def run_scalar_test(mesh, V, degree):
""" Manufactured Poisson problem, solving u = x[1]**p, where p is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
# Get quadrature degree for bilinear form integrand (ignores effect of non-affine map)
a = inner(grad(u), grad(v)) * dx(metadata={"quadrature_degree": -1})
a.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(a)
a = form(a)
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
f = -div(grad(u_exact))
# Set quadrature degree for linear form integrand (ignores effect of non-affine map)
L = inner(f, v) * dx(metadata={"quadrature_degree": -1})
L.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(L)
L = form(L)
u_bc = Function(V)
u_bc.interpolate(lambda x: x[1]**degree)
# Create Dirichlet boundary condition
facetdim = mesh.topology.dim - 1
mesh.topology.create_connectivity(facetdim, mesh.topology.dim)
bndry_facets = np.where(
np.array(compute_boundary_facets(mesh.topology)) == 1)[0]
bdofs = locate_dofs_topological(V, facetdim, bndry_facets)
bc = dirichletbc(u_bc, bdofs)
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
a = form(a)
A = assemble_matrix(a, bcs=[bc])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
uh = Function(V)
solver.solve(b, uh.vector)
uh.x.scatter_forward()
M = (u_exact - uh)**2 * dx
M = form(M)
error = mesh.comm.allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def test_manufactured_vector2(family, degree, filename, datadir):
"""Projection into H(div/curl) spaces"""
# Skip slowest tests
if "tetra" in filename and degree > 2:
return
with XDMFFile(MPI.comm_world, os.path.join(datadir, filename)) as xdmf:
mesh = xdmf.read_mesh(GhostMode.none)
# FIXME: these test are currently failing on unordered meshes
if "tetra" in filename:
if family == "N1curl":
Ordering.order_simplex(mesh)
V = FunctionSpace(mesh, (family, degree + 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = inner(u, v) * dx
xp = np.array([0.33, 0.33, 0.0])
tree = geometry.BoundingBoxTree(mesh, mesh.geometry.dim)
cells = geometry.compute_first_entity_collision(tree, mesh, xp)
# Source term
x = SpatialCoordinate(mesh)
u_ref = x[0]**degree
L = inner(u_ref, v[0]) * dx
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a)
A.assemble()
# Create LU linear solver (Note: need to use a solver that
# re-orders to handle pivots, e.g. not the PETSc built-in LU
# solver)
solver = PETSc.KSP().create(MPI.comm_world)
solver.setType("preonly")
solver.getPC().setType('lu')
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
up = uh.eval(xp, cells[0])
print("test0:", up)
print("test1:", xp[0]**degree)
u_exact = np.zeros(mesh.geometry.dim)
u_exact[0] = xp[0]**degree
assert np.allclose(up, u_exact)
def codeDG(self):
code = self._code()
u = self.trialFunction
ubar = Coefficient(u.ufl_function_space())
penalty = self.penalty
if penalty is None:
penalty = 1
if isinstance(penalty, Expr):
if penalty.ufl_shape == ():
penalty = as_vector([penalty])
try:
penalty = expand_indices(
expand_derivatives(expand_compounds(penalty)))
except:
pass
assert penalty.ufl_shape == u.ufl_shape
dmPenalty = as_vector([
replace(
expand_derivatives(diff(replace(penalty, {u: ubar}),
ubar))[i, i], {ubar: u})
for i in range(u.ufl_shape[0])
])
else:
dmPenalty = None
code.append(AccessModifier("public"))
x = SpatialCoordinate(self.space.cell())
predefined = {}
self.predefineCoefficients(predefined, x)
spatial = Variable('const auto', 'y')
predefined.update({
x:
UnformattedExpression(
'auto', 'entity().geometry().global( Dune::Fem::coordinate( x ) )')
})
generateMethod(code,
penalty,
'RRangeType',
'penalty',
args=['const Point &x', 'const DRangeType &u'],
targs=['class Point', 'class DRangeType'],
static=False,
const=True,
predefined=predefined)
generateMethod(code,
dmPenalty,
'RRangeType',
'linPenalty',
args=['const Point &x', 'const DRangeType &u'],
targs=['class Point', 'class DRangeType'],
static=False,
const=True,
predefined=predefined)
return code
def test_manufactured_vector1(family, degree, filename, datadir):
"""Projection into H(div/curl) spaces"""
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, (family, degree))
W = VectorFunctionSpace(mesh, ("CG", degree))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = inner(u, v) * dx
# Source term
x = SpatialCoordinate(mesh)
u_ref = x[0]**degree
L = inner(u_ref, v[0]) * dx
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a)
A.assemble()
# Create LU linear solver (Note: need to use a solver that
# re-orders to handle pivots, e.g. not the PETSc built-in LU
# solver)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType("preonly")
solver.getPC().setType('lu')
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
u_exact = Function(W)
u_exact.interpolate(lambda x: np.array([
x[0]**degree if i == 0 else 0 * x[0] for i in range(mesh.topology.dim)
]))
M = inner(uh - u_exact, uh - u_exact) * dx
M = fem.Form(M)
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def test_manufactured_vector1(family, degree, filename, datadir):
"""Projection into H(div/curl) spaces"""
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, (family, degree))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = inner(u, v) * dx
# Source term
x = SpatialCoordinate(mesh)
u_ref = x[0]**degree
L = inner(u_ref, v[0]) * dx
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a)
A.assemble()
# Create LU linear solver (Note: need to use a solver that
# re-orders to handle pivots, e.g. not the PETSc built-in LU
# solver)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType("preonly")
solver.getPC().setType('lu')
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
xp = np.array([0.33, 0.33, 0.0])
tree = geometry.BoundingBoxTree(mesh, mesh.geometry.dim)
cells = geometry.compute_first_entity_collision(tree, mesh, xp)
up = uh.eval(xp, cells[0])
print("test0:", up)
print("test1:", xp[0]**degree)
u_exact = np.zeros(mesh.geometry.dim)
u_exact[0] = xp[0]**degree
assert np.allclose(up, u_exact)
def run_vector_test(mesh, V, degree):
"""Projection into H(div/curl) spaces"""
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = inner(u, v) * dx
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[0]**degree
L = inner(u_exact, v[0]) * dx
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 (Note: need to use a solver that
# re-orders to handle pivots, e.g. not the PETSc built-in LU solver)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType("preonly")
solver.getPC().setType('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[0])**2 * dx
for i in range(1, mesh.topology.dim):
M += uh[i]**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 create_box_mesh(nxs, lxs, pxs, bcs):
assert (len(nxs) == len(lxs))
assert (len(nxs) == len(pxs))
assert (len(nxs) == len(bcs))
pxs = np.array(pxs, dtype=np.float32)
lxs = np.array(lxs, dtype=np.float32)
nxs = np.array(nxs, dtype=np.int32)
mesh = SingleBlockMesh(nxs, bcs)
xs = SpatialCoordinate(mesh)
newcoords = Function(mesh.coordinates.function_space(), name='newcoords')
xlist = []
for i in range(len(pxs)):
xlist.append(xs[i] * Constant(lxs[i]) + Constant(pxs[i]) -
Constant(lxs[i]) / 2.)
newcoords.interpolate(as_vector(xlist))
mesh.coordinates.assign(newcoords)
return mesh
def run_vector_test(mesh, V, degree):
"""Projection into H(div/curl) spaces"""
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(u, v) * dx)
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[0]**degree
L = form(inner(u_exact, v[0]) * dx)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a)
A.assemble()
# Create LU linear solver (Note: need to use a solver that
# re-orders to handle pivots, e.g. not the PETSc built-in LU solver)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType("preonly")
solver.getPC().setType('lu')
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.x.scatter_forward()
# Calculate error
M = (u_exact - uh[0])**2 * dx
for i in range(1, mesh.topology.dim):
M += uh[i]**2 * dx
M = form(M)
error = mesh.comm.allreduce(assemble_scalar(M), op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def initialize(self, obj):
A, P = obj.getOperators()
prefix = obj.getOptionsPrefix()
V = get_function_space(obj.getDM())
mesh = V.mesh()
family = str(V.ufl_element().family())
degree = V.ufl_element().degree()
if family != 'Raviart-Thomas' or degree != 1:
raise ValueError(
"Hypre ADS requires lowest order RT elements! (not %s of degree %d)"
% (family, degree))
P1 = FunctionSpace(mesh, "Lagrange", 1)
NC1 = FunctionSpace(mesh, "N1curl", 1)
# DiscreteGradient
G = Interpolator(grad(TestFunction(P1)), NC1).callable().handle
# DiscreteCurl
C = Interpolator(curl(TestFunction(NC1)), V).callable().handle
pc = PETSc.PC().create(comm=obj.comm)
pc.incrementTabLevel(1, parent=obj)
pc.setOptionsPrefix(prefix + "hypre_ads_")
pc.setOperators(A, P)
pc.setType('hypre')
pc.setHYPREType('ads')
pc.setHYPREDiscreteGradient(G)
pc.setHYPREDiscreteCurl(C)
V = VectorFunctionSpace(mesh, "Lagrange", 1)
linear_coordinates = interpolate(SpatialCoordinate(mesh),
V).dat.data_ro.copy()
pc.setCoordinates(linear_coordinates)
pc.setUp()
self.pc = pc
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 compute(space, epsilon, weakBnd, skeleton, mol=None):
u = TrialFunction(space)
v = TestFunction(space)
n = FacetNormal(space)
he = avg(CellVolume(space)) / FacetArea(space)
hbnd = CellVolume(space) / FacetArea(space)
x = SpatialCoordinate(space)
exact = uflFunction(space.gridView,
name="exact",
order=3,
ufl=sin(x[0] * x[1]))
uh = space.interpolate(exact, name="solution")
# diffusion factor
eps = Constant(epsilon, "eps")
# transport direction and upwind flux
b = as_vector([1, 0])
hatb = (dot(b, n) + abs(dot(b, n))) / 2.0
# characteristic function for left/right boundary
dD = conditional((1 + x[0]) * (1 - x[0]) < 1e-10, 1, 0)
# penalty parameter
beta = Constant(20 * space.order**2, "beta")
rhs = -(div(eps * grad(exact) - b * exact)) * v * dx
aInternal = dot(eps * grad(u) - b * u, grad(v)) * dx
aInternal -= eps * dot(grad(exact), n) * v * (1 - dD) * ds
diffSkeleton = eps*beta/he*jump(u)*jump(v)*dS -\
eps*dot(avg(grad(u)),n('+'))*jump(v)*dS -\
eps*jump(u)*dot(avg(grad(v)),n('+'))*dS
if weakBnd:
diffSkeleton += eps*beta/hbnd*(u-exact)*v*dD*ds -\
eps*dot(grad(exact),n)*v*dD*ds
advSkeleton = jump(hatb * u) * jump(v) * dS
if weakBnd:
advSkeleton += (hatb * u + (dot(b, n) - hatb) * exact) * v * dD * ds
if skeleton:
form = aInternal + diffSkeleton + advSkeleton
else:
form = aInternal
if weakBnd and skeleton:
strongBC = None
else:
strongBC = DirichletBC(space, exact, dD)
if space.storage[0] == "numpy":
solver = {
"solver": ("suitesparse", "umfpack"),
"parameters": {
"newton.verbose": True,
"newton.linear.verbose": False,
"newton.linear.tolerance": 1e-5,
}
}
else:
solver = {
"solver": "bicgstab",
"parameters": {
"newton.linear.preconditioning.method": "ilu",
"newton.linear.tolerance": 1e-13,
"newton.verbose": True,
"newton.linear.verbose": False
}
}
if mol == 'mol':
scheme = molSolutionScheme([form == rhs, strongBC], **solver)
else:
scheme = solutionScheme([form == rhs, strongBC], **solver)
eoc = []
info = scheme.solve(target=uh)
error = dot(uh - exact, uh - exact)
error0 = math.sqrt(integrate(gridView, error, order=5))
print(error0, " # output", flush=True)
for i in range(3):
gridView.hierarchicalGrid.globalRefine(1)
uh.interpolate(exact)
scheme.solve(target=uh)
error = dot(uh - exact, uh - exact)
error1 = math.sqrt(integrate(gridView, error, order=5))
eoc += [math.log(error1 / error0) / math.log(0.5)]
print(i, error0, error1, eoc, " # output", flush=True)
error0 = error1
# print(space.order,epsilon,eoc)
if (eoc[-1] - (space.order + 1)) < -0.1:
print("ERROR:", space.order, epsilon, eoc)
return eoc
from __future__ import print_function, division from ufl import as_vector, dot, grad, cos, pi, SpatialCoordinate, triangle from dune.grid import structuredGrid, gridFunction from dune.fem.space import lagrange, combined, product x = SpatialCoordinate(triangle) exact = as_vector([cos(2. * pi * x[0]) * cos(2. * pi * x[1]), dot(x, x)]) grid = structuredGrid([0, 0], [1, 1], [16, 16]) spc1 = lagrange(grid, dimRange=1, order=1) spc2 = lagrange(grid, dimRange=1, order=2) test1 = spc1.interpolate(exact[0], name="test") test2 = spc2.interpolate(exact[1], name="test") spc = combined(spc1, spc2) solution = spc.interpolate(exact, name="solution") space = product(spc1, spc2, components=["p", "s"]) df = space.interpolate(exact, name="df") # print(df.dofVector.size,solution.dofVector.size, # df.components[0].dofVector.size,df.p.dofVector.size,test1.dofVector.size) assert df.components[0].dofVector.size == test1.dofVector.size assert df.s.dofVector.size == test2.dofVector.size assert df.dofVector.size == solution.dofVector.size df.interpolate(solution) solution.interpolate(df) test1.interpolate(df.p) df.s.interpolate(test2) df.components[0].interpolate(solution[0]) df.p.interpolate(solution[0])
