def set_linear_dirichlet_constant(mesh,
dt,
var_name='Temperature',
bc_temperature=1300.,
temperature=300.,
source_value=0.,
axis=1,
right=True,
tol=1e-4,
conductivity=1.,
density=1.,
specific_heat=1.):
'''
Heat equation problem where an edge/surface is set at a given temperature
and everything else at another.
Parameters
----------
axis : int
Controls boundary to which Dirichlet bc is applied.
right : bool
Dirichlet bc is applied to the planar and perpendicular face found in
the max (True) or min (False) coordinate.
'''
# function space
V = FunctionSpace(mesh, 'Lagrange', 1)
# boundary conditions
bcs = [
set_dirichlet_bc_lim(V,
value=bc_temperature,
axis=axis,
right=right,
tol=tol)
]
# initial condition
u_initial = Constant(temperature)
u_n = Function(V, name=var_name)
u_n.interpolate(u_initial)
# problem formulation
f = Constant(source_value)
u_n, a, L = get_formulation_constant_props(V,
u_initial,
f,
dt,
conductivity,
density,
specific_heat,
var_name=var_name)
u = Function(V, name=var_name)
return u_n, a, L, u, bcs
def get_formulation(V, u_initial, f, dt, var_name='Temperature'):
# interpolate initial condition
u_n = Function(V, name=var_name)
u_n.interpolate(u_initial)
# function spaces
u_h = TrialFunction(V)
v = TestFunction(V)
# formulation
a = u_h * v * dx + dt * dot(grad(u_h), grad(v)) * dx
L = (u_n + dt * f) * v * dx
return u_n, a, L
def set_poisson_eq_2d(n=8):
'''
Basic example where only the mesh refinement is controlled.
'''
# mesh
mesh = UnitSquareMesh(n, n)
# function space
V = FunctionSpace(mesh, 'P', 1)
# bcs
u_D = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]', degree=2)
def boundary(x, on_boundary):
return on_boundary
bc = DirichletBC(V, u_D, boundary)
# variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(-6.0)
a = dot(grad(u), grad(v)) * dx
L = f * v * dx
# define unknown
u = Function(V)
return a, L, u, bc
def read_checkpoint(self, V, name: str, counter: int = -1) -> Function:
"""Read finite element Function from checkpointing format
Parameters
----------
V
Function space of saved function.
name
Name of function as saved into XDMF file.
counter : optional
Position of function in the file within functions of the same
name. Counter is used to read function saved as time series.
To get last saved function use counter=-1, or counter=-2 for
one before last, etc.
Note
----
Parameter `V: Function space` must be the same as saved function space
except for ordering of mesh entities.
Returns
-------
dolfin.function.function.Function
The finite element Function read from checkpoint file
"""
V_cpp = getattr(V, "_cpp_object", V)
u_cpp = self._cpp_object.read_checkpoint(V_cpp, name, counter)
return Function(V, u_cpp.vector())
def set_linear_equal_opposite(mesh,
dt,
var_name='Temperature',
axis=0,
bc_temperature=300.,
source_value=0.,
conductivity=1.,
density=1.,
specific_heat=1.,
V=None):
'''
Heat equation problem where two opposite edges/faces are set at the same
temperature and the points in between follow a quadratic variation.
Notes
-----
* assumes opposite faces of interest are planar and perpendicular to the
axis.
'''
# function space
if V is None:
V = FunctionSpace(mesh, 'Lagrange', 1)
# boundary conditions
x_min, x_max = get_mesh_axis_lims(mesh, axis)
bcs = [
set_dirichlet_bc(V, x, bc_temperature, axis) for x in [x_min, x_max]
]
# initial condition
c = (x_max - x_min) / 2
param = bc_temperature / (c**2)
u_initial = Expression('param * (x[axis] - c) * (x[axis] - c)',
degree=2,
param=param,
c=c,
name='T',
axis=axis)
# problem formulation
f = Constant(source_value)
u_n, a, L = get_formulation_constant_props(V,
u_initial,
f,
dt,
conductivity,
density,
specific_heat,
var_name=var_name)
u = Function(V, name=var_name)
return u_n, a, L, u, bcs
def load_last_checkpoint(xdmf_filename, V, var_name):
