A multiscale method for computing effective parameters of composite electromagnetic materials with memory effects

Authors:
V. A. Bokil, H. T. Banks, D. Cioranescu and G. Griso

Journal:
Quart. Appl. Math. **76** (2018), 713-738

MSC (2010):
Primary 74A40, 74A60, 78M40, 35L51, 35L52

DOI:
https://doi.org/10.1090/qam/1503

Published electronically:
March 12, 2018

MathSciNet review:
3855828

Full-text PDF

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Additional Information

Abstract: We consider the problem of computing (macroscopic) effective properties of composite materials that are mixtures of complex dispersive dielectrics described by polarization and magnetization laws. We assume that the micro-structure of the composite material is described by spatially periodic and deterministic parameters. Mathematically, the problem is to *homogenize* Maxwell’s equations along with constitutive laws that describe the material response of the micro-structure comprising the mixture, to obtain an equivalent effective model for the composite material with constant effective parameters. The novel contribution of this paper is the homogenization of a hybrid model consisting of the Maxwell partial differential equations along with ordinary (auxiliary) differential equations modeling the evolution of the polarization and magnetization, as a model for the complex dielectric material. This is in contrast to our previous work (2006) in which we employed a convolution in time of a susceptibility kernel with the electric field to model the delayed polarization effects in the dispersive material. In this paper, we describe the auxiliary differential equation approach to modeling material responses in the composite material and use the periodic unfolding method to construct a homogenized model.

References
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*Homogenization of the Maxwell equations in an anisotropic material*, Tech. Rep. LUTEDX/(TEAT-7104)/1-12/(2001), Department of Electroscience, Lund Institute of Technology, Sweden, 2001.
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*Homogenization of corrugated interfaces in electomagnetics*, Tech. Rep. LUTEDX/(TEAT-7122)/1-29/(2004), Department of Electroscience, Lund Institute of Technology, Sweden, 2004.
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*Effective constitutive parameters of structured chiral metamaterials*, Microwave and Optical Technology Letters, 48 (2006), pp. 1884–1886.
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*Stability and Phase Error Analysis of FD-TD in Dispersive Dielectrics*, IEEE Trans. Antennas Propagat., 42 (1994), pp. 62–69.
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*Homogenization of dispersive material parameters for Maxwell’s equations using a singular value decomposition*, Tech. Rep. LUTEDX/(TEAT-7124)/1-24/(2004), Department of Electroscience, Lund Institute of Technology, Sweden, 2004.
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*A floquet-bloch decomposition of Maxwell’s equations, applied to homogenization*, Tech. Rep. LUTEDX/(TEAT-7119)/1-27/(2003), Department of Electroscience, Lund Institute of Technology, Sweden, 2003.
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*Past achievements and future challenges in the development of three-dimensional photonic metamaterials*, Nature Photonics, 5 (2011), pp. 523–530.
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*Time-domain finite-difference and finite-element methods for Maxwell equations in complex media*, IEEE Trans. Antennas and Propagation **56** (2008), no. 8, 2150–2166. MR **2444879**, DOI https://doi.org/10.1109/TAP.2008.926767
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*Homogenization of the Maxwell equations at fixed frequency*, SIAM J. Appl. Math. **64** (2003), no. 1, 170–195. MR **2029130**, DOI https://doi.org/10.1137/S0036139902403366
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*A multiscale approach and a hybrid FE-BE algorithm for heterogeneous scattering of Maxwell’s equations*, J. Comput. Appl. Math. **319** (2017), 460–479. MR **3614238**, DOI https://doi.org/10.1016/j.cam.2017.01.017

