def interpolate(v, V):
"""
Return interpolation of a given function into a given finite element space.
*Arguments*
v
a :py:class:`Function <dolfin.functions.function.Function>` or
an :py:class:`Expression <dolfin.functions.expression.Expression>`
V
a :py:class:`FunctionSpace (standard, mixed, etc.)
<dolfin.functions.functionspace.FunctionSpaceBase>`
*Example of usage*
.. code-block:: python
v = Expression("sin(pi*x[0])")
V = FunctionSpace(mesh, "Lagrange", 1)
Iv = interpolate(v, V)
"""
# Check arguments
if not isinstance(V, FunctionSpaceBase):
cpp.dolfin_error(
"interpolation.py", "compute interpolation",
"Illegal function space for interpolation, not a FunctionSpace (%s)"
% str(v))
# Compute interpolation
Pv = Function(V)
Pv.interpolate(v)
return Pv
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