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

超材料是一种人工结构，由于其不寻常的电磁特性，具有广泛的潜在应用，如亚波长聚焦、隐形斗篷和天线小型化。解决涉及超材料的电磁问题对于分析这些结构及其与环境的相互作用至关重要。例如，超材料问题的精确解决方案可以在新设计实际实现之前提供基本信息，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非常困难的，因为它们是复合的多尺度结构，也就是说，它们由数千个具有小细节的单位细胞组成，而它们的总体尺寸相对于波长来说可能非常大。因此，使用全波解算器对超材料进行精确的数值分析可能需要解决涉及数百万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具有功能，在这些频率下，数值解可能变得不稳定。由于计算解决方案的限制，超材料无法进行足够深入的研究，并且大多数文献中的研究都是基于近似均匀化技术，无法提供对现实结构的严格和准确的分析。本研究的目的是开发一种基于并行实现强大算法的快速求解器，即多层快速多极算法（MLFMA），用于超材料分析。通过开发一个复杂的模拟环境，包括来自不同领域的不同组件，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，研究结果还显示了构建未来世界设备的新超材料设计的可行性，预计将在广义上对科学和技术产生重大影响。

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.