Development and Application of a Fast Multipole Method in a Hybrid FEM/MoM Field Solver

Hybrid FEM/MoM methods combine the finite element method (FEM) and the method of moments (MoM) to model inhomogeneous unbounded problems. These two methods are coupled by enforcing the continuity of tangential fields on the boundary that separates the FEM and MoM regions. When modeling complex geometries with many elements on the boundary, the MoM part of the problem is the bottleneck of the hybrid method since it requires O N memory and O N computation time. This paper presents a hybrid FEM/MoM formulation applying the fast multipole method (FMM) that greatly reduces the memory requirement associated with MoM part. Two practical electromagnetic problems are presented to validate this method. 2 ( ) 3 ( ) INTRODUCTION The hybrid finite-element-method/method-ofmoments (FEM/MoM) has been used to analyze a variety of electromagnetic scattering and radiation problems effectively. FEM is used to model detailed structures with complex inhomogeneities and MoM is used to model larger metallic structures and to provide an exact radiation boundary condition to terminate the FEM mesh. These two methods are coupled by enforcing tangential field continuity on the boundary separating the FEM and MoM regions. Both the FEM and MoM are powerful methods, but each of these methods has its own advantages and disadvantages. MoM handles unbounded problems very effectively but is less efficient when complex inhomogeneities are present. Inhomogeneities are easily handled by FEM. However, FEM is most suitable for bounded problems. Hence, methods that combine MoM and FEM are advantageous for treating electromagnetic problems involving unbounded, complex structures. The FEM part of the hybrid method produces a sparse matrix, which requires ( ) O N memory, where N is the total number of unknowns in the FEM region. On the other hand, the MoM part of the hybrid method produces a dense matrix, which requires ( ) 2 S O N memory and ( ) 3 S O N CPU time, where NS is the total number of unknowns on the MoM boundary. The final system of equations produced by the hybrid method consists of a partially full, partially sparse matrix. An iterative solver is usually preferred to solve this matrix equation. However, the computational effort primarily associated with the MoM part limits the size of the problems that can be solved. ( O N ( S O N log Rokhlin introduced a fast multipole method to speed up the matrix-vector multiplication that arises in the iterative solution of MoM equations [1]. This method has been applied to electromagnetic scattering computation by Engheta [2], Lu [3], and Song [4] et al. The memory required for matrixvector multiplications can be reduced from ( ) 2 S O N in MoM to ) 1.5 S by using a two-level FMM, and to ) N by using a multilevel version of the FMM method. S In this paper, a two-level FMM is implemented in a hybrid FEM/MoM method. Section II describes the formulation of the hybrid FEM/MoM and the related formulation using the FMM method. Preconditioning techniques to improve the condition of the resulting system of equations are also discussed. Section III presents numerical results using the FMM-enhanced hybrid FEM/MoM method. FORMULATION The FMM method provides an efficient technique for performing matrix-vector multiplications for MoM matrices. This section describes the hybrid FEM/MoM formulation with FMM applied to the evaluation of the MoM integrals. The Hybrid FEM/MoM Formulation In the hybrid FEM/MoM, an electromagnetic problem is divided into an interior equivalent part and an exterior equivalent part. The interior part is modeled using the FEM and the exterior part is 1054-4887 © 2004 ACES 126 ACES JOURNAL, VOL. 19, NO. 3, NOVEMBER 2004