傳統拓樸最佳化方法因設計變數多，設計空間複雜，導致難以求得全域極值，且因限制條件的束縛，使其執行效率無法提升。有鑑於此，本文提出多目標函數拓樸最佳化方法，突破以結構順從度為目標函數、材料使用量為限制函數之拓樸最佳化傳統作法，而改以結構順從度及材料使用量作為目標函數，利用多目標函數的導入，試圖改善設計空間的複雜程度，以獲得較傳統拓樸最佳化方法更佳且穩定之最佳拓樸圖形，另並進一步取消限制函數，使之成為無限制條件之拓樸最佳化問題，擴大最佳化過程中可能搜尋的範圍，增加求得全域極值的機率，同時亦可減少最佳化求解所需之電腦運算次數，大幅提升執行效率。
利用拓樸最佳化方法所求得之最佳拓樸圖形，其外部輪廓與內部孔洞並非以直線或平滑線條呈現，且拓樸最佳化方法一般並無考慮允許應力或允許位移等限制條件，無法滿足結構設計者的需求。為克服上述困難，本文發展影像前處理技術、兩種不同的圖形辨識技術及形狀最佳化變數定義技術，成功結合拓樸最佳化與形狀最佳化，並提出以特徵值判別法為基之全自動結構最佳設計系統及以多邊形模擬法為基之自動化結構最佳設計系統，同時進行實例驗證與比較，以提供滿足結構設計者需求且快速、簡便的結構設計工具。

With the traditional topology optimization method, the large number of design variables often results in complicated design domains. The constraint functions often complicate the searching domain, making it difficult to find the global optimum and to increase the efficiency. Therefore, this dissertation proposes the multicriterion topology optimization method to tackle the above-mentioned disadvantages. Instead of using compliance as the objective function, the objective function of this new method includes both compliance and volume as two multicriterion functions to simplify the multimodality of the model and ease of attaining a consistent good result. Meanwhile, this new method also cancels all constraint functions, expanding the searching domain to increase the probability of getting global optimum and reducing the computation time to increase the efficiency.
The resulting topological image obtained by the topology optimization method usually has indented interior holes and exterior boundary. Moreover, allowable stress or displacement is not under consideration, so the final configuration is not practical for direct application as a structural design. This research develops the image preprocessing technique, the image interpreting technique and the shape optimization variable defining technique to successfully integrate the topology optimization and the shape optimization into an autonomous procedure. Two automated structural optimization systems which are based on the characteristic value based image interpreting technique and the polygonal image interpreting technique are proposed. Two automated structural optimization systems are demonstrated and compared in several design examples.

[1] Michell, A. G. M., “The Limits of Economy of Material in Framed Structures,” Philosophy Magazine, Series 6, Vol. 8, pp. 589-597 (1904).
[2] Wagner, H., “Remarks on Airplane Struts and Girders under Compressive and Bending Stresses,” Aeronautical Research Council, Reports and Memoranda, No. 500 (1929).
[3] Zahorski, A., “Effects of Material Distribution on Strength of Panels,” Journal of Aeronautical Science, Vol. 11, No. 3, pp. 247-253 (1944).
[4] Cox, H. L. and Smith, H. E., “Structures of Minimum Weight, ” Aeronautical Research Council, Reports and Memoranda, No. 1923 (1945).
[5] Schmit, L. A., “Structural Design by Systematic Synthesis, ” Proceedings of the 2nd Conference on Electronic Computation, ASCE, New York, pp. 105-122 (1960).
[6] Kristensen, E. S., and Madsen, N. F., “On the Optimum Shape of Fillets in Plates Subjected to Multiple In-plane Loading Cases,” International Journal of Numerical Methods in Engineering, Vol. 10, No. 5, pp. 1007-1009 (1976).
[7] Bhavikatti, S. S., and Ramakrishnan, C. V., 1977, “Optimum Design of Fillets in Flat and Round Tension Bars,” Proceedings of the ASME 1977 Design Engineering Technical Conference, Chicago, Illinois, 77-DET-45.
[8] Dems, K., “Multiparameter Shape Optimization of Elastic Bars in Torsion,” International Journal of Numerical Methods in Engineering, Vol. 15, No. 10, pp. 1517-1539 (1980).
[9] Pedersen, P. and Laursen, C. L., “Design for Minimum Stress Concentration by Finite Elements and Linear Programming,” Journal of Structural Mechanics, Vol. 10, No. 4, pp. 375-391 (1983).
[10] Prasad, B., and Emerson, J. F., “Optimal Structural Remodeling of Multi-Objective System,” Computers and Structures, Vol. 18, No. 4, pp. 619-628 (1984).
