An integrated approach to shape optimization and mesh adaptivity based on material residual forces

This contribution is concerned with the coupling of finite element based shape optimization to methods of mesh adaptivity. Therein, the nodal points of a finite element mesh serve as design variables in an optimization problem that aims to minimize a cost functional with respect to different constraints. In order to avoid the occurrence of oscillating boundaries in the optimal design trials, we generate the respective design updates through solving a series of fictitious boundary value problems on an updated Lagrangian configuration. The design update process is accompanied by adaptive mesh refinement to obtain more accurate results for the structural analysis and the design sensitivity analysis. For the mesh adaptation process, we consider r-adaptive node relocation, h-adaptive mesh refinement and a combination of both approaches. In each case, the derivations rest upon the notion of material residual forces induced by finite element discretization and the coupling of shape optimization and mesh adaptivity is of intermittent type, i.e. mesh adaptivity is invoked once a certain number of design updates has been generated. We examine the benefits of the proposed method on the basis of some numerical examples in comparison to the same shape optimization method not involving adaptive mesh refinement where we evaluate the corresponding numerical costs, the gain in accuracy and the effects on the optimal shapes being obtained.