Shape optimization under uncertainty from a stochastic programming point of view [Elektronische Ressource] / von Harald Held

Harald Held

Sujets

• Mathematics

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2009
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

Shape Optimization under Uncertainty from a
Stochastic Programming Point of View
Vom Fachbereich Mathematik der
Universit at Duisburg-Essen
(Campus Duisburg)
zur Erlangung des akademischen Grades eines
Dr. rer. nat.
genehmigte Dissertation
von
Harald Held
aus Oberhausen
Referent: Prof. Dr. Rudiger Schultz
Korreferent: Prof. Dr. Martin Rumpf
Datum der mundlic hen Prufung: 02.02.2009iiAcknowledgments
I owe a great deal to my supervisors, colleagues, and friends who have always sup-
ported, encouraged and enlightened me through their own research, comments and
questions.
When I started as a freshman at the University of Duisburg nearly a decade
ago, the very rst lecture I attended was given by Prof. Dr. Rudiger Schultz.
Undoubtedly, it was his enthusiasm and passion for mathematics that kindled my
interests and ambitions at that time, which nally led to this thesis. For that, his
constant motivation and invaluable advice, his encouragement to pursue my own
ideas, and the faith he put in me, I thank him deeply.
I further thank Prof. Dr. Martin Rumpf for his support, invaluable advice, and
helpful ideas that proved useful in many di cult situations. I am also thankful to
him and his group for allowing me access to their excellent software library, which
was a great asset to my research.
I am grateful to all of my colleagues for their willingness to genuinely help and
discuss virtually everything at any time, providing the most pleasant work envi-
ronment. In particular, I would like to express my gratitude to Ralf Gollmer, Uwe
Gotzes, and Martin Pach for fruitful discussions and suggestions.
Not least I thank my wife, Karina, for her patience, love, and proofreading. I
spent many evenings and weekends writing and \bug squishing". For that I am in
her debt.
iiiivAbstract
We consider an elastic body subjected to internal and external forces which are un-
certain. Simply averaging the possible loadings will result in a structure that might
not be robust for the individual loadings at all. Instead, we apply techniques from
level set based shape optimization and two{stage stochastic programming: In the
rst stage, the non-anticipative decision on the shape has to be taken. Afterwards,
the realizations of the random forces are observed, and the variational formulation
of the elasticity system takes the role of the second-stage problem. Taking ad-
vantage of the PDE’s linearity, we are able to compute solutions for an arbitrary
number of scenarios without increasing the computational e ort signi cantly. The
deformations are described by PDEs that are solved e ciently by Composite Finite
Elements. The objective is, e.g., to minimize the compliance. A gradient method
using the shape derivative is used to solve the problem. Results for 2D instances are
shown. The obtained solutions strongly depend on the initial guess, in particular
its topology. To overcome this issue, we included the topological derivative into our
algorithm as well.
The stochastic programming perspective also allows us to incorporate risk mea-
sures into our model which might be a more appropriate objective in many practical
applications.
Parts of this thesis have been published in [32].
vviContents
1 Introduction 1
1.1 The Elasticity PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . 6
1.2 Shape Optimization Problems . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Two{Stage Stochastic Programming . . . . . . . . . . . . . . . . . . . 16
1.3.1 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Solution of the Elasticity PDE 25
2.1 Composite Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1.1 Construction for the Neumann Boundary . . . . . . . . . . . . 29
2.1.2 for the Dirichlet Boundary . . . . . . . . . . . . 36
2.1.3 Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . 43
2.1.4 Computation of the System Matrix and the Right-Hand Side
Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Stochastic Programming Perspective 47
3.1 Stochastic Shape Optimization Problem . . . . . . . . . . . . . . . . 48
3.1.1 Two{Stage Stochastic Shape Optimization Problem . . . . . . 50
3.1.2 Dual Problem and Saddle Point Formulation . . . . . . . . . . 52
3.2 Reformulation and Solution Plan for the Expectation based Model . . 59
3.3 Expected Excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Barrier Method . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.2 Smooth Approximation . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Excess Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
viiviii Contents
4 Solving Shape Optimization Problems 73
4.1 Level set Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.1 Computation of the Mean Curvature . . . . . . . . . . . . . . 76
4.2 Shape Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Topological Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Steepest Descent Algorithm . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.1 Regularized Descent Direction . . . . . . . . . . . . . . . . . . 93
5 Numerical Results 97
5.1 Deterministic and Expectation based Results . . . . . . . . . . . . . . 98
5.1.1 VSS and EVPI . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A Appendix 117
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.2 Important Facts and Theorems . . . . . . . . . . . . . . . . . . . . . 121
References 123Symbol Index
O The elastic body
Part of the boundary that is to be optimized0
The xed Dirichlet boundaryD
Neumann boundary where the surface loads act onN
; Lame coe cients
Level set function
Vector of probabilities
nR n-dimensional Euclidean space
! A scenario
J(O) =J(O;u(O)) Shape objective functional
A Elasticity tensor
e(u) Linearized strain tensor
thf i partial derivative of a scalar function f, see A.4 on page 119;i
1 2V Function space H (O;R )
D
D Working domain that contains all admissible shapes
A more detailed overview of the notations we used can be found in the Appen-
dix A on page 117.
ixx