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A Polynomial Chaos Method for Dispersive Electromagnetics

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Probabilistic methods, simulation and stochastic differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  23 November 2015

Nathan L. Gibson
Nathan L. Gibson*
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA.
*
*Corresponding author. Email address: [email protected] (N. L. Gibson)
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Abstract

Electromagnetic wave propagation in complex dispersive media is governed by the time dependent Maxwell's equations coupled to equations that describe the evolution of the induced macroscopic polarization. We consider “polydispersive” materials represented by distributions of dielectric parameters in a polarization model. The work focuses on a novel computational framework for such problems involving Polynomial Chaos Expansions as a method to improve the modeling accuracy of the Debye model and allow for easy simulation using the Finite Difference Time Domain (FDTD) method. Stability and dispersion analyzes are performed for the approach utilizing the second order Yee scheme in two spatial dimensions.

Keywords

Maxwell's equationsDebye polarizationrelaxation time distributionFDTD

MSC classification

Secondary: 35Q61: Maxwell equations 65C99: None of the above, but in this section 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1234 - 1263
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Albanese, R. A., Penn, J. W., and Medina, R. L.Short-rise-time microwave pulse propagation through dispersive biological media. J. Opt. Soc. Amer. A, 6:1441–1446, 1989.Google Scholar
[2]Armentrout, M. and Gibson, N. L. Electromagnetic relaxation time distribution inverse problems in the time-domain. In WAVES International Conference Proceedings, 2011.Google Scholar
[3]Banks, H and Bokil, VA computational and statistical framework for multidimensional domain acoustooptic material interrogation. Quarterly of Applied Mathematics, 63(1):156–200, 2005.Google Scholar
[4]Banks, H. T., Buksas, M. W., and Lin, T.Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts, volume FR21 of Frontiers in Applied Mathematics. SIAM, Philadelphia, PA, 2000.CrossRefGoogle Scholar
[5]Banks, H. T. and Gibson, N. L.Well-posedness in Maxwell systems with distributions of polarization relaxation parameters. Appl. Math. Let., 18(4):423–430, 2005.Google Scholar
[6]Banks, H. T. and Gibson, N. L.Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters. Quarterly of Appl. Math., 64(4):749–795, 2006.CrossRefGoogle Scholar
[7]Banks, HT, Gibson, Nathan L, and Winfree, William P.Gap detection with electromagnetic terahertz signals. Nonlinear Analysis: Real World Applications, 6(2):381–416, 2005.Google Scholar
[8]Barba, I., Cabeceira, A. C. L., Panizo, M., and Represa, J.Modelling dispersive dielectrics in TLM method. Int.J. Numer. Model., 14:15–30, 2001.Google Scholar
[9]Barrese, K. and Chugh, N.Approximating dispersive mechanisms using the Debye model with distributed dielectric parameters. In Gibson, N. L., editor, REU Program at Oregon State University Proceedings, 2008.Google Scholar
[10]Beezley, R.S. and Krueger, R.J.An electromagnetic inverse problem for dispersive media. Journal of mathematical physics, 26:317, 1985.Google Scholar
[11]Bela, E. and Hortsch, E.Generalized polynomial chaos and dispersive dielectric media. In Gibson, N. L., editor, REU Program at Oregon State University Proceedings, 2010.Google Scholar
[12]Bello, A., Laredo, E., and Grimau, M.Distribution of relaxation times from dielectric spec-troscopy using monte carlo simulated annealing: application to α-pvdf. Physical Review B, 60(18):12764, 1999.Google Scholar
[13]Bokil, V. A. and Gibson, N. L.Analysis of spatial high-order finite difference methods for Maxwell's equations in dispersive media. IMA Journal of Numerical Analysis, 32(3):926–956, 2012.Google Scholar
[14]Bokil, V. A. and Gibson, N. L.Convergence analysis of Yee schemes for Maxwell's equations in Debye and Lorentz dispersive media. International Journal of Numerical Analysis & Modeling, 11(4):657–687, 2014.Google Scholar
[15]Bokil, V. A. and Gibson, N. L.Stability and dispersion analysis of high order FDTD methods for Maxwell's equations in dispersive media. In Recent Advances in Scientific Computing and Applications, volume 586 of Contemporary Mathematics, pages 73–82. American Mathematical Soc., 2013.Google Scholar
[16]Bokil, V.A., Keefer, O.A., and Leung, A.C.-Y.Operator splitting methods for Maxwells equations in dispersive media with orientational polarization. Journal of Computational and Applied Mathematics, 263:160–188, 2014.Google Scholar