# TODO: file should be passed in order to open only once?
# get times
times = get_times_from_xdfm_checkpoints(xdmf_filename)
# load last checkpoint
with XDMFFile(MPI.comm_self, xdmf_filename) as file:
u = Function(V, name=var_name)
file.read_checkpoint(u, var_name, -1)
return u, times[-1], len(times) - 1
def get_formulation_constant_props(V, u_initial, f, dt, conductivity, density,
specific_heat, var_name='Temperature'):
# medium properties
k = Constant(conductivity)
rho = Constant(density)
c_p = Constant(specific_heat)
# interpolate initial condition
u_n = Function(V, name=var_name)
u_n.interpolate(u_initial)
# function spaces
u_h = TrialFunction(V)
v = TestFunction(V)
# formulation
param1 = rho * c_p
param2 = k / param1
a = u_h * v * dx + param2 * dt * dot(grad(u_h), grad(v)) * dx
L = (u_n + param1 * dt * f) * v * dx
return u_n, a, L
def initialize_data(self):
"""
Extract required objects for defining error control
forms. This will be stored and reused.
"""
# Developer's note: The UFL-FFC-DOLFIN--PyDOLFIN toolchain for
# error control is quite fine-tuned. In particular, the order
# of coefficients in forms is (and almost must be) used for
# their assignment. This means that the order in which these
# coefficients are defined matters and should be considered
# fixed.
# Primal trial element space
self._V = self.u.function_space()
# Primal test space == Dual trial space
Vhat = self.weak_residual.arguments()[0].function_space()
# Discontinuous version of primal trial element space
self._dV = tear(self._V)
# Extract cell and geometric dimension
mesh = self._V.mesh()
dim = mesh.topology().dim()
# Function representing improved dual
E = increase_order(Vhat)
self._Ez_h = Function(E)
# Function representing cell bubble function
B = FunctionSpace(mesh, "B", dim + 1)
self._b_T = Function(B)
self._b_T.vector()[:] = 1.0
# Function representing strong cell residual
self._R_T = Function(self._dV)
# Function representing cell cone function
C = FunctionSpace(mesh, "DG", dim)
self._b_e = Function(C)
# Function representing strong facet residual
self._R_dT = Function(self._dV)
# Function for discrete dual on primal test space
self._z_h = Function(Vhat)
# Piecewise constants for assembling indicators
self._DG0 = FunctionSpace(mesh, "DG", 0)
def initialize_data(self):
"""
Extract required objects for defining error control
forms. This will be stored and reused.
"""
# Developer's note: The UFL-FFC-DOLFIN--PyDOLFIN toolchain for
# error control is quite fine-tuned. In particular, the order
# of coefficients in forms is (and almost must be) used for
# their assignment. This means that the order in which these
# coefficients are defined matters and should be considered
# fixed.
# Primal trial element space
self._V = self.u.function_space()
# Primal test space == Dual trial space
Vhat = self.weak_residual.arguments()[0].function_space()
# Discontinuous version of primal trial element space
self._dV = tear(self._V)
# Extract cell and geometric dimension
mesh = self._V.mesh()
dim = mesh.topology().dim()
# Function representing improved dual
E = increase_order(Vhat)
self._Ez_h = Function(E)
# Function representing cell bubble function
B = FunctionSpace(mesh, "B", dim + 1)
self._b_T = Function(B)
self._b_T.vector()[:] = 1.0
# Function representing strong cell residual
self._R_T = Function(self._dV)
# Function representing cell cone function
C = FunctionSpace(mesh, "DG", dim)
self._b_e = Function(C)
# Function representing strong facet residual
self._R_dT = Function(self._dV)
# Function for discrete dual on primal test space
self._z_h = Function(Vhat)
# Piecewise constants for assembling indicators
self._DG0 = FunctionSpace(mesh, "DG", 0)
def interpolate(v, V):
"""Return interpolation of a given function into a given finite
element space.
*Arguments*
v
a :py:class:`Function <dolfin.function.function.Function>` or
a :py:class:`MultiMeshFunction
<dolfin.function.function.MultiMeshFunction>` or
an :py:class:`Expression <dolfin.functions.expression.Expression>`
V
a :py:class:`FunctionSpace (standard, mixed, etc.)
