An Adaptive and Robust Discrete Geometry Based Helmholtz Solver and Applications to Device Design

该奖项的研究目标是发展数学和算法，促进受亥姆霍兹方程支配的物理系统的设计和优化。虽然研究结果具有广泛的适用性，但这项工作的重点将限于声学和电磁系统。拟议工作的关键将依赖于开发能够进行多尺度分析的解决方案技术，并且在表面和函数表示方面都具有高度的灵活性和适应性。该研究依次经历了三个阶段：(I)可自适应细化的局部参数曲面描述符的发展，(Ii)表示场的函数框架和相应的积分方程解算器，以及(Iii)响应外部约束的曲面变形的分层算法。所开发的方法将用于三个应用领域：纳米光子学、微气泡的引导和反问题中的形状重建。成果包括高度灵活和适应性的建模和分析工具，通过应用程序、文档和研究成果的传播进行演示和验证，以及跨学科的学生教育。这项研究的结果将提供一个足够灵活和适应性的令人信服的建模工具，以便在Helmholtz系统的设计和优化框架中使用。为了突出它们的广泛适用性，开发的工具将在不同类型的应用中进行测试，从仿生显示器到太阳能电池，从微气泡到成像中的反问题。为了确保研究的传播，私人投资机构将创建一个专门的网站，并与密歇根工业和应用数学中心以及网络使能研究所合作。将利用密歇根州立大学现有的妇女和少数民族招聘渠道。本科生将通过高级设计项目和密歇根州立大学现有的REU项目参与进来。工程学、计算机科学和物理学的研究生和本科生将从课堂教学和参与研究中受益。

The research objective of this award is the development of mathematics and algorithms that facilitate design and optimization of physical systems that are governed by the Helmholtz equation. While the research results are broadly applicable, the focus of this effort will be restricted to acoustic and electromagnetic systems. The crux of the proposed effort will rely on developing solution techniques that are capable of multiscale analysis, and are highly flexible and adaptive in both surface and function representation. The research progresses in sequence through three stages; the development of (i) local parametric surface descriptors that can be adaptively refined, (ii) a functional framework to represent fields and corresponding integral equation solvers, and (iii) hierarchical algorithms for morphing surfaces in response to external constraints. The methods developed will be used in three application areas; nanophotonics, steering of microbubbles and shape reconstruction in inverse problems. Deliverables include a highly flexible and adaptive modeling and analysis tool, demonstration and validation via applications, documentation and dissemination of research results, and cross-disciplinary student education.Tthe results of this research will provide a cogent modeling tool that is sufficiently flexible and adaptive so as to be usable within a design and optimization framework for Helmholtz systems. To highlight their broad applicability, the tools developed will be tested in different types of applications ranging from biomimetic displays to solar cells to microbubbles to inverse problems in imaging. To ensure dissemination of the research, the PIs will create a dedicated website, as well as work with the Michigan Center of Industrial and Applied Mathematics and Institute of Cyber-Enabled Research. Existing channels in women and minority recruitment at MSU will be utilized. Undergraduate students will be involved through senior design projects and through existing REU programs at MSU. Graduate and Undergraduate Engineering, Computer Science and Physics students will benefit through classroom instruction and involvement in research.