Efficient Methods for Electromagnetic and Acoustic Problems

研究人员提出了一系列新的算法，以满足声学和电磁问题对计算效率和/或精度的日益增长的需求。该项目的主要思想在于充分结合有限元、积分方程组和区域分解方法等多种数值技术。研究人员计划利用Dirirchlet对Neumann算子的适当逼近，开发一种求解Maxwell方程的准最优非重叠区域分解方法。对于部分涂覆的介质目标，提出了如下建议：(1)设计条件良好的积分方程组；(2)利用这些新的积分公式，引入一种新的稳健区域分解方法，其中迭代算子仅定义在孔径界面上。还提出了探索飞机和卫星等大型复杂物体的混合算法。特别是，研究人员计划将有限元、Dirirchlet到Neumann映射的定位技术和子结构方法结合起来，以处理具有大型平台的散射体，其中连接了介电对象和深腔。并行计算和数学分析将被用于帮助实现这些目标。拟议的项目涉及改进所需的计算工具，以应对迅速增长的工程和工业需求。事实上，声波和电磁波的计算是一个广阔的研究领域。这在很大程度上是由于应用的广泛性，其中许多应用对系统提出了技术要求，如降噪、海洋散射、光纤、隐身技术、雷达设计、遥感等。在开发由许多非常不同的部件组成的现代飞机时，工程师们使用高性能计算和创新的数学算法来增强性能和优化乘客安全。这一程序降低了设计成本，并允许对新技术进步做出快速反应，同时将能耗降至最低。通过这一研究计划获得的成果将随时提供给航空航天行业的工程师和科学家，这将有助于加强美国在该领域的领导地位。此外，这项工作还可用于水文中的地下水流动、石油工程中的采油以及流体在人体组织中的流动。这个项目的几个方面将有利于本科生和研究生的教育，并将培训他们最先进的科学计算和数学分析。这将加强他们面对未来科技挑战的准备。

The investigator proposes a family of new algorithms to respond to the increasing demand for computational efficiency and/or accuracy for acoustic and electromagnetic problems. The main idea of this project consists in adequately combining several numerical techniques such as finite elements, integral equations, and domain decomposition methods. The investigator plans to develop a quasi-optimal non-overlapping domain decomposition method for Maxwell's equations using an appropriate approximation of the Dirirchlet to Neumann operator. In the case of partially coated dielectric objects, the following are proposed: (1) design well-conditioned integral equations; and (2) use these new integral formulations to introduce a novel robust domain decomposition method, where the iteration operator is only defined on the aperture interface. It is also proposed to explore hybrid algorithms for large and complex bodies such as aircraft and satellites. In particular, the investigator plans to couple finite elements, localization techniques of the Dirirchlet to Neumann map, and substructuring methods to deal with scatterers with large platforms where dielectric objects and deep cavities are attached. Parallel computing and mathematical analysis will be used to help achieve these goals. The proposed project is concerned with the improvement of computational tools required to face rapidly increasing engineering and industrial needs. Indeed, the computation of acoustic and electromagnetic waves is a vast area of research. This is largely due to breadth of applications, many of which have imposed technological requirements in systems, such as noise reduction, oceanic scattering, optical fibers, stealth technology, radar design, remote sensing, and many others. In developing modern aircraft, which consist of many very different components, engineers use high performance computing and innovative mathematical algorithms to enhance performance and optimize passenger safety. This procedure reduces the cost of the design, and allows to rapid response to new technological advances as well as minimization energy consumption. The results obtained through this research plan will be made readily available to engineers and scientists in the aerospace industry, which will contribute to enhancing U.S leadership in this field. In addition, this work can be used in the areas of underground water flow in hydrology, oil recovery in petroleum engineering and fluid flow through body tissues. Several aspects in this project will benefit the education of both undergraduate and graduate students, and will train them in state-of-art scientific computing and mathematical analysis. This will reinforce their preparation to face future challenges in science and technology.