Novel FDTD Technique Over Tetrahedral Grids for Conductive Media

Explicit geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids

“…Inverse Hodge operators play a vital role in many applications. Most notably, inverse Hodge operators enable consistent and explicit schemes to solve time-domain wave propagation problems [2], [14], [15], [16]. Other applications enabled by the inverse mass matrices comprise the explicit construction of the codifferential operator, the Laplace-de Rham operator [17] and compute the discrete Hodge decomposition of discrete fields [17], [18].…”

“…Yet, devising a recipe to construct sparse inverse Hodge operators on a barycentric dual grid appears to be a formidable task [23], [15], [17], [21] given that the conventional wisdom is that the barycentric dual grid "prohibits a sparse representation for their inverse operators" [24] and, consequently, only approximate constructions have been proposed [15]. By using a barycentric dual grid, in [14], [16], inverse mass matrices that map from dual faces to primal edges are constructed by assembling local contributions inside dual cells and then computing the algebraic inverse of the resulting local matrices. Yet, we note that the time needed to compute all local inverses is not negligible, because the rank of the local matrices is twenty or more.…”

“…The construction of the inverse mass matrix that maps from dual edges of a barycentric dual grid to primal faces of a tetrahedral grid proposed in this paper is entirely new. A key difference with respect to the approach proposed in [14], [16] in the construction of the other inverse mass matrix that maps from dual faces of the barycentric dual grid to primal edges of the simplicial grid is the explicit and geometric construction of the consistent term, without the necessity of computing local inverses. In addition, the recipe to construct the stabilization term adds more flexibility in the construction of the mass matrices.…”

“…The first one is a weighted averaging technique classically used in Finite Volumes Methods [20]. The second one is based on an extension of the approach proposed [14], [16].…”

High order geometric methods with splines: an analysis of discrete Hodge--star operators

“…Methods based on discrete exterior calculus [20,32,31], which explicitly construct a second dual mesh by taking centroids of elements of the first mesh and connecting them with edges [31,40], are very popular. Methods based on the integral formulation of Maxwell's equations such as the finite integration technique (FIT) [14,15], or the cell method [46,41,17,16] also fall into this second family. In general, for the differential form approximations, one proceeds by attaching discrete degrees of freedom to geometric entities of the mesh (vertices, edges, faces and volumes), and by repeatedly using an adapted version of the generalized Stokes theorem, which amounts to building incidence matrices for the geometric entities.…”