def discretize_elliptic_cg(analytical_problem,
diameter=None,
domain_discretizer=None,
grid=None,
boundary_info=None):
"""Discretizes an |EllipticProblem| using finite elements.
Parameters
----------
analytical_problem
The |EllipticProblem| to discretize.
diameter
If not `None`, `diameter` is passed 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 further arguments should be passed to the discretizer, use
:func:`functools.partial`. If `None`, |discretize_domain_default| is used.
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.
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, EllipticProblem)
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
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if diameter is None:
grid, boundary_info = domain_discretizer(analytical_problem.domain)
else:
grid, boundary_info = domain_discretizer(analytical_problem.domain,
diameter=diameter)
assert isinstance(grid, (OnedGrid, TriaGrid, RectGrid))
if isinstance(grid, RectGrid):
Operator = cg.DiffusionOperatorQ1
Functional = cg.L2ProductFunctionalQ1
else:
Operator = cg.DiffusionOperatorP1
Functional = cg.L2ProductFunctionalP1
p = analytical_problem
if p.diffusion_functionals is not None:
L0 = Operator(grid,
boundary_info,
diffusion_constant=0,
name='diffusion_boundary_part')
Li = [
Operator(grid,
boundary_info,
diffusion_function=df,
dirichlet_clear_diag=True,
name='diffusion_{}'.format(i))
for i, df in enumerate(p.diffusion_functions)
]
L = LincombOperator(operators=[L0] + Li,
coefficients=[1.] + list(p.diffusion_functionals),
name='diffusion')
else:
assert len(p.diffusion_functions) == 1
L = Operator(grid,
boundary_info,
diffusion_function=p.diffusion_functions[0],
name='diffusion')
F = Functional(grid,
p.rhs,
boundary_info,
dirichlet_data=p.dirichlet_data,
neumann_data=p.neumann_data)
if isinstance(grid, (TriaGrid, RectGrid)):
visualizer = PatchVisualizer(grid=grid,
bounding_box=grid.domain,
codim=2)
else:
visualizer = Matplotlib1DVisualizer(grid=grid, codim=1)
empty_bi = EmptyBoundaryInfo(grid)
l2_product = cg.L2ProductQ1(grid, empty_bi) if isinstance(
grid, RectGrid) else cg.L2ProductP1(grid, empty_bi)
h1_semi_product = Operator(grid, empty_bi)
products = {
'h1': l2_product + h1_semi_product,
'h1_semi': h1_semi_product,
'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='{}_CG'.format(p.name))
return discretization, {'grid': grid, 'boundary_info': boundary_info}
def discretize(grid_and_problem_data, polorder=1, solver_options=None):
logger = getLogger('discretize_elliptic_swipdg.discretize')
logger.info('discretizing ... ')
over_integrate = 2
grid, boundary_info = grid_and_problem_data['grid'], grid_and_problem_data[
'boundary_info']
_lambda, kappa, f = (grid_and_problem_data['lambda'],
grid_and_problem_data['kappa'],
grid_and_problem_data['f'])
lambda_bar, lambda_bar = grid_and_problem_data[
'lambda_bar'], grid_and_problem_data['lambda_bar']
mu_bar, mu_hat, parameter_range = (
grid_and_problem_data['mu_bar'], grid_and_problem_data['mu_hat'],
grid_and_problem_data['parameter_range'])
space = make_dg_space(grid)
# prepare operators and functionals
if isinstance(_lambda, dict):
system_ops = [
make_elliptic_swipdg_matrix_operator(lambda_func, kappa,
boundary_info, space,
over_integrate)
for lambda_func in _lambda['functions']
]
elliptic_ops = [
make_elliptic_matrix_operator(lambda_func, kappa, space,
over_integrate)
for lambda_func in _lambda['functions']
]
else:
system_ops = [
make_elliptic_swipdg_matrix_operator(_lambda, kappa, boundary_info,
space, over_integrate),
]
elliptic_ops = [
make_elliptic_matrix_operator(_lambda, kappa, space,
over_integrate),
]
if isinstance(f, dict):
rhs_functionals = [
make_l2_volume_vector_functional(f_func, space, over_integrate)
for f_func in f['functions']
]
else:
rhs_functionals = [
make_l2_volume_vector_functional(f, space, over_integrate),
]
l2_matrix_with_system_pattern = system_ops[0].matrix().copy()
