Fiche individuelle
Zuqi TANG  
Titre  Maître de conférences  
Equipe  Outils et Méthodes Numériques  
Adresse  Université Lille 1 Batiment P2 59655 VILLENEUVED'ASCQ  
Téléphone  +33 (0)320434280  
zuqi.tang@univlille.fr  
Site personnel  https://orcid.org/0000000254025858  
Réseau scientifique  https://www.researchgate.net/profile/Zuqi_Tang  
Observation / Thématique de recherche  Scientific computing/Numerical analysis/A posteriori error estimate/Deep learning for Electromagnetic modeling and computation, as well as Multiphysics modeling and computation;  
Publications 
ACLI Revue internationale avec comité de lecture 

[1] AutoGauging of Vector Potential by Parallel Sparse Direct Solvers–Numerical Observations IEEE Transactions on Magnetics, Vol. 55, N°. 6, pages. 14, 06/2019, URL, Abstract TANG Zuqi, ZHAO Yanpu, REN Zhuoxiang 
When using magnetic vector potential (MVP)based formulations for magnetostatic or eddycurrent problems, either gauge conditions specifying the divergence of the MVP or tree gauging by eliminating redundant degrees of freedom of the MVP is usually imposed to ensure uniqueness of solutions. Explicit gauging of the MVP is not always necessary since classical iterative solvers can automatically and implicitly fix the gauge as long as the righthand side vectors are consistent. Besides iterative solvers, implicit gauging is also observed when using stateoftheart parallel sparse direct solvers (PSDSs), thanks to the builtin functions of handling nullspaces of either real symmetric positive semidefinite matrix systems or those complex symmetric systems from eddycurrent problems. Both static and eddycurrent examples are solved by PSDS to demonstrate results of local physical quantities or global quantities such as magnetic energy or joule losses. Highorder edge/nodal elements are also considered in our numerical examples and it is observed that PSDS can also easily and correctly handle the delicate discrete null spaces. 
[2] Improved Equilibrated Error Estimates for Open Boundary Magnetostatic Problems Based on Dual A and H Formulations IEEE Transactions on Magnetics, Vol. 55, N°. 6, pages. 15, 06/2019, URL, Abstract ZHAO Yanpu, TANG Zuqi 
Calculating the bounds of global energy is an important issue in computational electromagnetism, which can provide guaranteed results when extracting inductance parameters. In this paper, an improved equilibrated type a posteriori error estimate for open boundary magnetostatic problems is proposed. We derive our error estimator based on vector dual formulations, which can be efficiently solved using parallel sparse direct solvers. The new estimator can provide a sharp and guaranteed estimate of the finiteelement spatial discretization error. Moreover, the computational cost is cheaper than using existing equilibrated error estimators. Numerical experiments are carried out to showcase the performance of our error estimator, including the modified TEAM workshop problem 13 and the benchmark IEEJ problem. 
[3] Accurate Extraction of Winding Inductances using Dual Formulations without Source Field Computation IEEE Transactions on Magnetics, Vol. 55, N°. 6, pages. 14, 06/2019, URL, Abstract ZHAO Yanpu, TANG Zuqi 
Dual formulations are accurate in use for computing energyrelated global quantities such as inductance and providing upper and lower bounds of the unknown true values of these global parameters, which is not possible if using a single formulation. Since traditional dual formulations result in totally different algebraic matrix equations, people have to develop two different finite element programs and solve the resultant two algebraic equation systems respectively. In this work two practical dual formulations
for open region magnetostatic problems, where the global finite element matrices are exactly the same, are adopted for extracting the winding inductances. Finite element formulation and implementation details are presented. Practical examples having complex windings are solved using the proposed methods to showcase the effectiveness and accuracy. High order FE basis functions are also used to enhance the solution accuracy. The proposed method is highly useful for mediumsized industrial applications by providing guaranteed inductance parameters. 
