, v) * dx - inner(grad(u), grad(v)) * dx
bc = dirichletbc(PETSc.ScalarType(1.0),
locate_dofs_geometrical(V, lambda x: np.logical_or(np.isclose(x[0], 0.0),
np.isclose(x[0], 1.0))), V)
# Create nonlinear problem
problem = NonlinearPDEProblem(F, u, bc)
# Create Newton solver and solve
solver = _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
bc.g.value[...] = PETSc.ScalarType(2.0)
n, converged = solver.solve(u.vector)
assert converged
assert n == 1
def test_overlapping_bcs():
"""Test that, when boundaries condition overlap, the last provided
boundary condition is applied"""
n = 23
mesh = create_unit_square(MPI.COMM_WORLD, n, n)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(u, v) * dx)
L = form(inner(1, v) * dx)
dofs_left = locate_dofs_geometrical(V, lambda x: x[0] < 1.0 / (2.0 * n))
dofs_top = locate_dofs_geometrical(V, lambda x: x[1] > 1.0 - 1.0 /
(2.0 * n))
dof_corner = np.array(list(set(dofs_left).intersection(set(dofs_top))),
dtype=np.int64)
# Check only one dof pair is found globally
assert len(set(np.concatenate(MPI.COMM_WORLD.allgather(dof_corner)))) == 1
bcs = [
dirichletbc(PETSc.ScalarType(0), dofs_left, V),
dirichletbc(PETSc.ScalarType(123.456), dofs_top, V)
]
A, b = create_matrix(a), create_vector(L)
assemble_matrix(A, a, bcs=bcs)
A.assemble()
# Check the diagonal (only on the rank that owns the row)
d = A.getDiagonal()
if len(dof_corner) > 0 and dof_corner[0] < V.dofmap.index_map.size_local:
assert np.isclose(d.array_r[dof_corner[0]], 1.0)
with b.localForm() as b_loc:
b_loc.set(0)
assemble_vector(b, L)
apply_lifting(b, [a], [bcs])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
b.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
if len(dof_corner) > 0:
with b.localForm() as b_loc:
assert b_loc[dof_corner[0]] == 123.456
def test_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 = function.FunctionSpace(mesh, ("Lagrange", 1))
u = function.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 = function.Function(V)
u_bc.vector.set(1.0)
u_bc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
bc = fem.DirichletBC(u_bc, fem.locate_dofs_geometrical(V, boundary))
# Create nonlinear problem
problem = NonlinearPDE_SNESProblem(F, u, bc)
u.vector.set(0.9)
u.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
b = dolfinx.cpp.la.create_vector(V.dofmap.index_map)
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
u_bc.vector.set(0.5)
u_bc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
snes.solve(None, u.vector)
assert snes.getConvergedReason() > 0
assert snes.getIterationNumber() < 6
def test_locate_dofs_geometrical():
"""Test that locate_dofs_geometrical, when passed two function
spaces, returns the correct degrees of freedom in each space"""
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)
W = FunctionSpace(mesh, P0 * P1)
V = W.sub(0).collapse()[0]
with pytest.raises(RuntimeError):
locate_dofs_geometrical(
W, lambda x: np.isclose(x.T, [0, 0, 0]).all(axis=1))
dofs = locate_dofs_geometrical(
(W.sub(0), V), lambda x: np.isclose(x.T, [0, 0, 0]).all(axis=1))
# Collect dofs (global indices) from all processes
dofs0_global = W.sub(0).dofmap.index_map.local_to_global(dofs[0])
dofs1_global = V.dofmap.index_map.local_to_global(dofs[1])
all_dofs0 = set(np.concatenate(MPI.COMM_WORLD.allgather(dofs0_global)))
