本文選題：連續(xù)體結構 + 拓撲優(yōu)化��； 參考：《長沙理工大學》2015年碩士論文

【摘要】：基于應力約束的拓撲優(yōu)化研究是拓撲優(yōu)化設計中的一個非常有應用前景,又具有很大難度的課題。與涉及柔順度的拓撲優(yōu)化的文獻相比,基于應力約束的拓撲優(yōu)化方面文獻很少。當前基于應力約束的拓撲優(yōu)化方面仍然存在許多問題,如應力奇異、應力集中和近似等效優(yōu)化模型等。因此,研究應力約束的拓撲優(yōu)化問題具有一定的理論價值和工程應用價值。針對應力約束、體積最小拓撲優(yōu)化問題,在已有方法和措施基礎上,本文完成了大量數(shù)值仿真,并分析了應力場及相關凝聚函數(shù)的變化規(guī)律。提出了基于應力梯度及應力約束凝聚化的連續(xù)體結構拓撲優(yōu)化技術,分別建立了相應的算法,給出了驗證算例。首先,本文采用RAMP過濾函數(shù)以及qp方法來解決應力奇異現(xiàn)象。在簡化的近似優(yōu)化模型中,將拓撲變量的離散條件和結構應力的1q范數(shù)作為目標體積的懲罰函數(shù),將最可能的主動單元應力約束及KS總體凝聚應力約束作為應力約束條件,減少了計算規(guī)模,有利于控制局部單元應力。參考MMA方法,構建了結構應力1q范數(shù)的近似再近似表示式,結合動態(tài)變化的應力約束限,建立一套高效的拓撲優(yōu)化算法。數(shù)值仿真研究了該方法的可行性和優(yōu)缺點。再者,定義了應力梯度近似度量值。將最可能的主動單元應力約束以及應力梯度總體凝聚函數(shù)作為約束條件,建立一個新的應力約束、體積最小拓撲優(yōu)化近似等效模型。結合MMA近似展開函數(shù)的數(shù)學二次規(guī)劃算法,給出了新的應力約束的連續(xù)體結構拓撲優(yōu)化方法,并完成了其數(shù)值仿真研究。然而,將一個或幾個應力約束凝聚化函數(shù)作為懲罰項,引入目標函數(shù)中具有經驗性,并不嚴謹,且近似優(yōu)化模型與原優(yōu)化問題模型的等效性和工程應用適用性等值得商討。本文為了合理解決應力集中問題,攻克近似等效優(yōu)化模型等關鍵技術難題,提出了一套優(yōu)化模型減縮和求解技術。第一,引入了幾個類似于正態(tài)分布的函數(shù),并將其作為結構應力的q范數(shù)函數(shù)的加權函數(shù),形成了幾個加權的應力約束總體凝聚函數(shù)。第二,引入應力梯度凝聚約束函數(shù),借鑒變約束限優(yōu)化方法思想和引進置信區(qū)間方案,構建了具有較好等效性的近似減縮優(yōu)化模型,解決了應力集中和應力控制問題。第三,提出了對偶子問題的閉式解的近似光滑函數(shù),構建了新的對偶求解方法。最后,形成了基于應力梯度及應力約束凝聚化的連續(xù)體結構拓撲優(yōu)化技術,分別建立了相應的算法,給出了驗證算例。算例結果表明本文提出的基于應力梯度及應力約束凝聚化的連續(xù)體結構拓撲優(yōu)化方法能夠解決應力約束的拓撲優(yōu)化問題,得到的最佳拓撲具有較好的0/1分布特征,所提出方法可靠和有效的,方法具有良好的理論價值及工程應用價值。
[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.
【學位授予單位】：長沙理工大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：TH122

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