On the trajectories of penalization methods for topology optimization

Abstract

We consider the discretized zero-one minimum compliance topology optimization problem of elastic continuum structures under multiple load conditions. The binary design variables indicate presence or absence of material in the finite elements. A common approach to solve these problems is to relax the binary constraints, i.e. allow the design variables to attain values between zero and one, and penalize intermediate values to obtain a “black and white” (zero-one) design. To avoid convergence to a local minimum, it has been suggested that a continuation method should be used, where the penalized problems are solved with increasing penalization.

In this paper, the trajectories associated with optimal solutions to the penalized problems, for continuously increasing penalization, are studied on some carefully chosen examples. Two different penalization techniques are used. The global trajectory is defined as the path followed by the global optimal solutions to the penalized problems, and we present examples for which the global trajectory is discontinuous even though the original zero-one problem has a unique solution. Furthermore, we present examples where the penalization method combined with a continuation approach fails to produce a black and white design, no matter how large the penalization becomes.