def demo_elasticity():
mesh = create_unit_square(MPI.COMM_WORLD, 10, 10)
V = fem.VectorFunctionSpace(mesh, ("Lagrange", 1))
# Generate Dirichlet BC on lower boundary (Fixed)
def boundaries(x):
return np.isclose(x[0], np.finfo(float).eps)
facets = locate_entities_boundary(mesh, 1, boundaries)
topological_dofs = fem.locate_dofs_topological(V, 1, facets)
bc = fem.dirichletbc(np.array([0, 0], dtype=PETSc.ScalarType),
topological_dofs, V)
bcs = [bc]
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
# Elasticity parameters
E = PETSc.ScalarType(1.0e4)
nu = 0.0
mu = fem.Constant(mesh, E / (2.0 * (1.0 + nu)))
lmbda = fem.Constant(mesh, E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
# Stress computation
def sigma(v):
return (2.0 * mu * sym(grad(v)) +
lmbda * tr(sym(grad(v))) * Identity(len(v)))
x = SpatialCoordinate(mesh)
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(sigma(u), grad(v)) * dx
rhs = inner(as_vector((0, (x[0] - 0.5) * 10**4 * x[1])), v) * dx
# Create MPC
def l2b(li):
return np.array(li, dtype=np.float64).tobytes()
s_m_c = {l2b([1, 0]): {l2b([1, 1]): 0.9}}
mpc = MultiPointConstraint(V)
mpc.create_general_constraint(s_m_c, 1, 1)
mpc.finalize()
# Solve Linear problem
petsc_options = {"ksp_type": "preonly", "pc_type": "lu"}
problem = LinearProblem(a, rhs, mpc, bcs=bcs, petsc_options=petsc_options)
u_h = problem.solve()
u_h.name = "u_mpc"
with XDMFFile(MPI.COMM_WORLD, "results/demo_elasticity.xdmf",
"w") as outfile:
outfile.write_mesh(mesh)
outfile.write_function(u_h)
# Solve the MPC problem using a global transformation matrix
# and numpy solvers to get reference values
bilinear_form = fem.form(a)
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
linear_form = fem.form(rhs)
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
fem.petsc.set_bc(L_org, bcs)
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A_org)
u_ = fem.Function(V)
solver.solve(L_org, u_.vector)
u_.x.scatter_forward()
u_.name = "u_unconstrained"
with XDMFFile(MPI.COMM_WORLD, "results/demo_elasticity.xdmf",
"a") as outfile:
outfile.write_function(u_)
outfile.close()
root = 0
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem.A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem.b, mpc, root=root)
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(u_h.vector, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T * A_csr * K
reduced_L = K.T @ L_np
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K @ d
assert np.allclose(uh_numpy, u_mpc)
# Print out master-slave connectivity for the first slave
master_owner = None
master_data = None
slave_owner = None
if mpc.num_local_slaves > 0:
slave_owner = MPI.COMM_WORLD.rank
bs = mpc.function_space.dofmap.index_map_bs
slave = mpc.slaves[0]
print("Constrained: {0:.5e}\n Unconstrained: {1:.5e}".format(
u_h.x.array[slave], u_.vector.array[slave]))
master_owner = mpc._cpp_object.owners.links(slave)[0]
_masters = mpc.masters
master = _masters.links(slave)[0]
glob_master = mpc.function_space.dofmap.index_map.local_to_global(
[master // bs])[0]
coeffs, offs = mpc.coefficients()
master_data = [
glob_master * bs + master % bs,
coeffs[offs[slave]:offs[slave + 1]][0]
]
# If master not on proc send info to this processor
if MPI.COMM_WORLD.rank != master_owner:
MPI.COMM_WORLD.send(master_data, dest=master_owner, tag=1)
else:
print("Master*Coeff: {0:.5e}".format(
coeffs[offs[slave]:offs[slave + 1]][0] *
u_h.x.array[_masters.links(slave)[0]]))
# As a processor with a master is not aware that it has a master,
# Determine this so that it can receive the global dof and coefficient
master_recv = MPI.COMM_WORLD.allgather(master_owner)
for master in master_recv:
if master is not None:
master_owner = master
break
if slave_owner != master_owner and MPI.COMM_WORLD.rank == master_owner:
dofmap = mpc.function_space.dofmap
bs = dofmap.index_map_bs
in_data = MPI.COMM_WORLD.recv(source=MPI.ANY_SOURCE, tag=1)
num_local = dofmap.index_map.size_local + dofmap.index_map.num_ghosts
l2g = dofmap.index_map.local_to_global(
np.arange(num_local, dtype=np.int32))
l_index = np.flatnonzero(l2g == in_data[0] // bs)[0]
print("Master*Coeff (on other proc): {0:.5e}".format(
u_h.x.array[l_index * bs + in_data[0] % bs] * in_data[1]))
def _compileUFL(integrands, form, *args, tempVars=True):
if isinstance(form, Equation):
form = form.lhs - form.rhs
if not isinstance(form, Form):
raise ValueError("ufl.Form or ufl.Equation expected.")
# added for dirichlet treatment same as conservationlaw model
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
uflExpr = [form] + [bc.ufl_value for bc in dirichletBCs]
if len(form.arguments()) < 2:
raise ValueError(
"Integrands model requires form with at least two arguments.")
x = SpatialCoordinate(form.ufl_cell())
n = FacetNormal(form.ufl_cell())
cellVolume = CellVolume(form.ufl_cell())
maxCellEdgeLength = MaxCellEdgeLength(form.ufl_cell())
minCellEdgeLength = MinCellEdgeLength(form.ufl_cell())
facetArea = FacetArea(form.ufl_cell())
maxFacetEdgeLength = MaxFacetEdgeLength(form.ufl_cell())
minFacetEdgeLength = MinFacetEdgeLength(form.ufl_cell())
phi, u = form.arguments()
ubar = Coefficient(u.ufl_function_space())
derivatives = gatherDerivatives(form, [phi, u])
derivatives_phi = derivatives[0]
derivatives_u = derivatives[1]
derivatives_ubar = map_expr_dags(Replacer({u: ubar}), derivatives_u)
try:
integrands.field = u.ufl_function_space().field
except AttributeError:
pass
integrals = splitForm(form, [phi])
dform = apply_derivatives(derivative(action(form, ubar), ubar, u))
linearizedIntegrals = splitForm(dform, [phi, u])
if not set(
integrals.keys()) <= {'cell', 'exterior_facet', 'interior_facet'}:
raise Exception('unknown integral encountered in ' +
str(set(integrals.keys())) + '.')
