def merge_meshtags(mts, dim):
""" Merge multiple MeshTags into one.
Parameters
----------
mts:
List of meshtags
dim:
Dimension of MeshTags which should be merged. Note it is
not possible to merge MeshTags with different dimensions into one
MeshTags object.
"""
mts = [(mt, name) for name, mt in mts.items() if mt.dim == dim]
if len(mts) == 0:
raise RuntimeError(f"Cannot find MeshTags of dimension {dim}")
indices = numpy.hstack([mt.indices for mt, name in mts])
values = numpy.hstack([mt.values for mt, name in mts])
keys = {}
for mt, name in mts:
comm = mt.mesh.comm
# In some cases this process could receive a MeshTags which are empty
# We need to return correct "keys" mapping on each process, so this
# communicates the value from processes which don't have empty meshtags
if len(mt.values) == 0:
value = -1
else:
if numpy.max(mt.values) < 0:
raise RuntimeError("Not expecting negative values for MeshTags")
value = int(mt.values[0])
value = comm.allreduce(value, op=MPI.MAX)
keys[name] = value
indices, pos = numpy.unique(indices, return_index=True)
mt = meshtags(mts[0][0].mesh, dim, indices, values[pos])
return mt, keys
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)
mesh.topology.create_entities(tdim - 1)
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 = form(v * ufl.ds(subdomain_data=marker, subdomain_id=j))
result = assemble_scalar(a)
out.append(result)
assert np.isclose(result, out[0])
def test_surface_integrals(get_assemblers): # noqa: F811
assemble_matrix, assemble_vector = get_assemblers
N = 4
mesh = create_unit_square(MPI.COMM_WORLD, N, N)
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
# Fixed Dirichlet BC on the left wall
def left_wall(x):
return np.isclose(x[0], np.finfo(float).eps)
fdim = mesh.topology.dim - 1
left_facets = locate_entities_boundary(mesh, fdim, left_wall)
bc_dofs = fem.locate_dofs_topological(V, 1, left_facets)
u_bc = fem.Function(V)
with u_bc.vector.localForm() as u_local:
u_local.set(0.0)
bc = fem.dirichletbc(u_bc, bc_dofs)
bcs = [bc]
# Traction on top of domain
def top(x):
return np.isclose(x[1], 1)
top_facets = locate_entities_boundary(mesh, 1, top)
arg_sort = np.argsort(top_facets)
mt = meshtags(mesh, fdim, top_facets[arg_sort], np.full(len(top_facets), 3, dtype=np.int32))
ds = ufl.Measure("ds", domain=mesh, subdomain_data=mt, subdomain_id=3)
g = fem.Constant(mesh, PETSc.ScalarType((0, -9.81e2)))
# Elasticity parameters
E = PETSc.ScalarType(1.0e4)
nu = 0.0
mu = fem.Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = fem.Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
# Stress computation
def sigma(v):
return (2.0 * mu * ufl.sym(ufl.grad(v))
+ lmbda * ufl.tr(ufl.sym(ufl.grad(v))) * ufl.Identity(len(v)))
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = ufl.inner(sigma(u), ufl.grad(v)) * ufl.dx
rhs = ufl.inner(fem.Constant(mesh, PETSc.ScalarType((0, 0))), v) * ufl.dx\
+ ufl.inner(g, v) * ds
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Setup LU solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
# Setup multipointconstraint
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {}
for i in range(1, N):
s_m_c[l2b([1, i / N])] = {l2b([1, 1]): 0.8}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, 1, 1)
mpc.finalize()
with Timer("~TEST: Assemble matrix old"):
A = assemble_matrix(bilinear_form, mpc, bcs=bcs)
with Timer("~TEST: Assemble vector"):
b = assemble_vector(linear_form, mpc)
dolfinx_mpc.apply_lifting(b, [bilinear_form], [bcs], mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(b, bcs)
solver.setOperators(A)
uh = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
# Write solution to file
# u_h = dolfinx.Function(mpc.function_space)
# u_h.vector.setArray(uh.array)
# u_h.name = "u_mpc"
# outfile = dolfinx.io.XDMFFile(MPI.COMM_WORLD, "output/uh.xdmf", "w")
# outfile.write_mesh(mesh)
# outfile.write_function(u_h)
# outfile.close()
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
# Generate reference matrices and unconstrained solution
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc)
list_timings(comm, [TimingType.wall])
fdim = mesh.topology.dim - 1
facets = locate_entities_boundary(mesh, fdim, dirichletboundary)
topological_dofs = fem.locate_dofs_topological(V, fdim, facets)
bc = fem.dirichletbc(u_bc, topological_dofs)
bcs = [bc]
def periodicboundary(x):
return np.isclose(x[0], 1)
facets = locate_entities_boundary(mesh, fdim, periodicboundary)
arg_sort = np.argsort(facets)
mt = meshtags(mesh, fdim, 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
mpc = MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V, mt, 2, periodic_relation, bcs)
mpc.finalize()
# Define variational problem
u = TrialFunction(V)
def plot_higher_order():
msh = create_unit_square(MPI.COMM_WORLD,
12,
12,
cell_type=CellType.quadrilateral)
# We continue using the mesh from the previous section, and find all
# cells satisfying the condition below
def in_circle(x):
"""Mark sphere with radius < sqrt(2)"""
return np.array((x.T[0] - 0.5)**2 + (x.T[1] - 0.5)**2 < 0.2**2,
dtype=np.int32)
# Create a dolfinx.MeshTag for all cells. If midpoint is inside the
# circle, it gets value 1, otherwise 0.
num_cells = msh.topology.index_map(msh.topology.dim).size_local
midpoints = compute_midpoints(msh, msh.topology.dim,
list(np.arange(num_cells, dtype=np.int32)))
cell_tags = meshtags(msh, msh.topology.dim, np.arange(num_cells),
in_circle(midpoints))
# We start by interpolating a discontinuous function into a second order
# discontinuous Lagrange space Note that we use the `cell_tags` from
# the previous section to get the cells for each of the regions
cells0 = cell_tags.indices[cell_tags.values == 0]
cells1 = cell_tags.indices[cell_tags.values == 1]
V = FunctionSpace(msh, ("Discontinuous Lagrange", 2))
u = Function(V, dtype=np.float64)
u.interpolate(lambda x: x[0], cells0)
u.interpolate(lambda x: x[1] + 1, cells1)