<dolfin.functions.functionspace.FunctionSpace>`
or a :py:class:`MultiMeshFunctionSpace
<dolfin.function.MultiMeshFunctionSpace>`.
*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, cpp.functionFunctionSpace):
# cpp.dolfin_error("interpolation.py",
# "compute interpolation",
# "Illegal function space for interpolation, not a FunctionSpace (%s)" % str(v))
if isinstance(V, MultiMeshFunctionSpace):
Pv = MultiMeshFunction(V)
else:
Pv = Function(V)
# Compute interpolation
if hasattr(v, "_cpp_object"):
Pv.interpolate(v._cpp_object)
else:
Pv.interpolate(v)
return Pv
def convert_xdmf_checkpoints_to_vtk(xdmf_filename,
V,
var_name,
vtk_filename=None):
# get times
times = get_times_from_xdfm_checkpoints(xdmf_filename)
# create vtk file
if vtk_filename is None:
vtk_filename = xdmf_filename.split('.')[0] + '.pvd'
vtkfile = File(vtk_filename)
# load variable and save as vtk
with XDMFFile(xdmf_filename) as file:
for i, t in enumerate(times):
u = Function(V, name=var_name)
file.read_checkpoint(u, var_name, i)
vtkfile << (u, t)
def read_function(self, V, name: str):
"""Read finite element Function from file
Parameters
----------
V
Function space of saved function.
name
Name of function as saved into HDF file.
Returns
-------
dolfin.function.function.Function
Function read from file
Note
----
Parameter `V: Function space` must be the same as saved function space
except for ordering of mesh entities.
"""
V_cpp = getattr(V, "_cpp_object", V)
u_cpp = self._cpp_object.read(V_cpp, name)
return Function(V, u_cpp.vector())
def r2_errornorm(u, uh, norm_type="l2", degree_rise=3, mesh=None ):
"""
This function is a modification of FEniCS's built-in errornorm function that adopts the :math:`r^2dr`
measure as opposed to the standard Cartesian :math:`dx` measure.
For documentation and usage, see the
original module <https://bitbucket.org/fenics-project/dolfin/src/master/python/dolfin/fem/norms.py>_.
"""
# Get mesh
if isinstance(u, cpp.function.Function) and mesh is None:
mesh = u.function_space().mesh()
if isinstance(uh, cpp.function.Function) and mesh is None:
mesh = uh.function_space().mesh()
# if isinstance(uh, MultiMeshFunction) and mesh is None:
# mesh = uh.function_space().multimesh()
if hasattr(uh, "_cpp_object") and mesh is None:
mesh = uh._cpp_object.function_space().mesh()
if hasattr(u, "_cpp_object") and mesh is None:
mesh = u._cpp_object.function_space().mesh()
if mesh is None:
raise RuntimeError("Cannot compute error norm. Missing mesh.")
# Get rank
if not u.ufl_shape == uh.ufl_shape:
raise RuntimeError("Cannot compute error norm. Value shapes do not match.")
shape = u.ufl_shape
rank = len(shape)
# Check that uh is associated with a finite element
if uh.ufl_element().degree() is None:
raise RuntimeError("Cannot compute error norm. Function uh must have a finite element.")
# Degree for interpolation space. Raise degree with respect to uh.
degree = uh.ufl_element().degree() + degree_rise
# Check degree of 'exact' solution u
degree_u = u.ufl_element().degree()
if degree_u is not None and degree_u < degree:
cpp.warning("Degree of exact solution may be inadequate for accurate result in errornorm.")
# Create function space
if rank == 0:
V = FunctionSpace(mesh, "Discontinuous Lagrange", degree)
elif rank == 1:
V = VectorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
dim=shape[0])
elif rank > 1:
V = TensorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
shape=shape)
# Interpolate functions into finite element space
pi_u = interpolate(u, V)
pi_uh = interpolate(uh, V)
# Compute the difference
e = Function(V)
e.assign(pi_u)
e.vector().axpy(-1.0, pi_uh.vector())
# Compute norm
return r2_norm(e, func_degree=degree, norm_type=norm_type, mesh=mesh )
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 _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 = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
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
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_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 _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 _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 = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
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
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_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 = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
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_.ufl_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 adaptivity 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
class DOLFINErrorControlGenerator(ErrorControlGenerator):
"""
This class provides a realization of
ffc.errorcontrol.errorcontrolgenerators.ErrorControlGenerator for
use with UFL forms defined over DOLFIN objects
"""
def __init__(self, F, M, u):
"""
*Arguments*
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.
