Asymptotic behaviour of the inductance coefficient for thin conductors

We study the asymptotic behaviour of the inductance coefficient for a thin toroidal inductor whose thickness depends on a small parameter \eps>0. We give an explicit form of the singular part of the corresponding potential u\ue which allows to construct the limit potential $u$ (as \eps\to 0) and an approximation of the inductance coefficient L\ue. We establish some estimates of the deviation u\ue-u and of the error of approximation of the inductance. We show that L\ue behaves asymptotically as \ln\eps, when \eps\to 0

An Object Oriented Finite Element Toolkit

We present an object oriented finite element library written in C++. We outline the main motivations in developing such a library. Through a simple example program we show a typical use of the library. We describe the main class categories and typical problems to solve using the library

A Two-dimensional eddy current model using thin inductors

International audienceWe derive a mathematical model for eddy currents in two dimensional geometries where the conductors are thin domains. We assume that the current flows in the $x_3$-direction and the inductors are domains with small diameters of order $O(\epsilon)$. The model is derived by taking the limit $\epsilon\to 0$. A convergence rate of $O(\epsilon^\alpha)$ with $0<\alpha<1/2$ in the $L^2$--norm is shown as well as weak convergence in the $W^{1,p}$ spaces for $1< p <2$

Asymptotic behaviour of the inductance coefficient for thin conductors

We study the asymptotic behaviour of the inductance coefficient for a thin toroidal inductor whose thickness depends on a small parameter \eps>0. We give an explicit form of the singular part of the corresponding potential u\ue which allows to construct the limit potential $u$ (as \eps\to 0) and an approximation of the inductance coefficient L\ue. We establish some estimates of the deviation u\ue-u and of the error of approximation of the inductance. We show that L\ue behaves asymptotically as \ln\eps, when \eps\to 0

Numerical simulations of thermally stratified flows in an open domain

We present two dimensional numerical simulations of circulation induced by a heat island in an unbounded domain. The °ow is thermally stratified in the vertical direction. Boussinesq equations in a closed domain are used to describe the flow variables. For this problem, very elongated computational domains have to be used in order to obtain accurate solutions. A term, whose effect is to smoothly damp the convective terms in a layer close to the vertical boundaries, is introduced in the temperature equation. Therefore, shorter domains can be considered. This method is investigated through the numerical simulations of stationary solutions at Rayleigh number $Ra = 10^5$ and $2.5x10^5$. Time periodic solutions at $Ra = 5x10^5$ and $10^6$ are also reported and analyzed

An accurate finite element method for elliptic interface problems

A finite element method for elliptic problems with discontinuous coefficients is presented. The discontinuity is assumed to take place along a closed smooth curve. The proposed method allows to deal with meshes that are not adapted to the discontinuity line. The (nonconforming) finite element space is enriched with local basis functions. We prove an optimal convergence rate in the $H^1$--norm. Numerical tests confirm the theoretical results

Self-inductance coefficient for toroidal thin conductors

We consider the inductance coefficient for a thin toroidal inductor whose thickness depends on a small parameter \eps>0. An explicit form of the singular part of the corresponding potential u\ue is given. This allows to construct the limit potential $u$ (as \eps\to 0) and an approximation of the inductance coefficient L\ue. We establish some estimates of the deviation u\ue-u and of the error of approximation of the inductance. The main result shows that L\ue behaves asymptotically as \ln\eps, when \eps\to 0

A perturbation method for the numerical solution of the Bernoulli problem

International audienceWe consider the numerical solution of the free boundary Bernoulli problem by employing level set formulations. Using a perturbation technique, we derive a second order method that leads to a fast iteration solver. The iteration procedure is adapted in order to work in the case of topology changes. Various numerical experiments confirm the efficiency of the derived numerical method