# Numerical solution of nonlinear stationary magnetic field problems

• View
216

• Category

## Documents

Embed Size (px)

### Text of Numerical solution of nonlinear stationary magnetic field problems

• JO H AN N E S KE P LE RUN IV E R S I T AT L I NZ

Ne t z w e r k f u r F o r s c h u n g , L e h r e u n d P r a x i s

Numerical solution of nonlinear stationarymagnetic field problems

Bakkalaureatsarbeit(Projektpraktikum)

Bakkalaureus der Technischen Wissenschaften

in der Studienrichtung

Technische Mathematik

Angefertigt am Institut fur Numerische Mathematik

Betreuung:

O. Univ. Prof. Dipl.-Ing. Dr. Ulrich Langer

Eingereicht von:

Alexander Lechner, Stefan Muhlbock

Mitbetreuung:

Dipl.-Ing. Clemens Pechstein

Linz, November 2008

Johannes Kepler UniversitatA-4040 Linz Altenbergerstrae 69 Internet: http://www.uni-linz.ac.at DVR 0093696

• Abstract

This thesis deals with the solution of two-dimensional nonlinear stationarymagnetic field problems. Starting point for this work is a project with theACCM (Austrian Center of Competence in Mechatronics). The group in theACCM uses a tool for the simulation of electric motors. Our task now is topresent mathematical background of such magnetic field problems.The starting point for deriving an appropriate mathematical model are theMaxwell equations. In the following we derive the variational formulation forour model and show the existence of a unique solution of the problem.For a better understanding we also discuss a one-dimensional model problem.We discretize the problem with the FEM (Finite Element Method). TheNewton procedure is used, to solve the nonlinear problem via a sequence oflinearized problems. Finally, we present some numerical results for our one-dimensional model problem. These numerical results were obtained using aself-developed C++ program.

Zusammenfassung

Diese Bakkalaureatsarbeit behandelt die Losung von zweidimensionalen nicht-linearen stationaren Magnetfeldproblemen. Ausgangspunkt dafur war ein indiesem Zusammenhang stehendes Projekt mit dem ACCM (Austrian Centerof Competence in Mechatronics). Dieses Team nutzt ein Tool zur Simulationvon Elektromotoren. Unsere Aufgabe ist es nun, den mathematischen Hin-tergrund fur solche Magnetfeldprobleme aufzubereiten.Wir leiten im ersten Schritt ein mathematisches Modell aus den MaxwellschenGleichungen dafur her. Im Anschluss wird fur das Modell die Variationsfor-mulierung hergeleitet und es wird die Existenz einer eindeutigen Losung desProblems gezeigt.Zum besseren Verstandnis wird weiters ein eindimensionales Modellprob-lem behandelt. Dieses Problem wird mit FEM (Finite Elemente Methode)diskretisiert. Mit Hilfe des Newtonverfahrens wird anschlieend das nichtlin-eare Problem durch eine Folge von linearisierten Problemen gelost. Zuletztprasentieren wir noch einige numerische Resultate von unserem eindimension-alen Modellproblem. Diese Resultate erhalten wir durch Anwendung einesselbst entwickelten C++ Programms.

i

• Acknowledgement

First of all, we would like to express our thanks to our supervisor Prof. UlrichLanger for giving us the chance to write this thesis, for various discussionsand for his time.Special thanks go to Dipl.-Ing. Clemens Pechstein for help in programmingand debugging.

This work has been carried out at the Institute of Computational Mathemat-ics, JKU Linz, in cooperation with ACCM (Austrian Center of Competencein Mechatronics), a K2-Center of the COMET/K2 program of the FederalMinistry of Transport, Innovation, and Technology, and the Federal Ministryof Economics and Labour, Austria.

Last but not least we acknowledge the Institute of Computational Math-ematics at the Johannes Kepler University of Linz for the technical support.

ii

• Contents

1 Introduction 1

2 Problem Formulation and Analysis 42.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Maxwells equations . . . . . . . . . . . . . . . . . . . 42.1.2 Material laws . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Reduction to 2D . . . . . . . . . . . . . . . . . . . . . 9

2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . 102.2.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . 102.2.2 The Variational Formulation . . . . . . . . . . . . . . . 11

2.3 Existence and Uniqueness of the Solution . . . . . . . . . . . . 162.3.1 Conditions on the Reluctivity . . . . . . . . . . . . . 162.3.2 Zarantonellos Theorem . . . . . . . . . . . . . . . . . 19

3 A 1D model problem and its discretization 213.1 1D problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 The Variational Formulation . . . . . . . . . . . . . . . 213.2 Finite Element Method for the Boundary Value Problem . . . 24

3.2.1 Computation of the stiffness matrix . . . . . . . . . . . 273.2.2 Computation of the load vector . . . . . . . . . . . . . 28

3.3 Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Computation of the Jacobian matrix . . . . . . . . . . 29

