An Unstructured Mesh Coordinate Transformation-Based FDTD Method

We propose a novel unstructured mesh finite-difference time-domain (FDTD) method for solving electromagnetics problems with complicated geometries. The method, which solves the TE-mode reduced form of Maxwell's equations, can handle both material interfaces and anisotropic material. Using the transformation optics principle, which describes how fields and material tensors change under coordinate transformations, we locally transform each cell in the mesh to a reference unit-square computational domain where the usual FDTD update is performed. This comes at a cost: employing unstructured grids and coordinate transformations requires more complicated data structures, a mesh orientation process, and potentially introduces an anisotropic material tensor at every mesh cell. Nonetheless, we find that the method maintains the same desirable properties of the classic FDTD method (explicit, divergence-free B-field, nondissapative, and second-order accuracy) while also gaining conforming material interfaces and boundaries in complicated geometry. Even further, we prove that the method is stable under a Courant condition and a fairly nonrestrictive mesh condition, hence defeating the late-time stability issue plaguing prior nonorthogonal FDTD methods. To verify the method, we conduct convergence studies on three electromagnetic cavity problems with known exact solutions. For these numerical studies, we find that the method maintains second-order convergence and stability.