Stéphane Clain

Numerical modeling in induction heating for axisymmetric geometries

This paper deals with numerical simulation of induction heating for axisymmetric geometries. A mathematical model is presented, together with a numerical scheme based on the Finite Element Method. A numerical simulation code was implemented using the model presented in this paper. A comparison between results given by the code and experimental measurements is provided.

NUMERICAL MODELING OF INDUCTION HEATING FOR TWO-DIMENSIONAL GEOMETRIES

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 and deals with the two different time scales of electromagnetic and thermal phenomena.

Monoslope and multislope MUSCL methods for unstructured meshes

We present new MUSCL techniques associated with cell-centered finite volume method on triangular meshes. The first reconstruction consists in calculating one vectorial slope per control volume by a minimization procedure with respect to a prescribed stability condition. The second technique we propose is based on the computation of three scalar slopes per triangle (one per edge) still respecting some stability condition. The resulting algorithm provides a very simple scheme which is extensible to higher dimensional problems. Numerical approximations have been performed to obtain the convergence order for the advection scalar problem whereas we treat a nonlinear vectorial example, namely the Euler system, to show the capacity of the new MUSCL technique to deal with more complex situations.

Numerical modelling of induction heating of long workpieces

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. >

L ∞ stability of the MUSCL methods

We present a general L∞ stability result for generic finite volume methods coupled with a large class of reconstruction for hyperbolic scalar equations. We show that the stability is obtained if the reconstruction respects two fundamental properties: the convexity property and the sign inversion property. We also introduce a new MUSCL technique named the multislope MUSCL technique based on the approximations of the directional derivatives in contrast to the classical piecewise reconstruction, the so-called monoslope MUSCL technique, based on the gradient reconstruction. We show that under specific constraints we shall detail, the two MUSCL reconstructions satisfy the convexity and sign inversion properties and we prove the L∞ stability.

Numerical modelling of thermal ablation phenomena due to a cathodic spot

A numerical simulation of the ablation problem for the cathode spot that is based on the enthalpy formulation is presented and solved with a finite-element method using the Euler explicit scheme. Vaporization latent heat and ablation phenomena constitute the main difficulties of the cathodic surface erosion. We deduce characteristic information related to the cathode spot such as the energy repartition in three phases and the ablation length for different energy fluxes.

First- and second-order finite volume methods for the one-dimensional nonconservative Euler system

Gas flow in porous media with a nonconstant porosity function provides a nonconservative Euler system. We propose a new class of schemes for such a system for the one-dimensional situations based on nonconservative fluxes preserving the steady-state solutions. We derive a second-order scheme using a splitting of the porosity function into a discontinuous and a regular part where the regular part is treated as a source term while the discontinuous part is treated with the nonconservative fluxes. We then present a classification of all the configurations for the Riemann problem solutions. In particularly, we carefully study the resonant situations when two eigenvalues are superposed. Based on the classification, we deal with the inverse Riemann problem and present algorithms to compute the exact solutions. We finally propose new Sod problems to test the schemes for the resonant situations where numerical simulations are performed to compare with the theoretical solutions. The schemes accuracy (first- and second-order) is evaluated comparing with a nontrivial steady-state solution with the numerical approximation and convergence curves are established.

Numerical Investigations on the Pressure Wave Absorption and the Gas Cooling Interacting in a Porous Filter, During an Internal arc Fault in a Medium-Voltage Cell

A mathematical model and a numerical method have been developed to simulate the mechanical and the thermal physical phenomena in a energy absorber like a porous filter, during an internal arc fault in a medium-voltage apparatus. A 1D gas flow model in porous medium with variable porosity is described. The main point is the introduction of a new numerical scheme to take accurately into account discontinuous variations of the porosity and correctly simulate the creation of the transmitted and reflected waves. Numerical simulations are compared to experimental measurements performed on apparatus specially adapted for tests to provide a better understanding of the physical phenomena involved, for instance, the gas cooling and the shock wave absorption by the porous medium of the filter.

Multi-dimensional modelling of electrostatic force distance curve over dielectric surface: Influence of tip geometry and correlation with experiment

Electric Force-Distance Curves (EFDC) is one of the ways whereby electrical charges trapped at the surface of dielectric materials can be probed. To reach a quantitative analysis of stored charge quantities, measurements using an Atomic Force Microscope (AFM) must go with an appropriate simulation of electrostatic forces at play in the method. This is the objective of this work, where simulation results for the electrostatic force between an AFM sensor and the dielectric surface are presented for different bias voltages on the tip. The aim is to analyse force-distance curves modification induced by electrostatic charges. The sensor is composed by a cantilever supporting a pyramidal tip terminated by a spherical apex. The contribution to force from cantilever is neglected here. A model of force curve has been developed using the Finite Volume Method. The scheme is based on the Polynomial Reconstruction Operator—PRO-scheme. First results of the computation of electrostatic force for different tip–sample distances (from 0 to 600 nm) and for different DC voltages applied to the tip (6 to 20 V) are shown and compared with experimental data in order to validate our approach.

A multislope MUSCL method on unstructured meshes applied to compressible Euler equations for axisymmetric swirling flows

A finite volume method for the numerical solution of axisymmetric inviscid swirling flows is presented. The governing equations of the flow are the axisymmetric compressible Euler equations including swirl (or tangential) velocity. A first-order scheme is introduced where the convective fluxes at cell interfaces are evaluated by the Rusanov or the HLLC numerical flux while the geometric source terms are discretizated to provide a well-balanced scheme i.e. the steady-state solutions with null velocity are preserved. Extension to the second-order space approximation using a multislope MUSCL method is then derived. To test the numerical scheme, a stationary solution of the fluid flow following the radial direction has been established with a zero and nonzero tangential velocity. Numerical and exact solutions are compared for classical Riemann problems where we employ different limiters and effectiveness of the multislope MUSCL scheme is demonstrated for strongly shocked axially symmetric flows like in spherical bubble compression problem. Two other tests with axisymmetric geometries are performed: the supersonic flow in a tube with a cone and the axisymmetric blunt body with a free stream.