Choice of the Numerical Flux in Discontinuous Galerkin Methods for TwoDimensional Magnetostatic Problems
Sebastian Straßer, HansGeorg Herzog
Technical University of Munich (TUM), Germany
In this paper different discontinuous Galerkin methods for magnetostatic ﬁeld problems are compared. The magnetostatic ﬁeld problem is described by the magnetic vector potential A in two dimensions and therefore by a secondorder elliptic equation. First of all the problem is formulated as a ﬁrstorder system using an auxiliary variable. Afterwards, the functions on the boundary of an element are expressed by an approximation, called the numerical ﬂux. One can then use trace operators to deﬁne different numerical ﬂuxes for the system of interest which will result in different discontinuous Galerkin methods. The aim of this paper is to determine practical guidelines for selecting the numerical ﬂux when applying a discontinuous Galerkin method to a secondorder elliptic equation. Therefore the implementation of different methods, the numerical tradeoffs between the choice of the numerical ﬂuxes and the properties of the system matrix are observed.
Studying a Linear Adjustable Stiffness Magnetic Spring using a Magnetic Charge Integral Approach
Md Emrad Hossain, Jonathan Bird
Portland State University, United States of America
A new type of adjustable stiffness magnetic spring is studied using an analytic magnetic charge modelling approach. An overview of the force calculation approach is presented, and the analytic model is used to demonstrate both the adjustability of the spring stiffness as well as the linear force position characteristics of the magnetic spring. The analytic based integral calculation approach is validated using finite element analysis. The full paper will utilize the analytic model to conduct a sizing analysis and in addition experiment validation of the modelling approach will be presented.
Large surface LCresonant metamaterials: from circuit model to modal theory and efficient numerical methods
Laurent Krähenbühl^{1,4}, Riccardo Scorretti^{1,4}, Arnaud Bréard^{1,4}, Christian Vollaire^{1,4}, JeanMichel Guichon^{2,4}, Olivier Chadebec^{2,4}, Gérard Meunier^{2,4}, Alberto UrdanetaCalzadilla^{2}, Viviane Cristine Silva^{3,4}, Carlos A.F. Sartori^{3,4}
^{1}Ampère Lab, CNRS, ECLyon, France; ^{2}G2Elab, CNRS, Univ Grenoble Alpes, France; ^{3}EPUSP, LMAG, São Paulo, Brazil; ^{4}LIAMaxwell, CNRSCNPq, FranceBrazil
We study the harmonic magnetodynamic behavior (without wave propagation) of a resonant surface metamaterial, made up of many identical and regularly arranged LC cells. The "circuit" model gives the exact solution, but is not numerically efficient for the simulation of very large structures (e.g. 1000x1000 cells). For the first time, we highlight the modal character of the solutions, which makes it possible to explain their frequency and spatial related properties. From these results, we show under what assumptions it is possible to homogenize the metamaterial, which opens the way for using this approach together with efficient numerical methods.
Eddy current modeling of an array of rods for the analysis of unidirectional composite materials
Denis Prémel^{1}, Gérard Granet^{2}
^{1}CEA LIST, France; ^{2}Université Clermont Auvergne,CNRS, Institut Pascal
This paper deals with the computation of quasistatic fields induced by a 3D Eddy Current (EC) probe in an array of conductor rods embedded in a dielectric host medium. The implemented numerical method is based on a modal approach, the Fourier Modal Method (FMM), widely used in the optical community for the analysis of diffraction gratings. The main difficulty to overcome in this contribution comes from the aperiodic behavior of the fields since the array of rods is excited by a 3D eddy current probe rather than a plane wave as it is usual for the analysis of gratings. The intented application is to carry out a homogenization method by integrating the fast semianalytical model in an iterative process.
Multiport Model Order Reduction Using Matrix Cauer Ladder Network
Tetsuji Matsuo^{1}, Takayuki Fujiwara^{1}, Kenta Kuriyama^{1}, Kengo Sugahara^{2}, Akihisa Kameari^{3}, Tadashi Tokumasu^{4}, Yuji Shindo^{5}
^{1}Kyoto University, Japan; ^{2}Kindai University; ^{3}Science Solutions International Laboratory, Inc; ^{4}Toshiba Infrastructure Systems Solutions Corporation; ^{5}Kawasaki Heavy Industries, Ltd.
