def create_subdm(dm, fields, *args, **kwargs):
"""Callback to create a sub-DM describing the specified fields.
:arg DM: The DM.
:arg fields: The fields in the new sub-DM.
.. note::
This should, but currently does not, transfer appropriately
split application contexts onto the sub-DMs.
"""
W = get_function_space(dm)
# TODO: Correct splitting of SNESContext for len(fields) > 1 case
if len(fields) == 1:
# Subspace is just a single FunctionSpace.
idx = fields[0]
subdm = W[idx].dm
iset = W._ises[idx]
return iset, subdm
else:
try:
# Look up the subspace in the cache
iset, subspace = W._subspaces[tuple(fields)]
return iset, subspace.dm
except KeyError:
pass
# Need to build an MFS for the subspace
subspace = firedrake.MixedFunctionSpace([W[f] for f in fields])
# Index set mapping from W into subspace.
iset = PETSc.IS().createGeneral(numpy.concatenate([W._ises[f].indices
for f in fields]),
comm=W.comm)
# Keep hold of strong reference to created subspace (given we
# only hold a weakref in the shell DM), and so we can
# reuse it later.
W._subspaces[tuple(fields)] = iset, subspace
return iset, subspace.dm
def construct_kronecker_matrix(self, interp_1d):
"""
Construct the tensorized interpolation matrix.
Do this by computing the kron product of the rows of
the 1d univariate interpolation matrices.
In the future, this may be done matrix-free.
"""
# this is one of the two bottlenecks that slow down initiating Bsplines
IFW = PETSc.Mat().create(self.comm)
IFW.setType(PETSc.Mat.Type.AIJ)
comm = self.comm
# owned part of global problem
local_N = self.N // comm.size + int(comm.rank < (self.N % comm.size))
(lsize, gsize) = interp_1d[0].getSizes()[0]
IFW.setSizes(((lsize, gsize), (local_N, self.N)))
# guess sparsity pattern from interp_1d[0]
for row in range(lsize):
row = self.lg_map_fe.apply([row])[0]
nnz_ = len(interp_1d[0].getRow(row)[0]) # lenght of nnz-array
self.IFWnnz = max(self.IFWnnz, nnz_**self.dim)
IFW.setPreallocationNNZ(self.IFWnnz)
IFW.setUp()
for row in range(lsize):
row = self.lg_map_fe.apply([row])[0]
M = [[A.getRow(row)[0],
A.getRow(row)[1],
A.getSize()[1]] for A in interp_1d]
M = reduce(self.vectorkron, M)
columns, values, lenght = M
IFW.setValues([row], columns, values)
IFW.assemble()
return IFW
def create_subdm(dm, fields, *args, **kwargs):
"""Callback to create a sub-DM describing the specified fields.
:arg DM: The DM.
:arg fields: The fields in the new sub-DM.
"""
W = get_function_space(dm)
ctx = get_appctx(dm)
coarsen = get_ctx_coarsener(dm)
if len(fields) == 1:
# Subspace is just a single FunctionSpace.
idx, = fields
subdm = W[idx].dm
iset = W._ises[idx]
if ctx is not None:
ctx, = ctx.split([(idx, )])
push_appctx(subdm, ctx)
push_ctx_coarsener(subdm, coarsen)
return iset, subdm
else:
# Need to build an MFS for the subspace
subspace = firedrake.MixedFunctionSpace([W[f] for f in fields])
# Pass any transfer operators over
prolong, restrict, inject = get_transfer_operators(dm)
push_transfer_operators(subspace.dm, prolong, restrict, inject)
# Index set mapping from W into subspace.
iset = PETSc.IS().createGeneral(numpy.concatenate(
[W._ises[f].indices for f in fields]),
comm=W.comm)
if ctx is not None:
ctx, = ctx.split([fields])
push_appctx(subspace.dm, ctx)
push_ctx_coarsener(subspace.dm, coarsen)
return iset, subspace.dm
def create_interpolation(dmc, dmf):
cctx = firedrake.dmhooks.get_appctx(dmc)
fctx = firedrake.dmhooks.get_appctx(dmf)
prolong, _, _ = firedrake.dmhooks.get_transfer_operators(dmc)
_, restrict, _ = firedrake.dmhooks.get_transfer_operators(dmf)
V_c = cctx._problem.u.function_space()
V_f = fctx._problem.u.function_space()
row_size = V_f.dof_dset.layout_vec.getSizes()
col_size = V_c.dof_dset.layout_vec.getSizes()
cfn = firedrake.Function(V_c)
ffn = firedrake.Function(V_f)
cbcs = cctx._problem.bcs
fbcs = fctx._problem.bcs
ctx = Interpolation(cfn, ffn, prolong, restrict, cbcs, fbcs)
mat = PETSc.Mat().create(comm=dmc.comm)
mat.setSizes((row_size, col_size))
mat.setType(mat.Type.PYTHON)
mat.setPythonContext(ctx)
mat.setUp()
return mat, None
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 _build_monolithic_basis(self):
r"""Build a basis for the complete mixed space.
The monolithic basis is formed by the cartesian product of the
bases forming each sub part.
"""
self._vecs = []
for idx, basis in enumerate(self):
if isinstance(basis, VectorSpaceBasis):
vecs = basis._vecs
if basis._constant:
vecs = vecs + (function.Function(
self._function_space[idx]).assign(1), )
for vec in vecs:
mvec = function.Function(self._function_space)
mvec.sub(idx).assign(vec)
self._vecs.append(mvec)
self._petsc_vecs = []
for v in self._vecs:
with v.dat.vec_ro as v_:
self._petsc_vecs.append(v_)
# orthonormalize:
basis = self._petsc_vecs
for i, vec in enumerate(basis):
alphas = []
for vec_ in basis[:i]:
alphas.append(vec.dot(vec_))
for alpha, vec_ in zip(alphas, basis[:i]):
vec.axpy(-alpha, vec_)
vec.normalize()
self._nullspace = PETSc.NullSpace().create(constant=False,
vectors=self._petsc_vecs,
comm=self.comm)
def plot_matrix(a_form, bcs=[], **kwargs):
"""Provides a plot of a matrix."""
fig, ax = plt.subplots(1, 1)
A = assemble(a_form, bcs=bcs, mat_type="aij")
petsc_mat = A.M.handle
size = petsc_mat.getSize()
Mij = PETSc.Mat()
petsc_mat.convert("aij", Mij)
n, m = size
Mnp = np.array(Mij.getValues(range(n), range(m)))
Am = np.ma.masked_values(Mnp, 0, rtol=1e-13)
# Plot the matrix
plot = ax.matshow(Am, cmap=my_cmap, **kwargs)
# Remove axis ticks and values
ax.tick_params(length=0)
ax.set_xticklabels([])
ax.set_yticklabels([])
return plot
def test_duplicate(a, bcs):
test, trial = a.arguments()
if test.function_space().shape == ():
rhs_form = inner(Constant(1), test)*dx
elif test.function_space().shape == (2, ):
rhs_form = inner(Constant((1, 1)), test)*dx
if bcs is not None:
Af = assemble(a, mat_type="matfree", bcs=bcs)
rhs = assemble(rhs_form, bcs=bcs)
else:
Af = assemble(a, mat_type="matfree")
rhs = assemble(rhs_form)
# matrix-free duplicate creates a matrix-free copy of Af
# we have not implemented the default copy = False
B_petsc = Af.petscmat.duplicate(copy=True)
ksp = PETSc.KSP().create()
ksp.setOperators(Af.petscmat)
ksp.setFromOptions()
solution1 = Function(test.function_space())
solution2 = Function(test.function_space())
# Solve system with original matrix A
with rhs.dat.vec_ro as b, solution1.dat.vec as x:
ksp.solve(b, x)
# Multiply with copied matrix B
with solution1.dat.vec_ro as x, solution2.dat.vec_ro as y:
B_petsc.mult(x, y)
# Check if original rhs is equal to BA^-1 (rhs)
assert np.allclose(rhs.vector().array(), solution2.vector().array())
def initialize(self, pc):
"""Set up the problem context. Take the original
mixed problem and reformulate the problem as a
hybridized mixed system.
A KSP is created for the Lagrange multiplier system.
"""
from firedrake import (FunctionSpace, Function, Constant,
TrialFunction, TrialFunctions, TestFunction,
DirichletBC, assemble)
from firedrake.assemble import (allocate_matrix,
create_assembly_callable)
from firedrake.formmanipulation import split_form
from ufl.algorithms.replace import replace
# Extract the problem context
prefix = pc.getOptionsPrefix() + "hybridization_"
_, P = pc.getOperators()
self.cxt = P.getPythonContext()
if not isinstance(self.cxt, ImplicitMatrixContext):
raise ValueError("The python context must be an ImplicitMatrixContext")
test, trial = self.cxt.a.arguments()
V = test.function_space()
mesh = V.mesh()
if len(V) != 2:
raise ValueError("Expecting two function spaces.")
if all(Vi.ufl_element().value_shape() for Vi in V):
raise ValueError("Expecting an H(div) x L2 pair of spaces.")
# Automagically determine which spaces are vector and scalar
for i, Vi in enumerate(V):
if Vi.ufl_element().sobolev_space().name == "HDiv":
self.vidx = i
else:
assert Vi.ufl_element().sobolev_space().name == "L2"
self.pidx = i
# Create the space of approximate traces.
W = V[self.vidx]
if W.ufl_element().family() == "Brezzi-Douglas-Marini":
tdegree = W.ufl_element().degree()
else:
try:
# If we have a tensor product element
h_deg, v_deg = W.ufl_element().degree()
tdegree = (h_deg - 1, v_deg - 1)
except TypeError:
tdegree = W.ufl_element().degree() - 1
TraceSpace = FunctionSpace(mesh, "HDiv Trace", tdegree)
# Break the function spaces and define fully discontinuous spaces
broken_elements = ufl.MixedElement([ufl.BrokenElement(Vi.ufl_element()) for Vi in V])
V_d = FunctionSpace(mesh, broken_elements)
# Set up the functions for the original, hybridized
# and schur complement systems
self.broken_solution = Function(V_d)
self.broken_residual = Function(V_d)
self.trace_solution = Function(TraceSpace)
self.unbroken_solution = Function(V)
self.unbroken_residual = Function(V)
# Set up the KSP for the hdiv residual projection
hdiv_mass_ksp = PETSc.KSP().create(comm=pc.comm)
hdiv_mass_ksp.setOptionsPrefix(prefix + "hdiv_residual_")
# HDiv mass operator
p = TrialFunction(V[self.vidx])
q = TestFunction(V[self.vidx])
mass = ufl.dot(p, q)*ufl.dx
# TODO: Bcs?
M = assemble(mass, bcs=None, form_compiler_parameters=self.cxt.fc_params)
M.force_evaluation()
Mmat = M.petscmat
hdiv_mass_ksp.setOperators(Mmat)
hdiv_mass_ksp.setUp()
hdiv_mass_ksp.setFromOptions()
self.hdiv_mass_ksp = hdiv_mass_ksp
# Storing the result of A.inv * r, where A is the HDiv
# mass matrix and r is the HDiv residual
self._primal_r = Function(V[self.vidx])
tau = TestFunction(V_d[self.vidx])
self._assemble_broken_r = create_assembly_callable(
ufl.dot(self._primal_r, tau)*ufl.dx,
tensor=self.broken_residual.split()[self.vidx],
form_compiler_parameters=self.cxt.fc_params)
# Create the symbolic Schur-reduction:
# Original mixed operator replaced with "broken"
# arguments
arg_map = {test: TestFunction(V_d),
trial: TrialFunction(V_d)}
Atilde = Tensor(replace(self.cxt.a, arg_map))
gammar = TestFunction(TraceSpace)
n = ufl.FacetNormal(mesh)
sigma = TrialFunctions(V_d)[self.vidx]
# We zero out the contribution of the trace variables on the exterior
# boundary. Extruded cells will have both horizontal and vertical
# facets
if mesh.cell_set._extruded:
trace_bcs = [DirichletBC(TraceSpace, Constant(0.0), "on_boundary"),
DirichletBC(TraceSpace, Constant(0.0), "bottom"),
DirichletBC(TraceSpace, Constant(0.0), "top")]
K = Tensor(gammar('+') * ufl.dot(sigma, n) * ufl.dS_h +
gammar('+') * ufl.dot(sigma, n) * ufl.dS_v)
else:
trace_bcs = [DirichletBC(TraceSpace, Constant(0.0), "on_boundary")]
K = Tensor(gammar('+') * ufl.dot(sigma, n) * ufl.dS)
# If boundary conditions are contained in the ImplicitMatrixContext:
if self.cxt.row_bcs:
raise NotImplementedError("Strong BCs not currently handled. Try imposing them weakly.")
