Publications by Jacques Rappaz

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A numerical model for the three-dimensional simulation of liquid-gas flows with free surfaces is presented. The incompressible Navier-Stokes equations are assumed to hold in the liquid domain. In the gas domain, the velocity is disregarded, the pressure is ...

A comprehensive micro-macroscopic model of the continuous hardening of 3d-axisymmetric steel components by induction heating has been developed. At the macroscopic scale, the Maxwell and heat flow equations are solved using a mixed numerical formulation : the inductor and the workpiece are enmeshed with finite elements (FE) but boundary elements (BE) are used for the solution of the electromagnetic equations in the ambient air. This method allows the inductor to be moved with respect to the workpiece withou

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stok

A numerical model for the simulation of three-dimensional liquid-gas flow with free surfaces is presented. The incompressible Navier-Stokes equations are assumed to hold in the liquid domain, while the surrounding gas is assumed to be compressible, with co ...

In the last decade, the cell productivity has been dramatically improved as demonstrated on AP30 technology with amperage increase from 350 kA to 440 kA. At the same time, available liquid bath volume in the pots has been drastically reduced by the anode surface enlargement. In order to achieve such impressive step change, management of alumina dissolution is a key point to optimize both cell design and process control algorithms. Rio Tinto Aluminium and EPFL have developed a 3D model taking into account th

A numerical model is presented for the simulation of complex fluid flows with free surfaces in three space dimensions. The model described in Maronnier et al. (J. Comput. Phys. 1999; 155(2):439) is extended to three dimensional situations. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different. A splitting method is used for the time discretization. At each time step, two advection problems-one for the predicted velocity f

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non-singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints ...

We present both a mathematical model and a numerical method for simulating induction heating processes. The geometry of the conductors is cylindrical and the magnetic field is assumed to be parallel to the invariance axis. The model equations have current tension as prescribed data rather than current intensity. In Particular, the formulation of the electromagnetic problem uses the magnetic field as the unknown function. The numerical method takes into account the time periodicity of the prescribed tension

A proof of existence is given for a stationary model of alloy solidification. The system is composed of heat equation, solute equation and Navier-Stokes equations. In rite latter Carman-Kozeny penalization of porous medium models the mushy zone. The problem is first regularized and a sequence of regularized solutions is built thanks to Leray-Schauder's fixed point Theorem. A solution is then extracted by compactness argument.

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discre ...

In an abstract framework similar to the one developed in Crouzeix-Rappaz [2], we present a formalism which makes precise the notions of consistency and stability for the finite element approximation of nonlinear elliptic problems. The consistency is connected to the approximation by interpolation, whereas the stability follows from discrete inf-sup conditions of the linearized problem. Moreover, contrary to the method developed in [2], this formalism does not require us to invert the principal part of the o

A system of partial differential equations describing the thermal behavior of aluminium cell coupled with magnetohydrodynamic effects is numerically solved. The thermal model is considered its a two-phases Stefan problem which consists of a non-linear conv ...

We investigate in this paper a numerical method to control the temperature of an induction heating process through electric current parameters such as frequency and current voltage imposed to the terminals of the inductor. We seek a sinusoidal voltage function whose amplitude varies in time. For this, we use a two-dimensional induction heating model. that involves a phase transition in which the magnetic field has only one nonvanishing component. The procedure can be extended to more sophisticated models wi

We consider in this paper an induction heating process. A mathematical model is presented, together with numerical methods used in order to describe the magnetic field, as well as the temperature field evolution. Experimental measurements were performed in order to validate the numerical simulation results. A comparison is presented for both ferromagnetic and non-ferromagnetic materials. An error discussion is provided.

We consider a nonlinear elliptic-parabolic system which stems from the modelling of heat induction processes in which a sinusoidal electromagnetic field is applied at the boundary of the inductor. The magnetic field satisfies an elliptic equation whereas the temperature equation is parabolic. These two equations are coupled due to the Joule effect and to the conductivity dependence on temperature. A Galerkin method for our problem is analyzed and shown to yield error estimates in L-2 and H-1.

In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in an aluminum production cell as well as its numerical approximation by a finite element method. We put the emphasis on the magnetic effects which live in the whole three dimensional space and which are solved numerically with a domain decomposition method.

A mixture model to take into account the flow of small carbon dioxide bubbles dissolved in a liquid is presented. The model describes the evolution of the velocity fields (mixture and gas), the pressure and the volume fraction of gas. The system of equatio ...

The time-dependent Stokes problem is solved using continuous, piecewise linear finite elements and a classical stabilization procedure. Four order-one methods are proposed for the time discretization. The first one is nothing but the Euler backward scheme and requires a large linear system involving the velocity and pressure unknowns to be solved. The other three schemes allow velocity and pressure computations to be decoupled, namely the pressure-matrix method, a method based on an inexact LU factorization

The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear elliptic problems arising in glaciology. The stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in a first two-dimensional approximation, the so-called infinite parallel sided slab model. The approximation of this model is made by a finite element method with piecewise polynomial functions of degree 1. Numerical results show that t

Jacques Rappaz

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Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas

Modelling of electromagnetic heating, cooling and phase transformations during surface hardening of steels

Numerical simulation of free surface flows

Numerical simulation of two-phase flow with interface tracking by adaptive Eulerian grid subdivision

Alumina Dissolution Modeling in Aluminium Electrolysis Cell Considering MHD Driven Convection and Thermal Impact

Numerical simulation of three-dimensional free surface flows

Numerical analysis of a non-singular boundary integral method: Part I. The circular case

Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

Numerical Modeling of Induction-Heating for 2-Dimensional Geometries

Existence for a Stationary Model of Binary Alloy Solidification

Finite element approximation of multi-scale elliptic problems using patches of elements

Consistency, Stability, a-Priori and a-Posteriori Errors for Nonlinear Problems

Numerical simulation of thermal problems coupled with magnetohydrodynamic effects in aluminium cell

Optimal control of an induction heating process for thixoforming

Numerical Modeling of Induction-Heating of Long Workpieces

A quasi-static two-dimensional induction heating problem II. Numerical analysis

Some Mathematical and Numerical Aspects in Aluminum Production

A mixture model for a dilute dispersion of gas in a liquid flow: Mathematical analysis and application to aluminium electrolysis

Stability of time-splitting schemes for the Stokes problem with stabilized finite elements

Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology