基于证据理论的结构非概率可靠性拓扑优化设计
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摘要
考虑数据信息较少以及认知水平有限情况下的不确定性对于结构拓扑优化具有重要意义.本文引入证据理论处理不精确的数据信息,采用证据理论的不确定测度克服精确概率约束模型建立的困难,并结合拓扑优化策略,形成了基于证据理论的可靠性拓扑优化设计模型.为了提高不确定测度的计算效率,提出了仿生智能优化算法和不精确极值思想相结合的改进优化算法来降低焦元数目和极限状态函数极值求解所导致的计算量.通过两个桁架算例对所提方法进行验证,结果表明,确定性优化结果可能是不确定情况下的失效解,虽然基于证据理论的最优设计在重量和拓扑形式上相对于确定性优化结果偏保守,但具备抵抗不确定波动的能力.
It is of great importance to consider uncertainty due to insufficient data or imprecise information in topology optimization design.In this paper,evidence theory is presented to handle the imprecise data situation.The plausibility measure based on evidence theory is introduced to overcome the difficulty of constructing the precise probabilistic constraint.Furthermore,the topology problem in this evidence-based optimization design is solved by combined strategy of the differential evolution and superior topology technique.In order to overcome the difficulty of intensive computational cost in calculating plausibility measure,an improved method of evolutionary optimization design integrating with imprecise extremum idea is proposed.Two truss examples are given to demonstrate the proposed approach.The research indicates that deterministic results may be the failure solutions under epistemic uncertainty.Although evidence-based optimum topology designs are more conservative than deterministic results in aspects of weight and optimal structural topology layout,it gains a more robust design under epistemic uncertainty.
It is of great importance to consider uncertainty due to insufficient data or imprecise information in topology optimization design.In this paper,evidence theory is presented to handle the imprecise data situation.The plausibility measure based on evidence theory is introduced to overcome the difficulty of constructing the precise probabilistic constraint.Furthermore,the topology problem in this evidence-based optimization design is solved by combined strategy of the differential evolution and superior topology technique.In order to overcome the difficulty of intensive computational cost in calculating plausibility measure,an improved method of evolutionary optimization design integrating with imprecise extremum idea is proposed.Two truss examples are given to demonstrate the proposed approach.The research indicates that deterministic results may be the failure solutions under epistemic uncertainty.Although evidence-based optimum topology designs are more conservative than deterministic results in aspects of weight and optimal structural topology layout,it gains a more robust design under epistemic uncertainty.
引文
1张卓群,李宏男,黄连状.基于蚁群算法的结构拓扑优化方法.应用力学学报,2011,28:226-232
2 Deb K,Gulati S.Design of truss-structures for minimum weight using genetic algorithms.Finite Elem Anal Des,2001,37:447-465
3 Miguel L F F,Lopez R H,Miguel L F F.Multimodal size,shape,and topology optimisation of truss structures using the Firefly algorithm.Adv Eng Software,2013,56:23-37
4陈建军,曹一波.基于可靠性的桁架结构拓扑优化设计.力学学报,1998,30:277-284
5徐斌,姜节胜,闫云聚.具有频率约束的桁架结构可靠性拓扑优化.应用力学学报,2001,18:45-50
6亢战,罗阳军.桁架结构非概率可靠性拓扑优化.计算力学学报,2008,25:589-594
7李东泽,于登云,马兴瑞.不确定载荷下的桁架结构拓扑优化.北京航空航天大学学报,2009,35:1171-1178
8乔心州,魏琳娟,仇原鹰.桁架结构概率-非概率混合可靠性拓扑优化.应用力学学报,2011,28:402-407
9 Helton J C,Johnson J D,Oberkampf W L.An exploration of alternative approaches to the representation of uncertainty in model predictions.Reliab Eng Syst Safe,2004,85:39-71
10 Dempster A P.Upper and lower probabilities induced by a multivalued mapping.Ann Math Statist,1967,38:325-339
11 Shafer G A.Mathematical Theory of Evidence.Princeton:Princeton University Press,1976.10-50
12 Agarwal H,Renaud J E,Preston E L,et al.Uncertainty quantification using evidence theory in multidisciplinary design optimization.Reliab Eng Syst Safe,2004,85:281-294
13 Mourelatos Z P,Zhou J.A design optimization method using evidence theory.J Mech Des,2006,128:901-908
14 Srivastava R K,Deb K,Tulshyan R.An evolutionary algorithm based approach to design optimization using evidence theory.J Mech Des,2013,135:081003