[4] A Symmetric Fieldcircuit Coupled Formulation for 3D Transient Fullwave Maxwell Problems IEEE Transactions on Magnetics, Vol. 55, N°. 6, pages. 14, 06/2019, URL, Abstract ZHAO Yanpu, TANG Zuqi 
In this paper, a symmetric fieldcircuit coupled finite element method (FEM) for lowfrequency (LF) fullwave Maxwell problems using a magnetic vector potential (MVP) formulation is proposed. The resultant fullydiscrete coefficient matrix is made symmetric for the first time by introducing the socalled source electric scalar potential (ESP) for solid conductors, where the terminal currents are converted from surface integrations of the current density vectors to volumetric integrations. Numerical examples, including a benchmark capacitor charging problem with external circuit connections, are solved and the numerical results match well with reference solutions. The proposed formulation is useful when analyzing electromagnetic fields with coupled inductivecapacitive effects and external circuit connections. 
[5] An Improved Newton Method Based on Choosing Initial Guess Applied to Scalar Formulation in Nonlinear Magnetostatics IEEE Transactions on Magnetics, Vol. 55, N°. 6, pages. 14, 06/2019, URL, Abstract CHERIF Riheb, TANG Zuqi, GUYOMARCH Frédéric, CHEVALLIER Loïc, LE MENACH Yvonnick 
An improved starting point Newton method applied to 3D scalar formulation in magnetostatics is proposed in this paper. Compared with the classical Newton method, the inexactNewton and quasiNewton methods are reported by testing on a benchmark problem as well as an industrial example. Remarkable convergence acceleration using the proposed strategy is observed, and thus, it significantly
reduces the computational time. 
[6] A Novel Gauged Potential Formulation for 3D Electromagnetic Field Analysis Including Both Inductive and Capacitive Effects IEEE Transactions on Magnetics, Vol. 55, N°. 6, 06/2019, URL, Abstract ZHAO Yanpu, TANG Zuqi 
In this paper, a novel potential formulation for lowfrequency (LF) applications taking into account both inductive and capacitive effects but without considering wave propagation is proposed. Both timedomain and frequencydomain formulations are presented.
The resultant fully discrete finiteelement matrix is made symmetric by incorporating a gauge condition and also rewriting the current continuity equation. To improve numerical accuracy and computational efficiency, highorder mixededge elements and nodal elements are adopted to approximate the vector and scalar unknown variables together with highorder timestepping schemes. Several numerical examples are solved to validate and showcase the accuracy of the proposed methods. The proposed formulations are stable in use for LF electromagnetic field computations by considering inductive and capacitive effects simultaneously, such as finding the resonant frequencies of wireless power transfer devices. 
[7] Multiscale modeling of magnetic distribution of ribbon magnetic cores CES Transactions on Electrical Machines and Systems, Vol. 2, N°. 4, pages. 425  429, 12/2018, URL LI Hailin, TANG Zuqi, WANG Shuhong, ZHU Jianguo 
[8] Adaptive inexact iterative algorithms based on polynomialdegreerobust a posteriori estimates for the Stokes problem Numerische Mathematik, Vol. 138, N°. 4, pages. 1027–1065, 04/2018, URL, Abstract ČERMÁK Martin, HECHT Frédéric, TANG Zuqi, VOHRALÍK Martin 
In this paper, we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear Stokes problem. We base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components, namely the discretization error component, the (inner) algebraic solver error component, and possibly the outer algebraic solver error component for algorithms of the Uzawa type. We prove that our estimate gives a guaranteed upper bound on the total error, as well as a polynomialdegreerobust local efficiency, and this on each step of the employed iterative algorithm. Our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error. The developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver. Implementation into the FreeFem++ programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies. As example, we choose here the unpreconditioned and preconditioned Uzawa algorithm and the preconditioned minimum residual algorithm, in combination with the Taylor–Hood discretization. 
[9] Modeling of MagneticInduced Deformation Using Computer Code Chaining and SourceTensor Projection IEEE Transactions on Magnetics, Vol. 53, N°. 6, pages. 1–4, 06/2017, URL, Abstract LIU Mingyong, TANG Zuqi, MININGER Xavier, BOUILLAULT Frédéric, HUBERT Olivier, BERNARD Laurent 
Source tensor projections are developed for the magnetoelastic coupled problems when magnetostrictioninduced force and magnetic force are considered. Comparisons with classical force density projection are first performed on a simple example. Then, it is investigated on an application of a multilayer transformer core with the consideration of material anisotropy and multilayer inhomogeneity. 
