Source code for pymor.discretizers.builtin.cg

# This file is part of the pyMOR project (http://www.pymor.org).
# Copyright 2013-2020 pyMOR developers and contributors. All rights reserved.
# License: BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause)

""" This module provides some operators for continuous finite element discretizations."""

from functools import partial

import numpy as np
from scipy.sparse import coo_matrix, csc_matrix

from pymor.algorithms.preassemble import preassemble as preassemble_
from pymor.algorithms.timestepping import ExplicitEulerTimeStepper, ImplicitEulerTimeStepper
from pymor.analyticalproblems.elliptic import StationaryProblem
from pymor.analyticalproblems.functions import Function, ConstantFunction, LincombFunction
from pymor.analyticalproblems.instationary import InstationaryProblem
from pymor.discretizers.builtin.domaindiscretizers.default import discretize_domain_default
from pymor.discretizers.builtin.grids.boundaryinfos import EmptyBoundaryInfo
from pymor.discretizers.builtin.grids.referenceelements import line, triangle, square
from pymor.discretizers.builtin.gui.visualizers import PatchVisualizer, OnedVisualizer
from pymor.models.basic import StationaryModel, InstationaryModel
from pymor.operators.constructions import LincombOperator
from pymor.operators.numpy import NumpyMatrixBasedOperator
from pymor.vectorarrays.numpy import NumpyVectorSpace


[docs]def CGVectorSpace(grid, id='STATE'): return NumpyVectorSpace(grid.size(grid.dim), id)
[docs]class L2ProductFunctionalP1(NumpyMatrixBasedOperator): """Linear finite element functional representing the inner product with an L2-|Function|. Parameters ---------- grid |Grid| for which to assemble the functional. function The |Function| with which to take the inner product. dirichlet_clear_dofs If `True`, set dirichlet boundary DOFs to zero. boundary_info |BoundaryInfo| determining the Dirichlet boundaries in case `dirichlet_clear_dofs` is set to `True`. name The name of the functional. """ sparse = False source = NumpyVectorSpace(1) def __init__(self, grid, function, dirichlet_clear_dofs=False, boundary_info=None, name=None): assert grid.reference_element(0) in {line, triangle} assert function.shape_range == () assert not dirichlet_clear_dofs or boundary_info self.__auto_init(locals()) self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # evaluate function at element centers F = self.function(g.centers(0), mu=mu) # evaluate the shape functions at the quadrature points on the reference # element -> shape = (number of shape functions, number of quadrature points) q, w = g.reference_element.quadrature(order=1) if g.dim == 1: SF = np.array((1 - q[..., 0], q[..., 0])) elif g.dim == 2: SF = np.array(((1 - np.sum(q, axis=-1)), q[..., 0], q[..., 1])) else: raise NotImplementedError # integrate the products of the function with the shape functions on each element # -> shape = (g.size(0), number of shape functions) SF_INTS = np.einsum('e,pi,e,i->ep', F, SF, g.integration_elements(0), w).ravel() # map local DOFs to global DOFs # FIXME This implementation is horrible, find a better way! SF_I = g.subentities(0, g.dim).ravel() I = coo_matrix((SF_INTS, (np.zeros_like(SF_I), SF_I)), shape=(1, g.size(g.dim))).toarray().ravel() if self.dirichlet_clear_dofs and bi.has_dirichlet: DI = bi.dirichlet_boundaries(g.dim) I[DI] = 0 return I.reshape((-1, 1))
[docs]class BoundaryL2ProductFunctional(NumpyMatrixBasedOperator): """Linear finite element functional representing the inner product with an L2-|Function| on the boundary. Parameters ---------- grid |Grid| for which to assemble the functional. function The |Function| with which to take the inner product. boundary_type The type of domain boundary (e.g. 'neumann') on which to assemble the functional. If `None` the functional is assembled over the whole boundary. dirichlet_clear_dofs If `True`, set dirichlet boundary DOFs to zero. boundary_info If `boundary_type` is specified or `dirichlet_clear_dofs` is `True`, the |BoundaryInfo| determining which boundary entity belongs to which physical boundary. name The name of the functional. """ sparse = False source = NumpyVectorSpace(1) def __init__(self, grid, function, boundary_type=None, dirichlet_clear_dofs=False, boundary_info=None, name=None): assert grid.reference_element(0) in {line, triangle, square} assert function.shape_range == () assert not (boundary_type or dirichlet_clear_dofs) or boundary_info self.