# To get a topology that has a 1-1 correspondence with the
# degrees-of-freedom in the function space, we call
# `dolfinx.plot.create_vtk_mesh`.
cells, types, x = plot.create_vtk_mesh(V)
# Create a pyvista mesh from the topology and geometry, and attach
# the coefficients of the degrees of freedom
grid = pyvista.UnstructuredGrid(cells, types, x)
grid.point_data["u"] = u.x.array
grid.set_active_scalars("u")
# We would also like to visualize the underlying mesh and obtain
# that as we have done previously
num_cells = msh.topology.index_map(msh.topology.dim).size_local
cell_entities = np.arange(num_cells, dtype=np.int32)
cells, types, x = plot.create_vtk_mesh(msh, msh.topology.dim,
cell_entities)
org_grid = pyvista.UnstructuredGrid(cells, types, x)
# We visualize the data
plotter = pyvista.Plotter()
plotter.add_text("Second-order (P2) discontinuous elements",
position="upper_edge",
font_size=14,
color="black")
sargs = dict(height=0.1,
width=0.8,
vertical=False,
position_x=0.1,
position_y=0,
color="black")
plotter.add_mesh(grid,
show_edges=False,
scalar_bar_args=sargs,
line_width=0)
plotter.add_mesh(org_grid, color="white", style="wireframe", line_width=5)
plotter.add_mesh(grid.copy(),
style="points",
point_size=15,
render_points_as_spheres=True,
line_width=0)
plotter.view_xy()
if pyvista.OFF_SCREEN:
plotter.screenshot(f"DG_{MPI.COMM_WORLD.rank}.png",
transparent_background=transparent,
window_size=[figsize, figsize])
else:
plotter.show()
def test_surface_integral_dependency(get_assemblers): # noqa: F811
assemble_matrix, assemble_vector = get_assemblers
N = 10
mesh = create_unit_square(MPI.COMM_WORLD, N, N)
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
def top(x):
return np.isclose(x[1], 1)
fdim = mesh.topology.dim - 1
top_facets = locate_entities_boundary(mesh, fdim, top)
indices = np.array([], dtype=np.intc)
values = np.array([], dtype=np.intc)
markers = {3: top_facets}
for key in markers.keys():
indices = np.append(indices, markers[key])
values = np.append(values, np.full(len(markers[key]), key, dtype=np.intc))
sort = np.argsort(indices)
mt = meshtags(mesh, mesh.topology.dim - 1, np.array(indices[sort], dtype=np.intc),
np.array(values[sort], dtype=np.intc))
ds = ufl.Measure("ds", domain=mesh, subdomain_data=mt)
g = fem.Constant(mesh, PETSc.ScalarType((2, 1)))
h = fem.Constant(mesh, PETSc.ScalarType((3, 2)))
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
a = ufl.inner(u, v) * ds(3) + ufl.inner(ufl.grad(u), ufl.grad(v)) * ds
rhs = ufl.inner(g, v) * ds + ufl.inner(h, v) * ds(3)
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Create multipoint constraint and assemble system
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {}
for i in range(1, N):
s_m_c[l2b([1, i / N])] = {l2b([1, 1]): 0.3}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, 1, 1)
mpc.finalize()
with Timer("~TEST: Assemble matrix"):
A = assemble_matrix(bilinear_form, mpc)
with Timer("~TEST: Assemble vector"):
b = assemble_vector(linear_form, mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
# Generate reference matrices and unconstrained solution
A_org = fem.petsc.assemble_matrix(bilinear_form)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
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)
list_timings(comm, [TimingType.wall])
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
mesh.topology.create_entities(tdim - 1)
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
normal = ufl.FacetNormal(mesh)
v = Function(V)
mesh.topology.create_entities(tdim - 1)
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 = form(
ufl.inner(v, normal) *
ufl.ds(subdomain_data=marker, subdomain_id=j))
result = assemble_scalar(a)
if np.isclose(result, 1):
ones += 1
else:
assert np.isclose(result, 0)
assert ones == 1
def plot_meshtags():
msh = create_unit_square(MPI.COMM_WORLD,
12,
12,
cell_type=CellType.quadrilateral)
# We continue using the mesh from the previous section, and find all
# cells satisfying the condition below
def in_circle(x):
"""True for points inside circle with radius 2"""
return np.array((x.T[0] - 0.5)**2 + (x.T[1] - 0.5)**2 < 0.2**2,
dtype=np.int32)
# Create a dolfinx.MeshTag for all cells. If midpoint is inside the
# circle, it gets value 1, otherwise 0.
num_cells = msh.topology.index_map(msh.topology.dim).size_local
midpoints = compute_midpoints(msh, msh.topology.dim,
list(np.arange(num_cells, dtype=np.int32)))
cell_tags = meshtags(msh, msh.topology.dim, np.arange(num_cells),
in_circle(midpoints))
cells, types, x = plot.create_vtk_mesh(msh, msh.topology.dim)
grid = pyvista.UnstructuredGrid(cells, types, x)
# As the dolfinx.MeshTag contains a value for every cell in the
# geometry, we can attach it directly to the grid
grid.cell_data["Marker"] = cell_tags.values
grid.set_active_scalars("Marker")
