广义有限差分法在含阻抗边界空腔声学分析中的应用

陈增涛; 王发杰; 王超

doi:10.6052/0459-1879-20-311

广义有限差分法在含阻抗边界空腔声学分析中的应用

doi: 10.6052/0459-1879-20-311

陈增涛^*,
王发杰^*,†,2), ,,
王超^*

^*青岛大学动力集成及储能系统工程技术中心, 青岛大学机电工程学院, 山东青岛 266071
^†青岛大学多功能材料与结构力学研究院, 山东青岛 266071

基金项目: 1)国家自然科学基金(11802151);山东省自然科学基金(ZR2019BA008);中国博士后科学基金(2019M652315);青岛市民生科技计划(19-6-1-88-nsh);青岛市民生科技计划(19-6-1-92-nsh)

详细信息

作者简介:
2)王发杰, 副教授, 主要研究方向: 力学建模、计算力学、数值仿真. E-mail: wfj1218@126.com

通讯作者:
王发杰

中图分类号: O302,O422.2
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出版历程
- 收稿日期: 2020-09-07
- 刊出日期: 2021-04-10

APPLICATION OF GENERALIZED FINITE DIFFERENCE METHOD IN ACOUSTIC ANALYSIS OF CAVITY WITH IMPEDANCE BOUNDARY

Chen Zengtao^*,
Wang Fajie^{*,†,2)
, ,},
Wang Chao^*

^*Power Integration and Energy Storage System Engineering Technology Center, Qingdao University. College of Mechanical and Electrical Engineering, Qingdao University, Qingdao 266071, Shandong, China
^†Institute of Multifunctional Materials and Structural Mechanics, Qingdao University, Qingdao 266071, Shandong, China

摘要

摘要: 声学分析在噪声控制、室内隔音等工程计算中有着重要的作用. 由于现实生活中的声学模型往往伴随着吸声材料, 因此分析含阻抗边界条件的声学问题显得十分必要. 广义有限差分法是一种新型区域型无网格数值离散方法, 该方法基于多元函数泰勒级数展开式和加权最小二乘拟合, 将控制方程中未知参量的各阶偏导数表示为相邻节点函数值的线性组合. 本文首次将广义有限差分法应用于含阻抗边界条件空腔声学问题的分析中, 建立了空腔声场问题的广义有限差分法数值离散格式. 与传统算法相比, 所建立的数值模型具有无需网格剖分和数值积分、计算精度高、适用于大规模声学分析等优点. 通过具有解析解的经典算例, 研究了总节点数目和局部支撑点数目对数值结果的影响, 得到了最大计算频率与节点间距之间关系的经验公式. 此外, 将广义有限差分法应用于无解析解的二维和三维复杂声学模型, 并与COMSOL Multiphysics软件所得的有限元结果进行了比较分析. 数值实验表明, 该算法是一种高效、精确、稳定、收敛的数值模拟方法, 在含阻抗边界空腔声学分析中具有广阔的应用前景.
- 无网格法 /
- 广义有限差分法 /
- 空腔声学问题 /
- 阻抗边界 /
- 有限元法
Abstract: Acoustic analysis plays significant role in engineering calculations such as noise control and indoor sound insulation. Since the practical acoustic problems usually involve sound-absorbing materials, it is very necessary to analyze acoustic problems with impedance boundary conditions. The generalized finite difference method is a new mesh-less numerical discretization method, this method is based on Taylor series expansion of multivariate function and weighted least square, the partial derivatives of unknown values in the governing equation are expressed as a linear combination of function values at supporting nodes. In this paper, the generalized finite difference method is applied to the analysis of cavity acoustics with impedance boundary conditions firstly, and the corresponding numerical discrete scheme is established. Compared with traditional algorithms, the developed numerical model is a local meshless method with the merits of being mathematically simple, numerically accurate and easy to large-scale acoustic analysis. A benchmark numerical example with analytical solution is examined to verify the influence of the total number of nodes and the number of local supporting nodes on the numerical results, and to obtain an empirical formula of the relationship between maximum computable frequency and node spacing. In addition, the generalized finite difference method is applied to two-dimensional and three-dimensional complex acoustic models without analytical solutions, and is compared with the FEM solutions obtained by COMSOL Multiphysics. Numerical experiments demonstrate that the generalized finite difference method is an efficient, accurate, stable and convergent numerical method, and has broad application prospects in the acoustic analysis of cavities with impedance boundaries.
- meshless method /
- generalized finite difference method /
- cavity acoustic problems /
- impedance boundary /
- finite element method

