Abstract
In this article, we propose a shape optimization algorithm which is able to handle large deformations while maintaining a high level of mesh quality. Based on the method of mappings, we introduce a nonlinear extension operator, which links a boundary control to domain deformations, ensuring admissibility of resulting shapes. The major focus is on comparisons between well-established approaches involving linear-elliptic operators for the extension and the effect of additional nonlinear advection on the set of reachable shapes. It is moreover discussed how the computational complexity of the proposed algorithm can be reduced. The benefit of the nonlinearity in the extension operator is substantiated by several numerical test cases of stationary, incompressible Navier–Stokes flows in 2d and 3d.
Highlights
In the field of optimization constrained by partial differential equations (PDEs), there is a large class of problems where optimal controls are to be found and an optimal shape of the experimental domain
We focus on shape optimization in fluid dynamics, which is one of the pioneering applications in this field [11,17,20]
The purpose is to illuminate features of the nonlinear extension operator S proposed in Sect
Summary
In the field of optimization constrained by partial differential equations (PDEs), there is a large class of problems where optimal controls are to be found and an optimal shape of the experimental domain. Based on the method of mappings (cf to [21]), the question for admissible shapes Gadm := {F( ) : F ∈ Fadm} in (1) is translated to the choice of appropriate function spaces, in which a deformation from reference to the optimal configuration is to be found. We discuss cases where the reference domain is not of circular shape and illustrate the performance of the extension and influence on the mesh quality in a deformed domain The intention of this experiment is to demonstrate that the set of shapes Gadm, which is constructed via the mappings from Fadm, can be extended significantly and its dependence on the choice of a reference domain is reduced.
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