# We create a plotter consisting of two windows, and add a plot of the
# Meshtags to the first window.
subplotter = pyvista.Plotter(shape=(1, 2))
subplotter.subplot(0, 0)
subplotter.add_text("Mesh with markers",
font_size=14,
color="black",
position="upper_edge")
subplotter.add_mesh(grid, show_edges=True, show_scalar_bar=False)
subplotter.view_xy()
# We can also visualize subsets of data, by creating a smaller topology,
# only consisting of those entities that has value one in the
# dolfinx.MeshTag
cells, types, x = plot.create_vtk_mesh(
msh, msh.topology.dim, cell_tags.indices[cell_tags.values == 1])
# We add this grid to the second plotter
sub_grid = pyvista.UnstructuredGrid(cells, types, x)
subplotter.subplot(0, 1)
subplotter.add_text("Subset of mesh",
font_size=14,
color="black",
position="upper_edge")
subplotter.add_mesh(sub_grid, show_edges=True, edge_color="black")
if pyvista.OFF_SCREEN:
subplotter.screenshot("2D_markers.png",
transparent_background=transparent,
window_size=[2 * figsize, figsize])
else:
subplotter.show()
def mesh_2D_dolfin(celltype: str, theta: float = 0):
"""
Create two 2D cubes stacked on top of each other,
and the corresponding mesh markers using dolfin built-in meshes
"""
def find_line_function(p0, p1):
"""
Find line y=ax+b for each of the lines in the mesh
https://mathworld.wolfram.com/Two-PointForm.html
"""
# Line aligned with y axis
if np.isclose(p1[0], p0[0]):
return lambda x: np.isclose(x[0], p0[0])
return lambda x: np.isclose(
x[1], p0[1] + (p1[1] - p0[1]) / (p1[0] - p0[0]) * (x[0] - p0[0]))
def over_line(p0, p1):
"""
Check if a point is over or under y=ax+b for each of the
lines in the mesh https://mathworld.wolfram.com/Two-PointForm.html
"""
return lambda x: x[1] > p0[1] + (p1[1] - p0[1]) / (p1[0] - p0[0]) * (x[
0] - p0[0])
# Using built in meshes, stacking cubes on top of each other
N = 15
if celltype == "quadrilateral":
ct = _mesh.CellType.quadrilateral
elif celltype == "triangle":
ct = _mesh.CellType.triangle
else:
raise ValueError("celltype has to be tri or quad")
if MPI.COMM_WORLD.rank == 0:
mesh0 = _mesh.create_unit_square(MPI.COMM_SELF, N, N, ct)
mesh1 = _mesh.create_unit_square(MPI.COMM_SELF, 2 * N, 2 * N, ct)
mesh0.geometry.x[:, 1] += 1
# Stack the two meshes in one mesh
r_matrix = _utils.rotation_matrix([0, 0, 1], theta)
points = np.vstack([mesh0.geometry.x, mesh1.geometry.x])
points = np.dot(r_matrix, points.T)
points = points[:2, :].T
# Transform topology info into geometry info
tdim0 = mesh0.topology.dim
num_cells0 = mesh0.topology.index_map(tdim0).size_local
cells0 = _cpp.mesh.entities_to_geometry(
mesh0, tdim0,
np.arange(num_cells0, dtype=np.int32).reshape((-1, 1)), False)
tdim1 = mesh1.topology.dim
num_cells1 = mesh1.topology.index_map(tdim1).size_local
cells1 = _cpp.mesh.entities_to_geometry(
mesh1, tdim1,
np.arange(num_cells1, dtype=np.int32).reshape((-1, 1)), False)
cells1 += mesh0.geometry.x.shape[0]
cells = np.vstack([cells0, cells1])
cell = ufl.Cell(celltype, geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 1))
mesh = _mesh.create_mesh(MPI.COMM_SELF, cells, points, domain)
tdim = mesh.topology.dim
fdim = tdim - 1
# Find information about facets to be used in meshtags
bottom_points = np.dot(
r_matrix,
np.array([[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0]]).T)
bottom = find_line_function(bottom_points[:, 0], bottom_points[:, 1])
bottom_facets = _mesh.locate_entities_boundary(mesh, fdim, bottom)
top_points = np.dot(
r_matrix,
np.array([[0, 1, 0], [1, 1, 0], [1, 2, 0], [0, 2, 0]]).T)
top = find_line_function(top_points[:, 2], top_points[:, 3])
top_facets = _mesh.locate_entities_boundary(mesh, fdim, top)
left_side = find_line_function(top_points[:, 0], top_points[:, 3])
left_facets = _mesh.locate_entities_boundary(mesh, fdim, left_side)
right_side = find_line_function(top_points[:, 1], top_points[:, 2])
right_facets = _mesh.locate_entities_boundary(mesh, fdim, right_side)
top_cube = over_line(bottom_points[:, 2], bottom_points[:, 3])
num_cells = mesh.topology.index_map(tdim).size_local
cell_midpoints = _cpp.mesh.compute_midpoints(mesh, tdim,
range(num_cells))
interface = find_line_function(bottom_points[:, 2], bottom_points[:,
3])
i_facets = _mesh.locate_entities_boundary(mesh, fdim, interface)
bottom_interface = []
top_interface = []
mesh.topology.create_connectivity(fdim, tdim)
facet_to_cell = mesh.topology.connectivity(fdim, tdim)
for facet in i_facets:
i_cells = facet_to_cell.links(facet)
assert (len(i_cells == 1))
i_cell = i_cells[0]
if top_cube(cell_midpoints[i_cell]):
top_interface.append(facet)
else:
bottom_interface.append(facet)
top_cube_marker = 2
cell_indices = []
cell_values = []
for cell_index in range(num_cells):
if top_cube(cell_midpoints[cell_index]):
cell_indices.append(cell_index)
cell_values.append(top_cube_marker)
ct = _mesh.meshtags(mesh, tdim, np.array(cell_indices, dtype=np.intc),
np.array(cell_values, dtype=np.intc))
# Create meshtags for facet data
markers: Dict[int, np.ndarray] = {
3: top_facets,
4: np.hstack(bottom_interface),
9: np.hstack(top_interface),
5: bottom_facets,
6: left_facets,
7: right_facets
}
all_indices = []
all_values = []
for key in markers.keys():
all_indices.append(markers[key])
all_values.append(np.full(len(markers[key]), key, dtype=np.intc))
arg_sort = np.argsort(np.hstack(all_indices))
mt = _mesh.meshtags(mesh, fdim,
np.hstack(all_indices)[arg_sort],
np.hstack(all_values)[arg_sort])
mt.name = "facet_tags" # type: ignore
with _io.XDMFFile(MPI.COMM_SELF,
f"meshes/mesh_{celltype}_{theta:.2f}.xdmf",
"w") as o_f:
o_f.write_mesh(mesh)
o_f.write_meshtags(ct)
o_f.write_meshtags(mt)
MPI.COMM_WORLD.barrier()
def test_3d(tempdir, cell_type, encoding):
filename = os.path.join(tempdir, "meshtags_3d.xdmf")
comm = MPI.COMM_WORLD
mesh = create_unit_cube(comm, 4, 4, 4, cell_type)
