An adaptive multilevel approach to the minimal compliance problem in topology optimization

This paper presents a new solution strategy for the minimal compliance problem in topology optimization. This problem contains the system of linear elasticity partial differential equations (PDEs) as a constraint resulting in a large scaled optimization problem after the finite element discretization. Due to the repeated solution of the direct field problem given by the PDE constraints, efficient solution techniques are required. In this paper we present a new solution method involving adaptive multilevel techniques combined with a multigrid approach for the direct problem. Topology optimization problems are ill-posed, so regularization is needed. In our algorithm we combine two regularization techniques, in fact filter methods, such that their disadvantages are eliminated and only their positive properties remain. Numerical experiments are performed with several benchmark problems, where our multilevel approach turns out to be quite efficient. For solving the optimization problems arising in each iteration step, the method of moving asymptotes is used. Copyright © 2005 John Wiley & Sons, Ltd.