Abstract
The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.
Similar content being viewed by others
References
Balanis CA (1989) Advanced engineering electromagnetics. Wiley, New York
Bérenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114: 185–200
Bos L, Taylor MA, Wingate BA (2000) Tensor product Gauss–Lobatto points are Fekete points for the cube. Math Comput 70: 1543–1547
Chew WC (2004) Computational electromagnetics: the physics of smooth versus oscillatory fields. Philos Trans Royal Soc Math Phys Eng Sci 362: 579–602
Cioni JP, Fezoui L, Steve H (1993) A parallel time–domain Maxwell solver using upwind schemes and triangular meshes. IMPACT Comput Sci Eng 5: 215–247
Cockburn B, Shu CW (2001) Runge–Kutta discontinuous Galerkin methods for convection dominated problems. J Sci Comput 16: 173–261
Cohen GC (2002) High–order numerical methods for transient wave equations. Springer, Berlin
Darve E, Havé P (2004) A fast multipole method for Maxwell’s equations stable at all frequencies. Philos Trans Roy Soc Math Phys Eng Sci 362: 603–628
Darve E, Löhner R (1997) Advanced structured–unstructured solver for electromagnetic scattering from multimaterial objects. AIAA Paper 97–0863. AIAA, Washington
Davies RW (2007) A hybrid spectral element method for the time domain solution of wave scattering problems. PhD Thesis, Swansea University
Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, Chichester
El Hachemi M, Hassan O, Morgan K, Rowse DP, Weatherill NP (2003) Hybrid methods for electromagnetic scattering simulations on overlapping grids. Commun Numer Methods Eng 19: 749–760
Hesthaven JS, Warburton T (2002) Nodal high–order methods on unstructured grids I: time–domain solution of Maxwell’s equations. J Comput Phys 181: 186–221
Hesthaven JS, Warburton T (2004) High order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem. Philos Trans Roy Soc Math Phys Eng Sci 362: 493–524
Komatitsch D, Vilotte JP, Vai R, Castillo–Covarrubias JM (1999) The spectral element method for elastic wave equations—application to 2D and 3D seismic problems. Int J Numer Methods Eng 45: 1139–1164
Komatitsch D, Martin R, Tromp J, Taylor MA, Wingate BA (2002) Wave propagation in 2D elastic media using a spectral element method with triangles and quadrangles. J Comput Acoust 92: 703–718
Kopriva DA, Woodruff SL, Hussaini MY (2002) Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. Int J Numer Methods Eng 53: 105–122
Ledger PD (2001) An hp-adaptive finite element procedure for electromagnetic wave scattering problems. PhD Thesis, University of Wales Swansea
Maday Y, Patera AT (1989) Spectral element methods for the incompressible Navier–Stokes equations. In: Noor AK (eds) State of the art survey in computational mechanics. ASME, New York, pp 71–143
Mercerat ED, Vilotte JP, Sanchez–Sesma FJ (2006) Triangular spectral element simulation of two–dimensional elastic wave propagation using unstructured triangular grids. Geophys J Int 166: 679–698
Mohammadian AH, Shankar V, Hall WF (1991) Computation of electromagnetic scattering and radiation using a time–domain finite–volume discretization procedure. Comput Phys Commun 68: 175–196
Morgan K, El hachemi M, Hassan O, Weatherill NP (2006) Explicit time domain methods for electromagnetics. In: Caughey DA, Hafez MM (eds) Frontiers of computational fluid dynamics. World Scientific, New Jersey, pp 161–181
Morgan K, Hassan O, Pegg NE, Weatherill NP (2000) The simulation of electromagnetic scattering in piecewise homogeneous media using unstructured grids. Comput Mech 25: 438–447
Morgan K, Hassan O, Peraire J (1994) An unstructured grid algorithm for the solution of Maxwell’s equations in the tgime domain. Int J Numer Methods Fluids 19: 849–863
Morgan K, Hassan O, Peraire J (1996) A time domain unstructured grid approach to the simulation of electromagnetic scattering in piecewise homogeneous media. Comput Methods Appl Mech Eng 134: 17–36
Morgan K, Hassan O, Peraire J (1998) Parallel processing for the simulation of problems involving scattering of electromagnetic waves. Comput Methods Appl Mech Eng 152: 157–174
Patera AT (1984) A spectral element method for fluid dynamics. Laminar flow in a channel expansion. J Comput Phys 54: 468–488
Šólin P, Segeth K, Doležel I (2003) Higher–order finite element methods. Chapman & Hall/CRC, Boca Raton
Peiró J (1999) Surface grid generation. In: Thompson JF, Soni BK, Weatherill NP (eds) Handbook of grid generation. CRC Press, Boca Raton, pp 19.1–19.20
Peraire J, Vahdati M, Morgan K, Zienkiewicz OC (1987) Adaptive remeshing for compressible flow computations. J Comput Phys 72: 449–466
Petitjean B, Löhner R (1992) Finite element solvers for radar cross section RCS calculations. AIAA Paper 92–0455, AIAA Washington
Roe PL (1981) Approximate Riemann solvers, parameter vectors and difference schemes. J Comput Phys 43: 357–372
Sherwin SJ, Karniadakis GE (1995) A triangular spectral element: applications to the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 123: 189–229
Taylor MA (2004) Private communication
Taylor MA, Wingate BA (2000) A generalized diagonal mass matrix spectral element method for non–quadrilateral elements. Appl Numer Math 33: 259–265
Taylor MA, Wingate BA, Vincent RE (2000) An algorithm for computing Fekete points in the triangle. SIAM J Numer Anal 38: 1707–1720
Tromp J, Komatitsch D, Liu Q (2008) Spectral–element and adjoint methods in seismology. Commun Comput Phys 3: 1–32
Warburton T (2004) Private communication
Yee KS (1966) Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans Antennas Propag 14: 302–307
Zienkiewicz OC, Morgan K (2006) Finite elements and approximation. Dover, Mineola
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Davies, R.W., Morgan, K. & Hassan, O. A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain. Comput Mech 44, 321–331 (2009). https://doi.org/10.1007/s00466-009-0377-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-009-0377-4