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.

翻译：一阶形状优化方法通常需要大量迭代才能达到局部最优设计。虽然高阶方法能显著减少迭代次数，但仅具有局部收敛特性，需要足够接近的初始猜测。本文提出一种非正则化形状牛顿法，将形状优化与同伦（或连续）方法相结合，使得即使初始设计远离解也能使用高阶方法。同伦方法的核心思想是将待求解问题与一个更简单的问题连续连接，并通过预测-校正方案跟踪对应的解路径。我们采用形状牛顿法作为校正器，并使用任意阶形状导数进行预测。此外，我们还将同伦方法应用于多目标形状优化场景，以高效获得帕累托前沿上均匀分布的点。最后，通过一系列数值实验验证了我们的结果。