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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 209

New Exact Analytical Solutions as Benchmarks for Numerical Topology Optimization

G.I.N. Rozvany1, T. Lewinski2, J. Lógó1 and V. Pomezanski1

1Department of Structural Mechanics, Research Group of Computational Structural Mechanics, Hungarian Academy of Science, Budapest University of Technology and Economics, Hungary
2Institute of Structural Mechanics, Faculty of Civil Engineering, Warsaw University of Technology, Poland

Full Bibliographic Reference for this paper
G.I.N. Rozvany, T. Lewinski, J. Lógó, V. Pomezanski, "New Exact Analytical Solutions as Benchmarks for Numerical Topology Optimization", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 209, 2006. doi:10.4203/ccp.83.209
Keywords: topology optimization, benchmark problems, least-weight structures, Michell trusses, grillages, optimality criteria, optimal design, minimum volume design.

Topology optimization has become one of the most rapidly expanding fields in structural mechanics and the number of research papers appearing in journals and conference proceedings on new methods in this field is also very large. The critical evaluation of these methods, on the other hand, is rather inadequate.

It is well known that the best benchmarks for numerical methods (e.g. by SIMP, Rozvany and Zhou [11], Rozvany [9]) are exact analytical results for the same boundary conditions and loading. In the case of topology optimization, it has been shown by several researchers (including the first author) that the optimal topology for perforated plates in plane stress and for three dimensional solids with cavities converges to that for least-weight (Michell [3]) trusses, if for the former the volume fraction tends to zero and the number of ground elements to infinity. The latter class of problems therefore provides reliable benchmark solutions for checking the validity, convergence and accuracy of various numerical methods in topology optimization. Some earlier exact analytical optimal topologies were presented in Prager and Rozvany [4], Lewinski et al. [2], Rozvany and Gollub [7], Rozvany [5,6,8] and more recent those in Lewinski [1] and Rozvany et al. [10].

The current paper will deal with the latest classes of exact analytical solutions for some popular problems used in examples of topology optimization. These will include:

new boundary and loading conditions for classical Michell trusses,
extended Michell trusses to new design conditions, such as
pre-existing members,
supports of non-zero cost,
several loading conditions,
prescribed displacements;
selfweight, as well as
grillages (beam systems).

It is to be remarked that the theory of exact least-weight grillages has advanced much further than that of least-weight trusses. At present, analytical, closed-formed optimal grillage solutions are available for almost any loading and support conditions.

T. Lewinski, "Variational proof of optimality criteria for Michell structures with pre-existing members", Struct. Multidisc. Optim. 32, 2006 (in press) doi:10.1007/s00158-006-0014-1
T. Lewinski, M. Zhou, G.I.N. Rozvany "Extended exact solutions for least-weight truss layouts. Part. I. Cantilever with a horizontal axis of symmetry. Part II. Unsymmetric cantilevers", Int. J. Mech. Sci. 36, 375-419, 1994. doi:10.1016/0020-7403(94)90043-4
A.G.M. Michell "The Limits of Economy of Material in Frames Structures", Phil. Mag. 8, pp.589-597, 1904.
W. Prager, G.I.N. Rozvany "Optimization of structural geometry" (invited lecture). In: Bednarek, A. R., Cesari, L. (eds.): Dynamical systems (Proc. Univ. Florida Int. Symposium, Gainesville, March 1976), pp. 265-294, 1977, Academic Press, New York.
G.I.N. Rozvany "Optimal design of flexural systems", Oxford: Pergamon, 1976.
G.I.N. Rozvany "Structural design via optimality criteria", Dordrecht: Kluwer, 1989.
G.I.N. Rozvany; W. Gollub "Michell layouts for various combinations of line supports, Part. I.", Int. J. Mech. Sci. 32, 1021-1043, 1990. doi:10.1016/0020-7403(90)90006-5
G.I.N. Rozvany "Exact analytical solutions for some popular benchmark problems in topology optimization", Struct. Optim. 15, 42-48, 1998. doi:10.1007/BF01197436
G.I.N. Rozvany "Aims, scope, methods, history, and unified terminology of computer-aided topology optimization in structural mechanics", Struct. Multidisc. Optim. 21, 90-108, 2001. doi:10.1007/s001580050174
G.I.N. Rozvany, O.M. Querin, J.; Lógó, V. Pomezanski "Exact analytical theory of topology optimization with some pre-existing members or elements", Struct. Multidisc. Optim. 31, 373-378, 2006. doi:10.1007/s00158-005-0594-1
G.I.N. Rozvany, M. Zhou "Applications of COC method in layout optimization" In: Eschenauer, H.; Mattheck, C.; Olhoff, N. (eds.) Proc. Conf. "Eng. Optim. in Design Processes" (held in Karlsruhe 1990), 59-70. Berlin, Heidelberg, New York, Springer-Verlag, 1991.

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