def __init__(self, a, L, u, bcs=None, form_compiler_parameters=None):
"""
Create linear variational problem a(u, v) = L(v).
An optional argument bcs may be passed to specify boundary
conditions.
Another optional argument form_compiler_parameters may be
specified to pass parameters to the form compiler.
"""
# Extract and check arguments
u = _extract_u(u)
bcs = _extract_bcs(bcs)
# Store input UFL forms and solution Function
self.a_ufl = a
self.L_ufl = L
self.u_ufl = u
# Store form compiler parameters
form_compiler_parameters = form_compiler_parameters or {}
self.form_compiler_parameters = form_compiler_parameters
# Wrap forms (and check if linear form L is empty)
if L.empty():
L = cpp.Form(1, 0)
else:
L = Form(L, form_compiler_parameters=form_compiler_parameters)
a = Form(a, form_compiler_parameters=form_compiler_parameters)
# Initialize C++ base class
cpp.LinearVariationalProblem.__init__(self, a, L, u, bcs)
def _solve_varproblem(*args, **kwargs):
"Solve variational problem a == L or F == 0"
# Extract arguments
eq, u, bcs, J, tol, M, form_compiler_parameters, petsc_options \
= _extract_args(*args, **kwargs)
# Solve linear variational problem
if isinstance(eq.lhs, ufl.Form) and isinstance(eq.rhs, ufl.Form):
a = Form(eq.lhs, form_compiler_parameters=form_compiler_parameters)
L = Form(eq.rhs, form_compiler_parameters=form_compiler_parameters)
A = PETScMatrix()
b = PETScVector()
assembler = SystemAssembler(a, L, bcs)
assembler.assemble(A, b)
comm = L.mesh().mpi_comm()
solver = PETScKrylovSolver(comm)
solver.set_options_prefix("dolfin_solve_")
for k, v in petsc_options.items():
PETScOptions.set("dolfin_solve_" + k, v)
solver.set_from_options()
solver.set_operator(A)
solver.solve(u.vector(), b)
# Solve nonlinear variational problem
else:
raise RuntimeError("Not implemented")
def __init__(self, a, L, u, bcs=None, form_compiler_parameters=None):
"""Create linear variational problem a(u, v) = L(v).
An optional argument bcs may be passed to specify boundary
conditions.
Another optional argument form_compiler_parameters may be
specified to pass parameters to the form compiler.
"""
if bcs is None:
bcs = []
elif not isinstance(bcs, (list, tuple)):
bcs = [bcs]
# Store input UFL forms and solution Function
self.a_ufl = a
self.L_ufl = L
self.u_ufl = u
# Store form compiler parameters
form_compiler_parameters = form_compiler_parameters or {}
self.form_compiler_parameters = form_compiler_parameters
# Wrap forms (and check if linear form L is empty)
if L.empty():
L = cpp.fem.Form(1, 0)
else:
L = Form(L, form_compiler_parameters=form_compiler_parameters)
a = Form(a, form_compiler_parameters=form_compiler_parameters)
# Initialize C++ base class
cpp.fem.LinearVariationalProblem.__init__(self, a, L, u._cpp_object, bcs)
def __init__(self, F, u, bcs=None, J=None, form_compiler_parameters=None):
"""Create nonlinear variational problem F(u; v) = 0.
Optional arguments bcs and J may be passed to specify boundary
conditions and the Jacobian J = dF/du.
Another optional argument form_compiler_parameters may be
specified to pass parameters to the form compiler.
"""
if bcs is None:
bcs = []
elif not isinstance(bcs, (list, tuple)):
bcs = [bcs]
# Store input UFL forms and solution Function
self.F_ufl = F
self.J_ufl = J
self.u_ufl = u
# Store form compiler parameters
form_compiler_parameters = form_compiler_parameters or {}
self.form_compiler_parameters = form_compiler_parameters
# Wrap forms
F = Form(F, form_compiler_parameters=form_compiler_parameters)
if J is not None:
J = Form(J, form_compiler_parameters=form_compiler_parameters)
# Initialize C++ base class
cpp.fem.NonlinearVariationalProblem.__init__(self, F, u._cpp_object, bcs, J)
def __init__(self,
a,
L=None,
solver_type=cpp.fem.LocalSolver.SolverType.LU):
"""Create a local (cell-wise) solver for a linear variational problem
a(u, v) = L(v).
