High-order surface impedance boundary conditions in three-dimensional boundary element modeling of eddy-current problems

This paper presents a three-dimensional boundary element method (BEM) formulation for eddy-current problems incorporating high-order surface impedance boundary conditions (SIBCs). We employ the Sun–Klaseboer boundary integral formulation, whose decoupled electric and magnetic field equations naturally accommodate surface impedance relations. The formulation implements three SIBC models of increasing asymptotic order: the Leontovich, Mitzner, and Rytov conditions. To evaluate the second-order tangential derivatives required by the Rytov SIBC, a discrete differential geometry (DDG) technique reconstructs the principal frames, curvatures, and tangential differential operators directly from quadratic surface meshes. The resulting block system is solved via a matrix-free Schur-complement reduction and an inexact nested GMRES solver, while retaining the (𝑁2) dense-BEM matrix–vector complexity. Validation includes a spheroidal benchmark, an ℎ-convergence study, and a 3D bent-rod geometry. The Rytov SIBC shows the clearest improvement in the moderate skin-effect regime, especially for the low-frequency 3D bent-rod benchmark.