def discretize_stationary_cg(analytical_problem,
diameter=None,
domain_discretizer=None,
grid_type=None,
grid=None,
boundary_info=None,
preassemble=True):
"""Discretizes an |StationaryProblem| using finite elements.
Parameters
----------
analytical_problem
The |StationaryProblem| to discretize.
diameter
If not `None`, `diameter` is passed as an argument to the `domain_discretizer`.
domain_discretizer
Discretizer to be used for discretizing the analytical domain. This has
to be a function `domain_discretizer(domain_description, diameter, ...)`.
If `None`, |discretize_domain_default| is used.
grid_type
If not `None`, this parameter is forwarded to `domain_discretizer` to specify
the type of the generated |Grid|.
grid
Instead of using a domain discretizer, the |Grid| can also be passed directly
using this parameter.
boundary_info
A |BoundaryInfo| specifying the boundary types of the grid boundary entities.
Must be provided if `grid` is specified.
preassemble
If `True`, preassemble all operators in the resulting |Discretization|.
Returns
-------
d
The |Discretization| that has been generated.
data
Dictionary with the following entries:
:grid: The generated |Grid|.
:boundary_info: The generated |BoundaryInfo|.
"""
assert isinstance(analytical_problem, StationaryProblem)
assert grid is None or boundary_info is not None
assert boundary_info is None or grid is not None
assert grid is None or domain_discretizer is None
assert grid_type is None or grid is None
p = analytical_problem
if not (p.nonlinear_advection == p.nonlinear_advection_derivative ==
p.nonlinear_reaction == p.nonlinear_reaction_derivative == None):
raise NotImplementedError
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if grid_type:
domain_discretizer = partial(domain_discretizer,
grid_type=grid_type)
if diameter is None:
grid, boundary_info = domain_discretizer(p.domain)
else:
grid, boundary_info = domain_discretizer(p.domain,
diameter=diameter)
assert grid.reference_element in (line, triangle, square)
if grid.reference_element is square:
DiffusionOperator = DiffusionOperatorQ1
AdvectionOperator = AdvectionOperatorQ1
ReactionOperator = L2ProductQ1
Functional = L2ProductFunctionalQ1
else:
DiffusionOperator = DiffusionOperatorP1
AdvectionOperator = AdvectionOperatorP1
ReactionOperator = L2ProductP1
Functional = L2ProductFunctionalP1
Li = [
DiffusionOperator(grid,
boundary_info,
diffusion_constant=0,
name='boundary_part')
]
coefficients = [1.]
# diffusion part
if isinstance(p.diffusion, LincombFunction):
Li += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=df,
dirichlet_clear_diag=True,
name='diffusion_{}'.format(i))
for i, df in enumerate(p.diffusion.functions)
]
coefficients += list(p.diffusion.coefficients)
elif p.diffusion is not None:
Li += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=p.diffusion,
dirichlet_clear_diag=True,
name='diffusion')
]
coefficients.append(1.)
# advection part
if isinstance(p.advection, LincombFunction):
Li += [
AdvectionOperator(grid,
boundary_info,
advection_function=af,
dirichlet_clear_diag=True,
name='advection_{}'.format(i))
for i, af in enumerate(p.advection.functions)
]
coefficients += list(p.advection.coefficients)
elif p.advection is not None:
Li += [
AdvectionOperator(grid,
boundary_info,
advection_function=p.advection,
dirichlet_clear_diag=True,
name='advection')
]
coefficients.append(1.)
# reaction part
if isinstance(p.reaction, LincombFunction):
Li += [
ReactionOperator(grid,
boundary_info,
coefficient_function=rf,
dirichlet_clear_diag=True,
name='reaction_{}'.format(i))
for i, rf in enumerate(p.reaction.functions)
]
coefficients += list(p.reaction.coefficients)
elif p.reaction is not None:
Li += [
ReactionOperator(grid,
boundary_info,
coefficient_function=p.reaction,
dirichlet_clear_diag=True,
name='reaction')
]
coefficients.append(1.)
# robin boundaries
if p.robin_data is not None:
if grid.reference_element is square:
raise NotImplementedError
Li += [
RobinBoundaryOperator(grid,
boundary_info,
robin_data=p.robin_data,
order=2,
name='robin')
]
coefficients.append(1.)
