Sensitivity and Uncertainty Analysis of Coupled Reactor Physics … · 2017-09-14 · Sensitivity...

Sensitivity and Uncertainty Analysisof Coupled Reactor Physics Problems

Method Development for Multi-Physics in Reactors

Zoltán Perkó

Sensitivity and Uncertainty A

nalysis of Coupled Reactor Physics Problems

Zoltán Perkó

DelftUniversity of Technology

R 0 R limit

σR

µR

Adjoint problem

Forward problem

Integration levels

Polynom

ial Chaos

Cubature poi

nts

Gauss

ian

dist

ribut

ion

Covariance matrix

Unce

rtain inputs

S&U tech

nique

Uncertain

out

put

Department of Nuclear Energy and Radiation Applications

These propositions are considered opposable and defendable and as such have been approved by the promotor, Pro. Dr. ir. T. H. J. J. van der Hagen.

Propositions

Belonging to the PhD thesis of Zoltán Perkó

Sensitivity and Uncertainty Analysis of Coupled Reactor Physics Problems – Method

Development for Multi-Physics in Reactors

1. The advantages of adjoint sensitivity analysis surpass its obvious shortcomings even for nonlinear

problems. Therefore adjoint methods should be implemented for nonlinear coupled problems as well.

(Chapters 2 and 3 of this thesis)

2. Adaptive Polynomial Chaos methods are computationally cheaper and more accurate than sampling based

methods even for problems with a relatively high (≈50) number of input parameters. (Chapters 4 and 5 of

this thesis)

3. Research on deterministic sensitivity and uncertainty analysis methods is most beneficial for

interdisciplinary applications. (C. M. H. Hartman, “Sensitivity and Uncertainty Analysis in Proton Therapy

Treatment Plans Using Polynomial Chaos Methods”, MSc. thesis, 2014)

4. In practice stochastic algorithms will always be favoured over deterministic ones for sensitivity and

uncertainty analysis due to their ease of use.

5. Method and user related uncertainties are generally larger than data related uncertainties in

computational sciences and should therefore get much more focus. (S. N. Aksan, F. D’Auria, H. Städtke,

“User effects on the thermal-hydraulic transient system code calculations”, Nuclear Engineering and

Design, 145, pp. 159-174, 1993)

6. Electricity generation by newly built nuclear power plants will only become economically competitive with

traditional (fossil-fuelled) power plants if the latter are also obliged to cover the full costs of their waste

management. (C. Mari, “The costs of generating electricity and the competitiveness of nuclear power”,

Progress in Nuclear Energy, 73, pp. 153-161, 2014)

7. In countries with low levels of foreign language proficiency the cheapest and most practical method for

increasing it is subtitling television programs instead of using dubbing or voice-over. (C. M. Koolstra, J. W. J.

Beentjes, “Children’s Vocabulary Acquisition in a Foreign Language Through Watching Subtitled Television

Programs at Home”, Education Technology Research and Development, 47, pp. 51-60, 1999)

8. Renewable energy sources alone cannot solve the problem of climate change. (Google “RE<C” project

(https://www.google.org/rec.html))

9. The scientifically most sound resolution of the Fermi paradox1 is that the contact cross section between

intelligent spacefaring civilizations is far less than what could be expected based on spatial-temporal

arguments. (G.D. Brin, “The ‘Great Silence’: the Controversy Concerning Extraterrestrial Intelligent Life”,

Quarterly Journal of the Royal Astronomical Society, 24 (3), pp. 283-309, 1983)

10. Focusing on proliferation concerns in nuclear fuel cycle policies is counterproductive, since it prevents the

nuclear community from applying already existing technologies (e.g. PUREX) and better directing research

to safeguard nuclear materials. (J. Mueller, “Simplicity and Spook: Terrorism and the Dynamics of Threat

Exaggeration”, International Studies Perspectives, 6, pp. 208-234, 2005)

1 The Fermi paradox refers to the apparent contradiction between the seemingly high estimates for the

probability of the existence of extra-terrestrial intelligent life forms and our lack of evidence for it.

Deze stellingen worden opponeerbaar en verdedigbaar geacht en zijn als zodanig goedgekeurd door de promotor, Prof. Dr. Ir. T. H. J. J. van der Hagen.

Stellingen

Behorende bij het proefschrift van Zoltán Perkó

Sensitivity and Uncertainty Analysis of Coupled Reactor Physics Problems – Method

Development for Multi-Physics in Reactors

1. De voordelen van geadjugeerde gevoeligheidsanalyse wegen zelfs voor non-lineaire problemen op tegen

de tekortkomingen. Daarom dienen ook voor niet-lineaire gekoppelde problemen geadjugeerde methoden

geïmplementeerd te worden. (Hoofdstukken 2 en 3 van dit proefschrift)

2. Adaptieve Polynoom-Chaos methoden kosten minder rekentijd en zijn preciezer dan

signaalbemonsteringsmethoden, zelfs voor problemen met relatief veel (≈50) invoerparameters.

(Hoofdstukken 4 en 5 van dit proefschrift)

3. Onderzoek naar deterministische gevoeligheids- en onzekerheidsanalyse methoden heeft vooral

voordelen voor interdisciplinaire toepassingen. (C. M. H. Hartman, “Sensitivity and Uncertainty Analysis in

Proton Therapy Treatment Plans Using Polynomial Chaos Methods”, MSc. thesis, 2014)

4. In de praktijk zullen stochastische algoritmes voor gevoeligheids- en onzekerheidsanalyse altijd verkozen

worden boven deterministische omdat ze makkelijker zijn in gebruik.

5. Methode- en gebruikergerelateerde onzekerheden in de computerwetenschappen zijn in het algemeen

groter dan datagerelateerde onzekerheden, en behoren daarom meer aandacht te krijgen. (S. N. Aksan, F.

D’Auria, H. Städtke, “User effects on the thermal-hydraulic transient system code calculations”, Nuclear

Engineering and Design, 145, pp. 159-174, 1993)

6. Nieuwe kerncentrales kunnen alleen concurreren met traditionele (fossiel-gestookte) centrales, als de

laatst genoemde ook verplicht worden op te draaien voor de volledige kosten die horen bij het

afvalmanagement. (C. Mari, “The costs of generating electricity and the competitiveness of nuclear

power”, Progress in Nuclear Energy, 73, pp. 153-161, 2014)

7. In landen waar vreemde talen slecht beheerst worden, is het ondertitelen van televisieprogramma’s, in

plaats van het nasynchroniseren daarvan, de goedkoopste en de meest praktische methode om het niveau

van de beheersing van vreemde talen op te krikken. (C. M. Koolstra, J. W. J. Beentjes, “Children’s

Vocabulary Acquisition in a Foreign Language Through Watching Subtitled Television Programs at Home”,

Education Technology Research and Development, 47, pp. 51-60, 1999)

8. De inzet van alleen hernieuwbare energiebronnen kan het probleem van klimaatverandering niet oplossen.

(https://www.google.org/rec.html))

9. De wetenschappelijk gezien meest logische oplossing van de Fermi paradox2 is dat de kans op contact

tussen intelligente ruimte-bevarende samenlevingen veel kleiner is dan wat verwacht mag worden op

basis van ruimtelijke en temporele argumenten. (G.D. Brin, “The ‘Great Silence’: the Controversy

Concerning Extraterrestrial Intelligent Life”, Quarterly Journal of the Royal Astronomical Society, 24 (3), pp.

283-309, 1983)

10. De aandacht voor proliferatiezorgen in nucleair brandstofcyclusbeleid is contraproductief, omdat het de

nucleaire gemeenschap belet om reeds bestaande technologieën (zoals PUREX) toe te passen en om

onderzoek te verrichten ten behoeve van het veilig beheer van nucleair materiaal. (J. Mueller, “Simplicity

and Spook: Terrorism and the Dynamics of Threat Exaggeration”, International Studies Perspectives, 6, pp.

208-234, 2005)

2 De Fermi paradox refereert aan de ogenschijnlijke tegenstelling tussen de schijnbare hoge schattingen van de

kans dat buitenaards intelligent leven bestaat, ondanks dat daar geen bewijs voor is.

Sensitivity and Uncertainty Analysisof Coupled Reactor Physics

Problems - Method Developmentfor Multi-Physics in Reactors

Sensitivity and Uncertainty Analysisof Coupled Reactor Physics

Problems - Method Developmentfor Multi-Physics in Reactors

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universtiteit Delft,

op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben,voorzitter van het College voor Promoties,

in het openbaar te verdedigenop 21 januari 2015 om 12:30 uur

door

Zoltán PERKÓ

Master of Science in Engineering Physics, Budapesti Műszaki Egyetem

geboren te Keszthely, Hungary

Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. T.H.J.J. van der Hagen

Samenstelling promotiecommissie:Rector Magnificus,Prof. dr. ir. T.H.J.J. van der Hagen,Dr. ir. J.L. Kloosterman,Prof. dr. ir. drs. H. Bijl,Prof. dr. A. W. Heemink,Prof. C. Pain,Prof. P. Ravetto,Dr. O.P. Le Maître,

voorzitterTechnische Universiteit Delft, promotorTechnische Universiteit Delft, supervisorTechnische Universiteit DelftTechnische Universiteit DelftImperial College London, Verenigd KoninkrijkPolitecnico di Torino, Italiaanse RepubliekLIMSI-CNRS, Orsay, Frankrijk.

Dr. Danny Lathouwers has provided invaluable guidance and support in thepreparation of this thesis.

c© 2014, Zoltán Perkó

All rights reserved. No part of this book may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, without priorpermission from the copyright owner.

ISBN 978-94-6295-071-9

Keywords: Sensitivity analysis, uncertainty quantification, adjoint, PolynomialChaos, coupled systems

The research described in this thesis was performed in the Physics of NuclearReactors (PNR)/Nuclear Energy and Radiation Applications (NERA) sectionof the department of Radiation, Radionuclides & Reactors (R3)/Radiation,Science and Technology (RST) of the Delft University of Technology, Delft,The Netherlands.

Cover design: Proefschriftmaken.nl || Uitgeverij BOXPressPrinted by: Proefschriftmaken.nl || Uitgeverij BOXPressPublished by: Uitgeverij BOXPress, ’s-Hertogenbosch

To my family

Financial supportThe research described in this thesis was partially funded by the GoFastRProject within the 7th Framework Program of the European Commission undercontract No. 249678.

Contents

Contents

1 Introduction 11.1 The Role of Computational Physics . . . . . . . . . . . . . . . 11.2 Importance of Uncertainties in Simulations . . . . . . . . . . 41.3 Traditional Methods for Sensitivity and Uncertainty Analysis 71.4 Contents and Overview of This Thesis . . . . . . . . . . . . . 11

2 Adjoint Techniques for Coupled Criticality Problems 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Solution of the Coupled Adjoint Problem . . . . . . . . . . . 252.4 Application to a One-Dimensional Slab . . . . . . . . . . . . . 272.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Practical Applicability of Adjoint Techniques to CoupledProblems 433.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Adaptive Polynomial Chaos Techniques for Multi-PhysicsProblems 714.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

vi

Contents

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Large Scale Application of Adaptive PCE 1215.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2 Computational Cost Reduction Techniques . . . . . . . . . . 1225.3 Post-Processing PCE . . . . . . . . . . . . . . . . . . . . . . . 1245.4 Application to a Gas Cooled Fast Reactor Transient . . . . . 1265.5 Sources of Uncertainties . . . . . . . . . . . . . . . . . . . . . 1325.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.7 Computational Costs and Cost Reduction Techniques . . . . 1555.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6 Conclusions and Recommendations 1636.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 165

Bibliography 167

A Notation 179

B Derivation of Perturbation Formulas and Adjoint Problems 181B.1 Derivation of the Power Perturbation Expressions . . . . . . . 181B.2 Derivation of the Eigenvalue Perturbation Expressions . . . . 183

C Constrained Quantities in Sensitivity and Uncertainty Ana-lysis 187C.1 Constrained Sensitivities . . . . . . . . . . . . . . . . . . . . . 187C.2 Covariance Matrix of Linearly Constrained Quantities . . . . 189C.3 Sensitivities with Polynomial Chaos Expansion . . . . . . . . 191

Summary 195

Samenvatting 197

Acknowledgements 201

List of Publications 203

Curriculum Vitae 207

vii

Contents

viii

Chapter 1

Introduction

This thesis is about the development of novel methods for the sensitivity anduncertainty (S&U) analysis of coupled multi-physics problems encountered inreactor physics. For understanding the main motivation behind the presentedwork in this chapter a general introduction to S&U analysis is given. Firstthe relevance of sensitivity and uncertainty analysis within the domain ofcomputational physics is briefly reviewed, then the most important traditionalmethods for performing S&U analysis are outlined. Finally, the main researchtopics - namely adjoint methods and adaptive Polynomial Chaos techniques -are shortly discussed and an overview of the contents is given.

1.1 The Role of Computational Physics

Traditionally physics is divided into two equally important, complementarybranches: theoretical and experimental physics. In theoretical physics funda-mental axioms are established, from which physical laws expressed as mathem-atical equations are derived to explain known phenomena and predict new ones.In reactor physics for example, the linear Boltzmann equation is obtained fromthe basic principle of particle conservation. The resulting transport equationseen in Figure 1.1 reflects this by stating that the change of the φ flux ofneutrons (or the equivalent density of neutrons) having energy E traveling indirection Ω at any r point in space is equal to the difference between the F φ

1

1. Introduction

source and the Lφ loss of such neutrons. By specifying the individual sourceand loss terms - usually including fission in the former, leakage, absorption andscatter in the latter case - the transport equation can then be used to predictthe neutron distribution in a nuclear reactor or the neutron dose beyond thesurrounding biological shielding for instance.

In experimental physics practical techniques and methods are used, andexperimental devices are constructed with which reproducible and quantitativemeasurements can be performed to check theoretical predictions, discover newphenomena and measure various physical quantities. For the nuclear field theassembly of cross section libraries is the most important example, which isan extensive experimental procedure. First, empirical data is collected andrecorded about the cross sections (or probabilities) of particular reactionsbetween neutrons and various nuclei, typically in multiple measurements, oftendone by several different experimentalists in independent laboratories. Thenthe raw data is analysed and combined with theoretical models, after whichthe results are reported in some structured format, such as the EvaluatedNuclear Data Files (ENDF) (ENDFB6 Format Manual). Finally, these filesare assembled into comprehensive cross section libraries, containing the energydependent cross sections of all types of reactions for a wide range of isotopes.In Figure 1.1 an example of the outcome of such a procedure can be seen, wherethe fission cross section of Pu-239 taken from the most recent ENDF/B-VII.1evaluation is depicted (IAEA, 2014).

With the rapid development of computers in the 20th century computationalphysics became a third, equally important branch of physics. It focuses ondeveloping and implementing fast, efficient and accurate numerical algorithmsto solve the mathematical equations describing the behaviour of physical sys-tems. On one hand computational physics only complements the traditionaldisciplines, since today performing state of the art theoretical or experimentalresearch is no longer possible without some - if not intensive - computationaleffort. Series expansions with hundreds of terms, calculating thousands ofintegrals and derivatives, or handling and processing vast amounts of exper-imental data are only a few of the examples where sophisticated numericaltools are required. On the other hand computational physics has a place of itsown, since it enables the solution of problems for which analytical approachesare inapplicable and experimental procedures are impractical or impossible.It is safe to say that practically all real life systems of interest fall into thiscategory, where performing measurements is too expensive and time consuming,

2

1.1. The Role of Computational Physics

), tΩ, E,r(Lφˆ−), tΩ, E,r(Fφ+ ˆ

), tΩ, E,r(φ∂t∂ =v

1

Incident Energy (MeV)1e-8 1e-6 1e-4 1e-2 1

0.1

1

10

100

1000 ENDF/B-VII.1

Pu-239 Fission Cross Section (barns)Theo

retic

al physics Experimental physics

Computational physics

Heat-

exchanger

Do

wn

-

com

er

Primarycircuit

He

Accumu-

lator

He

Upperplenum

DHR

loop

He

Secondary

circuit

He/N2

Water

H2O

Lowerplenum

Active

core

Figure 1.1: The three main branches of physics with examples from nuclearsystems

and where even if the physical laws for all relevant processes are known, thegoverning equations are so complicated that they seldom have an analyticalsolution. Engineering applications provide plenty of examples, the typical onewithin the domain of reactor physics is the transient behaviour of a wholenuclear power plant, which is determined by several interacting phenomena,all depending on different properties and acting on different spatial and tem-poral scales. The neutronic and thermal-hydraulic evolution of the active core,thermo-mechanical changes due to thermal expansion, chemical processes, theeffects of various primary, secondary or tertiary loop devices (such as pumps,heat-exchangers, turbines, valves, etc.) are all relevant for an accurate analysis,as are their delicate inter-dependencies. For such simulations usually several

3

1. Introduction

uni-physics computer codes have to be used, appropriate models have to bebuilt, coupling interfaces have to be designed, etc. The result is a complicatedmodel of the nuclear system (such as the one seen in Figure 1.1), which - ifwell built - captures all important details and can be used to calculate theanticipated response of the reactor to various incidents and accidental events.

1.2 Importance of Uncertainties in Simulations

Before the results obtained by simulations can be trusted it is imperativeto ensure that they are representative of reality, i.e. that under the givencircumstances reality would behave as predicted by our computational model.Two processes are aimed to address this issue, verification and validation (V&V)(Oberkampf and Trucano, 2008):

• Verification makes sure that the mathematical models are correctly pro-grammed in a computer code and are solved properly, hence it mainlyfocuses on the numerical approximations, discretization errors, trunca-tions, etc. The main tools of verification are the method of manufacturedsolutions, comparison with known analytical solutions, or comparisonwith highly accurate solutions of special or simplified cases.

• Validation makes sure that the computational model is physically accur-ate, i.e. that all relevant processes are included, the used mathematicalsimplifications and correlations are valid, and ultimately that the softwareis an acceptable representation of the real world from the perspectiveof its intended use. The key activities of validation are the quantitativecomparison of computational results with experimentally measured data,the assessment of accuracy, and the estimation of accuracy for extrapol-ated use of the code (i.e. under conditions that fell outside those of thevalidation experiments).

Uncertainties - or in other words our lack of knowledge about the exact valuesof certain parameters and variables, numerical imprecisions and hidden physicalprocesses - play an essential role throughout V&V activities. First, duringverification the numerical uncertainties must be quantified and proven to beacceptably low. Second, during validation the conducted experiments mustbe accompanied by a rigorous quantification of the uncertainties, uncertainty

4

1.2. Importance of Uncertainties in Simulations

information should be provided in terms of intervals, incomplete or completeprobability distributions for as many conditions and measured quantities aspossible. Third, when comparing computational results with experimentaldata the imprecise knowledge about the values of the input parameters have tobe taken into account by propagating the input uncertainties to the computedoutputs, this last action is called uncertainty propagation.

Uncertainties are commonly divided into two groups: aleatoric and epistemicuncertainties. Aleatoric uncertainties affect parameters which are inherentlystochastic and therefore their value is impossible to determine. For examplethe imprecise representation of numbers on computers, fluctuations in weatherconditions or neutron noise due to the stochasticity of nuclear reactions areall aleatoric uncertainties. Epistemic uncertainties affect parameters whosevalues are theoretically possible to know exactly, however they are not, mostoften due to the lack of experimental data. In reactor physics the most typicalexamples are cross section uncertainties caused by measurement uncertainties(see Figure 1.2). A key difference between the two classes is that whileepistemic uncertainties can be decreased with better measurements, aleatoricuncertainties can not due to their inherent stochastic nature.

In the terminology of verification and validation activities the general termsof uncertainty analysis (UA) and uncertainty quantification (UQ) encompassall of the aforementioned topics related to uncertainties. However, they arefrequently used in a narrower sense as synonyms for uncertainty propagation,focusing only on data type epistemic uncertainties and their effects on thecomputed outputs. Sensitivity analysis (SA) is a closely related, almostinseparable discipline, which studies the connection between the inputs of amodel or experiment and the calculated or measured outputs. It can provide adeeper understanding of the problems being investigated, can reveal the mainmechanisms in complex systems, furthermore it can identify the importanceof the different parameters and their contribution to the output uncertainties.Throughout this thesis these two phrases will be used with these meanings,i.e. sensitivity analysis referring to determining the connection between theoutputs and inputs, and uncertainty analysis referring to the propagation ofinput uncertainties to the computed outputs, respectively.

As computational physics plays an increasingly central role already in theplanning phase of most engineering projects, verification and validation activ-ities in general, and sensitivity and uncertainty analysis techniques in specific

5

1. Introduction

Incident Energy (MeV)

Pu

-23

9 F

issi

on

Cro

ss S

ect

ion

(b

arn

s)

10-8 10-6 10-4 10-2 110-1

1

10

102

103

ENDF/B-VII.12005 Rochman1980 Wagemans2010 Tovesson2010 Tovesson2006 Rochman2006 Rochman

Figure 1.2: Example of uncertainty for the Pu-239 fission cross section. Ex-perimental data from multiple measurements are shown together with theENDF/B-VII.1 evaluation. Data taken from (IAEA, 2014).

are expected to gain more importance as well. This is especially true for thenuclear field, where S&U analysis has long been an essential part of the designand operation nuclear installations (Usachev, 1964; Gandini, 1967; Stacey,1974; Williams, 1986; Ronen, 1988; Cacuci, 2003), for two main reasons. First,simulations are routinely used to ensure the safety of reactors during normaland off-normal conditions, and inaccurate predictions from such calculationscan clearly have severe consequences. Second, cross sections, the basic data forreactor calculations are characterized with significantly higher uncertaintiesthan most other physical quantities, as exemplified by Figure 1.2 (note thelogarithmic scale of the cross section). As in recent years regulatory bodiesstarted to shift their licensing approach from conservatism (i.e. the use of largeengineering margins to failure point in order to make up for hidden or unknown

6

1.3. Traditional Methods for Sensitivity and Uncertainty Analysis

processes) to Best Estimate Plus Uncertainty (BEPU) methodologies (Wilson,2013), where the best possible estimations are made augmented by rigoroussensitivity and uncertainty analysis, the emphasis on uncertainties has onlyintensified. Since most S&U analysis methodologies are either computationallyexpensive or involve significant additional modelling and coding efforts, thedevelopment of affordable, accurate and easily applicable S&U analysis tech-niques is expected to remain a prominent issue, and is the main topic of thisthesis.

1.3 Traditional Methods for Sensitivity andUncertainty Analysis

There are several well established methods for performing sensitivity and uncer-tainty analysis. The main distinction is usually made between stochastic anddeterministic methods. Stochastic, or in other words sampling based methodsuse repeated execution of the same simulation with different realizations ofthe uncertain input parameters. Therefore they are conceptually simple andrelatively easy to perform, however at the cost of significant computationaloverhead. Deterministic methods rely on solving a reformulated version ofthe problem at hand to incorporate uncertainties. As a result they are ingeneral theoretically more complicated than stochastic techniques and requiresubstantial analysis and additional code development, however they are com-putationally cheaper. In the rest of this section a brief overview of the mostoften used S&U analysis techniques is given.

1.3.1 Stochastic Methods

Stochastic techniques address uncertainties in a very straightforward way bysimulating different “possible scenarios” according to the probability of theiroccurrence and using the results to draw conclusions about the randomness ofthe quantities of interest. Correspondingly, all these methods follow the samethree steps:

• Generation of input parameters: during this step a sufficient number ofrandom realizations of the α input parameters are generated, accordingto their pα (α) joint probability density function (α ⊂ RN represents the

7

1. Introduction

N separate input parameters). The result is the αi : i = 1, ..., Ns set,containing Ns different values for each input parameter.

• Propagation of samples: during this step the same simulation is executedNs times, each time with a different αi realization of the inputs. Theoutcome is Ns realizations of the R (α) ⊂ RNR quantities of interest,constituting the αi → Ri = R (αi) : i = 1, ..., Ns mapping.

