def __init__(self, module, F, M, u):
"""
*Arguments*
module (Python module)
The module to use for specific form manipulations
(typically ufl or dolfin)
F (tuple or Form)
tuple of (bilinear, linear) forms or linear form
M (Form)
functional or linear form
u (Coefficient)
The coefficient considered as the unknown.
"""
# Store module
self.module = module
# Store solution Coefficient/Function
self.u = u
# Extract the lhs (bilinear form), rhs (linear form), goal
# (functional), weak residual (linear form)
linear_case = (isinstance(F, (tuple, list)) and len(F) == 2)
if linear_case:
self.lhs, self.rhs = F
try:
self.goal = action(M, u)
except:
self.goal = M # Allow functionals as input as well
self.weak_residual = self.rhs - action(self.lhs, u)
else:
self.lhs = self.module.derivative(F, u)
self.rhs = F
self.goal = M
self.weak_residual = -F
# At least check that final forms have correct rank
assert (len(self.lhs.arguments()) == 2)
assert (len(self.rhs.arguments()) == 1)
assert (len(self.goal.arguments()) == 0)
assert (len(self.weak_residual.arguments()) == 1)
# Get the domain, assuming there's only one
assert (len(self.weak_residual.domains()) == 1)
self.domain, = self.weak_residual.domains()
# Store map from identifiers to names for forms and generated
# coefficients
self.ec_names = {}
# Use predefined names for the forms in the primal problem
self.ec_names[id(self.lhs)] = "lhs"
self.ec_names[id(self.rhs)] = "rhs"
self.ec_names[id(self.goal)] = "goal"
# Initialize other required data
self.initialize_data()
def __init__(self, module, F, M, u):
"""
*Arguments*
module (Python module)
The module to use for specific form manipulations
(typically ufl or dolfin)
F (tuple or Form)
tuple of (bilinear, linear) forms or linear form
M (Form)
functional or linear form
u (Coefficient)
The coefficient considered as the unknown.
"""
# Store module
self.module = module
# Store solution Coefficient/Function
self.u = u
# Extract the lhs (bilinear form), rhs (linear form), goal
# (functional), weak residual (linear form)
linear_case = (isinstance(F, (tuple, list)) and len(F) == 2)
if linear_case:
self.lhs, self.rhs = F
try:
self.goal = action(M, u)
except Exception:
self.goal = M # Allow functionals as input as well
self.weak_residual = self.rhs - action(self.lhs, u)
else:
self.lhs = self.module.derivative(F, u)
self.rhs = F
self.goal = M
self.weak_residual = - F
# At least check that final forms have correct rank
assert(len(self.lhs.arguments()) == 2)
assert(len(self.rhs.arguments()) == 1)
assert(len(self.goal.arguments()) == 0)
assert(len(self.weak_residual.arguments()) == 1)
# Get the domain
self.domain = self.weak_residual.ufl_domain()
# Store map from identifiers to names for forms and generated
# coefficients
self.ec_names = {}
# Use predefined names for the forms in the primal problem
self.ec_names[id(self.lhs)] = "lhs"
self.ec_names[id(self.rhs)] = "rhs"
self.ec_names[id(self.goal)] = "goal"
# Initialize other required data
self.initialize_data()
def subtract_adjoint_derivative_actions(self, adj_x, nl_deps, dep_Bs):
for dep_index, dep_B in dep_Bs.items():
if dep_index not in self._adjoint_dF_cache:
dep = self.dependencies()[dep_index]
dF = derivative(self._F, dep)
dF = ufl.algorithms.expand_derivatives(dF)
dF = eliminate_zeros(dF)
if dF.empty():
dF = None
else:
dF = adjoint(dF)
self._adjoint_dF_cache[dep_index] = dF
dF = self._adjoint_dF_cache[dep_index]
if dF is not None:
if dep_index not in self._adjoint_action_cache:
if self._cache_rhs_assembly \
and isinstance(adj_x, backend_Function) \
and is_cached(dF):
# Cached matrix action
self._adjoint_action_cache[dep_index] = CacheRef()
elif self._defer_adjoint_assembly:
# Cached form, deferred assembly
self._adjoint_action_cache[dep_index] = None
else:
# Cached form, immediate assembly
self._adjoint_action_cache[dep_index] = unbound_form(
ufl.action(dF, coefficient=adj_x),
list(self.nonlinear_dependencies()) + [adj_x])
cache = self._adjoint_action_cache[dep_index]
if cache is None:
# Cached form, deferred assembly
replace_map = dict(zip(self.nonlinear_dependencies(),
nl_deps))
dep_B.sub(ufl.action(ufl.replace(dF, replace_map),
coefficient=adj_x))
elif isinstance(cache, CacheRef):
# Cached matrix action
mat_bc = cache()
if mat_bc is None:
replace_map = dict(zip(self.nonlinear_dependencies(),
nl_deps))
self._adjoint_action_cache[dep_index], (mat, _) = \
assembly_cache().assemble(
dF,
form_compiler_parameters=self._form_compiler_parameters, # noqa: E501
replace_map=replace_map)
else:
mat, _ = mat_bc
dep_B.sub(matrix_multiply(mat, function_vector(adj_x)))
else:
# Cached form, immediate assembly
assert isinstance(cache, ufl.classes.Form)
bind_form(cache, list(nl_deps) + [adj_x])
dep_B.sub(assemble(
cache,
form_compiler_parameters=self._form_compiler_parameters)) # noqa: E501
unbind_form(cache)
def __init__(self, form, target, source, rowbcs, colbcs):
self._twoform = form
self._rowbcs = rowbcs
self._target = target
self._source = source
self._colbcs = colbcs
self._x = Function(target, name='internalx')
self._y = Function(source, name='internaly')
self._form = ufl.action(form, self._x)
self._formT = ufl.action(adjoint(form), self._y)
self._assembled = False
if len(rowbcs) > 0:
self._xbcs = Function(target)
if len(colbcs) > 0:
self._ybcs = Function(source)
idx_kernels = compile_form(self._form)
self.local_assembly_idx = []
self.local_assembly_kernels = []
for idx, kernels in idx_kernels:
self.local_assembly_idx.append(idx)
self.local_assembly_kernels.append(kernels)
idx_kernels = compile_form(self._formT)
self.Tlocal_assembly_idx = []
self.Tlocal_assembly_kernels = []
for idx, kernels in idx_kernels:
self.Tlocal_assembly_idx.append(idx)
self.Tlocal_assembly_kernels.append(kernels)
# create matrix
mlist = []
nlist = []
for si1 in range(target.nspaces):
tspace = target.get_space(si1)
for ci1 in range(tspace.ncomp):
m = tspace.get_localndofs(ci1)
mlist.append(m)
for si2 in range(source.nspaces):
sspace = source.get_space(si2)
for ci2 in range(sspace.ncomp):
n = sspace.get_localndofs(ci2)
nlist.append(n)
M = np.sum(np.array(mlist, dtype=np.int32))
N = np.sum(np.array(nlist, dtype=np.int32))
self.petscmat = PETSc.Mat()
self.petscmat.create(PETSc.COMM_WORLD)
self.petscmat.setSizes(((M, None), (N, None)))
self.petscmat.setType('python')
self.petscmat.setPythonContext(self)
self.petscmat.setUp()
self.petscmat.assemblyBegin()
self.petscmat.assemblyEnd()
def _assemble_system(A_form,
b_form=None,
bcs=[],
form_compiler_parameters={},
*args,
**kwargs):
if isinstance(bcs, backend_DirichletBC):
bcs = (bcs, )
if b_form is None:
bind_forms(A_form)
else:
bind_forms(A_form, b_form)
A = _assemble(A_form,
bcs=bcs,
form_compiler_parameters=form_compiler_parameters,
*args,
**kwargs)
if len(bcs) > 0:
F = backend_Function(A_form.arguments()[0].function_space())
for bc in bcs:
bc.apply(F)
if b_form is None:
b = _assemble(-ufl.action(A_form, F),
bcs=bcs,
form_compiler_parameters=form_compiler_parameters,
*args,
**kwargs)
with b.dat.vec_ro as b_v:
if b_v.norm(norm_type=PETSc.NormType.NORM_INFINITY) == 0.0:
b = None
else:
b = _assemble(b_form - ufl.action(A_form, F),
bcs=bcs,
form_compiler_parameters=form_compiler_parameters,
*args,
**kwargs)
else:
if b_form is None:
b = None
else:
b = _assemble(b_form,
form_compiler_parameters=form_compiler_parameters,
*args,
**kwargs)
A._tlm_adjoint__lift_bcs = False
if b_form is None:
unbind_forms(A_form)
else:
unbind_forms(A_form, b_form)
return A, b
def __init__(self, a, row_bcs=[], col_bcs=[], fc_params=None, appctx=None):
self.a = a
self.aT = a.T if isinstance(self.a, slate.TensorBase) else adjoint(a)
self.fc_params = fc_params
self.appctx = appctx
self.row_bcs = row_bcs
self.col_bcs = col_bcs
# create functions from test and trial space to help
# with 1-form assembly
test_space, trial_space = [
a.arguments()[i].function_space() for i in (0, 1)
]
from firedrake import function
self._y = function.Function(test_space)
self._x = function.Function(trial_space)
