摘 要
本文主要針對含關連性離散變數、靜態和動態反應束制之結構輕量化設計問題，提出三種以離散拉格朗日法(Discrete Lagrangian Method, DLM)為基礎的搜尋策略，其中靜態反應束制包括位移、應力、挫屈應力及長細比束制，動態反應束制包括頻率及頻率反應振幅的束制。研究中，首先針對DLM中的拉格朗日乘子的更新公式提出修正，避免每次迭代可能必須反覆更新拉格朗日乘子的缺失。接著探討DLM應用於關連性離散變數以及動態反應束制條件下所遭遇的困難，並對此提出動態擴大鄰點搜尋及改善震盪現象策略。最後，為了確保DLM的求解品質，本研究亦提出一種藉由折減拉格朗日乘子啟動再搜尋的方法，使DLM有機會跳脫一個局部最佳解區域搜尋另一局部最佳解。數個結構輕量化設計問題將分別用來探討其適用性和影響求解品質與效率的相關參數，並藉由設計結果之比較，來探討本文所發展之三種搜尋法的優缺點。

Abstract
This research studies the minimum weight design of structures with linked discrete variables, static and dynamic response constraints. Three discrete Lagrangian based searching procedures are proposed in this report. The static response constraints include displacement, stress, buckling stress and slenderness ratio. The dynamic response constraints include frequency and frequency response amplitude. In this research, an update formula for the Lagrange multiplies is developed first. The difficulties in applying the DLM to solve for problems containing linked discrete variables and dynamic response constraints are then discussed. To resolve the difficulties, a dynamic extending neighborhood technique and an improving strategy for eliminating fluctuated searching trajectory are proposed. Finally, a restarting procedure for the DLM by scaling down the values of Lagrange multipliers is also proposed to help the search escaping from a local minimum to search for another one. The feasibility of three procedures is validated by several design examples. The results from comparative studies of the DLM against other discrete optimization algorithms are reported to show the solution quality of the proposed DLM procedures. The advantages and drawbacks of the three DLM algorithms are also discussed.

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