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参考文献(36)

[1]	陶文铨, 吴学红, 戴艳俊. 无网格数值求解方法. 中国电机工程学报, 2010,30(5):1-10 (Tao Wenquan, Wu Xuehong, Dai Yanjun. Numerical solutions of meshless methods. Proceedings of the Chinese Society of Electrical Engineering, 2010,30(5):1-10 (in Chinese))
[2]	张雄, 刘岩, 马上. 无网格法的理论及应用. 力学进展, 2009 ( 1):1-36 (Zhang Xiong, Liu Yan, Ma Shang. Meshfree methods and their applications. Advances in Mechanics 2009 ( 1):1-36 (in Chinese))
[3]	Atluri SN, Shen S. The meshless method. Encino, CA: Tech Science Press, 2002
[4]	杨建军, 文丕华. 无网格法: 理论与算法, 北京: 科学出版社, 2018: 1-11 (Yang Jianjun, Wen Pihua. Meshless method: theory and applications, Beijing: Science Press, 2018: 1-11(in Chinese))
[5]	王莉华, 李溢铭, 褚福运. 基于分区径向基函数配点法的大变形分析. 力学学报, 2019,51(3):743-753 (Wang Lihua, Li Yiming, Chu Fuyun. Finte subdomain radial basis collocation method for the large deformation analysis. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(3):743-753 (in Chinese))
[6]	覃霞, 刘珊珊, 谌亚菁等. 基于遗传算法的弹性地基加肋板肋梁无网格优化分析. 力学学报, 2020,52(1):93-110 (Qin Xia, Liu Shanshan, Shen Yajing, et al. Rib meshless optimization of stiffened plates resting on elastic foundation based on genetic algorithm. Chinese Journal of Theoretical and Applied Mechanics, 2020,52(1):93-110 (in Chinese))
[7]	Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, et al. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers & Structures, 2009,87(1-2):14-26
[8]	Chan HF, Fan CM, Kuo CW. Generalized finite difference method for solving two-dimensional non-linear obstacle problems. Engineering Analysis with Boundary Elements, 2013,37(9):1189-1196
[9]	Gu Y, Chen W, Gao HW, et al. A meshless singular boundary method for three-dimensional elasticity problems. International Journal for Numerical Methods in Engineering, 2016,107(2):109-126
[10]	Marin L. An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation. Computational Mechanics, 2010,45(6):665-677
[11]	Chen W, Shen LJ, Shen ZJ, et al. Boundary knot method for Poisson equations. Engineering Analysis with Boundary Elements, 2005,29(8):756-760
[12]	Wang FJ, Gu Y, Qu WZ, et al. Localized boundary knot method and its application to large-scale acoustic problems. Computer Methods in Applied Mechanics and Engineering, 2020,361(C):112729
[13]	Chen CS, Golberg MA, Hon YC. The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations. International Journal for Numerical Methods in Engineering, 2015,43(8):1421-1435
[14]	王发杰, 张耀明, 公颜鹏. 改进的基本解法在薄体各向异性位势Cauchy问题中的应用. 工程力学, 2016,33(2):18-24 (Wang Fajie, Zhang Yaoming, Gong Yanpeng, Inverse identification of boudary conditions for anisotropic potential problems of thin body using MFS. Engineering Mechanics, 2016,33(2):18-24 (in Chinese))
[15]	Young DL, Chen KH, Lee CW. Novel meshless method for solving the potential problems with arbitrary domain. Journal of Computational Physics, 2005,209(1):290-321
[16]	谷岩, 陈文. 改进的奇异边界法模拟三维位势问题. 力学学报, 2012, ( 2):351-360 (Gu Yan, Chen Wen. Improved singular boundary method for three dimensional potential problems. Chinese Journal of Theoretical and Applied Mechanics, 2012,44(2):351-360 (in Chinese))
[17]	李煜冬, 王发杰, 陈文. 瞬态热传导的奇异边界法及其MATLAB实现. 应用数学和力学, 2019,40(3):259-268 (Li Yudong, Wang Fajie, Chen Wen. Matlab implementation of a singular boundary method for transient heat conduction. Applied Mathematics and Mechanics, 2019,40(3):259-268 (in Chinese))
[18]	李珺璞, 陈文. 一种模拟大规模高频声场的双层奇异边界法. 力学学报, 2018,50(4):961-969 (Li Junpu, Chen Wen. A dual-level singular boundary method for large-scale high frequency sound field analysis. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(4):961-969 (in Chinese))
