def test_subdomains(compile_args):
cell = ufl.triangle
element = ufl.FiniteElement("Lagrange", cell, 1)
u, v = ufl.TrialFunction(element), ufl.TestFunction(element)
a0 = ufl.inner(u, v) * ufl.dx + ufl.inner(u, v) * ufl.dx(2)
a1 = ufl.inner(u, v) * ufl.dx(2) + ufl.inner(u, v) * ufl.dx
a2 = ufl.inner(u, v) * ufl.dx(2) + ufl.inner(u, v) * ufl.dx(1)
a3 = ufl.inner(u, v) * ufl.ds(210) + ufl.inner(u, v) * ufl.ds(0)
forms = [a0, a1, a2, a3]
compiled_forms, module, code = ffcx.codegeneration.jit.compile_forms(
forms,
parameters={'scalar_type': 'double'},
cffi_extra_compile_args=compile_args)
for f, compiled_f in zip(forms, compiled_forms):
assert compiled_f.rank == len(f.arguments())
form0 = compiled_forms[0]
ids = form0.integral_ids(module.lib.cell)
assert ids[0] == -1 and ids[1] == 2
form1 = compiled_forms[1]
ids = form1.integral_ids(module.lib.cell)
assert ids[0] == -1 and ids[1] == 2
form2 = compiled_forms[2]
ids = form2.integral_ids(module.lib.cell)
assert ids[0] == 1 and ids[1] == 2
form3 = compiled_forms[3]
assert form3.num_integrals(module.lib.cell) == 0
ids = form3.integral_ids(module.lib.exterior_facet)
assert ids[0] == 0 and ids[1] == 210
def test_additivity(mode):
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
f1 = Function(V)
f2 = Function(V)
f3 = Function(V)
f1.x.array[:] = 1.0
f2.x.array[:] = 2.0
f3.x.array[:] = 3.0
j1 = ufl.inner(f1, f1) * ufl.dx(mesh)
j2 = ufl.inner(f2, f2) * ufl.ds(mesh)
j3 = ufl.inner(ufl.avg(f3), ufl.avg(f3)) * ufl.dS(mesh)
# Assemble each scalar form separately
J1 = mesh.comm.allreduce(assemble_scalar(form(j1)), op=MPI.SUM)
J2 = mesh.comm.allreduce(assemble_scalar(form(j2)), op=MPI.SUM)
J3 = mesh.comm.allreduce(assemble_scalar(form(j3)), op=MPI.SUM)
# Sum forms and assemble the result
J12 = mesh.comm.allreduce(assemble_scalar(form(j1 + j2)), op=MPI.SUM)
J13 = mesh.comm.allreduce(assemble_scalar(form(j1 + j3)), op=MPI.SUM)
J23 = mesh.comm.allreduce(assemble_scalar(form(j2 + j3)), op=MPI.SUM)
J123 = mesh.comm.allreduce(assemble_scalar(form(j1 + j2 + j3)), op=MPI.SUM)
# Compare assembled values
assert (J1 + J2) == pytest.approx(J12)
assert (J1 + J3) == pytest.approx(J13)
assert (J2 + J3) == pytest.approx(J23)
assert (J1 + J2 + J3) == pytest.approx(J123)
def test_facet_integral(cell_type):
"""Test that the integral of a function over a facet is correct"""
for count in range(5):
mesh = unit_cell(cell_type)
tdim = mesh.topology.dim
V = FunctionSpace(mesh, ("Lagrange", 2))
v = Function(V)
map_f = mesh.topology.index_map(tdim - 1)
num_facets = map_f.size_local + map_f.num_ghosts
indices = np.arange(0, num_facets)
values = np.arange(0, num_facets, dtype=np.intc)
marker = MeshTags(mesh, tdim - 1, indices, values)
# Functions that will have the same integral over each facet
if cell_type == CellType.triangle:
root = 3 ** 0.25 # 4th root of 3
v.interpolate(lambda x: (x[0] - 1 / root) ** 2 + (x[1] - root / 3) ** 2)
elif cell_type == CellType.quadrilateral:
v.interpolate(lambda x: x[0] * (1 - x[0]) + x[1] * (1 - x[1]))
elif cell_type == CellType.tetrahedron:
s = 2 ** 0.5 * 3 ** (1 / 3) # side length
v.interpolate(lambda x: (x[0] - s / 2) ** 2 + (x[1] - s / 2 / np.sqrt(3)) ** 2
+ (x[2] - s * np.sqrt(2 / 3) / 4) ** 2)
elif cell_type == CellType.hexahedron:
v.interpolate(lambda x: x[0] * (1 - x[0]) + x[1] * (1 - x[1]) + x[2] * (1 - x[2]))
# assert that the integral of these functions over each face are equal
out = []
for j in range(num_facets):
a = v * ds(subdomain_data=marker, subdomain_id=j)
result = fem.assemble_scalar(a)
out.append(result)
assert np.isclose(result, out[0])
def test_additivity(mode):
mesh = dolfinx.UnitSquareMesh(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
V = dolfinx.FunctionSpace(mesh, ("CG", 1))
f1 = dolfinx.Function(V)
f2 = dolfinx.Function(V)
f3 = dolfinx.Function(V)
with f1.vector.localForm() as f1_local:
f1_local.set(1.0)
with f2.vector.localForm() as f2_local:
f2_local.set(2.0)
with f3.vector.localForm() as f3_local:
f3_local.set(3.0)
j1 = ufl.inner(f1, f1) * ufl.dx(mesh)
j2 = ufl.inner(f2, f2) * ufl.ds(mesh)
j3 = ufl.inner(ufl.avg(f3), ufl.avg(f3)) * ufl.dS(mesh)
# Assemble each scalar form separately
J1 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1), op=MPI.SUM)
J2 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j2), op=MPI.SUM)
J3 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j3), op=MPI.SUM)
# Sum forms and assemble the result
J12 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1 + j2), op=MPI.SUM)
J13 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1 + j3), op=MPI.SUM)
J23 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j2 + j3), op=MPI.SUM)
J123 = mesh.mpi_comm().allreduce(dolfinx.fem.assemble_scalar(j1 + j2 + j3), op=MPI.SUM)
# Compare assembled values
assert (J1 + J2) == pytest.approx(J12)
assert (J1 + J3) == pytest.approx(J13)
assert (J2 + J3) == pytest.approx(J23)
assert (J1 + J2 + J3) == pytest.approx(J123)
def test_facet_normals(cell_type):
"""Test that FacetNormal is outward facing"""
for count in range(5):
mesh = unit_cell(cell_type)
tdim = mesh.topology.dim
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
normal = FacetNormal(mesh)
v = Function(V)
map_f = mesh.topology.index_map(tdim - 1)
num_facets = map_f.size_local + map_f.num_ghosts
indices = np.arange(0, num_facets)
values = np.arange(0, num_facets, dtype=np.intc)
marker = MeshTags(mesh, tdim - 1, indices, values)
# For each facet, check that the inner product of the normal and
# the vector that has a positive normal component on only that facet
# is positive
for i in range(num_facets):
if cell_type == CellType.interval:
co = mesh.geometry.x[i]
v.interpolate(lambda x: x[0] - co[0])
if cell_type == CellType.triangle:
co = mesh.geometry.x[i]
# Vector function that is zero at `co` and points away from `co`
# so that there is no normal component on two edges and the integral
# over the other edge is 1
v.interpolate(lambda x: ((x[0] - co[0]) / 2,
(x[1] - co[1]) / 2))
