FAST MULTIPOLE SOLUTIONS FOR DIFFUSIVE SCALAR PROBLEMS

Abstract

Nondestructive evaluation (NDE) of airframe structures may involve finding eddy‐current distributions in complicated geometrical features including cracks, fasteners, sharp corners/edges, multi‐layered structures, complex ferrite‐cored probes, etc. The eddy‐current problem can be formulated in terms of boundary integral equations (BIE), which can be discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The Fast Multipole Method (FMM) is a well‐established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large‐scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N2) operations to compute the system of equations and another O(N3) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two‐level FMM can potentially reduce the operations and memory requirement to O(N3/2). This paper introduces the procedure of the FMM accelerated BIE method, which is not only efficient in meshing complicated geometries, accurate for solving singular fields or fields in infinite domains, but also practical and often superior to other methods in solving large‐scale problems. Computational tests of the numerical FMM solutions against the conventional BIE results are presented for the two‐dimensional Helmholtz equation with a complex wave number.