if 'cell' in integrals.keys():
arg = Variable(integrands.domainValueTuple, 'u')
predefined = {
derivatives_u[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.interior = generateUnaryCode(predefined,
derivatives_phi,
integrals['cell'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.linearizedInterior = generateUnaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('cell'),
tempVars=tempVars)
if 'exterior_facet' in integrals.keys():
arg = Variable(integrands.domainValueTuple, 'u')
predefined = {
derivatives_u[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[n] = integrands.facetNormal('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.boundary = generateUnaryCode(predefined,
derivatives_phi,
integrals['exterior_facet'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[n] = integrands.facetNormal('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.linearizedBoundary = generateUnaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('exterior_facet'),
tempVars=tempVars)
if 'interior_facet' in integrals.keys():
argIn = Variable(integrands.domainValueTuple, 'uIn')
argOut = Variable(integrands.domainValueTuple, 'uOut')
predefined = {
derivatives_u[i](s): arg[i]
for i in range(len(derivatives_u))
for s, arg in (('+', argIn), ('-', argOut))
}
predefined[x] = integrands.spatialCoordinate('xIn')
predefined[n('+')] = integrands.facetNormal('xIn')
predefined[cellVolume('+')] = integrands.cellVolume('Side::in')
predefined[cellVolume('-')] = integrands.cellVolume('Side::out')
predefined[maxCellEdgeLength('+')] = maxEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[maxCellEdgeLength('-')] = maxEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[minCellEdgeLength('+')] = minEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[minCellEdgeLength('-')] = minEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, True)
integrands.skeleton = generateBinaryCode(predefined,
derivatives_phi,
integrals['interior_facet'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i](s): arg[i]
for i in range(len(derivatives_u))
for s, arg in (('+', argIn), ('-', argOut))
}
predefined[x] = integrands.spatialCoordinate('xIn')
predefined[n('+')] = integrands.facetNormal('xIn')
predefined[cellVolume('+')] = integrands.cellVolume('Side::in')
predefined[cellVolume('-')] = integrands.cellVolume('Side::out')
predefined[maxCellEdgeLength('+')] = maxEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[maxCellEdgeLength('-')] = maxEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[minCellEdgeLength('+')] = minEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[minCellEdgeLength('-')] = minEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, True)
integrands.linearizedSkeleton = generateBinaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('interior_facet'),
tempVars=tempVars)
if dirichletBCs:
integrands.hasDirichletBoundary = True
predefined = {}
# predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + integrands.arg_x.name + ' ) )')
predefined[x] = UnformattedExpression(
'auto', 'intersection.geometry().center( )')
integrands.predefineCoefficients(predefined, False)
maxId = 0
codeDomains = []
bySubDomain = dict()
neuman = []
wholeDomain = None
for bc in dirichletBCs:
if bc.subDomain in bySubDomain:
raise Exception(
'Multiply defined Dirichlet boundary for subdomain ' +
str(bc.subDomain))
if not isinstance(bc.functionSpace,
(FunctionSpace, FiniteElementBase)):
raise Exception(
'Function space must either be a ufl.FunctionSpace or a ufl.FiniteElement'
)
if isinstance(bc.functionSpace, FunctionSpace) and (
bc.functionSpace != u.ufl_function_space()):
raise Exception(
'Space of trial function and dirichlet boundary function must be the same - note that boundary conditions on subspaces are not available, yet'
)
if isinstance(bc.functionSpace, FiniteElementBase) and (
bc.functionSpace != u.ufl_element()):
raise Exception(
'Cannot handle boundary conditions on subspaces, yet')
if isinstance(bc.value, list):
neuman = [i for i, x in enumerate(bc.value) if x == None]
else:
neuman = []
value = ExprTensor(u.ufl_shape)
for key in value.keys():
value[key] = Indexed(
bc.ufl_value,
MultiIndex(tuple(FixedIndex(k) for k in key)))
if bc.subDomain is None:
wholeDomain = value, neuman
elif isinstance(bc.subDomain, int):
bySubDomain[bc.subDomain] = value, neuman
maxId = max(maxId, bc.subDomain)
else:
domain = ExprTensor(())
for key in domain.keys():
domain[key] = Indexed(
bc.subDomain,
MultiIndex(tuple(FixedIndex(k) for k in key)))
codeDomains.append((value, neuman, domain))
defaultCode = []
if len(codeDomains) > 0:
defaultCode.append(Declaration(Variable('int', 'domainId')))
# defaultCode.append(Declaration(Variable('auto', 'x'),
# initializer=UnformattedExpression('auto','intersection.geometry().center()')))
for i, v in enumerate(codeDomains):
block = Block()
block.append(
generateDirichletDomainCode(predefined,
v[2],
tempVars=tempVars))
block.append('if (domainId)')
ifBlock = UnformattedBlock()
ifBlock.append(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ str(maxId + i + 2) + ' );')
if len(v[1]) > 0:
[
ifBlock.append('dirichletComponent[' + str(c) + '] = 0;')
for c in v[1]
]
ifBlock.append('return true;')
block.append(ifBlock)
defaultCode.append(block)
if wholeDomain is not None:
block = UnformattedBlock()
block.append(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ str(maxId + 1) + ' );')
if len(wholeDomain[1]) > 0:
[
block.append('dirichletComponent[' + str(c) + '] = 0;')
for c in wholeDomain[1]
]
block.append('return true;')
defaultCode.append(block)
defaultCode.append(return_(False))
bndId = Variable('const int', 'bndId')
getBndId = UnformattedExpression(
'int', 'BoundaryIdProviderType::boundaryId( ' +
integrands.arg_i.name + ' )')
# getBndId = UnformattedExpression('int', 'boundaryIdGetter_.boundaryId( ' + integrands.arg_i.name + ' )')
switch = SwitchStatement(bndId, default=defaultCode)
for i, v in bySubDomain.items():
code = []
if len(v[1]) > 0:
[
code.append('dirichletComponent[' + str(c) + '] = 0;')
for c in v[1]
]
code.append(return_(True))
switch.append(i, code)
integrands.isDirichletIntersection = [
Declaration(bndId, initializer=getBndId),
UnformattedBlock(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ bndId.name + ' );'), switch
]
predefined[x] = UnformattedExpression(
'auto', 'entity().geometry().global( Dune::Fem::coordinate( ' +
integrands.arg_x.name + ' ) )')
if wholeDomain is None:
defaultCode = assign(integrands.arg_r, construct("RRangeType", 0))
else:
defaultCode = generateDirichletCode(predefined,
wholeDomain[0],
tempVars=tempVars)
switch = SwitchStatement(integrands.arg_bndId, default=defaultCode)
for i, v in bySubDomain.items():
switch.append(
i, generateDirichletCode(predefined, v[0], tempVars=tempVars))
for i, v in enumerate(codeDomains):
switch.append(
i + maxId + 2,
generateDirichletCode(predefined, v[0], tempVars=tempVars))
integrands.dirichlet = [switch]
return integrands
def test_manufactured_poisson(degree, filename, datadir):
""" Manufactured Poisson problem, solving u = x[1]**p, where p is the
degree of the Lagrange function space.
