Keywords: topology optimization, probability, stochastic loading, optimality criteria method, optimal design, robust design.

The achievements of structural optimization and reliability based design optimization are tremendous and one can see that topology optimization as a research field has achieved significant results only in the last two decades. In spite of the clear concept, the solution of topology optimization problems poses significant technical challenges. Problems are typically large-scale and discrete, and often exhibit some numerical difficulty associated with underlying mechanics (such as instability of members and checkerboard patterns in continua). For these reasons, the majority of topology optimization research has focused on deterministic design problems, neglecting uncertainty that arises in most engineering applications. Until the end of the last century one could almost not find any publication on topology optimization considering uncertainties. Fortunately, this trend has changed and a great number of works were published during the last twelve years. A common feature among many of these different approaches is to reformulate the uncertainties and create an alternative deterministic formulation. The uncertain objective function or the uncertain constraint can be replaced by their statistical averages.

Lógó [1] and Lógó et al. [2,3] elaborated a rather powerful method for the stochastic topology optimization where the magnitude of the loads or the compliance bounds are given by their mean values, covariances and distribution functions. The loading positions as uncertain data in the topology optimization problem is considered in [4]. Here two computational models and the corresponding algorithms are elaborated. Both models use simple transformations to substitute the original load position problem with uncertain loading magnitude ones.

In this paper the uncertainties of the load positions are considered. By the use of a simple simulation technique and the stochastic upperbound theorem of Kataoka [5] a generalized compliance design problem is elaborated. The uncertain quantities are substituted by their statistical measures and a robust mathematical programming problem is derived. To solve this constrained deterministic mathematical programming problem an iterative solution technique is derived by the use of the optimality criteria method. Several numerical examples are presented and compared.

1

J. Lógó, "New Type of Optimality Criteria Method in Case of Probabilistic Loading Conditions", Mechanics Based Design of Structures and Machines, 35(2), 147-162, 2007. doi:10.1080/15397730701243066

2

J. Lógó, M. Ghaemi, A. Vásárhelyi, "Stochastic compliance constrained topology optimization based on optimality criteria method", Periodica Polytechnica-Civil Engineering, 51(2), 5-10, 2007. doi:10.3311/pp.ci.2007-2.02

3

J. Lógó, M. Ghaemi, M. Movahedi Rad, "Optimal topologies in case of probabilistic loading: The influence of load correlation", Mechanics Based Design of Structures and Machines, 37(3), 327-348, 2009. doi:10.1080/15397730902936328

4

J. Lógó, D.B. Merczel, L. Nagy, "On Optimal Topologies for the Case of Uncertain Load Positions", in B.H.V. Topping, Y. Tsompanakis, (Editors), "Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 92, 2011. doi:10.4203/ccp.96.92

5

S. Kataoka, "A Stochastic Programming Model", Econometria, 31(1-2), 181-196, 1963. doi:10.2307/1910956

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