Search

This contribution introduces a novel method for constructing mass matrices for general tetrahedral elements. By considering the construction of the reluctivity mass matrix as an example, a new derivation of a recipe to geometrically construct a symmetric positive semi-definite and consistent mass matrix is provided. It is shown why such a matrix can be used inside formulations where the mass matrix is right multiplied by the appropriate incidence matrix. Then, a second recipe is introduced to produce a consistent and symmetric positive definite matrix. The proposed method is based on an original algebraic technique that differs fundamentally from what is available in the literature. The proposed recipe opens also the possibility to improve the conditioning of the resulting elementary mass matrices. [ABSTRACT FROM AUTHOR]

This paper first introduces a simplified construction of the constitutive matrices for general polyhedral meshes. These constitutive matrices are geometrically defined by means of simple closed-form expressions involving the geometric elements of the primal and dual meshes. Then, we solve for the first time the Maxwell’s eigenvalue problem on general polyhedral meshes. In particular, we analyze the convergence of the eigenvalues when the mesh is adaptively refined by using the subgridding technique together with the constitutive inconsistency as error indicator. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

As a first step, we will investigate the extent of the consistency error for a hexahedral primal grid due to the use mass matrices of finite elements within the discrete geometric approach (DGA) as discrete counterparts of the constitutive relations. Such matrices are constructed by resorting to mixed elements edge and face vector functions. Then, we will propose a novel set of geometric vector basis functions for hexahedra that guarantee consistency and stability of the constitutive matrices for the DGA constructed according to the so-called energetic approach. [ABSTRACT FROM AUTHOR]

We examine the construction of reluctivity and Ohm's constitutive matrices for discrete geometric approaches using nonregular hexa-hedra as primal volumes. The effect of element deformation on the representation of uniform fields have been investigated for static fields. Then convergence and accuracy of the proposed matrices has been carried out using an eddy-current problem as working example. [ABSTRACT FROM AUTHOR]

We examine the construction of a symmetric positive definite conductance matrix for eddy-current problems, using a discrete approach. We construct a new set of piecewise uniform basis vector functions on both the primal and the dual complex. We define these vector functions for both tetrahedra and prisms. [ABSTRACT FROM AUTHOR]

Using a geometric formulation for eddy currents, we present a geometric approach to constructing approximations of the discrete magnetic and Ohm's constitutive matrices. In the case of Ohm's matrix, we also show how to make it symmetric. We compared the impact on the solution of the proposed Ohm's matrices, and an iterative technique to obtain a consistent right-hand-side term in the final system is described. [ABSTRACT FROM AUTHOR]