双尺度分级结构时域动刚度问题的并行拓扑优化

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振动与冲击 ›› 2023, Vol. 42 ›› Issue (15) : 31-41.

论文

江旭东1，马佳琪1，滕晓燕2

作者信息 +

Concurrent topology optimization for time-domain dynamic stiffness problem of two-scale hierarchical structure

JIANG Xudong1, MA Jiaqi1, TENG Xiaoyan2

Author information +

文章历史 +

摘要

提出双尺度分级结构时域动刚度问题的并行拓扑优化方法，实现结构宏观拓扑构型和材料微观分布协同优化。利用三场密度方法实施宏观与微观结构的设计，基于能量的均匀化方法(Energy-based Homogenization Method，EBHM)在微观结构上计算等效的宏观力学特性。针对比例阻尼系统，将HHT-α方法作为时间积分算法，求解多尺度结构的动力学有限元模型。融合先离散-后微分方法与伴随方法，在时间与空间离散的拓扑优化模型上实施敏度分析，避免了将时间作为连续变量而导致的灵敏度计算的一致性误差问题。以多尺度结构的动柔度最小化为目标，宏观与微观结构体积为约束，分别求解了正弦半波和余弦半波冲击载荷作用下多尺度结构的并行拓扑优化问题。最后，通过2种典型算例的数值结果验证了所提算法的有效性。

Abstract

This paper aims to develop an efﬁcient concurrent topology optimization approach with two-scale hierarchical structure for dynamic stiffness problem, which simultaneously realizes macroscopic and microscopic layout optimization. The three-ﬁeld density-based approach is employed for topological design of macrostructure and microstructure, with an energy-based homogenization method (EBHM) to evaluate the macroscopic effective properties of the microstructure. The HHT-α method is exploited as a time integration scheme to solve the multi-scale structural dynamics problem for the proportional damping system. Based on the discretize-then-differentiate approach with the adjoint method, the sensitivity analysis is conducted on the discretized (both in space and time) topology optimization statement. Accordingly, the consistency errors are avoided, which arise when the adjoint method is used while considering time as a continuous variable. A multiscale topology optimization model is built for minimizing the dynamic compliance under two volume constraints of macrostructure and microstructure. We performed topology optimization of multiscale structures subjected to dynamic loads, such as half-cycle sinusoidal load and half-cycle cosine load. Finally, numerical results obtained from two benchmark examples demonstrate the effectiveness of the proposed method.

导出引用

江旭东1，马佳琪1，滕晓燕2. 双尺度分级结构时域动刚度问题的并行拓扑优化[J]. 振动与冲击, 2023, 42(15): 31-41

JIANG Xudong1, MA Jiaqi1, TENG Xiaoyan2. Concurrent topology optimization for time-domain dynamic stiffness problem of two-scale hierarchical structure[J]. Journal of Vibration and Shock, 2023, 42(15): 31-41