[11] Bennett, J. A., and Botkin, M. E., “Structural Shape Optimization with Geometric Description and Adaptive Mesh Refinement,” AIAA Journal, Vol. 23, No. 3, pp. 458-464 (1985).
[12] Briabant, V., and Fluery, C., “Shape Optimum Design Using B-Splines,” Computer Methods in Applied Mechanics and Engineering, Vol. 44, No. 3, pp. 247-267 (1984).
[13] Haftka, R. T., and Grandhi, R. V., “Structural Shape Optimization – A Survey,” Computer Methods in Applied Mechanics and Engineering, Vol. 57, No. 1, pp. 91-106 (1986).
[14] Bendse, M. P., and Kikuchi, N., “Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 71, pp. 197-224 (1988).
[15] 范成瑋，「拓樸最佳化法於動態結構設計之應用」，碩士論文，國立中正大學，嘉義 (2003)。
[16] Mlejenk, H. P., “Some Aspects of the Genesis of Structures”, Structural Optimization, Vol. 5, pp. 64-69 (1992).
[17] Mlejnek, H. P., and Schirrmacher, R., “An Engineering Approach to Optimal Material Distribution and Shape Finding,” Computer Methods in Applied Mechanics and Engineering, Vol. 106, pp. 1-26 (1993).
[18] Bendse, M. P., and Haber, R. B., “The Michell Layout Problem as a Low Volume Fraction Limit of the Homogenization Method for Topology Design: an Asymptotic Study,” Structural Optimization, Vol. 6, pp. 263-267 (1993).
[19] Bendse, M. P., “Optimal Shape Design as a Material Distribution Problem, ” Structural Optimization, Vol. 1, pp. 193-202 (1989).
[20] Zhou, M., Rozvany, G. I. N., “The COC Algorithm, Part II：Topological, Geometry and Generalized Shape Optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 89, pp. 197-224 (1991).
[21] Yang, R. J., and Chuang, C. H., “Optimal Topology Design Using Linear Programming,” Computers and Structures, Vol. 52, No. 2, pp. 265-276 (1994).
[22] Yang, R. J., and Chahande, A. I., “Automotive Applications of Topology Optimization,” Structural Optimization, Vol. 9, No. 3-4, pp. 245-249 (1995).
[23] Rozvany, G. I. N., Zhou, N., and Sigmund, O., “Topology Optimization in Structural Design,” Structural Optimization Advances in Design Optimization, Adeli, ed., Chapman and Hall, London, pp. 340-399 (1994).
[24] Bendse, M. P., and Sigmund, O., “Material Interpolation Schemes in Topology Optimization,” Archives of Applied Mechanics, Vol. 69, No. 9-10, pp. 635-654 (1999).
[25] Rozvany, G. I. N., “Aims, Scope, Methods, History and Unified Terminology of Computer-aided Topology Optimization in Structural Mechanics,” Structural and Multidisciplinary Optimization, Vol. 21, pp. 90-108 (2001).
[26] Suzuki, K., and Kikuchi, N., “A Homogenization Method for Shape and Topology Optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 93, pp. 291-318 (1991).
[27] Diaz, A. R., and Kikuchi, N., “Solution to Shape and Topology Eigenvalue Optimization Problems Using a Homogenization Method,” International Journal for Numerical Methods in Engineering, Vol. 35, pp. 1487-1502 (1992).
[28] Ma, Z. D., Kikuchi, N., and Cheng, H. C., “ Topology Design for Vibrating Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 121, No. 1-4, pp. 259-280 (1995).
[29] Pedersen, N. L., “Maximization of Eigenvalues Using Topology Optimization,” Structural and Multidisciplinary Optimization, Vol. 20, No. 1, pp. 2-11 (2000).
[30] Mayer, R. R., Kikuchi, N., and Scott, R. A., “Application of Topological Optimization Techniques to Structural Crashworthiness,” International Journal for Numerical Methods in Engineering, Vol. 39, No. 8, pp. 1383-1403 (1996).
[31] Mayer, R. R., “Application of Topological Optimization Techniques to Automotive Structural Design,” Proceedings of the ASME 2001 International Mechanical Engineering Congress and Exposition, New York, NY, IMECE2001/AMD-25458 (2001).
[32] Luo, J., Gea, H. C., and Yang, R. J., “Topology Optimization for Crush Design,” Proceedings of the 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, AIAA Paper No. AIAA-2000-4770 (2000).
