Continuation Methods for Higher‐Order Topology Optimization

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 step‐size 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 satisfied. 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.