Efficient Numerical Methods for Time-harmonic Acoustic Wave Propagation

本研究发展并分析了时谐声波在流体和固体中传播问题的有效数值方法。特别考虑了非均匀介质中声散射和流固散射的外部二维和三维问题。在正交矩形网格上采用局部自适应边界和界面的有限元方法对控制模型进行离散化。研究并应用了特殊相位误差减小低阶有限元。其目的是解决计算域直径从几十到几百波长的三维时谐波传播问题。这些问题导致了非常大的线性方程组，其中可能有数百万到数十亿个未知数。对于这一频率范围的数值方法正在积极研究中，但似乎还没有有效的方法。采用schwarz型域分解预调节器迭代求解大型线性系统。采用域嵌入的方法，利用快速直接求解器构造了非常高效的子域预条件。这种新颖性使得所提出的方法非常快。对预调节系统的调节进行了分析，找出合适的模型问题。所得到的方法是这样构造的，迭代可以在与接口相关的小的稀疏子空间上进行。所研究的散射问题在许多学科中经常出现。由于现代方法和计算资源的限制，许多问题在实践中无法解决。本研究的目的是发展数值方法，使许多这类大问题的解决成为可能。第一类研究的模型问题是用于石油勘探的声学地质调查。目前有必要使用粗糙的方法进行调查，这些方法在许多情况下是不准确的。所开发的方法最终可以提高石油的采收率，减少对环境的影响。第二类模型问题描述了海底地雷等弹性物体的散射。水雷是海战中造成人员伤亡的最大单一原因。这部分研究是与海军研究中心的研究人员合作进行的，它与海军研究办公室资助的一个项目密切相关。这项活动的一个更广泛的目标是发展有助于提高海底地雷探测和识别能力的计算方法。

The proposed reserch develops and analyzes efficient numerical methodsfor time-harmonic acoustic wave propagation problems in fluids and solids.Particularly, exterior two-dimensional and three-dimensional problems foracoustic scattering in inhomogenous media and fluid-structure scatteringare considered. The discretization of the governing PDE models is performedusing finite elements on orthogonal rectangular meshes with local adaptationto boundaries and interfaces. Special phase error reducing low-order finiteelements are studied and employed. The aim is to solve three-dimensionaltime-harmonic wave propagation problems in which the diameter of thecomputational domain is from tens to hundreds wavelengths. These problemslead to very large systems of linear equations which can have from millionsto several billions unknowns. Numerical methods for this frequency rangeare under active research, and no efficient method still seem to exist.The large scale linear systems are solved iteratively with a Schwarz-typedomain decomposition preconditioner. Very efficient subdomain preconditionersare constructed using fast direct solvers by means of domain embedding.This novelty makes the proposed approach very fast. The conditioning ofpreconditioned systems are analyzed for suitable model problems. Theresulting methods are constructed in such a way that the iterations canbe carried out on a small sparse subspace related to the interfaces.The studied scattering problems arise routinely in many disciplines.Due to limitations of contemporary methods and computational resourcesmany of these problems cannot be solved in practice. The aim of thisresearch is to develop numerical methods which enable the solution ofmany of these large problems. The first class of studied model problemsare acoustical geological surveys employed in oil exploration. Currentlyit is necessary to use crude methods for surveys which are in manycases inaccurate. The developed methods can eventually lead to improvedyield in the extraction of oil and reduced environmental impact.The second class of model problems describe scattering by elastic objectslike mines in seabed. Mines have been the single largest cause of fatalitiesin naval warfare. This part of the research is conducted in cooperationwith researchers at a Navy research center and it is closely related toa project funded by the Office of Naval Research. A broader goal of thisactivity is develop computational methods which help to enhance thedetection and identification capabilities of mines in seabed.