[10] Residual a posteriori error estimation for a stochastic magnetostatic problem Journal of Computational and Applied Mathematics, Vol. 289, pages. 5167, 12/2015, URL, Abstract MAC Duy Hung, TANG Zuqi, CLENET Stéphane, CREUSE Emmanuel 
In this paper, we propose an a posteriori error estimator for the numerical approximation of a stochastic magnetostatic problem, whose solution depends on the spatial variable but also on a stochastic one. The spatial discretization is performed with finite elements and the stochastic one with a polynomial chaos expansion. As a consequence, the numerical error results from these two levels of discretization. In this paper, we propose an error estimator that takes into account these two sources of error, and which is evaluated from the residuals. 
[11] Residualbased a posteriori estimators for the potential formulations of electrostatic and timeharmonic eddy current problems with voltage or current excitation International Journal for Numerical Methods in Engineering, Vol. 107, N°. 5, pages. 377394, 12/2015, URL, Abstract CHEN Chao, CREUSE Emmanuel, NICAISE Serge, TANG Zuqi 
In this paper, we consider some potential formulations of electrostatic as well as timeharmonic eddy current problems with voltage or current excitation sources. The wellposedness of each formulation is first established. Then, the reliability of the corresponding residualbased a posteriori estimators is derived in the context of the finite element method approximation. Finally, the implementation in an industrial code is performed, and the obtained theoretical results are illustrated on an academic and on an industrial benchmark. 
[12] A posteriori residual error estimators with mixed boundary conditions for quasistatic electromagnetic problem The International Journal for Computation and Mathematics in Electrical and Electronic Engineering (COMPEL), Vol. 34, N°. 3, pages. 724739, 07/2015, URL TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis, NEMITZ Nicolas 
[13] Residual a posteriori estimator for magnetoharmonic potential formulations with global quantities for the source terms IEEE Transactions on Magnetics, Vol. 51, N°. 3, 06/2015, Abstract TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
In the modeling of eddy current problems, potential formulations are widely used in recent days. In this paper, the results of residualbased a posteriori error estimators, which evaluate the discretization error in the finiteelement computation, are extended to the case of several kinds of source terms for both A/φ and T/Ω harmonic formulations. The definitions of the estimators are given and some numerical examples are provided to show the behavior of the estimators. 
[14] Finite element mesh adaptation strategies from residual and hierarchical error estimators in eddy current problems IEEE Transactions on Magnetics, Vol. 51, N°. 3, 04/2015, URL, Abstract DULAR Patrick, LE MENACH Yvonnick, TANG Zuqi, CREUSE Emmanuel, PIRIOU Francis 
A strategy of mesh adaptation in eddy current finite element modeling is developed from both residual and hierarchical error estimators. Wished distributions of element sizes of adapted meshes are determined from the elementwise local contributions to the estimators and define constraints for the mesh generator. Uniform distributions of the local error are searched. 
[15] Energetic Galerkin Projection of Electromagnetic Fields Between Different Meshes IEEE Transactions on Magnetics, Vol. 50, N°. 2, pages. 613616, 02/2014, URL, Abstract WANG Zifu, TANG Zuqi, HENNERON Thomas, PIRIOU Francis, MIPO JeanClaude 
Meshtomesh field transfer arises frequently in finite element computations. Typical applications may concern remeshing, multigrid methods, domain decomposition and multiphysics problems. For electromagnetic fields, one of the essential constraints in such transfers is to conserve energetic quantities such as the magnetic energy and the joule heating. Within the framework of Galerkin projection on overlapping domains, we introduce the definition of energetic norms for electromagnetic fields. The corresponding formulations we propose, provide energyconserving projection of electromagnetic fields between different meshes.

[16] Comparison of Residual and Hierarchical Finite Element Error Estimators in Eddy Current Problems IEEE Transactions on Magnetics, Vol. 50, N°. 2, pages. 501504, 02/2014, URL, Abstract DULAR Patrick, TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, PIRIOU Francis 
The finite element computation of eddy current problems introduces numerical error. This error can only be estimated. Among all error estimators (EEs) already developed, two estimators, called residual and hierarchical EEs, proven to be reliable and efficient, are theoretically and numerically compared. Both estimators show similar behaviors and locations of the error. 
