Topology optimization with stress constraints considering uncertainty in applied loads

André T. Beck¹; Gustavo Assis da Silva¹

¹ University of São Paulo

doi:10.20906/CPS/CILAMCE2017-0694

Resumo

This paper proposes a formulation for solving continuum reliability-based topology optimization problems, of volume minimization subject to local stress constraints, considering uncertainty in applied loads. The local nature of stress constraints makes this problem very challenging to solve, even in a deterministic setting, since there is one constraint for every finite element in the mesh. The consideration of uncertainties in applied loading makes the problem even more challenging to solve, because stochastic problems are naturally more nonlinear than deterministic ones, and additional effort is needed for uncertainty quantification and propagation. The solution proposed in this paper avoids aggregation techniques, and imposes stress constraints by means of the augmented Lagrangian method. The solution includes a sensitivity analysis for the gradients of the augmented Lagrangian function. The paper also discusses some issues related to the density approach, namely: the singularity phenomenon and wrong stress computation due to jagged boundaries. Probabilistic stress constraints are handled via Performance Measure Approach (PMA), where a first-order approximation is considered for probability of failure. In order to alleviate the computational effort associated with solution of inverse reliability analyses at each step of outer optimization problem the principle of superposition is used. The proposed methodology leads to crisp black-and-white topologies, at a reasonable number of interactions. Through numerical examples it is verified that solutions of the reliability-based problems are different from solutions of deterministic problems, and that it is generally not possible to establish equivalences between these solutions.

Palavras-chave: topology optimization; RBDO; stress constraints