The Journal of Korean Institute of Electromagnetic Engineering and Science (한국전자파학회논문지)

The Korean Institute of Electromagnetic Engineering and Science (한국전자파학회)
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A Study for Improving Computational Efficiency in Method of Moments with Loop-Star Basis Functions and Preconditioner

루프-스타(Loop-Star) 기저 함수와 전제 조건(Preconditioner)을 이용한 모멘트법의 계산 효율 향상에 대한 연구

  • Yeom, Jae-Hyun (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology) ;
  • Park, Hyeon-Gyu (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology) ;
  • Lee, Hyun-Suck (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology) ;
  • Chin, Hui-Cheol (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology) ;
  • Kim, Hyo-Tae (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology) ;
  • Kim, Kyung-Tae (Department of Electronic and Electrical Engineering, Pohang University of Science and Technology)
  • 염재현 (포항공과대학교 전자전기공학과) ;
  • 박현규 (포항공과대학교 전자전기공학과) ;
  • 이현석 (포항공과대학교 전자전기공학과) ;
  • 진희철 (포항공과대학교 전자전기공학과) ;
  • 김효태 (포항공과대학교 전자전기공학과) ;
  • 김경태 (포항공과대학교 전자전기공학과)
  • Received : 2011.12.01
  • Accepted : 2011.12.29
  • Published : 2012.02.29
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Abstract

This paper uses loop-star basis functions to overcome the low frequency breakdown problem in method of moments (MoM) based on electric field integral equation(EFIE). In addition, p-Type Multiplicative Schwarz preconditioner (p-MUS) technique is employed to reduce the number of iterations required for the conjugate gradient method(CGM). Low frequency instability with Rao Wilton Glisson(RWG) basis functions in EFIE can be resolved using loop-start basis functions and frequency normalized techniques. However, loop-star basis functions, consisting of irrotational and solenoidal components of RWG basis functions, require a large number of iterations to calculate a solution through iterative methods, such as conjugate gradient method(CGM), due to high condition number. To circumvent this problem, in this paper, the pMUS preconditioner technique is proposed to reduce the number of iterations in CGM. Simulation results show that pMUS preconditioner is much faster than block diagonal preconditioner(BDP) when the sparsity of pMUS is the same as that of BDP.

본 논문에서는 전계 적분 방정식 (Electric Field Integral Equation: EFIE)을 사용하는 모멘트 법의 저주파 오차(low frequency breakdown) 문제를 해결하기 위한 방법으로 루프-스타(loop-star) 기저 함수를 사용하였다. 또한, 모멘트 법의 해를 계산하기 위하여 conjugate gradient method(CGM)과 같은 반복법을 적용할 경우 반복 횟수를 줄이기 위한 기법으로 p-Type Multiplicative Schwarz preconditioner(pMUS)를 이용하였다. 헬름홀쯔 정리(Helmholtz theorem)에 기반한 루프-스타(loop-star) 기저 함수와 주파수 정규화 기법을 이용하여 전계 적분 방정식에서 Rao-Wilton-Glisson(RWG) 기저 함수를 사용하였을 때 발생하는 저주파 오차(low frequency instability) 문제를 해결할 수 있다. 하지만, RWG 기저 함수를 비발산(solenoidal) 성분과 비회전성(irroatational) 성분으로 분해함으로써 발생하는 행렬 방정식의 높은 조건 수(condition number)로 인하여 CGM과 같은 반복법을 사용할 경우 해를 계산하기 위하여 많은 반복 횟수가 요구된다. 본 논문에서는 이러한 문제점을 해결하기 위한 방안으로 pMUS 전제 조건 기법을 이용하여 CGM의 반복 횟수를 줄였다. 수치 해석 결과, pMUS와 같은 희소성(sparsity)을 가진 블럭 대각 전제 조건(Block Diagonal Precondtioner: BDP)과 비교하였을 때 pMUS는 BDP보다 빠르게 해를 계산할 수 있다.

