Sharp interface limit for a phase field model in structural optimization

We formulate a general shape and topology optimization problem in structural optimization by using a phase field approach. This problem is considered in view of well-posedness and we derive optimality conditions. We relate the diffuse interface problem to a perimeter penalized sharp interface shape optimization problem in the sense of $\Gamma$-convergence of the reduced objective functional. Additionally, convergence of the equations of the first variation can be shown. The limit equations can also be derived directly from the problem in the sharp interface setting. Numerical computations demonstrate that the approach can be applied for complex structural optimization problems

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

International audienceThis chapter proposes a framework for dealing with two problems related to the analysis of shapes: the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [8], we consider the characteristic functions of the subsets of ℝ2 and their distance functions. The L 2 norm of the difference of characteristic functions and the L∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular that induced by the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of ℝ2 of positive reach in the sense of Federer [12], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem. We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational de.nitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples

Linear Quadratic Zero-Sum Two-Person Differential Games

International audienceAs in optimal control theory, linear quadratic (LQ) differential games (DG) can be solved, even in high dimension, via a Riccati equation. However, contrary to the control case, existence of the solution of the Riccati equation is not necessary for the existence of a closed-loop saddle point. One may " survive " a particular, non generic, type of conjugate point. An important application of LQDG's is the so-called H∞-optimal control, appearing in the theory of robust control

Formal asymptotic limit of a diffuse-interface tumor-growth model

We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the diffuse-interface model to the identified sharp-interface limit

Moving frames applied to shell elasticity

Exterior calculus and moving frames are used to describe curved elastic shells. The kinematics follow from the Lie-derivative on forms whereas the dynamics via stress-forms.Comment: 20 pages, 1 figur

hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated

Le paléolithique inférieur et moyen en Midi toulousain : nouvelles données et perspectives de l’archéologie préventive

Depuis quelques années, les opérations d’archéologie préventive menées en Midi toulousain ont permis de réactualiser certaines données relatives à la géomorphologie des sites paléolithiques inférieur et moyen. Ainsi, plusieurs contextes peuvent être considérés comme plus favorables que d’autres pour la conservation du matériel. Par ailleurs, les résultats préliminaires d’études typo-technologiques des industries mises au jour récemment, offrent de nouvelles perspectives de recherche pour la constitution d’un cadre chrono-culturel plus précis.In the last years, preventive archeology projects conducted in the Midi toulousain Region (southern France) allowed to update data concerning the geomorphology of Early and Middle Palaeolithic sites. It became apparent that certain contexts can be considered as more favorable to vestige conservation than others. Besides, preliminary results of typo-technological studies of recently discovered artifacts offer new research perspectives for the definition of a more precise chrono-cultural frame

Le paléolithique inférieur et moyen en Midi toulousain : nouvelles données et perspectives de l’archéologie préventive

Depuis quelques années, les opérations d’archéologie préventive menées en Midi toulousain ont permis de réactualiser certaines données relatives à la géomorphologie des sites paléolithiques inférieur et moyen. Ainsi, plusieurs contextes peuvent être considérés comme plus favorables que d’autres pour la conservation du matériel. Par ailleurs, les résultats préliminaires d’études typo-technologiques des industries mises au jour récemment, offrent de nouvelles perspectives de recherche pour la constitution d’un cadre chrono-culturel plus précis.In the last years, preventive archeology projects conducted in the Midi toulousain Region (southern France) allowed to update data concerning the geomorphology of Early and Middle Palaeolithic sites. It became apparent that certain contexts can be considered as more favorable to vestige conservation than others. Besides, preliminary results of typo-technological studies of recently discovered artifacts offer new research perspectives for the definition of a more precise chrono-cultural frame