"""
# Store input UFL forms and solution Function
self.a_ufl = a
self.L_ufl = L
# Wrap as DOLFIN forms
a = Form(a)
if L is None:
# Initialize C++ base class
cpp.fem.LocalSolver.__init__(self, a, solver_type)
else:
if L.empty():
L = cpp.fem.Form(1, 0)
else:
L = Form(L)
# Initialize C++ base class
cpp.fem.LocalSolver.__init__(self, a, L, solver_type)
def __init__(self, F, u, bcs=None, J=None, form_compiler_parameters=None):
"""
Create nonlinear variational problem F(u; v) = 0.
Optional arguments bcs and J may be passed to specify boundary
conditions and the Jacobian J = dF/du.
Another optional argument form_compiler_parameters may be
specified to pass parameters to the form compiler.
"""
# Extract and check arguments
u = _extract_u(u)
bcs = _extract_bcs(bcs)
# Store input UFL forms and solution Function
self.F_ufl = F
self.J_ufl = J
self.u_ufl = u
# Store form compiler parameters
form_compiler_parameters = form_compiler_parameters or {}
self.form_compiler_parameters = form_compiler_parameters
# Wrap forms
F = Form(F, form_compiler_parameters=form_compiler_parameters)
if J is not None:
J = Form(J, form_compiler_parameters=form_compiler_parameters)
# Initialize C++ base class
cpp.NonlinearVariationalProblem.__init__(self, F, u, bcs, J)
def __init__(self, a, L, u, bcs=None, form_compiler_parameters=None):
"""Create mixed linear variational problem a(u, v) = L(v).
An optional argument bcs may be passed to specify boundary
conditions.
Another optional argument form_compiler_parameters may be
specified to pass parameters to the form compiler.
"""
# Extract and check arguments (u is a list of Function)
u_comps = [u[i]._cpp_object for i in range(len(u))]
bcs = dolfin.fem.solving._extract_bcs(bcs)
# Store form compiler parameters
form_compiler_parameters = form_compiler_parameters or {}
self.form_compiler_parameters = form_compiler_parameters
# Update rhs if we don't have a consistent number of blocks
if len(L) != len(u):
L_tmp = [None for i in range(len(u))]
for Li in L:
L_tmp[Li.arguments()[0].part()] = Li
L = L_tmp
# Check number of blocks in lhs, rhs are consistent
assert(len(a) == len(u) * len(u))
assert(len(L) == len(u))
# Create list of forms/blocks
a_list = list()
L_list = list()
for Li in L:
if Li is None:
L_list.append([cpp.fem.Form(1, 0)]) # single-elt list
elif Li.empty():
L_list.append([cpp.fem.Form(1, 0)]) # single-elt list
else:
Ls = [] # List of Li subforms
for Lsub in sub_forms_by_domain(Li):
if Lsub is None:
Ls.append(cpp.fem.Form(1, 0))
elif Lsub.empty():
Ls.append(cpp.fem.Form(1, 0))
else:
Ls.append(Form(Lsub, form_compiler_parameters=form_compiler_parameters))
L_list.append(Ls)
for ai in a:
if ai is None:
a_list.append([cpp.fem.Form(2, 0)])
else:
As = []
for Asub in sub_forms_by_domain(ai):
As.append(Form(Asub, form_compiler_parameters=form_compiler_parameters))
a_list.append(As)
# Initialize C++ base class
cpp.fem.MixedLinearVariationalProblem.__init__(self, a_list, L_list, u_comps, bcs)
def generate_error_control(problem, goal):
"""
Create suitable ErrorControl object from problem and the goal
*Arguments*
problem (:py:class:`LinearVariationalProblem <dolfin.fem.solving.LinearVariationalProblem>`
or :py:class:`NonlinearVariationalProblem <dolfin.fem.solving.NonlinearVariationalProblem>`)
The (primal) problem
goal (:py:class:`Form <dolfin.fem.form.Form>`)
The goal functional
*Returns*
:py:class:`ErrorControl <dolfin.cpp.ErrorControl>`
"""
# Generate UFL forms to be used for error control
(ufl_forms, is_linear) = generate_error_control_forms(problem, goal)
# Compile generated forms
p = problem.form_compiler_parameters
forms = [Form(form, form_compiler_parameters=p) for form in ufl_forms]
# Create cpp.ErrorControl object
forms += [is_linear] # NOTE: Lingering design inconsistency.
