Please wait a minute...
浙江大学学报(工学版)
浙江大学学报(工学版)  2019, Vol. 53 Issue (3): 435-443    DOI: 10.3785/j.issn.1008-973X.2019.03.004
机械工程     
稀疏混合不确定变量优化方法及应用
张鹏(),刘晓健*(),张树有,裘乐淼,伊国栋
浙江大学 流体动力与机电系统国家重点实验室，浙江 杭州 310027
Sparse hybrid uncertain variable optimization method and application
Peng ZHANG(),Xiao-jian LIU*(),Shu-you ZHANG,Le-miao QIU,Guo-dong YI
State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, China
 全文: PDF(879 KB)   HTML
摘要：

针对复杂产品设计中稀疏混合不确定变量导致的设计边界识别困难、计算结果失真等问题，提出一种基于Chebyshev逼近的稀疏混合不确定变量优化方法. 首先采用极大似然估计方法构造稀疏混合不确定变量在给定分布下的概率密度函数，初步确定其在给定分布下对应的分布参数；其次基于贝叶斯信息准则计算对应分布的信息损失，进一步确定稀疏混合不确定变量的最合适的分布及分布参数. 再次，为解决传统区间分析方法中区间扩张导致的计算失真问题，采用Chebyshev逼近优化目标函数并利用改进的HL-RF算法求解，获取可靠性指标及失效概率，在满足设计需求的同时，有效实现产品轻量化、刚度保持的设计目标. 最后，以数值算例及高速压力机滑块的优化设计验证了所提方法的有效性.
关键词： 稀疏混合不确定;  优化设计;  贝叶斯信息准则;  分布参数;  Chebyshev逼近    
Abstract:

Sparse hybrid uncertainties in the distribution of random variables brings the problem of hard border detection and calculation results distortion in complex product design. A sparse hybrid uncertain variable optimization method based on Chebyshev approximation was proposed. Firstly, the maximum likelihood estimation was utilized to construct the probability density function of a sparse hybrid uncertain variable under a given distribution, and its distribution parameter corresponding to the given distribution was preliminarily determined. Secondly, based on the Bayesian information criterion, the information loss of the corresponding distribution was calculated, and the most suitable distribution and parameters of the sparse mixed uncertain variables were further determined. Thirdly, in order to solve the computational distortion caused by interval expansion of the traditional interval analysis method, Chebyshev approximation was used to optimize the objective function and the improved HL-RF algorithm was used to obtain the reliability index and the failure probability value; while meeting the design requirements, the design goal of light weight and rigidity retention was effectively realized. Finally, the effectiveness of the proposed method is verified by numerical examples and the optimization design of a high-speed press slider.
Key words: sparse hybrid uncertainty    optimization design    Bayesian information criterion    distribution parameters    Chebyshev approximation
收稿日期: 2018-02-09 出版日期: 2019-03-04
CLC:  TH 122  
通讯作者: 刘晓健     E-mail: absent1353@163.com;liuxj@zju.edu.cn
作者简介: 张鹏（1988—），男，博士生，从事产品数字化设计、最优化方法研究. orcid.org/0000-0002-3090-5025. E-mail: absent1353@163.com
服务　　
把本文推荐给朋友
加入引用管理器
E-mail Alert
作者相关文章　　
张鹏
刘晓健
张树有
裘乐淼
伊国栋

引用本文:

张鹏,刘晓健,张树有,裘乐淼,伊国栋. 稀疏混合不确定变量优化方法及应用[J]. 浙江大学学报(工学版), 2019, 53(3): 435-443.

Peng ZHANG,Xiao-jian LIU,Shu-you ZHANG,Le-miao QIU,Guo-dong YI. Sparse hybrid uncertain variable optimization method and application. Journal of ZheJiang University (Engineering Science), 2019, 53(3): 435-443.

链接本文:

http://www.zjujournals.com/eng/CN/10.3785/j.issn.1008-973X.2019.03.004    或    http://www.zjujournals.com/eng/CN/Y2019/V53/I3/435

