Direct and Inverse Scattering Problems in Elastic Waves: Analysis and Computation

弹性波中的正向和逆向散射问题：分析与计算

• 批准号: 1912704
• 负责人: Peijun Li
• 金额: $ 14.98万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1912704&HistoricalAwards=false
• 关键词: Direct; Inverse; Scattering; Problems; Elastic

Scattering problems are concerned with the effect that an inhomogeneous medium has on an incident field. Driven by significant applications in diverse scientific areas such as radar and sonar, geophysical exploration, nondestructive testing, medical imaging, near-field optical microscopy, and nano-optics,the scattering problems have been extensively studied by many researchers, especially for acoustic and electromagnetic waves. However, many theoretical analysis and numerical computation are left undone for elastic waves due to the complexity of the underlying model equations. The research is multidisciplinary by nature and lies at the interface of mathematics, physics, engineering, and materials sciences. It will contribute towards better understandings of the complex physical and mathematical problems in scattering theory of elasticity. It has significant potential for advancing the frontiers of applied and computational mathematics, and for evolving new mathematics and science. The results of the proposed research activities will be disseminated through publications, seminars, minisymposia, conferences, and workshops. The PI will introduce an advanced graduate course and a graduate seminar series. These will aid in the recruitment and retention of talented students with diverse backgrounds throughout the academic pipeline. The software codes and new course materials developed in the project will be disseminated on a public website and will be available for download by the scientific community. The research and educational components will be integrated together to help to train a new generation of researchers and foster greater awareness and interests in applied and computational mathematics with particular applications to scattering theory among graduate students and postdocs. This project outlines a three-year research plan for developing effective mathematical models, examining fundamental mathematical issues, and designing efficient computational methods for new and important classes of direct and inverse scattering problems in elastic waves. The proposed research builds on the PI?s prior research accomplishments in the area of scattering theory for acoustic and electromagnetic waves. It concerns the following three topics: (1) time-domain obstacle scattering problem; (2) time-harmonic medium scattering problem; (3) inverse random source scattering problem. The mathematical modeling and analysis techniques and computational methods developed in this project will address several key scientific challenges and open problems in direct and inverse scattering theory for elastic waves, which include modeling and computation of the elastic wave propagation in an inhomogeneous medium, numerical solution of the elastic wave equations and well-posedness of the associated model, uniqueness and stability of stochastic inverse source scattering problem. The proposed computational models and tools are highly promising for quantitative study of the complex physical and mathematical problems in elasticity. They have great potentials to provide inexpensive and easily controllable virtual prototypes of the structures in the design and fabrication of novel elastic devices.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.

散射问题涉及非均匀介质对入射场的影响。在雷达和声纳、地球物理勘探、无损检测、医学成像、近场光学显微镜和纳米光学等科学领域的重要应用的推动下，散射问题得到了许多研究人员的广泛研究，特别是声波和电磁波的散射问题。然而，由于基本模型方程的复杂性，许多关于弹性波的理论分析和数值计算都没有完成。这项研究本质上是多学科的，位于数学、物理、工程和材料科学的交界处。这将有助于更好地理解弹性散射理论中复杂的物理和数学问题。它在推进应用数学和计算数学的前沿领域，以及发展新的数学和科学方面具有巨大的潜力。拟议研究活动的结果将通过出版物、研讨会、小型研讨会、会议和讲习班传播。PI将推出高级研究生课程和研究生研讨会系列。这些将有助于在整个学术过程中招聘和留住具有不同背景的有才华的学生。在该项目中开发的软件代码和新的课程材料将在一个公共网站上散发，供科学界下载。研究和教育部分将被整合在一起，以帮助培训新一代研究人员，并培养对应用数学和计算数学的更大意识和兴趣，特别是在研究生和博士后中应用分散理论。该项目概述了一项为期三年的研究计划，该计划旨在开发有效的数学模型，研究基本的数学问题，并为新的和重要的弹性波中的正反向散射问题设计有效的计算方法。建议的研究建立在皮埃尔？S先前在声波和电磁波散射理论领域的研究成果的基础上。它涉及以下三个问题：(1)时域障碍物散射问题；(2)时间谐和介质散射问题；(3)随机源逆散射问题。该项目中发展的数学建模和分析技术和计算方法将解决弹性波正、逆散射理论中的几个关键科学挑战和公开问题，包括弹性波在非均匀介质中传播的建模和计算、弹性波方程的数值解及其相关模型的适定性、随机逆来源散射问题的唯一性和稳定性。所提出的计算模型和工具对于定量研究弹性力学中复杂的物理和数学问题是非常有前途的。在新型弹性设备的设计和制造中，它们具有提供廉价且易于控制的结构虚拟原型的巨大潜力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLift

• DOI: 10.1088/1361-6420/abe6f0
• 发表时间: 2020-12
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Yuxuan Gong;Peijun Li;Xu Wang;Xiang Xu

An inverse random source problem for the one-dimensional Helmholtz equation with attenuation

• DOI: 10.1088/1361-6420/abcd43
• 发表时间: 2020-09
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Peijun Li;Xu Wang

Inverse Random Source Scattering for the Helmholtz Equation with Attenuation

• DOI: 10.1137/19m1309456
• 发表时间: 2019-11
• 期刊: SIAM J. Appl. Math.
• 作者: Peijun Li;Xu Wang

An adaptive finite element DtN method for the elastic wave scattering problem

• DOI: 10.1007/s00211-022-01273-4
• 发表时间: 2022-03
• 期刊: Numerische Mathematik
• 影响因子: 2.1
• 作者: Peijun Li;Xiaokai Yuan

An adaptive finite element DtN method for the open cavity scattering problems

• DOI: 10.4208/csiam-am.2020-0013
• 发表时间: 2020-04
• 期刊: ArXiv
• 作者: Xiaokai Yuan;Gang Bao;Peijun Li