基于梯度和海森矩阵计算考虑压杆稳定的桁架动力响应形状优化
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摘要
提出一种在动力作用下,以获取重量最小化设计同时满足强度、刚度和稳定性约束的桁架形状优化设计方法。①在Newmark-β法的基础上导出动力响应及其对设计变量一阶导数和二阶导数的计算方法。②通过积分型罚函数将含时间参数的不等式约束问题转变为一系列不含时间参数的无约束问题,并利用动力响应的一阶导数和二阶导数计算罚函数的梯度和海森矩阵。③使用充分利用梯度和海森矩阵的Marquardt方法,求解无约束优化问题。最后,演示一个十杆桁架的动力响应形状优化设计。结果表明,文中所提方法是一种考虑压杆稳定的桁架在动力作用下有效的形状优化设计方法。
A shape optimization method of truss structures subjected to dynamic loads is developed.The objective is to achieve the minimum weight design.The constraints include the strength,stiffness and stability of the truss.The work is arranged as: Firstly,the dynamic responses,their first and second derivatives with respect to design variables are calculated based on Newmarkβ method.Secondly,the inequality time-dependent constraint problem is converted into a sequence of appropriately formed time-independent unconstrained problems using the integral penalty function method.Gradient and Hessian matrix of the penalty function are calculated using the first and second derivatives of dynamic responses with respect to design variables.Thirdly,Marquardt's method which makes fully use of gradient and Hessian matrix is employed to solve unconstrained problems.Finally,finding optimum shape design of a ten-bar truss subjected to dynamic loads is demonstrated.The results show that optimization methods presented in this paper are an effective approach for minimum weight design with strength,stiffness and stability constraints.
A shape optimization method of truss structures subjected to dynamic loads is developed.The objective is to achieve the minimum weight design.The constraints include the strength,stiffness and stability of the truss.The work is arranged as: Firstly,the dynamic responses,their first and second derivatives with respect to design variables are calculated based on Newmarkβ method.Secondly,the inequality time-dependent constraint problem is converted into a sequence of appropriately formed time-independent unconstrained problems using the integral penalty function method.Gradient and Hessian matrix of the penalty function are calculated using the first and second derivatives of dynamic responses with respect to design variables.Thirdly,Marquardt's method which makes fully use of gradient and Hessian matrix is employed to solve unconstrained problems.Finally,finding optimum shape design of a ten-bar truss subjected to dynamic loads is demonstrated.The results show that optimization methods presented in this paper are an effective approach for minimum weight design with strength,stiffness and stability constraints.
引文
[1]朱朝艳,刘斌,郭鹏飞,等.离散变量桁架结构拓扑优化的混合遗传算法[J].机械强度,2004,26(6):656-661.ZHU ChaoYan,LIU Bin,GUO PengFei,et al.Hybrid genetic algorithmfor discrete topology optimization of trusses[J].Journal of MechanicalStrength,2004,26(6):656-661(In Chinese).
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[14]刘齐茂,燕柳斌,邓朗妮.桁架形状优化的一种改进模拟退火算法[J].计算机工程与应用,2007,43(27):218-221.LIU QiMao,YAN LiuBin,DENG LangNi.Improved simulated annealingalgorithm study on truss shape optimal design[J].Computer Engineeringand Applications,2007,43(27):218-221(In Chinese).
[15]刘齐茂,燕柳斌.强震作用下框架结构的优化设计[J].西北地震学报,2008,30(2):107-112.LIU QiMao,YAN LiuBin.An optimal design method for spacial framestructures under severe earthquake loading[J].Northwestern Seismologi-cal Journal,2008,30(2):107-112(In Chinese).
[16]Fletcher R,Reeves C M.Function minimization by Conjugate Gradients[J].Computer Journal,1964,6:163-168.
[17]Broydon C G.Quasi-newton methods and their application to functionminimization[J].Mathematics of Computation,1967,21:368-381.
[18]陈建军,车建文,崔明涛,等.结构动力优化设计述评与展望[J].力学进展,2001,31(2):181-192.CHEN JianJun,CHE JianWen,CUI MingTao,et al.A review on struc-tural dynamic optimum design[J].Advances in Mechanics,2001,31(2):182-192(In Chinese).
