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- 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.
Highlights
Adjoint-based local optimization has been matured toward an efficient industrially applied strategy, e.g., Othmer (2014), Papoutsis-Kiachagias and Giannakoglou (2016)
Different gradients are associated with different transformations applied to the shape derivative, and several techniques have been proposed to increase the regularity of the shape updates: (a) CAD-related shape definitions connect the node-based shape derivatives to the CAD parameterization using the chain rule of differentiation, cf
We introduce the mathematical setting for our hydrodynamic shape optimization problem, where we refer to Fig. 1 for the geometrical setting and the notation
Summary
Adjoint-based local optimization has been matured toward an efficient industrially applied strategy, e.g., Othmer (2014), Papoutsis-Kiachagias and Giannakoglou (2016). The shape gradient is identified by the Riesz representation of the directional derivative of the shape functional Even though this leads to smoother deformations, the approach is algorithmic challenging due to solving a PDE on a hyperplane, and mathematically questionable in the general case, see Allaire et al (2021) and cf Sect. (c) A more rigor approach of Jameson and Vassberg (2005) and Vassberg and Jameson (2006a, b) applies an implicit, continuous smoothing operator to either the shape derivative J′ or the deformation field , based on an extended definition of the inner product, frequently labeled ‘Sobolev-gradient’. 2 outlines the mathematical framework and the rationale that leads to the p-Laplace problem to approximate the steepest descent direction within the W1,∞ - topology together with a discussion of our fluid dynamic shape optimization problem.
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