def test_vector_assemble_matrix_interior():
mesh = create_unit_square(MPI.COMM_WORLD, 3, 3)
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(ufl.inner(ufl.jump(u), ufl.jump(v)) * ufl.dS)
A = assemble_matrix(a)
A.assemble()
def xtest_tetrahedron_integral(space_type, space_order):
domain = ufl.Mesh(ufl.VectorElement("Lagrange", "tetrahedron", 1))
temp_points = np.array([[-1., 0., -1.], [0., 0., 0.], [1., 0., 1.],
[0., 1., 0.], [0., 0., 1.]])
for repeat in range(10):
order = [i for i, j in enumerate(temp_points)]
shuffle(order)
points = np.zeros(temp_points.shape)
for i, j in enumerate(order):
points[j] = temp_points[i]
cells = []
for cell in [[0, 1, 3, 4], [1, 2, 3, 4]]:
# Randomly number the cell
cell_order = list(range(4))
shuffle(cell_order)
cells.append([order[cell[i]] for i in cell_order])
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
V = FunctionSpace(mesh, (space_type, space_order))
Vvec = VectorFunctionSpace(mesh, ("P", 1))
dofs = [i for i in V.dofmap.cell_dofs(0) if i in V.dofmap.cell_dofs(1)]
for d in dofs:
v = Function(V)
v.vector[:] = [1 if i == d else 0 for i in range(V.dim)]
if space_type in ["RT", "BDM"]:
# Hdiv
def normal(x):
values = np.zeros((3, x.shape[1]))
values[0] = [1 for i in values[0]]
return values
n = Function(Vvec)
n.interpolate(normal)
form = ufl.inner(ufl.jump(v), n) * ufl.dS
elif space_type in ["N1curl", "N2curl"]:
# Hcurl
def tangent1(x):
values = np.zeros((3, x.shape[1]))
values[1] = [1 for i in values[1]]
return values
def tangent2(x):
values = np.zeros((3, x.shape[1]))
values[2] = [1 for i in values[2]]
return values
t1 = Function(Vvec)
t1.interpolate(tangent1)
t2 = Function(Vvec)
t2.interpolate(tangent1)
form = ufl.inner(ufl.jump(v), t1) * ufl.dS
form += ufl.inner(ufl.jump(v), t2) * ufl.dS
else:
form = ufl.jump(v) * ufl.dS
value = fem.assemble_scalar(form)
assert np.isclose(value, 0)
def xtest_quadrilateral_integral(space_type, space_order):
domain = ufl.Mesh(ufl.VectorElement("Lagrange", "quadrilateral", 1))
temp_points = np.array([[-1., -1.], [0., 0.], [1., 0.], [-1., 1.],
[0., 1.], [2., 2.]])
for repeat in range(10):
order = [i for i, j in enumerate(temp_points)]
shuffle(order)
points = np.zeros(temp_points.shape)
for i, j in enumerate(order):
points[j] = temp_points[i]
connections = {0: [1, 2], 1: [0, 3], 2: [0, 3], 3: [1, 2]}
cells = []
for cell in [[0, 1, 3, 4], [1, 2, 4, 5]]:
# Randomly number the cell
start = choice(range(4))
cell_order = [start]
for i in range(2):
diff = choice([
i for i in connections[start] if i not in cell_order
]) - cell_order[0]
cell_order += [c + diff for c in cell_order]
cells.append([order[cell[i]] for i in cell_order])
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
V = FunctionSpace(mesh, (space_type, space_order))
Vvec = VectorFunctionSpace(mesh, ("P", 1))
dofs = [i for i in V.dofmap.cell_dofs(0) if i in V.dofmap.cell_dofs(1)]
for d in dofs:
v = Function(V)
v.vector[:] = [1 if i == d else 0 for i in range(V.dim)]
if space_type in ["RTCF"]:
# Hdiv
def normal(x):
values = np.zeros((2, x.shape[1]))
values[0] = [1 for i in values[0]]
return values
n = Function(Vvec)
n.interpolate(normal)
form = ufl.inner(ufl.jump(v), n) * ufl.dS
elif space_type in ["RTCE"]:
# Hcurl
def tangent(x):
values = np.zeros((2, x.shape[1]))
values[1] = [1 for i in values[1]]
return values
t = Function(Vvec)
t.interpolate(tangent)
form = ufl.inner(ufl.jump(v), t) * ufl.dS
else:
form = ufl.jump(v) * ufl.dS
value = fem.assemble_scalar(form)
assert np.isclose(value, 0)
def test_vector_assemble_matrix_interior():
mesh = dolfinx.UnitSquareMesh(MPI.COMM_WORLD, 3, 3)
V = dolfinx.VectorFunctionSpace(mesh, ("CG", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = ufl.inner(ufl.jump(u), ufl.jump(v)) * ufl.dS
A = dolfinx.fem.assemble_matrix(a)
A.assemble()
def test_integral(cell_type, space_type, space_order):
if cell_type == "hexahedron" and space_order >= 3:
pytest.skip("Skipping expensive test on hexahedron")
random.seed(4)
for repeat in range(10):
mesh = random_evaluation_mesh(cell_type)
tdim = mesh.topology.dim
V = FunctionSpace(mesh, (space_type, space_order))
Vvec = VectorFunctionSpace(mesh, ("P", 1))
dofs = [i for i in V.dofmap.cell_dofs(0) if i in V.dofmap.cell_dofs(1)]
for d in dofs:
v = Function(V)
v.vector[:] = [
1 if i == d else 0 for i, _ in enumerate(v.vector[:])
]
if space_type in ["RT", "BDM", "RTCF", "NCF", "BDMCF", "AAF"]:
# Hdiv
def normal(x):
values = np.zeros((tdim, x.shape[1]))
values[0] = [1 for i in values[0]]
return values
n = Function(Vvec)
n.interpolate(normal)
_form = ufl.inner(ufl.jump(v), n) * ufl.dS
elif space_type in [
"N1curl", "N2curl", "RTCE", "NCE", "BDMCE", "AAE"
]:
# Hcurl
def tangent(x):
values = np.zeros((tdim, x.shape[1]))
values[1] = [1 for i in values[1]]
return values
t = Function(Vvec)
t.interpolate(tangent)
_form = ufl.inner(ufl.jump(v), t) * ufl.dS
if tdim == 3:
def tangent2(x):
values = np.zeros((3, x.shape[1]))
values[2] = [1 for i in values[2]]
return values
t2 = Function(Vvec)
t2.interpolate(tangent2)
_form += ufl.inner(ufl.jump(v), t2) * ufl.dS
else:
_form = ufl.jump(v) * ufl.dS
value = assemble_scalar(form(_form))
assert np.isclose(value, 0)
def test_integral(cell_type, space_type, space_order):
# TODO: Fix jump integrals in FFC by passing in full info for both cells, then re-enable these tests
pytest.xfail()
random.seed(4)
for repeat in range(10):
mesh = random_evaluation_mesh(cell_type)
tdim = mesh.topology.dim
V = FunctionSpace(mesh, (space_type, space_order))