mesh.topology.create_entities(2)
bottom_facets = locate_entities(mesh, 2, lambda x: np.isclose(x[1], 0.0))
bottom_values = np.full(bottom_facets.shape, 1, dtype=np.int32)
left_facets = locate_entities(mesh, 2, lambda x: np.isclose(x[0], 0.0))
left_values = np.full(left_facets.shape, 2, dtype=np.int32)
indices, pos = np.unique(np.hstack((bottom_facets, left_facets)),
return_index=True)
mt = meshtags(mesh, 2, indices,
np.hstack((bottom_values, left_values))[pos])
mt.name = "facets"
top_lines = locate_entities(mesh, 1, lambda x: np.isclose(x[2], 1.0))
top_values = np.full(top_lines.shape, 3, dtype=np.int32)
right_lines = locate_entities(mesh, 1, lambda x: np.isclose(x[0], 1.0))
right_values = np.full(right_lines.shape, 4, dtype=np.int32)
indices, pos = np.unique(np.hstack((top_lines, right_lines)),
return_index=True)
mt_lines = meshtags(mesh, 1, indices,
np.hstack((top_values, right_values))[pos])
mt_lines.name = "lines"
mesh.topology.create_connectivity(1, 3)
mesh.topology.create_connectivity(2, 3)
with XDMFFile(comm, filename, "w", encoding=encoding) as file:
file.write_mesh(mesh)
file.write_meshtags(mt)
file.write_meshtags(mt_lines)
file.write_information("units", "mm")
with XDMFFile(comm, filename, "r", encoding=encoding) as file:
mesh_in = file.read_mesh()
tdim = mesh_in.topology.dim
mesh_in.topology.create_connectivity(tdim - 1, tdim)
mesh_in.topology.create_connectivity(1, tdim)
mt_in = file.read_meshtags(mesh_in, "facets")
mt_lines_in = file.read_meshtags(mesh_in, "lines")
units = file.read_information("units")
assert units == "mm"
assert mt_in.name == "facets"
assert mt_lines_in.name == "lines"
with XDMFFile(comm,
os.path.join(tempdir, "meshtags_3d_out.xdmf"),
"w",
encoding=encoding) as file:
file.write_mesh(mesh_in)
file.write_meshtags(mt_lines_in)
file.write_meshtags(mt_in)
# Check number of owned and marked entities
lines_local = comm.allreduce(
(mt_lines.indices < mesh.topology.index_map(1).size_local).sum(),
op=MPI.SUM)
lines_local_in = comm.allreduce(
(mt_lines_in.indices < mesh_in.topology.index_map(1).size_local).sum(),
op=MPI.SUM)
assert lines_local == lines_local_in
# Check that only owned data is written to file
facets_local = comm.allreduce(
(mt.indices < mesh.topology.index_map(2).size_local).sum(), op=MPI.SUM)
parser = ElementTree.XMLParser()
tree = ElementTree.parse(os.path.join(tempdir, "meshtags_3d_out.xdmf"),
parser)
num_lines = int(
tree.findall(".//Grid[@Name='lines']/Topology")[0].get(
"NumberOfElements"))
num_facets = int(
tree.findall(".//Grid[@Name='facets']/Topology")[0].get(
"NumberOfElements"))
assert num_lines == lines_local
assert num_facets == facets_local
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 demo_periodic3D(celltype: CellType):
# Create mesh and finite element
if celltype == CellType.tetrahedron:
# Tet setup
N = 10
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)
V = fem.VectorFunctionSpace(mesh, ("CG", 1))
else:
# Hex setup
N = 10
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
V = fem.VectorFunctionSpace(mesh, ("CG", 2))
def dirichletboundary(x: NDArray[np.float64]) -> NDArray[np.bool_]:
return np.logical_or(
np.logical_or(np.isclose(x[1], 0), np.isclose(x[1], 1)),
np.logical_or(np.isclose(x[2], 0), np.isclose(x[2], 1)))
# Create Dirichlet boundary condition
zero = PETSc.ScalarType([0, 0, 0])
geometrical_dofs = fem.locate_dofs_geometrical(V, dirichletboundary)
bc = fem.dirichletbc(zero, geometrical_dofs, V)
bcs = [bc]
def PeriodicBoundary(x):
return np.isclose(x[0], 1)
facets = locate_entities_boundary(mesh, mesh.topology.dim - 1,
PeriodicBoundary)
arg_sort = np.argsort(facets)
mt = meshtags(mesh, mesh.topology.dim - 1, facets[arg_sort],
np.full(len(facets), 2, dtype=np.int32))
def periodic_relation(x):
out_x = np.zeros(x.shape)
out_x[0] = 1 - x[0]
out_x[1] = x[1]
out_x[2] = x[2]
return out_x
with Timer("~~Periodic: Compute mpc condition"):
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V.sub(0), mt, 2,
periodic_relation, bcs, 1)
mpc.finalize()
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx
x = SpatialCoordinate(mesh)
dx_ = x[0] - 0.9
dy_ = x[1] - 0.5
dz_ = x[2] - 0.1
f = as_vector((x[0] * sin(5.0 * pi * x[1]) +
1.0 * exp(-(dx_ * dx_ + dy_ * dy_ + dz_ * dz_) / 0.02),
0.1 * dx_ * dz_, 0.1 * dx_ * dy_))
rhs = inner(f, v) * dx
petsc_options: Dict[str, Union[str, float, int]]
if complex_mode:
rtol = 1e-16
petsc_options = {"ksp_type": "preonly", "pc_type": "lu"}
else:
rtol = 1e-8
petsc_options = {
"ksp_type": "cg",
"ksp_rtol": rtol,
"pc_type": "hypre",
"pc_hypre_typ": "boomeramg",
"pc_hypre_boomeramg_max_iter": 1,
"pc_hypre_boomeramg_cycle_type": "v",
"pc_hypre_boomeramg_print_statistics": 1
}
problem = LinearProblem(a, rhs, mpc, bcs, petsc_options=petsc_options)
u_h = problem.solve()
# --------------------VERIFICATION-------------------------
print("----Verification----")
u_ = fem.Function(V)
u_.x.array[:] = 0
org_problem = fem.petsc.LinearProblem(a,
rhs,
u=u_,
bcs=bcs,
petsc_options=petsc_options)
with Timer("~Periodic: Unconstrained solve"):
org_problem.solve()
it = org_problem.solver.getIterationNumber()
print(f"Unconstrained solver iterations: {it}")
# Write solutions to file
ext = "tet" if celltype == CellType.tetrahedron else "hex"
u_.name = "u_" + ext + "_unconstrained"
# NOTE: Workaround as tabulate dof coordinates does not like extra ghosts
u_out = fem.Function(V)
old_local = u_out.x.map.size_local * u_out.x.bs
old_ghosts = u_out.x.map.num_ghosts * u_out.x.bs
mpc_local = u_h.x.map.size_local * u_h.x.bs
assert (old_local == mpc_local)
u_out.x.array[:old_local + old_ghosts] = u_h.x.array[:mpc_local +
old_ghosts]
u_out.name = "u_" + ext
fname = f"results/demo_periodic3d_{ext}.bp"
out_periodic = VTXWriter(MPI.COMM_WORLD, fname, u_out)
out_periodic.write(0)
