Fast and Accurate Electromagnetic Analysis of Metamaterials With Parallel Multilevel Fast Multipole Algorithm

超材料是一种人工结构，由于其独特的电磁特性，具有广泛的潜在应用，如亚波长聚焦、隐身隐身和天线小型化。超材料电磁学问题的解决对于分析这些结构及其与环境的相互作用是极其重要的。例如，超材料问题的准确解决方案可以在新设计实际实现之前就提供必要的信息，防止在原型制造过程中浪费资源和时间。不幸的是，超材料的精确模拟是极其困难的，因为它们是复合的多尺度结构，即它们由数千个细节很小的单元组成，而它们的总尺寸相对于波长可能非常大。因此，使用全波求解器对超材料进行精确的数值分析可能需要求解涉及数百万个未知数的巨大矩阵方程。此外，超材料通常在某些共振频率下是泛函的，在那里数值解可能会变得不稳定。由于其计算解的局限性，超材料不能被深入研究，而且文献中的大多数研究都是基于近似的均匀化技术，无法提供对现实结构的严格和准确的分析。这项研究的目的是开发一个快速而准确的求解器，基于一种强大的分析超材料的算法，即多层快速多极子算法(MLFMA)的并行实现。通过开发一个由来自不同领域的不同组件组成的复杂模拟环境，如数值技术、迭代方法、快速算法、并行化和并行计算机，涉及复杂超材料的现实问题将以前所未有的精度和细节水平得到解决。除了在计算机科学和高性能计算方面的学术影响外，这些结果还有望在广泛意义上对科学和技术产生重大影响，因为它表明了构建未来世界设备的新超材料设计的可行性。

Metamaterials are artificial structures having a vast variety of potential applications, such as sub-wavelength focusing, invisibility cloaking, and miniaturization of the antennas, due to their unusual electromagnetic properties. Solutions of electromagnetics problems involving metamaterials are extremely important to analyze these structures and their interactions with the environment. For example, accurate solutions of metamaterial problems can provide essential information on novel designs even before their actual realizations, preventing the waste of sources and time during the manufacturing of the prototypes. Unfortunately accurate simulations of metamaterials are extremely difficult since they are composite multi-scale structures, i.e., they consist of thousands of unit cells with small details whereas their overall dimensions can be very large with respect to the wavelength. Hence, accurate numerical analysis of metamaterials using full-wave solvers may require the solution of huge matrix equations involving millions of unknowns. In addition, metamaterials are usually functional at some resonance frequencies, where the numerical solutions may become unstable. Due to the limitations in their computational solutions, metamaterials could not be investigated in sufficient depth, and most of the studies in the literature are based on approximate homogenization techniques that are unable to provide rigorous and accurate analysis of realistic structures. The purpose of this study is to develop a fast and accurate solver based on a parallel implementation of a powerful algorithm, namely, the multilevel fast multipole algorithm (MLFMA), for the analysis of metamaterials. By developing a sophisticated simulation environment consisting of diverse components from different areas, such as numerical techniques, iterative method, fast algorithms, parallelization, and parallel computers, real-life problems involving complex metamaterials will be solved with unprecedented levels of accuracy and detail. In addition to academic impacts in computer science and high performance computing, the results are expected have high impacts in science and technology in a broad sense by showing the feasibility of new metamaterial designs for constructing the devices of the future's world.