Vvec = VectorFunctionSpace(mesh, ("P", 1))
dofs = [i for i in V.dofmap.cell_dofs(0) if i in V.dofmap.cell_dofs(1)]
for d in dofs:
v = Function(V)
v.vector[:] = [1 if i == d else 0 for i in range(V.dim)]
if space_type in ["RT", "BDM", "RTCF", "NCF"]:
# Hdiv
def normal(x):
values = np.zeros((tdim, x.shape[1]))
values[0] = [1 for i in values[0]]
return values
n = Function(Vvec)
n.interpolate(normal)
form = ufl.inner(ufl.jump(v), n) * ufl.dS
elif space_type in ["N1curl", "N2curl", "RTCE", "NCE"]:
# Hcurl
def tangent(x):
values = np.zeros((tdim, x.shape[1]))
values[1] = [1 for i in values[1]]
return values
t = Function(Vvec)
t.interpolate(tangent)
form = ufl.inner(ufl.jump(v), t) * ufl.dS
if tdim == 3:
def tangent2(x):
values = np.zeros((3, x.shape[1]))
values[2] = [1 for i in values[2]]
return values
t2 = Function(Vvec)
t2.interpolate(tangent2)
form += ufl.inner(ufl.jump(v), t2) * ufl.dS
else:
form = ufl.jump(v) * ufl.dS
value = fem.assemble_scalar(form)
assert np.isclose(value, 0)
def test_interior_facet_integral(mode):
cell = ufl.triangle
element = ufl.FiniteElement("Lagrange", cell, 1)
u, v = ufl.TrialFunction(element), ufl.TestFunction(element)
a0 = ufl.inner(ufl.jump(ufl.grad(u)), ufl.jump(ufl.grad(v))) * ufl.dS
forms = [a0]
compiled_forms, module = ffc.codegeneration.jit.compile_forms(
forms, parameters={'scalar_type': mode})
for f, compiled_f in zip(forms, compiled_forms):
assert compiled_f.rank == len(f.arguments())
ffi = cffi.FFI()
form0 = compiled_forms[0][0]
ids = np.zeros(form0.num_interior_facet_integrals, dtype=np.int32)
form0.get_interior_facet_integral_ids(ffi.cast('int *', ids.ctypes.data))
assert ids[0] == -1
ffi = cffi.FFI()
c_type, np_type = float_to_type(mode)
integral0 = form0.create_interior_facet_integral(-1)
A = np.zeros((6, 6), dtype=np_type)
w = np.array([], dtype=np_type)
c = np.array([], dtype=np.float64)
facets = np.array([0, 2], dtype=np.int32)
orients = np.array([1, 1], dtype=np.int32)
coords = np.array(
[[0.0, 0.0, 1.0, 0.0, 0.0, 1.0], [1.0, 0.0, 0.0, 1.0, 1.0, 1.0]],
dtype=np.float64)
integral0.tabulate_tensor(ffi.cast('{} *'.format(c_type), A.ctypes.data),
ffi.cast('{} *'.format(c_type), w.ctypes.data),
ffi.cast('{} *'.format(c_type), c.ctypes.data),
ffi.cast('double *', coords.ctypes.data),
ffi.cast('int *', facets.ctypes.data),
ffi.cast('int *', orients.ctypes.data))
print(A)
def test_interior_facet_integral(mode, compile_args):
cell = ufl.triangle
element = ufl.FiniteElement("Lagrange", cell, 1)
u, v = ufl.TrialFunction(element), ufl.TestFunction(element)
a0 = ufl.inner(ufl.jump(ufl.grad(u)), ufl.jump(ufl.grad(v))) * ufl.dS
forms = [a0]
compiled_forms, module, code = ffcx.codegeneration.jit.compile_forms(
forms,
parameters={'scalar_type': mode},
cffi_extra_compile_args=compile_args)
for f, compiled_f in zip(forms, compiled_forms):
assert compiled_f.rank == len(f.arguments())
ffi = cffi.FFI()
form0 = compiled_forms[0]
ffi = cffi.FFI()
c_type, np_type = float_to_type(mode)
integral0 = form0.integrals(module.lib.interior_facet)[0]
A = np.zeros((6, 6), dtype=np_type)
w = np.array([], dtype=np_type)
c = np.array([], dtype=np.float64)
facets = np.array([0, 2], dtype=np.intc)
perms = np.array([0, 1], dtype=np.uint8)
coords = np.array([[0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0, 0.0],
[1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0, 1.0, 0.0]],
dtype=np.float64)
integral0.tabulate_tensor(ffi.cast('{} *'.format(c_type), A.ctypes.data),
ffi.cast('{} *'.format(c_type), w.ctypes.data),
ffi.cast('{} *'.format(c_type), c.ctypes.data),
ffi.cast('double *', coords.ctypes.data),
ffi.cast('int *', facets.ctypes.data),
ffi.cast('uint8_t *', perms.ctypes.data))
def b(tau_S, v):
n = FacetNormal(mesh)
return inner(tau_S, grad(grad(v))) * dx \
- ufl.dot(ufl.dot(tau_S('+'), n('+')), n('+')) * jump(grad(v), n) * dS \
- ufl.dot(ufl.dot(tau_S, n), n) * ufl.dot(grad(v), n) * ds
)
source = lambda u1, u2, u3, v: -1 / spiral_eps * u1 * (1 - u2) * (u3 - ustar(v)
)
xForm = inner(diffusiveFlux(u, grad(u)), grad(phi)) * dx
xForm += ufl.conditional(uh_n < ustar(vh_n), source(u, uh_n, uh_n, vh_n),
source(uh_n, u, uh_n, vh_n)) * phi * dx
# <markdowncell>
# To handle the possible non conforming intersection we add DG type
# skeleton terms:
# <codecell>
penalty = 5 * (order * (order + 1)) * spiral_D
if nonConforming:
xForm -= ( inner( outer(jump(u), n('+')), avg(diffusiveFlux(u,grad(phi)))) +\
inner( avg(diffusiveFlux(u,grad(u))), outer(jump(phi), n('+'))) ) * dS
xForm += penalty / hS * inner(jump(u), jump(phi)) * dS
# <markdowncell>
# After adding the time discretization (a simple backward Euler scheme),
# the model is now completely implemented and the scheme can be created:
# <codecell>
form = (inner(u, phi) - inner(uh_n, phi)) * dx + dt * xForm
solverParameters =\
{"newton.tolerance": 1e-8,
"newton.linear.tolerance": 1e-12,
"newton.linear.preconditioning.method": "amg-ilu",
"newton.linear.maxiterations":1000,
"newton.verbose": False,
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
drho_integral = self.tdrho * self.wrho * self.dx
dU_integral = self.tdU * self.wU * self.dx
self.A = fe.assemble(drho_integral + dU_integral)
# if self.solver_type == 'lu':
# self.solver = fe.LUSolver(
# self.A,
# method=self.solver_type
# )
# self.solver.parameters['reuse_factorization'] = True
# else:
# self.solver = fe.KrylovSolver(
# self.A,
# self.solver_type,
# self.preconditioner_type
# )
self.dsol = Function(self.VS)
self.drho, self.dU = self.dsol.sub(0), self.dsol.sub(1)
