Hybrid Algorithms for Wave Propagation

该项目专注于开发创新和高效的算法，致力于解决声波和电磁波传播问题。该策略包括利用区域分解设计先进的数值技术，以获得较高的计算效率，并提高收敛性和精度。该方法还允许适当地利用并行计算。研究者关心两类问题。第一种方法是利用区域分解方法将有限元方法与边界元方法相结合，以及将有限元方法与渐近技术相结合。在第二类中，研究者提出了将区域分解方法与特定的积分方程方法相结合的方法来解决高频区域的多散射体问题。由此产生的算法绕过了在波长尺度上解析的需要，同时保留了误差可控性。本文将研究一种新的krylov -子空间方法，该方法能显著提高迭代过程的收敛性。这种方法将减少获得给定精度所需的计算时间。将进行仔细的数学分析，以帮助实现这些目标。本提案的主要焦点是开发创新和高效的算法，专门用于声学和电磁波传播问题。该策略包括仔细的数学分析和高效算法的设计。这项工作是高度跨学科的，对先进的技术应用有影响，包括在电信、飞机设计和海洋学等领域出现的问题。例如，大型几何复杂结构对入射电磁辐射的响应是航空航天工业非常感兴趣的话题，航空航天工业寻求提高机载飞行器的隐身能力。工程师经常使用计算工具来预测飞机的雷达特征，这有助于最大限度地降低设计成本。显然，这种努力将大大有助于发展计算效率高和严格的数值方法。这个项目的某些方面将得到工程师们积极合作的支持。

This project is focused on the development of innovative and efficient algorithms dedicated to solving problems of acoustic and electromagnetic wave propagation. The strategy consists of using domain decomposition to design advanced numerical techniques for obtaining high computational efficiency, and improved convergence and accuracy properties. The proposed approach also allows for suitable utilization of parallel computing. The investigator is concerned with two classes of problems. The first consists of using domain decomposition methods to suitably combine, (1) finite element methods with boundary element methods, and (2) finite element methods with asymptotic techniques. In the second class, the investigator proposes to couple domain decomposition methods with a specific integral equation method for problems concerning multiple scatterers in the high frequency regime. The resulting algorithm bypasses the need to resolve at the wavelength scale while retaining error-controllability. A new Krylov-subspace method that significantly improves convergence of the iterative procedure will be investigated. This approach will decrease the computational time required to obtain a given accuracy. A careful mathematical analysis will be conducted to help achieve these goals.The main focus of this proposal is the development of innovative and efficient algorithms dedicated to problems of acoustic and electromagnetic wave propagation. The strategy consists of careful mathematical analysis and design of high efficiency algorithms. The work is highly interdisciplinary and has impact on advanced technological applications including problems arising in areas such as telecommunications, aircraft design, and oceanography. For example, the response of large, geometrically complex structures to incoming electromagnetic radiation is a topic of great interest to the aerospace industry, which seeks to improve stealth capabilities of airborne vehicles. Engineers regularly employ computational tools to predict the radar signature of aircraft, which helps to minimize design costs. Clearly, such efforts will be greatly aided by the development of computationally efficient and rigorous numerical methods. Some aspects of this project will be supported by an active collaboration with engineers.