大规模宽频弹性动力学分析的快速定向压缩边界元法

doi:10.19596/j.cnki.1001-246x.7866

摘要/Abstract

摘要： 发展一种大规模宽频弹性动力学分析的快速定向压缩边界元法.证明弹性动力学核函数具有定向低秩特性，为采用快速定向压缩算法提供理论基础.根据S波的波数，将节点之间的相互作用划分为低频相互作用和高频相互作用，并将后者进一步划分为与多个楔形区.在楔形区上，可以采用核函数的定向低秩特性进行快速计算.低频相互作用与核无关快速多极边界元法中计算方法相同，不同方向楔形区上的变换矩阵可以采用坐标系旋转的方法进行快速计算.可对任意频率进行快速谐响应分析.数值算例表明：该方法可以将宽频弹性动力学问题计算复杂度降低到O（N log^αN）.与卷积求积边界元法相结合，也可以应用于弹性动力学瞬态分析.

关键词: 边界元法, 弹性动力学, 快速定向压缩算法, 宽频

Abstract: A fast directional boundary element method for large scale wideband elastodynamic analysis is developed. Directional low rank property of elastodynamic kernels is shown which serves as the theoretical basis of its fast directional algorithm. By only considering S-wave number, interactions of different nodes are divided into low-frequency interactions and high-frequency interactions, and the latter is further divided into interactions with directional wedges on which the directional low rank property is applied. Low-frequency interactions are computed in same manner with that in kernel independent fast multipole BEM for elastodynamics, and translation matrices for different directional wedges are calculated efficiently by coordinate frame rotations. Thus harmonic responses for any frequencies can be computed efficiently. Numerical examples show that the computational complexity for wideband elastodynamic problems are successfully brought down to O(N log^αN). It can also be applied to transient elastodynamic analysis combined with convolution quadrature method.

Key words: boundary element method, elastodynamics, fast directional algorithm, wideband

中图分类号:

O242.1

曹衍闯, 校金友, 文立华, 王政. 大规模宽频弹性动力学分析的快速定向压缩边界元法[J]. 计算物理, 2019, 36(3): 305-316.

CAO Yanchuang, XIAO Jinyou, WEN Lihua, WANG Zheng. Fast Directional Boundary Element Method for Large Scale Wideband Elastodynamic Analysis[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2019, 36(3): 305-316.

