Adaptive finite element method for the numerical simulation of electric, magnetic and acoustic fields [Elektronische Ressource] / vorgelegt von Elena Zhelezina

Sujets

• Mathematics

Informations

• Publié par: friedrich-alexander-universitat_erlangen-nurnberg
• Publié le: 01 janvier 2005
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

Adaptive Finite Element Method
for the Numerical Simulation of
Electric, Magnetic and Acoustic
Fields
Der Technischen Fakult˜at der
Universitat˜ Erlangen-Nurn˜ berg
zur Erlangung des Grades
DOKTOR - INGENIEUR
vorgelegt von
Dipl.-Math. Elena Zhelezina
Erlangen, 2005Als Dissertation genehmigt von
der Technischen Fakult˜at der
Universitat Erlangen-Nurnberg˜ ˜
Tag der Einreichung: 14. Juni 2005
Tag der Promotion: 3. August 2005
Dekan: Prof. Dr. A. Winnacker
Berichterstatter: Priv. Doz. Dr. techn. M. Kaltenbacher
Prof. Dr. rer. nat. habil. U. Rude˜Adaptive
Finite-Elemente-Methode fur die˜
numerische Simulation
elektrischer, magnetischer und
akustischer Felder
Der Technischen Fakultat˜ der
Universitat˜ Erlangen-Nurn˜ berg
zur Erlangung des Grades
DOKTOR - INGENIEUR
vorgelegt von
Dipl.-Math. Elena Zhelezina
Erlangen, 2005Als Dissertation genehmigt von
der Technischen Fakult˜at der
Universitat Erlangen-Nurnberg˜ ˜
Tag der Einreichung: 14. Juni 2005
Tag der Promotion: 3. August 2005
Dekan: Prof. Dr. A. Winnacker
Berichterstatter: Priv. Doz. Dr. techn. M. Kaltenbacher
Prof. Dr. rer. nat. habil. U. Rude˜v
Acknowledgements
The research presented in this thesis was carried out at the Department of
Sensor Technology, University of Nuremberg-Erlangen, Germany, under the su-
pervision of PD Dr. techn. Manfred Kaltenbacher. I wish to express my deepest
gratitude to him for his unbounded personal generosity and for the invaluable
help and encouragement he has given me throughout my work. Undoubtedly,
without his constant expert advice this work would not be what it is.
Sincere thanks are directed to all my former and present colleagues at the
Department of Sensor Technology in Erlangen for their invaluable help in many
aspects during my work.
I gratefully acknowledge the flnancial support provided by the SFB 603 Mo-
dellbasierte Analyse und Visualisierung komplexer Szenen und Sensordaten in
Erlangen funded by the German science foundation DFG, which made this rese-
arch possible.
Finally, I owe my deepest thanks to my parents for their encouragement and
help.vi
To my parentsContents
Abstract xi
Notations xiii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 The flnite element method . . . . . . . . . . . . . . . . . . 2
1.2.2 Errorestimatorsandadaptivityfortheflniteelementmethod 2
1.2.3 Reflnement strategy . . . . . . . . . . . . . . . . . . . . . 5
1.3 Overview of the work . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Innovative features of the work . . . . . . . . . . . . . . . . . . . 6
2 Space adaptivity 9
2.1 A posteriori error estimation . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Superconvergent Patch Recovery Method . . . . . . . . . 9
2.1.2.1 Elements patch . . . . . . . . . . . . . . . . . . 10
2.1.2.2 Recovery procedure . . . . . . . . . . . . . . . . 10
2.1.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Remeshing strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Design of an optimal mesh for prescribed tolerance . . . . 13
2.3 Mesh reflnement techniques . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 2D mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1.1 Triangular grid . . . . . . . . . . . . . . . . . . . 16
2.3.1.2 Rectangular grid . . . . . . . . . . . . . . . . . . 19
2.3.2 3D mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2.1 Tetrahedric grid . . . . . . . . . . . . . . . . . . 21
3 Electrostatic fleld computation 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Formulation of problem . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . 28
viiviii
3.4 Adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Validation of the adaptive algorithm . . . . . . . . . . . . . . . . 30
3.5.1 2D numerical example: voltage drived bar. . . . . . . . . . 30
3.5.2 3D n examples: domain with corner singularity. . . 33
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Electromagnetic problems 39
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Problem: formulation and flnite element analysis . . . . . . . . . . 40
4.2.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . 40
4.2.2 Magnetostatic case . . . . . . . . . . . . . . . . . . . . . . 41
4.2.3 Boundary and interface conditions. . . . . . . . . . . . . . 42
4.2.4 Magnetic vector potential . . . . . . . . . . . . . . . . . . 43
4.2.5 Boundary and interface conditions for magnetic vector po-
tential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.6 Variational formulation of magnetostatic problem . . . . . 44
4.3 Finite element analysis of the problem . . . . . . . . . . . . . . . 46
4.3.1 Vector edge flnite element . . . . . . . . . . . . . . . . . . 46
4.3.1.1 Construction of edge flnite element for tetrahedron 47
4.3.2 Discretization with edge flnite elements . . . . . . . . . . . 49
4.4 Adaptive procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4.1 Recovery procedure . . . . . . . . . . . . . . . . . . . . . . 50
4.4.2 Reflnement strategy . . . . . . . . . . . . . . . . . . . . . 52
4.4.3 Conduction of adaptive procedure . . . . . . . . . . . . . . 53
4.5 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5.1 3D model problem : magnetic assembly . . . . . . . . . . . 53
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Harmonic analysis of acoustic equation 61
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 The formulation of the problem . . . . . . . . . . . . . . . . . . . 61
5.3 The flnite element formulation . . . . . . . . . . . . . . . . . . . . 63
5.4 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.6 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.1 Example: impedance channel problem. 2D . . . . . . . . . 66
5.6.2 standing waves in a channel: forced vibrations. 2D 72
6 Industrial application 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Voltage bar (3D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Standing wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4 3D Magnetostatic Force Problem: TEAM Problem 20 . . . . . . 87ix
7 Computational implementation 95
7.1 The code CFS++ . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2 The structure of the code. . . . . . . . . . . . . . . . . . . . . . . 96
8 Conclusions and future developments 101
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . 102
A Standard linear shape functions 103
References 105x