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.
