A Novel $p$-Harmonic Descent Approach Applied to Fluid Dynamic Shape Optimization

Abstract

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 $$p-$$\n \n p\n -\n \n Laplacian for $$p > 2$$\n \n p\n >\n 2\n \n . This approach is closely related to the computation of steepest descent directions of the shape functional in the $$W^{1,\\infty }-$$\n \n \n W\n \n 1\n ,\n ∞\n \n \n -\n \n topology and refers to the recent publication Deckelnick et al. (A novel $$W^{1,\\infty}$$\n \n W\n \n 1\n ,\n ∞\n \n \n 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 $$W^{1,\\infty }$$\n \n W\n \n 1\n ,\n ∞\n \n \n -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.