• Post-processing of results: in this step the corresponding input-outputpairs are used to compute response moments (mean, standard deviation,skewness, etc.), correlations (correlation and covariance matrices), com-plete probability density functions or cumulative distribution functions(PDFs or CDFs), draw scatterplots, perform regression analysis andstatistical tests, determine sensitivity coefficients (Pearson or partialcorrelation coefficients), etc.

The most basic of the stochastic S&U analysis techniques is the brute forceMonte Carlo (MC) approach, which uses standard methods for sampling thepα (α) joint probability density function of the inputs. In this case it is easyto construct unbiased estimators for the quantities of interest, for examplethe mean and the variance of any particular response R can be calculated

as µR = 1/Ns

Ns∑i

R (αi) and σ2R = 1/ (Ns − 1)

Ns∑i

(R (αi)− µR)2, respectively.

The main drawback is that the convergence of these estimators is slow, the errortypically decreases only according to 1/

√Ns. Therefore several techniques

have been designed to improve the convergence rate of the brute force MCapproach via “smarter” sampling. Latin Hypercube Sampling (LHS) andStratified Sampling for example achieve this by ensuring a better coverage ofthe input phase space (Helton and Davis, 2003; Helton et al., 2005; Gajev,2012), while adaptive Monte Carlo methods use the information from previoussamples to adaptively draw quasi-random samples of the inputs (Karaivanovaand Dimov, 1998; Dimov and Georgieva, 2010).

The slow convergence of MC techniques is not an issue if one is only interestedin confidence intervals, i.e. the ranges where output quantities lie with a givenprobability. The method to produce confidence intervals with relatively fewrandom samples of the inputs was first introduced by Wilks (Wilks, 1942), andlater was extended and is known today as the GRS methodology (Wald, 1943;Macian et al., 2007; Vinai et al., 2007; Glaeser, 2008). While this approach is

8

1.3. Traditional Methods for Sensitivity and Uncertainty Analysis

very helpful for regulatory purposes, unfortunately it provides little informationabout anything other than the constructed confidence intervals.

The major advantage all stochastic methods share is that the convergencerate of the moments of quantities of interest does not depend strongly onthe N number of input parameters. This, together with their relatively easyimplementation makes them very attractive for practical applications, despitetheir slow convergence. Moreover, for truly large scale problems with thousandsof input parameters and strong nonlinearities they often represent the onlyviable solution for performing S&U analysis. As a result they are regularly usedin the industry, as well as in research, where they are frequently considered ascomputational benchmarks against which other methods are compared andverified.

1.3.2 Deterministic methods

Deterministic methods are most often based on the assumption that the outputquantities of interest can be represented (at least to a certain accuracy) as someanalytical function of the uncertain input parameters. Once this assumptionis made a corresponding new mathematical problem is constructed with whichthe functional dependence can be determined.

The most significant work has probably been done on perturbation techniques(Ronen, 1988; Williams, 1986; Cacuci, 2003), where the assumption is thatthe particular response of interest (R) is well approximated by a Taylor seriesaround the α0 reference values of the input parameters, i.e. that:

R (α) = R(α0)

+N∑j=1

∂R

∂αj

∣∣∣∣∣0

(αj − α0

j

)+

N∑j=1

N∑k=1

∂2R

∂αj∂αk

∣∣∣∣∣0

(αj − α0

j

) (αk − α0

k

)+ ... (1.1)

Naturally the expansion has to be truncated, and the accuracy of this ap-proximation, as well as the associated computational costs strongly dependon how many terms are included. First order perturbation techniques onlytake into account the linear terms, whereas higher order techniques are usuallytruncated at the second order.

9

1. Introduction

Several methods are available for obtaining first order information. The moststraightforward applies one-at-a-time perturbations of the individual inputparameters (OAT method) and uses finite differences to calculate the firstorder derivatives (Ronen, 1988; Saltelli et al., 2000). A different approach isthe Forward Sensitivity Analysis Procedure (FSAP), where a linearized versionof the problem is solved for a perturbed set of input parameters (Cacuci, 2003;Zhao and Mousseau, 2012). For small systems this can be derived analytically,while for large scale applications automatic differentiation algorithms can beused, although these provide some challenges from the implementation point ofview (Anitescu et al., 2007; Alexe et al., 2010). From all first order perturbationmethods adjoint techniques (or ASAP, standing for Adjoint Sensitivity AnalysisProcedure) are by far the most successful (Williams, 1986; Cacuci, 2003). Theirmain advantage is that the number of calculations depends on the NR numberof output quantities of interest instead of the N number of input parameters.This makes them ideal for many applications, especially if the underlyingproblem is linear or only mildly nonlinear (which explains their great successin the nuclear field). They are, however, conceptually more difficult than othertechniques and may involve significant code development efforts, particularlyfor nonlinear systems.

A few papers have been published on higher order perturbation methods(Gandini, 1978; Moore and Turinsky, 1998; Gilli et al., 2013a). The generalconclusion of these studies is that while the accuracy of predictions can beimproved, it comes at the cost of significantly higher code development andcomputational times. Higher order perturbation techniques have thereforeremained an academic interest only, so far their challenging application tolarger scale problems has mostly prevented them from practical use.

Recently, a very promising new approach has been presented, which is basedon efficient subspace methods (ESM) and aims at identifying the perturbationmodes which are most essential for the responses of interest (Abdel-khaliket al., 2008). This is done by the extensive use of matrix rank revealingdecompositions at each level of the (multi-physics) simulations to find lowrank approximations of the large dense matrices associated with the problems,without having to first assemble these. The works published so far show greatpotential (Wang and Abdel-Khalik, 2011; Kennedy et al., 2012; Bang andAbdel-Khalik, 2013; Wang and Abdel-Khalik, 2013; Abdel-Khalik et al., 2013),but their practical usefulness is still to be proved.

10

1.4. Contents and Overview of This Thesis

Finally, spectral methods represent another set of deterministic techniqueswhich have received a lot of attention lately. The original idea was developedby Wiener already in 1938 (Wiener, 1938), but their practical application influid dynamics (Xiu and Karniadakis, 2003), structural mechanics (Ghanemand Spanos, 1991) or neutron transport (Williams, 2007) only started 10-15years ago. Spectral methods are based on the idea of representing all stochasticquantities (i.e. both inputs and outputs) as spectral expansions, similar toclassical Fourier series. For this one has to define appropriate basis functionsand needs to determine the expansion coefficients of the series. Once this isdone the original problem can be substituted by the constructed expansions,giving a direct connection between inputs and outputs. Spectral methodscan be applied in both an intrusive and a non-intrusive fashion: the formerinvolves dedicated code development and the definition of a new mathematicalproblem to be solved (i.e. different from the original problem), while the latter- similarly to statistical approaches - requires repeated execution of the samecomputer program with different values of the inputs.

1.4 Contents and Overview of This Thesis

The focus of this thesis is on the development of efficient sensitivity and un-certainty analysis techniques specifically for coupled multi-physics problemsencountered in the nuclear field. The primary motivation for further researchon S&U analysis comes from the need to accommodate the expanding useof simulations in both traditional and non-traditional fields. High resolutiontightly coupled multi-physics and multi-scale calculations are becoming pro-gressively widespread in engineering (Ragusa and Mahadevan, 2009; Gastonet al., 2009; Hales et al., 2012; Guo et al., 2013; Keyes et al., 2013) andcomputational tools are increasingly used in interdisciplinary applications aswell (e.g. in the medical field). Although the available computational capab-ilities continue to advance, the growing complexity and nonlinearity of suchcalculations often make existing stochastic methods prohibitively expensive, atthe same time known deterministic approaches inaccurate or not applicable.Therefore research on appropriate sensitivity and uncertainty analysis methodstargeted to these kinds of simulations is not only of scientific interest, but alsoof significant practical relevance.

The first half of this thesis focuses on coupled criticality calculations, asthey represent one of the most frequent problems to be solved in the nuclear

11

1. Introduction

field. In such simulation usually neutron transport or diffusion is coupled tothermal-hydraulics, but other phenomena can be taken into account as well,such as fission product poisoning, crud formation, etc. To accommodate thesensitivity and uncertainty analysis of these system this thesis presents anextension of classical adjoint methods. The main rationale behind doing so isthe immense success of adjoint techniques for pure transport problems, thealready available computer programs capable of solving the adjoint transportequation and the recent and anticipated future code development efforts foradjoint capable thermal-hydraulics codes (Fang et al., 2011; Farrell et al.,2013).

In the second half of this thesis transients (i.e. the time-dependent behaviourof reactors) are targeted, since they present the second most common typeof coupled calculations in reactor physics. Our method of choice is the non-intrusive spectral projection approach, since it is easily applicable to arbitraryproblems and recent studies have shown promising convergence properties invarious systems (Blatman and Sudret, 2011; Gilli et al., 2013b). These twoattributes make it potentially highly useful for the S&U analysis of accidentalevents, especially in cases where legacy or proprietary codes are used to simulatetransients, therefore code modification is not possible.

The structure of this thesis is as follows. In Chapter 2 the theoretical basisof the adjoint sensitivity analysis of coupled criticality problems is laid out. Itis shown that the effects of perturbations can be interpreted in three differentways: either by letting the steady state power to change, or by constrainingthe power and letting the eigenvalue differ from unity, or by adjusting a controlparameter in parallel to preserve the criticality of the core at the desired powerlevel. To derive sensitivities a coupled adjoint problem needs to be solved. Thenumerical aspects of this problem are also addressed and a solution recipe isgiven which relies on Krylov techniques and employing the individual neutronicsand augmenting codes as preconditioners. Finally a proof of principle study ispresented using a homogeneous one-dimensional slab with two-group diffusiontheory, coupled to heat conduction and xenon poisoning.

In Chapter 3 the larger scale applicability of the coupled adjoint theory isdemonstrated. For this an in-house developed, general purpose finite elementmultigroup discrete ordinates transport code is used, coupled to a purposemade adjoint capable thermal-hydraulics code. As a more realistic system aninfinite array of fuel pins is simulated, where neutron transport determines the

12

1.4. Contents and Overview of This Thesis

power distribution in the pin, and the temperature distribution has a feedbackon the multigroup cross sections. The major merit of this chapter is that itaddresses the main implementation issues of the coupled adjoint theory, andreveals that only minor modifications are needed in the transport and theaugmenting codes, as for most needed actions the already available functionsof the separate codes can be reused. The most significant effort lies in thedevelopment of the coupling scheme, and not in the development of the coupledadjoint scheme.

In Chapter 4 we turn our attention to the development of novel adaptivespectral methods, namely grid and basis adaptive Polynomial Chaos Expansion(PCE) techniques. The novelty of this chapter is twofold. First, an adaptivenumerical integration technique is devised based on the Gerstner algorithm,which aims at efficiently calculating all multidimensional integrals defining theexpansion coefficients of the Polynomial Chaos (PC) basis vectors. Second, theissue of basis adaptivity is investigated and it is shown that by adaptively choos-ing the basis vectors to be included in the PCE the associated computationalcosts can be decreased significantly. The chapters finishes with the applicationof the developed Fully Adaptive Non-Intrusive Spectral Projection (FANISP)algorithm to three demonstrational problems, each showing its effectivenesscompared to both MC techniques as well as non-adaptive PC methods.

In Chapter 5 the truly large scale applicability of the FANISP algorithm isdemonstrated by performing the S&U analysis of a Gas Cooled Fast Reactor(GFR) transient, with a completely detailed system model of the GFR2400reactor design and 42 input uncertainties. To address so many input parametersfirst two cost reduction techniques are presented, one based on an incrementalincrease of the global polynomial order of the used expansion, the other basedon an automatic dimensionality reduction using the information about theindividual effects of the inputs on the output of interest. Next the usedsystem model of the reactor is presented, with a detailed description of theincorporation of the uncertainties. Finally the results are analysed, showingthe versatility and excellent performance of the FANISP algorithm.

As a closure of this thesis, Chapter 6 contains some concluding remarksabout the prospects of practical applications of the developed adjoint andspectral methods, together with some recommendations for future research.

13

1. Introduction

14

Chapter 2

Adjoint Techniques forCoupled Criticality

Problems

2.1 Introduction

The adjoint based sensitivity analysis of neutron transport problems is apowerful tool to investigate changes in responses of interest due to variationsof input parameters characterizing a system. Starting from the early works ofUsachev (1964) and Gandini (1967), the theory of calculating perturbations inthe critical eigenvalue of reactors and functionals of the flux has been beingdeveloped continuously and is well established today (Stacey, 1974; Williams,1986). In parallel the implementation of traditional first order perturbationtheory capabilities has become common practice, with an increasing numberof commercial codes offering options for calculating various sensitivities usinggeneralized perturbation theory (GPT) (Rimpault et al., 2002; Jessee et al.,2009). The sensitivity analysis of coupled problems is however still not a dailypractice, despite the extensive work done on various coupled systems.

One of the most researched topics regarding coupled reactor physics problemsis fuel depletion. Gandini was the first to consider adjoint techniques for thesensitivity analysis of burn-up, without taking into account the coupling

15

2. Adjoint Techniques for Coupled Criticality Problems

between the flux and the changing nuclide concentrations (Gandini, 1975). Hiswork was later extended by Williams (1979) to include this feedback, applyinga quasi-static approximation for the forward problem to decouple the neutronand nuclide fields. The theory has been further developed to accommodateother circumstances, such as changing microscopic cross sections typical inlight water reactors (Downar, 1992), constant power depletion following reallife situations more closely than constant flux depletion (Yang and Downar,1988), or fuel shuffling and reloading for example (Gandini, 1987; Choi andDownar, 1992).

Transient situations represent another area where coupling is evidentlynecessary. In such works most often point-kinetics coupled to simplifiedthermal-hydraulics is considered and adjoint techniques are used to calculatesensitivities (Parks and Maudlin, 1981; Gilli et al., 2011). Some larger-scaleapplications of Adjoint Sensitivity Analysis Procedure in the field have alsobeen published (Cacuci and Ionescu-bujor, 2005; Ionescu-Bujor et al., 2005).

A few papers deal with the sensitivity analysis of shielding problems, inwhich coupled neutron-gamma transport is performed and the adjoint transportequation is used to generate sensitivities (Roussin and Schmidt, 1971; Bartineet al., 1972; Williams and Goluoglu, 2007). However, these works only considerone-way coupling, i.e. secondary gamma particles induced by neutrons aretaken into account, gamma-neutron interactions - such as photo-fission orneutron separation caused by high energy photons - are neglected.

In this chapter the sensitivity analysis of coupled criticality calculations isdiscussed with two-way coupling between neutronics and one or more aug-menting systems. The most relevant work to this research was done by Mooreand Turinsky (1998), who used generalized perturbation theory to efficientlycalculate boiling water reactor loading pattern characteristics with thermal-hydraulic feedbacks. Here however, all variables describing the augmentingsystems are taken into account and no a priori assumption is made on thestrength of feedback from the different augmenting unknowns. This becomesespecially important when the derived sensitivities are used to accurately calcu-late uncertainties on various reactor parameters, since it enables all uncertaininput data to be accounted for, i.e. the uncertainty of both the neutronics andthe augmenting parameters can be propagated to the final responses. Moreover,for the calculation of the necessary adjoint functions a procedure is introducedhere, which relies on using the neutronics and the augmenting codes separately.

16

2.2. Theory

In Section 2.2 the theoretical background is presented with three differentways for interpreting perturbations, namely the power-, the eigenvalue- andthe control parameter perturbation approaches. In Section 2.3 the numericalaspects of calculating the appropriate adjoint functions are investigated anda technique is proposed which uses the neutronics and augmenting codes asa preconditioner to Krylov methods applied to the system of coupled adjointequations. Section 2.4 presents a one-dimensional slab problem with thermaland fission product feedback for which the derived methods were applied, whilein Section 2.5 some results are shown. Concluding remarks are included inSection 2.6.

2.2 Theory

2.2.1 Formulation of the Coupled Criticality Problem

Criticality problems in reactor physics can be described by Equation 2.1:

L (αn)φ(x) = λF (αn)φ(x). (2.1)

Here x is a vector representing the independent variables (point in space, energy,direction), φ (x) and λ are the neutron flux and the critical eigenvalue (thedependent variables), while L and F are the usual loss and fission operators. αnstands for the different input parameters (cross sections, geometrical boundaries,etc.). Equation 2.1 can describe any form of the neutron transport equation (SN,PL, diffusion) with any discretization scheme, x and φ (x) simply take the formof the appropriate independent and dependent variables (e.g. x representingspatial points and energy groups, φ (x) group fluxes in the volumes in case ofa multigroup diffusion approximation with finite volume discretization).

The solution of Equation 2.1 is the critical eigenvalue and the correspondingeigenfunction φ (x), which can be arbitrarily normalized. This normalizationcan be expressed as

〈Cf (x) , φ (x)〉φC

= 1,

where Cf (x) is a chosen constraint function, C is a pre-set constraint value,while 〈 , 〉φ indicates integration over the phase space. Most commonlythe flux is normalized to a certain power level P and only heat from fission istaken into account, in this case the constraint value is the power (C = P ) andthe constraint function takes the form of Cf (x) = Pf (x) = QΣf (x), where Q

17


is the energy released per fission. Other normalizations are also possible, e.g.to a detector response, to one fission-neutron, etc.

In a coupled criticality problem physical processes other than neutrontransport are taken into account as well. The description of these extraphenomena requires additional input parameters (αT ), dependent variables (T )depending on independent variables y possibly different from x, and equations(N). There can be mutual feedback between the systems being coupled,hence some of the original neutronic parameters can become functions of theadditional parameters and dependent variables. For example, if coupling tothermal-hydraulics is considered, T can represent the temperature field, coolantdensity, void fraction, N would stand for the equations describing heat transfer,two-phase flow, etc., while y would simply be the point in space. In the workof Cacuci and Ionescu-bujor (2005) a very accurate and detailed methodologyis presented how to transform an uncoupled model into an augmented one.Here a simpler, but equally accurate approach is used: a complete set of inputparameters (α) is considered containing all parameters necessary to describeneutron transport, the additional phenomena and the feedbacks, moreoverthe neutronics operators are allowed to depend explicitly on the augmentingdependent variables. Without loss of generality in the rest of the theoreticaldescription we will consider a single augmenting phenomenon, requiring oneadditional dependent variable T and equation N .

The coupled criticality problem can be formulated by Equations 2.2-2.4:

L(α, T

(y))φ (x) = λF

(α, T

(y))φ (x) (2.2)

N(α, T

(y), φ (x)

)= 0 (2.3)⟨

Cf(α, T

(y)), φ (x)

⟩φ

C= 1. (2.4)

The loss and fission operators, as well as the constraint function now also dependon the augmenting dependent variables, whereas the augmenting equationsare only indicated by a general operator N acting linearly or nonlinearly onthe augmenting dependent variables, the flux and the input parameters. Forsimplicity in the rest of this chapter the dependence of φ and T on independentvariables x and y will not be explicitly shown.

With feedback between the augmenting system and neutronics, Equa-tions 2.2-2.3 have a unique solution: the real physical steady state for which

18

2.2. Theory

λ = 1. In this case the flux normalization is no longer arbitrary and theconstraint value C cannot be freely chosen, but is simply given by Equation 2.4.For example, in case of coupling to thermal-hydraulics, this would correspondto the total thermal power of the reactor at steady state for the given config-uration. When Equations 2.2-2.4 are considered and the flux is normalizedto the chosen constraint value the eigenvalue may differ from 1, showing theoff-criticality of the system at the prescribed conditions. Again thinking ofthermal coupling this would translate to whether the reactor can be operatedat a given power level or not.

Based on the above, perturbations in coupled criticality problems can beconsidered in three different ways. Either the perturbed steady state is searchedfor and no arbitrary flux normalization is allowed (power perturbation); or theflux is constrained to the unperturbed constraint value and the k-effective isallowed to change (eigenvalue perturbation); or (multiple) perturbations aremade so that the constraint value and the eigenvalue are both unchanged (con-trol parameter perturbation). Power perturbation corresponds to the situationwhen feedbacks adjust the system to a new steady state after perturbations aremade, which can be used for optimization purposes in the design of reactors.With eigenvalue perturbation the difference in the k-effective of the perturbedand the reference configuration is calculated making it suitable to determinereactivity worths taking into account feedbacks. At last, control parameterperturbation is most useful for calculating sensitivities in operational conditionswhen the power is constrained and the reactor has to be critical. In the nextthree sections these three possibilities are examined.

2.2.2 Power Perturbation

For any (sensible) set of input parameters α Equations 2.2-2.3 always havea unique solution due to the feedbacks present between neutronics and theaugmenting system. This is the true physical steady state, for which λ = 1.Therefore the problem can be formulated as

L(α, T )φ = F (α, T )φN(α, T, φ) = 0,

and for a given parameter set α0 the solution is φ0 and T 0 describing thesteady state. The corresponding unique power (constraint) value is

P 0 = C0 =⟨Cf (α0, T 0), φ0

⟩φ

=⟨Pf (α0, T 0), φ0

⟩φ.

19


Perturbations made to the input parameters (∆α) result in a differentsteady state, a possibly different power (constraint) value and changes in otherresponses of interest as well. These variations can be efficiently calculated byapplying the standard adjoint sensitivity analysis procedure (Cacuci, 2003).Here only the final equations that need to be solved and the formulas forcalculating the response changes are presented, however in Section B.1 a shortsummary of the full derivation is given. For notation see Chapter A.

The relative change in the power (constraint) value - up to first order - isgiven by Equation 2.5, where the ∆P direct∆α /P 0 = 1/P 0 ⟨∂Pf/∂α|0 ∆α, φ0⟩

φdirect effect has been separated:

∆P∆αP 0 =

∆P direct∆αP 0 + 1

P 0

⟨(∂Pf∂T

∣∣∣∣0

)∗φ0,∆T

⟩T

+ 1P 0

⟨P 0f ,∆φ

⟩φ. (2.5)

As usual, we are concerned about the indirect terms containing ∆φ and ∆T inEquation 2.5 and want to avoid recalculating the coupled problem for everyinput parameter perturbation. This can be achieved by solving the followingadjoint problem for w0

φ and w0T :

(M0

)∗w0φ +

(∂N

∂φ

∣∣∣∣∣0

)∗w0T = w0

P

P 0 P0f (2.6)

(J0)∗w0φ +

(∂N

∂T

∣∣∣∣∣0

)∗w0T = w0

P

P 0

(∂Pf∂T

∣∣∣∣0

)∗φ0. (2.7)

In Equations 2.6-2.7 w0P is an arbitrarily chosen constant and we introduced

M0 = L0 − λ0F 0 and J0∆T = ∂M/∂T∣∣∣0

∆Tφ0 (with λ0 = 1 in both ex-pressions). The coupled adjoint operator on the left is basically identical tothat used by Moore and Turinsky (1998), with the exception that here no apriori assumption is made on the strength of the feedback from the differentaugmenting variables. The boundary conditions usually need a more detaileddiscussion, here it is only emphasized that they have to be chosen in a waythat the bilinear terms coming from the definition of the adjoint operatorsvanish, or at least become dependent only on the unperturbed solution, theknown parameters and the parameter changes (Cacuci, 2003).

Once the appropriate adjoint functions w0φ and w0

T are available, the relative

20

2.2. Theory

change in the power (constraint) value can be calculated by Equation 2.8:

∆P∆αP 0 =

∆P direct∆αP 0 − 1

w0P

⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

− 1w0P

⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

.

(2.8)

The steady state power is not the only quantity we are interested in. Forresponse functionals R (α, T, φ) the change caused by the perturbation of theinput parameters can be written as

∆R∆α = ∂R

∂α

∣∣∣∣0

∆α+⟨∂R

∂T

∣∣∣∣0,∆T

⟩T

+⟨∂R

∂φ

∣∣∣∣0,∆φ

⟩φ

= ∆Rdirect∆α +∆Rindirect∆α .

