Uncertainty Propagation Through Large Nonlinear Models William Becker

225 Pages·2015·34.52 MB·English

Uncertainty Propagation Through Large
Nonlinear Models
William Becker
A thesis submitted to the University of Sheffield for
the degree of Doctor of Philosophy in the faculty of
Engineering.
Department of Mechanical Engineering
January 2011
University of Sheffield
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Abstract
Uncertainty analysis in computer models has seen a rise in interest in recent years as
a result of the increased complexity of (and dependence on) computer models in the
design process. A major problem however, is that the computational cost of propagating
uncertainty through large nonlinear models can be prohibitive using conventional methods
(such as Monte Carlo methods). A powerful solution to this problem is to use an emulator,
which is a mathematical representation of the model built from a small set of model runs
at specified points in input space. Such emulators are massively cheaper to run and
can be used to mimic the "true" model, with the result that uncertainty analysis and
sensitivity analysis can be performed for a greatly reduced computational cost. The work
here investigates the use of an emulator known as a Gaussian process (GP), which is
an advanced probabilistic form of regression, hitherto relatively unknown in engineering.
The GP is used to perform uncertainty and sensitivity analysis on nonlinear finite element
models of a human heart valve and a novel airship design. Aside from results specific to
these models, it is evident that a limitation of the GP is that non-smooth model responses
cannot be accurately represented. Consequently, an extension to the GP is investigated,
which uses a classification and regression tree to partition the input space, such that
non-smooth responses, including bifurcations, can be modelled at boundaries. This new
emulator is applied to a simple nonlinear problem, then a bifurcating finite element model.
The method is found to be successful, as well as actually reducing computational cost,
although it is noted that bifurcations that are not axis-aligned cannot realistically be dealt
with.
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Contents
Abstract iii
Acknowledgements ix
1 Introduction 1
1.0.1 A mack Box View 2
1.1 Uncertainty in Computer Models 3
1.2 Sources of Uncertainty 4
1.3 Uncertainty Analysis. 5
1.3.1 Computational Expense 5
· .....
1.3.2 Solutions 5
1.3.3 Sensitivity Analysis 6
1.4 Objectives of this Thesis . 6
1.5 Summary of Chapters .. 7
2 Uncertainty Analysis: An Overview 11
2.1 Issues in Uncertainty Analysis. 11
2.1.1 Quantification 12
2.1.2 Fusion .. · . 21
2.1.3 Propagation . 22
2.1.4 Summary · . 23
2.2 Sensitivity Analysis: an Extension of Uncertainty Analysis. 24
2.2.1 1Iotivations ......... . 24
2.2.2 Levels of Sensitivity Analysis 25
2.2.3 Summary........... 30
2.3 Propagating Probabilistic Uncertainty 30
2.3.1 General Approaches ..... . 30
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CONTENTS vi
2.3.2 The Stochastic Finite Element Method. 33
2.3.3 Random l\fatrices. . . . . . . . . . 36
2.3.4 Summary of Propagation Methods 36
2.4 Emulators for Uncertainty Analysis. 37
2.4.1 About Emulators . 37
2.4.2 Types of Emulator 38
2.5 Conclusions . . . . . . . . 41
3 Gaussian Processes and Bayesian Sensitivity Analysis 43
3.1 Gaussian Processes . . . . 44
3.1.1 Prior Specification 46
3.1.2 Conditioning on the Training Data 48
3.1.3 Marginalising Hyperparameters . . 50
3.1.4 Estimating Roughness Parameters 55
3.2 Inference for Sensitivity Analysis ..... 56
3.2.1 Mean, Variance and Main Effects. 56
3.2.2 Sensitivity Indices · ........ 59
3.2.3 Derivation of Integrals for Sensitivity Indices 61
3.3 Conclusions . . . . . . . . · .............. 66
4 Modelling the Aortic Valve 69
4.1 The Aortic Valve . . . . . · ..... 70
4.1.1 Valve Failure and Prostheses 70
4.1.2 Material Properties. 72
4.2 Simulating the Aortic Valve 73
4.2.1 Existing Work 73
4.2.2 Uncertainties 74
4.3 The Finite Element Models 75
4.3.1 The Dry Model . 76
4.3.2 The Wet Model . 78
4.4 Conclusions . . . . . . . 84
5 Uncertainty Analysis of the Aortic Valve Models 87
.....
