Numerical Approximation of the Magnetoquasistatic Model with Uncertainties: Applications in Magnet Design

128 Pages·2016·1.76 MB·English

by  Ulrich Römer

Springer Theses
Recognizing Outstanding Ph.D. Research
Ulrich Römer

Numerical
Approximation of the
Magnetoquasistatic
Model with
Uncertainties
Applications in Magnet Design
Springer Theses
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ö
Ulrich R mer
Numerical Approximation
of the Magnetoquasistatic
Model with Uncertainties
Applications in Magnet Design
Doctoral Thesis accepted by
ä
Technische Universit t Darmstadt, Germany
123
Author Supervisor
Dr.-Ing Ulrich Römer Prof. ThomasWeiland
Institut für Theorie Elektromagnetischer Institut für Theorie Elektromagnetischer
Felder Felder
Technische UniversitätDarmstadt Technische UniversitätDarmstadt
Darmstadt Darmstadt
Germany Germany
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-3-319-41293-1 ISBN978-3-319-41294-8 (eBook)
DOI 10.1007/978-3-319-41294-8
LibraryofCongressControlNumber:2016943807
©SpringerInternationalPublishingSwitzerland2016
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’
Supervisor s Foreword
Uncertainties and manufacturing tolerances play an important role in accelerator
physics, in view of extreme design requirements and high production costs.
Quantifying uncertainties in magnetic fields is paramount for the design of accel-
erator magnets and other engineering applications such as machines and trans-
formers. This thesis proposes a thorough and general mathematical approach for
modeling and solving stochastic magnetic field problems. Variability in material
properties and the geometry is discussed with emphasis on the preservation of
structural physical and mathematical properties. The latter is used to show the
unique solvability of the resulting nonlinear stochastic equations. Also, efficient
discretization methods for random inputs are described, which are new in this
context. The stochastic partial differential equations are solved using adjoint based
perturbation methods as well as recently emerged higher order spectral approxi-
mation schemes. Thereby, this thesis gives both theoretical results on convergence
rates and error estimation and addresses numerical examples with practical rele-
vance, in particular recently designed accelerator magnets.
Darmstadt, Germany Prof. Thomas Weiland
December 2015
v
Parts of this thesis have been published in the following documents:
Journal articles
[1] Andreas Bartel, Herbert De Gersem, Timo Hülsmann, Ulrich Römer,
Sebastian Schöps, and Thomas Weiland. Quantification of uncertainty in the
field quality of magnets originating from material measurements. IEEE
Transactions on Magnetics, 49(5):2367–2370, 2013.
[2] Ulrich Römer, Sebastian Schöps, and Thomas Weiland. Approximation of
moments for the nonlinear magnetoquasistatic problem with material uncer-
tainties. IEEE Transactions on Magnetics, 50(2), 2014.
[3] Ulrich Römer, Sebastian Schöps, and Thomas Weiland. Stochastic modeling
and regularityof thenonlinear elliptic curl-curl equation.SIAM/ASA Journal
on Uncertainty Quantification (in press).
vii
Acknowledgments
IamverygratefulforthesupportandencouragementthatIreceivedduringmytime
as a Ph.D. student. In particular I would like to thank:
(cid:129) Prof. Thomas Weiland for the supervision and support and for providing and
excellent working atmosphere at the Institut für Theorie Elektromagnetischer
Felder
(cid:129) Prof. Stefan Ulbrich for kindly accepting to referee the thesis and for very
valuable comments and discussions on nonlinear partial differential equations
(cid:129) Prof. Sebastian Schöps for the supervision and support and for numerous
valuable discussions on uncertainties and numerics
(cid:129) Dr.-Ing. Stephan Koch for the supervision at the beginning of the thesis
(cid:129) My (former) colleagues at TEMF, in particular, Jens Trommler, Martin
Lilienthal, Klaus Klopfer and Cong Liu
(cid:129) Uwe Niedermayer, Prof. Johann Baumeister and Prof. Peter Kloeden for com-
ments on the thesis
(cid:129) Michaela Arnold for the very good cooperation on magnet design
(cid:129) My family and friends
I would also like to acknowledge the support of the Deutsche
Forschungsgemeinschaft through CRC 634.
ix
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