out_periodic.close()
root = 0
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(org_problem.A,
problem.A,
mpc,
root=root)
dolfinx_mpc.utils.compare_mpc_rhs(org_problem.b,
problem.b,
mpc,
root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(org_problem.A, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(org_problem.b, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(u_h.vector, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc, rtol=rtol)
def demo_stacked_cubes(outfile: XDMFFile,
theta: float,
gmsh: bool = True,
quad: bool = False,
compare: bool = False,
res: float = 0.1):
log_info(
f"Run theta:{theta:.2f}, Quad: {quad}, Gmsh {gmsh}, Res {res:.2e}")
celltype = "quadrilateral" if quad else "triangle"
if gmsh:
mesh, mt = gmsh_2D_stacked(celltype, theta)
mesh.name = f"mesh_{celltype}_{theta:.2f}_gmsh"
else:
mesh_name = "mesh"
filename = f"meshes/mesh_{celltype}_{theta:.2f}.xdmf"
mesh_2D_dolfin(celltype, theta)
with XDMFFile(MPI.COMM_WORLD, filename, "r") as xdmf:
mesh = xdmf.read_mesh(name=mesh_name)
mesh.name = f"mesh_{celltype}_{theta:.2f}"
tdim = mesh.topology.dim
fdim = tdim - 1
mesh.topology.create_connectivity(tdim, tdim)
mesh.topology.create_connectivity(fdim, tdim)
mt = xdmf.read_meshtags(mesh, name="facet_tags")
# Helper until meshtags can be read in from xdmf
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
r_matrix = rotation_matrix([0, 0, 1], theta)
g_vec = np.dot(r_matrix, [0, -1.25e2, 0])
g = Constant(mesh, PETSc.ScalarType(g_vec[:2]))
def bottom_corner(x):
return np.isclose(x, [[0], [0], [0]]).all(axis=0)
# Fix bottom corner
bc_value = np.array((0, ) * mesh.geometry.dim, dtype=PETSc.ScalarType)
bottom_dofs = locate_dofs_geometrical(V, bottom_corner)
bc_bottom = dirichletbc(bc_value, bottom_dofs, V)
bcs = [bc_bottom]
# Elasticity parameters
E = PETSc.ScalarType(1.0e3)
nu = 0
mu = Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
# Stress computation
def sigma(v):
return (2.0 * mu * sym(grad(v)) +
lmbda * tr(sym(grad(v))) * Identity(len(v)))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(sigma(u), grad(v)) * dx
ds = Measure("ds", domain=mesh, subdomain_data=mt, subdomain_id=3)
rhs = inner(Constant(mesh, PETSc.ScalarType(
(0, 0))), v) * dx + inner(g, v) * ds
def left_corner(x):
return np.isclose(x.T, np.dot(r_matrix, [0, 2, 0])).all(axis=1)
# Create multi point constraint
mpc = MultiPointConstraint(V)
with Timer("~Contact: Create contact constraint"):
nh = create_normal_approximation(V, mt, 4)
mpc.create_contact_slip_condition(mt, 4, 9, nh)
with Timer("~Contact: Add non-slip condition at bottom interface"):
bottom_normal = facet_normal_approximation(V, mt, 5)
mpc.create_slip_constraint(V, (mt, 5), bottom_normal, bcs=bcs)
with Timer("~Contact: Add tangential constraint at one point"):
vertex = locate_entities_boundary(mesh, 0, left_corner)
tangent = facet_normal_approximation(V, mt, 3, tangent=True)
mtv = meshtags(mesh, 0, vertex, np.full(len(vertex), 6,
dtype=np.int32))
mpc.create_slip_constraint(V, (mtv, 6), tangent, bcs=bcs)
mpc.finalize()
rtol = 1e-9
petsc_options = {
"ksp_rtol": 1e-9,
"pc_type": "gamg",
"pc_gamg_type": "agg",
"pc_gamg_square_graph": 2,
"pc_gamg_threshold": 0.02,
"pc_gamg_coarse_eq_limit": 1000,
"pc_gamg_sym_graph": True,
"mg_levels_ksp_type": "chebyshev",
"mg_levels_pc_type": "jacobi",
"mg_levels_esteig_ksp_type": "cg"
# , "help": None, "ksp_view": None
}
# Solve Linear problem
problem = LinearProblem(a, rhs, mpc, bcs=bcs, petsc_options=petsc_options)
# Build near nullspace
null_space = rigid_motions_nullspace(mpc.function_space)
problem.A.setNearNullSpace(null_space)
u_h = problem.solve()
it = problem.solver.getIterationNumber()
if MPI.COMM_WORLD.rank == 0:
print("Number of iterations: {0:d}".format(it))
unorm = u_h.vector.norm()
if MPI.COMM_WORLD.rank == 0:
print(f"Norm of u: {unorm}")
# Write solution to file
ext = "_gmsh" if gmsh else ""
u_h.name = "u_mpc_{0:s}_{1:.2f}{2:s}".format(celltype, theta, ext)
outfile.write_mesh(mesh)
outfile.write_function(u_h, 0.0,
f"Xdmf/Domain/Grid[@Name='{mesh.name}'][1]")
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
if not compare:
return
log_info("Solving reference problem with global matrix (using numpy)")
with Timer("~MPC: Reference problem"):
# Generate reference matrices and unconstrained solution
A_org = assemble_matrix(form(a), bcs)
A_org.assemble()
L_org = assemble_vector(form(rhs))
apply_lifting(L_org, [form(a)], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(L_org, bcs)
root = 0
with Timer("~MPC: Verification"):
compare_mpc_lhs(A_org, problem.A, mpc, root=root)
compare_mpc_rhs(L_org, problem.b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = gather_PETScMatrix(A_org, root=root)
K = gather_transformation_matrix(mpc, root=root)
L_np = gather_PETScVector(L_org, root=root)
u_mpc = gather_PETScVector(u_h.vector, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc, rtol=rtol)
def test_cell_domains(get_assemblers): # noqa: F811
"""
Periodic MPC conditions over integral with different cell subdomains
"""
assemble_matrix, assemble_vector = get_assemblers
N = 5
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 15, N)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
def left_side(x):
return x[0] < 0.5
tdim = mesh.topology.dim
num_cells = mesh.topology.index_map(tdim).size_local
cell_midpoints = compute_midpoints(mesh, tdim, range(num_cells))
values = np.ones(num_cells, dtype=np.intc)
# All cells on right side marked one, all other with 1
values += left_side(cell_midpoints.T)
ct = meshtags(mesh, mesh.topology.dim, np.arange(num_cells,
dtype=np.int32), values)
# Solve Problem without MPC for reference
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
x = ufl.SpatialCoordinate(mesh)
c1 = fem.Constant(mesh, PETSc.ScalarType(2))
