Approximation of topology optimization problems using sizing optimization problems

Doktorsavhandling, 2004

The present work is devoted to approximation techniques for singular extremal problems arising from optimal design problems in structural and fluid mechanics. The thesis consists of an introductory part and four independent papers, which however are united by the common idea of approximation and the related application areas. In the first half of the thesis we are concerned with finding the optimal topology of truss-like structures. This class of optimal design problems arises when in order to find the optimal truss not only are we allowed to redistribute the material among the structural members (bars), but also to completely remove some parts altering the connectivity (topology) of the structure. The other half of the thesis addresses the question of the optimal topological design of flow domains for Stokes and Navier-Stokes fluids. For flows, optimizing topology means finding the optimal partition of the given design domain into disjoint parts occupied by the fluid and the impenetrable walls, given the in-flow and the out-flow boundaries. In particular, impenetrable walls change the shape and the connectivity of the flow domain. In the first paper we construct an example demonstrating the singular behaviour of truss topology optimization problems including a linearized global buckling (linear elastic stability) constraint. This singularity phenomenon has not been known before and affects the choice of numerical methods that can be applied to the optimization problem. We propose a simple approximation strategy and establish the convergence of globally optimal solutions to perturbed problems towards globally optimal solutions to the original singular problem. In the second paper we are concerned with the construction of finer approximating problems that allow us to reconstruct the local behaviour of a general class of singular truss topology optimization problems, namely to approximate stationary points to the limiting problem with sequences of stationary points to the regular approximating problems. We do so on the classic problem of weight minimization under stress constraints for trusses in unilateral contact with rigid obstacles. In the third paper we extend a design parametrization previously proposed for the topological design of flow domains for Stokes flows to also include the limiting case of porous materials - completely impenetrable walls. We demonstrate that, in general, the resulting design-to-flow mapping is not closed, yet under mild assumptions it is possible to approximate globally optimal minimal-power-dissipation domains using porous materials with diminishing permeability. In the fourth and last paper we consider the optimal design of flow domains for Navier-Stokes flows. We illustrate the discontinuous behaviour of the design-to-flow mapping caused by the topological changes in the design, and propose "minor" changes to the design parametrization and the equations that allow us to rigorously establish the closedness of the design-to-flow mapping. The existence of optimal solutions as well as the convergence of approximation schemes then easily follows from the closedness result.