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

In the expanding field of topology optimization the majority of papers deal with deterministic problems. The first author elaborated a rather powerful optimality criteria method for stochastic topology optimization, where the magnitudes of the loads or the compliance limit [1] are given by their mean values, covariances and distribution functions. Applying an appropriate approximation for the loading uncertainties, the stochastic expressions are substituted by equivalent deterministic ones. This work is a continuation of the above cited papers. Load positions are taken as probabilistic variables and all the other data are deterministic. To make the optimization method robust an equivalent deterministic problem is derived.

Two problem classes are elaborated. The first method, named "adjoint design", can be summarized as follows: the support reaction forces are calculated as stochastic expressions (uncertainty of the magnitudes [2]) and used as loads while the original load positions are considered as supports at the location of the mean values. This technique can be applied if the structure is supported by a few (two or three) simple supports. According to the second technique, each load is considered and handled with "3sigma" rules and the original problem is substituted by extended loading within the [-"3sigma", +"3sigma"] interval. The method is named "distribution of the loads." Because the positions of the loads are not known precisely, an equivalent loading system should be created. The distribution assumption is based on the equivalent work theory where a single load is substituted by seven forces. Those forces act symmetrically in an interval [-"3sigma", +"3sigma"] around the expected location of the given force. The magnitudes of the forces are approximated using continuous beam theory. Depending on the support rigidities, the magnitudes of the forces start at approximately 14% of the original force. The design problem becomes a deterministic one after this transformation. Several numerical examples are presented and compared.

1

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

2

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

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