To realize efficient multiport modelorder reduction, a multiport Cauer ladder network (CLN) method is formulated that directly yields resistance and inductance matrices giving the elements of a ladder network in the matrix Cauer form. The eddycurrent field driven by multiple power sources is accurately reconstructed by a small number of network elements. The matrix Cauer form achieves faster convergence of the transfer function than a 1port CLN method
Subgridding in volume integral formulations for eddy currents: cohomology computation and exploitation of cyclic symmetry
Mauro Passarotto, Ruben Specogna
Polytechnic Department of Engineering and Architecture, University of Udine, Italy
This contribution addresses for the first time the problem of solving eddy current problems on grids built with subgridding. In particular, an algorithm to compute a set of suitable cohomology generators needed when the conductors are not simply connected is introduced. Beside being purely combinatorial, with lineartime worstcase complexity, and suitable with meshes with subgridding, it reuses a code that computes generators for triangular surface meshes, with obvious advantages concerning the implementation effort. Finally, the formulation and the algorithm for cohomology computation are tweaked to be able to solve eddy current problems with cyclic symmetry. Preliminary results validate the proposed method.
MSFEM and MOR to Minimize the Computational Costs of Nonlinear eddy Current Problems in Laminated Iron Cores
Karl Hollaus, Joachim Schöberl, Markus Schöbinger
Vienna University of Technology, Austria
The multiscale finite element method (MSFEM) reduces the computational costs for the simulation of eddy currents (ECs) in
laminated iron cores compared to the standard finite element method (SFEM) essentially. Nevertheless, the complexity of the resulting
problem is still too large to solve it conveniently. The idea is to additionally exploit model order reduction (MOR). Snapshots for a
reduced basis are calculated by MSFEM cheaply. Numerical simulations of a small transformer show an exceptional performance.
High Effective Calculation and 3D Modeling of Ion Flow Field near the Crossing of HVDC Transmission Lines
Liming Hao^{1}, Bin Bai^{1}, Donglai Wang^{1}, Li Xie^{2}, Tiebing Lu^{1}, Qian Zhang^{1}, Xuebao Li^{1}
^{1}State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power University, Beijing, China; ^{2}China Electric Power Research Institute, Beijing, China
3D calculation of ion flow field is necessary to the crossing issue on HVDC transmission lines. An appropriate artificial boundary of ion flow field is vital for calculation. A numerical method called Upstream CSMFEM (Charge Simulation MethodFinite Element Method) is adopted in this paper to calculate the ion flow field near the crossing of HVDC transmission lines. An improved CSM using quadratic charges is proposed to solve the nominal electric field generated by HVDC lines. Based on improved CSM, the algorithm ensures high accuracy and provides a basis for artificial boundary conditions. The error of different methods is evaluated by the analytical solution of a 3D coaxial cylinder, and relative error of CSMFEM is evidently lower than that of FEM under the same number of meshes. Therefore, the proposed model and method are helpful in analyzing ion flow field near the crossing of HVDC transmission lines.
3D BEM Formulations for Eddy Current Problems with Multiply Connected Domains and Circuit Coupling
Quang Anh Phan, Olivier Chadebec, Gerard Meunier, JeanMichel Guichon, Bertrand Bannwarth
University Grenoble Alpes, CNRS, Grenoble INP, G2Elab, 38000 Grenoble, France
Quasistatic linear problems can be solved efficiently with Boundary Element Method (BEM). This method is based on surface integral equations dealing with equivalent magnetic and electric surface current densities. Many works have shown the potentiality of BEM especially for the modeling of nondestructive testing devices. In this paper, after selecting formulations enabling the modeling of multiplyconnected regions, an original coupling is proposed in order to take into account external electric circuit in the problem.
A 2D/1D Multiscale Finite Element Method Using the BiotSavart Field for Synchronous Machines
Markus Schöbinger^{1}, Joachim Schöberl^{1}, Paavo Rasilo^{2}, Karl Hollaus^{1}
^{1}TU Wien, Austria; ^{2}Tampere University, Finland
A 2D/1D method for the Aformulation of the eddy current problem in a thin iron sheet is presented. It allows for the reduction of a three dimensional problem to a two dimensional one, where the coupling to the remaining one dimensional problem is already integrated implicitly into the two dimensional system. Similar to the classical multiscale finite element method this is achieved by separating the reference solution into what can be resolved on the two dimensional mesh and a rest which is resolved by an expansion using predefined polynomial shape functions. The presented method utilizes a modification to the previously presented version, which
allows for the correct treatment of edge effects. Important novelties are the extension of the previously linear method to nonlinear materials and the application of systems which are driven by a prescribed BiotSavart field. The proposed method is tested numerically in the context of a synchronous machine.