ec = cpp.adaptivity.ErrorControl(*forms)
# Return generated ErrorControl
return ec
def _create_cpp_form(form: typing.Union[Form, cpp.fem.Form]) -> cpp.fem.Form:
"""Create a compiled Form from a UFL form"""
if form is None:
return None
elif isinstance(form, cpp.fem.Form):
return form
else:
# FIXME: Attach cpp Form to UFL Form to avoid re-processing
form = Form(form)
return form._cpp_object
def _create_cpp_form(form, form_compiler_parameters=None):
"""Create a C++ Form from a UFL form"""
if isinstance(form, cpp.fem.Form):
if form_compiler_parameters is not None:
cpp.warning(
"Ignoring form_compiler_parameters when passed a dolfin Form!")
return form
elif isinstance(form, ufl.Form):
form = Form(form, form_compiler_parameters=form_compiler_parameters)
return form._cpp_object
elif form is None:
return None
else:
raise TypeError("Invalid form type: {}".format(type(form)))
def _create_cpp_form(form, form_compiler_parameters=None):
# First check if we got a cpp.Form
if isinstance(form, cpp.fem.Form):
# Check that jit compilation has already happened
if not hasattr(form, "_compiled_form"):
raise TypeError(
"Expected a dolfin form to have a _compiled_form attribute.")
# Warn that we don't use the parameters if we get any
if form_compiler_parameters is not None:
cpp.warning(
"Ignoring form_compiler_parameters when passed a dolfin Form!")
return form
elif isinstance(form, ufl.Form):
form = Form(form, form_compiler_parameters=form_compiler_parameters)
return form._cpp_object
else:
raise TypeError("Invalid form type %s" % (type(form), ))
def __init__(self, problem, goal):
"""
Create AdaptiveNonlinearVariationalSolver
*Arguments*
problem (:py:class:`NonlinearVariationalProblem <dolfin.fem.solving.NonlinearVariationalProblem>`)
"""
# Store problem
self.problem = problem
# Generate error control object
ec = generate_error_control(self.problem, goal)
# Compile goal functional separately
p = self.problem.form_compiler_parameters
M = Form(goal, form_compiler_parameters=p)
# Initialize C++ base class
cpp.AdaptiveNonlinearVariationalSolver.__init__(self, problem, M, ec)
def _butcher_scheme_generator_adm(a, b, c, time, solution, rhs_form, adj):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
adj (_Function_)
The derivative of the functional with respect to y_n+1
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
kbar = [Function(solution.function_space(), name="kbar_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
ydot = solution.copy()
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
forward_forms = []
stage_solutions = []
jacobian_indices = []
# The recomputation of the forward run:
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(\
time_dep_expressions, time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v) * DX
stage_forms = [stage_form_implicit, derivative(\
stage_form_implicit, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
for i, kbari in reversed(list(enumerate(kbar))):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# And now the adjoint linearisation:
stage_form_adm = ufl.inner(dt * b[i] * adj, v)*DX + sum(\
[dt * float(a[j,i]) * safe_action(safe_adjoint(derivative(\
forward_forms[j], y_)), kbar[j]) for j in range(i, size)])
if explicit:
stage_forms_adm = [stage_form_adm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_adm -= ufl.inner(kbar[i], v) * DX
stage_forms_adm = [
stage_form_adm,
derivative(stage_form_adm, kbari)
]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_adm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_adm])
stage_solutions.append(kbari)
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(adj, v)*DX + sum(\
[safe_action(safe_adjoint(derivative(forward_forms[i], y_)), kbar[i]) \
for i in range(size)]))
else:
raise Exception("Not sure what to do here")
human_form = "unimplemented"
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage,\