__auto_init(locals()) self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info NI = bi.boundaries(self.boundary_type, 1) if self.boundary_type else g.boundaries(1) if g.dim == 1: I = np.zeros(self.range.dim) I[NI] = self.function(g.centers(1)[NI]) else: F = self.function(g.centers(1)[NI], mu=mu) q, w = line.quadrature(order=1) # remove last dimension of q, as line coordinates are one dimensional q = q[:, 0] SF = np.array([1 - q, q]) SF_INTS = np.einsum('e,pi,e,i->ep', F, SF, g.integration_elements(1)[NI], w).ravel() SF_I = g.subentities(1, 2)[NI].ravel() I = coo_matrix((SF_INTS, (np.zeros_like(SF_I), SF_I)), shape=(1, g.size(g.dim))).toarray().ravel() if self.dirichlet_clear_dofs and bi.has_dirichlet: DI = bi.dirichlet_boundaries(g.dim) I[DI] = 0 return I.reshape((-1, 1))
[docs]class BoundaryDirichletFunctional(NumpyMatrixBasedOperator): """Linear finite element functional for enforcing Dirichlet boundary values. Parameters ---------- grid |Grid| for which to assemble the functional. dirichlet_data |Function| providing the Dirichlet boundary values. boundary_info |BoundaryInfo| determining the Dirichlet boundaries. name The name of the functional. """ sparse = False source = NumpyVectorSpace(1) def __init__(self, grid, dirichlet_data, boundary_info, name=None): assert grid.reference_element(0) in {line, triangle, square} self.__auto_init(locals()) self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info I = np.zeros(self.range.dim) DI = bi.dirichlet_boundaries(g.dim) I[DI] = self.dirichlet_data(g.centers(g.dim)[DI], mu=mu) return I.reshape((-1, 1))
[docs]class L2ProductFunctionalQ1(NumpyMatrixBasedOperator): """Bilinear finite element functional representing the inner product with an L2-|Function|. Parameters ---------- grid |Grid| for which to assemble the functional. function The |Function| with which to take the inner product. dirichlet_clear_dofs If `True`, set dirichlet boundary DOFs to zero. boundary_info |BoundaryInfo| determining the Dirichlet boundaries in case `dirichlet_clear_dofs` is set to `True`. name The name of the functional. """ sparse = False source = NumpyVectorSpace(1) def __init__(self, grid, function, dirichlet_clear_dofs=False, boundary_info=None, name=None): assert grid.reference_element(0) in {square} assert function.shape_range == () assert not dirichlet_clear_dofs or boundary_info self.__auto_init(locals()) self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # evaluate function at all quadrature points -> shape = (g.size(0), number of quadrature points) F = self.function(g.centers(0), mu=mu) # evaluate the shape functions at the quadrature points on the reference # element -> shape = (number of shape functions, number of quadrature points) q, w = g.reference_element.quadrature(order=1) if g.dim == 2: SF = np.array(((1 - q[..., 0]) * (1 - q[..., 1]), (1 - q[..., 1]) * (q[..., 0]), (q[..., 0]) * (q[..., 1]), (q[..., 1]) * (1 - q[..., 0]))) else: raise NotImplementedError # integrate the products of the function with the shape functions on each element # -> shape = (g.size(0), number of shape functions) SF_INTS = np.einsum('e,pi,e,i->ep', F, SF, g.integration_elements(0), w).ravel() # map local DOFs to global DOFs # FIXME This implementation is horrible, find a better way! SF_I = g.subentities(0, g.dim).ravel() I = coo_matrix((SF_INTS, (np.zeros_like(SF_I), SF_I)), shape=(1, g.size(g.dim))).toarray().ravel() if self.dirichlet_clear_dofs and bi.has_dirichlet: DI = bi.dirichlet_boundaries(g.dim) I[DI] = 0 return I.reshape((-1, 1))