Keywords

References

  1. Roger F. Harrington, Field Computation by Moment Methods, IEEE Press, 1993.
  2. Walton C. Gibson, The Method of Moments in Electromagnetics, Chapman & Hall/CRC, 2008.
  3. Andrew F. Peterson, Scott L. Ray, and Raj Mittra, Computational Methods for Electromagnetics, IEEE Press, 1998.
  4. Weng Cho Chew, Mei Song Tong, and Bin Hu, "Integral equation methods for electromagnetic and elastic waves", Morgan & Claypool, pp. 43-107, 2009.
  5. Weng Cho Chew, C. P. Davis, K. F. Warnick, Z. P. Nie, J. Hu, S. Yan, and L. Gurel, "EFIE and MFIE, Why the difference?", in Proc. IEEE Int. Symp. Antennas Propagation/USNC/URSI Nat. Radio Sci. Meeting, San Diego, CA, pp. 1-2, 2008.
  6. Giuseppe Vecchi, "Loop-star decomposition of basis functions in the discretization of the EFIE", IEEE Trans. on Antenna and Propagation, vol. 47, no. 2, pp. 339-346, Feb. 1999. https://doi.org/10.1109/8.761074
  7. Jun-Sheng Zhao, Weng Cho Chew, "Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies", IEEE Trans. on Antenna and Propagation, vol. 48, no. 10, pp. 1635-1645, Oct. 2000. https://doi.org/10.1109/8.899680
  8. Jin-Fa Lee, Robert Lee, and Robert J. Burkholder, "Loop star basis functions and a robust preconditioner for EFIE scattering problems", IEEE Trans. on Antenna and Propagation, vol. 51, no. 8, pp. 1855-1863, Aug. 2003. https://doi.org/10.1109/TAP.2003.814736
  9. Ozgur Ergul, Levent Gurel, "The use of curl-conforming basis functions for the magnetic-field integral equation", IEEE Trans. on Antenna and Propagation, vol. 54, no. 7, pp. 1917-1926, Jul. 2006. https://doi.org/10.1109/TAP.2006.877159
  10. Eduard Ubeda, Juan M. Rius, "Novel monopolar MFIE MoM-discretization for the scattering analysis of small objects", IEEE Trans. on Antenna and Propagation, vol. 54, no. 1, pp. 50-57, Jan. 2006. https://doi.org/10.1109/TAP.2005.861529
  11. Ozgur Ergul, Levent Gurel, "Improving the accuracy of the magnetic field integral equation with the linear-linear basis functions", Radio Science, vol. 41, pp. 1-15, Jul. 2006.
  12. Ozgur Ergul, Levent Gurel, "Linear-linear basis functions for MLFMA solution of magnetic-field and combined-field integral equations", IEEE Trans. on Antenna and Propagation, vol. 55, no. 4, pp. 1103-1110, Apr. 2007. https://doi.org/10.1109/TAP.2007.893393
  13. Roberto D. Graglia, Donald Wilton, and Andrew F. Peterson, "Higher order interpolatory vector bases for computational electromagnetics", IEEE Trans. on Antenna and Propagation, vol. 45, no 3. pp. 329-342, Mar. 1997. https://doi.org/10.1109/8.558649
  14. John L. Volakis, Arindam Chatterjee, and Leo C. Kempel, Finite Element Method for Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications, IEEE Press, 1998.
  15. Seung-Cheol Lee, Vineet Rawat, and Jin-Fa Lee, "A hybrid finite/boundary element method for periodic structures on non-periodic meshes using an interior penalty formulation for Maxwell's equations", Journal of Computational Physics, vol. 229, issue 13, pp. 4934-4951, Jul. 2010. https://doi.org/10.1016/j.jcp.2010.03.014
  16. Constantine A. Balanis, Advanced Engineering Electromagnetics, 2nd Ed., John Wiley and Sons, 1989.
  17. Y. Saad, Iterative Methods for Sparse Linear Systems, 1st Ed., PWS, 1996.