Solving approximate cloaking problems using finite element methods

Abstract

Motivated by the approximate cloaking problem, we consider a variable coefficient Helmholtz equation with a fixed wave number. We use finite element methods to discretize the equation. Numerical results are shown to exhibit cloaking behaviour. References A. Alu and N. Engheta. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72 :016623, 2005. doi:10.1103/PhysRevE.72.016623 W. Cai, U. K. Chettiar, A. V. Kildishev and V. M. Shalaev. Optical cloaking with metamaterials. Nature Photonics 1 :224–227, 2007. doi:10.1038/nphoton.2007.28 A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann. Cloaking devices, electromagnetic wormholes and transformation optics. SIAM Rev. 51 :3–33, 2009. doi:10.1137/080716827 R. V. Kohn, D. Onofrei, M. S. Vogelius and M. I. Weinstein. Cloaking via change of variables for the Helmholtz equation. Comm. Pure Appl. Math. 63 :973–1016, 2010. doi:10.1002/cpa.20341 R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein. Cloaking via change of variables in electric impedance tomography. Inverse Problems 24 :015016, 2008. doi:10.1088/0266-5611/24/1/015016 U. Leonhardt. Optical conformal mapping. Science 312 :1777–1780, 2006. doi:10.1126/science.1126493 M. Maischak. Book of Numerical Experiments (B.O.N.E) . http://www.ifam.uni-hannover.de/ maiprogs/ J. B. Pendry, D. Schurig and D. R. Smith. Controlling electromagnetic fields. Science 312 :1780–1782, 2006. doi:10.1126/science.1125907 D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr and D. Smith. Metamaterial electromagnetic cloak at microwave frequencies. Science 314 :977–980, 2006. doi:10.1126/science.1133628