基于梯度和海森矩阵计算在地震作用下桁架的形状优化
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摘要
大型复杂桁架地震响应的形状优化需要大量的计算量,非梯度类算法由于效率低下通常很难成功解决该类问题.本文提出一种在地震作用下以获取质量最小化的二阶优化设计同时满足应力和位移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和二阶导数计算罚函数的梯度和海森矩阵;3)充分利用梯度和海森矩阵的Marquardt方法求解无约束优化问题;演示了一个45杆桁架的形状优化设计.结果表明本文方法是一种桁架在地震作用下有效和高效的形状优化设计方法.
This paper developed a shape and cross-section optimization method of truss subjected to earthquake excitation for achieving minimum weight design with normal stress and nodal displacement constraints.First,the dynamic responses,their first and second derivatives with respect to design variables are calculated based on Newmark-β method.Second,the inequality constraint problem with time parameter is converted into a sequence of appropriately formed unconstrained problems without time parameter by using the integral penalty function method.The gradient and Hessian matrix of penalty function are calculated by using dynamic response first and second derivatives.Third,Marquardt's method which makes fully use of gradient and Hessian matrix is employed to solve unconstrained problems. Finally,finding optimum design of a 45-bar truss is demonstrated.The results show that optimization methods presented in this paper are an effective and highly efficient approach for minimum weight design.
This paper developed a shape and cross-section optimization method of truss subjected to earthquake excitation for achieving minimum weight design with normal stress and nodal displacement constraints.First,the dynamic responses,their first and second derivatives with respect to design variables are calculated based on Newmark-β method.Second,the inequality constraint problem with time parameter is converted into a sequence of appropriately formed unconstrained problems without time parameter by using the integral penalty function method.The gradient and Hessian matrix of penalty function are calculated by using dynamic response first and second derivatives.Third,Marquardt's method which makes fully use of gradient and Hessian matrix is employed to solve unconstrained problems. Finally,finding optimum design of a 45-bar truss is demonstrated.The results show that optimization methods presented in this paper are an effective and highly efficient approach for minimum weight design.
引文
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[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 frequency constraints[J].Engineering Mechanics,2007,24(4):129-134.(in Chinese)
[9]PEREZA R E,BEHDINAN K.Particle swarm approach for structural design optimization[J].Computers and Structures,2007,85:1579-1588.
[10]AZID I A,KWAN A S K,SEETHARAMU K N.A GA-based technique for layout optimization of truss with stress anddisplacement constraints[J].International Journal for Numerical Methods in Engineering,2002,53:1641-1674.
[11]KALYANMOY D,SURENDRA G.Design of truss-structures for minimum weight using genetic algorithms[J].FiniteElements in Analysis and Design,2001,37:447-465.
[12]VEDAT T,AYSE T D.Optimization of3d trusses with adaptive approach in genetic algorithms[J].Engineering Structures,2006,28:1019-1027.
[13]OGUZHAN H,FUAT E.Layout optimisation of trusses using simulated annealing[J].Advances in Engineering Software,2002,33:681-696.
[14]REKLAITIS G V,RAVINDRAN A,RAGSDELL K M.Engineering optimization methods and applications[M].New York:John Wiley&Sons,1983.
[2]WANG D,ZHANG WH,JIANG J S.Truss shape optimization with multiple displacement constraints[J].Computer Methodsin Applied Mechanics and Engineering,2002,191:3597-3612.
[3]LLUS G,ANTONI A.Shape and cross-section optimisation of a truss structure[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 Yun-kang,GAO Feng,LONG Lian-chun,et al.Shape optimization for truss structures based on decomposition method[J].Chinese Journal of Computational Mechanics,2006,23(1):46-51.(in Chinese)
[5]MAKOTO O.Simultaneous optimization of topology and geometry of a regular plane truss[J].Computers&Structures,1998,66(1):69-71.
[6]TSUNEYOSHI N,MAKOTO O.A natural generator of optimum topology of plane trusses for specified fundamental frequency[J].Computer 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 Jun-wei,JIANG Jie-sheng.Shape optimization for truss dynamics with unified design variables[J].Journal of VibrationEngineering,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 frequency constraints[J].Engineering Mechanics,2007,24(4):129-134.(in Chinese)
[9]PEREZA R E,BEHDINAN K.Particle swarm approach for structural design optimization[J].Computers and Structures,2007,85:1579-1588.
[10]AZID I A,KWAN A S K,SEETHARAMU K N.A GA-based technique for layout optimization of truss with stress anddisplacement constraints[J].International Journal for Numerical Methods in Engineering,2002,53:1641-1674.
[11]KALYANMOY D,SURENDRA G.Design of truss-structures for minimum weight using genetic algorithms[J].FiniteElements in Analysis and Design,2001,37:447-465.
[12]VEDAT T,AYSE T D.Optimization of3d trusses with adaptive approach in genetic algorithms[J].Engineering Structures,2006,28:1019-1027.
[13]OGUZHAN H,FUAT E.Layout optimisation of trusses using simulated annealing[J].Advances in Engineering Software,2002,33:681-696.
[14]REKLAITIS G V,RAVINDRAN A,RAGSDELL K M.Engineering optimization methods and applications[M].New York:John Wiley&Sons,1983.
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