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

Simultaneous Shape and Mesh Quality Optimization using Pre-Shape Calculus

“…( 55) leads to a linear system, which is solvable by use of standard techniques such the CG-method. It is reported in [13], that the p-Laplacian metric has particular advantages in resolution of sharp edges or kinks of optimal shapes. Illustration of this is not the goal of this paper.…”

“…To be more specific, we test two bilinear forms and four regularizations of gradients for a standard gradient descent algorithm with a backtracking line search. The two bilinear forms are given by the linear elasticity as found in [17] and the p-Laplacian inspired by [13] and studies found in [4]. Different gradients tested will be the unregularized, the shape regularized, the shape and volume regularized, and the shape and volume regularized one with varying outer boundary.…”

“…As a standard approach, we recommend the linear elasticity metric as found in [17]. To illustrate the general applicability of our regularization method, we also test the very reasonable p-Laplacian metrics as inspired by [13] to represent gradients.…”

Combining Sobolev Smoothing with Parameterized Shape Optimization

“…An more recent paper by Müller, Kühl, et.al. [7] suggests solving a p-Laplace problem as a relaxation for the steepest descent step. This is done in a free node optimization setting and can deal effectively with situations where the optimum might contain points or edges in the shape.…”

“…Hence, using a coarser parameterization, which only produces shapes of sufficient regularity, circumvents this problem [5]. The most popular solution to solve this problem is to apply an elliptic partial differential equation to smooth the gradient [6,7]. This is equivalent to a reinterpretation of the gradient in a search space of higher regularity.…”