图 1  区间分析方法与Chebyshev逼近的误差对比
对比方法 β Pf δ/%
区间分析方法 ?0.65 0.742 2 1.41
本文方法 ?0.676 4 0.750 6 0.29
Rosenbrock精确解 ?0.683 0.752 8 —
表 1  数值算例可靠性分析对比
图 2  宽台面超精密高速压力机及滑块数字样机
分布 参数1 参数2 BIC值
θ1：正态分布 600 5 29.34
θ2：均匀分布 520 687 35.45
θ3：极值I型分布 622 9 38.36
θ4：对数正态分布 6.42 2.48 33.82
θ5：F分布 5 2 42.23
θ6：威布尔分布 590 85 27.25
θ7：指数分布 600 — 38.56
θ8：伽马分布 600 1.58 36.85
表 2  分布类型及分布参数最优解
图 3  威布尔条件下变量x3的概率密度分布
最优设计变量 β Pf 设计目标
区间分析
方法 		x1=412 mm 1.982 0.023
x2=266 mm m=1 287 kg,
Pmax=53.48 MPa
x3=572 mm
E=1.45×105 MPa
v=0.25
本文方法 x1=366 mm 2.10 0.018
x2=272 mm m=1 201 kg,
Pmax=53.44 MPa
x3=564 mm
E=1.44×105 MPa
v=0.24
表 3  宽台面超精密高速压力机滑块设计优化对比
图 5  滑块载荷约束施加示意图
图 4  Chebyshev逼近及区间分析方法的可靠性求解迭代过程
1 JENSEN H A, MAYORGA F, VALDEBENITO M A Reliability sensitivity estimation of nonlinear structural systems under stochastic excitation: a simulation-based approach[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 289 (1): 1- 23
2 SESHADRI P, CONTINE P, IACARINO G, et al A density-matching approach for optimization under uncertainty[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 305 (15): 562- 578
3 LIN T P, GEA H C, JALURIA Y A modified reliability index approach for reliability-based design optimization[J]. Journal of Mechanical Design, 2011, 133 (4): 044501
4 DEB K, GUPTA S, DAUM D, et al Reliability-based optimization using evolutionary algorithms[J]. IEEE Transactions On Evolutionary Computation, 2009, 13 (5): 1054- 1074
doi: 10.1109/TEVC.2009.2014361
5 SINHA K Reliability-based multi-objective optimization for automotive crashworthiness and occupant safety[J]. Structural and Multidisciplinary Optimization, 2007, 33 (3): 255- 268
6 姜潮. 基于区间的不确定性优化理论与算法[D]. 长沙: 湖南大学,2008.
JIANG Chao. Theories and algorithms of uncertain optimization based on interval [D]. Changsha: Hunan University, 2008.
7 CHENG J, TANG M Y, LIU Z Y, et al Direct reliability-based design optimization of uncertain structures with interval parameters[J]. Journal of Zhejiang University-Science A: Applied Physics and Engineering, 2016, 17 (11): 841- 854
8 KUNDU A, ADHIKARI S, FRISWELL M I Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty[J]. International Journal for Numerical Methods in Engineering, 2014, 100 (3): 183- 221
doi: 10.1002/nme.v100.3
9 CHENG J, LIU Z Y, TANG M Y, et al Robust optimization of uncertain structures based on normalized violation degree of interval constraint[J]. Computers and Structures, 2017, 182 (1): 41- 54
10 JIANG C, LI W X, HAN X Structural reliability analysis based on random distributions with interval parameters[J]. Computers and Structures, 2011, 89 (23/24): 2292- 2302
11 JIANG C, LI W X, HAN X A hybrid reliability approach based on probability and interval for uncertain structures[J]. Journal of Mechanical Design, 2012, 134 (3): 031001
12 姜潮, 韩旭, 谢慧超. 区间不确定性优化设计理论有方法: 第一版[M]. 北京: 科学出版社, 2017: 22−29.
13 JIANG C, HAN X, LIU G P A sequential nonlinear interval number programming method for uncertain structures[J]. Computer Methods in Applied Mechanics and Engineering, 2008, 197 (49-50): 4250- 4265
doi: 10.1016/j.cma.2008.04.027
14 张德权, 韩旭, 姜潮, 等 时变可靠性的区间PHI2分析方法[J]. 中国科学: 物理学力学天文学, 2015, 45 (5): 54- 61
ZHANG De-quan, HAN Xu, JIANG Chao, et al The interval PHI2 analysis method for time dependent reliability[J]. SCIENTIA SINICA: Physica, Mechanica and Astronomica, 2015, 45 (5): 54- 61
