Choice of the Numerical Flux in Discontinuous Galerkin Methods for Two-Dimensional Magnetostatic Problems
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 second-order elliptic equation. First of all the problem is formulated as a ﬁrst-order 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 second-order elliptic equation. Therefore the implementation of different methods, the numerical trade-offs 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
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 LC-resonant metamaterials: from circuit model to modal theory and efficient numerical methods
1Ampère Lab, CNRS, ECLyon, France; 2G2Elab, CNRS, Univ Grenoble Alpes, France; 3EP-USP, LMAG, São Paulo, Brazil; 4LIA-Maxwell, CNRS-CNPq, France-Brazil
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
1CEA LIST, France; 2Université Clermont Auvergne,CNRS, Institut Pascal
This paper deals with the computation of quasi-static 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 semi-analytical model in an iterative process.
Multi-port Model Order Reduction Using Matrix Cauer Ladder Network
1Kyoto University, Japan; 2Kindai University; 3Science Solutions International Laboratory, Inc; 4Toshiba Infrastructure Systems Solutions Corporation; 5Kawasaki Heavy Industries, Ltd.
To realize efficient multi-port model-order reduction, a multi-port 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 eddy-current 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 1-port CLN method
Subgridding in volume integral formulations for eddy currents: cohomology computation and exploitation of cyclic symmetry
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 linear-time worst-case 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
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
1State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power University, Beijing, China; 2China 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 CSM-FEM (Charge Simulation Method-Finite 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 CSM-FEM 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
University Grenoble Alpes, CNRS, Grenoble INP, G2Elab, 38000 Grenoble, France
Quasi-static 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 non-destructive testing devices. In this paper, after selecting formulations enabling the modeling of multiply-connected 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 Biot-Savart Field for Synchronous Machines
1TU Wien, Austria; 2Tampere University, Finland
A 2D/1D method for the A-formulation 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 Biot-Savart 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
1Xi'an Jiaotong University, China, People's Republic of,; 2Politecnico di Milano, Italy
This paper presents a non-intrusive 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 ill-posedness of the inverse problem.
A novel collapsing based algorithm to determine the generalized source magnetic fields
Università degli studi di Udine, Italy
A technique based on a tree-cotree decomposition, called Spanning Tree Technique (STT), has been shown to be the most efficient way to compute the generalized source magnetic fields for h-oriented 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 A-H Formulation for Magneto-Thermal Transients in High-Temperature Superconducting Magnets
1CERN, Switzerland; 2Technische Universität Darmstadt, Darmstadt, Germany
The application of high-temperature 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 A-H formulation for the analysis of magneto-thermal 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 non-conducting 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 field-coupling conditions, are discussed and discretized as well as illustrated with numerical results.
Efficient Simulation of Field/Circuit Coupled Systems with Parallelised Waveform Relaxation
1Technische Universität Darmstadt, Germany; 2Institut für Teilchenbeschleunigung und Elektromagnetische Felder, Germany
This paper proposes an efficient parallelised computation of field/circuit coupled systems co-simulated with the Waveform Relaxation (WR) technique. The main idea of the introduced approach lies in application of the parallel-in-time method Parareal to the WR framework. Acceleration obtained by the time-parallelisation is further increased in the framework of Micro-Macro 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 single-phase isolation transformer coupled to a rectifier circuit.
Homogenization Method Based on Cauer circuit via Unit Cell Approach
Hokkaido university, Japan
This paper proposes a novel homogenization method which provides the continued-fraction and, equivalently, Cauer-circuit representations of the complex permeability of multi-scale materials. The proposed method makes it possible to perform the homogenization analysis of nonlinear materials in the time-domain. Moreover, the proposed method can be applied to homogenization of any unit cell which contain arbitrary shaped-conductors with any permeabilities.
Statistical Analysis of the Effect of 3D Conducting Structures on the Axisymmetric Evolution of Fusion Plasmas
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 next-generation fusion reactor DEMO is presented.
In the code CarMa0NL, the plasma is assumed axisymmetric: it is enough to study the evolution of a poloidal cross-section 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 three-dimensional. 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 DEMO-relevant 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 code-predicted value. A linearized model allow to propagate the uncertainty to plasma-wall gaps, leading to values near 1 cm.
Parallel-in-Time Simulation of Transient Electro-Quasistatic Time-Harmonic Nonlinear Field Problems
University of Wuppertal, Germany
Two variants of the Parareal algorithm are presented for parallel-in-time simulations of transient nonlinear time-periodic electro-quasistatic field problems. As a numerical test problem, both variants of the Parareal algorithm are used to simulate a three-dimensional metal-oxide surge arrester model. The simulation results are verified by comparing the Parareal results to a sequential reference solution.
Subdomain Perturbation Finite-Element Method for Quasistatic Darwin Approximation
1Tensor Research, LLC, Andover, MA, United States of America; 2Budapest 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 low-frequency stable A-V 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 run-time improvement compared to the full FE solution.
Simplified and Generalized Corona Charge Injection Scheme for Analyzing Corona Electric Field under HVDC Transmission Lines
1Korea Electric Power Research Institute; 2Kyungpook 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 sub-conductors. This newly proposed method has shown good agreement with the experimental results.