[17] Helmholtz decomposition of vector fields with mixed boundary conditions and an application to a posteriori finite element error analysis of the Maxwell system Mathematical Methods in the Applied Sciences, Vol. 38, N°. 4, pages. 738750, 02/2014, URL, Abstract CREUSE Emmanuel, NICAISE Serge, TANG Zuqi 
This paper is devoted to the derivation of a Helmholtz decomposition of vector fields in the case ofmixed boundary conditions imposed on the boundary of the domain. This particular decomposition allows to obtain a residual a posteriori error estimator for the approximation ofmagnetostatic problems given in the socalled Aformulation, for which the reliability can be established. Numerical tests confirm the obtained theoretical predictions. 
[18] Residual and equilibrated error estimators for magnetostatic problems solved by finite element method IEEE Transactions on Magnetics, Vol. 49, N°. 12, pages. 5715  5723, 12/2013, URL, Abstract TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
In finite element computations, the choice of the mesh is crucial to obtain accurate solutions. In order to evaluate the quality of the mesh, a posteriori error estimators can be used. In this paper, we develop residualbased error estimators for magnetostatic problems with both classical formulations in term of potentials used, as well as the equilibrated error estimator. We compare their behaviors on some numerical applications, to understand the interest of each of them in the remeshing process. 
[19] A posteriori error estimator for harmonic APhi formulation The international journal for computation and mathematics in electrical and electronic engineering, Vol. 32, N°. 4, pages. 1219  1229, 07/2013, URL TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
[20] Residual based a posteriori error estimators for harmonic A/Phi and T/Omega formulations in eddy current problems IEEE Transactions on Magnetics, Vol. 49, N°. 5, 05/2013, URL TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
[21] Residualbased a posteriori estimators for the T/Omega magnetodynamic harmonic formulation of the Maxwell system International Journal of Numerical Analysis and Modeling, Vol. 10, N°. 2, pages. 411429, 02/2013, URL, Abstract CREUSE Emmanuel, NICAISE Serge, TANG Zuqi, LE MENACH Yvonnick, NEMITZ Nicolas, PIRIOU Francis 
In this paper, we focus on an a posteriori residualbased error estimator for the T/Omega
magnetodynamic harmonic formulation of the Maxwell system. Similarly to the A/Phi formulation, the weak continuous and discrete formulations are established, and the wellposedness of
both of them is addressed. Some useful analytical tools are derived. Among them, an adhoc
Helmholtz decomposition for the T/Omega
case is derived, which allows to pertinently split the error.
Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally
efficient. Finally, numerical tests confirm the theoretical results. 
[22] Residualbased a posteriori estimators for the A/Phi magnetodynamic harmonic formulation of the Maxwell system Mathematical Models and Methods in Applied Sciences, Vol. 22, N°. 5, pages. 30, 05/2012, URL, Abstract CREUSE Emmanuel, NICAISE Serge, TANG Zuqi, LE MENACH Yvonnick, NEMITZ Nicolas, PIRIOU Francis 
This paper is devoted to the derivation of an a posteriori residualbased error estimator for the APhi magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the wellposedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results. 
