The use of hybrid meshes to improve the efficiency of a discontinuous Galerkin method for the solution of Maxwell’s equations

Abstract This paper investigates the efficiency of a high-order nodal discontinuous Galerkin method for the numerical solution of Maxwell’s equations using hybrid meshes. An unstructured triangular or tetrahedral mesh is used near curved boundaries and a structured Cartesian mesh is used to fill the remainder of the domain. A quadrature-free implementation is employed for the regular quadrilateral and hexahedral elements which, together with the reduction in the number of internal faces, leads to a reduction in the cpu time requirements. Numerical examples in two and three dimensions are used to illustrate the benefits of using hybrid meshes.

[1]  Antonio Huerta,et al.  Comparison of high‐order curved finite elements , 2011 .

[2]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[3]  Harold L. Atkins,et al.  QUADRATURE-FREE IMPLEMENTATION OF DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS , 1996 .

[4]  J Peiró.  Surface Grid Generation , 1998 .

[5]  R. Cárdenas.  NURBS-Enhanced Finite Element Method (NEFEM) , 2009 .

[6]  Antonio Huerta,et al.  NURBS‐enhanced finite element method for Euler equations , 2008 .

[7]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[8]  S. Wandzurat,et al.  Symmetric quadrature rules on a triangle , 2003 .

[9]  R. Harrington.  Time-Harmonic Electromagnetic Fields , 1961 .

[10]  Kenneth Morgan,et al.  Tailoring unstructured meshes for use with a 3D time domain co‐volume algorithm for computational electromagnetics , 2011 .

[11]  Mark Ainsworth,et al.  Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods , 2004 .

[12]  Oubay Hassan,et al.  A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain , 2009 .

[13]  H. Fahs.  Development of a hp-like discontinuous Galerkin time-domain method on non-conforming simplicial meshes for electromagnetic wave propagation , 2009 .

[14]  Jan S. Hesthaven,et al.  High-order accurate methods in time-domain computational electromagnetics: A review , 2003 .

[15]  Haoxiang Luo,et al.  A Lobatto interpolation grid in the tetrahedron , 2006 .

[16]  C. Bernardi.  Optimal finite-element interpolation on curved domains , 1989 .

[17]  J. S. Chen,et al.  The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell's equations , 1997 .

[18]  Bharat K. Soni,et al.  Handbook of Grid Generation , 1998 .

[19]  Fredrik Edelvik,et al.  Explicit Hybrid Time Domain Solver for the Maxwell Equations in 3D , 2000, J. Sci. Comput..

[20]  Marc Duruflé,et al.  Higher-order Finite Elements for Hybrid Meshes Using New Nodal Pyramidal Elements , 2010, J. Sci. Comput..

[21]  J. Hesthaven,et al.  Staircase-free finite-difference time-domain formulation for general materials in complex geometries , 2001 .

[22]  N. Weatherill,et al.  Efficient three‐dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints , 1994 .

[23]  Hui,et al.  A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA , 2009 .

[24]  Explicit Time Domain Finite Element Methods for Electromagnetics , 2005 .

[25]  L. Demkowicz,et al.  CONTROL OF GEOMETRY INDUCED ERROR IN hp FINITE ELEMENT(FE) SIMULATIONS I. EVALUATION OF FE ERROR FOR CURVILINEAR GEOMETRIES , 2005 .

[26]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[27]  Antonio Huerta,et al.  NURBS-Enhanced Finite Element Method ( NEFEM ) A Seamless Bridge Between CAD and FEM , 2012 .

[28]  Francesca Rapetti,et al.  Recent Achievements on a DGTD Method for Time-Domain Electromagnetics , 2010, IEEE Transactions on Magnetics.

[29]  I. Doležel,et al.  Higher-Order Finite Element Methods , 2003 .

[30]  Bengt Fornberg,et al.  A split step approach for the 3-D Maxwell's equations , 2003 .

[31]  C. Balanis.  Advanced Engineering Electromagnetics , 1989 .

[32]  Oubay Hassan,et al.  A low–order unstructured–mesh approach for computational electromagnetics in the time domain , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[33]  Chi-Wang Shu,et al.  Locally divergence-free discontinuous Galerkin methods for the Maxwell equations , 2004, Journal of Computational Physics.

[34]  D. A. Dunavant.  Efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube , 1986 .

[35]  D. A. Dunavant.  Economical symmetrical quadrature rules for complete polynomials over a square domain , 1985 .

[36]  Mark A. Taylor,et al.  An Algorithm for Computing Fekete Points in the Triangle , 2000, SIAM J. Numer. Anal..

[37]  Eloi Ruiz Gironés.  Automatic hexahedral meshing algorithms: from structured to unstructured meshes , 2011 .

[38]  Pingwen Zhang,et al.  Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions , 2004 .

[39]  Vijaya Shankar,et al.  Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations , 2004, J. Sci. Comput..

[40]  Xia Ji,et al.  High‐order DGTD methods for dispersive Maxwell's equations and modelling of silver nanowire coupling , 2007 .

[41]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[42]  D. J. Riley,et al.  VOLMAX: a solid-model-based, transient volumetric Maxwell solver using hybrid grids , 1997 .

[43]  M. König,et al.  The Discontinuous Galerkin Time-Domain method for Maxwell’s equations with anisotropic materials , 2010 .

[44]  Xevi Roca Navarro.  Paving the path towards automatic hexahedral mesh generation , 2009 .

[45]  Fernando Reitich,et al.  High-order RKDG Methods for Computational Electromagnetics , 2005, J. Sci. Comput..

[46]  Nigel P. Weatherill,et al.  Hybrid methods for electromagnetic scattering simulations on overlapping grids , 2003 .

[47]  K. Morgan,et al.  The generation of arbitrary order curved meshes for 3D finite element analysis , 2013 .