Abstract
In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic complicance or minimum total elastic energy at equilibrium.
The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem.
Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.
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References
Allaire, G. 1994: Explicit lamination parameters for three-dimensional shape optimization.Control & Cybernetics 23, 311–326
Allaire, G.; Bonnetier, E.; Francfort, G.; Jouve, F. 1996: Shape optimization by the homogenization method.Report, 43 pp., Lab. d'Analyse Numerique, Université Paris 6
Bendsøe, M.P. 1989: Optimal shape design as a material distribution problem.Struct. Optim. 1, 193–202
Bendsøe, M.P., 1995:Optimization of structural topology, shape, and material. Berlin, Heidelberg, New York: Springer
Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method.Comput. Meth. Appl. Mech. Engng. 71, 197–224
Cheng, K.T.; Olhoff, N. 1981: An investigation concerning optimal design of solid elastic plates.Int. J. Solids Struct. 17, 305–323
Cheng, K.T.; Olhoff, N. 1982: Regularized formulation for optimal design of axisymmetric plates.Int. J. Solids Struct. 18, 153–169
Cherkaev, A.V.; Palais, R. 1996: Optimal design of three-dimensional axisymmetric elastic structures.Struct. Optim. 12, 35–45
Diaz, A.; Lipton, R. 1997: Optimal material layout for 3D elastic structures.Struct. Optim. 13, 60–64
Diaz, A.; Lipton, R.; Soto, C.A. 1994: A new formulation of the problem of optimum reinforcment of Reissner-Mindlin plates.Comp. Meth. Appl. Mech. Engrg.
Gibiansky, L.V.; Cherkaev A.V. 1987: Microstructures of composites of extremal rigidity and exact estimates of provided energy density (in Russian).Report No. 1155, A.F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad. English translation in: Kohn, R.V.; Cherkaev, A.V. (eds.), 1997:Topics in the mathematical modelling of composite materials. New York: Birkhaüser
Jacobsen, J.B.; Olhoff, N.; Rønholt, E. 1997 Generalized shape optimization of three-dimensional structures using materials with optimum microstructures.Mech. Mat. (to appear)
Kohn, R.V.; Strang, G. 1982: Structural design optimization, homogenization and regularization of vibrational problems.Lecture Notes in Physics,154, 131–147. Berlin, Heidelberg, New York: Springer
Krog, L.A. 1996:Layout optimization of disk, plate, and shell structures. Ph.D. Thesis, Special Report No. 27, Institute of Mechanical Engineering, Aalborg University, Denmark
Lipton, R.; Diaz, A. 1997: Reinforced Mindlin plates with extremal stiffness.Int. J. Solids Struct. 34, 3691–3704
Lund, E. 1994:Finite element based design sensitivity analysis and optimization. Ph.D. Thesis, Special Report No. 23, Institute of Mechanical Engineering, Aalborg University, Denmark
Lurie, K.A. 1993:Applied optimal control theory of distributed systems. New York: Plenum press
Lurie, K.A.; Cherkaev, A.V.; Fedorov, A.V. 1982: Regularization of optimal design problems for bars and plates.JOTA 37, 499–522 (Part 1), 523–543 (Part 2)
Masur, E.F. 1970: Optimum stiffness and strength of elastic structures.J. Eng. Mech. Div. ASCE EM5, 621–640
Olhoff, N.; Lund, E. 1995: Finite element based engineering design sensitivity analysis and optimization. In: Herskovits, J. (ed.)Advances in structural optimization, pp. 1–45. Dordrecht: Kluwer
Olhoff, N.; Bendsøe, M.P.; Rasmussen, J. 1991: On CAD-integrated structural topology and design optimization.Comput. Math. Appl. Mech. Engrg. 89, 259–279
Olhoff, N.; Lund, E.; Rasmussen, J. 1993: Concurrent engineering design optimization in a CAD environment. In Haug, E.J. (ed.)Concurrent engineering: tools and technologies for mechanical system design, pp. 523–586. Berlin, Heidelberg, New York: Springer
Olhoff, N.; Lurie, K.A.; Cherkaev, A.V.; Fedorov, A.V. 1981: Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates.Int. J. Solids Struct. 17, 931–948
Olhoff, N.; Thomsen, J.; Rasmussen, J. 1993: Topology optimization of bi-material structures. In: Pedersen, P. (ed.)Proc. IUTAM Symp. Optimal design with advanced materials, pp. 191–206. Berlin, Heidelberg, New York: Springer
Pedersen, P. 1990: Bounds on elastic energy in solids of orthotropic materials.Struct. Optim. 2, 55–63
Rasmussen, J.; Lund, E.; Olhoff, N. 1993: Integration of parametric modeling and structural analysis for optimum design. In: Gilmore et al. (eds.)Proc. Advances in design automation. Albuquerque, New Mexico: ASME
Rozvany, G.I.N.; Bendsøe, M.P.; Kirsch, U. 1995: Layout optimization of structures.Appl. Mech. Rev. 48, 41–119
Sigmund, O. 1994:Design of material structures using topology optimization. Ph.D. Thesis, DCAMM Report S 69, Technical University of Denmark
Soto, C.A. 1993:Shape optimization of plate structures using plate homogenization with applications in mechanical design. Ph.D. Thesis, Department of Mechanical Engineering, Michigan State University
Soto, C.A.; Diaz, A. 1993a: On modelling of ribbed plates for shape optimization.Struct. Optim. 6, 175–188
Soto, C.A.; Diaz, A. 1993b: Optimum layout of plate structures using homogenization. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design of structures, pp. 407–420. Dordrecht: Kluwer
Soto, C.A.; Diaz, A. 1993c: A model for layout optimization of plate structures. In: Pedersen, P. (ed.)Optimal design with advanced materials, pp. 337–350. Amsterdam: Elsevier
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Olhoff, N., Rønholt, E. & Scheel, J. Topology optimization of three-dimensional structures using optimum microstructures. Structural Optimization 16, 1–18 (1998). https://doi.org/10.1007/BF01213995
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DOI: https://doi.org/10.1007/BF01213995