Unified Regularization of 2D Singular Integrals for Axisymmetric Galerkin BEM in Eddy-Current Evaluation

This paper presents an axisymmetric Galerkin boundary element method (BEM) for modeling eddy-current interactions between excitation coils and conductive objects. The formulation derives boundary integral equations from the Stratton-Chu representation for the azimuthal component of the vector potential in both air and conductive regions. The central contribution is a unified regularization framework for the two-dimensional (2D) singular integrals arising in Galerkin BEM. This framework handles both logarithmic and Cauchy singularities through a common set of integral transformations, eliminating the need for case-by-case analytical singularity extraction and enabling straightforward numerical quadrature. The regularization and quadrature stability are proved and verified numerically. The method is validated on several representative axisymmetric geometries, including cylindrical, conical, and spherical shells. Numerical experiments demonstrate consistently high accuracy and computational efficiency across broad frequency ranges and coil lift-off distances. The results confirm that the proposed axisymmetric Galerkin BEM, combined with the integral transformation technique, provides a robust and efficient framework for axisymmetric eddy-current nondestructive evaluation.

💡 Research Summary

This paper introduces a robust axisymmetric Galerkin boundary element method (BEM) for eddy‑current nondestructive evaluation (EC‑NDE) and, more importantly, a unified regularization technique for the two‑dimensional singular integrals that inevitably appear in Galerkin formulations. Starting from the Stratton‑Chu representation, the authors derive boundary integral equations (BIEs) for the azimuthal component of the magnetic vector potential both in free space (Laplace kernel) and in the conductive region (Helmholtz kernel). By exploiting the axisymmetry, the three‑dimensional problem collapses to a set of one‑dimensional meridional curves, but the resulting BIEs still contain double integrals with logarithmic (ln |x‑y|) or Cauchy (1/(x‑y)) singularities when source and observation points coincide or share a common endpoint.

The core contribution is a coordinate‑transformation based regularization framework that treats both singularity types in a single, element‑order‑independent manner. The authors classify element pairs into two configurations: (1) coincident elements (identical panels) and (2) touching elements (sharing a vertex). For coincident panels, the unit square