wedge cells and elements

[gdmcbain]

Would it be worth adding wedge cells and elements? I recently received a mesh of these (too big to attach here, but I think a simple one could be constructed by extruding a triangular mesh in the third dimension).

Four isoparametric wedge elements with 6, 15, 9, and 12 nodes, respectively, are listed in

• Kardestuncer, H. (ed.) (1988). Finite Element Handbook. New York: McGraw-Hill, p. 2.130

[gdmcbain]

A difficulty: the facets of a wedge aren't all of the same topological type, there are three quadrilaterals and two triangles. (I don't know the name for the subclass of polyhedra whose facets are the same kind of polygon, like tetrahedra and hexahedra, or for its complement.) This complicates the Mesh.facets attribute; it can't be simple a rectangular array. (Unless padded with a sentinel value like the −1 in Mesh.f2t for facets on the boundary?)

[gdmcbain]

Ern & Guermond (2004, §1.2.5) refer to wedge elements as ‘prismatic Lagrange finite elements’ and (table 1.3) depict ℙℝ_k for k = 1, 2, and 3.

• Ern, A., Guermond, J.-L. (2004). Theory and Practice of Finite Elements. Springer, §1.2.5 ‘Prismatic Lagrange finite elements’

[gdmcbain]

Zienkiewicz & Taylor (2000, §8.13) call them ‘triangular prism elements’ and draw the 6, 15, and 26-node ‘serendipity’ elements (figure. 8.23).

• Zienkiewicz, O. C., Taylor, R. L. (2000). The Finite Element Method. I: The Basis. Butterworth-Heinemann.

[gdmcbain]

There's some discussion of ‘vtkWedge’ and Wedge6 in nschloe/meshio#863.

[kinnala]

I've been thinking about this every now and then and still a proper solution escapes me. If only CellBasis assembly is needed for nodal elements then I think this can be implemented by hacking Mesh.facets, essentially disabling it.

[gdmcbain]

Hmm...Would the boundary of a mesh of homogeneous wedges or pyramids be something like a mixed mesh #412 ? Could it be handled with something like a MultiBasis #728 ?

[kinnala]

I think we could technically implement a separate FacetMeshTri and FacetMeshQuad for which CellBasis gives an integral over the boundary. Regarding wedge elements, I'm making some progress in #740.