Our main concern is again the evaluation of the indirect term, for which theappropriate adjoint problem that needs to be solved takes the form of

(M0

)∗wPφ +

(∂N

∂φ

∣∣∣∣∣0

)∗wPT = ∂R

∂φ

∣∣∣∣0

(2.9)

(J0)∗wPφ +

(∂N

∂T

∣∣∣∣∣0

)∗wPT = ∂R

∂T

∣∣∣∣0. (2.10)

Having obtained the solution of Equations 2.9-2.10, the indirect term of theresponse change can be evaluated, leading to the following expression for thetotal perturbation:

∆R∆α = ∆Rdirect∆α −⟨wPT ,

∂N

∂α

∣∣∣∣∣0

∆α⟩T

−⟨wPφ ,

∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

. (2.11)

As in practice reactors are usually operated at a given power level it is worthcalculating sensitivities when the power is constrained (or a chosen constraintvalue has to be met). This can be done if parallel to the perturbation of theinput parameters for which the sensitivities are to be calculated (∆α) oneor more other parameters are perturbed as well, such that the power in theperturbed state is equal to that in the unperturbed state. This is very similarto the traditional k-reset procedure used in standard GPT (Williams, 1986).When a single parameter is used to reset the power, this can be designatedas a control parameter (which will be denoted as αc in the remainder of thischapter) and its value can be calculated by requiring that the power change

21


caused by the ∆α perturbation and the ∆αc control parameter change canceleach other, i.e. that

∆P∆αP 0 = −∆P∆αc

P 0 .

Equation 2.8 can be used to evaluate the power change of both perturbations,leading to the following formula for the change in the control parameter:

∆αc = −

w0P

∆P direct∆αP 0 −

⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

−⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

w0P

P 0

⟨∂Pf∂αc

∣∣∣∣0, φ0

⟩φ

−⟨w0T ,

∂N

∂αc

∣∣∣∣∣0

⟩T

−⟨w0φ,∂M

∂αc

∣∣∣∣∣0φ0⟩φ

. (2.12)

Once the changes in all input parameters are known the power-constrainedsensitivities can be calculated by using Equation 2.11 to evaluate the responsechange due to both the original perturbations and the control parametervariation (∆Rdirect∆α+∆αc = ∆Rdirect∆α + ∆Rdirect∆αc is the total direct change):

∆RP−reset∆α = ∆Rdirect∆α+∆αc −[⟨wPT ,

∂N

∂α

∣∣∣∣∣0

∆α⟩T

+⟨wPT ,

∂N

∂αc

∣∣∣∣∣0

∆αc

⟩T

]

−

⟨wPφ , ∂M∂α∣∣∣∣∣0

∆αφ0⟩φ

+⟨wPφ ,

∂M

∂αc

∣∣∣∣∣0

∆αcφ0⟩φ

. (2.13)

2.2.3 Eigenvalue Perturbation

In the sensitivity analysis of reactor physics problems one of the prime interestsis the change of the critical eigenvalue due to perturbations of the inputparameters, and this is also the case for coupled criticality calculations. Whenfeedbacks are present the eigenvalue only differs from unity if the flux isconstrained, hence the problem can be described by

L(α, T )φ = λF (α, T )φN(α, T, φ) = 0

〈Pf (α, T ), φ〉φP 0 = 1,

22

2.2. Theory

where P 0 is the pre-set power (constraint) value. In the unperturbed system(characterized by a parameter set α0) the eigenvalue is λ0 = 1 and the corres-ponding steady state solution is φ0 and T 0. Perturbing the input parameterswhile constraining the flux to the power of P 0, the eigenvalue may differ fromλ0, this difference can be interpreted as the reactivity worth of the perturbationand can be calculated as

∆λ =

−w0P

∆P direct∆αP 0 +

⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

+⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ⟨

w0φ, F

0φ0⟩φ

. (2.14)

In Equation 2.14 w0φ and w0

T are the solutions to Equations 2.6-2.7 for thearbitrarily chosen w0

P . The derivation of Equation 2.14 and the correspondingadjoint problem is quite similar to that presented by Williams (1986) forpure criticality problems, and is fully detailed in Section B.2. As expected,Equation 2.14 reduces to the standard eigenvalue perturbation formula whenno coupling is present.

When functional responses other than the critical eigenvalue are considered,the adjoint problem that needs to be solved is

(M0

)∗wλφ +

(∂N

∂φ

∣∣∣∣∣0

)∗wλT = wλP

P 0 P0f + ∂R

∂φ

∣∣∣∣0

(2.15)

(J0)∗wλφ +

(∂N

∂T

∣∣∣∣∣0

)∗wλT = wλP

P 0

(∂Pf∂T

∣∣∣∣0

)∗φ0 + ∂R

∂T

∣∣∣∣0, (2.16)

with the auxillary condition that

⟨wλφ, F

0φ0⟩φ

= 0.

The latter condition is needed so that the term ∆λ⟨wλφ, F

0φ0⟩φin the de-

rivation disappears. As in the coupled case the neutronic adjoint is unique(unless the coupled adjoint operator is singular), this can be achieved by theproper choice of wλP (see Section B.2). Having obtained the solution of Equa-tions 2.15-2.16 the response variation due to the perturbation in the input

23


parameters can be calculated according to Equation 2.17:

∆R∆α = ∆Rdirect∆α + wλPP 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

−⟨wλT ,

∂N

∂α

∣∣∣∣∣0

∆α⟩T

−⟨wλφ,

∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

. (2.17)

Just like in standard GPT, one can apply a k-reset procedure to obtaincriticality constrained sensitivities. Equation 2.14 can be used to evaluate bothsides of

∂λ

∂α

∣∣∣∣0

∆α = − ∂λ

∂αc

∣∣∣∣0

∆αc,

leading to the same formula for the control parameter change as Equation 2.12.Finally, Equation 2.17 can be used to evaluate the change in the response bothdue to the input parameter perturbations and the control parameter change,leading to the following expression for the criticality-constrained sensitivities(which coincide with the power-constrained sensitivities):

∆Rk−reset∆α = ∆Rdirect∆α + ∆Rdirect∆αc + wλPP 0

⟨[∂Pf∂α

∣∣∣∣0

∆α+ ∂Pf∂αc

∣∣∣∣0

∆αc], φ0

⟩φ

−⟨wλT ,

[∂N

∂α

∣∣∣∣∣0

∆α+ ∂N

∂αc

∣∣∣∣∣0

∆αc

]⟩T

−⟨wλφ,

[∂M

∂α

∣∣∣∣∣0

∆α+ ∂M

∂αc

∣∣∣∣∣0

∆αc

]φ0⟩φ

. (2.18)

2.2.4 Control Parameter Perturbation

A direct way to calculate constrained sensitivities is to consider the problemwith the constraint from the beginning and make simultaneous perturbationsto the desired α parameters and the αc control parameter in order to preservecriticality at the unperturbed power (constraint) value. Applying the standardASAP again the obtained formula for the control parameter change is the sameas Equation 2.12 and the adjoint problem needed to be solved is identical toEquations 2.6-2.7. For responses other than the control parameter the adjointproblem is similar to Equations 2.9-2.10, but the sources are more complicatedas they contain the extra terms coming from the constraint, furthermore there isan extra equation to solve to obtain the adjoint corresponding to the constraint

24

2.3. Solution of the Coupled Adjoint Problem

equation. The resulting response changes are however exactly the same asthe power-reset and k-reset response variations in the power and eigenvalueperturbation procedures, hence the remainder of this chapter focuses on thesetwo approaches, for the details see (Perkó et al., 2013).

2.3 Solution of the Coupled Adjoint ProblemThe different perturbation procedures lead to very similar coupled adjointproblems. With the exception of the control parameter perturbation, for agiven value of wP they are all of the form of Equations 2.19-2.20:

(M0

)∗wφ +

(∂N

∂φ

∣∣∣∣∣0

)∗wT = S∗φ (2.19)

(J0)∗wφ +

(∂N

∂T

∣∣∣∣∣0

)∗wT = S∗T . (2.20)

One way of dealing with the above problem is to develop new coupled neutronicscodes which are capable of directly solving Equations 2.19-2.20. However froma programming point of view it is much easier to use existing neutronics andaugmenting codes capable of solving the separate adjoint problems and takecare of the coupling via source terms in an iterative way. Thus only the propercalculation of the adjoint sources is needed and the lengthy and complicateddevelopment of new codes can be avoided.

At first glance it seems tempting to introduce the following block iterativescheme for solving Equations 2.19-2.20 (k is the iteration index):

(M0

)∗w

(k+1)φ = S∗φ −

(∂N

∂φ

∣∣∣∣∣0

)∗w

(k)T(

∂N

∂T

∣∣∣∣∣0

)∗w

(k+1)T = S∗T −

(J0)∗w

(k+1)φ .

One problem with this approach is that the(M0

)∗adjoint transport op-

erator is singular. This issue is well known in traditional GPT and limitsits use to a certain class of responses (Williams, 1986). From elementarylinear algebra it follows that the inhomogeneous adjoint transport equationcan only be solved if the source is orthogonal to the forward solution

(φ0).25


As a consequence the block iteration described above could only be conver-gent if

⟨S∗φ −

(∂N/∂φ

∣∣∣0

)∗w

(k)T , φ0

⟩φ

= 0 held in every step of the iteration,which would most probably not be the case. To overcome the obstacle Equa-tions 2.19-2.20 can be recasted in the following form:

(M0ζ

)∗wφ +

(∂N

∂φ

∣∣∣∣∣0

)∗wT = S∗φ + λ0ζ

(F 0)∗wφ

(J0)∗wφ +

(∂N

∂T

∣∣∣∣∣0

)∗wT = S∗T .

Here M0ζ = D0 − λ0(1− ζ)F 0 was introduced, which is no longer singular for

ζ ∈ (0, 1], since λ0 is the fundamental eigenvalue and for all other eigenvaluesof Equation 2.1 λ0 < λ1 < λ2 < ... stands. With the above form the naturalblock iteration scheme to follow is(

M0ζ

)∗w

(k+ 12 )

φ = S∗φ −(∂N

∂φ

∣∣∣∣∣0

)∗w

(k)T + λ0ζ

(F 0)∗w

(k)φ (2.21)(

∂N

∂T

∣∣∣∣∣0

)∗w

(k+ 12 )

T = S∗T −(J0)∗w

(k+ 12 )

φ (2.22)

w(k+1)φ = w

(k)φ + rφ

(w

(k+ 12 )

φ − w(k)φ

)w

(k+1)T = w

(k)T + rT

(w

(k+ 12 )

T − w(k)T

).

Here rφ and rT are chosen relaxation constants. Such schemes are convergentwith strong enough under-relaxation, regardless of the value of ζ, but arerather unstable and not very robust (see Section 2.5.3).

As in most practical situations the coupled adjoint operator is represented bya sparse matrix, Krylov algorithms (Saad, 2003) are a natural and promisingchoice for solving the adjoint system. For a generic linear system described asAx = b, Krylov methods search for the approximate solution xm in the Krylovsubspace Km(A, r0) = spanr0, Ar0, A

2r0, . . . , Am−1r0, where r0 = b − Ax0

is the initial residual vector for some initial guess x0. This means that theimplementation of a Krylov solver around Equations 2.19-2.20 only requiresthe ability of multiplying with the coupled adjoint matrix, which takes the

26

2.4. Application to a One-Dimensional Slab

place of A in our case. This multiplication can be done easily: a neutrontransport and the augmenting solver can be modified without major effortsto only multiply vectors with

(M0

)∗and

(∂N/∂T

∣∣∣0

)∗respectively, moreover

the multiplications with(J0)∗

and(∂N/∂φ

∣∣∣0

)∗are required anyway when

the sources need to be calculated for the separate adjoint problems. As will bedemonstrated in Section 2.5.3 Krylov algorithms built this way are capableof solving the coupled adjoint problem, however the best results are obtainedwhen they are preconditioned with the block iteration scheme described byEquations 2.21-2.22.

One problem encountered when the preconditioned Krylov solver was usedto obtain the adjoint functions is that for certain responses and perturbationsthe values calculated using the perturbation expressions were slightly off, evenfor responses depending linearly on input parameters (i.e. the slope of thepredicted linear change was incorrect). The reason for this is that the coupledadjoint problem is usually very badly conditioned, for example when thermalor fission product coupling is considered there can as many as 20 orders ofmagnitude difference between the elements of

(J0)∗

and(∂N/∂φ

∣∣∣0

)∗. This

issue can be circumvented if instead of the original physical quantities φ and T ,scaled variables φ = φ/τφ and T = T/τT are used as unknowns in the equations.For example the flux vector can be normalized with a chosen flux value typicalfor the configuration being investigated, considering fission product poisoningxenon number densities can be normalized with the Avogadro number, etc.Whenever the unscaled adjoint system provided not perfect predictions, theuse of such scaling factors proved to eliminate the problem, though typicallythe number of needed Krylov iterations increased slightly (to 5-6 instead of2-3, see Section 2.5.3).

2.4 Application to a One-Dimensional Slab

For illustration, the theory presented in Section 2.2 was applied to a one-dimensional coupled slab problem. The neutronics part is described by atwo-group diffusion model and there are two augmenting phenomena: heatconduction through the slab and fission product poisoning. Thermal feedbackis present due to the dependence of the total (absorption) cross section of thefast group on the average temperature of the slab, while xenon concentrationmainly affects the absorption in the thermal group. In case there is only

27


thermal coupling the problem has an analytical solution, this is presented inSection 2.4.1, while details of the complete model are given in Section 2.4.2.

2.4.1 Coupled Neutronics-Thermal-Hydraulics Model

The two-group diffusion equation for a homogeneous slab of 2a width takesthe usual form (Duderstadt and Hamilton, 1976):

−D1d2

dx2φ1(x) +(Σt

1 (T (x))− Σtr1→1

)φ1(x)

− Σtr2→1φ2(x)− λχ1

[ν1Σf

1φ1(x) + ν2Σf2φ2(x)

]= 0 (2.23)

−D2d2

dx2φ2(x) +(Σt

2 − Σtr2→2

)φ2(x)

− Σtr1→2φ1(x)− λχ2

[ν1Σf

1φ1(x) + ν2Σf2φ2(x)

]= 0. (2.24)

Neglecting the difference between the extrapolated and real boundaries thezero boundary condition reads φi(±a) = 0 for i = 1, 2. The thermal feedbackis given by

Σt1 (T (x)) = Σt,1

1 + Σt,21 ·

(T − Tref

), (2.25)

where T = 12a∫ a−a T (x)dx is the average temperature of the slab and Tref is

some reference temperature. For simplicity the temperature dependence of Σt1

will be omitted from the notation in the remainder of this chapter.

The augmenting phenomenon is thermal-hydraulics, which in our case isdescribed by pure heat conduction in the slab, subject to zero boundaryconditions at ±a (with h being the thermal conductivity):

hd2

dx2T (x) +Q[Σf

1φ1(x) + Σf2φ2(x)

]= 0 (2.26)

T (−a) = T (a) = 0.

Finally, the power constraint is given by∫ a

−aQ[Σf

1φ1(x) + Σf2φ2(x)

]dx = P. (2.27)

All used model parameters (including the ones describing the fission productfeedback) are summarized in Table 2.1. Note that though in the actual

28


numerical calculations all fission neutrons fell in the fast group and no up-scattering was present the formulas presented here correspond to the generalcase.

Table 2.1: Input parameter values of the slab problemParameter Value Parameter Value Parameter ValueΣt,1

1 [cm−1] 0.04 Σt2[cm−1] 0.139 h [Wcm−1K−1] 1

Σt,21 [cm−1] 0.00001 a [cm] 50

Σf1 [cm−1] 0.005 Σf

2 [cm−1] 0.05 Q [MeV] 200D1 [cm] 1.25 D2 [cm] 0.5 Tref [K] 450

Σtr1→1 [cm−1] 0.0075 Σtr

2→1 [cm−1] 0.0 YXe 0.00228Σtr

1→2 [cm−1] 0.02 Σtr2→2 [cm−1] 0.015 λXe [h−1] 0.0756

χ1 1.0 χ2 0.0 YI 0.06386ν1 3 ν2 2.5 λI [h−1] 0.105

σXe1 [Mb] 2.275 σXe2 [Mb] 20.475

The solution of the coupled system described by Equations 2.23-2.27 caneasily be derived analytically and only involves elementary steps, here only thefinal solution is given, for the details see Perkó et al. (2013):

φ1(x) =TB3

+ha

Q

[Σf

1 + Σf2

Σ1→2D2B2

+ + Σ2

] cos(B+x) T (x) = TB+a cos(B+x)

φ2(x) =TB3

+ha

Q

[Σf

2 + Σf1D2B

2+ + Σ2

Σ1→2

] cos(B+x) P = 2TB2+ha.

The steady state of the system is characterized by λ = 1. In this casethe average temperature (T ) is given by the criticality condition that thegeometrical buckling has to satisfy B+ = π

2a , which in turn fixes the fluxesand the power as well. When the power is given, the average temperature iscalculated from the power equation and the condition that B+ = π

2a yieldsthe corresponding eigenvalue for the system.

For the steady state power and eigenvalue responses the adjoint problem

29


also has an analytical solution. Using the usual inner products, i.e.

〈f, g〉φ =∫ a

−a[f1(x)g1(x) + f2(x)g2(x)] dx 〈f, g〉T =

∫ a

−af(x)g(x)dx,

the adjoint problem that has to be solved is

−D1d2

dx2w0φ1(x) + Σ1w

0φ1(x)− Σ1→2w

0φ2(x) + QΣf

1wT (x) = w0P

P 0 QΣf1

−D2d2

dx2w0φ2(x) + Σ2w

0φ2(x)− Σ2→1w

0φ1(x) + QΣf

2wT (x) = w0P

P 0 QΣf2

Σt,21

2a

∫ a

−awφ,1(x)φ0

1(x)dx + hd2

dx2w0T (x) = 0.

It can be shown that the solution of this adjoint problem is of the form of:

w0φ,1(x) = v1 cos (B+x) + v2ch (|B−|x) + v3x

2 + v4w0φ,2(x) = v5 cos (B+x) + v6ch (|B−|x) + v7x

2 + v8w0T (x) = v9

(x2 − a2) ,

where vi are analytically determined constants. For more complicated responsesthe above adjoint problem with the appropriate source on the right hand sidehas to be solved numerically.

In this model the effective multiplication factor, the steady state power leveland factors of the six-factor formula were investigated as responses, togetherwith the average and maximum temperature of the slab. Since the analyticalsolution is known, these quantities and their change due to the variation ofthe input parameters are easy to calculate to verify the results obtained byapplying the perturbation technique.

30


2.4.2 Fully Coupled Model Including Thermal-Hydraulicsand Fission Product Poisoning

When coupling to fission products is also present Equations 2.23-2.24 have tobe modified to include the effects of xenon poisoning:

−D1d2

dx2φ1(x) +(Σt

1 (T,Xe)− Σtr1→1

)φ1(x)

− Σtr2→1φ2(x)− λχ1

[ν1Σf

1φ1(x) + ν2Σf2φ2(x)

]= 0 (2.28)

−D2d2

dx2φ2(x) +(Σt

2 (Xe)− Σtr2→2

)φ2(x)

− Σtr1→2φ1(x)− λχ2

[ν1Σf

1φ1(x) + ν2Σf2φ2(x)

]= 0. (2.29)

The full feedback takes the following form:

Σt1 (T,Xe) = Σt,1

1 + Σt,21 ·

(T − Tref

)+ σXe1 Xe (x)

Σt2 (Xe) = Σt

2 + σXe2 Xe (x) .

The thermal-hydraulics model is unchanged, the temperature is again givenby Equation 2.26. Finally, the number of Xe atoms in equilibrium is given byEquation 2.30 (Duderstadt and Hamilton, 1976):

Xe (x) =(YI + YXe)

[Σf

1φ1 (x) + Σf2φ2 (x)

]λXe + σXe1 φ1 (x) + σXe2 φ2 (x)

. (2.30)

In the fully coupled model the average slab temperature and the maximumxenon concentration were examined to highlight the interdependency of threesystems, moreover the numerical aspects of the perturbation method wereinvestigated.

2.4.3 Solution of the Forward and Adjoint Problems

As mentioned before when only thermal feedback is present the solution ofthe homogeneous forward problem (i.e. when all parameters are constant inthe slab) can be derived analytically. In this case the responses are availableas functions of the input parameters and were calculated by their analyticalexpressions (moreover they were checked against the numerical solution). Forsimple responses the adjoint problem can also be solved analytically, hence the

31


perturbation integrals can be calculated exactly. However for more complicatedresponses and in case of the fully coupled model both the forward and theadjoint problems have to be solved numerically. This was done by using afinite difference discretization scheme and solving the resulting equations inMATLAB with LU factorization. The coupled forward problem was solved byiteratively solving the separate two-group diffusion, thermal-hydraulics andfission product build-up equations, while the solution of the coupled adjointproblem is detailed in Section 2.5.3 as this is one of the main issues of thischapter.

2.5 Results

2.5.1 Sample Responses for the Slab With Pure ThermalFeedback

Figure 2.1 shows the change of the steady state power due to variations in thethermal neutron yield (ν2), the thermal absorption cross section (Σt

2) and thethermal conductivity (h). When the average number of fission neutrons for thethermal group (ν2) is increased the power increases as well, as the criticalitycondition requires a higher absorption to counterbalance the higher number offission neutrons, which can only be ensured at a higher average temperatureand higher power. When the heat-conduction coefficient (h) is increased thecriticality condition is not changed, the same average temperature is required,which is reached at a higher power level due to the increased conduction. Inthese two cases the perturbation theory prediction is exact, as the power is alinear function of the average temperature, which linearly depends on bothquantities. In contrast, when the thermal absorption (Σt

2) is increased thishas to be counterbalanced by a lower absorption in the fast group to reachcriticality, requiring lower temperatures and a lower power level. As can beseen, the dependence of the power on the thermal absorption is nonlinear,hence the perturbation theory prediction is only accurate up to first order.

In Figure 2.2 the reactivity worth of various perturbations can be seen. Firstan example of the difference between the coupled and uncoupled models isshown, i.e. the k-effective when it is calculated according to Equations 2.23-2.27taking into account the temperature dependence of the absorption in thefast group and when it is simply calculated from Equations 2.23-2.24 withthe unperturbed average temperature profile

(T = T 0

). In both cases the

32

2.5. Results

−10 −8 −6 −4 −2 0 2 4 6 8 10−40

−30

−20

−10

0

10

20

30

40

50

Perturbation [%]

Po

we

r p

ert

urb

ati

on

[%

]

tν2

− exac

ν2

− perturbed

Σ2

t,1 − exact

Σ2

t,1 − perturbed

h − exact

h − perturbed

Figure 2.1: The change of the steady state power level due to the perturbationof various input parameters. Parameters increasing the neutron productionincrease, parameters increasing neutron destruction decrease the steady statepower level, respectively.

increase of the fast fission cross-section(Σf

1

)in the central region of the slab

(−a/2 ≤ x ≤ a/2) has a positive reactivity effect, however when the couplingis present the k-effective increase is slightly lower due to the negative thermalfeedback. Though in this case only a small difference was found, it is obviousthat the sensitivities in the coupled model can differ from those in a pureneutronics model.

Two more examples are displayed in Figure 2.2: the reactivity effect of thethermal conductivity (h) and the strength of the coupling

(Σt,2

1

). When the

thermal conductivity increases, the constrained power level results in lowertemperature, hence the absorption in the fast group decreases, increasing thek-effective. When the thermal feedback

(Σt,2

1

)is made stronger, the same

average temperature corresponds to higher absorption decreasing the reactivity.

