Connections: Interpolating between Discretizations

Base classes

class meshmode.discretization.connection.DiscretizationConnection(from_discr, to_discr, is_surjective)

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

from_discr
to_discr
is_surjective

A bool indicating whether every output degree of freedom is set by the connection.

__call__(queue, vec)

Call self as a function.

class meshmode.discretization.connection.ChainedDiscretizationConnection(connections, from_discr=None)

Aggregates multiple DiscretizationConnection instances into a single one.

connections
class meshmode.discretization.connection.L2ProjectionInverseDiscretizationConnection(conn, is_surjective=False)

Creates an inverse DiscretizationConnection from an existing connection to allow transporting from the original connection’s to_discr to from_discr.

from_discr
to_discr
is_surjective
conn
__call__(queue, vec)

Call self as a function.

class meshmode.discretization.connection.DirectDiscretizationConnection(from_discr, to_discr, groups, is_surjective)

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 in to_discr.

is_surjective

A bool indicating whether every output degree of freedom is set by the connection.

__call__(queue, vec)

Call self as a function.

full_resample_matrix(queue)

Build a dense matrix representing this discretization connection.

Warning

On average, this will be exceedingly expensive (\(O(N^2)\) in the number N of discretization points) in terms of memory usage and thus not what you’d typically want.

Same-mesh connections

meshmode.discretization.connection.make_same_mesh_connection(to_discr, from_discr)

Restriction to faces

meshmode.discretization.connection.FACE_RESTR_INTERIOR = <class 'meshmode.discretization.connection.face.FACE_RESTR_INTERIOR'>

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'>

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(discr, group_factory, boundary_tag, per_face_groups=False)

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 – The boundary tag for which to create a face restriction. May be meshmode.discretization.connection.FRESTR_INTERIOR_FACES to indicate interior faces, or meshmode.discretization.connection.FRESTR_ALL_FACES to make a discretization consisting of all (interior and boundary) faces.

  • per_face_groups –

    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 the meshmode.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(faces_connection, all_faces_discr, from_discr=None)

Return a meshmode.discretization.connection.DiscretizationConnection connecting a discretization containing some faces of a discretization to one containing all faces.

Parameters
meshmode.discretization.connection.make_opposite_face_connection(volume_to_bdry_conn)

Given a boundary restriction connection volume_to_bdry_conn, return a DirectDiscretizationConnection that performs data exchange across opposite faces.

Mesh partitioning

meshmode.discretization.connection.make_partition_connection(local_bdry_conn, i_local_part, remote_bdry, remote_adj_groups, remote_from_elem_faces, remote_from_elem_indices)

Connects local_bdry_conn to a neighboring partition.

Parameters

  • local_bdry_conn – A DiscretizationConnection of the local partition.

  • i_local_part – The partition number of the local partition.

  • remote_adj_groups – A list of InterPartitionAdjacency` of the remote partition.

  • remote_bdry – A Discretization of the boundary of the remote partition.

  • remote_from_elem_faces – remote_from_elem_faces[igrp][idx] gives the face that batch idx interpolates from in group igrp.

  • remote_from_elem_indices – remote_from_elem_indices[igrp][idx] gives a np.array of element indices that batch idx interpolates from in group igrp.

Returns

A DirectDiscretizationConnection that performs data exchange across faces from the remote partition to partition i_local_part.

New in version 2017.1.

Warning

Interface is not final.

Refinement

meshmode.discretization.connection.make_refinement_connection(refiner, coarse_discr, group_factory)

Return a meshmode.discretization.connection.DiscretizationConnection connecting coarse_discr to a discretization on the fine mesh.

Parameters

  • refiner – An instance of meshmode.mesh.refinement.Refiner

  • coarse_discr – An instance of meshmode.mesh.discretization.Discretization associated with the mesh given to the refiner

  • group_factory – An instance of meshmode.mesh.discretization.ElementGroupFactory. Used for discretizing the fine mesh.

Flattening a ChainedDiscretizationConnection

meshmode.discretization.connection.flatten_chained_connection(queue, connection)

Collapse a connection into a direct connection.

If the given connection is already a DirectDiscretizationConnection nothing is done. However, if the connection is a ChainedDiscretizationConnection, a new direct connection is constructed that transports from connection.from_discr to connection.to_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
Returns

An instance of DirectDiscretizationConnection.

Implementation details

class meshmode.discretization.connection.InterpolationBatch(from_group_index, from_element_indices, to_element_indices, result_unit_nodes, to_element_face)

One interpolation batch captures how a batch of elements within an element group should be an interpolated. Note that while it’s possible that an interpolation batch takes care of interpolating an entire element group from source to target, that’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

element_id_t [nelements]. (a pyopencl.array.Array) This contains the (group-local) element index (relative to from_group_index from which this “to” element’s data will be interpolated.

to_element_indices

element_id_t [nelements]. (a pyopencl.array.Array) This contains the (group-local) element index to which this “to” element’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. (a pyopencl.array.Array if existent) 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 “from” unit nodes per batch, one batch will always go to a single face index.)

class meshmode.discretization.connection.DiscretizationConnectionElementGroup(batches)
batches

A list of InterpolationBatch instances.