ACT Conférence internationale avec acte 
[1] Adaptive Stopping Criterion of PGD for Edge Elements based on Equilibrated Error Estimates in Magnetostatic Problems COMPUMAG 2019, Paris, France, 07/2019 YAN Shuai, TANG Zuqi, HENNERON Thomas, REN Zhuoxiang 
[2] StructurePreserved POD for Parametric Low Frequency Fullwave Problems with Gauged Potential Formulations COMPUMAG 2019, Paris, France, 07/2019 YAN Shuai, TANG Zuqi, HENNERON Thomas, REN Zhuoxiang 
[3] Convolutional Neural Network Unet applied in Transformer Multiphysics Analysis COMPUMAG 2019, Paris, France, 07/2019 GONG Ruohan, TANG Zuqi 
[4] Residual Type a posteriori Error Estimates for 3D Low Frequency Stable Maxwell Formulations in Both Frequency and Time Domains COMPUMAG 2019, Paris, France, 07/2019 TANG Zuqi, ZHAO Yanpu 
[5] 3D coupled electromagneticfluidthermal analysis and experiment of 10kV oilimmersed triangular wound core transformer 2019 Joint MMMIntermag, January 1418, 2019 Washington, DC, 01/2019 GONG Ruohan, TANG Zuqi, WANG Shuhong, HENNERON Thomas, RUAN Jiangjun 
[6] An Improved Newton Method Based on choosing Initial Guess Applied to Scalar Potential Formulation in Nonlinear Magnetostatics CEFC 2018, Hangzhou, China, 10/2018 CHERIF Riheb, TANG Zuqi, GUYOMARCH Frédéric, CHEVALLIER Loïc, LE MENACH Yvonnick 
[7] Calibration of a Miniature Single Sheet Tester with Guaranteed FEsimulation CEFC 2018, Hangzhou, China, 10/2018 TANG Zuqi, BENABOU Abdelkader, ZHAO Yanpu 
[8] Application of UNet Network and Training Strategy to Optimal Mesh Refinement in Computational Electromagnetism CEFC 2018, Hangzhou, China, 10/2018 TANG Zuqi, SHEN Xi, HENNERON Thomas 
[9] Application of FreeFem++ Programming Language for 3D Electromagnetic Field Simulations CEFC 2018, Hangzhou, China, 10/2018 TANG Zuqi, ZHAO Yanpu, HECHT Frédéric 
[10] An improved starting point for Newton’s method solving 3D nonlinear magnetostatic problems EPNC 2018, Arras, France, 06/2018 CHERIF Riheb, TANG Zuqi, GUYOMARCH Frédéric, CHEVALLIER Loïc, LE MENACH Yvonnick 
[11] Gauged Dual Formlations for Fast and Accurate Computation of Inductance Parameters of Magnetostatics Problems EMF 2018, Darmstadt, Germany, 04/2018 ZHAO Yanpu, TANG Zuqi 
[12] Dual Regularized Formulations for Open Boundary Magnetostatic Problems EMF 2018, Darmstadt, Germany, 04/2018 ZHAO Yanpu, TANG Zuqi 
[13] 3D modeling of magnetoelastic behavior using simplified multiscale model: Application on power transformer core COMPUMAG 2017, Daejeon, Korea, 06/2017 TANG Zuqi, LIU Mingyong, BOUILLAULT Frédéric, MININGER Xavier, HUBERT Olivier 
[14] Vibration Prediction of NonOriented Silicon Iron Power Transformer Core under DC Bias COMPUMAG 2017, Daejeon, Korea, 06/2017 LIU Mingyong, HUBERT Olivier, TANG Zuqi, BOUILLAULT Frédéric, MININGER Xavier, BERNARD Laurent 
[15] Adaptive Stopping Criteria for Iterative Solver Applied to Potential Formulations in Magnetostatic Problems COMPUMAG 2017, Daejeon, Korea, 06/2017 TANG Zuqi 
[16] A posteriori error estimators for A and Ω magnetostatic formulations based on equilibrated fluxes reconstructions COMPUMAG 2017, Daejeon, Korea, 06/2017 TANG Zuqi 
[17] Error estimation in the computation of induced currents of human body 2013 CIGRE SCC3 & EMFELF Colloquium, Japan, 10/2013 LELONG Thomas, TANG Zuqi, SCORRETTI Riccardo, THOMAS Pierre, LE MENACH Yvonnick, CREUSE Emmanuel 
[18] Energetic Galerkin projection of electromagnetic fields between different meshes
COMPUMAG 2013 Budapest, Hungary, 07/2013, Abstract WANG Zifu, TANG Zuqi, HENNERON Thomas, PIRIOU Francis, MIPO JeanClaude 
The Galerkin projection provides an useful tool to transfer electromagnetic fields between different meshes.
Given an electromagnetic field calculated on the source mesh, the transfer to a different mesh can be employed for modelcoupling, domain decomposition, remeshing, visualization and similar proposes.
The Galerkin projection consists of calculating a target field which minimizes the interpolation error between two discretized fields.