"""
with XDMFFile(MPI.comm_world, os.path.join(datadir, filename)) as xdmf:
mesh = xdmf.read_mesh(GhostMode.none)
V = FunctionSpace(mesh, ("Lagrange", degree))
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
# Get quadrature degree for bilinear form integrand (ignores effect
# of non-affine map)
a = inner(grad(u), grad(v)) * dx(metadata={"quadrature_degree": -1})
a.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(a)
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
f = -div(grad(u_exact))
# Set quadrature degree for linear form integrand (ignores effect of
# non-affine map)
L = inner(f, v) * dx(metadata={"quadrature_degree": -1})
L.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(L)
t0 = time.time()
L = fem.Form(L)
t1 = time.time()
print("Linear form compile time:", t1 - t0)
u_bc = Function(V)
u_bc.interpolate(lambda x: x[1]**degree)
# Create Dirichlet boundary condition
mesh.create_connectivity_all()
facetdim = mesh.topology.dim - 1
bndry_facets = np.where(
np.array(mesh.topology.on_boundary(facetdim)) == 1)[0]
bdofs = locate_dofs_topological(V, facetdim, bndry_facets)
assert (len(bdofs) < V.dim())
bc = DirichletBC(u_bc, bdofs)
t0 = time.time()
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
t1 = time.time()
print("Vector assembly time:", t1 - t0)
t0 = time.time()
a = fem.Form(a)
t1 = time.time()
print("Bilinear form compile time:", t1 - t0)
t0 = time.time()
A = assemble_matrix(a, [bc])
A.assemble()
t1 = time.time()
print("Matrix assembly time:", t1 - t0)
# Create LU linear solver
solver = PETSc.KSP().create(MPI.comm_world)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
t0 = time.time()
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
t1 = time.time()
print("Linear solver time:", t1 - t0)
M = (u_exact - uh)**2 * dx
t0 = time.time()
M = fem.Form(M)
t1 = time.time()
print("Error functional compile time:", t1 - t0)
t0 = time.time()
error = assemble_scalar(M)
error = MPI.sum(mesh.mpi_comm(), error)
t1 = time.time()
print("Error assembly time:", t1 - t0)
assert np.absolute(error) < 1.0e-14
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
plt.savefig("stretchesVisco.png")
plt.close()
# stabilization parameters
h = FacetArea(mesh)
h_avg = avg(h)
# new variable name to take derivatives
FF = Identity(3) + grad(u)
CC = FF.T * FF
Fv = variable(FF)
S = diff(freeEnergy(Fv.T * Fv, CCv), Fv) # first PK stress
dl_interp(CC, C)
dl_interp(CC, Cn)
my_identity = grad(SpatialCoordinate(mesh))
dl_interp(my_identity, CCv)
dl_interp(my_identity, Cvn)
dl_interp(my_identity, C_quart)
dl_interp(my_identity, C_thr_quart)
dl_interp(my_identity, C_half)
a_uv = (derivative(freeEnergy(CC, CCv), u, v) * dx +
qvals / h_avg * dot(jump(u), jump(v)) * dS)
jac = derivative(a_uv, u, du)
# assign DirichletBC
left_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, left)
right_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, right)
bottom_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, bottom)
[30, 30, 60])
from dune.alugrid import aluSimplexGrid as leafGridView3d
gridView3d = leafGridView3d(domain3d)
# <markdowncell>
# As before we solve a simple Laplace problem
# <codecell>
from dune.fem.space import lagrange as solutionSpace
from dune.fem.scheme import galerkin as solutionScheme
from ufl import TrialFunction, TestFunction, SpatialCoordinate, dot, grad, dx, conditional, sqrt
space3d = solutionSpace(gridView3d, order=1)
u = TrialFunction(space3d)
v = TestFunction(space3d)
x = SpatialCoordinate(space3d)
scheme3d = solutionScheme((dot(grad(u), grad(v)) + u * v) *
dx == conditional(dot(x, x) < .01, 100, 0) * v * dx,
solver='cg')
uh3d = space3d.interpolate([0], name="solution")
info = scheme3d.solve(target=uh3d)
# <markdowncell>
# Instead of plotting this using paraview we want to only study the
# solution along a single line. This requires findings points
# $x_i = x_0+\frac{i}{N}(x1-x0)$ for $i=0,\dots,N$ within the unstructured
# grid. This would be expensive to compute on the Python so we implement
# this algorithm in C++ using the `LineSegmentSampler` class available in
# `Dune-Fem`. The resulting `algorithm` returns a pair of two lists with
# coordinates $x_i$ and the values of the grid function at these points:
#
order = 2
grid = create.grid("ALUCube",
constructor=cartesianDomain([0, 0], [3, 1], [30, 10]))
spcU = create.space("lagrange",
grid,
dimRange=grid.dimension,
order=order,
storage="fem")
spcP = create.space("lagrange",
grid,
dimRange=1,
order=order - 1,
storage="fem")
cell = spcU.cell()
x = SpatialCoordinate(cell)
mu = Constant(0, "mu")
nu = Constant(0, "nu")
u = TrialFunction(spcU)
v = TestFunction(spcU)
p = TrialFunction(spcP)
q = TestFunction(spcP)
exact_u = as_vector([x[1] * (1. - x[1]), 0])
exact_p = as_vector([(-2 * x[0] + 2) * mu])
f = as_vector([
0,
] * grid.dimension)
f += nu * exact_u
mainModel = (nu * dot(u, v) + mu * inner(grad(u) + grad(u).T, grad(v)) -
dot(f, v)) * dx
gradModel = -inner(p[0] * Identity(grid.dimension), grad(v)) * dx
from matplotlib import pyplot
# <markdowncell>
# ## Setting up the Mesh
# <codecell>
from dune.grid import structuredGrid as leafGridView
gridView = leafGridView([0, 0], [1, 1], [4, 4])
# <markdowncell>
# ## Grid Functions
# We can integrate grid function
# <codecell>
from ufl import SpatialCoordinate, triangle
x = SpatialCoordinate(triangle)
exact = 1/2*(x[0]**2+x[1]**2) - 1/3*(x[0]**3 - x[1]**3) + 1
from dune.fem.function import integrate
mass = integrate(gridView, exact, order=5)
print(mass)
# <markdowncell>
# and plot them using matplotlib or write a vtk file for postprocessing
# <codecell>
from dune.fem.plotting import plotPointData as plot
plot(exact, grid=gridView)
gridView.writeVTK('exact', pointdata={'exact': exact})
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_div_grad_then_integrate_over_cells_and_boundary():
# Define 2D geometry
n = 10
mesh = RectangleMesh(
[numpy.array([0.0, 0.0, 0.0]),
numpy.array([2.0, 3.0, 0.0])], 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-ffcx-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
def test_diff_then_integrate():
# Define 1D geometry
n = 21
mesh = UnitIntervalMesh(MPI.COMM_WORLD, n)
# Shift and scale mesh
x0, x1 = 1.5, 3.14
mesh.coordinates()[:] *= (x1 - x0)
mesh.coordinates()[:] += x0
x = SpatialCoordinate(mesh)[0]
xs = 0.1 + 0.8 * x / x1 # scaled to be within [0.1,0.9]
# 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-ffcx-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([x**x])
reg([x**(x**2)], 8)
reg([x**(x**3)], 6)
reg([x**(x**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)
try:
import scipy
except ImportError:
scipy = None
if hasattr(math, 'erf') or scipy is not None:
reg([erf(xs)])
else:
print(
"Warning: skipping test of erf, old python version and no scipy.")
