pymor.discretizers.builtin.grids.interfaces

Module Contents

class pymor.discretizers.builtin.grids.interfaces.BoundaryInfo

Bases: pymor.core.cache.CacheableObject

Provides boundary types for the boundaries of a given Grid.

For every boundary type and codimension a mask is provided, marking grid entities of the respective type and codimension by their global index.

boundary_types

set of all boundary types the grid has.

Methods

boundaries
check_boundary_types
dirichlet_boundaries
dirichlet_mask
mask	Return mask.
neumann_boundaries
neumann_mask
no_boundary_type_mask	Return no boundary type mask.
robin_boundaries
robin_mask
unique_boundary_type_mask	Return unique boundary type mask.

cache_region = 'memory'

boundaries(boundary_type, codim)

check_boundary_types(assert_unique_type=(1,), assert_some_type=())

dirichlet_boundaries(codim)

dirichlet_mask(codim)

mask(boundary_type, codim)

Return mask.

retval[i] is True if the codim-codim entity of global index i is associated to the boundary type boundary_type.

neumann_boundaries(codim)

neumann_mask(codim)

no_boundary_type_mask(codim)

Return no boundary type mask.

retval[i] is True if the codim-codim entity of global index i is associated to no boundary type.

robin_boundaries(codim)

robin_mask(codim)

unique_boundary_type_mask(codim)

Return unique boundary type mask.

retval[i] is True if the codim-codim entity of global index i is associated to one and only one boundary type.

class pymor.discretizers.builtin.grids.interfaces.Grid

Bases: pymor.core.cache.CacheableObject

Affine grid.

Topological grid with geometry where each codim-0 entity is affinely mapped to the same ReferenceElement.

The grid is completely determined via the subentity relation given by subentities and the embeddings given by embeddings. In addition, only size and reference_element have to be implemented.

Methods

boundaries	Returns the global indices of all codim-codim boundary entities.
boundary_mask	Return boundary mask.
bounding_box	Returns a (2, dim)-shaped array containing lower/upper bounding box coordinates.
centers	retval[e] is the barycenter of the codim-codim entity with global index e.
diameters	retval[e] is the diameter of the codim-codim entity with global index e.
embeddings	Return embeddings.
integration_elements	retval[e] is given as sqrt(det(A^T*A)), where A = embeddings(codim)[0][e].
jacobian_inverse_transposed	Return the inverse of transposed Jacobian.
neighbors	Maps entity index and local neighbor index to global neighbor index.
quadrature_points	Returns the quadrature points.
reference_element	The ReferenceElement of the codim-codim entities.
size	The number of entities of codimension codim.
subentities	Return subentities.
superentities	Return superentities.
superentity_indices	Return indices of superentities.
unit_outer_normals	Return unit outer normals.
volumes	Return volumes.
volumes_inverse	retval[e] = 1 / volumes(codim)[e].

MAX_DOMAIN_RATIO = 1000000.0

MAX_DOMAIN_WIDTH = 10000000000.0

MIN_DOMAIN_WIDTH = 1e-10

cache_region = 'memory'

boundaries(codim)

Returns the global indices of all codim-codim boundary entities.

By definition, a codim-1 entity is a boundary entity if it has only one codim-0 superentity. For codim != 1, a codim-codim entity is a boundary entity if it has a codim-1 sub/super-entity.

boundary_mask(codim)

Return boundary mask.

retval[e] is true iff the codim-codim entity with global index e is a boundary entity.

By definition, a codim-1 entity is a boundary entity if it has only one codim-0 superentity. For codim != 1, a codim-codim entity is a boundary entity if it has a codim-1 sub/super-entity.

bounding_box()

Returns a (2, dim)-shaped array containing lower/upper bounding box coordinates.

centers(codim)

retval[e] is the barycenter of the codim-codim entity with global index e.

diameters(codim)

retval[e] is the diameter of the codim-codim entity with global index e.

abstract embeddings(codim)

Return embeddings.

Returns tuple (A, B) where A[e] and B[e] are the linear part and the translation part of the map from the reference element of e to e.

For codim > 0, we provide a default implementation by taking the embedding of the codim-1 parent entity e_0 of e with lowest global index and composing it with the subentity_embedding of e into e_0 determined by the reference element.

integration_elements(codim)

retval[e] is given as sqrt(det(A^T*A)), where A = embeddings(codim)[0][e].

jacobian_inverse_transposed(codim)

Return the inverse of transposed Jacobian.

retval[e] is the transposed (pseudo-)inverse of the Jacobian of embeddings(codim)[e].

neighbors(codim, neighbor_codim, intersection_codim=None)

Maps entity index and local neighbor index to global neighbor index.

retval[e,n] is the global index of the n-th codim-neighbor_codim entity of the codim-codim entity e that shares with e a subentity of codimension intersection_codim.

If intersection_codim == None, it is set to codim + 1 if codim == neighbor_codim and to min(codim, neighbor_codim) otherwise.