However, the $L^2$ Galerkin projection suffers from nonconservation of the electromagnetic energy.
In this paper, we present an energetic approach for Galerkin projections. 
[19] Comparison of Residual and Hierarchical Finite Element Error
Estimators in Eddy Current Problems COMPUMAG 2013 Budapest, Hungary, 07/2013, URL, Abstract TANG Zuqi, DULAR Patrick, LE MENACH Yvonnick, CREUSE Emmanuel, PIRIOU Francis 
The finite element computation of eddy current
problems gives numerical error. This error cannot be calculated,
but can only be estimated. Among all error estimators already
developed, it is proposed to compare two proven estimators
called residual and hierarchical error estimators. 
[20] Error estimation in the Computation of Induced Current of Human Body in the Case of Low frequency Magnetic Field Excitation COMPUMAG 2013 Budapest, Hungary, 07/2013 LELONG Thomas, TANG Zuqi, SCORRETTI Riccardo, THOMAS Pierre, LE MENACH Yvonnick, CREUSE Emmanuel 
[21] Residual based a posteriori error estimator for harmonic APhi and TOmega formulation in eddy current problems CEFC 2012 Oita, Japan, 11/2012, Abstract TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
For the eddy current problem, the potential formulations are widely used today. In this communication, residual based a posteriori error estimators are introduced to evaluate the discretization error in the finite element calculation in both case of APhi and TOmega harmonic formulations. An example is carried out to show the behavior of our estimators. 
[22] A posteriori error estimator for harmonic APhi formulation EPNC 2012 Pula, Croatia, 06/2012, Abstract TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
In this paper a residualbased error estimator is proposed to evaluate the numerical error induced by the computation of the electromagnetic systems using a Finite Element Method in the case of the harmonic APhi formulation. This estimator is based on the evaluation of quantities weakly verified by the formulation. Furthermore as this estimator verifies the notions of reliability and efficiency it allows to estimate the qualities of local and global solutions. Two examples of electromagnetic systems with current density induced are used to show the efficiency of the proposed estimator. 
[23] A TIMEDOMAIN IMPLICITSCHEMA DIRECT SOLVER: APPLICATION TO FINITE INTEGRATION SOLUTION
ISEF 2011, 09/2011, Abstract WANG Zifu, LE MENACH Yvonnick, TANG Zuqi, KORECKI Julien, HENNERON Thomas 
In timedomain electromagnetic fields computation, numerical methods (such as Finite Element
Method (FEM), Finite Integration Technique (FIT) [13] and etc.) have been applied. For the time
domain integration solution, explicit and implicit schemas have been widely used.
In comparison with implicit methods, the explicit methods are easier to realize in terms of
computation complexity, however, they are constrained by the stability condition. This condition
may require a small time step and therefore a prohibitive computing time. Another possibility is to
use an implicit schema which ensures the numerical stability. Unfortunately the implicit methods
require equation solved at each time step [4]. As a consequence, despite of a free choice on time
step, the computation time using the full implicit methods increases.
In this paper, fixedpoint explicit calculation is introduced to an implicit schema. This method
combines the two advantages of implicit and explicit methods: no stability condition and no
equation solving. The solver is then applied to a timedomain eddy current problem. Using
orthogonal mesh cells and FIT, the massmatrices in discrete formulations are diagonal. The fixed
point explicit method allows direct calculations without matrix inversion or decomposition.

[24] Comparison of Residual and Equilibrated Error Estimators for FEM Applied to Magnetostatic Problems COMPUMAG 2011 Sydney, Australie, 07/2011, Abstract TANG Zuqi, LE MENACH Yvonnick, CREUSE Emmanuel, NICAISE Serge, PIRIOU Francis 
In finite element computations, the choice of the mesh is crucial to obtain an accurate solution. In order to evaluate
the quality of the mesh, a posteriori error estimators can be used. In this paper, we analyze and compare the residual and equilibrated error estimators for magnetostatic problems. 
