Optimal (topology) design under stochastic uncertainty

For optimal (topology) design of mechanical structures, e.g. trusses, frames, there are many well known equivalent problem formulations, one of them is based on the linear yield/strength condition and the equilibrium equation (of the ground structure). Having to take into account in many cases stochastic variations of the model parameters and external loads of a mechanical structure, the basic optimization problem under stochastic uncertainty must be replaced by a certain deterministic substitute problem. Appropriate substitute problems may be obtained by considering the displacements or the strength/volume "reserves" of the elements of the underlying structure for the evaluation of the violation of the basic yield/strength condition. In each case a stochastic linear program (SLP) "with complete fixed recourse", hence, a linear optimization problem with a special structure is obtained. Also in case that for topology optimization a ground structure is given, e.g, by all possible nonoverlapping connections between the chosen nodal points, the topology design problem under stochastic uncertainty can be formulated as a stochastic linear program "with complete fixed recourse". The numerical solution of the resulting deterministic substitute problems is discussed, and some numerical examples are given. Especially, in case of discrete parameter distributions, the performance of special purpose LP-solvers exploiting the "dual block angular structure" of the resulting SLP with complete fixed recourse is considered.