l2_operator = make_l2_matrix_operator(l2_matrix_with_system_pattern, space)
# assemble everything in one grid walk
system_assembler = make_system_assembler(space)
for op in system_ops:
system_assembler.append(op)
for op in elliptic_ops:
system_assembler.append(op)
for func in rhs_functionals:
system_assembler.append(func)
system_assembler.append(l2_operator)
system_assembler.walk()
# wrap everything
if isinstance(_lambda, dict):
op = LincombOperator([
DuneXTMatrixOperator(o.matrix(),
dof_communicator=space.dof_communicator)
for o in system_ops
], _lambda['coefficients'])
elliptic_op = LincombOperator(
[DuneXTMatrixOperator(o.matrix()) for o in elliptic_ops],
_lambda['coefficients'])
else:
op = DuneXTMatrixOperator(system_ops[0].matrix())
elliptic_op = DuneXTMatrixOperator(elliptic_ops[0].matrix())
if isinstance(f, dict):
rhs = LincombOperator([
VectorFunctional(op.range.make_array([func.vector()]))
for func in rhs_functionals
], f['coefficients'])
else:
rhs = VectorFunctional(
op.range.make_array([rhs_functionals[0].vector()]))
operators = {
'l2':
DuneXTMatrixOperator(l2_matrix_with_system_pattern),
'elliptic':
elliptic_op,
'elliptic_mu_bar':
DuneXTMatrixOperator(elliptic_op.assemble(mu=mu_bar).matrix)
}
d = StationaryDiscretization(op,
rhs,
operators=operators,
visualizer=DuneGDTVisualizer(space))
d = d.with_(parameter_space=CubicParameterSpace(
d.parameter_type, parameter_range[0], parameter_range[1]))
return d, {'space': space}
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_elliptic_fv(analytical_problem,
diameter=None,
domain_discretizer=None,
grid=None,
boundary_info=None):
"""Discretizes an |EllipticProblem| using the finite volume method.
Parameters
----------
analytical_problem
The |EllipticProblem| to discretize.
diameter
If not `None`, `diameter` is passed 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 further arguments should be passed to the discretizer, use
:func:`functools.partial`. If `None`, |discretize_domain_default| is used.
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.
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, EllipticProblem)
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
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if diameter is None:
grid, boundary_info = domain_discretizer(analytical_problem.domain)
else:
grid, boundary_info = domain_discretizer(analytical_problem.domain,
diameter=diameter)
p = analytical_problem
if p.diffusion_functionals is not None:
Li = [
fv.DiffusionOperator(grid,
boundary_info,
diffusion_function=df,
name='diffusion_{}'.format(i))
for i, df in enumerate(p.diffusion_functions)
]
L = LincombOperator(operators=Li,
coefficients=list(p.diffusion_functionals),
name='diffusion')
F0 = fv.L2ProductFunctional(grid,
p.rhs,
boundary_info=boundary_info,
neumann_data=p.neumann_data)
if p.dirichlet_data is not None:
Fi = [
fv.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 = LincombOperator(operators=[F0] + Fi,
coefficients=[1.] +
list(p.diffusion_functionals),
name='rhs')
else:
F = F0
else:
assert len(p.diffusion_functions) == 1
L = fv.DiffusionOperator(grid,
boundary_info,
diffusion_function=p.diffusion_functions[0],
name='diffusion')
F = fv.L2ProductFunctional(grid,
p.rhs,
boundary_info=boundary_info,
dirichlet_data=p.dirichlet_data,
diffusion_function=p.diffusion_functions[0],
neumann_data=p.neumann_data)
if isinstance(grid, (TriaGrid, RectGrid)):
visualizer = PatchVisualizer(grid=grid,
bounding_box=grid.domain,
codim=0)
elif isinstance(grid, (OnedGrid)):
visualizer = Matplotlib1DVisualizer(grid=grid, codim=0)
else:
visualizer = None
l2_product = fv.L2Product(grid)
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))
return discretization, {'grid': grid, 'boundary_info': boundary_info}
def _discretize_fenics(xblocks, yblocks, grid_num_intervals, element_order):