# if 0:
# print("Warning: skipping tests of bessel functions, doesn't build on all platforms.")
# elif scipy is None:
# print("Warning: skipping tests of bessel functions, missing scipy.")
# else:
# for nu in (0, 1, 2):
# # Many of these are possibly more accurately integrated,
# # but 4 covers all and is sufficient for this test
# reg([bessel_J(nu, xs), bessel_Y(nu, xs), bessel_I(nu, xs), bessel_K(nu, xs)], 4)
# 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]])
x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4])
cc = as_matrix([[2, 3], [4, 5]])
reg2([xx])
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 * xx, 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
for F, acc in F_list:
# Apply UFL differentiation
f = diff(F, SpatialCoordinate(mesh))[..., 0]
if debug:
print(F)
print(x)
print(f)
# Apply integration with DOLFINx
# (also passes through form compilation and jit)
M = f * dx
f_integral = assemble_scalar(M) # noqa
f_integral = mesh.mpi_comm().allreduce(f_integral, op=MPI.SUM)
# Compute integral of f manually from anti-derivative F
# (passes through pybind11 interface and uses UFL evaluation)
F_diff = F((x1, )) - F((x0, ))
# Compare results. Using custom relative delta instead
# of decimal digits here because some numbers are >> 1.
delta = min(abs(f_integral), abs(F_diff)) * 10**-acc
assert f_integral - F_diff <= delta
def reference_periodic(tetra: bool,
r_lvl: int = 0,
out_hdf5: h5py.File = None,
xdmf: bool = False,
boomeramg: bool = False,
kspview: bool = False,
degree: int = 1):
# Create mesh and finite element
if tetra:
# Tet setup
N = 3
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)
for i in range(r_lvl):
mesh.topology.create_entities(mesh.topology.dim - 2)
mesh = refine(mesh, redistribute=True)
N *= 2
else:
# Hex setup
N = 3
for i in range(r_lvl):
N *= 2
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
V = FunctionSpace(mesh, ("CG", degree))
# Create Dirichlet boundary condition
def dirichletboundary(x):
return np.logical_or(
np.logical_or(np.isclose(x[1], 0), np.isclose(x[1], 1)),
np.logical_or(np.isclose(x[2], 0), np.isclose(x[2], 1)))
mesh.topology.create_connectivity(2, 1)
geometrical_dofs = locate_dofs_geometrical(V, dirichletboundary)
bc = dirichletbc(PETSc.ScalarType(0), geometrical_dofs, V)
bcs = [bc]
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx
x = SpatialCoordinate(mesh)
dx_ = x[0] - 0.9
dy_ = x[1] - 0.5
dz_ = x[2] - 0.1
f = x[0] * sin(5.0 * pi * x[1]) + 1.0 * exp(
-(dx_ * dx_ + dy_ * dy_ + dz_ * dz_) / 0.02)
rhs = inner(f, v) * dx
# Assemble rhs, RHS and apply lifting
bilinear_form = form(a)
linear_form = form(rhs)
A_org = assemble_matrix(bilinear_form, bcs)
A_org.assemble()
L_org = assemble_vector(linear_form)
apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES,
mode=PETSc.ScatterMode.REVERSE)
set_bc(L_org, bcs)
# Create PETSc nullspace
nullspace = PETSc.NullSpace().create(constant=True)
PETSc.Mat.setNearNullSpace(A_org, nullspace)
# Set PETSc options
opts = PETSc.Options()
if boomeramg:
opts["ksp_type"] = "cg"
opts["ksp_rtol"] = 1.0e-5
opts["pc_type"] = "hypre"
opts['pc_hypre_type'] = 'boomeramg'
opts["pc_hypre_boomeramg_max_iter"] = 1
opts["pc_hypre_boomeramg_cycle_type"] = "v"
# opts["pc_hypre_boomeramg_print_statistics"] = 1
else:
opts["ksp_type"] = "cg"
opts["ksp_rtol"] = 1.0e-12
opts["pc_type"] = "gamg"
opts["pc_gamg_type"] = "agg"
opts["pc_gamg_sym_graph"] = True
# Use Chebyshev smoothing for multigrid
opts["mg_levels_ksp_type"] = "richardson"
opts["mg_levels_pc_type"] = "sor"
# opts["help"] = None # List all available options
# opts["ksp_view"] = None # List progress of solver
# Initialize PETSc solver, set options and operator
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setFromOptions()
solver.setOperators(A_org)
# Solve linear problem
u_ = Function(V)
start = perf_counter()
with Timer("Solve"):
solver.solve(L_org, u_.vector)
end = perf_counter()
u_.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
if kspview:
solver.view()
it = solver.getIterationNumber()
num_dofs = V.dofmap.index_map.size_global * V.dofmap.index_map_bs
if out_hdf5 is not None:
d_set = out_hdf5.get("its")
d_set[r_lvl] = it
d_set = out_hdf5.get("num_dofs")
d_set[r_lvl] = num_dofs
d_set = out_hdf5.get("solve_time")
d_set[r_lvl, MPI.COMM_WORLD.rank] = end - start
if MPI.COMM_WORLD.rank == 0:
print("Rlvl {0:d}, Iterations {1:d}".format(r_lvl, it))
# Output solution to XDMF
if xdmf:
ext = "tet" if tetra else "hex"
fname = "results/reference_periodic_{0:d}_{1:s}.xdmf".format(
r_lvl, ext)
u_.name = "u_" + ext + "_unconstrained"
with XDMFFile(MPI.COMM_WORLD, fname, "w") as out_periodic:
out_periodic.write_mesh(mesh)
out_periodic.write_function(
u_, 0.0,
"Xdmf/Domain/" + "Grid[@Name='{0:s}'][1]".format(mesh.name))
from ufl import SpatialCoordinate, dot
from dune.grid import cartesianDomain
from dune.alugrid import aluConformGrid as leafGridView
from dune.fem.view import filteredGridView
from dune.fem.space import lagrange
gridView = leafGridView( cartesianDomain([0,0],[1,1],[16,16]) )
filteredView = filteredGridView(gridView, lambda e: e.geometry.center.two_norm > 0.5, domainId=1)
space = lagrange(filteredView, order=2)
x = SpatialCoordinate(space)
solution = space.interpolate(dot(x,x),name="solution")
solution.plot()
print("number of dofs:", solution.size,\
"integral over filtered domain",solution.integrate())
filteredView = filteredGridView(gridView, lambda e: e.geometry.center.two_norm < 0.5, domainId=1,
useFilteredIndexSet=True)
space = lagrange(filteredView, order=2)
x = SpatialCoordinate(space)
solution = space.interpolate(dot(x,x),name="solution")
solution.plot()
print("number of dofs:", solution.size,\
"integral over filtered domain",solution.integrate())
space = lagrange(gridView, order=2)
solution = space.interpolate(dot(x,x),name="solution")
solution.plot()
print("number of dofs:", solution.size,\
"integral over filtered domain",solution.integrate())