#
# assemble RHS (has to be done for each time point)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
self.v = -ufl.grad(self.V(self.iU, self.irho))
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = (self.vn('+') * self.irho('+') -
self.vn('-') * self.irho('-'))
self.rho_flux_jump = -self.facet_flux * ufl.jump(
self.wrho) * self.dS
self.rho_grad_move = ufl.dot(self.flux, ufl.grad(
self.wrho)) * self.dx
self.rho_penalty = -(
(self.rhopen * self.degree**2 / self.havg) * ufl.dot(
ufl.jump(self.irho, self.n), ufl.jump(self.wrho, self.n)) *
self.dS)
# self.facet_flux = (
# self.vn('+')*self.rho('+') - self.vn('-')*self.rho('-')
# )
# self.rho_flux_jump = -self.facet_flux*ufl.jump(self.wrho)*self.dS
# self.rho_grad_move = ufl.dot(self.flux, ufl.grad(self.wrho))*self.dx
self.jump_grhow = (
self.s2 * ufl.jump(self.wrho * ufl.grad(self.irho), self.n) *
self.dS)
self.rho_diffusion = -self.s2 * ufl.dot(ufl.grad(
self.irho), ufl.grad(self.wrho)) * self.dx
# self.rho_penalty = -(
# (self.rhopen * self.degree**2 / self.havg) *
# ufl.dot(ufl.jump(self.rho, self.n),
# ufl.jump(self.wrho, self.n)) * self.dS
# )
self.grho_penalty = -(self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) *
ufl.jump(ufl.grad(self.wrho), self.n)) *
self.dS)
self.rho_terms = (
# advection terms
self.rho_flux_jump + self.rho_grad_move +
# diffusive terms
self.rho_diffusion + self.jump_grhow +
# penalty terms (to enforce continuity)
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.gamma = self.params['gamma']
self.s = self.params['s']
self.D = self.params['D']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = -self.gamma * self.iU * self.wU * self.dx
self.U_secretion = self.s * self.irho * self.wU * self.dx
self.jump_gUw = (self.D *
ufl.jump(self.wU * ufl.grad(self.iU), self.n) *
self.dS)
self.U_diffusion = -self.D * ufl.dot(ufl.grad(self.iU),
ufl.grad(self.wU)) * self.dx
self.U_penalty = -(
(self.Upen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.iU, self.n), ufl.jump(self.wU, self.n)) *
self.dS)
self.gU_penalty = -(self.gUpen * self.degree**2 *
(ufl.jump(ufl.grad(self.iU), self.n) *
ufl.jump(ufl.grad(self.wU), self.n)) *
self.dS)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
drho_integral = vectotal([
tdrho * wrho * self.dx
for tdrho, wrho in zip(self.tdrhos, self.wrhos)
])
dU_integral = vectotal(
[tdU * wU * self.dx for tdU, wU in zip(self.tdUs, self.wUs)])
self.A = fe.assemble(drho_integral + dU_integral)
for bc in self.bcs:
bc.apply(self.A)
self.dsol = Function(self.VS)
self.drhos = self.dsol.split()[:2**self.dim]
self.dUs = self.dsol.split()[2**self.dim:]
#
# These are the values of rho and U themselves (not their
# symmetrized versions) on all subdomains of the original
# domain.
#
if not hasattr(self, 'rhosds'):
self.rhosds = matmul(self.eomat, self.irhos)
# self.Usds is a list of nligands lists. Sublist i is of
# length 2**dim and lists the value of ligand i on each of the
# 2**dim subdomains.
#
if not hasattr(self, 'Usds'):
self.Usds = [
matmul(self.eomat,
self.iUs[i * 2**self.dim:(i + 1) * 2**self.dim])
for i in range(self.nligands)
]
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
#
# Compute fluxes on subdomains.
# Vsds is a list of length 2**dim, the value of V on each
# subdomain.
#
self.Vsds = []
for Usd, rhosd in zip(zip(*self.Usds), self.rhosds):
self.Vsds.append(self.V(Usd, rhosd))
#
# I may need to adjust the signs of the subdomain vs by
# the symmetries of the combinations
#
self.vsds = [
-ufl.grad(Vsd) - (self.s2 * ufl.grad(rhosd) /
ufl.max_value(rhosd, self.rho_min))
for Vsd, rhosd in zip(self.Vsds, self.rhosds)
]
self.fluxsds = [
vsd * rhosd for vsd, rhosd in zip(self.vsds, self.rhosds)
]
self.vnsds = [
ufl.max_value(ufl.dot(vsd, self.n), 0) for vsd in self.vsds
]
self.facet_fluxsds = [
(vnsd('+') * ufl.max_value(rhosd('+'), 0.0) -
vnsd('-') * ufl.max_value(rhosd('-'), 0.0))
for vnsd, rhosd in zip(self.vnsds, self.rhosds)
]
#
# Now combine the subdomain fluxes to get the fluxes for
# the symmetrized functions
#
self.fluxs = matmul((2.0**-self.dim) * self.eomat, self.fluxsds)
self.facet_fluxs = matmul((2.0**-self.dim) * self.eomat,
self.facet_fluxsds)
self.rho_flux_jump = vectotal([
-facet_flux * ufl.jump(wrho) * self.dS
for facet_flux, wrho in zip(self.facet_fluxs, self.wrhos)
])
self.rho_grad_move = vectotal([
ufl.dot(flux, ufl.grad(wrho)) * self.dx
for flux, wrho in zip(self.fluxs, self.wrhos)
])
self.rho_penalty = vectotal([
-(self.rhopen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(rho, self.n), ufl.jump(wrho, self.n)) *
self.dS for rho, wrho in zip(self.irhos, self.wrhos)
])
self.grho_penalty = vectotal([
-self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(rho), self.n) *
ufl.jump(ufl.grad(wrho), self.n)) * self.dS
for rho, wrho in zip(self.irhos, self.wrhos)
])
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = 0.0
self.U_secretion = 0.0
self.jump_gUw = 0.0
self.U_diffusion = 0.0
self.U_penalty = 0.0
self.gU_penalty = 0.0
for j, lig in enumerate(self.ligands.ligands()):
sl = slice(j * 2**self.dim, (j + 1) * 2**self.dim)
self.U_decay += -lig.gamma * sum([
iUi * wUi * self.dx
for iUi, wUi in zip(self.iUs[sl], self.wUs[sl])
])
self.U_secretion += lig.s * sum([
rho * wU * self.dx
for rho, wU in zip(self.irhos, self.wUs[sl])
])
self.jump_gUw += lig.D * sum([
ufl.jump(wU * ufl.grad(U), self.n) * self.dS
for wU, U in zip(self.wUs[sl], self.iUs[sl])
])
self.U_diffusion += -lig.D * sum([
ufl.dot(ufl.grad(U), ufl.grad(wU)) * self.dx
