Analysis of a FEM–BEM model posed on the conducting domain for the time-dependent eddy current problem

The three-dimensional eddy current time-dependent problem is considered. We formulate it in terms of two variables, one lying only on the conducting domain and the other on its boundary. We combine finite elements (FEM) and boundary elements (BEM) to obtain a FEM–BEM coupled variational formulation. We establish the existence and uniqueness of the solution in the continuous and the fully discrete case. Finally, we investigate the convergence order of the fully discrete scheme.     

一类带有铁磁材料参数的非线性涡流问题的A-φ有限元法

针对带有铁磁材料的非线性涡流问题,其非线性性通常体现在磁场强度和磁感应强度的关系上.本文提出了一种全离散的有限元A-φ格式,分别在时间和空间上采用向后欧拉公式以及节点有限元进行离散.首先,在合适的函数空间里给出时间上的半离散格式,通过考察其弱形式建立相应的适定性理论,并证明近似解收敛于弱解.其次,给出全离散格式并讨论其误差估计.最后,给出两个数值算例以验证理论结果.     

A transient eddy current problem on a moving domain. Numerical analysis

The aim of this paper is to introduce and analyze a numerical method to solve a transient eddy current problem which arises from the modeling of electromagnetic forming in the axisymmetric case. The resulting problem is degenerate parabolic with the time derivative acting on a moving subdomain. This paper is the sequel of Bermúdez et al. (SIAM J. Math. Anal. 45, 3629–3650, 2013), where a weak formulation of this problem was proved to be well posed and additional regularity of the solution was also established. In the present paper, we propose a finite element method in space combined with a backward Euler time scheme for its numerical solution. We obtain error estimates and report numerical results which allow us to assess the performance of the proposed method.     

Analytical and numerical investigations of a batch crystallization model

This article is concerned with the analytical and numerical investigations of a one-dimensional population balance model for batch crystallization processes. We start with a one-dimensional batch crystallization model and prove the local existence and uniqueness of the solution of this model. For this purpose Laplace transformation is used as a basic tool. A semi-discrete high resolution finite volume scheme is proposed for the numerical solution of the current model. The issues of positivity (monotonicity), consistency, stability and convergence of the proposed scheme for the current model are analyzed and proved. Finally, we give a numerical test problem. The numerical results of the proposed high resolution scheme are compared with the solution of the reduced four-moments model and the first-order upwind scheme.     

Solution of the 2-D eddy current problem via the domain decomposition methods on serial and parallel computers

Domain decomposition algorithms are applied to the solution of a time harmonic two-dimensional eddy current problem. The system of differential equations describing this problem is considered as a singularly perturbed problem. An iterative domain decomposition algorithm suitable for parallelization is described, and convergence of this algorithm is established. The implementation on a shared memory multiprocessor is described, and numerical experiments are presented.     

Results for a turbulent system with unbounded viscosities: Weak formulations,existence of solutions,boundedness and smoothness

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solution are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution.     

Existence of a solution for a thermoelectric model with several phase changes and a Carathéodory thermal conductivity

We present an existence result for the time domain eddy current model for electromagnetic field, coupled with a Stefan problem for temperature. We have extended the existence result proved by Bermúdez et al. (2005) [40] to the case of materials with several phase changes and thermal conductivity depending on both position and temperature.The new proof starts from a totally implicit time discretization of a truncated system. The thermal part can be rewritten as a variational inequality of the second kind. Then, a priori estimates independent of the truncation parameter are obtained for the solution of the truncated problem, using a technique that adapts the method of Boccardo and Gallouët (1989)  [28] to the case of materials with several phase changes.     

Numerical analysis of the electric field formulation of an eddy current problem

In this paper we analyze a finite element method for the numerical solution of an eddy current problem in a bounded conducting domain. We use a weak formulation in terms of the electric field and impose non-local non-standard boundary conditions. The unique data are the input current intensities which are imposed by means of some special curves lying on the boundary of the domain. Optimal error estimates are shown and implementation issues are discussed. To cite this article: A. Bermúdez et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A combined boundary value and transmission problem arising from the calculation of eddy currents: Well-posedness and numerical treatment

The problem of calculating eddy current losses is formulated as a boundary value and transmission problem for a complex Helmholtz equation and Laplace's equation. By deriving an equivalent system of integral equations, we show the existence of a unique solution depending continuously on the data. Furthermore, we propose a numerical method based on decomposing the problem into two boundary value problems, where part of the boundary values have to be determined by a system of linear equations.

