Continuation methods for higher-order topology optimization

TL;DR

This paper introduces a continuation method combined with a barrier strategy to effectively solve nonconvex topology optimization problems, ensuring feasible solutions and convergence to local optima without prior solution knowledge.

Contribution

It proposes a homotopy-based continuation approach integrated with a barrier method for improved convergence in density-based topology optimization.

Findings

The combined method maintains feasibility of the density function.

It converges to a local optimum without needing an initial guess close to the solution.

Numerical results demonstrate effectiveness in PDE-constrained compliance minimization.

Abstract

We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a density function. To obtain candidates for local minima, we want to solve the first order optimality system via Newton's method. This requires the initial guess to be sufficiently close to the a priori unknown solution. Introducing a stepsize rule often allows for less restrictions on the initial guess while still preserving convergence. In topology optimization one typically encounters nonconvex problems where this approach might fail. We therefore opt for a homotopy (continuation) approach which is based on solving a sequence of parametrized problems to approach the solution of the original problem. In the density based framework the values of the design variable are constrained by 0 from below and 1 from above. Coupling the homotopy method with a barrier strategy enforces these constraints to be satisified. The numerical results for a PDE-constrained compliance minimization problem demonstrate that this combined approach maintains feasibility of the density function and converges to a (candidate for a) locally optimal design without a priori knowledge of the solution.