A one shot approach to topology optimization with local stress constraints

We consider the topology optimization problem with local stress constraints. In the basic formulation we have a pde-constrained optimization problem, where the finite element and design analysis are solved simultaneously. Here we introduce a new relaxation scheme based on a phase-field method. The starting point of this relaxation is a reformulation of the constraints of the optimization problem involving only linear and 0-1 constraints. The 0-1 constraints are then relaxed and approximated by a Cahn-Hillard type penalty in the objective functional. As the corresponding penalty parameter decreases to zero, it yields convergence of minimizers to 0-1 designs. A major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter. After the relaxation we end up with a large-scale optimization problem with a high number of linear inequality constraints. Discretization is done by usual finite elements and for solving the resulting finite-dimensional programming problems an interior-point method is used. Numerical experiments based on different stress criteria attest the success of the new approach. To speed up computational times we investigated the construction of an optimal solver for the arising subproblem in the interior-point formulation.