all_dofs1 = set(np.concatenate(MPI.COMM_WORLD.allgather(dofs1_global)))
# Check only one dof pair is found globally
assert len(all_dofs0) == 1
assert len(all_dofs1) == 1
# On process with the dof pair
if len(dofs) == 1:
# Check correct dof returned in W
coords_W = W.tabulate_dof_coordinates()
assert np.isclose(coords_W[dofs[0][0]], [0, 0, 0]).all()
# Check correct dof returned in V
coords_V = V.tabulate_dof_coordinates()
assert np.isclose(coords_V[dofs[0][1]], [0, 0, 0]).all()
def test_newton_solver_inheritance_override_methods():
import functools
called_methods = {}
def check_is_called(method):
@functools.wraps(method)
def wrapper(*args, **kwargs):
called_methods[method.__name__] = True
return method(*args, **kwargs)
return wrapper
class CustomNewtonSolver(dolfinx.cpp.nls.NewtonSolver):
def __init__(self, comm):
super().__init__(comm)
@check_is_called
def update_solution(self, x, dx, relaxation, problem, it):
return super().update_solution(x, dx, relaxation, problem, it)
@check_is_called
def converged(self, r, problem, it):
return super().converged(r, problem, it)
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 12, 12)
V = function.FunctionSpace(mesh, ("Lagrange", 1))
u = function.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 = function.Function(V)
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 = CustomNewtonSolver(MPI.COMM_WORLD)
n, converged = solver.solve(problem, u.vector)
assert called_methods[CustomNewtonSolver.converged.__name__]
assert called_methods[CustomNewtonSolver.update_solution.__name__]
def test_nonlinear_pde_snes():
"""Test Newton solver for a simple nonlinear PDE"""
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 12, 15)
V = FunctionSpace(mesh, ("Lagrange", 1))
u = 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
u_bc = Function(V)
u_bc.x.array[:] = 1.0
bc = dirichletbc(
u_bc,
locate_dofs_geometrical(
V, lambda x: np.logical_or(np.isclose(x[0], 0.0),
np.isclose(x[0], 1.0))))
# Create nonlinear problem
problem = NonlinearPDE_SNESProblem(F, u, bc)
u.x.array[:] = 0.9
b = la.create_petsc_vector(V.dofmap.index_map, V.dofmap.index_map_bs)
J = create_matrix(problem.a)
# 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.solve(None, u.vector)
assert snes.getConvergedReason() > 0
assert snes.getIterationNumber() < 6
# Modify boundary condition and solve again
u_bc.x.array[:] = 0.6
snes.solve(None, u.vector)
assert snes.getConvergedReason() > 0
assert snes.getIterationNumber() < 6
def test_nonlinear_pde():
"""Test Newton solver for a simple nonlinear PDE"""
# Create mesh and function space
mesh = dolfinx.generation.UnitSquareMesh(dolfinx.MPI.comm_world, 12, 5)
V = function.FunctionSpace(mesh, ("Lagrange", 1))
u = dolfinx.function.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 = function.Function(V)
u_bc.vector.set(1.0)
u_bc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
bc = fem.DirichletBC(u_bc, fem.locate_dofs_geometrical(V, boundary))
# Create nonlinear problem
problem = NonlinearPDEProblem(F, u, bc)
# Create Newton solver and solve
u.vector.set(0.9)
u.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
solver = dolfinx.cpp.nls.NewtonSolver(dolfinx.MPI.comm_world)
n, converged = solver.solve(problem, u.vector)
assert converged
assert n < 6
# Modify boundary condition and solve again
u_bc.vector.set(0.5)