The default implementation is to compute the result from subentities(codim, intersection_codim) and superentities(intersection_codim, neighbor_codim).

quadrature_points(codim, order=None, npoints=None, quadrature_type='default')

Returns the quadrature points.

retval[e] is an array of quadrature points in global coordinates for the codim-codim entity with global index e.

The quadrature is of order order or has npoints integration points. To integrate a function f over e one has to form

np.dot(f(quadrature_points(codim, order)[e]),
    reference_element(codim).quadrature(order)[1]) *
integration_elements(codim)[e].  # NOQA

abstract reference_element(codim)

The ReferenceElement of the codim-codim entities.

abstract size(codim)

The number of entities of codimension codim.

abstract subentities(codim, subentity_codim)

Return subentities.

retval[e,s] is the global index of the s-th codim-subentity_codim subentity of the codim-codim entity with global index e.

The ordering of subentities(0, subentity_codim)[e] has to correspond, w.r.t. the embedding of e, to the local ordering inside the reference element.

For codim > 0, we provide a default implementation by calculating the subentities of e as follows:

1. Find the codim-1 parent entity e_0 of e with minimal global index

2. Lookup the local indices of the subentities of e inside e_0 using the reference element.

3. Map these local indices to global indices using subentities(codim - 1, subentity_codim).

This procedures assures that subentities(codim, subentity_codim)[e] has the right ordering w.r.t. the embedding determined by e_0, which agrees with what is returned by embeddings(codim)

superentities(codim, superentity_codim)

Return superentities.

retval[e,s] is the global index of the s-th codim-superentity_codim superentity of the codim-codim entity with global index e.

retval[e] is sorted by global index.

The default implementation is to compute the result from subentities(superentity_codim, codim).

superentity_indices(codim, superentity_codim)

Return indices of superentities.

retval[e,s] is the local index of the codim-codim entity e in the codim-superentity_codim superentity superentities(codim, superentity_codim)[e,s].

unit_outer_normals()

Return unit outer normals.

retval[e,i] is the unit outer normal to the i-th codim-1 subentity of the codim-0 entity with global index e.

volumes(codim)

Return volumes.

retval[e] is the (dim-codim)-dimensional volume of the codim-codim entity with global index e.

volumes_inverse(codim)

retval[e] = 1 / volumes(codim)[e].

class pymor.discretizers.builtin.grids.interfaces.GridWithOrthogonalCenters

Bases: Grid

Grid with an additional orthogonal_centers method.

Methods

orthogonal_centers

Return orthogonal centers.

abstract orthogonal_centers()

Return orthogonal centers.

retval[e] is a point inside the codim-0 entity with global index e such that the line segment from retval[e] to retval[e2] is always orthogonal to the codim-1 entity shared by the codim-0 entities with global index e and e2.

(This is mainly useful for gradient approximation in finite volume schemes.)

class pymor.discretizers.builtin.grids.interfaces.ReferenceElement

Bases: pymor.core.cache.CacheableObject

Defines a reference element.

All reference elements have the property that all subentities of a given codimension are of the same type. I.e. a three-dimensional reference element cannot have triangles and rectangles as faces at the same time.

dim

The dimension of the reference element

volume

The volume of the reference element

Methods

center	Coordinates of the barycenter.
mapped_diameter	Return the diameter after transformation.
quadrature	Return quadrature points and weights.
quadrature_info	Return quadrature information.
quadrature_types
size	Number of subentities of codimension codim.
sub_reference_element	Returns the reference element of the codim-codim subentities.
subentities	Return subentities.
subentity_embedding	Return subentity embedding.
unit_outer_normals	retval[e] is the unit outer-normal vector to the codim-1 subentity with index e.

cache_region = 'memory'

abstract center()

Coordinates of the barycenter.

abstract mapped_diameter(A)

Return the diameter after transformation.

The diameter of the reference element after transforming it with the matrix A (vectorized).

abstract quadrature(order=None, npoints=None, quadrature_type='default')

Return quadrature points and weights.

Returns tuple (P, W) where P is an array of quadrature points with corresponding weights W.

The quadrature is of order order or has npoints integration points.

abstract quadrature_info()

Return quadrature information.

Returns a tuple of dicts (O, N) where O[quadrature_type] is a list of orders which are implemented for quadrature_type and N[quadrature_type] is a list of the corresponding numbers of integration points.

quadrature_types()

abstract size(codim)

Number of subentities of codimension codim.

abstract sub_reference_element(codim)

Returns the reference element of the codim-codim subentities.

abstract subentities(codim, subentity_codim)

Return subentities.

subentities(c,sc)[i,j] is, with respect to the indexing inside the reference element, the index of the j-th codim-subentity_codim subentity of the i-th codim-codim subentity of the reference element.

abstract subentity_embedding(subentity_codim)

Return subentity embedding.

Returns a tuple (A, B) which defines the embedding of the codim-subentity_codim subentities into the reference element.

For subentity_codim > 1', the embedding is by default given recursively via `subentity_embedding(subentity_codim - 1) and sub_reference_element(subentity_codim - 1).subentity_embedding(1) choosing always the superentity with smallest index.

abstract unit_outer_normals()

retval[e] is the unit outer-normal vector to the codim-1 subentity with index e.