A Multipole Galerkin Boudary Element Method And Its Application Inacoustic Problems

With the rapid development of computational technology, numerical method iswidely used to solve acoustic problems in many areas of engineering, such astransportation, aerospace, mechanical and military fields. For the acoustic problems,the boundary element method (BEM) is one of the most developed and acceptednumerical methods. Since it process many advantages, for instances, reduce thedimensionalities of the problems by one, high accuracy and suitable to solveunbounded domain problems. However, one disadvantage makes the BEM falling tosolve large scale acoustic problems, that is, it leads to linear system equations withfully populated, sometime ill conditioned coefficient matrices. The storagerequirements and computational time to solve this kind of system equations arerelatively high. As a result, it is necessary to develop a new method, which inherits allthe advantages of the BEM and at the same time processes higher computationalefficiency. In this dissertation, the fast multipole method is introduced into Galerkinboundary element method, and a wideband multipole Galerkin boundary elementmethod is developed to solve large scale acoustic problems. The implementation ofthe wideband multipole Galerkin boundary element is investigated in detail. The mainresearch works and results of this dissertation are listed as follows:(1) The fast multipole method is introduced into the Galerkin BEM, a newwideband multipole Galerkin BEM is presented for solving two-dimensional acousticproblems. In order to remove the non-uniqueness problems associated withconventional BEM when solving exterior acoustic problems, the Burton-Millerformulation–a linear combination of the boundary integral equation and its normalderivative equation is employed. The hyper-singular integral is desingularized usingsome properties of Laplace equation and the singular subtraction approach, the newformulation of the hypersingular integral can be calculated efficiency. Based on thepartial wave expansion method and the plane wave expansion method, twoformulations of the fast method are developed for the low and high frequency acousticproblems repectively. In order to obtain overall computational efficiency in widebandfrequencies, a wideband multipole Galerkin BEM is proposed. It unifies previousexisting fast multipole method for low and high frequencies into an algorithm, which is accurate and efficient for any frequency. This wideband multipole Galerkin BEMhaving a CPU time of O(n) if low frequency computations dominate, or O(nlog2n) ifthe high frequency computations dominate. A brief procedure of this widebandmultipole Galerkin BEM is presented. During the process of construction of the quadtree, a modified definition of the interaction list, which can reduce the multipoleexpansion coefficient to local expansion coefficient translations is suitably adopted.Furthermore, an efficient preconditioning technique–spares approximate inverse pre-conditioner is employed to improve the convergence of the generalized minimalresidual (GMRES) solver. Finally, the numerical result of the rigid cylinder scatteringproblem demonstrates the accuracy and efficiency of the wideband multipole GalerkinBEM for two-dimensional acoustic problems. The sparse approximate inversepreconditioning technique dramatically reduces the iteration steps required by theGREMS solver, the total computational efficiency is further improved. A multi-bodyscattering problem with34cylinders is solved effectively. This example clearly showsthe great potential of the wideband multipole Galerkin BEM for engineeringapplications.(2) Based on the Burton-Miller formulation, a wideband multipole Galerkin BEMcombining with the low frequency and high frequency method is proposed to solve thethree-dimensional acoustic problems. The method evaluating the hyper-singularintegral in three dimension problem is differ from the method used in chapter two. Byusing the idea of regularization in sense of distributions, the double normalderivatives for hyper-singular integral are shifted to the boundary rotations of trialfunction and test function, then the hyper-singular integral is transformed to a weaklysingular form. To determine the number of terms in multipole and local expansions,an improved empirical formula is applied. Finally, several examples including an Lshaped box radiation problem, a sphere radiation problem and a sphere scatteringproblem are studied to investigate the accuracy and efficiency of the threedimensional wideband multipole Galerkin BEM.(3) The half space fundamental solution of the boundary integral equation isemployed, a new multipole Galerkin BEM for solving half space acoustic problems isproposed. Compared with full space method, a tree structure of the boundary elementsonly for the boundaries of the real domain need to be constructed. The implementationof the half space multipole Galerkin BEM is simplified, only the local expansion isdifferent from the full space method. Finally, the two-dimensional and three-dimensional examples validate the accuracy and efficiency of the half spacewideband multipole Galerkin BEM. By using the half space method, the half spaceacoustic problems can be solved with less CPU time.(4) The half space multipole Galerkin BEM is employed to predict the acousticperformance of three dimensional finite noise barrier, and the accuracy of the methodis validated. Firstly, a three dimensional upright noise barrier with finite length isestablished. The simulation result is coincidence with the result given in the referencebook, and the accuracy of the half space method is validated again. Secondly, underdifferent frequencies, the performance of T shape noise barrier is considered.Examination of the source and receiver location, the geometry parameters andtreatment of the T top surface is investigated. Finally, more engineering applicationsare expected to be simulated by the wideband multipole Galerkin BEM algorithm inthe future.