stage_solutions, dt, human_form, adj
def _butcher_scheme_generator_tlm(a, b, c, time, solution, rhs_form,
perturbation):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
perturbation (_Function_)
The perturbation in the initial condition of the solution
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
kdot = [Function(solution.function_space(), name="kdot_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
ydot = solution.copy()
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
forward_forms = []
stage_solutions = []
jacobian_indices = []
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
# The recomputation of the forward run:
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v) * DX
stage_forms = [
stage_form_implicit,
derivative(stage_form_implicit, ki)
]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
# And now the tangent linearisation:
stage_form_tlm = safe_action(derivative(stage_form, y_), perturbation) + \
sum([dt*float(a[i,j]) * safe_action(derivative(\
forward_forms[j], y_), kdot[j]) for j in range(i+1)])
if explicit:
stage_forms_tlm = [stage_form_tlm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_tlm -= ufl.inner(kdot[i], v) * DX
stage_forms_tlm = [
stage_form_tlm,
derivative(stage_form_tlm, kdot[i])
]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_tlm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_tlm])
stage_solutions.append(kdot[i])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(perturbation + sum(\
[dt*float(bi)*kdoti for bi, kdoti in zip(b, kdot)], zero_), v)*DX)
else:
raise Exception("Not sure what to do here")
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
kdotterm = " + ".join("%(a)sh*action(derivative(f(t_n%(cih)s, y_n + "\
"%(kterm)s), kdot_%(i)s" % \
{"a": ("" if a[i,j] == 1.0 else "%s*"% a[i,j], j),
"i": i,
"cih": cih,
"kterm": kterm} \
for j in range(size) if a[i,j] != 0)
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {
"i": i,
"cih": cih
})
human_form.append("kdot_%(i)s = action(derivative("\
"f(t_n%(cih)s, y_n), y_n), ydot_n)" % \
{"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
human_form.append("kdot_%(i)s = action(derivative(f(t_n%(cih)s, "\
"y_n + %(kterm)s), y_n) + %(kdotterm)s" % \
{"i": i, "cih": cih, "kterm": kterm, "kdotterm": kdotterm})
parentheses = "(%s)" if np.sum(b > 0) > 1 else "%s"
human_form.append("ydot_{n+1} = ydot_n + h*" + parentheses % (" + ".join(\
"%skdot_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
stage_solutions, dt, human_form, perturbation
def _butcher_scheme_generator(a, b, c, time, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
jacobian_indices = []
# Create the stage forms
y_ = solution
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v) * DX
stage_forms = [stage_form, derivative(stage_form, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b, k)], zero_), v)*DX)
else:
# FIXME: Add support for addaptivity in RKSolver and MultiStageScheme
last_stage = [Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[0,:], k)], zero_), v)*DX),
Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[1,:], k)], zero_), v)*DX)]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {
"i": i,
"cih": cih
})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b > 0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
k, dt, human_form, None
def _rush_larsen_scheme_generator(rhs_form, solution, time, order, generalized):
"""Generates a list of forms and solutions for a given Butcher
tableau
*Arguments*
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
solution (_Function_)
The prognostic variable
time (_Constant_)
A Constant holding the time at the start of the time step
order (int)
The order of the scheme
generalized (bool)
If True generate a generalized Rush Larsen scheme, linearizing all
components.
"""
DX = _check_form(rhs_form)
if DX != ufl.dP:
raise TypeError("Expected a form with a Pointintegral.")