参考文献

[1] CHATTERJEE J, MA F, HENRY D P, et al. Two- and three-dimensional transient heat conduction and thermoelastic analyses by BEM via efficient time convolution[J]. Computer Methods in Applied Mechanics and Engineering., 2007, 196(29-30):2828-2838.
[2] WU H J, LIU Y J, JIANG W K. A fast multipole boundary element method for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation[J]. Engineering Analysis with Boundary Elements, 2012, 36(5):779-788.
[3] WU H J, LIU Y J, JIANG W K, et al. A fast multipole boundary element method for three-dimensional half-space acoustic wave problems over an impedance plane[J]. International Journal of Computational Methods, 2015, 12(1):1350090.
[4] HUANG S, XIAO J Y, HU Y C, et al. GPU-accelerated boundary element method for Burton-Miller equation in acoustics[J]. Chinese Journal of Computational Physics, 2011, 28(4):481-487.
[5] ZHANG R, WEN L H, XIAO J Y. GPU-accelerated boundary element method for large-scale problems in acoustics[J]. Chinese Journal of Computational Physics, 2015, 32(3):299-309.
[6] YAN Z Y, GAO X W. The development of the pFFT accelerated BEM for 3-D acoustic scattering problems based on the Burton and Miller's integral formulation[J]. Engineering Analysis with Boundary Elements, 2013, 37(2):409-418.
[7] SHEN L, LIU Y. An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton-Miller formulation[J]. Computational Mechanics, 2007, 40(3):461-472.
[8] MOLINET F, STÈVE H, ADAM J P. State of the art in computational methods for the prediction of radar cross-sections and antenna-platform interactions[J]. C R Physique, 2005, 6(6):626-639.
[9] TONG Meisong, CHEW Wengcho. Multilevel fast multipole acceleration in the Nyström discretization of surface electromagnetic integral equations for composite objects[J]. IEEE Transactions on Antennas and Propagation, 2010, 58(10):3411-3416.
[10] WILKERSON S A. Boundary integral technique for explosion bubble collapse analysis[R]. Army Research Laboratory, 1993, ARL-TR-184.
[11] SOLTANI N, ESFAHANIAN V, HADDADPOUR H, et al. Unsteady supersonic aerodynamics based on BEM, including thickness effects in aeroelastic analysis[J]. Journal of Fluids and Structures, 2004, 19(6):801-813.
[12] AYDIN M, FENNER R T. Boundary element analysis of driven cavity flow for low and moderate Reynolds numbers[J]. International Journal for Numerical Methods in Fluids, 2001, 37(1):45-64.
[13] CHAILLAT S, BONNET M. Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics[J]. Wave Motion, 2013, 50(7):1090-1104.
[14] GREENGARD L, ROKHLIN V. A fast algorithm for particle simulations[J]. Journal of Computational Physics, 1987, 73(2):325-348.
[15] GREENGARD L, ROKHLIN V. A new version of the fast multipole method for the Laplace equation in three dimensions[J]. Acta Numerical, 1997, 6(1):229-269.
[16] DARVE E. The fast multipole method:Numerical implementation[J]. Journal of Computational Physics, 2000, 160(1):195-240
[17] YING L, BIROS G, ZORIN D. A kernel-independent adaptive fast multipole algorithm in two and three dimensions[J]. Journal of Computational Physics, 2004, 196(2):591-626.
[18] FONG W, DARVE E. The black-box fast multipole method[J]. Journal of Computational Physics, 2009, 228(23):8712-8725.
[19] HACKBUSCH W. A sparse matrix arithmetic based on H-matrices Part I:Introduction to H-matrices[J]. Computing, 1999, 62(2):89-108.
[20] BEBENDORF M. Approximation of boundary element matrices[J]. Numer Math, 2000, 86(4):565-589.
[21] BEBENDORF M, RJASANOW S. Adaptive low-rank approximation of collocation matrices[J]. Computing, 2003, 70(1):1-24.
[22] BEYLKIN G, COIFMAN R, ROKHLIN V. Fast wavelet transforms and numerical algorithms I[J]. Communications on Pure and Applied Mathematics, 1991, 44(2):141-183.
[23] TAUSCH J. A variable order wavelet method for the sparse representation of layer potentials in the non-standard form[J]. Journal of Numerical Mathematics, 2004, 12(3):233-254.
[24] HAWKINS S C, CHEN K, HARRIS P J. On the influence of the wavenumber on compression in a wavelet boundary element method for the Helmholtz equation[J]. International Journal of Numerical Analysis and Modeling, 2007, 4(1):48-62.
[25] CHENG H, CRUTCHFIELD W Y, GIMBUTAS Z, et al. A wideband fast multipole method for the Helmholtz equation in three dimensions[J]. Journal of Computational Physics, 2006, 216(1):300-325.
[26] GUMEROV N A, DURAISWAMI R. A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation[J]. Journal of the Acoustical Society of America, 2009, 125(1):191-205.
[27] ZHENG C J, MATSUMOTO T, TAKAHASHI T, et al. A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method[J]. Engineering Analysis with Boundary Elements, 2012, 36(3):361-371.
[28] TONG M, CHEW W. Multilevel fast multipole algorithm for elastic wave scattering by large three-dimensional objects[J]. Journal of Computational Physics, 2009, 228(3):921-932.
[29] MESSNER M, SCHANZ M, DARVE E. Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation[J]. Journal of Computational Physics, 2012, 231(4):1175-1196.
[30] BEBENDORF M, KUSKE C, VENN R. Wideband nested cross approximation for Helmholtz problems[R]. SFB 611 Preprint, 2012:536.
[31] ENGQUIST B, YING L. Fast directional multilevel algorithms for oscillatory kernels[J]. SIAM Journal on Scientific Computing, 2007, 29(4):1710-1737.
[32] ENGQUIST B, YING L. Fast directional algorithms for the Helmholtz kernel[J]. Journal of Computational and Applied Mathematics, 2010, 234(6):1851-1859.
[33] CAO Y, WEN L, XIAO J, et al. A fast directional BEM for large-scale acoustic problems based on the Burton-Miller formulation[J]. Engineering Analysis with Boundary Elements, 2015, 50:47-58.
[34] CHAILLAT S, BONNET M, SEMBLAT J-F. A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain[J]. Computer Methods in Applied Mechanics and Engineering, 2008, 197(49-50):4233-4249.
[35] CHAILLAT S, BONNET M. A new fast multipole formulation for the elastodynamic half-space Green's tensor[J]. Journal of Computational Physics, 2014, 258:787-808.
[36] TSUJI P H. Fast algorithms for frequency-domain wave propagation[D]. The University of Texas at Austin, PhD thesis, 2012.
[37] CAO Y, RONG J, WEN L, et al. A kernel-independent fast multipole BEM for large-scale elastodynamic analysis[J]. Engineering Computations, 2015, 32(8):2391-2418.
[38] SANZ J A, BONNET M, DOMINGUEZ J. Fast multipole method applied to 3-D frequency domain elastodynamics[J]. Engineering Analysis with Boundary Elements, 2008, 32(10):787-795.
[39] CANINO L F, OTTUSCH J J, STALZER M A, et al. Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nyström discretization[J]. Journal of Computational Physics, 1998, 146(2):627-663.
[40] RONG J, WEN L, XIAO J. Efficiency improvement of the polar coordinate transformation for evaluating BEM singular integrals on curved elements[J]. Engineering Analysis with Boundary Elements, 38(1):83-93.
[41] LUBICH C. Convolution quadrature and discretized operational calculus I[J]. Numerische Mathematik, 1988, 52(2):129-145.
[42] BETCKE T, SALLES N, SMIGAJ W. Solution of time-domain problems using convolution quadrature methods and BEM++[C]. International Conference on Boundary Element and Meshless Techniques, 2014.
[43] SCHANZ M, YE W, XIAO J. Comparison of the convolution quadrature method and enhanced inverse FFT with application in elastodynamic boundary element method[J]. Computational Mechanics, 2016, 57(4):523-536.
[44] BANJAI L. Multistep and multistage convolution quadrature for the wave equation:Algorithms and experiments[J]. SIAM Journal on Scientific Computing, 2010, 32(5):2946-2994.