Free Surface Fluid Mechanics and Electromagnetic Scattering: Stable, High-Order Perturbation Techniques

摘要/ Abstract摘要：自由表面流体力学和电磁散射：稳定的高阶微扰技术首席研究员（PI）提出了研究自由表面理想流体流动的基本现象，以及电磁和声散射。这些应用包括高效、稳定、高阶的三维运动、毛细管重力水波的计算，以及它们的动力稳定性的数值研究。PI将解决的另一个问题是对无界域上的微分方程施加“非反射”边界条件，特别是在电磁和声散射的情况下。PI还提出了一个综合他的研究的两个主要领域，通过研究从行进的海浪电磁辐射的后向散射返回。这个项目的一个统一元素，也是PI将使用的主要数值工具，是一类边界摄动方法，该方法首先由O. Bruno和F. Reitich在声学和电磁散射问题的数值模拟中引入。这些方法随后被Reitich和PI用于计算Dirichlet-Neumann算子和近似散射构型的bvpp进行了显著的改进和稳定。在这些改进中，PI &amp； Reitich开发了“变换场展开”（TFE）方法，该方法能够可靠，高阶，稳定地计算BVP和FBP。虽然这种方法在解决模拟问题方面非常成功，但与其他技术（例如边界积分/元素）相比，它在计算复杂性方面有些劣势。PI提出的最后一个项目是对该方法的两种改进进行研究，以提高其效率。固定的和自由的边界问题出现在工程和科学的所有领域。在这个建议中有两个特殊的相关实例，一个是表面波在大水体（如湖泊、海洋或海洋）中运动的经典自由边界问题，另一个是电磁波或声波从不规则表面散射的固定边界问题。表面波的精确和可靠的模拟不仅用于模拟开放海洋结构（如石油平台）的能力，而且还用于研究污染物和其他对环境有兴趣的物质在湖泊、海洋和海洋上的运输。电磁散射和声散射的应用出现在雷达、成像和传感问题中，仅举几例。许多数值技术可用于模拟这些问题，本提案的目标之一是利用和改进由首席研究员（PI）和合作者发现和改进的一类技术。特别是在全三维模拟时，计算量变得非常大，真正成为高性能计算的问题。

Abstract 0406007, Nicholls, University of Notre DameFree Surface Fluid Mechanics and Electromagnetic Scattering: Stable, High-Order Perturbation Techniques The Principal Investigator (PI) proposes the investigation offundamental phenomena in free-surface ideal fluid flows, andelectromagnetic and acoustic scattering. Among the applications arethe efficient, stable, high-order computation of travelingthree-dimensional, capillary-gravity water waves, and a numericalinvestigation of their dynamic stability. Another problem the PI willaddress is the imposition of "non-reflecting" boundary conditions fordifferential equations posed on unbounded domains, particularly in thesetting of electromagnetic and acoustic scattering. The PI alsoproposes a synthesis of two major areas of his research through thestudy of backscattering returns of electromagnetic radiation fromtraveling ocean waves. A unifying element in this project, and theprincipal numerical tool the PI will employ, is a class of boundaryperturbation methods first introduced by O. Bruno & F. Reitich in thecontext of numerical simulation of acoustic and electromagneticscattering problems. These methods have subsequently beensignificantly refined and stabilized by Reitich and the PI for the BVPof computing Dirichlet-Neumann operators, and approximating scatteringconfigurations. Among these refinements, the PI & Reitich developedthe method of "Transformed Field Expansions" (TFE) which enables thereliable, high-order, stable perturbative computation of BVP and FBP.While this method is extremely successful in resolving simulationswell outside the reach of competing methods, it is somewhatdisadvantaged in terms of computational complexity in comparison toother techniques (e.g. boundary integrals/elements). A final projectthat the PI proposes is the investigation of two refinements of thisTFE approach to increase its efficiency.Fixed and free boundary problems arise in all areas of engineering andthe sciences. Two particular instances of relevance in this proposalare the classic free boundary problem of the motion of surface waveson a large body of water (e.g. a lake, sea, or ocean), and the fixedboundary problem of scattering of electromagnetic or acoustic wavesfrom an irregular surface. The accurate and reliable simulation ofsurface waves is used not only to model the capabilities of open-oceanstructures (e.g. oil platforms), but also in the study of transport ofpollutants and other substances of environmental interest acrosslakes, seas, and oceans. Applications of electromagnetic and acousticscattering come in problems of radar, imaging, and sensing to namejust a few. Many numerical techniques are available for thesimulation of these problems and one of the goals of this proposal isthe utilization and improvement of a class of techniques discoveredand refined by the Principal Investigator (PI) and collaborators. Inparticular, when the problems mentioned above are simulated in fullthree dimensions, the computations become quite intensive and theissues truly become those of high-performance computing.