Reconstruction of Electric Arc Current Density in a Miniature Circuit Breaker from the Magnetic Field
Jinlong Dong^{1,2}, Luca Di Rienzo^{2}, Jianhua Wang^{1}, Guogang Zhang^{1}
^{1}Xi'an Jiaotong University, China, People's Republic of,; ^{2}Politecnico di Milano, Italy
This paper presents a nonintrusive magnetic diagnostic method able to reconstruct the electric arc current density from its magnetic field by solving a magnetic inverse problem. The focus is on the formulation and solution of the inverse problem using a realistic geometry of a miniature circuit breaker. In order to guarantee that the divergence of the reconstructed current density is zero, Whitney face elements are used. Tikhonov regularization is applied to tackle the illposedness of the inverse problem.
A novel collapsing based algorithm to determine the generalized source magnetic fields
Silvano Pitassi, Ruben Specogna
Università degli studi di Udine, Italy
A technique based on a treecotree decomposition, called Spanning Tree Technique (STT), has been shown to be the most efficient way to compute the generalized source magnetic fields for horiented magnetostatic or eddy current formulations starting from solenoidal source electric currents over mesh faces. Yet, it is known that STT may fail in practice, by getting stuck in an infinite loop. We demonstrate that the failure of STT is related to a topological property of the given mesh called collapsibility. Based on this discovery, a new algorithm to construct a spanning tree suitable for STT is developed. Numerical examples show that this new algorithm has the same performance of the previous STT and is able to terminate in configurations where breadth first search (BFS) trees usually fail. An additional geometrical construction, which does not worsen the overall performance, extends the technique for handling the most common case of failure in which only one critical edge is present.
A Coupled AH Formulation for MagnetoThermal Transients in HighTemperature Superconducting Magnets
Lorenzo Bortot^{1,2}, Bernhard Auchmann^{1}, Herbert De Gersem^{2}, Michal Maciejewski^{1}, Matthias Mentink^{1}, Sebastian Schöps^{2}, Arjan Peter Verweij^{1}
^{1}CERN, Switzerland; ^{2}Technische Universität Darmstadt, Darmstadt, Germany
The application of hightemperature superconductors to accelerator magnets is under study. Numerical methods are crucial for a careful evaluation of the complex dynamical behavior of the magnets, especially concerning the magnetic field quality. We present a coupled AH formulation for the analysis of magnetothermal transients in accelerator magnets. The magnetic field strength H accounts for the eddy current problem in the conducting regions, while the magnetic vector potential represents the magnetostatic problem in the nonconducting domains. Furthermore, we introduce a slab approximation for the conductors, making the formulation suitable for large scale models composed of thousands of tapes. The relevant equations, with emphasis on the fieldcoupling conditions, are discussed and discretized as well as illustrated with numerical results.
Efficient Simulation of Field/Circuit Coupled Systems with Parallelised Waveform Relaxation
Idoia Cortes Garcia^{1,2}, Iryna KulchytskaRuchka^{1,2}, Sebastian Schöps^{1,2}
^{1}Technische Universität Darmstadt, Germany; ^{2}Institut für Teilchenbeschleunigung und Elektromagnetische Felder, Germany
This paper proposes an efficient parallelised computation of field/circuit coupled systems cosimulated with the Waveform Relaxation (WR) technique. The main idea of the introduced approach lies in application of the parallelintime method Parareal to the WR framework. Acceleration obtained by the timeparallelisation is further increased in the framework of MicroMacro Parareal. Here, the field system is replaced by a lumped model in the circuit environment for the sequential computations of Parareal. The introduced algorithm is tested with a model of a singlephase isolation transformer coupled to a rectifier circuit.
Homogenization Method Based on Cauer circuit via Unit Cell Approach
Shingo Hiruma, Hajime Igarashi
Hokkaido university, Japan
This paper proposes a novel homogenization method which provides the continuedfraction and, equivalently, Cauercircuit representations of the complex permeability of multiscale materials. The proposed method makes it possible to perform the homogenization analysis of nonlinear materials in the timedomain. Moreover, the proposed method can be applied to homogenization of any unit cell which contain arbitrary shapedconductors with any permeabilities.