[docs]class L2ProductP1(NumpyMatrixBasedOperator): """|Operator| representing the L2-product between linear finite element functions. Parameters ---------- grid The |Grid| for which to assemble the product. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. dirichlet_clear_rows If `True`, set the rows of the system matrix corresponding to Dirichlet boundary DOFs to zero. dirichlet_clear_columns If `True`, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero. dirichlet_clear_diag If `True`, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise, if either `dirichlet_clear_rows` or `dirichlet_clear_columns` is `True`, the diagonal entries are set to one. coefficient_function Coefficient |Function| for product with `shape_range == ()`. If `None`, constant one is assumed. solver_options The |solver_options| for the operator. name The name of the product. """ sparse = True def __init__(self, grid, boundary_info, dirichlet_clear_rows=True, dirichlet_clear_columns=False, dirichlet_clear_diag=False, coefficient_function=None, solver_options=None, name=None): assert grid.reference_element in (line, triangle) self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # our shape functions if g.dim == 2: SF = [lambda X: 1 - X[..., 0] - X[..., 1], lambda X: X[..., 0], lambda X: X[..., 1]] q, w = triangle.quadrature(order=2) elif g.dim == 1: SF = [lambda X: 1 - X[..., 0], lambda X: X[..., 0]] q, w = line.quadrature(order=2) else: raise NotImplementedError # evaluate the shape functions on the quadrature points SFQ = np.array(tuple(f(q) for f in SF)) self.logger.info('Integrate the products of the shape functions on each element') # -> shape = (g.size(0), number of shape functions ** 2) if self.coefficient_function is not None: C = self.coefficient_function(self.grid.centers(0), mu=mu) SF_INTS = np.einsum('iq,jq,q,e,e->eij', SFQ, SFQ, w, g.integration_elements(0), C).ravel() del C else: SF_INTS = np.einsum('iq,jq,q,e->eij', SFQ, SFQ, w, g.integration_elements(0)).ravel() del SFQ self.logger.info('Determine global dofs ...') SF_I0 = np.repeat(g.subentities(0, g.dim), g.dim + 1, axis=1).ravel() SF_I1 = np.tile(g.subentities(0, g.dim), [1, g.dim + 1]).ravel() self.logger.info('Boundary treatment ...') if bi.has_dirichlet: if self.dirichlet_clear_rows: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I0], 0, SF_INTS) if self.dirichlet_clear_columns: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I1], 0, SF_INTS) if not self.dirichlet_clear_diag and (self.dirichlet_clear_rows or self.dirichlet_clear_columns): SF_INTS = np.hstack((SF_INTS, np.ones(bi.dirichlet_boundaries(g.dim).size))) SF_I0 = np.hstack((SF_I0, bi.dirichlet_boundaries(g.dim))) SF_I1 = np.hstack((SF_I1, bi.dirichlet_boundaries(g.dim))) self.logger.info('Assemble system matrix ...') A = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) del SF_INTS, SF_I0, SF_I1 A = csc_matrix(A).copy() # See DiffusionOperatorP1 for why copy() is necessary return A
[docs]class L2ProductQ1(NumpyMatrixBasedOperator): """|Operator| representing the L2-product between bilinear finite element functions. Parameters ---------- grid The |Grid| for which to assemble the product. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. dirichlet_clear_rows If `True`, set the rows of the system matrix corresponding to Dirichlet boundary DOFs to zero. dirichlet_clear_columns If `True`, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero. dirichlet_clear_diag If `True`, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise, if either `dirichlet_clear_rows` or `dirichlet_clear_columns` is `True`, the diagonal entries are set to one. coefficient_function Coefficient |Function| for product with `shape_range == ()`. If `None`, constant one is assumed. solver_options The |solver_options| for the operator. name The name of the product. """ sparse = True def __init__(self, grid, boundary_info, dirichlet_clear_rows=True, dirichlet_clear_columns=False, dirichlet_clear_diag=False, coefficient_function=None, solver_options=None, name=None): assert grid.reference_element in {square} self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # our shape functions if g.dim == 2: SF = [lambda X: (1 - X[..., 0]) * (1 - X[..., 1]), lambda X: (1 - X[..., 1]) * (X[..., 0]), lambda X: (X[..., 0]) * (X[..., 1]), lambda X: (1 - X[..., 0]) * (X[..., 1])] else: raise NotImplementedError q, w = square.quadrature(order=2) # evaluate the shape functions on the quadrature points SFQ = np.array(tuple(f(q) for f in SF)) self.logger.info('Integrate the products of the shape functions on each element') # -> shape = (g.size(0), number of shape functions ** 2) if self.coefficient_function is not None: C = self.coefficient_function(self.grid.centers(0), mu=mu) SF_INTS = np.einsum('iq,jq,q,e,e->eij', SFQ, SFQ, w, g.integration_elements(0), C).ravel() del C else: SF_INTS = np.einsum('iq,jq,q,e->eij', SFQ, SFQ, w, g.integration_elements(0)).ravel() del SFQ self.logger.info('Determine global dofs ...') SF_I0 = np.repeat(g.subentities(0, g.dim), 4, axis=1).ravel() SF_I1 = np.tile(g.subentities(0, g.dim), [1, 4]).ravel() self.logger.info('Boundary treatment ...') if bi.has_dirichlet: if self.dirichlet_clear_rows: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I0], 0, SF_INTS) if self.dirichlet_clear_columns: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I1], 0, SF_INTS) if not self.dirichlet_clear_diag and (self.dirichlet_clear_rows or self.dirichlet_clear_columns): SF_INTS = np.hstack((SF_INTS, np.ones(bi.dirichlet_boundaries(g.dim).size))) SF_I0 = np.hstack((SF_I0, bi.dirichlet_boundaries(g.dim))) SF_I1 = np.hstack((SF_I1, bi.dirichlet_boundaries(g.dim))) self.logger.info('Assemble system matrix ...') A = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) del SF_INTS, SF_I0, SF_I1 A = csc_matrix(A).copy() # See DiffusionOperatorP1 for why copy() is necessary return A
[docs]class DiffusionOperatorP1(NumpyMatrixBasedOperator): """Diffusion |Operator| for linear finite elements. The operator is of the form :: (Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ] The function `d` can be scalar- or matrix-valued. Parameters ---------- grid The |Grid| for which to assemble the operator. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. diffusion_function The |Function| `d(x)` with `shape_range == ()` or `shape_range = (grid.dim, grid.dim)`. If `None`, constant one is assumed. diffusion_constant The constant `c`. If `None`, `c` is set to one. dirichlet_clear_columns If `True`, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero. dirichlet_clear_diag If `True`, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one. solver_options The |solver_options| for the operator. name Name of the operator. """ sparse = True def __init__(self, grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None): assert grid.reference_element(0) in {triangle, line}, 'A simplicial grid is expected!' assert diffusion_function is None \ or (isinstance(diffusion_function, Function) and diffusion_function.dim_domain == grid.dim and diffusion_function.shape_range == () or diffusion_function.shape_range == (grid.dim,) * 2) self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # gradients of shape functions if g.dim == 2: SF_GRAD = np.array(([-1., -1.], [1., 0.], [0., 1.])) elif g.dim == 1: SF_GRAD = np.array(([-1.], [1., ])) else: raise NotImplementedError self.logger.info('Calulate gradients of shape functions transformed by reference map ...') SF_GRADS = np.einsum('eij,pj->epi', g.jacobian_inverse_transposed(0), SF_GRAD) self.logger.info('Calculate all local scalar products beween gradients ...') if self.diffusion_function is not None and self.diffusion_function.shape_range == (): D = self.diffusion_function(self.grid.centers(0), mu=mu) SF_INTS = np.einsum('epi,eqi,e,e->epq', SF_GRADS, SF_GRADS, g.volumes(0), D).ravel() del D elif self.diffusion_function is not None: D = self.diffusion_function(self.grid.centers(0), mu=mu) SF_INTS = np.einsum('epi,eqj,e,eij->epq', SF_GRADS, SF_GRADS, g.volumes(0), D).ravel() del D else: SF_INTS = np.einsum('epi,eqi,e->epq', SF_GRADS, SF_GRADS, g.volumes(0)).ravel() del SF_GRADS if self.diffusion_constant is not None: SF_INTS *= self.diffusion_constant self.logger.info('Determine global dofs ...') SF_I0 = np.repeat(g.subentities(0, g.dim), g.dim + 1, axis=1).ravel() SF_I1 = np.tile(g.subentities(0, g.dim), [1, g.dim + 1]).ravel() self.logger.info('Boundary treatment ...') if bi.has_dirichlet: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I0], 0, SF_INTS) if self.dirichlet_clear_columns: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I1], 0, SF_INTS) if not self.dirichlet_clear_diag: SF_INTS = np.hstack((SF_INTS, np.ones(bi.dirichlet_boundaries(g.dim).size))) SF_I0 = np.hstack((SF_I0, bi.dirichlet_boundaries(g.dim))) SF_I1 = np.hstack((SF_I1, bi.dirichlet_boundaries(g.dim))) self.logger.info('Assemble system matrix ...') A = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) del SF_INTS, SF_I0, SF_I1 A = csc_matrix(A).copy() # The call to copy() is necessary to resize the data arrays of the sparse matrix: # During the conversion to crs_matrix, entries corresponding with the same # coordinates are summed up, resulting in shorter data arrays. The shortening # is implemented by calling self.prune() which creates the view self.data[:self.nnz]. # Thus, the original data array is not deleted and all memory stays allocated. return A