elif cell_type == CellType.tetrahedron:
co = mesh.geometry.x[i]
# Vector function that is zero at `co` and points away from `co`
# so that there is no normal component on three faces and the integral
# over the other edge is 1
v.interpolate(lambda x: ((x[0] - co[0]) / 3, (x[1] - co[1]) /
3, (x[2] - co[2]) / 3))
elif cell_type == CellType.quadrilateral:
# function that is 0 on one edge and points away from that edge
# so that there is no normal component on three edges
v.interpolate(lambda x: tuple(x[j] - i % 2 if j == i // 2 else
0 * x[j] for j in range(2)))
elif cell_type == CellType.hexahedron:
# function that is 0 on one face and points away from that face
# so that there is no normal component on five faces
v.interpolate(lambda x: tuple(x[j] - i % 2 if j == i // 3 else
0 * x[j] for j in range(3)))
# assert that the integrals these functions dotted with the normal over a face
# is 1 on one face and 0 on the others
ones = 0
for j in range(num_facets):
a = inner(v, normal) * ds(subdomain_data=marker,
subdomain_id=j)
result = fem.assemble_scalar(a)
if np.isclose(result, 1):
ones += 1
else:
assert np.isclose(result, 0)
assert ones == 1
def robin_bcs(self, v, v_old):
w, w_old = as_ufl(0), as_ufl(0)
for r in self.bc_dict['robin']:
if r['type'] == 'dashpot':
if r['dir'] == 'xyz':
for i in range(len(r['id'])):
ds_ = ds(
subdomain_data=self.io.mt_b1,
subdomain_id=r['id'][i],
metadata={'quadrature_degree': self.quad_degree})
w += self.vf.deltaP_ext_robin_dashpot(
v, r['visc'], ds_)
w_old += self.vf.deltaP_ext_robin_dashpot(
v_old, r['visc'], ds_)
elif r['dir'] == 'normal': # reference normal
for i in range(len(r['id'])):
ds_ = ds(
subdomain_data=self.io.mt_b1,
subdomain_id=r['id'][i],
metadata={'quadrature_degree': self.quad_degree})
w += self.vf.deltaP_ext_robin_dashpot_normal(
v, r['visc'], ds_)
w_old += self.vf.deltaP_ext_robin_dashpot_normal(
v_old, r['visc'], ds_)
else:
raise NameError("Unknown dir option for Robin BC!")
else:
raise NameError("Unknown type option for Robin BC!")
return w, w_old
def test_subdomains(compile_args):
cell = ufl.triangle
element = ufl.FiniteElement("Lagrange", cell, 1)
u, v = ufl.TrialFunction(element), ufl.TestFunction(element)
a0 = ufl.inner(u, v) * ufl.dx + ufl.inner(u, v) * ufl.dx(2)
a1 = ufl.inner(u, v) * ufl.dx(2) + ufl.inner(u, v) * ufl.dx
a2 = ufl.inner(u, v) * ufl.dx(2) + ufl.inner(u, v) * ufl.dx(1)
a3 = ufl.inner(u, v) * ufl.ds(210) + ufl.inner(u, v) * ufl.ds(0)
forms = [a0, a1, a2, a3]
compiled_forms, module = ffcx.codegeneration.jit.compile_forms(
forms,
parameters={'scalar_type': 'double'},
cffi_extra_compile_args=compile_args)
for f, compiled_f in zip(forms, compiled_forms):
assert compiled_f.rank == len(f.arguments())
ffi = cffi.FFI()
form0 = compiled_forms[0][0]
ids = np.zeros(form0.num_cell_integrals, dtype=np.int32)
form0.get_cell_integral_ids(ffi.cast('int *', ids.ctypes.data))
assert ids[0] == -1 and ids[1] == 2
form1 = compiled_forms[1][0]
ids = np.zeros(form1.num_cell_integrals, dtype=np.int32)
form1.get_cell_integral_ids(ffi.cast('int *', ids.ctypes.data))
assert ids[0] == -1 and ids[1] == 2
form2 = compiled_forms[2][0]
ids = np.zeros(form2.num_cell_integrals, dtype=np.int32)
form2.get_cell_integral_ids(ffi.cast('int *', ids.ctypes.data))
assert ids[0] == 1 and ids[1] == 2
form3 = compiled_forms[3][0]
ids = np.zeros(form3.num_cell_integrals, dtype=np.int32)
form3.get_cell_integral_ids(ffi.cast('int *', ids.ctypes.data))
assert len(ids) == 0
ids = np.zeros(form3.num_exterior_facet_integrals, dtype=np.int32)
form3.get_exterior_facet_integral_ids(ffi.cast('int *', ids.ctypes.data))
assert ids[0] == 0 and ids[1] == 210
def error_indicators(self):
"""
Generate and return linear form defining error indicators
"""
# Extract these to increase readability
R_T = self._R_T
R_dT = self._R_dT
z = self._Ez_h
z_h = self._z_h
# Define linear form for computing error indicators
v = self.module.TestFunction(self._DG0)
eta_T = (v * inner(R_T, z - z_h) * dx(self.domain) +
avg(v)*(inner(R_dT('+'), (z - z_h)('+')) +
inner(R_dT('-'), (z - z_h)('-'))) * dS(self.domain) +
v * inner(R_dT, z - z_h) * ds(self.domain))
return eta_T
def facet_residual(self):
"""
Generate and return (bilinear, linear) forms defining linear
variational problem for the strong facet residual(s)
"""
# Define trial and test functions for the facet residuals on
# discontinuous version of primal trial space
R_e = self.module.TrialFunction(self._dV)
v = self.module.TestFunction(self._dV)
# Extract original test function in the weak residual
v_h = self.weak_residual.arguments()[0]
# Define forms defining linear variational problem for facet
# residual
v_e = self._b_e*v
a_R_dT = ((inner(v_e('+'), R_e('+')) + inner(v_e('-'), R_e('-')))*dS(self.domain)
+ inner(v_e, R_e)*ds(self.domain))
L_R_dT = (replace(self.weak_residual, {v_h: v_e})
- inner(v_e, self._R_T)*dx(self.domain))
return (a_R_dT, L_R_dT)
def facet_residual(self):
"""
Generate and return (bilinear, linear) forms defining linear
variational problem for the strong facet residual(s)
"""
# Define trial and test functions for the facet residuals on
# discontinuous version of primal trial space
R_e = self.module.TrialFunction(self._dV)
v = self.module.TestFunction(self._dV)
# Extract original test function in the weak residual
v_h = self.weak_residual.arguments()[0]
# Define forms defining linear variational problem for facet
# residual
v_e = self._b_e * v
a_R_dT = ((inner(v_e('+'), R_e('+')) + inner(v_e('-'), R_e('-'))) *
dS(self.domain) + inner(v_e, R_e) * ds(self.domain))
L_R_dT = (replace(self.weak_residual, {v_h: v_e}) -
inner(v_e, self._R_T) * dx(self.domain))
return (a_R_dT, L_R_dT)
def Mesh(arg, **kwargs):
"""
Overload Firedrake's ``Mesh`` constructor to
endow the output mesh with useful quantities.
The following quantities are computed by default:
* cell size;
* facet area.
The argument and keyword arguments are passed to
Firedrake's ``Mesh`` constructor, modified so
that the argument could also be a mesh.
