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An Adaptive Perfectly Matched Layer Method for Multiple Cavity Scattering Problems

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  01 February 2016

Xinming Wu  and
Weiying Zheng
Xinming Wu*
Affiliation:
The Key Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China
Weiying Zheng
Affiliation:
NCMIS, LSEC, ICMSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P.R. China
*
*Corresponding author. Email addresses:wuxinming@fudan.edu.cn (X. Wu), zwy@lsec.cc.ac.cn (W. Zheng)
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Abstract

A uniaxial perfectly matched layer (PML) method is proposed for solving the scattering problem with multiple cavities. By virtue of the integral representation of the scattering field, we decompose the problem into a system of single-cavity scattering problems which are coupled with Dirichlet-to-Neumann maps. A PML is introduced to truncate the exterior domain of each cavity such that the computational domain does not intersect those for other cavities. Based on the a posteriori error estimates, an adaptive finite element algorithm is proposed to solve the coupled system. The novelty of the proposed method is that its computational complexity is comparable to that for solving uncoupled single-cavity problems. Numerical experiments are presented to demonstrate the efficiency of the adaptive PML method.

Keywords

Uniaxial perfectly matched layermultiple cavity scatteringadaptive finite elementa posteriori error estimate

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N50: Mesh generation and refinement
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 2 , February 2016 , pp. 534 - 558
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Ammari, H., Bao, G. and Wood, A., An integral equation method for the electromagnetic scattering from cavities, Math. Meth. Appl. Sci. 23 (2000), 1057–1072.Google Scholar
[2]Ammari, H., Bao, G. and Wood, A., Analysis of the electromagnetic scattering from a cavity, Jpn. J. Indus. Appl. Math. 19 (2001), 301–308.CrossRefGoogle Scholar
[3]Babuška, I. and Aziz, A., Survey lectures on mathematical foundations of the finite element method. in The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, ed. by Aziz, A., Academic Press, New York, 1972,5–359.Google Scholar
[4]Bao, G., Gao, J. and Li, P., Analysis of direct and inverse cavity scattering problems, Numer. Math. Theory Methods Appl. 4 (2011), 419–442.Google Scholar
[5]Bao, G., Gao, J., Lin, J. and Zhang, W., Mode matching for the electromagnetic scattering from three dimensional large cavities, IEEE Trans. Antennas Propag. 60 (2012), 1–7.CrossRefGoogle Scholar
[6]Bao, G. and Sun, W., A fast algorithm for the electromagnetic scattering from a large cavity, SIAM J. Sci. Comput. 27 (2005), 553–574.CrossRefGoogle Scholar
[7]Bao, G. and Wu, H., Convergence analysis of the PML problems for time-harmonic Maxwell's equations, SIAM J. Numer. Anal. 43 (2005), 2121–2143.Google Scholar
[8]Bao, G. and Zhang, W., An improved mode matching method for large cavities, IEEE Antennas Wirel. Propag. Lett. 4 (2005), 393–396.Google Scholar
[9]Bérenger, J.P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), 185–200.Google Scholar
[10]Bramble, J. and Pasciak, J., Analysis of a Cartesian PML approximation to acoustic scattering problems in ℝ2 and ℝ3, J. Comput. Appl. Math. 247 (2013), 209–230.Google Scholar
[11]Chen, J. and Chen, Z., An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems, Math. Comp. 77 (2008), 673–698.CrossRefGoogle Scholar
[12]Chen, Z. and Liu, X., An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal. 43 (2005), 645–671.Google Scholar
[13]Chen, Z. and Wu, H., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAMJ. Numer. Anal. 41 (2003), 799C826.Google Scholar
[14]Chen, Z. and Wu, X., An adaptive uniaxial perfectly matched layer method for time-harmonic scattering problems, Numer. Math. Theor. Meth. Appl. 1 (2008), 113–137.Google Scholar
[15]Chen, Z. and Zheng, W, Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media, SIAM J. Numer. Anal. 48 (2010), 2158–2185.Google Scholar
[16]Du, K., Two transparent boundary conditions for the electromagnetic scattering from two-dimensional overfilled cavities, J. Comput. Phys. 230 (2011), 5822–5835.CrossRefGoogle Scholar
[17]Grote, M. and Kirsch, C., Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys. 201 (2004), 630–650.Google Scholar
[18]Huang, J. and Wood, A., Numerical simulation of electromagnetic scattering induced by an overfilled cavity in the ground plane, IEEE Antennas Wirel. Propag. Lett. 4 (2005), 224–228.Google Scholar
[19]Huang, J., Wood, A. and Havrilla, M., A hybrid finite element-Laplace transform method for the analysis of transient electromagnetic scattering by an overfilled cavity in the ground plane, Commun. Comput. Phys. 5 (2009), 126–141.Google Scholar
[20]Jiang, X. and Zheng, W., Adaptive perfectly matched layer method for multiple scattering problems, Comput. Methods Appl. Mech. Engrg. 201 (2012), 42–52.Google Scholar
[21]Jin, J.M., Liu, J., Lou, Z. and Liang, S., A fully high-order finite-element simulation of scattering by deep cavities, IEEE Trans. Antennas Propag. 51 (2003), 2420–2429.Google Scholar
[22]Jin, J.M. and Volakis, J.L., A hybrid finite element method for scattering and radiation by micro strip patch antennas and arrays residing in a cavity, IEEE Trans. Antennas Propag. 39 (1991), 1598–1604.Google Scholar
[23]Li, P. and Wood, A., A two-dimensional Helmholtz equation solution for the multiple cavity scattering problem, J. Comput. Phys. 240 (2013), 100–120.Google Scholar
[24]Li, P., Wu, H. and Zheng, W., An overfilled cavity problem for Maxwells equations, Math. Methods Appl. Sci. 35 (2012), 1951–1979.Google Scholar
[25]Liu, J. and Jin, J.M., A special higher order finite-element method for scattering by deep cavities, IEEE Trans. Antennas Propag. 48 (2000), 694–703.Google Scholar
[26]Schatz, A.H., An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959–962.Google Scholar
[27]Teixeira, F.L. and Chew, W.C., Advances in the theory of perfectly matched layers, in: Chew, W.C.et al. (Eds.), Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, 2001, 283C346.Google Scholar
[28]Turkel, E. and Yefet, A., Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math. 27 (1998), 533C557.CrossRefGoogle Scholar
[29]Van, T. and Wood, A., Finite element analysis for 2-D cavity problem, IEEE Trans. Antennas Propag. 51 (2003), 1C8.CrossRefGoogle Scholar
[30]Wang, Y., Du, K. and Sun, W., A second-order method for the electromagnetic scattering from a large cavity, Numer. Math. Theory Methods Appl. 1 (2008), 357–382.Google Scholar
[31]Wood, A., Analysis of electromagnetic scattering from an overfilled cavity in the ground plane, J. Comput. Phys. 215 (2006), 630–641.CrossRefGoogle Scholar
[32]Wood, W. and Wood, A., Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane, IEEE Trans. Antennas Propag. 47 (1999), 1318–1322.Google Scholar
[33]Zhang, D., Ma, F. and Dong, H., A finite element method with rectangular perfectly matched layers for the scattering from cavities, J. Comput. Math. 27 (2009), 812–834.Google Scholar
[34]Zhao, M., Qiao, Z. and Tang, T., A fast high order method for electromagnetic scattering by large open cavities, J. Comput. Math. 29 (2011), 287–304.CrossRefGoogle Scholar