volume | PIER Journals

Abstract

The contrast current density volume integral equation, discretized with piecewise constant spatial basis and test functions and Dirac-delta temporal test functions and the piecewise polynomial temporal basis functions, results in a causal implicit marching-on-in-time scheme that we refer to as the marching-on-in-time contrast current density volume integral equation (MOT-JVIE). The companion matrix stability analysis of the MOT-JVIE solver shows that for a fixed spatial and temporal step size, the stability is independent of the scatterer's dielectric contrast for quadratic spline temporal basis functions. Whereas, Lagrange and cubic spline exhibit instabilities at higher contrast. We relate this stability performance to the expansion and testing procedure in time. We further illustrate the capabilities of the MOT-JVIE based on quadratic spline temporal basis functions by: comparing the MOT-JVIE solution to time-domain results from literature and frequency-domain results from a commercial combined field integral equation solver. Finally, we present a long time sequence for a high-contrast scatterer discretized with 24,000 spatial unknowns.

1. Ergül, O., New Trends in Computational Electromagnetics, 1st Ed., Institution of Engineering and Technology (the IET), 2009.

2. Jin, Jian-Ming and Su Yan, "Multiphysics modeling in electromagnetics," IEEE Antennas and Propagation Magazine, Vol. 61, No. 2, 14-26, Apr. 2019.
doi:10.1109/MAP.2019.2895623

3. Zhang, Huan Huan, Pan Pan Wang, Shuai Zhang, Long Li, Ping Li, Wei E. I. Sha, and Li Jun Jiang, "Electromagnetic-circuital-thermal multiphysics simulation method: A review," Progress in Electromagnetics Research, Vol. 169, 87-101, 2020.
doi:10.2528/PIER20112801

4. Sankaran, Krishnaswamy, "Are you using the right tools in computational electromagnetics?," Engineering Reports, Vol. 1, No. 3, 1-19, Oct. 2019.
doi:10.1002/eng2.12041

5. De Hoop, Adrianus T., "28 electromagnetic reciprocity theorems and their applications," Handbook of Radiation and Scattering of Waves: Acoustic Waves in Fluids, Elastic Waves in Solids, Electromagnetic Waves, 851-857, 2008.

6. Gres, N. T., A. A. Ergin, E. Michielssen, and B. Shanker, "Volume-integral-equation-based analysis of transient electromagnetic scattering from three-dimensional inhomogeneous dielectric objects," Radio Science, Vol. 36, No. 3, 379-386, May 2001.
doi:10.1029/2000RS002342

7. Shanker, B., K. Aygün, and E. Michielssen, "Fast analysis of transient scattering from lossy inhomogeneous dielectric bodies," Radio Science, Vol. 39, No. 2, Mar. 2004.
doi:10.1029/2003RS002877

8. Kobidze, G., J. Gao, B. Shanker, and E. Michielssen, "A fast time domain integral equation based scheme for analyzing scattering from dispersive objects," IEEE Transactions on Antennas and Propagation, Vol. 53, No. 3, 1215-1226, Mar. 2005.
doi:10.1109/TAP.2004.841295

9. Shi, Yan and Jian-Ming Jin, "A marching-on-in-degree solution of volume integral equations for transient electromagnetic scattering by bi-isotropic objects," Electromagnetics, Vol. 31, No. 3, 159-172, 2011.
doi:10.1080/02726343.2011.558449

10. Shi, Yan and Jian-Ming Jin, "A time-domain volume integral equation and its marching-on-in-degree solution for analysis of dispersive dielectric objects," IEEE Transactions on Antennas and Propagation, Vol. 59, No. 3, 969-978, Mar. 2011.
doi:10.1109/TAP.2010.2103038

11. Shi, Yan and Jian-Ming Jin, "Marching-on-in-degree solution of volume integral equations for analysis of transient electromagnetic scattering by inhomogeneous dielectric bodies with conduction loss," Microwave and Optical Technology Letters, Vol. 53, No. 5, 1104-1109, May 2011.
doi:10.1002/mop.25897

12. Bin Sayed, Sadeed, Huseyin Arda Ulku, and Hakan Bagci, "A stable marching on-in-time scheme for solving the time-domain electric field volume integral equation on high-contrast scatterers," IEEE Transactions on Antennas and Propagation, Vol. 63, No. 7, 3098-3110, Jul. 2015.
doi:10.1109/TAP.2015.2429736

13. Al-Jarro, Ahmed, Mohamed A. Salem, Hakan Bagci, Trevor M. Benson, Phillip Sewell, and Ana Vukovic, "Explicit solution of the time domain volume integral equation using a stable predictor-corrector scheme," IEEE Transactions on Antennas and Propagation, Vol. 60, No. 11, 5203-5214, Nov. 2012.
doi:10.1109/TAP.2012.2207691

14. Liu, Yang, Ahmed Al-Jarro, Hakan Bagci, and Eric Michielssen, "Parallel pwtd-accelerated explicit solution of the time-domain electric field volume integral equation," IEEE Transactions on Antennas and Propagation, Vol. 64, No. 6, 2378-2388, Jun. 2016.
doi:10.1109/TAP.2016.2546964

15. Hu, Y. L., J. Li, D. Z. Ding, and R. S. Chen, "Analysis of transient em scattering from penetrable objects by time domain nonconformal vie," IEEE Transactions on Antennas and Propagation, Vol. 64, No. 1, 360-365, Jan. 2016.
doi:10.1109/TAP.2015.2501437

16. Cao, Jun, Dazhi Ding, Guangshang Cheng, and Rushan Chen, "A higher order nyström td-vie method for scattering from magnetized plasma objects," IEEE Antennas and Wireless Propagation Letters, Vol. 16, 408-411, 2016.
doi:10.1109/LAWP.2016.2641019

