Jean-Paul Zolésio

Shape sensitivity analysis via min max differentiability

The object of this paper is twofold. We introduce a new theorem on the differentiability of a Min Max with respect to a parameter and we show how such a theorem can be applied to compute the material derivative in shape sensitivity analysis problems. We consider the Min Max of a functional which is parametrized by t. We show that, under appropriate conditions, the derivative of the Min Max with respect to t is the Min Max with respect to the points solution of the Min Max problem of the derivative of the original functional with respect to t. To illustrate the use of this theorem, we apply it to the control of an elliptic equation with a nondifferentiable observation and to shape design problems.

Moving shape analysis and control : applications to fluid structure interactions

Introduction Classical and Moving Shape Analysis Fluid-Structure Interaction Problems Plan of the Book Detailed Overview of the Book An Introductory Example: The Inverse Stefan Problem The Mechanical and Mathematical Settings The Inverse Problem Setting The Eulerian Derivative and the Transverse Field The Eulerian Material Derivative of the State The Eulerian Partial Derivative of the State The Adjoint State and the Adjoint Transverse Field Weak Evolution of Sets and Tube Derivatives Introduction Weak Convection of Characteristic Functions Tube Evolution in the Context of Optimization Problems Tube Derivative Concepts A First Example: Optimal Trajectory Problem Shape Differential Equation and Level Set Formulation Introduction Classical Shape Differential Equation Setting The Shape Control Problem The Asymptotic Behavior Shape Differential Equation for the Laplace Equation Shape Differential Equation in Rd+1 The Level Set Formulation Dynamical Shape Control of the Navier-Stokes Equations Introduction Problem Statement Elements of Noncylindrical Shape Calculus Elements of Tangential Calculus State Derivative Strategy Min-Max and Function Space Parameterization Min-Max and Function Space Embedding Conclusion Tube Derivative in a Lagrangian Setting Introduction Evolution Maps Navier-Stokes Equations in Moving Domain Sensitivity Analysis for a Simple Fluid-Solid Interaction System Introduction Mathematical Settings Well-Posedness of the Coupled System Inverse Problem Settings KKT Optimality Conditions Conclusion Sensitivity Analysis for a General Fluid-Structure Interaction System Introduction Mechanical Problem and Main Result KKT Optimality Conditions Appendix A: Functional Spaces and Regularity of Domains Appendix B: Distribution Spaces Appendix C: The Fourier Transform Appendix D: Sobolev Spaces References Index

Approximation of Nonlinear Problems Associated with Radiating Bodies in Space

This paper presents an efficient and reliable method to approximate the solution of nonlinear boundary value problems describing the temperature distribution in a radiating body in space. A variational formulation is constructed for an arbitrary radiation law in

Uniform stability in structural acoustic models with flexible curved walls

T^p

Control of Moving Domains, Shape Stabilization and Variational Tube Formulations

,

Hidden Boundary Shape Derivative for the Solution to Maxwell Equations and Non Cylindrical Wave Equations

p \geqq 1

Galerkin Strategy for Level Set Shape Analysis: Application to Geodesic Tube

an integer. The minimizing element of an appropriate functional is shown to coincide with the solution of the initial problem. The solution is approximated by the finite element method. Nonlinear programming techniques are discussed and a new fast and stable iterative method is presented.

ON THE CONTROLLED EVOLUTION OF LEVEL SETS AND LIKE METHODS: THE SHAPE AND CONTRAST RECONSTRUCTION

Abstract The aim of this paper is twofold. First, we develop an explicit extension of the Kirchhoff model for thin shells, based on the model developed by Michel Delfour and Jean-Paul Zolesio. This model relies heavily on the oriented distance function which describes the geometry. Once this model is established, we investigate the uniform stability of a structural acoustic model with structural damping. The result no longer requires that the active wall be a plate. It can be virtually any shell, provided that the shell is thin enough to accommodate the curvatures.

Oriented Distance Functions in Shape Analysis and Optimization

This paper deals with the control of a moving dynamical domain in which a non cylindrical dynamical boundary value problem is considered. We consider weak Eulerian evolution of domains through the convection of a measurable set by (non necessarily smooth) vector field V. We introduce the concept of tubes by “product space” and we show a closure result leading to existence results for a variational shape principle. We illustrate this by new results: heat equation and wave equation in moving domains with various boundary conditions and also the geodesic characterisation for two Eulerian shape metrics leading to the Euler equation through the transverse field considerations. We consider the non linear Hamilton-Jacobi like equation associated with level set parametrization of the moving domain and give new existence result of possible topological change in finite time in the solution.

Complete Shape Metric and Geodesic

This paper deals with the shape derivative of boundary shape functionals governed by electromagnetic 3D time-depending Maxwell equations solution E, H (or non cylindrical wave equation to begin with) involving derivatives terms at the boundary whose existence are related to hiddenlike regularity results. Under weak regularity of data we compute the shape derivative of L 2 norms at the boundary of the electrical field E and magnetic field H. solution to the 3D Maxwell system. This analysis is obtained via MinMax parameter derivative under saddle point existence, so that no shape sensitivity analysis of the solution E, H is needed, see [1]. The results are completely new concerning the non-cylindrical wave equation as well as for Maxwell equations. Of course this technic is true for all classical shape derivative of quasi convex functionals governed by classical linear problems such as elliptic or parabolic problems (including elasticity). For these situations the result as been developed in many former papers after [12] and later the book [11]. So that the strong material and shape derivative of the “state” is still necessary only for non linear problems for which the Langrangian is definitively not convex-concave, then we use the weak form of the Implicit function theorem (in order to deal with the minimum regularity), see [14] R. Dziri and J.P. Zolesio, but there is still hope to extend this analysis to a larger class of “local saddle points”.