On the discrete variant of the traction method in parameter-free shape optimization

One of the major challenges in the parameter-free approach to computational shape optimization is the avoidance of oscillating (i.e. non-smooth) boundaries in the optimal design trials. In order to achieve this, we investigate a method for regularization that corresponds to the discrete variant of the so-called traction method. In this approach, the design updates are generated in terms of a displacement field, which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain. The main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function. We propose an algorithm in which constraint functions will be taken into account by using an augmented Lagrangian formulation and a step-length control ensures a sufficient decrease condition in terms of the objective function within each iteration. We examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest-descent algorithm.