Robust topology optimization design of a multi-material structure considering load uncertainty

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Journal of Vibration and Shock ›› 2019, Vol. 38 ›› Issue (19) : 182-190.

ZHAO Qinghai1,2, ZHANG Hongxin2, JIANG Rongchao2, HUA Qingsong1,2, YUAN Lin2

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Abstract

The traditional topology optimization design is generally based on single-material and deterministic conditions, it is difficult to consider the robustness of structural performance. Here, aiming at the load uncertainty in practical engineering, the robust topology optimization design methodwas studied. The multi-material interpolation model was characterized based on the ordered-solid isotropic microstructures with penalization (Ordered-SIMP). The weighted objective function forthe mean and standard deviation of structural flexibility under the load probability distribution was constructedand assisted by volume constraints. When load satisfied the random field distribution, the load random field was transformed into a weighted sum of finite uncorrelated load random variables using Karhunen-Loèveexpansion, and the sparse grid numerical integration method was employed to convert the robust topology optimization of multi-material structure into solving a set of multi-condition weighted multi-objective deterministic topology optimization design problems. The effectiveness of the proposed method and the robustness of optimization resultswere verified with numerical examples. The results demonstrated that good topological configurationscan be effectively achieved for combination schemes of different materials;compared with deterministic designs, robust designs can have different material layout schemes and more stable structural performance.

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ZHAO Qinghai1,2, ZHANG Hongxin2, JIANG Rongchao2, HUA Qingsong1,2, YUAN Lin2. Robust topology optimization design of a multi-material structure considering load uncertainty[J]. Journal of Vibration and Shock, 2019, 38(19): 182-190

