本文利用結構之運動方程式，提出結構拓樸最佳化的新目標函數，以期最佳化後結構能於輕量化同時改善振動特性，達到減振的功效。本文基於演進式結構最佳化方法(Evolutionary structural optimization method)與由其改善而得的雙向演進式拓樸最佳化方法(Bi-evolutionary structural optimization method)，並參考前人最佳化結構單一自然頻率與最大化結構自然頻率之目標函數，提出用於雙向演進式結構最佳化方法之目標函數，用於處理自然頻率與強迫振動最佳化問題。自然頻率最佳化方面，新目標函數主要用於使結構自然頻率避開特定振動頻率，以改善結構之振動特性，並避免共振現象之產生。強迫振動最佳化方面，新目標函數用於最小化強迫振動情況下目標節點之穩態振幅。最後更將自然頻率與強迫振動問題結合，提出新目標函數，在最佳化目標節點穩態振幅，同時還能使增大自然頻率與外加頻率之差值，使結構在目標節點振動降低時，還因為自然頻率與外加頻率差值增大，使結構其他處振動同時減低。並同時就收斂性較低之目標函數，提出新收斂準則用於增加其收斂性。整體而言，本研究根據自由與強迫振動問題，提出五個新目標函數，最後結果皆能有效達成結構減振之目標。

This study presents several new objective functions for topology optimization based on the bi-directional evolutionary structural optimization (BESO) method. The objective is to minimize vibration under free and forced vibration conditions. In free vibration problems, the proposed method can avoid resonance by: (1) increasing one specific natural frequency of the analyzed structure, and (2) maximizing the gap of two adjacent natural frequencies to one specific frequency. In forced vibration problems, the proposed method can minimize the forced vibration amplitudes of the targeted nodes. Finally, the free and forced vibration conditions are both considered and the proposed method is used to minimize the forced vibration amplitudes of the targeted nodes and to maximize the gap of two adjacent natural frequencies to one specific frequency. In this study, five analysis cases are provided as the illustrative examples. The results show that the proposed topology optimization methods can meet the design objectives effectively.

[1] M. P. Bendsøe and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Computer methods in applied mechanics and engineering, vol. 71, pp. 197-224, 1988.
[2] A. R. Díaaz and N. Kikuchi, "Solutions to shape and topology eigenvalue optimization problems using a homogenization method," International Journal for Numerical Methods in Engineering, vol. 35, pp. 1487-1502, 1992.
[3] L. Shu, M. Y. Wang, Z. Fang, Z. Ma, and P. Wei, "Level set based structural topology optimization for minimizing frequency response," Journal of Sound and Vibration, vol. 330, pp. 5820-5834, 2011.
[4] Y. Xie and G. P. Steven, "A simple evolutionary procedure for structural optimization," Computers & structures, vol. 49, pp. 885-896, 1993.
[5] Y. Xie and G. Steven, "A simple approach to structural frequency optimization," Computers & structures, vol. 53, pp. 1487-1491, 1994.
[6] D. N. Chu, Y. Xie, A. Hira, and G. Steven, "Evolutionary structural optimization for problems with stiffness constraints," Finite Elements in Analysis and Design, vol. 21, pp. 239-251, 1996.
[7] Y. Xie and G. Steven, "Evolutionary structural optimization for dynamic problems," Computers & Structures, vol. 58, pp. 1067-1073, 1996.
[8] Y.-M. Xie and G. P. Steven, Evolutionary structural optimization: Springer, 1997.
[9] O. Querin, G. Steven, and Y. Xie, "Evolutionary structural optimisation (ESO) using a bidirectional algorithm," Engineering Computations, vol. 15, pp. 1031-1048, 1998.
[10] X. Y. Yang, Y. M. Xei, G. P. Steven, and O. M. Querin, "Bidirectional evolutionary method for stiffness optimization," AIAA Journal, vol. 37, pp. 1483-1488, 1999.
[11] V. Young, O. Querin, G. Steven, and Y. Xie, "3D and multiple load case bi-directional evolutionary structural optimization (BESO)," Structural optimization, vol. 18, pp. 183-192, 1999.
[12] O. Querin, G. Steven, and Y. Xie, "Evolutionary structural optimisation using an additive algorithm," Finite Elements in Analysis and Design, vol. 34, pp. 291-308, 2000.
[13] X. Huang and Y. M. Xie, "Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials," Computational Mechanics, vol. 43, pp. 393-401, 2008.
[14] X. Huang, Z. H. Zuo, and Y. M. Xie, "Evolutionary topological optimization of vibrating continuum structures for natural frequencies," Computers & Structures, vol. 88, pp. 357-364, 2010.
[15] A. Rietz, "Sufficiency of a finite exponent in SIMP (power law) methods," Structural and Multidisciplinary Optimization, vol. 21, pp. 159-163, 2001.
[16] Q. Li, G. Steven, and Y. Xie, "A simple checkerboard suppression algorithm for evolutionary structural optimization," Structural and Multidisciplinary Optimization, vol. 22, pp. 230-239, 2001.
[17] X. Huang and Y. M. Xie, "Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method," Finite Elements in Analysis and Design, vol. 43, pp. 1039-1049, 2007.
[18] O. Sigmund and J. Petersson, "Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima," Structural optimization, vol. 16, pp. 68-75, 1998.
[19] 蔡文基, "生長變形法在機械元件形狀最佳設計之研究," 碩士, 機械工程研究所, 國立成功大學, 台南市, 1992.
[20] 黃崇銘, "自己組織化方法在機械結構元件拓撲最佳設計之研究," 碩士, 機械工程研究所, 國立成功大學, 台南市, 1996.
[21] 洪英達, "高階有限單元方法模擬生物成長的形狀最佳設計研究," 碩士, 機械工程學系, 國立成功大學, 台南市, 1997.
[22] 朱祚群, "拓樸最佳化方法在撓性機構合成設計上的應用," 碩士, 機械工程學系, 國立成功大學, 台南市, 1998.
[23] 韓乃智, "三維機械結構拓樸最佳化設計之研究," 碩士, 機械工程學系, 國立成功大學, 台南市, 1998.
[24] 張祥唐, "拓樸最佳化設計方法在車架設計之研究," 碩士, 機械工程學系, 國立成功大學, 台南市, 2000.
[25] 劉至行, "多重領域拓樸最佳化設計方法於點銲位置、結構振動與熱應力問題之研究," 碩士, 機械工程學系, 國立成功大學, 台南市, 2001.
[26] R. Grandhi, "Structural optimization with frequency constraints - A review," AIAA Journal, vol. 31, pp. 2296-2303, 1993.