An adaptive edge finite element method for electromagnetic cloaking simulation

In this paper we develop an adaptive edge finite element method based on a reliable and efficient recovery type a posteriori error estimator for time-harmonic Maxwell equations. The asymptotically exact a posteriori error estimator is based on the superconvergence result proved for the lowest-order edge element on triangular grids, where most pairs of triangles sharing a common edge form approximate parallelograms. The efficiency and robustness of the proposed method is demonstrated by extensive numerical experiments for electromagnetic cloaking problems with highly anisotropic permittivity and permeability.

[1]  Wei Yang,et al.  Developing a time-domain finite-element method for modeling of electromagnetic cylindrical cloaks , 2012, J. Comput. Phys..

[2]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..

[3]  Yunqing Huang,et al.  Superconvergence analysis for time‐dependent Maxwell's equations in metamaterials , 2012 .

[4]  Zhonghua Qiao,et al.  Superconvergence and Extrapolation Analysis of a Nonconforming Mixed Finite Element Approximation for Time-Harmonic Maxwell’s Equations , 2011, J. Sci. Comput..

[5]  Tie Jun Cui,et al.  Finite-element Analysis of Three-dimensional Axisymmetrical Invisibility Cloaks and Other Metamaterial Devices , 2010 .

[6]  Long Chen,et al.  Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations , 2011, Math. Comput..

[7]  J. Willis,et al.  On cloaking for elasticity and physical equations with a transformation invariant form , 2006 .

[8]  U. Leonhardt.  Optical Conformal Mapping , 2006, Science.

[9]  Ningning Yan,et al.  Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes , 2001 .

[10]  Robert V. Kohn,et al.  Cloaking via change of variables for the Helmholtz equation , 2010 .

[11]  Gang Bao,et al.  An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures , 2010, Math. Comput..

[12]  Yunqing Huang,et al.  Averaging for superconvergence: Verification and application of 2D edge elements to Maxwell’s equations in metamaterials☆ , 2013 .

[13]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[14]  I. Doležel,et al.  Adaptive hp-FEM with Arbitrary-Level Hanging Nodes for Maxwell's Equations , 2010 .

[15]  Yunqing Huang,et al.  Centroidal Voronoi tessellation‐based finite element superconvergence , 2008 .

[16]  Peijun Li,et al.  Numerical Solution of Acoustic Scattering by an Adaptive DtN Finite Element Method , 2013 .

[17]  Matti Lassas,et al.  Cloaking Devices, Electromagnetic Wormholes, and Transformation Optics , 2009, SIAM Rev..

[18]  Yunqing Huang,et al.  Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence , 2011 .

[19]  R. Hoppe,et al.  A review of unified a posteriori finite element error control , 2012 .

[20]  P. Sheng,et al.  Transformation optics and metamaterials. , 2010, Nature materials.

[21]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[22]  Weiying Zheng,et al.  An Adaptive Multilevel Method for Time-Harmonic Maxwell Equations with Singularities , 2007, SIAM J. Sci. Comput..

[23]  P. Deuflhard,et al.  Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations , 1997 .

[24]  Zhimin Zhang,et al.  Analysis of recovery type a posteriori error estimators for mildly structured grids , 2003, Math. Comput..

[25]  Leszek Demkowicz,et al.  Numerical simulations of cloaking problems using a DPG method , 2013 .

[26]  Gunzburger,et al.  Advances in Studies and Applications of Centroidal Voronoi Tessellations , 2010 .

[27]  Yunqing Huang,et al.  Mathematical Simulation of Cloaking Metamaterial Structures , 2012 .

[28]  S. Guenneau,et al.  The colours of cloaks , 2011 .

[29]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[30]  Jun Zou,et al.  Convergence analysis of an adaptive edge element method for Maxwell's equations , 2009 .

[31]  Ting Zhou,et al.  On Approximate Electromagnetic Cloaking by Transformation Media , 2010, SIAM J. Appl. Math..

[32]  Weidong Zhao,et al.  Adaptive Finite Element Methods for Elliptic PDEs Based on Conforming Centroidal Voronoi-Delaunay Triangulations , 2006, SIAM J. Sci. Comput..

[33]  Lewei Li,et al.  Material parameters characterization for arbitrary N-sided regular polygonal invisible cloak , 2009 .

[34]  Mark Ainsworth,et al.  Hierarchic hp-edge element families for Maxwell's equations on hybrid quadrilateral/triangular meshes , 2001 .

[35]  David R. Smith,et al.  Metamaterial Electromagnetic Cloak at Microwave Frequencies , 2006, Science.

[36]  David R. Smith,et al.  Controlling Electromagnetic Fields , 2006, Science.

[37]  Robert V. Kohn,et al.  Erratum: Cloaking via change of variables for the Helmholtz equation , 2010 .

[38]  Jichun Li,et al.  Computational Partial Differential Equations Using MATLAB , 2008 .