Hierarchical Divergence-Conforming Vector Bases for Pyramid Cells

Abstract

Divergence-conforming hierarchical vector bases for the pyramid consist of face-and volume-based functions obtained by a simple procedure that uses a new paradigm recently introduced by this author to produce pyramid bases. In order to define the bases' order, the procedure starts by mapping the pyramids into a cube of a new Cartesian space, which we call the grandparent space, where the basis functions and their divergences take on polynomial form. Then we get the face-based functions of zero polynomial order and the volumebased functions of the first order. Functions of arbitrarily high order are obtained by multiplying the vector functions of the lowest order by independent scalar polynomials of higher order. Our face-based functions conform to those of other differently shaped elements to allow the use of hybrid meshes, while the multiplicative construction technique generates right away the volume-based basis functions. The completeness of the bases is demonstrated and all the basis functions we obtain are suitably normalized; their expression involves orthogonal polynomials which are easy to implement and alleviate the loss of linear independence Index Terms -Electromagnetic fields, finite-element methods, higher order vector elements, pyramidal elements, numerical analysis.

Assessing Curl-Conforming Bases for Pyramid Cells

“…Almost all practitioners of computational electromagnetics (CEM) avoid using pyramidal cells because the related vector bases are very complicated and never extensively tested; in fact very few authors have used pyramids so far, as can be seen from [1], [2], [3] and references therein. This has hindered the development of codes that use hybrid meshes smoothly; that is, meshes that employ higher-order pyramidal elements in addition to tetrahedra, bricks, and prisms, despite the fact that a reasonably extensive scientific literature on higher-order pyramidal elements has developed over the past twenty years [4]- [11].…”

“…This has hindered the development of codes that use hybrid meshes smoothly; that is, meshes that employ higher-order pyramidal elements in addition to tetrahedra, bricks, and prisms, despite the fact that a reasonably extensive scientific literature on higher-order pyramidal elements has developed over the past twenty years [4]- [11]. Things may now change, firstly because the method in [1], [2] to obtain conforming bases of arbitrarily high order for pyramids is, in our opinion, decidedly simpler than those previously available in the literature. Secondly because this paper fills the lack of useful test cases for validating numerical codes that use pyramid cells, or at least their curl-conforming bases for Finite Element Method (FEM) applications.…”

“…Recall that all elements have shape functions and basis functions of polynomial form in the so-called "parent" space [12], with the exception of pyramid elements which must have shape-functions and basis functions of fractional form for conformity with the other cells and to guarantee cell-to-cell tangential continuity, even in the case of curved cells [1]. (In the parent space, the divergence-conforming basis functions of the pyramid element have fractional form to guarantee the continuity of the normal component of the field from cell to cell [2]). The upper part of the Table shows the first wavenumbers; i.e., the first six non-zero eigenvalues (EV) of the pyramidal cavity with edges of equal length, calculated with bases of order p from 0 to 6.…”

“…For pyramids, the shape functions and basis functions take polynomial form in a different space, called "grandparent" space, where all the pyramid cells are mapped by a cube of unit side [1], [2]. The polynomial order of the functions in the grandparent domain for the pyramid elements and in the parent domain for the other non-pyramidal elements is denoted by the integer p. For any polynomial order p, the curl-conforming functions are tangentially continuous on the boundaries in common to adjacent cells of different shape [1].…”

“…In summary, in this paper we use meshes formed only by quadrangularbased pyramidal cells; unlike what happens when pyramids are used (albeit rarely) as "fillers" of hybrid meshes; that is, as gluing elements inserted between other non-pyramidal cells. Readers may find it helpful to review [1], [2], for a detailed introduction to the notation and other background information. Preliminar results of this work were presented in [14].…”