u_bc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
n, converged = solver.solve(problem, u.vector)
assert converged
assert n < 6
def test_nonlinear_pde():
"""Test Newton solver for a simple nonlinear PDE"""
# Create mesh and function space
mesh = dolfinx.generation.UnitSquareMesh(MPI.COMM_WORLD, 12, 5)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
u = dolfinx.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 = NonlinearPDEProblem(F, u, bc)
# Create Newton solver and solve
with u.vector.localForm() as u_local:
u_local.set(0.9)
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 < 6
# Modify boundary condition and solve again
with u_bc.vector.localForm() as u_local:
u_local.set(0.5)
n, converged = solver.solve(u.vector)
assert converged
assert n < 6
# Driving velocity condition u = (1, 0) on top boundary (y = 1)
lid_velocity = Function(W0)
lid_velocity.interpolate(lid_velocity_expression)
facets = locate_entities_boundary(mesh, 1, lid)
dofs = locate_dofs_topological((W.sub(0), V), 1, facets)
bc1 = DirichletBC(lid_velocity, dofs, W.sub(0))
# Since for this problem the pressure is only determined up to a
# constant, we pin the pressure at the point (0, 0)
zero = Function(Q)
with zero.vector.localForm() as zero_local:
zero_local.set(0.0)
dofs = locate_dofs_geometrical((W.sub(1), Q),
lambda x: np.isclose(x.T, [0, 0, 0]).all(axis=1))
bc2 = DirichletBC(zero, dofs, W.sub(1))
# Collect Dirichlet boundary conditions
bcs = [bc0, bc1, bc2]
# Define variational problem
(u, p) = ufl.TrialFunctions(W)
(v, q) = ufl.TestFunctions(W)
f = Function(W0)
zero = dolfinx.Constant(mesh, 0.0)
a = (inner(grad(u), grad(v)) + inner(p, div(v)) + inner(div(u), q)) * dx
L = inner(f, v) * dx
# Assemble LHS matrix and RHS vector
A = dolfinx.fem.assemble_matrix(a, bcs)
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])
# 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))
# 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)
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)
#FOR IMPOSED DISPLACEMENT :
if parameters['loading']['type'] == 'ID':
def ID_points(x):
return np.logical_and(
np.equal(x[1], h),
np.logical_and(np.greater_equal(x[0], -1 * n),
np.less_equal(x[0], n)))
ID_entities = dolfinx.mesh.locate_entities_boundary(mesh, 0, ID_points)
ID_dofs = dolfinx.fem.locate_dofs_topological(V_u, 0, ID_entities)
u_imposed.interpolate(lambda x: (np.zeros_like(x[0]), -1 * parameters[
'loading']['max'] * np.ones_like(x[1])))
bcs_u = [dirichletbc(u_, BC_dofs), dirichletbc(u_imposed, ID_dofs)]
dofs_alpha_left = locate_dofs_geometrical(V_alpha,
lambda x: np.isclose(x[0], -L / 2))
dofs_alpha_right = locate_dofs_geometrical(V_alpha,
lambda x: np.isclose(x[0], L / 2))
BC_dofs_alpha = dolfinx.fem.locate_dofs_topological(V_alpha, 0, BC_entities)
if parameters['loading']['type'] == 'IF':
bcs_alpha = [
dirichletbc(np.array(0., dtype=PETSc.ScalarType), BC_dofs_alpha,
V_alpha)
]
if parameters['loading']['type'] == 'ID':
ID_dofs_alpha = dolfinx.fem.locate_dofs_topological(
V_alpha, 0, ID_entities)
bcs_alpha = [
dirichletbc(
np.array(0., dtype=PETSc.ScalarType),
np.concatenate([
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])
def test_eigen_assembly(tempdir): # noqa: F811
"""Compare assembly into scipy.CSR matrix with PETSc assembly"""
def compile_eigen_csr_assembler_module():
