Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T02:12:30.618Z Has data issue: false hasContentIssue false
  • English
  • Français

A High-Order Discontinuous Galerkin Method for the Two-Dimensional Time-Domain Maxwell’s Equations on Curved Mesh

Published online by Cambridge University Press:  21 December 2015

Hongqiang Lu ,
Yida Xu ,
Yukun Gao ,
Wanglong Qin  and
Qiang Sun
Hongqiang Lu*
Affiliation:
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Yida Xu
Affiliation:
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Yukun Gao
Affiliation:
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Wanglong Qin
Affiliation:
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Qiang Sun
Affiliation:
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
*
*Corresponding author. Email:[email protected] (H. Q. Lu)
Get access

Abstract

In this paper, a DG (Discontinuous Galerkin) method which has been widely employed in CFD (Computational Fluid Dynamics) is used to solve the two-dimensional time-domain Maxwell’s equations for complex geometries on unstructured mesh. The element interfaces on solid boundary are treated in both curved way and straight way. Numerical tests are performed for both benchmark problems and complex cases with varying orders on a series of grids, where the high-order convergence in accuracy can be observed. Both the curved and the straight solid boundary implementation can give accurate RCS (Radar Cross-Section) results with sufficiently small mesh size, but the curved solid boundary implementation can significantly improve the accuracy when using relatively large mesh size. More importantly, this CFD-based high-order DG method for the Maxwell’s equations is very suitable for complex geometries.

Keywords

65Z05Maxwell’s equationsdiscontinuous Galerkin methodcurved meshradar cross-section
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 1 , February 2016 , pp. 104 - 116
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Yee, K. S., Numerical solution of initial value problems involving Maxwell’s equation in isotropic media, IEEE Trans. Antennas Prop., 14(3) (1966), pp. 302–307.Google Scholar
[2]Taflove, A. and Hagness, S. C., Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second ed., Artech, Norwood, MA, 2000.Google Scholar
[3]Cangellaris, A. C. and Wright, D. B., Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena, IEEE Trans. Antennas Prop., 39(10) (1991), pp. 1518–1525.CrossRefGoogle Scholar
[4]Cohen, G. and Monk, P., Mur-Nedelec finite element schemes for Maxwell’s equations, Comput. Meth. Appl. Mech. Eng., 169(3-4) (1999).Google Scholar
[5]Piperno, S., Remaki, M. and Fezoui, L., A non-diffusive finite volume scheme for the 3D Maxwell euqations on unstructured meshes, SIAM J. Numer. Anal., 39(6) (2002), pp. 2089–2108.CrossRefGoogle Scholar
[6]Gao, Y. K. and Chen, H. Q., Electromagnetic scattering simulation based on cell-vertex unstructured-grid FVTD algorithm, J. Nanjing University of Aeronautics and Astronautics, 45(3) (2013), pp. 415–423.Google Scholar
[7]Cockburn, B., Karniadakis, G. E. and Shu, C. W., Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer, Berlin, 1999.Google Scholar
[8]Lu, Tiao, Zhang, Pingwen and Cai, Wei, Discontinuous Galerkin methods for dispersive and lossy Maxwells equations and PML boundary conditions, J. Comput. Phys., 200 (2004), pp. 549–580.Google Scholar
[9]Cohen, G., Ferrieres, X. and Pernet, S., A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell’s equations in time domain, J. Comput. Phys., 217 (2006), pp. 340–363.Google Scholar
[10]Sevilla, Ruben, Hassan, Oubay and Morgan, Kenneth, The use of hybrid meshes to improve the efficiency of a discontinuous Galerkin method for the solution of Maxwells equations, Comput. Structures, 137 (2014), pp. 2–13.Google Scholar
[11]Kyle Anderson, W., Wang, Li, Kapadia, Sagar, Tanis, Craig and Hilbert, Bruce, Petrov-Galerkin and discontinuous-Galerkin methods for time-domain and frequency-domain electromagnetic simulations, Contents lists available at SciVerse ScienceDirect, 230 (2011), pp. 8360–8385.Google Scholar
[12]Durochat, Clement, Lanteri, Stephane and Scheid, Claire, High order non-conforming multi-element discontinuous Galerkin method for time domain electromagnetics, Appl. Math. Comput., 224 (2013), pp. 681–704.Google Scholar
[13]Nguyen, N. C., Peraire, J. and Cockburn, B., Hybridezable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations, J. Comput. Phys., 230 (2011), pp. 7151–7175.CrossRefGoogle Scholar
[14]Chung, E. T., Jr, P. C. and Yu, T. F., Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids, J. Comput. Phys., 235 (2013), pp. 14–31.CrossRefGoogle Scholar
[15]Li, Jichun, Development of discontinuous Galerkin methods for Maxwells equations in metamaterials and perfectly matched layers, J. Comput. Appl. Math., 236 (2011), pp. 950–961.Google Scholar
[16]Li, Jichun, Waters, Jiajia Wang and Machorro, Eric A., An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwells equations in metamaterials, Comput. Methods Appl. Mech. Eng., 223–224 (2012), pp. 43–54.Google Scholar
[17]Konig, Michael, Busch, Kurt and Niegemann, Jens, The discontinuous Galerkin Time-Domain method for Maxwells equations with anisotropic materials, Photonics Nanostructures-Fundamentals Appl., 8 (2010), pp. 303–309.Google Scholar
[18]Dolean, Victorita, Lanteri, Stephane and Perrussel, Ronan, A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods, J. Comput. Phys., 227 (2008), pp. 2044–2072.Google Scholar
[19]Schnepp, Sascha M. and Weiland, Thomas, Efficient large scale electromagnetic simulations using dynamically adapted meshes with discontinuous Galerkin method, J. Comput. Appl. Math., 236 (2012), pp. 4909–4924.CrossRefGoogle Scholar
[20]Bouajaji, Mohamed El and Lanteri, Stephane, High order discontinuous Galerkin method for the solution of 2D time-harmonic Maxwells equations, Appl. Math. Comput., 219 (2013), pp. 7241–7251.Google Scholar
[21]Lübon, C., Kessler, M., Wagner, S. and Krmer, E., High-order boundary discretization for discontinuous Galerkin codes, 25th AIAA Applied Aerodynamics Conference, 05C08 June 2006, San Francisco, California.Google Scholar
[22]Atkins, H. L. and Shu, C. H., Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations, AIAA J., 36(5) (1998).Google Scholar
[23]Chen, H. Q. and Huang, M. K., Computations of scattering waves in 2-D complicated fields by using exact controllability approach, Chinese J. Comput. Phys., 17(4) (2000), pp. 414–420.Google Scholar