Direct and Inverse Scattering in Biharmonic Waves: Analysis and Computation

散射问题涉及到非均匀介质对入射场的影响，是许多科学领域的基础，包括地球物理检测、医学成像、隐身技术和无损检测。散射问题是现代数学物理中的重要课题之一，人们对散射问题进行了广泛的研究，得到了大量的数学和数值结果，特别是声波、弹性波和电磁波的散射问题。近年来，由于双谐波散射问题在弹性薄板中的重要应用，如柏拉图衍射光栅的设计和超宽带弹性隐身等，引起了工程界和数学界的广泛关注。这个项目的目标是解决双谐波板波方程的散射问题所带来的科学挑战。拟议研究的性质是多学科的，拟议工作的结果将与数学，物理，工程和材料科学的其他研究人员积极分享。教育计划围绕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、弹性波和电磁波方程相比，四阶双谐波方程的正散射和逆散射问题还没有得到很好的理解。这个项目将进一步的双谐波板波方程的直接和逆散射问题的建模，理论和算法的发展，解决在周期性结构中的散射，多个腔体的散射，和随机源的逆散射。具体而言，主要研究者将开发有效的数学模型，并研究周期性结构中的双谐波板波方程的数学问题，设计多个腔中双谐波传播的有效计算方法，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。该项目的结果旨在帮助我们理解薄板弹性散射理论中复杂的物理和数学问题。这项研究有可能发展新的科学，并为工业界提供指导，以设计和制造新的弹性装置在薄板。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.