# These are temporary storage for holding the BC
# values during matvec application. _xbc is for
# the action and ._ybc is for transpose.
if len(self.row_bcs) > 0:
self._xbc = function.Function(trial_space)
if len(self.col_bcs) > 0:
self._ybc = function.Function(test_space)
# Get size information from template vecs on test and trial spaces
trial_vec = trial_space.dof_dset.layout_vec
test_vec = test_space.dof_dset.layout_vec
self.col_sizes = trial_vec.getSizes()
self.row_sizes = test_vec.getSizes()
self.block_size = (test_vec.getBlockSize(), trial_vec.getBlockSize())
if isinstance(self.a, slate.TensorBase):
self.action = self.a * slate.AssembledVector(self._x)
self.actionT = self.aT * slate.AssembledVector(self._y)
else:
self.action = action(self.a, self._x)
self.actionT = action(self.aT, self._y)
from firedrake.assemble import create_assembly_callable
self._assemble_action = create_assembly_callable(
self.action,
tensor=self._y,
form_compiler_parameters=self.fc_params)
self._assemble_actionT = create_assembly_callable(
self.actionT,
tensor=self._x,
form_compiler_parameters=self.fc_params)
def __init__(self, a, L, u, bcs=None, aP=None,
form_compiler_parameters=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to ``False``.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
# Jacobian is checked in superclass, but let's check L here.
if not isinstance(L, ufl.Form):
raise TypeError("Provided RHS is a '%s', not a Form" % type(L).__name__)
if len(L.arguments()) != 1:
raise ValueError("Provided RHS is not a linear form")
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J, aP,
form_compiler_parameters=form_compiler_parameters)
self._constant_jacobian = constant_jacobian
def laplace_coeff_action_forms(mesh, el, coeff_el):
Q = dolfin.function.functionspace.FunctionSpace(mesh, el)
u = dolfin.function.argument.TrialFunction(Q)
v = dolfin.function.argument.TestFunction(Q)
"""
def boundary(x):
return np.sum(np.logical_or(x < DOLFIN_EPS, x > 1.0 - DOLFIN_EPS), axis=1) > 0
"""
# u_bc = dolfin.function.constant.Constant(50.0)
# bc = dolfin.fem.dirichletbc.DirichletBC(Q, u_bc, boundary)
c = dolfin.function.expression.Expression("3.14*x[0]", element=coeff_el)
f = dolfin.function.expression.Expression("0.4*x[1]*x[2]", element=el)
a = inner(c * grad(u), grad(v)) * dx
L = f * v * dx
w = dolfin.Function(Q)
w.vector().set(1.2)
L = ufl.action(a, w)
return None, L, None
def test_mass_action(form, cell, order):
degrees = numpy.arange(4, 10)
flops = [
count_flops(action(form(cell, int(degree)))) for degree in degrees
]
rates = numpy.diff(numpy.log(flops)) / numpy.diff(numpy.log(degrees + 1))
assert (rates < order).all()
def __init__(self, a, L, u, bcs=None, aP=None,
form_compiler_parameters=None,
nest=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param nest: indicate if matrices on mixed spaces should be
built as monolithic operators (suitable for direct
solves), or as nested blocks (suitable for fieldsplit
preconditioning). If not provided, uses the default
given by :data:`parameters["matnest"]`.
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to :data:`False`.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J, aP,
form_compiler_parameters=form_compiler_parameters, nest=nest)
self._constant_jacobian = constant_jacobian
def __init__(self, a, L, u, bcs=None, aP=None,
form_compiler_parameters=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to ``False``.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
# Jacobian is checked in superclass, but let's check L here.
if not isinstance(L, ufl.Form):
raise TypeError("Provided RHS is a '%s', not a Form" % type(L).__name__)
if len(L.arguments()) != 1:
raise ValueError("Provided RHS is not a linear form")
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J, aP,
form_compiler_parameters=form_compiler_parameters)
self._constant_jacobian = constant_jacobian
def error_estimate(self):
"""
Generate and return functional defining error estimate
"""
# Error estimate defined as r(Ez_h):
eta_h = action(self.weak_residual, self._Ez_h)
return eta_h
def verify_assembly(J, rhs, J_mat, b, bcs, form_compiler_parameters,
linear_solver_parameters, J_tolerance, b_tolerance):
if J_mat is not None and not np.isposinf(J_tolerance):
J_mat_debug = backend_assemble(
J, bcs=bcs, **assemble_arguments(2,
form_compiler_parameters,
linear_solver_parameters))
assert J_mat.petscmat.assembled
J_error = J_mat.petscmat.copy()
J_error.axpy(-1.0, J_mat_debug.petscmat)
assert J_error.assembled
assert J_error.norm(norm_type=PETSc.NormType.NORM_INFINITY) \
<= J_tolerance * J_mat.petscmat.norm(norm_type=PETSc.NormType.NORM_INFINITY) # noqa: E501
if b is not None and not np.isposinf(b_tolerance):
F = backend_Function(rhs.arguments()[0].function_space())
for bc in bcs:
bc.apply(F)
b_debug = backend_assemble(
rhs - ufl.action(J, F), bcs=bcs,
form_compiler_parameters=form_compiler_parameters)
b_error = b.copy(deepcopy=True)
with b_error.dat.vec as b_error_v, b_debug.dat.vec_ro as b_debug_v:
b_error_v.axpy(-1.0, b_debug_v)
with b_error.dat.vec_ro as b_error_v, b.dat.vec_ro as b_v:
assert b_error_v.norm(norm_type=PETSc.NormType.NORM_INFINITY) \
<= b_tolerance * b_v.norm(norm_type=PETSc.NormType.NORM_INFINITY) # noqa: E501
def __init__(self, a, L, u, bcs=None, aP=None,
form_compiler_parameters=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to :data:`False`.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J, aP,
form_compiler_parameters=form_compiler_parameters)
self._constant_jacobian = constant_jacobian
def adjoint_derivative_action(self, nl_deps, dep_index, adj_x):