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.U_penalty += -self.Upen * self.degree**2 * sum([
(1.0 / self.havg) * ufl.dot(ufl.jump(U, self.n),
ufl.jump(wU, self.n)) * self.dS
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.gU_penalty += -self.gUpen * self.degree**2 * sum([
ufl.jump(ufl.grad(U), self.n) *
ufl.jump(ufl.grad(wU), self.n) * self.dS
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
# Error estimator
# <codecell>
fvspace = dune.fem.space.finiteVolume(uh.space.grid)
estimate = fvspace.interpolate([0], name="estimate")
estimate_pm1 = fvspace.interpolate([0], name="estimate_pm1")
chi = ufl.TestFunction(fvspace)
hT = ufl.MaxCellEdgeLength(fvspace.cell())
he = ufl.MaxFacetEdgeLength(fvspace.cell())('+')
n = ufl.FacetNormal(fvspace.cell())
residual = (u-uh_n)/dt - div(diffusiveFlux(u,grad(u))) + source(u,u,u,vh)
estimator_ufl = hT**2 * residual**2 * chi * dx +\
he * inner( jump(diffusiveFlux(u,grad(u))), n('+'))**2 * avg(chi) * dS +\
1/he * jump(u)**2 * avg(chi) * dS
estimator = dune.fem.operator.galerkin(estimator_ufl)
# <markdowncell>
# Time loop
# <codecell>
nextSaveTime = saveInterval
count = 0
levelFunction = dune.fem.function.levelFunction(gridView)
gridView.writeVTK("spiral", pointdata=[uh,vh], number=count, celldata=[estimate,levelFunction])
count += 1
@gridFunction(gridView,name="pEstimate")
def pEstimator(e,x):
def xtest_hexahedron_integral(space_type, space_order):
domain = ufl.Mesh(ufl.VectorElement("Lagrange", "hexahedron", 1))
temp_points = np.array([[-1., 0., -1.], [0., 0., 0.], [1., 0., 1.],
[-1., 1., 1.], [0., 1., 0.], [1., 1., 1.],
[-1., 0., 0.], [0., 0., 1.], [1., 0., 2.],
[-1., 1., 2.], [0., 1., 1.], [1., 1., 2.]])
for repeat in range(10):
order = [i for i, j in enumerate(temp_points)]
shuffle(order)
points = np.zeros(temp_points.shape)
for i, j in enumerate(order):
points[j] = temp_points[i]
connections = {
0: [1, 2, 4],
1: [0, 3, 5],
2: [0, 3, 6],
3: [1, 2, 7],
4: [0, 5, 6],
5: [1, 4, 7],
6: [2, 4, 7],
7: [3, 5, 6]
}
cells = []
for cell in [[0, 1, 3, 4, 6, 7, 9, 10], [1, 2, 4, 5, 7, 8, 10, 11]]:
# Randomly number the cell
start = choice(range(8))
cell_order = [start]
for i in range(3):
diff = choice([
i for i in connections[start] if i not in cell_order
]) - cell_order[0]
cell_order += [c + diff for c in cell_order]
cells.append([order[cell[i]] for i in cell_order])
mesh = create_mesh(MPI.COMM_WORLD, cells, points, domain)
V = FunctionSpace(mesh, (space_type, space_order))
Vvec = VectorFunctionSpace(mesh, ("P", 1))
dofs = [i for i in V.dofmap.cell_dofs(0) if i in V.dofmap.cell_dofs(1)]
for d in dofs:
v = Function(V)
v.vector[:] = [
1 if i == d else 0 for i in range(v.vector.local_size)
]
if space_type in ["NCF"]:
# Hdiv
def normal(x):
values = np.zeros((3, x.shape[1]))
values[0] = [1 for i in values[0]]
return values
n = Function(Vvec)
n.interpolate(normal)
form = ufl.inner(ufl.jump(v), n) * ufl.dS
elif space_type in ["NCE"]:
# Hcurl
def tangent1(x):
values = np.zeros((3, x.shape[1]))
values[1] = [1 for i in values[1]]
return values
def tangent2(x):
values = np.zeros((3, x.shape[1]))
values[2] = [1 for i in values[2]]
return values
t1 = Function(Vvec)
t1.interpolate(tangent1)
t2 = Function(Vvec)
t2.interpolate(tangent1)
form = ufl.inner(ufl.jump(v), t1) * ufl.dS
form += ufl.inner(ufl.jump(v), t2) * ufl.dS
else:
form = ufl.jump(v) * ufl.dS
value = fem.assemble_scalar(form)
assert np.isclose(value, 0)
1) # No normal displacement for solid on left side
bcd3 = DirichletBC(W.sub(0).sub(0), Constant(0.0), boundaries,
3) # No normal displacement for solid on right side
bcd4 = DirichletBC(W.sub(0).sub(1), Constant(0.0), boundaries,
4) # No normal displacement for solid on bottom side
bcs = BlockDirichletBC([[bcd1, bcd3, bcd4], []])
a = inner(2 * mu_l * strain(u) + lmbda_l * div(u) * I, sym(grad(v))) * dx
b = inner(-alpha * p * I, sym(grad(v))) * dx
c = rho * alpha * div(u) * q * dx
d = (rho * ct * p * q * dx +
dt * dot(rho * k / vis * grad(p), grad(q)) * dx - dt *
dot(avg_w(rho * k / vis * grad(p), weight_e(k, n)), jump(q, n)) * dS -
theta * dt *
dot(avg_w(rho * k / vis * grad(q), weight_e(k, n)), jump(p, n)) * dS +
dt * penalty1 / h_avg * avg(rho) * k_e(k, n) / avg(vis) *
dot(jump(p, n), jump(q, n)) * dS -
dt * dot(rho * k / vis * grad(p), q * n) * ds(2) -
dt * dot(rho * k / vis * grad(q), p * n) * ds(2) + dt *
(penalty2 / h * rho / vis * dot(dot(n, k), n) * dot(p * n, q * n)) *
ds(2))
lhs = [[a, b], [c, d]]
f_u = (inner(f, v) * dx + dot(f_stress_y * n, v) * ds(2))
f_p = (
rho * alpha * div(u0) * q * dx + rho * ct * p0 * q * dx +
def setup_problem(self, t, debug=False):
self.set_time(t)
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
logVARIABLE('making matrix A')
self.drho_integral = sum([
tdrho * wrho * self.dx
for tdrho, wrho in zip(self.tdrhos, self.wrhos)
])
self.dU_integral = sum(
[tdU * wU * self.dx for tdU, wU in zip(self.tdUs, self.wUs)])
logVARIABLE('assembling A')
self.A = fe.PETScMatrix()
logVARIABLE('self.A', self.A)
fe.assemble(self.drho_integral + self.dU_integral, tensor=self.A)
logVARIABLE('A assembled. Applying BCs')
pA = fe.as_backend_type(self.A).mat()
Adiag = pA.getDiagonal()
logVARIABLE('Adiag.array', Adiag.array)
# self.A = fe.assemble(self.drho_integral + self.dU_integral +
# self.dP_integral)
for bc in self.bcs:
bc.apply(self.A)
Adiag = pA.getDiagonal()
logVARIABLE('Adiag.array', Adiag.array)
self.dsol = Function(self.VS)
dsolsplit = self.dsol.split()
self.drhos, self.dUs = (dsolsplit[:2**self.dim],
dsolsplit[2**self.dim:])