"""
ErrorControlGenerator.__init__(self, __import__("dolfin"), F, M, u)
def initialize_data(self):
"""
Extract required objects for defining error control
forms. This will be stored and reused.
"""
# Developer's note: The UFL-FFC-DOLFIN--PyDOLFIN toolchain for
# error control is quite fine-tuned. In particular, the order
# of coefficients in forms is (and almost must be) used for
# their assignment. This means that the order in which these
# coefficients are defined matters and should be considered
# fixed.
# Primal trial element space
self._V = self.u.function_space()
# Primal test space == Dual trial space
Vhat = self.weak_residual.arguments()[0].function_space()
# Discontinuous version of primal trial element space
self._dV = tear(self._V)
# Extract cell and geometric dimension
mesh = self._V.mesh()
dim = mesh.topology().dim()
# Function representing improved dual
E = increase_order(Vhat)
self._Ez_h = Function(E)
# Function representing cell bubble function
B = FunctionSpace(mesh, "B", dim + 1)
self._b_T = Function(B)
self._b_T.vector()[:] = 1.0
# Function representing strong cell residual
self._R_T = Function(self._dV)
# Function representing cell cone function
C = FunctionSpace(mesh, "DG", dim)
self._b_e = Function(C)
# Function representing strong facet residual
self._R_dT = Function(self._dV)
# Function for discrete dual on primal test space
self._z_h = Function(Vhat)
# Piecewise constants for assembling indicators
self._DG0 = FunctionSpace(mesh, "DG", 0)
def project(v, V=None, bcs=None, mesh=None,
function=None,
solver_type="lu",
preconditioner_type="default",
form_compiler_parameters=None):
"""Return projection of given expression *v* onto the finite element
space *V*.
*Arguments*
v
a :py:class:`Function <dolfin.functions.function.Function>` or
an :py:class:`Expression <dolfin.functions.expression.Expression>`
bcs
Optional argument :py:class:`list of DirichletBC
<dolfin.fem.bcs.DirichletBC>`
V
Optional argument :py:class:`FunctionSpace
<dolfin.functions.functionspace.FunctionSpace>`
mesh
Optional argument :py:class:`mesh <dolfin.cpp.Mesh>`.
solver_type
see :py:func:`solve <dolfin.fem.solving.solve>` for options.
preconditioner_type
see :py:func:`solve <dolfin.fem.solving.solve>` for options.
form_compiler_parameters
see :py:class:`Parameters <dolfin.cpp.Parameters>` for more
information.
*Example of usage*
.. code-block:: python
v = Expression("sin(pi*x[0])")
V = FunctionSpace(mesh, "Lagrange", 1)
Pv = project(v, V)
This is useful for post-processing functions or expressions
which are not readily handled by visualization tools (such as
for example discontinuous functions).
"""
# Try figuring out a function space if not specified
if V is None:
# Create function space based on Expression element if trying
# to project an Expression
if isinstance(v, dolfin.function.expression.Expression):
if mesh is not None and isinstance(mesh, cpp.mesh.Mesh):
V = FunctionSpace(mesh, v.ufl_element())
# else:
# cpp.dolfin_error("projection.py",
# "perform projection",
# "Expected a mesh when projecting an Expression")
else:
# Otherwise try extracting function space from expression
V = _extract_function_space(v, mesh)
# Projection into a MultiMeshFunctionSpace
if isinstance(V, MultiMeshFunctionSpace):
# Create the measuresum of uncut and cut-cells
dX = ufl.dx() + ufl.dC()
# Define variational problem for projection
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv) * dX
L = ufl.inner(w, v) * dX
# Assemble linear system
A = assemble_multimesh(a, form_compiler_parameters=form_compiler_parameters)
b = assemble_multimesh(L, form_compiler_parameters=form_compiler_parameters)
# Solve linear system for projection
if function is None:
function = MultiMeshFunction(V)
cpp.la.solve(A, function.vector(), b, solver_type, preconditioner_type)
return function
# Ensure we have a mesh and attach to measure
if mesh is None:
mesh = V.mesh()
dx = ufl.dx(mesh)
# Define variational problem for projection
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv) * dx
L = ufl.inner(w, v) * dx
# Assemble linear system
A, b = assemble_system(a, L, bcs=bcs,
form_compiler_parameters=form_compiler_parameters)
# Solve linear system for projection
if function is None:
function = Function(V)
cpp.la.solve(A, function.vector(), b, solver_type, preconditioner_type)
return function
def project(v, V=None, func_degree=None, bcs=None, mesh=None,
function=None,
solver_type="lu",
preconditioner_type="default",
form_compiler_parameters=None):
"""
This function is a modification of FEniCS's built-in project function that adopts the :math:`r^2dr`
measure as opposed to the standard Cartesian :math:`dx` measure.
