Keywords: topology optimization, stochastic programming, optimality criteria, compliance, optimal design, minimum volume design, probability, joint distribution.

The aim of this paper is to introduce a new type of probabilistic optimal topology design method where the elements of the loading are given randomly and they can have linear relationship. This pre-design condition can be considered as a load combination. In the proposed probabilistic topology optimization method the minimum penalized weight design of the structure is subjected to a compliance constraint which has uncertainties and side constraints. The calculation of compliance value is based on the assumption that external loads have uncertainties where their joint normal distribution function, mean values and covariances are known. If the probability of this compliance value is given by the use of recommendation of Prekopa [1] this probabilistic expression can be substituted with an equivalent deterministic one and used as a deterministic constraint in the original problem.

The object of the design (ground structure) is a rectangular disk in plane stress with given loading and support conditions. The material is linearly elastic. The design variables are the thickness of the finite elements. To obtain the correct optimal topology some filtering method (the ground elements are subdivided into further elements) has to be applied to avoid the so-called "checker-board pattern" [2]. By the use of the first order optimality conditions a redesign formula of the stochastically constrained topology optimization problem can be derived which is an improved one of the previously [3] presented iterative expression. The new class of optimal topologies with their numerical confirmation are presented. The standard finite element computer program with quadrilateral membrane elements is applied in the numerical calculation. Through the numerical examples the paper investigates the deterministic optimal topologies and optimal topologies obtained for the case of uncertain situations. The effects of the covariances of the loads and the given probability are particularly investigated. The present stage of the research shows that to compute the "stochastic" topology takes more computational time. The algorithm is rather stable and provides the convergence to reach the optimum. The applied method gives a wider possibility to the designer to take into consideration a more realistic loading description than the deterministic topology design.

1

A. Prekopa, "Stochastic Programming", Akademia Kiado and Kluwer, Budapest, Dordrecht, 1995.

2

J. Lógó "New Type of Optimal Topologies by Iterative Method", Mechanics Based Design of Structures and Machines,33(2), 149-172, 2005. doi:10.1081/SME-200067035

3

J. Lógó, M. Ghaemi and A. Vásárhelyi, "Topology Optimization in the Case of Uncertain Loading Conditions", in Proceedings of the Eighth International Conference on Computational Structures Technology, B.H.V. Topping, G. Montero and R. Montenegro, (Editors), Civil-Comp Press, Stirlingshire, United Kingdom, paper 215, 2006. doi:10.4203/ccp.83.215

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