In Figure 2.3 the changes in the factors of the six-factor formula are showndue to a +10% perturbation of different input parameters, with and withoutapplying power reset (for the exact definition of the factors see Perkó et al.(2013)). As the perturbation results are only accurate up to first order,the precision of the perturbation predictions depends on the nonlinearity ofthe individual factors with respect to the different input parameters. The

33


−10 −8 −6 −4 −2 0 2 4 6 8 100.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

Perturbation [%]

k−

e!

ect

ive

Σ1

f − exact, with feedback

Σ1

f − perturbed, with feedback

Σ1

f − exact, no feedback

Σ1

f − perturbed, no feedback

h − exact

h − perturbed

Σ1

t,2 − exact

Σ1

t,2 − perturbed

Figure 2.2: The change of the effective multiplication factor due to the per-turbation of various input parameters. Taking into account feedbacks affectthe calculated sensitivities, moreover the reactivity effect of non-neutronicparameters can be calculated.

results show many examples when a traditional sensitivity analysis would yieldno sensitivity, whereas the coupled does. For example, when the diffusioncoefficient in the fast group (D1) is increased (see in Figure 2.3a) usually onlythe P fast

NL fast non-leakage probability would be affected due to the increasedleakage. In contrast, in the coupled case the p resonance escape probabilityincreases as the higher leakage is compensated by lower power and averagetemperature, decreasing the absorption in the fast group. When the poweris reset (by adjusting the Σt

2 thermal absorption) this dependence disappearsas the average temperature returns to its unperturbed value (see the last two“rows” of Figure 2.3a), however the η thermal utilization factor increases (asthe thermal absorption decreases), whereas the ε fast fission factor decreases.The situation is similar in case of the thermal group diffusion (see Figure 2.3b),with the exception of the fast fission factor, which increases in case of noreset and is not affected when reset is used. The increase of the fast fissioncross section (Figure 2.3c) naturally makes the fast fission factor higher, atthe same time it decreases the resonance escape probability when no resetis used, and the thermal utilization factor when the thermal absorption ischanged to reset the power. The effect of the thermal conductivity is a goodexample of the interdependency of neutronics and thermal-hydraulics: as canbe seen in Figure 2.3d the factors are unchanged when the power is let toadjust to a higher level, since in this case the average temperature remains the

34

2.5. Results

same and the spectrum is unchanged, whereas when the thermal absorptionis increased to decrease the power level all responses depending on Σt

2 arechanged. The perturbation of the transport cross section from the fast tothe thermal group is an interesting example of the effects of power reset (seeFigure 2.3e). Without the reset procedure the spectrum softens, hence the fastfission factor decreases, whereas the resonance escape probability increases.When the power is decreased to its unperturbed level, the spectrum returnsto its unperturbed shape, however the resonance escape probability is evenhigher due to the smaller absorption in the fast group as a result of lowertemperatures. Finally, in Figure 2.3f the effects of the control parameter areshown, which was chosen to be the thermal absorption cross section. Theincrease of Σt

2 decreases the thermal neutron yield and hardens the spectrum,at the same time decreases the power level, the average temperature and thefast absorption, increasing the resonance escape probability.

2.5.2 Sample Responses for the Slab with Thermal andFission Product Feedback

When fission product feedback (xenon poisoning) is also included we dealwith a three component system, and the cross-dependencies are even morecomplicated. However, the theory described in Section 2.2 can be appliedwithout a problem yielding correct first order changes in responses of interest.Here only two illustrative examples are provided. Figure 2.4 shows the changein the average temperature of the slab (T ) due to the perturbation of inputparameters in the three different components. The increase of the diffusioncoefficient in the fast group (D1) and the thermal absorption cross section ofxenon

(σXe2

)result in higher neutron leakage and higher absorption in xenon

respectively. Both effects have to be counterbalanced by a smaller absorption inthe fast group and smaller xenon poisoning to maintain criticality. This resultsin lower power and fluxes, hence lower temperatures and xenon concentration.When the thermal conductivity (h) is increased, the power increases togetherwith the flux and the xenon concentration. The net effect is that criticalityis reached at a higher power level, but slightly lower average temperature asthere is higher absorption in xenon.

Figure 2.5 shows the change in the maximum xenon concentration (whichis reached at the center of the slab) due to different perturbations, with(Figure 2.5a) and without (Figure 2.5b) employing the power-reset procedure.

35


Exact no reset

Perturbed no reset

Exact reset

Perturbed reset

−0.5

0

0.5

1

P fastNL

P thermalNL

p

η

Pe

rtu

rba

tio

n [

%]

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

(a) +10% perturbation of D1

Exact no reset

Perturbed no reset

Exact reset

Perturbed reset

−0.04

−0.02

0

0.02

0.04

P fastNL

P thermalNL

p

η

Pe

rtu

rba

tio

n [

%]

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

(b) +10% perturbation of D2

Exact no reset

Perturbed no reset

Exact reset

Perturbed reset

−10

−5

0

5

10

P fastNL

P thermalNL

p

η

Pe

rtu

rba

tio

n [

%]

−6

−4

−2

0

2

4

6

8

(c) +10% perturbation of Σf1

Exact no reset

Perturbed no reset

Exact reset

Perturbed reset

−3

−2

−1

0

1

2

P fastNL

P thermalNL

p

η

Pe

rtu

rba

tio

n [

%]

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

(d) +10% perturbation of h

Exact no reset

Perturbed no reset

Exact reset

Perturbed reset

−20

−10

0

10

20

P fastNL

P thermalNL

p

η

Pe

rtu

rba

tio

n [

%]

−10

−8

−6

−4

−2

0

2

4

6

8

10

(e) +10% perturbation of Σtr1→2

Exact no reset

Perturbed no reset

Exact reset

Perturbed reset

−15

−10

−5

0

5

10

P fastNL

P thermalNL

p

η

Pe

rtu

rba

tio

n [

%]

−10

−8

−6

−4

−2

0

2

4

6

(f) +10% perturbation of Σt,12

Figure 2.3: The change in the factors of the six-factor formula due to +10%perturbation of different variables. Many examples are shown where traditionalsensitivity analysis would yield no sensitivities in contrast to the coupled case.

36

2.5. Results

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Perturbation [%]

Av

era

ge

te

mp

era

ture

pe

rtu

rba

tio

n [

%]

D1

− exact

D1

− perturbed

h − exact

h− perturbed

σ2

Xe − exact

σ2

Xe − perturbed

Figure 2.4: The change of the average temperature of the slab (T ) due to theperturbation of various input parameters. With increased neutron absorptionthe power and the temperatures have to decrease to reach criticality, whilewith better heat conduction the higher power level still results in slightly loweraverage temperatures due to the increased absorption in xenon.

When the power is allowed to change, an increase in the fast fission crosssection

(Σf

1

)and the thermal conductivity (h) leads to a higher power level

and higher xenon concentrations. In contrast, when the absorption cross sectionof xenon is increased, criticality can only be reached at lower power with lowertemperatures and xenon concentration. When power-reset is used and thethermal absorption cross section of the slab (Σt

2) is perturbed according toEquation 2.12, the sensitivities are much smaller as most of the reactivityworth of the perturbations is counterbalanced by the change in the thermalabsorption, hence smaller changes are needed in the xenon concentration. Incase of perturbing the thermal conductivity the change is negligible, as thesame power leads to almost the same flux in the perturbed state, yieldingidentical xenon poisoning, and criticality is reached with the higher thermalabsorption of the slab being counterbalanced by the lower average temperaturesdue to the higher conductivity.

2.5.3 Solution of the Adjoint Problem

The main issue when calculating sensitivities is obtaining the appropriate ad-joint functions, i.e. solving Equations 2.19-2.20. The block iterative proceduredescribed in Section 2.3 was successfully applied to the slab problem and the

37


−10 −8 −6 −4 −2 0 2 4 6 8 10−20

−15

−10

−5

0

5

10

15

20

Perturbation [%]

Ma

xim

um

Xe

co

nc

en

tra

tio

n p

ert

urb

ati

on

[%

]

Σ1

f − exact

Σ1

f − perturbed

h − exact

h − perturbed

σ2

Xe − exact

σ2

Xe − perturbed

(a) Change without power-reset

−10 −8 −6 −4 −2 0 2 4 6 8 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Perturbation [%]

Ma

xim

um

Xe

co

nc

en

tra

tio

n p

ert

urb

ati

on

[%

]

Σ1

f − exact

Σ1

f − perturbed

h − exact

h − perturbed

σ2

Xe − exact

σ2

Xe − perturbed

(b) Change with power-reset

Figure 2.5: The change in the maximum xenon concentration due to theperturbation of various input parameters

solution could be calculated typically within 100-200 iterations depending onthe response dependent adjoint source and the value of ζ. However, the numer-ical stability of the procedure was far from satisfactory. For certain values ofζ divergence occurred, while for other values only very slow convergence wasexperienced. Moreover 100 iterations means that for a single response boththe adjoint of the neutronics and the augmenting equation need to be solved100 times, which in certain cases is comparable to the computational cost ofdirect recalculation of the coupled problem, and might even be comparable tothe effort needed to develop one code capable of solving the coupled adjointproblem as a whole.

Hence in the next step a Krylov solver was employed to solve the adjointsystem and the procedure described in Section 2.3 was supplied to it as apreconditioner without any relaxation (for comparison calculations were donewithout preconditioning as well). As the coupled adjoint problem is non-symmetric, the GMRES algorithm was chosen, which provided significantimprovements. In Figure 2.6 the number of Krylov iterations is shown fordifferent values of ζ and k (the number of iterations in the preconditioningstep). Increasing the value of ζ distorts the adjoint transport operator andmakes it easier to solve the adjoint transport problem, but increases the numberof needed Krylov iterations. In contrast, increasing the number of iterationsduring preconditioning (k) decreases the number of steps in the Krylov solver

38

2.5. Results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92

3

4

5

6

7

8

Value of

Nu

mb

er

of

Kry

lov

ite

rati

on

s

k=1

k=2

k=3

k=4

ζ

Figure 2.6: The number of Krylov iterations needed to calculate the adjointfunctions for the maximum temperature in the fully coupled model. The morethe transport operator is distorted, the more iterations are needed but theeasier it gets to solve the adjoint transport problem. The more steps are usedin preconditioning the fewer are needed in the Krylov iterations.

(in the extreme case of k 1 only one Krylov step is needed and only theprocedure described in Section 2.3 is used). It is clear that there is a trade-offbetween the number of Krylov steps, preconditioning steps and the value of ζ.

Figure 2.7 shows the number of separate (uncoupled) transport and augment-ing adjoint calculations needed to obtain the adjoint functions for differentresponses in the fully coupled model. This value is simply the number ofiterations during each preconditioning step (k) multiplied by the number ofKrylov iterations. Increasing the number of iterations in preconditioning doesnot lead to a decrease in the overall computational time since the number ofKrylov iterations does not change substantially. The results suggest that thebest choice of k is one, i.e. only one transport and one augmenting adjointcalculations are needed in every preconditioning step. The ideal choice ofζ is more complicated since the time needed for the solution of the adjointtransport problem depends on it as well.

Finally, in Figure 2.8 a comparison is made between the convergence of thevarious approaches for the solution of the coupled adjoint problem for thepower and eigenvalue perturbations in the fully coupled model. When the blockiteration presented in Section 2.3 is used as a standalone method the residualdecreases in every step of the iteration and convergence is reached with roughly

39


4

6

8

10

12

14

16

18

20

Nu

mb

er

of

sep

ara

te a

djo

int

calc

ula

tio

ns

λ R1 R2 T η p P thermalNL P fast

NLTmaxXemax

k=1k=2k=3k=4

Figure 2.7: The number of separate adjoint calculations needed for variousresponses as a function of the number of preconditioning steps in the fullycoupled model (with ζ = 0.3). Increasing the number of preconditioning stepsusually does not lead to a decrease of calculation time as the number of Kryloviterations does not decrease substantially. R1 is the fast fission reaction rate,while R2 is the thermal to fast fission reaction rate ratio.

100 iterations. When the GMRES algorithm is used to the full problem withoutpreconditioning, more than 1000 iterations are needed, moreover when restartedGMRES method is used convergence is not reached. However when GMRESwith the block preconditioning is applied only 8 iterations are necessary.All results presented in this section correspond to the scaled adjoint problems,since the unscaled problems - though usually needed slightly less Kryloviterations (typically 2-3) - proved to provide incorrect variations for certainperturbations and responses.

2.6 Summary

In this chapter a technique for calculating sensitivities in coupled criticalityproblems was presented. The developed method enables the efficient calcula-tion of first order changes in responses of interest with respect to variations ofboth the neutronics input parameters (e.g. cross sections) and those describingaugmenting phenomena (e.g. thermal-hydraulics). Responses include thesteady state power level, the reactivity worth of perturbations and functionalsof the flux and the augmenting variables. Just like in standard generalizedperturbation theory sensitivities can be obtained with or without reset pro-

40

2.6. Summary

100

101

102

103

10−7

10−5

10−3

10−1

101

Iteration number

Residual

Block iteration

GMRES

Rest. GMRES

Prec. GMRES

Figure 2.8: The residual of the adjoint problem with different iterative ap-proaches. The block iterative method is clearly better than the Krylov meth-ods without preconditioning, however the preconditioned GMRES algorithmprovides superior convergence properties.

cedures constraining the power level or criticality, and the power- and k-resetyield identical results. The obtained sensitivities make it possible to quantifyuncertainties on important safety parameters (like maximum cladding temper-ature), taking into account sources other than uncertain neutronic parameters(for example an uncertain thermal conductivity or gap conductance).

The sensitivities in the coupled model can be calculated by using appropri-ate neutronics and augmenting adjoint functions. A possible procedure forobtaining these functions was also presented. The method makes use of theneutronics and augmenting codes capable of solving the respective (decoupled)adjoint problems and obtains the solution of the coupled system by iterativelysolving the separate adjoint problems and taking care of the coupling throughappropriate source terms. This eliminates the need of developing complicatednew computer programs to deal with the coupled adjoint problem and requiresonly modest changes to the neutronics and the augmenting codes.

The developed method was demonstrated on a one-dimensional slab withthermal-hydraulic feedback and fission product poisoning. Examples of re-sponse sensitivities were presented with special interest to multi-physics de-pendencies, e.g. sensitivities of reaction rates to thermal-hydraulics parametersor sensitivities of temperatures to cross-sections. The viability of the proposediterative procedures was shown, providing promising results when used as a

41


preconditioner for Krylov algorithms applied to the coupled adjoint problem.

In the next chapter the larger scale implementation issues of the adjointtheory are investigated on the coupled neutronic thermal-hydraulic model ofa fuel pin. As will be seen if both the neutronics and the augmenting codesare adjoint capable this requires only minor code modifications, therefore themethod has significant potential for practical application as well.

42

Chapter 3

Practical Applicability ofAdjoint Techniques to

Coupled Problems

In Chapter 2 the theory of the adjoint based sensitivity analysis of coupledcriticality problems was discussed. The approaches for calculating and interpret-ing sensitivities were demonstrated on the example of a one-dimensional slab,where two-group diffusion theory was coupled with heat conduction and fissionproduct poisoning. While this was suitable for a proof of principle study, itspractical usefulness is clearly limited. Therefore in this chapter the large scaleimplementation issues of the coupled adjoint theory are discussed. It is shownthat existing adjoint capable neutron transport and thermal-hydraulics (andother augmenting) codes can relatively easily be modified in order to solve thecoupled adjoint problems, therefore the true effort lies in the implementationof the coupling strategy itself, rather than on the code development.

In Section 3.1 first a deeper insight is given into the exact form of the requiredadjoint operators and their possible implementation into neutron transport andaugmenting codes. Then in Section 3.2 an application is discussed to an infinitearray of fuel pins, where a multigroup discrete ordinates neutron transportcode is coupled to a purpose made adjoint capable thermal-hydraulics code.This code is suited for calculating the steady state temperature distribution

43

3. Practical Applicability of Adjoint Techniques to Coupled Problems

in a GFR2400 fuel pin, as well as solving the linearized heat-conductionequation (i.e. performing the Forward Sensitivity Analysis Procedure) andits adjoint (performing Adjoint Sensitivity Analysis Procedure). Finally, inSection 3.3 some sample results are shown, including the eigenvalue of thesystem, traditional reaction rate ratios, as well as the average temperatures inthe pin. The chapter finishes with some concluding remarks in Section 3.4.

3.1 TheoryIn Chapter 2 it was shown that the different perturbation approaches lead tovery similar adjoint problems. Therefore in this chapter we only focus on theeigenvalue perturbation approach.

3.1.1 The Perturbed Coupled Criticality Problem

First let us recall the perturbed coupled system in case the power is constrainedto the unperturbed value of P 0 (for a more detailed derivation of the equationssee Section B.2). The linearized problem around the unperturbed solutioncharacterized by φ0, T 0 and λ0 takes the form of

M0∆φ + ∂M

∂T

∣∣∣∣∣0

∆Tφ0 + ∂M

∂α

∣∣∣∣∣0

∆αφ0 = ∆λF 0φ0 (3.1)

∂N

∂φ

∣∣∣∣∣0

∆φ + ∂N

∂T

∣∣∣∣∣0

∆T + ∂N

∂α

∣∣∣∣∣0

∆α = 0 (3.2)

1P 0

⟨P 0f ,∆φ

⟩φ+ 1

P 0

⟨∂Pf∂T

∣∣∣∣0

∆T, φ0⟩φ

+ 1P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

= 0. (3.3)

Since the neutron transport equation (without feedback) is linear, its effecton the change of the flux, i.e. the M0∆φ term is similar to its effect on theflux itself (M0φ0). The effect of the augmenting system on neutronics, i.e. the∂M/∂T

∣∣∣0

∆Tφ0 term is more complicated. In the transport equation the crosssections, fission spectra, etc. are the input parameters and each Σk neutronicdata can depend on the T augmenting variables. This dependence is most oftenthrough some T l averaged quantities (for example an average fuel temperature,or an average coolant density, etc.), therefore it can be calculated as

∂M

∂T

∣∣∣∣∣0

∆Tφ0 =∑k

∂M

∂Σk

∣∣∣∣∣0

∂Σk

∂T

∣∣∣∣0

∆Tφ0 =∑k

∑l

∂M

∂Σk

∣∣∣∣∣0

∂Σk

∂T l

∣∣∣∣0

∂T l∂T

∣∣∣∣∣0

∆Tφ0.

44

3.1. Theory

Introducing the notation of DlM0 =

∑k ∂M/∂Σk

∣∣∣0∂Σk/∂T l

∣∣∣0for the deriv-

ative of the transport operator with respect to the T l averaged augmentingquantities, the effect of the augmenting system can be written as

∂M

∂T

∣∣∣∣∣0

∆Tφ0 =∑l

DlM0 ∂T l∂T

∣∣∣∣∣0

∆Tφ0 =∑l

DlM0φ0 ∂T l

∂T

∣∣∣∣∣0

∆T.

In the second step we make use of the fact that the derivative transportoperator acts on the φ0 unperturbed flux and its action can be separated fromthe ∆T change of the augmenting variables. As a result, the J0 couplingoperator can be identified as J0 =

∑lDlM

0φ0 ∂T l/∂T∣∣∣0. Since the perturbed

coupled system is linear, each operator in Equations 3.1-3.3 can be representedby matrices. For J0 its corresponding matrix, specifically the (i,j)-th elementcan be written as [

J0]i,j

=∑l

[DlMφ0

]i

∂T l∂Tj

∣∣∣∣∣0,

where[DlM

0φ0]iis the i-th element of the flux-vector resulting from the

operation of the derivative transport operator on the unperturbed flux. Intro-ducing an

[L0T

]l,j

= ∂T l/∂Tj∣∣∣0operator linking the average quantities and the

individual augmenting dependent variables the J0 matrix can be calculated as[J0]i,j

=∑l

[DlM

0φ0]i

[L0T

]l,j

=∑l

[L0φ

]i,l

[L0T

]l,j. (3.4)

The construction given by Equation 3.4 is very advantageous from the im-plementation point of view. The matrix of the L0

T operator purely dependson the discretization of the augmenting system and its entries are typicallyweights, e.g. in case of a finite volume code the volume fraction of a spatialmesh element in different material regions. Furthermore in most cases it isa sparse matrix, since only a few of the augmenting variables determine theaverage values of the quantities the cross sections depend on. Consideringthermal-hydraulics in a light water reactor for instance, the most relevantquantities would be the fuel temperature and the coolant density, thereforethe matrix of L0

T would contain two rows, and only the entries belonging tothe fuel and the water around the pins would differ from zero. DlM

0φ0 areflux-like vectors (i.e. vectors of the same size as the flux) and are obtained as

45


the effect of the DlM0 derivative transport operators on the φ0 unperturbed

flux. Assembling the L0φ matrix as

L0φ =

[D1M

0φ0 D2M0φ0 D3M

0φ0 . . . DLM0φ0,

]

the coupling operator is easily obtained as a product:

J0 = L0φL

0T .

3.1.2 The Derivative Transport Operator

The DlM0 derivative transport operators are very similar to the normal

M0 transport operator, since each Σk neutronic data depends only on itself,therefore their derivatives with respect to all other parameters are zero, i.e.∂Σk′/∂Σk|0 = δk′,k (we neglect the implicit effect one cross section has on theothers through self-shielding). This means that an ∂M/∂T l

∣∣∣0φ0 operation can

be performed by any traditional neutron transport code with minimal modific-ations. To demonstrate this consider the steady state transport operator:

M0φ0 (r, E,Ω) = Ω · ∇φ0 (r, E,Ω) + Σt (r, E)φ0 (r, E,Ω) (3.5)

−∫

4π

∫ ∞0

Σtr (r, E′ → E,Ω′ → Ω)φ0 (r, E′,Ω′) dE′dΩ′

− λ0χ (r, E)4π

∫4π

∫ ∞0

ν(E′)

Σf (r, E′)φ0 (r, E′,Ω) dE′dΩ′.

The terms are as usual, the first two represent streaming and total removal,the second two transfer between directions and energies (i.e. traditionalscattering plus possible n,2n, n,3n, etc. reactions) and fission. Differentiatingall neutronics data with respect to some averaged augmenting quantity T l

46

3.1. Theory

yields

DlM0φ0 (r, E,Ω) = ∂Σt (r, E)

∂T l

∣∣∣∣∣0φ0 (r, E,Ω)

−∫

4π

∫ ∞0

∂Σtr(r, E′ → E,Ω′ → Ω

)∂T l

∣∣∣∣∣0φ0 (r, E′,Ω′) dE′dΩ′

− λ0

4π∂χ (r, E)∂T l

∣∣∣∣0

∫4π

∫ ∞0

ν(E′)

Σf (r, E′)φ0 (r, E′,Ω) dE′dΩ′

− λ0χ (r, E)4π

∫4π

∫ ∞0

∂ν (E′)∂T l

∣∣∣∣0

Σf (r, E′)φ0 (r, E′,Ω) dE′dΩ′

− λ0χ (r, E)4π

∫4π

∫ ∞0

ν(E′) ∂Σf (r, E′)

∂T l

∣∣∣∣∣0φ0 (r, E′,Ω) dE′dΩ′. (3.6)

Therefore the derivative transport operator is very similar to the traditionalone, with only three differences. First, there is no streaming, since it containsno cross sections and therefore its derivative is zero. Second, the total removaland transfer terms are as usual, with the normal cross sections replaced bytheir derivatives. Last, the fission term consists of three separate terms due tothe chain rule of differentiation. However, these can easily be calculated byroutines being able to calculate the fission source, with an artificial use of thecross sections: the routine has to be called three times, where the normal crosssections and one derivative cross section is used in place of the fission spectrum,the number of fission neutrons and the fission cross section respectively (thelast three terms in Equation 3.6).

