The goal of the proposed project is to develop innovative numerical approaches to produce fourth order accurate simulation of electromagnetic waves in inhomogeneous media with complex geometries, by using only simple Cartesian grids. The rapid growth of computer capability in the past few decades not withstanding, our ability to model three-dimensional wave propagation and scattering involving geometrically complicated dielectric interfaces is severely limited. Mathematically, the wave solutions are usually non-smooth or even discontinuous across the material interfaces, so that our effort in designing efficient algorithms is easily foiled, unless the complex interfaces are properly treated. The complex interfaces and geometries are commonly tackled by using body-fitted grids in the literature. Even though considerable progress has been made in grid generation, the formation of a good quality body-fitted grid system in geometrically complex domain remains a difficult and time-consuming task. Alternatively, in this project, the investigator will explore how to accommodate dielectric interfaces with complex geometries by using Cartesian grids including the staggered Yee grids. The resulting Cartesian grid methods, which in some sense fit the numerical differentiation operators to the complicated geometries, are less well studied in the literature, in contrast to the body-fitted grid methods. The development of high order Cartesian grid methods with complex interfaces being accurately treated, is of imminent practical importance to efficient wave simulations, but remains unsolved. In this project, innovative fourth order Cartesian grid approaches will be constructed based on the matched interface and boundary (MIB) method newly developed by the investigator and his collaborators for solving partial differential equations (PDEs) involving material interfaces or inhomogeneous media. To address a widespread variety of electromagnetic applications, a complete set of fourth order MIB methods will be developed for different electromagnetic formulations including the Helmholtz equation, the wave equation, and Maxwell's equations, and for different scenarios including the transverse magnetic mode, the transverse electric mode, and fully three-dimensional mode.
Computational electromagnetics (CEM), an interdisciplinary field where one witnesses mutual contributions from mathematicians and engineers is of paramount importance for a wide range of applications, including analysis and synthesis of antenna, calculation of radar cross section (RCS), simulation of ground or surface penetrating radar, to name only a few. The proposed numerical approaches aim to address challenging CEM applications involving large-scale and irregularly shaped structures, for which currently existing methods encounter great difficulties. By delivering more accurate and efficient wave simulations, the proposed methods will lead to breakthroughs in resolving long-standing problems in the real CEM applications. Moreover, the proposed methods will have considerable impact on other challenging interface problems in scientific computing, such as the immersed interface and moving interface problems in fluid dynamics, electrostatic interface problems for structural prediction of large biomolecules in computational biology.
