Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids

We study the shape differentiability of a cost function for the flow of an incompressible viscous fluid of power-law type. The fluid is confined to a bounded planar domain surrounding an obstacle. For smooth perturbations of the shape of the obstacle we express the shape gradient of the cost function which can be subsequently used to improve the initial design

Approximation of boundary control problems on curved domains

In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ωh (typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in Ωh, and we study the influence of the replacement of Ω by Ωh on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = ∂Ω with those defined on Γh = ∂Ωh and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates

On shape optimization for compressible isothermal Navier-Stokes equations

The steady state system of isothermal Navier-Stokes equations is considered in two dimensional domain including an obstacle. The shape optimisation problem of drag minimisation with respect to the admissible shape of the obstacle is defined. The generalized solutions for the Navier-Stokes equations are introduced. The existence of an optimal shape is proved in the class of admissible domains. In general the solution to the problem under consideration is not unique

Displacement Derivatives in Shape Optimization of Thin Shells

In the present paper the framework for the shape sensitivity analysis of systems of equations defined on a surface in \bbbr^3 is established. The model of thin shell presented in Koiter, 1970 is considered. The formulation of the model in a reference domain has been chosen for our analysis. The shape gradients and shape Hessians of associated shape functionals are defined and evaluated using the so-called displacement derivatives

Analysis of crack singularities in an aging elastic material

Version 10.03.2005International audienceWe consider a quasistatic system involving a Volterra kernel modelling an hereditarily-elastic aging body. We are concerned with the behavior of displacement and stress fields in the neighborhood of cracks. In this paper, we investigate the case of a straight crack in a two-dimensional domain with a possibly anisotropic material law. We study the asymptotics of the time dependent solution near the crack tips. We prove that, depending on the regularity of the material law and the Volterra kernel, these asymptotics contain singular functions which are simple homogeneous functions of degree \frac12 or have a more complicated dependence on the distance variable r to the crack tips. In the latter situation, we observe a novel behavior of the singular functions, incompatible with the usual fracture criteria, involving super polynomial functions of \ln(r) growing in time

Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spactial dimensions

Topological derivatives for elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions of elasticity boundary value problems in domains with small holes, and determine the explicit asymptotic expansions of solutions with respect to small parameter which describes the radius of internal hole. The elastic potentials of Muskhelishvili gives us an explicite solution in the ring $C(\rho,R)=\{\rho < |x| < R \}$ in the form of complex valued series. The series depends on the small parameter, the radius $\rho$ of the ring, and we are interested in the behavior of the series for the passage $\rho\to 0$. Such analysis leads to the expansion of the elastic energy in the form $$\mathcal{E}(\rho,R)=\mathcal{E}(0,R)+\rho^2\mathcal{E}^1(R)+\rho^4\mathcal{E}^2(R)+\dots\ ,$$ where $\mathcal{E}^1(R)$ is used to determine the first order topological derivatives of shape functionals, and $\mathcal{E}^2(R)$ can be used to determine the second order topological derivatives of shape functionals. In the paper the first order term $\mathcal{E}^1(R)$ is given, however the method is general and can be used to determine the subsequent terms of the energy expansion and the topological derivatives of higher order

Optimal Shells Formed on a Sphere. The Topological Derivative Method

The subject of the paper is the analysis of sensitivity of a thin elastic spherical shell to the change of its shape associated with forming a small circular opening, far from the loading applied. The analysis concerns the elastic potential of the shell. The sensitivity of this functional is measured as a topological derivative, introduced for the plane elasticity problem by Sokolowski and \buildrel . \over {\hbox{Z}}ochowski (1997) and extended here to the case of a spherical shell. A proof is given that : i) the first derivative of the functional with respect to the radius of the opening vanishes, and : ii) the second derivative does not blow up. A partially constructive formula for the second derivative or for the topological derivative is put forward. The theoretical considerations are confirmed by the analysis of a special case of a shell loaded rotationally symmetric, weakened by an opening at its north-pole. The whole treatment is based on the Niordson-Koiter theory of spherical shells, belonging to the family of correct first order shell models of Love

Topological Derivative for Nucleation of Non-Circular Voids

The hitherto existing literature concerning the topological derivative of shape functionals concerned perturbations of domains caused by introduction of circular or ball openings. In the present study the notion of the topologic- al derivative is generalized to the case of openings of arbitrary shape. The report is concerned with energy functionals related to the Neumann problem and the 2D elasticity system

Mathematical challenges in shape optimization

International audienceThis paper covers some theoretical investigations performed in France, in the framework of the CNRS programme GDR {\it Applications Nouvelles de l'Optimisation de Forme}. The programme included also some activities in Poland. We do not restrict the presentation to the French community in the research field, the list of references includes all recent monographs on the shape optimization. The outline of the paper is the following. First we present some main fields of the activity in shape optimization. To present some precise results, from mathematical point of view, we include two sections. The first is devoted to the eigenvalues, the second to the drag minimization. Many theoretical questions related to these problems are still open