Abstract
In this work a review of different formulations for topology design of structures and materials is presented. Using the concept of material distribution introduced in 1988 by Bendsøe and Kikuchi, different treatments have been developed in the last decade not only for the design of structural components but also to applications in material design namely of cellular, composite and/or piezoelectric materials and to problems in biomechanics, wave propagation, adaptive structures, fluid flow etc. In this paper we review topology design formulations developed for periodic material design and extend them to the simultaneous design of the structure and the material used in its production. Examples of application are presented for elasticity and heat transfer transient problems.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bendsøe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenisation method. Comput. Methods Appl. Mech. Eng., 1988; 71: 197–224.
Allaire G. Shape Optimization by the Homogenization Method. Springer, New York, Berlin Heidelberg, 2002.
Bendsøe MP, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer, Berlin, Heidelberg, New York, 2003.
Bendsøe MP, Olhoff N, Sigmund O eds. Topological Design Optimization of Structures Machines and Materials, IUTAM Symposium. Springer, Dordrecht, 2006.
Bendsøe MP. Computational Challenges for multi-physics topology optimization. in Mota Soares CA, Martins JAC, Rodrigues HC, Ambrósio JAC eds. Computational Mechanics: Solids, Structures and Coupled Problems. Springer, 2006.
Lions JL. Some Methods in Mathematical Analysis of Systems and Their Control. Science Press, Beijing, China, Gordon Breach, NY, USA, 1981.
Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Springer, Berlin, Germany, 1980.
Guedes JM, Kikuchi N. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite elements methods. Comput. Meths. Appl. Mech. Engng., 1990; 83: 143–198.
Rozvany GIN, Zhou M, Birker T. Generalized shape optimization without homogenization. Struct Multidiscipl Optim., 1992; 4: 250–254.
Gibianski LV, Cherkaev, AV. Microstructures of composites of extremal rigidity and exact bounds on the associated energy density. in Cherkaev A, Kohn RU eds. Topics in Mathematical Modeling of Composite Materials, Birkhauser, New York, USA, 1997, pp. 273–317.
Guedes JM, Rodrigues H, Bendsøe MP. A material optimization model to approximate energy bounds for cellular materials under multiple load conditions. Structural and Multidisciplinary Optimization, 2003; 25: 446–452.
Bendsøe MP, Guedes JM, Neves MM, Rodrigues HC, Sigmund O. Aspects of the design of microstructures by computational Means. In Carbone L, Arcangelis R eds, Gakuto International Series Mathematical Sciences and Applications, vol. 18: Homogenization 2001, Gakkotosho, Tokyo, Japan, 2003.
Sigmund O. Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. of Solids and Structures, 1994; 31(17): 2313–2329.
Diaz, AR, Bénard A. Designing materials with prescribed elastic properties using polygonal cells. Int. J. Numer. Meth. Engng, 2003; 57: 301–314.
Rodrigues H, Guedes JM, Bendsøe MP. Hierarchical optimization of material and structure. Structural Optimization, 2002; 24(1): 1–10.
Coelho P, Fernandes PR, Guedes JM, Rodrigues HC. Hierarchical model for topology and material optimization of three-dimensional structures. Structural and Multidisciplinary Optimization, 2006 (submitted).
Rodrigues H, Jacobs C, Guedes JM, Bendsøe MP. Global and local material optimization models applied to anisotropic bone adaptation. in Pedersen P, Bendsøe MP eds. Synthesis in Bio Solid Mechanics, IUTAM Symposium, Kluwer, London, UK, 1999.
Turteltaub S. Optimal material properties for transient problems. Structural and Multidisciplinary Optimization, 2001; 22(2): 157–166.
Guedes JM, Lubrano E, Rodrigues HC, Turteltaub S. Hierarchical optimization of material and structure for transient thermal problems. in Bendsøe MP, Olhoff N, Sigmund O eds. Topological Design Optimization of Structures Machines and Materials, IUTAM Symposium. Springer, Dordrecht, 2006.
Coelho PG, Fernandes PR, Cardoso JB, Guedes JM, Rodrigues HC. A three-dimensional hierarchical model for topology optimization of structures. in Mota Soares CA et al eds. III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, Lisbon, June 5–8, 2006. Book of Abstracts, Springer, 2006.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2006 Tsinghua University Press & Springer
About this paper
Cite this paper
Rodrigues, H.C. (2006). Topology Optimization of Structures: Applications in the Simulation and Design of Cellular Materials. In: Computational Methods in Engineering & Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48260-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-48260-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-48259-8
Online ISBN: 978-3-540-48260-4
eBook Packages: EngineeringEngineering (R0)