Jean Roche

University of Lorraine | UdL · IEC - Institut Elie Cartan

A numerical method is proposed for the solution of the inverse scattering problem. This problem consists of determining the location and shape of an unknown number of inclusions composed by a homogeneous material with known mechanical properties different that those of the surrounding medium. The information available to solve the inverse problem a...

A new gas radiation model based on the principle of weighted-sum-of-gray-gases (WSGG) is proposed. It is shown that expressing the weighting factors as linear functions of the temperature is more suitable than the fourth-order polynomial dependence to the temperature that is commonly adopted in the WSGG model. Using an efficient inverse algorithm,...

Purpose In this paper the authors review the recent numerical techniques proposed to solve the forward and inverse problems concerning the electromagnetic casting and electromagnetic levitation techniques of the metallurgical industry. In addition, the authors present a new topology optimization method to solve the inverse axisymmetric electromagne...

This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if...

A new method based on topology optimization is proposed for solving a three-dimensional axisymmetric inverse problem regarding the configuration of electric currents used in the electromagnetic casting technique of the metallurgical industry. These electric currents must generate an electromagnetic field such that when certain mass of liquid metal...

We present a new feasible direction interior point algorithm to compute a numerical solution of a normalized Generalized Nash Equilibrium Problem, (GNEP). The GNEP is an extension of the Nash Equilibrium Problem, (NEP). This one is an equilibrium problem that involves two or more players, each player is associated with a feasible strategy set and a...

A new method is proposed for solving the equilibrium problem related to the electromagnetic casting technique. The method is appropriate for the case when part of the metal is in solid state. It is shown that in the presence of solid inclusions the equilibrium equation of the free liquid boundary can be formulated as a complementarity equation. A N...

This article deals with a finite dimensional reduced order state observer for a class of nonlinear partial differential equations (PDEs) described by a radiative transfer equation (RTE) coupled with a nonlinear heat equation (NHE) in two-dimensional domains. First, the original plant is approximated by an N-dimensional ordinary differential equatio...

This paper deals with the convergence of numerical scheme for combined nonlinear radiation–conduction heat transfer system in a gray, absorbing and non-scattering two-dimensional medium. The radiative transfer equation is solved using a Discontinuous Galerkin method with upwind fluxes. The conductive equation is discretized using the finite element...

In this paper we deal with an inverse electromagnetic casting problem, which consists in designing a set of inductors in such a way that a liquid metal achieves a given shape. The inductors are assumed to be made of single solid-core wires with a negligible cross-section area. The inverse problem is rewritten in the form of an optimization problem....

A new reconstruction algorithm for fluorescence diffuse optical tomography of biological tissues is proposed. The radiative transport equation in the frequency domain is used to model light propagation. The adjoint method studied in this work provides an efficient way for solving the inverse problem. The methodology is applied to a 2D tissue-like p...

This paper deals with existence and uniqueness results for a transient nonlinear radiative–conductive system in three-dimensional case. This system describes the heat transfer for a grey, semi-transparent and non-scattering medium with general boundary conditions. We reformulate the full transient state system as a fixed-point problem. The existenc...

The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the Ndimensional case, N ≥ 1, and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm...

This paper deals with nonlinear smooth optimization problems with equality and inequality constraints, as well as semidefinite constraints on nonlinear symmetric matrix-valued functions. A new semidefinite programming algorithm that takes advantage of the structure of the matrix constraints is presented. This one is relevant in applications where t...

We consider a model for the propagation and absorption of electromagnetic waves (in the time-harmonic regime) in a magnetised plasma. We present a rigorous derivation of the model and several boundary conditions modelling wave injection into the plasma. Then we propose several variational formulations, mixed and non-mixed, and prove their well-pose...

This contribution deals with state observer design for a class of non linear coupled PDE that describe radiative-conductive heat transfer systems. This approach uses first a stable spatial discretization technique that is the Galerkin method to obtain a large scale but finite dimensional system in a suitable form. Thanks to the special structure of...

We present a new algorithm for nonlinear semidefinite programming, based on the iterative solution in the primal and dual variables of Karush-Kuhn-Tucker optimality conditions, which generates a feasible decreasing sequence. At each iteration, two linear systems with the same matrix are solved to compute a feasible descent direction and then an ine...

We propose a new iterative method for the topology design of the inductors in electromagnetic casting. The method is based on a level-set representation of the solution together with first and second order topological derivatives. The optimal design is found by minimizing a Kohn–Vogelius-type functional for the problem. The complete topological exp...

This contribution deals with state observer design for a class of PDE non linear systems described by Radia-tive transfer equation (RTE) coupled with nonlinear heat equation (NHE) in two dimensional domain. Observations are made though sensors placed at the upper boundary of the two dimensional domain. We explored the Galerkin method for a semi-dis...

A new optimization method is proposed for solving an inverse problem concerning the shape and topology of the inductors used in the electromagnetic casting technique of the metallurgical industry. The method is based on an sparse convex quadratic programming version of a recently proposed topology optimization formulation of the inverse electromagn...

