Enriched finite element approach for modeling discontinuous electric field in multimaterial problems

TL;DR

This paper introduces an enriched finite element method that efficiently captures electric field discontinuities in multi-material problems by adding a single degree of freedom per interface-crossed element, maintaining computational simplicity.

Contribution

The novel approach employs hat-type shape functions within elements to accurately model electric field discontinuities with minimal additional computational cost.

Findings

Effective in capturing electric field discontinuities

Maintains the same system matrix graph as standard FEM

Works on structured and unstructured grids

Abstract

This work is devoted to the development of an efficient and robust technique for accurate capturing of the electric field in multi-material problems. The formulation is based on the finite element method enriched by the introduction of hat-type shape function within the elements crossed by the material interface. The peculiar feature of the proposed method consists in the direct employment of the hat-function that requires solely one additional degree of freedom per cut element for capturing the discontinuity in the electric potential gradient and, thus, the electric field. This additional degree of freedom is subsequently statically condensed element-wise prior to the assembly of the global discrete system. As a consequence, the graph of the system matrix remains the same as that of the standard finite element method. In order to guarantee the robust performance of the proposed method for a wide range of electrical material property ratios, it also accounts for the possible discontinuities among the neighboring cut elements that arise due to employing additional degrees of freedom fully local to the element. The method is tested using several examples solved on structured and unstructured grids. The proposed approach constitutes a basis for enriched FEM applicable to a wide range of electromagnetic problems.