# Create time step
dt = Constant(0.1)
# Get test function
# arguments = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
# Get time dependent expressions
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
# Extract rhs expressions from form
rhs_integrand = rhs_form.integrals()[0].integrand()
rhs_exprs, v = extract_tested_expressions(rhs_integrand)
vector_rhs = len(v.ufl_shape) > 0 and v.ufl_shape[0] > 1
system_size = v.ufl_shape[0] if vector_rhs else 1
# Fix for indexing of v for scalar expressions
v = v if vector_rhs else [v]
# Extract linear terms if not using generalized Rush Larsen
if not generalized:
linear_terms = _find_linear_terms(rhs_exprs, solution)
else:
linear_terms = [True for _ in range(system_size)]
# Wrap the rhs expressions into a ufl vector type
rhs_exprs = ufl.as_vector([rhs_exprs[i] for i in range(system_size)])
rhs_jac = ufl.diff(rhs_exprs, solution)
# Takes time!
if vector_rhs:
diff_rhs_exprs = [expand_indices(expand_derivatives(rhs_jac[ind, ind]))
for ind in range(system_size)]
else:
diff_rhs_exprs = [expand_indices(expand_derivatives(rhs_jac[0]))]
solution = [solution]
ufl_stage_forms = []
dolfin_stage_forms = []
dt_stage_offsets = []
# Stage solutions (3 per order rhs, linearized, and final step)
# If 2nd order the final step for 1 step is a stage
if order == 1:
stage_solutions = []
rl_ufl_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms,
system_size, solution, None, dt, time, 1.0,
0.0, v, DX, time_dep_expressions)
elif order == 2:
# Stage solution for order 2
if vector_rhs:
stage_solutions = [Function(solution.function_space(), name="y_1/2")]
else:
stage_solutions = [Function(solution[0].function_space(), name="y_1/2")]
stage_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs,
linear_terms, system_size,
solution, None, dt, time, 0.5,
0.0, v, DX,
time_dep_expressions)
rl_ufl_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs,
linear_terms, system_size,
solution, stage_solutions[0],
dt, time, 1.0, 0.5, v, DX,
time_dep_expressions)
ufl_stage_forms.append([stage_form])
dolfin_stage_forms.append([Form(stage_form)])
# Get last stage form
last_stage = Form(rl_ufl_form)
human_form = "%srush larsen %s" % ("generalized " if generalized else "",
str(order))
return rhs_form, linear_terms, ufl_stage_forms, dolfin_stage_forms, last_stage, \
stage_solutions, dt, dt_stage_offsets, human_form, None
def _rush_larsen_scheme_generator_adm(rhs_form, solution, time, order,
generalized, perturbation):
"""Generates a list of forms and solutions for the adjoint
linearisation of the Rush-Larsen scheme
*Arguments*
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
solution (_Function_)
The prognostic variable
time (_Constant_)
A Constant holding the time at the start of the time step
order (int)
The order of the scheme
generalized (bool)
If True generate a generalized Rush Larsen scheme, linearizing all
components.
perturbation (Function)
The vector on which we compute the adjoint action.
"""
DX = _check_form(rhs_form)
if DX != ufl.dP:
raise TypeError("Expected a form with a Pointintegral.")
# Create time step
dt = Constant(0.1)
# Get test function
# arguments = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
# Get time dependent expressions
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
# Extract rhs expressions from form
rhs_integrand = rhs_form.integrals()[0].integrand()
rhs_exprs, v = extract_tested_expressions(rhs_integrand)
vector_rhs = len(v.ufl_shape) > 0 and v.ufl_shape[0] > 1
system_size = v.ufl_shape[0] if vector_rhs else 1
# Fix for indexing of v for scalar expressions
v = v if vector_rhs else [v]
# Extract linear terms if not using generalized Rush Larsen
if not generalized:
linear_terms = _find_linear_terms(rhs_exprs, solution)
else:
linear_terms = [True for _ in range(system_size)]
# Wrap the rhs expressions into a ufl vector type
rhs_exprs = ufl.as_vector([rhs_exprs[i] for i in range(system_size)])
rhs_jac = ufl.diff(rhs_exprs, solution)
