volume | PIER Journals

Abstract

A new numerical method is proposed for uncertainty quantification in the two-dimensional finite-difference frequency-domain (FDFD) method. The method is based on an intrusive polynomial chaos expansion (PCE) of the Helmholtz equation in terms of the material properties. The resulting PCE-FDFD method is validated against Monte-Carlo simulations for an electromagnetic scattering problem at 1.0 GHz. Good agreement is found between the statistics of the electric fields computed using the proposed method and the Monte-Carlo results, with a factor 15-120 reduction in the computational costs. The PCE-FDFD method is also applied to estimate the material properties from exterior measurements by formulating an objective function and applying constrained optimisation techniques. A maximum 1.7% error in the material properties was observed for a test geometry with six unknowns and 20 sample points.

1. Rappaport, C. M., M. Kilmer, and E. Miller, "Accuracy considerations in using the PML ABC with FDFD Helmholtz equation computation," International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 13, No. 5, 471-482, 2000.
doi:10.1002/1099-1204(200009/10)13:5<471::AID-JNM378>3.0.CO;2-A

2. Rumpf, R. C., "Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain," Progress In Electromagnetics Research B, Vol. 36, 221-248, 2012.
doi:10.2528/PIERB11092006

3. Masumnia-Bisheh, K., K. Forooraghi, and M. Ghaffari-Miab, "Electromagnetic uncertainty analysis using stochastic FDFD method," IEEE Transactions on Antennas and Propagation, Vol. 67, No. 5, 3268-3277, 2019.
doi:10.1109/TAP.2019.2896771

4. Dong, Q. and C. M. Rappaport, "Microwave subsurface imaging using direct finite-difference frequency-domain-based inversion," IEEE Transactions on Geoscience and Remote Sensing, Vol. 47, No. 11, 3664-3670, 2009.
doi:10.1109/TGRS.2009.2028740

5. Sun, S., B. J. Kooij, and A. G. Yarovoy, "A linear model for microwave imaging of highly conductive scatterers," IEEE Transactions on Microwave Theory and Techniques, Vol. 66, No. 3, 1149-1164, 2017.
doi:10.1109/TMTT.2017.2772795

6. Layek, M. K. and P. Sengupta, "Forward modeling of GPR data by unstaggered finite difference frequency domain (FDFD) method: An approach towards an appropriate numerical scheme," Journal of Environmental and Engineering Geophysics, Vol. 24, No. 3, 487-496, 2019.
doi:10.2113/JEEG24.3.487

7. Masumnia-Bisheh, K. and C. Furse, "Bioelectromagnetic uncertainty analysis using geometrically stochastic FDFD method," IEEE Transactions on Antennas and Propagation, 2020, doi: 10.1109/TAP.2020.3025238.

8. Austin, A. C. M. and C. D. Sarris, "Efficient analysis of geometrical uncertainty in the FDTD method using polynomial chaos with application to microwave circuits," IEEE Transactions on Microwave Theory and Techniques, Vol. 61, No. 12, 4293-4301, 2013.
doi:10.1109/TMTT.2013.2281777

9. Litvinenko, A., A. C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen, and R. Tempone, "Computation of electromagnetic fields scattered from objects with uncertain shapes using multilevel Monte Carlo method," IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 4, 37-50, 2019.
doi:10.1109/JMMCT.2019.2897490

10. Edwards, R. S., A. C. Marvin, and S. J. Porter, "Uncertainty analyses in the finite-difference time-domain method," IEEE Transactions on Electromagnetic Compatibility, Vol. 52, No. 1, 155-163, Feb. 2010.
doi:10.1109/TEMC.2009.2034645

11. Nguyen, B. T., A. Samimi, S. E. W. Vergara, C. D. Sarris, and J. J. Simpson, "Analysis of electromagnetic wave propagation in variable magnetized plasma via polynomial chaos expansion," IEEE Transactions on Antennas and Propagation, Vol. 67, No. 1, 438-449, 2018.
doi:10.1109/TAP.2018.2879676

12. Nguyen, B. T., S. E. W. Vergara, C. D. Sarris, and J. J. Simpson, "Ionospheric variability effects on impulsive ELF antipodal propagation about the earth sphere," IEEE Transactions on Antennas and Propagation, Vol. 11, No. 66, 6244-6254, 2018.
doi:10.1109/TAP.2018.2874478

13. Gorniak, P., "An effective FDTD algorithm for simulations of stochastic EM fields in 5G frequency band," 2018 IEEE 29th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), 1417-1421, Bologna, 2018.

14. Zygiridis, T., A. Papadopoulos, N. Kantartzis, C. Antonopoulos, E. N. Glytsis, and T. D. Tsiboukis, "Intrusive polynomial-chaos approach for stochastic problems with axial symmetry," IET Microwaves, Antennas and Propagation, Vol. 13, No. 6, 782-788, 2019.
doi:10.1049/iet-map.2018.5306

15. Cheng, X., W. Shao, K. Wang, and B.-Z. Wang, "Uncertainty analysis in dispersive and lossy media for ground-penetrating radar modeling," IEEE Antennas and Wireless Propagation Letters, Vol. 18, No. 9, 1931-1935, 2019.
doi:10.1109/LAWP.2019.2933777

16. Liu, J., H. Li, and X. Xi, "General polynomial chaos-based expansion finite-difference time-domain method for analysing electromagnetic wave propagation in random dispersive media," IET Microwaves, Antennas and Propagation, Vol. 15, No. 2, 221-228, 2021.
doi:10.1049/mia2.12040

17. Wang, K. C., Z. He, D. Z. Ding, and R. S. Chen, "Uncertainty scattering analysis of 3-D objects with varying shape based on method of moments," IEEE Transactions on Antennas and Propagation, Vol. 67, No. 4, 2835-2840, 2019.
doi:10.1109/TAP.2019.2896456

18. Xiu, D., "Efficient collocational approach for parametric uncertainty analysis," Communications in Computational Physics, Vol. 2, No. 2, 293-309, 2007.

19. Giraldi, L., A. Litvinenko, D. Liu, H. G. Matthies, and A. Nouy, "To be or not to be intrusive? The solution of parametric and stochastic equations - The plain vanilla Galerkin case," SIAM Journal on Scientific Computing, Vol. 36, No. 6, A2720-A2744, 2014.
doi:10.1137/130942802

20. Austin, A. C. M., "Wireless channel characterization in burning buildings over 100-1000 MHz," IEEE Transactions on Antennas and Propagation, Vol. 64, No. 7, 3265-3269, 2016.
doi:10.1109/TAP.2016.2562671

21. Xiu, D., Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010.

22. Crestaux, T., O. Le Maıtre, and J.-M. Martinez, "Polynomial chaos expansion for sensitivity analysis," Reliability Engineering & System Safety, Vol. 94, No. 7, 1161-1172, 2009.
doi:10.1016/j.ress.2008.10.008

Dr. Andrew C. M. Austin