Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns

When unknowns are spatially grouped for electrically large bodies, the blocked method of moments (MoM) Z matrix and its LU factors are comprised of blocks which can be well approximated by matrices of low rank expressed as the outer product of a column matrix times a row matrix. This approximation is called matrix compression and can save significant memory and operations count. And for monostatic scattering, where there are many right-hand sides (RHS) for electrically large bodies, the blocked RHS V and current solution J are also well approximated by this low rank outer product form. The blocked system equation ZJ = LUJ = V is LU factored and solved using this compressed block form where each outer product approximant is computed using the adaptive cross approximation (ACA). This approach has been applied to a frequency domain EFIE RWG Galerkin integral equation for 3-D PEC surfaces. Preliminary results for this compressed block LU factor and solve technique are presented and compared to measured data, BOR code predictions, and the same code without matrix compression. Complexity of this approach is studied numerically for an open pipe geometry with specific error tolerances, using a PC Workstation, with unknowns N ranging from 2592 to 1 025 109 and the number of RHS varying from 310 to 6162. Matrix fill time scales as N1.34, LU matrix memory storage scales as N1.5, LU factor time scales as N2.0, and the time per RHS solve scales as N1.67. Complete solution wall time scales as N1.8.

[1]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[2]  B. Cranganu-Cretu,et al.  Fast BEM-solution of Laplace problems with H-matrices and ACA , 2006, IEEE Transactions on Magnetics.

[3]  Mario Bebendorf,et al.  Approximate Inverse Preconditioning of Finite Element Discretizations of Elliptic Operators with Nonsmooth Coefficients , 2005, SIAM J. Matrix Anal. Appl..

[4]  M. Vouvakis,et al.  The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems , 2005, IEEE Transactions on Electromagnetic Compatibility.

[5]  Adaptive cross approximation for MOM matrix fill for PC problem sizes to 157000 unknowns , 2005, IEEE/ACES International Conference on Wireless Communications and Applied Computational Electromagnetics, 2005..

[6]  Mario Bebendorf,et al.  Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients , 2004, Math. Comput..

[7]  D. Wilton,et al.  Issues and Methods Concerning the Evaluation of Hypersingular and Near-Hypersingular Integrals in BEM Formulations , 2005 .

[8]  R. Burkholder,et al.  Fast dual-MGS block-factorization algorithm for dense MoM matrices , 2004, IEEE Transactions on Antennas and Propagation.

[9]  Stefan Kurz,et al.  Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems , 2003 .

[10]  Mario Bebendorf,et al.  Mathematik in den Naturwissenschaften Leipzig Existence of H-Matrix Approximants to the Inverse FE-Matrix of Elliptic Operators with L ∞-Coefficients , 2003 .

[11]  Stefan Kurz,et al.  The adaptive cross-approximation technique for the 3D boundary-element method , 2002 .

[12]  Andrew J. Gatesman,et al.  VHF/UHF imagery and RCS measurements of ground targets in forested terrain , 2002, SPIE Defense + Commercial Sensing.

[13]  Mario Bebendorf,et al.  Approximation of boundary element matrices , 2000, Numerische Mathematik.

[14]  Sergej Rjasanow,et al.  Adaptive Cross Approximation of Dense Matrices , 2000 .

[15]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

[16]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[17]  Kapur,et al.  IES/sup 3/: a fast integral equation solver for efficient 3-dimensional extraction , 1997, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[18]  S. Kapur,et al.  IES3: a fast integral equation solver for efficient 3-dimensional extraction , 1997, ICCAD.

[19]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[20]  Parallel CARLOS-3D code development , 1996 .

[21]  Gene H. Golub,et al.  Matrix Computations, Third Edition , 1996 .

[22]  A. C. Woo,et al.  Benchmark radar targets for the validation of computational electromagnetics programs , 1993 .

[23]  M. J. Schuh,et al.  EM programmer's notebook-benchmark plate radar targets for the validation of computational electroma , 1992 .

[24]  L. N. Medgyesi-Mitschang,et al.  Combined field integral equation formulation for inhomogeneous two- and three-dimensional bodies: the junction problem , 1991 .

[25]  D. Wilton,et al.  Electromagnetic scattering by surfaces of arbitrary shape , 1980 .

[26]  Allen W. Glisson,et al.  Electromagnetic Scattering by Surface of Arbitrary Shape. , 1980 .