For documentation and usage, see the
`original module <https://bitbucket.org/fenics-project/dolfin/src/master/python/dolfin/fem/projection.py>`_.
.. note:: Note the extra argument func_degree: this is used to interpolate the :math:`r^2`
Expression to the same degree as used in the definition of the Trial and Test function
spaces.
"""
# Try figuring out a function space if not specified
if V is None:
# Create function space based on Expression element if trying
# to project an Expression
if isinstance(v, Expression):
# FIXME: Add handling of cpp.MultiMesh
if mesh is not None and isinstance(mesh, cpp.mesh.Mesh):
V = FunctionSpace(mesh, v.ufl_element())
# else:
# cpp.dolfin_error("projection.py",
# "perform projection",
# "Expected a mesh when projecting an Expression")
else:
# Otherwise try extracting function space from expression
V = _extract_function_space(v, mesh)
# Ensure we have a mesh and attach to measure
if mesh is None:
mesh = V.mesh()
dx = ufl.dx(mesh)
# Define variational problem for projection
# DS: HERE IS WHERE I MODIFY
r2 = Expression('pow(x[0],2)', degree=func_degree)
w = TestFunction(V)
Pv = TrialFunction(V)
a = ufl.inner(w, Pv) * r2 * dx
L = ufl.inner(w, v) * r2 * dx
# Assemble linear system
A, b = assemble_system(a, L, bcs=bcs,
form_compiler_parameters=form_compiler_parameters)
# Solve linear system for projection
if function is None:
function = Function(V)
cpp.la.solve(A, function.vector(), b, solver_type, preconditioner_type)
return function
def part(self, i, deepcopy=False):
f = Function(self._cpp_object.part(i, deepcopy))
f.rename(self._cpp_object.name(), self._cpp_object.label())
return f
class DOLFINErrorControlGenerator(ErrorControlGenerator):
"""
This class provides a realization of
ffc.errorcontrol.errorcontrolgenerators.ErrorControlGenerator for
use with UFL forms defined over DOLFIN objects
"""
def __init__(self, F, M, u):
"""
*Arguments*
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.
"""
ErrorControlGenerator.__init__(self, __import__("dolfin"), F, M, u)
def initialize_data(self):
"""
Extract required objects for defining error control
forms. This will be stored and reused.
"""
# Developer's note: The UFL-FFC-DOLFIN--PyDOLFIN toolchain for
# error control is quite fine-tuned. In particular, the order
# of coefficients in forms is (and almost must be) used for
# their assignment. This means that the order in which these
# coefficients are defined matters and should be considered
# fixed.
# Primal trial element space
self._V = self.u.function_space()
# Primal test space == Dual trial space
Vhat = self.weak_residual.arguments()[0].function_space()
# Discontinuous version of primal trial element space
self._dV = tear(self._V)
# Extract cell and geometric dimension
mesh = self._V.mesh()
dim = mesh.topology().dim()
# Function representing improved dual
E = increase_order(Vhat)
self._Ez_h = Function(E)
# Function representing cell bubble function
B = FunctionSpace(mesh, "B", dim + 1)
self._b_T = Function(B)
self._b_T.vector()[:] = 1.0
# Function representing strong cell residual
self._R_T = Function(self._dV)
# Function representing cell cone function
C = FunctionSpace(mesh, "DG", dim)
self._b_e = Function(C)
# Function representing strong facet residual
self._R_dT = Function(self._dV)
# Function for discrete dual on primal test space
self._z_h = Function(Vhat)
# Piecewise constants for assembling indicators
self._DG0 = FunctionSpace(mesh, "DG", 0)
def errornorm(u, uh, norm_type="l2", degree_rise=3, mesh=None):
"""
Compute and return the error :math:`e = u - u_h` in the given norm.