References
- Grégoire Allaire,
*Homogenization and two-scale convergence*, SIAM J. Math. Anal. **23** (1992), no. 6, 1482–1518. MR **1185639**
- P. Baldus, M. Jansen, and D. Sporn,
*Ceramic fibers for matrix composites in high-temperature engine applications*, Science, 285 (1999), pp. 699–703.
- H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara,
*Homogenization of periodically varying coefficients in electromagnetic materials*, J. Sci. Comput. **28** (2006), no. 2-3, 191–221. MR **2272629**
- H. T. Banks, M. W. Buksas, and T. Lin,
*Electromagnetic material interrogation using conductive interfaces and acoustic wavefronts*, Frontiers in Applied Mathematics, vol. 21, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR **1787981**
- H. T. Banks, J. Catenacci, and A. Criner,
*Quantifying the degradation in thermally treated ceramic matrix composites, CRSC-TR15-10, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, September, 2015*, International Journal of Applied Electromagnetics and Mechanics, 52 (2016), pp. 1–22.
- H. Thomas Banks, Jared Catenacci, and Shuhua Hu,
*Estimation of distributed parameters in permittivity models of composite dielectric materials using reflectance*, J. Inverse Ill-Posed Probl. **23** (2015), no. 5, 491–509. MR **3403489**
- H. T. Banks, J. Catenacci, and S. Hu,
*Method comparison for estimation of distributed parameters in permittivity models using reflectance, CRSC-TR15-06, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, May, 2015*, Eurasian Journal of Mathematical and Computer Application, 3 (2015), pp. 4–23.
- Alain Bossavit, Georges Griso, and Bernadette Miara,
*Modelling of periodic electromagnetic structures bianisotropic materials with memory effects*, J. Math. Pures Appl. (9) **84** (2005), no. 7, 819–850 (English, with English and French summaries). MR **2144646**
- Liqun Cao, Keqi Li, Jianlan Luo, and Yaushu Wong,
*A multiscale approach and a hybrid FE-FDTD algorithm for 3D time-dependent Maxwell’s equations in composite materials*, Multiscale Model. Simul. **13** (2015), no. 4, 1446–1477. MR **3432141**
- Patrick Ciarlet Jr., Sonia Fliss, and Christian Stohrer,
*On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous multiscale method for Maxwell’s equations*, Comput. Math. Appl. **73** (2017), no. 9, 1900–1919. MR **3634959**
- D. Cioranescu, A. Damlamian, and G. Griso,
*The periodic unfolding method in homogenization*, SIAM J. Math. Anal. **40** (2008), no. 4, 1585–1620. MR **2466168**
- Doina Cioranescu and Patrizia Donato,
*An introduction to homogenization*, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR **1765047**
- P. Debye,
*Polar Molecules*, Chemical Catalog Co., New York, 1929.
- A. M. Efimov,
*Quantitative ir spectroscopy: applications to studying glass structure and properties*, Journal of Non-Crystalline Solids, 203 (1996), pp. 1–11.
- A. M. Efimov,
*Vibrational spectra, related properties, and structure of inorganic glasses*, Journal of Non-Crystalline Solids, 253 (1999), pp. 95–118.
- C. Engström and D. Sjöberg,
*A comparison of two numerical methods for homogenization of Maxwell’s equations*, Tech. Rep. LUTEDX/(TEAT-7121)/1-10/(2004), Department of Electroscience, Lund Institute of Technology, Sweden, 2004.
- E. C. Fear, P. M. Meaney, and M. A. Stuchly,
*Microwaves for breast cancer detection*, IEEE Potentials, (2003), pp. 12–18.
- T. Kashiwa and I. Fukai,
*A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization*, Microwave Opt. Technol. Lett., 3 (1990), pp. 203–205.
- T. Kashiwa, N. Yoshida, and I. Fukai,
*A treatment by the finite-difference time domain method of the dispersive characteristics associated with orientational polarization*, IEEE Trans. IEICE, 73 (1990), pp. 1326–1328.
- G. Kristensson,
*Homogenization of the Maxwell equations in an anisotropic material*, Tech. Rep. LUTEDX/(TEAT-7104)/1-12/(2001), Department of Electroscience, Lund Institute of Technology, Sweden, 2001.
- G. Kristensson,
*Homogenization of corrugated interfaces in electomagnetics*, Tech. Rep. LUTEDX/(TEAT-7122)/1-29/(2004), Department of Electroscience, Lund Institute of Technology, Sweden, 2004.
- Jichun Li and Yunqing Huang,