# Assemble the Schur complement operator and right-hand side
self.schur_rhs = Function(TraceSpace)
self._assemble_Srhs = create_assembly_callable(
K * Atilde.inv * self.broken_residual,
tensor=self.schur_rhs,
form_compiler_parameters=self.cxt.fc_params)
schur_comp = K * Atilde.inv * K.T
self.S = allocate_matrix(schur_comp, bcs=trace_bcs,
form_compiler_parameters=self.cxt.fc_params)
self._assemble_S = create_assembly_callable(schur_comp,
tensor=self.S,
bcs=trace_bcs,
form_compiler_parameters=self.cxt.fc_params)
self._assemble_S()
self.S.force_evaluation()
Smat = self.S.petscmat
# Nullspace for the multiplier problem
nullspace = create_schur_nullspace(P, -K * Atilde,
V, V_d, TraceSpace,
pc.comm)
if nullspace:
Smat.setNullSpace(nullspace)
# Set up the KSP for the system of Lagrange multipliers
trace_ksp = PETSc.KSP().create(comm=pc.comm)
trace_ksp.setOptionsPrefix(prefix)
trace_ksp.setOperators(Smat)
trace_ksp.setUp()
trace_ksp.setFromOptions()
self.trace_ksp = trace_ksp
split_mixed_op = dict(split_form(Atilde.form))
split_trace_op = dict(split_form(K.form))
# Generate reconstruction calls
self._reconstruction_calls(split_mixed_op, split_trace_op)
# NOTE: The projection stage *might* be replaced by a Fortin
# operator. We may want to allow the user to specify if they
# wish to use a Fortin operator over a projection, or vice-versa.
# In a future add-on, we can add a switch which chooses either
# the Fortin reconstruction or the usual KSP projection.
# Set up the projection KSP
hdiv_projection_ksp = PETSc.KSP().create(comm=pc.comm)
hdiv_projection_ksp.setOptionsPrefix(prefix + 'hdiv_projection_')
# Reuse the mass operator from the hdiv_mass_ksp
hdiv_projection_ksp.setOperators(Mmat)
# Construct the RHS for the projection stage
self._projection_rhs = Function(V[self.vidx])
self._assemble_projection_rhs = create_assembly_callable(
ufl.dot(self.broken_solution.split()[self.vidx], q)*ufl.dx,
tensor=self._projection_rhs,
form_compiler_parameters=self.cxt.fc_params)
# Finalize ksp setup
hdiv_projection_ksp.setUp()
hdiv_projection_ksp.setFromOptions()
self.hdiv_projection_ksp = hdiv_projection_ksp
def nonlocal_integral_eq(
mesh,
scatterer_bdy_id,
outer_bdy_id,
wave_number,
options_prefix=None,
solver_parameters=None,
fspace=None,
vfspace=None,
true_sol_grad_expr=None,
actx=None,
dgfspace=None,
dgvfspace=None,
meshmode_src_connection=None,
qbx_kwargs=None,
):
r"""
see run_method for descriptions of unlisted args
args:
gamma and beta are used to precondition
with the following equation:
\Delta u - \kappa^2 \gamma u = 0
(\partial_n - i\kappa\beta) u |_\Sigma = 0
"""
# make sure we get outer bdy id as tuple in case it consists of multiple ids
if isinstance(outer_bdy_id, int):
outer_bdy_id = [outer_bdy_id]
outer_bdy_id = tuple(outer_bdy_id)
# away from the excluded region, but firedrake and meshmode point
# into
pyt_inner_normal_sign = -1
ambient_dim = mesh.geometric_dimension()
# {{{ Build src and tgt
# build connection meshmode near src boundary -> src boundary inside meshmode
from meshmode.discretization.poly_element import \
InterpolatoryQuadratureSimplexGroupFactory
from meshmode.discretization.connection import make_face_restriction
factory = InterpolatoryQuadratureSimplexGroupFactory(
dgfspace.finat_element.degree)
src_bdy_connection = make_face_restriction(actx,
meshmode_src_connection.discr,
factory, scatterer_bdy_id)
# source is a qbx layer potential
from pytential.qbx import QBXLayerPotentialSource
disable_refinement = (fspace.mesh().geometric_dimension() == 3)
qbx = QBXLayerPotentialSource(src_bdy_connection.to_discr,
**qbx_kwargs,
_disable_refinement=disable_refinement)
# get target indices and point-set
target_indices, target = get_target_points_and_indices(
fspace, outer_bdy_id)
# }}}
# build the operations
from pytential import bind, sym
r"""
..math:
x \in \Sigma
grad_op(x) =
\nabla(
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
)
"""
grad_op = pyt_inner_normal_sign * sym.grad(
ambient_dim,
sym.D(HelmholtzKernel(ambient_dim),
sym.var("u"),
k=sym.var("k"),
qbx_forced_limit=None))
r"""
..math:
x \in \Sigma
op(x) =
i \kappa \cdot
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
"""
op = pyt_inner_normal_sign * 1j * sym.var("k") * (sym.D(
HelmholtzKernel(ambient_dim),
sym.var("u"),
k=sym.var("k"),
qbx_forced_limit=None))
# bind the operations
pyt_grad_op = bind((qbx, target), grad_op)
pyt_op = bind((qbx, target), op)
# }}}
class MatrixFreeB(object):
def __init__(self, A, pyt_grad_op, pyt_op, actx, kappa):
"""
:arg kappa: The wave number
"""
self.actx = actx
self.k = kappa
self.pyt_op = pyt_op
self.pyt_grad_op = pyt_grad_op
self.A = A
self.meshmode_src_connection = meshmode_src_connection
# {{{ Create some functions needed for multing
self.x_fntn = Function(fspace)
# CG
self.potential_int = Function(fspace)
self.potential_int.dat.data[:] = 0.0
self.grad_potential_int = Function(vfspace)
self.grad_potential_int.dat.data[:] = 0.0
self.pyt_result = Function(fspace)
self.n = FacetNormal(mesh)
self.v = TestFunction(fspace)
# some meshmode ones
self.x_mm_fntn = self.meshmode_src_connection.discr.empty(
self.actx, dtype='c')
# }}}
def mult(self, mat, x, y):
# Copy function data into the fivredrake function
self.x_fntn.dat.data[:] = x[:]
# Transfer the function to meshmode
self.meshmode_src_connection.from_firedrake(project(
self.x_fntn, dgfspace),
out=self.x_mm_fntn)
# Restrict to boundary
x_mm_fntn_on_bdy = src_bdy_connection(self.x_mm_fntn)
# Apply the operation
potential_int_mm = self.pyt_op(self.actx,
u=x_mm_fntn_on_bdy,
k=self.k)
grad_potential_int_mm = self.pyt_grad_op(self.actx,
u=x_mm_fntn_on_bdy,
k=self.k)
# Store in firedrake
self.potential_int.dat.data[target_indices] = potential_int_mm.get(
)
for dim in range(grad_potential_int_mm.shape[0]):
self.grad_potential_int.dat.data[
target_indices, dim] = grad_potential_int_mm[dim].get()
# Integrate the potential
r"""
Compute the inner products using firedrake. Note this
will be subtracted later, hence appears off by a sign.
.. math::
\langle
n(x) \cdot \nabla(
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
), v
\rangle_\Sigma
- \langle
i \kappa \cdot
\int_\Gamma(
u(y) \partial_n H_0^{(1)}(\kappa |x - y|)
)d\gamma(y), v
\rangle_\Sigma
"""
self.pyt_result = assemble(
inner(inner(self.grad_potential_int, self.n), self.v) *
ds(outer_bdy_id) -
inner(self.potential_int, self.v) * ds(outer_bdy_id))
# y <- Ax - evaluated potential
self.A.mult(x, y)
with self.pyt_result.dat.vec_ro as ep:
y.axpy(-1, ep)
# {{{ Compute normal helmholtz operator
u = TrialFunction(fspace)
v = TestFunction(fspace)
r"""
.. math::
\langle
\nabla u, \nabla v
\rangle
- \kappa^2 \cdot \langle
u, v
\rangle
- i \kappa \langle
u, v
\rangle_\Sigma
"""
a = inner(grad(u), grad(v)) * dx \
- Constant(wave_number**2) * inner(u, v) * dx \
- Constant(1j * wave_number) * inner(u, v) * ds(outer_bdy_id)
# get the concrete matrix from a general bilinear form
A = assemble(a).M.handle
# }}}
# {{{ Setup Python matrix
B = PETSc.Mat().create()
# build matrix context
Bctx = MatrixFreeB(A, pyt_grad_op, pyt_op, actx, wave_number)
# set up B as same size as A
B.setSizes(*A.getSizes())
B.setType(B.Type.PYTHON)
B.setPythonContext(Bctx)
B.setUp()
# }}}
# {{{ Create rhs
# Remember f is \partial_n(true_sol)|_\Gamma
# so we just need to compute \int_\Gamma\partial_n(true_sol) H(x-y)
sigma = sym.make_sym_vector("sigma", ambient_dim)
r"""
..math:
x \in \Sigma
grad_op(x) =
\nabla(
\int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
)
"""
grad_op = pyt_inner_normal_sign * \
sym.grad(ambient_dim, sym.S(HelmholtzKernel(ambient_dim),
sym.n_dot(sigma),
k=sym.var("k"), qbx_forced_limit=None))
r"""
..math:
x \in \Sigma
op(x) =
i \kappa \cdot
\int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
)
"""
op = 1j * sym.var("k") * pyt_inner_normal_sign * \
sym.S(HelmholtzKernel(ambient_dim),
sym.n_dot(sigma),
k=sym.var("k"),
qbx_forced_limit=None)
rhs_grad_op = bind((qbx, target), grad_op)
rhs_op = bind((qbx, target), op)
# Transfer to meshmode
metadata = {'quadrature_degree': 2 * fspace.ufl_element().degree()}
dg_true_sol_grad = project(true_sol_grad_expr,
dgvfspace,
form_compiler_parameters=metadata)
true_sol_grad_mm = meshmode_src_connection.from_firedrake(dg_true_sol_grad,
actx=actx)
true_sol_grad_mm = src_bdy_connection(true_sol_grad_mm)
# Apply the operations
f_grad_convoluted_mm = rhs_grad_op(actx,
sigma=true_sol_grad_mm,
k=wave_number)
f_convoluted_mm = rhs_op(actx, sigma=true_sol_grad_mm, k=wave_number)
# Transfer function back to firedrake
f_grad_convoluted = Function(vfspace)
f_convoluted = Function(fspace)
f_grad_convoluted.dat.data[:] = 0.0
f_convoluted.dat.data[:] = 0.0
for dim in range(f_grad_convoluted_mm.shape[0]):
f_grad_convoluted.dat.data[target_indices,
dim] = f_grad_convoluted_mm[dim].get()
f_convoluted.dat.data[target_indices] = f_convoluted_mm.get()
r"""
\langle
f, v
\rangle_\Gamma
+ \langle
i \kappa \cdot \int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y), v
\rangle_\Sigma
- \langle
n(x) \cdot \nabla(
\int_\Gamma(
f(y) H_0^{(1)}(\kappa |x - y|)
)d\gamma(y)
), v
\rangle_\Sigma
"""
rhs_form = inner(inner(true_sol_grad_expr, FacetNormal(mesh)),
v) * ds(scatterer_bdy_id, metadata=metadata) \
+ inner(f_convoluted, v) * ds(outer_bdy_id) \
- inner(inner(f_grad_convoluted, FacetNormal(mesh)),
v) * ds(outer_bdy_id)
rhs = assemble(rhs_form)
# {{{ set up a solver:
solution = Function(fspace, name="Computed Solution")
# {{{ Used for preconditioning
if 'gamma' in solver_parameters or 'beta' in solver_parameters:
gamma = complex(solver_parameters.pop('gamma', 1.0))
import cmath
beta = complex(solver_parameters.pop('beta', cmath.sqrt(gamma)))
p = inner(grad(u), grad(v)) * dx \
- Constant(wave_number**2 * gamma) * inner(u, v) * dx \
- Constant(1j * wave_number * beta) * inner(u, v) * ds(outer_bdy_id)
P = assemble(p).M.handle
else:
P = A
# }}}
# Set up options to contain solver parameters:
ksp = PETSc.KSP().create()
if solver_parameters['pc_type'] == 'pyamg':
del solver_parameters['pc_type'] # We are using the AMG preconditioner
pyamg_tol = solver_parameters.get('pyamg_tol', None)
if pyamg_tol is not None:
pyamg_tol = float(pyamg_tol)
pyamg_maxiter = solver_parameters.get('pyamg_maxiter', None)
if pyamg_maxiter is not None:
pyamg_maxiter = int(pyamg_maxiter)
ksp.setOperators(B)
ksp.setUp()
pc = ksp.pc
pc.setType(pc.Type.PYTHON)
pc.setPythonContext(
AMGTransmissionPreconditioner(wave_number,
fspace,
A,
tol=pyamg_tol,
maxiter=pyamg_maxiter,
use_plane_waves=True))
# Otherwise use regular preconditioner
else:
ksp.setOperators(B, P)
options_manager = OptionsManager(solver_parameters, options_prefix)
options_manager.set_from_options(ksp)
import petsc4py.PETSc
petsc4py.PETSc.Sys.popErrorHandler()
with rhs.dat.vec_ro as b:
with solution.dat.vec as x:
ksp.solve(b, x)
# }}}
return ksp, solution
def get_boundary_masks(mesh, key, finat_element):
"""Get masks for facet dofs.
