Fast and Accurate Integral Equation Solvers for Mixed-scale Electromagnetic Simulation

在真实结构的电磁仿真中，被分析区域的空间表示不仅取决于感兴趣的频率，而且取决于捕捉可能的精细几何特征的需要。这种混合尺度在三个方面对基于标准积分方程的求解器造成了严重破坏：(I)随着单元尺寸的变小，离散化的积分方程变得条件差，(Ii)所使用的函数空间不能最好地表示潜在的物理，以及(Iii)总的计算负担非常大。这在很大程度上限制了现有方法的适用性。拟议的项目旨在开发一种明显统一、健壮和准确的求解方法，该方法在广泛的频率范围内都有良好的条件，同时具有处理复杂(可能接近奇异的)几何图形的灵活性。这是通过以下方式实现的：(I)发展一个条件良好的积分方程格式(即第二类Fredholm型方程)，在收敛速度和精度上具有可证明的界，以求在很大的空间频率范围内求解电磁量；(Ii)扩大用于表示未知量的近似空间，以包括局部物理；(Iii)设计一种允许各种基函数之间无缝相互作用的方案，以模拟与上述积分方程格式一起使用的未知量；(4)推导出这些格式的误差界和收敛估计，以证明用户对误差的清晰和容易的控制，以及(4)发展一个区域分解框架，使这些格式可以与经典积分方程法和有限元方法无缝结合来求解电大问题。教育目标是在这项研究的基础上开发一套公开可用的教程/教学模块。模拟方法的快速发展与摩尔定律相一致，使得在简单的台式计算机上分析电大问题成为可能，计算时间合理。如此之多，以至于对现实设备的全波或严格模拟都是可能的。然而，随着人们趋向于这一目标，新的和更具挑战性的问题出现了。在混合尺度物理建模中，需要正确地表示局部物理，开发克服条件问题的方法，并开发在多尺度上加速计算的方法。该项目旨在解决这些问题。这里开发的方法将具有广泛的足迹，从国家安全(共形天线的设计)到传感器技术(表面增强的拉曼和等离子)，到超材料，到纳米技术(纳米结构晶体生长动力学)，再到分子动力学。除了对研究生进行工程和数学方面的培训外，还利用现有渠道招募妇女和少数族裔，并通过高级设计项目和潜在的REU补充方案让本科生参与进来。

In the electromagnetic simulation of realistic structures, the spatial representation of the domain being analyzed depends not only on the frequency of interest but also on the need to capture possible fine geometric features. Such mixed scales cause havoc in standard integral equation based solvers on three fronts; (i) discretized integral equations become poorly conditioned as the size of the element becomes smaller, (ii) the function spaces used do not optimally represent the underlying physics, and (iii) the overall computational burden is exceedingly large. This largely limits the applicability of the existing methods. The proposed project seeks to develop a demonstrably unified, robust and accurate solution methodology that is well conditioned over a wide range of frequencies and, at the same time, has the flexibility to handle complicated (and possibly near singular) geometries. This is achieved by (i) developing a well conditioned integral equation scheme (that are Fredholm equations of the second kind) with provable bounds on convergence rates and accuracy to solve for electromagnetic quantities over a large range of spatial frequencies; (ii) enlarging the approximation space used for representing the unknown quantity so as to include the local physics; (iii) designing a scheme that permits seamless interplay between a variety of basis functions to model the unknown quantities to be used with the above integral equation scheme; (iv) deriving error bounds and convergence estimates on these schemes to demonstrate clear and easy usercontrol over the error, and (iv) developing a domain decomposition framework so that these schemes can be integrated seamlessly with classical integral equation and finite element methods to solve electrically large problems. The educational objective is to develop a publicly available set of tutorials/teaching modules based on this research. The rapid progress in simulation methods in concert with the Moore's law has made the analysis of electrically large problems possible on simple desktop machines in reasonable computational times. So much so that fullwave or rigorous simulation of realistic devices are within the realm of possibility. However, as one tends towards this goal, new and more challenging problems arise. In modeling mixed scale physics, it is necessary to correctly represent local physics, develop methods to overcome conditioning issues, and develop means to accelerate computation over multiple scales. This project addresses the resolution of these problems. The methods developed herein will have a wide footprint ranging from national security (design of conformal antennas) to sensor technology (surface enhanced raman and plasmonics) to metamaterials to nanotechnology (nano-structure crystal growth dynamics) to molecular dynamics. In addition to training graduate students in engineering and mathematics, existing channels are utilized to recruit women and minorities and undergraduate students are involved through senior design projects and potential REUsupplements.