dolfinx_pc = dolfinx.pkgconfig.parse("dolfinx")
eigen_dir = dolfinx.pkgconfig.parse("eigen3")["include_dirs"]
cpp_code_header = f"""
<%
setup_pybind11(cfg)
cfg['include_dirs'] = {dolfinx_pc["include_dirs"] + [petsc4py.get_include()]
+ [pybind11.get_include()] + [str(pybind_inc())] + eigen_dir}
cfg['compiler_args'] = ["-std=c++17", "-Wno-comment"]
cfg['libraries'] = {dolfinx_pc["libraries"]}
cfg['library_dirs'] = {dolfinx_pc["library_dirs"]}
%>
"""
cpp_code = """
#include <pybind11/pybind11.h>
#include <pybind11/eigen.h>
#include <pybind11/stl.h>
#include <vector>
#include <Eigen/Sparse>
#include <petscsys.h>
#include <dolfinx/fem/assembler.h>
#include <dolfinx/fem/DirichletBC.h>
#include <dolfinx/fem/Form.h>
template<typename T>
Eigen::SparseMatrix<T, Eigen::RowMajor>
assemble_csr(const dolfinx::fem::Form<T>& a,
const std::vector<std::shared_ptr<const dolfinx::fem::DirichletBC<T>>>& bcs)
{
std::vector<Eigen::Triplet<T>> triplets;
auto mat_add
= [&triplets](const xtl::span<const std::int32_t>& rows,
const xtl::span<const std::int32_t>& cols,
const xtl::span<const T>& v)
{
for (std::size_t i = 0; i < rows.size(); ++i)
for (std::size_t j = 0; j < cols.size(); ++j)
triplets.emplace_back(rows[i], cols[j], v[i * cols.size() + j]);
return 0;
};
dolfinx::fem::assemble_matrix(mat_add, a, bcs);
auto map0 = a.function_spaces().at(0)->dofmap()->index_map;
int bs0 = a.function_spaces().at(0)->dofmap()->index_map_bs();
auto map1 = a.function_spaces().at(1)->dofmap()->index_map;
int bs1 = a.function_spaces().at(1)->dofmap()->index_map_bs();
Eigen::SparseMatrix<T, Eigen::RowMajor> mat(
bs0 * (map0->size_local() + map0->num_ghosts()),
bs1 * (map1->size_local() + map1->num_ghosts()));
mat.setFromTriplets(triplets.begin(), triplets.end());
return mat;
}
PYBIND11_MODULE(eigen_csr, m)
{
m.def("assemble_matrix", &assemble_csr<PetscScalar>);
}
"""
path = pathlib.Path(tempdir)
open(pathlib.Path(tempdir, "eigen_csr.cpp"),
"w").write(cpp_code + cpp_code_header)
rel_path = path.relative_to(pathlib.Path(__file__).parent)
p = str(rel_path).replace("/", ".") + ".eigen_csr"
return cppimport.imp(p)
def assemble_csr_matrix(a, bcs):
"""Assemble bilinear form into an SciPy CSR matrix, in serial."""
module = compile_eigen_csr_assembler_module()
A = module.assemble_matrix(a, bcs)
if a.function_spaces[0] is a.function_spaces[1]:
for bc in bcs:
if a.function_spaces[0].contains(bc.function_space):
bc_dofs, _ = bc.dof_indices()
# See https://github.com/numpy/numpy/issues/14132
# for why we copy bc_dofs as a work-around
dofs = bc_dofs.copy()
A[dofs, dofs] = 1.0
return A
mesh = create_unit_square(MPI.COMM_SELF, 12, 12)
Q = FunctionSpace(mesh, ("Lagrange", 1))
u = ufl.TrialFunction(Q)
v = ufl.TestFunction(Q)
a = form(ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx)
bdofsQ = locate_dofs_geometrical(
Q,
lambda x: np.logical_or(np.isclose(x[0], 0.0), np.isclose(x[0], 1.0)))
bc = dirichletbc(PETSc.ScalarType(1), bdofsQ, Q)
A1 = assemble_matrix(a, [bc])
A1.assemble()
A2 = assemble_csr_matrix(a, [bc])
assert np.isclose(A1.norm(), scipy.sparse.linalg.norm(A2))
def create_dictionary_constraint(
V: fem.FunctionSpace,
slave_master_dict: typing.Dict[bytes, typing.Dict[bytes, float]],
subspace_slave: int = None,
subspace_master: int = None):
"""
Returns a multi point constraint for a given function space
and dictionary constraint.