"""
try:
mesh = firedrake.Mesh(arg, **kwargs)
except TypeError:
mesh = firedrake.Mesh(arg.coordinates, **kwargs)
P0 = firedrake.FunctionSpace(mesh, "DG", 0)
P1 = firedrake.FunctionSpace(mesh, "CG", 1)
dim = mesh.topological_dimension()
# Facet area
boundary_markers = sorted(mesh.exterior_facets.unique_markers)
one = firedrake.Function(P1).assign(1.0)
bnd_len = OrderedDict(
{i: firedrake.assemble(one * ufl.ds(int(i))) for i in boundary_markers}
)
if dim == 2:
mesh.boundary_len = bnd_len
else:
mesh.boundary_area = bnd_len
# Cell size
if dim == 2 and mesh.coordinates.ufl_element().cell() == ufl.triangle:
mesh.delta_x = firedrake.interpolate(ufl.CellDiameter(mesh), P0)
return mesh
def set_variational_forms_and_jacobians(self):
# add constant Neumann terms for large scale problem (trigger pressures)
self.neumann_funcs = []
w_neumann = as_ufl(0)
for n in range(len(self.pbsmall.surface_p_ids)):
self.neumann_funcs.append(Function(self.pblarge.Vd_scalar))
for i in range(len(self.pbsmall.surface_p_ids[n])):
ds_ = ds(
subdomain_data=self.pblarge.io.mt_b1,
subdomain_id=self.pbsmall.surface_p_ids[n][i],
metadata={'quadrature_degree': self.pblarge.quad_degree})
# we apply the pressure onto a fixed configuration of the G&R trigger point, determined by the displacement field u_set
# in the last G&R cycle, we assure that growth falls below a tolerance and hence the current and the set configuration coincide
w_neumann += self.pblarge.vf.deltaW_ext_neumann_true(
self.pblarge.ki.J(self.pblarge.u_set),
self.pblarge.ki.F(self.pblarge.u_set),
self.neumann_funcs[-1], ds_)
self.pblarge.weakform_u -= w_neumann
def test_facet_integral(cell_type):
"""Test that the integral of a function over a facet is correct"""
for count in range(5):
mesh = unit_cell(cell_type)
V = FunctionSpace(mesh, ("Lagrange", 2))
num_facets = mesh.num_entities(mesh.topology.dim - 1)
v = Function(V)
facet_function = MeshFunction("size_t", mesh, mesh.topology.dim - 1, 1)
facet_function.values[:] = range(num_facets)
# Functions that will have the same integral over each facet
if cell_type == CellType.triangle:
root = 3**0.25 # 4th root of 3
v.interpolate(lambda x: (x[0] - 1 / root)**2 +
(x[1] - root / 3)**2)
elif cell_type == CellType.quadrilateral:
v.interpolate(lambda x: x[0] * (1 - x[0]) + x[1] * (1 - x[1]))
elif cell_type == CellType.tetrahedron:
s = 2**0.5 * 3**(1 / 3) # side length
v.interpolate(lambda x: (x[0] - s / 2)**2 +
(x[1] - s / 2 / np.sqrt(3))**2 +
(x[2] - s * np.sqrt(2 / 3) / 4)**2)
elif cell_type == CellType.hexahedron:
v.interpolate(lambda x: x[0] * (1 - x[0]) + x[1] *
(1 - x[1]) + x[2] * (1 - x[2]))
# assert that the integral of these functions over each face are equal
out = []
for j in range(num_facets):
a = v * ds(subdomain_data=facet_function, subdomain_id=j)
result = fem.assemble_scalar(a)
out.append(result)
assert np.isclose(result, out[0])
def ref_elasticity(tetra: bool = True, r_lvl: int = 0, out_hdf5: h5py.File = None,
xdmf: bool = False, boomeramg: bool = False, kspview: bool = False, degree: int = 1):
if tetra:
N = 3 if degree == 1 else 2
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)
else:
N = 3
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
for i in range(r_lvl):
# set_log_level(LogLevel.INFO)
N *= 2
if tetra:
mesh = refine(mesh, redistribute=True)
else:
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
# set_log_level(LogLevel.ERROR)
N = degree * N
fdim = mesh.topology.dim - 1
V = VectorFunctionSpace(mesh, ("Lagrange", int(degree)))
# Generate Dirichlet BC on lower boundary (Fixed)
u_bc = Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
def boundaries(x):
return np.isclose(x[0], np.finfo(float).eps)
facets = locate_entities_boundary(mesh, fdim, boundaries)
topological_dofs = locate_dofs_topological(V, fdim, facets)
bc = dirichletbc(u_bc, topological_dofs)
bcs = [bc]
# Create traction meshtag
def traction_boundary(x):
return np.isclose(x[0], 1)
t_facets = locate_entities_boundary(mesh, fdim, traction_boundary)
facet_values = np.ones(len(t_facets), dtype=np.int32)
arg_sort = np.argsort(t_facets)
mt = meshtags(mesh, fdim, t_facets[arg_sort], facet_values[arg_sort])
# Elasticity parameters
E = PETSc.ScalarType(1.0e4)
nu = 0.1
mu = Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
g = Constant(mesh, PETSc.ScalarType((0, 0, -1e2)))
x = SpatialCoordinate(mesh)
f = Constant(mesh, PETSc.ScalarType(1e4)) * \
as_vector((0, -(x[2] - 0.5)**2, (x[1] - 0.5)**2))
# 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
rhs = inner(g, v) * ds(domain=mesh, subdomain_data=mt, subdomain_id=1) + inner(f, v) * dx
num_dofs = V.dofmap.index_map.size_global * V.dofmap.index_map_bs
if MPI.COMM_WORLD.rank == 0:
print("Problem size {0:d} ".format(num_dofs))
# Generate reference matrices and unconstrained solution
bilinear_form = form(a)
A_org = assemble_matrix(bilinear_form, bcs)
A_org.assemble()
null_space_org = rigid_motions_nullspace(V)
A_org.setNearNullSpace(null_space_org)
linear_form = form(rhs)
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)
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_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"] = 0.02
# opts["help"] = None # List all available options
# opts["ksp_view"] = None # List progress of solver
# Create solver, set operator and options
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setFromOptions()
solver.setOperators(A_org)
# Solve linear problem
u_ = Function(V)
start = perf_counter()
with Timer("Ref solve"):
solver.solve(L_org, u_.vector)
end = perf_counter()
u_.x.scatter_forward()
if kspview:
solver.view()
it = solver.getIterationNumber()
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("Refinement level {0:d}, Iterations {1:d}".format(r_lvl, it))
# List memory usage
mem = sum(MPI.COMM_WORLD.allgather(
resource.getrusage(resource.RUSAGE_SELF).ru_maxrss))
if MPI.COMM_WORLD.rank == 0:
print("{1:d}: Max usage after trad. solve {0:d} (kb)"
.format(mem, r_lvl))
if xdmf:
# Name formatting of functions
u_.name = "u_unconstrained"
fname = "results/ref_elasticity_{0:d}.xdmf".format(r_lvl)
with XDMFFile(MPI.COMM_WORLD, fname, "w") as out_xdmf:
out_xdmf.write_mesh(mesh)
out_xdmf.write_function(u_, 0.0, "Xdmf/Domain/Grid[@Name='{0:s}'][1]".format(mesh.name))
