Topology optimization of continuum structures with local and global stress constraints
Tóm tắt
Topology structural optimization problems have been usually stated in terms of a maximum stiffness (minimum compliance) approach. The objective of this type of approach is to distribute a given amount of material in a certain domain, so that the stiffness of the resulting structure is maximized (that is, the compliance, or energy of deformation, is minimized) for a given load case. Thus, the material mass is restricted to a predefined percentage of the maximum possible mass, while no stress or displacement constraints are taken into account. This paper presents a different strategy to deal with topology optimization: a minimum weight with stress constraints Finite Element formulation for the topology optimization of continuum structures. We propose two different approaches in order to take into account stress constraints in the optimization formulation. The local approach of the stress constraints imposes stress constraints at predefined points of the domain (i.e. at the central point of each element). On the contrary, the global approach only imposes one global constraint that gathers the effect of all the local constraints by means of a certain so-called aggregation function. Finally, some application examples are solved with both formulations in order to compare the obtained solutions.
Tài liệu tham khảo
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsøe MP (1995) Optimization of structural topology, shape, and material. Springer, Heidelberg
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224
Cheng GD, Guo X (1997) ε-relaxed approach in structural topology optimization. Struct Optim 13:258–266
Cheng G, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20:129–148
Duysinx P (1998) Topology optimization with different stress limits in tension and compression. International report: robotics and automation. Institute of Mechanics, University of Liege, Liege
Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43:1453–1478
Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distributions. In: 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary design optimization, Saint Louis, 2–4 September 1998
Fletcher R (1987) Practical methods of optimization. Wiley, Edinburgh
Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs
Kirsch U (1990) On singular topologies in optimum structural design. Struct Optim 2:133–142
Kirsch U, Topping BHV (1992) Minimum weight design of structural topologies. J Struct Eng, ASCE 118:1770–1785
Kohn RV, Strang G (1986) Optimal design and relaxation of variational problems. Commun Pure Appl Math 39:1–25 (Part I), 139–182 (Part II), 353–377 (Part III)
Liang QQ et al (1999) Optimal selection of topologies for the minimum-weight design of continuum structures with stress constraints. Proc Inst Mech Eng, C, J Mech Eng Sci (UK) 213(8):755–762
Lewinsky T, Rozvany GIN (2008) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Struct Multidisc Optim 35:165–174
Martins JRRA, Poon NMK (2005) On structural optimization using constraint aggregation. In: (WCSMO6) Proc. VI world congress on structural and multidisciplinary optimization, Rio de Janeiro, 30 May–3 June 2005
Muíños I (2001) Optimización topológica de estructuras: una formulación de elementos finitos para la minimización del peso con restricciones en tensión (in Spanish). Technical Report, ETSICCP, Universidade da Coruña, A Coruña
Muíños I et al (2002) Una formulación de mínimo peso con restricciones en tensión para la optimización topológica de estructuras (in Spanish). Métodos Numéricos en Ingeniería y Ciencias Aplicadas, CIMNE, 399–408, Barcelona
Navarrina F, Casteleiro M (1991) A general methodologycal analysis for optimum design. Int J Numer Methods Eng 31:85–111
Navarrina F et al (2000) High order shape design sensitivity: a unified approach. Comput Methods Appl Mech Eng 188:681–696
Navarrina F et al (2001) An efficient MP algorithm for structural shape optimization problems. In: Hernández S, Brebbia CA (eds) Computer aided optimum design of structures VII. WIT, Southampton, pp 247–256
Navarrina F et al (2002) Optimización topológica de estructuras: una formulación de mínimo peso con restricciones en tensión (in Spanish). Métodos Numéricos en Ingeniería V (Book and CD-ROM, ISBN: 84-95999-03-X), SEMNI, Barcelona
Navarrina F et al (2004) Topology optimization of structures: a minimum weight approach with stress constraints. Adv Eng Softw 36:599–606
París et al (2005) A minimum weight FEM formulation for structural topological optimization with local stress constraints. In: (WCSMO6) Proc. VI world congress on structural and multidisciplinary optimization, Rio de Janeiro, 30 May–3 June 2005
Pereira JT et al (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidisc Optim 26(1–2):50–66
Ramm E et al (2000) Advances in structural optimization including nonlinear mechanics. In: Proc. of the European congress on computational methods in applied sciences and engineering [ECCOMAS 2000] (CD-ROM, ISBN: 84-89925-70-4), ECCOMAS, Barcelona, 11–14 September 2000
Rozvany GIN, Zhou M (1991) Applications of the COC algorithm in layout optimization. In: Eschenauer H, Matteck C, Olhoff N (eds) Engineering optimization in design processes, Proc. int. conf. held in Karlsruhe, Germany, Sept. 1990, pp. 59–70, Springer, Berlin
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogeneization. Struct Optim 4:250–254
Schmit LA (1960) Structural design by systematic synthesis. In: Proc. of the second ASCE conference on electronic computation. ASCE, Pittsburgh, pp 105–122
Stolpe M, Svanberg K (2001) On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization. Struct Multidisc Optim 21(2):140–151
Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Optim 12:98–105
Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336