IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY2003 1107 Comparison Between Different Approaches for Fast and Efficient 3-D BEM Computations André Buchau, Wolfgang M. Rucker, Oliver Rain, Volker Rischmüller, Stefan Kurz, and Sergej Rjasanow Abstract—Fast methods like the fast multipole method or the adaptive cross approximation technique reduce the memory re- quirements and the computational costs of the boundary-element method (BEM) to approximately . In this paper, both fast methods are applied in combination with BEM–finite-element method coupling to nonlinear magnetostatic problems. Index Terms—Adaptive cross approximation (ACA) technique, boundary element methods (BEMs), fast multipole method, finite element methods (FEMs), iterative solution methods. I. INTRODUCTION B OUNDARY-ELEMENT methods (BEMs) are very pop- ular for the numerical solution of electrostatic or magne- tostatic problems with linear, homogeneous media embedded in free space. Then, only the surfaces of the considered bodies must be discretized and the surrounding space is taken into ac- count exactly. To handle nonlinear material, the BEM can be extended with a volume integral equation or coupled with the finite-element method (FEM) [1], [2]. In this paper, the second case is considered to combine the advantages of the BEM and FEM, where the focus is on the efficient solution in the BEM domain. Among many advantages, the BEM has one significant dis- advantage—the fully populated matrix of the system of linear equations. Hence, in the past, only relatively small problems were solved with the BEM. However, this disadvantage of the BEM can be overcome if the linear system of equations is solved with an iterative solution method in combination with a fast and efficient approximation technique for the matrix. An important property of many iterative solvers is that only the product of the system matrix with a vector has to be computed. Therefore, the matrix needs not to be explicitly available and can be rep- resented by an approximation technique. Good approximation techniques reduce the computational costs and, particularly, the memory requirements to approximately , where is the number of unknowns. In this paper, two of such approximation techniques are compared, the fast-multipole method (FMM) [3] and the adaptive cross approximation (ACA) technique [4]. Manuscript received June 18, 2002. A. Buchau and W. M. Rucker are with the Institute for Theory of Electrical Engineering, University of Stuttgart, 70569 Stuttgart, Germany (e-mail: andre.buchau@ite.uni-stuttgart.de). O. Rain, V. Rischmüller, and S. Kurz are with the Robert Bosch GmbH, 70049 Stuttgart, Germany. S. Rjasanow is with the Universität des Saarlandes, 66041 Saarbrücken, Germany. Digital Object Identifier 10.1109/TMAG.2003.810167 II. THEORETICAL BACKGROUND AND FORMULATION A. Problem and Solution With BEM–FEM Coupling The magnetic flux density inside magnetic material shall be computed. The nonlinear material lies in free space and is excited with current driven coils. As mentioned previously, for a numerical solution of such kind of problems, the BEM–FEM coupling is very advantageous. Only the magnetic material and the coils have to be discretized, e.g., with second-order hexahedral elements. By restriction of the mesh to the bound- aries, second-order quadrilateral elements on the surfaces are obtained. To describe an eddy current problem, an formula- tion is applied [1], [2]. A Newton method is used to take into account the nonlinearity of the material. In each iteration step of the Newton method, a linear problem is solved within the framework of an iterative solver that exploits the principle of domain decomposition [5]. In the BEM domain, only the mag- netic vector potential is needed and the governing Poisson equation (1) can be split into three independent equations, if Cartesian co- ordinates are used. is the density of impressed currents. To exclude the FEM domain from the BEM computations, a di- rect formulation based on Green’s theorem is used. A system of linear equations is obtained by application of the collocation method. For each component of the vector potential, the dis- cretized equation (2) has to be fulfilled, where . Note that the matrices and are the same for all three components of . The left-hand side in (2) corresponds to the computation of a single-layer po- tential and the right-hand side to the computation of a double- layer potential. Since both the - and the -matrix are fully populated, a solution with the conventional BEM is limited to relatively small problems. B. Fast Multipole Method Meanwhile, the fast multipole method (FMM) is a popular technique to compute particle interactions or electrostatic prob- lems efficiently [3], [6]. With some modifications, the FMM can also be applied to magnetic field computations with nonlinear media in combination with BEM–FEM coupling. 0018-9464/03$17.00 © 2003 IEEE