Hierarchical Curl-Conforming Vector Bases for Pyramid Cells

Abstract

Advanced applications of the finite element method use hybrid meshes of differently shaped elements that need transition cells between quadrilateral and triangular faced elements. The greatest ease of construction is obtained when, in addition to triangular prisms, one uses also pyramids with a quadrilateral base, as these are the transition elements with the fewest possible faces and edges. A distinctive geometric feature of the pyramid is that its vertex is the point in common with four of its faces, while the other canonical elements have vertices in common with three edges and three faces, and that is why pyramids' vector bases have hitherto been obtained with complex procedures. Here we present a much simpler and more straightforward procedure by shifting to a new paradigm that requires mapping the pyramidal cell into a cube and then directly enforcing the conformity of the vector bases with those used on adjacent differently shaped cells (tetrahedra, hexahedra and triangular prisms). The hierarchical curl-conforming vector bases derived here have simple and easy to implement mathematical expressions, including those of their curls. Bases completeness is demonstrated for the first time, and results confirming avoidance of spurious modes and faster convergence are also reported.

Hierarchical Divergence-Conforming Vector Bases for Pyramid Cells

“…As said, a brief historical review of research devoted to the development of basis functions for pyramidal cells can be found in the Introduction of [12]. Here we just reiterate that, in our opinion, the main problem that researchers have had so far in building the bases for the pyramid is to find the simplest and most direct way to build the volume-based vector functions that here, as in [12], we get thanks to simple analytical and geometric considerations.…”

“…On the contrary, the few methods so far introduced in [3]- [11] to build the pyramidal bases require specialized knowledge and skills to be understood and have produced higher-order bases different from each other. Although the literature on this topic is not consolidated, as already discussed in the Introduction of [12], the results in [10] nevertheless deserve particular attention.…”

“…At the same time, there is also a lot of interest in increasing the order of the elements, whatever their shape, because 1) models that use higher-order elements use fewer degrees of freedom (DoF), i.e., fewer unknowns [1]; 2) sophisticated parallel solution strategies can benefit from the use of higher-order elements [13]; 3) mesh refinement occurs more naturally when the mesh cells are defined by higher order shape functions; 4) hor p-adaptive techniques provide faster convergence as the order of the elements increases [14], [15], [16]. Given the growing interest in hybrid models that use differently shaped cells, and in consideration of all the difficulties encountered so far in building conforming higher-order pyramids, the new paradigm recently proposed in [12] seems to be a turning point as it allows to easily produce higher order pyramidal bases having simple and easily implementable expressions. Indeed, [12] discusses in depth new hierarchical curl-conforming bases for the pyramid that complement and are compatible with the families reported in [1], with continuous tangential components across adjacent cells in the mesh; that is, the families first presented individually in [17], [18], [19].…”

“…Given the growing interest in hybrid models that use differently shaped cells, and in consideration of all the difficulties encountered so far in building conforming higher-order pyramids, the new paradigm recently proposed in [12] seems to be a turning point as it allows to easily produce higher order pyramidal bases having simple and easily implementable expressions. Indeed, [12] discusses in depth new hierarchical curl-conforming bases for the pyramid that complement and are compatible with the families reported in [1], with continuous tangential components across adjacent cells in the mesh; that is, the families first presented individually in [17], [18], [19]. Similarly, the present article uses the same paradigm as [12] to build hierarchical divergence-conforming bases for pyramids that complement the families reported in [20] (as well as in [1]), with continuous normal components across adjacent cells in the mesh.…”

“…Indeed, [12] discusses in depth new hierarchical curl-conforming bases for the pyramid that complement and are compatible with the families reported in [1], with continuous tangential components across adjacent cells in the mesh; that is, the families first presented individually in [17], [18], [19]. Similarly, the present article uses the same paradigm as [12] to build hierarchical divergence-conforming bases for pyramids that complement the families reported in [20] (as well as in [1]), with continuous normal components across adjacent cells in the mesh.…”