[docs]class DiffusionOperatorQ1(NumpyMatrixBasedOperator): """Diffusion |Operator| for bilinear finite elements. The operator is of the form :: (Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ] The function `d` can be scalar- or matrix-valued. Parameters ---------- grid The |Grid| for which to assemble the operator. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. diffusion_function The |Function| `d(x)` with `shape_range == ()` or `shape_range = (grid.dim, grid.dim)`. If `None`, constant one is assumed. diffusion_constant The constant `c`. If `None`, `c` is set to one. dirichlet_clear_columns If `True`, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero. dirichlet_clear_diag If `True`, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one. solver_options The |solver_options| for the operator. name Name of the operator. """ sparse = True def __init__(self, grid, boundary_info, diffusion_function=None, diffusion_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None): assert grid.reference_element(0) in {square}, 'A square grid is expected!' assert diffusion_function is None \ or (isinstance(diffusion_function, Function) and diffusion_function.dim_domain == grid.dim and diffusion_function.shape_range == () or diffusion_function.shape_range == (grid.dim,) * 2) self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # gradients of shape functions if g.dim == 2: q, w = g.reference_element.quadrature(order=2) SF_GRAD = np.array(([q[..., 1] - 1., q[..., 0] - 1.], [1. - q[..., 1], -q[..., 0]], [q[..., 1], q[..., 0]], [-q[..., 1], 1. - q[..., 0]])) else: raise NotImplementedError self.logger.info('Calulate gradients of shape functions transformed by reference map ...') SF_GRADS = np.einsum('eij,pjc->epic', g.jacobian_inverse_transposed(0), SF_GRAD) self.logger.info('Calculate all local scalar products beween gradients ...') if self.diffusion_function is not None and self.diffusion_function.shape_range == (): D = self.diffusion_function(self.grid.centers(0), mu=mu) SF_INTS = np.einsum('epic,eqic,c,e,e->epq', SF_GRADS, SF_GRADS, w, g.integration_elements(0), D).ravel() del D elif self.diffusion_function is not None: D = self.diffusion_function(self.grid.centers(0), mu=mu) SF_INTS = np.einsum('epic,eqjc,c,e,eij->epq', SF_GRADS, SF_GRADS, w, g.integration_elements(0), D).ravel() del D else: SF_INTS = np.einsum('epic,eqic,c,e->epq', SF_GRADS, SF_GRADS, w, g.integration_elements(0)).ravel() del SF_GRADS if self.diffusion_constant is not None: SF_INTS *= self.diffusion_constant self.logger.info('Determine global dofs ...') SF_I0 = np.repeat(g.subentities(0, g.dim), 4, axis=1).ravel() SF_I1 = np.tile(g.subentities(0, g.dim), [1, 4]).ravel() self.logger.info('Boundary treatment ...') if bi.has_dirichlet: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I0], 0, SF_INTS) if self.dirichlet_clear_columns: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I1], 0, SF_INTS) if not self.dirichlet_clear_diag: SF_INTS = np.hstack((SF_INTS, np.ones(bi.dirichlet_boundaries(g.dim).size))) SF_I0 = np.hstack((SF_I0, bi.dirichlet_boundaries(g.dim))) SF_I1 = np.hstack((SF_I1, bi.dirichlet_boundaries(g.dim))) self.logger.info('Assemble system matrix ...') A = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) del SF_INTS, SF_I0, SF_I1 A = csc_matrix(A).copy() # The call to copy() is necessary to resize the data arrays of the sparse matrix: # During the conversion to crs_matrix, entries corresponding with the same # coordinates are summed up, resulting in shorter data arrays. The shortening # is implemented by calling self.prune() which creates the view self.data[:self.nnz]. # Thus, the original data array is not deleted and all memory stays allocated. return A