3.4 B-H-Curve Approximation . . . . . . . . . . . . . . . . . . . . 31

4 Computer implementation and numerical results 334.1 1D- Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 FEMAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 ParNFB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Bibliography 40

iii

• Chapter 1

Introduction

This work is closely related to a cooperation with the ACCM (Austrian Cen-ter of Competence in Mechatronics) within the Project Seminar NumerikSS 2008. The group in the ACCM uses a program called FEMAG for theanalysis and simulation of electric machines. One of their applications is theshape optimization of electric motors by simulating for a large set of param-eters, such as rotor diameters etc., and then choosing the paramter config-uration with the best degree of efficiency. This method of operation works,but is very expensive. The main task now was to optimize this process bydifferent methods of optimization. One way was discussed by Markus Koll-mann, namely shape optimization by shape derivatives. Another method, thederivative free optimization, was described by Mykhaylo Yudytskiy. Gradi-ent based optimization was discussed by Elisabeth Frank. For more detailswe refer to the final reports of these methods, which can be found at theInstitute of Computational Mathematics, Johannes Kepler University.The major task of our group was to formulate a concrete model of the be-havior of an electric motor from the magnetic point of view. Furthermorethe direct field simulation of underlying partial differential equations is dis-cussed in this thesis. Another goal of the seminar was to get the idea of thesimulation done by the ACCM with FEMAG.

Starting point of our discussion was a model problem of an electric mo-tor sketched in Figure 1.1, which was provided by the group of the ACCM.The area A7 is the permanent magnet of the motor. Areas A1 up to A6 arethe coils. Between the coils and the permanent magnet there is a little airgap. Area A8 is an iron layer.From the famous Maxwell equations, which are used to describe electromag-netic phenomena, we derive the stationary magnetic field formulation. Undercertain assumptions the system of nonlinear partial differential equations canbe reduced to a single nonlinear differential equation in two dimensions.

1

• Figure 1.1: Motor

The nonlinearity occurs in a material relation between the magnetic fieldH and the magnetic induction B. The relation can be expressed by the so-called B-H-curve, or the closely related reluctivity . In general, none of thesemappings are given analytically and must be approximated from measureddata. With suitable boundary conditions, the magnetostatic 2D - problemcan be formulated in a nonlinear variational formulation.Existence and uniqueness is shown by the theorem of Zarantonello. Usuallyone uses Newtons method to obtain the solution of the nonlinear variationalproblem.To demonstrate how such a simulation works in principle, we discuss a one-dimensional model problem and we develop a program to get the solution ofthis problem. The main part of this implementation was the Finite ElementMethod and the Newtons Method. Finally we take a look at the numericalresults. We compare the results of our own 1D program and the results ob-tained by FEMAG and ParNFB, a program developed by Dipl.-Ing. ClemensPechstein.

2

• The organization of this thesis:

Chapter 2 - Problem Formulation and Analysis:Starting from the physical model for magnetostatics, i.e. Maxwellsequations, the mathematical formulation is derived, namely the bound-ary value problem for a nonlinear partial differential equation (PDE).We discuss the solvability of this PDE under suitable conditions im-posed on the B-H-curve.

Chapter 3 - A 1D model problem and its discretization:For disretization we introduce a one-dimensional problem. After a shortintroduction on the Finite Element Method (cf. [2, 6]), FEM and New-tons method are applied to the variational formulation of the problem.At the end of this chapter we discuss the approximation of the B-H-curve (cf. [3, 4]).

Chapter 4 - Computer implementation and numerical results:The theoretical results are used in numerical studies, tested on a 1D -model for a electric motor. Finally we compare the results of the two2D- programs: FEMAG and ParNFB.

Since this thesis was written by a group of two people, we want to clarifywhich part of this work was mainly discussed by which person.

Alexander Lechner:

The derivation of the variational formulation of the nonlinear partialdifferential equation. Here, a detailed reflection of different interfacesis a crucial point.

The proof of existence and uniqueness of the solution. The simulation by FEMAG.

Stefan Muhlbock:

The de

Recommended ##### Nonlinear Phenomena during Bulk CO Electrooxidation ?· PHYSIK-DEPARTMENT Nonlinear Phenomena during…
Documents ##### Numerical Investigation of Air-Side Heat Transfer and ... ?· Numerical Investigation of Air-Side Heat…
Documents ##### A Semigroup Approach to the Numerical Solution of ... ?· Numerical Solution of Parabolic Differential…
Documents ##### Nonlinear Model Reduction for Constrained Me- chanical Systems Nonlinear Model Reduction for Constrained
Documents ##### SAVE THE DATE! + CALL FOR PROBLEMS Digitalwerkstatt SAVE THE DATE! + CALL FOR PROBLEMS . Digitalwerkstatt
Documents ##### Magnetic interactions in martensitic Ni-Mn based ?· Magnetic interactions in martensitic Ni-Mn based…
Documents ##### Forschungsbericht 201 7-35 Numerical Simulation Concept ... ?· Forschungsbericht 201 7-35 Numerical…
Documents ##### Nonlinear Thermoacoustic Stability Analysis of Gas ?· Nonlinear Thermoacoustic Stability Analysis of…
Documents ##### Modified cyclodextrins as chiral stationary phases Modified cyclodextrins as chiral stationary phases
Documents ##### Nonlinear Analysis of Pneumatic Membranes: “From Subgrid ... ?· Nonlinear Analysis of Pneumatic Membranes:…
Documents Documents