# Derived from EquationSolver.derivative_action (see dolfin-adjoint
# reference below). Code first added 2017-12-07.
# Re-written 2018-01-28
# Updated to adjoint only form 2018-01-29
eq_deps = self.dependencies()
if dep_index < 0 or dep_index >= len(eq_deps):
raise EquationException("dep_index out of bounds")
elif dep_index == 0:
return adj_x
dep = eq_deps[dep_index]
dF = derivative(self._rhs, dep)
dF = ufl.algorithms.expand_derivatives(dF)
dF = eliminate_zeros(dF)
if dF.empty():
return None
dF = ufl.replace(dF, dict(zip(self.nonlinear_dependencies(), nl_deps)))
if self._rank == 0:
dF = assemble(
dF, form_compiler_parameters=self._form_compiler_parameters)
return (-real_function_value(adj_x), dF)
else:
assert self._rank == 1
dF = assemble(
ufl.action(adjoint(dF), adj_x),
form_compiler_parameters=self._form_compiler_parameters)
return (-1.0, dF)
def adjoint_derivative_action(self, nl_deps, dep_index, adj_x):
# Derived from EquationSolver.derivative_action (see dolfin-adjoint
# reference below). Code first added 2017-12-07.
# Re-written 2018-01-28
# Updated to adjoint only form 2018-01-29
eq_deps = self.dependencies()
if dep_index < 0 or dep_index >= len(eq_deps):
raise EquationException("dep_index out of bounds")
elif dep_index == 0:
return adj_x
dep = eq_deps[dep_index]
dF = derivative(self._rhs, dep)
dF = ufl.algorithms.expand_derivatives(dF)
dF = eliminate_zeros(dF)
if dF.empty():
return None
dF = self._nonlinear_replace(dF, nl_deps)
if self._rank == 0:
dF = ufl.Form([integral.reconstruct(integrand=ufl.conj(integral.integrand())) # noqa: E501
for integral in dF.integrals()]) # dF = adjoint(dF)
dF = assemble(
dF, form_compiler_parameters=self._form_compiler_parameters)
return (-function_scalar_value(adj_x), dF)
else:
assert self._rank == 1
dF = assemble(
ufl.action(adjoint(dF), coefficient=adj_x),
form_compiler_parameters=self._form_compiler_parameters)
return (-1.0, dF)
def error_estimate(self):
"""
Generate and return functional defining error estimate
"""
# Error estimate defined as r(Ez_h):
eta_h = action(self.weak_residual, self._Ez_h)
return eta_h
def __init__(self, a, L, u, bcs=None, aP=None,
form_compiler_parameters=None,
nest=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param nest: indicate if matrices on mixed spaces should be
built as monolithic operators (suitable for direct
solves), or as nested blocks (suitable for fieldsplit
preconditioning). If not provided, uses the default
given by :data:`parameters["matnest"]`.
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to :data:`False`.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J, aP,
form_compiler_parameters=form_compiler_parameters, nest=nest)
self._constant_jacobian = constant_jacobian
def test_rhs(cell, order):
degrees = list(range(3, 8))
if cell == TensorProductCell(triangle, interval):
degrees = list(range(3, 6))
flops = [count_flops(action(helmholtz(cell, degree)))
for degree in degrees]
rates = numpy.diff(numpy.log(flops)) / numpy.diff(numpy.log(degrees))
assert (rates < order).all()
def __init__(self, a, row_bcs=[], col_bcs=[],
fc_params=None, appctx=None):
self.a = a
self.aT = adjoint(a)
self.fc_params = fc_params
self.appctx = appctx
self.row_bcs = row_bcs
self.col_bcs = col_bcs
# create functions from test and trial space to help
# with 1-form assembly
test_space, trial_space = [
a.arguments()[i].function_space() for i in (0, 1)
]
from firedrake import function
self._y = function.Function(test_space)
self._x = function.Function(trial_space)
# These are temporary storage for holding the BC
# values during matvec application. _xbc is for
# the action and ._ybc is for transpose.
if len(self.row_bcs) > 0:
self._xbc = function.Function(trial_space)
if len(self.col_bcs) > 0:
self._ybc = function.Function(test_space)
# Get size information from template vecs on test and trial spaces
trial_vec = trial_space.dof_dset.layout_vec
test_vec = test_space.dof_dset.layout_vec
self.col_sizes = trial_vec.getSizes()
self.row_sizes = test_vec.getSizes()
self.block_size = (test_vec.getBlockSize(), trial_vec.getBlockSize())
self.action = action(self.a, self._x)
self.actionT = action(self.aT, self._y)
from firedrake.assemble import create_assembly_callable
self._assemble_action = create_assembly_callable(self.action, tensor=self._y,
form_compiler_parameters=self.fc_params)
self._assemble_actionT = create_assembly_callable(self.actionT, tensor=self._x,
form_compiler_parameters=self.fc_params)
def test_vector_laplace_action(cell, order):
degrees = numpy.arange(3, 8)
if cell == TensorProductCell(triangle, interval):
degrees = numpy.arange(3, 6)
flops = [[count_flops(action(form))
for form in split_vector_laplace(cell, int(degree))]
for degree in degrees]
rates = numpy.diff(numpy.log(flops).T) / numpy.diff(numpy.log(degrees))
assert (rates < order).all()
def vjp_solve_eval_impl(
g: np.array,
fenics_solution: fenics.Function,
fenics_residual: ufl.Form,
fenics_inputs: List[FenicsVariable],
bcs: List[fenics.DirichletBC],
) -> Tuple[np.array]:
"""Computes the gradients of the output with respect to the inputs."""
# Convert tangent covector (adjoint) to a FEniCS variable
adj_value = numpy_to_fenics(g, fenics_solution)
adj_value = adj_value.vector()
F = fenics_residual
u = fenics_solution
V = u.function_space()
dFdu = fenics.derivative(F, u)
adFdu = ufl.adjoint(
dFdu, reordered_arguments=ufl.algorithms.extract_arguments(dFdu)
)
u_adj = fenics.Function(V)
adj_F = ufl.action(adFdu, u_adj)
adj_F = ufl.replace(adj_F, {u_adj: fenics.TrialFunction(V)})
adj_F_assembled = fenics.assemble(adj_F)
if len(bcs) != 0:
for bc in bcs:
bc.homogenize()
hbcs = bcs
for bc in hbcs:
bc.apply(adj_F_assembled)
bc.apply(adj_value)
fenics.solve(adj_F_assembled, u_adj.vector(), adj_value)
fenics_grads = []
for fenics_input in fenics_inputs:
if isinstance(fenics_input, fenics.Function):
V = fenics_input.function_space()
dFdm = fenics.derivative(F, fenics_input, fenics.TrialFunction(V))
adFdm = fenics.adjoint(dFdm)
result = fenics.assemble(-adFdm * u_adj)
if isinstance(fenics_input, fenics.Constant):
fenics_grad = fenics.Constant(result.sum())
else: # fenics.Function
fenics_grad = fenics.Function(V, result)
fenics_grads.append(fenics_grad)
# Convert FEniCS gradients to jax array representation
jax_grads = (
None if fg is None else np.asarray(fenics_to_numpy(fg)) for fg in fenics_grads
)
jax_grad_tuple = tuple(jax_grads)
return jax_grad_tuple
def _rhs(self):
from firedrake.assemble import create_assembly_callable
u = function.Function(self.trial_space)
b = function.Function(self.test_space)
if isinstance(self.A.a, slate.TensorBase):
expr = -self.A.a * slate.AssembledVector(u)
else:
expr = -ufl.action(self.A.a, u)
return u, create_assembly_callable(expr, tensor=b), b
def _Abcs(self):
"""A function storing the action of the operator on a zero Function
satisfying the BCs.
