复杂结构声学特性预测的快速多极子边界元法研究
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摘要
边界元法是求解复杂结构声学问题的一种有效数值方法。由于传统边界元法(CBEM)形成满秩的系数矩阵，其计算量、存储量和计算时间为O(N2)，限制了其在大尺度声学问题中的应用。相比之下，快速多极子边界元法(FMBEM)具有较高的计算效率，但是无法直接应用于带有薄壁结构以及含有吸声材料的复杂结构内部声场问题的计算。鉴于此，本文发展了两种适用于大尺度声学问题计算的快速算法：(1)适用于阻抗复合结构内部声场计算的子结构快速多极子边界元法(Sub-FMBEM)，(2)适用于具有多薄壁元件的复杂抗性结构内部声场计算的快速多极子双重边界元法(FMDBEM)。所做的具体工作如下：
详细阐述了多级快速多极子边界元法(MLFMBEM)计算内部三维声场问题的理论基础和数值实施过程。研究表明，MLFMBEM的计算量和内存量均能达到O(NlnN)。应用三级FMBEM对简单结构消声器的传递损失进行了数值计算，通过和CBEM比较验证了FMBEM的计算精度及在高网格数时的计算效率。
将子结构技术应用到FMBEM中，形成了适用于阻抗复合结构内部声场计算的Sub-FMBEM，详细介绍了该方法的基本原理和总体矩阵向量积的计算过程，对不同吸声材料下单级和多级传递关系计算进行了考察。研究表明，展开式截断项数计算参数的选取在一定程度上影响了格林函数展开式的计算，而吸声材料的选取并非是影响其计算精度的主要因素。复波数的引入使展开式计算产生偏差，当复波数虚部与展开点间距的积k i i rLM大于13.5时，展开式值逐渐偏离理论值。因此，当k i i rLM值处于失真范围时，可将展开式中转移因子的值以0来近似，或者应用Sub-FMBEM将大尺寸结构模型划分为若干小尺寸结构分别计算。本文采用ILUT预条件处理技术和Bi-CGSTAB(l)迭代求解器进行Sub-FMBEM的迭代计算。在实际的编程计算中，未知量列向量的构建次序及离散节点编号顺序对迭代法的收敛速度有着重要影响，结合多子结构模型的声场计算给出了Sub-FMBEM整体系数矩阵的构建方法和原则。应用Sub-FMBEM对抗性和阻抗复合型消声器的声学性能进行了计算，通过与Sub-BEM计算结果以及实验测量结果的比较验证了该方法的正确性和计算精度。在计算效率方面，Sub-FMBEM在高网格数时有明显优势，而在低网格数时则略低于Sub-BEM。在迭代收敛特性方面，Sub-FMBEM的预处理时间和迭代计算时间均随着频率的增大而增加，而Sub-BEM随频率基本不变。网格数越多，Sub-FMBEM相对于Sub-BEM能体现高效性的频率范围也越大。无论是预处理阶段还是迭代阶段，Sub-BEM的计算时间随网格数的增长率远远大于Sub-FMBEM，从而验证了FMBEM在大尺度中高频声学问题计算中的优势。此外，吸声材料的引入不但在一定程度上降低了Sub-FMBEM的迭代次数，而且使之随着频率的变化更为稳定。因此，Sub-FMBEM更适合于阻性结构声场的计算。
由双重边界积分方程出发，推导出了适用于复杂抗性结构内部声场计算的边界积分控制方程，并结合快速多极子算法加速求解，创建了适用于含多薄壁结构的复杂抗性消声器内部声场计算的FMDBEM，给出了具体的求解过程以及超奇异积分的处理方法。FMDBEM的主要优点在于：无需针对薄壁结构划分子结构，并对薄壁边界和虚拟交界面重复离散，只需对所有薄壁结构离散一次，从而减少了网格数量；其缺点在于：不能直接应用于含有吸声材料的阻性消声器声场计算，而且要求抗性消声器中薄壁结构的厚度必须足够小到可以忽略不计。此外，由于控制方程中未知量系数计算存在超奇异积分，需要在计算中进行适当地弱奇异化处理。通过应用FMDBEM、Sub-FMBEM和Sub-BEM对复杂抗性消声器传递损失进行计算，验证了FMDBEM的计算精度和计算效率。在同样的网格尺寸下，由于FMDBEM减少了网格数量，从而在一定程度上节省了计算时间。
为进一步检验本文所发展的数值方法的正确性，使用阻抗管测量系统结合两负载法测量了直通穿孔管消声器和阻抗复合型消声器的传递损失。比较了实验测量结果与FMBEM的计算结果，它们在较宽频域内吻合良好，从而进一步验证了本文数值方法的正确性和在复杂结构内部声场计算中的适用性。
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.