def run_dg_test(mesh, V, degree):
""" Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
# Exact solution
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
# Coefficient
k = Function(V)
k.vector.set(2.0)
k.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Source term
f = -div(k * grad(u_exact))
# Mesh normals and element size
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2.0
# Penalty parameter
alpha = 32
dx_ = dx(metadata={"quadrature_degree": -1})
ds_ = ds(metadata={"quadrature_degree": -1})
dS_ = dS(metadata={"quadrature_degree": -1})
with common.Timer("Compile forms"):
a = inner(k * grad(u), grad(v)) * dx_ \
- k("+") * inner(avg(grad(u)), jump(v, n)) * dS_ \
- k("+") * inner(jump(u, n), avg(grad(v))) * dS_ \
+ k("+") * (alpha / h_avg) * inner(jump(u, n), jump(v, n)) * dS_ \
- inner(k * grad(u), v * n) * ds_ \
- inner(u * n, k * grad(v)) * ds_ \
+ (alpha / h) * inner(k * u, v) * ds_
L = inner(f, v) * dx_ - inner(k * u_exact * n, grad(v)) * ds_ \
+ (alpha / h) * inner(k * u_exact, v) * ds_
for integral in a.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
a)
for integral in L.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
L)
with common.Timer("Assemble vector"):
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
with common.Timer("Assemble matrix"):
A = assemble_matrix(a, [])
A.assemble()
with common.Timer("Solve"):
# 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
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
with common.Timer("Error functional compile"):
# Calculate error
M = (u_exact - uh)**2 * dx
M = fem.Form(M)
with common.Timer("Error assembly"):
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
common.list_timings(MPI.COMM_WORLD, [common.TimingType.wall])
assert np.absolute(error) < 1.0e-14
def truth_solve(mu_unkown):
print("Performing truth solve at mu =", mu_unkown)
(mesh, subdomains, boundaries, restrictions) = read_mesh()
# (mesh, subdomains, boundaries, restrictions) = create_mesh()
dx = Measure('dx', subdomain_data=subdomains)
ds = Measure('ds', subdomain_data=boundaries)
W = generate_block_function_space(mesh, restrictions)
# Test and trial functions
block_v = BlockTestFunction(W)
v, q = block_split(block_v)
block_du = BlockTrialFunction(W)
du, dp = block_split(block_du)
block_u = BlockFunction(W)
u, p = block_split(block_u)
# gap
# V2 = FunctionSpace(mesh, "CG", 1)
# gap = Function(V2, name="Gap")
# obstacle
R = 0.25
d = 0.15
x_0 = mu_unkown[0]
y_0 = mu_unkown[1]
obstacle = Expression("-d+(pow(x[0]-x_0,2)+pow(x[1]-y_0, 2))/2/R", d=d, R=R , x_0 = x_0, y_0 = y_0, degree=0)
# Constitutive parameters
E = Constant(10.0)
nu = Constant(0.3)
mu, lmbda = Constant(E/(2*(1 + nu))), Constant(E*nu/((1 + nu)*(1 - 2*nu)))
B = Constant((0.0, 0.0, 0.0)) # Body force per unit volume
T = Constant((0.0, 0.0, 0.0)) # Traction force on the boundary
# Kinematics
# -----------------------------------------------------------------------------
mesh_dim = mesh.topology().dim() # Spatial dimension
I = Identity(mesh_dim) # Identity tensor
F = I + grad(u) # Deformation gradient
C = F.T*F # Right Cauchy-Green tensor
J = det(F) # 3rd invariant of the deformation tensor
# Strain function
def P(u): # P = dW/dF:
return mu*(F - inv(F.T)) + lmbda*ln(J)*inv(F.T)
def eps(v):
return sym(grad(v))
def sigma(v):
return lmbda*tr(eps(v))*Identity(3) + 2.0*mu*eps(v)
# Definition of The Mackauley bracket <x>+
def ppos(x):
return (x+abs(x))/2.
# Define the augmented lagrangian
def aug_l(x):
return x + pen*(obstacle-u[2])
pen = Constant(1e4)
# Boundary conditions
# bottom_bc = DirichletBC(W.sub(0), Constant((0., 0., 0.)), boundaries, 2)
# left_bc = DirichletBC(W.sub(0), Constant((0., 0., 0.)), boundaries, 3)
# right_bc = DirichletBC(W.sub(0), Constant((0., 0., 0.)), boundaries, 4)
# front_bc = DirichletBC(W.sub(0), Constant((0., 0., 0.)), boundaries, 5)
# back_bc = DirichletBC(W.sub(0), Constant((0., 0., 0.)), boundaries, 6)
# # sym_x_bc = DirichletBC(W.sub(0).sub(0), Constant(0.), boundaries, 2)
# # sym_y_bc = DirichletBC(W.sub(0).sub(1), Constant(0.), boundaries, 3)
# # bc = BlockDirichletBC([bottom_bc, sym_x_bc, sym_y_bc])
# bc = BlockDirichletBC([bottom_bc, left_bc, right_bc, front_bc, back_bc])
bottom_bc = DirichletBC(W.sub(0), Constant((0., 0., 0.)), boundaries, 2)
left_bc_x = DirichletBC(W.sub(0).sub(0), Constant(0.), boundaries, 3)
left_bc_y = DirichletBC(W.sub(0).sub(1), Constant(0.), boundaries, 3)
right_bc_x = DirichletBC(W.sub(0).sub(0), Constant(0.), boundaries, 4)
right_bc_y = DirichletBC(W.sub(0).sub(1), Constant(0.), boundaries, 4)
front_bc_x = DirichletBC(W.sub(0).sub(0), Constant(0.), boundaries, 5)
front_bc_y = DirichletBC(W.sub(0).sub(1), Constant(0.), boundaries, 5)
back_bc_x = DirichletBC(W.sub(0).sub(0), Constant(0.), boundaries, 6)
back_bc_y = DirichletBC(W.sub(0).sub(1), Constant(0.), boundaries, 6)
# sym_x_bc = DirichletBC(W.sub(0).sub(0), Constant(0.), boundaries, 2)
# sym_y_bc = DirichletBC(W.sub(0).sub(1), Constant(0.), boundaries, 3)
# bc = BlockDirichletBC([bottom_bc, sym_x_bc, sym_y_bc])
bc = BlockDirichletBC([bottom_bc, left_bc_x, left_bc_y, \
right_bc_x, right_bc_y, front_bc_x, front_bc_y, \
back_bc_x, back_bc_y])
# Variational forms
# F = inner(sigma(u), eps(v))*dx + pen*dot(v[2], ppos(u[2]-obstacle))*ds(1)
# F = [inner(sigma(u), eps(v))*dx - aug_l(l)*v[2]*ds(1) + ppos(aug_l(l))*v[2]*ds(1),
# (obstacle-u[2])*v*ds(1) - (1/pen)*ppos(aug_l(l))*v*ds(1)]
# F_a = inner(sigma(u), eps(v))*dx
# F_b = - aug_l(p)*v[2]*ds(1) + ppos(aug_l(p))*v[2]*ds(1)
# F_c = (obstacle-u[2])*q*ds(1)
# F_d = - (1/pen)*ppos(aug_l(p))*q*ds(1)
#
# block_F = [[F_a, F_b],
# [F_c, F_d]]
F_a = inner(P(u), grad(v))*dx - dot(B, v)*dx - dot(T, v)*ds \
- aug_l(p)*v[2]*ds(1) + ppos(aug_l(p))*v[2]*ds(1)
F_b = (obstacle-u[2])*q*ds(1) - (1/pen)*ppos(aug_l(p))*q*ds(1)
block_F = [F_a,
F_b]
J = block_derivative(block_F, block_u, block_du)
# Setup solver
problem = BlockNonlinearProblem(block_F, block_u, bc, J)
solver = BlockPETScSNESSolver(problem)
solver.parameters.update({
"linear_solver": "mumps",
"absolute_tolerance": 1E-4,
"relative_tolerance": 1E-4,
"maximum_iterations": 50,
"report": True,
"error_on_nonconvergence": True
})
