以密度等高線法整合二維及三維之拓樸最佳化與型態最佳化

Integrating 2-D and 3-D topology and shape optimization using the density contour approach

許銘修(Ming-Hsiu Hsu)

指導教授 ： 徐業良

元智大學/工程學院/機械工程學系/博士(2003年)

摘要

本研究提出了一以密度等高線法整合二維及三維之拓樸最佳化與型態最佳化的自動化整合程序，此整合程序採用了三階段式的整合程序：第一階段為拓樸最佳化程序，第二階段為拓樸最佳化結果闡釋程序，第三階段則為細部設計程序，型態最佳化以及尺寸最佳化均在此程序中進行。
為了降低整合拓樸最佳化與型態最佳化時的複雜度，首先與拓樸最佳化結果品質相關的mesh-dependence問題、中間密度值(intermediate density)問題、棋盤狀模式(checkerboard pattern)問題等都在本論文中進行討論。在二維整合程序中，本研究提出了連續性分析(continuity analysis)、刪除無價值結構與空孔(trivial solid and void filter)，以及密度等高線法(density contour approach)等方法，以使得用來拓樸最佳化結果能被順利轉換成為型態最佳化的初始設計。文中所提出的整合程序，不但可用於進行二維結構的整合，同時也可擴展至三維結構上。本研究同時提出了代表性截面方法(representative cross-section selection operation)，用以將三維結構改為一序列的二維結面來討論，亦可應用二維結構之密度等高線法來闡釋拓樸最佳化之結果，最後重建三維幾何模型。本論文中共進行了14個二維整合範例，以及6個三維整合範例，用以展示本研究所發展出的整合方法。

關鍵字

拓樸最佳化 ； 型態最佳化 ； 三階段式的整合程序 ； 密度等高線法

並列摘要

An automated process for integrating two- and three-dimensional structural topology optimization and shape optimization using the density contour approach is developed in this research. The three-phase integration process is adopted. Phase I of the integration process is the topology generation process, Phase II is the topology interpretation process, and Phase III is the detail design phase. The shape and size optimizations are implemented in Phase III.
In order to reduce the complexity of integrating topology and shape optimization, a clear topology optimization result is a basic requirement. Thus, the fundamental issues such as mesh-dependence problem, intermediate density problem, and checkerboard pattern problem are considered in the topology generation process. In the topology interpretation process, which interprets the topology optimization result into a CAD representation that can be used in shape optimization, the continuity analysis, trivial solid and void filter, and density contour approach are proposed. Same topology generation and interpretation process can be easily extended to three-dimensional structures. The three-dimensional CAD model can be reconstructed based on the boundaries of the representative cross-sections interpreted from topology optimization results using the same density contour approach. Fourteen two-dimensional and six three-dimensional examples are demonstrated in this research.

並列關鍵字

Topology optimization ； shape optimization ； three-phase integrating process ； density contour approach

參考文獻

Ambrosio, L., Buttazzo, G., 1993, “An Optimal Design Problem with Perimeter Penalization,” Calculus of Variations and Partial Differential Equations, Vol. 1, pp. 55-69.

Beckers, M., 1999, “Topology optimization using a dual method with discretevariables,” Structural Optimization, Vol. 17, pp. 14-24.

Bendsfe, M. P., Kikuchi, N., 1988, “Generating Optimal Topologies in Structural Design Using Homogenization Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 71, pp. 197-244.

Bendsfe, M. P., Rodrigues, H. C., 1991, “Integrated Topology and Boundary Shape Optimization of 2-D Solid,” Computer Methods in Applied Mechanics and Engineering, Vol. 87, pp.15-34.

Bremicker, M., Chirehdast, M., Kikuchi, N., Papalambros, P. Y., 1991, “Integrated Topology and Shape Optimization in Structural Design,” Mechanics of Structures and Machines, Vol. 19, pp.551-587.