Efficient and stable spectral methods for acoustic scattering over bounded obstacles

Qirong Fang, Purdue University

Abstract

The original work of this thesis consists of three parts: Part A. Acoustic scattering over 3-D bounded-obstacles. An efficient and high–order algorithm for three–dimensional bounded obstacle scattering is developed (cf. [1]). The method is a non–trivial extension of recent work of Nicholls and Shen [2] for two–dimensional bounded obstacle scattering, and is based on a Boundary Perturbation technique coupled to a well–conditioned high–order Spectral–Galerkin solver. This Boundary Perturbation approach is justified by rigorous theoretical results on analyticity of the scattered field with respect to boundary variations which show that, in fact, the domain of analyticity can be extended to a neighborhood of the entire real axis. The numerical method is augmented by Padé approximation techniques to access this region of extended analyticity so that configurations which are large deformations of the base (spherical) geometry can be simulated. Several numerical results are presented to exemplify the accuracy, stability, and versatility of the proposed method. Part B. Acoustic scattering over an elongated obstacles. While in principle the algorithms in part A and [2] can be applied to elongated scatterers (e.g., submarines and airfoils), which are found in many important applications, it may not be computationally efficient to do so due to the fact that large artificial boundaries are needed to enclosed the elongated obstacles. In such cases, it is more appropriate to use elliptic and ellipsoidal artificial surfaces to truncate the unbounded computational domains. The purpose of this part is to develop an efficient and accurate numerical method for the acoustic scattering from an elongated obstacle. The basic idea is to consider an elongated obstacle as a perturbation of ellipse in 2-D and of ellipsoid in 3-D, use a larger ellipse or ellipsoid to enclose the obstacle and reduce the problem to a bounded domain through the Dirichlet-to-Neumann mapping, and then develop an efficient and accurate spectral method for the reduced equation in the separated elliptic domain. Several numerical results are presented for ellipse in 2-D case and we leave the more complicated 3-D ellipsoid case for future work. Part C. Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains. An error analysis is presented here for the spectral-Galerkin method to the Helmholtz equation in 2-D exterior domains. The problem in unbounded domain is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. A main difficulty here is to obtain error estimates with explicit dependence on the high wave number. In order to illustrate the essential procedures for error analysis of spectral methods, we will only consider the case where the wave number k is sufficient small. The error analysis can be extended for general wave numbers by using the usual techniques used in [3].

Degree

Ph.D.

Advisors

Shen, Purdue University.

Subject Area

Mathematics

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