Used in the presence of BCs.
"""
b = function.Function(self._W)
for bc in self.A.bcs:
bc.apply(b)
return assemble._assemble(ufl.action(self.A.a, b))
def _Matrix_mul(self, other):
return_value = backend_Matrix._tlm_adjoint__orig___mul__(self, other)
if hasattr(self, "_tlm_adjoint__form") \
and hasattr(other, "_tlm_adjoint__function") \
and len(self._tlm_adjoint__bcs) == 0:
return_value._tlm_adjoint__form = ufl.action(
self._tlm_adjoint__form, coefficient=other._tlm_adjoint__function)
return_value._tlm_adjoint__bcs = []
return_value._tlm_adjoint__form_compiler_parameters \
= self._tlm_adjoint__form_compiler_parameters
return return_value
def _Abcs(self):
"""A function storing the action of the operator on a zero Function
satisfying the BCs.
Used in the presence of BCs.
"""
b = function.Function(self._W)
for bc in self.A.bcs:
bc.apply(b)
from firedrake.assemble import _assemble
return _assemble(ufl.action(self.A.a, b))
def forward(x):
u = fn.TrialFunction(V)
w = fn.TestFunction(V)
sigma = lmbda * tr(sym(grad(u))) * Identity(2) + 2 * G * sym(
grad(u)) # Stress
R = simp(x) * inner(sigma, grad(w)) * dx - dot(b, w) * dx
a, L = ufl.lhs(R), ufl.rhs(R)
u = fn.Function(V)
F = L - ufl.action(a, u)
fn.solve(a == L, u, bcs)
return u, F, bcs
def action(form, coefficient):
"""Compute the action of a form on a coefficient.
:arg form: A UFL form, or a Slate tensor.
:arg coefficient: The :class:`~.Function` to act on.
:returns: a symbolic expression for the action.
"""
if isinstance(form, firedrake.slate.TensorBase):
if form.rank == 0:
raise ValueError("Can't take action of rank-0 tensor")
return form * firedrake.AssembledVector(coefficient)
else:
return ufl.action(form, coefficient)
def _form_action(self, u):
"""Assemble the form action of this :class:`Matrix`' bilinear form
onto the :class:`Function` ``u``.
.. note::
This is the form **without** any boundary conditions."""
if not hasattr(self, '_a_action'):
self._a_action = ufl.action(self._a, u)
if hasattr(self, '_a_action_coeff'):
self._a_action = ufl.replace(self._a_action, {self._a_action_coeff: u})
self._a_action_coeff = u
# Since we assemble the cached form, the kernels will already have
# been compiled and stashed on the form the second time round
return assemble._assemble(self._a_action)
def action(form, coefficient):
"""Compute the action of a form on a coefficient.
:arg form: A UFL form, or a Slate tensor.
:arg coefficient: The :class:`~.Function` to act on.
:returns: a symbolic expression for the action.
"""
if isinstance(form, firedrake.slate.TensorBase):
if form.rank == 0:
raise ValueError("Can't take action of rank-0 tensor")
return form * firedrake.AssembledVector(coefficient)
else:
return ufl.action(form, coefficient)
def _Abcs(self):
"""A function storing the action of the operator on a zero Function
satisfying the BCs.
Used in the presence of BCs.
"""
b = function.Function(self._W)
for bc in self.A.bcs:
bc.apply(b)
from firedrake.assemble import _assemble
if isinstance(self.A.a, slate.TensorBase):
return _assemble(self.A.a * b)
else:
return _assemble(ufl.action(self.A.a, b))
def _Abcs(self):
"""A function storing the action of the operator on a zero Function
satisfying the BCs.
Used in the presence of BCs.
"""
b = function.Function(self._W)
for bc in self.A.bcs:
bc.apply(b)
from firedrake.assemble import _assemble
if isinstance(self.A.a, slate.TensorBase):
return _assemble(self.A.a * slate.AssembledVector(b))
else:
return _assemble(ufl.action(self.A.a, b))
def __init__(self,
a,
L,
u,
bcs=None,
aP=None,
form_compiler_parameters=None,
nest=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param nest: indicate if matrices on mixed spaces should be
built as monolithic operators (suitable for direct
solves), or as nested blocks (suitable for fieldsplit
preconditioning). If not provided, uses the default
given by ``parameters["matnest"]``.
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to ``False``.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
# Jacobian is checked in superclass, but let's check L here.
if not isinstance(L, ufl.Form):
raise TypeError("Provided RHS is a '%s', not a Form" %
type(L).__name__)
if len(L.arguments()) != 1:
raise ValueError("Provided RHS is not a linear form")
F = ufl.action(J, u) - L
super(LinearVariationalProblem,
self).__init__(F,
u,
bcs,
J,
aP,
form_compiler_parameters=form_compiler_parameters,
nest=nest)
self._constant_jacobian = constant_jacobian
def _form_action(self, u):
"""Assemble the form action of this :class:`Matrix`' bilinear form
onto the :class:`Function` ``u``.
.. note::
This is the form **without** any boundary conditions."""
if not hasattr(self, '_a_action'):
self._a_action = ufl.action(self._a, u)
if hasattr(self, '_a_action_coeff'):
self._a_action = ufl.replace(self._a_action,
{self._a_action_coeff: u})
self._a_action_coeff = u
# Since we assemble the cached form, the kernels will already have
# been compiled and stashed on the form the second time round
from firedrake.assemble import _assemble
return _assemble(self._a_action)
def __init__(self, a, L, u, bcs=None,
form_compiler_parameters=None):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J,
form_compiler_parameters)
def hyperelasticity_action_forms(mesh, vec_el):
cell = mesh.ufl_cell()
Q = dolfin.FunctionSpace(mesh, vec_el)
# Coefficients
v = dolfin.function.argument.TestFunction(Q) # Test function
du = dolfin.function.argument.TrialFunction(Q) # Incremental displacement
u = dolfin.Function(Q) # Displacement from previous iteration
u.vector().set(0.5)
B = dolfin.Constant((0.0, -0.5, 0.0), cell) # Body force per unit volume
T = dolfin.Constant((0.1, 0.0, 0.0), cell) # Traction force on the boundary
# Kinematics
d = u.geometric_dimension()
F = ufl.Identity(d) + grad(u) # Deformation gradient
C = F.T * F # Right Cauchy-Green tensor
# Invariants of deformation tensors
Ic = tr(C)
J = det(F)
# Elasticity parameters
E, nu = 10.0, 0.3
mu = dolfin.Constant(E / (2 * (1 + nu)), cell)
lmbda = dolfin.Constant(E * nu / ((1 + nu) * (1 - 2 * nu)), cell)
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu / 2) * (Ic - 3) - mu * ln(J) + (lmbda / 2) * (ln(J)) ** 2
# Total potential energy
Pi = psi * dx - dot(B, u) * dx - dot(T, u) * ds
# Compute first variation of Pi (directional derivative about u in the direction of v)
F = ufl.derivative(Pi, u, v)
# Compute Jacobian of F
J = ufl.derivative(F, u, du)
w = dolfin.Function(Q)
w.vector().set(1.2)
L = ufl.action(J, w)
return None, L, None
def jvp_solve_eval(
fenics_function: Callable,
fenics_templates: Iterable[FenicsVariable],
primals: Tuple[np.array],
tangents: Tuple[np.array],
) -> Tuple[np.array]:
"""Computes the tangent linear model
"""
(
numpy_output_primal,
fenics_solution_primal,
residual_form,
fenics_primals,
bcs,
) = solve_eval(fenics_function, fenics_templates, *primals)