*Time-domain finite element methods for Maxwell’s equations in metamaterials*, Springer Series in Computational Mathematics, vol. 43, Springer, Heidelberg, 2013. MR **3013583**
- K. B. Liaskos, I. G. Stratis, and A. N. Yannacopoulos,
*A priori estimates for a singular limit approximation of the constitutive laws for chiral media in the time domain*, J. Math. Anal. Appl. **355** (2009), no. 1, 288–302. MR **2514468**
- Gabriel Nguetseng,
*A general convergence result for a functional related to the theory of homogenization*, SIAM J. Math. Anal. **20** (1989), no. 3, 608–623. MR **990867**
- H. Ohnabe, S. Masaki, M. Onozuka, K. Miyahara, and T. Sasa,
*Potential application of ceramic matrix composites to aero-engine components*, Composites Part A: Applied Science and Manufacturing, 30 (1999), pp. 489–496.
- O. Ouchetto, C.-W. Qiu, S. Zouhdi, L.-W. Li, and A. Razek,
*Homogenization of 3-D periodic bianisotropic metamaterials*, IEEE trans. Microwave Theory and Techniques, 54 (2006), pp. 3893–3898.
- O. Ouchetto, S. Zouhdi, A. Bossavit, G. Griso, and B. Miara,
*Effective constitutive parameters of periodic composites*, in 2005 European Microwave Conference, vol. 2, IEEE, 2005, pp. 2–pp.
- O. Ouchetto, S. Zouhdi, A. Razek, and B. Miara,
*Effective constitutive parameters of structured chiral metamaterials*, Microwave and Optical Technology Letters, 48 (2006), pp. 1884–1886.
- P. G. Petropoulos,
*Stability and Phase Error Analysis of FD-TD in Dispersive Dielectrics*, IEEE Trans. Antennas Propagat., 42 (1994), pp. 62–69.
- A. Sihvola,
*Electromagnetic mixing formulae and applications*, IEE Electromagnetic Waves Series, 47 (1999).
- A. Sihvola,
*Effective Permittivity of Mixtures: Numerical Validation by the FDTD method*, IEEE Transactions on Geosciences and Remote Sensing, 38 (2000), pp. 1303–1308.
- D. Sjöberg,
*Homogenization of dispersive material parameters for Maxwell’s equations using a singular value decomposition*, Tech. Rep. LUTEDX/(TEAT-7124)/1-24/(2004), Department of Electroscience, Lund Institute of Technology, Sweden, 2004.
- D. Sjöberg, C. Engström, G. Kristensson, D. J. N. Wall, and N. Wellander,
*A floquet-bloch decomposition of Maxwell’s equations, applied to homogenization*, Tech. Rep. LUTEDX/(TEAT-7119)/1-27/(2003), Department of Electroscience, Lund Institute of Technology, Sweden, 2003.
- C. M. Soukoulis and M. Wegener,
*Past achievements and future challenges in the development of three-dimensional photonic metamaterials*, Nature Photonics, 5 (2011), pp. 523–530.
- Fernando L. Teixeira,
*Time-domain finite-difference and finite-element methods for Maxwell equations in complex media*, IEEE Trans. Antennas and Propagation **56** (2008), no. 8, 2150–2166. MR **2444879**
- Niklas Wellander and Gerhard Kristensson,
*Homogenization of the Maxwell equations at fixed frequency*, SIAM J. Appl. Math. **64** (2003), no. 1, 170–195. MR **2029130**
- Ya Zhang, Liqun Cao, Walter Allegretto, and Yanping Lin,
*Multiscale numerical algorithm for 3-D Maxwell’s equations with memory effects in composite materials*, Int. J. Numer. Anal. Model. Ser. B **1** (2010), no. 1, 41–57. MR **2837399**
- Yongwei Zhang, Liqun Cao, Yangde Feng, and Wu Wang,
*A multiscale approach and a hybrid FE-BE algorithm for heterogeneous scattering of Maxwell’s equations*, J. Comput. Appl. Math. **319** (2017), 460–479. MR **3614238**

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Additional Information

**V. A. Bokil**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

MR Author ID:
741294

Email:
bokilv@math.oregonstate.edu

**H. T. Banks**

Affiliation:
CRSC, North Carolina State University, Raleigh, North Carolina 27695-8205

MR Author ID:
194993

Email:
htbanks@ncsu.edu

**D. Cioranescu**

Affiliation:
Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, 4, Place Jussieu, 75252 Paris, Cedex 05, France

MR Author ID:
49540

Email:
cioran@ann.jussieu.fr

**G. Griso**

Affiliation:
Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, 4, Place Jussieu, 75252 Paris, Cedex 05, France

MR Author ID:
250326

Email:
griso@ann.jussieu.fr

Keywords:
Maxwell’s equations,
periodic unfolding,
homogenization,
dispersive media,
auxiliary differential equation.

Received by editor(s):
February 5, 2018

Published electronically:
March 12, 2018

Article copyright:
© Copyright 2018
Brown University