References.. .... .... .... ..... .... .... .... .... .... ..... .... 3
2 Magnetoquasistatic Approximation of Maxwell’s Equations,
Uncertainty Quantification Principles .... .... .... .... ..... .... 5
2.1 Maxwell’s Equations . ..... .... .... .... .... .... ..... .... 5
2.2 Magnetoquasistatic Approximation.... .... .... .... ..... .... 6
2.3 Magnetoquasistatic Model .. .... .... .... .... .... ..... .... 8
2.4 Uncertainty Quantification, Verification and Validation ..... .... 12
2.5 Conclusion. .... .... ..... .... .... .... .... .... ..... .... 14
References.. .... .... .... ..... .... .... .... .... .... ..... .... 14
3 Magnetoquasistatic Model and its Numerical Approximation .. .... 17
3.1 Weak Formulation ... ..... .... .... .... .... .... ..... .... 17
3.2 Reformulation as a Minimization Problem.. .... .... ..... .... 21
3.3 Space Discretization.. ..... .... .... .... .... .... ..... .... 23
3.3.1 Higher Order Whitney Forms.. .... .... .... ..... .... 24
3.3.2 Spline Finite Elements ... .... .... .... .... ..... .... 25
3.3.3 Finite Element Formulation.... .... .... .... ..... .... 26
3.3.4 Finite Element a Priori Error Analysis ... .... ..... .... 27
3.4 Linearization.... .... ..... .... .... .... .... .... ..... .... 28
3.5 A Posteriori Error Analysis of Linearization and Discretization
Error.. .... .... .... ..... .... .... .... .... .... ..... .... 31
3.6 Temporal Discretization .... .... .... .... .... .... ..... .... 35
3.7 Conclusion. .... .... ..... .... .... .... .... .... ..... .... 36
References.. .... .... .... ..... .... .... .... .... .... ..... .... 36
xi
xii Contents
4 Parametric Model, Continuity and First Order Sensitivity
Analysis ... .... .... .... ..... .... .... .... .... .... ..... .... 39
4.1 Abstract Mathematical Reformulation of the Model... ..... .... 39
4.2 Definition of the Model Inputs and Parametrization... ..... .... 40
4.2.1 The Material Coefficient as a Model Input .... ..... .... 41
4.2.2 Approximation of the Magnetic Material Coefficient.. .... 42
4.2.3 The Shape of the Interface as a Model Input .. ..... .... 43
4.2.4 Approximation and Representation of Shapes.. ..... .... 45
4.2.5 The Source Current Density as a Model Input . ..... .... 47
4.2.6 Conductor Models... .... .... .... .... .... ..... .... 47
4.3 Continuity with Respect to the Input Data .. .... .... ..... .... 48
4.4 Sensitivity Analysis, Direct Approach . .... .... .... ..... .... 50
4.4.1 Magnetic Material Coefficient Sensitivity . .... ..... .... 51
4.4.2 Interface Sensitivity.. .... .... .... .... .... ..... .... 54
4.4.3 Source Current Sensitivity. .... .... .... .... ..... .... 58
4.5 Sensitivity Analysis, Adjoint Approach .... .... .... ..... .... 59
4.6 Sensitivity Analysis for the Time Transient Case. .... ..... .... 60
4.7 Conclusion. .... .... ..... .... .... .... .... .... ..... .... 61
References.. .... .... .... ..... .... .... .... .... .... ..... .... 62
5 Uncertainty Quantification..... .... .... .... .... .... ..... .... 65
5.1 Uncertainty Modeling. ..... .... .... .... .... .... ..... .... 65
5.1.1 Probabilistic Description of Uncertainties . .... ..... .... 66
5.1.2 Karhunen-Loève Expansion ... .... .... .... ..... .... 66
5.1.3 Stochastic Formulation and KL Modeling Error ..... .... 70
5.2 Model Dimension Reduction and Uncertainty Propagation... .... 73
5.2.1 Dimension Reduction .... .... .... .... .... ..... .... 74
5.2.2 Monte Carlo Sampling ... .... .... .... .... ..... .... 76
5.2.3 Perturbation Methods for the Statistical Moments.... .... 77
5.2.4 Collocation Based Polynomial Chaos Method.. ..... .... 83
5.2.5 Worst-Case Scenario. .... .... .... .... .... ..... .... 86
5.3 Conclusion. .... .... ..... .... .... .... .... .... ..... .... 88
References.. .... .... .... ..... .... .... .... .... .... ..... .... 88
6 Uncertainty Quantification for Magnets .. .... .... .... ..... .... 91
6.1 Field Equations for Magnets and Multipole Coefficients..... .... 92
6.2 Numerical Examples.. ..... .... .... .... .... .... ..... .... 95
References.. .... .... .... ..... .... .... .... .... .... ..... .... 103
7 Conclusion and Outlook.. ..... .... .... .... .... .... ..... .... 105
7.1 Conclusion. .... .... ..... .... .... .... .... .... ..... .... 105
7.2 Outlook ... .... .... ..... .... .... .... .... .... ..... .... 106