15 JIANG C, LU G Y, HAN X, et al A new reliability analysis method for uncertain structures with random and interval variables[J]. International Journal of Mechanics and Materials in Design, 2012, 8 (2): 169- 182
doi: 10.1007/s10999-012-9184-8
16 HAN X, JIANG C, LIU L X, et al Response-surface-based structural reliability analysis with random and interval mixed uncertainties[J]. Science China Technological Sciences, 2014, 57 (7): 1322- 1334
doi: 10.1007/s11431-014-5581-6
17 LIM W, JANG J, KIM S, et al Reliability-based design optimization of an automotive structure using a variable uncertainty[J]. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 2016, 230 (10): 1314- 1323
doi: 10.1177/0954407015606825
18 PENG X, LI J Q, JIANG S F Unified uncertainty representation and quantification based on insufficient input data[J]. Structural and Multidisciplinary Optimization, 2017, 56 (6): 1305- 1317
doi: 10.1007/s00158-017-1722-4
19 PENG X, WU T J, LI J Q, et al Hybrid reliability analysis with uncertain statistical variables, sparse variables and interval variables[J]. Engineering Optimization, 2018, 50 (4): 1- 17
20 SANKARARAMAN S, MAHADEVAN S Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data[J]. Reliability Engineering and System Safety, 2011, 96 (7): 814- 824
doi: 10.1016/j.ress.2011.02.003
21 SANKARARAMAN S, MAHADEVAN S Distribution type uncertainty due to sparse and imprecise data[J]. Mechanical Systems and Signal Processing, 2013, 37 (1-2): 182- 198
doi: 10.1016/j.ymssp.2012.07.008
22 DAVID F F Counterexamples to parsimony and BIC[J]. Annals of the Institute of Statistical Mathematics, 1991, 43 (3): 505- 514
doi: 10.1007/BF00053369
23 WU J L, ZHANG Y Q, CHEN L P A Chebyshev interval method for nonlinear dynamic systems under uncertainty[J]. Applied Mathematical Modelling, 2013, 37 (6): 4578- 4591
doi: 10.1016/j.apm.2012.09.073
24 BERNARDO J M, SMITH A F. Bayesian theory [M]. 1st ed. NewYork: JohnWiley & Sons, 1994.
25 MADSEN H O, KRENK S, LIND N C. Methods of structural safety [M]. 1st ed. Englewood Cliffs: Prentice-Hall, 1986.
[1] 王忠宇,王玲,王艳丽,吴兵. 基于网络变结构优化的大型活动交通拥堵预防方法[J]. 浙江大学学报(工学版), 2021, 55(2): 358-366.
[2] 黄凯,孙志坚,钱文瑛,杨继虎,俞自涛,胡亚才. 中间介质型烟气换热器无量纲材料成本模型[J]. 浙江大学学报(工学版), 2020, 54(7): 1362-1368.
[3] 喻伯平,李高华,谢亮,王福新. 基于代理模型的旋翼翼型动态失速优化设计[J]. 浙江大学学报(工学版), 2020, 54(4): 833-842.
[4] 张雷,张立华,王家序,李俊阳,肖科. 基于响应面的柔轮应力和刚度分析[J]. 浙江大学学报(工学版), 2019, 53(4): 638-644.
[5] 周朝君,黄明辉,陆新江. 基于低维约束嵌入的分布参数系统建模[J]. 浙江大学学报(工学版), 2019, 53(11): 2154-2162.
[6] 刘国梁, 李新, 伍梁, 李振宇, 陈国柱. 宽范围输入输出电压LCC谐振变换器的分析设计[J]. 浙江大学学报(工学版), 2018, 52(9): 1762-1770.
[7] 宋伟, 姜红建, 王滔, 高振飞, 杜镇韬, 朱世强. 爬壁机器人磁吸附组件优化设计与试验研究[J]. 浙江大学学报(工学版), 2018, 52(10): 1837-1844.
[8] 唐伟佳, 江道灼, 尹瑞, 梁一桥, 王玉芬. 模块化隔离型直流变换器的同轴变压器设计[J]. 浙江大学学报(工学版), 2017, 51(8): 1646-1652.
[9] 张玄武, 郑耀, 杨波威, 张继发. 基于级联前向网络的翼型优化设计[J]. 浙江大学学报(工学版), 2017, 51(7): 1405-1411.
[10] 蒋卓华, 蒋焕煜, 童俊华. 穴盘苗自动移栽机末端执行器的优化设计[J]. 浙江大学学报(工学版), 2017, 51(6): 1119-1125.
[11] 田燕萍,金肖玲,王永. 强化现象启发的随机振动能量收集器优化设计[J]. 浙江大学学报(工学版), 2016, 50(5): 934-940.
[12] 赵琼,童水光,钟崴,葛俊旭. 基于GA-FEA的门座起重机变幅机构优化设计[J]. 浙江大学学报(工学版), 2015, 49(5): 880-886.
[13] 胡友瑞,刘彦,汪洋,刘建忠,周俊虎,胡巍,李洪伟. 高湿氢氧喷注器数值分析与正交设计[J]. 浙江大学学报(工学版), 2015, 49(12): 2403-2409.
[14] 陈进,庆飞,庞晓平. 基于组合挖掘的反铲液压挖掘机工作装置优化设计[J]. 浙江大学学报(工学版), 2014, 48(9): 1654-1660.
[15] 王异凡, 张 曙,陈国柱. MCR型动态无功补偿装置优化设计[J]. 浙江大学学报(工学版), 2014, 48(9): 1676-1681.