交換式最佳演進法於多材料之結構拓樸最佳化設計

Structural Topology Optimization with Multiple Materials Using the Evolutionary Switching Algorithm

施宏璋(Hung-Chang Shih)

指導教授 ： 黃仲偉

中原大學/工學院/土木工程學系/碩士(2008年)

https://doi.org/10.6840/cycu200900233

摘要

摘要
近幾年來隨著電腦計算能力與圖形處理的進步，拓樸最佳化開始引起
學界與業的注意。文獻上的研究多集中在單一材料之結構拓樸最佳化，然
而多重材料之複合結構具有眾多的潛在用途，應不容忽視。本文研究提出
交換式最佳化演進法（evolutionary switching algorithm）用以探討多重材料
之結構拓樸最佳化設計，並推導其對應的數學模型。該法是以元素的彈性
模數為設計變數，目標在固定載重下最小化結構物應變能，同時滿足材料
體積限制式。
交換式最佳化演進法包含演進階段與檢查階段兩個過程。前者是透過
逐步交換元素彈性模數的方式得到滿足各材料體積限制式的基本解；後者
則是檢查基本解中元素應變能的排序與材料彈性模數是否相符合，用以判
定結構拓樸形狀是否收斂。該法具有觀念上淺顯易懂，且容易與有限元素
相結合的特性。若在演進階段與檢查階段的每次迭代過程中都只改變最少
的元素（對稱問題的最少元素數目為2，非對稱問題的最少元素數目為1），
該法會等價於最佳化方法中求解非線性問題的循序線性規劃。
數值實例顯示無論是雙重或多重材料拓樸最佳化問題，該法均可迅速
收斂至結構拓樸最佳化的狀態；且將初始材料設定為最硬材料，逐步替換
變能較低元素的材料性質為較軟材料的演進結果最優。另一方面，對於雙
重材料拓樸最佳化問題而言，材料彈性模數的比值會影響最佳化的拓樸形
狀；而多重材料拓樸最佳化可透過材料群組的方式來增加演進次數，俾利
得到較優的基本解。

關鍵字

材料群組 ； 彈性模數比值 ； 演進階段 ； 交換式最佳化演進法 ； 檢查階段 ； 循序線性規劃 ； 拓樸最佳化

並列摘要

Abstract
In this research, an evolutionary switching algorithm (ESA) has been
developed on the basis of the evolutionary structural optimization method
(ESO) to obtain an optimal topology of a multiple-material structure. The
proposed method has two processes: the evolutionary process and the checking
process. The former is used to provide a basic solution of topology which is
satisfied with the material volume constraints and the latter is used to ascertain
whether the solution of topology is converged. The objective function of the
proposed method is to minimize the compliance of the structure and the
corresponding design variables are the elastic modulus of each element. From
the viewpoint of mathematics, the proposed method is proven to be equivalent
to the sequential linear programming method if the smallest value of changing
elements is adopted in the process.
The evolutionary switching algorithm can deal with traditional two-phase
(solid and void) topology optimization problem and can be extended to solve
topology optimization with multiple materials with the aid of the concept of
material grouping. Numerical examples demonstrate that different modului
combination result in different optimal topology for two-material topology
optimization. The smaller value of the ratio of elastic modului intends to
generate a truss-like topology. On the other hand, the stiffer material intends to
concentrate on local regions where forces apply for two materials with
approximate elastic modului. Finally, the contours of optimal topologies for
multiple materials are similar with those of two-material. However, the
skeletons are made of the stiffest material and other materials cover the
skeleton in the sequence of the elastic modulus.

並列關鍵字

Multiple materials ； Topology optimization ； Evolutionary switching

參考文獻

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Bendsøe, M. P., 1995, Optimization of Structural Topology, Shape and Material,