The ∂Pf/∂T |0 ∆T term appearing in the constraint equation (Equation 3.3)is similar to the removal term in the derivative transport operator, since theconstraint most often contains some cross section (i.e. QΣf in case of a powerconstraint, νΣf in case of constraining to one fission neutron, or some Σdet

in case of constraining to a detector response). Hence the derivative power(constraint) function usually takes the form of

∂Pf∂T

∣∣∣∣0

∆T =∑l

∂ΣCf

∂T l

∣∣∣∣∣0

∂T l∂T

∣∣∣∣∣0

∆T,

where ΣCf stands for the particular cross section corresponding to the chosenconstraint.

47


Producing Derivative Cross Sections The dependence of the nucleardata on the augmenting variables is usually complicated and problem de-pendent. Typically multigroup cross sections are determined by elaborateself-shielding methods and interpolation algorithms for the desired conditions(i.e. material compositions, temperatures, densities, etc.). This makes theanalytical determination of the derivatives - at least in the scope of this thesis- unfeasible. Since the cross section preparation processes themselves can takesignificant amount of time, in coupled calculations it is not common practiceto generate a library at each time or iteration step for the exact conditionsof the system. Instead cross sections are prepared at the beginning of thecalculations for a number of predefined conditions (e.g. at temperatures with acertain step size, different coolant densities, etc.), and then these pre-generatedlibraries are interpolated by “mixing” them with appropriate weights. Thisapproach provides an approximate, but easy way to calculate the derivativecross sections at the nominal conditions (at T 0) by using finite differences:

∂Σk

∂T l

∣∣∣∣0≈

Σk|T 0l+∆T l

− Σk|T 0l−∆T l

2∆T l.

The above calculation can easily be carried out with standard cross sectionpreparation tools: one simply has to produce two sets of libraries at conditionsT

0l + ∆T l and T

0l −∆T l, then mix them with weights 1/2∆T l and −1/2∆T l

respectively. The mixing can be done by a simple subtraction, furthermoremost transport codes offer specific modules for such operations (the ICE moduleof the SCALE code package is an excellent example for instance (SCALE,2011)).

3.1.3 The Derivative Augmenting Operator

The derivative of the augmenting operator with respect to the flux is the lastpart we need for formulating the perturbed coupled problem. Without detailedknowledge about the augmenting system little can be said, therefore we willfocus on two important cases: thermal feedback and fission product poisoning.In the former case the flux represents a heat source as

P(∫ ∞

0

∫4πQ (r) Σf (r, E)φ (r, E,Ω) dΩdE, r → r′

),

where P ( . , r → r′) is a projection operator projecting the spatial domainof neutronics (r ∈ Ωφ) onto the spatial domain of the augmenting system

48

3.1. Theory

(r′ ∈ ΩT ), with Ωφ and ΩT being the spatial domains of the two systems.If the domain of the neutronics overlaps that of the augmenting system, atleast within the fissile zones the projection is simply unity and one only hasto replace r with r′ in the fission power density. If this is not the case, forexample because the neutronics calculation is done with homogenized regions,the projection is more complicated. However, most often it would be a lineartransformation of the form of

P(f (r) , r → r′

)= Φ

(r′) ∫

Ωφf (r) δ

(r − r′

)dr

where Φ (r′) represents a shape function reconstructing the heterogeneouspower distribution from the distribution calculated with the homogenizedgeometries. In such cases the derivative of the augmenting operator withrespect to the flux is therefore

∂N

∂φ

∣∣∣∣∣0

∆φ = Φ(r′) ∫

Ωφ

∫ ∞0

∫4πδ(r − r′

)Q (r) Σf (r, E) ∆φ (r, E,Ω) dΩdEdr.

The case of fission product poisoning is rather similar, and due to theoverlapping spatial domains somewhat even simpler. In the correspondingequations the flux represents both a source through the fission process and aloss due to absorption in the fission product nuclei. In case of the most frequentxenon poisoning the augmenting operator in steady state is (Duderstadt andHamilton, 1976)

N (φ,Xe, α) = (YI + YXe)∫ ∞

0

∫4π

Σf (r, E)φ (r, E,Ω) dΩdE−∫ ∞0

∫4πσa,Xe (r, E)Xe (r)φ (r, E,Ω) dΩdE − λXeXe (r) ,

where YI and YXe stand for the fission yields of iodine (135I) and xenon (235Xe),σa,Xe (r, E) and λXe are the microscopic absorption cross section and decayconstant of xenon, and finally Xe (r) is the number density of xenon atoms.Hence the derivative operator takes the form of

∂N

∂φ

∣∣∣∣∣0

∆φ = (YI + YXe)∫ ∞

0

∫4π

Σf (r, E) ∆φ (r, E,Ω) dΩdE−∫ ∞0

∫4πσa,Xe (r, E)Xe (r) ∆φ (r, E,Ω) dΩdE.

49


3.1.4 The Coupled Adjoint Operator

With all operators defined in the perturbed system we can move on to define itsadjoint. For this let us recall that during the derivation of the adjoint systemthe neutronics and the augmenting equations are multiplied with neutronicsand augmenting weight functions, then they are integrated over their respectivephase spaces. To find the exact form of the adjoint operators appearing in thecoupled adjoint system these inner products have to be investigated.

Adjoint of the Neutron Transport and the Augmenting CouplingLet us start with the transport equation (Equation 3.1). When taking itsinner product with the w0

φ neutronics weight function there are four resultingterms. The first is the inner product with the traditional neutron transportpart, and the adjoint operator defined by

⟨w0φ, M

0∆φ⟩φ

=⟨(M0

)∗w0φ,∆φ

⟩φ

is the well-known adjoint transport operator taking the form of(M0

)∗w0φ (r, E,Ω) = −Ω · ∇w0

φ (r, E,Ω) + Σt (r, E)w0φ (r, E,Ω) (3.7)

−∫

4π

∫ ∞0

Σtr (r, E → E′,Ω→ Ω′)w0φ

(r, E′,Ω′

)dE′dΩ′

− λ0 ν (E) Σf (r, E)4π

∫4π

∫ ∞0

χ(r, E′

)w0φ

(r, E′,Ω

)dE′dΩ′.

The second term is the inner product with the derivative transport operator,which can be written as

⟨w0φ, J

0∆T⟩φ

=∫

Ωφ

∫4π

∫ ∞0

w0φ (r, E,Ω)

∑l

DlM0φ0 (r, E,Ω) ∂T l

∂T

∣∣∣∣∣0

∆TdEdΩdr

The dependence of the average augmenting quantities on the augmentingvariables only involves the augmenting spatial domain (ΩT ), and in most casescan be written as

T l = 1Vl

∫ΩT

T(r′)δ(r′ ∈ Vl

)dr′,

where Vl is the volume over which the average is calculated and δ (r′ ∈ Vl) isthe usual Dirac delta (or indicator) function. Again thinking of a light waterreactor, Vl would be the volume of the fuel or the water coolant around the pinin the unit cell, over which the integration has to be performed to calculate the

50

3.1. Theory

average fuel temperature or coolant density on which the cross sections dependthe most. The change of the averaged quantities can therefore be calculated as

∂T l∂T

∣∣∣∣∣0

∆T = 1Vl

∫ΩT

∆T(r′)δ(r′ ∈ Vl

)dr′. (3.8)

The adjoint operator defined by⟨w0φ, J

0∆T⟩φ

=⟨(J0)∗w0φ,∆T

⟩T

can beidentified by writing out the integrations as⟨

w0φ, J

0∆T⟩φ

=∑l

∫4π

∫ ∞0

∫Ωφw0φ (r, E,Ω)DlM

0φ0 (r, E,Ω) drdEdΩ ·

1Vl

∫ΩT

∆T(r′)δ(r′ ∈ Vl

)dr′ =∫

ΩT∆T

(r′)∑

l

δ (r′ ∈ Vl)Vl

⟨w0φ, DlM

0φ0⟩φ

dr′.

From the above one can see that the(J0)∗

adjoint operator is given by

(J0)∗w0φ =

∑l

δ (r′ ∈ Vl)Vl

⟨w0φ, DlM

0φ0⟩φ, (3.9)

therefore it involves taking inner products in the neutronics phase space withthe derivative transport operators and a projection of constant terms onto the(spatial) basis vectors of the augmenting space. This is very advantageousfrom the implementation point of view: routines to perform inner productsin the neutronics space are present in all neutron transport codes equippedwith Generalized Perturbation Theory capabilities, and as was shown in Sec-tion 3.1.2, the effect of the DlM

0 derivative transport operators can also beeasily computed with existing routines. Furthermore the projection of constantterms onto the augmenting space is necessary anyway for performing coupledcalculations and is simple: the discretization of the δ

(r′ ∈ Vl

)/Vl terms would

only involve rows of the L0T matrix.

The remaining two⟨w0φ, ∂M

0/∂α∣∣∣0

∆αφ0⟩φand ∆λ

⟨w0φ, F

0φ0⟩φterms are

very similar to the previous ones. In the former case again an inner product isneeded with the perturbed transport operator, whereas the latter case involvesthe inner product with the fission source. Both are needed for traditionalGeneralized Perturbation Theory, the former in the calculation of the response

51


variations, the latter when performing fundamental mode removal to obtainthe adjoint corresponding to responses other than the eigenvalue (Williams,1986). Therefore we can conclude that the adjoint of the neutron transportpart of the coupled system can easily be constructed using codes with GPTcapabilities, since the existing routines can perform all necessary inner productcalculations as well as calculate the effect of the derivative transport operator.

Adjoint of the Augmenting System and the Neutron Transport Coup-ling Taking the inner product of the augmenting equation (Equation 3.2)with the w0

T augmenting weight function yields three terms. In case of thermal-hydraulics coupling the inner product with the derivative of the augmentingoperator with respect to the flux can be written as⟨

w0T ,∂N

∂φ

∣∣∣∣∣0

∆φ⟩T

=∫

ΩTdr′w0

T

(r′)

Φ(r′)·∫

Ωφ

∫ ∞0

∫4πδ(r − r′

)Q (r) Σf (r, E) ∆φ (r, E,Ω) dΩdEdr =

∫Ωφ

∫ ∞0

∫4π

drdΩdE[∆φ (r, E,Ω)Q (r) Σf (r, E) ·

∫ΩT

w0T

(r′)

Φ(r′)δ(r − r′

)dr′]

=⟨(

∂N

∂φ

∣∣∣∣∣0

)∗w0T ,∆φ

⟩φ

.

From the above the adjoint of the coupling operator between the augmentingsystem and neutron transport can be identified as(

∂N

∂φ

∣∣∣∣∣0

)∗w0T = Q (r) Σf (r, E)

∫ΩT

w0T

(r′)

Φ(r′)δ(r − r′

)dr′. (3.10)

The inner product with the derivative of the augmenting operator withrespect to the augmenting dependent variables depends entirely on the aug-menting system, therefore little can be said in general. However, any ad-joint capable code contains routines for such calculations, since they areneeded in the calculation of the response variations, the main rationale ofhaving an adjoint capable code. As a result the adjoint operator defined by⟨w0T , ∂N/∂T

∣∣∣0

∆T⟩T

=⟨⟨∂N/∂T

∣∣∣0

⟩∗w0T ,∆T

⟩Trepresents no implementa-

tion issues if the augmenting code is adjoint capable, i.e. it would automaticallycontain routines for such inner products as well as matrix-vector multiplications.

52

3.1. Theory

The inner product with the last term in Equation 3.2,⟨w0T , ∂N/∂α

∣∣∣0

∆α⟩T

is similar to the previous case, the only difference is that the augmentingoperator has to be differentiated with respect to the input data instead of thedependent variables. Again, such routines are needed in every adjoint capableaugmenting code and it should pose no difficulty to use these routines.

The Source Terms in the Coupled Adjoint System The last issue toinvestigate before the coupled adjoint system can be assembled is the operatorsappearing in the constraint equation (Equation 3.3). Multiplying Equation 3.3with −w0

P yields three terms, the first two of which provide the source for theadjoint equations. In case of a power constraint the first term is

−w0P

P 0 〈Pf ,∆φ〉φ = −w0P

P 0

∫Ωφ

∫ ∞0

∫4πQ (r) Σf (r, E) ∆φ (r, E,Ω) dΩdEdr,

from which the source in the neutronics adjoint equation is simply identified asw0P /P

0Q (r) Σf (r, E), which in fact is identical to a traditional GPT source fora response where Σdet = w0

P /P0QΣf , and the detector is the whole neutronics

spatial domain (Ωφ).

The second term is similar to the derivative transport operator, since itcontains the derivative of the power (constraint) function with respect to theT l averaged augmenting quantities:

−w0P

P 0

⟨∂Pf∂T

∣∣∣∣0

∆T, φ0⟩φ

=

−w0P

P 0

∫Ωφ

∫ ∞0

∫4π

∑l

∂ (Q (r) Σf (r, E))∂T l

∣∣∣∣0

∂T l∂T

∣∣∣∣∣0

∆Tφ0 (r, E,Ω) dΩdEdr.

Again using Equation 3.8 the above expression can be written as

w0P

P 0

⟨∂Pf∂T

∣∣∣∣0

∆T, φ0⟩φ

=−w0P

P 0

∫ΩT

∆T(r′)∑

l

δ (r′ ∈ Vl)Vl

⟨∂Pf

∂Tl, φ0

⟩φ

dr′ =

−w0P

P 0

⟨∆T,

(∂Pf∂T

∣∣∣∣0

)∗φ0⟩T

and the source in the augmenting adjoint equation can be identified as

−w0P

P 0

∑l

δ (r′ ∈ Vl)Vl

⟨∂Pf

∂Tl, φ0

⟩φ

.

53


The term therefore involves an inner product in the neutronic space with thederivative power (constraint) operator and the unperturbed flux, and projectiononto the augmenting spatial space.

Finally, the −w0P /P

0⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

term is simply an inner product of

the perturbed power (constraint) operator and the unperturbed flux, purely inthe neutronic phase space. It is only different from zero if perturbations in thefission cross section or the energy release per fission (Q (r)) are made.

When considering responses other than the eigenvalue or the power thesources in the adjoint equations have to be supplemented with the responsedependent parts, i.e. the ∂R/∂φ|0 and ∂R/∂T |0 terms. The derivation ofthese terms however involve very similar procedures as presented before in thischapter. For example for a typical R =

⟨Σdet,1, φ

⟩φ/⟨

Σdet,2, φ⟩φreaction rate

ratio response the source in the adjoint neutron transport equation is identicalto the source when using standard GPT:

S∗φ,R = ∂R

∂φ

∣∣∣∣0

= Σdet,1

〈Σdet,2, φ0〉φ−

⟨Σdet,1, φ0

⟩φ

〈Σdet,2, φ0〉φΣdet,2

〈Σdet,2, φ0〉φ.

The augmenting source again contains the derivative cross sections with respectto the averaged augmenting quantities and reads

S∗T,R = ∂R

∂T

∣∣∣∣0

=

∑l

δ (r′ ∈ Vl)Vl

⟨∂Σdet,1/∂T l

∣∣∣0, φ0

⟩φ

〈Σdet,2, φ0〉φ−

⟨Σdet,1, φ0

⟩φ

〈Σdet,2, φ0〉φ

⟨∂Σdet,2/∂T l

∣∣∣0, φ0

⟩φ

〈Σdet,2, φ0〉φ

.Last, for an average or maximum augmenting quantity response there isno neutronics source and the augmenting source is simply δ

(r′ ∈ Vl

)/Vl or

δ (r′ − r′max) respectively, where rmax is the position where the the maximumof the augmenting quantity occurs (Cacuci et al., 2005).

3.1.5 Solution Strategy for the Coupled Adjoint Problem

As was detailed in Section 2.3 a very efficient method to solve the coupledadjoint equation is to use a Krylov solver together with the individual codes

54

3.1. Theory

to perform the necessary matrix-vector multiplications as well as inversionsduring the preconditioning steps. To illustrate this procedure consider thefinal form of the coupled adjoint problem (Equations 2.19-2.20):

(M0

)∗wφ +

(∂N

∂φ

∣∣∣∣∣0

)∗wT = S∗φ

(J0)∗wφ +

(∂N

∂T

∣∣∣∣∣0

)∗wT = S∗T .

First the sources have to be calculated. As was shown in Section 3.1.4 thisstep involves calculating traditional Generalized Perturbation Theory sourcesfor different responses, taking inner products in the neutronics phase spaceand possibly sources in the augmenting equations which are spatially uniformover certain parts of the augmenting spatial domain or which are representedby a Dirac delta.

Once the sources have been calculated the Krylov iteration can start, in-volving two types of actions: matrix-vector multiplications and inversions.During matrix-vector multiplications four separate multiplications have to beperformed: (i) multiplication with the adjoint transport operator (involvingonly the neutron transport part), (ii) multiplication with the adjoint of theneutronics coupling to the augmenting system (involving integration in theaugmenting spatial domain and projection to the neutronics spatial domain ac-cording to Equation 3.10), (iii) multiplication with the adjoint of the derivativetransport operator (involving inner products in the neutronic phase space andprojection to the augmenting spatial domain) and (iv) multiplication with theadjoint augmenting operator (purely involving the augmenting system). Allthese operations can be easily carried out with the already available routines inadjoint capable neutron transport and augmenting codes, the most significanteffort is in the projection of the neutronic and augmenting spatial domains,which is needed for any coupling strategy.

During the preconditioning steps the separate codes are used to invert theadjoint transport and augmenting operators with the sources given by theKrylov algorithm. As was shown in Section 2.5.3 an effective preconditioningstrategy is to perform a simple block iteration, i.e. solving

(M0ζ

)∗wk+1φ = S∗,Kφ −

(∂N

∂φ

∣∣∣∣∣0

)∗wkT + λ0ζ

(F 0)∗wkφ

55


and (∂N

∂T

∣∣∣∣∣0

)∗wk+1T = S∗,KT −

(J0)∗wk+1φ

using a single iteration (k = 0, with S∗,Kφ and S∗,KT being the sources providedby the Krylov iteration). For calculating the sources coming from the couplingthe same routines can be used as in matrix-vector multiplication, while theinversion of the adjoint transport equation and the adjoint augmenting equationcan be done with the individual codes. One only has to ensure the necessaryinterfaces between them, i.e. passing an external source from one to theother and requesting a solution with that source, then passing back the finalneutronics and augmenting weight functions.

3.2 ApplicationTo demonstrate the practical, larger scale applicability of the theory introducedin Chapter 2 and detailed in Section 3.1 the sensitivity analysis of a coupledmodel of an infinite pin array has been performed. For the calculations neutrontransport was coupled to thermal-hydraulics, where the two way couplingwas present due to the dependence of the cross sections on the temperaturedistribution in the pin and the neutron distribution providing the heat source.The calculations were performed using an in-house developed neutron transportcode and a purpose made thermal-hydraulics code, together with a Krylovsolver wrapped around them. In Section 3.2.1 and Section 3.2.2 these twoprograms are discussed, whereas Section 3.2.3 provides some details on thecoupled model.

3.2.1 Neutronics Code

For the neutronics calculations the in-house developed PHANTOM code wasused. PHANTOM is a general purpose multigroup discrete ordinates codewith finite element spatial discretization using discontinuous Galerkin methods(Lathouwers, 2011a,b; Kópházi and Lathouwers, 2012), equipped with goaloriented spatial and angular adaptivity as well. The code is interfaced withthe open-source GMSH mesh generator (Geuzaine and Remacle, 2009) and iscapable of solving the transport equation in one, two and three-dimensionalCartesian geometries. To be able to solve the coupled adjoint problem tra-ditional Generalized Perturbation Theory capabilities were implemented in

56

3.2. Application

the code, both for the eigenvalue as well as reaction rate and reaction rateratio responses. Since PHANTOM employs Krylov methods for solving theneutron transport equation, routines were already available for performingmatrix-vector multiplications, as well as inversion of the transport operatorwith fixed sources. Therefore apart from the traditional GPT routines (i.e.functions to calculate sources for specific reaction rate and reaction rate ratioresponses, perform inner products with the perturbed transport operator)only one additional routine had to be implemented, which calculates the in-ner product of an adjoint and the result of the derivative transport operatoroperating on a flux (according to Equation 3.9).

3.2.2 Thermal-Hydraulics Code

For the thermal-hydraulics calculations a purpose made program was employedthat uses the KINSOL C package, providing both nonlinear and linear solvers,as well as convenient data structures for matrices and vectors (Hindmarsh et al.,2005). This code solves the heat-conduction equation specifically for the fuel pinof the GFR2400 Gas Cooled Fast Reactor design (Stainsby et al., 2011; Perkóet al., 2014b) with finite volume discretization in r-z geometry. Both radialand axial conduction is taken into account, and appropriate correlation lawsare used for the UPuC fuel thermal conductivity, gap conductance, claddingthermal conductivity, heat transfer coefficient and properties of the coolanthelium (Petersen, 1970; Zabiego et al., 2011; Mikityuk, 2012). For a schematicview of the fuel pin see Figure 3.1.

In the direct mode the code solves the nonlinear system of equations resultingfrom the discretization, where the nonlinearity comes from the dependenceof the thermal conductivities on the temperatures, the inclusion of the heatradiation in the gap and the dependence of the heat transfer correlation ontemperatures, Reynolds number, etc. In the forward sensitivity mode (FSAP)the code solves the linearized system for specific perturbations of the inputthermal data. Finally, in the adjoint sensitivity mode (ASAP) it solves theadjoint of the heat transfer, via transposing the matrix of the linearizedsystem and using appropriate adjoint sources corresponding to average andmaximum responses. For the coupled use minimal modifications of the codewere needed, namely two extra execution modes and one new routine. Thefirst additional execution mode involves reading a temperature distribution(augmenting variables) then multiplying it with the adjoint matrix (matrix-

57


Fuel pellet diameter: 6.71 mm

Pin He gap:

145 mm

Internal liner:

40 mm W14Re,

10 mm Re

Clad:1mm

External li

ner:

30 mm SiC

Pin

diam

eter

: 9.1

6 m

m

(a) Cross sectional view

Tin

, Pin

, m

P

Tout

, Pin

- Pdrop

.

(b) Schematic view

Figure 3.1: Schematics of the GFR2400 pin. Thin metallic liners on theinner side of the ceramic cladding ensure the leak tightness of the pins. Thethermal-hydraulics code calculates the temperature distribution correspondingto the specified inlet temperature, inlet pressure and pressure drop, mass flowand power distribution.

58

3.2. Application

vector multiplication for the Krylov iteration), the second solving the adjointequations with an external source (inversion during preconditioning). Theextra routine was needed to calculate the integrals appearing in Equation 3.10.

3.2.3 Coupled Neutronics Thermal-Hydraulics Model of aPin

Preliminary Steps For the coupled model of the pin first preliminaryneutron transport calculations were carried out with the SCALE code package(SCALE, 2011). The purpose of these was to homogenize the heterogeneousgeometry of the pin and to generate few group cross sections from the standard238 group ENDF/B-VII library. Both actions were done using classic techniquesimplemented in the XSDRN one-dimensional finite difference discrete ordinatestransport routine of SCALE. To homogenize the pin containing the fuel pellets,gap, liners, cladding and the surrounding helium coolant (constituting the unitcell of the reaction) one-dimensional calculations were employed in cylindricalgeometry, these provided the cross sections for the active zone. To collapsethe cross sections for the plenums and reflectors above and below the activezone the calculations were carried out in slab geometry, using the homogenizedcross sections for the active zone. The resulting cross section library thereforecontained the few group cross sections for a one-dimensional homogenized axialmodel of the infinite pin array.

In the second step, the dependence of the cross sections on the temperaturedistribution of the pin was examined. The calculations detailed in the previousparagraph were carried out for several different fuel, liner and cladding temper-atures and it was found that only the fuel temperature has noticeable effectson the effective multiplication factor. Therefore the final cross sections wereonly considered to be a function of the fuel temperature and were generatedon a 100 K mesh.

