Abstract
A method to minimize the compliance of structures subject to multiple load cases is presented. Firstly, the material distribution in design domain is optimized to form a truss-like continuum. The anisotropic composite is employed as the material model to simulate the constitutive relation of the truss-like continuum. The member densities and orientations at the nodes are taken as design variables. The member densities and orientations at any point in an element vary continuously. Then, parts of members, which are formed according to the member distribution field, are chosen to form the nearly optimum discrete structure. Lastly, the positions of the nodes and the cross-sectional areas of the members are optimized. In the above process, numerical instabilities such as checkerboard and mesh dependencies disappear without any additional technique. The sensitivities of the compliance are derived. Examples are presented to demonstrate the capability of the proposed method.
Similar content being viewed by others
References
Bendsϕe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Meth Appl Mech Eng 71:197–224
Bendsϕe MP, Guedes JM, Haber RB, Pedersen P, Taylor JE (1994) An analytical model to predict optimal material properties in the context of optimal structural design. J Appl Mech 61:930–937
Haber RB, Jog CS, Bendsϕe MP (1996) A new approach to variable-topology design using a constraint on the perimeter. Struct Multidisc Optim 11:1–12
Hörnlein HREM, Kočvara M, Werner R (2001) Material optimization: bridging the gap between conceptual and preliminary design. Aerosp Sci Technol 5:541–554
Hsu Y, Sho M, Chen C (2001) Interpreting results from topology optimization using density contours. Comput Struct 79:1049–1058
Matsui K, Terada K (2004) Continuous approximation of material distribution for topology optimization. Int J Numer Methods Eng 59:1925–1944
Michell AGM (1904) The limits of economy of material in framestructure. Phil Mag 8:589–597
Niordson F (1982) Optimal design of plate with a constraint on the slop of the thickness function. Int J Solids Struct 19:141–151
Prager W (1974) A note on discretized Michell structures. Comput Methods Appl Mech Eng 3:349–355
Prager W, Rozvany GIN (1976) Optimization of the structural geometry. In: Bednarek AR, Cesari L (eds) Dynamical systems (Proceedings of the International Conference in Gainesville, Florida, March, 1976). Academic, New York, pp 265–293
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. Proceedings of the International Conference held in Karlsruhe, Germany, Sept. 1990). Springer, Berlin, pp 59–70
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Multidisc Optim 4:250–254
Sigmund O (1994) Material with prescribed continuative parameters: an inverse homogenization problem. Int J Solids Struct 31:2313–2329
Xie YM, Steven GP (1993) A simple evolutionary procedure for structures optimization. Comput Struct 49:885–896
Zhou K, Li J (2004) The exact weight of discretized Michell trusses for a central point load. Struct Multidisc Optim 28:69–72
Zhou K, Hu Y (2002) A method of constructing Michell truss using finite element method. Acta Mech Sinica 34:935–940
Zhou K, Li J (2005a) Forming Michell truss in three-dimensions by finite element method. Appl Math Mech 26:381–388
Zhou K, Li X (2005b) Topology optimization of structures under multiple load cases using a fiber-reinforced composite material model. Comput Mech 38:163–170
Zhou K, Li X (2006) Influence of finite element mesh on topology optimization based on continuous distribution of members. The Fourth China–Japan–Korea Joint Symposium on Optimization of Structural and Mechanical Systems. Kunming, China, Nov. 6–9 2006, pp 215–218
Zhou M, Rozvany GIN (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, K., Li, X. Topology optimization for minimum compliance under multiple loads based on continuous distribution of members. Struct Multidisc Optim 37, 49–56 (2008). https://doi.org/10.1007/s00158-007-0214-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-007-0214-3