An Augmented Lagrangian Approach to Plane Truss Layout Optimization Under Multiple Loading Conditions

Singular topologies constitute one of the major computational obstacles in stress-constrained layout optimization, especially in the presence of multiple loading conditions. In order to deal with the issue posed by singular topologies, many numerical strategies have been proposed in the literature, such as the adoption of smooth envelope functions for stress constraint or the ε-relaxation method. However, it has been shown that such approaches may not converge to the correct global optimal solution. In this contribution, we propose a novel approach for two-dimensional stress-constrained truss layout optimization under multiple loading conditions based on an elastic formulation that enforces both equilibrium and compatibility, and the adoption of a suitable implementation of the augmented Lagrangian (AL) method, which allows to overcome the computational challenges arising from the existence of singular optimal topologies. To ensure convergence, a polynomial expression for stress-constraints is adopted, which implies a significantly higher penalty on constraint violation with respect to traditional stress-constraints. As it will be demonstrated, the global minimum of the objective function often corresponds to a singular infinitesimal region enclosed in the infeasibility domain of the design space, making it difficult for most optimization algorithms to reach. To demonstrate the effectiveness of the proposed method, the results obtained by applying the algorithm to the problem of minimizing the volume of a truss subjected to multiple loading conditions will be presented. They were compared with those reported in the literature, highlighting the algorithm’s ability to identify optimal solutions in the singular regions of the infeasible domain.