The last preparatory step was the investigation of the flux distribution withinthe pin to determine the Φ (r′) flux reconstruction function. It was foundthat the power distribution has only minor radial dependence in the fuel andchanges are much more significant in the axial direction. Therefore Φ (r′) = 1was chosen for a given axial height and all subsequent transport calculationswere effectively carried out in one dimension with the homogenized pin model.

59


Model Development Once all preliminary calculations have been carriedout traditional convergence studies were performed with the two separatecodes (i.e. PHANTOM and the purpose made thermal-hydraulics code) todetermine appropriate spatial meshes, SN set, scattering expansion, etc. Itwas found that an S8 set and first order scattering is sufficient, with 161structured quadrilateral elements and linear basis functions in the neutronicsspace, and roughly 1500 finite volume elements in the thermal-hydraulic mesh.The two meshes were chosen such that in the fuel region they overlap, whichmakes the coupling strategy easy. The calculations were carried out with thesespecifications and the power of the system was constrained to the average pinpower of the GFR2400 design, i.e. 2400 MW/ (516 · 217) = 21433.93 kW.

3.3 ResultsIn this section some sample results are shown for the coupled system, highlight-ing the interdependencies between neutronics and thermal-hydraulics. Theseare specific for the investigated GFR2400 fuel pin and give estimates of themost important parameters influencing the neutronics of the active core and thetemperature distribution within the pin. The results indicate which parametersneed to be determined with high precision in order to limit uncertainties forthe GFR in question, however the major merits of these results are that theydemonstrate the versatility and the relatively easy applicability of the adjointtechnique. Furthermore they also prove that obtaining the solution of thecoupled adjoint problems is possible with low computational costs in realisticsystems as well.

3.3.1 Eigenvalue

In Figure 3.2 the eigenvalue of the system can be seen as a function of varioustraditional cross section perturbations. Figure 3.2a shows typical behaviour, theeigenvalue of the system decreases with increasing the fission cross section (Σf

1)or the number of neutrons born in the fast group (χ1), whereas it increases whenonly the capture cross section (Σc

1) is increased, since in this case more neutronsare absorbed in the fuel. The predictions from the coupled adjoint theory arenaturally only first order accurate, therefore result in some underpredictionsin this case for large (±20%) perturbations of the inputs. Figure 3.2b showsthe effects of the upper reflector on the system, for both energy groups. Onecan see that the system is clearly a fast system, since even in the reflector

60

3.3. Results

−20 −10 0 10 20

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Perturbation [%]

Eig

en

va

lue

(λ

)

Σ1

f − exact

Σ1

f − perturbed

Σ1

c − exact

Σ1

c − perturbed

χ1

− exact

χ1

− perturbed

(a) The effect of fast group fuel crosssections.

−20 −10 0 10 200.83645

0.83650

0.83655

0.83660

0.83665

0.83670

Perturbation [%]

Eig

en

va

lue

(λ

)

Σ1

t − exact

Σ1

t − pert.

Σ2

t − exact

Σ2

t − pert.

Σ1

s − exact

Σ1

s − pert.

Σ1

c − exact

Σ1

c − pert.

(b) The effect of top reflector cross sec-tions

Figure 3.2: The dependence of the eigenvalue on different cross sections. Asexpected the coupled adjoint technique (depicted with straight lines) accuratelypredicts the first order changes (exact values depicted with symbols).

region, where the spectrum is the softest there is hardly any effect from theperturbations of the thermal cross sections (highlighted by the Σt

2 total crosssection), in fact all group two results overlap each other at the unperturbedlevel of the eigenvalue. Capture (Σc

1) and scattering (Σs1) compensate each

other, the former clearly increases neutron absorption therefore increases theeigenvalue, whereas the latter reflects neutrons back to the active core anddecreases the eigenvalue. The two effects are almost identical in magnitude,with scattering being slightly stronger, resulting in a slight negative sensitivityof the eigenvalue to the total fast cross section of the top reflector.

Figure 3.3 highlights some examples of the interdependency of neutrontransport and thermal-hydraulics. As can be seen in Figure 3.3a increasing thethermal conductivity of the fuel or the cladding both decrease the eigenvalue,since the power generated by the pin is conducted more efficiently to the heliumcoolant, therefore the fuel temperature is lower, which in turn decreases theabsorption due to the Doppler effect. The effect of the heat transfer coefficientbetween the cladding and the coolant is similar and even stronger, by increasingits value the same amount of heat can be transferred to coolant with lowercladding outer temperature, which decreases the temperatures in the wholepin. Figure 3.3b shows similar trends as a function of different helium thermalproperties. Increasing the specific heat of the coolant or the gap conductance

61


−20 −10 0 10 200.8364

0.8364

0.8365

0.8365

0.8366

0.8366

0.8367

0.8367E

ige

nv

alu

e

(λ)

Perturbation [%]

Fuel cond. − exact

Fuel cond. − perturbed

Clad cond. − exact

Clad cond. − perturbed

Heat transfer − exact

Heat transfer − pert.

(a) The effect of the thermal propertiesof the pin

−20 −10 0 10 200.8362

0.8364

0.8366

0.8368

0.837

0.8372

Perturbation [%]

Eig

en

va

lue

(λ

)

He sp. heat − exact

He sp. heat − perturbed

He viscosity − exact

He viscosity − perturbed

Gap conductance − exact

Gap conductance − pert.

(b) The effect of the helium thermalproperties

Figure 3.3: The dependence of the eigenvalue on different thermal properties.The coupled adjoint theory accurately predicts the first order variations of theneutronic responses due to perturbations in the augmenting system.

(i.e. perturbing the properties of the helium in the gap between the fuel andthe tungsten inner liner) both decrease the eigenvalue since both result lowerfuel temperatures. In contrast, increasing the viscosity of helium increases theeigenvalue, as it results in less effective heat transfer between the pin and thecoolant.

3.3.2 Reaction Rate Ratios

To demonstrate the applicability of the coupled adjoint theory for responsesother than the eigenvalue four classic reaction rate ratios were chosen andare shown in Figure 3.4. For easy comparison in each case the effect of threethermal-hydraulic parameters is highlighted: the fuel and cladding thermalconductivity, as well as the helium specific heat. Naturally, the effects ofcross section perturbations can also be calculated using the adjoint functions,however these are of less novelty than the effects of the augmenting system,therefore are not shown.

Figure 3.4a shows the ratio of the fission to total reaction rate ratio in theactive zone. As can be seen it increases with better conduction as well as withhigher helium specific heat, as all these result in a lower fuel temperature. Thesmaller temperature decreases the Doppler broadening of both the capture and

62

3.3. Results

−20 −10 0 10 20

0.012160

0.012162

0.012164

0.012166

0.012168

Perturbation [%]

Fis

sio

n/t

ota

l ra

tio







0.012158

(a) Fuel fission to total ratio

−20 −10 0 10 200.829750

0.829752

0.829754

0.829756

0.829758

0.829760

0.829762

0.829764

0.829766

0.829768

0.829770

Perturbation [%]

Fu

el/

tota

l a

bso

rpti

on







(b) Fuel to total absorption ratio

−20 −10 0 10 209035

9040

9045

9050

9055

9060

9065

Perturbation [%]

Fa

st/t

he

rma

l n

eu

tro

n y

ield







(c) Fast to thermal neutron yield ratio

−20 −10 0 10 201.2547

1.2548

1.2549

1.255

1.2551

1.2552

1.2553

1.2554

1.2555

Perturbation [%]

Ass

ym

me

try

fa

cto

r







(d) Top to bottom total ratio

Figure 3.4: The dependence of various reaction rate ratio responses as afunction of thermal properties. For small perturbations adjoint techniques canbe more accurate than direct calculations.

the fission resonances, however the former decrease relatively more. The lowabsolute value of fissions compared to all reactions is due to the high fraction ofscattering in the homogenized fuel region and the fast spectrum of the reactor.

In Figure 3.4b the fuel utilization factor can be seen, i.e. the ratio of theabsorption in the fuel and the total absorption in the pin array (integrated overthe energy groups). Not surprisingly it is relatively high, over 80%, since theplenums above and below the core contain only He and the reflectors mainlyscatter. When the fuel temperature decreases, it reduces the fuel utilizationas well, since the absorption in the fuel decreases whereas it is not affected inthe other parts. These changes are however very small, and highlight another

63


advantage of adjoint techniques, namely that for small changes they can bemore accurate than direct methods. In any calculation the iterations continueuntil a certain convergence criteria (or the maximum number of iterations) ismet. In our case for example both neutron transport and thermal-hydraulicswere solved for a relative change of less than 10−7, whereas the coupledcalculations were stopped when the relative change between two iterationswere below 10−5. From Figure 3.4b it can clearly be seen that this accuracyis comparable to the changes in the fuel utilization factor (10−6), making thedirect results scatter around the predictions by the adjoints. As the changesbecome more significant the direct results get smoother as well, and the adjointresults correctly give their tangent.

In Figure 3.4c the fast to thermal fission neutron yield ratio, i.e.

ν1Σf1∫Vfφ (r) dr

ν2Σf2∫Vfφ (r) dr

is depicted. Its high value clearly shows that the investigated GFR2400 designis a fast reactor, there is a difference of almost 4 orders of magnitude betweenthe number of fast and thermal neutrons produced by fast and thermal fission.The decrease of the fuel temperature decreases both reaction rates, however thefast fission yield decreases relatively more, resulting in the declining reactionrate ratio.

Finally, Figure 3.4d shows an asymmetry factor, where the total reactionrate in the top reflector and plenum is compared to that in the bottom reflectorand plenum. Since the top plenum is 85 cm high and the bottom is only 50cm, the asymmetry factor is higher than 1. The difference in the plenums alsocauses and asymmetry in the flux distribution, which is slightly tilted to thebottom due to the closer reflector. With the increase of the conductivitiesor the specific heat the fuel temperature decreases again, making the activezone less absorbing and therefore more transparent. This slightly decreasesthe asymmetry of the flux as well, causing the asymmetry factor to decreasesomewhat.

3.3.3 Temperatures

As a last example of responses the average fuel temperature is investigated.Figure 3.5 shows its dependence on the heat conduction and helium properties

64

3.3. Results

−20 −10 0 10 201330

1340

1350

1360

1370

1380

Perturbation [%]

Av

era

ge

fu

el

tem

pe

ratu

re [

K]





Heat transfer − exact

Heat transfer − pert.

(a) The effect of the thermal propertiesof the pin

−20 −10 0 10 201300

1325

1350

1375

1400

1425

1450

Perturbation [%]

Av

era

ge

fu

el

tem

pe

ratu

re [

K]



He viscosity − exact

He viscosity − perturbed

Gap conductance − exact

Gap conductance − pert.

(b) The effect of the helium thermalproperties

Figure 3.5: The dependence of the average fuel temperature on the differentthermal properties

and has a direct correlation with Figure 3.3, since the average fuel temperatureprovides the feedback to neutronics. As one expects therefore the propertiesthat decrease (or increase) the eigenvalue in Figure 3.3 are the same as theones that decrease (or increase) the fuel temperature in Figure 3.5. The formerones include the thermal conductivities of the fuel and the cladding, the heattransfer coefficient, the gap conductance and the helium specific heat, andsolely the helium viscosity belongs to the latter group from the parametersshown in Figure 3.5. The only other two parameters significant for the fueltemperature can be seen in Figure 3.6. As expected increasing the mass flowaround the pin and the inlet temperature of the coolant result in a lower anda higher fuel temperature respectively. Since eigenvalue perturbation is done,the power is constrained in each case to its unperturbed value, therefore noneof the cross sections have a significant effect on the average temperatures.

3.3.4 Performance Evaluation

To evaluate the performance of the coupled adjoint theory the computationalcosts associated to obtaining the necessary adjoint functions are investig-ated. Figure 3.7 shows the relative residual as the function of the numberof Krylov iterations corresponding to the adjoint problems that were neces-sary for obtaining the neutronics and the thermal-hydraulics adjoint functionsin Sections 3.3.1-3.3.3. As can be seen the convergence of the iterations is

65


−20 −10 0 10 201200

1250

1300

1350

1400

1450

1500

Perturbation [%]

Av

era

ge

fu

el t

em

pe

ratu

re [

K]

Inlet temp. − exact

Inlet temp. − pert.

Mass !ow − exact

Mass !ow − perturbed

Figure 3.6: The average fuel temperature as a function of different thermalparameters

very similar, therefore we can conclude that regardless of the source and thesize of the system the used GMRES algorithm preconditioned with the blockiteration is a very effective way for solving the coupled adjoint system. 8to 12 iterations are sufficient to achieve a relative residual of less than thespecified convergence criteria (10−14 and 10−11 for the coarsely and finelymeshed cases respectively). One needs to remember however that for responsesother than the eigenvalue the adjoint problem cannot be solved directly, sincedue to the presence of the ∆λ

⟨wλφ, F

0φ0⟩φterm the source is not known at

the beginning. The term associated to the change of the eigenvalue can bemade disappear by adjusting the wλP parameter during the iteration accord-ing to Equation B.22. This can be done for example with a special Krylovsolver where the source can be updated during the iteration, however it issignificantly easier to simply solve the problem twice: once with wλP = w0

P ,i.e. using the same value as was used for the eigenvalue problem, and oncewith wλP = wλP = w0

P −⟨wλφ, F

0φ0⟩φ/⟨w0φ, F

0φ0⟩φ, where w0

φ is the neutronicsadjoint function for the eigenvalue . Therefore the total number of Kryloviterations needed for responses other than the eigenvalue, i.e. for all responsesdiscussed in Section 3.3.2 and in Section 3.3.3 is twice of those indicatedin Figure 3.7, since the coupled adjoint system has to be solved twice fortwo different sources. Both of these however converge similarly as shown inFigure 3.7.

66

3.3. Results

1 2 3 4 5 6 7 8 9 10 11

10−15

10−10

10−5

100

Iteration number

Relative residual

Lambda adjoint

R1 adjoint

R2 adjoint

R3 adjoint

R4 adjoint

(a) Coarse mesh

2 4 6 8 10 1210−15

10−10

10−5

100

Iteration number

Relative residual

Lambda adjoint

R1 adjoint

R2 adjoint

R3 adjoint

R4 adjoint

(b) Fine mesh

Figure 3.7: The relative residuals (compared to the initial residual) as afunction of the number of Krylov iterations in various adjoint problems, witha coarse and a fine mesh. The same trends are seen as was suggested byFigure 2.8.

As can be seen the coupled adjoint theory can predict the first order changesin responses of interest at the expense of roughly 20 Krylov iterations. Thismeans that the neutron transport and the augmenting system also has tobe solved 20 times due to preconditioning. Compared to the costs of tradi-tional GPT for a reaction rate ratio response this is an increase of an orderof magnitude (assuming two adjoint neutron transport calculations due tofundamental mode removal). However, the fair comparison is not against a pureneutronic problem, but against a coupled problem, where both neutron trans-port and the augmenting system has to be solved multiple times to convergeto the true coupled solution. For the current problem direct recalculationstypically need 5 to 10 iterations. Furthermore when direct calculation is usedneutron transport is an eigenvalue problem, needing power iterations to besolved, whereas during the Krylov iterations in the coupled adjoint theorythe neutron transport problem is only a fixed source multigroup problem.For the investigated system power iterations typically used 15-20 multigroupiterations, therefore the total cost of a direct recalculation for one set of inputparameters is 75-200 multigroup iterations, compared to the 20 multigroupiterations needed in the coupled adjoint problem. Last, one has to keep inmind that the coupled adjoint theory provides the first order changes for allperturbations, whereas the direct approach needs the coupled problem to be

67


solved repeatedly for each perturbation.

3.4 Summary

In this chapter the main implementation issues of the coupled adjoint theoryare discussed. The deeper look into the exact form of the adjoint operatorsreveals that for the most common cases of coupling neutron transport tothermal-hydraulics and fission product poisoning the effects of the operatorscan easily be calculated by routines able to calculate fission sources, innerproducts and inner products with the derivative transport operator. For thelatter it is shown that only very little code modification is needed, as routinesable to calculate the fission source can be used with a smart use of the normalcross sections and the derivative ones. The main challenge from the applicationpoint of view therefore lies in the projection of spatial meshes used in neutrontransport and the augmenting system, however this is needed anyway for acoupled calculation.

The applicability of the coupled adjoint theory to more realistic systems wasdemonstrated on an infinite array of fuel pins. For the temperature distributionin the investigated GFR design the most important parameters were found tobe the thermal conductivity of the fuel and the cladding, and the heat transfercoefficient between the cladding outer wall and the coolant. Each of theseparameters had their respective effects on all neutronic responses, since eachinfluences the average fuel temperature which provides the feedback to neutrontransport. Both neutronics and thermal-hydraulics parameters were shown tohave an effect on the eigenvalue of the system and the coupled adjoint theoryproved to be capable of providing the first order changes in all cases.

The main conclusion of this chapter is that adjoint techniques can notonly be used for pure neutronics calculations, but can also be applied tocoupled transport problems with relatively little code modifications, once theaugmenting code (and the used neutron transport code) is adjoint capable. Insuch calculations the number of input parameters is typically in the hundredsand even in the thousands, as neutron transport itself has a high number ofinputs (if many group calculations are used a single isotope alone can representmore than a thousand input parameters). Since the great advantage of usingadjoints is the fact that once the adjoint functions are available, they providethe values of the responses for all parameter perturbations, it is highly useful

68

3.4. Summary

to expand traditional adjoint techniques for coupled cases as well. This chapterproved that this is not only possible, but can be done relatively easily.

The challenge for the practical usefulness of the theory presented in Chapter 2and in this chapter is the development of adjoint capable augmenting codes,most significantly adjoint capable thermal-hydraulics solvers. However thereare already such projects underway (Fang et al., 2011; Farrell et al., 2013),once these reach maturity the coupled adjoint theory can a have significantcontribution to more accurate reactor calculations.

69


70

Chapter 4

Adaptive Polynomial ChaosTechniques for

Multi-Physics Problems

4.1 Introduction

In the previous chapters adjoint techniques were discussed for the sensitivityanalysis of coupled criticality problems. While the presented methods canbe used to determine the inter-dependencies between the different physicalphenomena present in reactors, they are tailored to the steady state operationof nuclear systems, furthermore they are intrusive, i.e. they need dedicated‘code development effort before they can be applied for large scale problems.Therefore in this chapter we focus on developing non-intrusive PolynomialChaos (PC) techniques, which can be used to perform the sensitivity anduncertainty analysis of any problem for which a computer program is alreadyavailable to solve it.

In simple terms the essence of PC schemes is nothing more than approx-imating a model output (i.e. a response of our interest) as a polynomialfunction of the model input parameters. Such a function, the PolynomialChaos Expansion (PCE) of the response basically constitutes a metamodelof the original problem and can be used to describe the stochastic nature of

71

4. Adaptive Polynomial Chaos Techniques for Multi-Physics Problems

the output in terms of its mean value, variance, covariance, distribution, etc.The idea was first introduced by Wiener to represent Gaussian processes byHermite polynomials (see Wiener, 1938), but later it was extended so thatother types of stochastic processes could also be addressed using polynomialsof the Askey family in the scope of generalized Polynomial Chaos (gPC) (Xiuand Karniadakis, 2002; Eldred et al., 2008).

PC methods belong to a wider family of spectral techniques aimed at recon-structing the solution of a stochastic problem by a Fourier series like expansion(Le Maître and Knio, 2010). Like any expansion such representation usesbasis vectors and expansion coefficients. The basis vectors are predefinedfunctionals of the random variables representing the stochastic input data,therefore they are random variables themselves. The expansion coefficients aredeterministic and their efficient computation is the main issue. In traditionalPC the basis vectors are multidimensional polynomials up to a certain order,hence the method inherently contains two limitations. On one hand, responsesnot smooth enough in the stochastic domain (i.e. being highly nonlinear incertain input variables) might require high polynomial orders to be properlyreconstructed, moreover discontinuous responses are even impossible to repres-ent with polynomials. On the other hand, even if the response is smooth withrespect to all input variables, the PC metamodel is restricted by the predefinedorder of the expansion and it is difficult to judge a priori what order is neededfor an adequate representation.

The former difficulty can be overcome by employing local basis vectorsinstead of global polynomials. In multi-element gPC (ME-gPC) the stochasticspace is decomposed into disjoint domains and on each of them polynomialswith a local support are used as basis vectors (Wan and Karniadakis, 2005a,b).The multi-element probabilistic collocation method (ME-PCM) is a variantof the same idea where a separate grid of points is used on each sub-domainof the stochastic space to interpolate the solution (Foo et al., 2008; Foo andKarniadakis, 2010). Employing wavelets is another promising method, thebasic idea was demonstrated by Le Maître et al. using wavelets of the Haarfamily in the Wiener-Haar expansion (Le Maître et al., 2004b), then later it wasgeneralized (Le Maître et al., 2004a). Ma and Zabaras introduced an adaptivecollocation algorithm in 2009 applying piecewise multi-linear hierarchical basisfunctions with which the response is interpolated on a locally refined mesh(Ma and Zabaras, 2009). All these techniques are mainly concerned withproblems which experience sharp changes or discontinuities in the stochastic

72

4.1. Introduction

space. In many engineering problems however such rapid response variationsare not encountered, hence the focus of this chapter is on the other limitationassociated with PC methods.

The predefined order of a traditional, full PCE confines the dependenceof the response to a maximum mixed polynomial order (and therefore to amaximum order in any of the input variables) and to a maximum dimension,i.e. a maximum number of interacting parameters. Though as the sparsityof effects principle (see Montgomery, 2001) suggests responses are generallydominated by only a handful of inputs and low order interactions, usuallyit is not possible to know a priori what that polynomial order is and howmany interacting parameters there are. Moreover with the increase of inputparameters the full PC basis grows rapidly, hence typically only low order PCexpansions of 2nd or 3rd order are used, obviously unable to catch higher orderdependencies and interactions.

These problems can be alleviated by using an adaptively constructed poly-nomial basis instead of a predefined one. A brief discussion of such basisadaptivity was already presented by Li and Ghanem in (1998), and lateressentially the same idea was revived by Lucor and Karniadakis in (2004),both papers dealing with a nonlinear oscillator subject to stochastic excitation.In these two articles the random variables used in the discretization of theexcitation are separated at each time step based on their linear contribution tothe solution (i.e. the norm of their linear component in the PCE) and only themost important ones are retained to produce the higher-order (nonlinear) terms.Another example of basis adaptivity set can be found in the works of Todorand Schwab (2007) and Bieri and Schwab (2009), both papers consideringelliptic stochastic partial differential equations with a PC representation wherethe maximum polynomial order and dimension of the retained basis vectorsare chosen such that the error associated with the truncation of the PCEis comparable to the errors associated with the finite representation of thestochastic inputs and the spatial discretization of the problem.

The most relevant research to this work with respect to adaptively con-structing the PC basis was done in the scope of stochastic point collocationmethods (Blatman and Sudret, 2010a,b, 2011). These techniques rely on anumber of model evaluations and use interpolation to calculate the coefficientsof the PCE. The use of a full PC basis (i.e. the traditional basis containingall multidimensional polynomials up to a certain mixed order) results in a

73


high number of needed simulations for the interpolation matrix to be wellconditioned. In contrast when a sparse PCE is applied the computationalcost can be decreased due to the smaller number of PC basis vectors withwhich the interpolation is done. Moreover the PC basis can be adaptivelybuilt up until the overall convergence of the sparse PCE is reached and theexperimental design (i.e. the stochastically chosen input parameters with whichthe model is run) can be expanded on-the-fly as needed by the algorithm forthe interpolations to be well conditioned.