#
# assemble RHS (for each time point, but compile only once)
#
#
# These are the values of rho and U themselves (not their
# symmetrized versions) on all subdomains of the original
# domain.
#
if not hasattr(self, 'rhosds'):
self.rhosds = matmul(self.eomat, self.irhos)
# self.Usds is a list of nligands lists. Sublist i is of
# length 2**dim and lists the value of ligand i on each of the
# 2**dim subdomains.
#
if not hasattr(self, 'Usds'):
self.Usds = [
matmul(self.eomat,
self.iUs[i * 2**self.dim:(i + 1) * 2**self.dim])
for i in range(self.nligands)
]
if not hasattr(self, 'rho_terms'):
logVARIABLE('making rho_terms')
self.sigma = self.iparams['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rhomin = self.iparams['rhomin']
self.rhopen = self.iparams['rhopen']
self.grhopen = self.iparams['grhopen']
#
# Compute fluxes on subdomains.
# Vsds is a list of length 2**dim, the value of V on each
# subdomain.
#
self.Vsds = []
for Usd, rhosd in zip(zip(*self.Usds), self.rhosds):
self.Vsds.append(self.V(Usd, ufl.max_value(rhosd,
self.rhomin)))
self.vsds = [
-ufl.grad(Vsd) -
(self.s2 * ufl.grad(rhosd) / ufl.max_value(rhosd, self.rhomin))
for Vsd, rhosd in zip(self.Vsds, self.rhosds)
]
self.fluxsds = [
vsd * rhosd for vsd, rhosd in zip(self.vsds, self.rhosds)
]
self.vnsds = [
ufl.max_value(ufl.dot(vsd, self.n), 0) for vsd in self.vsds
]
self.facet_fluxsds = [
(vnsd('+') * ufl.max_value(rhosd('+'), 0.0) -
vnsd('-') * ufl.max_value(rhosd('-'), 0.0))
for vnsd, rhosd in zip(self.vnsds, self.rhosds)
]
#
# Now combine the subdomain fluxes to get the fluxes for
# the symmetrized functions
#
self.fluxs = matmul((2.0**-self.dim) * self.eomat, self.fluxsds)
self.facet_fluxs = matmul((2.0**-self.dim) * self.eomat,
self.facet_fluxsds)
self.rho_flux_jump = sum([
-facet_flux * ufl.jump(wrho) * self.dS
for facet_flux, wrho in zip(self.facet_fluxs, self.wrhos)
])
self.rho_grad_move = sum([
ufl.dot(flux, ufl.grad(wrho)) * self.dx
for flux, wrho in zip(self.fluxs, self.wrhos)
])
self.rho_penalty = sum([
-(self.degree**2 / self.havg) *
ufl.dot(ufl.jump(rho, self.n),
ufl.jump(self.rhopen * wrho, self.n)) * self.dS
for rho, wrho in zip(self.irhos, self.wrhos)
])
self.grho_penalty = sum([
self.degree**2 *
(ufl.jump(ufl.grad(rho), self.n) *
ufl.jump(ufl.grad(-self.grhopen * wrho), self.n)) * self.dS
for rho, wrho in zip(self.irhos, self.wrhos)
])
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
logVARIABLE('rho_terms made')
if not hasattr(self, 'U_terms'):
logVARIABLE('making U_terms')
self.Umin = self.iparams['Umin']
self.Upen = self.iparams['Upen']
self.gUpen = self.iparams['gUpen']
self.U_decay = 0.0
self.U_secretion = 0.0
self.jump_gUw = 0.0
self.U_diffusion = 0.0
self.U_penalty = 0.0
self.gU_penalty = 0.0
for j, lig in enumerate(self.iligands.ligands()):
sl = slice(j * 2**self.dim, (j + 1) * 2**self.dim)
self.U_decay += sum([
-lig.gamma * iUi * wUi * self.dx
for iUi, wUi in zip(self.iUs[sl], self.wUs[sl])
])
self.U_secretion += sum([
lig.s * rho * wU * self.dx
for rho, wU in zip(self.irhos, self.wUs[sl])
])
self.jump_gUw += sum([
ufl.jump(lig.D * wU * ufl.grad(U), self.n) * self.dS
for wU, U in zip(self.wUs[sl], self.iUs[sl])
])
self.U_diffusion += sum([
-lig.D * ufl.dot(ufl.grad(U), ufl.grad(wU)) * self.dx
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.U_penalty += sum([
(-self.degree**2 / self.havg) *
ufl.dot(ufl.jump(U, self.n),
ufl.jump(self.Upen * wU, self.n)) * self.dS
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.gU_penalty += sum([
-self.degree**2 * ufl.jump(ufl.grad(U), self.n) *
ufl.jump(ufl.grad(self.gUpen * wU), self.n) * self.dS
for U, wU in zip(self.iUs[sl], self.wUs[sl])
])
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
logVARIABLE('U_terms made')
if not hasattr(self, 'all_terms'):
logVARIABLE('making all_terms')
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
logVARIABLE('making J_terms')
self.J_terms = fe.derivative(self.all_terms, self.sol)
def setup_problem(self, t, debug=False):
self.set_time(t)
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
self.drho_integral = self.tdrho * self.wrho * self.dx
self.dU_integral = sum([
tdUi * wUi * self.dx for tdUi, wUi in zip(self.tdUs, self.wUs)
])
logVARIABLE('assembling A')
self.A = PETScMatrix()
logVARIABLE('self.A', self.A)
fe.assemble(self.drho_integral + self.dU_integral, tensor=self.A)
logVARIABLE('A assembled. Applying BCs')
self.dsol = Function(self.VS)
dsolsplit = self.dsol.split()
self.drho, self.dUs = dsolsplit[0], dsolsplit[1:]
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.iparams['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rhomin = self.iparams['rhomin']
self.rhopen = self.iparams['rhopen']
self.grhopen = self.iparams['grhopen']
self.v = -ufl.grad(
self.V(self.iUs, ufl.max_value(self.irho, self.rhomin)) -
(self.s2 * ufl.grad(self.irho) /
ufl.max_value(self.irho, self.rhomin)))
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = ufl.jump(self.vn * ufl.max_value(self.irho, 0.0))
self.rho_flux_jump = -self.facet_flux * ufl.jump(
self.wrho) * self.dS
self.rho_grad_move = ufl.dot(self.flux, ufl.grad(
self.wrho)) * self.dx
self.rho_penalty = -(
(self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.irho, self.n),
ufl.jump(self.rhopen * self.wrho, self.n)) * self.dS)
self.grho_penalty = -(
self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) * ufl.jump(
