FEM and BEM Implementations of a High Order Surface Impedance Boundary Condition for Three-Dimensional Eddy Current Problems

We implement a three-dimensional formulation for eddy current problems based on the reduced magnetic scalar potential enforcing a high order surface impedance boundary condition (SIBC) which takes into account the curvatures of the conductor surface. Based on perturbation theory, the formulation reduces to three Laplace boundary value problems with Neumann boundary conditions and hence can be easily implemented in any existing Finite Element Method (FEM) or Boundary Element Method (BEM) code for the Laplace equation. The appropriate choice of the small parameter in the perturbation approach correctly represents the order of accuracy of the SIBC. The validation is carried out by comparison with full FEM solutions of a canonical test problem and of a more realistic example of a non-destructive testing probe. The validity of the extension of a high order SIBC to lower frequencies is verified and the fields can be obtained at any frequency in the range of interest once the formulation is solved only once.