Efficient Numerical Methods for Time-harmonic Acoustic Wave Propagation

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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.