基于广义模态截断扩增方法的结构频响拓扑变量灵敏度分析
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摘要
基于结构拓扑优化设计中的变密度法,采用伴随法推导结构在简谐激励下的频响振幅对单元设计变量的解析灵敏度列式。针对灵敏度数值求解中传统模态位移法引发的低精度问题,通过引入计算成本较低的广义模态截断扩增方法提升灵敏度的计算精度。数值算例将该文方法与全局有限差分法和其他灵敏度计算方法进行了比较。结果证明了该文方法在不同的激励频率及有限元网格密度下高效求解高精度灵敏度的有效性。
Based on the variable density method for structural topology optimization, the analytical sensitivity formulation of the frequency response displacement amplitude of structures under harmonic excitations is proposed using the adjoint method. The generalized modal truncation augmentation method is introduced to obtain high accuracy without high computational cost in contrast to the poor accuracy of the traditional modal displacement method in sensitivity computation. Numerical examples are presented to compare the proposed method with the global finite difference method and other computation methods. The computational results demonstrate the effectiveness of the proposed method in computing accurate sensitivities with high efficiency under different excitation frequencies and different densities of finite element meshes.
Based on the variable density method for structural topology optimization, the analytical sensitivity formulation of the frequency response displacement amplitude of structures under harmonic excitations is proposed using the adjoint method. The generalized modal truncation augmentation method is introduced to obtain high accuracy without high computational cost in contrast to the poor accuracy of the traditional modal displacement method in sensitivity computation. Numerical examples are presented to compare the proposed method with the global finite difference method and other computation methods. The computational results demonstrate the effectiveness of the proposed method in computing accurate sensitivities with high efficiency under different excitation frequencies and different densities of finite element meshes.
引文
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[2]李东泽,于登云,马兴瑞.基频约束下的桁架结构半定规划法拓扑优化[J].工程力学,2011,28(2):181-185.Li Dongze,Yu Dengyun,Ma Xingrui.Truss topology optimization with fundamental frequency constraints via wsemi-defi·nite programming[J].Engineering Mechanics,2011,28(2):181-185.(in Chinese)
[3]Ma Z D,Kikuchi N,Hagiwara I.Structural topology and shape optimization for a frequency response problem[J].Computational Mechanics,1993,13(3):157-174.
[4]Jog C S.Topology design of structures subjected to periodic loading[J].Journal of Sound and vibration,2002,253(3):687-709.
[5]Jensen J S.Topology optimization of dynamic problems with Padéapproximants[J].International Journal Numerical Methods in Engineering,2007,72(13):1605-1630.
[6]Yoon G H.Structural topology optimization for frequency response problem using model reduction schemes[J].Computer Methods in Applied Mechanics and Engineering,2010,199(25-28):1744-1763.
[7]Olhoff Niels,Du Jianbin.Topological design for minimum dynamic compliance of structures under forced vibration[M]//Rozvany G I N,Lewiński T.Topology Optimization in Structural and Continuum Mechanics,CISM International Centre for Mechanical Sciences,vol549.Vienna:Springer,2014:325-329.
[8]Olhoff Niels,Du Jianbin.Generalized incremental frequency method for topological design of continuum structures for minimum dynamic compliance subject to forced vibration at a prescribed low or high value of the excitation frequency[J].Structural and Multidisciplinary Optimization,2016,54(5):1113-1141.
[9]Kang Z,Zhang X,Jiang S,et al.On topology optimization of damping layer in shell structures underharmonic excitations[J].Structural and Multidisciplinary Optimization,2012,46(1):51-67.
[10]叶红玲,李耀明,张颜明,等.基于对数型Heaviside近似函数作为过滤函数的动力响应结构拓扑优化ICM方法[J].工程力学,2014,31(6):13-20.Ye Hongling,Li Yaoming,Zhang Yanming,et al.Structural topology optimization of frequency response problem by approximately logarithmic Heaviside function based on ICM method[J].Engineering Mechanics,2014,31(6):13-20.(in Chinese)
[11]Hu Liu,Weihong Zhang,Tong Gao.A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations[J].Structural and Multidisciplinary Optimization,2015,51(6):1321-1333.
[12]Williams D E.Dynamic loads in aeroplanes under given impulsive loads with particular reference to landing and gust loads on a large flying boat[R].UK:Great Britain Royal Aeronautical Establishment Reports.SME,1945:3309-3316.
[13]Daniel Rixen.Generalized mode acceleration methods and modal truncation augmentation[C].Seattle,WA,U.S.A.:42nd AIAA/ASME/ASCE/AHS/ASC Structures,Structrual Dynamics and Material Conference and Exhibition,AIAA Paper 2001-1300,2001.