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A High-Order Time Domain Discontinuous Galerkin Method with Orthogonal Tetrahedral Basis for Electromagnetic Simulations in 3-D Heterogeneous Conductive Media

Part of: Partial differential equations, boundary value problems Numerical linear algebra

Published online by Cambridge University Press:  08 March 2017

Jun Yang ,
Wei Cai  and
Xiaoping Wu
Jun Yang
Affiliation:
Laboratory of Seismology and Physics of Earth's Interior, School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Wei Cai*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, Chin Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Xiaoping Wu
Affiliation:
Laboratory of Seismology and Physics of Earth's Interior, School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China
*
*Corresponding author. Email addresses:[email protected], [email protected] (W. Cai)
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Abstract

We present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

Keywords

Maxwell's equationstime domaindiscontinuous Galerkin methodorthonormal polynomial basis functionstetrahedral elementsUPMLparallelization

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65F35: Matrix norms, conditioning, scaling 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 4 , April 2017 , pp. 1065 - 1089
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bohren, C. F. and Huffman, D. R.. Absorption and Scattering of Light by Small Particles, John Wiley & Sons, 2008.Google Scholar
[2] Cai, W.. Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport, Cambridge University Press, 2013.Google Scholar
[3] Cockburn, B. and Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comp., 52 (1989), pp. 411–435.Google Scholar
[4] Cockburn, B. and Shu, C.-W.. The Runge-Kutta local projection P1-Discontinuous Galerkin finite element method for scalar conservation laws, Mathematical Modelling and Numerical Analysis, 25 (1991), pp. 337–361.Google Scholar
[5] Cockburn, B., Hou, S. and Shu, C.-W.. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comp., 54 (1990), pp. 545–581.Google Scholar
[6] Cockburn, B. and Shu, C.-W.. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199–224.CrossRefGoogle Scholar
[7] Dolean, V., Fahs, H., Fezoui, L., and Lanteri, S.. Locally implicit discontinuous Galerkin method for time domain electromagnetics, J. Comput. Phys., 229 (2010), pp. 512–526.CrossRefGoogle Scholar
[8] Gedney, S. D.. An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices, IEEE Trans. Antennas Propagat., 44 (1996), pp. 1630–1639.Google Scholar
[9] Geuzaine, C. and Remacle, J. F.. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numr. Meth. Engng, 79 (2009), pp. 1309–1331.Google Scholar
[10] Ji, X., Lu, T., Cai, W. and Zhang, P.. Discontinuous Galerkin time domain (DGTD) methods for the study of 2-D waveguide-coupled microring resonators, Journal of lightwave technology, 23 (2005), pp. 3864–3874.Google Scholar
[11] Jin, J.-M.. The Finite Element Method in Electromagnetics, John Wiley & Sons, 2014.Google Scholar
[12] Kabakian, A. V., Shankar, V. and Hall, W. F.. Unstructured grid-based discontinuous Galerkin method for broadband electromagnetic simulations, J. Sci. Comput., 20 (2004), pp. 405–431.CrossRefGoogle Scholar
[13] Keast, P.. Moderate-degree tetrahedral quadrature formulas, Computer Methods in Applied Mechanics and Engineering, 55 (1986), pp. 339–348.CrossRefGoogle Scholar
[14] Kim, J. and Gopinath, A.. Simulation of a metamaterial containing cubic high dielectric resonators, Phys. Rev. B, 76 (2007), 115126 CrossRefGoogle Scholar
[15] Kopriva, D.A., Woodruff, S. L. and Hussaini, M. Y.. Computation of electromagnetic scattering with a non-onforming discontinuous spectral element method, Int. J. Numr. Meth. Engng, 53 (2002), pp. 105–122.CrossRefGoogle Scholar
[16] Li, J., Chen, Y. and Liu, Y.. Mathematical simulation of metamaterial solar cells, Advances in Applied Mathematics and Mechanics, 3 (2011), pp. 702–715.CrossRefGoogle Scholar
[17] Li, J., Waters, J.W. and Machorro, E. A.. An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwell's equations in metamaterials, Computer Methods in Applied Mechanics and Engineering, 223 (2012), pp. 43–54.CrossRefGoogle Scholar
[18] Li, J. and Hesthaven, J. S.. Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258(2014), pp. 915–930.Google Scholar
[19] Lu, T., Cai, W. and Zhang, P.. Discontinuous Galerkin Time-Domain Method for GPR Simulation in Dispersive Media, IEEE Trans Geoscience and Remote Sensing, 43 (2005), pp. 72–80.Google Scholar
[20] Lu, T., Zhang, P. and Cai, W.. Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions, J. Comput. Phys., 200 (2004), pp. 549–580.CrossRefGoogle Scholar
[21] Nédélec, J. C.. Mixed finite elements in , Numerische Mathematik, 35 (1980), pp. 315–341.CrossRefGoogle Scholar
[22] Reed, W. H. and Hill, T. R.. Triangular mesh methods for the neutron transport equation, Technical Report, LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[23] Russer, P.. Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, Artech House, 2003.Google Scholar
[24] Sacks, Z. S., Kingsland, D. M., Lee, R. and Lee, J. F.. A perfectly matched anisotropic absorber for use as an absorbing boundary condition, IEEE trans. Antennas Propagat., 43 (1995), pp. 1460–1463.Google Scholar
[25] Simpson, J. J.. Current and future applications of 3-D global earth-ionosphere models based on the full-vector Maxwell's equations FDTD method, Surveys in Geophysics, 30 (2009), pp. 105–130.Google Scholar
[26] Streich, R.. Controlled-source electromagnetic approaches for hydrocarbon exploration and monitoring on land, Surveys in Geophysics, 37 (2016), pp. 47–80.CrossRefGoogle Scholar
[27] Taflove, A. and Hagness, S. C.. Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech house publishers, 2005.Google Scholar
[28] Warburton, T.. Application of the discontinuous Galerkin method to Maxwell's equations using unstructured polymorphic hp-finite elements, In Discontinuous Galerkin Methods (pp. 451–458), Springer Berlin Heidelberg, 2000.CrossRefGoogle Scholar
[29] Ward, S.H. and Hohmann, G.W.. Electromagnetic theory for geophysical applications, in Electromagnetic Methods in Applied Geophysics, Vol. 1, chap. 4, pp. 131–311, ed. Nabighian, M.N., Society of Exploration Geophysicists, 1991.Google Scholar
[30] Xie, Z., Wang, B. and Zhang, Z.. Space-Time discontinuous Galerkin method for Maxwell's equations, Commun. Comput. Phys., 14 (2013), pp. 916–939.CrossRefGoogle Scholar
[31] Xie, Z., Wang, J., Wang, B. and Chen, C.. Solving Maxwell's equation in meta-materials by a CG-DG method, Commun. Comput. Phys., 19 (2016), pp. 1242–1264.Google Scholar
[32] Xin, J. and Cai, W.. Well-conditioned orthonormal hierarchical bases on simplicial elements, J. Sci. Comput., 50 (2012), 446–461.Google Scholar
[33] Xin, J., Guo, N. and Cai, W.. On the construction of well-conditioned hierarchical bases for tetrahedral H(curl)-conforming Nédélec elements, J. Comput. Math. 29 (2011), 526–542.CrossRefGoogle Scholar
[34] Yee, K. S.. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propagat., 14 (1966), pp. 302–307.Google Scholar
[35] Youssef, N. N.. Radar cross section of complex targets, Proceedings of the IEEE, 77 (1989), pp. 722–734.Google Scholar