:arg mesh: The mesh to use.
:arg key: Canonicalised entity_dofs (see :func:`entity_dofs_key`).
:arg finat_element: The FInAT element.
:returns: A dict mapping ``"topological"`` and ``"geometric"``
keys to boundary nodes or ``None``. If not None, the entry in
the mask dict 3-tuple of a Section, an array of indices, and
an array indicating which points in the Section correspond to
the facets of the cell. If section.getDof(p) is non-zero,
then there are ndof basis functions topologically associated
with points in the closure of
point p (for "topological", ndof basis functions with non-zero
support on points in the closure of p for "geometric"). The
basis function indices are in the index array, starting at
section.getOffset(p).
"""
if not mesh.cell_set._extruded:
return None
masks = {}
_, kind = key
assert kind in {"cell", "interior_facet"}
dim = finat_element.cell.get_spatial_dimension()
ecd = finat_element.entity_closure_dofs()
try:
esd = finat_element.entity_support_dofs()
except NotImplementedError:
# 4-D cells
esd = None
# Number of entities on cell excepting the cell itself.
chart = sum(map(len, ecd.values())) - 1
closure_section = PETSc.Section().create(comm=PETSc.COMM_SELF)
support_section = PETSc.Section().create(comm=PETSc.COMM_SELF)
# Double up for interior facets.
if kind == "cell":
ncell = 1
else:
ncell = 2
closure_section.setChart(0, ncell * chart)
support_section.setChart(0, ncell * chart)
closure_indices = []
support_indices = []
facet_points = []
p = 0
offset = finat_element.space_dimension()
for cell in range(ncell):
for ent in sorted(ecd.keys()):
# Never need closure of cell
if sum(ent) == dim:
continue
for key in sorted(ecd[ent].keys()):
closure_section.setDof(p, len(ecd[ent][key]))
vals = numpy.asarray(sorted(ecd[ent][key]), dtype=IntType)
closure_indices.extend(vals + cell * offset)
if esd is not None:
support_section.setDof(p, ncell * len(esd[ent][key]))
vals = numpy.asarray(sorted(esd[ent][key]), dtype=IntType)
support_indices.extend(vals + cell * offset)
if sum(ent) == dim - 1:
facet_points.append(p)
p += 1
closure_section.setUp()
support_section.setUp()
closure_indices = numpy.asarray(closure_indices, dtype=IntType)
support_indices = numpy.asarray(support_indices, dtype=IntType)
facet_points = numpy.asarray(facet_points, dtype=IntType)
masks["topological"] = (closure_section, closure_indices, facet_points)
masks["geometric"] = (support_section, support_indices, facet_points)
return masks
def __init__(self, problem, **kwargs):
r"""
:arg problem: A :class:`NonlinearVariationalProblem` to solve.
:kwarg nullspace: an optional :class:`.VectorSpaceBasis` (or
:class:`.MixedVectorSpaceBasis`) spanning the null
space of the operator.
:kwarg transpose_nullspace: as for the nullspace, but used to
make the right hand side consistent.
:kwarg near_nullspace: as for the nullspace, but used to
specify the near nullspace (for multigrid solvers).
:kwarg solver_parameters: Solver parameters to pass to PETSc.
This should be a dict mapping PETSc options to values.
:kwarg appctx: A dictionary containing application context that
is passed to the preconditioner if matrix-free.
:kwarg options_prefix: an optional prefix used to distinguish
PETSc options. If not provided a unique prefix will be
created. Use this option if you want to pass options
to the solver from the command line in addition to
through the ``solver_parameters`` dict.
:kwarg pre_jacobian_callback: A user-defined function that will
be called immediately before Jacobian assembly. This can
be used, for example, to update a coefficient function
that has a complicated dependence on the unknown solution.
:kwarg pre_function_callback: As above, but called immediately
before residual assembly
Example usage of the ``solver_parameters`` option: to set the
nonlinear solver type to just use a linear solver, use
.. code-block:: python
{'snes_type': 'ksponly'}
PETSc flag options (where the presence of the option means something) should
be specified with ``None``.
For example:
.. code-block:: python
{'snes_monitor': None}
To use the ``pre_jacobian_callback`` or ``pre_function_callback``
functionality, the user-defined function must accept the current
solution as a petsc4py Vec. Example usage is given below:
.. code-block:: python
def update_diffusivity(current_solution):
with cursol.dat.vec_wo as v:
current_solution.copy(v)
solve(trial*test*dx == dot(grad(cursol), grad(test))*dx, diffusivity)
solver = NonlinearVariationalSolver(problem,
pre_jacobian_callback=update_diffusivity)
"""
assert isinstance(problem, NonlinearVariationalProblem)
parameters = kwargs.get("solver_parameters")
if "parameters" in kwargs:
raise TypeError("Use solver_parameters, not parameters")
nullspace = kwargs.get("nullspace")
nullspace_T = kwargs.get("transpose_nullspace")
near_nullspace = kwargs.get("near_nullspace")
options_prefix = kwargs.get("options_prefix")
pre_j_callback = kwargs.get("pre_jacobian_callback")
pre_f_callback = kwargs.get("pre_function_callback")
super(NonlinearVariationalSolver,
self).__init__(parameters, options_prefix)
# Allow anything, interpret "matfree" as matrix_free.
mat_type = self.parameters.get("mat_type")
pmat_type = self.parameters.get("pmat_type")
matfree = mat_type == "matfree"
pmatfree = pmat_type == "matfree"
appctx = kwargs.get("appctx")
ctx = solving_utils._SNESContext(problem,
mat_type=mat_type,
pmat_type=pmat_type,
appctx=appctx,
pre_jacobian_callback=pre_j_callback,
pre_function_callback=pre_f_callback,
options_prefix=self.options_prefix)
# No preconditioner by default for matrix-free
if (problem.Jp is not None and pmatfree) or matfree:
self.set_default_parameter("pc_type", "none")
elif ctx.is_mixed:
# Mixed problem, use jacobi pc if user has not supplied
# one.
self.set_default_parameter("pc_type", "jacobi")
self.snes = PETSc.SNES().create(comm=problem.dm.comm)
self._problem = problem
self._ctx = ctx
self._work = problem.u.dof_dset.layout_vec.duplicate()
self.snes.setDM(problem.dm)
ctx.set_function(self.snes)
ctx.set_jacobian(self.snes)
ctx.set_nullspace(nullspace,
problem.J.arguments()[0].function_space()._ises,
transpose=False,
near=False)
ctx.set_nullspace(nullspace_T,
problem.J.arguments()[1].function_space()._ises,
transpose=True,
near=False)
ctx.set_nullspace(near_nullspace,
problem.J.arguments()[0].function_space()._ises,
transpose=False,
near=True)
ctx._nullspace = nullspace
ctx._nullspace_T = nullspace_T
ctx._near_nullspace = near_nullspace
# Set from options now, so that people who want to noodle with
# the snes object directly (mostly Patrick), can. We need the
# DM with an app context in place so that if the DM is active
# on a subKSP the context is available.
dm = self.snes.getDM()
with dmhooks.appctx(dm, self._ctx):
self.set_from_options(self.snes)
# Used for custom grid transfer.
self._transfer_operators = ()
self._setup = False
def initialize(self, pc):
_, P = pc.getOperators()
assert P.type == "python"
context = P.getPythonContext()
(self.J, self.bcs) = (context.a, context.row_bcs)
test, trial = self.J.arguments()
if test.function_space() != trial.function_space():
raise NotImplementedError("test and trial spaces must be the same")
Pk = test.function_space()
element = Pk.ufl_element()
shape = element.value_shape()
mesh = Pk.ufl_domain()
if len(shape) == 0:
P1 = firedrake.FunctionSpace(mesh, "CG", 1)
elif len(shape) == 2:
P1 = firedrake.VectorFunctionSpace(mesh, "CG", 1, dim=shape[0])
else:
P1 = firedrake.TensorFunctionSpace(mesh,
"CG",
1,
shape=shape,
symmetry=element.symmetry())
# TODO: A smarter low-order operator would also interpolate
# any coefficients to the coarse space.
mapper = ArgumentReplacer({
test: firedrake.TestFunction(P1),
trial: firedrake.TrialFunction(P1)
})
self.lo_J = map_integrands.map_integrand_dags(mapper, self.J)
lo_bcs = []
for bc in self.bcs:
# Don't actually need the value, since it's only used for
# killing parts of the restriction matrix.
lo_bcs.append(
firedrake.DirichletBC(P1,
firedrake.zero(P1.shape),
bc.sub_domain,
method=bc.method))
self.lo_bcs = tuple(lo_bcs)
mat_type = PETSc.Options().getString(
pc.getOptionsPrefix() + "lo_mat_type",
firedrake.parameters["default_matrix_type"])
self.lo_op = firedrake.assemble(self.lo_J,
bcs=self.lo_bcs,
mat_type=mat_type)
self.lo_op.force_evaluation()
A, P = pc.getOperators()
nearnullsp = P.getNearNullSpace()
if nearnullsp.handle != 0:
# Actually have a near nullspace
tmp = firedrake.Function(Pk)
low = firedrake.Function(P1)
vecs = []
for vec in nearnullsp.getVecs():
with tmp.dat.vec as v:
vec.copy(v)
low.interpolate(tmp)
with low.dat.vec_ro as v:
vecs.append(v.copy())
nullsp = PETSc.NullSpace().create(vectors=vecs, comm=pc.comm)
self.lo_op.petscmat.setNearNullSpace(nullsp)
lo = PETSc.PC().create(comm=pc.comm)
lo.incrementTabLevel(1, parent=pc)
lo.setOperators(self.lo_op.petscmat, self.lo_op.petscmat)
lo.setOptionsPrefix(pc.getOptionsPrefix() + "lo_")
lo.setFromOptions()
self.lo = lo
self.restriction = restriction_matrix(Pk, P1, self.bcs, self.lo_bcs)
self.work = self.lo_op.petscmat.createVecs()
if len(self.bcs) > 0:
bc_nodes = numpy.unique(
numpy.concatenate([bc.nodes for bc in self.bcs]))
bc_nodes = bc_nodes[bc_nodes < Pk.dof_dset.size]
bc_iset = PETSc.IS().createBlock(numpy.prod(shape),
bc_nodes,
comm=PETSc.COMM_SELF)
self.bc_indices = bc_iset.getIndices()
bc_iset.destroy()
else:
self.bc_indices = numpy.empty(0, dtype=numpy.int32)
def createSubMatrix(self, mat, row_is, col_is, target=None):
if target is not None:
# Repeat call, just return the matrix, since we don't
# actually assemble in here.
target.assemble()
return target
# These are the sets of ISes of which the the row and column
# space consist.
row_ises = self._y.function_space().dof_dset.field_ises
col_ises = self._x.function_space().dof_dset.field_ises
row_inds = find_sub_block(row_is, row_ises)
if row_is == col_is and row_ises == col_ises:
col_inds = row_inds
else:
col_inds = find_sub_block(col_is, col_ises)
splitter = ExtractSubBlock()
asub = splitter.split(self.a, argument_indices=(row_inds, col_inds))
Wrow = asub.arguments()[0].function_space()
Wcol = asub.arguments()[1].function_space()
row_bcs = []
col_bcs = []
for bc in self.bcs:
if isinstance(bc, DirichletBC):
bc_temp = bc.reconstruct(field=row_inds,
V=Wrow,
g=bc.function_arg,
sub_domain=bc.sub_domain,
use_split=True)
elif isinstance(bc, EquationBCSplit):
bc_temp = bc.reconstruct(field=row_inds,
V=Wrow,
row_field=row_inds,
col_field=col_inds,
use_split=True)
if bc_temp is not None:
row_bcs.append(bc_temp)
if Wrow == Wcol and row_inds == col_inds and self.bcs == self.bcs_col:
col_bcs = row_bcs
else:
for bc in self.bcs_col:
if isinstance(bc, DirichletBC):
bc_temp = bc.reconstruct(field=col_inds,
V=Wcol,
g=bc.function_arg,
sub_domain=bc.sub_domain,
use_split=True)
elif isinstance(bc, EquationBCSplit):
bc_temp = bc.reconstruct(field=col_inds,
V=Wcol,
row_field=row_inds,
col_field=col_inds,
use_split=True)
if bc_temp is not None:
col_bcs.append(bc_temp)
submat_ctx = ImplicitMatrixContext(asub,
row_bcs=row_bcs,
col_bcs=col_bcs,
fc_params=self.fc_params,
appctx=self.appctx)
submat_ctx.on_diag = self.on_diag and row_inds == col_inds
submat = PETSc.Mat().create(comm=mat.comm)
submat.setType("python")
submat.setSizes((submat_ctx.row_sizes, submat_ctx.col_sizes),
bsize=submat_ctx.block_size)
submat.setPythonContext(submat_ctx)
submat.setUp()
return submat
# Define Weak form
a = ( (Ub*phi - 1./(Ro*k2)*psi + 1./Ro*Hb*vphi)*u )*dx \
+ ( ((dUb - psi/Ro)*phi + Ub*psi - 1./Ro*Hb*vphi.dx(0))*v )*dx \
+ ( 1./Ro*phi*eta - 1./(Ro*k2)*psi*eta.dx(0) + vphi*Ub )*dx
m = (phi * u + psi * v + vphi * eta) * dx
# Build Petsc operators
petsc_a = assemble(a, mat_type='aij', bcs=bc).M.handle
petsc_m = assemble(m, mat_type='aij', bcs=bc).M.handle
# Define Petsc options
opts = PETSc.Options()
opts.setValue("eps_gen_non_hermitian", None)
opts.setValue("st_pc_factor_shift_type", "NONZERO")
#opts.setValue("eps_type", "lapack")
opts.setValue("eps_type", "krylovschur")
opts.setValue("eps_largest_imaginary", None)
opts.setValue("eps_tol", 1e-10)
# Define Solver options
es = SLEPc.EPS().create(comm=COMM_WORLD)
es.setDimensions(num_eigenvalues)
es.setOperators(petsc_a, petsc_m)
es.setFromOptions()
es.solve()
def assemble_mixed_mass_matrix(V_A, V_B):
"""
Construct the mixed mass matrix of two function spaces,
using the TrialFunction from V_A and the TestFunction
from V_B.
"""
if len(V_A) > 1 or len(V_B) > 1:
raise NotImplementedError(
"Sorry, only implemented for non-mixed spaces")
if V_A.ufl_element().mapping() != "identity" or V_B.ufl_element().mapping(
) != "identity":
msg = """
Sorry, only implemented for affine maps for now. To do non-affine, we'd need to
import much more of the assembly engine of UFL/TSFC/etc to do the assembly on
each supermesh cell.
"""
raise NotImplementedError(msg)
mesh_A = V_A.mesh()
mesh_B = V_B.mesh()
dim = mesh_A.geometric_dimension()
assert dim == mesh_B.geometric_dimension()
assert dim == mesh_A.topological_dimension()
assert dim == mesh_B.topological_dimension()
(mh_A, level_A) = get_level(mesh_A)
(mh_B, level_B) = get_level(mesh_B)
if mesh_A is mesh_B:
def likely(cell_A):
return [cell_A]
else:
if (mh_A is None or mh_B is None) or (mh_A is not mh_B):
# No mesh hierarchy structure, call libsupermesh for
# intersection finding
intersections = intersection_finder(mesh_A, mesh_B)
likely = intersections.__getitem__
else:
# We do have a mesh hierarchy, use it
if abs(level_A - level_B) > 1:
raise NotImplementedError(
"Only works for transferring between adjacent levels for now."
)
# What are the cells of B that (probably) intersect with a given cell in A?
if level_A > level_B:
cell_map = mh_A.fine_to_coarse_cells[level_A]
def likely(cell_A):
return cell_map[cell_A]
elif level_A < level_B:
cell_map = mh_A.coarse_to_fine_cells[level_A]
def likely(cell_A):
return cell_map[cell_A]
assert V_A.value_size == V_B.value_size
orig_value_size = V_A.value_size
if V_A.value_size > 1:
V_A = firedrake.FunctionSpace(mesh_A,
V_A.ufl_element().sub_elements()[0])
if V_B.value_size > 1:
V_B = firedrake.FunctionSpace(mesh_B,
V_B.ufl_element().sub_elements()[0])
assert V_A.value_size == 1
assert V_B.value_size == 1
preallocator = PETSc.Mat().create(comm=mesh_A.comm)
preallocator.setType(PETSc.Mat.Type.PREALLOCATOR)
rset = V_B.dof_dset
cset = V_A.dof_dset
nrows = rset.layout_vec.getSizes()
ncols = cset.layout_vec.getSizes()
preallocator.setLGMap(rmap=rset.scalar_lgmap, cmap=cset.scalar_lgmap)
preallocator.setSizes(size=(nrows, ncols), bsize=1)
preallocator.setUp()
zeros = numpy.zeros((V_B.cell_node_map().arity, V_A.cell_node_map().arity),
dtype=ScalarType)
for cell_A, dofs_A in enumerate(V_A.cell_node_map().values):
for cell_B in likely(cell_A):
dofs_B = V_B.cell_node_map().values_with_halo[cell_B, :]
preallocator.setValuesLocal(dofs_B, dofs_A, zeros)
preallocator.assemble()
dnnz, onnz = get_preallocation(preallocator, nrows[0])
# Unroll from block to AIJ
dnnz = dnnz * cset.cdim
dnnz = numpy.repeat(dnnz, rset.cdim)
onnz = onnz * cset.cdim
onnz = numpy.repeat(onnz, cset.cdim)
preallocator.destroy()
assert V_A.value_size == V_B.value_size
rdim = V_B.dof_dset.cdim
cdim = V_A.dof_dset.cdim
#
# Preallocate M_AB.
#
mat = PETSc.Mat().create(comm=mesh_A.comm)
mat.setType(PETSc.Mat.Type.AIJ)
rsizes = tuple(n * rdim for n in nrows)
csizes = tuple(c * cdim for c in ncols)
mat.setSizes(size=(rsizes, csizes), bsize=(rdim, cdim))
mat.setPreallocationNNZ((dnnz, onnz))
mat.setLGMap(rmap=rset.lgmap, cmap=cset.lgmap)
