Connections: Interpolating between Discretizations#
Base classes#
- class meshmode.discretization.connection.DiscretizationConnection(from_discr: Discretization, to_discr: Discretization, is_surjective: bool)[source]#
Abstract interface for transporting a DOF vector from one
meshmode.discretization.Discretization
to another. Possible applications include:upsampling/downsampling on the same mesh
restricition to the boundary
interpolation to a refined/coarsened mesh
interpolation onto opposing faces
computing modal data from nodal coefficients
computing nodal coefficients from modal data
- from_discr#
- to_discr#
- abstract __call__(ary: ArrayOrContainerT) ArrayOrContainerT [source]#
Apply the connection. If applicable, may return a view of the data instead of a copy, i.e. changes to ary may or may not appear in the result returned by this method, and vice versa.
- class meshmode.discretization.connection.IdentityDiscretizationConnection(discr: Discretization)[source]#
A no-op connection from a
Discretization
to the same discretization that returns the same data unmodified.
- class meshmode.discretization.connection.ChainedDiscretizationConnection(connections, from_discr=None)[source]#
Aggregates multiple
DiscretizationConnection
instances into a single one.- connections#
- class meshmode.discretization.connection.L2ProjectionInverseDiscretizationConnection(connections, is_surjective=False)[source]#
Creates an inverse
DiscretizationConnection
from an existing connection to allow transporting from the original connectionâ€™s to_discr to from_discr.- from_discr#
- to_discr#
- is_surjective#
- conn#
- __call__(ary)[source]#
- Parameters:
ary â€“ a
DOFArray
, or anarraycontext.ArrayContainer
of them, containing nodal coefficient data onfrom_discr
.
- class meshmode.discretization.connection.DirectDiscretizationConnection(from_discr: Discretization, to_discr: Discretization, groups: Sequence[DiscretizationConnectionElementGroup], is_surjective: bool)[source]#
A concrete
DiscretizationConnection
supported by interpolation data.- from_discr#
- to_discr#
- groups#
a list of
DiscretizationConnectionElementGroup
instances, with a one-to-one correspondence to the groups into_discr
.
- is_permutation(tol_multiplier: Optional[float] = None) bool [source]#
Return True if no interpolation is used in applying this connection, i.e. if the source unit nodes in the connection (cf.
InterpolationBatch.result_unit_nodes
) match up with the unit nodes in the source element, up to machine epsilon multiplied by tol_multiplier (or an internally specified tolerance if none is given).
- __call__(ary: ArrayOrContainerT, *, _force_use_loopy: bool = False, _force_no_merged_batches: bool = False) ArrayOrContainerT [source]#
- Parameters:
ary â€“ a
DOFArray
, or anarraycontext.ArrayContainer
of them, containing nodal coefficient data onfrom_discr
.
Mapping between modal and nodal representations#
- class meshmode.discretization.connection.NodalToModalDiscretizationConnection(from_discr, to_discr, allow_approximate_quad=False)[source]#
A concrete subclass of
DiscretizationConnection
, which maps nodal data to its modal representation. For interpolatory (unisolvent) element groups, the mapping from nodal to modal representations is performed via:\[y = V^{-1} [\text{nodal basis coefficients}]\]where \(V_{i,j} = \phi_j(x_i)\) is the generalized Vandermonde matrix, \(\phi_j\) is a nodal basis, and \(x_i\) are nodal points on the reference element defining the nodal discretization.
For non-interpolatory element groups (for example,
QuadratureSimplexElementGroup
), modal coefficients are computed using the underlying quadrature rule \((w_q, x_q)\), and an orthonormal basis \(\psi_i\) spanning the modal discretization space. The modal coefficients are then obtained via:\[y = V^T W [\text{nodal basis coefficients}]\]where \(V_{i, j} = \psi_j(x_i)\) is the Vandermonde matrix constructed from the orthonormal basis evaluated at the quadrature nodes \(x_i\), and \(W = \text{Diag}(w_q)\) is a diagonal matrix containing the quadrature weights \(w_q\).
Note
This connection requires that both nodal and modal discretizations are defined on the same mesh.
- from_discr#
An instance of
meshmode.discretization.Discretization
containingNodalElementGroupBase
element groups
- to_discr#
An instance of
meshmode.discretization.Discretization
containingModalElementGroupBase
element groups
- groups#
A list of
DiscretizationConnectionElementGroup
instances, with a one-to-one correspondence to the groups into_discr
.