# solver.parameters.update({
# "linear_solver": "cg",
# "absolute_tolerance": 1E-4,
# "relative_tolerance": 1E-4,
# "maximum_iterations": 50,
# "report": True,
# "error_on_nonconvergence": True
# })
# Perform a fake loop over time. Note how up will store the solution at the last time.
# Q. for?
# A. You can remove it, since your problem is stationary. The template was targeting
# a final application which was transient, but in which the ROM should have only
# described the final solution (when reaching the steady state).
# for _ in range(2):
# solver.solve()
a1 = solver.solve()
print(a1)
# save all the solution here as a function of time
# Return the solution at the last time
# Q. block_u or block
# A. I think block_u, it will split split among the components elsewhere
return block_u
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_div_grad_then_integrate_over_cells_and_boundary():
# Define 2D geometry
n = 10
mesh = RectangleMesh(Point(0.0, 0.0), Point(2.0, 3.0), 2 * n, 3 * n)
x, y = SpatialCoordinate(mesh)
xs = 0.1 + 0.8 * x / 2 # scaled to be within [0.1,0.9]
# ys = 0.1 + 0.8 * y / 3 # scaled to be within [0.1,0.9]
n = FacetNormal(mesh)
# Define list of expressions to test, and configure accuracies
# these expressions are known to pass with. The reason some
# functions are less accurately integrated is likely that the
# default choice of quadrature rule is not perfect
F_list = []
def reg(exprs, acc=10):
for expr in exprs:
F_list.append((expr, acc))
# FIXME: 0*dx and 1*dx fails in the ufl-ffc-jit framework somewhere
# reg([Constant(0.0, cell=cell)])
# reg([Constant(1.0, cell=cell)])
monomial_list = [x**q for q in range(2, 6)]
reg(monomial_list)
reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list])
reg([xs**xs])
reg(
[xs**(xs**2)], 8
) # Note: Accuracies here are from 1D case, not checked against 2D results.
reg([xs**(xs**3)], 6)
reg([xs**(xs**4)], 2)
# Special functions:
reg([atan(xs)], 8)
reg([sin(x), cos(x), exp(x)], 5)
reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3)
reg([asin(xs), acos(xs)], 1)
reg([tan(xs)], 7)
# To handle tensor algebra, make an x dependent input tensor
# xx and square all expressions
def reg2(exprs, acc=10):
for expr in exprs:
F_list.append((inner(expr, expr), acc))
xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]])
xxs = as_matrix([[2 * xs**2, 3 * xs**3], [11 * xs**5, 7 * xs**4]])
x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4])
cc = as_matrix([[2, 3], [4, 5]])
reg2(
[xx]
) # TODO: Make unit test for UFL from this, results in listtensor with free indices
reg2([x3v])
reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))])
reg2([xx.T])
reg2([tr(xx)])
reg2([det(xx)])
reg2([dot(xx, 0.1 * xx)])
reg2([outer(xx, xx.T)])
reg2([dev(xx)])
reg2([sym(xx)])
reg2([skew(xx)])
reg2([elem_mult(7 * xx, cc)])
reg2([elem_div(7 * xx, xx + cc)])
reg2([elem_pow(1e-3 * xxs, 1e-3 * cc)])
reg2([elem_pow(1e-3 * cc, 1e-3 * xx)])
reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate...
# FIXME: Add tests for all UFL operators:
# These cause discontinuities and may be harder to test in the
# above fashion:
# 'inv', 'cofac',
# 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not',
# 'conditional', 'sign',
# 'jump', 'avg',
# 'LiftingFunction', 'LiftingOperator',
# FIXME: Test other derivatives: (but algorithms for operator
# derivatives are the same!):
# 'variable', 'diff',
# 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative',
# Run through all operators defined above and compare integrals
debug = 0
if debug:
F_list = F_list[1:]
for F, acc in F_list:
if debug:
print('\n', "F:", str(F))
# Integrate over domain and its boundary
int_dx = assemble(div(grad(F)) * dx(mesh)) # noqa
int_ds = assemble(dot(grad(F), n) * ds(mesh)) # noqa
if debug:
print(int_dx, int_ds)
# Compare results. Using custom relative delta instead of
# decimal digits here because some numbers are >> 1.
delta = min(abs(int_dx), abs(int_ds)) * 10**-acc
assert int_dx - int_ds <= delta
# Copyright (C) 2008 Anders Logg
#
# This file is part of UFL.
#
# UFL is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# UFL is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with UFL. If not, see <http://www.gnu.org/licenses/>.
#
# This example illustrates how to define a form over a
# given subdomain of a mesh, in this case a functional.
from ufl import (Coefficient, FiniteElement, TestFunction, TrialFunction, ds,
dx, tetrahedron)
element = FiniteElement("CG", tetrahedron, 1)
v = TestFunction(element)
u = TrialFunction(element)
f = Coefficient(element)
M = f * dx(2) + f * ds(5)
def set_variational_forms_and_jacobians(self):
self.cq, self.cq_old, self.dcq, self.dforce = [], [], [], []
self.coupfuncs, self.coupfuncs_old = [], []
if self.coupling_type == 'monolithic_lagrange':
# Lagrange multiplier stiffness matrix (currently treated with FD!)
self.K_lm = PETSc.Mat().createAIJ(size=(self.num_coupling_surf,
self.num_coupling_surf),
bsize=None,
nnz=None,
csr=None,
comm=self.comm)
self.K_lm.setUp()
# Lagrange multipliers
self.lm, self.lm_old = self.K_lm.createVecLeft(
), self.K_lm.createVecLeft()
# 3D fluxes
self.constr, self.constr_old = [], []
self.power_coupling, self.power_coupling_old = as_ufl(0), as_ufl(0)
# coupling variational forms and Jacobian contributions
for n in range(self.num_coupling_surf):
self.pr0D = expression.template()
self.coupfuncs.append(Function(
self.pbs.Vd_scalar)), self.coupfuncs_old.append(
Function(self.pbs.Vd_scalar))
self.coupfuncs[-1].interpolate(
self.pr0D.evaluate), self.coupfuncs_old[-1].interpolate(
self.pr0D.evaluate)
cq_, cq_old_ = as_ufl(0), as_ufl(0)
for i in range(len(self.surface_vq_ids[n])):
ds_vq = ds(
subdomain_data=self.pbs.io.mt_b1,
subdomain_id=self.surface_vq_ids[n][i],
metadata={'quadrature_degree': self.pbs.quad_degree})
if self.coupling_params['coupling_quantity'] == 'flux':
assert (self.coupling_type == 'monolithic_direct')
cq_ += self.pbs.vf.flux(self.pbs.v, ds_vq)
elif self.coupling_params['coupling_quantity'] == 'pressure':
assert (self.coupling_type == 'monolithic_lagrange')
cq_ += self.pbs.vf.flux(self.pbs.v, ds_vq)
else:
raise NameError(
"Unknown coupling quantity! Choose flux or pressure!")
