Shape Optimization under Uncertainty from a Stochastic Programming Point of View / Stochastic Programming (PDF)

4 Solving Shape Optimization Problems (S. 77-78)

This chapter is dedicated to the actual numerical solution techniques we implemented to solve the (random) shape optimization problems described in Chapter 3. As noted in the beginning, we employed a steepest descent algorithm (see Section 4.4) together with a level set method (see Section 4.1).

The necessary function evaluations are done according to Algorithm 3.16, whereas the computation of the descent direction is described here in this chapter, making use of the shape derivative (see Section 4.2) and also the topological derivative (see Section 4.3). There are various methods that aim to solve shape optimization problems, and before we start describing our particular level set approach, we brie?y mention some of these methods. For example, there is the homogenization method (cf. Allaire [All02]) whose physical idea in principle consists of averaging heterogeneous media in order to derive effective properties. In [All02, Chapter 4], the method is applied to optimal design problems with linear elasticity in form of two-phase optimization problems.

The task is then to ?nd an optimal distribution of two elastic materials, i.e. there are no void areas. This results in an ill-posed optimization problem, which, however, homogenization theory provides a relaxation to by introducing generalized designs. Numerical examples can also be found in [HN97]. Another approach, namely topology optimization by the material distribution method, is described in the book by Bendsøe and Sigmund [BS03]. Each point in the design can have material or not1. In a discrete setting, there is a grid where each grid cell, or “pixel”, is either ?lled with material, or there is none.

This leads to nonlinear optimization problems with binary variables which indicate the presence or absence of material in the grid cells, respectively. In [SS03] for example, they show that certain nonlinear 0-1 topology optimization

Dr. Harald Held completed his doctoral thesis at the Department of Mathematics at the University of Duisburg-Essen. He is now a Research Scientist at Siemens AG, Corporate Technology.