# assemble system matrices - FEniCS code
########################################
import dolfin as df
mesh = df.UnitSquareMesh(grid_num_intervals, grid_num_intervals, 'crossed')
V = df.FunctionSpace(mesh, 'Lagrange', element_order)
u = df.TrialFunction(V)
v = df.TestFunction(V)
diffusion = df.Expression(
'(lower0 <= x[0]) * (open0 ? (x[0] < upper0) : (x[0] <= upper0)) *' +
'(lower1 <= x[1]) * (open1 ? (x[1] < upper1) : (x[1] <= upper1))',
lower0=0.,
upper0=0.,
open0=0,
lower1=0.,
upper1=0.,
open1=0,
element=df.FunctionSpace(mesh, 'DG', 0).ufl_element())
def assemble_matrix(x, y, nx, ny):
diffusion.user_parameters['lower0'] = x / nx
diffusion.user_parameters['lower1'] = y / ny
diffusion.user_parameters['upper0'] = (x + 1) / nx
diffusion.user_parameters['upper1'] = (y + 1) / ny
diffusion.user_parameters['open0'] = (x + 1 == nx)
diffusion.user_parameters['open1'] = (y + 1 == ny)
return df.assemble(
df.inner(diffusion * df.nabla_grad(u), df.nabla_grad(v)) * df.dx)
mats = [
assemble_matrix(x, y, xblocks, yblocks) for x in range(xblocks)
for y in range(yblocks)
]
mat0 = mats[0].copy()
mat0.zero()
h1_mat = df.assemble(df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx)
l2_mat = df.assemble(u * v * df.dx)
f = df.Constant(1.) * v * df.dx
F = df.assemble(f)
bc = df.DirichletBC(V, 0., df.DomainBoundary())
for m in mats:
bc.zero(m)
bc.apply(mat0)
bc.apply(h1_mat)
bc.apply(F)
# wrap everything as a pyMOR discretization
###########################################
# FEniCS wrappers
from pymor.gui.fenics import FenicsVisualizer
from pymor.operators.fenics import FenicsMatrixOperator
from pymor.vectorarrays.fenics import FenicsVector
# generic pyMOR classes
from pymor.discretizations.basic import StationaryDiscretization
from pymor.operators.constructions import LincombOperator, VectorFunctional
from pymor.parameters.functionals import ProjectionParameterFunctional
from pymor.parameters.spaces import CubicParameterSpace
from pymor.vectorarrays.list import ListVectorArray
# define parameter functionals (same as in pymor.analyticalproblems.thermalblock)
def parameter_functional_factory(x, y):
return ProjectionParameterFunctional(
component_name='diffusion',
component_shape=(yblocks, xblocks),
coordinates=(yblocks - y - 1, x),
name='diffusion_{}_{}'.format(x, y))
parameter_functionals = tuple(
parameter_functional_factory(x, y) for x in range(xblocks)
for y in range(yblocks))
# wrap operators
ops = [FenicsMatrixOperator(mat0, V, V)
] + [FenicsMatrixOperator(m, V, V) for m in mats]
op = LincombOperator(ops, (1., ) + parameter_functionals)
rhs = VectorFunctional(ListVectorArray([FenicsVector(F, V)]))
h1_product = FenicsMatrixOperator(h1_mat, V, V, name='h1_0_semi')
l2_product = FenicsMatrixOperator(l2_mat, V, V, name='l2')
# build discretization
visualizer = FenicsVisualizer(V)
parameter_space = CubicParameterSpace(op.parameter_type, 0.1, 1.)
d = StationaryDiscretization(op,
rhs,
products={
'h1_0_semi': h1_product,
'l2': l2_product
},
parameter_space=parameter_space,
visualizer=visualizer)
return d
def discretize_stationary_from_disk(parameter_file):
"""Generates stationary discretization only based on data loaded from files.
The path and further specifications to these objects are given in an '.ini' parameter file (see example below).
Suitable for discrete problems given by::
L(u, w) = F(w)
with an operator L and a linear functional F with a parameter w given as system matrices and rhs vectors in
an affine decomposition on the hard disk.
Parameters
----------
parameterFile
String containing the path to the .ini parameter file.
Returns
-------
discretization
The |Discretization| that has been generated.
Example
-------
Following parameter file is suitable for a discrete elliptic problem with
L(u, w) = (f_1(w)*K1 + f_2(w)*K2+...)*u and F(w) = g_1(w)*L1+g_2(w)*L2+... with
parameter w_i in [a_i,b_i], where f_i(w) and g_i(w) are strings of valid python
expressions.
