FAST MONOSTATIC RCS COMPUTATIONIN FEM BASED SOLVERS USING QR DECOMPOSITION Neelakantam V. Venkatarayalu 1 , Yeow -Beng Gan 1 , Kezhong Zhao 2 , and Jin -Fa Lee 2 1 Temasek Laboratories, National University of Singapore, 5, Sports Drive 2, Singapore 2 ElectroScience Laboratory, Ohio State University, 1320 Kinnear Road, Columbus, Ohio, U.S.A. ABSTRACT A technique to reduce computational time in problems in- volving monostatic RCS computation using the finite ele- ment method is presented. In scattering problems, the ex- citation vectors for different angle of incidence are highly linearly dependent. Using this fact, the excitation matrix assembled over all incident angles can be approximated with a specified tolerance by QR decomposition with col- umn rank that is much less than the number of incident angles of interest. The number of matrix solutions re- quired is reduced to the effective column rank, resulting in significant reduction in computational time. Numerical examples highlighting the effectiveness of the proposed acceleration technique is presented. Key words: FEM; Monostatic Radar Cross Section; QR Decomposition. 1. INTRODUCTION Computation of radar cross section is a vital step in the design of low observable antenna arrays or shape opti- mization of electromagnetically invisible bodies. Though integral equations based methods are often more efficient for such problems, they require special treatment when the geometry involved is inhomogeneous, which is often the case in practical design problems. The finite element method (FEM) with its inherent capability to model in- homogeneous media is therefore an attractive alternative [1]. In FEM with total field formulation for scattering prob- lems, plane wave excitation is imposed on the boundary of the computational domain enclosing the body whose scattered fields are to be computed. Such an excitation is represented on the right hand side (excitation vector) of the resultant matrix system. For excitation at different angles of incidence, the FEM system matrix remains the same so long as the finite element meshes are unchanged. Based on this fact, we demonstrate a simple and efficient technique to reduce the CPU time involved in the compu- tation of monostatic RCS over a range of incident angles. It is essentially based on the compression of the excitation matrix assembled over all planewave incident angles of interest using a low rank QR decomposition. Such com- pression techniques have recently been successfully ap- plied to system matrix in integral equations based solvers [4]. Based on this technique, the number of required ma- trix solutions can be significantly reduced depending on the rank of the excitation matrix. The organization of this paper is as follows. In section 2, we present the total field FEM formulation for scattering problems. In section 3, we present the QR decomposition algorithm and its use in reducing computational time for monostatic RCS over a range of incident angles. Finally, in section 4, some numerical examples are presented to illustrate the advan- tages of the proposed technique. 2. FEM FORMULATION In FEM using using curl-conforming basis functions, we start with the vector Helmholtz’s equation given by ∇× 1 µ r ∇×  E − k 2 0 ε r  E = −jkη  J imp in Ω (1) γ t  E =0 on Γ pec (2) ˆ n ×∇×  E − jk 0 γ t  E =0 on Γ abc (3) where γ t u =ˆ n × u × ˆ n is the tangential trace operator. γ t u is the tangential component of the vector field u on a surface with outward normal ˆ n.  J imp is the impressed current density which excites the electromagnetic field. In scattering problems, there are no sources in the domain Ω, thus  J imp =0. However, the incident plane wave  E inc excites the electromagnetic fields inside Ω. Thus the total electric field  E consists of two components viz., incident (  E inc ) and scattered (  E sca ) fields.  E =  E inc +  E sca (4) The incident electric field is given as  E inc =  E o e −j  k inc · r . The radiation boundary condition Eq.(3) on Γ abc , in the scattering case applies only to the scat- tered field and not the incident plane wave. Thus, for the _____________________________________________________ Proc. ‘EuCAP 2006’, Nice, France 6–10 November 2006 (ESA SP-626, October 2006)