In this chapter a similar approach is proposed in the scope of Non-IntrusiveSpectral Projection (NISP). The principle idea is to tailor the construction ofthe PC basis to the adaptive sparse grid algorithm used for the calculation of themultidimensional integrals encountered in the definition of the PC coefficients.In Section 4.2 the theory is presented, covering a general introduction toPolynomial Chaos, the details of the adaptive sparse grid algorithm and theadaptive build-up of the basis, finally concluding with the developed grid andbasis adaptive PC technique termed as Fully Adaptive Non-Intrusive SpectralProjection (FANISP). Section 4.3 introduces three models which are used totest the performance of the FANISP algorithm compared to traditional fullPCE and standard Monte Carlo sampling. Results are presented in Section 4.4,while in Section 4.5 conclusions are summarized.

4.2 Theory

As mentioned before the main idea behind spectral techniques is that stochasticquantities - like the uncertain input data or the corresponding random solutionof a problem - can be represented by a Fourier series like expansion. Sucha representation can take many forms depending on the chosen set of basisvectors. Among these Polynomial Chaos Expansion (PCE) is one of themost prominent ones using multidimensional polynomials. In this section themathematical background of PC techniques is presented (largely relying onthe excellent book of Le Maître and Knio (2010)), the construction of the PCbasis set is detailed and numerical procedures for calculating the expansioncoefficients are introduced. The theoretical description is concluded with theexplanation of the developed FANISP algorithm.

74

4.2. Theory

4.2.1 Generalized Polynomial Chaos Expansion

The mathematical basis of PC techniques is a probability space (Θ,Σ,P), whereΘ is a sample space containing the random events θ ∈ Θ (representing theuncertain input data), while Σ and P are the usual σ-algebra and probabilitymeasure respectively. The quantities of interest are real valued stochasticresponses R (θ) : Θ → R and processes U (x, θ) : Ω × Θ → R, where theprocesses are indexed by a vector x ∈ Ω ⊂ Rd containing the independentspatial, temporal, etc. variables, and U (x, θ) is a random variable for any fixedx ∈ Ω (just like R (θ)). Defining the L2 (Θ,P) space of second order randomvariables as

L2 (Θ,P) = R (θ) : [R (θ) : Θ→ R] ∧ [〈R,R〉 <∞] ,

where the inner product is given by

〈Q,R〉 =∫

ΘQ (θ)R (θ) dP (θ) ,

one usually restricts quantities to responses R (θ) ∈ L2 (Θ,P) and processesU (x, θ) ∈ L2 (Θ,P) ∀x ∈ Ω. In this study only scalar responses are consideredsince in many practical applications they are more relevant than the entirecorresponding stochastic process. Just like above, in the rest of this chapter ∧is used as a shorthand notation for “logical and” conditions.

Each random event θ ∈ Θ can be characterized by a random vector

ξ (θ) =(ξ1 (θ) , ξ2 (θ) , ..., ξNξ (θ)

)T.

In many cases independent ξj (θ) variables are sufficient and this thesis onlyconsiders such problems, but this is not the most general case. The numberof random variables

(Nξ

)determines the dimension of the problem and as

many are chosen as needed to describe the random event. Typically this equalsthe number of stochastic input parameters, i.e. Nξ = N and this is the casethroughout this chapter. In general it should correspond to the number ofdistinct sources of uncertainties, which can be the number of terms in theKarhunen-Loéve expansion of random fields (see Ghanem and Spanos, 1991),or the number non-zero eigenvalues of the covariance matrix of the inputswhen reduced order modelling is used (see Rising et al., 2013), etc. (a similarcase is also discusseed in Chapter 5 and Chapter C). The distribution of the

75


variables, i.e. the pξ(ξ)joint probability density function (PDF) is chosen to

best approximate the stochastic nature of the input data and in the case ofindependent variables it is simply the product of the PDFs of the individualvariables, i.e. pξ

(ξ)

=∏Nj=1 pξj (ξj). In the rest of this chapter the dependence

of the random variables on the random event θ will not be explicitly shownunless it is necessary for understanding.

The spectral expansion of a stochastic quantity R (θ) belonging to L2 (Θ,P)takes the form of

R (θ) = R(ξ (θ)

)=∞∑k=0

rkΨk

(ξ),

where Ψk

(ξ)are the chosen basis vectors (depending on the value of the

random variables, therefore being random variables themselves) and rk arethe expansion coefficients. For technical reasons the infinite sum has to betruncated, hence in practice the quantity of interest is approximated by:

R (θ) = R(ξ)≈

P∑k=0

rkΨk

(ξ).

It is clearly desirable to define the basis vectors in such a way that the smallestnumber of terms is needed in the expansions. In PCE (and in gPC in thegeneral case) this is done by choosing the basis vectors as multidimensionalpolynomials constructed by the tensorization of one-dimensional polynomials(i.e. uni-variate polynomials each depending on a single random variable ξj). Asa result traditional gPC is most appropriate for responses of interest varyingrather smoothly with the uncertain input variables, sharply changing anddiscontinuous quantities require special techniques (see Wan and Karniadakis,2005a; Le Maître et al., 2004b; Ma and Zabaras, 2009). Secondly, certaintypes of polynomials are more appropriate than others to represent differentstochastic quantities. The problem is that one rarely has a priori informationabout the distribution of R

(ξ)

to choose the best polynomials, we onlyknow the (supposed) PDFs of the ξj input parameters themselves. It hasbeen shown however that for the most common distributions, polynomialsfrom the Askey family provide an optimal trial basis (Xiu and Karniadakis,2002). Probabilists’ Hermite polynomials Heo are best suited for Gaussianinput variables, Legendre polynomials Po and Laguerre polynomials Lao aremost appropriate for variables with uniform and exponential distributions

76

4.2. Theory

respectively, etc. (o denotes the order of the polynomials). Hence by choosingthe trial basis vectors according to the Wiener-Askey scheme one can expectto achieve optimal convergence with respect to the number of terms in thePCE of the stochastic quantities.

This basis construction also has the added benefit that the resulting PCbasis vectors are orthogonal with respect to the inner product of L2 (θ,P), i.e.that

〈Ψk,Ψl〉 =∫Θ

Ψk (θ) Ψl (θ) dP (θ) =∫

D(Θ)

Ψk

(ξ)

Ψl

(ξ)pξ(ξ)

dξ = h2kδk,l.

In the above δk,l denotes the usual Kronecker delta, D (Θ) stands for thedomain of the random variables describing the Θ sample space and hk is thenorm of the k-th basis vector. By enumerating the basis vectors such thatΨ0(ξ)

= 1, both the mean and the variance of stochastic quantities can easilybe calculated from the expansion as

µR =∫R(ξ)pξ(ξ)

dξ = r0

and

σ2R =

∫ (R(ξ)− µR

)2pξ(ξ)

dξ =∞∑k=1

r2kh

2k ≈

P∑k=1

r2kh

2k.

4.2.2 Polynomial Chaos Basis Sets

Full Polynomial Chaos Basis Sets

Let us now turn our attention to the details of the PC basis set construction.Once the distribution of the ξ random variables have been selected, theappropriate polynomial families are chosen following the Wiener-Askey scheme(see Xiu and Karniadakis, 2002). The final PC basis vectors are built fromthe one-dimensional polynomials by tensorization, i.e. as a product of theN uni-variate polynomials, each depending on a different random variable ξj .Introducing a multi-index γ = (γ1, ..., γN ) each basis vector can be written as

Ψ(ξ)

=N∏j=1

ψj,γj (ξj) ,

77


where ψj,γj (ξj) ∈Heγj (ξj) , Pγj (ξj) , Laγj (ξj) , ...

. The double indexing is

needed to differentiate between the different polynomials corresponding to thedifferent random variables (first index) and the different orders (second index).Introducing the multi-index set

λ (o) =

γ :N∑j=1

γj = o

,the o-th order multidimensional polynomials can be written as

Γo =

⋃

γ∈λ(o)

N∏j=1

ψj,γj (ξj)

.Traditionally the truncation of the PCE is done by retaining polynomials upto a certain mixed order. If the maximum allowed order is O, the full Oth

order PC basis set is defined by the multi-index set

L (O) =⋃

o∈[0,1,...,O]λ (o) =

γ :N∑j=1

γj ≤ O

,whereas the basis vectors themselves are given by the set

Γ (O) =⋃

o∈[0,1,...,O]Γo =

⋃

γ∈L(O)

N∏j=1

ψj,γj (ξj)

The number of basis vectors in the full PC basis is

P + 1 = (N +O)!N !O!

and the corresponding truncated PCE is

R(ξ)≈

P∑k=0

rkΨk

(ξ),

with Ψk

(ξ)∈ Γ (O) ∀k.

78

4.2. Theory

Sparse Polynomial Chaos Basis Sets

More general PC basis sets can also be defined. Choosing a set of multi-indicesγ according to some (yet not defined) criterion g

(γ)

= 0 and denoting it by

G =γ : g

(γ)

= 0, the general O-th order PC basis set is given by

ΓG (O) =

⋃

γ∈G⋂L(O)

N∏j=1

ψj,γj (ξj)

.The number of basis vectors in ΓG (O) is simply given by the

PG = card (ΓG (O))

cardinality of the set (i.e. the number of elements in the set), while thecorresponding truncated PCE is

R (θ) ≈PG∑k=0

rkΨk

(ξ)

with Ψk

(ξ)∈ ΓG (O) ∀k. The main motivation behind using such more general

sets is the expectation that by adaptively building up the PC basis the resultingexpansion will contain less terms than a full Oth order basis, at the same timewould reduce the computational cost of calculating the corresponding PCEcoefficients. Following the terminology used in the works of Blatman and Sudret(2010a; 2010b; 2011) such a representation will be called “sparse polynomialchaos expansion”, and as a measure of its “efficiency” the index of sparsity ofL (O) will be defined as

IS (L (O)) = PGP.

4.2.3 Non-Intrusive Spectral Projection

The next step of PC techniques is the calculation of the rk expansion coefficients.Since the Ψk

(ξ)polynomial basis functions are orthogonal with respect to

the pξ(ξ)joint probability density function of the input variables they can be

determined as follows:

rk =

⟨R(ξ),Ψk

(ξ)⟩

⟨Ψk

(ξ),Ψk

(ξ)⟩ =

∫ ∫...∫R(ξ)

Ψk

(ξ)pξ(ξ)

dξ1dξ2...dξN⟨Ψk

(ξ),Ψk

(ξ)⟩

79


= 1h2k

∫ ∫...

∫R(ξ) N∏j=1

ψj, γk,j (ξj) pξj (ξj) dξ1dξ2...dξN . (4.1)

In the above the multi-index γk

= (γk,1, ..., γk,N )T defining the k-th basis vectorΨk

(ξ)was introduced and the domains of the individual random variables

were not explicitly written. The only difficulty in evaluating such expressions isthe determination of the numerator, since it contains the unknown dependenceof the response on the input parameters. In the context of Non-IntrusiveSpectral Projection (NISP) these integrals are evaluated by using the model(i.e. the computer program calculating the R

(ξ)response corresponding to

the different realizations of the ξ random input variables) as a black box.Stochastic techniques use standard Monte Carlo simulation (or some enhancedversion of it), while the deterministic approach relies on numerical integrationtechniques.

The numerator is a multidimensional integral which can be approximated bya cubature formula (i.e. a finite sum), usually constructed from one-dimensionalquadratures. For a general function f (ξj) depending on a single variable ξj aquadrature formula for calculating its integral has the following form:

I(1)f =∫ b

af (ξj) pξj (ξj) dξj ≈ Q(1)

levf =nlev∑i=1

f(ξ

(i)j,lev

)w

(i)lev,

where ξ(i)j,lev ∈ [a, b] and w(i)

lev ∈ R are predefined quadrature points and weightsaccording to the weight function pξj (ξj) and the chosen quadrature rule.The level index lev is used to distinguish between quadratures of differentaccuracy, the higher it is, the more precise the approximation of the integralbecomes, however the higher the nlev number of function evaluations gets.Many different rules are available in the literature (Gauss, Clenshaw-Curtis,trapezoid, etc.), each with its own advantages and disadvantages. Traditionallythey are distinguished based upon polynomial exactness (i.e. up to whichorder of polynomials they integrate accurately) and whether they are nestedor not. In case of nested rules the quadratures of each level contain all pointsof the lower level quadratures as well, whereas this is not the case of notnested rules. In this work Gauss quadratures were used, since they offer highpolynomial exactness up to order 2nlev − 1, their drawback being only a lowlevel of nestedness (including a single point). The level index lev correspondsto a quadrature rule with nlev = 2 · lev− 1 points throughout this thesis, hence

80

4.2. Theory

is accurate for polynomials up to order 2 (2 · lev − 1)−1 = 4 · lev−3 and needsnnewlev = 2 · lev − 2 number of new function evaluations (except for the lev = 1level 1 grid needing only nnew1 = 1 evaluation).

The cubature formulas are constructed from the one-dimensional quadraturesby tensorization. For a general function f

(ξ)depending on N independent

variables a cubature formula for calculating its integral has the following form:

I(N)f =∫ b1

a1

∫ b2

a2...

∫ bN

aN

f(ξ)pξ(ξ)

dξ1dξ2...dξN ≈ Q(N)lev f

=nlev1∑i1=1

nlev2∑i2=1

...

nlevN∑iN=1

f(ξ

(i1)1,lev1

, ξ(i2)2,lev2

, ..., ξ(iN )N,levN

)w

(i1)lev1

w(i2)lev2

...w(iN )levN

. (4.2)

Equation 4.2 constitutes a full tensorization, i.e. in each direction j a quad-rature rule of level levj is used which results in a summation over all possiblecombination of quadrature points. One can immediately recognize the problemof this approach: the number of needed function evaluations

(∏Nj=1 nlevj

)grows exponentially with the dimension of the problem. This is the “curse ofdimensionality” and traditionally sparse grids first introduced by Smolyak areused to alleviate it (Le Maître and Knio, 2010).

4.2.4 Sparse Grids and Adaptive Sparse Grids

Smolyak Sparse Grids for Integration

Sparse grids are based on difference formulas of the form ∆(1)levf = Q

(1)levf −

Q(1)lev−1f , with Q

(1)0 f = 0. Since the quadrature rules can be regained as a sum

of the difference formulas, i.e. Q(1)levf =

∑levl=1 ∆(1)

l f , the cubature formula usingfull tensorization (Equation 4.2) can be written as

Q(N)lev f =

lev1∑l1=1

lev2∑l2=1

...levN∑lN=1

(∆(1)l1⊗∆(1)

l2⊗ ...⊗∆(1)

lN

)f,

where lev = (lev1, ..., levN )T is a vector containing the different quadraturelevels used along the different directions. The idea behind the sparse gridconstruction of cubatures is that not all grids contribute significantly to theintegral, therefore many of the multidimensional difference formulas can be

81


discarded from the sum. Introducing a multi-index l = (l1, l2, ..., lN )T to dis-tinguish the different grids more general cubature formulas can be constructedas

Q(N)lev f =

∑l∈I(lev)

(∆(1)l1⊗∆(1)

l2⊗ ...⊗∆(1)

lN

)f =

∑l∈I(lev)

∆(N)l f, (4.3)

where I (lev) is the set of included multi-indices depending on the general levelindex lev. With this notation the full tensorization corresponds to

IFull (lev) = l : lj ≤ lev ∀ j ∈ [1, ..., N ] ,

supposing that the quadrature levels in the different directions are all the same(levj = lev). The original Smolyak construction is given by

ISmolyak (lev) =

l :N∑j=1

lj ≤ lev +N − 1

,meaning that each direction is integrated at least with a lev = 1 quadratureand lev − 1 extra integration levels can be distributed. Hence the maximumintegration level in any direction is lev and the maximum dimension of theincluded grids is lev − 1. While the use of sparse grids can significantlyreduce computational costs, further improvements are possible via introducingadaptivity into the calculation scheme. The underlying hope of course isthat with a suitable algorithm one can discard grids (i.e. multi-indices fromISmolyak (lev)) having negligible contribution to the integral in question andfocus computational efforts where needed.

Adaptive Sparse Grids for Integration - The Gerstner Algorithm

One such adaptive sparse grid method was introduced by Gerstner and Griebel(1998). Before presenting the details of the Gerstner algorithm some usefulconcepts have to be discussed. The forward neighbourhood of a grid l is theset of multi-indices given by

Fl =l + ej : j ∈ [1, ..., N ]

.

Here ej is the j-th unit vector, therefore each grid has exactly card(Fl)

= Nforward neighbours. The backward neighbourhood of a grid l is defined as

Bl =l − ej : [j ∈ [1, ..., N ]] ∧ [lj − 1 > 0]

,

82

4.2. Theory

therefore each grid has card(Bl)∈ [1, ..., N ] backward neighbours (except for

the initial 1 grid which has no backward neighbours). We also define thedirections of multi-indices as

D (l) = j ∈ [1, ..., N ] : lj > 1 ,

which is simply the set of indices where the integration level is higher than 1.The interaction order or dimension of a multi-index is

IO (l) = card (D (l)) ,

while the degree or general order of a multi-index l is

GO (l) =N∑j=1

lj −N.

The multi-index l is the vector describing grid l in case of the integrationalgorithms, but the same notation can be used to define the directions and theorder of the polynomial basis vectors characterized by multi-indices γ, usingγ + 1 in the arguments of D and GO. Finally we define the total backwardneighbourhood of grid l as

BTl (S) =l′ ∈ S : D

(l′)⊆ D (l)

∪ 1,

which is a set of multi-indices in a general multi-index set S with any of thedirections of l, plus the 1 grid.

Let us now return to the details of Gerstner’s method for calculating multi-dimensional integrals using adaptive sparse grids. Recognizing that not eachdirection (i.e. not each independent variable) is equally important one aimsto adaptively choose (and calculate) those grids in Equation 4.3 which have asignificant contribution to the integral and discard the rest. To do so an errorindicator is defined for each grid l as

gl = max

α∣∣∣∣∣∣∆(N)l f

∆(N)1 f

∣∣∣∣∣∣ , (1− α)n1nl

,with α ∈ [0, 1] and nl being the number of cubature points corresponding tothe grid. The value of gl combines the contribution of a grid to the value ofthe multidimensional integral with its computational cost (i.e. the number

83


of function evaluations it needs), both relative to the first grid denoted by1. The algorithm is based on two sets of grids: an old (O) and an active set(A). In each step the grid with the highest error indicator in the active set(i.e. l ∈ A : gl = maxk∈A gk) is put into the old set and next those grids arecalculated from its Fl forward neighbourhood, for which all their backwardneighbours are included in the old set (in other words such that the index setO ∪A is admissible (see Le Maître et al., 2004b)). Hence the computed gridset is FAl =

l′ ∈ Fl : Bl′ ⊂ O

, and the iteration continues while the global

error indicatorη =

∑l∈A

gl

is above a predefined tolerance level ε or until the full ISmolyak (lev) Smolyaksparse grid set is calculated (i.e. when the general multi-index set I (lev)coincides with the Smolyak set and no saving in computational cost is providedby the adaptive algorithm).

Adaptive Sparse Grids for Polynomial Chaos Expansion - TheOriginal Gerstner Method

Gerstner’s adaptive sparse grid algorithm detailed in Section 4.2.4 is a generaltechnique to calculate multidimensional integrals. Since for our purposes asmany such integrals have to be computed as many polynomial chaos basisvectors there are in the expansion, the use of Gerstner’s method is not straight-forward. A relatively simple idea would be to specify a stopping condition withrespect to the convergence of all expansion coefficients. This would howeverinclude defining as many active and old sets as the number of PC basis vec-tors, moreover the contribution of the active sets to all the highest order PCcoefficients would have to fall below the specified tolerance limit. The resultwould be a significant computational burden which is neither necessary norviable from the practical point of view.

The simplest idea is to ensure the convergence of only one integral withoutany regard to the others. In this case the obvious choice is f = µR = r0, withwhich the error indicator takes the form of

gOGl = max

α∣∣∣∣∣∣∆(N)l µR

∆(N)1 µR

∣∣∣∣∣∣ , (1− α)n1nl

84

4.2. Theory

and the iteration continues till

ηOG =∑l∈A

gOGl > εµ.

This way the algorithm steers the computation towards grids which significantlycontribute to the mean value of our response or which are easy to calculate. Inthis work α = 1 was used to ensure the convergence of the mean regardless ofthe computational cost and this approach will be called the original Gerstnermethod in the rest of this thesis (with abbreviation OG). While this techniqueis certainly easy, unfortunately it provides no assurance about the accuracy ofthe higher moments or the probability density function of the response. Formany applications however the method proves to be surprisingly applicablewhen used together with basis adaptivity. Nevertheless, in the subsequentsection advanced variations of the algorithm are presented which at least alwaysensure the convergence of the variance as well.

Adaptive Sparse Grids for Polynomial Chaos Expansion -Advanced Gerstner-like Methods

In a straightforward manner one can easily construct an improved version ofthe original Gerstner method by not only taking into account the contributionsto the mean, but also to the variance (or even the higher moments). Theproposed form of the error indicator is

gIGl

=

∣∣∣∣∣∣∆(N)l µR

µR

∣∣∣∣∣∣ ,∣∣∣∣∣∣δ

(N)l σ2

R

σ2R

∣∣∣∣∣∣ ∈ R2,

where µR and σ2R are the most up to date estimates of the mean and the

variance, i.e.µR = r0 =

∑l∈O∪A

∆(N)l r0

and

σ2R =

P∑k=1

r2kh

2k =

P∑k=1

1h2k

∑l∈O∪A

∆(N)l rk

2

h2k

respectively. ∆(N)l µR and δ(N)

l σ2R stand for the contribution of grid l to the

mean and the variance. The former is easily calculated as

∆(N)l µR = ∆(N)

l r0,

85


while a heuristic estimate of the latter is

δ(N)l σ2

R ≈ δ(N),hl σ2

R =P∑k=1

r2k −

(rk −

1h2k

∆(N)l rk

)2h2

k.

This way the contribution to the variance is also taken into account, moreoverthe importance of each grid is determined based on the most up to date estim-ates of the mean and the variance. By separately registering the contributionof the grids to µR and σ2

R, several criteria can be formulated for stopping thealgorithm and adding new grids to the active set.

The improved Gerstner method (which will be abbreviated with IG) ischaracterized by putting the grids from the A active set to the O old setaccording to the gIGl = max gIG

limproved error indicator and stopping the

algorithm when both elements of the global error indicator

ηIG =∑l∈A

gIGl∈ R2

fall below the predefined tolerance levels, i.e. when ηIG1 < εµ and ηIG2 < εσ2 .As a results the method ensures proper convergence of both the mean and thevariance. It is possible however to reach a similar accuracy with a significantlydecreased computational cost.

The main idea behind the simplification is that at later stages of the cal-culation all grids in the active set have only a small contribution to boththe mean and the variance, however there are many of them, causing theglobal error indicator to be above the defined tolerance limit. Since the wholeadaptive algorithm is based on the assumption that if certain grids have smallcontribution to the integrals then their forward neighbours would be evenless significant, one can exclude several grids from the calculation. Whenadding grid l to the old set, instead of computing the grid set FAl one can onlycalculate the reduced set

FA,Rl =

l′ ∈ FAl : 1card

(Bl′) ∑m∈Bl′

max gIGm

> min εµ, εσ2

.This way only those grids are added to the active set for which their backwardneighbourhoods have a high enough average contribution. This approach willbe called the simplified Gerstner method (with abbreviation SG).

86

4.2. Theory

Following the same logic one can further reduce the computational costs byrelaxing the stopping criterion with respect to the variance toηRG1 =

∑l∈A

gIGl,1 < εµ

∧[ηRG2 = max

l∈AgIGl,2 < εσ2

].