15 Bae H R,Grandhi R V,Canfield R A.An approximation approach for uncertainty quantification using evidence theory.Reliab Eng Syst Safe,2004,86:215-225
16 Vasile M.Robust mission design through evidence theory and multiagent collaborative search.Ann New York Acad Sci,2005,1065:152-173
17 Hou L,Cai Y,Zhang R,et al.Evidence theory based multidisciplinary robust optimization for micro mars entry probe design.In:Emmerich M,Deutz A,Schuetze O,et al.EVOLVE-A Bridge Between Probability,Set Oriented Numerics,and Evolutionary Computation IV.Heidelberg:Springer,2013.307-22
18郭慧昕,刘德顺,胡冠昱,等.证据理论和区间分析相结合的可靠性优化设计方法.机械工程学报,2008,44:36-41
19姜潮,张哲,韩旭,等.一种基于证据理论的结构可靠性分析方法.力学学报,2013,45:103-115
20李晓斌,张为华,王中伟.基于证据理论的固体火箭发动机不确定性分析.弹箭与制导学报,2006,26:420-425
21唐和生,苏瑜,薛松涛,等.结构可靠性优化设计的证据理论和微分演化算法.湖南大学学报:自然科学版,2014,41:33-38
22 Hu S,Luo J.Uncertainty quantification for structural optimal design based on evidence theory.J Shanghai Jiaotong Univ(Sci),2015,20:338-343
23 Storn R,Price K.Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces.J Glob Optimiz,1997,11:341-359
24黄冀卓,王湛.基于遗传算法的离散型结构拓扑优化设计.工程力学,2008,25:32-38
25 Rahami H,Kaveh A,Gholipour Y.Sizing,geometry and topology optimization of trusses via force method and genetic algorithm.Eng Struct,2008,30:2360-2369
26 Soh C K,Yang J.Optimal layout of bridge trusses by genetic algorithms.Comp-aided Civil Eng,1998,13:247-254
2 Deb K,Gulati S.Design of truss-structures for minimum weight using genetic algorithms.Finite Elem Anal Des,2001,37:447-465
3 Miguel L F F,Lopez R H,Miguel L F F.Multimodal size,shape,and topology optimisation of truss structures using the Firefly algorithm.Adv Eng Software,2013,56:23-37
4陈建军,曹一波.基于可靠性的桁架结构拓扑优化设计.力学学报,1998,30:277-284
5徐斌,姜节胜,闫云聚.具有频率约束的桁架结构可靠性拓扑优化.应用力学学报,2001,18:45-50
6亢战,罗阳军.桁架结构非概率可靠性拓扑优化.计算力学学报,2008,25:589-594
7李东泽,于登云,马兴瑞.不确定载荷下的桁架结构拓扑优化.北京航空航天大学学报,2009,35:1171-1178
8乔心州,魏琳娟,仇原鹰.桁架结构概率-非概率混合可靠性拓扑优化.应用力学学报,2011,28:402-407
9 Helton J C,Johnson J D,Oberkampf W L.An exploration of alternative approaches to the representation of uncertainty in model predictions.Reliab Eng Syst Safe,2004,85:39-71
10 Dempster A P.Upper and lower probabilities induced by a multivalued mapping.Ann Math Statist,1967,38:325-339
11 Shafer G A.Mathematical Theory of Evidence.Princeton:Princeton University Press,1976.10-50
12 Agarwal H,Renaud J E,Preston E L,et al.Uncertainty quantification using evidence theory in multidisciplinary design optimization.Reliab Eng Syst Safe,2004,85:281-294
13 Mourelatos Z P,Zhou J.A design optimization method using evidence theory.J Mech Des,2006,128:901-908
14 Srivastava R K,Deb K,Tulshyan R.An evolutionary algorithm based approach to design optimization using evidence theory.J Mech Des,2013,135:081003
15 Bae H R,Grandhi R V,Canfield R A.An approximation approach for uncertainty quantification using evidence theory.Reliab Eng Syst Safe,2004,86:215-225
16 Vasile M.Robust mission design through evidence theory and multiagent collaborative search.Ann New York Acad Sci,2005,1065:152-173
17 Hou L,Cai Y,Zhang R,et al.Evidence theory based multidisciplinary robust optimization for micro mars entry probe design.In:Emmerich M,Deutz A,Schuetze O,et al.EVOLVE-A Bridge Between Probability,Set Oriented Numerics,and Evolutionary Computation IV.Heidelberg:Springer,2013.307-22
18郭慧昕,刘德顺,胡冠昱,等.证据理论和区间分析相结合的可靠性优化设计方法.机械工程学报,2008,44:36-41
19姜潮,张哲,韩旭,等.一种基于证据理论的结构可靠性分析方法.力学学报,2013,45:103-115
20李晓斌,张为华,王中伟.基于证据理论的固体火箭发动机不确定性分析.弹箭与制导学报,2006,26:420-425
21唐和生,苏瑜,薛松涛,等.结构可靠性优化设计的证据理论和微分演化算法.湖南大学学报:自然科学版,2014,41:33-38
22 Hu S,Luo J.Uncertainty quantification for structural optimal design based on evidence theory.J Shanghai Jiaotong Univ(Sci),2015,20:338-343
23 Storn R,Price K.Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces.J Glob Optimiz,1997,11:341-359
24黄冀卓,王湛.基于遗传算法的离散型结构拓扑优化设计.工程力学,2008,25:32-38
25 Rahami H,Kaveh A,Gholipour Y.Sizing,geometry and topology optimization of trusses via force method and genetic algorithm.Eng Struct,2008,30:2360-2369
26 Soh C K,Yang J.Optimal layout of bridge trusses by genetic algorithms.Comp-aided Civil Eng,1998,13:247-254