- __init__(from_discr, to_discr, allow_approximate_quad=False)[source]#
- Parameters:
from_discr â€“ a
meshmode.discretization.Discretization
containingNodalElementGroupBase
element groups.to_discr â€“ a
meshmode.discretization.Discretization
containingModalElementGroupBase
element groups.allow_approximate_quad â€“ an optional
bool
flag indicating whether to proceed with numerically approximating (via quadrature) modal coefficients, even when the underlying quadrature method is not exact. The default value is False.
- __call__(ary)[source]#
Computes modal coefficients data from a functions nodal coefficients.
- Parameters:
ary â€“ a
DOFArray
, or anarraycontext.ArrayContainer
of them, containing nodal coefficient data.
- class meshmode.discretization.connection.ModalToNodalDiscretizationConnection(from_discr, to_discr)[source]#
A concrete subclass of
DiscretizationConnection
, which maps modal data back to its nodal representation. This is computed via:\[y = V [\text{modal basis coefficients}]\]where \(V_{i,j} = \phi_j(x_i)\) is the generalized Vandermonde matrix, \(\phi_j\) is an orthonormal (modal) basis, and \(x_i\) are nodal points on the reference element defining the nodal discretization.
Note
This connection requires that both nodal and modal discretizations are defined on the same mesh.
- from_discr#
An instance of
meshmode.discretization.Discretization
containingModalElementGroupBase
element groups
- to_discr#
An instance of
meshmode.discretization.Discretization
containingNodalElementGroupBase
element groups
- groups#
A list of
DiscretizationConnectionElementGroup
instances, with a one-to-one correspondence to the groups into_discr
.
- __init__(from_discr, to_discr)[source]#
- Parameters:
from_discr â€“ a
meshmode.discretization.Discretization
containingModalElementGroupBase
element groups.to_discr â€“ a
meshmode.discretization.Discretization
containingNodalElementGroupBase
element groups.
- __call__(ary)[source]#
Computes nodal coefficients from modal data.
- Parameters:
ary â€“ a
DOFArray
, or anarraycontext.ArrayContainer
of them, containing modal coefficient data.
Same-mesh connections#
Restriction to faces#
- meshmode.discretization.connection.FACE_RESTR_INTERIOR = <class 'meshmode.discretization.connection.face.FACE_RESTR_INTERIOR'>[source]#
A special value to pass to
meshmode.discretization.connection.make_face_restriction()
to produce a discretization consisting of all interior faces in a discretization.
- meshmode.discretization.connection.FACE_RESTR_ALL = <class 'meshmode.discretization.connection.face.FACE_RESTR_ALL'>[source]#
A special value to pass to
meshmode.discretization.connection.make_face_restriction()
to produce a discretization consisting of all faces (interior and boundary) faces in a discretization.
- meshmode.discretization.connection.make_face_restriction(actx, discr, group_factory, boundary_tag, per_face_groups=False)[source]#
Create a mesh, a discretization and a connection to restrict a function on discr to its values on the edges of element faces denoted by boundary_tag.
- Parameters:
boundary_tag â€“ The boundary tag for which to create a face restriction. May be
FACE_RESTR_INTERIOR
to indicate interior faces, orFACE_RESTR_ALL
to make a discretization consisting of all (interior and boundary) faces.per_face_groups â€“
If True, the resulting discretization is guaranteed to have groups organized as:
(grp0, face0), (grp0, face1), ... (grp0, faceN), (grp1, face0), (grp1, face1), ... (grp1, faceN), ...
each with the elements in the same order as the originating group. If False, volume and boundary groups correspond with each other one-to-one, and an interpolation batch is created per face.
- Returns:
a
meshmode.discretization.connection.DirectDiscretizationConnection
representing the new connection. The new boundary discretization can be obtained from themeshmode.discretization.connection.DirectDiscretizationConnection.to_discr
attribute of the return value, and the corresponding new boundary mesh from that.
- meshmode.discretization.connection.make_face_to_all_faces_embedding(actx, faces_connection, all_faces_discr, from_discr=None)[source]#
Return a
meshmode.discretization.connection.DiscretizationConnection
connecting a discretization containing some faces of a discretization to one containing all faces.- Parameters:
faces_connection â€“ must be the (connection) result of calling
meshmode.discretization.connection.make_face_restriction()
withmeshmode.discretization.connection.FACE_RESTR_INTERIOR
or a boundary tag.all_faces_discr â€“ must be the (discretization) result of calling
meshmode.discretization.connection.make_face_restriction()
withmeshmode.discretization.connection.FACE_RESTR_ALL
for the same volume discretization as the one from which faces_discr was obtained.from_discr â€“ Allows substituting in a different origin discretization for the returned connection. This discretization must use the same mesh as
faces_connection.to_discr
.