self.cq.append(cq_), self.cq_old.append(cq_old_)
self.dcq.append(derivative(self.cq[-1], self.pbs.v, self.pbs.dv))
df_ = as_ufl(0)
for i in range(len(self.surface_p_ids[n])):
ds_p = ds(subdomain_data=self.pbs.io.mt_b1,
subdomain_id=self.surface_p_ids[n][i],
metadata={'quadrature_degree': self.pbs.quad_degree})
df_ += self.pbs.timefac * self.pbs.vf.surface(ds_p)
# add to fluid rhs contributions
self.power_coupling += self.pbs.vf.deltaP_ext_neumann_normal(
self.coupfuncs[-1], ds_p)
self.power_coupling_old += self.pbs.vf.deltaP_ext_neumann_normal(
self.coupfuncs_old[-1], ds_p)
self.dforce.append(df_)
# minus sign, since contribution to external power!
self.pbs.weakform_u += -self.pbs.timefac * self.power_coupling - (
1. - self.pbs.timefac) * self.power_coupling_old
# add to fluid Jacobian
self.pbs.jac_uu += -self.pbs.timefac * derivative(
self.power_coupling, self.pbs.v, self.pbs.dv)
if self.coupling_type == 'monolithic_lagrange':
# old Lagrange multipliers - initialize with initial pressures
self.pbf.cardvasc0D.initialize_lm(
self.lm, self.pbf.time_params['initial_conditions'])
self.pbf.cardvasc0D.initialize_lm(
self.lm_old, self.pbf.time_params['initial_conditions'])
def bench_elasticity_edge(tetra: bool = True, r_lvl: int = 0, out_hdf5=None, xdmf: bool = False,
boomeramg: bool = False, kspview: bool = False, degree: int = 1, info: bool = False):
N = 3
for i in range(r_lvl):
N *= 2
ct = CellType.tetrahedron if tetra else CellType.hexahedron
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, ct)
# Get number of unknowns on each edge
V = VectorFunctionSpace(mesh, ("Lagrange", int(degree)))
# Generate Dirichlet BC (Fixed)
u_bc = Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
def boundaries(x):
return np.isclose(x[0], np.finfo(float).eps)
fdim = mesh.topology.dim - 1
facets = locate_entities_boundary(mesh, fdim, boundaries)
topological_dofs = locate_dofs_topological(V, fdim, facets)
bc = dirichletbc(u_bc, topological_dofs)
bcs = [bc]
def PeriodicBoundary(x):
return np.logical_and(np.isclose(x[0], 1), np.isclose(x[2], 0))
def periodic_relation(x):
out_x = np.zeros(x.shape)
out_x[0] = x[0]
out_x[1] = x[1]
out_x[2] = x[2] + 1
return out_x
with Timer("~Elasticity: Initialize MPC"):
edim = mesh.topology.dim - 2
edges = locate_entities_boundary(mesh, edim, PeriodicBoundary)
arg_sort = np.argsort(edges)
periodic_mt = meshtags(mesh, edim, edges[arg_sort], np.full(len(edges), 2, dtype=np.int32))
mpc = MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V, periodic_mt, 2, periodic_relation, bcs, scale=0.5)
mpc.finalize()
# Create traction meshtag
def traction_boundary(x):
return np.isclose(x[0], 1)
t_facets = locate_entities_boundary(mesh, fdim, traction_boundary)
facet_values = np.ones(len(t_facets), dtype=np.int32)
arg_sort = np.argsort(t_facets)
mt = meshtags(mesh, fdim, t_facets[arg_sort], facet_values)
# Elasticity parameters
E = PETSc.ScalarType(1.0e4)
nu = 0.1
mu = Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
g = Constant(mesh, PETSc.ScalarType((0, 0, -1e2)))
x = SpatialCoordinate(mesh)
f = Constant(mesh, PETSc.ScalarType(1e3)) * as_vector((0, -(x[2] - 0.5)**2, (x[1] - 0.5)**2))
# Stress computation
def epsilon(v):
return sym(grad(v))
def sigma(v):
return (2.0 * mu * epsilon(v) + lmbda * tr(epsilon(v)) * Identity(len(v)))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(sigma(u), grad(v)) * dx
rhs = inner(g, v) * ds(domain=mesh, subdomain_data=mt, subdomain_id=1) + inner(f, v) * dx
# Setup MPC system
if info:
log_info(f"Run {r_lvl}: Assembling matrix and vector")
bilinear_form = form(a)
linear_form = form(rhs)
with Timer("~Elasticity: Assemble LHS and RHS"):
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
b = assemble_vector(linear_form, mpc)
# Create nullspace for elasticity problem and assign to matrix
null_space = rigid_motions_nullspace(mpc.function_space)
A.setNearNullSpace(null_space)
# Apply boundary conditions
apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, bcs)
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_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"] = 0.02
# opts["help"] = None # List all available options
# opts["ksp_view"] = None # List progress of solver
# Setup PETSc solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setFromOptions()
if info:
log_info(f"Run {r_lvl}: Solving")
with Timer("~Elasticity: Solve problem") as timer:
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)
solver_time = timer.elapsed()
if kspview:
solver.view()
mem = sum(MPI.COMM_WORLD.allgather(resource.getrusage(resource.RUSAGE_SELF).ru_maxrss))
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("num_slaves")
d_set[r_lvl, MPI.COMM_WORLD.rank] = mpc.num_local_slaves
d_set = out_hdf5.get("solve_time")
d_set[r_lvl, MPI.COMM_WORLD.rank] = solver_time[0]
if info:
log_info(f"Lvl: {r_lvl}, Its: {it}, max Mem: {mem}, dim(V): {num_dofs}")
if xdmf:
# Write solution to file
u_h = Function(mpc.function_space)
u_h.vector.setArray(uh.array)
u_h.name = "u_mpc"
fname = f"results/bench_elasticity_edge_{r_lvl}.xdmf"
with XDMFFile(MPI.COMM_WORLD, fname, "w") as outfile:
outfile.write_mesh(mesh)
outfile.write_function(u_h)
def test_assemble_functional_ds(mode):
mesh = create_unit_square(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
M = form(1.0 * ds(domain=mesh))
value = assemble_scalar(M)
value = mesh.comm.allreduce(value, op=MPI.SUM)
assert value == pytest.approx(4.0, 1e-12)
def neumann_bcs(self, V, V_real):
w, w_old = as_ufl(0), as_ufl(0)
for n in self.bc_dict['neumann']:
if n['dir'] == 'xyz':
func, func_old = Function(V), Function(V)
if 'curve' in n.keys():
load = expression.template_vector()
load.val_x, load.val_y, load.val_z = self.ti.timecurves(
n['curve'][0])(self.ti.t_init), self.ti.timecurves(
n['curve'][1])(self.ti.t_init), self.ti.timecurves(
n['curve'][2])(self.ti.t_init)
func.interpolate(load.evaluate), func_old.interpolate(
load.evaluate)
self.ti.funcs_to_update_vec.append({
func: [
self.ti.timecurves(n['curve'][0]),
self.ti.timecurves(n['curve'][1]),
self.ti.timecurves(n['curve'][2])
]
})
self.ti.funcs_to_update_vec_old.append({
func_old: [
self.ti.timecurves(n['curve'][0]),
self.ti.timecurves(n['curve'][1]),
self.ti.timecurves(n['curve'][2])
]
})
else:
func.vector.set(
n['val']
) # currently only one value for all directions - use constant load function otherwise!