References

[1] Eschenauer HA,Olhoff N. Topology optimization of continuum structures: A review.Applied Mechanics Reviews, 2001, 54(4):331-390.
[2] LeeJO, YangYS, Ruy WS. A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Computers & Structures, 2002, 80(3-4): 257-269.
[3] Kharmanda G, Olhoff N, Mohamed A, LemaireM. Reliability-based topology optimization.Structural and Multidisciplinary Optimization, 2004, 26(5): 295-307.
[4] Zhao Q H, Chen X K, Ma Z-D, Lin Y. A comparison of deterministic, reliability-based topology optimization under uncertainties. ActaMechanicaSolidaSinica, 2016, 29(1): 31-45.
[5] 宋宗凤, 陈建军. 随机参数平面连续体结构的频率拓扑优化设计. 振动与冲击, 2009, 28(12):30-34. (Song Zongfeng,Chen Jianjun. Frequency topology optimization design of random parameters continuum structure. Journal of Vibration and Shock, 2009, 28(12):30-34(in Chinese))
[6] 尤芳, 陈建军, 曹鸿钧, 谢永强. 稳态热传导结构非概率可靠性拓扑优化设计. 振动与冲击, 2015 , 34(3): 118-122. (You Fang, Chen Jian-jun, Cao Hong-jun, Xie Yong-qiang. Topology optimization design of steady-state heat conduction structures considering non-probabilistic reliability. Journal of Vibration and Shock, 2015 , 34(3): 118-122 (in Chinese))
[7] Zhao Q H, Chen X K, Ma Z-D, Lin Y. Robust topology optimization based on stochastic collocation methods under loading uncertainties. Mathematical Problems in Engineering, 2015, (2015): 580980.
[8] Amir O, Sigmund O, Lazarov BS, Schevenels M. Efficient reanalysis techniques for robust topology optimization.ComputerMethods in Applied Mechanics and Engineering, 2012, 245: 217-231.
[9] Park S H. Robust design and analysis for quality engineering. London: Chapman and Hall, 1996.
[10] Park G, Lee T, Lee K H, et al. Robust optimization: An overview. AIAA Journal, 2006, 44(1): 181-191.
[11] Thore CJ, Holmberg E, Klarbring A. A general framework for robust topology optimization under load-uncertainty including stress constraints. Computer Methods in Applied Mechanics and Engineering, 2017, 319: 1-18.
[12] Zhao J P, Wang C J. Robust topology optimization of structures under loading uncertainty. AIAA Journal, 2014, 52(2): 398-407.
[13] Martinez-Frutos J, Herrero-Perez D. Large-scale robust topology optimization using multi-GPU systems. Computer Methods in Applied Mechanics and Engineering, 2016, 311: 393-414.
[14] 付志方, 王春洁. 载荷不确定的周期性结构稳健拓扑优化. 北京航空航天大学学报, 2017, 43(4): 747-753. (Fu Zhifang, Wang Chunjie. Robust topology optimization of periodic structures under uncertain loading. Journal of Beijing University of Aeronautics and Astronautics, 2017, 43(4):747-753 (in Chinese))
[15] Holmberg E, Thore CJ, Klarbring A. Game theory approach to robust topology optimization with uncertain loading. Structural and Multidisciplinary Optimization, 2017, 55(4): 1383-1397.
[16] 李涤尘, 贺健康, 田小永等. 增材制造:实现宏微结构一体化制造. 机械工程学报, 2013, 49(6):129-135. (LiDichen, HeJiankang, TianXiaoyong, et al. Additive manufacturing: integrated fabrication of macro/microstructures. Journal of Mechanical Engineering, 2013 , 49 (6) :129-135 (in Chinese))
[17] 林鑫,黄卫东. 高性能金属构件的激光增材制造. 中国科学:信息科学, 2015, 45(9):1111-1126. (Lin Xin, HuangWeidong. Laser additive manufacturing of high-performance metal components. ScientiaSinicaInformations, 2015, 45(9): 1111-1126 (in Chinese))
[18] 王华明. 高性能大型金属构件激光增材制造:若干材料基础问题. 航空学报, 2014, 35(10): 2690-2698. (WangHuaming. Materials’ Fundamental Issues of Laser Additive Manufacturing for High-performance Large Metallic Components.ActaAeronautica et AstronauticaSinica, 2014, 35(10): 2690-2698 (in Chinese))
[19] Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids, 1997, 45(6): 1037-1067.
[20] Wang M Y, Wang X. ‘‘Color’’ level sets: a multi-phase method forstructural topology optimization with multiple materials. Computer Methods in Applied Mechanics and Engineering, 2004, 193(6-8): 469-496.
[21] Wang Y,Luo Z,Kang Z, Zhang N.A multi-material level set-based topology and shape optimization method. Computer Methods in Applied Mechanics & Engineering, 2015, 283:1570-1586.
[22] Zhou S W, Wang M Y. Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Structural and Multidisciplinary Optimization, 2007, 33(2): 89-111.
[23] Tavakoli R, Mohseni SM. Alternating active-phase algorithm for multimaterialtopology optimization problems: a 115-line MATLABimplementation. Structural and Multidisciplinary Optimization, 2014, 49(4): 621-642.
[24] Zuo W, Saitou K. Multi-material topology optimization using ordered SIMP interpolation. Structural and Multidisciplinary Optimization, 2017, 55(2): 477-491.
[25] Bendsøe M P, Sigmund O. Material interpolations in topology optimization. Archive of Applied Mechanics, 1999, 69(9-10): 635-654.
[26] Stolpe M,Svanberg K. An alternative interpolation scheme for minimum compliance topology optimization. Structural and Multidisciplinary Optimization, 2001, 22(2): 116-124.
[27] Ghanem R G, Spanos P D. Stochastic finite elements: a spectral approach. New York: Springer,1991.
[28] Kang Z, Liu P. Reliability-based topology optimization against geometricimperfections with random threshold model[J].International Journal for Numerical Methods in Engineering, 2018, 115:99-116.
[29] Haldar A, Mahadevan S. Reliability assessment using stochastic finite element analysis. New York : Wiley, 2000.
[30] Ghanem R G, Doostan A. On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data. Journal of Computational Physics, 2006, 217(1): 63-81.
[31] Smolyak S A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Mathematics, 1963, 4:240-243.
[32] Novak E, Ritter K. Simple cubature formulas with high polynomial exactness. Journal of Constructive Approximation, 1999, 15(4):499-522.
[33] Barthelmann V, Novak E, Ritter K. High dimensional polynomial interpolation on sparse grids. Journal of Advances in Computational Mathematics, 2000, 12(4):273-288.
[34] Klimke A, Wohlmuth B. Piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Transactions on Mathematical Software, 2005, 31(4):561-579.
[35] Gerstner T, Griebel M. Numerical integration using sparse grids. Number Algorithms, 1998, 18(3-4):209-232.
[36] Klimke A, Wohlmuth B. Computing expensive multivariate functions of fuzzy numbers using sparse grids. Journal of Fuzzy Sets and Systems, 2005, 154(3):432-453.
[37] Xiu D, Hesthaven J. High order collocation method for differential equations with random inputs. Society for Industrial and Applied Mathematics, 2005, 27(3):1118-1139.
[38] Behrooz H and Ernest H. Homogenization and structural topology optimization. London: Springer-Verlag Limited, 1999.
[39] 张志飞, 倪新帅, 徐中明, 贺岩松, 袁琼, 李耀光. 基于优化准则法的自由阻尼材料拓扑优化.振动与冲击, 2013, 32(14): 98-102. (Zhang Zhifei, Ni Xinshuai, Xu Zhongming, He Yansong, Yuan Qiong, Li Yaoguang. Topologic optimization of a free damping material based on optimal criteria method. Journal of Vibration and Shock, 2013, 32(14): 98-102(in Chinese))
[40] Bendsøe M P and Sigmund O. Topology Optimization Theory, Methods and Application. Berlin: Springer, 2003.