孔洞於結構中之最佳化設計

Optimal Design of Structures with Openings

曾彥淩(Yen-Ling Tseng)

指導教授 ： 呂良正

國立臺灣大學/工學院/土木工程學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.02052

摘要

結構物無可避免需要開孔，無論是開窗、開門，或是配置管線須穿越結構體，亦或是需要減輕結構重量刻意設置孔洞，開孔必會對結構造成影響。本研究期望能透過最佳化方式找尋對結構物強度影響最低之孔洞位置。本研究分為兩部分：第一部分是在結構中置放形狀、大小固定之孔洞，尋找其中心座標最佳解；第二部分則將孔洞位置最佳化結合拓樸最佳化，期望能有更廣及更彈性之應用。
孔洞位置最佳化設計中，給予孔洞初始位置，藉由孔洞移動準則迭代尋找最終孔洞位置座標。孔洞移動準則是參考置入物件於拓樸最佳化中之移動（Qian and Ananthasuresh 2004）進行延伸應用，設置孔洞相當於置入勁度為零之物件，並透過最陡梯度法使孔洞一步步往較好的方向演進。
孔洞位置最佳化結合到拓樸最佳化中，透過多層最佳化的概念來實現此問題。其中，拓樸最佳化使用元素交換法（Rouhi et al. 2010）作為演算法核心，此方法有一限制，不論在第幾個迭代步，材料體積皆固定與目標體積一致，此限制對於孔洞位置最佳化的結合影響很大，且初始拓樸不易設計，故參考本研究團隊過去提出之變體積函數（施可葳2013），避免此阻礙。
目前孔洞位置最佳化僅能找到局域最佳解，對於選擇孔洞開口位置時，使用者可先設置幾種不同的初始位置，均勻分布於結構中，透過此方法，可避免將孔洞設置於對結構影響較大之處，並且尋求相對初始位置較好之解。
孔洞位置最佳化結合拓樸最佳化問題中，可看出孔洞對於拓樸造成之影響，並且透過例題結果，進一步驗證多層最佳化之可靠性，可看出孔洞確實朝向拓樸過程中挖除材料的位置移動。

關鍵字

結構最佳化 ； 結構開孔 ； 拓樸最佳化 ； 多層最佳化

並列摘要

Prescribed openings are embedded to structures for many reasons like windows, doors, room for pipes, or reducing weight. Stiffness of structures will decrease because of openings. This research is aimed at deciding the positions of openings that will affect the stiffness as little as possible. This research can be divided into two parts. Firstly, the positions of openings are optimized within the design domain. Secondly, topology optimization is considered in conjunction with the first part.
The optimization of structures with openings is similar to an embedding problem in topology optimization (Qian and Ananthasuresh 2004), but the stiffness of the embedded objects is zero instead. The steepest descent method is used to determine the positions of openings in such kind of problems.
Next, the above problem combining with topology optimization is solved using the multi-level optimization technique, where the element exchange method (EEM; Rouhi et al. 2010) is applied to carry out topology optimization. A restriction of EEM is that the volume at each iteration is same as the objective volume. Because of this, it would be very difficult to design the initial topology with openings. A volume-changeable function (Shih 2013) is employed avoid this situation.
Because the steepest descent method is a kind of gradient-based optimization method, the results could be different when different initial position are used. To avoid this situation, different initial arrangements of the openings are adopted. Although the best solution may be not be reached, a better one can always be obtained, where the openings are placed in the inefficient region of the design domain.
In the mixed application of topology and the opening optimization, the effect of openings in the process of topology optimization is clearly demonstrated. The multi-level optimization algorithms shown to be very reliable when searching for optimal results.

並列關鍵字

Structural Optimization ； Openings ； Topology Optimization ； Multi-level Optimization

參考文獻

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