Geometric Aspects of Shape Optimization

Abstract

We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report (Plotnikov and Sokolowski, Gradient flow for Kohn–Vogelius functional).

Asymptotic Representation of Vorticity and Dissipation Energy in the Flux Problem for the Navier–Stokes Equations in Curved Pipes

“…In fluid dynamics, the resistance force and the energy required for fluid propulsion are significant [6,8]. For a viscous fluid, the resistance force is computed through the energy dissipation, which is represented by an integral over the flow region of the square of the vorticity vector modulus [15][16][17][18].…”

On second-order tensor representation of derivatives in shape optimization

“…Shape sensitivity analysis is an important part of shape optimization, providing optimality conditions from which properties of minimizers can be extracted. Moreover, it is the basis for numerical techniques employed in approximating optimal shapes [18][19][20][21] and for shape gradient flows [22][23][24][25]. Shape derivatives [2,3,26] are a key tool for the shape sensitivity analysis of PDEs, and can usually be written in strong form or in weak form.…”