We introduce a novel method for the implementation of shape optimization for non-parameterized shapes in fluid dynamics applications, where we propose to use the shape derivative to determine deformation fields with the help of the \(p-\) Laplacian for \(p > 2\). This approach is closely related to the computation of steepest descent directions of the shape functional in the \(W^{1,\infty }-\) topology and refers to the recent publication Deckelnick et al. (A novel \(W^{1,\infty}\) approach to shape optimisation with Lipschitz domains, 2021), where this idea is proposed. Our approach is demonstrated for shape optimization related to drag-minimal free floating bodies. The method is validated against existing approaches with respect to convergence of the optimization algorithm, the obtained shape, and regarding the quality of the computational grid after large deformations. Our numerical results strongly indicate that shape optimization related to the \(W^{1,\infty }\)-topology—though numerically more demanding—seems to be superior over the classical approaches invoking Hilbert space methods, concerning the convergence, the obtained shapes and the mesh quality after large deformations, in particular when the optimal shape features sharp corners.

我们介绍了一种在流体动力学应用中实现非参数化形状的形状优化的新方法，我们建议在\(p-\)拉普拉斯算子的帮助下使用形状导数来确定变形场\(p > 2\)。这种方法与\(W^{1,\infty }-\)拓扑中形状泛函的最陡下降方向的计算密切相关，并参考了最近发表的 Deckelnick 等人。（一本小说\(W^{1,\infty}\)使用 Lipschitz 域进行形状优化的方法，2021 年），其中提出了这个想法。我们的方法被证明用于与阻力最小的自由浮动体相关的形状优化。该方法在优化算法的收敛性、获得的形状以及大变形后计算网格的质量方面针对现有方法进行了验证。我们的数值结果强烈表明与\(W^{1,\infty }\)拓扑相关的形状优化——尽管在数值上要求更高——似乎优于调用希尔伯特空间方法的经典方法，关于收敛性，获得的大变形后的形状和网格质量，特别是当最佳形状具有尖角时。