Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing

TL;DR

This paper introduces a homotopy-augmented shape-Newton method for higher order shape optimization, enabling global convergence from distant initial guesses and efficient Pareto front tracing.

Contribution

It combines shape-Newton methods with homotopy techniques to improve convergence and applicability in multi-objective shape optimization tasks.

Findings

Homotopy methods enable global convergence from far initial guesses.

Higher order shape derivatives improve optimization efficiency.

Numerical experiments demonstrate effectiveness in Pareto front tracing.

Abstract

First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.