17. Chen, Rui, Sadeed B. Sayed, H. Arda Ulku, and Hakan Bagci, "An explicit time marching scheme for efficient solution of the magnetic field integral equation at low frequencies," IEEE Transactions on Antennas and Propagation, Vol. 69, No. 2, 1213-1218, Feb. 2021.
doi:10.1109/TAP.2020.3010997

18. Tong, Mei Song and Jian Zhang, "Numerical solution of time-domain volume integral equations for transient electromagnetic scattering by dielectric objects," IEEE Transactions on Antennas and Propagation, Vol. 64, No. 10, 4487-4492, Oct. 2016.
doi:10.1109/TAP.2016.2587748

19. Bin Sayed, Sadeed, Huseyin Arda Ulku, and Hakan Bagci, "Explicit time marching schemes for solving the magnetic field volume integral equation," IEEE Transactions on Antennas and Propagation, Vol. 68, No. 3, 2224-2237, Mar. 2020.
doi:10.1109/TAP.2019.2949381

20. Ülkü, H. Arda, Sadeed Bin Sayed, and Hakan Bağci, "An explicit MOT-TDVIE scheme for analyzing electromagnetic field interactions on nonlinear scatterers," 2015 IEEE International Conference on Computational Electromagnetics, 101-103, Hong Kong, China, Feb. 2015.

21. Sayed, Sadeed B., H. Arda Ulku, and Hakan Bagci, "An explicit MOT-TD-VIE solver for time varying media," 2016 IEEE/ACES International Conference on Wireless Information Technology and Systems (ICWITS) and Applied Computational Electromagnetics (ACES), Honolulu, HI, USA, Mar. 2016.

22. Tao, Shifei, Jun Cao, Zhenhong Fan, and R. S. Chen, "A novel td-vie based on mot scheme for analysis of dispersive objects," IEEE Transactions on Antennas and Propagation, Vol. 65, No. 10, 5387-5395, Oct. 2017.
doi:10.1109/TAP.2017.2742547

23. Sayed, Sadeed Bin, Rui Chen, Hüseyin Arda Ülkü, and Hakan Bagci, "A time domain volume integral equation solver to analyze electromagnetic scattering from nonlinear dielectric objects," IEEE Transactions on Antennas and Propagation, Vol. 71, No. 12, 9255-9267, 2023.
doi:10.1109/TAP.2023.3245301

24. Polimeridis, A. G., J. F. Villena, L. Daniel, and J. K. White, "Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects," Journal of Computational Physics, Vol. 269, 280-296, Jul. 2014.
doi:10.1016/j.jcp.2014.03.026

25. Van Beurden, Martijn Constant, "Integro-differential equations for electromagnetic scattering: Analysis and computation for objects with electric contrast," Ph.D. dissertation, 2003.

26. Van Beurden, M. C. and S. J. L. Van Eijndhoven, "Gaps in present discretization schemes for domain integral equations," 2007 International Conference on Electromagnetics in Advanced Applications, Turin, Italy, Sep. 2007.

27. Van Beurden, M. C. and S. J. L. Van Eijndhoven, "Well-posedness of domain integral equations for a dielectric object in homogeneous background," Journal of Engineering Mathematics, Vol. 62, No. 3, 289-302, Nov. 2008.
doi:10.1007/s10665-008-9218-2

28. Poggio, A. J. and E. K. Miller, Integral Equation Solutions of Three-dimensional Scattering Problems, 159-264, Computer Techniques for Electromagnetics, 1st ed., Mittra, R., Ed., Pergamon Press, Ch. 4, 1973.

29. Shanker, Balasubramaniam, Mingyu Lu, Jun Yuan, and Eric Michielssen, "Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields," IEEE Transactions on Antennas and Propagation, Vol. 57, No. 5, 1506-1520, May 2009.
doi:10.1109/TAP.2009.2016700

30. Van't Wout, Elwin, Duncan R. van der Heul, Harmen van der Ven, and Cornelis Vuik, "The influence of the exact evaluation of radiation fields in finite precision arithmetic on the stability of the time domain integral equation method," IEEE Transactions on Antennas and Propagation, Vol. 61, No. 12, 6064-6074, Dec. 2013.
doi:10.1109/TAP.2013.2281365

31. Golub, G. H. and J. H. Welsch, "Calculation of gauss quadrature rules," Mathematics of Computation, Vol. 23, No. 106, 221-230, 1969.
doi:10.2307/2004418

32. Dodson, S. J., S. P. Walker, and M. J. Bluck, "Implicitness and stability of time domain integral equation scattering analyses," The Applied Computational Electromagnetics Society, Vol. 13, No. 3, 291-301, 1998.

33. Wang, P., M. Y. Xia, J. M. Jin, and L. Z. Zhou, "Time-domain integral equation solvers using quadratic B-spline temporal basis functions," Microwave and Optical Technology Letters, Vol. 49, No. 5, 1154-1159, May 2007.
doi:10.1002/mop.22385

34. Van't Wout, Elwin, Duncan R. van der Heul, Harmen van der Ven, and Cornelis Vuik, "Design of temporal basis functions for time domain integral equation methods with predefined accuracy and smoothness," IEEE Transactions on Antennas and Propagation, Vol. 61, No. 1, 271-280, Jan. 2013.
doi:10.1109/TAP.2012.2220318

35. Rohatgi, A., "Webplotdigitizer," https://apps.automeris.io/wpd/, 2021.

36. "Cst studio suite 2023," https://ww w.3ds.com/products-services/simulia/products/cst-studio-suite/, 2023.

Mr. Petrus Wilhelmus Nicolaas (Pieter) Van Diepen

Prof. Martijn Constant van Beurden

Mr. Roeland Johannes Dilz