# Now tangent evaluation!
F = residual_form
u = fenics_solution_primal
V = u.function_space()
if len(bcs) != 0:
for bc in bcs:
bc.homogenize()
hbcs = bcs
fenics_tangents = convert_all_to_fenics(fenics_primals, *tangents)
fenics_output_tangents = []
for fp, ft in zip(fenics_primals, fenics_tangents):
dFdu = fenics.derivative(F, u)
dFdm = fenics.derivative(F, fp, ft)
u_tlm = fenics.Function(V)
tlm_F = ufl.action(dFdu, u_tlm) + dFdm
tlm_F = ufl.replace(tlm_F, {u_tlm: fenics.TrialFunction(V)})
fenics.solve(ufl.lhs(tlm_F) == ufl.rhs(tlm_F), u_tlm, bcs=hbcs)
fenics_output_tangents.append(u_tlm)
jax_output_tangents = (fenics_to_numpy(ft)
for ft in fenics_output_tangents)
jax_output_tangent = sum(jax_output_tangents)
return numpy_output_primal, jax_output_tangent
def __init__(self, a, L, u, bcs=None, aP=None,
form_compiler_parameters=None,
nest=None,
constant_jacobian=True):
"""
:param a: the bilinear form
:param L: the linear form
:param u: the :class:`.Function` to solve for
:param bcs: the boundary conditions (optional)
:param aP: an optional operator to assemble to precondition
the system (if not provided a preconditioner may be
computed from ``a``)
:param dict form_compiler_parameters: parameters to pass to the form
compiler (optional)
:param nest: indicate if matrices on mixed spaces should be
built as monolithic operators (suitable for direct
solves), or as nested blocks (suitable for fieldsplit
preconditioning). If not provided, uses the default
given by ``parameters["matnest"]``.
:param constant_jacobian: (optional) flag indicating that the
Jacobian is constant (i.e. does not depend on
varying fields). If your Jacobian can change, set
this flag to ``False``.
"""
# In the linear case, the Jacobian is the equation LHS.
J = a
# Jacobian is checked in superclass, but let's check L here.
if not isinstance(L, ufl.Form):
raise TypeError("Provided RHS is a '%s', not a Form" % type(L).__name__)
if len(L.arguments()) != 1:
raise ValueError("Provided RHS is not a linear form")
F = ufl.action(J, u) - L
super(LinearVariationalProblem, self).__init__(F, u, bcs, J, aP,
form_compiler_parameters=form_compiler_parameters, nest=nest)
self._constant_jacobian = constant_jacobian
#
# Author: Martin Sandve Alnes
# Date: 2008-10-30
#
from ufl import (Coefficient, FiniteElement, action, adjoint, derivative, dx,
grad, inner, triangle)
element = FiniteElement("Lagrange", triangle, 1)
w = Coefficient(element)
# H1 semi-norm
f = inner(grad(w), grad(w)) / 2 * dx
# grad(w) : grad(v)
b = derivative(f, w)
# stiffness matrix, grad(u) : grad(v)
a = derivative(b, w)
# adjoint, grad(v) : grad(u)
astar = adjoint(a)
# action of adjoint, grad(v) : grad(w)
astaraction = action(astar)
forms = [f, b, a, astar, astaraction]
def __init__(self, eq, x, bcs=[], J=None, form_compiler_parameters={},
solver_parameters={}, adjoint_solver_parameters=None,
tlm_solver_parameters=None, initial_guess=None,
cache_jacobian=None, cache_adjoint_jacobian=None,
cache_tlm_jacobian=None, cache_rhs_assembly=None,
match_quadrature=None, defer_adjoint_assembly=None):
if isinstance(bcs, backend_DirichletBC):
bcs = (bcs,)
else:
bcs = tuple(bcs)
if cache_jacobian is None:
if not parameters["tlm_adjoint"]["EquationSolver"]["enable_jacobian_caching"]: # noqa: E501
cache_jacobian = False
if cache_rhs_assembly is None:
cache_rhs_assembly = parameters["tlm_adjoint"]["EquationSolver"]["cache_rhs_assembly"] # noqa: E501
if match_quadrature is None:
match_quadrature = parameters["tlm_adjoint"]["EquationSolver"]["match_quadrature"] # noqa: E501
if defer_adjoint_assembly is None:
defer_adjoint_assembly = parameters["tlm_adjoint"]["EquationSolver"]["defer_adjoint_assembly"] # noqa: E501
if match_quadrature and defer_adjoint_assembly:
raise EquationException("Cannot both match quadrature and defer adjoint assembly") # noqa: E501
lhs, rhs = eq.lhs, eq.rhs
del eq
lhs = ufl.classes.Form(lhs.integrals())
linear = isinstance(rhs, ufl.classes.Form)
if linear:
rhs = ufl.classes.Form(rhs.integrals())
if J is not None:
J = ufl.classes.Form(J.integrals())
if linear:
if x in lhs.coefficients() or x in rhs.coefficients():
raise EquationException("Invalid non-linear dependency")
F = ufl.action(lhs, coefficient=x) + form_neg(rhs)
nl_solve_J = None
J = lhs
else:
F = lhs
if rhs != 0:
raise EquationException("Invalid right-hand-side")
nl_solve_J = J
J = derivative(F, x)
J = ufl.algorithms.expand_derivatives(J)
deps, nl_deps = extract_dependencies(F)
if nl_solve_J is not None:
for dep in nl_solve_J.coefficients():
if is_function(dep):
dep_id = function_id(dep)
if dep_id not in deps:
deps[dep_id] = dep
if initial_guess is not None:
warnings.warn("'initial_guess' argument is deprecated",
DeprecationWarning, stacklevel=2)
if initial_guess == x:
initial_guess = None
else:
initial_guess_id = function_id(initial_guess)
if initial_guess_id not in deps:
deps[initial_guess_id] = initial_guess
deps = list(deps.values())
if x in deps:
deps.remove(x)
deps.insert(0, x)
nl_deps = tuple(nl_deps.values())
hbcs = tuple(homogenized_bc(bc) for bc in bcs)
if cache_jacobian is None:
cache_jacobian = is_cached(J) and bcs_is_cached(bcs)
if cache_adjoint_jacobian is None:
cache_adjoint_jacobian = cache_jacobian
if cache_tlm_jacobian is None:
cache_tlm_jacobian = cache_jacobian
if nl_solve_J is None:
(solver_parameters, linear_solver_parameters,
ic, J_ic) = process_solver_parameters(solver_parameters, linear)
else:
(solver_parameters, _,
ic, _) = process_solver_parameters(solver_parameters, linear)
(_, linear_solver_parameters,
_, J_ic) = process_solver_parameters(solver_parameters, linear)
if adjoint_solver_parameters is None:
adjoint_solver_parameters = process_adjoint_solver_parameters(linear_solver_parameters) # noqa: E501
adj_ic = J_ic
else:
(_, adjoint_solver_parameters,
adj_ic, _) = process_solver_parameters(adjoint_solver_parameters, linear=True) # noqa: E501
if tlm_solver_parameters is not None:
(_, tlm_solver_parameters,
_, _) = process_solver_parameters(tlm_solver_parameters, linear=True) # noqa: E501
form_compiler_parameters_ = copy_parameters_dict(parameters["form_compiler"]) # noqa: E501
update_parameters_dict(form_compiler_parameters_,
form_compiler_parameters)
form_compiler_parameters = form_compiler_parameters_
if match_quadrature:
update_parameters_dict(
form_compiler_parameters,
form_form_compiler_parameters(F, form_compiler_parameters))
super().__init__(x, deps, nl_deps=nl_deps,
ic=initial_guess is None and ic,
adj_ic=adj_ic)
self._F = F
self._lhs, self._rhs = lhs, rhs
self._bcs = bcs
self._hbcs = hbcs
self._J = J
self._nl_solve_J = nl_solve_J
self._form_compiler_parameters = form_compiler_parameters
self._solver_parameters = solver_parameters
self._linear_solver_parameters = linear_solver_parameters
self._adjoint_solver_parameters = adjoint_solver_parameters
self._tlm_solver_parameters = tlm_solver_parameters
if initial_guess is None:
self._initial_guess_index = None
else:
self._initial_guess_index = deps.index(initial_guess)
self._linear = linear
self._cache_jacobian = cache_jacobian
self._cache_adjoint_jacobian = cache_adjoint_jacobian
self._cache_tlm_jacobian = cache_tlm_jacobian
self._cache_rhs_assembly = cache_rhs_assembly
self._defer_adjoint_assembly = defer_adjoint_assembly
self._forward_eq = None
self._forward_J_solver = CacheRef()
self._forward_b_pa = None
self._adjoint_dF_cache = {}
self._adjoint_action_cache = {}
self._adjoint_J_solver = CacheRef()
self._adjoint_J = None
def _compileUFL(integrands, form, *args, tempVars=True):
if isinstance(form, Equation):
form = form.lhs - form.rhs
if not isinstance(form, Form):
raise ValueError("ufl.Form or ufl.Equation expected.")