# Takes time!
if vector_rhs:
diff_rhs_exprs = [expand_indices(expand_derivatives(rhs_jac[ind, ind]))
for ind in range(system_size)]
soln = solution
else:
diff_rhs_exprs = [expand_indices(expand_derivatives(rhs_jac[0]))]
solution = [solution]
soln = solution[0]
ufl_stage_forms = []
dolfin_stage_forms = []
dt_stage_offsets = []
trial = TrialFunction(soln.function_space())
# Stage solutions (3 per order rhs, linearized, and final step)
# If 2nd order the final step for 1 step is a stage
if order == 1:
stage_solutions = []
# Fetch the original step
rl_ufl_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs,
linear_terms, system_size,
solution, None, dt, time, 1.0,
0.0, v, DX,
time_dep_expressions)
# If this is commented out, we don't get NaNs. Yhy is
# solution a list of length zero anyway?
rl_ufl_form = safe_action(safe_adjoint(derivative(rl_ufl_form, soln, trial)),
perturbation)
elif order == 2:
# Stage solution for order 2
fn_space = soln.function_space()
stage_solutions = [Function(fn_space, name="y_1/2"),
Function(fn_space, name="y_1"),
Function(fn_space, name="y_bar_1/2")]
y_half_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs,
linear_terms, system_size,
solution, None, dt, time, 0.5,
0.0, v, DX,
time_dep_expressions)
y_one_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs,
linear_terms, system_size,
solution, stage_solutions[0],
dt, time, 1.0, 0.5, v, DX,
time_dep_expressions)
y_bar_half_form = safe_action(safe_adjoint(derivative(y_one_form,
stage_solutions[0], trial)), perturbation)
rl_ufl_form = safe_action(safe_adjoint(derivative(y_one_form, soln, trial)), perturbation) + \
safe_action(safe_adjoint(derivative(y_half_form, soln, trial)), stage_solutions[2])
ufl_stage_forms.append([y_half_form])
ufl_stage_forms.append([y_one_form])
ufl_stage_forms.append([y_bar_half_form])
dolfin_stage_forms.append([Form(y_half_form)])
dolfin_stage_forms.append([Form(y_one_form)])
dolfin_stage_forms.append([Form(y_bar_half_form)])
# Get last stage form
last_stage = Form(rl_ufl_form)
human_form = "%srush larsen %s" % ("generalized " if generalized else "",
str(order))
return rhs_form, linear_terms, ufl_stage_forms, dolfin_stage_forms, last_stage, \
stage_solutions, dt, dt_stage_offsets, human_form, perturbation
def __init__(self, F, u, bcs=None, J=None, form_compiler_parameters=None):
"""Create nonlinear variational problem F(u; v) = 0.
An optional argument bcs may be passed to specify boundary
conditions.
Another optional argument form_compiler_parameters may be
specified to pass parameters to the form compiler.