[33] Mayer, R. R., Maurer, D., and Bottcher, C., “Application of Topological Optimization Program to the Danner Test Simulation,” Proceedings of the ASME 2000 Design Engineering and Technical Conferences, Baltimore, Maryland, DETC2000/DAC-14292 (2000).
[34] Gea, H. C., and Luo, J., “Design for Energy Absorption: A Topology Optimization Approach,” Proceedings of the ASME 2001 Design Engineering and Technical Conferences, Pittsburgh, PA, DETC2001/DAC-21060 (2001).
[35] Soto, C. A., “Optimal Structural Topology Design for Energy Absorption: A Heurtistic Approach,” Proceedings of the ASME 2001 Design Engineering and Technical Conferences, Pittsburgh, PA, DETC2001/DAC-21126 (2001).
[36] Soto, C. A., “Structural Topology for Crashworthiness Design by Matching Plastic Strain and Stress Levels,” Proceedings of the ASME 2001 International Mechanical Engineering Congress and Exposition, New York, NY, IMECE2001/AMD-25455 (2001).
[37] Bae, K. K., Wang, S. W., and Choi, K. K., “Reliability-Based Topology Optimization with Uncertainties,” Proceedings of the Second China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems (CJK-OSM 2), Busan, Korea, pp. 647-653 (2002).
[38] Kharmanda, G., Olhoff, N., Mohamed, A., and Lemaire, M., “Reliability-Based Topology Optimization,” Structural and Multidisciplinary Optimization, Vol. 26, No. 5, pp. 295-307 (2004).
[39] Mattheck, C., and Burkhardt, S., ”A New Method of Structural Shape Optimization Based on Biological Growth,” International Journal of Fatigue, Vol. 12, No. 3, pp. 185-190 (1990).
[40] Chen, J. L., and Tsai, W. C., “Shape Optimization by Using Simulated Biological Growth Approach,” AIAA Journal, Vol. 31, No. 11, pp. 2143-2147 (1993).
[41] Weinans, H., Huiskes, R., and Grootenboer, H. J., ”The Behavior of Adaptive Bone-remodeling Simulation Models,” Journal of Biomechanics, Vol. 25, No. 12, pp. 1425-1441 (1992).
[42] Baumgartner, A., Harzheim, L., and Mattheck, C., ”SKO(Soft Kill Option): The Biological Way to Find an Optimum Structure Topology,” International Journal of Fatigue, Vol. 14, pp. 387-393 (1992).
[43] Mattheck, C., ”Design in Nature,” Structural Engineering International, Vol. 6, No. 3, pp. 177-180 (1996).
[44] Mattheck, C., ”Design in Nature: Learning from trees,” Nature, Vol. 392, No. 6673, pp. 0-242 (1998).
[45] Liu, J. S., Parks, G. T., and Clarkson, P. J., ”Metamorphic Device for Varying the Number of Cavities in an Optimized Topology Using Evolutionary Structural Optimization,” Structural and Multidisciplinary Optimization, Vol. 19, No. 2, pp. 140-147 (2000).
[46] Xie, Y. M., and Steven, G. P., “A Simple Evolutionary Procedure for Structural Optimization,” Computers and Structures, Vol. 49, No. 5, pp. 885-896 (1993).
[47] Zhao, C., Steven, G. P., and Xie, Y. M., ”Evolutionary Optimization of maximizing the Difference Between Two Natural Frequencies of a Vibrating Structure,” Structural Optimization, Vol. 13, No. 2-3, pp. 148-154 (1997).
[48] Chu, D. N., Xie, Y. M., Hira, A., and Steven, G. P., “On Various Aspects of Evolutionary Structural Optimization for Problems with Stiffness Constraints,” Finite Element in Analysis and Design, Vol. 24, pp. 197-212 (1997).
[49] Kim, H., Querin, O. M., Steven, G. P., and Xie, Y. M., ”Method for Varying the Number of Cavities in an Optimized Topology Using Evolutionary Structural Optimization,” Structural and Multidisciplinary Optimization, Vol. 19, No. 2, pp. 140-147 (2000).
[50] Proos, K. A., Steven, G. P., Querin, O. M., Xie, Y. M., ”Multicriterion Evolutionary Structural Optimization Using the Weighting and the Global Criterion Methods,” AIAA Journal, Vol. 39, No. 10, pp. 2006-2012 (2001).
[51] Li, Q., Steven, G. P., and Xie, Y. M., ”A Simple Checkerboard Suppression Algorithm for Evolutionary Structural Optimization,” Structural and Multidisciplinary Optimization, Vol. 22, No. 3, pp. 230-239 (2001).