详细阐述了多级快速多极子边界元法(MLFMBEM)计算内部三维声场问题的理论基础和数值实施过程。研究表明，MLFMBEM的计算量和内存量均能达到O(NlnN)。应用三级FMBEM对简单结构消声器的传递损失进行了数值计算，通过和CBEM比较验证了FMBEM的计算精度及在高网格数时的计算效率。
将子结构技术应用到FMBEM中，形成了适用于阻抗复合结构内部声场计算的Sub-FMBEM，详细介绍了该方法的基本原理和总体矩阵向量积的计算过程，对不同吸声材料下单级和多级传递关系计算进行了考察。研究表明，展开式截断项数计算参数的选取在一定程度上影响了格林函数展开式的计算，而吸声材料的选取并非是影响其计算精度的主要因素。复波数的引入使展开式计算产生偏差，当复波数虚部与展开点间距的积k i i rLM大于13.5时，展开式值逐渐偏离理论值。因此，当k i i rLM值处于失真范围时，可将展开式中转移因子的值以0来近似，或者应用Sub-FMBEM将大尺寸结构模型划分为若干小尺寸结构分别计算。本文采用ILUT预条件处理技术和Bi-CGSTAB(l)迭代求解器进行Sub-FMBEM的迭代计算。在实际的编程计算中，未知量列向量的构建次序及离散节点编号顺序对迭代法的收敛速度有着重要影响，结合多子结构模型的声场计算给出了Sub-FMBEM整体系数矩阵的构建方法和原则。应用Sub-FMBEM对抗性和阻抗复合型消声器的声学性能进行了计算，通过与Sub-BEM计算结果以及实验测量结果的比较验证了该方法的正确性和计算精度。在计算效率方面，Sub-FMBEM在高网格数时有明显优势，而在低网格数时则略低于Sub-BEM。在迭代收敛特性方面，Sub-FMBEM的预处理时间和迭代计算时间均随着频率的增大而增加，而Sub-BEM随频率基本不变。网格数越多，Sub-FMBEM相对于Sub-BEM能体现高效性的频率范围也越大。无论是预处理阶段还是迭代阶段，Sub-BEM的计算时间随网格数的增长率远远大于Sub-FMBEM，从而验证了FMBEM在大尺度中高频声学问题计算中的优势。此外，吸声材料的引入不但在一定程度上降低了Sub-FMBEM的迭代次数，而且使之随着频率的变化更为稳定。因此，Sub-FMBEM更适合于阻性结构声场的计算。
由双重边界积分方程出发，推导出了适用于复杂抗性结构内部声场计算的边界积分控制方程，并结合快速多极子算法加速求解，创建了适用于含多薄壁结构的复杂抗性消声器内部声场计算的FMDBEM，给出了具体的求解过程以及超奇异积分的处理方法。FMDBEM的主要优点在于：无需针对薄壁结构划分子结构，并对薄壁边界和虚拟交界面重复离散，只需对所有薄壁结构离散一次，从而减少了网格数量；其缺点在于：不能直接应用于含有吸声材料的阻性消声器声场计算，而且要求抗性消声器中薄壁结构的厚度必须足够小到可以忽略不计。此外，由于控制方程中未知量系数计算存在超奇异积分，需要在计算中进行适当地弱奇异化处理。通过应用FMDBEM、Sub-FMBEM和Sub-BEM对复杂抗性消声器传递损失进行计算，验证了FMDBEM的计算精度和计算效率。在同样的网格尺寸下，由于FMDBEM减少了网格数量，从而在一定程度上节省了计算时间。
为进一步检验本文所发展的数值方法的正确性，使用阻抗管测量系统结合两负载法测量了直通穿孔管消声器和阻抗复合型消声器的传递损失。比较了实验测量结果与FMBEM的计算结果，它们在较宽频域内吻合良好，从而进一步验证了本文数值方法的正确性和在复杂结构内部声场计算中的适用性。
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.
引文
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