Optional products can be provided to introduce a dict of inner products on
the discrete space. The content of the file is then given as::
[system-matrices]
# path_to_object: parameter_functional_associated_with_object
K1.mat: f_1(w_1,...,w_n)
K2.mat: f_2(w_1,...,w_n)
...
[rhs-vectors]
L1.mat: g_1(w_1,...,w_n)
L2.mat: g_2(w_1,...,w_n)
...
[parameter]
# Name: lower_bound,upper_bound
w_1: a_1,b_1
...
w_n: a_n,b_n
[products]
# Name: path_to_object
Prod1: S.mat
Prod2: T.mat
...
"""
assert ".ini" == parameter_file[-4:], "Given file is not an .ini file"
base_path = os.path.dirname(parameter_file)
# Get input from parameter file
config = configparser.ConfigParser()
config.optionxform = str
config.read(parameter_file)
# Assert that all needed entries given
assert 'system-matrices' in config.sections()
assert 'rhs-vectors' in config.sections()
assert 'parameter' in config.sections()
system_mat = config.items('system-matrices')
rhs_vec = config.items('rhs-vectors')
parameter = config.items('parameter')
# Dict of parameters types and ranges
parameter_type = {}
parameter_range = {}
# get parameters
for i in range(len(parameter)):
parameter_name = parameter[i][0]
parameter_list = tuple(float(j) for j in parameter[i][1].replace(" ", "").split(','))
parameter_range[parameter_name] = parameter_list
# Assume scalar parameter dependence
parameter_type[parameter_name] = 0
# Create parameter space
parameter_space = CubicParameterSpace(parameter_type=parameter_type, ranges=parameter_range)
# Assemble operators
system_operators, system_functionals = [], []
# get parameter functionals and system matrices
for i in range(len(system_mat)):
path = os.path.join(base_path, system_mat[i][0])
expr = system_mat[i][1]
parameter_functional = ExpressionParameterFunctional(expr, parameter_type=parameter_type)
system_operators.append(NumpyMatrixOperator.from_file(path))
system_functionals.append(parameter_functional)
system_lincombOperator = LincombOperator(system_operators, coefficients=system_functionals)
# get rhs vectors
rhs_operators, rhs_functionals = [], []
for i in range(len(rhs_vec)):
path = os.path.join(base_path, rhs_vec[i][0])
expr = rhs_vec[i][1]
parameter_functional = ExpressionParameterFunctional(expr, parameter_type=parameter_type)
op = NumpyMatrixOperator.from_file(path)
assert isinstance(op._matrix, np.ndarray)
op = op.with_(matrix=op._matrix.reshape((1, -1)))
rhs_operators.append(op)
rhs_functionals.append(parameter_functional)
rhs_lincombOperator = LincombOperator(rhs_operators, coefficients=rhs_functionals)
# get products if given
if 'products' in config.sections():
product = config.items('products')
products = {}
for i in range(len(product)):
product_name = product[i][0]
product_path = os.path.join(base_path, product[i][1])
products[product_name] = NumpyMatrixOperator.from_file(product_path)
else:
products = None
# Create and return stationary discretization
return StationaryDiscretization(operator=system_lincombOperator, rhs=rhs_lincombOperator,
parameter_space=parameter_space, products=products)
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
def discretize_stationary_from_disk(parameter_file):
"""Load a linear affinely decomposed |StationaryDiscretization| from file.
The discretization is defined via an `.ini`-style file as follows ::
[system-matrices]
L_1.mat: l_1(μ_1,...,μ_n)
L_2.mat: l_2(μ_1,...,μ_n)
...
[rhs-vectors]
F_1.mat: f_1(μ_1,...,μ_n)
F_2.mat: f_2(μ_1,...,μ_n)
...
[parameter]
μ_1: a_1,b_1
...
μ_n: a_n,b_n
[products]
Prod1: P_1.mat
Prod2: P_2.mat
...
