Olivier Bodart

Asymptotic Approximation of the Solution of the Laplace Equation in a Domain with Highly Oscillating Boundary

We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of whose boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities whose size depends on a small parameter,

Numerical investigation of optimal control of induction heating processes

\varepsilon >0.

Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity

The assumption of sharp asperities is made; that is, the height of the asperities is fixed. Using a boundary layer corrector, we derive and analyze a nonoscillating approximation of the solution at order

A Local Result on Insensitizing Controls for a Semilinear Heat Equation with Nonlinear Boundary Fourier Conditions

{\cal O}(\varepsilon^{3/2})

Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary

for the H1 -norm.

Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity

Abstract We consider optimal control problems arising in induction heating processes. We are mainly concerned with two classes of these processes: uniform heating and metal hardening. The cost functions are chosen according to these classes. The control parameters are the inductor shape (assumed to be thin), the frequency, the current voltage and the heating duration. The induction heating model is a two-dimensional cartesian geometry. The numerical scheme is given as well as numerical experiments for practical applications.

Sheared sheet intrusions as mechanism for lateral flank displacement on basaltic volcanoes: Applications to Réunion Island volcanoes

Abstract In this paper we consider a semilinear heat equation (in a bounded domain Ω of ℝ N ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ⊂ Ω, that insensitizes the L 2 − norm of the observation of the solution in another open subset 𝒪 ⊂ Ω when ω ∩ 𝒪 ≠ ∅, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L r -controls (r large enough) starting from insensitizing controls in L 2.

Numerical approximation of laminar flows over rough walls with sharp asperities

In this paper we present a local result on the existence of insensitizing controls for a semilinear heat equation when nonlinear boundary conditions of the form

Stokes equations with interface condition in an unbounded domain

\partial_n y + f(y) = 0

XFEM-Based Fictitious Domain Method for Linear Elasticity Model with Crack

are considered. The problem leads to an analysis of a special type of nonlinear null controllability problem. A sharp study of the linear case and a later application of an appropriate fixed point argument constitute the scheme of the proof of the main result. The boundary conditions we are dealing with lead us to seek a fixed point, and thus also control functions, in certain Holder spaces. The main strategy in this paper is the construction of controls with Holderian regularity starting from L2-controls in the linear case. Sufficient regularity in the data and appropriate assumptions on the right-hand side term