# added for dirichlet treatment same as conservationlaw model
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
uflExpr = [form] + [bc.ufl_value for bc in dirichletBCs]
if len(form.arguments()) < 2:
raise ValueError(
"Integrands model requires form with at least two arguments.")
x = SpatialCoordinate(form.ufl_cell())
n = FacetNormal(form.ufl_cell())
cellVolume = CellVolume(form.ufl_cell())
maxCellEdgeLength = MaxCellEdgeLength(form.ufl_cell())
minCellEdgeLength = MinCellEdgeLength(form.ufl_cell())
facetArea = FacetArea(form.ufl_cell())
maxFacetEdgeLength = MaxFacetEdgeLength(form.ufl_cell())
minFacetEdgeLength = MinFacetEdgeLength(form.ufl_cell())
phi, u = form.arguments()
ubar = Coefficient(u.ufl_function_space())
derivatives = gatherDerivatives(form, [phi, u])
derivatives_phi = derivatives[0]
derivatives_u = derivatives[1]
derivatives_ubar = map_expr_dags(Replacer({u: ubar}), derivatives_u)
try:
integrands.field = u.ufl_function_space().field
except AttributeError:
pass
integrals = splitForm(form, [phi])
dform = apply_derivatives(derivative(action(form, ubar), ubar, u))
linearizedIntegrals = splitForm(dform, [phi, u])
if not set(
integrals.keys()) <= {'cell', 'exterior_facet', 'interior_facet'}:
raise Exception('unknown integral encountered in ' +
str(set(integrals.keys())) + '.')
if 'cell' in integrals.keys():
arg = Variable(integrands.domainValueTuple, 'u')
predefined = {
derivatives_u[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.interior = generateUnaryCode(predefined,
derivatives_phi,
integrals['cell'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.linearizedInterior = generateUnaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('cell'),
tempVars=tempVars)
if 'exterior_facet' in integrals.keys():
arg = Variable(integrands.domainValueTuple, 'u')
predefined = {
derivatives_u[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[n] = integrands.facetNormal('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.boundary = generateUnaryCode(predefined,
derivatives_phi,
integrals['exterior_facet'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i]: arg[i]
for i in range(len(derivatives_u))
}
predefined[x] = integrands.spatialCoordinate('x')
predefined[n] = integrands.facetNormal('x')
predefined[cellVolume] = integrands.cellVolume()
predefined[maxCellEdgeLength] = maxEdgeLength(
integrands.cellGeometry())
predefined[minCellEdgeLength] = minEdgeLength(
integrands.cellGeometry())
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, False)
integrands.linearizedBoundary = generateUnaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('exterior_facet'),
tempVars=tempVars)
if 'interior_facet' in integrals.keys():
argIn = Variable(integrands.domainValueTuple, 'uIn')
argOut = Variable(integrands.domainValueTuple, 'uOut')
predefined = {
derivatives_u[i](s): arg[i]
for i in range(len(derivatives_u))
for s, arg in (('+', argIn), ('-', argOut))
}
predefined[x] = integrands.spatialCoordinate('xIn')
predefined[n('+')] = integrands.facetNormal('xIn')
predefined[cellVolume('+')] = integrands.cellVolume('Side::in')
predefined[cellVolume('-')] = integrands.cellVolume('Side::out')
predefined[maxCellEdgeLength('+')] = maxEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[maxCellEdgeLength('-')] = maxEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[minCellEdgeLength('+')] = minEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[minCellEdgeLength('-')] = minEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, True)
integrands.skeleton = generateBinaryCode(predefined,
derivatives_phi,
integrals['interior_facet'],
tempVars=tempVars)
predefined = {
derivatives_ubar[i](s): arg[i]
for i in range(len(derivatives_u))
for s, arg in (('+', argIn), ('-', argOut))
}
predefined[x] = integrands.spatialCoordinate('xIn')
predefined[n('+')] = integrands.facetNormal('xIn')
predefined[cellVolume('+')] = integrands.cellVolume('Side::in')
predefined[cellVolume('-')] = integrands.cellVolume('Side::out')
predefined[maxCellEdgeLength('+')] = maxEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[maxCellEdgeLength('-')] = maxEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[minCellEdgeLength('+')] = minEdgeLength(
integrands.cellGeometry('Side::in'))
predefined[minCellEdgeLength('-')] = minEdgeLength(
integrands.cellGeometry('Side::out'))
predefined[facetArea] = integrands.facetArea()
predefined[maxFacetEdgeLength] = maxEdgeLength(
integrands.facetGeometry())
predefined[minFacetEdgeLength] = minEdgeLength(
integrands.facetGeometry())
integrands.predefineCoefficients(predefined, True)
integrands.linearizedSkeleton = generateBinaryLinearizedCode(
predefined,
derivatives_phi,
derivatives_u,
linearizedIntegrals.get('interior_facet'),
tempVars=tempVars)
if dirichletBCs:
integrands.hasDirichletBoundary = True
predefined = {}
# predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + integrands.arg_x.name + ' ) )')
predefined[x] = UnformattedExpression(
'auto', 'intersection.geometry().center( )')
integrands.predefineCoefficients(predefined, False)
maxId = 0
codeDomains = []
bySubDomain = dict()
neuman = []
wholeDomain = None
for bc in dirichletBCs:
if bc.subDomain in bySubDomain:
raise Exception(
'Multiply defined Dirichlet boundary for subdomain ' +
str(bc.subDomain))
if not isinstance(bc.functionSpace,
(FunctionSpace, FiniteElementBase)):
raise Exception(
'Function space must either be a ufl.FunctionSpace or a ufl.FiniteElement'
)
if isinstance(bc.functionSpace, FunctionSpace) and (
bc.functionSpace != u.ufl_function_space()):
raise Exception(
'Space of trial function and dirichlet boundary function must be the same - note that boundary conditions on subspaces are not available, yet'
)
if isinstance(bc.functionSpace, FiniteElementBase) and (
bc.functionSpace != u.ufl_element()):
raise Exception(
'Cannot handle boundary conditions on subspaces, yet')
if isinstance(bc.value, list):
neuman = [i for i, x in enumerate(bc.value) if x == None]
else:
neuman = []
value = ExprTensor(u.ufl_shape)
for key in value.keys():
value[key] = Indexed(
bc.ufl_value,
MultiIndex(tuple(FixedIndex(k) for k in key)))
if bc.subDomain is None:
wholeDomain = value, neuman
elif isinstance(bc.subDomain, int):
bySubDomain[bc.subDomain] = value, neuman
maxId = max(maxId, bc.subDomain)