c2 = fem.Constant(mesh, PETSc.ScalarType(10))
dx = ufl.Measure("dx", domain=mesh, subdomain_data=ct)
a = c1 * ufl.inner(ufl.grad(u), ufl.grad(v)) * dx(1) +\
c2 * ufl.inner(ufl.grad(u), ufl.grad(v)) * dx(2)\
+ 0.01 * ufl.inner(u, v) * dx(1)
rhs = ufl.inner(x[1], v) * dx(1) + \
ufl.inner(fem.Constant(mesh, PETSc.ScalarType(1)), v) * dx(2)
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Generate reference matrices
A_org = fem.petsc.assemble_matrix(bilinear_form)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {}
for i in range(0, N + 1):
s_m_c[l2b([1, i / N])] = {l2b([0, i / N]): 1}
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c)
mpc.finalize()
# Setup MPC system
with Timer("~TEST: Assemble matrix old"):
A = assemble_matrix(bilinear_form, mpc)
with Timer("~TEST: Assemble vector"):
b = assemble_vector(linear_form, mpc)
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
uh = b.copy()
uh.set(0)
solver.solve(b, uh)
uh.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
mpc.backsubstitution(uh)
root = 0
comm = mesh.comm
with Timer("~TEST: Compare"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc)
list_timings(comm, [TimingType.wall])
Ue = ufl.VectorElement("Lagrange", msh.ufl_cell(), 2)
S = FunctionSpace(msh, Se)
U = FunctionSpace(msh, Ue)
# Get local dofmap sizes for later local tensor tabulations
Ssize = S.element.space_dimension
Usize = U.element.space_dimension
sigma, tau = ufl.TrialFunction(S), ufl.TestFunction(S)
u, v = ufl.TrialFunction(U), ufl.TestFunction(U)
# Locate all facets at the free end and assign them value 1. Sort the
# facet indices (requirement for constructing MeshTags)
free_end_facets = np.sort(locate_entities_boundary(msh, 1, lambda x: np.isclose(x[0], 48.0)))
mt = meshtags(msh, 1, free_end_facets, 1)
ds = ufl.Measure("ds", subdomain_data=mt)
# Homogeneous boundary condition in displacement
u_bc = Function(U)
u_bc.x.array[:] = 0.0
# Displacement BC is applied to the left side
left_facets = locate_entities_boundary(msh, 1, lambda x: np.isclose(x[0], 0.0))
bdofs = locate_dofs_topological(U, 1, left_facets)
bc = dirichletbc(u_bc, bdofs)
# Elastic stiffness tensor and Poisson ratio
E, nu = 1.0, 1.0 / 3.0
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 mesh_3D_dolfin(theta=0,
ct=CellType.tetrahedron,
ext="tetrahedron",
num_refinements=0,
N0=5):
timer = Timer("Create mesh")
def find_plane_function(p0, p1, p2):
"""
Find plane function given three points:
http://www.nabla.hr/CG-LinesPlanesIn3DA3.htm
"""
v1 = np.array(p1) - np.array(p0)
v2 = np.array(p2) - np.array(p0)
n = np.cross(v1, v2)
D = -(n[0] * p0[0] + n[1] * p0[1] + n[2] * p0[2])
return lambda x: np.isclose(0, np.dot(n, x) + D)
def over_plane(p0, p1, p2):
"""
Returns function that checks if a point is over a plane defined
by the points p0, p1 and p2.
"""
v1 = np.array(p1) - np.array(p0)
v2 = np.array(p2) - np.array(p0)
n = np.cross(v1, v2)
D = -(n[0] * p0[0] + n[1] * p0[1] + n[2] * p0[2])
return lambda x: n[0] * x[0] + n[1] * x[1] + D > -n[2] * x[2]
tmp_mesh_name = "tmp_mesh.xdmf"
r_matrix = rotation_matrix([1 / np.sqrt(2), 1 / np.sqrt(2), 0], -theta)
if MPI.COMM_WORLD.rank == 0:
# Create two coarse meshes and merge them
mesh0 = create_unit_cube(MPI.COMM_SELF, N0, N0, N0, ct)
mesh0.geometry.x[:, 2] += 1
mesh1 = create_unit_cube(MPI.COMM_SELF, 2 * N0, 2 * N0, 2 * N0, ct)
tdim0 = mesh0.topology.dim
num_cells0 = mesh0.topology.index_map(tdim0).size_local
cells0 = entities_to_geometry(
mesh0, tdim0,
np.arange(num_cells0, dtype=np.int32).reshape((-1, 1)), False)
tdim1 = mesh1.topology.dim
num_cells1 = mesh1.topology.index_map(tdim1).size_local
cells1 = entities_to_geometry(
mesh1, tdim1,
np.arange(num_cells1, dtype=np.int32).reshape((-1, 1)), False)
cells1 += mesh0.geometry.x.shape[0]
# Concatenate points and cells
points = np.vstack([mesh0.geometry.x, mesh1.geometry.x])
cells = np.vstack([cells0, cells1])
cell = Cell(ext, geometric_dimension=points.shape[1])
domain = Mesh(VectorElement("Lagrange", cell, 1))
# Rotate mesh
points = np.dot(r_matrix, points.T).T
mesh = create_mesh(MPI.COMM_SELF, cells, points, domain)
with XDMFFile(MPI.COMM_SELF, tmp_mesh_name, "w") as xdmf:
xdmf.write_mesh(mesh)
MPI.COMM_WORLD.barrier()
with XDMFFile(MPI.COMM_WORLD, tmp_mesh_name, "r") as xdmf:
mesh = xdmf.read_mesh()
# Refine coarse mesh
for i in range(num_refinements):
mesh.topology.create_entities(mesh.topology.dim - 2)
mesh = refine(mesh, redistribute=True)
tdim = mesh.topology.dim
fdim = tdim - 1
# Find information about facets to be used in meshtags
bottom_points = np.dot(
r_matrix,
np.array([[0, 0, 0], [1, 0, 0], [0, 1, 0], [1, 1, 0]]).T)
bottom = find_plane_function(bottom_points[:, 0], bottom_points[:, 1],
bottom_points[:, 2])
bottom_facets = locate_entities_boundary(mesh, fdim, bottom)
top_points = np.dot(
r_matrix,
np.array([[0, 0, 2], [1, 0, 2], [0, 1, 2], [1, 1, 2]]).T)
top = find_plane_function(top_points[:, 0], top_points[:, 1],
top_points[:, 2])
top_facets = locate_entities_boundary(mesh, fdim, top)
# Determine interface facets
if_points = np.dot(
r_matrix,
np.array([[0, 0, 1], [1, 0, 1], [0, 1, 1], [1, 1, 1]]).T)
interface = find_plane_function(if_points[:, 0], if_points[:, 1],
if_points[:, 2])
i_facets = locate_entities_boundary(mesh, fdim, interface)
mesh.topology.create_connectivity(fdim, tdim)
top_interface = []
bottom_interface = []
facet_to_cell = mesh.topology.connectivity(fdim, tdim)
num_cells = mesh.topology.index_map(tdim).size_local
# Find top and bottom interface facets
cell_midpoints = compute_midpoints(mesh, tdim, range(num_cells))
top_cube = over_plane(if_points[:, 0], if_points[:, 1], if_points[:, 2])
for facet in i_facets:
i_cells = facet_to_cell.links(facet)
assert (len(i_cells == 1))
i_cell = i_cells[0]
if top_cube(cell_midpoints[i_cell]):
top_interface.append(facet)