L = LincombOperator(operators=Li,
coefficients=coefficients,
name='ellipticOperator')
rhs = p.rhs or ConstantFunction(0., dim_domain=p.domain.dim)
F = Functional(grid,
rhs,
boundary_info,
dirichlet_data=p.dirichlet_data,
neumann_data=p.neumann_data)
if grid.reference_element in (triangle, square):
visualizer = PatchVisualizer(grid=grid,
bounding_box=grid.bounding_box(),
codim=2)
elif grid.reference_element is line:
visualizer = OnedVisualizer(grid=grid, codim=1)
else:
visualizer = None
Prod = L2ProductQ1 if grid.reference_element is square else L2ProductP1
empty_bi = EmptyBoundaryInfo(grid)
l2_product = Prod(grid, empty_bi, name='l2')
l2_0_product = Prod(grid,
boundary_info,
dirichlet_clear_columns=True,
name='l2_0')
h1_semi_product = DiffusionOperator(grid, empty_bi, name='h1_semi')
h1_0_semi_product = DiffusionOperator(grid,
boundary_info,
dirichlet_clear_columns=True,
name='h1_0_semi')
products = {
'h1': l2_product + h1_semi_product,
'h1_semi': h1_semi_product,
'l2': l2_product,
'h1_0': l2_0_product + h1_0_semi_product,
'h1_0_semi': h1_0_semi_product,
'l2_0': l2_0_product
}
parameter_space = p.parameter_space if hasattr(p,
'parameter_space') else None
d = StationaryDiscretization(L,
F,
products=products,
visualizer=visualizer,
parameter_space=parameter_space,
name='{}_CG'.format(p.name))
data = {'grid': grid, 'boundary_info': boundary_info}
if preassemble:
data['unassembled_d'] = d
d = preassemble_(d)
return d, data
def discretize_stationary_cg(analytical_problem,
diameter=None,
domain_discretizer=None,
grid_type=None,
grid=None,
boundary_info=None,
preassemble=True):
"""Discretizes a |StationaryProblem| using finite elements.
Parameters
----------
analytical_problem
The |StationaryProblem| to discretize.
diameter
If not `None`, `diameter` is passed as an argument to the `domain_discretizer`.
domain_discretizer
Discretizer to be used for discretizing the analytical domain. This has
to be a function `domain_discretizer(domain_description, diameter, ...)`.
If `None`, |discretize_domain_default| is used.
grid_type
If not `None`, this parameter is forwarded to `domain_discretizer` to specify
the type of the generated |Grid|.
grid
Instead of using a domain discretizer, the |Grid| can also be passed directly
using this parameter.
boundary_info
A |BoundaryInfo| specifying the boundary types of the grid boundary entities.
Must be provided if `grid` is specified.
preassemble
If `True`, preassemble all operators in the resulting |Model|.
Returns
-------
m
The |Model| that has been generated.
data
Dictionary with the following entries:
:grid: The generated |Grid|.
:boundary_info: The generated |BoundaryInfo|.
:unassembled_m: In case `preassemble` is `True`, the generated |Model|
before preassembling operators.
"""
assert isinstance(analytical_problem, StationaryProblem)
assert grid is None or boundary_info is not None
assert boundary_info is None or grid is not None
assert grid is None or domain_discretizer is None
assert grid_type is None or grid is None
p = analytical_problem
if not (p.nonlinear_advection == p.nonlinear_advection_derivative ==
p.nonlinear_reaction == p.nonlinear_reaction_derivative is None):
raise NotImplementedError
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if grid_type:
domain_discretizer = partial(domain_discretizer,
grid_type=grid_type)
if diameter is None:
grid, boundary_info = domain_discretizer(p.domain)
else:
grid, boundary_info = domain_discretizer(p.domain,
diameter=diameter)
assert grid.reference_element in (line, triangle, square)
if grid.reference_element is square:
DiffusionOperator = DiffusionOperatorQ1
AdvectionOperator = AdvectionOperatorQ1
ReactionOperator = L2ProductQ1
L2Functional = L2ProductFunctionalQ1
BoundaryL2Functional = BoundaryL2ProductFunctional
else:
DiffusionOperator = DiffusionOperatorP1
AdvectionOperator = AdvectionOperatorP1
ReactionOperator = L2ProductP1
L2Functional = L2ProductFunctionalP1
BoundaryL2Functional = BoundaryL2ProductFunctional
Li = [
DiffusionOperator(grid,
boundary_info,
diffusion_constant=0,
name='boundary_part')
]
coefficients = [1.]
# diffusion part
if isinstance(p.diffusion, LincombFunction):
Li += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=df,
dirichlet_clear_diag=True,
name=f'diffusion_{i}')
for i, df in enumerate(p.diffusion.functions)
]
coefficients += list(p.diffusion.coefficients)
elif p.diffusion is not None:
Li += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=p.diffusion,
dirichlet_clear_diag=True,
name='diffusion')
]
coefficients.append(1.)