*Arguments*
u, uh
:py:class:`Functions <dolfin.functions.function.Function>`
norm_type
Type of norm. The :math:`L^2` -norm is default.
For other norms, see :py:func:`norm <dolfin.fem.norms.norm>`.
degree_rise
The number of degrees above that of u_h used in the
interpolation; i.e. the degree of piecewise polynomials used
to approximate :math:`u` and :math:`u_h` will be the degree
of :math:`u_h` + degree_raise.
mesh
Optional :py:class:`Mesh <dolfin.cpp.Mesh>` on
which to compute the error norm.
In simple cases, one may just define
.. code-block:: python
e = u - uh
and evalute for example the square of the error in the :math:`L^2` -norm by
.. code-block:: python
assemble(e**2*dx(mesh))
However, this is not stable w.r.t. round-off errors considering that
the form compiler may expand(#) the expression above to::
e**2*dx = u**2*dx - 2*u*uh*dx + uh**2*dx
and this might get further expanded into thousands of terms for
higher order elements. Thus, the error will be evaluated by adding
a large number of terms which should sum up to something close to
zero (if the error is small).
This module computes the error by first interpolating both
:math:`u` and :math:`u_h` to a common space (of high accuracy),
then subtracting the two fields (which is easy since they are
expressed in the same basis) and then evaluating the integral.
(#) If using the tensor representation optimizations.
The quadrature represenation does not suffer from this problem.
"""
# Check argument
# if not isinstance(u, cpp.function.GenericFunction):
# cpp.dolfin_error("norms.py",
# "compute error norm",
# "Expecting a Function or Expression for u")
# if not isinstance(uh, cpp.function.Function):
# cpp.dolfin_error("norms.py",
# "compute error norm",
# "Expecting a Function for uh")
# Get mesh
if isinstance(u, cpp.function.Function) and mesh is None:
mesh = u.function_space().mesh()
if isinstance(uh, cpp.function.Function) and mesh is None:
mesh = uh.function_space().mesh()
if hasattr(uh, "_cpp_object") and mesh is None:
mesh = uh._cpp_object.function_space().mesh()
if hasattr(u, "_cpp_object") and mesh is None:
mesh = u._cpp_object.function_space().mesh()
if mesh is None:
cpp.dolfin_error("norms.py",
"compute error norm",
"Missing mesh")
# Get rank
if not u.ufl_shape == uh.ufl_shape:
cpp.dolfin_error("norms.py",
"compute error norm",
"Value shapes don't match")
shape = u.ufl_shape
rank = len(shape)
# Check that uh is associated with a finite element
if uh.ufl_element().degree() is None:
cpp.dolfin_error("norms.py",
"compute error norm",
"Function uh must have a finite element")
# Degree for interpolation space. Raise degree with respect to uh.
degree = uh.ufl_element().degree() + degree_rise
# Check degree of 'exact' solution u
degree_u = u.ufl_element().degree()
if degree_u is not None and degree_u < degree:
cpp.warning("Degree of exact solution may be inadequate for accurate result in errornorm.")
# Create function space
if rank == 0:
V = FunctionSpace(mesh, "Discontinuous Lagrange", degree)
elif rank == 1:
V = VectorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
dim=shape[0])
elif rank > 1:
V = TensorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
shape=shape)
# Interpolate functions into finite element space
pi_u = interpolate(u, V)
pi_uh = interpolate(uh, V)
# Compute the difference
e = Function(V)
e.assign(pi_u)
e.vector().axpy(-1.0, pi_uh.vector())
# Compute norm
return norm(e, norm_type=norm_type, mesh=mesh)
def part(self, i, deepcopy=False):
f = Function(self._cpp_object.part(i, deepcopy))
f.rename(self._cpp_object.name(), self._cpp_object.label())
return f