PIER B
 
Progress In Electromagnetics Research B
ISSN: 1937-6472
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 49 > pp. 197-213

THE SUBGRID MODELING FOR MAXWELL'S EQUATIONS WITH MULTISCALE ISOTROPIC RANDOM CONDUCTIVITY AND PERMITTIVITY

By E. P. Kurochkina and O. N. Soboleva

Full Article PDF (268 KB)

Abstract:
The effective coefficients for Maxwell's equations in the frequency domain are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The correlated fields of conductivity and permeability are approximated by Kolmogorov's multiplicative continuous cascades with a lognormal probability distribution. The wavelength is assumed to be large as compared with the scale of heterogeneities of the medium. The permittivity ε(x) and the electric conductivity σ(x) satisfy the condition σ(x)/(ωε(x)) < 1, where ω is the cyclic frequency. The theoretical results obtained in the paper are compared with the results from direct 3D numerical simulation.

Citation:
E. P. Kurochkina and O. N. Soboleva, "The Subgrid Modeling for Maxwell's Equations with Multiscale Isotropic Random Conductivity and Permittivity," Progress In Electromagnetics Research B, Vol. 49, 197-213, 2013.
doi:10.2528/PIERB12120308

References:
1. Taflove, A. and S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method, Artech House, Norwood, 2005.

2. Mikhailenko, B. G. and O. N. Soboleva, "Mathematical modeling of seismomagnetic effects arising in the seismic wave motion in Earthes's constant magnetic field," Applied Mathematics Letters, Vol. 10, 47-55, 1997.
doi:10.1016/S0893-9659(97)00033-5

3. Mastryukov, A. F. and B. G. Mikhilenko, "Numerical solution of Maxwell's equations for anisotropic media using the Laguerre transform," Russion Geology and Geophysics, Vol. 49, 621-627, 2008.
doi:10.1016/j.rgg.2007.12.011

4. Wellander, N., "Homogenization of the Maxwell equations," Application of Mathematics,, Vol. 46, No. 1, 29-51, 2001.
doi:10.1023/A:1013727504393

5. Jones, J. and B. Lee, "A multigrid method for variable coefficient Maxwell's equations," SIAM J. Sci. Comput., Vol. 27, No. 5, 1689-1708, 2006.
doi:10.1137/040608283

6. Dagan, G., "Higher-oder correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity Distribution," Transport in Porous Media, Vol. 12, 279-290, 1993.
doi:10.1007/BF00624462

7. Yukalov, V. I., "Self-semilar approximations for strongly interacting systems," Physica A, Vol. 167, 833-860, 1990.
doi:10.1016/0378-4371(90)90294-3

8. Yukalov, V. I. and S. Gluzman, "Self-semilar bootstrap of divergent series," Phys. Rev. E, Vol. 55, 6552-6570, 1997.
doi:10.1103/PhysRevE.55.6552

9. Gluzman, S. and D. Sornette, "Self-similar approximants of the permeability in heterogeneous porous media from moment equation expansions," Transport in Porous Media, Vol. 71, 75-97, 2008.
doi:10.1007/s11242-007-9112-9

10. Germano, M., P. Moin, W. Piomelly, and H. Cabot, "A dynamic subgrid scale eddy viscosity model," Phys. Fluids A, Vol. 3, 1760-1765, 1991.
doi:10.1063/1.857955

11. Germano, M. and P. Sagat, arge Eddy Simmulation for Incompressible Flow, Sprimger, Berlin Heidelberg, 1998.

12. Hoffman, J., "Dynamic subgrid modelling for time dependent convection-di®usion-reaction equations with fractal solutions," International Journal for Numerical Methods in Fluids, Vol. 40, No. 3-4, 583-592, 2002.
doi:10.1002/fld.304

13. Sahimi, M., "Flow phenomena in rocks: From continuum models, to fractals, percolation, cellular automata, and simulated annealing," Reviews of Modern Physics, Vol. 65, No. 4, 1393-1534, 1993.
doi:10.1103/RevModPhys.65.1393

14. Krylov, S. S. and V. A. Lyubchich, "The apparent resistivity scaling and fractal structure of an iron formation," Izvestiya Physics of the Solid Earth, Vol. 38, 1006-1012, 2002.

15. Bekele, A., H. W. Hudnall, J. J. Daigle, A. Prudente, and M. Wolcott, "Scale dependent variability of soil electrical conductivity by indirect measures of soil properties," Journal of Terramechanics, Vol. 42, 339-351, 2005.
doi:10.1016/j.jterra.2004.12.004

16. Kurochkina, E. P. and O. N. Soboleva, "Effective coefficients of quasi-steady Maxwell's equations with multiscale isotropic random conductivity," Physica A, Vol. 39, 231-244, 2011.
doi:10.1016/j.physa.2010.09.028

17. Kuz'min, G. A. and O. N. Soboleva, "Subgrid modeling of filtration in porous self-similar media," Journal Appl. Mech. Tech. Phys., Vol. 43, 583-592, 2002.
doi:10.1023/A:1016057832296

18. Kolmogorov, A. N., "A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number," Journal Fluid Mech., Vol. 13, No. 1, 82-85, 1962.
doi:10.1017/S0022112062000518

19. Gnedenko, B. V. and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, English Trans., K. L. Chung, Addison-Wesley, Cambridge , 1954.

20. Feller, W., An Introduction to Probability Theory and Its Applications, 3rd Ed., Vol. 1, Wiley, New York, 1968.

21. Koshljakov, N. S., M. M. Smirnov, and E. B. Gliner, Differential Equations of Mathematical Physics, Fizmatgiz, Moscow, 1962.

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

23. Ogorodnikov, V. A. and S. M. Prigarin, Numerical Modeling of Random Processes and Fields: Algorithms and Applications, VSP, Utrecht, 1996.

24. Lebedev, V. I., "Difference analogies of orthogonal decompositions of basic di®eretial operators and some boundary value problems," Journal Comut. Maths. Math. Phys., Vol. 3, No. 3, 449-465, 1964 (in Russia).

25. Davydycheva, S., V. Drushkin, and T. Habashy, "An efficient finite-difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media," Geophysics, Vol. 68, No. 5, 1525-1530, 2003.
doi:10.1190/1.1620626

26. Modersitzki, J., G. Sleijpen, and H. van der Vorst, "Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems," Matrix Anal. Appl., Vol. 22, No. 3, 726-751, 2000.


© Copyright 2010 EMW Publishing. All Rights Reserved