[docs]class AdvectionOperatorP1(NumpyMatrixBasedOperator): """Linear advection |Operator| for linear finite elements. The operator is of the form :: (Lu)(x) = c ∇ ⋅ [ v(x) u(x) ] The function `v` is vector-valued. Parameters ---------- grid The |Grid| for which to assemble the operator. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. advection_function The |Function| `v(x)` with `shape_range = (grid.dim, )`. If `None`, constant one is assumed. advection_constant The constant `c`. If `None`, `c` is set to one. dirichlet_clear_columns If `True`, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero. dirichlet_clear_diag If `True`, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one. solver_options The |solver_options| for the operator. name Name of the operator. """ sparse = True def __init__(self, grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None): assert grid.reference_element(0) in {triangle, line}, 'A simplicial grid is expected!' assert advection_function is None \ or (isinstance(advection_function, Function) and advection_function.dim_domain == grid.dim and advection_function.shape_range == (grid.dim,)) self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # gradients of shape functions if g.dim == 2: SF_GRAD = np.array(([-1., -1.], [1., 0.], [0., 1.])) SF = [lambda X: 1 - X[..., 0] - X[..., 1], lambda X: X[..., 0], lambda X: X[..., 1]] # SF_GRAD(function, component) else: raise NotImplementedError q, w = g.reference_element.quadrature(order=1) self.logger.info('Calulate gradients of shape functions transformed by reference map ...') SF_GRADS = np.einsum('eij,pj->epi', g.jacobian_inverse_transposed(0), SF_GRAD) # SF_GRADS(element, function, component) SFQ = np.array(tuple(f(q) for f in SF)) # SFQ(function, quadraturepoint) self.logger.info('Calculate all local scalar products beween gradients ...') D = self.advection_function(self.grid.centers(0), mu=mu) SF_INTS = - np.einsum('pc,eqi,c,e,ec->eqp', SFQ, SF_GRADS, w, g.integration_elements(0), D).ravel() del D del SF_GRADS if self.advection_constant is not None: SF_INTS *= self.advection_constant self.logger.info('Determine global dofs ...') SF_I0 = np.repeat(g.subentities(0, g.dim), g.dim + 1, axis=1).ravel() SF_I1 = np.tile(g.subentities(0, g.dim), [1, g.dim + 1]).ravel() self.logger.info('Boundary treatment ...') if bi.has_dirichlet: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I0], 0, SF_INTS) if self.dirichlet_clear_columns: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I1], 0, SF_INTS) if not self.dirichlet_clear_diag: SF_INTS = np.hstack((SF_INTS, np.ones(bi.dirichlet_boundaries(g.dim).size))) SF_I0 = np.hstack((SF_I0, bi.dirichlet_boundaries(g.dim))) SF_I1 = np.hstack((SF_I1, bi.dirichlet_boundaries(g.dim))) self.logger.info('Assemble system matrix ...') A = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) del SF_INTS, SF_I0, SF_I1 A = csc_matrix(A).copy() # The call to copy() is necessary to resize the data arrays of the sparse matrix: # During the conversion to crs_matrix, entries corresponding with the same # coordinates are summed up, resulting in shorter data arrays. The shortening # is implemented by calling self.prune() which creates the view self.data[:self.nnz]. # Thus, the original data array is not deleted and all memory stays allocated. return A
[docs]class AdvectionOperatorQ1(NumpyMatrixBasedOperator): """Linear advection |Operator| for bilinear finite elements. The operator is of the form :: (Lu)(x) = c ∇ ⋅ [ v(x) u(x) ] The function `v` has to be vector-valued. Parameters ---------- grid The |Grid| for which to assemble the operator. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. advection_function The |Function| `v(x)` with `shape_range = (grid.dim, )`. If `None`, constant one is assumed. advection_constant The constant `c`. If `None`, `c` is set to one. dirichlet_clear_columns If `True`, set columns of the system matrix corresponding to Dirichlet boundary DOFs to zero to obtain a symmetric system matrix. Otherwise, only the rows will be set to zero. dirichlet_clear_diag If `True`, also set diagonal entries corresponding to Dirichlet boundary DOFs to zero. Otherwise they are set to one. solver_options The |solver_options| for the operator. name Name of the operator. """ sparse = True def __init__(self, grid, boundary_info, advection_function=None, advection_constant=None, dirichlet_clear_columns=False, dirichlet_clear_diag=False, solver_options=None, name=None): assert grid.reference_element(0) in {square}, 'A square grid is expected!' assert advection_function is None \ or (isinstance(advection_function, Function) and advection_function.dim_domain == grid.dim and advection_function.shape_range == (grid.dim,)) self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info # gradients of shape functions if g.dim == 2: q, w = g.reference_element.quadrature(order=2) SF_GRAD = np.array(([q[..., 1] - 1., q[..., 0] - 1.], [1. - q[..., 1], -q[..., 0]], [q[..., 1], q[..., 0]], [-q[..., 1], 1. - q[..., 0]])) # SF_GRAD(function, component, quadraturepoint) SF = [lambda X: (1 - X[..., 0]) * (1 - X[..., 1]), lambda X: (1 - X[..., 1]) * (X[..., 0]), lambda X: (X[..., 0]) * (X[..., 1]), lambda X: (1 - X[..., 0]) * (X[..., 1])] else: raise NotImplementedError self.logger.info('Calulate gradients of shape functions transformed by reference map ...') SF_GRADS = np.einsum('eij,pjc->epic', g.jacobian_inverse_transposed(0), SF_GRAD) # SF_GRADS(element,function,component,quadraturepoint) SFQ = np.array(tuple(f(q) for f in SF)) # SFQ(function, quadraturepoint) self.logger.info('Calculate all local scalar products beween gradients ...') D = self.advection_function(self.grid.centers(0), mu=mu) SF_INTS = - np.einsum('pc,eqic,c,e,ei->eqp', SFQ, SF_GRADS, w, g.integration_elements(0), D).ravel() del D del SF_GRADS if self.advection_constant is not None: SF_INTS *= self.advection_constant self.logger.info('Determine global dofs ...') SF_I0 = np.repeat(g.subentities(0, g.dim), 4, axis=1).ravel() SF_I1 = np.tile(g.subentities(0, g.dim), [1, 4]).ravel() self.logger.info('Boundary treatment ...') if bi.has_dirichlet: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I0], 0, SF_INTS) if self.dirichlet_clear_columns: SF_INTS = np.where(bi.dirichlet_mask(g.dim)[SF_I1], 0, SF_INTS) if not self.dirichlet_clear_diag: SF_INTS = np.hstack((SF_INTS, np.ones(bi.dirichlet_boundaries(g.dim).size))) SF_I0 = np.hstack((SF_I0, bi.dirichlet_boundaries(g.dim))) SF_I1 = np.hstack((SF_I1, bi.dirichlet_boundaries(g.dim))) self.logger.info('Assemble system matrix ...') A = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) del SF_INTS, SF_I0, SF_I1 A = csc_matrix(A).copy() # The call to copy() is necessary to resize the data arrays of the sparse matrix: # During the conversion to crs_matrix, entries corresponding with the same # coordinates are summed up, resulting in shorter data arrays. The shortening # is implemented by calling self.prune() which creates the view self.data[:self.nnz]. # Thus, the original data array is not deleted and all memory stays allocated. return A
[docs]class RobinBoundaryOperator(NumpyMatrixBasedOperator): """Robin boundary |Operator| for linear finite elements. The operator represents the contribution of Robin boundary conditions to the stiffness matrix, where the boundary condition is supposed to be given in the form :: -[ d(x) ∇u(x) ] ⋅ n(x) = c(x) (u(x) - g(x)) `d` and `n` are the diffusion function (see :class:`DiffusionOperatorP1`) and the unit outer normal in `x`, while `c` is the (scalar) Robin parameter function and `g` is the (also scalar) Robin boundary value function. Parameters ---------- grid The |Grid| over which to assemble the operator. boundary_info |BoundaryInfo| for the treatment of Dirichlet boundary conditions. robin_data Tuple providing two |Functions| that represent the Robin parameter and boundary value function. If `None`, the resulting operator is zero. solver_options The |solver_options| for the operator. name Name of the operator. """ sparse = True def __init__(self, grid, boundary_info, robin_data=None, solver_options=None, name=None): assert robin_data is None or (isinstance(robin_data, tuple) and len(robin_data) == 2) assert robin_data is None or all([isinstance(f, Function) and f.dim_domain == grid.dim and (f.shape_range == () or f.shape_range == (grid.dim,)) for f in robin_data]) self.