else:
bottom_interface.append(facet)
# Create cell tags
num_cells = mesh.topology.index_map(tdim).size_local
cell_midpoints = compute_midpoints(mesh, tdim, range(num_cells))
top_cube_marker = 2
indices = []
values = []
for cell_index in range(num_cells):
if top_cube(cell_midpoints[cell_index]):
indices.append(cell_index)
values.append(top_cube_marker)
ct = meshtags(mesh, tdim, np.array(indices, dtype=np.intc),
np.array(values, dtype=np.intc))
# Create meshtags for facet data
markers = {
3: top_facets,
4: bottom_interface,
9: top_interface,
5: bottom_facets
} # , 6: left_facets, 7: right_facets}
indices = np.array([], dtype=np.intc)
values = np.array([], dtype=np.intc)
for key in markers.keys():
indices = np.append(indices, markers[key])
values = np.append(values,
np.full(len(markers[key]), key, dtype=np.intc))
sorted_indices = np.argsort(indices)
mt = meshtags(mesh, fdim, indices[sorted_indices], values[sorted_indices])
mt.name = "facet_tags"
fname = f"meshes/mesh_{ext}_{theta:.2f}.xdmf"
with XDMFFile(MPI.COMM_WORLD, fname, "w") as o_f:
o_f.write_mesh(mesh)
o_f.write_meshtags(ct)
o_f.write_meshtags(mt)
timer.stop()
def test_pipeline(u_from_mpc):
# Create mesh and function space
mesh = create_unit_square(MPI.COMM_WORLD, 5, 5)
V = fem.FunctionSpace(mesh, ("Lagrange", 1))
# Solve Problem without MPC for reference
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
d = fem.Constant(mesh, PETSc.ScalarType(0.01))
x = ufl.SpatialCoordinate(mesh)
f = ufl.sin(2 * ufl.pi * x[0]) * ufl.sin(ufl.pi * x[1])
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx - d * ufl.inner(u, v) * ufl.dx
rhs = ufl.inner(f, v) * ufl.dx
bilinear_form = fem.form(a)
linear_form = fem.form(rhs)
# Generate reference matrices
A_org = fem.petsc.assemble_matrix(bilinear_form)
A_org.assemble()
L_org = fem.petsc.assemble_vector(linear_form)
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE)
# Create multipoint constraint
def periodic_relation(x):
out_x = np.copy(x)
out_x[0] = 1 - x[0]
return out_x
def PeriodicBoundary(x):
return np.isclose(x[0], 1)
facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, PeriodicBoundary)
arg_sort = np.argsort(facets)
mt = meshtags(mesh, mesh.topology.dim - 1, facets[arg_sort], np.full(len(facets), 2, dtype=np.int32))
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V, mt, 2, periodic_relation, [], 1)
mpc.finalize()
if u_from_mpc:
uh = fem.Function(mpc.function_space)
problem = dolfinx_mpc.LinearProblem(bilinear_form, linear_form, mpc, bcs=[], u=uh,
petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
problem.solve()
root = 0
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem.A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem.b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh.vector, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc)
else:
uh = fem.Function(V)
with pytest.raises(ValueError):
problem = dolfinx_mpc.LinearProblem(bilinear_form, linear_form, mpc, bcs=[], u=uh,
petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
problem.solve()
def test_assembly_ds_domains(mode):
mesh = create_unit_square(MPI.COMM_WORLD, 10, 10, ghost_mode=mode)
V = FunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
def bottom(x):
return np.isclose(x[1], 0.0)
def top(x):
return np.isclose(x[1], 1.0)
def left(x):
return np.isclose(x[0], 0.0)
def right(x):
return np.isclose(x[0], 1.0)
bottom_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1,
bottom)
bottom_vals = np.full(bottom_facets.shape, 1, np.intc)
top_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, top)
top_vals = np.full(top_facets.shape, 2, np.intc)
left_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, left)
left_vals = np.full(left_facets.shape, 3, np.intc)
right_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, right)
right_vals = np.full(right_facets.shape, 6, np.intc)
indices = np.hstack((bottom_facets, top_facets, left_facets, right_facets))
values = np.hstack((bottom_vals, top_vals, left_vals, right_vals))
indices, pos = np.unique(indices, return_index=True)
marker = meshtags(mesh, mesh.topology.dim - 1, indices, values[pos])
ds = ufl.Measure('ds', subdomain_data=marker, domain=mesh)
w = Function(V)
w.x.array[:] = 0.5
bc = dirichletbc(Function(V), range(30))
# Assemble matrix
a = form(w * ufl.inner(u, v) * (ds(1) + ds(2) + ds(3) + ds(6)))
A = assemble_matrix(a)
A.assemble()
norm1 = A.norm()
a2 = form(w * ufl.inner(u, v) * ds)
A2 = assemble_matrix(a2)
A2.assemble()
norm2 = A2.norm()
assert norm1 == pytest.approx(norm2, 1.0e-12)
# Assemble vector
L = form(ufl.inner(w, v) * (ds(1) + ds(2) + ds(3) + ds(6)))
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
L2 = form(ufl.inner(w, v) * ds)
b2 = assemble_vector(L2)
apply_lifting(b2, [a2], [[bc]])
b2.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b2, [bc])
assert b.norm() == pytest.approx(b2.norm(), 1.0e-12)
# Assemble scalar
L = form(w * (ds(1) + ds(2) + ds(3) + ds(6)))
s = assemble_scalar(L)
s = mesh.comm.allreduce(s, op=MPI.SUM)
L2 = form(w * ds)
s2 = assemble_scalar(L2)
s2 = mesh.comm.allreduce(s2, op=MPI.SUM)
assert (s == pytest.approx(s2, 1.0e-12)
and 2.0 == pytest.approx(s, 1.0e-12))
def mesh_3D_dolfin(theta: float = 0,
ct: _mesh.CellType = _mesh.CellType.tetrahedron,
ext: str = "tetrahedron",
res: float = 0.1):
timer = _common.Timer("~~Contact: Create mesh")
def find_plane_function(p0, p1, p2):
"""
Find plane function given three points:
http://www.nabla.hr/CG-LinesPlanesIn3DA3.htm
"""
v1 = np.array(p1) - np.array(p0)
v2 = np.array(p2) - np.array(p0)
n = np.cross(v1, v2)
D = -(n[0] * p0[0] + n[1] * p0[1] + n[2] * p0[2])
return lambda x: np.isclose(0, np.dot(n, x) + D)
def over_plane(p0, p1, p2):
"""
Returns function that checks if a point is over a plane defined
by the points p0, p1 and p2.