Parameters
----------
V
The function space
slave_master_dict
The dictionary.
subspace_slave
If using mixed or vector space, and only want to use dofs from
a sub space as slave add index here.
subspace_master
Subspace index for mixed or vector spaces
Example
-------
If the dof D located at [d0,d1] should be constrained to the dofs E and
F at [e0,e1] and [f0,f1] as
D = alpha E + beta F
the dictionary should be:
{np.array([d0, d1], dtype=numpy.float64).tobytes():
{numpy.array([e0, e1], dtype=numpy.float64).tobytes(): alpha,
numpy.array([f0, f1], dtype=numpy.float64).tobytes(): beta}}
"""
dfloat = np.float64
comm = V.mesh.comm
bs = V.dofmap.index_map_bs
local_size = V.dofmap.index_map.size_local * bs
index_map = V.dofmap.index_map
owned_entities = {}
ghosted_entities = {}
non_local_entities = {}
slaves_local = {}
slaves_ghost = {}
slave_point_nd = np.zeros((3, 1), dtype=dfloat)
for i, slave_point in enumerate(slave_master_dict.keys()):
num_masters = len(list(slave_master_dict[slave_point].keys()))
# Status for current slave, -1 if not on proc, 0 if ghost, 1 if owned
slave_status = -1
# Wrap slave point as numpy array
sp = np.frombuffer(slave_point, dtype=dfloat)
for j, coord in enumerate(sp):
slave_point_nd[j] = coord
slave_point_nd[len(sp):] = 0
if subspace_slave is None:
slave_dofs = fem.locate_dofs_geometrical(V,
close_to(slave_point_nd))
else:
Vsub = V.sub(subspace_slave).collapse()[0]
slave_dofs = fem.locate_dofs_geometrical(
(V.sub(subspace_slave), Vsub), close_to(slave_point_nd))[0]
if len(slave_dofs) == 1:
# Decide if slave is ghost or not
if slave_dofs[0] < local_size:
slaves_local[i] = slave_dofs[0]
owned_entities[i] = {
"masters": np.full(num_masters, -1, dtype=np.int64),
"coeffs": np.full(num_masters, -1, dtype=np.float64),
"owners": np.full(num_masters, -1, dtype=np.int32),
"master_count": 0,
"local_index": []
}
slave_status = 1
else:
slaves_ghost[i] = slave_dofs[0]
ghosted_entities[i] = {
"masters": np.full(num_masters, -1, dtype=np.int64),
"coeffs": np.full(num_masters, -1, dtype=np.float64),
"owners": np.full(num_masters, -1, dtype=np.int32),
"master_count": 0,
"local_index": []
}
slave_status = 0
elif len(slave_dofs) > 1:
raise RuntimeError("Multiple slaves found at same point. " +
"You should use sub-space locators.")
