Implicit parametrizations in shape optimization: boundary observation

We present first a brief review of the existing literature on shape optimization, stressing the recent use of Hamiltonian systems in topology optimization. In the second section, we collect some preliminaries on the implicit parametrization theorem, especially in dimension two, which is a case of interest in shape optimization. The formulation of the problem is also discussed. The approximation via penalization and its differentiability properties are analyzed in Section 3. Next, we investigate the discretization process in Section 4. The last section is devoted to numerical experiments

Topology optimization and boundary observation for clamped plates

We indicate a new approach to the optimization of the clamped plates with holes. It is based on the use of Hamiltonian systems and the penalization of the performance index. The alternative technique employing the penalization of the state system, cannot be applied in this case due to the (two) Dirichlet boundary conditions. We also include numerical tests exhibiting both shape and topological modifications, both creating and closing holes

Penalization of stationary Navier-Stokes equations and applications

We consider the steady Navier-Stokes system with mixed boundary conditions, in subdomains of a holdall domain. We study, via the penalization method, its approximation properties. Error estimates and the uniqueness of the solution, obtained in a non standard manner, are also discussed. Numerical tests, including topological optimization applications, are presented

La méthode du Lagrangien augmenté pour une probleme d'interaction fluide-structure

In [3, p. 91] is presented a not-conditionally stable numerical method for solving a time-depend fluid structure interaction problem. This method consists in solving at each time step the mixed hybrid system (1) and to find out: $v$ the velocity of the fluid, $\nu$ the velocity of the structure, $p$ the pressure of the fluid and $\lambda$ the force on the fluid-structure interface. In order to solve the system (1), we can use many algorithms just like: Uzawa's algorithm or the Augmented Lagrangien Method, but these algorithms don't permit to solve the fluid-structure problem in a decoupled way (via partitioned procedures), more exactly the fluid and structure problems are not solved separately. Consequently, we can't use the well established theories and software for the fluid and respectively for the structure.  Alternatively, we can use the iterative method presented in [4] in order to solve the fluid-structure linear system via partitioned procedures. Unfortunely, this method converges slowly. Based on the method used in [4], we present in this paper an augmented algorithm, where the continuity of the fluid and structure velocities on the contact surface is penalized in order to improve the convergence rate. Numerically, the continuity of the fluid and structure velocities on the contact surface has the form $B_{21}v + B_{22}\nu =0$. The convergence of the method is proved in the third section

Approximation of a simply supported plate with obstacle

We discuss an algorithm for the solution of variational inequalities associated to simply supported plates in contact with a rigid obstacle. Our approach has a fixed domain character, uses just linear equations and approximates both the solution and the corresponding coincidence set. Numerical examples are also provided

Poleni curves on surfaces of constant curvature

{aeres : ACL}International audienceIn the euclidean plane, a regular curve can be defined through its intrinsic equation which relates its curvature $k$ to the arc length $s$. Elastic plane curves were determined this way. If $k(s)=\frac {2\alpha}{\cosh \left(\alpha s\right)}$, the curve is known under the name ''la courbe des for\c cats'', introduced in 1729 by Giovanni Poleni in relation with the tractrix \cite{Palais1976}. The above equation is yet meaningful on a surface if one interprets $k$ as the geodesic curvature of the curve. In this paper we solve the above equation on a surface of constant curvature

Extension theorems related to a fluid-structure interaction problem

The aim of this paper is to prove the existence of an approximate weak solution for a steady fluid-structure interaction problem. A fictitious domain approach with penalization is used. One of the main ingredients is an extension theorem for domains with Lipschitz boundaries. The fluid and structure domains are not necessarily double connected and the structure is not completely surrounded by the fluid. These assumptions are more realistic for some engineering and medical applications