Statistical Analysis of the Effect of 3D Conducting Structures on the Axisymmetric Evolution of Fusion Plasmas
Nicola Isernia, Fabio Villone
Consorzio CREATE, Università degli Studi di Napoli Federico II, Italy
A statistical approach is presented for the quantification of the effects of three dimensional conducting structures on the time evolution of plasmas in fusion devices. Available codes either disregard 3D effects or consider a spatial average at most; in the present paper also higher order statistical moments are considered. The tools of statistical analysis are adapted and implemented in the CarMa0NL code. A test case referring to nextgeneration fusion reactor DEMO is presented.
In the code CarMa0NL, the plasma is assumed axisymmetric: it is enough to study the evolution of a poloidal crosssection of the plasma column, all the physical quantities of interest will be constant along the toroidal angle. On the other hand, currents in surrounding structures are fully 3D, numerically modelled by an integral formulation, using an electric vector potential together with edge shape elements. The effect of the 3D eddy currents on the plasma would be in principle threedimensional. Hence, the magnetic quantities related to eddy currents are averaged along the toroidal angle to evaluate their effect on the plasma, in order to keep the convenient axisymmetric assumption in the solution of the plasma problem. A significant amount of information is hence purposely discarded in the evaluation of the effect eddy currents have on the plasma column evolution.
The idea of the present work is to quantify the amount of discarded information through statistical tools. Namely two distinct approaches, based on random variables and random functions respectively are proposed and implemented on a DEMOrelevant study case.
Results relative to the study case show the possibility to consider few spatial harmonics of the random functions to describe the fluctuation of the poloidal flux per radian from the codepredicted value. A linearized model allow to propagate the uncertainty to plasmawall gaps, leading to values near 1 cm.
ParallelinTime Simulation of Transient ElectroQuasistatic TimeHarmonic Nonlinear Field Problems
MarvinLucas Henkel, Fotios Kasolis, Markus Clemens
University of Wuppertal, Germany
Two variants of the Parareal algorithm are presented for parallelintime simulations of transient nonlinear timeperiodic electroquasistatic field problems. As a numerical test problem, both variants of the Parareal algorithm are used to simulate a threedimensional metaloxide surge arrester model. The simulation results are verified by comparing the Parareal results to a sequential reference solution.
Subdomain Perturbation FiniteElement Method for Quasistatic Darwin Approximation
Zsolt Badics^{1}, Sandor Bilicz^{2}, Jozsef Pavo^{2}, Szabolcs Gyimothy^{2}
^{1}Tensor Research, LLC, Andover, MA, United States of America; ^{2}Budapest University of Technology and Economics, Budapest, Hungary
The subdomain perturbation (SDP) finite element (FE) method is a very efficient numerical scheme and it speeds up the FE solution of magneto and electroquasistatic problems significantly. Additionally, the authors have recently developed a lowfrequency stable AV FE formulation to solve quasistatic Darwin models that incorporate both the capacitive and inductive effects, thereby allowing solving for resonances. This work combines the Darwin FE formulation with the SDP strategy at the first time. It successfully validates the combined technique and demonstrates a significant runtime improvement compared to the full FE solution.
Simplified and Generalized Corona Charge Injection Scheme for Analyzing Corona Electric Field under HVDC Transmission Lines
Yoonho Park^{1}, MinHee Kim^{2}, JaeHyun Lee^{1}, SeHee Lee^{2}
^{1}Korea Electric Power Research Institute; ^{2}Kyungpook national university, Korea, Republic of (South Korea)
A simplified and generalized charge injection method has been proposed for analyzing the corona electric field and current density from the positive and negative ion flows under HVDC transmission lines. Until now, various methods have been introduced to calculate the corona electric field such as the method of characteristics (MOC), the finite element method (FEM), and the charge simulation method (CSM). To adopt these numerical methods, it is critical to estimate the initial charge injection value on the conductor surface due to the electric field intensity. The charge injection value has usually been calculated by considering the geometric structure, surface roughness, and applied potential based on the Peek’s law, and then analytical derivation with simple geometry. However, it is difficult to derive a general expression for charge injection with complicated situations such as the number of bundles and poles, roughness, and twisted wire. Here, the value of corona charge injection was estimated based on the electric field strength around the conductor surface. Therefore, a simple and generalized charge injection method is proposed that is expressed using the electric field strength, surface roughness, and air density. The results have been compared with those of previous studies with various transmission line specifications such as the number of subconductors. This newly proposed method has shown good agreement with the experimental results.