# advection part
if isinstance(p.advection, LincombFunction):
Li += [
AdvectionOperator(grid,
boundary_info,
advection_function=af,
dirichlet_clear_diag=True,
name=f'advection_{i}')
for i, af in enumerate(p.advection.functions)
]
coefficients += list(p.advection.coefficients)
elif p.advection is not None:
Li += [
AdvectionOperator(grid,
boundary_info,
advection_function=p.advection,
dirichlet_clear_diag=True,
name='advection')
]
coefficients.append(1.)
# reaction part
if isinstance(p.reaction, LincombFunction):
Li += [
ReactionOperator(grid,
boundary_info,
coefficient_function=rf,
dirichlet_clear_diag=True,
name=f'reaction_{i}')
for i, rf in enumerate(p.reaction.functions)
]
coefficients += list(p.reaction.coefficients)
elif p.reaction is not None:
Li += [
ReactionOperator(grid,
boundary_info,
coefficient_function=p.reaction,
dirichlet_clear_diag=True,
name='reaction')
]
coefficients.append(1.)
# robin boundaries
if p.robin_data is not None:
assert isinstance(p.robin_data, tuple) and len(p.robin_data) == 2
if isinstance(p.robin_data[0], LincombFunction):
for i, rd in enumerate(p.robin_data[0].functions):
robin_tuple = (rd, p.robin_data[1])
Li += [
RobinBoundaryOperator(grid,
boundary_info,
robin_data=robin_tuple,
name=f'robin_{i}')
]
coefficients += list(p.robin_data[0].coefficients)
else:
Li += [
RobinBoundaryOperator(grid,
boundary_info,
robin_data=p.robin_data,
name=f'robin')
]
coefficients.append(1.)
L = LincombOperator(operators=Li,
coefficients=coefficients,
name='ellipticOperator')
# right-hand side
rhs = p.rhs or ConstantFunction(0., dim_domain=p.domain.dim)
Fi = []
coefficients_F = []
if isinstance(p.rhs, LincombFunction):
Fi += [
L2Functional(grid,
rh,
dirichlet_clear_dofs=True,
boundary_info=boundary_info,
name=f'rhs_{i}')
for i, rh in enumerate(p.rhs.functions)
]
coefficients_F += list(p.rhs.coefficients)
else:
Fi += [
L2Functional(grid,
rhs,
dirichlet_clear_dofs=True,
boundary_info=boundary_info,
name='rhs')
]
coefficients_F.append(1.)
if p.neumann_data is not None and boundary_info.has_neumann:
if isinstance(p.neumann_data, LincombFunction):
Fi += [
BoundaryL2Functional(grid,
-ne,
boundary_info=boundary_info,
boundary_type='neumann',
dirichlet_clear_dofs=True,
name=f'neumann_{i}')
for i, ne in enumerate(p.neumann_data.functions)
]
coefficients_F += list(p.neumann_data.coefficients)
else:
Fi += [
BoundaryL2Functional(grid,
-p.neumann_data,
boundary_info=boundary_info,
boundary_type='neumann',
dirichlet_clear_dofs=True)
]
coefficients_F.append(1.)
if p.robin_data is not None and boundary_info.has_robin:
if isinstance(p.robin_data[0], LincombFunction):
Fi += [
BoundaryL2Functional(grid,
rob * p.robin_data[1],
boundary_info=boundary_info,
boundary_type='robin',
dirichlet_clear_dofs=True,
name=f'robin_{i}')
for i, rob in enumerate(p.robin_data[0].functions)
]
coefficients_F += list(p.robin_data[0].coefficients)
else:
Fi += [
BoundaryL2Functional(grid,
p.robin_data[0] * p.robin_data[1],
boundary_info=boundary_info,
boundary_type='robin',
dirichlet_clear_dofs=True)
]
coefficients_F.append(1.)
if p.dirichlet_data is not None and boundary_info.has_dirichlet:
if isinstance(p.dirichlet_data, LincombFunction):
Fi += [
BoundaryDirichletFunctional(grid,
di,
boundary_info,
name=f'dirichlet{i}')
for i, di in enumerate(p.dirichlet_data.functions)
]
coefficients_F += list(p.dirichlet_data.coefficients)
else:
Fi += [
BoundaryDirichletFunctional(grid, p.dirichlet_data,
boundary_info)
]
coefficients_F.append(1.)
