Abstract
This paper presents an extension of the hierarchical model for topology optimisation to three-dimensional structures. The problem addressed covers the simultaneous characterisation of the optimal topology of the structure and the optimal design of the cellular material used in its construction. In this study, hierarchical suggests that the optimisation model works at two interconnected levels, the global and local levels identified, respectively, with the structure and its material. The class of cellular materials, defining the material microstructure, is restricted to single scale cellular materials, with the cell geometry locally optimised for the given objective function and constraints. The model uses the asymptotic homogenisation model to obtain the equivalent material properties for the specific local microstructures designed using a SIMP based approach. The necessary optimality conditions for the hierarchical optimal design problem are discussed and approximated numerically by a proper finite element discretisation of the global and local analysis and design problems. Examples to explore and demonstrate the model developed are presented.
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References
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Meth Appl Mech Eng 71:197–224
Bendsøe MP, Sigmund O (2003) Topology optimization theory, methods and applications. Springer, Berlin Heidelberg New York
Brown RE (2004) Engineering a Beowulf-style computer cluster. Duke University
Díaz AR, Bendsøe MP (1992) Shape optimization of structures for multiple loading conditions using a homogenization method. Struct Multidisc Optim 4:17–22
Fletcher R (1987) Practical methods for optimization. Wiley, Chichester
Grama A, Karypis G, Kumar V, Gupta A (2003) An introduction to parallel computing: design and analysis of algorithms. Addison-Wesley, Reading
Guedes JM, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element method. Comput Meth Appl Mech Eng 83:143–198
Malard J (1996) MPI: a message-passing interface standard. History, overview and current status. Technology Watch Report, Edinburgh Parallel Computing Center
Rodrigues H, Jacobs C, Guedes JM, Bendsøe MP (1999) Global and local material optimization models applied to anisotropic bone adaptation. In: Pedersen P, Bendsøe MP (eds) Synthesis in biosolid mechanics. Kluwer, pp 209–220
Rodrigues H, Guedes JM, Bendsøe MP (2002) Hierarchical optimization of material and structure. Struct Multidisc Optim 24:1–10
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Multidisc Optim 16:68–75
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Coelho, P.G., Fernandes, P.R., Guedes, J.M. et al. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidisc Optim 35, 107–115 (2008). https://doi.org/10.1007/s00158-007-0141-3
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DOI: https://doi.org/10.1007/s00158-007-0141-3