Direct and Inverse Scattering in Biharmonic Waves: Analysis and Computation

Purdue University, West Lafayette IN

Investigators

Abstract

Scattering problems, which are concerned with the effect that an inhomogeneous medium has on an incident field, are fundamental in many scientific areas, including geophysical inspection, medical imaging, stealth technology, and nondestructive testing. As one of the key topics in modern mathematical physics, scattering problems have been widely investigated, and a large number of mathematical and numerical results are available, especially for acoustic, elastic, and electromagnetic waves. Recently, scattering problems for biharmonic waves have attracted much attention in the engineering and mathematical communities due to significant applications in thin plate elasticity, such as the design of platonic diffraction gratings and ultra-broadband elastic cloaking. The goal of this project is to address scientific challenges posed by scattering problems of the biharmonic plate wave equation. The nature of the proposed research is multidisciplinary, and the results of the proposed work will be actively shared with other researchers in mathematics, physics, engineering, and materials science. The educational plan is centered around providing interdisciplinary student training as well as developing an integrated curriculum from the undergraduate level to the graduate level. Compared with the second-order acoustic, elastic, and electromagnetic wave equations, many direct and inverse scattering problems for the fourth-order biharmonic wave equation are not well understood. This project will further the modeling, theory, and algorithmic development of the direct and inverse scattering problems of the biharmonic plate wave equation, addressing scattering in periodic structures, scattering by multiple cavities, and inverse scattering for random sources. Specifically, the principal investigator will develop effective mathematical models and examine mathematical issues for the biharmonic plate wave equation in periodic structures, design an efficient computational approach for biharmonic wave propagation in multiple cavities, and establish mathematical theory on the uniqueness and stability of the inverse problems for the stochastic biharmonic wave equation. Results of this project are intended to contribute to our understanding of complex physical and mathematical problems in the scattering theory of thin plate elasticity. The research has the potential for evolving new science and providing the industry with guidance to design and fabricate new elastic devices in thin plates. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.