F = LincombOperator(operators=Fi,
coefficients=coefficients_F,
name='rhsOperator')
if grid.reference_element in (triangle, square):
visualizer = PatchVisualizer(grid=grid,
bounding_box=grid.bounding_box(),
codim=2)
elif grid.reference_element is line:
visualizer = OnedVisualizer(grid=grid, codim=1)
else:
visualizer = None
Prod = L2ProductQ1 if grid.reference_element is square else L2ProductP1
empty_bi = EmptyBoundaryInfo(grid)
l2_product = Prod(grid, empty_bi, name='l2')
l2_0_product = Prod(grid,
boundary_info,
dirichlet_clear_columns=True,
name='l2_0')
h1_semi_product = DiffusionOperator(grid, empty_bi, name='h1_semi')
h1_0_semi_product = DiffusionOperator(grid,
boundary_info,
dirichlet_clear_columns=True,
name='h1_0_semi')
products = {
'h1': l2_product + h1_semi_product,
'h1_semi': h1_semi_product,
'l2': l2_product,
'h1_0': l2_0_product + h1_0_semi_product,
'h1_0_semi': h1_0_semi_product,
'l2_0': l2_0_product
}
# assemble additionals output functionals
if p.outputs:
if any(v[0] not in ('l2', 'l2_boundary') for v in p.outputs):
raise NotImplementedError
outputs = [
L2Functional(grid, v[1], dirichlet_clear_dofs=False).H
if v[0] == 'l2' else BoundaryL2Functional(
grid, v[1], dirichlet_clear_dofs=False).H for v in p.outputs
]
if len(outputs) > 1:
from pymor.operators.block import BlockColumnOperator
output_functional = BlockColumnOperator(outputs)
else:
output_functional = outputs[0]
else:
output_functional = None
parameter_space = p.parameter_space if hasattr(p,
'parameter_space') else None
m = StationaryModel(L,
F,
output_functional=output_functional,
products=products,
visualizer=visualizer,
parameter_space=parameter_space,
name=f'{p.name}_CG')
data = {'grid': grid, 'boundary_info': boundary_info}
if preassemble:
data['unassembled_m'] = m
m = preassemble_(m)
return m, data
def discretize_stationary_fv(analytical_problem,
diameter=None,
domain_discretizer=None,
grid_type=None,
num_flux='lax_friedrichs',
lxf_lambda=1.,
eo_gausspoints=5,
eo_intervals=1,
grid=None,
boundary_info=None,
preassemble=True):
"""Discretizes an |StationaryProblem| using the finite volume method.
Parameters
----------
analytical_problem
The |StationaryProblem| to discretize.
diameter
If not `None`, `diameter` is passed as an argument to the `domain_discretizer`.
domain_discretizer
Discretizer to be used for discretizing the analytical domain. This has
to be a function `domain_discretizer(domain_description, diameter, ...)`.
If `None`, |discretize_domain_default| is used.
grid_type
If not `None`, this parameter is forwarded to `domain_discretizer` to specify
the type of the generated |Grid|.
num_flux
The numerical flux to use in the finite volume formulation. Allowed
values are `'lax_friedrichs'`, `'engquist_osher'`, `'simplified_engquist_osher'`
(see :mod:`pymor.operators.fv`).
lxf_lambda
The stabilization parameter for the Lax-Friedrichs numerical flux
(ignored, if different flux is chosen).
eo_gausspoints
Number of Gauss points for the Engquist-Osher numerical flux
(ignored, if different flux is chosen).
eo_intervals
Number of sub-intervals to use for integration when using Engquist-Osher
numerical flux (ignored, if different flux is chosen).
grid
Instead of using a domain discretizer, the |Grid| can also be passed directly
using this parameter.
boundary_info
A |BoundaryInfo| specifying the boundary types of the grid boundary entities.
Must be provided if `grid` is specified.
preassemble
If `True`, preassemble all operators in the resulting |Discretization|.
Returns
-------
discretization
The |Discretization| that has been generated.
data
Dictionary with the following entries:
:grid: The generated |Grid|.
:boundary_info: The generated |BoundaryInfo|.
"""
assert isinstance(analytical_problem, StationaryProblem)
assert grid is None or boundary_info is not None
assert boundary_info is None or grid is not None
assert grid is None or domain_discretizer is None
assert grid_type is None or grid is None
p = analytical_problem
if analytical_problem.robin_data is not None:
raise NotImplementedError
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if grid_type:
domain_discretizer = partial(domain_discretizer,
grid_type=grid_type)
if diameter is None:
grid, boundary_info = domain_discretizer(analytical_problem.domain)
else:
grid, boundary_info = domain_discretizer(analytical_problem.domain,
diameter=diameter)
L, L_coefficients = [], []
F, F_coefficients = [], []
if p.rhs is not None or p.neumann_data is not None:
F += [
L2ProductFunctional(grid,
p.rhs,
boundary_info=boundary_info,
neumann_data=p.neumann_data)
]
F_coefficients += [1.]