__auto_init(locals()) self.source = self.range = CGVectorSpace(grid) def _assemble(self, mu=None): g = self.grid bi = self.boundary_info if g.dim > 2: raise NotImplementedError if bi is None or not bi.has_robin or self.robin_data is None: return coo_matrix((g.size(g.dim), g.size(g.dim))).tocsc() RI = bi.robin_boundaries(1) if g.dim == 1: robin_c = self.robin_data[0](g.centers(1)[RI], mu=mu) I = coo_matrix((robin_c, (RI, RI)), shape=(g.size(g.dim), g.size(g.dim))) return csc_matrix(I).copy() else: xref = g.centers(1)[RI] # xref(robin-index, quadraturepoint-index) if self.robin_data[0].shape_range == (): robin_c = self.robin_data[0](xref, mu=mu) else: robin_elements = g.superentities(1, 0)[RI, 0] robin_indices = g.superentity_indices(1, 0)[RI, 0] normals = g.unit_outer_normals()[robin_elements, robin_indices] robin_values = self.robin_data[0](xref, mu=mu) robin_c = np.einsum('ei,eqi->eq', normals, robin_values) # robin_c(robin-index, quadraturepoint-index) q, w = line.quadrature(order=2) # remove last dimension of q, as line coordinates are one dimensional q = q[:, 0] SF = np.array([1 - q, q]) SF_INTS = np.einsum('e,pi,pj,e,p->eij', robin_c, SF, SF, g.integration_elements(1)[RI], w).ravel() SF_I0 = np.repeat(g.subentities(1, g.dim)[RI], 2).ravel() SF_I1 = np.tile(g.subentities(1, g.dim)[RI], [1, 2]).ravel() I = coo_matrix((SF_INTS, (SF_I0, SF_I1)), shape=(g.size(g.dim), g.size(g.dim))) return csc_matrix(I).copy()
[docs]class InterpolationOperator(NumpyMatrixBasedOperator): """Vector-like Lagrange interpolation |Operator| for continuous finite element spaces. Parameters ---------- grid The |Grid| on which to interpolate. function The |Function| to interpolate. """ source = NumpyVectorSpace(1) linear = True def __init__(self, grid, function): assert function.dim_domain == grid.dim assert function.shape_range == () self.__auto_init(locals()) self.range = CGVectorSpace(grid) def _assemble(self, mu=None): return self.function.evaluate(self.grid.centers(self.grid.dim), mu=mu).reshape((-1, 1))
[docs]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 m = StationaryModel(L, F, output_functional=output_functional, products=products, visualizer=visualizer, name=f'{p.name}_CG') data = {'grid': grid, 'boundary_info': boundary_info} if preassemble: data['unassembled_m'] = m m = preassemble_(m) return m, data
[docs]def discretize_instationary_cg(analytical_problem, diameter=None, domain_discretizer=None, grid_type=None, grid=None, boundary_info=None, num_values=None, time_stepper=None, nt=None, preassemble=True): """Discretizes an |InstationaryProblem| with a |StationaryProblem| as stationary part using finite elements. Parameters ---------- analytical_problem The |InstationaryProblem| 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. num_values The number of returned vectors of the solution trajectory. If `None`, each intermediate vector that is calculated is returned. time_stepper The :class:`time-stepper <pymor.algorithms.timestepping.TimeStepper>` to be used by :class:`~pymor.models.basic.InstationaryModel.solve`. nt If `time_stepper` is not specified, the number of time steps for implicit Euler time stepping. 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, InstationaryProblem) assert isinstance(analytical_problem.stationary_part, 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 (time_stepper is None) != (nt is None) p = analytical_problem m, data = discretize_stationary_cg(p.stationary_part, diameter=diameter, domain_discretizer=domain_discretizer, grid_type=grid_type, grid=grid, boundary_info=boundary_info) if p.initial_data.parametric: I = InterpolationOperator(data['grid'], p.initial_data) else: I = p.initial_data.evaluate(data['grid'].centers(data['grid'].dim)) I = m.solution_space.make_array(I) if time_stepper is None: if p.stationary_part.diffusion is None: time_stepper = ExplicitEulerTimeStepper(nt=nt) else: time_stepper = ImplicitEulerTimeStepper(nt=nt) mass = m.l2_0_product m = InstationaryModel(operator=m.operator, rhs=m.rhs, mass=mass, initial_data=I, T=p.T, products=m.products, output_functional=m.output_functional, time_stepper=time_stepper, visualizer=m.visualizer, num_values=num_values, name=f'{p.name}_CG') if preassemble: data['unassembled_m'] = m m = preassemble_(m) return m, data