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

在现实结构的电磁仿真中，被分析域的空间表示不仅取决于感兴趣的频率，还取决于捕获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造成了严重破坏；(i)离散化的积分方程随着元素的尺寸变小而变得条件不佳，（ii）所使用的函数空间不能最佳地代表底层物理，以及（iii）总体计算负担非常大。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的、健壮的和准确的解决方法，这种方法可以很好地适应广泛的频率范围，同时具有处理复杂（可能接近单一）几何形状的灵活性。这是通过(i)开发一个条件良好的积分方程方案（即第二类Fredholm方程）来实现的，该方案具有可证明的收敛速率和精度界限，以解决大范围空间频率范围内的电磁量；（ii）扩大用于表示未知量的近似空间，使其包括局部物理；（iii）设计一种方案，使各种基函数之间能够无缝地相互作用，以模拟上述积分方程方案所使用的未知量；（iv）推导这些方案的误差边界和收敛估计，以证明用户对误差的清晰和容易的控制，以及（iv）开发一个域分解框架，使这些方案可以与经典积分方程和有限元方法无缝集成，以解决电力大问题。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。与摩尔定律相一致的模拟方法的迅速发展，使得在合理的计算时间内，在简单的台式机器上分析大型电气问题成为可能。因此，对现实设备的全波或严格模拟是可能的。然而，当一个人趋向于这个目标时，新的和更具挑战性的问题就会出现。在混合尺度物理建模中，需要正确地表示局部物理，开发克服条件反射问题的方法，以及开发加速多尺度计算的方法。本项目旨在解决这些问题。本文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（表面增强拉曼和等离子体），从超材料到纳米技术（纳米结构晶体生长动力学）到分子动力学。除了训练工程和数学方面的研究生外，还利用现有渠道招收妇女和少数民族学生，并通过高级设计项目和可能的再利用补充项目招收本科生。

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.