# diffusion part
if isinstance(p.diffusion, LincombFunction):
L += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=df,
name='diffusion_{}'.format(i))
for i, df in enumerate(p.diffusion.functions)
]
L_coefficients += p.diffusion.coefficients
if p.dirichlet_data is not None:
F += [
L2ProductFunctional(grid,
None,
boundary_info=boundary_info,
dirichlet_data=p.dirichlet_data,
diffusion_function=df,
name='dirichlet_{}'.format(i))
for i, df in enumerate(p.diffusion.functions)
]
F_coefficients += p.diffusion.coefficients
elif p.diffusion is not None:
L += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=p.diffusion,
name='diffusion')
]
L_coefficients += [1.]
if p.dirichlet_data is not None:
F += [
L2ProductFunctional(grid,
None,
boundary_info=boundary_info,
dirichlet_data=p.dirichlet_data,
diffusion_function=p.diffusion,
name='dirichlet')
]
F_coefficients += [1.]
# advection part
if isinstance(p.advection, LincombFunction):
L += [
LinearAdvectionLaxFriedrichs(grid,
boundary_info,
af,
name='advection_{}'.format(i))
for i, af in enumerate(p.advection.functions)
]
L_coefficients += list(p.advection.coefficients)
elif p.advection is not None:
L += [
LinearAdvectionLaxFriedrichs(grid,
boundary_info,
p.advection,
name='advection')
]
L_coefficients.append(1.)
# nonlinear advection part
if p.nonlinear_advection is not None:
if num_flux == 'lax_friedrichs':
L += [
nonlinear_advection_lax_friedrichs_operator(
grid,
boundary_info,
p.nonlinear_advection,
dirichlet_data=p.dirichlet_data,
lxf_lambda=lxf_lambda)
]
elif num_flux == 'upwind':
L += [
nonlinear_advection_upwind_operator(
grid,
boundary_info,
p.nonlinear_advection,
p.nonlinear_advection_derivative,
dirichlet_data=p.dirichlet_data)
]
elif num_flux == 'engquist_osher':
L += [
nonlinear_advection_engquist_osher_operator(
grid,
boundary_info,
p.nonlinear_advection,
p.nonlinear_advection_derivative,
gausspoints=eo_gausspoints,
intervals=eo_intervals,
dirichlet_data=p.dirichlet_data)
]
elif num_flux == 'simplified_engquist_osher':
L += [
nonlinear_advection_simplified_engquist_osher_operator(
grid,
boundary_info,
p.nonlinear_advection,
p.nonlinear_advection_derivative,
dirichlet_data=p.dirichlet_data)
]
else:
raise NotImplementedError
L_coefficients.append(1.)
# reaction part
if isinstance(p.reaction, LincombFunction):
raise NotImplementedError
elif p.reaction is not None:
L += [ReactionOperator(grid, p.reaction, name='reaction')]
L_coefficients += [1.]
# nonlinear reaction part
if p.nonlinear_reaction is not None:
L += [
NonlinearReactionOperator(grid, p.nonlinear_reaction,
p.nonlinear_reaction_derivative)
]
L_coefficients += [1.]
# system operator
if len(L_coefficients) == 1 and L_coefficients[0] == 1.:
L = L[0]
else:
L = LincombOperator(operators=L,
coefficients=L_coefficients,
name='elliptic_operator')
# rhs
if len(F_coefficients) == 0:
F = ZeroOperator(L.range, NumpyVectorSpace(1))
elif len(F_coefficients) == 1 and F_coefficients[0] == 1.:
F = F[0]
else:
F = LincombOperator(operators=F,
coefficients=F_coefficients,
name='rhs')
if grid.reference_element in (triangle, square):
visualizer = PatchVisualizer(grid=grid,
bounding_box=grid.bounding_box(),
codim=0)
elif grid.reference_element is line:
visualizer = OnedVisualizer(grid=grid, codim=0)
else:
visualizer = None
l2_product = L2Product(grid, name='l2')
products = {'l2': l2_product}
parameter_space = p.parameter_space if hasattr(p,
'parameter_space') else None
discretization = StationaryDiscretization(L,
F,
products=products,
visualizer=visualizer,
parameter_space=parameter_space,
name='{}_FV'.format(p.name))
data = {'grid': grid, 'boundary_info': boundary_info}
if preassemble:
data['unassembled_discretization'] = discretization
discretization = preassemble_(discretization)
return discretization, data