兩點及三點保守近似法設計最佳化

Two/Three points Conservative Approximation Approach for Design Optimization

郭于鴻(Yu Hung Guo)

指導教授 ： 史建中

淡江大學/工學院/機械與機電工程學系碩士班/碩士(2010年)

https://doi.org/10.6846/TKU.2010.00288

摘要

本研究旨在發展兩點及三點保守近似方法，建構替代目標函數或限制條件中無顯函數的最佳化問題，發展數學模型及探討求解策略。在近似函數考慮保守度的情況下，以修正凸線性近似為基礎，改善其準確性不佳特性，以泰勒級數展開至對角三次項，可得準三次兩點保守近似法(QcTCA)數學模型，在三點近似法裡沒有考慮保守度的情況下，以進階新型三點近似法為基礎，同時增加保守性的考慮，發展保守三點指數近似法(CTEA)數學模型，增加迭代過程穩定性，提升最佳化求解效率。
本研究含準三次兩點保守近似函數及保守三點指數近似函數的準確性測試分析，並分別以修正凸線性近似及進階新型三點近似法做比較，討論其誤差性及準確性。再對前述兩種近似法，分別發展合適的序列近似最佳化流程，其中包括參考點的選取方式及收斂策略，以及探討求解的合理性與有效性。
本研究結合有限元素分析軟體ANSYS及最佳化數值求解工具Visual Doc，用文獻中的範例做測試，經研究比較得知，本文提出的近似方法及求解策略，能以較少的數值迭代次數得到最佳化收斂結果，驗證本文所提出的準三次兩點保守近似法及保守三點指數近似法於最佳化設計的有效性與應用性。

關鍵字

局部近似法 ； 最佳化 ； 保守近似方法

並列摘要

One, two and three point approximation of a function are currently recognized as formal approaches that applies to large-scale structural optimization problems or to implicit functions, such as a function value computed by finite element analysis. To achieve the stable and fewer convergence times, the conservative function approach is a popular usage in solution process.
This thesis develop a Quasi-cubic two-point conservative approximation (QcTCA) based on modified convex approximation (MCA) that include quasi-cubic term in Tayler series approximation. It is predicable that not only a limited and stable iteration times can be maintained but also a relative precise result can be obtained in QcTCA optimization. There is not any conservative approach in the category of tree-point approximation optimization so far. The recent Advance new Three Point Approximation (AnTPA) optimization was proposed that results in a better optimum, however it contains complicate coefficients computation and, more important, without the consideration of conservative strategy. This thesis refers to AnTPA and develops a conservative three-point exponential approximation (CTEA) optimization approach.
The above QcTCA and CTEA are examined by analytical examples in the developing process. The optimization process containing those two approximation techniques are proposed and applied to structural optimization problems. QcTCA optimization shows the efficiency and
stability in the solution process. CTEA approach shows the conservation in the optimization solution process. Those two proposed conservative approximation based optimization required further study for more structural optimization problems, particularly for large-scale design problems with implicit function.

並列關鍵字

local approximation ； optimization ； conservative approximation method

參考文獻

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