基于应力约束凝聚化的连续体结构拓扑优化技术研究

发布时间：2018-04-14 01:00

本文选题：连续体结构 + 拓扑优化　；参考：《长沙理工大学》2015年硕士论文

【摘要】：基于应力约束的拓扑优化研究是拓扑优化设计中的一个非常有应用前景,又具有很大难度的课题。与涉及柔顺度的拓扑优化的文献相比,基于应力约束的拓扑优化方面文献很少。当前基于应力约束的拓扑优化方面仍然存在许多问题,如应力奇异、应力集中和近似等效优化模型等。因此,研究应力约束的拓扑优化问题具有一定的理论价值和工程应用价值。针对应力约束、体积最小拓扑优化问题,在已有方法和措施基础上,本文完成了大量数值仿真,并分析了应力场及相关凝聚函数的变化规律。提出了基于应力梯度及应力约束凝聚化的连续体结构拓扑优化技术,分别建立了相应的算法,给出了验证算例。首先,本文采用RAMP过滤函数以及qp方法来解决应力奇异现象。在简化的近似优化模型中,将拓扑变量的离散条件和结构应力的1q范数作为目标体积的惩罚函数,将最可能的主动单元应力约束及KS总体凝聚应力约束作为应力约束条件,减少了计算规模,有利于控制局部单元应力。参考MMA方法,构建了结构应力1q范数的近似再近似表示式,结合动态变化的应力约束限,建立一套高效的拓扑优化算法。数值仿真研究了该方法的可行性和优缺点。再者,定义了应力梯度近似度量值。将最可能的主动单元应力约束以及应力梯度总体凝聚函数作为约束条件,建立一个新的应力约束、体积最小拓扑优化近似等效模型。结合MMA近似展开函数的数学二次规划算法,给出了新的应力约束的连续体结构拓扑优化方法,并完成了其数值仿真研究。然而,将一个或几个应力约束凝聚化函数作为惩罚项,引入目标函数中具有经验性,并不严谨,且近似优化模型与原优化问题模型的等效性和工程应用适用性等值得商讨。本文为了合理解决应力集中问题,攻克近似等效优化模型等关键技术难题,提出了一套优化模型减缩和求解技术。第一,引入了几个类似于正态分布的函数,并将其作为结构应力的q范数函数的加权函数,形成了几个加权的应力约束总体凝聚函数。第二,引入应力梯度凝聚约束函数,借鉴变约束限优化方法思想和引进置信区间方案,构建了具有较好等效性的近似减缩优化模型,解决了应力集中和应力控制问题。第三,提出了对偶子问题的闭式解的近似光滑函数,构建了新的对偶求解方法。最后,形成了基于应力梯度及应力约束凝聚化的连续体结构拓扑优化技术,分别建立了相应的算法,给出了验证算例。算例结果表明本文提出的基于应力梯度及应力约束凝聚化的连续体结构拓扑优化方法能够解决应力约束的拓扑优化问题,得到的最佳拓扑具有较好的0/1分布特征,所提出方法可靠和有效的,方法具有良好的理论价值及工程应用价值。
[Abstract]:The study of topological optimization based on stress constraints is a very promising and difficult subject in topology optimization design.Compared with the literature on topological optimization of flexibility, there are few papers on topological optimization based on stress constraints.At present, there are still many problems in topology optimization based on stress constraints, such as stress singularity, stress concentration and approximate equivalent optimization model.Therefore, the study of topological optimization problem with stress constraints has certain theoretical value and engineering application value.On the basis of existing methods and measures, a large number of numerical simulations have been completed to solve the problem of stress constraint and volume minimization topology optimization, and the variation of stress field and related condensing function has been analyzed.The topology optimization technique of continuum structure based on stress gradient and stress constraint condensation is proposed. The corresponding algorithms are established and the verification examples are given.Firstly, RAMP filter function and QP method are used to solve the stress singularity phenomenon.In the simplified approximate optimization model, the discrete conditions of topological variables and the 1q norm of structural stress are taken as the penalty function of the target volume, and the most probable stress constraints of active element and KS aggregate stress are taken as stress constraints.The calculation scale is reduced and the local element stress is controlled.Referring to the MMA method, the approximate reapproximate expression of the structural stress 1q norm is constructed, and a set of efficient topology optimization algorithm is established combining with the dynamic variation stress constraint limit.The feasibility, advantages and disadvantages of this method are studied by numerical simulation.Furthermore, the approximate measure of stress gradient is defined.Taking the most probable stress constraints of active elements and the aggregate function of stress gradient as the constraint conditions, a new stress constraint, approximate equivalent model of volume minimization topology optimization, is established.Combined with the mathematical quadratic programming algorithm of MMA approximate expansion function, a new topology optimization method for continuum structure with stress constraints is presented, and its numerical simulation is completed.However, the introduction of one or more stress-constrained condensing functions into the objective function is empirical and not rigorous, and the equivalence between the approximate optimization model and the original optimization model and the applicability of the engineering application are worth discussing.In order to solve the stress concentration problem reasonably and overcome the key technical problems such as approximate equivalent optimization model, a set of optimization model reduction and solution techniques are proposed in this paper.Firstly, several functions similar to normal distribution are introduced, which are used as weighting functions of Q norm function of structural stress, and several weighted aggregate condensate functions with stress constraints are formed.Secondly, by introducing the stress gradient condensation constraint function, using the idea of variable constraint limit optimization method and the introduction of confidence interval scheme, an approximate reduction optimization model with good equivalence is constructed, which solves the problems of stress concentration and stress control.Thirdly, the approximate smooth function of the closed solution of the dual subproblem is proposed, and a new dual solution method is constructed.Finally, the topology optimization technology of continuum structure based on stress gradient and stress constraint condensation is formed, and the corresponding algorithms are established, and the verification examples are given.The numerical results show that the proposed topology optimization method based on stress gradient and stress constraint condensation can solve the topological optimization problem of stress constraints, and the obtained optimal topology has a good 0 / 1 distribution characteristic.The proposed method is reliable and effective, and has good theoretical value and engineering application value.
【学位授予单位】：长沙理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TH122

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