ufl.grad(self.grhopen * self.wrho), self.n)) * self.dS)
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.Umin = self.iparams['Umin']
self.Upen = self.iparams['Upen']
self.gUpen = self.iparams['gUpen']
self.U_decay = sum([
-lig.gamma * iUi * wUi * self.dx for lig, iUi, wUi in zip(
self.iligands.ligands(), self.iUs, self.wUs)
])
self.U_secretion = sum([
lig.s * self.irho * wUi * self.dx
for lig, wUi in zip(self.iligands.ligands(), self.wUs)
])
self.jump_gUw = sum([
ufl.jump(lig.D * wUi * ufl.grad(iUi), self.n) * self.dS
for lig, wUi, iUi in zip(self.iligands.ligands(), self.wUs,
self.iUs)
])
self.U_diffusion = sum([
-lig.D * ufl.dot(ufl.grad(iUi), ufl.grad(wUi)) * self.dx
for lig, iUi, wUi in zip(self.iligands.ligands(), self.iUs,
self.wUs)
])
self.U_penalty = sum([
-(self.degree**2 / self.havg) * ufl.dot(
ufl.jump(iUi, self.n), ufl.jump(self.Upen * wUi, self.n)) *
self.dS for iUi, wUi in zip(self.iUs, self.wUs)
])
self.gU_penalty = sum([
-self.degree**2 * ufl.jump(ufl.grad(iUi), self.n) *
ufl.jump(ufl.grad(self.gUpen * wUi), self.n) * self.dS
for iUi, wUi in zip(self.iUs, self.wUs)
])
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
def model(space, epsilon, weakBnd, skeleton, useMol):
u = TrialFunction(space)
v = TestFunction(space)
n = FacetNormal(space)
he = avg(CellVolume(space)) / FacetArea(space)
hbnd = CellVolume(space) / FacetArea(space)
x = SpatialCoordinate(space)
#exact = sin(x[0]*x[1]) # atan(1*x[1])
exact = uflFunction(space.gridView,
name="exact",
order=3,
ufl=sin(x[0] * x[1]))
# diffusion factor
eps = 1 # 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(10 * space.order**2 if space.order > 0 else 1, "beta")
rhs = (-div(eps * grad(exact) - b * exact) + exact) * v * dx
aInternal = (dot(eps * grad(u) - b * u, grad(v)) + dot(u, v)) * dx
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
diffSkeleton -= eps * dot(grad(exact), n) * v * (1 - dD) * 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 = None # DirichletBC(space,exact,dD)
if space.storage[0] == "fem":
solver = {"solver": ("suitesparse", "umfpack")}
else:
solver = {
"solver": "bicgstab",
"parameters": {
"newton.linear.preconditioning.method": "jacobi",
"newton.linear.tolerance": 1e-13
}
}
if useMol:
scheme = solutionMolScheme([form == rhs, strongBC], **solver)
else:
scheme = solutionScheme([form == rhs, strongBC], **solver)
uh = space.interpolate(exact, name="solution")
A = linear(scheme)
return scheme, uh, A, exact
hbnd = CellVolume(space) / FacetArea(space)
x = SpatialCoordinate(space)
# diffusion factor
eps = Constant(0.1,"eps")
# transport direction and upwind flux
b = as_vector([1,0])
hatb = (dot(b, n) + abs(dot(b, n)))/2.0
# boundary values (for left/right boundary)
dD = conditional((1+x[0])*(1-x[0])<1e-10,1,0)
g = conditional(x[0]<0,atan(10*x[1]),0)
# penalty parameter
beta = 10*order*order
aInternal = dot(eps*grad(u) - b*u, grad(v)) * dx
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
diffSkeleton += eps*beta/hbnd*(u-g)*v*dD*ds -\
eps*dot(grad(u),n)*v*dD*ds
advSkeleton = jump(hatb*u)*jump(v)*dS
advSkeleton += ( hatb*u + (dot(b,n)-hatb)*g )*v*dD*ds
form = aInternal + diffSkeleton + advSkeleton
scheme = solutionScheme(form==0, solver="gmres",
parameters={"newton.linear.preconditioning.method":"jacobi"})
uh = space.interpolate(0, name="solution")
scheme.solve(target=uh)
uh.plot()
# <markdowncell>
def run_dg_test(mesh, V, degree):
""" Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
u, v = TrialFunction(V), TestFunction(V)
# Exact solution
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
# Coefficient
k = Function(V)
k.vector.set(2.0)
k.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Source term
f = -div(k * grad(u_exact))
# Mesh normals and element size
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2.0
# Penalty parameter
alpha = 32
dx_ = dx(metadata={"quadrature_degree": -1})
ds_ = ds(metadata={"quadrature_degree": -1})
dS_ = dS(metadata={"quadrature_degree": -1})
with common.Timer("Compile forms"):
a = inner(k * grad(u), grad(v)) * dx_ \
- k("+") * inner(avg(grad(u)), jump(v, n)) * dS_ \
- k("+") * inner(jump(u, n), avg(grad(v))) * dS_ \
+ k("+") * (alpha / h_avg) * inner(jump(u, n), jump(v, n)) * dS_ \
- inner(k * grad(u), v * n) * ds_ \
- inner(u * n, k * grad(v)) * ds_ \
+ (alpha / h) * inner(k * u, v) * ds_
L = inner(f, v) * dx_ - inner(k * u_exact * n, grad(v)) * ds_ \
+ (alpha / h) * inner(k * u_exact, v) * ds_
for integral in a.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
a)
for integral in L.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
L)
with common.Timer("Assemble vector"):
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD,
mode=PETSc.ScatterMode.REVERSE)
with common.Timer("Assemble matrix"):
A = assemble_matrix(a, [])
A.assemble()
with common.Timer("Solve"):
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
with common.Timer("Error functional compile"):
# Calculate error
M = (u_exact - uh)**2 * dx
M = fem.Form(M)
with common.Timer("Error assembly"):
error = mesh.mpi_comm().allreduce(assemble_scalar(M), op=MPI.SUM)
common.list_timings(MPI.COMM_WORLD, [common.TimingType.wall])
assert np.absolute(error) < 1.0e-14
def test_manufactured_poisson_dg(degree, filename, datadir):
""" Manufactured Poisson problem, solving u = x[component]**n, where n is the
degree of the Lagrange function space.
