Fast Simulation of Wave Scattering and Propagation in Inhomogeneous Media with Complex Geometries

该项目的目标是开发创新的数值方法，通过仅使用简单的笛卡尔网格来产生复杂几何非均匀介质中的电磁波的四阶精确模拟。在过去的几十年里，计算机能力的快速增长无法抵挡住，我们对涉及几何复杂介质界面的三维波传播和散射进行建模的能力受到严重限制。在数学上，波解在材料界面上通常是非光滑的，甚至是不连续的，因此我们设计有效算法的努力很容易被挫败，除非对复杂的界面进行适当的处理。在文献中，通常使用贴体网格来处理复杂的界面和几何形状。尽管在网格生成方面已经取得了很大的进展，但在几何复杂区域中形成一个高质量的贴体网格系统仍然是一个困难和耗时的任务。或者，在这个项目中，研究人员将探索如何通过使用笛卡尔网格(包括交错的Yee网格)来适应具有复杂几何形状的介电界面。由此产生的笛卡尔网格法，在某种意义上适合于复杂几何的数值微分算子，但与贴体网格法相比，文献中对其研究较少。精确处理复杂界面的高阶笛卡尔网格方法的发展，对于有效的波浪模拟具有迫在眉睫的现实意义，但仍未得到解决。在这个项目中，创新的四阶笛卡尔网格方法将基于匹配界面和边界(MIB)方法，该方法是由研究者和他的合作者开发的，用于求解包含材料界面或非均匀介质的偏微分方程组(PDE)。为了解决广泛的电磁应用，将开发一套完整的四阶MIB方法，用于不同的电磁公式，包括亥姆霍兹方程、波动方程和麦克斯韦方程，并用于不同的场景，包括横向磁模、横向电模和全三维模式。计算电磁学(CEM)是数学家和工程师共同贡献的一个交叉学科领域，对于天线的分析和综合、雷达散射截面(RCS)的计算、地面或地面探地雷达的模拟等广泛的应用都是至关重要的。所提出的数值方法旨在解决涉及大规模和不规则形状结构的具有挑战性的CEM应用，现有的方法在这方面遇到了很大的困难。通过提供更准确和高效的波浪模拟，所提出的方法将在解决实际CEM应用中长期存在的问题方面取得突破。此外，所提出的方法将对科学计算中其他具有挑战性的界面问题，如流体动力学中的浸没界面和运动界面问题，计算生物学中用于生物大分子结构预测的静电界面问题，都将产生相当大的影响。

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.