- meshmode.discretization.connection.make_opposite_face_connection(actx, volume_to_bdry_conn)[source]#
Given a boundary restriction connection volume_to_bdry_conn, return a
DirectDiscretizationConnection
that performs data exchange across opposite faces.
Refinement#
- meshmode.discretization.connection.make_refinement_connection(actx, refiner, coarse_discr, group_factory)[source]#
Return a
meshmode.discretization.connection.DiscretizationConnection
connecting coarse_discr to a discretization on the fine mesh.- Parameters:
refiner â€“ An instance of
meshmode.mesh.refinement.RefinerWithoutAdjacency
coarse_discr â€“ An instance of
meshmode.discretization.Discretization
associated with the mesh given to the refinergroup_factory â€“ An instance of
meshmode.discretization.ElementGroupFactory
. Used for discretizing the fine mesh.
Flattening a ChainedDiscretizationConnection
#
- meshmode.discretization.connection.flatten_chained_connection(actx, connection)[source]#
Collapse a connection into a direct connection.
If the given connection is already a
DirectDiscretizationConnection
nothing is done. However, if the connection is aChainedDiscretizationConnection
, a new direct connection is constructed that transports fromfrom_discr
toto_discr
.The new direct connection will have a number of groups and batches that is, at worse, the product of all the connections in the chain. For example, if we consider a connection between a discretization and a two-level refinement, both levels will have \(n\) groups and \(m + 1\) batches per group, where \(m\) is the number of subdivisions of an element (exact number depends on implementation details in
make_refinement_connection()
). However, a direct connection from level \(0\) to level \(2\) will have at worst \(n^2\) groups and each group will have \((m + 1)^2\) batches.Warning
If a large number of connections is chained, the number of groups and batches can become very large.
- Parameters:
actx â€“ An instance of
arraycontext.ArrayContext
.connection â€“ An instance of
DiscretizationConnection
.
- Returns:
An instance of
DirectDiscretizationConnection
.
Implementation details#
- class meshmode.discretization.connection.InterpolationBatch(from_group_index: int, from_element_indices: ArrayT, to_element_indices: ArrayT, result_unit_nodes: ndarray, to_element_face: Optional[int])[source]#
One interpolation batch captures how a batch of elements within an element group should be an interpolated. Note that while itâ€™s possible that an interpolation batch takes care of interpolating an entire element group from source to target, thatâ€™s not necessarily the case. Consider the case of extracting boundary values of a discretization. For, say, a triangle, at least three different interpolation batches are needed to cover boundary edges that fall onto each of the three edges of the unit triangle.
- from_group_index#
An integer indicating from which element group in the from discretization the data should be interpolated.
- from_element_indices#
An array of dtype/shape
element_id_t [nelements]
. This contains the (group-local) element index (relative tofrom_group_index
from which this â€śtoâ€ť elementâ€™s data will be interpolated.
- to_element_indices#
An array of dtype/shape
element_id_t [nelements]
. This contains the (group-local) element index to which this â€śtoâ€ť elementâ€™s data will be interpolated.
- result_unit_nodes#
A
numpy.ndarray
of shape(from_group.dim,to_group.nunit_nodes)
storing the coordinates of the nodes (in unit coordinates of the from reference element) from which the node locations of this element should be interpolated.
- nelements#
- to_element_face#
int or None. If this interpolation batch targets interpolation to a face, then this number captures the face number (on all elements referenced by
from_element_indices
to which this batch interpolates. (Since there is a fixed set of â€śfromâ€ť unit nodes per batch, one batch will always go to a single face index.)Note
This attribute is not required. It exists only to carry along metadata from
make_face_restriction()
to routines that build upon its output, such asmake_opposite_face_connection()
. If you are not building or consuming face restrictions, it is safe to leave this unset and/or ignore it. This attribute probably belongs in a subclass, but that refactoring hasnâ€™t happened yet. (Sorry!)
- class meshmode.discretization.connection.DiscretizationConnectionElementGroup(batches)[source]#
- batches#
A list of
InterpolationBatch
instances.