In this paper, we prove the convergence of a numerical scheme for one-dimensional coupled system of nonlinear partial and ordinary integro-differential equations. This system describes the steady-state coupled radiative-conductive heat transfer for a non-grey anisotropically absorbing, emitting and scattering medium, with axial symmetry and nonhomo...

The industrial technique of electromagnetic casting allows for contactless heating, shaping and controlling of chemical aggressive, hot melts. The inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given m...

In this communication we give a result of existence and present a numerical analysis of weak solutions for the following quasi-linear elliptic problem in one and two dimensions: \begin{equation} \;\;\;\;\;\left\{ \begin{array}{l} -{{A} }u(x)+G(x,{{D}} u(x))=F(x,u(x))+f(x)\hbox{\ \ in }\Omega\,, \\ u(x)=0\hbox{\ \ on }\partial \Omega \end{array} \ri...

The inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape. In this paper we propose a new method for the topology design of such inductors. The inverse el...

The inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape. In this paper we propose a new method for the topology design of such inductors. The inverse el...

In this paper we present an algorithm for inverse optimization problems concerning electromagnetic casting of molten metals. We are interested in locating suitable inductors around the molten metal so that the equilibrium shape be as near as possible to a desired target shape. A Simultaneous Analysis and Design (SAND) mathematical programming formu...

We propose a result of local existence and uniqueness of a mild solution to the one-dimensional Vlasov–Poisson system. We establish the result for an initial condition lying in the space W1,1(ℝ2), then we extend it to initial conditions lying in the space BV(ℝ2), without any assumption of continuity, boundedness or compact support. Copyright © 2010...

The design of inductors in electromagnetic shaping of molten metals consists in looking for the position and the shape of a set of electric wires such that the induced electromagnetic field makes a given mass of liquid metal acquire a predefined shape. In this paper we formulate an inverse optimization problem where the position and shape of the in...

In the continuation of the works led in cylindrical geometry [2], a full toroidal description for an arbitrary poloidal cross-section of the plasma has been developed. For simulation purpose a mixed augmented variational formulation (MAVF), which is particularly well suited for solving Maxwell equations, is considered [4]. The discretization of the...

Topological Derivatives for Semilinear Elliptic Equations The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L∞ norm are obtained. The results of numerical...

The inverse problem concerning electromagnetic casting of molten metals consists of looking for an electric current density distribution such that the induced electromagnetic field makes a given mass of liquid metal acquire a predefined shape. This problem is formulated here as an optimization problem where the positions of a finite set of inductor...

In this paper we show the existence of weak solutions for a periodic equation with critical growth non linearity with respect to the gradient. A numerical algorithm to compute a numerical approximation of the weak solution is described. The algorithm is based on the Schwarz overlapping domain decomposition method, combined with finite element metho...

We propose a result of local existence and uniqueness of a mild solution to the one-dimensional Vlasov-Poisson system. We establish the result for an initial condition lying in the Sobolev space of integrable functions with integrable derivatives, then we extend it to initial conditions lying in the space of functions of bounded variation, without...

To understand the nickel-iron electrodeposition process better, we have developed a one-dimensional numerical model. This model addresses dissociation, diffusion, electromigration, convection and deposition of multiple ion species. The reaction mechanism in this model differs in that Ni(2+) and Fe(2+) are the electroactive species and NiOH(+) and F...

Electromagnetic Casting (EMC) and Magnetic Suspension Melt Processing (MSMP) are very important technologies in the metallurgical industry. They make use of an electromagnetic field for contactless heating, shaping and control of solidification of hot melts. Advantages of these techniques are high surface quality, high cleanness, low contamination...

The aim of this paper is to present an algorithm for inverse optimization problems concerning electromagnetic casting of molten metals. We are interested in locating suitable inductors around the molten metal so that the equilibrium shape be as near as possible to a desired one. In this paper a Simultaneous Analysis and Design (SAND) mathematical p...

This paper deals with a coupled system of non-linear elliptic differential equations arising in electrodeposition modelling process. We show the existence and uniqueness of the solution. A numerical algorithm to compute an approximation of the weak solution is described. We introduce a domain decomposition method to take in account the anisotropy o...

The aim of this paper is to show the existence and present a numerical analysis of weak solutions for a quasi-linear elliptic problem with Dirichlet boundary conditions in a domain Ω and data belonging to L 1(Ω). A numerical algorithm to compute a numerical approximation of the weak solution is described and analyzed. Numerical examples are present...

To understand the nickel iron electrodeposition process, we have developed an one dimensional numerical method. This model addresses dissociation, diffusion, electromigration, convection and deposition of multiple ion species. We consider a system of steady-state transport equations of five dissolved species with simultaneous homogeneous reactions...

The principal objective of this work is to introduce an adaptive strategy to monitor the convergence rate of a Newton-like method in shape optimization. Shape optimization problems are characterized by a cost function and a partial differential equation (P.D.E.) which depend both of the geometrical domain. Typically we want to compute a shape * suc...