# TODO: Boundary conditions not handled.
mat.setOption(mat.Option.IGNORE_OFF_PROC_ENTRIES, False)
mat.setOption(mat.Option.NEW_NONZERO_ALLOCATION_ERR, True)
mat.setOption(mat.Option.KEEP_NONZERO_PATTERN, True)
mat.setOption(mat.Option.UNUSED_NONZERO_LOCATION_ERR, False)
mat.setOption(mat.Option.IGNORE_ZERO_ENTRIES, True)
mat.setUp()
evaluate_kernel_A = compile_element(ufl.Coefficient(V_A),
name="evaluate_kernel_A")
evaluate_kernel_B = compile_element(ufl.Coefficient(V_B),
name="evaluate_kernel_B")
# We only need one of these since we assume that the two meshes both have CG1 coordinates
to_reference_kernel = to_reference_coordinates(
mesh_A.coordinates.ufl_element())
if dim == 2:
reference_mesh = UnitTriangleMesh(comm=COMM_SELF)
else:
reference_mesh = UnitTetrahedronMesh(comm=COMM_SELF)
evaluate_kernel_S = compile_element(ufl.Coefficient(
reference_mesh.coordinates.function_space()),
name="evaluate_kernel_S")
V_S_A = FunctionSpace(reference_mesh, V_A.ufl_element())
V_S_B = FunctionSpace(reference_mesh, V_B.ufl_element())
M_SS = assemble(inner(TrialFunction(V_S_A), TestFunction(V_S_B)) * dx)
M_SS = M_SS.M.handle[:, :]
node_locations_A = utils.physical_node_locations(
V_S_A).dat.data_ro_with_halos
node_locations_B = utils.physical_node_locations(
V_S_B).dat.data_ro_with_halos
num_nodes_A = node_locations_A.shape[0]
num_nodes_B = node_locations_B.shape[0]
to_reference_kernel = to_reference_coordinates(
mesh_A.coordinates.ufl_element())
supermesh_kernel_str = """
#include "libsupermesh-c.h"
#include <petsc.h>
%(to_reference)s
%(evaluate_S)s
%(evaluate_A)s
%(evaluate_B)s
#define complex_mode %(complex_mode)s
#define PrintInfo(...) do { if (PetscLogPrintInfo) printf(__VA_ARGS__); } while (0)
static void print_array(PetscScalar *arr, int d)
{
for(int j=0; j<d; j++)
PrintInfo(stderr, "%%+.2f ", arr[j]);
}
static void print_coordinates(PetscScalar *simplex, int d)
{
for(int i=0; i<d+1; i++)
{
PrintInfo("\t");
print_array(&simplex[d*i], d);
PrintInfo("\\n");
}
}
#if complex_mode
static void seperate_real_and_imag(PetscScalar *simplex, double *real_simplex, double *imag_simplex, int d)
{
for(int i=0; i<d+1; i++)
{
for(int j=0; j<d; j++)
{
real_simplex[d*i+j] = creal(simplex[d*i+j]);
imag_simplex[d*i+j] = cimag(simplex[d*i+j]);
}
}
}
static void merge_back_to_simplex(PetscScalar* simplex, double* real_simplex, double* imag_simplex, int d)
{
print_coordinates(simplex,d);
for(int i=0; i<d+1; i++)
{
for(int j=0; j<d; j++)
{
simplex[d*i+j] = real_simplex[d*i+j]+imag_simplex[d*i+j]*_Complex_I;
}
}
}
#endif
int supermesh_kernel(PetscScalar* simplex_A, PetscScalar* simplex_B, PetscScalar* simplices_C, PetscScalar* nodes_A, PetscScalar* nodes_B, PetscScalar* M_SS, PetscScalar* outptr, int num_ele)
{
#define d %(dim)s
#define num_nodes_A %(num_nodes_A)s
#define num_nodes_B %(num_nodes_B)s
double simplex_ref_measure;
PrintInfo("simplex_A coordinates\\n");
print_coordinates(simplex_A, d);
PrintInfo("simplex_B coordinates\\n");
print_coordinates(simplex_B, d);
int num_elements = num_ele;
if (d == 2) simplex_ref_measure = 0.5;
else if (d == 3) simplex_ref_measure = 1.0/6;
PetscScalar R_AS[num_nodes_A][num_nodes_A];
PetscScalar R_BS[num_nodes_B][num_nodes_B];
PetscScalar coeffs_A[%(num_nodes_A)s] = {0.};
PetscScalar coeffs_B[%(num_nodes_B)s] = {0.};
PetscScalar reference_nodes_A[num_nodes_A][d];
PetscScalar reference_nodes_B[num_nodes_B][d];
#if complex_mode
double real_simplex_A[d*(d+1)];
double imag_simplex_A[d*(d+1)];
seperate_real_and_imag(simplex_A, real_simplex_A, imag_simplex_A, d);
double real_simplex_B[d*(d+1)];
double imag_simplex_B[d*(d+1)];
seperate_real_and_imag(simplex_B, real_simplex_B, imag_simplex_B, d);
double real_simplices_C[num_elements*d*(d+1)];
double imag_simplices_C[num_elements*d*(d+1)];
for (int ii=0; ii<num_elements*d*(d+1); ++ii) imag_simplices_C[ii] = 0.;
%(libsupermesh_intersect_simplices)s(real_simplex_A, real_simplex_B, real_simplices_C, &num_elements);
merge_back_to_simplex(simplex_A, real_simplex_A, imag_simplex_A, d);
merge_back_to_simplex(simplex_B, real_simplex_B, imag_simplex_B, d);
for(int s=0; s<num_elements; s++)
{
PetscScalar* simplex_C = &simplices_C[s * d * (d+1)];
double* real_simplex_C = &real_simplices_C[s * d * (d+1)];
double* imag_simplex_C = &imag_simplices_C[s * d * (d+1)];
merge_back_to_simplex(simplex_C, real_simplex_C, imag_simplex_C, d);
}
#else
%(libsupermesh_intersect_simplices)s(simplex_A, simplex_B, simplices_C, &num_elements);
#endif
PrintInfo("Supermesh consists of %%i elements\\n", num_elements);
// would like to do this
//PetscScalar MAB[%(num_nodes_A)s][%(num_nodes_B)s] = (PetscScalar (*)[%(num_nodes_B)s])outptr;
// but have to do this instead because we don't grok C
PetscScalar (*MAB)[num_nodes_A] = (PetscScalar (*)[num_nodes_A])outptr;
PetscScalar (*MSS)[num_nodes_A] = (PetscScalar (*)[num_nodes_A])M_SS; // note the underscore
for ( int i = 0; i < num_nodes_B; i++ ) {
for (int j = 0; j < num_nodes_A; j++) {
MAB[i][j] = 0.0;
}
}
for(int s=0; s<num_elements; s++)
{
PetscScalar* simplex_S = &simplices_C[s * d * (d+1)];
double simplex_S_measure;
#if complex_mode
double real_simplex_S[d*(d+1)];
double imag_simplex_S[d*(d+1)];
seperate_real_and_imag(simplex_S, real_simplex_S, imag_simplex_S, d);
%(libsupermesh_simplex_measure)s(real_simplex_S, &simplex_S_measure);
merge_back_to_simplex(simplex_S, real_simplex_S, imag_simplex_S, d);
#else
%(libsupermesh_simplex_measure)s(simplex_S, &simplex_S_measure);
#endif
PrintInfo("simplex_S coordinates with measure %%f\\n", simplex_S_measure);
print_coordinates(simplex_S, d);
PrintInfo("Start mapping nodes for V_A\\n");
PetscScalar physical_nodes_A[num_nodes_A][d];
for(int n=0; n < num_nodes_A; n++) {
PetscScalar* reference_node_location = &nodes_A[n*d];
PetscScalar* physical_node_location = physical_nodes_A[n];
for (int j=0; j < d; j++) physical_node_location[j] = 0.0;
pyop2_kernel_evaluate_kernel_S(physical_node_location, simplex_S, reference_node_location);
PrintInfo("\\tNode ");
print_array(reference_node_location, d);
PrintInfo(" mapped to ");
print_array(physical_node_location, d);
PrintInfo("\\n");
}
PrintInfo("Start mapping nodes for V_B\\n");
PetscScalar physical_nodes_B[num_nodes_B][d];
for(int n=0; n < num_nodes_B; n++) {
PetscScalar* reference_node_location = &nodes_B[n*d];
PetscScalar* physical_node_location = physical_nodes_B[n];
for (int j=0; j < d; j++) physical_node_location[j] = 0.0;
pyop2_kernel_evaluate_kernel_S(physical_node_location, simplex_S, reference_node_location);
PrintInfo("\\tNode ");
print_array(reference_node_location, d);
PrintInfo(" mapped to ");
print_array(physical_node_location, d);
PrintInfo("\\n");
}
PrintInfo("==========================================================\\n");
PrintInfo("Start pulling back dof from S into reference space for A.\\n");
for(int n=0; n < num_nodes_A; n++) {
for(int i=0; i<d; i++) reference_nodes_A[n][i] = 0.;
to_reference_coords_kernel(reference_nodes_A[n], physical_nodes_A[n], simplex_A);
PrintInfo("Pulling back ");
print_array(physical_nodes_A[n], d);
PrintInfo(" to ");
print_array(reference_nodes_A[n], d);
PrintInfo("\\n");
}
PrintInfo("Start pulling back dof from S into reference space for B.\\n");
for(int n=0; n < num_nodes_B; n++) {
for(int i=0; i<d; i++) reference_nodes_B[n][i] = 0.;
to_reference_coords_kernel(reference_nodes_B[n], physical_nodes_B[n], simplex_B);
PrintInfo("Pulling back ");
print_array(physical_nodes_B[n], d);
PrintInfo(" to ");
print_array(reference_nodes_B[n], d);
PrintInfo("\\n");
}
PrintInfo("Start evaluating basis functions of V_A at dofs for V_A on S\\n");
for(int i=0; i<num_nodes_A; i++) {
coeffs_A[i] = 1.;
for(int j=0; j<num_nodes_A; j++) {
R_AS[i][j] = 0.;
pyop2_kernel_evaluate_kernel_A(&R_AS[i][j], coeffs_A, reference_nodes_A[j]);
}
print_array(R_AS[i], num_nodes_A);
PrintInfo("\\n");
coeffs_A[i] = 0.;
}
PrintInfo("Start evaluating basis functions of V_B at dofs for V_B on S\\n");
for(int i=0; i<num_nodes_B; i++) {
coeffs_B[i] = 1.;
for(int j=0; j<num_nodes_B; j++) {
R_BS[i][j] = 0.;
pyop2_kernel_evaluate_kernel_B(&R_BS[i][j], coeffs_B, reference_nodes_B[j]);
}
print_array(R_BS[i], num_nodes_B);
PrintInfo("\\n");
coeffs_B[i] = 0.;
}
PrintInfo("Start doing the matmatmat mult\\n");
for ( int i = 0; i < num_nodes_B; i++ ) {
for (int j = 0; j < num_nodes_A; j++) {
for ( int k = 0; k < num_nodes_B; k++) {
for ( int l = 0; l < num_nodes_A; l++) {
MAB[i][j] += (simplex_S_measure/simplex_ref_measure) * R_BS[i][k] * MSS[k][l] * R_AS[j][l];
}
}
}
}
}
return num_elements;
}
""" % {
"evaluate_S":
str(evaluate_kernel_S),
"evaluate_A":
str(evaluate_kernel_A),
"evaluate_B":
str(evaluate_kernel_B),
"to_reference":
str(to_reference_kernel),
"num_nodes_A":
num_nodes_A,
"num_nodes_B":
num_nodes_B,
"libsupermesh_simplex_measure":
"libsupermesh_triangle_area"
if dim == 2 else "libsupermesh_tetrahedron_volume",
"libsupermesh_intersect_simplices":
"libsupermesh_intersect_tris_real"
if dim == 2 else "libsupermesh_intersect_tets_real",
"dim":
dim,
"complex_mode":
1 if complex_mode else 0
}
dirs = get_petsc_dir() + (sys.prefix, )
includes = ["-I%s/include" % d for d in dirs]
libs = ["-L%s/lib" % d for d in dirs]
libs = libs + ["-Wl,-rpath,%s/lib" % d
for d in dirs] + ["-lpetsc", "-lsupermesh"]
lib = load(supermesh_kernel_str,
"c",
"supermesh_kernel",
cppargs=includes,
ldargs=libs,
argtypes=[
ctypes.c_voidp, ctypes.c_voidp, ctypes.c_voidp,
ctypes.c_voidp, ctypes.c_voidp, ctypes.c_voidp,
ctypes.c_voidp
],
restype=ctypes.c_int)
ammm(V_A, V_B, likely, node_locations_A, node_locations_B, M_SS,
ctypes.addressof(lib), mat)
if orig_value_size == 1:
return mat
else:
(lrows, grows), (lcols, gcols) = mat.getSizes()
lrows *= orig_value_size
grows *= orig_value_size
lcols *= orig_value_size
gcols *= orig_value_size
size = ((lrows, grows), (lcols, gcols))
context = BlockMatrix(mat, orig_value_size)
blockmat = PETSc.Mat().createPython(size,
context=context,
comm=mat.comm)
blockmat.setUp()
return blockmat
def print_at_exit(cls):
"""Print citations when exiting."""
# We devolve to PETSc for actually printing citations.
PETSc.Options()["citations"] = None
def __init__(self,
A,
P=None,
solver_parameters=None,
nullspace=None,
transpose_nullspace=None,
near_nullspace=None,
options_prefix=None):
"""A linear solver for assembled systems (Ax = b).
:arg A: a :class:`~.MatrixBase` (the operator).
:arg P: an optional :class:`~.MatrixBase` to construct any
preconditioner from; if none is supplied ``A`` is
used to construct the preconditioner.
:kwarg parameters: (optional) dict of solver parameters.
:kwarg nullspace: an optional :class:`~.VectorSpaceBasis` (or
:class:`~.MixedVectorSpaceBasis` spanning the null space
of the operator.
:kwarg transpose_nullspace: as for the nullspace, but used to
make the right hand side consistent.
:kwarg near_nullspace: as for the nullspace, but used to set
the near nullpace.
:kwarg options_prefix: an optional prefix used to distinguish
PETSc options. If not provided a unique prefix will be
created. Use this option if you want to pass options
to the solver from the command line in addition to
through the ``solver_parameters`` dict.
.. note::
Any boundary conditions for this solve *must* have been
applied when assembling the operator.
"""
if not isinstance(A, matrix.MatrixBase):
raise TypeError("Provided operator is a '%s', not a MatrixBase" %
type(A).__name__)
if P is not None and not isinstance(P, matrix.MatrixBase):
raise TypeError(
"Provided preconditioner is a '%s', not a MatrixBase" %
type(P).__name__)
self.A = A
self.comm = A.comm
self.P = P if P is not None else A
# Set up parameters mixin
super(LinearSolver, self).__init__(solver_parameters, options_prefix)
self.A.petscmat.setOptionsPrefix(self.options_prefix)
self.P.petscmat.setOptionsPrefix(self.options_prefix)
# Set some defaults
self.set_default_parameter("ksp_rtol", "1e-7")
# If preconditioning matrix is matrix-free, then default to no
# preconditioning.
if isinstance(self.P, matrix.ImplicitMatrix):
self.set_default_parameter("pc_type", "none")
elif self.P.block_shape != (1, 1):
# Otherwise, mixed problems default to jacobi.
self.set_default_parameter("pc_type", "jacobi")
self.ksp = PETSc.KSP().create(comm=self.comm)
W = self.test_space
# DM provides fieldsplits (but not operators)
self.ksp.setDM(W.dm)
self.ksp.setDMActive(False)
if nullspace is not None:
nullspace._apply(self.A)
if P is not None:
nullspace._apply(self.P)
if transpose_nullspace is not None:
transpose_nullspace._apply(self.A, transpose=True)
if P is not None:
transpose_nullspace._apply(self.P, transpose=True)
if near_nullspace is not None:
near_nullspace._apply(self.A, near=True)
if P is not None:
near_nullspace._apply(self.P, near=True)
self.nullspace = nullspace
self.transpose_nullspace = transpose_nullspace
self.near_nullspace = near_nullspace
# Operator setting must come after null space has been
# applied
# Force evaluation here
self.A.force_evaluation()
self.P.force_evaluation()
self.ksp.setOperators(A=self.A.petscmat, P=self.P.petscmat)
# Set from options now (we're not allowed to change parameters
# anyway).
self.set_from_options(self.ksp)
def initialize(self, pc):
from firedrake import TrialFunction, TestFunction, dx, \
assemble, inner, grad, split, Constant, parameters
from firedrake.assemble import allocate_matrix, create_assembly_callable
if pc.getType() != "python":
raise ValueError("Expecting PC type python")
prefix = pc.getOptionsPrefix() + "pcd_"
# we assume P has things stuffed inside of it
_, P = pc.getOperators()
context = P.getPythonContext()
test, trial = context.a.arguments()
if test.function_space() != trial.function_space():
raise ValueError("Pressure space test and trial space differ")
Q = test.function_space()
p = TrialFunction(Q)
q = TestFunction(Q)
mass = p * q * dx
# Regularisation to avoid having to think about nullspaces.
stiffness = inner(grad(p), grad(q)) * dx + Constant(1e-6) * p * q * dx
opts = PETSc.Options()
# we're inverting Mp and Kp, so default them to assembled.
# Fp only needs its action, so default it to mat-free.
# These can of course be overridden.
# only Fp is referred to in update, so that's the only
# one we stash.
default = parameters["default_matrix_type"]
Mp_mat_type = opts.getString(prefix + "Mp_mat_type", default)
Kp_mat_type = opts.getString(prefix + "Kp_mat_type", default)
self.Fp_mat_type = opts.getString(prefix + "Fp_mat_type", "matfree")
Mp = assemble(mass,
form_compiler_parameters=context.fc_params,
mat_type=Mp_mat_type,
options_prefix=prefix + "Mp_")
Kp = assemble(stiffness,
form_compiler_parameters=context.fc_params,
mat_type=Kp_mat_type,
options_prefix=prefix + "Kp_")
Mp.force_evaluation()
Kp.force_evaluation()
# FIXME: Should we transfer nullspaces over. I think not.
Mksp = PETSc.KSP().create(comm=pc.comm)
Mksp.incrementTabLevel(1, parent=pc)
Mksp.setOptionsPrefix(prefix + "Mp_")
Mksp.setOperators(Mp.petscmat)
Mksp.setUp()
Mksp.setFromOptions()
self.Mksp = Mksp
Kksp = PETSc.KSP().create(comm=pc.comm)
Kksp.incrementTabLevel(1, parent=pc)
Kksp.setOptionsPrefix(prefix + "Kp_")
Kksp.setOperators(Kp.petscmat)
Kksp.setUp()
Kksp.setFromOptions()
self.Kksp = Kksp
state = context.appctx["state"]
Re = context.appctx.get("Re", 1.0)
velid = context.appctx["velocity_space"]
u0 = split(state)[velid]
fp = 1.0 / Re * inner(grad(p), grad(q)) * dx + inner(u0,
grad(p)) * q * dx
self.Re = Re
self.Fp = allocate_matrix(fp,
form_compiler_parameters=context.fc_params,
mat_type=self.Fp_mat_type,
options_prefix=prefix + "Fp_")
self._assemble_Fp = create_assembly_callable(
fp,
tensor=self.Fp,
form_compiler_parameters=context.fc_params,
mat_type=self.Fp_mat_type)
self._assemble_Fp()
self.Fp.force_evaluation()
Fpmat = self.Fp.petscmat
self.workspace = [Fpmat.createVecLeft() for i in (0, 1)]
def __init__(self, problem, **kwargs):
"""
Solve a :class:`NonlinearVariationalProblem` on a hierarchy of meshes.
:arg problem: A :class:`NonlinearVariationalProblem` or
iterable thereof (if specifying the problem on each level
by hand).
:kwarg nullspace: an optional :class:`.VectorSpaceBasis` (or
:class:`MixedVectorSpaceBasis`) spanning the null space of the
operator.
:kwarg solver_parameters: Solver parameters to pass to PETSc.
This should be a dict mapping PETSc options to values.
PETSc flag options should be specified with `bool`
values (:data:`True` for on, :data:`False` for off).
:kwarg options_prefix: an optional prefix used to distinguish
PETSc options. If not provided a unique prefix will be
created. Use this option if you want to pass options
to the solver from the command line in addition to
through the :data:`solver_parameters` dict.
.. note::
This solver is set up for use with geometric multigrid,
that is you can use :data:`"snes_type": "fas"` or
:data:`"pc_type": "mg"` transparently.
"""
# Do this first so __del__ doesn't barf horribly if we get an
# error in __init__
parameters, nullspace, options_prefix \
= firedrake.solving_utils._extract_kwargs(**kwargs)
if options_prefix is not None:
self._opt_prefix = options_prefix
self._auto_prefix = False
else:
self._opt_prefix = "firedrake_nlvsh_%d_" % NLVSHierarchy._id
self._auto_prefix = True
NLVSHierarchy._id += 1
if isinstance(problem, firedrake.NonlinearVariationalProblem):
# We just got a single problem so coarsen up the hierarchy
problems = []
while True:
if problem:
problems.append(problem)
else:
break
problem = coarsen_problem(problem)
problems.reverse()
else:
# User has provided list of problems
problems = problem
ctx = firedrake.solving_utils._SNESContext(problems)
if nullspace is not None:
raise NotImplementedError(
"Coarsening nullspaces no yet implemented")
snes = PETSc.SNES().create()
snes.setDM(problems[-1].dm)
self.problems = problems
self.snes = snes
self.ctx = ctx
self.ctx.set_function(self.snes)
self.ctx.set_jacobian(self.snes)
self.snes.setOptionsPrefix(self._opt_prefix)
# Allow command-line arguments to override dict parameters
opts = PETSc.Options()
for k, v in opts.getAll().iteritems():
if k.startswith(self._opt_prefix):
parameters[k[len(self._opt_prefix):]] = v
self.parameters = parameters
def initialize(self, pc):
"""Set up the problem context. Take the original
mixed problem and reformulate the problem as a
hybridized mixed system.
A KSP is created for the Lagrange multiplier system.
"""
from firedrake import (FunctionSpace, Function, Constant,
TrialFunction, TrialFunctions, TestFunction,
DirichletBC)
from firedrake.assemble import (allocate_matrix,
create_assembly_callable)
from firedrake.formmanipulation import split_form
from ufl.algorithms.replace import replace
# Extract the problem context
prefix = pc.getOptionsPrefix() + "hybridization_"
_, P = pc.getOperators()
self.ctx = P.getPythonContext()
if not isinstance(self.ctx, ImplicitMatrixContext):
raise ValueError(
"The python context must be an ImplicitMatrixContext")
test, trial = self.ctx.a.arguments()
V = test.function_space()
mesh = V.mesh()
if len(V) != 2:
raise ValueError("Expecting two function spaces.")
if all(Vi.ufl_element().value_shape() for Vi in V):
raise ValueError("Expecting an H(div) x L2 pair of spaces.")
# Automagically determine which spaces are vector and scalar
for i, Vi in enumerate(V):
if Vi.ufl_element().sobolev_space().name == "HDiv":
self.vidx = i
else:
assert Vi.ufl_element().sobolev_space().name == "L2"
self.pidx = i
# Create the space of approximate traces.
W = V[self.vidx]
if W.ufl_element().family() == "Brezzi-Douglas-Marini":
tdegree = W.ufl_element().degree()
else:
try:
# If we have a tensor product element
h_deg, v_deg = W.ufl_element().degree()
tdegree = (h_deg - 1, v_deg - 1)
except TypeError:
tdegree = W.ufl_element().degree() - 1
TraceSpace = FunctionSpace(mesh, "HDiv Trace", tdegree)
# Break the function spaces and define fully discontinuous spaces
broken_elements = ufl.MixedElement(
[ufl.BrokenElement(Vi.ufl_element()) for Vi in V])
V_d = FunctionSpace(mesh, broken_elements)
# Set up the functions for the original, hybridized
# and schur complement systems
self.broken_solution = Function(V_d)
self.broken_residual = Function(V_d)
self.trace_solution = Function(TraceSpace)
self.unbroken_solution = Function(V)
self.unbroken_residual = Function(V)
shapes = (V[self.vidx].finat_element.space_dimension(),
np.prod(V[self.vidx].shape))
domain = "{[i,j]: 0 <= i < %d and 0 <= j < %d}" % shapes
instructions = """
for i, j
w[i,j] = w[i,j] + 1
end
"""
self.weight = Function(V[self.vidx])
par_loop((domain, instructions),
ufl.dx, {"w": (self.weight, INC)},
is_loopy_kernel=True)
instructions = """
for i, j
vec_out[i,j] = vec_out[i,j] + vec_in[i,j]/w[i,j]
end
"""
self.average_kernel = (domain, instructions)
# Create the symbolic Schur-reduction:
# Original mixed operator replaced with "broken"
# arguments
arg_map = {test: TestFunction(V_d), trial: TrialFunction(V_d)}
Atilde = Tensor(replace(self.ctx.a, arg_map))
gammar = TestFunction(TraceSpace)
n = ufl.FacetNormal(mesh)
sigma = TrialFunctions(V_d)[self.vidx]
if mesh.cell_set._extruded:
Kform = (gammar('+') * ufl.jump(sigma, n=n) * ufl.dS_h +
gammar('+') * ufl.jump(sigma, n=n) * ufl.dS_v)
else:
Kform = (gammar('+') * ufl.jump(sigma, n=n) * ufl.dS)
# Here we deal with boundaries. If there are Neumann
# conditions (which should be enforced strongly for
# H(div)xL^2) then we need to add jump terms on the exterior
# facets. If there are Dirichlet conditions (which should be
# enforced weakly) then we need to zero out the trace
# variables there as they are not active (otherwise the hybrid
# problem is not well-posed).
# If boundary conditions are contained in the ImplicitMatrixContext:
if self.ctx.row_bcs:
# Find all the subdomains with neumann BCS
# These are Dirichlet BCs on the vidx space
neumann_subdomains = set()
for bc in self.ctx.row_bcs:
if bc.function_space().index == self.pidx:
raise NotImplementedError(
"Dirichlet conditions for scalar variable not supported. Use a weak bc"
)
if bc.function_space().index != self.vidx:
raise NotImplementedError(
"Dirichlet bc set on unsupported space.")