ACN Conférence nationale avec acte 
[1] Estimateurs d’erreur a posteriori pour les équations de la magnétodynamique en formulation potentielle (A/φ) harmonique Journées du GDR Calcul 2011 Paris, 07/2011 TANG Zuqi 
INV Conférence invité 
[1] Residualbased Error Estimator for Finite Element Method Applied to Quasi Static fields monag, Curitiba, 09/2014 LE MENACH Yvonnick, TANG Zuqi, TITTARELLI Roberta 
TH Thèse 
[1] Estimateurs d’erreur a posteriori résiduels en éléments finis pour la résolution de problèmes d'électromagnétisme en formulations potentielles Université Lille 1, 11/2012, URL, Abstract TANG Zuqi 
Résumé
Ce travail s’intéresse à la résolution numérique par éléments finis des équations de Maxwell en régime quasistationnaire et en formulations potentielles. L’objectif poursuivi consiste à développer des estimateurs d’erreur a posteriori résiduels, afin de contrôler l’erreur de discrétisation spatiale, dans le cadre d’applications en régime statique ou en régime dynamique harmonique.La première partie de cette thèse est composée de deux chapitres. Le premier est consacré à la modélisation des phénomènes physiques étudiés et à l’obtention des équations mathématiques en résultant. Dans le second, on présente les estimateurs a posteriori et leur intérêt dans le cadre de la mise en oeuvre de la méthode des éléments finis. On détaille notamment les notions de fiablité et d’efficacité d’un estimateur. La deuxième partie se décompose en trois chapitres. Le premier développe l’estimateur a posteriori dans le cas de la magnétostatique en formulation potentielle vecteur A. Les outils mathématiques nécessaires à l’étude sont en particulier détaillés. L’estimateur obtenu est alors validé sur quelques cas tests académiques. Le deuxième traite de l’estimateur a posteriori pour la formulation magnétodynamique en potentiel A/φ en régime harmonique. Un soin particulier est apporté pour générer une décomposition de Helmholtz ad hoc permettant d’obtenir la fiabilité de l’estimateur. Plusieurs configurations sont traitées en fonction de la position du domaine conducteur dans le domaine de calcul et des conditions aux limites associées. Un test numérique est ensuite effectué. Le troisième chapitre est consacré à l’estimateur d’erreur a posteriori pour la formulation T/Ω en régime harmonique pour le problème de la magnétodynamique, en supposant le domaine conducteur simplement connexe. Similairement à la formulation A/φ, une décomposition de Helmholtz est développée pour établir la fiabilité. Une validation numérique est proposée. Enfin, la troisième partie présente une batterie de tests numériques applicatifs et industriels permettant de tester les estimateurs développés dans des conditions réelles. Celleci se termine notamment par une application de EDF R&D ayant pour objet le contrôle non destructif par courant de Foucault de tubes générateurs de vapeur.
abstract
We are interested in resolving the Maxwell equations in the case of quasistationary and potential formulations when the finite element method is used. The aim of this work is to develop residualbased a posteriori estimators to control the spatial discretization error in magnetostatic and magnetodynamic problems. The first part is decomposed in two chapters. In the first one, the modeling of the physical phenomena involved are proposed and the mathematical equations are derived. Then, in the second one, the definition of the a posteriori estimators and their interest are presented in the context of the finite element method. The particular notions of reliability and efficiency of an estimator are presented. The second part can be decomposed into three chapters. In the first one, a residualbased a posteriori estimator for the vector potential formulation A in the case of magnetostatic problems is developed. Some necessary mathematical tools for the study are particularly detailed. The estimator is then validated by some academic tests. In the second chapter, a residualbased a posteriori estimator for the A/φ magnetodynamic harmonic formulation is developed. An adhoc Helmholtz decomposition is derived to obtain the reliability of the estimator. Several configurations are considered according to the position of the conductor domain in the computational domain as well as boundary conditions used. A numerical test is then performed. In the third chapter, a residualbased a posteriori estimator is derived for the T/Ω magnetodynamic harmonic formulation, when the conductor domain is simply connected. Similarly to the A/φ formulation, an adhoc Helmholtz decomposition is developed to establish the reliability. A numerical validation is proposed.Finally, in the third part, a set of numerical experiments and industrial applications are presented to evaluate our estimators. It ends with a particular application of EDF R&D focusing on the eddy current nondestructive evaluation of steam generator tubes. 
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