Enriched finite element approach for modeling discontinuous electric field in multi-material problems

This work is devoted to developing an efficient and robust technique for accurately capturing 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 of 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 discrete global 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.