def run_scalar_test(mesh, V, degree):
""" Manufactured Poisson problem, solving u = x[1]**p, where p is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx
# Get quadrature degree for bilinear form integrand (ignores effect of non-affine map)
a = inner(grad(u), grad(v)) * dx(metadata={"quadrature_degree": -1})
a.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(a)
# Source term
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
f = -div(grad(u_exact))
# Set quadrature degree for linear form integrand (ignores effect of non-affine map)
L = inner(f, v) * dx(metadata={"quadrature_degree": -1})
L.integrals()[0].metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(L)
with common.Timer("Linear form compile"):
L = fem.Form(L)
with common.Timer("Function interpolation"):
u_bc = Function(V)
u_bc.interpolate(lambda x: x[1]**degree)
# Create Dirichlet boundary condition
mesh.topology.create_connectivity_all()
facetdim = mesh.topology.dim - 1
bndry_facets = np.where(
np.array(cpp.mesh.compute_boundary_facets(mesh.topology)) == 1)[0]
bdofs = locate_dofs_topological(V, facetdim, bndry_facets)
bc = DirichletBC(u_bc, bdofs)
with common.Timer("Vector assembly"):
b = assemble_vector(L)
apply_lifting(b, [a], [[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
set_bc(b, [bc])
with common.Timer("Bilinear form compile"):
a = fem.Form(a)
with common.Timer("Matrix assembly"):
A = assemble_matrix(a, [bc])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
with common.Timer("Solve"):
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
with common.Timer("Error functional compile"):
M = (u_exact - uh)**2 * dx
M = fem.Form(M)
with common.Timer("Error assembly"):
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
common.list_timings(MPI.COMM_WORLD, [common.TimingType.wall])
assert np.absolute(error) < 1.0e-14
try:
grid = create.grid("ALUConform",
dune.grid.cartesianDomain([0, 0], [1, 1], [9, 9]),
dimgrid=2)
except:
grid = create.grid("Yasp",
dune.grid.cartesianDomain([0, 0], [1, 1], [9, 9]),
dimgrid=2)
d = 0.001
p = 1.7
uflSpace = Space((2, 2), 1)
u = TrialFunction(uflSpace)
v = TestFunction(uflSpace)
x = SpatialCoordinate(uflSpace.cell())
rhs = (x[0] + x[1]) * v[0]
a = (pow(d + inner(grad(u), grad(u)), (p - 2) / 2) * inner(grad(u), grad(v)) +
grad(u[0])[0] * v[0]) * dx + 10 * inner(u, v) * ds
b = rhs * dx + 10 * rhs * ds
model = create.model("integrands", grid, a == b)
def test(space):
if test_numpy:
numpySpace = create.space(space,
grid,
dimRange=1,
order=1,
storage='numpy')
def compileUFL(form, patch, *args, **kwargs):
if isinstance(form, Equation):
form = form.lhs - form.rhs
if not isinstance(form, Form):
raise Exception("ufl.Form expected.")
if len(form.arguments()) < 2:
raise Exception("ConservationLaw model requires form with at least two arguments.")
phi_, u_ = form.arguments()
if phi_.ufl_function_space().scalar:
phi = TestFunction(phi_.ufl_function_space().toVectorSpace())
form = replace(form,{phi_:phi[0]})
else:
phi = phi_
if u_.ufl_function_space().scalar:
u = TrialFunction(u_.ufl_function_space().toVectorSpace())
form = replace(form,{u_:u[0]})
else:
u = u_
_, coeff_ = extract_arguments_and_coefficients(form)
coeff_ = set(coeff_)
# added for dirichlet treatment same as conservationlaw model
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
# remove the dirichletBCs
arg = [arg for arg in args if not isinstance(arg, DirichletBC)]
for dBC in dirichletBCs:
_, coeff__ = extract_arguments_and_coefficients(dBC.ufl_value)
coeff_ |= set(coeff__)
if patch is not None:
for a in patch:
try:
_, coeff__ = extract_arguments_and_coefficients(a)
coeff_ |= set(coeff__)
except:
pass # a might be a float/int and not a ufl expression
coeff = {c : c.toVectorCoefficient()[0] for c in coeff_ if len(c.ufl_shape) == 0 and not c.is_cellwise_constant()}
form = replace(form,coeff)
for bc in dirichletBCs:
bc.ufl_value = replace(bc.ufl_value, coeff)
if patch is not None:
patch = [a if not isinstance(a, Expr) else replace(a,coeff) for a in patch]
phi = form.arguments()[0]
dimRange = phi.ufl_shape[0]
u = form.arguments()[1]
du = Grad(u)
d2u = Grad(du)
ubar = Coefficient(u.ufl_function_space())
dubar = Grad(ubar)
d2ubar = Grad(dubar)
dimDomain = u.ufl_shape[0]
x = SpatialCoordinate(form.ufl_cell())
try:
field = u.ufl_function_space().field
except AttributeError:
field = "double"
# if exact solution is passed in subtract a(u,.) from the form
if "exact" in kwargs:
b = replace(form, {u: as_vector(kwargs["exact"])} )
form = form - b
dform = apply_derivatives(derivative(action(form, ubar), ubar, u))
source, flux, boundarySource = splitUFLForm(form)
linSource, linFlux, linBoundarySource = splitUFLForm(dform)
fluxDivergence, _, _ = splitUFLForm(inner(source.as_ufl() - div(flux.as_ufl()), phi) * dx(0))
# split linNVSource off linSource
# linSources = splitUFL2(u, du, d2u, linSource)
# linNVSource = linSources[2]
# linSource = linSources[0] + linSources[1]
if patch is not None:
model = ConservationLawModel(dimDomain, dimRange, u, modelSignature(form,*patch,*args))
else:
model = ConservationLawModel(dimDomain, dimRange, u, modelSignature(form,None,*args))
model._replaceCoeff = coeff
model.hasNeumanBoundary = not boundarySource.is_zero()
#expandform = expand_indices(expand_derivatives(expand_compounds(equation.lhs)))
#if expandform == adjoint(expandform):
# model.symmetric = 'true'
model.field = field
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
# deprecated
# if "dirichlet" in kwargs:
# dirichletBCs += [DirichletBC(u.ufl_function_space(), as_vector(value), bndId) for bndId, value in kwargs["dirichlet"].items()]
uflCoefficients = set(form.coefficients())
for bc in dirichletBCs:
_, c = extract_arguments_and_coefficients(bc.ufl_value)
uflCoefficients |= set(c)
if patch is not None:
for a in patch:
if isinstance(a, Expr):
_, c = extract_arguments_and_coefficients(a)
uflCoefficients |= set(c)
constants = dict()
coefficients = dict()
for coefficient in uflCoefficients:
try:
name = getattr(coefficient, "name")
except AttributeError:
name = str(coefficient)
if coefficient.is_cellwise_constant():
try:
parameter = getattr(coefficient, "parameter")
except AttributeError:
parameter = None
if len(coefficient.ufl_shape) == 0:
constants[coefficient] = model.addConstant('double', name=name, parameter=parameter)
elif len(coefficient.ufl_shape) == 1:
constants[coefficient] = model.addConstant('Dune::FieldVector< double, ' + str(coefficient.ufl_shape[0]) + ' >', name=name, parameter=parameter)
else:
Exception('Currently, only scalars and vectors are supported as constants')
else:
shape = coefficient.ufl_shape[0]
try:
coefficients[coefficient] = model.addCoefficient(
shape,
coefficient.cppTypeName,
name=name,
field=coefficient.ufl_function_space().field)
except AttributeError:
coefficients[coefficient] = model.addCoefficient(
shape,
coefficient.cppTypeName,
name=name)
model.coefficients = coefficients
model.constants = constants
tempVars = kwargs.get("tempVars", True)
predefined = {u: model.arg_u, du: model.arg_du, d2u: model.arg_d2u}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
model.source = generateCode(predefined, source, tempVars=tempVars)
model.flux = generateCode(predefined, flux, tempVars=tempVars)
predefined.update({ubar: model.arg_ubar, dubar: model.arg_dubar, d2ubar: model.arg_d2ubar})
model.linSource = generateCode(predefined, linSource, tempVars=tempVars)
model.linFlux = generateCode(predefined, linFlux, tempVars=tempVars)
# model.linNVSource = generateCode({u: arg, du: darg, d2u: d2arg, ubar: argbar, dubar: dargbar, d2ubar: d2argbar}, linNVSource, model.coefficients, tempVars)
predefined = {u: model.arg_u}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
model.alpha = generateCode(predefined, boundarySource, tempVars=tempVars)
predefined.update({ubar: model.arg_ubar})
model.linAlpha = generateCode(predefined, linBoundarySource, tempVars=tempVars)
predefined = {u: model.arg_u, du: model.arg_du, d2u: model.arg_d2u}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
model.fluxDivergence = generateCode(predefined, fluxDivergence, tempVars=tempVars)
if dirichletBCs:
model.hasDirichletBoundary = True
predefined = {}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
maxId = 0
codeDomains = []
bySubDomain = dict()
neuman = []
for bc in dirichletBCs:
if bc.subDomain in bySubDomain:
raise Exception('Multiply defined Dirichlet boundary for subdomain ' + str(bc.subDomain))
if not isinstance(bc.functionSpace, (FunctionSpace, FiniteElementBase)):
raise Exception('Function space must either be a ufl.FunctionSpace or a ufl.FiniteElement')
if isinstance(bc.functionSpace, FunctionSpace) and (bc.functionSpace != u.ufl_function_space()):
raise Exception('Space of trial function and dirichlet boundary function must be the same - note that boundary conditions on subspaces are not available, yet')
if isinstance(bc.functionSpace, FiniteElementBase) and (bc.functionSpace != u.ufl_element()):
raise Exception('Cannot handle boundary conditions on subspaces, yet')
if isinstance(bc.value, list):
neuman = [i for i, x in enumerate(bc.value) if x == None]
else:
neuman = []
value = ExprTensor(u.ufl_shape)
for key in value.keys():
value[key] = Indexed(bc.ufl_value, MultiIndex(tuple(FixedIndex(k) for k in key)))
if isinstance(bc.subDomain,int):
bySubDomain[bc.subDomain] = value,neuman
maxId = max(maxId, bc.subDomain)
else:
domain = ExprTensor(())
for key in domain.keys():
domain[key] = Indexed(bc.subDomain, MultiIndex(tuple(FixedIndex(k) for k in key)))
codeDomains.append( (value,neuman,domain) )
defaultCode = []
defaultCode.append(Declaration(Variable('int', 'domainId')))
defaultCode.append(Declaration(Variable('auto', 'tmp0'),
initializer=UnformattedExpression('auto','intersection.geometry().center()')))
for i,v in enumerate(codeDomains):
block = Block()
defaultCode.append(
generateDirichletDomainCode(predefined, v[2], tempVars=tempVars))
defaultCode.append('if (domainId)')
block = UnformattedBlock()
block.append('std::fill( dirichletComponent.begin(), dirichletComponent.end(), ' + str(maxId+i+1) + ' );')
if len(v[1])>0:
[block.append('dirichletComponent[' + str(c) + '] = 0;') for c in v[1]]
block.append('return true;')
defaultCode.append(block)
defaultCode.append(return_(False))
bndId = Variable('const int', 'bndId')
getBndId = UnformattedExpression('int', 'BoundaryIdProviderType::boundaryId( ' + model.arg_i.name + ' )')
switch = SwitchStatement(bndId, default=defaultCode)
for i,v in bySubDomain.items():
code = []
if len(v[1])>0:
[code.append('dirichletComponent[' + str(c) + '] = 0;') for c in v[1]]
code.append(return_(True))
switch.append(i, code)
model.isDirichletIntersection = [Declaration(bndId, initializer=getBndId),
UnformattedBlock('std::fill( dirichletComponent.begin(), dirichletComponent.end(), ' + bndId.name + ' );'),
switch
]
switch = SwitchStatement(model.arg_bndId, default=assign(model.arg_r, construct("RRangeType", 0)))
for i, v in bySubDomain.items():
switch.append(i, generateDirichletCode(predefined, v[0], tempVars=tempVars))
for i,v in enumerate(codeDomains):
switch.append(i+maxId+1, generateDirichletCode(predefined, v[0], tempVars=tempVars))
model.dirichlet = [switch]
return model
la.orthonormalize(ns)
assert la.is_orthonormal(ns)
return PETSc.NullSpace().create(vectors=ns)
mesh = create_box(MPI.COMM_WORLD,
[np.array([0.0, 0.0, 0.0]),
np.array([2.0, 1.0, 1.0])], [12, 12, 12],
CellType.tetrahedron, GhostMode.shared_facet)
# Rotation rate and mass density
omega = 300.0
rho = 10.0
# Loading due to centripetal acceleration (rho*omega^2*x_i)
x = SpatialCoordinate(mesh)
f = as_vector((rho * omega**2 * x[0], rho * omega**2 * x[1], 0.0))
# Elasticity parameters
E = 1.0e9
nu = 0.0
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
def sigma(v):
return 2.0 * mu * sym(grad(v)) + lmbda * tr(sym(grad(v))) * Identity(
len(v))
# Create function space
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