# Wrap as list to ensure order later
master_points = list(slave_master_dict[slave_point].keys())
master_points_nd = np.zeros((3, len(master_points)), dtype=dfloat)
for (j, master_point) in enumerate(master_points):
# Wrap bytes as numpy array
for k, coord in enumerate(np.frombuffer(master_point,
dtype=dfloat)):
master_points_nd[k, j] = coord
if subspace_master is None:
master_dofs = fem.locate_dofs_geometrical(
V, close_to(master_points_nd[:, j:j + 1]))
else:
Vsub = V.sub(subspace_master).collapse()[0]
master_dofs = fem.locate_dofs_geometrical(
(V.sub(subspace_master), Vsub),
close_to(master_points_nd[:, j:j + 1]))[0]
# Only add masters owned by this processor
master_dofs = master_dofs[master_dofs < local_size]
if len(master_dofs) == 1:
master_block = master_dofs[0] // bs
master_rem = master_dofs % bs
glob_master = index_map.local_to_global([master_block])[0]
if slave_status == -1:
if i in non_local_entities.keys():
non_local_entities[i]["masters"].append(glob_master *
bs +
master_rem)
non_local_entities[i]["coeffs"].append(
slave_master_dict[slave_point][master_point])
non_local_entities[i]["owners"].append(comm.rank),
non_local_entities[i]["local_index"].append(j)
else:
non_local_entities[i] = {
"masters": [glob_master * bs + master_rem],
"coeffs":
[slave_master_dict[slave_point][master_point]],
"owners": [comm.rank],
"local_index": [j]
}
elif slave_status == 0:
ghosted_entities[i]["masters"][
j] = glob_master * bs + master_rem
ghosted_entities[i]["owners"][j] = comm.rank
ghosted_entities[i]["coeffs"][j] = slave_master_dict[
slave_point][master_point]
ghosted_entities[i]["local_index"].append(j)
elif slave_status == 1:
owned_entities[i]["masters"][
j] = glob_master * bs + master_rem
owned_entities[i]["owners"][j] = comm.rank
owned_entities[i]["coeffs"][j] = slave_master_dict[
slave_point][master_point]
owned_entities[i]["local_index"].append(j)
else:
raise RuntimeError(
"Invalid slave status: {0:d} (-1,0,1 are valid options)"
.format(slave_status))
elif len(master_dofs) > 1:
raise RuntimeError(
"Multiple masters found at same point. You should use sub-space locators."
)
# Send the ghost and owned entities to processor 0 to gather them
data_to_send = [owned_entities, ghosted_entities, non_local_entities]
if comm.rank != 0:
comm.send(data_to_send, dest=0, tag=1)
del owned_entities, ghosted_entities, non_local_entities
# Gather all info on proc 0 and sort data
owned_slaves, ghosted_slaves = None, None
if comm.rank == 0:
recv = {0: data_to_send}
for proc in range(1, comm.size):
recv[proc] = comm.recv(source=proc, tag=1)
for proc in range(comm.size):
# Loop through all masters
other_procs = np.arange(comm.size)
other_procs = other_procs[other_procs != proc]
# Loop through all owned slaves and ghosts, and update
# the master entries
for pair in [[0, 1], [1, 0]]:
i, j = pair
for slave in recv[proc][i].keys():
for o_proc in other_procs:
# If slave is ghost on other proc add local masters
if slave in recv[o_proc][j].keys():
# Update master with possible entries from ghost
o_masters = recv[o_proc][j][slave]["local_index"]
for o_master in o_masters:
recv[proc][i][slave]["masters"][
o_master] = recv[o_proc][j][slave][
"masters"][o_master]
recv[proc][i][slave]["coeffs"][
o_master] = recv[o_proc][j][slave][
"coeffs"][o_master]
recv[proc][i][slave]["owners"][
o_master] = recv[o_proc][j][slave][
"owners"][o_master]
# If proc only has master, but not the slave
if slave in recv[o_proc][2].keys():
o_masters = recv[o_proc][2][slave]["local_index"]
# As non owned indices only store non-zero entries
for k, o_master in enumerate(o_masters):
recv[proc][i][slave]["masters"][
