| 1d45022f | 04-Aug-2016 |
Toby Isaac <tisaac@uchicago.edu> |
Merge branch 'tisaac/dualspace-feature-symmetry'
In finite element assembly, to compute a residual or assemble a Jacobian
on an element, the dual space functionals of a reference finite element are
Merge branch 'tisaac/dualspace-feature-symmetry'
In finite element assembly, to compute a residual or assemble a Jacobian
on an element, the dual space functionals of a reference finite element are related
(i.e., pushed forward) to the functionals that make of the degrees of
freedom of the discretized system.
In doing so, we have to draw a correspondence between the boundary
points of the reference element (its edges and vertices) and the
boundary points of each cell in the mesh. Sometimes, this
correspondence is not perfect, because the orientations of the boundary
points of a cell may be different than the orientations of the boundary
points of the reference element. For example, an edge that runs
clockwise around the reference element may map onto an edge that runs
counter-clockwise around the real cell.
For some finite elements (such as the most common P1 Lagrange finite
elements) this doesn't matter. But for others---high-order elements,
H-div and H-curl conforming elements---a difference in orientation
affects the mapping between reference functionals and degrees of
freedom.
The simplest example is a high-order nodal Lagrange finite element in
2D: the degrees of freedom on an edge of the reference element are
numbered left-to-right, but if the orientation of the real edge is
reverse, then we must read/write degrees of freedom right-to-left.
Things get more complicated in 3D: a quadrilateral face between
hexahedra may have any of 8 different orientations (that make of the
dihedral symmetries), and each one corresponds to a different reordering
of the 2D grid of nodes supported on that quadrilateral.
Things also get more complicated for H-div elements. Functionals on the
interfaces between cells represent the flux from one cell to another,
and thus have direction. Reversing the orientation of an edge changes
the sign of the mapping between the reference functional and the degree
of freedom.
To assemble generic finite elements, we thus need to accommodate
transforms between reference functionals and degrees of freedom that
encompass (1) permutations, and (2) scalar multipliers.
This branch accomplishes this in two steps:
- We add PetscDualSpaceGetSymmetries() to the interface, to allow a dual
space to describe how symmetries of the mesh points affect
referent-to-real maps of functionals.
- We add PetscSectionSym, an object that encapsulates symmetries of the
degrees of freedom whose layout is described by a PetscSection.
When a section is created from a DM that has a finite element
discretization in its PetscDS, it will automatically construct the
appropriate PetscSectionSym when it creates a PetscSection to describe a
discretized function space.
If a PetsFE finite element discretization is not used, the user can
create a PetscSectionSym for their degrees of freedom for themselves.
We give an example of this for spectal elements in
src/dm/impls/plex/examples/tutoerials/ex6.c.
* tisaac/dualspace-feature-symmetry: (22 commits)
cast memzeros to void to avoid compiler complaints about const qualifiers
DMPlex: fix stray PetscScalar to PetscReal
DMPlex: fix stray PetscScalar to PetscReal
Plex tutorials ex6: use symmetries in SEM example
Plex tests ex3: added high order tests
Plex tests ex3: add matrix-free near null space test
DMPlex: fix PetscInt %D PetscPrintf character
DMPlexTree: use symmetries when computing constraint matrices
DMPlex: use symmetries in DMPlexGetIndicesPoint{Fields}_Internal() and DMPlexAnchorsModifyMat()
DMPlex: use PetscSectionGet/RestorePointSyms() in DMPlexVecGet/SetClosure()
DMPlex: added DMPlexGet/RestoreCompressedClosure()
DMPlex: added DMPlexGetPointDualSpaceFEM()
DMPlex: set PetscSectionSym in DMPlexCreateSectionInitial()
DMLabel: added PETSCSECTIONSYMLABEL
PetscSection: add PetscSectionSym
PetscDualSpace: added PetscDualSpaceGetSymmetries()
PetscDualSpace_Lagrange: setup recursively on dimension
PetscDualSpace_Lagrange: order functions now go after tuple used
PetscDualSpace_Lagrange: use lexicographic order for nodes
DMPlex: added DMPlexComputePointGeometry_Internal()
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