"""
# Extract and check arguments (u is a list of Function)
u_comps = [u[i]._cpp_object for i in range(len(u))]
bcs = dolfin.fem.solving._extract_bcs(bcs)
# Store form compiler parameters
form_compiler_parameters = form_compiler_parameters or {}
self.form_compiler_parameters = form_compiler_parameters
# Store input UFL forms and solution Function
self.F_ufl = F
self.J_ufl = J
self.u_ufl = u
# Update rhs if we don't have a consistent number of blocks
if len(F) != len(u):
F_tmp = [None for i in range(len(u))]
for Fi in F:
F_tmp[Fi.arguments()[0].part()] = Fi
F = F_tmp
# Check number of blocks in the residual and solution are coherent
assert(len(J) == len(u) * len(u))
assert(len(F) == len(u))
# Create list of forms/blocks
F_list = list()
for Fi in F:
if Fi is None:
F_list.append([cpp.fem.Form(1, 0)])
elif Fi.empty():
F_list.append([cpp.fem.Form(1, 0)]) # single-elt list
else:
Fs = []
for Fsub in sub_forms_by_domain(Fi):
if Fsub is None:
Fs.append(cpp.fem.Form(1, 0))
elif Fsub.empty():
Fs.append(cpp.fem.Form(1, 0))
else:
Fs.append(Form(Fsub, form_compiler_parameters=form_compiler_parameters))
F_list.append(Fs)
print("[problem] create list of residual forms OK")
J_list = None
if J is not None:
J_list = list()
for Ji in J:
if Ji is None:
J_list.append([cpp.fem.Form(2, 0)])
elif Ji.empty():
J_list.append([cpp.fem.Form(2, 0)])
else:
Js = []
for Jsub in sub_forms_by_domain(Ji):
Js.append(Form(Jsub, form_compiler_parameters=form_compiler_parameters))
J_list.append(Js)
print("[problem] create list of jacobian forms OK, J_list size = ", len(J_list))
# Initialize C++ base class
cpp.fem.MixedNonlinearVariationalProblem.__init__(self, F_list, u_comps, bcs, J_list)
def _butcher_scheme_generator(a, b, c, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
solution (_Function_)
The prognastic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
if not (isinstance(a, np.ndarray) and (len(a) == 1 or \
(len(a.shape)==2 and a.shape[0] == a.shape[1]))):
raise TypeError("Expected an m x m numpy array as the first argument")
if not (isinstance(b, np.ndarray) and len(b.shape) in [1,2]):
raise TypeError("Expected a 1 or 2 dimensional numpy array as the second argument")
if not (isinstance(c, np.ndarray) and len(c.shape) == 1):
raise TypeError("Expected a 1 dimensional numpy array as the third argument")
# Make sure a is a "matrix"
if len(a) == 1:
a.shape = (1, 1)
# Get size of system
size = a.shape[0]
# If b is a matrix we expect it to have two rows
if len(b.shape) == 2:
if not (b.shape[0] == 2 and b.shape[1] == size):
raise ValueError("Expected a 2 row matrix with the same number "\
"of collumns as the first dimension of the a matrix.")
elif len(b) != size:
raise ValueError("Expected the length of the b vector to have the "\
"same size as the first dimension of the a matrix.")
if len(c) != size:
raise ValueError("Expected the length of the c vector to have the "\
"same size as the first dimension of the a matrix.")
# Check if tableau is fully implicit
for i in range(size):
for j in range(i):
if a[j, i] != 0:
raise ValueError("Does not support fully implicit Butcher tableau.")
if not isinstance(rhs_form, ufl.Form):
raise TypeError("Expected a ufl.Form as the 5th argument.")
# Check if form contains a cell or point integral
if "cell" in rhs_form.integral_groups():
DX = ufl.dx
elif "point" in rhs_form.integral_groups():
DX = ufl.dP
else:
raise ValueError("Expected either a cell or point integral in the form.")
# Get test function
arguments, coefficients = ufl.algorithms.extract_arguments_and_coefficients(rhs_form)
if len(arguments) != 1:
raise ValueError("Expected the form to have rank 1")
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [solution.copy(deepcopy=True) for i in range(size)]
# Create the stage forms
y_ = solution
for i, ki in enumerate(k):
# Check wether the stage is explicit
explicit = a[i,i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], ufl.zero(*y_.shape()))
stage_form = ufl.replace(rhs_form, {y_:evalargs})
if explicit:
stage_forms = [stage_form]
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v)*DX
stage_forms = [stage_form, derivative(stage_form, ki)]
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b, k)])
else:
# FIXME: Add support for addaptivity in RKSolver and MultiStageScheme
last_stage = [cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b[0,:], k)]),
cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b[1,:], k)])]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b>0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, last_stage, k, dt, human_form