Transformation optics based local mesh refinement for solving Maxwell's equations

In this paper, a novel local mesh refinement algorithm based on transformation optics (TO) has been developed for solving the Maxwell@?s equations of electrodynamics. The new algorithm applies transformation optics to enlarge a small region so that it can be resolved by larger grid cells. The transformed anisotropic Maxwell@?s equations can be stably solved by an anisotropic FDTD method, while other subgridding or adaptive mesh refinement FDTD methods require time-space field interpolations and often suffer from the late-time instability problem. To avoid small time steps introduced by the transformation optics approach, an additional application of the mapping of the material matrix to a dispersive material model is employed. Numerical examples on scattering problems of dielectric and dispersive objects illustrate the performance and the efficiency of the transformation optics based FDTD method.

[1]  M. Okoniewski,et al.  Three-dimensional subgridding algorithm for FDTD , 1997 .

[2]  R. Lee,et al.  Optimization of subgridding schemes for FDTD , 2002, IEEE Microwave and Wireless Components Letters.

[3]  F. Teixeira,et al.  Improved FDTD subgridding algorithms via digital filtering and domain overriding , 2005, IEEE Transactions on Antennas and Propagation.

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

[5]  Zhizhang Chen,et al.  A hybrid ADI‐FDTD subgridding scheme for efficient electromagnetic computation , 2004 .

[6]  Jean-Pierre Bérenger,et al.  The Huygens subgridding for the numerical solution of the Maxwell equations , 2011, J. Comput. Phys..

[7]  Allen Taflove,et al.  Application of the Finite-Difference Time-Domain Method to Sinusoidal Steady-State Electromagnetic-Penetration Problems , 1980, IEEE Transactions on Electromagnetic Compatibility.

[8]  Y. Gan,et al.  A Stable FDTD Subgridding Method Based on Finite Element Formulation With Hanging Variables , 2007, IEEE Transactions on Antennas and Propagation.

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

[10]  I. S. Kim,et al.  A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations , 1990 .

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

[12]  Richard W. Ziolkowski,et al.  A Three-Dimensional Modified Finite Volume Technique for Maxwell's Equations , 1990 .

[13]  Qiang Cheng,et al.  Shrinking an arbitrary object as one desires using metamaterials , 2011 .

[14]  B. Donderici,et al.  Domain-overriding and digital filtering for 3-D FDTD subgridded simulations , 2006, IEEE Microwave and Wireless Components Letters.

[15]  A new stable hybrid three-dimensional generalized finite difference time domain algorithm for analyzing complex structures , 2005, IEEE Transactions on Antennas and Propagation.

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

[17]  A. Taflove,et al.  Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's Equations , 1975 .

[18]  T. Fouquet,et al.  Conservative space-time mesh refinement methods for the FDTD solution of Maxwell's equations , 2006 .

[19]  L. Cristoforetti,et al.  A robust and efficient subgridding algorithm for finite-difference time-domain simulations of Maxwell's equations , 2004 .

[20]  V. P. Cable,et al.  FDTD local grid with material traverse , 1997 .

[21]  Raj Mittra,et al.  A NEW SUBGRIDDING METHOD FOR THE FINITE-DIFFERENCE TIME-DOMAIN (FDTD) ALGORITHM , 1999 .

[22]  U. Leonhardt,et al.  Transformation Optics and the Geometry of Light , 2008, 0805.4778.

[23]  Thomas Weiland,et al.  A CONSISTENT SUBGRIDDING SCHEME FOR THE FINITE DIFFERENCE TIME DOMAIN METHOD , 1996 .

[24]  S. Gedney,et al.  Numerical stability of nonorthogonal FDTD methods , 2000 .

[25]  John R. Cary,et al.  A stable FDTD algorithm for non-diagonal, anisotropic dielectrics , 2007, J. Comput. Phys..

[26]  K. Yee,et al.  A subgridding method for the time-domain finite-difference method to solve Maxwell's equations , 1991 .

[27]  John R. Cary,et al.  A more accurate, stable, FDTD algorithm for electromagnetics in anisotropic dielectrics , 2012, J. Comput. Phys..

[28]  Melinda Piket-May,et al.  9 – Computational Electromagnetics: The Finite-Difference Time-Domain Method , 2005 .

[29]  M. Brio,et al.  Stability of 2D FDTD algorithms with local mesh refinement for Maxwell's equations , 2006 .

[30]  J. Bérenger,et al.  Extension of the FDTD Huygens Subgridding Algorithm to Two Dimensions , 2009, IEEE Transactions on Antennas and Propagation.

[31]  M. Brio,et al.  FDTD based second-order accurate local mesh refinement method for Maxwell's equations in two space dimensions , 2004 .

[32]  S. Gedney.  An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices , 1996 .

[33]  Naoki Okada,et al.  FDTD Modeling of a Cloak with a Nondiagonal Permittivity Tensor , 2012 .

[34]  Zhizhang David Chen,et al.  A hybrid ADI-FDTD subgridding scheme for efficient electromagnetic computation: Research Articles , 2004 .

[35]  Chi Hou Chan,et al.  An explicit fourth-order orthogonal curvilinear staggered-grid FDTD method for Maxwell's equations , 2002 .

[36]  Y. Hao,et al.  Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures. , 2008, Optics express.

[37]  N. Madsen.  Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids , 1995 .

[38]  J. Bérenger.  A Huygens Subgridding for the FDTD Method , 2006, IEEE Transactions on Antennas and Propagation.

[39]  M. McCall,et al.  A spacetime cloak, or a history editor , 2011 .

[40]  S. Gedney,et al.  Full wave analysis of microwave monolithic circuit devices using a generalized Yee-algorithm based on an unstructured grid , 1996 .

[41]  R. Lee,et al.  Conservative and Provably Stable FDTD Subgridding , 2007, IEEE Transactions on Antennas and Propagation.

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

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

[44]  N. Shuley,et al.  A method for incorporating different sized cells into the finite-difference time-domain analysis technique , 1992, IEEE Microwave and Guided Wave Letters.

[45]  R. Holland.  Finite-Difference Solution of Maxwell's Equations in Generalized Nonorthogonal Coordinates , 1983, IEEE Transactions on Nuclear Science.

[46]  Raj Mittra,et al.  Modeling three-dimensional discontinuities in waveguides using nonorthogonal FDTD algorithm , 1992 .

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