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A leap-frog meshless method with radial basis functions for simulating electromagnetic wave splitter and rotator. (English) Zbl 1464.78025

Summary: In this paper, we develop a meshless method by using radial basis functions (RBFs) to solve time domain Maxwell’s equations resulting from simulating wave propagation in electromagnetic wave splitter and rotator devices. To simulate wave propagation in these devices, we introduce a perfectly matched layer (PML) to reduce the unbounded physical domain problem to a bounded domain problem. Using PML leads to a multi-physics problem with different governing equations in different subdomains, which makes the simulation quite challenging. By following the idea of leap-frog FDTD scheme, we develop the leap-frog RBF meshless method to solve these complicated coupled modeling equations. Extensive numerical results by using multiquadric RBF and Gaussian RBF are presented to demonstrate the effectiveness of our meshless method.

MSC:

78M22 Spectral, collocation and related methods applied to problems in optics and electromagnetic theory
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
65D12 Numerical radial basis function approximation
78A25 Electromagnetic theory (general)

Software:

Matlab
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[1] Bokil, V. A.; Gibson, N. L., Analysis of spatial high-order fnite difference methods for Maxwell’s equations in dispersive media, IMA J Numer Anal, 32, 926-956 (2012) · Zbl 1253.78048
[2] Brenner, S. C.; Gedicke, J.; Sung, L. Y., An adaptive P1 finite element method for two-dimensional transverse magnetic time harmonic Maxwell’s equations with general material properties and general boundary conditions, J Sci Comput, 68, 2, 848-863 (2016) · Zbl 1373.78416
[3] Buhmann, M. D., Radial basis functions: theory and implementations (2003), Cambridge University Press · Zbl 1038.41001
[4] Cassier, M.; Joly, P.; Kachanovska, M., Mathematical models for dispersive electromagnetic waves: an overview, Comput Math Appl, 74, 2792-2830 (2017) · Zbl 1397.78004
[5] Castaldi, G.; Savoia, S.; Galdi, V.; Alu, A.; Engheta, N., PT metamaterials via complexcoordinate transformation optics, Phys Rev Lett, 110, 173901 (2013)
[6] Chen, H.; Chen, C. T., Electromagnetic wave manipulation by layered systems using the transformation media concept, Phys Rev B, 78, 054204 (2008)
[7] Chen, H.; Hou, B.; Chen, S.; Ao, X.; Wen, W.; Chan, C. T., Design and experimental realization of a broadband transformationmedia field rotator atmicrowave frequencies, Phys Rev Lett, 102, 18, 183903 (2009)
[8] Chen, W.; Fu, Z.-J.; Chen, C. S., Recent advances in radial basis function collocation methods (2014), Springer · Zbl 1282.65160
[9] Chen, Y.; Gottlieb, S.; Heryudono, A.; Narayan, A., A reduced radial basis function method for partial differential equations on irregular domains, J Sci Comput, 66, 67-90 (2016) · Zbl 1338.65252
[10] Dehghan, M.; Abbaszadeh, M., Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method, Appl Numer Math, 137, 252-273 (2019) · Zbl 1412.65070
[11] Dehghan, M.; Abbaszadeh, M., The solution of nonlinear Green-Naghdi equation arising in water sciences via a meshless method which combines moving kriging interpolation shape functions with the weighted essentially non oscillatory method, Commun Nonlinear Sci Numer Simul, 68, 220-239 (2019) · Zbl 1524.65452
[12] Dehghan, M.; Abbaszadeh, M., The use of proper orthogonal decomposition (POD) meshless RBF FD technique to simulate the shallow water equations, J Comput Phys, 351, 478-510 (2017) · Zbl 1380.65301
[13] Dehghan, M.; Haghjoo-Saniji, M., The local radial point interpolation meshless method for solving Maxwell equations, Eng Comput, 33, 4, 897-918 (2017)
[14] Dehghan, M.; Mohammadi, V., A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrdinger equations using an explicit time discretization: runge kutta method, Comput Phys Comm, 217, 23-34 (2017) · Zbl 1411.65108
[15] Dehghan, M.; Salehi, R., A meshless local Petrov-Galerkin method for the time-dependent Maxwell equations, J Comput Appl Math, 268, 93-110 (2014) · Zbl 1293.65128
[16] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math Comput Simul, 79, 3, 700-715 (2008) · Zbl 1155.65379
[17] Fasshauer, G. E., Meshfree approximation methods with MATLAB (2007), World Scientific · Zbl 1123.65001
[18] Fornberg, B.; Flyer, N., Solving PDEs with radial basis functions, Acta Numer, 24, 215-258 (2015) · Zbl 1316.65073
[19] Fornberg, B.; Flyer, N., A primer on radial basis functions with applications to the geosciences, Philadelphia, PA (2015), Society for Industrial and Applied Mathematics · Zbl 1358.86001
[20] Gao, W. W.; Wu, Z. M., Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation, Appl Math Comput, 253, 377-386 (2015) · Zbl 1338.65205
[21] Greenleaf, A.; Kurylev, Y.; Lassas, M.; Uhlmann, G., Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev, 51, 3-33 (2009) · Zbl 1158.78004
[22] Guo, J.; Jung, J. H., A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method, Appl Numer Math, 112, 27-50 (2017) · Zbl 1354.65177
[23] Hao, Y.; Mittra, R., FDTD modeling of metamaterials: theory and applications, Artech House Publishers (2008)
[24] Hon, Y. C.; Cheung, K. F.; Mao, X. Z.; Kansa, E. J., Multiquadric solution for shallow water equations, J Hydraul Eng, 125, 5, 524-533 (1999)
[25] Huang, Z.; Xiao, J.; Boyd, J. P., Adaptive radial basis function and hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter, J Comput Phys, 281, 269-284 (2015) · Zbl 1352.65566
[26] Iske, A., Multiresolution methods in scattered data modelling (2004), Springer · Zbl 1057.65004
[27] Jiang, W. X.; Cui, T. J.; Cheng, Q.; Chin, J. Y.; Yang, X. M.; Liu, R.; Smith, D. R., Design of arbitrarily shaped concentrators based on conformally optical transformation of nonuniform rational b-spline surfaces, Appl Phys Lett, 92, 264101 (2008)
[28] Kansa, E. J., Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics, part II: solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput Math Appl, 19, 147-161 (1990) · Zbl 0850.76048
[29] Lai, S. J.; Wang, B. Z.; Duan, Y., Meshless radial basis function method for transient electromagnetic computations, IEEE Trans Magn, 44, 10, 2288-2295 (2008)
[30] Gia, Q. T.L.; Sloan, I.; Wendland, H., Multiscale RBF collocation for solving PDEs on spheres, Numer Math, 121, 99-125 (2012) · Zbl 1246.65222
[31] Li, J., Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions, Adv Comput Math, 23, 21-30 (2005) · Zbl 1069.65107
[32] Li, J.; Chen, Y., Computational partial differential equations using MATLAB (2008), CRC Press
[33] Li, J.; Cheng, A. H.-D.; Chen, C. S., A comparison of effciency and error convergence of multiquadric collocation method and finite element method, Eng Anal Bound Elem, 27, 251-257 (2003) · Zbl 1044.76050
[34] Li, J.; Hesthaven, J. S., Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J Comput Phys, 258, 915-930 (2014) · Zbl 1349.74327
[35] Li, J.; Huang, Y., Time-domain finite element methods for Maxwell’sequations in metamaterials, Springer Ser. Comput. Math., 43 (2013), Springer: Springer New York · Zbl 1304.78002
[36] Li, J.; Huang, Y.; Yang, W.; Wood, A., Mathematical analysis and time-domain finite element simulation of carpet cloak, SIAM J Appl Math, 74, 4, 1136-1151 (2014) · Zbl 1302.65252
