volume | PIER Journals

Abstract

Hierarchical (H-) matrix based fast direct and iterative algorithms are presented for acceleration of the Method of Moment (MoM) solution of the Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE) formulated for scattering and radiation problems on homogeneous dielectric objects. As the SVS-EFIE features the product of the integral operator mapping the tangential equivalent electric current on the surface of the scatterer to the volume polarization current and the integral operator mapping the volume polarization current to the tangential component of the scattered electric field, its MoM discretization produces the product of non-square matrices. Formation of the non-square H-matrices for the MoM discretized integral operators is described. The algorithms for arithmetics pertinent to the product of the non-square H-matrices are explained. The memory and CPU time complexity scaling of the required H-matrix operations are analyzed in details and verified numerically. The numerical validation of the proposed algorithm is provided for both the lowloss dielectric objects as well as for the high-loss biological tissues found in the bioelectromagnetics applications. The numerical experiments demonstrate a signicant reduction of memory usage and a considerable speedup for CPU time compared to nave MoM, thus, enabling solution of the large-scale scattering and radiation problems with the SVS-EFIE.

1. Wu, T., T. S. Rappaport, and C. M. Collins, "The human body and millimeter-wave wireless communication systems: Interactions and implications," IEEE Int. Conf. Commun. (ICC), 2423-2429, Jun. 2015.

2. Fichte, L. O., "Interaction of biological tissue with electromagnetic waves in the RF range," Asia-Pacific Conf. Env. Electromag. (CEEM), 10, Nov. 2015.

3. Agarwal, K. and Y. X. Guo, "Interaction of electromagnetic waves with humans in wearable and biomedical implant antennas," Asia-Pacific Symp. on Electromag. Compat. (APEMC), 154-157, May 2015.
doi:10.1109/APEMC.2015.7175377

4. Ostadrahimi, M., P. Mojabi, S. Noghanian, L. Shafai, S. Pistorius, and J. LoVetri, "A novel microwave tomography system based on the scattering probe technique," IEEE Trans. Instrum. Meas., Vol. 61, No. 2, 379-390, Feb. 2012.
doi:10.1109/TIM.2011.2161931

5. Ferreira, D., P. Pires, R. Rodrigues, and R. F. S. Caldeirinha, "Wearable textile antennas: examining the effect of bending on their performance," IEEE Antennas Propag. Mag., Vol. 59, No. 3, 54-59, Jun. 2017.

6. De Santis, V., M. Feliziani, and F. Maradei, "Safety assessment of UWB radio systems for body area network by the FD²TD method," IEEE Trans. Mag., Vol. 46, No. 8, 3245-3248, Aug. 2010.

7. Aguirre, E., J. Arpn, L. Azpilicueta, S. de Migue, V. Ramos, and F. Falcone, "Evaluation of electromagnetic dosimetry of wireless systems in complex indoor scenarios with human body interaction," Progress In Electromagnetics Research B, Vol. 43, 189-209, Sep. 2012.

8. Ojaroudiparchin, N., M. Shen, S. Zhang, and G. F. Pedersen, "A switchable 3-D-coverage-phased array antenna package for 5G mobile terminals," IEEE Antennas Wireless Propag. Lett., Vol. 15, 1747-1750, Feb. 2016.

9. Thotahewa, K. M. S., J. M. Redoute, and M. R. Yuce, "SAR SA and temperature variation in the human head caused by IR-UWB implants operating at 4 GHz," IEEE Trans. Microw. Theory Techn., Vol. 61, No. 5, 2161-2169, May 2013.

10. Mittra, M., Computational Electromagnetics: Recent Advances and Engineering Applications, 3rd Ed., Springer, New York, 2014.

11. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd Ed., Artech House, Boston, MA, 2005.

12. Jin, J. M., The Finite Element Method in Electromagnetics, 3rd Ed., Wiley-IEEE Press, 2015.

13. Ansoft HFSS 11.1 user's guide, ANSYS Inc., Pittsburgh, PA, USA, 2009.

14. XFDTD 7.3 user's manual, Remcom Inc., State College, PA, USA, 2012.

15. Chew, W. C., E. Michielssen, J. M. Song, and J. M. Jin, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Norwood, MA, 2001.

