Marinos N. Vouvakis

The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems

This paper presents the adaptive cross approximation (ACA) algorithm to reduce memory and CPU time overhead in the method of moments (MoM) solution of surface integral equations. The present algorithm is purely algebraic; hence, its formulation and implementation are integral equation kernel (Greens function) independent. The algorithm starts with a multilevel partitioning of the computational domain. The interactions of well-separated partitioning clusters are accounted through a rank-revealing LU decomposition. The acceleration and memory savings of ACA come from the partial assembly of the rank-deficient interaction submatrices. It has been demonstrated that the ACA algorithm results in O(NlogN) complexity (where N is the number of unknowns) when applied to static and electrically small electromagnetic problems. In this paper the ACA algorithm is extended to electromagnetic compatibility-related problems of moderate electrical size. Specifically, the ACA algorithm is used to study compact-range ground planes and electromagnetic interference and shielding in vehicles. Through numerical experiments, it is concluded that for moderate electrical size problems the memory and CPU time requirements for the ACA algorithm scale as N/sup 4/3/logN.

A symmetric FEM-IE formulation with a single-level IE-QR algorithm for solving electromagnetic radiation and scattering problems

This paper presents, for the first time in the engineering community, a symmetric coupling between the finite element and integral equation methods (FEM-IE) for solving three-dimensional unbounded radiation and scattering problems. The proposed FEM-IE is based on the E-field vector Helmholtz equation. Curl-conforming vector finite elements are used to discretize the interior region, whereas the divergence-conforming surface elements are utilized in the IE truncation surface. The symmetry in the IE part is restored through the application of the Calderon-projector. Moreover, the IE computations are accelerated using a single level QR algorithm. This reduces both memory and computational time. Furthermore it allows the use of different Greens functions for the exterior problem, with only minor modifications on the algorithm. The resulted system of equations is solved with a very efficient preconditioned conjugate gradient (PCCG) with a p-Multiplicative Schwarz preconditioner.

A fast non-conforming DP-FETI domain decomposition method for the solution of large EM problems

The paper presents an efficient finite element based domain decomposition method (DDM) (Benamou, J-D and Despre/spl grave/s, B., J Comput. Phys., vol.136, p.68-82. 1997; Stupfel, B. and Mognot, M., IEEE Trans. Antennas Propagat., vol.48, p.653-60, 2000) for the analysis of large electromagnetic problems with a finite number of different building blocks. Such problems arise in important engineering applications such as antenna arrays, frequency selective surfaces, photonic crystals and metamaterials. The method is general enough to analyze arbitrary geometries. The proposed method is based on three key ingredients: (a) the optimized Robin transmission condition across interfaces; (b) the new mortar method for nonconforming triangulations; (c) a fast dual-primal finite element tearing and interconnecting (DP-FETI) algorithm similar to that of Wolfe, C.T. et al. (see IEEE Trans. Antennas Propagat., vol.48, no.2, p.278-84, 2000).

Application of the multilevel adaptive cross-approximation on ground plane designs

In this paper, the multilevel adaptive cross approximation (MLACA) technique is presented. This method reduces the numerical complexity of both memory requirement and matrix-vector multiplication of integral equation with asymptotically smooth kernels. The method does not require explicitly the knowledge of the integral kernel and therefore can be integrated into existing boundary integral equation (BIE) and/or MoM codes easily. This paper extends the ACA to oscillatory integral kernel and applies it to the ground plane designs. We have demonstrated through numerical examples that for moderate-sized problems (less than few hundred thousand unknowns) the multilevel ACA algorithm is still very effective and reliable for computer modeling of electromagnetic radiation and scattering problems.

A finite element domain decomposition technique for the analysis of large electromagnetic problems

The main motivation of this paper lays in a simple yet crucial observation that most real-world electromagnetic problems all exhibit certain degrees of redundancies, locally and/or globally. Take for example a vehicle: its geometry is symmetric with respect to a mid-plane; for an antenna array or frequency selective surface the redundancies are more obvious since all elements are identical, The present paper proposes a novel approach for analyzing electrically large, geometrically complicated structures that involve complex materials, by systematically exploiting local or global mirror, translational or rotational symmetries in the geometry.

Modeling large almost periodic structures using a non-overlapping domain decomposition method

Domain decomposition is a tool that is introduced artificially to ease large scale computation or that is natural in some situations. Domain decomposition methods are a very natural way to exploit the possibilities of multiprocessor computers, but such algorithms are very useful even when used on single process PC environment. The idea is to decompose the computational domain into smaller subdomains. The equations are solved on each subdomain. We first describe a domain decomposition method, a non-overlapping Schwarz algorithm, for solving large almost periodic electromagnetic problems. For large almost periodic structures, when using domain decomposition approach, the FEM matrix only needs to be evaluated once for all domains that are of the same building block. The solving process of domain decomposition is iterative and can become prohibitively slow. We describe a procedure that results in superior speed-up of the solving process.

A Domain Decomposition Approach for Non-conformal Couplings between Finite and Boundary Elements for Electromagnetic Scattering Problems in R3

To solve electromagetic scattering problems in R, the popular approach is to combine and couple finite and boundary elements. Common engineering practises in coupling finite and boundary elements usually result in non-symmetric and nonvariational formulations [5, 8]. The symmetric coupling between finite and boundary elements was first proposed by Costabel [2] in 1987. Since then, quite a few papers have been published on the topic of symmetric couplings. Among them, we list references [3, 4, 12, 7]. In particular, references [4, 12, 7] deal with variational formulations for solving electromagnetic wave radiation and scattering problems. Although the formulations detailed in [4, 12, 7] result in symmetric couplings between finite and boundary elements, they still suffer the notorious internal resonances. The purpose of this chapter is to present a variational formulation, which couples finite and boundary elements through non-conformal meshes. The formulation results in matrix equations that are symmetric, coercive, and free of internal resonances. Our plan for this chapter is as follows. Section 2 details the proposed variational formulation for non-conformal couplings between finite and boundary elements. In section 3, we show that, through a box-shaped computational domain, the proposed formulation is free of internal resonances and it satisfies the C.B.S inequality [1]. Moreover, in section 3 we validate the accuracy of the proposed formulation by a complex scattering problem. A brief conclusion is provided in section 4.

Application of DP-FETI domain decomposition method for the negative index of refraction materials

In this paper, NIM are modeled by a fast nonconforming dual primal-finite element tearing interconnecting (DP-FETI) domain decomposition method (DDM). We apply the DP-FETI-DDM method to model both positive and negative index materials, refer to as PIM and NIM. Since NIM are generally periodic in nature, the proposed method is capable of taking advantage of such periodicity and only requires small amount of memory.

Speed up the hybrid FEM+IE formulation using a low-rank matrix approximation

The paper addresses a methodology for hybridizing the finite element method with the integral equation method, FEM/IE, for unbounded scattering or radiation problems in electromagnetics. The methodology also speeds up the IE portion of the computation using a single-level low-rank matrix approximation. The motivation for this research has been the challenging problem of analyzing electrically large EM problems involving heterogeneous bodies, such as scattering by aircraft and ships and radiation by large antenna arrays.

A symmetric domain decomposition formulation of hybrid FEM-BEM coupling for electromagnetic analysis

This paper presents a novel symmetric FEM-BEM formulation for solving unbounded electromagnetic problems. The beauties of the proposed method include: (1) symmetry, (2) modularity, (3) non-conformity between FEM and BEM domains, (4) free of internal resonance, and (5) natural and very effective preconditioning schemes that guarantee spectral radius less or equal to one