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Zusammenfassung Das Problem der Berechnung von Wirbelstromverlusten wird als Randwert- und Übergangsproblem für eine komplexe Helmholtz- und die Laplacegleichung formuliert. Wir führen dieses Problem in ein äquivalentes Integralgleichungssystem über und zeigen die Existenz einer eindeutigen, stetig von den Daten abhängenden Lösung. Weiter schlagen wir eine numerische Lösungsmethode vor, die auf der Dekomposition des Problems in zwei Randwertprobleme beruht, wobei ein Teil der Randwerte durch ein lineares Gleichungssystem zu bestimmen ist.

     

Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition

In this study, we consider a coefficient problem of a quasi-linear two-dimensional parabolic inverse problem with periodic boundary and integral over determination conditions. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also, we consider numerical solution for this inverse problem by using linearization and the implicit finite-difference scheme.     

B-spline finite-element method in polar coordinates

A numerical solution method for two-dimensional electromagnetic field problems is presented using the B-spline finite-element expression based on polar coordinates. The technique has two main advantages: (1) to avoid the truncation errors at some curved boundaries and (2) to improve the accuracy of singular boundary-value problems with a sharp corner. The B-spline finite-element formulation in polar coordinates is derived and its numerical applications are illustrated by an eddy current problem and several waveguide eigenvalue problems.     

Existence and smoothness of continuous and discrete solutions of a two-dimensional shallow water problem over movable beds

We consider a two-dimensional shallow water system over movable beds. We begin with a continuous system and prove the existence of the solutions, and then we investigate their smoothness. Then, we employ a Galerkin method to obtain a finite-dimensional problem which is solved using a Brouwer fixed point theorem. Therefore, we show that the limits of the resulting solution sequences satisfy the model equations.After solving the continuous problem, we focus on the corresponding discrete problem. We employ a local discontinuous Galerkin scheme for numerical solution of the discrete system and conduct an error analysis of the numerical scheme. We prove that the method is convergent and that the error is bounded according to a specific norm defined herein.     

Solution of elliptic optimal control problem with pointwise and non-local state constraints

We study an optimal control problem of a system governed by a linear elliptic equation, with pointwise control constraints and pointwise and non-local (integral) state constraints. We construct a finite-difference approximation of the problem, we prove the existence and the convergence of the approximate solutions to the exact solution. We construct and study mesh saddle point problem and its iterative solution method and analyze the results of numerical experiments.     

A frictionless contact problem for elastic–viscoplastic materials with normal compliance: Numerical analysis and computational experiments

Summary. In this paper we consider a frictionless contact problem between an elastic–viscoplastic body and an obstacle. The process is assumed to be quasistatic and the contact is modeled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using strongly monotone operators arguments and Banach's fixed point theorem. We also study the numerical approach to the problem using spatially semi-discrete and fully discrete finite elements schemes with implicit and explicit discretization in time. We show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions. Finally, we present some numerical results involving examples in one, two and three dimensions. Received May 20, 2000 / Revised version received January 8, 2001 / Published online June 7, 2001     

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.     

Structural analysis of electrical circuits including magnetoquasistatic devices

Modeling electric circuits that contain magnetoquasistatic (MQS) devices leads to a coupled system of differential-algebraic equations (DAEs). In our case, the MQS device is described by the eddy current problem being already discretized in space (via edge-elements). This yields a DAE with a properly stated leading term, which has to be solved in the time domain. We are interested in structural properties of this system, which are important for numerical integration. Applying a standard projection technique, we are able to deduce topological conditions such that the tractability index of the coupled problem does not exceed two. Although index-2, we can conclude that the numerical difficulties for this problem are not severe due to a linear dependency on index-2 variables.     

Recovery of the unknown coefficients in a two-dimensional hyperbolic equation

In this paper, an inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples.     

Solution with an internal transition layer for a singularly perturbed second-order equation with an advancing and a retarded argument

We consider a singularly perturbed boundary value problem for a differential equation with a retarded and a deviating argument. By using the method of boundary functions and the sewing method, we find not only a continuous but also a smooth solution of the problem. We prove the existence of a solution with an internal transition layer. A graphical numerical example is presented.     

A numerical solution of a two-dimensional IHCP

In this paper, we will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e., the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite-difference method and the least-squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition (SVD) method.     

An efficient algorithm for bounded spectral matching with affine constraint

Graph matching problem appears frequently in the applications of computer vision and machine learning. In this work, based on the spectral matching with affine constraint (SMAC) formulation, we present a new formulation, named bounded SMAC (BSMAC), for the graph matching problem by adding an upper‐bound constraint on the solution norm. We demonstrate the existence of a unique solution with BSMAC, whereas SMAC needs not to have any meaningful solution in general. We develop an effective numerical method to solve the BSMAC formulation as an optimization problem. Numerical experiments are presented to verify feasibility and to show the performance of the proposed numerical method.