else:
domain = ExprTensor(())
for key in domain.keys():
domain[key] = Indexed(
bc.subDomain,
MultiIndex(tuple(FixedIndex(k) for k in key)))
codeDomains.append((value, neuman, domain))
defaultCode = []
if len(codeDomains) > 0:
defaultCode.append(Declaration(Variable('int', 'domainId')))
# defaultCode.append(Declaration(Variable('auto', 'x'),
# initializer=UnformattedExpression('auto','intersection.geometry().center()')))
for i, v in enumerate(codeDomains):
block = Block()
block.append(
generateDirichletDomainCode(predefined,
v[2],
tempVars=tempVars))
block.append('if (domainId)')
ifBlock = UnformattedBlock()
ifBlock.append(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ str(maxId + i + 2) + ' );')
if len(v[1]) > 0:
[
ifBlock.append('dirichletComponent[' + str(c) + '] = 0;')
for c in v[1]
]
ifBlock.append('return true;')
block.append(ifBlock)
defaultCode.append(block)
if wholeDomain is not None:
block = UnformattedBlock()
block.append(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ str(maxId + 1) + ' );')
if len(wholeDomain[1]) > 0:
[
block.append('dirichletComponent[' + str(c) + '] = 0;')
for c in wholeDomain[1]
]
block.append('return true;')
defaultCode.append(block)
defaultCode.append(return_(False))
bndId = Variable('const int', 'bndId')
getBndId = UnformattedExpression(
'int', 'BoundaryIdProviderType::boundaryId( ' +
integrands.arg_i.name + ' )')
# getBndId = UnformattedExpression('int', 'boundaryIdGetter_.boundaryId( ' + integrands.arg_i.name + ' )')
switch = SwitchStatement(bndId, default=defaultCode)
for i, v in bySubDomain.items():
code = []
if len(v[1]) > 0:
[
code.append('dirichletComponent[' + str(c) + '] = 0;')
for c in v[1]
]
code.append(return_(True))
switch.append(i, code)
integrands.isDirichletIntersection = [
Declaration(bndId, initializer=getBndId),
UnformattedBlock(
'std::fill( dirichletComponent.begin(), dirichletComponent.end(), '
+ bndId.name + ' );'), switch
]
predefined[x] = UnformattedExpression(
'auto', 'entity().geometry().global( Dune::Fem::coordinate( ' +
integrands.arg_x.name + ' ) )')
if wholeDomain is None:
defaultCode = assign(integrands.arg_r, construct("RRangeType", 0))
else:
defaultCode = generateDirichletCode(predefined,
wholeDomain[0],
tempVars=tempVars)
switch = SwitchStatement(integrands.arg_bndId, default=defaultCode)
for i, v in bySubDomain.items():
switch.append(
i, generateDirichletCode(predefined, v[0], tempVars=tempVars))
for i, v in enumerate(codeDomains):
switch.append(
i + maxId + 2,
generateDirichletCode(predefined, v[0], tempVars=tempVars))
integrands.dirichlet = [switch]
return integrands
def _action(A, x):
A = ufl.algorithms.expand_derivatives(A)
if A.integrals() != (): # form is not empty:
return ufl.action(A, x)
else:
return A # form is empty, doesn't matter
coord_element = VectorElement("Lagrange", triangle, 1)
mesh = Mesh(coord_element)
# Function Space
element = FiniteElement("Lagrange", triangle, 2)
V = FunctionSpace(mesh, element)
# Trial and test functions
u = TrialFunction(V)
v = TestFunction(V)
# Define a constant RHS
f = Constant(V)
# Define the bilinear and linear forms according to the
# variational formulation of the equations::
a = inner(grad(u), grad(v)) * dx
L = inner(f, v) * dx
# Define linear form representing the action of the form "a" on
# the coefficient "ui"
ui = Coefficient(V)
M = action(a, ui)
# Define form to compute the L2 norm of the error
usol = Coefficient(V)
uexact = Coefficient(V)
E = inner(usol - uexact, usol - uexact) * dx
forms = [M, L, E]
def compileUFL(form, patch, *args, **kwargs):
if isinstance(form, Equation):
form = form.lhs - form.rhs
if not isinstance(form, Form):
raise Exception("ufl.Form expected.")
if len(form.arguments()) < 2:
raise Exception("ConservationLaw model requires form with at least two arguments.")
phi_, u_ = form.arguments()
if phi_.ufl_function_space().scalar:
phi = TestFunction(phi_.ufl_function_space().toVectorSpace())
form = replace(form,{phi_:phi[0]})
else:
phi = phi_
if u_.ufl_function_space().scalar:
u = TrialFunction(u_.ufl_function_space().toVectorSpace())
form = replace(form,{u_:u[0]})
else:
u = u_
_, coeff_ = extract_arguments_and_coefficients(form)
coeff_ = set(coeff_)
# added for dirichlet treatment same as conservationlaw model
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
# remove the dirichletBCs
arg = [arg for arg in args if not isinstance(arg, DirichletBC)]
for dBC in dirichletBCs:
_, coeff__ = extract_arguments_and_coefficients(dBC.ufl_value)
coeff_ |= set(coeff__)
if patch is not None:
for a in patch:
try:
_, coeff__ = extract_arguments_and_coefficients(a)
coeff_ |= set(coeff__)
except:
pass # a might be a float/int and not a ufl expression
coeff = {c : c.toVectorCoefficient()[0] for c in coeff_ if len(c.ufl_shape) == 0 and not c.is_cellwise_constant()}
form = replace(form,coeff)
for bc in dirichletBCs:
bc.ufl_value = replace(bc.ufl_value, coeff)
if patch is not None:
patch = [a if not isinstance(a, Expr) else replace(a,coeff) for a in patch]
phi = form.arguments()[0]
dimRange = phi.ufl_shape[0]
u = form.arguments()[1]
du = Grad(u)
d2u = Grad(du)
ubar = Coefficient(u.ufl_function_space())
dubar = Grad(ubar)
d2ubar = Grad(dubar)
dimDomain = u.ufl_shape[0]
x = SpatialCoordinate(form.ufl_cell())
try:
field = u.ufl_function_space().field
except AttributeError:
field = "double"
# if exact solution is passed in subtract a(u,.) from the form
if "exact" in kwargs:
b = replace(form, {u: as_vector(kwargs["exact"])} )
form = form - b
dform = apply_derivatives(derivative(action(form, ubar), ubar, u))
source, flux, boundarySource = splitUFLForm(form)
linSource, linFlux, linBoundarySource = splitUFLForm(dform)
fluxDivergence, _, _ = splitUFLForm(inner(source.as_ufl() - div(flux.as_ufl()), phi) * dx(0))
# split linNVSource off linSource
# linSources = splitUFL2(u, du, d2u, linSource)
# linNVSource = linSources[2]
# linSource = linSources[0] + linSources[1]
if patch is not None:
model = ConservationLawModel(dimDomain, dimRange, u, modelSignature(form,*patch,*args))
else:
model = ConservationLawModel(dimDomain, dimRange, u, modelSignature(form,None,*args))
model._replaceCoeff = coeff
model.hasNeumanBoundary = not boundarySource.is_zero()
#expandform = expand_indices(expand_derivatives(expand_compounds(equation.lhs)))