Here, `L_1.mat`, `L_2.mat`, ..., `F_1.mat`, `F_2.mat`, ... are files
containing matrices `L_1`, `L_2`, ... and vectors `F_1.mat`, `F_2.mat`, ...
which correspond to the affine components of the operator and right-hand
side functional. The respective coefficient functionals, are given via the
string expressions `l_1(...)`, `l_2(...)`, ..., `f_1(...)` in the
(scalar-valued) |Parameter| components `w_1`, ..., `w_n`. The allowed lower
and upper bounds `a_i, b_i` for the component `μ_i` are specified in the
`[parameters]` section. The resulting operator and right-hand side are
then of the form ::
L(μ) = l_1(μ)*L_1 + l_2(μ)*L_2+ ...
F(μ) = f_1(μ)*F_1 + f_2(μ)*L_2+ ...
In the `[products]` section, an optional list of inner products `Prod1`, `Prod2`, ..
with corresponding matrices `P_1.mat`, `P_2.mat` can be specified.
Example::
[system-matrices]
matrix1.mat: 1.
matrix2.mat: 1. - theta**2
[rhs-vectors]
rhs.mat: 1.
[parameter]
theta: 0, 0.5
[products]
h1: h1.mat
l2: mass.mat
Parameters
----------
parameter_file
Path to the parameter file.
Returns
-------
discretization
The |StationaryDiscretization| that has been generated.
"""
assert ".ini" == parameter_file[
-4:], 'Given file is not an .ini file: {}'.format(parameter_file)
assert os.path.isfile(parameter_file)
base_path = os.path.dirname(parameter_file)
# Get input from parameter file
config = configparser.ConfigParser()
config.optionxform = str
config.read(parameter_file)
# Assert that all needed entries given
assert 'system-matrices' in config.sections()
assert 'rhs-vectors' in config.sections()
assert 'parameter' in config.sections()
system_mat = config.items('system-matrices')
rhs_vec = config.items('rhs-vectors')
parameter = config.items('parameter')
# Dict of parameters types and ranges
parameter_type = {}
parameter_range = {}
# get parameters
for i in range(len(parameter)):
parameter_name = parameter[i][0]
parameter_list = tuple(
float(j) for j in parameter[i][1].replace(" ", "").split(','))
parameter_range[parameter_name] = parameter_list
# Assume scalar parameter dependence
parameter_type[parameter_name] = 0
# Create parameter space
parameter_space = CubicParameterSpace(parameter_type=parameter_type,
ranges=parameter_range)
# Assemble operators
system_operators, system_functionals = [], []
# get parameter functionals and system matrices
for i in range(len(system_mat)):
path = os.path.join(base_path, system_mat[i][0])
expr = system_mat[i][1]
parameter_functional = ExpressionParameterFunctional(
expr, parameter_type=parameter_type)
system_operators.append(
NumpyMatrixOperator.from_file(path,
source_id='STATE',
range_id='STATE'))
system_functionals.append(parameter_functional)
system_lincombOperator = LincombOperator(system_operators,
coefficients=system_functionals)
# get rhs vectors
rhs_operators, rhs_functionals = [], []
for i in range(len(rhs_vec)):
path = os.path.join(base_path, rhs_vec[i][0])
expr = rhs_vec[i][1]
parameter_functional = ExpressionParameterFunctional(
expr, parameter_type=parameter_type)
op = NumpyMatrixOperator.from_file(path, source_id='STATE')
assert isinstance(op._matrix, np.ndarray)
op = op.with_(matrix=op._matrix.reshape((1, -1)))
rhs_operators.append(op)
rhs_functionals.append(parameter_functional)
rhs_lincombOperator = LincombOperator(rhs_operators,
coefficients=rhs_functionals)
# get products if given
if 'products' in config.sections():
product = config.items('products')
products = {}
for i in range(len(product)):
product_name = product[i][0]
product_path = os.path.join(base_path, product[i][1])
products[product_name] = NumpyMatrixOperator.from_file(
product_path, source_id='STATE', range_id='STATE')
else:
products = None
# Create and return stationary discretization
return StationaryDiscretization(operator=system_lincombOperator,
rhs=rhs_lincombOperator,
parameter_space=parameter_space,
products=products)
def discretize_elliptic_cg(analytical_problem,
diameter=None,
domain_discretizer=None,
grid=None,
boundary_info=None):
"""Discretizes an |EllipticProblem| using finite elements.
Parameters
----------
analytical_problem
The |EllipticProblem| to discretize.
diameter
If not `None`, `diameter` is passed 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 further arguments should be passed to the discretizer, use
:func:`functools.partial`. If `None`, |discretize_domain_default| is used.
