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

TL;DR

This work addresses axisymmetric eddy-current evaluation by developing a Galerkin boundary element method that uses a unified regularization framework for 2D singular integrals. Central to the approach is a Duffy-type coordinate transformation that regularizes both logarithmic and Cauchy singularities, allowing smooth integrands and straightforward Gauss-Legendre quadrature for elements of arbitrary order. The method is derived from a Stratton-Chu formulation for the azimuthal vector potential, leading to coupled BIEs in air and conductor that are discretized Galerkin-style and coupled via interface conditions; impedance changes are computed with Auld’s formula. Numerical validation on cylindrical, conical, and spherical geometries shows high accuracy and efficiency across wide frequency ranges, with linear elements showing second-order convergence and quadratic elements achieving ~$10^{-5}$ relative errors even on coarse meshes. The framework is compact, generalizable beyond axisymmetry, and amenable to acceleration via fast multipole or $ ext{H}$-matrix techniques, making it a robust tool for axisymmetric eddy-current nondestructive evaluation.

Abstract

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.

Figures (6)

• Figure 1: Side view of a stainless-steel tube and a coaxial coil with center at $z_0$.
• Figure 2: Comparison of $\Delta R/X_0$ and $\Delta X/X_0$ obtained by the Galerkin BEM and FEM benchmark for the S30400 tube at two lift-off positions.
• Figure 3: Side view of a conical conductive tube with bottom radii $(a_1,a_2)$ at $z=-l/2$ and top radii $(a_3,a_4)$ at $z=+l/2$, and a coaxial coil centered at $z_0$.
• Figure 4: Comparison of $\Delta R/X_0$ and $\Delta X/X_0$ obtained by the Galerkin BEM and the FEM benchmark for the conical 7075–T6 tube at different axial positions of the coil.
• Figure 5: Side view of a spherical shell and a coil centered at $z_0$.