#if expandform == adjoint(expandform):
# model.symmetric = 'true'
model.field = field
dirichletBCs = [arg for arg in args if isinstance(arg, DirichletBC)]
# deprecated
# if "dirichlet" in kwargs:
# dirichletBCs += [DirichletBC(u.ufl_function_space(), as_vector(value), bndId) for bndId, value in kwargs["dirichlet"].items()]
uflCoefficients = set(form.coefficients())
for bc in dirichletBCs:
_, c = extract_arguments_and_coefficients(bc.ufl_value)
uflCoefficients |= set(c)
if patch is not None:
for a in patch:
if isinstance(a, Expr):
_, c = extract_arguments_and_coefficients(a)
uflCoefficients |= set(c)
constants = dict()
coefficients = dict()
for coefficient in uflCoefficients:
try:
name = getattr(coefficient, "name")
except AttributeError:
name = str(coefficient)
if coefficient.is_cellwise_constant():
try:
parameter = getattr(coefficient, "parameter")
except AttributeError:
parameter = None
if len(coefficient.ufl_shape) == 0:
constants[coefficient] = model.addConstant('double', name=name, parameter=parameter)
elif len(coefficient.ufl_shape) == 1:
constants[coefficient] = model.addConstant('Dune::FieldVector< double, ' + str(coefficient.ufl_shape[0]) + ' >', name=name, parameter=parameter)
else:
Exception('Currently, only scalars and vectors are supported as constants')
else:
shape = coefficient.ufl_shape[0]
try:
coefficients[coefficient] = model.addCoefficient(
shape,
coefficient.cppTypeName,
name=name,
field=coefficient.ufl_function_space().field)
except AttributeError:
coefficients[coefficient] = model.addCoefficient(
shape,
coefficient.cppTypeName,
name=name)
model.coefficients = coefficients
model.constants = constants
tempVars = kwargs.get("tempVars", True)
predefined = {u: model.arg_u, du: model.arg_du, d2u: model.arg_d2u}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
model.source = generateCode(predefined, source, tempVars=tempVars)
model.flux = generateCode(predefined, flux, tempVars=tempVars)
predefined.update({ubar: model.arg_ubar, dubar: model.arg_dubar, d2ubar: model.arg_d2ubar})
model.linSource = generateCode(predefined, linSource, tempVars=tempVars)
model.linFlux = generateCode(predefined, linFlux, tempVars=tempVars)
# model.linNVSource = generateCode({u: arg, du: darg, d2u: d2arg, ubar: argbar, dubar: dargbar, d2ubar: d2argbar}, linNVSource, model.coefficients, tempVars)
predefined = {u: model.arg_u}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
model.alpha = generateCode(predefined, boundarySource, tempVars=tempVars)
predefined.update({ubar: model.arg_ubar})
model.linAlpha = generateCode(predefined, linBoundarySource, tempVars=tempVars)
predefined = {u: model.arg_u, du: model.arg_du, d2u: model.arg_d2u}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
model.fluxDivergence = generateCode(predefined, fluxDivergence, tempVars=tempVars)
if dirichletBCs:
model.hasDirichletBoundary = True
predefined = {}
predefined[x] = UnformattedExpression('auto', 'entity().geometry().global( Dune::Fem::coordinate( ' + model.arg_x.name + ' ) )')
model.predefineCoefficients(predefined,'x')
maxId = 0
codeDomains = []
bySubDomain = dict()
neuman = []
for bc in dirichletBCs:
if bc.subDomain in bySubDomain:
raise Exception('Multiply defined Dirichlet boundary for subdomain ' + str(bc.subDomain))
if not isinstance(bc.functionSpace, (FunctionSpace, FiniteElementBase)):
raise Exception('Function space must either be a ufl.FunctionSpace or a ufl.FiniteElement')
if isinstance(bc.functionSpace, FunctionSpace) and (bc.functionSpace != u.ufl_function_space()):
raise Exception('Space of trial function and dirichlet boundary function must be the same - note that boundary conditions on subspaces are not available, yet')
if isinstance(bc.functionSpace, FiniteElementBase) and (bc.functionSpace != u.ufl_element()):
raise Exception('Cannot handle boundary conditions on subspaces, yet')
if isinstance(bc.value, list):
neuman = [i for i, x in enumerate(bc.value) if x == None]
else:
neuman = []
value = ExprTensor(u.ufl_shape)
for key in value.keys():
value[key] = Indexed(bc.ufl_value, MultiIndex(tuple(FixedIndex(k) for k in key)))
if isinstance(bc.subDomain,int):
bySubDomain[bc.subDomain] = value,neuman
maxId = max(maxId, bc.subDomain)
else:
domain = ExprTensor(())
for key in domain.keys():
domain[key] = Indexed(bc.subDomain, MultiIndex(tuple(FixedIndex(k) for k in key)))
codeDomains.append( (value,neuman,domain) )
defaultCode = []
defaultCode.append(Declaration(Variable('int', 'domainId')))
defaultCode.append(Declaration(Variable('auto', 'tmp0'),
initializer=UnformattedExpression('auto','intersection.geometry().center()')))
for i,v in enumerate(codeDomains):
block = Block()
defaultCode.append(
generateDirichletDomainCode(predefined, v[2], tempVars=tempVars))
defaultCode.append('if (domainId)')
block = UnformattedBlock()
block.append('std::fill( dirichletComponent.begin(), dirichletComponent.end(), ' + str(maxId+i+1) + ' );')
if len(v[1])>0:
[block.append('dirichletComponent[' + str(c) + '] = 0;') for c in v[1]]
block.append('return true;')
defaultCode.append(block)
defaultCode.append(return_(False))
bndId = Variable('const int', 'bndId')
getBndId = UnformattedExpression('int', 'BoundaryIdProviderType::boundaryId( ' + model.arg_i.name + ' )')
switch = SwitchStatement(bndId, default=defaultCode)
for i,v in bySubDomain.items():
code = []
if len(v[1])>0:
[code.append('dirichletComponent[' + str(c) + '] = 0;') for c in v[1]]
code.append(return_(True))
switch.append(i, code)
model.isDirichletIntersection = [Declaration(bndId, initializer=getBndId),
UnformattedBlock('std::fill( dirichletComponent.begin(), dirichletComponent.end(), ' + bndId.name + ' );'),
switch
]
switch = SwitchStatement(model.arg_bndId, default=assign(model.arg_r, construct("RRangeType", 0)))
for i, v in bySubDomain.items():
switch.append(i, generateDirichletCode(predefined, v[0], tempVars=tempVars))
for i,v in enumerate(codeDomains):
switch.append(i+maxId+1, generateDirichletCode(predefined, v[0], tempVars=tempVars))
model.dirichlet = [switch]
return model
def test_mass_action(form, cell, order):
degrees = numpy.arange(4, 10)
flops = [count_flops(action(form(cell, int(degree)))) for degree in degrees]
rates = numpy.diff(numpy.log(flops)) / numpy.diff(numpy.log(degrees + 1))
assert (rates < order).all()
