發(fā)布時(shí)間：2018-05-12 21:22

  本文選題：機(jī)械設(shè)計(jì) + 優(yōu)化設(shè)計(jì)��； 參考：《齊魯工業(yè)大學(xué)》2012年碩士論文

【摘要】：建立優(yōu)化設(shè)計(jì)問題的數(shù)學(xué)模型和選擇合適的優(yōu)化方法是機(jī)械優(yōu)化設(shè)計(jì)兩方面主要內(nèi)容。為了掌握優(yōu)化設(shè)計(jì)方法,需要在優(yōu)化理論、建模和計(jì)算機(jī)應(yīng)用等方面進(jìn)行知識(shí)更新；特別是CAD/CAM以及CIMS(計(jì)算機(jī)集成制造系統(tǒng))的發(fā)展,使優(yōu)化設(shè)計(jì)成為當(dāng)代不可缺少的技術(shù)和環(huán)節(jié)。由于機(jī)械優(yōu)化設(shè)計(jì)應(yīng)用數(shù)學(xué)方法來尋求機(jī)械設(shè)計(jì)的最佳方案,因此首先要根據(jù)實(shí)際的機(jī)械設(shè)計(jì)問題建立相應(yīng)的數(shù)學(xué)模型,也就是說應(yīng)有數(shù)學(xué)形式來描述實(shí)際設(shè)計(jì)問題。在建立數(shù)學(xué)模型時(shí),需要應(yīng)用專業(yè)知識(shí)確定設(shè)計(jì)的限制條件和所追求的目標(biāo),確立各設(shè)計(jì)變量之間的相互關(guān)系等。數(shù)學(xué)模型一旦建立,機(jī)械優(yōu)化設(shè)計(jì)問題就變成一個(gè)數(shù)學(xué)求解問題。應(yīng)用數(shù)學(xué)規(guī)劃方法的理論,根據(jù)數(shù)學(xué)模型的特點(diǎn)可以選擇適當(dāng)?shù)膬?yōu)化方法。信賴域方法是求解無約束優(yōu)化問題的一類重要方法,它不要求Hessian矩陣在每個(gè)迭代點(diǎn)處均正定,并適合于求解一些病態(tài)問題,而且它還具有較強(qiáng)的收斂性和魯棒性。由于這些優(yōu)點(diǎn),對(duì)信賴域方法的研究成為當(dāng)今非線性優(yōu)化領(lǐng)域內(nèi)一個(gè)重要研究方向。本文首先研究了一類自適應(yīng)的信賴域算法：具體給出了算法模型；在合理的假設(shè)條件下,對(duì)它的全局收斂性和超線性收斂性進(jìn)行了論證；并把它應(yīng)用于機(jī)械領(lǐng)域。本論文的結(jié)構(gòu)如下： 1.給出了一類新的信賴域方法。首先在非單調(diào)技巧的幫助下,構(gòu)造了一類自調(diào)節(jié)信賴域半徑的方法。在算法的實(shí)現(xiàn)的過程中,不要求函數(shù)值在每一步都下降,特別是對(duì)于目標(biāo)函數(shù)存在彎曲峽谷的情形,非單調(diào)性能加快算法的收斂速度。在每個(gè)迭代點(diǎn)處,我們充分利用當(dāng)前迭代點(diǎn)和先前迭代點(diǎn)的信息來構(gòu)造信賴域半徑,使得二次函數(shù)模型和目標(biāo)函數(shù)在當(dāng)前信賴域內(nèi)具有更好的一致性。這種自適應(yīng)的方法不但克服了初始信賴域半徑選取的盲目性,而且降低了問題的復(fù)雜性,使算法的速度得以提高；最后在某些假設(shè)下,對(duì)所給的算法收斂性質(zhì)進(jìn)行了證明。 2.在二次模型信賴域子問題的為基礎(chǔ),研究了一種帶線搜索的非單調(diào)信賴域方法。在當(dāng)前迭代點(diǎn)處,我們利用信賴域技巧來尋找下一個(gè)迭代點(diǎn),若子問題的近似解不能被接受,使用非單調(diào)線搜索尋找下一個(gè)迭代點(diǎn)。于是在當(dāng)前迭代點(diǎn)處,僅僅需要求解一次子問題就可以找到下一個(gè)成功的迭代點(diǎn),這樣可以避免過大的計(jì)算量,加快算法的收斂速度。在合適的條件下,證明了這種算法是全局收斂的。 3.通過實(shí)例給出了機(jī)械優(yōu)化設(shè)計(jì)的一般步驟,并研究了非單調(diào)自適應(yīng)的信賴域方法在機(jī)械優(yōu)化設(shè)計(jì)中的應(yīng)用,對(duì)該方法和存在的優(yōu)化方法進(jìn)行了比較,從而表明非單調(diào)自適應(yīng)的信賴域方法對(duì)于解決機(jī)械優(yōu)化問題是有效的和可行的。
[Abstract]:The establishment of mathematical model and the selection of appropriate optimization methods are two main contents of mechanical optimization design. In order to master the optimization design method, it is necessary to update the knowledge in optimization theory, modeling and computer application, especially the development of CAD/CAM and CIMS (computer Integrated Manufacturing system). Make the optimization design become the contemporary indispensable technology and link. Due to the application of mathematical method to seek the best scheme of mechanical design, it is necessary to establish the corresponding mathematical model according to the actual mechanical design problem, that is to say, there should be mathematical form to describe the practical design problem. In the process of establishing mathematical model, we need to apply professional knowledge to determine the limit condition and target of design, and to establish the relationship between design variables and so on. Once the mathematical model is established, the mechanical optimization design