o_master] = recv[o_proc][2][slave][
"masters"][k]
recv[proc][i][slave]["coeffs"][
o_master] = recv[o_proc][2][slave][
"coeffs"][k]
recv[proc][i][slave]["owners"][
o_master] = recv[o_proc][2][slave][
"owners"][k]
if proc == comm.rank:
owned_slaves = recv[proc][0]
ghosted_slaves = recv[proc][1]
else:
# If no owned masters, do not send masters
if len(recv[proc][0].keys()) > 0:
comm.send(recv[proc][0], dest=proc, tag=55)
if len(recv[proc][1].keys()) > 0:
comm.send(recv[proc][1], dest=proc, tag=66)
else:
if len(slaves_local.keys()) > 0:
owned_slaves = comm.recv(source=0, tag=55)
if len(slaves_ghost.keys()) > 0:
ghosted_slaves = comm.recv(source=0, tag=66)
# Flatten slaves (local)
slaves, masters, coeffs, owners, offsets = [], [], [], [], [0]
for slave_index in slaves_local.keys():
slaves.append(slaves_local[slave_index])
masters.extend(owned_slaves[slave_index]["masters"]) # type: ignore
owners.extend(owned_slaves[slave_index]["owners"]) # type: ignore
coeffs.extend(owned_slaves[slave_index]["coeffs"]) # type: ignore
offsets.append(len(masters))
for slave_index in slaves_ghost.keys():
slaves.append(slaves_ghost[slave_index])
masters.extend(ghosted_slaves[slave_index]["masters"]) # type: ignore
owners.extend(ghosted_slaves[slave_index]["owners"]) # type: ignore
coeffs.extend(ghosted_slaves[slave_index]["coeffs"]) # type: ignore
offsets.append(len(masters))
return (np.asarray(slaves,
dtype=np.int32), np.asarray(masters, dtype=np.int64),
np.asarray(coeffs, dtype=PETSc.ScalarType),
np.asarray(owners,
dtype=np.int32), np.asarray(offsets, dtype=np.int32))
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)
dx = Measure("dx", domain=mesh, subdomain_data=ct)
a = inner(sigma(u), grad(v)) * dx
x = SpatialCoordinate(mesh)
rhs = inner(Constant(mesh, PETSc.ScalarType((0, 0))), v) * dx
# Set boundary conditions
u_push = np.array([0.1, 0], dtype=PETSc.ScalarType)
dofs = locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 0))
bc_push = dirichletbc(u_push, dofs, V)
u_fix = np.array([0, 0], dtype=PETSc.ScalarType)
bc_fix = dirichletbc(
u_fix, locate_dofs_geometrical(V, lambda x: np.isclose(x[0], 2.1)), V)
bcs = [bc_push, bc_fix]
def gather_dof_coordinates(V: FunctionSpace, dofs: np.ndarray):
"""
Distributes the dof coordinates of this subset of dofs to all processors
"""
x = V.tabulate_dof_coordinates()
local_dofs = dofs[dofs < V.dofmap.index_map.size_local *
V.dofmap.index_map_bs]
coords = x[local_dofs]
u_ = Function(V_u, name="Boundary Displacement")
zero_u = Function(V_u, name=" Boundary Displacement")
alpha = Function(V_alpha, name="Damage")
zero_alpha = Function(V_alpha, name="Damage Boundary Field")
state = {"u": u, "alpha": alpha}
# need upper/lower bound for the damage field
alpha_lb = Function(V_alpha, name="Lower bound")
alpha_ub = Function(V_alpha, name="Upper bound")
# Measures
dx = ufl.Measure("dx", domain=mesh)
ds = ufl.Measure("ds", domain=mesh)
dofs_alpha_left = locate_dofs_geometrical(V_alpha,
lambda x: np.isclose(x[0], 0.0))
dofs_alpha_right = locate_dofs_geometrical(V_alpha,
lambda x: np.isclose(x[0], Lx))
dofs_u_left = locate_dofs_geometrical(V_u, lambda x: np.isclose(x[0], 0.0))
dofs_u_right = locate_dofs_geometrical(V_u, lambda x: np.isclose(x[0], Lx))
# Set Bcs Function
zero_u.interpolate(lambda x: (np.zeros_like(x[0]), np.zeros_like(x[1])))
zero_alpha.interpolate((lambda x: np.zeros_like(x[0])))
u_.interpolate(lambda x: (np.ones_like(x[0]), 0 * np.ones_like(x[1])))
alpha_lb.interpolate(lambda x: np.zeros_like(x[0]))
alpha_ub.interpolate(lambda x: np.ones_like(x[0]))
for f in [zero_u, zero_alpha, u_, alpha_lb, alpha_ub]:
f.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
def reference_periodic(tetra: bool,
r_lvl: int = 0,
out_hdf5: h5py.File = None,
xdmf: bool = False,
boomeramg: bool = False,
kspview: bool = False,
degree: int = 1):
# Create mesh and finite element
if tetra:
# Tet setup
N = 3
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)
for i in range(r_lvl):
mesh.topology.create_entities(mesh.topology.dim - 2)
mesh = refine(mesh, redistribute=True)
N *= 2
else:
# Hex setup
N = 3
for i in range(r_lvl):
N *= 2
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
V = FunctionSpace(mesh, ("CG", degree))
# Create Dirichlet boundary condition
def dirichletboundary(x):
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)))
mesh.topology.create_connectivity(2, 1)
geometrical_dofs = locate_dofs_geometrical(V, dirichletboundary)
bc = dirichletbc(PETSc.ScalarType(0), geometrical_dofs, V)
bcs = [bc]
# 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 = x[0] * sin(5.0 * pi * x[1]) + 1.0 * exp(
-(dx_ * dx_ + dy_ * dy_ + dz_ * dz_) / 0.02)
rhs = inner(f, v) * dx
# Assemble rhs, RHS and apply lifting
bilinear_form = form(a)
linear_form = form(rhs)
A_org = assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = assemble_vector(linear_form)
apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(L_org, bcs)
# Create PETSc nullspace
nullspace = PETSc.NullSpace().create(constant=True)
PETSc.Mat.setNearNullSpace(A_org, nullspace)
# Set PETSc options
opts = PETSc.Options()
if boomeramg:
opts["ksp_type"] = "cg"
opts["ksp_rtol"] = 1.0e-5
opts["pc_type"] = "hypre"
opts['pc_hypre_type'] = 'boomeramg'
opts["pc_hypre_boomeramg_max_iter"] = 1
opts["pc_hypre_boomeramg_cycle_type"] = "v"
# opts["pc_hypre_boomeramg_print_statistics"] = 1
else:
opts["ksp_type"] = "cg"
opts["ksp_rtol"] = 1.0e-12
opts["pc_type"] = "gamg"
opts["pc_gamg_type"] = "agg"
opts["pc_gamg_sym_graph"] = True
# Use Chebyshev smoothing for multigrid
opts["mg_levels_ksp_type"] = "richardson"
opts["mg_levels_pc_type"] = "sor"
# opts["help"] = None # List all available options
# opts["ksp_view"] = None # List progress of solver
# Initialize PETSc solver, set options and operator
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setFromOptions()
solver.setOperators(A_org)
# Solve linear problem
u_ = Function(V)
start = perf_counter()
with Timer("Solve"):
solver.solve(L_org, u_.vector)
end = perf_counter()
u_.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
if kspview:
solver.view()
it = solver.getIterationNumber()
num_dofs = V.dofmap.index_map.size_global * V.dofmap.index_map_bs
if out_hdf5 is not None:
d_set = out_hdf5.get("its")
d_set[r_lvl] = it
d_set = out_hdf5.get("num_dofs")
d_set[r_lvl] = num_dofs
d_set = out_hdf5.get("solve_time")
d_set[r_lvl, MPI.COMM_WORLD.rank] = end - start
if MPI.COMM_WORLD.rank == 0:
print("Rlvl {0:d}, Iterations {1:d}".format(r_lvl, it))
# Output solution to XDMF
if xdmf:
ext = "tet" if tetra else "hex"
fname = "results/reference_periodic_{0:d}_{1:s}.xdmf".format(
r_lvl, ext)
u_.name = "u_" + ext + "_unconstrained"
with XDMFFile(MPI.COMM_WORLD, fname, "w") as out_periodic:
out_periodic.write_mesh(mesh)
out_periodic.write_function(
u_, 0.0,
"Xdmf/Domain/" + "Grid[@Name='{0:s}'][1]".format(mesh.name))
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)
# Create function space
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = form(inner(sigma(u), grad(v)) * dx)
L = form(inner(f, v) * dx)
# Set up boundary condition on inner surface
bc = dirichletbc(
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])
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
