Catmull-Clark细分曲面边界元法的结构声学拓扑优化分析

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振动与冲击 ›› 2020, Vol. 39 ›› Issue (20) : 97-105.

论文

陈磊磊，王中王，卢闯，高春华，刘林超

作者信息 +

Structural acoustic topology optimization analysis of a Catmull-Clark subdivision surface boundary element method

CHEN Leilei，WANG Zhongwang，LU Chuang，GAO Chunhua，LIU Linchao

Author information +

文章历史 +

摘要

细分曲面克服了NURBS方法进行曲面片拼接时出现缝隙的困难，并可以构建带有任意拓扑形状的光滑结构模型。对于声学问题，频率越高，波长越短，为了满足计算精度要求，用于数值分析的离散网格就需要更密，传统算法往往需要对原始结构模型重新进行网格划分，耗费了大量的计算时间。细分曲面法只需对初始离散网格进行一定的细分操作，即可提供多层次多分辨率控制网格，避免了复杂耗时的前处理操作。将Catmull-Clark细分曲面与边界元法相结合，采用高阶双三次B样条基函数进行几何与物理场的插值近似，不仅提供高计算精度结果，而且自适应满足宽频段网格要求。对于粘附吸声材料结构声散射问题，引入声阻抗边界条件，建立以吸声材料单元密度为设计变量，以吸声材料的体积分数为约束的数学优化模型。采用伴随变量法计算目标函数对设计变量的敏感度，移动渐近线法对设计变量进行更新，最终获得最优吸声材料分布。若干实际问题算例验证了算法的正确性和有效性。

Abstract

The use of subdivision surface overcomes the difficulty of the gap in the surface slice splicing of NURBS, and can construct the smooth and continuous overall surfaces with arbitrary free-form topology. In the acoustic field, the higher the frequency, the shorter the wavelength and the larger the number of meshes to meet the calculation accuracy required. The traditional methods require reconstruct the mesh of the original structure model, which takes a lot of time. A subdivision surface method only requires refinement operation of the initial discrete mesh, which can provide multi-level-resolution control grid to avoid complex and time-consuming pre-processing. Combining the Catmull-Clark subdivision surface with the boundary element method, the interpolation approximation of geometry and physical field was carried out by using the high-order bi-cubic B-spline basis function, which not only provides the result of higher calculation accuracy, but also satisfies the requirement of broadband mesh. For the acoustic scattering problem of the structure of the adhesive sound absorbing material, the acoustic impedance boundary condition was introduced. The mathematical optimization model was established, which takes the density of the sound absorbing material element as the design variable and the volume fraction of the sound absorbing material as the constraint. The sensitivity of the objective function to the design variables was calculated by using the adjoint variable method, and the design variables were updated by the method of moving asymptotes. Finally the optimal distribution of sound absorbing materials was obtained. The correctness and validity of the algorithm were verified.

导出引用

陈磊磊，王中王，卢闯，高春华，刘林超. Catmull-Clark细分曲面边界元法的结构声学拓扑优化分析[J]. 振动与冲击, 2020, 39(20): 97-105

CHEN Leilei，WANG Zhongwang，LU Chuang，GAO Chunhua，LIU Linchao. Structural acoustic topology optimization analysis of a Catmull-Clark subdivision surface boundary element method[J]. Journal of Vibration and Shock, 2020, 39(20): 97-105

