雙向結構最佳化演進法及多重材料拓樸最佳化之探討

Abstract

本研究擴展李宗豪所開發的最佳化設計系統（李宗豪2005）作為最佳化設計工具。研究內容著重於結構最佳化中的拓樸最佳化，主要可以分為「雙向結構最佳化演進法」與「多重材料拓樸最佳化」兩個部分。 在本研究團隊相關的研究中，拓樸最佳化的部分，主要使用結構最佳化演進法（Evolutionary Structural Optimization，Xie et al. 1993），相較於其他一般常見的拓樸最佳化方法，本方法使用離散設計變數0與1，非其他方法之連續設計變數，因此在最終拓樸結果可以輕鬆界定材料範圍；並且本方法流程簡單、觀念容易，計算時間與負荷亦較其他方法為小，容易於實際工程設計應用。但此方法為單純移除之單向過程，對於許多邊界條件變更等問題可能最終導致失敗。 「雙向結構最佳化演進法」針對前述問題加以改良，以平滑化及投影策略之概念，使結構最佳化演進法除了移除外亦能夠做填回之動作，使方法具有更大之自由程度，並也允許不同的初始設計，不必要以填滿全設計領域做為初始條件，可以減少不少計算時間，並且若將之應用於徐千泰（2009）提出之支承位置最佳化上，可以取得更合理、更優良之最佳化支承位置結果。 篩選半徑為雙向結構最佳化演進法之重要預設參數，本研究提出其決定應首先考量欲設計桿件之尺寸大小，篩選半徑約定於之欲設計桿件寬度之一半大小。若欲對拓樸做初始設計，注意初始設計之桿件寬度大小應避免小於兩倍篩選半徑大小，以免造成拓樸破碎失敗之結果，且須注意過大之篩選半徑可能產生不良之拓樸結果。 本研究亦將雙向結構最佳化演進法加以變更與討論，融合反式投影策略達到控制拓樸桿件尺寸之目的，具有相當之效果，惟其結果與使用較大篩選半徑之拓樸相似，使用時可考慮以原始之雙向結構最佳化演進法使用較大之篩選半徑代替反式投影策略中繁複之計算及判斷；本研究亦將原雙向結構最佳化演進法原先繁複之計算流程加以簡化，成為簡易式雙向結構最佳化演進法，經實例比較，其方法與原雙向結構最佳化演進法所得之拓樸略有不同但大致相似，而於原雙向結構最佳化演進法之各特點亦可達到相當之結果。 「多重材料拓樸最佳化」的部分，本研究以結構最佳化演進法之相關方法為主軸，主要針對施宏璋（2008）與徐千泰（2009）之研究內容與成果及多重材料的集中化處理加以探討、比較。於雙重材料於軟材料為起始條件之拓樸問題，本研究提出可以兩段式交換收斂、平滑化處理與移除比例方法之相互配合，達到一個較為穩定之結果；而對於多重材料拓樸之演進方法，本研究亦加以比較探討，於演進方法上，不宜先決定硬材料之配置，以免造成過度差異之拓樸，而於研究考量之方法中，徐千泰（2009）提出逐漸將材料降階變軟之方法所得之結果略優於其他方法；最後在材料的集中化處理部分，本研究使用材料介面長度加入最佳化目標函數中，確實可有效形成材料集中化之效果。

This research expands and uses the optimization design system developed by Lee (2005) to carry out optimizations. It focuses on the topology optimization and takes two parts: “Bi-directional Evolutionary Optimization” and “Multi-material Topology.” In researches of our reseach group, we majorly use Evolutionary Structural Opti-mization method (Xie et al. 1993) on topology optimizations. The ESO method uses discrete design variables 0 or 1, so it easier ditiguish the material and the void than other methods with continuous design variable. And it’s simpler, easier and more time effi-cient. But because of its simple procedure with only removing elements, it may fail in some boundary condition changed cases. The “Bi-directional Evolutionary Structural Optimization” method is so developed. The method allows filling back elements with combing the concept of smoothing and projection, so it has more freedom than the original method. It also has allowance in ini-tial design to decrease the comsuming time of the optimization, and when applying it on optimization of support positions published by Hsu (2009), the result will be better than the former research and fit with the paper results. The filtering radius is the most important parameter in BESO method. This research makes lots of discussions about it by examples and suggests that it should be set about half of the desired member size you want, and if the initial design exists, the member width should be larger than twice the filtering redius to avoid crashing. And too large filtering radius may cause bad topology, check in smaller radius if you feel it bad. This research also tries to make modifications on the BESO method. First try is combing it with the inverse projection scheme to control the member and the void size. The try is successful control the member size. But the result goes similar with the to-pology with a larger filtering radius and the original BESO method. It may be more discussions about using it or not. Another try is to simplify the complex computions and procedures of the BESO method. The results of the simplified method make only a little difference with results of the original BESO method, and the method can work in BESO features. In part of “Multi-material Topology,” this research works on ESO based methods and focuses on the researches of Shih (2008) and Hsu (2009). First, in two-material to-pology problem with initial condition of soft material in the full design domain, the combination of two-step converging, smoothing technique and the rejection ratio meth-od can make more stable results. Second, in part of the evolutionary method in multiple materials, this research finds that the method with earlier decisidng on distribution of harder material will cause worse topology, and in all methods this research compared, Hsu’s Method makes better results. Last, this research takes the material interface length into the objective function and it can make clearly material concentrated topology.