三维高频声波的矩阵压缩边界节点法模拟
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摘要
本文采用边界节点法(Boundary Knot Method, BKM)求解三维高频声场.由于高频赫姆霍兹方程的解是振荡的,极大影响了数值求解的精确度,需要在计算区域增加离散点,这会增加计算量.同时对于大规模声学问题,依靠边界节点法形成的插值矩阵为满秩,导致计算量过高和存储量过大.所以本文采用矩阵压缩技术(Matrix Compression, MC),在有效继承边界节点法高精确度的基础上减少计算内存需求和时间,从而提高计算效率.数值实验表明,MC-BKM求解精度高、收敛速度快、计算时间少,在高频大规模声波问题中应用前景广泛.
In this paper, the boundary knot method is used to solve three-dimensional high frequency sound field. Because the solution of high frequency Helmholtz equation is oscillatory, which greatly affects the accuracy of numerical solution, it is necessary to add discrete points in the computational region, leading to the increase of calculation cost. At the same time, for large-scale acoustic problems, the interpolation matrix formed by the boundary knot method is full rank, resulting in excessive computation and excessive storage. The introduction of matrix compression technology can effectively reduce the memory requirement and time while at the same time effectively inherits the high accuracy of the boundary node method, leading to a significant improvement of the computational efficiency. Numerical experiments show that MC-BKM has high accuracy, fast convergence speed and less computation time, and has a wide application prospect in high frequency large scale acoustic wave problems.
In this paper, the boundary knot method is used to solve three-dimensional high frequency sound field. Because the solution of high frequency Helmholtz equation is oscillatory, which greatly affects the accuracy of numerical solution, it is necessary to add discrete points in the computational region, leading to the increase of calculation cost. At the same time, for large-scale acoustic problems, the interpolation matrix formed by the boundary knot method is full rank, resulting in excessive computation and excessive storage. The introduction of matrix compression technology can effectively reduce the memory requirement and time while at the same time effectively inherits the high accuracy of the boundary node method, leading to a significant improvement of the computational efficiency. Numerical experiments show that MC-BKM has high accuracy, fast convergence speed and less computation time, and has a wide application prospect in high frequency large scale acoustic wave problems.
引文
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[11]HON Y C,CHEN W.Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry[J].International Journal for Numerical Methods in Engineering,2003,56(13):1931-1948.
[12]HO K L,GREENGARD L.A fast direct solver for structured linear systems by recursive skeletonization[J].Siam Journal on Scientific Computing,2012,34(5):A2507-A2532.
[13]HO K L,YING L.Hierarchical interpolative factorization for elliptic operators:integral equations[J].Communications on Pure and Applied Mathematics,2016,69(7):1314-1353.
[2]毛崎波,徐柏龄.利用高分子压电膜设计声辐射模态传感器[J].声学学报:中文版,2003,28(3):262-266.
[3]周静雷,沈勇,邹建.长阻抗管过渡过程对于测量声学参数的影响研究[J].南京大学学报:自然科学版,2005,41(4):398-406.
[4]姜欣荣,陈文.基本解方法与边界节点法求解Helmholtz方程的比较研究[J].计算力学学报,2011,3:338-344.
[5]URAS R A,CHANG C T,CHEN Y,et al.Multiresolution reproducing kernel particle methods in acoustics[J].Journal of Computational Acoustics,1997,5(1):71-94.
[6]IHLENBURG F.Finite element analysis of acoustic scattering[M].New York:Springer-Verlag,1998.
[7]BOUILLARD P,LACROIX V,BEL E D.A meshless approach for 2D vibroacoustic problems[J].Revue Européenne DeséléMents Finis,2002,11(7-8):947-964.
[8]王福章,陈文.边界节点法求解高波数声波问题[C]//中国计算力学大会2010(CCCM2010)暨第八届南方计算力学学术会议(SCCM8)论文集,2010.
[9]潘一力.高频波在声波导中传播的计算[D].杭州:浙江大学,2011.
[10]陈文,傅卓佳,魏星.科学与工程计算中的径向基函数方法[M].北京:科学出版社,2014.
[11]HON Y C,CHEN W.Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry[J].International Journal for Numerical Methods in Engineering,2003,56(13):1931-1948.
[12]HO K L,GREENGARD L.A fast direct solver for structured linear systems by recursive skeletonization[J].Siam Journal on Scientific Computing,2012,34(5):A2507-A2532.
[13]HO K L,YING L.Hierarchical interpolative factorization for elliptic operators:integral equations[J].Communications on Pure and Applied Mathematics,2016,69(7):1314-1353.