Gabriel Caloz

Numerical analysis for nonlinear and bifurcation problems

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 …

Integral method for numerical simulation of MRI artifacts induced by metallic implants

Numerical simulation is a valuable tool for the study of magnetic susceptibility artifacts from metallic implants. A major difficulty in the simulation lies in the computation of the magnetic field induced by the metallic implant. A new method has been designed and implemented to compute the magnetic field induced by metallic objects of arbitrary shape. The magnetic field is expressed pointwise in terms of a surface integral. Efficient quadrature schemes are proposed to evaluate this integral. Finally, the method is linked to an artifact reconstruction model to simulate the images. Magn Reson Med 45:724–727, 2001.

Magnetic susceptibility artifacts in magnetic resonance imaging: calculation of the magnetic field disturbances

In magnetic resonance imaging, inhomogeneities of the static magnetic field lead to perturbations in the resulting images, called artifacts. The authors goal is to compute numerically the disturbances induced by a material having magnetic properties different from that of the surrounding tissues. The method is linked to an artifact reconstruction model to get simulated images. This model needs very accurate results in a fine three-dimensional grid around the implant. The use of standard methods would lead to solve very large systems. The authors method is based on a surface integral representation of the magnetic field. To implement it in an efficient way, an analytical expression is derived when the boundary of the domain can be meshed in flat panels. For curved surfaces a numerical quadrature scheme is implemented.

Mathematical Modeling and Numerical Simulation of Magnetic Susceptibility Artifacts in Magnetic Resonance Imaging.

Abstract The technique used to recognise information in Magnetic Resonance Imaging (MRI) is based on electromagnetic fields. A linearly varying field (around 10−2 Tesla per meter) is added to a strong homogeneous magnetic field (order of magnitude of approximately one Tesla). When these fields are disturbed by the presence of a paramagnetic material, in the sample for instance, the resulting image is usually distorted, these distortions being termed artifacts. Our goal is to present a method, assuming the field disturbances are known, to construct the resulting images. A mathematical model of the MRI process is developed. The way the images are distorted in intensity and shape is explained and an algorithm to simulate magnetic susceptibility artifacts is deduced.

Magnetostatic field computations based on the coupling of finite-element and integral representation methods

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.

Induced magnetic field computations using a boundary integral formulation

A method to compute the magnetic field induced by a metallic body embedded in a uniform external field is presented. It is based on boundary integral representation formulae for the magnetic induction . A computational procedure is proposed which consists of using analytic expressions to compute the integral over the flat panels of the boundary and a piecewise quadratic interpolation of the surface for the curved panels. Superconvergence occurs in the latter case. The method supplies both high accuracy and low computation time, requirements that are not fulfilled when using standard numerical methods.

Cancellation errors in an integral for calculating magnetic field from reduced scalar potential

In computation of magnetostatic fields in regions containing current sources, it is classical to write the corresponding magnetostatic problem in terms of the reduced scalar magnetic potential /spl phi/. Usually numerical differentiation is used to obtain the magnetic field H from the potential values, which implies loss in accuracy. An alternative is to compute H from /spl phi/ by an integral formula. In fact, the formula does not give a straightforward solution because of a cancellation in the integral. In this paper, we investigate the mathematical reason why the formula is not suited for numerical purposes. We carry out a careful numerical analysis with illustrations on a test example and propose a way to circumvent this difficulty by using a sort of decomposition method.