Gabriel Caloz - Academia.edu (original) (raw)

Papers by Gabriel Caloz

International Journal of Numerical Modelling-electronic Networks Devices and Fields, 2007

Practical three-dimensional magnetic field problems usually involve regions containing current so... more Practical three-dimensional magnetic field problems usually involve regions containing current sources as well as regions with magnetic materials. For computational purposes the use of the reduced scalar potential (RSP) as unknown has the advantage to transform a problem for a vector field throughout the space into a problem for a scalar function, thus reducing the number of degrees of freedom in the discretization. However in regions with high magnetic permeability the use of the RSP alone usually results in severe loss in accuracy and it is recommended to use both the reduced scalar potential and the total scalar potential. Using an asymptotic expansion, we investigate theoretically the underlying reasons for this lack of accuracy in permeable regions when using the RSP as a unique potential. Moreover this investigation leads to an efficient numerical method to compute the magnetic field in regions with high magnetic permeability.

arXiv (Cornell University), Oct 6, 2009

HAL (Le Centre pour la Communication Scientifique Directe), 2006

ESAIM: Mathematical Modelling and Numerical Analysis, 1991

Handbook of Numerical Analysis, 1997

PREFACE Computational applications generally involve nonlinear problems and often contain paramet... more PREFACE Computational applications generally involve nonlinear problems and often contain parameters. They may represent properties of the physical system they describe or quantities which can be varied. A basic problem in approximation consists in studying existence and convergence of approximated solutions for a given nonlinear problem, for instance when the parameters are xed. Another problem is to represent the families or manifolds of solutions under variations of some parameters. Apart from a theoretical approach, such representations are computed and continuation methods are concerned with generating the solution manifolds. By varying one parameter, we can follow a path of solutions. Then to study the eeects of change of parameters on a system, it is of prime interest to know the eeects of numerical approximation on its behavior. The goal of this article is to present a general framework in which approximations of nonlinear problems and approximations of solution manifolds can be studied. We will consider regular solutions, regular solution families, and singular solutions. Even though we will illustrate the general theory only with elementary nite element approximations of model boundary value problems, it can be applied to a much wider range of problems in connection with approximation methods. Our presentation is a remodelling of the one proposed by Crouzeix and Rappaz 1989] taking its origin in Descloux and The general problem we will handle and which covers a lot of applications is the following: nd x 2 X such that F(x) = 0 where X and Z are Banach spaces, F : X ! Z is a smooth nonlinear mapping. Of particular interest is the case where the space X has the form R m Y , where R m with m 1 is the parameter space and the Banach space Y is the state space. We will work under the assumption that the derivative of F is a Fredholm operator of index n 0. Both cases n = 0 and n 1 with a surjective derivative are studied separately. Note that when n is positive, the family of solutions to F(x) = 0 is a diierentiable manifold. The singular situation with a not surjective derivative is also studied. In the general setting, the approximation schemes are written in the form F h (x) = 0 where h is a parameter in (0; 1] and F h : X ! Z is an approximation of F. The family fF h g …

We study a transmission problem in high contrast media. The 3D case of the Maxwell equations in h... more We study a transmission problem in high contrast media. The 3D case of the Maxwell equations in harmonic regime is considered. We derive an asymptotic expansion with respect to a small parameter related to high conductivity. This expansion is theoretically justified at any order. Numerical simulations in axisymmetric geometry highlight the skin effect and the expansion accuracy.

We study a three-dimensional model for the skin effect in electromagnetism. The 3-D case of the M... more We study a three-dimensional model for the skin effect in electromagnetism. The 3-D case of the Maxwell equations in harmonic regime on a domain composed of a dielectric and of a highly conducting material is considered. We derive an asymptotic expansion with respect to a small parameter related to high conductivity. This expansion is theoretically justified at any order. The asymptotic expansion and numerical simulations in axisymmetric geometry exhibit the influence of the geometry of the interface on the skin effect.

SIAM Journal on Numerical Analysis, 1994

In this paper the approximate solution of a class of second order elliptic equations with rough c... more In this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered. Problems of the type considered arise in the analysis of unidi- rectional composites, where the coefficients represent the properties of the material. Several methods for this class of problems are presented, and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients. The methods are referred to as special finite element methods because they are of finite element type but employ special shape functions, chosen to accurately model the unknown solution.

Journal of Mathematical Analysis and Applications, 2010

IMA Journal of Numerical Analysis, 1997

. In this paper we extend the so called inf-sup conditions to non linear problems having regular ... more . In this paper we extend the so called inf-sup conditions to non linear problems having regular solution branches. The inf-sup conditions for non linear problems are introduced for instance in Caloz and Rappaz [1994]. Here we present a more general approach to include turning points on a solution branch. Our abstract results are applied to model examples. 1. Introduction Let X, Y be two reflexive Banach spaces and F : X ! Y 0 be of class C p , p 1. Our goal is to construct a framework to study approximations of the problem: find x 2 X such that (1.1) F (x) = 0 or equivalently 8y 2 Y hF (x); yi Y 0 Y = 0: We shall focus on the following type of approximations. Let fX h g 0!h1 be a family of finite dimensional subspaces of X and fY h g 0!h1 be a family of finite dimensional subspaces of Y . Then an approximation of problem (1.1) reads: find x h 2 X h such that (1.2) 8y h 2 Y h hF (x h ); y h i Y 0 Y = 0: This approximation is called a Petrov-Galerkin approximation of (1.1). L...

IMA Journal of Applied Mathematics, 1990

We are interested in the model plasma model −Δu=λu + in Ω, u=−d on ∂Ω, λ∫ Ω u + dx=j, where Ω is ... more We are interested in the model plasma model −Δu=λu + in Ω, u=−d on ∂Ω, λ∫ Ω u + dx=j, where Ω is a bounded domain in R 2 with boundary ∂Ω; here, j is a given positive number, the function u and the positive number λ are the unknows of the problem, and d is a real parameter. Using a variant of the implicit function theorem, we can prove the existence of a global solution branch parametrized by d. The method has the advantage that it can be used for analysing the approximation of the above problem by a finite-element method

IMA Journal of Applied Mathematics, 1998

IMA Journal of Applied Mathematics, 2001

IEEE Transactions on Magnetics, 1996

IEEE Transactions on Magnetics, 2002

We present an original method to compute the magnetic field generated by some electromagnetic dev... more We present an original method to compute the magnetic field generated by some electromagnetic device through the coupling of an integral representation formula and a finite-element method (FEM). The unbounded three-dimensional magnetostatic problem is formulated in terms of the reduced scalar potential. Through an integral representation formula, an equivalent problem is set in a bounded domain and discretized using a standard FEM. As a byproduct, an integral representation formula is proposed to compute the magnetic field in any point of the space from the reduced scalar potential without numerical differentiation.