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.
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, EllipticProblem)
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
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if diameter is None:
grid, boundary_info = domain_discretizer(analytical_problem.domain)
else:
grid, boundary_info = domain_discretizer(analytical_problem.domain,
diameter=diameter)
assert grid.reference_element in (line, triangle, square)
if grid.reference_element is square:
DiffusionOperator = cg.DiffusionOperatorQ1
AdvectionOperator = cg.AdvectionOperatorQ1
ReactionOperator = cg.L2ProductQ1
Functional = cg.L2ProductFunctionalQ1
else:
DiffusionOperator = cg.DiffusionOperatorP1
AdvectionOperator = cg.AdvectionOperatorP1
ReactionOperator = cg.L2ProductP1
Functional = cg.L2ProductFunctionalP1
p = analytical_problem
if p.diffusion_functionals is not None or p.advection_functionals is not None or p.reaction_functionals is not None:
# parametric case
Li = [
DiffusionOperator(grid,
boundary_info,
diffusion_constant=0,
name='boundary_part')
]
coefficients = [1.]
# diffusion part
if p.diffusion_functionals is not None:
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_functionals)
elif p.diffusion_functions is not None:
assert len(p.diffusion_functions) == 1
Li += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=p.diffusion_functions[0],
dirichlet_clear_diag=True,
name='diffusion')
]
coefficients.append(1.)
# advection part
if p.advection_functionals is not None:
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_functionals)
elif p.advection_functions is not None:
assert len(p.advection_functions) == 1
Li += [
AdvectionOperator(grid,
boundary_info,
advection_function=p.advection_functions[0],
dirichlet_clear_diag=True,
name='advection')
]
coefficients.append(1.)
# reaction part
if p.reaction_functionals is not None:
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_functionals)
elif p.reaction_functions is not None:
assert len(p.reaction_functions) == 1
Li += [
ReactionOperator(grid,
boundary_info,
coefficient_function=p.reaction_functions[0],
dirichlet_clear_diag=True,
name='reaction')
]
coefficients.append(1.)
# robin boundaries
if p.robin_data is not None:
Li += [
cg.RobinBoundaryOperator(grid,
boundary_info,
robin_data=p.robin_data,
order=2,
name='robin')
]
coefficients.append(1.)
L = LincombOperator(operators=Li,
coefficients=coefficients,
name='ellipticOperator')
else:
# unparametric case, not operator for boundary treatment
Li = []
# only one operator has diagonal values, all subsequent operators have clear_diag
dirichlet_clear_diag = False
# diffusion part
if p.diffusion_functions is not None:
assert len(p.diffusion_functions) == 1
Li += [
DiffusionOperator(grid,
boundary_info,
diffusion_function=p.diffusion_functions[0],
dirichlet_clear_diag=dirichlet_clear_diag,
name='diffusion')
]
dirichlet_clear_diag = True
# advection part
if p.advection_functions is not None:
assert len(p.advection_functions) == 1
Li += [
AdvectionOperator(grid,
boundary_info,
advection_function=p.advection_functions[0],
dirichlet_clear_diag=dirichlet_clear_diag,
name='advection')
]
dirichlet_clear_diag = True
# reaction part
if p.reaction_functions is not None:
assert len(p.reaction_functions) == 1
Li += [
ReactionOperator(grid,
boundary_info,
coefficient_function=p.reaction_functions[0],
dirichlet_clear_diag=dirichlet_clear_diag,
name='reaction')
]
dirichlet_clear_diag = True
# robin boundaries
if p.robin_data is not None:
Li += [
cg.RobinBoundaryOperator(grid,
boundary_info,
robin_data=p.robin_data,
order=2,
name='robin')
]
if len(Li) == 1:
L = Li[0]
else:
L = LincombOperator(operators=Li,
coefficients=[1.] * len(Li),
name='ellipticOperator')
F = Functional(grid,
p.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 = Matplotlib1DVisualizer(grid=grid, codim=1)
else:
visualizer = None
Prod = cg.L2ProductQ1 if grid.reference_element is square else cg.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
discretization = StationaryDiscretization(L,
F,
products=products,
visualizer=visualizer,
parameter_space=parameter_space,
name='{}_CG'.format(p.name))
return discretization, {'grid': grid, 'boundary_info': boundary_info}