In this paper we show the existence of weak solutions for a nonlinear elliptic equations with arbitrary growth of the non linearity and data measure. A numerical algorithm to compute a numerical approximation of the weak solution is discribed and analysed. In a first step a super-solution is computed using a domain decomposition method. Numerical e...

In this paper, we introduce a new PIC method based on an adaptive multi-resolution scheme for solving the one dimensional Vlasov–Poisson equation. Our approach is based on a description of the solution by particles of unit weight and on a reconstruction of the density at each time step of the numerical scheme by an adaptive wavelet technique: the d...

The aim of this work is to introduce an adaptive strategy to monitor the rate of convergence of a Newtonlike method in shape optimization. Numerical solution of this problem involves numerical representation of the domain, optimization algorithms and numerical solution of the state equation, a partial di#erential equation. Newtons like algorithms a...

Transient radiative and conductive heat transfer in a fibrous medium with anisotropic optical properties is investigated. Two different kinds of boundary conditions are treated: when the temperatures imposed on the boundaries vary with time and when the medium is subject to a radiation source which varies with time. A one dimensional case is consid...

We show the existence of weak solutions for a nonlinear elliptic equations with arbitrary growth of the non linearity and data measure. A numerical algorithm to compute a numerical approximation of the weak solution is discribed and analysed. In a first step a super-solution is computed using a domain decomposition method. A numerical example is pr...

In this paper, we prove the existence, uniqueness and regularity results for a one-dimensional coupled system of nonlinear partial and ordinary integro-differential equations. This system describes the steady-state coupled radiative–conductive heat transfer for a non-grey anisotropically absorbing, emitting and scattering medium, with axial symmetr...

A numerical method for solving a system of partial differential equations modelling steady-state coupled radiative-conductive heat transfer in semi-transparent media is proposed. The radiative transfer equation is coupled with a nonlinear heat conduction equation. A simulation on a real insulator composed of silica fibers is illustrated.

Coupled radiative and conductive heat transfer in a fibrous medium formed by silica fibres is investigated in this paper by not taking account of the axial symmetry for the distribution of fibres or the boundary conditions. Radiative properties of the medium are calculated by using the Mie theory. The model obtained depends only on optical paramete...

A finite difference solution for a system of non-linear integro–differential equations modelling the steady-state combined radiative–conductive heat transfer is proposed. A new backward–forward finite difference scheme is formulated for the Radiative Transfer Equation. The non-linear heat conduction equation is solved using the Kirchhoff transforma...

A new way of solving the steady-state coupled radiative-conductive problem in semi-transparent media is proposed. An angular discretization technique is applied in order to express the radiative transfer equation (RTE) in an inhomogeneous system of linear differential equations associated with Dirichlet boundary conditions. The system is solved by...

Our goal is to introduce a Newton method in computing the stationary points of a total energy with respect to the shape. We formulated a precise description of the second order shape derivative. It is given by a symmetrical boundary integral operator, useful for numerical calculations. This method is applied to a particular shape optimisation probl...

We study the heat diffusion in a domain with an ob- stacle inside. More precisely, we are interested in the quantity of heat in so far as a function of the position of the heat source at time 0. This quantity is also equal to the expectation of the sojourn time of the Brownian motion, reflected on the boundary of a small disk contained in the unit...

.-- We describe a Newton method applied to the evaluation of a critical point of a total energy associated to a shape optimization problem. The key point of these methods is the Hessian of the shape functional. We give an expression of the Hessian as well as the relation with the secondorder Eulerian semi-derivative. An application to the electroma...

The numerical resolution of kinetic equations and, in particular, of Vlasov-type equations is performed most of the time using particle in cell methods which consist in describing the time evolution of the equation through a finite number of particles which follow the characteristic curves of the equation, the interaction with the external and self...

Our goal is to introduce a Newton method to compute the stationary points of a total energy with respect to the shape. We produce a precise description of the second order shape derivative which is given by a symmetrical boundary integral operator, useful for numerical calculations. We apply this method to a particular shape optimization problem, t...

The goal of this paper is to study the differences between two approaches to gradient computation in shape optimization. Simulation methods in shape optimization require computation of a local minimum of an energy function by descent methods, which involves the gradient computation with respect to shape perturbations. Two different approaches are m...

The paper concerns the class of shape optimization problems for linear partial differential equations. A small set inside the domain of integration of an elliptic equation is identified by minimization of an integral cost functional. In the two-dimensional case an existence result for the problem is given. The material derivative method of shape op...

We describe a Newton method applied to the evaluation of a critical point of a total energy associated to a shape optimization problem. The key point of this methods is the Hessian of the shape functional. We give an expression of the Hessian as well as the relation with the second-order Eulerian semi-derivative. An application to the electromagnet...

We describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals. We use an energetic variational formulation and optimization techniques to compute, a critical point. The surfaces are represented by piecewise linear finite elements. Each step of the algorithm requires solving an elliptic boundary v...

We describe a numerical method to compute free surfaces in the electromagnetic shaping of liquid metals. It is based on an energetic variational formulation for the equilibrium so that optimization algorithms of a quasi-Newton type are used like the so-called BFGS method. The elliptic problem in an exterior domain involved in the computation of the...