Efficient Numerical Methods for Time-harmonic Acoustic Wave Propagation

该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，特别是考虑了非均匀介质中声散射的二维和三维外部问题以及流体-结构散射问题。控制PDE模型的离散化是在局部适应边界和界面的正交矩形网格上使用有限元进行的。研究并采用了特殊的减小相位误差的低阶有限元。其目的是解决计算区域直径从几十到几百个波长的三维时间谐波传播问题。这些问题导致了非常大的线性方程系统，其中可能有数百万到数十亿个未知数。针对这一频率范围的数值方法尚处于研究阶段，目前尚无有效的方法。大规模线性系统采用Schwarz型区域分解预条件迭代求解。通过区域嵌入的方法，利用快速直接求解器构造了非常有效的子域预条件，这使得所提出的方法非常快速。对于适当的模型问题，分析了预条件系统的条件性。计算方法的构造使得迭代可以在与界面相关的一个小的稀疏子空间上进行。所研究的散射问题经常出现在许多学科中，由于现有方法和计算资源的限制，许多问题在实践中无法解决。这项研究的目的是发展数值方法，使许多这样的大问题能够得到解决。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，使用粗略的方法进行调查是必要的，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开采的产量并减少对环境的影响。第二类模型问题描述了海底中像地雷这样的弹性目标的散射。在海战中，地雷一直是造成人员伤亡的最大原因。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开发有助于加强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.