Direct and Inverse Scattering in Biharmonic Waves: Analysis and Computation

散射问题与非均匀介质对入射场的影响有关，是许多科学领域的基础问题，包括地球物理检测、医学成像、隐身技术和无损检测。散射问题作为现代数学物理的重要课题之一，得到了广泛的研究，并得到了大量的数学和数值结果，特别是对于声波、弹性波和电磁波。近年来，由于双谐波在薄板弹性中的重要应用，如柏拉图衍射光栅和超宽带弹性隐身的设计，双谐波的散射问题引起了工程界和数学界的广泛关注。本项目旨在解决双谐板波方程的散射问题所带来的科学挑战。建议的研究性质是多学科的，建议的工作结果将与数学、物理、工程和材料科学领域的其他研究人员积极分享。教育计划的中心是提供跨学科的学生培训，以及开发从本科到研究生水平的综合课程。与二阶声波方程、弹性波方程和电磁波方程相比，四阶双谐波方程的许多正散射和逆散射问题还没有得到很好的理解。本项目将进一步发展双谐板波方程的正散射和逆散射问题的建模、理论和算法，解决周期结构中的散射、多腔散射和随机源的逆散射问题。具体而言，主要研究者将建立有效的数学模型并研究周期结构中双谐板波方程的数学问题，设计一种有效的双谐波在多腔中传播的计算方法，并建立随机双谐波方程反问题的唯一性和稳定性的数学理论。本课题的研究结果将有助于我们对薄板弹性散射理论中复杂的物理和数学问题的理解。这项研究具有发展新科学的潜力，并为工业设计和制造薄板上的新型弹性装置提供指导。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.