[17]Böttcher, C. J. F. and Bordewijk, P.Theory of electric polarization, volume II. Elsevier Amsterdam, 1978.Google Scholar
[18]Cai, W.Computational methods for electromagnetic phenomena: electrostatics in solvation, scattering, and electron transport. Cambridge University Press, 2013.Google Scholar
[19]Causley, M.F., Petropoulos, P.G., and Jiang, S. Incorporating the Havriliak-Negami dielectric model in the FD-TD method. Journal of Computational Physics, 2011.Google Scholar
[20]Chauvière, C., Hesthaven, J. S., and Lurati, L.Computational modeling of uncertainty in time-domain electromagnetics. SIAM Journal on Scientific Computing, 28(2):751–775, 2006.Google Scholar
[21]Chen, W., Li, X., and Liang, D.Symmetric energy-conserved splitting fdtd scheme for the Maxwell's equations. Communications in Computational Physics, 6:804–825, 2009.Google Scholar
[22]Cole, K. S. and Cole, R. H.Dispersion and absorption in dielectrics I. alternating current characteristics. J. Chem. Phy., 9:341–351, 1941.Google Scholar
[23]Debye, P.J.W.Polar molecules. The Chemical Catalog Company, Inc., 1929.Google Scholar
[24]Ellingsrud, S., Eidesmo, T., Johansen, S., Sinha, M.C., MacGregor, L.M., and Constable, S.Remote sensing of hydrocarbon layers by seabed logging (SBL): Results from a cruise offshore Angola. The Leading Edge, 21(10):972, 2002.CrossRefGoogle Scholar
[25]Fear, E. C., Meaney, P. M., and Stuchly, M. A.Microwaves for breast cancer detection. IEEE Potentials, pages 12–18, 2003.Google Scholar
[26]Foster, K. R. and Schwan, H. P.Dielectric properties of tissues and biological materials: A critical review. Critical Rev. in Biomed. Engr., 17:25–104, 1989.Google Scholar
[27]Foster, K. R. and Schwan, H. P.Dielectric properties of tissues. Handbook of biological effects of electromagnetic fields, 2:25–102, 1996.Google Scholar
[28]Fuks, P., Karlsson, A., and Larson, G.Direct and inverse scattering from dispersive media. Inverse Problems, 10:555, 1994.Google Scholar
[29]Gabriel, C.Compilation of the dielectric properties of body tissues at RF and microwave frequencies. Technical Report AL/OE-TR-1996-0037, USAF Armstrong Laboratory, Brooks AFB, TX, 1996.Google Scholar
[30]Gabriel, C., Gabriel, S., and Corthout, E.The dielectric properties of biological tissues: I. Literature survey. Physics in medicine and biology, 41:2231–2249, 1996.Google Scholar
[31]Gabriel, S., Lau, R. W., and Gabriel, C.The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz. Physics in medicine and biology, 41:2251–2269, 1996.CrossRefGoogle Scholar
[32]Gabriel, S., Lau, R. W., and Gabriel, C.The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Physics in medicine and biology, 41:2271–2293, 1996.Google Scholar
[33]Gao, L. and Liang, D.New energy-conserved identities and super-convergence of the symmetric ec-s-fdtd scheme for Maxwell's equations in 2d. Communications in Computational Physics, 11(5):1673, 2012.Google Scholar
[34]Golub, G. H. and Welsch, J. H.Calculation of gauss quadrature rules. Mathematics of Computation, 23(106):221–230, 1969.Google Scholar
[35]Hurt, W. D.Multiterm Debye dispersion relations for permittivity of muscle. Biomedical Engineering, IEEE Transactions on, (1):60–64, 1985.Google Scholar
[36]Joseph, R. M., Hagness, S. C., and Taflove, A.Direct time integration of Maxwell's equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses. Optics Lett., 16(18):1412–1414, 1991.Google Scholar
[37]Joseph, R. M. and Taflove, A.FDTD Maxwell's equations models for nonlinear electrodynamics and optics. Antennas and Propagation, IEEE Transactions on, 45(3):364–374, 1997.Google Scholar
[38]Kashiwa, T. and Fukai, I.A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization. Microwave Opt. Technol. Lett., 3(6):203–205, 1990.Google Scholar
[39]Kashiwa, T., Yoshida, N., and Fukai, I.A treatment by the finite-difference time domain method of the dispersive characteristics associated with orientational polarization. IEEE Transactions of the IEICE, 73(8):1326–1328, 1990.Google Scholar
[40]Kelley, D. F. and Luebbers, R. J.Debye function expansions of empirical models of complex permittivity for use in FDTD solutions. In Antennas and Propagation Society International Symposium, 2003. IEEE, volume 4, pages 372–375. IEEE, 2003.Google Scholar
[41]Lanteri, S. and Scheid, C. Convergence of a discontinuous Galerkin scheme for the mixed time-domain Maxwell's equations in dispersive media. IMA Journal of Numerical Analysis, 2012.CrossRefGoogle Scholar
[42]Li, J.Unified analysis of leap-frog methods for solving time-domain Maxwell's equations in dispersive media. Journal of Scientific Computing, 47(1):1–26, 2011.Google Scholar