for i in range(len(n['id'])):
ds_ = ds(subdomain_data=self.io.mt_b1,
subdomain_id=n['id'][i],
metadata={'quadrature_degree': self.quad_degree})
w += self.vf.deltaP_ext_neumann(func, ds_)
w_old += self.vf.deltaP_ext_neumann(func_old, ds_)
elif n['dir'] == 'normal': # reference normal
func, func_old = Function(V_real), Function(V_real)
if 'curve' in n.keys():
load = expression.template()
load.val = self.ti.timecurves(n['curve'])(self.ti.t_init)
func.interpolate(load.evaluate), func_old.interpolate(
load.evaluate)
self.ti.funcs_to_update.append(
{func: self.ti.timecurves(n['curve'])})
self.ti.funcs_to_update_old.append(
{func_old: self.ti.timecurves(n['curve'])})
else:
func.vector.set(n['val'])
for i in range(len(n['id'])):
ds_ = ds(subdomain_data=self.io.mt_b1,
subdomain_id=n['id'][i],
metadata={'quadrature_degree': self.quad_degree})
w += self.vf.deltaP_ext_neumann_normal(func, ds_)
w_old += self.vf.deltaP_ext_neumann_normal(func_old, ds_)
else:
raise NameError("Unknown dir option for Neumann BC!")
return w, w_old
bc_local.set(5.0)
u0 = Function(V)
with u0.vector.localForm() as bc_local:
bc_local.set(0.0)
# Define Dirichlet boundary conditions at top and bottom boundaries
bcs = [
DirichletBC(V, u5,
np.where(mf_line.values == tag_info['TOP'])[0]),
DirichletBC(V, u0,
np.where(mf_line.values == tag_info['BOTTOM'])[0])
]
dx = dx(subdomain_data=mf_triangle)
ds = ds(subdomain_data=mf_line)
# Define variational form
F = (inner(a0 * grad(u), grad(v)) * dx(tag_info['DOMAIN']) +
inner(a1 * grad(u), grad(v)) * dx(tag_info['OBSTACLE']) -
g_L * v * ds(tag_info['LEFT']) - g_R * v * ds(tag_info['RIGHT']) -
f * v * dx(tag_info['DOMAIN']) - f * v * dx(tag_info['OBSTACLE']))
# Separate left and right hand sides of equation
a, L = lhs(F), rhs(F)
# Solve problem
u = Function(V)
solve(a == L, u, bcs)
with XDMFFile(dolfin.MPI.comm_world, "output.xdmf") as xdmf_outfile:
def set_variational_forms_and_jacobians(self):
self.cq, self.cq_old, self.dcq, self.dforce = [], [], [], []
self.coupfuncs, self.coupfuncs_old = [], []
# Lagrange multiplier stiffness matrix (most likely to be zero!)
self.K_lm = PETSc.Mat().createAIJ(size=(self.num_coupling_surf,
self.num_coupling_surf),
bsize=None,
nnz=None,
csr=None,
comm=self.comm)
self.K_lm.setUp()
# Lagrange multipliers
self.lm, self.lm_old = self.K_lm.createVecLeft(
), self.K_lm.createVecLeft()
# 3D constraint variable (volume or flux)
self.constr, self.constr_old = [], []
self.work_coupling, self.work_coupling_old, self.work_coupling_prestr = as_ufl(
0), as_ufl(0), as_ufl(0)
# coupling variational forms and Jacobian contributions
for n in range(self.num_coupling_surf):
self.pr0D = expression.template()
self.coupfuncs.append(Function(
self.pbs.Vd_scalar)), self.coupfuncs_old.append(
Function(self.pbs.Vd_scalar))
self.coupfuncs[-1].interpolate(
self.pr0D.evaluate), self.coupfuncs_old[-1].interpolate(
self.pr0D.evaluate)
cq_, cq_old_ = as_ufl(0), as_ufl(0)
for i in range(len(self.surface_c_ids[n])):
ds_vq = ds(
subdomain_data=self.pbs.io.mt_b1,
subdomain_id=self.surface_c_ids[n][i],
metadata={'quadrature_degree': self.pbs.quad_degree})
# currently, only volume or flux constraints are supported
if self.coupling_params['constraint_quantity'] == 'volume':
cq_ += self.pbs.vf.volume(self.pbs.u,
self.pbs.ki.J(self.pbs.u),
self.pbs.ki.F(self.pbs.u), ds_vq)
cq_old_ += self.pbs.vf.volume(
self.pbs.u_old, self.pbs.ki.J(self.pbs.u_old),
self.pbs.ki.F(self.pbs.u_old), ds_vq)
elif self.coupling_params['constraint_quantity'] == 'flux':
cq_ += self.pbs.vf.flux(self.pbs.vel,
self.pbs.ki.J(self.pbs.u),
self.pbs.ki.F(self.pbs.u), ds_vq)
cq_old_ += self.pbs.vf.flux(self.pbs.v_old,
self.pbs.ki.J(self.pbs.u_old),
self.pbs.ki.F(self.pbs.u_old),
ds_vq)
else:
raise NameError(
"Unknown constraint quantity! Choose either volume or flux!"
)
self.cq.append(cq_), self.cq_old.append(cq_old_)
self.dcq.append(derivative(self.cq[-1], self.pbs.u, self.pbs.du))
df_ = as_ufl(0)
for i in range(len(self.surface_p_ids[n])):
ds_p = ds(subdomain_data=self.pbs.io.mt_b1,
subdomain_id=self.surface_p_ids[n][i],
metadata={'quadrature_degree': self.pbs.quad_degree})
df_ += self.pbs.timefac * self.pbs.vf.surface(
self.pbs.ki.J(self.pbs.u), self.pbs.ki.F(self.pbs.u), ds_p)
# add to solid rhs contributions
self.work_coupling += self.pbs.vf.deltaW_ext_neumann_true(
self.pbs.ki.J(self.pbs.u), self.pbs.ki.F(self.pbs.u),
self.coupfuncs[-1], ds_p)
self.work_coupling_old += self.pbs.vf.deltaW_ext_neumann_true(
self.pbs.ki.J(self.pbs.u_old),
self.pbs.ki.F(self.pbs.u_old), self.coupfuncs_old[-1],
ds_p)
# for prestressing, true loads should act on the reference, not the current configuration
if self.pbs.prestress_initial:
self.work_coupling_prestr += self.pbs.vf.deltaW_ext_neumann_refnormal(
self.coupfuncs_old[-1], ds_p)
self.dforce.append(df_)
# minus sign, since contribution to external work!
self.pbs.weakform_u += -self.pbs.timefac * self.work_coupling - (
1. - self.pbs.timefac) * self.work_coupling_old
# add to solid Jacobian
self.pbs.jac_uu += -self.pbs.timefac * derivative(
self.work_coupling, self.pbs.u, self.pbs.du)
def test_manufactured_poisson_dg(degree, filename, datadir):
""" Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
with XDMFFile(MPI.COMM_WORLD,
os.path.join(datadir, filename),
"r",
encoding=XDMFFile.Encoding.ASCII) as xdmf:
mesh = xdmf.read_mesh(name="Grid")
V = FunctionSpace(mesh, ("DG", degree))
u, v = TrialFunction(V), TestFunction(V)
# Exact solution
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
# Coefficient
k = Function(V)
k.vector.set(2.0)
k.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Source term
f = -div(k * grad(u_exact))
# Mesh normals and element size
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2.0
# Penalty parameter
alpha = 32
dx_ = dx(metadata={"quadrature_degree": -1})
ds_ = ds(metadata={"quadrature_degree": -1})
dS_ = dS(metadata={"quadrature_degree": -1})
a = inner(k * grad(u), grad(v)) * dx_ \
- k("+") * inner(avg(grad(u)), jump(v, n)) * dS_ \
- k("+") * inner(jump(u, n), avg(grad(v))) * dS_ \
+ k("+") * (alpha / h_avg) * inner(jump(u, n), jump(v, n)) * dS_ \
- inner(k * grad(u), v * n) * ds_ \
- inner(u * n, k * grad(v)) * ds_ \
+ (alpha / h) * inner(k * u, v) * ds_
L = inner(f, v) * dx_ - inner(k * u_exact * n, grad(v)) * ds_ \
+ (alpha / h) * inner(k * u_exact, v) * ds_
for integral in a.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
a)
for integral in L.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
L)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a, [])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
error = mesh.mpi_comm().allreduce(assemble_scalar((u_exact - uh)**2 * dx),
op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def clement_interpolant(source, target_space=None, boundary_tag=None):
r"""
Compute the Clement interpolant of a :math:`\mathbb P0`
source field, i.e. take the volume average over
neighbouring cells at each vertex.