5.1 Approach 87
5.1.1 Dry Model 88
CONTENTS vii
5.1.2 Wet model 89
5.2 Results....... 91
5.2.1 Model Outputs 91
5.2.2 Dry Model Results 92
5.2.3 \Vet l\Iodel Results . 98
5.3 Discussion. .108
5.4 Conclusions · 110
6 Modelling an Airship 113
6.1 Airships and Applications · 113
6.2 The Elettra Twin Flyers · 114
6.2.1 Design. . ... · 114
6.2.2 l\laterials and Uncertainties · 116
6.3 The Airship Models ...... . · 119
6.3.1 Nonlinear Dry Simulation · 121
6.3.2 Nonlinear Wet Simulation · 123
6.4 Conclusions ........... . · 127
7 Uncertainty Analysis of the Airship Model 129
7.1 Dry Model. . . . . . . . 129
7.1.1 VA Approach. . 129
7.1.2 Results . 131
7.2 Wet Model . 135
7.3 Conclusions . 138
8 Tree-Structured Gaussian Processes for Uncertainty Analysis 141
8.1 Classification and Regression Trees . . . . . . 142
8.1.1 Growing Trees - Greedy Algorithms . 143
8.2 Bayesian CART. . 144
8.2.1 Overview . 144
8.2.2 Prior Specification . 145
8.2.3 Searching the Posterior Distribution . 148
8.2.4 Comments on the Posterior Search . 151
8.2.5 Extension to Tree-Structured GPs · 151
8.2.6 Application to VA/SA ...... . · 155
CONTENTS viii
8.2.7 Summary of Tree-Structured CPs · 155
8.3 Example: Duffing Oscillators . . · 156
8.3.1 About Duffing Oscillators · 156
8.3.2 Results .. · 156
8.3.3 Discussion. · 163
8.4 Conclusions.... · 164
9 Emulating a Bifurcating System 167
9.1 Dry Leaflet Model · 167
9.2 Results ....... · 168
9.2.1 Raw Data. · 168
9.2.2 Fitting Emulators · 170
9.2.3 Reduced Training Data .175
9.3 Conclusions . . . . . . . . . . . · 178
10 Conclusions and Further Work 179
10.1 Emulator-based Uncertainty Analysis. · 179
10.2 Practical Observations . . · 181
10.3 Heart Valves and Airships · 182
10.4 Further Work · 182
References 184
A Publications 195
D Practical Aspects of Bayesian Uncertainty Analysis 197
C The Aortic Valve Model 205
Acknow ledgements
This thesis is the product of some great ideas from some very clever people, and some hard
work on my part, particularly inspired by very tight deadlines towards the end. First of
all, I would like to thank Keith Worden, who provided the ideas for the vast majority of
the work presented here, took the patience to explain them to me when necessary (quite
often, in fact!) and was always kind enough to take me and other students for curries,
meals and pub visits in Sheffield and beyond. It truly has been a privilege to work with
such a hugely intelligent but extremely friendly and approachable man such as Keith, who
will always do his utmost to help out ailing PhD students. I will very much miss working
with him when I move away from the group. I also express my thanks to Graeme Manson,
who as my other supervisor was always available for help when needed.
On the academic front, thanks must go to Jeremy Oakley, who was one of the main
creators of much of the theory in Chapter 3, also for his patience in attempting to explain
Bayesian statistics to a newly-graduated student with an A-level statistics education. I
also gratefully received much help from James Hensman concerning Gaussian processes,
although many times it went over my head! For the latter stages of my PhD I would like
to thank the members of "Bayes Club", which also included Rob Barthorpe and Lizzy
Cross. These discussions helped me enormously with the latter chapters of this thesis and
were also very enjoyable.
For help on the FE side of things, thanks go to Alaster Yoxall for his advice on heart valves
and modelling, and Jen Rowson for help with LS-Dyna. I must also thank James Kennedy
from KBS2 Inc. for his extremely generous assistance via the LS-Dyna forum on numerous
aspects of the use of LS-Dyna, which, incidentally, is woefully under-documented.
A special thanks goes to Jem Rongong, who firstly helped with funding in the latter stages
of the PhD, and secondly was very understanding about my time restrictions in the final
two months.
Thanks must inevitably go to all my friends in the department who have helped me over
the years with endless tips and assistance with various software packages and matters
relating to general PhD life. But much more valuably, they have made working in this
research group a wonderful experience. I can only hope that my next working environment
will be comparable. Specific thanks must go to the members of Lunch Club: the leaders
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Vaggelis (Blue) and Giovanni (Pink) for being my best friends over the past three years,
plus the more peripheral members Pete, Rob and Dorit for plenty of good trips to the
pub. Lunch Club alumni must also be mentioned - Frank Stolze provided endless help
and jokes in the office and is very much missed, as well as gli Italiani. All these people
have made working in the DRG a great experience. Thanks to all the people in my office
for putting up with my grumpiness in the last months of thesis-writing!
I must also thank the people who I met in Torino. Cecilia Surace and David Storer for their
warm welcome and endless help, including several excellent ski outings. All the Italians
for inviting me for excellent dinners of vast proportions, particularly Marco Gherlone
and Maria De Stefano. I must also thank Melegcim for being such an excellent friend
throughout the last few months of the visit and beyond.
Thanks also go to all the people who were not directly involved in the thesis, but have
been great friends and family throughout, which inevitably helps. My Mum and Dad have
always been there for me and I love them dearly. My brother Alex and Dave (members of
Magiklegz) are my oldest and best friends and I can count on them for anything. I should
particularly thank Valeria Amenta, who was probably the only thing stopping me going
crazy at a few points in the last months. Grazie mille Tortigliona!
I would like to thank Ove Arup for the use of LS-Dyna. The activity presented in Chapters
6 and 7 was part of the project entitled "Innovative Solutions for Control Systems, Electric
Plant, Materials and Technologies for a Non-Conventional Remotely-Piloted Aircraft"
funded by the Piedmont Region of Italy. A large part of the work here was funded by
a Platform Grant for uncertainty management in nonlinear systems. Funding also came
from the "Bridging the Gaps" initiative.
-p.s. Although I can't exactly thank my dog for any academic input, I would like to
dedicate this thesis to Bess, who is sorely missed.