# append the set of sub domains
subdom = bc.sub_domain
if isinstance(subdom, str):
neumann_subdomains |= set([subdom])
else:
neumann_subdomains |= set(
as_tuple(subdom, numbers.Integral))
# separate out the top and bottom bcs
extruded_neumann_subdomains = neumann_subdomains & {
"top", "bottom"
}
neumann_subdomains = neumann_subdomains - extruded_neumann_subdomains
integrand = gammar * ufl.dot(sigma, n)
measures = []
trace_subdomains = []
if mesh.cell_set._extruded:
ds = ufl.ds_v
for subdomain in sorted(extruded_neumann_subdomains):
measures.append({
"top": ufl.ds_t,
"bottom": ufl.ds_b
}[subdomain])
trace_subdomains.extend(
sorted({"top", "bottom"} - extruded_neumann_subdomains))
else:
ds = ufl.ds
if "on_boundary" in neumann_subdomains:
measures.append(ds)
else:
measures.extend((ds(sd) for sd in sorted(neumann_subdomains)))
markers = [int(x) for x in mesh.exterior_facets.unique_markers]
dirichlet_subdomains = set(markers) - neumann_subdomains
trace_subdomains.extend(sorted(dirichlet_subdomains))
for measure in measures:
Kform += integrand * measure
trace_bcs = [
DirichletBC(TraceSpace, Constant(0.0), subdomain)
for subdomain in trace_subdomains
]
else:
# No bcs were provided, we assume weak Dirichlet conditions.
# We zero out the contribution of the trace variables on
# the exterior boundary. Extruded cells will have both
# horizontal and vertical facets
trace_subdomains = ["on_boundary"]
if mesh.cell_set._extruded:
trace_subdomains.extend(["bottom", "top"])
trace_bcs = [
DirichletBC(TraceSpace, Constant(0.0), subdomain)
for subdomain in trace_subdomains
]
# Make a SLATE tensor from Kform
K = Tensor(Kform)
# Assemble the Schur complement operator and right-hand side
self.schur_rhs = Function(TraceSpace)
self._assemble_Srhs = create_assembly_callable(
K * Atilde.inv * AssembledVector(self.broken_residual),
tensor=self.schur_rhs,
form_compiler_parameters=self.ctx.fc_params)
mat_type = PETSc.Options().getString(prefix + "mat_type", "aij")
schur_comp = K * Atilde.inv * K.T
self.S = allocate_matrix(schur_comp,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type,
options_prefix=prefix)
self._assemble_S = create_assembly_callable(
schur_comp,
tensor=self.S,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type)
with timed_region("HybridOperatorAssembly"):
self._assemble_S()
Smat = self.S.petscmat
nullspace = self.ctx.appctx.get("trace_nullspace", None)
if nullspace is not None:
nsp = nullspace(TraceSpace)
Smat.setNullSpace(nsp.nullspace(comm=pc.comm))
# Set up the KSP for the system of Lagrange multipliers
trace_ksp = PETSc.KSP().create(comm=pc.comm)
trace_ksp.setOptionsPrefix(prefix)
trace_ksp.setOperators(Smat)
trace_ksp.setUp()
trace_ksp.setFromOptions()
self.trace_ksp = trace_ksp
split_mixed_op = dict(split_form(Atilde.form))
split_trace_op = dict(split_form(K.form))
# Generate reconstruction calls
self._reconstruction_calls(split_mixed_op, split_trace_op)
def __del__(self):
if self._auto_prefix and hasattr(self, '_opt_prefix'):
opts = PETSc.Options()
for k in self.parameters.iterkeys():
del opts[self._opt_prefix + k]
delattr(self, '_opt_prefix')
def lgmap(self, V, bcs, lgmap=None):
assert len(V) == 1, "lgmap should not be called on MixedFunctionSpace"
V = V.topological
if bcs is None or len(bcs) == 0:
return lgmap or V.dof_dset.lgmap
# Boundary condition list *must* be collectively ordered already.
# Key is a sorted list of bc subdomain, bc method, bc component.
bc_key = []
for bc in bcs:
fs = bc.function_space()
while fs.component is not None and fs.parent is not None:
fs = fs.parent
if fs.topological != V:
raise RuntimeError("DirichletBC defined on a different FunctionSpace!")
bc_key.append(bc._cache_key)
def key(a):
tpl, *rest = a
if len(tpl) == 1 and isinstance(tpl[0], str):
# tpl = ("some_string", )
return (True, tpl[0], (), tuple(rest))
else:
# Ex:
# tpl = ((facet_dim, ((1,), (2,), (3,))),
# (edge_dim, ((1, 3), (1, 4))),
# (vert_dim, ((1, 3, 4), )))
return (False, "", tpl, tuple(rest))
bc_key = tuple(sorted(bc_key, key=key))
node_set = V.node_set
key = (node_set, V.value_size, lgmap is None, bc_key)
try:
return self.map_cache[key]
except KeyError:
pass
unblocked = any(bc.function_space().component is not None for bc in bcs)
if lgmap is None:
lgmap = V.dof_dset.lgmap
if unblocked:
indices = lgmap.indices.copy()
bsize = 1
else:
indices = lgmap.block_indices.copy()
bsize = lgmap.getBlockSize()
assert bsize == V.value_size
else:
# MatBlock case, LGMap is already unrolled.
indices = lgmap.block_indices.copy()
bsize = lgmap.getBlockSize()
unblocked = True
nodes = []
for bc in bcs:
if bc.function_space().component is not None:
nodes.append(bc.nodes * V.value_size + bc.function_space().component)
elif unblocked:
tmp = bc.nodes * V.value_size
for i in range(V.value_size):
nodes.append(tmp + i)
else:
nodes.append(bc.nodes)
nodes = numpy.unique(numpy.concatenate(nodes))
indices[nodes] = -1
return self.map_cache.setdefault(key, PETSc.LGMap().create(indices, bsize=bsize, comm=lgmap.comm))
def create_interpolation(dmc, dmf):
_, clvl = utils.get_level(dmc)
_, flvl = utils.get_level(dmf)
cctx = dmc.getAppCtx()
fctx = dmf.getAppCtx()
V_c = dmc.getAttr("__fs__")()
V_f = dmf.getAttr("__fs__")()
nrow = sum(x.dof_dset.size * x.dof_dset.cdim for x in V_f)
ncol = sum(x.dof_dset.size * x.dof_dset.cdim for x in V_c)
cfn = firedrake.Function(V_c)
ffn = firedrake.Function(V_f)
cbcs = cctx._problems[clvl].bcs
fbcs = fctx._problems[flvl].bcs
class Interpolation(object):
def __init__(self, cfn, ffn, cbcs=None, fbcs=None):
self.cfn = cfn
self.ffn = ffn
self.cbcs = cbcs or []
self.fbcs = fbcs or []
def mult(self, mat, x, y, inc=False):
with self.cfn.dat.vec as v:
x.copy(v)
firedrake.prolong(self.cfn, self.ffn)
for bc in self.fbcs:
bc.zero(self.ffn)
with self.ffn.dat.vec_ro as v:
if inc:
y.axpy(1.0, v)
else:
v.copy(y)
def multAdd(self, mat, x, y, w):
if y.handle == w.handle:
self.mult(mat, x, w, inc=True)
else:
self.mult(mat, x, w)
w.axpy(1.0, y)
def multTranspose(self, mat, x, y, inc=False):
with self.ffn.dat.vec as v:
x.copy(v)
firedrake.restrict(self.ffn, self.cfn)
for bc in self.cbcs:
bc.zero(self.cfn)
with self.cfn.dat.vec_ro as v:
if inc:
y.axpy(1.0, v)
else:
v.copy(y)
def multTransposeAdd(self, mat, x, y, w):
if y.handle == w.handle:
self.multTranspose(mat, x, w, inc=True)
else:
self.multTranspose(mat, x, w)
w.axpy(1.0, y)
ctx = Interpolation(cfn, ffn, cbcs, fbcs)
mat = PETSc.Mat().create()
mat.setSizes(((nrow, None), (ncol, None)))
mat.setType(mat.Type.PYTHON)
mat.setPythonContext(ctx)
mat.setUp()
return mat, None
def run_steady_turbine(**model_options):
"""
Consider a simple test case with two turbines positioned in a channel. The mesh has been adapted
with respect to fluid speed and so has strong anisotropy in the direction of flow.
If the default SIPG parameter is used, this steady state problem fails to converge. However,
using the automatic SIPG parameter functionality, it should converge.
"""
# Load an anisotropic mesh from file
plex = PETSc.DMPlex().create()
abspath = os.path.realpath(__file__)
plex.createFromFile(
abspath.replace('test_anisotropic.py', 'anisotropic_plex.h5'))
mesh2d = Mesh(plex)
x, y = SpatialCoordinate(mesh2d)
# Create steady state solver object
solver_obj = solver2d.FlowSolver2d(mesh2d, Constant(40.0))
options = solver_obj.options
options.timestep = 20.0
options.simulation_export_time = 20.0
options.simulation_end_time = 18.0
options.timestepper_type = 'SteadyState'
options.timestepper_options.solver_parameters = {
'mat_type': 'aij',
'snes_type': 'newtonls',
'snes_rtol': 1e-8,
'snes_monitor': None,
'pc_type': 'lu',
'pc_factor_mat_solver_type': 'mumps',
}
options.output_directory = 'outputs'
options.fields_to_export = ['uv_2d', 'elev_2d']
options.use_grad_div_viscosity_term = False
options.element_family = 'dg-dg'
options.horizontal_viscosity = Constant(1.0)
options.quadratic_drag_coefficient = Constant(0.0025)
options.use_lax_friedrichs_velocity = True
options.lax_friedrichs_velocity_scaling_factor = Constant(1.0)
options.use_grad_depth_viscosity_term = False
options.update(model_options)
solver_obj.create_equations()
# Apply boundary conditions
solver_obj.bnd_functions['shallow_water'] = {
1: {
'uv': Constant([3.0, 0.0])
},
2: {
'elev': Constant(0.0)
},
3: {
'un': Constant(0.0)
},
}
def bump(fs, locs, scale=1.0):
"""Scaled bump function for turbines."""
i = 0
for j in range(len(locs)):
x0 = locs[j][0]
y0 = locs[j][1]
r = locs[j][2]
expr1 = (x - x0) * (x - x0) + (y - y0) * (y - y0)
expr2 = scale * exp(1 - 1 /
(1 - (x - x0) *
(x - x0) / r**2)) * exp(1 - 1 /
(1 - (y - y0) *
(y - y0) / r**2))
i += conditional(lt(expr1, r * r), expr2, 0)
return i
# Set up turbine array
L = 1000.0 # domain length
W = 300.0 # domain width
D = 18.0 # turbine diameter
A = pi * (D / 2)**2 # turbine area
locs = [(L / 2 - 8 * D, W / 2, D / 2),
(L / 2 + 8 * D, W / 2, D / 2)] # turbine locations
# NOTE: We include a correction to account for the fact that the thrust coefficient is based
# on an upstream velocity, whereas we are using a depth averaged at-the-turbine velocity
# (see Kramer and Piggott 2016, eq. (15)).
correction = 4 / (1 + sqrt(1 - A / (40.0 * D)))**2
scaling = len(locs) / assemble(
bump(solver_obj.function_spaces.P1DG_2d, locs) * dx)
farm_options = TidalTurbineFarmOptions()
farm_options.turbine_density = bump(solver_obj.function_spaces.P1DG_2d,
locs,
scale=scaling)
farm_options.turbine_options.diameter = D
farm_options.turbine_options.thrust_coefficient = 0.8 * correction
solver_obj.options.tidal_turbine_farms['everywhere'] = farm_options
# Apply initial guess of inflow velocity and solve
solver_obj.assign_initial_conditions(uv=Constant([3.0, 0.0]))
solver_obj.iterate()
def initialize(self, pc):
from firedrake import TestFunction, parameters
from firedrake.assemble import allocate_matrix, create_assembly_callable
from firedrake.interpolation import Interpolator
from firedrake.solving_utils import _SNESContext
from firedrake.matrix_free.operators import ImplicitMatrixContext
_, P = pc.getOperators()
appctx = self.get_appctx(pc)
fcp = appctx.get("form_compiler_parameters")
if pc.getType() != "python":
raise ValueError("Expecting PC type python")
ctx = dmhooks.get_appctx(pc.getDM())
if ctx is None:
raise ValueError("No context found.")
