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

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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.