- 學位論文
考量自重之單一材料及雙材料改良式雙向結構最佳化演進法之研究
Improved Bi-directional Evolutionary Structural Optimization Method of Single Material and Dual Material Considering Self-Weight
摘要
本研究著重於結構最佳化中的拓樸最佳化，以雙向結構最佳化演進法(Bi-directional Evolutionary Structural Optimization, BESO)為主要演算法，針對於本研究團隊過去的問題，提出改進與方法的融合，並從單一材料延伸於雙材料，同時進行多種類型之案例分析。 過去本研究團隊在使用BESO進行結構最佳化分析，使用各式不同的改進後都能得出良好的拓樸結果，但仍具有部分問題。首先由於BESO必須使用篩選投影之平滑化方法，但該方法在計算各元素間的權重時耗費大量時間，導致分析時會受限使用較小的設計領域或較大的元素去進行結構最佳化分析。其次因為過去演算法中材料內插方法皆是使用冪次內插方法，導致演算法無法考量到自重所造成的影響。再者雙材料的材料分配上多是基於材料彈性模數，容易使不同材料混合在同一桿件中，但實際中較少會只考量彈性模數的將材料進行混合使用，而是透過整體設計將桿件分為受拉與受壓再搭配相應的材料進行桿件設計。最後從過去本研究團隊分析結果可知，初始元素分佈對於結構最佳化演進法系列方法結果影響顯著，同時可知使用全填滿設計領域能夠在各種案例中得到相對較優的結果。 針對以上問題本研究依序透過不同的方式進行改進與延伸。首先透過Python中名為"Numba"的模組(Module)中的"JIT"，將Python源代碼直接編譯成機器碼，即電腦硬體能夠直接運作之代碼形式，進而大幅縮短計算篩選投影權重的時長，使整體分析時間縮短許多，同時分析大型案例時能夠使用較為細緻的元素大小進行分析。其次透過將材料內插方法改以使用替代內插方法(Alternative interpolation scheme)使BESO在進行結構最佳化分析時能夠考量重力場，使結構在進行最佳化時更貼近實務上會使用的力量加載。再者根據前一段所述之材料分配的問題，本研究引入以應力不變量I1分配抗拉與抗壓材料的雙材料方式，並將其與能量移除原則方法相結合，將敏感度因子進行必要修正後，即能夠拓樸出具有抗拉桿件與抗壓桿件之結果。最後根據前一段所述之起始元素分布的影響，對於根據材料彈性模數分配之雙材料BESO，即使是使用全填滿設計領域，兩種材料間不同的分佈與比例都會顯著的影響結果，故本研究根據單材料結果與目標體積提出三種結果較為穩定的初始元素分佈作為起始元素分佈的參考。 但即使是前段所提出較穩定之初始元素分佈，其拓樸結果與結構順從度仍有差距。為此本研究提出「目標體積變化(Target Volume Variation)」，透過BESO能夠將被移除元素進行回填的特性，在迭代過程中變化目標體積，使結構最佳化結果能夠在減少目標體積時移除部分重要性較低的桿件、在增加目標體積時得到更高的自由度，進而突破原有結構形式，並同時保持或是降低其結構順從度，最後使三種結果較優的初始元素分佈與兩種基礎全填滿材料1與材料2得到相像的拓樸結果和結構順從度。 融合以上之改進於演算法中，透過大量且多樣的案例進行分析與驗證，確保其改進方法對於演算法之有效性與對於各式案例之效果，最後將所有提出之改進方法與案例分析進行整理與總結。
並列摘要
This research focuses on topology optimization within structural optimization, using the bi-directional evolutionary structural optimization (BESO) as the primary algorithm. Building upon the previous issues for our research team, this study proposes improvements and methodological integrations, extends from single-material to dual-material applications, and conducts various case analyses. Initially, our team employed BESO for structural optimization analyses and achieved favorable topology results after various enhancements. However, several issues remained: Firstly, BESO requires a filtering scheme to smooth the sensitivity number, which is time-consuming when calculating the weights between elements and limits the structural optimization analysis to smaller design domains or larger elements. Secondly, our past algorithms utilized a power-law material interpolation scheme for the material interpolation method, which failed to account for the impact of self-weight. Thirdly, the allocation of dual-material is primarily based on the material elastic modulus, leading to the mix of different materials within the same structural member. In practice, however, it is rare to take elastic modulus as the sole consideration when using mixed materials; rather, structural members are divided into tension members and compression members by the overall design and then applying appropriate materials for the members design. Lastly, previous analyses indicated that the initial element distribution significantly influenced the outcomes of structural optimization evolutionary methods, and fully filled design domain consistently yielded relatively superior results across various cases. To address these issues, this research commits to improvements and extensions sequentially. Firstly, by using “JIT” from the “Numba” module in Python, the Python source code is directly compiled into machine code, which enables direct execution by computer hardware. This measure significantly reduces the time for calculating filter scheme weights, thus shortening the overall analysis duration and allowing for finer element sizes in large case analyses. Secondly, the material interpolation method is replaced with alternative interpolation scheme, enabling BESO to take gravitational fields into account during structural optimization and making the optimized structure more practical in terms of real-world force applications. Thirdly, to address the material allocation issue mentioned earlier, this study introduces a dual-material approach that distributes tensile and compressive materials based on the sum of principal stresses, combining it with the energy removal principle method. After appropriate adjustment of the sensitivity numbers, the topology results with tensile and compressive members are achieved. Lastly, considering the initial element distribution’s impact, even with a fully filled design domain, the different distributions and proportions of the two materials significantly influence the results. Therefore, this study proposes three relatively stable initial element distributions by single-material results and target volumes as references. However, a discrepancy still exists between the topology results for the proposed stabler initial element distributions and structural compliance. As a solution, the study introduces the “Target Volume Variation”: Taking advantage of BESO’s ability to refill removed elements, the target volume will be varied during iterations, enabling the optimized structure to remove less critical members when decreasing the target volume and reach higher degrees of freedom when increasing. “Target Volume Variation” allows the structure to break away from the original form while maintaining or reducing structural compliance, resulting in similar topology results and structural compliance for the three initial element distributions of optimal outcomes and two basics fully filled materials 1 and 2. The above improvements are integrated into the algorithm, and its effectiveness for various scenarios is ensured by massive and diverse case analyses and verifications. Finally, all proposed improvements and case analyses are organized and summarized.