"""
with XDMFFile(MPI.COMM_WORLD,
os.path.join(datadir, filename),
"r",
encoding=XDMFFile.Encoding.ASCII) as xdmf:
mesh = xdmf.read_mesh(name="Grid")
V = FunctionSpace(mesh, ("DG", degree))
u, v = TrialFunction(V), TestFunction(V)
# Exact solution
x = SpatialCoordinate(mesh)
u_exact = x[1]**degree
# Coefficient
k = Function(V)
k.vector.set(2.0)
k.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
# Source term
f = -div(k * grad(u_exact))
# Mesh normals and element size
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2.0
# Penalty parameter
alpha = 32
dx_ = dx(metadata={"quadrature_degree": -1})
ds_ = ds(metadata={"quadrature_degree": -1})
dS_ = dS(metadata={"quadrature_degree": -1})
a = inner(k * grad(u), grad(v)) * dx_ \
- k("+") * inner(avg(grad(u)), jump(v, n)) * dS_ \
- k("+") * inner(jump(u, n), avg(grad(v))) * dS_ \
+ k("+") * (alpha / h_avg) * inner(jump(u, n), jump(v, n)) * dS_ \
- inner(k * grad(u), v * n) * ds_ \
- inner(u * n, k * grad(v)) * ds_ \
+ (alpha / h) * inner(k * u, v) * ds_
L = inner(f, v) * dx_ - inner(k * u_exact * n, grad(v)) * ds_ \
+ (alpha / h) * inner(k * u_exact, v) * ds_
for integral in a.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
a)
for integral in L.integrals():
integral.metadata(
)["quadrature_degree"] = ufl.algorithms.estimate_total_polynomial_degree(
L)
b = assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
A = assemble_matrix(a, [])
A.assemble()
# Create LU linear solver
solver = PETSc.KSP().create(MPI.COMM_WORLD)
solver.setType(PETSc.KSP.Type.PREONLY)
solver.getPC().setType(PETSc.PC.Type.LU)
solver.setOperators(A)
# Solve
uh = Function(V)
solver.solve(b, uh.vector)
uh.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT,
mode=PETSc.ScatterMode.FORWARD)
error = mesh.mpi_comm().allreduce(assemble_scalar((u_exact - uh)**2 * dx),
op=MPI.SUM)
assert np.absolute(error) < 1.0e-14
def 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
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)
# <markdowncell>
# Error estimator
# <codecell>
fvspace = dune.fem.space.finiteVolume(uh.space.grid)
estimate = fvspace.interpolate([0], name="estimate")
chi = ufl.TestFunction(fvspace)
hT = ufl.MaxCellEdgeLength(fvspace.cell())
he = ufl.MaxFacetEdgeLength(fvspace.cell())('+')
n = ufl.FacetNormal(fvspace.cell())
residual = (u - uh_n) / dt - div(diffusiveFlux) + source(u, u, u, vh)
estimator_ufl = hT**2 * residual**2 * chi * dx +\
he * inner( jump(diffusiveFlux), n('+'))**2 * avg(chi) * dS
estimator = dune.fem.operator.galerkin(estimator_ufl)
# <markdowncell>
# Time loop
# <codecell>
nextSaveTime = saveInterval
count = 0
levelFunction = dune.fem.function.levelFunction(gridView)
gridView.writeVTK("spiral",
pointdata=[uh, vh],
number=count,
celldata=[estimate, levelFunction])
count += 1
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
self.drho_integral = self.tdrho * self.wrho * self.dx
self.dU_integral = self.tdU * self.wU * self.dx
self.A = fe.assemble(self.drho_integral + self.dU_integral)
# if self.solver_type == 'lu':
# self.solver = fe.LUSolver(
# self.A,
# )
# self.solver.parameters['reuse_factorization'] = True
# else:
# self.solver = fe.KrylovSolver(
# self.A,
# self.solver_type,
# self.preconditioner_type
# )
# self.solver.parameters.add('linear_solver', self.solver_type)
# kparams = fe.Parameters('krylov_solver')
# kparams.add('report', True)
# kparams.add('nonzero_initial_guess', True)
# self.solver.parameters.add(kparams)
# lparams = fe.Parameters('lu_solver')
# lparams.add('report', True)
# lparams.add('reuse_factorization', True)
# lparams.add('verbose', True)
# self.solver.parameters.add(lparams)
self.dsol = Function(self.VS)
self.drho, self.dU = self.dsol.sub(0), self.dsol.sub(1)
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
self.v = -ufl.grad(self.V(self.iU, self.irho)) - (
self.s2 * ufl.grad(self.irho) /
ufl.max_value(self.irho, self.rho_min))
self.flux = self.v * self.irho
self.vn = ufl.max_value(ufl.dot(self.v, self.n), 0)
self.facet_flux = (
self.vn('+') * ufl.max_value(self.irho('+'), 0.0) -
self.vn('-') * ufl.max_value(self.irho('-'), 0.0))
self.rho_flux_jump = -self.facet_flux * ufl.jump(
self.wrho) * self.dS
self.rho_grad_move = ufl.dot(self.flux, ufl.grad(
self.wrho)) * self.dx
self.rho_penalty = -(
(self.rhopen * self.degree**2 / self.havg) * ufl.dot(
ufl.jump(self.irho, self.n), ufl.jump(self.wrho, self.n)) *
self.dS)
self.grho_penalty = -(self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(self.irho), self.n) *
ufl.jump(ufl.grad(self.wrho), self.n)) *
self.dS)
self.rho_terms = (self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.gamma = self.params['gamma']
self.s = self.params['s']
self.D = self.params['D']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = -self.gamma * self.iU * self.wU * self.dx
self.U_secretion = self.s * self.irho * self.wU * self.dx
self.jump_gUw = (self.D *
ufl.jump(self.wU * ufl.grad(self.iU), self.n) *
self.dS)
self.U_diffusion = -self.D * ufl.dot(ufl.grad(self.iU),
ufl.grad(self.wU)) * self.dx
self.U_penalty = -(
(self.Upen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(self.iU, self.n), ufl.jump(self.wU, self.n)) *
self.dS)
self.gU_penalty = -(self.gUpen * self.degree**2 *
(ufl.jump(ufl.grad(self.iU), self.n) *
ufl.jump(ufl.grad(self.wU), self.n)) *
self.dS)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
# The bilinear form a(v, u) and linear form L(v) for
# Poisson's equation in a discontinuous Galerkin (DG)
# formulation.
from ufl import (Coefficient, Constant, FacetNormal, FiniteElement,
TestFunction, TrialFunction, avg, dot, dS, ds, dx, grad,
inner, jump, triangle)
element = FiniteElement("Discontinuous Lagrange", triangle, 1)
v = TestFunction(element)
u = TrialFunction(element)
f = Coefficient(element)
n = FacetNormal(triangle)
h = Constant(triangle)
gN = Coefficient(element)
alpha = 4.0
gamma = 8.0
a = inner(grad(v), grad(u)) * dx \
- inner(avg(grad(v)), jump(u, n)) * dS \
- inner(jump(v, n), avg(grad(u))) * dS \
+ alpha / h('+') * dot(jump(v, n), jump(u, n)) * dS \
- inner(grad(v), u * n) * ds \
- inner(v * n, grad(u)) * ds \
+ gamma / h * v * u * ds
L = v * f * dx + v * gN * ds
def setup_problem(self, debug=False):
#
# assemble the matrix, if necessary (once for all time points)
#
if not hasattr(self, 'A'):
drho_integral = vectotal(
[tdrho*wrho*self.dx for tdrho,wrho in
zip(self.tdrhos, self.wrhos)]
)
dU_integral = vectotal(
[tdU*wU*self.dx
for tdU,wU in zip(self.tdUs, self.wUs)
]
)
self.A = fe.assemble(drho_integral + dU_integral)
for bc in self.bcs:
bc.apply(self.A)
# if self.solver_type == 'lu':
# self.solver = fe.LUSolver(
# self.A,
# )
# self.solver.parameters['reuse_factorization'] = True
# else:
# self.solver = fe.KrylovSolver(
# self.A,
# self.solver_type,
# self.preconditioner_type
# )
self.dsol = Function(self.VS)
self.drhos = self.dsol.split()[: 2**self.dim]
self.dUs = self.dsol.split()[2**self.dim :]
#
# These are the values of rho and U themselves (not their
# symmetrized versions) on all subdomains of the original
# domain.
#
if not hasattr(self, 'rhosds'):
self.rhosds = matmul(self.eomat, self.irhos)
if not hasattr(self, 'Usds'):
self.Usds = matmul(self.eomat, self.iUs)
#
# assemble RHS (for each time point, but compile only once)
#
if not hasattr(self, 'rho_terms'):
self.sigma = self.params['sigma']
self.s2 = self.sigma * self.sigma / 2
self.rho_min = self.params['rho_min']
self.rhopen = self.params['rhopen']
self.grhopen = self.params['grhopen']
#
# Compute fluxes on subdomains.
#
self.Vsds = [self.V(Usd, rhosd) for Usd,rhosd in
zip(self.Usds, self.rhosds)]
#
# I may need to adjust the signs of the subdomain vs by
# the symmetries of the combinations
#
self.vsds = [-ufl.grad(Vsd) - (
self.s2*ufl.grad(rhosd)/ufl.max_value(rhosd, self.rho_min)
) for Vsd,rhosd in zip(self.Vsds, self.rhosds)]
self.fluxsds = [vsd * rhosd for vsd,rhosd in
zip(self.vsds, self.rhosds)]
self.vnsds = [ufl.max_value(ufl.dot(vsd, self.n), 0)
for vsd in self.vsds]
self.facet_fluxsds = [(
vnsd('+')*ufl.max_value(rhosd('+'), 0.0) -
vnsd('-')*ufl.max_value(rhosd('-'), 0.0)
) for vnsd,rhosd in zip(self.vnsds, self.rhosds)]
#
# Now combine the subdomain fluxes to get the fluxes for
# the symmetrized functions
#
self.fluxs = matmul((2.0**-self.dim)*self.eomat,
self.fluxsds)
self.facet_fluxs = matmul((2.0**-self.dim)*self.eomat,
self.facet_fluxsds)
self.rho_flux_jump = vectotal(
[-facet_flux*ufl.jump(wrho)*self.dS
for facet_flux,wrho in
zip(self.facet_fluxs, self.wrhos)]
)
self.rho_grad_move = vectotal(
[ufl.dot(flux, ufl.grad(wrho))*self.dx
for flux,wrho in
zip(self.fluxs, self.wrhos)]
)
self.rho_penalty = vectotal(
[-(self.rhopen * self.degree**2 / self.havg) *
ufl.dot(ufl.jump(rho, self.n),
ufl.jump(wrho, self.n)) * self.dS
for rho,wrho in zip(self.irhos, self.wrhos)]
)
self.grho_penalty = vectotal(
[-self.grhopen * self.degree**2 *
(ufl.jump(ufl.grad(rho), self.n) *
ufl.jump(ufl.grad(wrho), self.n)) * self.dS
for rho,wrho in zip(self.irhos, self.wrhos)]
)
self.rho_terms = (
self.rho_flux_jump + self.rho_grad_move +
self.rho_penalty + self.grho_penalty
)
if not hasattr(self, 'U_terms'):
self.U_min = self.params['U_min']
self.gamma = self.params['gamma']
self.s = self.params['s']
self.D = self.params['D']
self.Upen = self.params['Upen']
self.gUpen = self.params['gUpen']
self.U_decay = vectotal(
[-self.gamma * U * wU * self.dx
for U,wU in zip(self.iUs, self.wUs)]
)
self.U_secretion = vectotal(
[self.s * rho * wU * self.dx
for rho, wU in zip(self.irhos, self.wUs)]
)
self.jump_gUw = vectotal(
[self.D * ufl.jump(wU * ufl.grad(U), self.n) * self.dS
for wU, U in zip(self.wUs, self.iUs)
]
)
self.U_diffusion = vectotal(
[-self.D
* ufl.dot(ufl.grad(U), ufl.grad(wU))*self.dx
for U,wU in zip(self.iUs, self.wUs)
]
)
self.U_penalty = vectotal(
[-(self.Upen * self.degree**2 / self.havg)
* ufl.dot(ufl.jump(U, self.n), ufl.jump(wU, self.n))*self.dS
for U,wU in zip(self.iUs, self.wUs)
]
)
self.gU_penalty = vectotal(
[-self.gUpen * self.degree**2 *
ufl.jump(ufl.grad(U), self.n) *
ufl.jump(ufl.grad(wU), self.n) * self.dS
for U,wU in zip(self.iUs, self.wUs)
]
)
self.U_terms = (
# decay and secretion
self.U_decay + self.U_secretion +
# diffusion
self.jump_gUw + self.U_diffusion +
# penalties (to enforce continuity)
self.U_penalty + self.gU_penalty
)
if not hasattr(self, 'all_terms'):
self.all_terms = self.rho_terms + self.U_terms
if not hasattr(self, 'J_terms'):
self.J_terms = fe.derivative(self.all_terms, self.sol)
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)
back_facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, back)
left_dofs = fem.locate_dofs_topological(V.sub(0), mesh.topology.dim - 1,
left_facets)
right_dofs = fem.locate_dofs_topological(V.sub(0), mesh.topology.dim - 1,
right_facets)
back_dofs = fem.locate_dofs_topological(V.sub(2), mesh.topology.dim - 1,
back_facets)
bottom_dofs = fem.locate_dofs_topological(V.sub(1), mesh.topology.dim - 1,
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, int))
# 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)
self._assemble_S()
self.S.force_evaluation()
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)
