Existence and continuity of optimal solutions to some structural topology optimization problems including unilateral constraints and stochastic loads

• We consider a general discrete structural optimization problem including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. The loads applied (and, in principle, also other data such as the initial distances to the supports), are allowed to be stochastic, which we handle through a discretization of the probability space. The existence of optimal solutions to the resulting problem is established, as well as the continuity properties of the equilibrium displacements and forces with respect to the lower bounds on the design variables. The latter feature is important in topology optimization, in which one includes the possibility of vanishing structural parts by setting design variable values to zero. In design optimization computations, one usually replaces the zero lower design bound by a strictly positive number, hence rewriting the problem into a sizing form. For several such perturbations, we prove that the global optimal designs and equilibrium states converge to the correct ones as the lower bound converges to zero.

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