Stefan-Signorini moving boundary problem arisen from thermal plasma cutting [Elektronische Ressource] : mathematical modelling, analysis and numerical solution / von Arsen Narimanyan

Sujets

• Mathematics
• Physik

Informations

• Publié par: universitat_bremen
• Publié le: 01 janvier 2006
• Nombre de lectures: 38
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Stefan-Signorini Moving Boundary Problem
Arisen From Thermal Plasma Cutting:
Mathematical Modelling,
Analysis and Numerical Solution
von Arsen Narimanyan
Dissertation
zur Erlangung des Grades eines Doktors der Naturwissenschaften
- Dr.rer.nat. -
Vorgelegt im Fachbereich 3 (Mathematik & Informatik)
der Universit¨at Bremen
12. Juni 2006.Datum des Promotionskolloquiums: 25.07.2006
Gutachter: Prof. Dr. Alfred Schmidt (Universit¨at Bremen)
Prof. Dr. Gurgen Hakobyan (Staatliche Universit¨at Jerewan, Armenien)Acknowledgments
IfeelmostfortunatetohavehadtheopportunitytodomyPhDintheUniversityofBremen
and enjoy the company of wonderful people I have met there.
Completing this doctoral work has been a wonderful and often overwhelming experience. I
have been very privileged to have a smart and supportive supervisor and teacher, namely
Prof. Dr. Alfred Schmidt. He has an ability to cut through reams of numerical PDEs that
I will always admire. With his help I have learned a great deal of numerical analysis and
gained a lot of programmingskills. I thank Alfred Schmidt also for his invaluable time that
he provided for discussions.
Many thanks goes to Prof. Dr. Michael Bo¨hm for his remarks and advises concerning the
aspects of the mathematical analysis of the present study.
I appreciate all my friends and colleagues for their kindness and support, in particular
Ronald St¨over, a really nice person and an excellent friend. It is Ronald’s contribution
that I have got integrated in the German society very fast and learned how one should
speak correct German. It has also been my pleasure to work with (and hang out with)
Jenny Niebsch, Jorg Benke, Bettina Suhr,Thilo Moshagen, SergueiDachkovski andAdrian
Muntean. Ithank ThiloandBettina for their tips ontheimprovement of my program. The
fun that we experienced with Adrian while writing our first joint paper has been one of the
greatest ones during my stay in Bremen.
Last, but not least, I would like to thank my entire family, especially my parents, for their
love and support. My wife, Astghik, has been my guiding light and big love over all these
years. Shehas seen my best and my worst, and provided support,hugs and patience. Even
whenmy emotional andresearch brainsbecamesohopelessly entwined thatI dreamedthat
thetwo ofushavenocommon edgesonthehugetriangulation oftheworld,shestillforgave
me. Thank you, honey. I thank also my two sweet children Tatevik and Mane, who are
always atmysidetosharemyjoysandsorrowsandwithoutwhosepatiencethisworkwould
have remained just a dream.Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
1 Introduction 1
1.1 General: the plasma cutting process . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Problem statement and physical modelling 6
2.1 Device description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Thermal cutting process and industrial problems . . . . . . . . . . . . . . . 8
2.3 Physical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Mathematical modelling 12
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Mathematical modelling – one dimensional case . . . . . . . . . . . . . . . . 15
3.4 Mathematical modelling – higher dimensional case . . . . . . . . . . . . . . 18
3.5 Heat flux due to the plasma beam . . . . . . . . . . . . . . . . . . . . . . . 21
4 Definitions, functional analysis 26
4.1 Review of basic functional spaces . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.1 Banach spaces and Hilbert spaces. . . . . . . . . . . . . . . . . . . . 26
4.1.2 Basic concepts of Lebesgue spaces . . . . . . . . . . . . . . . . . . . 29
4.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 Weak (generalized) derivatives . . . . . . . . . . . . . . . . . . . . . 30
4.2.2 Introduction to Sobolev spaces . . . . . . . . . . . . . . . . . . . . . 31
4.2.3 Some useful properties of Sobolev spaces . . . . . . . . . . . . . . . . 31
4.3 Spaces of vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Weak formulation of the problem 37
5.1 Variational inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1.1 Signorini problem and variational inequalities . . . . . . . . . . . . . 38
5.2 Level set formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.1 Distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.2 Stefan condition as level-set equation . . . . . . . . . . . . . . . . . . 46
5.3 Weak formulation of Stefan-Signorini problem . . . . . . . . . . . . . . . . . 49
iiContents iii
6 Analysis of the Model 50
6.1 Existence and uniqueness of classical solution –
one dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.1.1 Regularity of the free boundary . . . . . . . . . . . . . . . . . . . . . 52
6.2 Existence and uniqueness of the weak solution –
higher dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2.1 Higher dimensional model . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2.2 The abstract theory of penalty method . . . . . . . . . . . . . . . . 53
6.2.3 Existence and uniqueness of the weak solution of Signorini problem. 55
6.2.4 Further regularity results . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.5 Existence and uniqueness of the weak solution of level-set equation . 62
6.2.6 Method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2.7 The coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7 Numerical Results 67
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.2 Discretization of the cutting model . . . . . . . . . . . . . . . . . . . . . . . 68
7.2.1 Heat equation with Signorini boundary data on a time dependent
domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Nonlinear solver for the algebraic system . . . . . . . . . . . . . . . 72
Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2.2 Discretization of the level set equation . . . . . . . . . . . . . . . . . 75
Viscosity solution of the level set equation . . . . . . . . . . . . . . . 76
Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Solver for the algebraic system . . . . . . . . . . . . . . . . . . . . . 78
Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.2.3 Coupling of sub-problems . . . . . . . . . . . . . . . . . . . . . . . . 82
Some remarks on distance function . . . . . . . . . . . . . . . . . . . 84
Numerical example for the coupled system . . . . . . . . . . . . . . . 86
7.3 Adaptive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.3.1 Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A priori error estimates . . . . . . . . . . . . . . . . . . . . . . . . . 90
A posteriori error estimates . . . . . . . . . . . . . . . . . . . . . . . 90
7.3.2 Adaptive refinement strategies. Equidistribution strategy . . . . . . 91
Adaptive refinement for elliptic problems . . . . . . . . . . . . . . . 92
Adaptive refinement for parabolic problems . . . . . . . . . . . . . . 94
A recursive approach to mesh refinement and coarsening . . . . . . . 95
7.4 Adaptive method for cutting model . . . . . . . . . . . . . . . . . . . . . . . 97
Temperature controlled adaptive refinement . . . . . . . . . . . . . . 97
Level set based adaptive refinement . . . . . . . . . . . . . . . . . . 98
Combined adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101iv Contents
7.5.1 Thermal cutting of a workpiece . . . . . . . . . . . . . . . . . . . . . 101
7.5.2 Flattening effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.5.3 Sensitivity to numerical parameters . . . . . . . . . . . . . . . . . . 103
7.5.4 Sensitivity to model parameters . . . . . . . . . . . . . . . . . . . . . 108
7.5.5 Topological changes . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Conclusions 114
8.1 Summary of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2 Remarks on further developments . . . . . . . . . . . . . . . . . . . . . . . . 115
Appendices 117
A Viscosity solution method 117
B Finite element method 121
B.0.1 Galerkin discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C ALBERTA - An adaptive finite element toolbox 128
Bibliography 129Chapter 1
Introduction
1.1 General: the plasma cutting process
There is a wide range of thermal cutting techniques available for the shaping of materials.
One example is the plasma cutting. The origin of plasma-arc process goes