A Study of Singularity and Near-Singularity Problems in Volume Integral Equation Discretized by Curvilinear Tetrahedrons

Le résumé fourni par la source

This article proposes a method for handling singularity/near-singularity integrals generated in the calculation of impedance matrix elements in the method of moments of volume integral equations (VIE) discretized by curvilinear tetrahedral elements. In order to reduce the VIE unknowns for electromagnetic scattering, the curved triangular elements and higher order hierarchical vector (HOHV) basis functions are used to VIE. However, the subtracted term is often challenging to singularity and near-singularity problems in VIE, where the issue is particularly prominent in the high-order VIE. In comparison with the singularity cancellation method, the singularity subtraction method for handling near-singular integrals shows more high accuracy and flexibility. This paper proposed a singularity subtraction method to cancel or weaken the weak singularity$(1 / R)$and strong singularity$(\nabla(1 / R))$in HO-VIE. In this method, a suitable projection planar tetrahedron is observed by the corresponding projection point of the curvilinear tetrahedral elements. Note that the projection point coincides with the field point for the case of the field/source tetrahedron coinciding, while the projection point is the closest point from the field point to the curved tetrahedron for the case of the field/source tetrahedron being close. By subtracting a singular term from the integrand, the remaining terms of impedance matrix elements become smooth over the integration domain and the subtracted term can be integrated analytically. Numerical results are given to verify the validity of this method in dealing with singularity/near-singularity problems in VIE, which is discretized by curvilinear tetrahedrons and higher order hierarchical vector (HOHV) basis functions.

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