[19]Reklaitis G V,Ravindran A,Ragsdell K M.Engineering optimizationmethods and applications[M].New York:John Wiley&Sons,1983:185-251.
[2]Wang D,Zhang W H,Jiang J S.Truss shape optimization with multipledisplacement constraints[J].Computer Methods in Applied Mechanicsand Engineering,2002,191:3597-3612.
[3]Lluís Gil,Antoni Andreu.Shape and cross-section optimization of a trussstructure[J].Computers and Structures,2001,79:681-689.
[4]隋允康,高峰,龙连春,等.基于层次分解方法的桁架结构形状优化[J].计算力学学报,2006,23(1):46-51.SUI YunKang,GAO Feng,LONG LianChun,et al.Shape optimizationfor truss structures based on decomposition method[J].Chinese Journal ofComputational Mechanics,2006,23(1):46-51(In Chinese).
[5]Makoto Ohsaki.Simultaneous optimization of topology and geometry of aregular plane truss[J].Computers and Structures,1998,66(1):69-71.
[6]Tsuneyoshi Nakamura,Makoto Ohsaki.A natural generator of optimumtopology of plane trusses for specified fundamental frequency[J].Com-puter Methods in Applied Mechanics and Engineering,1992,94(1):113-129.
[7]刘军伟,姜节胜.桁架动力学形状优化的统一设计变量方法[J].振动工程学报,2000,13(1):84-88.LIU JunWei,JIANG JieSheng.Shape optimization for truss dynamics withunified design variables[J].Journal of Vibration Engineering,2000,13(1):84-88(In Chinese).
[8]王栋,李晶.空间桁架结构动力学形状优化设计[J].工程力学,2007,24(4):129-134.WANG Dong,LI Jing.Shape optimization of space trusses subject to fre-quency constraints[J].Engineering Mechanics,2007,24(4):129-134(In Chinese).
[9]Pereza R E,Behdinan K.Particle swarm approach for structural designoptimization[J].Computers and Structures,2007,85:1579-1588.
[10]Azid I A,Kwan A S K,Seetharamu K N.A GA-based technique forlayout optimization of truss with stress and displacement constraints[J].International Journal for Numerical Methods in Engineering,2002,53:1641-1674.
[11]Kalyanmoy Deb,Surendra Gulati.Design of truss-structures for minimumweight using genetic algorithms[J].Finite Elements in Analysis and De-sign,2001,37:447-465.
[12]Vedat Togan,Ayse T Daloglu.Optimization of 3D trusses with adaptiveapproach in genetic algorithms[J].Engineering Structures,2006,28:1019-1027.
[13]Oguzhan Hasancebi,Fuat Erbatur.Layout optimization of trusses usingsimulated annealing[J].Advances in Engineering Software,2002,33:681-696.
[14]刘齐茂,燕柳斌,邓朗妮.桁架形状优化的一种改进模拟退火算法[J].计算机工程与应用,2007,43(27):218-221.LIU QiMao,YAN LiuBin,DENG LangNi.Improved simulated annealingalgorithm study on truss shape optimal design[J].Computer Engineeringand Applications,2007,43(27):218-221(In Chinese).
[15]刘齐茂,燕柳斌.强震作用下框架结构的优化设计[J].西北地震学报,2008,30(2):107-112.LIU QiMao,YAN LiuBin.An optimal design method for spacial framestructures under severe earthquake loading[J].Northwestern Seismologi-cal Journal,2008,30(2):107-112(In Chinese).
[16]Fletcher R,Reeves C M.Function minimization by Conjugate Gradients[J].Computer Journal,1964,6:163-168.
[17]Broydon C G.Quasi-newton methods and their application to functionminimization[J].Mathematics of Computation,1967,21:368-381.
[18]陈建军,车建文,崔明涛,等.结构动力优化设计述评与展望[J].力学进展,2001,31(2):181-192.CHEN JianJun,CHE JianWen,CUI MingTao,et al.A review on struc-tural dynamic optimum design[J].Advances in Mechanics,2001,31(2):182-192(In Chinese).
[19]Reklaitis G V,Ravindran A,Ragsdell K M.Engineering optimizationmethods and applications[M].New York:John Wiley&Sons,1983:185-251.
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