Accurate Simulation of Circular and Elliptic Cylindrical Invisibility Cloaks (Communications in Computational Physics, 2018)
The coordinate transformation offers a remarkable way to design cloaks thatcan steer electromagnetic fields so as to prevent waves from penetrating into the cloakedregion (denoted by $Ω_0$, where the objects inside are invisible to observers outside).The ideal circular and elliptic cylindrical cloaked regions are blown up from a pointand a line segment, respectively, so the transformed material parameters and the correspondingcoefficients of the resulted equations are highly singular at the cloakingboundary $∂Ω_0$. The electric field or magnetic field is not continuous across $∂Ω_0$. Theimposition of appropriate cloaking boundary conditions (CBCs) to achieve perfect concealmentis a crucial but challenging issue.Based upon the principle that a well-behaved electromagnetic field in the originalspace must be well-behaved in the transformed space as well, we obtain CBCs thatintrinsically relate to the essential "pole" conditions of a singular transformation. Wealso find that for the elliptic cylindrical cloak, the CBCs should be imposed differentlyfor the cosine-elliptic and sine-elliptic components of the decomposed fields. Withthese at our disposal, we can rigorously show that the governing equation in $Ω_0$ canbe decoupled from the exterior region $Ω^c_0$, and the total fields in the cloaked regionvanish under mild conditions. We emphasize that our proposal of CBCs is differentfrom any existing ones.Using the exact circular (resp., elliptic) Dirichlet-to-Neumann (DtN) non-reflectingboundary conditions to reduce the unbounded domain $Ω^c_0$ to a bounded domain, weintroduce an accurate and efficient Fourier-Legendre spectral-element method (FLSEM)(resp., Mathieu-Legendre spectral-element method (MLSEM)) to simulate the circularcylindrical cloak (resp., elliptic cylindrical cloak). We provide ample numerical resultsto demonstrate that the perfect concealment of waves can be achieved for the idealcircular/elliptic cylindrical cloaks under our proposed CBCs and accurate numericalsolvers.