参考文献

[1] Wu J, Sigmund O, Groen J P. Topology optimization of multi-scale structures: a review [J]. Structural and Multidisciplinary Optimization, 2021, 63: 1455-1480.
[2] Murphy R, Imediegwu C, Hewson R, et al. Multiscale structural optimization with concurrent coupling between scales [J]. Structural and Multidisciplinary Optimization, 2021, 63, 1721-1741.
[3] Bertolino G, Montemurro M. Two-scale topology optimisation of cellular materials under mixed boundary conditions [J]. International Journal of Mechanical Sciences, 2022, 216, 106961.
[4] 白影春, 景文秀. 基于滤波/映射边界描述的壳-填充结构多尺度拓扑优化方法[J]. 机械工程学报, 2021, 57(4): 121-129.
Bai Yingchun, Jing Wenxiu. Multiscale topology optimization method for shell-infill structures based on filtering/projection boundary description [J]. Journal of Mechanical Engineering, 2021, 57(4): 121-129.
[5] Gao J, Luo Z, Xia L, Gao L. Concurrent topology optimization of multiscale composite structures in Matlab [J]. Structural and Multidisciplinary Optimization, 2019, 60: 2621-2651.
[6] Gangwar T, Schillinger D. Concurrent material and structure optimization of multiphase hierarchical systems within a continuum micromechanics framework [J]. Structural and Multidisciplinary Optimization, 2021, (2-3):1-23.
[7] Xiao Mi, Liu Xiliang, Zhang Yan, et al. Design of graded lattice sandwich structures by multiscale topology optimization [J]. Computer Methods in Applied Mechanics and Engineering, 2021, 384: 113949.
[8] Zhang Yan, Gao Liang, Xiao Mi. Maximizing natural frequencies of inhomogeneous cellular structures by Kriging-assisted multiscale topology optimization [J]. Computers and Structures, 2020, 230, 106197.
[9] 付君健, 张跃, 杜义贤, 等.周期性多孔结构特征值拓扑优化[J].振动与冲击, 2022, 41(3): 73-81.
Fu Junjian, Zhang Yue, Du Yixian, et al. Eigenvalue topology optimization of periodic cellular structures [J]. Journal of Vibration and Shock, 2022, 41(3): 73-81.
[10] 倪维宇, 张横, 姚胜卫. 考虑阻尼性能的复合结构多尺度拓扑优化设计[J]. 航空学报, 2021, 42(3): 338-348.
Ni Weiyu, Zhang Heng, Yao Shengwei. Concurrent topology optimization of composite structures for considering structural damping [J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(3): 338-348.
[11] Ali A M, Shimoda M. Toward multiphysics multiscale concurrent topology optimization for lightweight structures with high heat conductivity and high stiffness using MATLAB [J]. Structural and Multidisciplinary Optimization, 2022, 65(7): 2017.
[12] Zhao J, Yoon H, Youn B D. An efficient concurrent topology optimization approach for frequency response problems [J]. Computer Methods in Applied Mechanics and Engineering, 2019, 347(APR.15): 700-734.
[13] Niu B, Wadbro E. Multiscale design of coated structures with periodic uniform inﬁll for vibration suppression [J]. Computers and Structures, 2021, 255: 106622.
[14] Zhang Y, Zhang L, Ding Z, et al. A multiscale topological design method of geometrically asymmetric porous sandwich structures for minimizing dynamic compliance [J]. Materials & Design, 2022, 214: 110404.
[15] Zhang C, Long K, Yang A, et al. A transient topology optimization with time-varying deformation restriction via augmented Lagrange method [J]. International Journal of Mechanics and Materials in Design, 2022.
[16] 文桂林, 陈高锡, 王洪鑫, 等. 含自重载荷的功能梯度材料结构时域动力学拓扑优化设计[J]. 中国机械工程, 2022.
Wen Guilin, Chen Gaoxi, Wang Hongxin, et al. Time domain dynamic topology optimization of functionally gradient material structures with self-weight load [J]. China Mechanical Engineering, 2022.
[17] Jensen J S, Nakshatrala P B, Tortorelli D A. On the consistency of adjoint sensitivity analysis for structural optimization of linear dynamic problems [J]. Structural & Multidisciplinary Optimization, 2014, 49(5): 831-837.
[18] 胡智强, 马海涛. 结构动力响应灵敏度分析伴随法一致性问题研究[J]. 振动与冲击, 2015, 34(20): 167-173.
Hu Zhiqiang, Ma Haitao. On the consistency issue of adjoint methods for sensitivity of dynamic responses [J]. Journal of Vibration and Shock, 2015, 34(20): 167-173.
[19] Ding Zhe, Zhang Lei, Gao Qiang, et al. State-space based discretize-then-differentiate adjoint sensitivity method for transient responses of non-viscously damped systems [J]. Computers and Structures, 2021, 250, 106540.
[20] Giraldo-Londono O, Paulino G H. PolyDyna: a Matlab implementation for topology optimization of structures subjected to dynamic loads [J]. Structural and Multidisciplinary Optimization, 2021, 1-34.
[21] Zhao J, Yoon H, Youn B D. Concurrent topology optimization with uniform microstructure for minimizing dynamic response in the time domain [J]. Computers and Structures, 2019, 222: 98-117.
[22] Xu S, Cai Y, Cheng G. Volume preserving nonlinear density ﬁlter based on heaviside functions [J]. Structural and Multidisciplinary Optimization, 2010, 41(4): 495-505.
[23] Sigmund O, Maute K. Topology optimization approaches [J]. Structural and Multidisciplinary Optimization, 2013, 48(6): 1031-55.
[24] Bourdin B. Filters in topology optimization [J]. International Journal for Numerical Methods in Engineering, 2001, 50(9): 2143-58.
[25] Wang F, Lazarov B S, Sigmund O. On projection methods, convergence and robust formulations in topology optimization [J]. Structural and Multidisciplinary Optimization, 2011, 43(6): 767-84.
[26] Bendsøe M P. Optimal shape design as a material distribution problem [J]. Structural and Multidisciplinary Optimization, 1989,1(4): 193-202.
[27] Liu L, Yan J, Cheng G. Optimum structure with homogeneous optimum truss-like material [J]. Computers and Structures, 2008, 86(13-14): 1417-1425.
[28] Niu B, Yan J, Cheng G. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency [J]. Structural and Multidisciplinary Optimization, 2009, 39(2): 115-32.
[29] Xia L, Breitkopf P. Design of materials using topology optimization and energy-based homogenization approach in Matlab [J]. Structural and Multidisciplinary Optimization, 2015, 52(6): 1229-1241.
[30] Zhao J, Wang C. Topology optimization for minimizing the maximum dynamic response in the time domain using aggregation functional method [J]. Computers and Structures, 2017, 190: 41-60.
[31] Svanberg K. The method of moving asymptotes-a new method for structural optimization [J]. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359-73.
[32] 张磊, 张严, 丁喆. 黏性阻尼系统时域响应灵敏度及其一致性研究[J]. 力学学报, 2022, 54(4): 1116-1127.
Zhang Lei, Zhang Yan, Ding Lei. Adjoint senility methods for transient responses of viscously damped systems and their consistency issues [J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(4): 1116-1127.