"""
v1 = np.array(p1) - np.array(p0)
v2 = np.array(p2) - np.array(p0)
n = np.cross(v1, v2)
D = -(n[0] * p0[0] + n[1] * p0[1] + n[2] * p0[2])
return lambda x: n[0] * x[0] + n[1] * x[1] + D > -n[2] * x[2]
N = int(1 / res)
if MPI.COMM_WORLD.rank == 0:
mesh0 = _mesh.create_unit_cube(MPI.COMM_SELF, N, N, N, ct)
mesh1 = _mesh.create_unit_cube(MPI.COMM_SELF, 2 * N, 2 * N, 2 * N, ct)
mesh0.geometry.x[:, 2] += 1
# Stack the two meshes in one mesh
r_matrix = _utils.rotation_matrix([1 / np.sqrt(2), 1 / np.sqrt(2), 0],
-theta)
points = np.vstack([mesh0.geometry.x, mesh1.geometry.x])
points = np.dot(r_matrix, points.T).T
# Transform topology info into geometry info
tdim0 = mesh0.topology.dim
num_cells0 = mesh0.topology.index_map(tdim0).size_local
cells0 = _cpp.mesh.entities_to_geometry(
mesh0, tdim0,
np.arange(num_cells0, dtype=np.int32).reshape((-1, 1)), False)
tdim1 = mesh1.topology.dim
num_cells1 = mesh1.topology.index_map(tdim1).size_local
cells1 = _cpp.mesh.entities_to_geometry(
mesh1, tdim1,
np.arange(num_cells1, dtype=np.int32).reshape((-1, 1)), False)
cells1 += mesh0.geometry.x.shape[0]
cells = np.vstack([cells0, cells1])
cell = ufl.Cell(ext, geometric_dimension=points.shape[1])
domain = ufl.Mesh(ufl.VectorElement("Lagrange", cell, 1))
mesh = _mesh.create_mesh(MPI.COMM_SELF, cells, points, domain)
tdim = mesh.topology.dim
fdim = tdim - 1
# Find information about facets to be used in meshtags
bottom_points = np.dot(
r_matrix,
np.array([[0, 0, 0], [1, 0, 0], [0, 1, 0], [1, 1, 0]]).T)
bottom = find_plane_function(bottom_points[:, 0], bottom_points[:, 1],
bottom_points[:, 2])
bottom_facets = _mesh.locate_entities_boundary(mesh, fdim, bottom)
top_points = np.dot(
r_matrix,
np.array([[0, 0, 2], [1, 0, 2], [0, 1, 2], [1, 1, 2]]).T)
top = find_plane_function(top_points[:, 0], top_points[:, 1],
top_points[:, 2])
top_facets = _mesh.locate_entities_boundary(mesh, fdim, top)
# left_side = find_line_function(top_points[:, 0], top_points[:, 3])
# left_facets = _mesh.locate_entities_boundary(
# mesh, fdim, left_side)
# right_side = find_line_function(top_points[:, 1], top_points[:, 2])
# right_facets = _mesh.locate_entities_boundary(
# mesh, fdim, right_side)
if_points = np.dot(
r_matrix,
np.array([[0, 0, 1], [1, 0, 1], [0, 1, 1], [1, 1, 1]]).T)
interface = find_plane_function(if_points[:, 0], if_points[:, 1],
if_points[:, 2])
i_facets = _mesh.locate_entities_boundary(mesh, fdim, interface)
mesh.topology.create_connectivity(fdim, tdim)
top_interface = []
bottom_interface = []
facet_to_cell = mesh.topology.connectivity(fdim, tdim)
num_cells = mesh.topology.index_map(tdim).size_local
cell_midpoints = _cpp.mesh.compute_midpoints(mesh, tdim,
range(num_cells))
top_cube = over_plane(if_points[:, 0], if_points[:, 1], if_points[:,
2])
for facet in i_facets:
i_cells = facet_to_cell.links(facet)
assert (len(i_cells == 1))
i_cell = i_cells[0]
if top_cube(cell_midpoints[i_cell]):
top_interface.append(facet)
else:
bottom_interface.append(facet)
num_cells = mesh.topology.index_map(tdim).size_local
cell_midpoints = _cpp.mesh.compute_midpoints(mesh, tdim,
range(num_cells))
top_cube_marker = 2
indices = []
values = []
for cell_index in range(num_cells):
if top_cube(cell_midpoints[cell_index]):
indices.append(cell_index)
values.append(top_cube_marker)
ct = _mesh.meshtags(mesh, tdim, np.array(indices, dtype=np.int32),
np.array(values, dtype=np.int32))
# Create meshtags for facet data
markers: Dict[int, np.ndarray] = {
3: top_facets,
4: np.hstack(bottom_interface),
9: np.hstack(top_interface),
5: bottom_facets
} # , 6: left_facets, 7: right_facets}
all_indices = []
all_values = []
for key in markers.keys():
all_indices.append(np.asarray(markers[key], dtype=np.int32))
all_values.append(np.full(len(markers[key]), key, dtype=np.int32))
arg_sort = np.argsort(np.hstack(all_indices))
sorted_vals = np.asarray(np.hstack(all_values)[arg_sort],
dtype=np.int32)
sorted_indices = np.asarray(np.hstack(all_indices)[arg_sort],
dtype=np.int32)
mt = _mesh.meshtags(mesh, fdim, sorted_indices, sorted_vals)
mt.name = "facet_tags" # type: ignore
fname = f"meshes/mesh_{ext}_{theta:.2f}.xdmf"
with _io.XDMFFile(MPI.COMM_SELF, fname, "w") as o_f:
o_f.write_mesh(mesh)
o_f.write_meshtags(ct)
o_f.write_meshtags(mt)
timer.stop()
def bench_elasticity_one(r_lvl: int = 0,
out_hdf5: h5py.File = None,
xdmf: bool = False,
boomeramg: bool = False,
kspview: bool = False):
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)
fdim = mesh.topology.dim - 1
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
# 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])
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
# Elasticity parameters
E = 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, (0, 0, -1e2))
# 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)
# Create MPC
with Timer("~Elasticity: Init constraint"):
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {l2b([1, 0, 0]): {l2b([1, 0, 1]): 0.5}}
mpc = MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, 2, 2)
mpc.finalize()
# Setup MPC system
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)
# 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)
# Create functionspace and function for mpc vector
# Solve Linear problem
solver = PETSc.KSP().create(MPI.COMM_WORLD)
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-10
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["help"] = None # List all available options
# opts["ksp_view"] = None # List progress of solver
with Timer("~Elasticity: Solve problem") as timer:
null_space = rigid_motions_nullspace(mpc.function_space)
A.setNearNullSpace(null_space)
solver.setFromOptions()
solver.setOperators(A)
# Solve linear problem
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()
it = solver.getIterationNumber()
if kspview:
solver.view()
# Print max usage of summary
mem = sum(
MPI.COMM_WORLD.allgather(
resource.getrusage(resource.RUSAGE_SELF).ru_maxrss))
num_dofs = V.dofmap.index_map.size_global * V.dofmap.index_map_bs
if MPI.COMM_WORLD.rank == 0:
print(f"Rlvl {r_lvl}, Iterations {it}")
print(f"Rlvl {r_lvl}, Max usage {mem} (kb), #dofs {num_dofs}")
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] = solver_time[0]
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_{r_lvl}.xdmf"
with XDMFFile(MPI.COMM_WORLD, fname, "w") as outfile:
outfile.write_mesh(mesh)
outfile.write_function(u_h)