16. Harrington, R. F., Field Computation by Moment Methods, Wiley-IEEE Press, 1993.

17. Massey, J. W., C. Liu, A. Menshov, and A. E. Yilmaz, "Bioelectromagnetic benchmarks,", 2016, [Online]. Available: http://bit.ly/BioEM-Benchmarks.

18. Saad, Y. and M. H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 7, No. 3, 856-869, Jul. 1986.

19. Van der Vorst, H. A., "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 13, No. 2, 631-644, 1992.

20. Coifman, R., V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: A pedestrian prescription," IEEE Antennas Propag. Mag., Vol. 35, No. 3, 7-12, Jun. 1993.

21. Ergul, O. and L. Gurel, The Multilevel Fast Multipole Algorithm (MLFMA) for Solving Large-Scale Computational Electromagnetics Problems, Wiley-IEEE Press, 2014.

22. Gumerov, N. A. and R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, Elsevier, Amsterdam, The Netherlands, 2006.

23. Aronsson, J. and V. Okhmatovski, "Vectorial low-frequency MLFMA for the combined field integral equation," IEEE Antennas Wireless Propag. Lett., Vol. 10, 532-535, 2011.

24. Catedra, M. F., R. F. Torres, J. Basterrechea, and E. Gago, The CG-FFT Method: Application of Signal Processing Techniques to Electromagnetics, Artech House, Boston, MA, 1995.

25. Bleszynski, E., M. Bleszynski, and T. Jaroszewicz, "AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems," Radio Sci., Vol. 31, 1255-1251, Sep.–Oct. 1996.

26. Yang, K. and A. Yilmaz, "A three-dimensional adaptive integral method for scattering from structures embedded in layered media," IEEE Trans. Geo. Remote Sens., Vol. 50, No. 4, 1130-1139, Apr. 2012.

27. Bebendorf, M. and S. Rjasanow, "Adaptive low-rank approximation of collocation matrices," Computing, Vol. 70, No. 1, 1-24, Mar. 2003.

28. Zhao, K., M. N. Vouvakis, and J.-F. Lee, "The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems," IEEE Trans. Electromagn. Compat., Vol. 47, No. 4, 763-773, Nov. 2005.

29. Mitra, R. and C. Li, "The art and science of matrix preconditioning --- A review," IEEE Int. Conf. Comput. EM (ICCEM), 17-19, Feb. 2016.

30. Vico, F., L. Greengard, and M. Ferrando, "Decoupled field integral equations for electromagnetic scattering from homogeneous penetrable obstacles,", Apr. 2017, [Online]. Available: http://arxiv.org/abs/1704.06741.

31. Hackbusch, W., "A sparse matrix arithmetic based on H-matrices. Part I. Introduction to H-matrices," Computing, Vol. 62, No. 2, 89-108, 1999.

32. Grasedyck, L. and W. Hackbusch, "Construction and arithmetics of H-matrices," Computing, Vol. 70, No. 4, 295-334, Aug. 2003.

33. Martinsson, P. G. and V. Rokhlin, "A fast direct solver for boundary integral equations in two dimensions," J. Comput. Phyis., Vol. 205, No. 1, 1-23, May 2005.

34. Guo, H., J. Hu, and E. Michielssen, "On MLMDA/Butterfly compressibility of inverse integral operators," IEEE Antennas Wireless Propag. Lett., Vol. 12, 31-34, 2013.

35. Shaeffer, J., "Direct solve of electrically large integral equations for problem sizes to 1 M unknowns," IEEE Trans. Antennas Propag., Vol. 56, No. 8, 2306-2313, Aug. 2008.

36. Brick, Y., V. Lomakin, and A. Boag, "Fast direct solver for essentially convex scatterers using multilevel non-uniform grids," IEEE Trans. Antennas Propag., Vol. 62, No. 8, 4314-4324, Aug. 2014.

37. Corona, E., A. Rahimian, and D. Zorin, "A Tensor-Train accelerated solver for integral equations in complex geometries," J. Comput. Phys., Vol. 334, 145-169, Apr. 2017.

