Study On Fast Multipole Boundary Element Methods For Acoustic Characteristics Prediction Of Complex Structures

The boundary element method is a powerful numerical tool for solving the acousticproblems of complex structures. However, the conventional boundary element method(CBEM) is not suitable for the largle acoustic problems since its full-populated systemmatrices and the arising computing cost, memory requirements and computational time oforder N2. By contrast, the fast multipole boundary element method (FMBEM) showsexcellent computational efficiency, but it can not be applied directly to calculate the internalsound field of complex structures with thin wall components and absorbing materials filled.In view of the above facts, two fast approaches are developed for computation of thelarge-scale acustic problems in the present paper:(1) the substructure fast multipole boundaryelement method (Sub-FMBEM) which is suitable for the internal sound field calculation ofhybrid structures,(2) the fast multipole dual boundary element method (FMDBEM) which isapplicable to calculate the inernal sound field of complex reactive structures with thin wallcomponents. The detailed works are described as below:The theoretical foundamentals and numerical procedure of the multi-level FMBEM(MLFMBEM) for the calculation of internal three-dimensional sound filed are introduced indetail. The studies indicated that, the computing cost and memory requirements of FMBEMcan achieve O(NlnN). The FMBEM of three levels is employed to calculate the transmissionloss of silencers with simple structure, and its computational accuracy and efficiency for largenumber of meshes are confirmed by comparing with CBEM.The Sub-FMBEM which is suitable for the calculation of internal sound fields withhybrid structures is formed by applying the substructure technique to FMBEM, and the basicprinciple and the computational process of entier matrix-vector multiplication are introducedin detail, the single level and multi-level translation relationship calculation for differentsound-absorbing materials are investigated. The studies demonstrated that, the evaluation ofparameter used in calculation of truncation number has influence on the computation ofGreen’s function expansion to a certain degree, and the kind of sound-absorbing materials isnot the main factor to affact the computational accuracy of Green’s function expansion. Theintroduction of complex wavenumber leads to the calculation error of the expansion, and theexpansion value diverge from the theoretical value when the productk i i rLMof imaginary partof wavenumber and distance between expansion points is bigger than13.5. So when the valueofk i i rLMfalls into the distortion range, the value of transfer factor in the expansion should be approximated as zero, or to divide the large structure into several small substructures whichcan be handled respectively. The ILUT preconditioning technique and Bi-CGSTAB(l) solverare employed for the iterative computation of Sub-FMBEM in present paper. For thecalculation in the practical programing, the arrangement order of unknown column vector andthe numbering of nodes have great influence on the speed of convergence, a principle wasproposed for composition of the global coefficient matrix of Sub-FMBEM considering thestudy of the sound field for a multi-substructure model. The Sub-FMBEM is applied topredict the acoustic performance of reactive and hybrid silensers, and its computationalaccuracy is validated by comparing the numerical results from Sub-BEM and experiments. Interms of computational efficiency, the Sub-FMBEM shows obvious advantage at largenumber of meshes, but its computational efficiency is slightly lower than Sub-BEM at thesmall number of meshes. In terms of the character of iterative convergence, the precomputingtime and iterative computational time of Sub-FMBEM are increased as the frequency arising,while the time of Sub-BEM keeps unchanged almost. The freqency range in whichSub-FMBEM exhibits higher efficiency than Sub-BEM will become wider as number ofmeshes arising. Either in the pretreatment or the iteration stage, the growth rate of computingtime versus number of meshes for Sub-BEM is much higher than Sub-FMBEM, so theadvantage in the computational efficiency of FMBEM for large-scale acoustic problem atmedium-high frequency is validated. Besides, the introduction of sound-absorbing materialsnot only decrease number of iterative steps in some degree, but also make the variation moresteady versus the frenquency. Therefore, the Sub-FMBEM is considered as a suitable methodfor the calculation of the sound field in dispative structures.The boundary integral governing equations came from the dual boundary integralequations are derived to calculate the internal sound field in the complex reactive structures,and the FMM is used to accelerate the computation, therefor the FMDBEM is created tocalculate the internal sound field in complex reactive silencers with thin wall components,and the detail solving process and the regularization of hypersingular integral equations aregiven. The major advantage of FMDBEM is that it is not necessary to divide subdomain forthe thin wall structure and repeat discretizing the thin boundaries and virtual interfaces, thethin wall components only need to be discretized once (one side surface only), so number ofmeshes is reduced. The disadvantage of FMDBEM is that it is impossible to be applieddirectly to the calculation of sound filed in dissipative silencers with absorbing materials, andit is not suitable if the thickness of the wall components in the reactive silencers is not thinenough to be neglected. In addition, the singularity of hypersingular integrals existed in the governing equations has to be treated properly. By applying the FMDBEM, Sub-FMBEMand Sub-BEM to calculate the transmisson loss of complex reactive silencers, thecomputational accuracy and efficiency of FMDBEM are validated. For the same size ofmeshes, the FMDBEM may save the computational time in some degree since it reducednumber of meshes.In order to further examine the correctness of the numetical methods developed in thepresent paper, the transmission loss of straight-through perforated tube silencer and hybridsilencers are measured using the impendance tube measurement system combined withtwo-load method. Comparison of the experimental measurements and the numerical resultsfrom FMBEM demonstrated good agrements in wide frequency range, so the accuracy andapplicability of the numerical methods developed in this paper are further confirmed for thecalculation of internal sound field in complex structures.