[52] Lin, C. Y., and Chao, L. S., “Constant Weight Fully Stressed Methods for Topological Design of Continuum Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 51, pp. 6867-6879 (2001).
[53] Eschenauer, H. A., and Olhoff, N., “Topology Optimization of Continuum Structures: A Review,” Applied Mechanics Reviews, Vol. 54, No. 4, pp. 331-390 (2001).
[54] Bendse, M. P., and Sigmund, O., Topology Optimization: Theory, Methods, and Applications, Springer, Berlin, Germany, ISBN-3540429921.
[55] Rozvany, G. I. N., “A Survey of Structural Optimization in Mechanical Product Development,” Journal of Computing and Information Science in Engineering, Vol. 5, pp. 214-226 (2005).
[56] Papalambros, P. Y., and Chirehdast, M., “An Integrated Environment for Structural Configuration Design,” Journal of Engineering Design, Vol. 1, No. 1, pp. 73-96 (1990).
[57] Bendse, M. P., and Rodrigues, H. C., “Integrated Topology and Boundary Shape Optimization of 2-D Solids,” Computer Methods in Applied Mechanics and Engineering, Vol. 87, No. 1, pp. 15-34 (1991).
[58] Bremicker, M., Chirehdast, M., Kikuchi, N., and Papalambros, P. Y., “Integrated Topology and Shape Optimization in Structural Design,” Mechanics of Structures and Machines, Vol. 19, No. 4, pp. 551-587 (1991).
[59] Olhoff, N., Bendse, M. P., and Rasmussen, J., “On CAD-Integrated Structural Topology and Design Optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 89, No. 4, pp. 259-279 (1992).
[60] Rosen, D. W., and Grosse, I. R., “A Feature Based Shape Optimization Technique for the Configuration and Parametric Design of Flat Plates,” Engineering with Computers, Vol. 8, No. 2, pp. 81-91 (1992).
[61] Chirehdast, M., Gea, H. C., Kikuchi, N., and Papalambros, P. Y., “Structural Configuration Examples of an Integrated Optimal Design Process,” Journal of Mechanical Design, Vol. 116, pp. 997-1004 (1994).
[62] Maute, K., and Ramm, E., “Adaptive Topology Optimization of Shell Structures,” AIAA Journal, Vol. 35, No. 11, pp. 1767-1773 (1997).
[63] Bletzinger, K. U., and Maute, K., “Towards Generalized Shape and Topology Optimization,” Engineering Optimization, Vol. 29, No. 1-4, pp. 201-216 (1997).
[64] Maute, K., and Ramm, E., “Adaptive Topology Optimization,” Structural Optimization, Vol. 10, No. 2, pp. 100-112 (1998).
[65] Lin, C. Y., and Chao, L. S., “Automated Image Interpretation for Integrated Topology and Shape Optimization,” Structural and Multidisciplinary Optimization, Vol. 20, No. 2, pp. 125-137 (2000).
[66] Hsu, Y. L., Hsu, M. S., and Chen, C. T., ”Interpreting Results from Topology Optimization Using Density Contours,” Computers and Structures, Vol. 79, No. 10, pp. 1049-1058 (2001).
[67] Tang, P. S., and Chang, K. H., ”Integration of Topology and Shape Optimization for Design of Structural Components,” Structural and Multidisciplinary Optimization, Vol. 22, No. 1, pp. 65-82 (2001).
[68] Lin, C. Y., and Lin, S. H., “Artificial Neural Network Based Hole Image Interpretation Techniques for Integrated Topology and Shape Optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 36-38, pp. 3817-3837 (2005).
[69] Bulman, S., Sienz, J., and Hinton, E., ”Comparisons Between Algorithms for Structural Topology Optimization Using a Series of Benchmark Studies,” Computers and Structures, Vol. 79, No. 12, pp. 1203-1218 (2001).
[70] Diaz, A. R., and Sigmund, O., “Checkerboard Patterns in Layout Optimization,” Structural Optimization, Vol. 10, No. 1, pp. 40-45 (1995).
[71] Jog, C. S., and Haber, R. B., “Stability of Finite Element Models for Distributed-parameter Optimization and Topology Design,” Computer Methods in Applied Mechanics and Engineering, Vol. 130, No. 3-4, pp. 203-226 (1996).
[72] Lin, C. Y., and Chou, J. N., “A Two-stage Approach for Structural Topology Optimization,” Advances in Engineering Software, Vol. 30, No. 4, pp. 261-271 (1999).
[73] 鄒宇新，「以最佳結構拓樸為基之形狀最佳化技術」，碩士論文，國立台灣科技大學，台北 (2000)。