[43]Li, J. and Hesthaven, J. S.Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials. Journal of Computational Physics, 258:915–930, 2014.Google Scholar
[44]Li, J. and Huang, Y.Time-domain finite element methods for Maxwell's equations in metamaterials, volume 43. Springer, 2012.Google Scholar
[45]Li, J., Huang, Y., and Lin, Y.Developing finite element methods for Maxwell's equations in a Cole-Cole dispersive medium. SIAM Journal on Scientific Computing, 33(6):3153–3174, 2011.Google Scholar
[46]Luebbers, R., Hunsberger, F. P., Kunz, K. S., Standler, R. B., and Schneider, M.A frequency-dependent finite-difference time-domain formulation for dispersive materials. IEEE Trans. Electromagnetic Compatibility, 32(3):222–227, 1990.Google Scholar
[47]Metternicht, G.I. and Zinck, J.A.Remote sensing of soil salinity: potentials and constraints. Remote Sensing of Environment, 85(1):1–20, 2003.Google Scholar
[48]Milne, M. and Wedde, D.Simulating polydisperse materials with distributions of the Debye model. In Gibson, N. L., editor, REU Program at Oregon State University Proceedings, 2009.Google Scholar
[49]Miura, N., Yagihara, S., and Mashimo, S.Microwave dielectric properties of solid and liquid foods investigated by time-domain reflectometry. Journal of food science, 68(4):1396–1403, 2003.Google Scholar
[50]Monk, P.A comparison of three mixed methods for the time-dependent Maxwell's equations. SIAM Sci. Stat. Comput., 13:1097–1122, 1992.Google Scholar
[51]Monk, P.Finite Element Methods for Maxwell's Equations. Oxford University Press, 2003.Google Scholar
[52]Monk, P.A convergence analysis of Yee's scheme on nonuniform grids. SIAM J. Numer. Anal., 31(2):393–412, April 1994.Google Scholar
[53]Petropoulos, P. G.Stability and phase error analysis of FD-TD in dispersive dielectrics. IEEE Trans. Antennas Propagat., 42(1):62–69, 1994.Google Scholar
[54]Petropoulos, P. G.On the time-domain response of Cole-Cole dielectrics. Antennas and Propagation, IEEE Transactions on, 53(11):3741–3746, 2005.Google Scholar
[55]Shea, J.D., Kosmas, P., Van Veen, B.D., and Hagness, S.C.Contrast-enhanced microwave imaging of breast tumors: a computational study using 3D realistic numerical phantoms. Inverse problems, 26:074009, 2010.Google Scholar
[56]Shea, J.D., Van Veen, B.D., and Hagness, S.C.A TSVD analysis of microwave inverse scattering for breast imaging. Biomedical Engineering, IEEE Transactions on, (99):1–1, 2011.Google Scholar
[57]Siushansian, R. and LoVetri, J.A comparison of numerical techniques for modeling electromagnetic dispersive media. IEEE Microwave Guided Wave Lett., 5:426–428, 1995.Google Scholar
[58]Strikverda, J. C.Finite Difference Schemes and Partial Differential Equations. SIAM, 2004.Google Scholar
[59]Taflove, A. and Hagness, S. C.Computational Electrodynamics: The Finite-Difference Time-Domain method. Artech House, Norwood, MA, 3rd edition, 2005.Google Scholar
[60]Tarasov, A. and Titov, K.Relaxation time distribution from time domain induced polarization measurements. Geophysical Journal International, 170(1):31–43, 2007.Google Scholar
[61]Tofighi, M.-R.FDTD modeling of biological tissues cole-cole dispersion for 0.5-30 GHz using relaxation time distribution samples-novel and improved implementations. Microwave Theory and Techniques, IEEE Transactions on, 57(10):2588–2596, 2009.Google Scholar
[62]Schweidler, E. R. v.Studien über anomalien im verhalten der dielektrika. Annalen der Physik, 329(14):711–770, 1907.Google Scholar
[63]Wagner, K.W.Zur theorie der unvollkommenen dielektrika. Annalen der Physik, 345(5):817–855, 1913.Google Scholar
[64]Wolfersdorf, L.V.On an electromagnetic inverse problem for dispersive media. Quarterly of Applied Mathematics, 49:237–246, 1991.Google Scholar
[65]Xiu, D. and Karniadakis, G.E.The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24(2):619–644, 2003.Google Scholar
[66]Xiu, D.Fast numerical methods for stochastic computations: A review. Communications in computational physics, 5(2-4):242–272, 2009.Google Scholar
[67]Xiu, D.Numerical Methods for Stochastic Computations. Princeton University Press, 2010.Google Scholar
[68]Yee, K.Numerical solution of inital boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas Propagat., 14(3):302–307, 1966.Google Scholar
[69]Young, J. L. and Nelson, R. O.A summary and systematic analysis of FDTD algorithms for linearly dispersive media. IEEE Antennas and Propagation Magazine, 43:61–77, 2001.Google Scholar
[70]Zhong, X.M., Liao, C., Chen, W., Yang, Z.B., Liao, Y., and Meng, F.B.Image reconstruction of arbitrary cross section conducting cylinder using UWB pulse. Journal of Electromagnetic Waves and Applications, 21(1):25–34, 2007.Google Scholar