:arg source: the :math:`\mathbb P0` source field
:kwarg target_space: the :math:`\mathbb P1` space to
interpolate into
:boundary_tag: optional boundary tag to compute the
Clement interpolant over.
"""
V = source.function_space()
assert V.ufl_element().family() == "Discontinuous Lagrange"
assert V.ufl_element().degree() == 0
rank = len(V.ufl_element().value_shape())
mesh = V.mesh()
dim = mesh.topological_dimension()
P1 = firedrake.FunctionSpace(mesh, "CG", 1)
dX = ufl.dx if boundary_tag is None else ufl.ds(boundary_tag)
if target_space is None:
if rank == 0:
target_space = P1
elif rank == 1:
target_space = firedrake.VectorFunctionSpace(mesh, "CG", 1)
elif rank == 2:
target_space = firedrake.TensorFunctionSpace(mesh, "CG", 1)
else:
raise ValueError(f"Rank-{rank} tensors are not supported.")
else:
assert target_space.ufl_element().family() == "Lagrange"
assert target_space.ufl_element().degree() == 1
target = firedrake.Function(target_space)
# Compute the patch volume at each vertex
if boundary_tag is None:
P0 = firedrake.FunctionSpace(mesh, "DG", 0)
dx = ufl.dx(domain=mesh)
volume = firedrake.assemble(firedrake.TestFunction(P0) * dx)
else:
volume = get_facet_areas(mesh)
patch_volume = firedrake.Function(P1)
kernel = "for (int i=0; i < p.dofs; i++) p[i] += v[0];"
keys = {
"v": (volume, op2.READ),
"p": (patch_volume, op2.INC),
}
firedrake.par_loop(kernel, dX, keys)
# Volume average
keys = {
"s": (source, op2.READ),
"v": (volume, op2.READ),
"t": (target, op2.INC),
}
if rank == 0:
firedrake.par_loop(
"""
for (int i=0; i < t.dofs; i++) {
t[i] += s[0]*v[0];
}
""",
dX,
keys,
)
elif rank == 1:
firedrake.par_loop(
"""
int d = %d;
for (int i=0; i < t.dofs; i++) {
for (int j=0; j < d; j++) {
t[i*d + j] += s[j]*v[0];
}
}
"""
% dim,
dX,
keys,
)
elif rank == 2:
firedrake.par_loop(
"""
int d = %d;
int Nd = d*d;
for (int i=0; i < t.dofs; i++) {
for (int j=0; j < d; j++) {
for (int k=0; k < d; k++) {
t[i*Nd + j*d + k] += s[j*d + k]*v[0];
}
}
}
"""
% dim,
dX,
keys,
)
else:
raise ValueError(f"Rank-{rank} tensors are not supported.")
target.interpolate(target / patch_volume)
if boundary_tag is not None:
target.dat.data_with_halos[:] = np.nan_to_num(target.dat.data_with_halos)
return target
u5 = Function(V)
with u5.vector().localForm() as bc_local:
bc_local.set(5.0)
u0 = Function(V)
with u0.vector().localForm() as bc_local:
bc_local.set(0.0)
# Define Dirichlet boundary conditions at top and bottom boundaries
bcs = [
DirichletBC(V, u5, boundaries.where_equal(boundary['TOP'][0])),
DirichletBC(V, u0, boundaries.where_equal(boundary['BOTTOM'][0]))
]
dx = dx(subdomain_data=domains)
ds = ds(subdomain_data=boundaries)
# Define variational form
F = (inner(a0 * grad(u), grad(v)) * dx(boundary['DOMAIN'][0]) +
inner(a1 * grad(u), grad(v)) * dx(boundary['OBSTACLE'][0]) -
g_L * v * ds(boundary['LEFT'][0]) - g_R * v * ds(boundary['RIGHT'][0]) -
f * v * dx(boundary['DOMAIN'][0]) - f * v * dx(boundary['OBSTACLE'][0]))
# Separate left and right hand sides of equation
a, L = lhs(F), rhs(F)
# Solve problem
u = Function(V)
solve(a == L, u, bcs)
bb_tree = cpp.geometry.BoundingBoxTree(mesh, 2)
# Copyright (C) 2008 Anders Logg and Kristian B. Oelgaard
#
# This file is part of UFL.
#
# UFL is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# UFL is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with UFL. If not, see <http://www.gnu.org/licenses/>.
#
# This simple example illustrates how forms can be defined on different sub domains.
# It is supported for all three integral types.
from ufl import (FiniteElement, TestFunction, TrialFunction, ds, dS, dx,
tetrahedron)
element = FiniteElement("CG", tetrahedron, 1)
v = TestFunction(element)
u = TrialFunction(element)
a = v * u * dx(0) + 10.0 * v * u * dx(1) + v * u * ds(0) + 2.0 * v * u * ds(1)\
+ v('+') * u('+') * dS(0) + 4.3 * v('+') * u('+') * dS(1)
def test_assemble_functional_ds(mode):
mesh = UnitSquareMesh(MPI.COMM_WORLD, 12, 12, ghost_mode=mode)
M = 1.0 * ds(domain=mesh)
value = dolfinx.fem.assemble_scalar(M)
value = mesh.mpi_comm().allreduce(value, op=MPI.SUM)
assert value == pytest.approx(4.0, 1e-12)
u5 = Function(V)
with u5.vector().localForm() as bc_local:
bc_local.set(5.0)
u0 = Function(V)
with u0.vector().localForm() as bc_local:
bc_local.set(0.0)
# Define Dirichlet boundary conditions at top and bottom boundaries
bcs = [
DirichletBC(V, u5, boundaries.where_equal(boundary["TOP"][0])),
DirichletBC(V, u0, boundaries.where_equal(boundary["BOTTOM"][0])),
]
dx = dx(subdomain_data=domains)
ds = ds(subdomain_data=boundaries)
# Define variational form
F = (inner(a0 * grad(u), grad(v)) * dx(boundary["DOMAIN"][0]) +
inner(a1 * grad(u), grad(v)) * dx(boundary["OBSTACLE"][0]) -
g_L * v * ds(boundary["LEFT"][0]) - g_R * v * ds(boundary["RIGHT"][0]) -
f * v * dx(boundary["DOMAIN"][0]) - f * v * dx(boundary["OBSTACLE"][0]))
# Separate left and right hand sides of equation
a, L = lhs(F), rhs(F)
# Solve problem
u = Function(V)
solve(a == L, u, bcs)
bb_tree = cpp.geometry.BoundingBoxTree(mesh, 2)