参考文献

[1] Hughes T J, Cottrell J A, Bazilevs Y, et al. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194: 4135-4195.
[2] Cirak F, Ortiz M, Schroder P, et al. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis[J]. International Journal for Numerical Methods in Engineering, 2000, 47(12): 2039-2072.
[3] Simpson R N, Bordas S, Trevelyan J, et al. A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2012: 87-100.
[4] Simpson R N, Scott M A, Taus M, et al. Acoustic isogeometric boundary element analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2014: 265-290.
[5] Peake M J, Trevelyan J, Coates G, et al. Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems[J]. Computer Methods in Applied Mechanics and Engineering, 2013: 93-102.
[6] Catmull E, Clark J. Recursively generated B-spline surfaces on arbitrary topological meshes[J]. Computer-aided Design, 1978, 10(6):350-355.
[7] Dyn N, Levin D, Gregory J A. A butterfly subdivision scheme for surface interpolation with tension control[J]. ACM Transactions on Graphics. 1990,9(2):160-169.
[8] Kobbelt L. -subdivision[C]. Proceedings of ACM Siggraph, 2000,18(1): 103-112.
[9] Liu Z, Majeed M, Cirak F, et al. Isogeometric FEM-BEM coupled structural-acoustic analysis of shells using subdivision surfaces[J]. International Journal for Numerical Methods in Engineering, 2018, 113(9): 1507-1530.
[10] Bandara K, Ruberg T, Cirak F, et al. Shape optimisation with multiresolution subdivision surfaces and immersed finite elements[J]. Computer Methods in Applied Mechanics and Engineering, 2016: 510-539.
[11] 文立华, 王卫祥, 张敏玉. 表面细分技术在二维声辐射和声散射中的应用[J]. 应用声学, 2006, 25(1):13-18.
    Wen L, Wang W, Zhang M. Application of the subdivision surface technique to acoustic radiation and scattering[J]. Applied Acoustics, 2006, 25(1):13-18.
[12] Zhao W, Zheng C, Chen H, et al. Acoustic topology optimization of porous material distribution based on an adjoint variable FMBEM sensitivity analysis[J]. Engineering Analysis With Boundary Elements, 2019: 60-75.
[13] 杜建镔,宋先凯,董立立. 基于拓扑优化的声学结构材料分布设计[J]. 力学学报,2011,02:306-315.
    Du J, Song X, Dong L. Design of material distribution of acoustic structure using topology optimization [J]. Chinese Journal of Theoretical and Applied Mechanics, 2011,02:306-315.
[14] 陈磊磊, 申晓伟, 刘程, et al. 基于等几何边界元法的声屏障结构形状优化分析[J]. 振动与冲击, 2019, 38(6):114-120.
    Chen L, Shen X, Liu C, et al. Shape optimization analysis of sound barriers based on the isogeometric boundary element method[J] JOURNAL OF VIBRATION AND SHOCK, 2019, 38(6):114-120.
[15] 刘海，高行山，王佩艳，et al. 基于拓扑优化的结构加强筋布局降噪方法研究[J]. 振动与冲击. 2013,13:62-65.
    Liu H, Gao H, Wang P, et al. Stiffeners layout design for noise reduction using topology optimization [J]. JOURNAL OF VIBRATION AND SHOCK, 2013,13:62-65.
[16] Stam J. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values[C]. International conference on computer graphics and interactive techniques, 1998: 395-404.
[17] Burton AJ and Miller GF. The application of integral equation methods to the numerical solution of some exterior boundary value problems[J]. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1971,323:201-210.
[18] Berardi U, Iannace G. Predicting the sound absorption of natural materials: Best-fit inverse laws for the acoustic impedance and the propagation constant[J]. Applied Acoustics, 2017, 115: 131-138
[19] 刘书田, 贺丹. 基于SIMP 插值模型的渐进结构优化方法[J]. 计算力学学报, 2009, 26(6): 761-765.
Liu S, He D. SIMP-based evolutionary structural optimization method for topology optimization[J]. Computational Mechanics, 2009, 26(6): 761-765.
[20] Zhao W, Chen L, Zheng C, et al. Design of absorbing material distribution for sound barrier using topology optimization[J]. Structural and Multidisciplinary Optimization, 2017, 56(2): 315-329.
[21] Junger M, Feit D. Sound, structures, and their interaction [M]. London: MIT Press, 1985: 135-155.
[22] 陈磊磊，袁晓辉，赵文畅. 基于Burton-Miller边界元法的不同类型单元计算精度对比[J]. 信阳师范学院学报, 2017, 30(2): 331-335.
Chen L eilei, Yuan Xiaohui, Zhao Wenchang. Comparison of Different Types of
Boundary Elements Based on Burton-Miller Method[J]. Journal of Xinyang Normal University , 2017, 30(2): 331-335.