Efficient High Frequency Integral Equations and Iterative Methods

该项目以其巨大的潜在科学应用和广泛的教育活动而闻名。事实上，在研究活动中开发的数值算法适用于物理学，声学/电磁学和其他学科的现实配置。这些领域中的主要理论和计算困难是由于复杂因素的存在，例如复杂的几何形状（包括飞机、卫星、雷达、天线等），高频散射等等。所获得的方法将提供准确模拟此类系统的能力，以便应用于工程车辆和设备的设计，包括军用和非军用雷达，遥感卫星，降噪，隐形技术以及许多其他将受到该提案结果积极影响的技术。获得的求解器将随时提供给工业科学家，这将有助于保持他们的竞争力，特别是在航空航天工业。教育影响将在几个领域产生重大影响。研究生和本科生将在科学计算和数学分析方面接受严格的培训，以使他们能够面对未来的科学和技术挑战。 他们将获得最先进的应用数值方法所需的技能，这将为他们提供加入高科技行业的绝佳机会，并在事业成功的同时为进一步推进美国技术做出贡献。研究人员计划开发高效和准确的算法，用于复杂结构中的声波/电磁波传播问题。新提出的研究活动将对数值方法和数学的进步产生重大影响，并将产生一个新的数值算法家族，其功能将比目前可用的数值算法更强。研究人员计划开发一个强大的非重叠区域分解方法的亥姆霍兹方程的基础上利用优化的传输条件的人工接口和适当使用的自适应辐射条件技术。这也将允许耦合有限元和边界元的有效算法的设计。在高频区域的情况下，研究者建议（1）使用Helmholtz方程的解的渐近展开，即总场的法向导数，严格地开发O（1）高频积分方程求解器，（2）分析所得算法的稳定性和收敛性，以及（3）将高性能计算与新提出的方法适当地结合以有效地解决现实生活中的问题。并行计算和数学分析将用于帮助实现这些目标。

The project is distinguished by its great wealth of potential scientific applications and broad educational activities. Indeed, the numerical algorithms to be developed in the research activities are applicable to realistic configurations in physics, acoustic/electromagnetic, and other disciplines. The major theoretical and computational difficulties in these fields result from the presence of complicating factors such as complex geometries (including aircraft, satellites, radars, antennas, etc.), high-frequency scattering, amongst others. The obtained methods will provide the ability to simulate such systems accurately in order to be applied to the design of engineering vehicles and devices, including military and non-military radar, remote sensing satellites, noise reduction, stealth technology, and many others that will be positively impacted by the results of this proposal. The solvers obtained will be made readily available to industrial scientists, which will contribute to maintaining their competitiveness in particular in the aerospace industry. The educational impact will be significant in several areas. Graduate and undergraduate students will be rigorously trained in both scientific computing and mathematical analysis in order to enable them to face future challenges in science and technology. They will acquire the skills needed in state-of-the-art in applied numerical methods, and this will provide them great opportunities to join high technological industries and contribute in further advancing the U.S technology while having a successful career.The investigator plans to develop efficient and accurate algorithms for acoustic/electromagnetic wave propagation problems in complex structures. The new proposed research activities will have a significant impact in enabling advances in numerical methods and mathematics, and will result in a new family of numerical algorithms with enhanced capabilities over those currently available. The investigator plans to develop a robust non-overlapping domain decomposition method for the Helmholtz equation based on the utilization of optimized transmission conditions on the artificial interfaces and appropriate use of the adaptive radiation condition technique. This also will allow the design of an effective algorithm coupling finite and boundary elements. In the case of the high frequency regime, the investigator proposes to (1) use asymptotic expansions of solutions of the Helmholtz equation, namely the normal derivative of the total field, to rigorously develop a O(1) high frequency integral equations solver, (2) analyze the stability and convergence of the resulting algorithms, and (3) suitably combine high performance computing with the new proposed methods to efficiently tackle real-life problems. Parallel computing and mathematical analysis will be used to help achieve these goals.