if not isinstance(ctx, _SNESContext):
raise ValueError("Don't know how to get form from %r", ctx)
prefix = pc.getOptionsPrefix()
options_prefix = prefix + self._prefix
opts = PETSc.Options()
# Handle the fine operator if type is python
if P.getType() == "python":
ictx = P.getPythonContext()
if ictx is None:
raise ValueError("No context found on matrix")
if not isinstance(ictx, ImplicitMatrixContext):
raise ValueError("Don't know how to get form from %r", ictx)
fine_operator = ictx.a
fine_bcs = ictx.row_bcs
if fine_bcs != ictx.col_bcs:
raise NotImplementedError("Row and column bcs must match")
fine_mat_type = opts.getString(options_prefix + "mat_type",
parameters["default_matrix_type"])
self.fine_op = allocate_matrix(fine_operator,
bcs=fine_bcs,
form_compiler_parameters=fcp,
mat_type=fine_mat_type,
options_prefix=options_prefix)
self._assemble_fine_op = create_assembly_callable(
fine_operator,
tensor=self.fine_op,
bcs=fine_bcs,
form_compiler_parameters=fcp,
mat_type=fine_mat_type)
self._assemble_fine_op()
fine_petscmat = self.fine_op.petscmat
else:
fine_petscmat = P
# Transfer fine operator null space
fine_petscmat.setNullSpace(P.getNullSpace())
fine_transpose_nullspace = P.getTransposeNullSpace()
if fine_transpose_nullspace.handle != 0:
fine_petscmat.setTransposeNullSpace(fine_transpose_nullspace)
# Handle the coarse operator
coarse_options_prefix = options_prefix + "mg_coarse"
coarse_mat_type = opts.getString(coarse_options_prefix + "mat_type",
parameters["default_matrix_type"])
get_coarse_space = appctx.get("get_coarse_space", None)
if not get_coarse_space:
raise ValueError(
"Need to provide a callback which provides the coarse space.")
coarse_space = get_coarse_space()
get_coarse_operator = appctx.get("get_coarse_operator", None)
if not get_coarse_operator:
raise ValueError(
"Need to provide a callback which provides the coarse operator."
)
coarse_operator = get_coarse_operator()
coarse_space_bcs = appctx.get("coarse_space_bcs", None)
# These should be callbacks which return the relevant nullspaces
get_coarse_nullspace = appctx.get("get_coarse_op_nullspace", None)
get_coarse_transpose_nullspace = appctx.get(
"get_coarse_op_transpose_nullspace", None)
self.coarse_op = allocate_matrix(coarse_operator,
bcs=coarse_space_bcs,
form_compiler_parameters=fcp,
mat_type=coarse_mat_type,
options_prefix=coarse_options_prefix)
self._assemble_coarse_op = create_assembly_callable(
coarse_operator,
tensor=self.coarse_op,
bcs=coarse_space_bcs,
form_compiler_parameters=fcp)
self._assemble_coarse_op()
coarse_opmat = self.coarse_op.petscmat
# Set nullspace if provided
if get_coarse_nullspace:
nsp = get_coarse_nullspace()
coarse_opmat.setNullSpace(nsp.nullspace(comm=pc.comm))
if get_coarse_transpose_nullspace:
tnsp = get_coarse_transpose_nullspace()
coarse_opmat.setTransposeNullSpace(tnsp.nullspace(comm=pc.comm))
interp_petscmat = appctx.get("interpolation_matrix", None)
if interp_petscmat is None:
# Create interpolation matrix from coarse space to fine space
fine_space = ctx.J.arguments()[0].function_space()
interpolator = Interpolator(TestFunction(coarse_space), fine_space)
interpolation_matrix = interpolator.callable()
interp_petscmat = interpolation_matrix.handle
# We set up a PCMG object that uses the constructed interpolation
# matrix to generate the restriction/prolongation operators.
# This is a two-level multigrid preconditioner.
pcmg = PETSc.PC().create(comm=pc.comm)
pcmg.incrementTabLevel(1, parent=pc)
pcmg.setType(pc.Type.MG)
pcmg.setOptionsPrefix(options_prefix)
pcmg.setMGLevels(2)
pcmg.setMGCycleType(pc.MGCycleType.V)
pcmg.setMGInterpolation(1, interp_petscmat)
pcmg.setOperators(A=fine_petscmat, P=fine_petscmat)
# Create new appctx
self._ctx_ref = self.new_snes_ctx(pc, coarse_operator,
coarse_space_bcs, coarse_mat_type,
fcp)
coarse_solver = pcmg.getMGCoarseSolve()
coarse_solver.setOperators(A=coarse_opmat, P=coarse_opmat)
# coarse space dm
coarse_dm = coarse_space.dm
coarse_solver.setDM(coarse_dm)
coarse_solver.setDMActive(False)
pcmg.setDM(coarse_dm)
pcmg.setFromOptions()
self.pc = pcmg
self._dm = coarse_dm
with dmhooks.add_hooks(coarse_dm,
self,
appctx=self._ctx_ref,
save=False):
coarse_solver.setFromOptions()
bytes = info["memory"]
nz = info["nz_used"]
return rows, cols, bytes, nz
first = True
workaround_flop_counting_bug = True
if workaround_flop_counting_bug:
# Prior to c91eb2e, PyOP2 overcounted flops by this factor
scaling = 3.0
else:
scaling = 1.0
sizeof_int = PETSc.IntType().dtype.itemsize
sizeof_double = PETSc.ScalarType().dtype.itemsize
PETSc.Sys.Print("Int Type has %d bytes, Scalar Type has %d bytes" %
(sizeof_int, sizeof_double))
def aij_matvec_bytes(rows, cols, nz, rbs=1, cbs=1):
# Gropp et al. 2000
if rbs == cbs and rbs != 1:
pass
else:
rbs = 1
cbs = 1
return ((cols + rows)*sizeof_double # Vec read/write
+ (rows / rbs)*sizeof_int # Row pointer
def __init__(self, Q, fixed_bids=[], extra_bcs=[], direct_solve=False):
if isinstance(extra_bcs, fd.DirichletBC):
extra_bcs = [extra_bcs]
self.direct_solve = direct_solve
self.fixed_bids = fixed_bids # fixed parts of bdry
self.params = self.get_params() # solver parameters
self.Q = Q
"""
V: type fd.FunctionSpace
I: type PETSc.Mat, interpolation matrix between V and ControlSpace
"""
(V, I_interp) = Q.get_space_for_inner()
free_bids = list(V.mesh().topology.exterior_facets.unique_markers)
self.free_bids = [int(i) for i in free_bids] # np.int->int
for bid in self.fixed_bids:
self.free_bids.remove(bid)
# Some weak forms have a nullspace. We import the nullspace if no
# parts of the bdry are fixed (we assume that a DirichletBC is
# sufficient to empty the nullspace).
nsp = None
if len(self.fixed_bids) == 0:
nsp_functions = self.get_nullspace(V)
if nsp_functions is not None:
nsp = fd.VectorSpaceBasis(nsp_functions)
nsp.orthonormalize()
bcs = []
# impose homogeneous Dirichlet bcs on bdry parts that are fixed.
if len(self.fixed_bids) > 0:
dim = V.value_size
if dim == 2:
zerovector = fd.Constant((0, 0))
elif dim == 3:
zerovector = fd.Constant((0, 0, 0))
else:
raise NotImplementedError
bcs.append(fd.DirichletBC(V, zerovector, self.fixed_bids))
if len(extra_bcs) > 0:
bcs += extra_bcs
if len(bcs) == 0:
bcs = None
a = self.get_weak_form(V)
A = fd.assemble(a, mat_type='aij', bcs=bcs)
ls = fd.LinearSolver(A,
solver_parameters=self.params,
nullspace=nsp,
transpose_nullspace=nsp)
self.ls = ls
self.A = A.petscmat
self.interpolated = False
# If the matrix I is passed, replace A with transpose(I)*A*I
# and set up a ksp solver for self.riesz_map
if I_interp is not None:
self.interpolated = True
ITAI = self.A.PtAP(I_interp)
from firedrake.petsc import PETSc
import numpy as np
zero_rows = []
# if there are zero-rows, replace them with rows that
# have 1 on the diagonal entry
for row in range(*ITAI.getOwnershipRange()):
(cols, vals) = ITAI.getRow(row)
valnorm = np.linalg.norm(vals)
if valnorm < 1e-13:
zero_rows.append(row)
for row in zero_rows:
ITAI.setValue(row, row, 1.0)
ITAI.assemble()
# overwrite the self.A created by get_impl
self.A = ITAI
# create ksp solver for self.riesz_map
Aksp = PETSc.KSP().create(comm=V.comm)
Aksp.setOperators(ITAI)
Aksp.setType("preonly")
Aksp.pc.setType("cholesky")
Aksp.pc.setFactorSolverType("mumps")
Aksp.setFromOptions()
Aksp.setUp()
self.Aksp = Aksp
def __init__(self, dm, section):
super(Halo, self).__init__()
# Use a DM to create the halo SFs
self.dm = PETSc.DMShell().create(dm.comm)
self.dm.setPointSF(dm.getPointSF())
self.dm.setDefaultSection(section)
def __init__(self, problem, **kwargs):
"""
:arg problem: A :class:`NonlinearVariationalProblem` to solve.
:kwarg nullspace: an optional :class:`.VectorSpaceBasis` (or
:class:`.MixedVectorSpaceBasis`) spanning the null
space of the operator.
:kwarg solver_parameters: Solver parameters to pass to PETSc.
This should be a dict mapping PETSc options to values. For
example, to set the nonlinear solver type to just use a linear
solver:
:kwarg options_prefix: an optional prefix used to distinguish
PETSc options. If not provided a unique prefix will be
created. Use this option if you want to pass options
to the solver from the command line in addition to
through the ``solver_parameters`` dict.
.. code-block:: python
{'snes_type': 'ksponly'}
PETSc flag options should be specified with `bool` values. For example:
.. code-block:: python
{'snes_monitor': True}
"""
parameters, nullspace, options_prefix = solving_utils._extract_kwargs(
**kwargs)
# Do this first so __del__ doesn't barf horribly if we get an
# error in __init__
if options_prefix is not None:
self._opt_prefix = options_prefix
self._auto_prefix = False
else:
self._opt_prefix = 'firedrake_snes_%d_' % NonlinearVariationalSolver._id
self._auto_prefix = True
NonlinearVariationalSolver._id += 1
assert isinstance(problem, NonlinearVariationalProblem)
ctx = solving_utils._SNESContext(problem)
self.snes = PETSc.SNES().create()
self.snes.setOptionsPrefix(self._opt_prefix)
# Mixed problem, use jacobi pc if user has not supplied one.
if ctx.is_mixed:
parameters.setdefault('pc_type', 'jacobi')
# Allow command-line arguments to override dict parameters
opts = PETSc.Options()
for k, v in opts.getAll().iteritems():
if k.startswith(self._opt_prefix):
parameters[k[len(self._opt_prefix):]] = v
self._problem = problem
self._ctx = ctx
self.snes.setDM(problem.dm)
ctx.set_function(self.snes)
ctx.set_jacobian(self.snes)
ctx.set_nullspace(nullspace,
problem.J.arguments()[0].function_space()._ises)
self.parameters = parameters