應用雙向結構最佳化演進法於單一材料及雙材料結構之拓樸最佳化設計

Abstract

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(BESO)作為演算法核心，進行各種方法融合與改良，並從單一材料拓展至雙材料，應用於多種案例。

結構最佳化演進法(ESO)雖然簡單有效，但在演進過程中只能單向的移除元素，若某次演進能量分布變化過大，就易形成糟糕的拓樸結果。針對此問題，雙向結構最佳化演進法(BESO)便被提出，使被移除元素能夠進行回填的動作。

為了讓「雙向結構最佳化演進法」有更好的最佳化結果與更高的適用性，於本研究中融合並改良體積限制、對稱性、影響區法、隨機交換、材料集中化以及適用任意元素分割等方法。體積限制將元素敏感度各自排序並限制每步迭代元素移除、回填及交換的數量，避免兩次迭代間元素大量更動。對稱性於迭代時考慮對稱元素或使用對稱模型做模擬，使每步迭代皆能使結構趨向或保持對稱。影響區法給予高應變能區域內之元素加權，減少初始拓樸相依性及避免高應變能元素在傳遞力量的過程中被過度考量。隨機交換於初期隨機改變部分元素，使拓樸能不被初始情形局限，並儘早趨向理想結構與材料配置。材料集中化將材料介面長度加入最佳化目標函數中，以解決材料分布較零散的問題。適用任意元素分割是在進行元素交換時考慮體積大小、取對稱元素時考慮對稱之一定範圍內的所有元素，使結構離散成不同形狀大小之元素時也可進行拓樸最佳化。

為了驗證改良後的雙向結構最佳化演進法，本研究進行多種案例探討。首先分別探討單一材料與雙材料的例題，針對單一材料的拓樸最佳化，探討了體積交換比例、影響區法在集中載重及均佈載重下對不同初始情形最佳化的影響；針對雙材料的拓樸最佳化，探討了隨機交換、材料集中化在集中載重及均佈載重下對不同初始情形最佳化的影響。接著針對改變設計區域的應用，探討了使用1/2對稱模型、結構離散呈不規則元素以及三種改變設計區域方式對最佳化的影響，並用簡化的兩跨橋設計證實改變設計區域形狀、設定不可設計區域以及設定對稱區域的可行性。最後整理出各方法的適用情形和使用建議、模型設定建議，以及各案例的最佳結果與發現。

This research focuses on topology optimization in structural optimization, using the Bi-directional Evolutionary Structural Optimization (BESO) as the core algorithm, and combing and improving various methods, expanding from single material to dual materials, and applying them to various cases.

Although the Evolutionary Structural Optimization (ESO) is simple and effective, it can only remove elements in one direction during evolution. If the energy distribution changes too much in one evolution, it can easily lead to poor topology results. To solve this problem, the Bi-directional Evolutionary Structural Optimization (BESO) was proposed, allowing the removed elements to be filled back.

In order to achieve better optimization results and higher applicability for the "Bi-directional Evolutionary Structural Optimization," this research combines and improves methods such as volume constraint, symmetry, influence zone, random exchange, material concentration, and applying arbitrary element segmentation. The volume constraint sorts the element sensitivity seperately and limits the number of elements removed, filled, and exchanged in each iteration to avoid excessive changes between two iterations. Symmetry considers symmetric elements or uses a symmetric model for simulation during iteration to make the structure tend towards or maintain symmetry in each iteration. The influence zone method gives weight to the elements in the high strain energy area, reduces the initial topology dependency, and avoids overemphasis on high strain energy elements in the process of force transmission. Random exchange randomly changes some elements in the early stage to avoid limiting the topology to the initial situation, and to approach the ideal structure and material configuration as soon as possible. Material concentration adds the material interface length to the optimization objective function to solve the problem of scattered material distribution. Applying arbitrary element segmentation considers the volume size when exchanging elements and considers all elements within a certain range of symmetry when selecting symmetric elements, so that topological optimization can be performed even when the structure is discretized into elements of different shapes and sizes.

To verify the improved bi-directional structural optimization evolution method, this research conducted multiple case studies. Firstly, single-material and dual-material examples were explored separately. For the topology optimization of single-material, the effects of volume exchange ratio and influence zone method on different initial optimization under concentrated load and uniform load were investigated. For the topology optimization of dual-material, the effects of random exchange and material concentration on different initial optimization under concentrated load and uniform load were explored. Then, the application of changing the design area was studied, and the effects of using a 1/2 symmetric model, structure which is discretized into elements of different shapes and sizes, and three ways of changing the design area on optimization were investigated. The feasibility of changing the design area shape, setting non-designable areas, and setting symmetric areas was confirmed by a simplified two-span bridge design. Finally, the applicable situations and usage recommendations of each method, model setting recommendations, and optimal results and findings of each method were summarized.