Volume 50, Number 3, May-June 2016
Special Issue – Polyhedral discretization for PDE
|Page(s)||677 - 698|
|Published online||23 May 2016|
Geometrically defined basis functions for polyhedral elements with applications to computational electromagnetics
Politecnico di Milano, Piazza Leonardo da Vinci 32,
2 Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
Received: 30 March 2015
Revised: 22 July 2015
In the recent years, reformulating the mathematical description of physical laws in an algebraic form using tools from algebraic topology gained popularity in computational physics. Physical variables are defined as fluxes or circulations on oriented geometric elements of a pair of dual interlocked cell complexes, while physical laws are expressed in a metric-free fashion with incidence matrices. The metric and the material information are encoded in the discrete counterpart of the constitutive laws of materials, also referred to as constitutive or material matrices. The stability and consistency of the method is guaranteed by precise properties (symmetry, positive definiteness, consistency) that material matrices have to fulfill. The main advantage of this approach is that material matrices, even for arbitrary star-shaped polyhedral elements, can be geometrically defined, by simple closed-form expressions, in terms of the geometric elements of the primal and dual grids. That is why this original technique may be considered as a “Discrete Geometric Approach” (DGA) to computational physics. This paper first details the set of vector basis functions associated with the edges and faces of a polyhedral primal grid or of a dual grid. Then, it extends the construction of constitutive matrices for bianisotropic media.
Mathematics Subject Classification: 65N06 / 65N30 / 78-08
Key words: Discrete Geometric Approach (DGA) / discrete constitutive equations / discrete hodge star operator / non-orthogonal polyhedral dual grids / bianisotropic media
© EDP Sciences, SMAI 2016
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