Efficient Methods for Electromagnetic and Acoustic Problems

研究人员提出了一系列新的算法，以应对日益增长的需求，计算效率和/或准确性的声学和电磁问题。 该项目的主要思想是充分结合几种数值技术，如有限元，积分方程和区域分解方法。研究人员计划开发一个准最佳的非重叠区域分解方法的麦克斯韦方程使用适当的近似的Dirirchlet诺依曼算子。在部分涂覆的电介质对象的情况下，提出了以下内容：（1）设计良好条件的积分方程;和（2）使用这些新的积分公式来引入一种新的鲁棒区域分解方法，其中迭代算子仅定义在孔径界面上。它还建议探索混合算法的大型和复杂的机构，如飞机和卫星。特别是，研究人员计划耦合有限元，局部化技术的Dirirchlet诺依曼地图，和子结构的方法来处理散射体与大型平台的电介质对象和深腔连接。并行计算和数学分析将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，以满足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度，其中许多应用对系统提出了技术要求，例如降噪、海洋散射、光纤、隐身技术、雷达设计、遥感和许多其他技术。 在开发由许多非常不同的部件组成的现代飞机时，工程师使用高性能计算和创新的数学算法来提高性能并优化乘客安全。这一过程降低了设计成本，并允许对新技术进步做出快速反应，以及最大限度地减少能耗。通过这项研究计划获得的结果将随时提供给航空航天工业的工程师和科学家，这将有助于加强美国在这一领域的领导地位。此外，这项工作还可用于水文学中的地下水流动、石油工程中的石油开采以及人体组织中的流体流动等领域。 这个项目的几个方面将有利于本科生和研究生的教育，并将培养他们在最先进的科学计算和数学分析。这将加强他们面对未来科学和技术挑战的准备。

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.