problem becomes a mathematical solution problem. Based on the theory of mathematical programming, the appropriate optimization method can be selected according to the characteristics of mathematical model. Trust region method is an important method for solving unconstrained optimization problems. It does not require the Hessian matrix to be positive definite at every iteration point, and is suitable for solving some ill-conditioned problems. Moreover, it has strong convergence and robustness. Because of these advantages, the research of trust region method has become an important research direction in the field of nonlinear optimization. In this paper, we first study a kind of adaptive trust region algorithm: we give the algorithm model in detail, prove its global convergence and superlinear convergence under reasonable assumptions, and apply it to the mechanical field. The structure of this thesis is as follows: 1. A new trust region method is given. Firstly, with the help of nonmonotone technique, a method of self adjusting trust region radius is constructed. In the implementation of the algorithm, the value of the function is not required to decrease at every step, especially in the case of the objective function with curved canyons, the non-monotonicity can speed up the convergence of the algorithm. At each iteration point, we make full use of the information of the current iteration point and the previous iteration point to construct the trust region radius, so that the quadratic function model and the objective function are more consistent in the current trust region. This adaptive method not only overcomes the blindness of selecting the initial trust region radius, but also reduces the complexity of the problem and improves the speed of the algorithm. Finally, under some assumptions, the convergence property of the proposed algorithm is proved. 2. Based on the trust region subproblem of quadratic model, a nonmonotone trust region method with line search is studied. At the current iteration point, we use the trust region technique to find the next iteration point. If the approximate solution of the subproblem is not acceptable, we use the non-monotone line search to find the next iteration point. Therefore, at the current iteration point, the next successful iteration point can be found only by solving one subproblem, which can avoid too much computation and speed up the convergence of the algorithm. Under suitable conditions, it is proved that the algorithm is globally convergent. 3. The general steps of mechanical optimization design are given through an example, and the application of nonmonotone adaptive trust region method in mechanical optimization design is studied, and the comparison between this method and the existing optimization method is given. It is shown that the nonmonotone adaptive trust region method is effective and feasible for solving mechanical optimization problems.
【學(xué)位授予單位】：齊魯工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2012
【分類號(hào)】：TH122

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