[19]	Chen KH, Chen JT, Kao JH. Regularized meshless method for solving acoustic eigenproblem with multiply-connected domain. Computer Modeling in Engineering and Sciences, 2006,16(1):27-39
[20]	Chen JT, Chen IL, Chen KH, et al. A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function. Engineering Analysis with Boundary Elements, 2004,28(5):535-545
[21]	Sadat H, Wang CA, Dez VL. Meshless method for solving coupled radiative and conductive heat transfer in complex multi-dimensional geometries. Applied Mathematics & Computation, 2012,218(20):10211-10225
[22]	Wang CA, Sadat H, Prax C. A new meshless approach for three dimensional fluid flow and related heat transfer problems. Computers & Fluids, 2012,69:136-146
[23]	Lu YY, Belytschko T, Gu L. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics & Engineering, 1994,113(3-4):397-414
[24]	邓立克, 王东东, 王家睿等. 薄板分析的线性基梯度光滑伽辽金无网格法. 力学学报, 2019,51(3):688-702 (Deng Like, Wang Dongdong, Wang Jiarui, et al. A gradient smoothing galerkin meshfree method for thin plate analysis with linear basis function. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(3):688-702 (in Chinese))
[25]	Benito JJ, Urena F, Gavete L, et al. An h-adaptive method in the generalized finite differences. Computer Methods in Applied Mechanics & Engineering, 2003,192(5-6):735-759
[26]	李艾伦, 傅卓佳, 李柏纬等. 含肿瘤皮肤组织传热分析的广义有限差分法. 力学学报, 2018,50(5):1198-1205 (Li Ailun, Fu Zhuojia, Li Powei, et al. Generalized finite difference method for bioheat transfer analysis on skin tissue with tumors. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(5):1198-1205 (in Chinese))
[27]	汤卓超, 傅卓佳, 范佳铭. 广义有限差分法求解Kirchhoff和Winkler薄板弯曲问题. 固体力学学报, 2018,39(4):419-428 (Tang Zhuochao, Fu Zhuojia, Fan Jiaming. Generalized finite difference method for solving Kirchhoff plate and Winkler plate bending problems. Chinese Journal of Solid Mechanics, 2018,39(4):419-428 (in Chinese))
[28]	Gu Y, Sun HG. A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives. Applied Mathematical Modelling, 2020,78:539-549
[29]	Gu Y, Qu WZ, Chen W, et al. The generalized finite difference method for long-time dynamic modeling of three-dimensional coupled thermoelasticity problems. Journal of Computational Physics, 2019,384:42-59
[30]	Li PW, Fu ZJ, Gu Y, et al. The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity. Solids Struct, 2019,174:69-84
[31]	朱小松. 基于无网格方法的声学预测研究. [博士论文]. 哈尔滨: 哈尔滨工程大学, 2018, 56-60 (Zhu Xiaosong. Study on the acoustic prediction based on meshless methods. [Doctor Thesis]. Harbin: Harbin Engineering University, 2018, 56-60 (in Chinese))
[32]	Benito JJ, Urena F, Gavete L, et al. A posteriori error estimator and indicator in generalized finite differences. Application to improve the approximated solution of elliptic PDEs. International Journal of Computer Mathematics, 2008,85(3-4):359-370
[33]	Gavete L, Gavete ML, Benito JJ. Improvements of generalized finite difference method and comparison with other meshless method. Applied Mathematical Modelling, 2003,27(10):831-847
[34]	Benito JJ, Urena F, Gavete L. Influence of several factors in the generalized finite difference method. Applied Mathematical Modelling, 2001,25(12):1039-1053
[35]	王刚. 基于节点积分的车身结构振动噪声计算方法研究. [博士论文]. 长沙: 湖南大学, 2016, 87-91 (Wang Gang. The research of calculational method of vehicle body structure vibration and noise based on node integral. [Doctor Thesis]. Changsha: Hunan University, 2016, 87-91 (in Chinese))
[36]	王双. 基于径向基配点型无网格方法的内部声学问题研究. [博士论文]. 武汉: 华中科技大学, 2013,100 (Wang Shuang. Study on the interior acoustic problem based on the radial basis collocation meshless method. Wuhan: Huazhong University of Science and Technology, 2013,100(in Chinese)