[37] Li, J.; Nan, B., Simulating backward wave propagation in metamaterial with radial basis functions, Results Appl Math, 2, 100009 (2019) · Zbl 1451.78043
[38] Li, J.; Feng, X.; He, Y., RBF-based meshless local Petrov- Galerkin method for the multi-dimensional convection-diffusion-reaction equation, Eng Anal Bound Elem, 98, 46-53 (2019) · Zbl 1404.65176
[39] Li, J.; Gao, Z.; Feng, X.; He, Y., Method of order reduction for the high-dimensional convection-diffusion-reaction equation with robin boundary conditions based on MQ RBF-FD, Int J Comput Methods (2019)
[40] Li, J.; Shi, C.; Shu, C. W., Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials, Comput Math Appl, 73, 1760-1780 (2017) · Zbl 1370.74143
[41] Li, J.; Zhai, S.; Weng, Z.; Feng, X., H-adaptive RBF-FD method for the high-dimensional convection-diffusion equation, Int Commun Heat Mass Transf, 89, 139-146 (2017)
[42] Li, N.; Su, H.; Gui, D.; Feng, X., Multiquadric RBF-FD method for the convection-dominated diffusion problems base on Shishkin nodes, Int J Heat Mass Transf, 118, 734-745 (2018)
[43] Petras, A.; Ling, L.; Ruuth, S. J., An RBF-FD closest point method for solving PDEs on surfaces, J Comput Phys, 370, 43-57 (2018) · Zbl 1395.65029
[44] Qiao, Y.; Zhao, J.; Feng, X., A compact integrated RBF method for time fractional convection-diffusion-reaction equations, Comput Math Appl, 77, 9, 2263-2278 (2019) · Zbl 1442.65297
[45] Rashidinia, J.; Fasshauer, G. E.; Khasi, M., A stable method for the evaluation of gaussian radial basis function solutions of interpolation and collocation problems, Comput Math Appl, 72, 1, 178-193 (2016) · Zbl 1443.65110
[46] Schaback, R.; Wendland, H., Kernel techniques: from machine learning to meshless methods, Acta Numer, 15, 543-639 (2006) · Zbl 1111.65108
[47] Shankar, V.; Wright, G. B.; Kirby, R. M.; Fogelson, A. L., A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces, J Sci Comput, 63, 3, 745-768 (2014) · Zbl 1319.65079
[48] Wang, W.; Lin, L.; Ma, J.; Wang, C.; Cui, J.; Du, C., Electromagnetic concentrators with reduced material parameters based on coordinate transformation, Opt Express, 16, 11431-11437 (2008)
[49] Wendland, H., Scattered data approximation (2005), Cambridge University Press · Zbl 1075.65021
[50] Werner, D. H.; Kwon, D. H., Transformation electromagnetics and metamaterials: fundamental principles and applications (2013), Springer: Springer New York
[51] Yang, S.; Yu, Y.; Chen, Z.; Ponomarenko, S., A time-domain collocation meshless method with local radial basis functions for electromagnetic transient analysis, IEEE Trans Antennas Prop, 62, 10, 5334-5338 (2014) · Zbl 1371.78328
[52] Yang, W.; Li, J.; Huang, Y.; He, B., Developing finite element methods for simulating transformation optics devices with metamaterials, Commun Comput Phys, 25, 135-154 (2019) · Zbl 1473.78012
[53] Young, D. L.; Chen, C. S.; Wong, T. K., Solution of Maxwell’s equations using the MQ method, CMC-Comput Mater Contin, 2, 4, 267-276 (2005)
[54] Zhang, Y.; Nguyen, D. D.; Du, K.; Xu, J.; Zhao, S., Time-domain numerical solutions of Maxwell interface problems with discontinuous electromagnetic waves, Adv Appl Math Mech, 8, 353-385 (2016) · Zbl 1488.65318
[55] Zheng, H.; Yao, G.; Kuo, L. H.; Li, X., On the selection of a good shape parameter of the localized method of approximated particular solutions, Adv Appl Math Mech, 10, 4, 896-911 (2018) · Zbl 1488.65696
[56] Zhou, X.; Hon, Y. C.; Li, J., Overlapping domain decomposition method by radial basis functions, Appl Numer Math, 44, 241-255 (2003) · Zbl 1013.65134
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