38. Oseledets, I. V., "Tensor-Train decomposition," SIAM J. Sci. Comput., Vol. 33, No. 5, 2295-2317, 2011.

39. Menshov, A. and V. Okhmatovski, "New single-source surface integral equations for scattering on penetrable cylinders and current flow modeling in 2-D conductors," IEEE Trans. Microw. Theory Techn., Vol. 61, No. 1, 341-350, Jan. 2013.

40. Hosseini, F. L. S., A. Menshov, R. Gholami, J. Mojolagbe, and V. Okhmatovski, "Novel single-source integral equation for scattering problems by 3D dielectric objects," IEEE Trans. Antennas Propag., Vol. 66, No. 2, 797-807, Feb. 2018.

41. Zheng, S., R. Gholami, and V. Okhmatovski, "Surface-volume-surface electric field integral equation for solution of scattering problems on 3-D dielectric objects in multilayered media," IEEE Trans. Microw. Theory Tech., 1-16, 2018.

42. Chen, Z., R. Gholami, J. Mojolagbe, and V. Okhmatovski, "Formulation of Surface-Volume-Surface-EFIE for solution of 3D scattering problems on composite dielectric objects," IEEE Antennas Wireless Propag. Lett., Vol. 17, No. 6, 1043-1047, Jun. 2018.

43. Mojolagbe, J., R. Gholami, and V. Okhmatovski, "On complexity reduction in solution of scattering problems on well-conducting 3D objects with Surface-Volume-Surface EFIE," Appl. Comput. Electromag. Conf. (ACES), 1-2, May 2018.

44. Swatek, D. Investigation of single source surface integral equation for electromagnetic wave scattering by dielectric bodies, Ph.D. dissertation, Univ. Manitoba, Winnipeg, Canada, 1999.

45. Qian, Z. G., W. C. Chew, and R. Suaya, "Generalized impedance boundary condition for conductor modeling in surface integral equation," IEEE Trans. Microw. Theory Techn., Vol. 55, No. 11, 2354-2364, Nov. 2007.

46. Muller, C., Foundations of the Mathematical Theory of Electromagnetic Waves, Springer-Verlag, Berlin, Heidelberg, New York, 1969.

47. Kishk, A. and L. Shafai, "Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions," IEEE Trans. Antennas Propag., Vol. 34, No. 5, 666-673, May 1986.

48. Rao, S. M., D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag., Vol. 30, No. 3, 409-418, May 1982.

49. Martinsson, P.-G., V. Rokhlin, and M. Tygert, "A randomized algorithm for the decomposition of matrices," Appl. Comput. Harmon. Anal., Vol. 30, No. 1, 47-68, Jan. 2011.

50. Brick, Y. and A. Yilmaz, "ast multilevel computation of low-rank representation of H-matrix blocks," IEEE Trans. Antennas Propag., Vol. 64, No. 12, 5326-5334, Dec. 2016.

51. Borm, S., L. Grasedyck, and W. Hackbusch, Hierarchical matrices, Technical Report 21, Max Planck Institute for Mathematics in the Sciences, 2006.

52. Chew, W. C., Waves and Fields in Inhomogeneous Media, Wiley-IEEE Press, 1999.

53. FEKO user's manual, EM Software & Syst. Inc., Stellenbosch 7600, South Africa, 2014.

54. Woo, A. C., H. T. G. Wang, and M. J. Schuh, "Benchmark radar targets for the validation of computational electromagnetics programs," IEEE Antennas Propag. Mag., Vol. 35, No. 1, 84-89, Feb. 1993.

55., IEEE Recommended Practice for Determining the Peak Spatial-Average Specific Absorption Rate (SAR) in the Human Head from Wireless Communications Devices: Measurement Techniques, IEEE Std 1528-2013 (Rev. IEEE Std 1528-2003), Sep. 2013.

Dr. Reza Gholami

Dr. Jamiu Mojolagbe

Dr. Anton Menshov

Dr. Farhad Sheikh Hosseini Lori

Prof. Vladimir I. Okhmatovski