This way only those grids are added to the old set which individually havesignificant contribution, and the reasoning behind is that for any grid havingnegligible contribution its forward neighbours will have even less effect. Obvi-ously one loses some accuracy with the approach, however saves computationaltime. This method will be called the relaxed Gerstner method (RG) and isvery similar to the one presented in Gilli’s papers (Gilli et al., 2012, 2013b),except that in his original work the grids are calculated in a structured way,i.e. increasing the interaction order of grids from 1 to a specified maximum levand in each step increasing the degree of grids till lev. Here the order in whichthe grids are calculated is less obvious and is steered by the error indicator.

4.2.5 Adaptive Basis Construction

The Advantage of Basis Adaptivity

The use of adaptive sparse grids significantly reduces the computational burdenof calculating the expansion coefficients. Further improvement can be achievedby using an adaptive set of basis vectors instead of a fixed one. The problemof using a predefined PC basis set is that even if a certain multidimensionalpolynomial is not present in the expansion of the response its (near) zeroPC coefficient still has to be calculated in order for the PCE to be correct.Since the coefficients are given by Equation 4.1, the polynomial basis vectorsthemselves introduce dependence on ξ into the integrands making it necessaryto use cubatures which are accurate for their integration. Moreover the gridswhich finally properly compute the zero coefficient of these unimportant PCbasis vectors will have a non-zero error indicator due to causing a change inthe value of the coefficients, and as a result (some) of their forward neighbourswill be calculated unnecessarily.

To highlight this issue let us consider a simple problem involving 2 Gaussianvariables and a response that only depends on one of them as R

(ξ)

= 1 + ξ1.Let us also assume that not knowing the dependence of the response on the ξinput parameters a priori we choose to use a full 2nd order (O = 2) PC basis set.

87


When calculating the coefficient of Ψ5 = He2 (ξ2) =(ξ2

2 − 1)the integral that

has to be determined is r5 = 1h2

5

∫ ∫(1 + ξ1)

(ξ2

2 − 1)

dξ1dξ2. In the first step

in any of the adaptive sparse grid algorithms the (1, 1) grid containing a singlepoint

(ξ = (0, 0)

)is calculated, which is only accurate for linear polynomials.

Hence after this initial step the value of the r5 expansion coefficient is −12 .

When the (1, 2) grid corresponding to a 3 point Gauss quadrature in ξ2 iscalculated it correctly integrates the zero expansion coefficient of Ψ5

(ξ)since

it is accurate for 5th order polynomials. The problem is that it causes a+1

2 change in the value of r5, therefore its error indicator will be significant,causing its (1, 3) and (2, 2) forward neighbours to be calculated. Since thesedo not change the value of any of the expansion coefficients they are obviouslyunnecessary to be computed. Should we know that no basis vector involvingξ2 is needed in the PCE, we could eliminate the superfluous calculations ofthese two grids. From this simple example one can immediately realize theadvantage of an adaptive PC basis set: by not having PC basis vectors whichare unnecessary to reconstruct the response of interest, the computation ofgrids purely needed for correctly integrating their zero expansion coefficientscan be spared. The final discovery to make is that the

(ξ(i), R

(ξ(i)))

pairscalculated by the adaptive sparse grid algorithm can not only serve as thecubature points for the numerical integration but also as an experimentaldesign to investigate the dependence of the response on the input parameters.Therefore the adaptive construction of the PC basis set, i.e. the selection ofthe important multidimensional polynomials for the PCE can be done withminimal extra computational effort.

The above example highlighted the essence of our approach to basis ad-aptivity. Tailoring the build-up of the PC basis set to the adaptive sparsegrid algorithm provides the advantage of not having to use high level gridsfor accurately calculating the expansion coefficients of high order polynomialswhich are not needed in the PCE. Furthermore less basis vectors have to besampled during the sensitivity analysis and uncertainty quantification stepswhen post-processing the results. In the rest of this section the details ofadaptively building up the PC basis set are presented.

88

4.2. Theory

Adaptive PC Basis Set Construction

In this work the PC basis set was built up based on interpolations. Since everysparse grid is constructed by tensorizing one-dimensional quadratures theyprecisely calculate integrals of polynomials up to certain orders. At most a gridcharacterized with multi-index l = (l1, l2, ..., lN )T is accurate for polynomialsof the form

∏Nj=1 ξ

4lj−3j . The integrals we are trying to calculate (Equation 4.1)

contain the product of the PC basis function and the unknown response. As aconsequence if a polynomial is truly present in the PCE of the response withan order higher than 2lj−2 in direction j, its expansion coefficient is surely notcalculated correctly, since the integrand contains a polynomial with an orderof 4lj − 2 at least. At best we can accurately calculate the coefficients of basisvectors up to orders 2lj − 2, assuming that R does not contain polynomialswith orders higher than 2lj − 1 and that the rest of the multidimensionalintegral is already computed correctly.

In our adaptive basis construction approach the basis vectors are chosenfrom a predefined Γ (O) set, and each vector for which the cubature is supposedto be accurate is put into the ΓGC (O) candidate basis set. Let us assume thatat a certain point our algorithm has already calculated the grids described byan I ⊂ I (lev) set of multi-indices, where I (lev) ⊂ ISmolyak (lev) stands forthe final set of multi-indices the adaptive sparse grid algorithm constructs. Atthis point the criterion for the multi-indices describing the basis vectors in thecandidate set is given by

gC(γ)

=

0 if ∃l ∈ I : γj ≤ 2lj − 2 ∀j ∈ [1, ..., N ]1 otherwise.

This criterion produces the GC =γ : gC

(γ)

= 0multi-index set and the

candidate basis set

ΓGC (O) =

⋃

γ∈GC⋂L(O)

N∏j=1

ψj,γj (ξj)

.Just like in case of the sparse grids, the chosen basis functions are groupedinto two disjoint sets. The ΓGO (O) ⊆ ΓGC (O) old basis set contains all vectorswhich are deemed as important, while the not needed basis vectors are putinto the ΓGA (O) ⊂ ΓGC (O) active basis set (ΓGO (O) ∩ ΓGA (O) = ∅ and

89


ΓGO (O) ∪ ΓGA (O) = ΓGC (O)). The old and active basis sets are defined withthe gO

(γ)and gA

(γ)criteria, respectively, however since their closed form is

difficult to formulate only their construction mechanism will be given later. Asthe algorithm progresses in theory there could be two way exchange betweenthe two sets, since basis vectors which seem to be unimportant based on theearly interpolation results can become important, and the coefficient of basisvectors in the old set can decrease to near zero values at later stages as thecubature becomes more and more accurate. As an example let us considera response which is a pure 4th order Hermite polynomial (R (ξ) = He4 (ξ)).It is easy to see how this would seem to be a simple 2nd order dependencebased on only the 3 point level 2 quadrature. In such a case once the level3 quadrature is calculated one could realize that the 4th order polynomialperfectly describes the response dependence and that He2 (ξ) is unnecessary.Situations like this are however rare in real world applications hence in thisthesis only a one way approach is investigated, where once a PC basis vectoris added to the old basis set, it is not put back into the active set. The roughidea is that by ensuring that both the integration error of the coefficients of theimportant basis vectors and the contribution of the unimportant basis vectorsare below a tolerance limit the constructed sparse PCE sufficiently reconstructsthe response of interest, while being computationally cheaper than a full PCE.

Let us now discuss the specific method of adaptively building up the PCbasis. After calculating grid l the candidate basis set is updated by settingI = I∪ l and an interpolation is done to distinguish between the important andunimportant basis vectors. For the interpolation all cubature points belongingto any of the grids in BTl (I) are used, hence the used input data consists ofthe

(ξ,R

(ξ))

pairs, where ξj = 0 ∀ j : lj = 1. Following the same logicthe interpolating basis vectors are the ones with at least order 1 in any of thedirections of l (and naturally with order 0 in all other directions). Thereforethe interpolation is done with the vector set ΓGI (O), where the criterion isgiven by

gI(γ)

=

0 if[D(γ + 1

)⊆ D (l)

]∧

[γ ∈ GC (O)

]1 otherwise.

This basically means that the response in a subdomain of the input parametersis approximated with the R

(ξ)≈∑PGIk=0 r

Ikψk

(ξ)sum, where ψk

(ξ)∈ ΓGI (O)

and rIk are the estimates of the PC coefficients from the interpolation. Due

90

4.2. Theory

to our choice of using Gauss quadratures corresponding to nlev = 2 · lev − 1points and the fact that each grid has all cubature points belonging to allits backward neighbours already calculated, any new grid l needs nnewl =N∏j=1

[δlj ,1 + (2lj − 2)

(1− δlj ,1

)]number of new function evaluations. On the

other hand the number of new basis vectors added to the candidate set when

processing a new grid l is given byN∏j=1

(δlj ,1 + 2 ·

(1− δlj ,1

)), since each new

quadrature level introduces 2 extra orders of polynomial accuracy. As a resultthe number of interpolation points is either equal to or larger than the numberof interpolating basis vectors, moreover equality only occurs for the level 2grids. This means that the interpolation can always be performed. In thiswork the rIk interpolation coefficients were determined using least square fitsand two options were investigated with regards to building up the basis set.

One-Step Approach to Basis Adaptivity

In the first approach (denoted as the one-step approach) a single fit is done usingall basis vectors in ΓGI (O), all vectors which contribute less to the variancethan a predefined limit are put to the ΓGA (O) active basis set, whereas therest are put into the ΓGO (O) old basis set. The criterion for the multi-indicesof basis vectors to be added to the important basis set ΓO (O) is given by

g+O

(γ)

=

0 if

(rI)2h2 ≥ ε+σ2

PGO∑k=0

r2kh

2k

∧[γ /∈ GO

]1 otherwise

,

where rI is the coefficient of the basis vector described by the multi-index γcoming from the interpolation, h is its norm, rk is the most up to date valueof the basis vectors in ΓGO (O) from the cubature and ε+σ2 is a fixed coefficient(given later, see Section 4.2.6). Consequently the criterion for the multi-indicesof basis vectors to be added to the unimportant basis set ΓGA (O) is given by

g+A

(γ)

=

0 if

(rI)2h2 < ε+σ2

PGO∑k=0

r2kh

2k

∧[γ /∈ GO

]1 otherwise

.

91


The second criterion is only needed so that basis vectors previously deemedas important are not put into the unimportant set based on the interpolationresults (this is the “one way” technique followed in the basis construction).

Two-Step Approach to Basis Adaptivity

The second option is a two-step approach. First the basis vectors in ΓGI (O) areseparated into two sets: the ΓGP (O) = ΓGI (O)

⋂ΓGO (O) set of basis vectors

already present in the old basis set and the ΓGN (O) = ΓGI (O) \ ΓGO (O) setof basis vectors, which either have been previously deemed as unimportantor have just been added to the candidate set ΓGC (O). In the forward stepthe contributions from the basis vectors in ΓGN (O) are checked one by one,i.e. an interpolation is done using the basis set ΓGP (O) ∪ ψ

(ξ)for each

ψ(ξ)∈ ΓGN (O). Each vector satisfying criterion gA+

(γ)is considered as

unimportant and is put into the active basis set, vectors failing the criterion(or in other words vectors satisfying the criterion g+

P

(γ)

= g+O

(γ)) are added

to ΓGP (O), i.e. ΓGP (O) := ΓGP (O) ∪ ψ(ξ). In the backward step one final

interpolation is done with the basis set ΓGP (O) and again g+O

(γ)and g+

A

(γ)

are used to put them into the old and active basis sets.

The two-step approach has one advantage over the first option: the interpol-ations have to be done with fewer basis vectors, hence the interpolation matrixis better conditioned (though in none of the examples was ill-conditioningdetected by Matlab when using the one-step method). The disadvantage isthat the number of needed interpolations is much higher than in the one-stepapproach, at each grid evaluation card (ΓGN )+1 interpolations have to be donevs. the 1 needed in the first option. However, since typically an interpolation isorders of magnitude faster than a model evaluation, especially as only low orderinterpolations are needed, this extra computational cost is not prohibiting. Aswill be seen later in most cases the resulting algorithm performs slightly worsethan the one-step approach, i.e. results in more model evaluations and basisvectors, but the differences are not significant.

As the algorithm proceeds - regardless of the used approach for buildingup the PC basis - the active basis set grows as more and more basis vectorsare added to it. To provide a sufficient accuracy of the variance the totalcontribution of the active basis set is required to remain below a predefined

92

4.2. Theory

level. Hence at the end of each step (i.e. after calculating the new grids inthe active set and performing the basis addition) a check is made on the totalcontribution of ΓA (O) and vectors are put to the old basis set ΓO (O) untilthis contribution is below the defined tolerance limit.

4.2.6 Error Estimates and Stopping Criteria

The mean and variance of our response can be readily obtained from its PCE:

µR =∫R(ξ)pξ(ξ)

dξ = r0

σ2R =

∫ ( ∞∑k=1

rkΨk

(ξ))2

pξ(ξ)

dξ =∞∑k=1

r2kh

2k.

Should we accurately know all the expansion coefficients of the infinite series,the precise estimate of both quantities would be at hand. However, weare bounded by the accuracy of the numerical integration method and thetruncation of the expansion. Controlling the error coming from both sourcesis essential for a proper representation of stochastic variables by their PCE.

Integration Errors

Let us first discuss the integration errors. All PCE coefficients contain a multi-dimensional integral which is approximated by a cubature formula. WritingEquation 4.1 in a compact form this means that

rk = 1h2k

∫ ∫...

∫R(ξ) N∏j=1

ψj,γk,j (ξj) pξjdξ1dξ2...dξN ≈1h2k

Q(N)lev fk,

where the integrand for the k-th coefficient was simply indicated by fk. Theerror estimates of most numerical integration methods are based on the as-sumption that the integrand belongs to a particular class of functions, typicallyhaving bounded derivatives up to a certain mixed order, in which case ana-lytical formulas can be derived for the errors. Sparse grids are no different,substantial literature is available about their error and efficiency (see Novakand Ritter, 1996, 1998, 1999; Petras, 2000). In practice however, one mostoften has “only a finite number of values of the integrand and possibly ofsome of its derivatives”, but has no further information that would allow thecalculation of exact error estimates (Davis and Rabinowitz, 1984). What can

93


be done is choosing a series of integration rules Q(N)l f belonging to some family

which is assumed to converge to I(N)f , the true value of the integral (i.e.liml→∞

Q(N)l f = I(N)f), and approximating the integration error as

E(N)l f =

∣∣∣I(N)f − Q(N)l f

∣∣∣ ≈ ∣∣∣Q(N)l f − Q(N)

l−1f∣∣∣ .

Strictly speaking this approximation of E(N)l f is an error estimate of the

integral value given by Q(N)l−1f , however since the work has already been put

into calculating the more accurate value of Q(N)l f , one usually accepts it as

the best estimate of I(N)f and unwillingly overestimates the error by E(N)l f

in most cases. Neither the one-dimensional quadratures of different types andlevels, nor the traditional cubatures using a full tensorization or the originalSmolyak sparse grids are exception from this practice. Their error estimatesare of the form of E(N)

lev f ≈∣∣∣(Q(N)

lev −Q(N)lev−1

)f∣∣∣ and the best estimate of the

integral is taken as I(N)f ≈ Q(N)lev f .

The case of adaptive integration techniques is more complicated and thetheoretical basis of the accuracy of such algorithms is much less developed. AsGerstner’s paper suitably expresses it one can only hope that “the error forgeneralized sparse grid quadrature formulas is at least as good as in the caseof conventional sparse grids“ (Gerstner and Griebel, 2003). Hence when usingthe adaptive sparse grids detailed in Section 4.2.4 for calculating integralsone expects that the integral value given by Q(N)

lev f =∑

l∈I(lev)∆lf is similarly

accurate as a Smolyak sparse grid would be using the full ISmolyak (lev) set.Moreover, the assumption is made that the error of the adaptive algorithm iswell approximated by the contribution of the grids in the active set, i.e.

E(N)lev f ≈

∣∣∣(Q(N)lev −Q

(N)lev−1

)f∣∣∣ ≈

∣∣∣∣∣∣∑l∈A

∆(N)l f

∣∣∣∣∣∣ ,and therefore it is controlled by the global error indicator.

Let us return now to the errors of the first two moments of our response ofinterest. The error of the mean only depends on the accuracy of the chosen

94

4.2. Theory

integration scheme and can be written as

|µR − µR| =

∣∣∣∣∣∣µR −∑

l∈I(lev)∆(N)l r0

∣∣∣∣∣∣ ≈∣∣∣∣∣∣∑l∈A

∆(N)l µR

∣∣∣∣∣∣ <∑l∈A

∣∣∣∆(N)l µR

∣∣∣ .Therefore by specifying one of the stopping criteria of our algorithm as

ηOG =∑l∈A

gOGl < εµ or ηIG1 = ηRG1 =∑l∈A

gIGl,1 < εµ, (4.4)

the convergence of the mean is ensured with a relative tolerance of εµ (withthe above mentioned assumptions):

|µR − µR| < cµεµµR.

In the above cµ is a proportionality constant, which is 1 in the ideal case.Obviously the original and the improved Gerstner methods will provide a valueclose to 1, while the simplified and the relaxed methods can end up with avalue slightly higher than that, consistent with more grids in the former twoand less in the latter two options.

Basis Addition Errors

Let us now turn our attention to the accuracy of the PC metamodel. The issueinvolves the truncation of the PCE (i.e. the number of PC basis vectors) andthe precision of the expansion coefficients. As was already mentioned beforeone could obviously employ a very high polynomial order and specify a stoppingcondition with respect to the convergence of all the values of the expansioncoefficients to ensure the most accurate representation of the model responseby its PCE. This is however neither necessary nor viable. In most practicalsituations - as the ”sparsity of effects“ principle dictates (Montgomery, 2001) -the response is mainly influenced by only a handful of input variables and highorder interactions between the different inputs are sufficiently weak or are notpresent at all. Hence the task at hand is how to choose the basis vectors toinclude and how to check the convergence of the expansion coefficients.

In this work the convergence criterion was on the variance of the response.The

∣∣σ2R − σ2

R

∣∣ error of the variance is given by∣∣∣∣∣∣P∑k=1

(r2k − r2

k

)h2k +

∞∑k=P+1

r2kh

2k

∣∣∣∣∣∣ ≤∣∣∣∣∣P∑k=1

(r2k − r2

k

)h2k

∣∣∣∣∣+∣∣∣∣∣∣∞∑

k=P+1r2kh

2k

∣∣∣∣∣∣ =

95


∣∣∣∣∣∣∣P∑k=1

r2k −

1h2k

∑l∈O∪A

∆(N)l rk

2h2

k

∣∣∣∣∣∣∣+∣∣∣∣∣∣∞∑

k=P+1r2kh

2k

∣∣∣∣∣∣ ,where the integration error (first term) and the truncation error (second term)can clearly be separated. A heuristic estimate of the integration error can begiven as ∣∣∣∣∣∣∣

P∑k=1

r2k −

1h2k

∑l∈O∪A

∆(N)l rk

2h2

k

∣∣∣∣∣∣∣ ≈∣∣∣∣∣∣∑l∈A

P∑k=1

r2k −

(rk −

1h2k

∆(N)l rk

)2h2

k

∣∣∣∣∣∣ ≤∑l∈A

∣∣∣δ(N),hl σ2

R

∣∣∣ ,where the contribution of the individual grids to the variance is calculatedaccording to

δ(N),hl =

P∑k=1

r2k −

(rk −

1h2k

∆(N)l rk

)2 =

=P∑k=1

1h2k

∑l′∈O∪A

∆(N)l′

rk

2

−

1h2k

∑l′∈O∪A\l

∆(N)l′

rk

2h2

k.

Obviously there are more rigorous estimates one could use, however since thereis little knowledge about the convergence of adaptive sparse grids in generalthe issue is not addressed further in this thesis.

Again we will assume that the integration error of the variance is boundedby the contribution of the active grids and specify the second stopping criterionof our algorithm as

ηIG2 =∑l∈A

gIGl,2 < β1εσ2 or ηRG2 = maxl∈A

gIGl,2 < β1εσ2 (4.5)

with β1 ∈ (0, 1]. When basis adaptivity is not used (i.e. all PC basis vectorsin Γ (O) are included) the error of the variance is purely supposed to be dueto the integration error, hence β1 is set to unity (β1 = 1). However when thePC basis set is adaptively built up, in order to ensure that the error of thevariance with the sparse PCE is at least as low as with the full PCE, β1 < 1 isused to allow the unimportant basis vectors to account for (1− β1) fraction

96

4.2. Theory

of the error of the variance. The tolerance level for adding basis vectors iscorrespondingly set to ε+σ2 = εσ2 (1− β1).

Therefore when basis adaptivity is used the contribution of the unimportantPC basis vectors to the variance is required to satisfy∣∣∣∣∣∣

∞∑k=P+1

r2kh

2k

∣∣∣∣∣∣ ≈∣∣∣∣∣∣

PGC∑k=PGO +1

(rIk

)2h2k

∣∣∣∣∣∣ < (1− β1) εσ2 σ2R. (4.6)

Equation 4.6 means that the contribution of the unimportant basis vectorsto the variance is approximated by the vectors in the active basis set ΓA (O),since these are the only vectors in the full PC basis (Γ (O)) for which we havean estimate of their PCE coefficients from the interpolations (namely rIk) andwhich are not in the old basis set ΓO (O).

As a result the convergence of the variance is ensured with a relative toleranceof εσ2 (with the above mentioned assumptions):∣∣∣σ2

R − σ2R

∣∣∣ < cσ2εσ2 σ2R.

Again, cσ2 is a proportionality constant, close to 1 in the improved Gerstnermethod, and possibly somewhat higher when using the simplified or relaxedoptions.

4.2.7 Fully Adaptive Non-Intrusive Spectral Projection

At this point all pieces required for our grid and basis adaptive non-intrusivespectral projection - which will be called Fully Adaptive Non-Intrusive SpectralProjection (FANISP) - are at hand. What one has to define are the tolerancelevels εµ and εσ2 on the mean and the variance, a grid inclusion method, abasis addition technique, the lev maximum allowed order of the grids and theO maximum polynomial order.

The different grid inclusion methods that are investigated in this thesis arethe following:

a: The original Gerstner method (OG) with α = 1 in the error indicatorgOGl and [

ηOG < εµ]

97


being the stopping criterion according to Equation 4.4. Hence no ex-plicit check is made for the convergence of the variance (and the PCEcoefficients) in this case.

b: The improved Gerstner method (IG) with the full FAl admissible forwardgrid set calculated and the stopping criterion being[

ηIG1 < εµ]∧

[ηIG2 < β1εσ2

]in accordance with in Equations 4.4-4.5.

c: The simplified Gerstner method (SG) with only the FA,Rl reduced ad-missible forward grid set calculated, and the stopping criterion being thesame as for the improved Gerstner method.

d: The relaxed Gerstner method (RG) being the same as the simplifiedmethod, but with the stopping relaxed to[

ηRG1 < εµ]∧

[ηRG2 < β1εσ2

].

This method basically assumes that if all grids in the active grid set havecontributions to the variance below the tolerance level, the combined con-tribution of their forward neighbours would also fall below the tolerancelevel.

With respect to the basis addition three methods are discussed:

• No basis adaptivity, using a predefined PC basis set Γ (O) and settingβ1 = 1.

• Basis adaptivity with one-step approach and setting β1 = 0.95.

• Basis adaptivity with two-step approach and setting β1 = 0.95.

The value of β1 was chosen on a purely empirical basis as a result of initial testruns and the applications detailed in Section 4.3, β1 = 0.95 provided the leastnumber of model evaluations in most cases, hence was a good compromise touse a single value for all options.

The flow chart of the algorithm is shown in Figure 4.1. During initializationthe input of the algorithm is processed (grid inclusion method, maximum PC

98

4.2. Theory

order, etc.), the ξil abscissas and wil weights are generated according to theGauss quadrature rules corresponding to the distribution of the stochastic inputvariables (up to the predefined maximum integration level lev, i.e. for levelsl ≤ lev) and the difference formulas are created. If no basis adaptivity is usedthe full PC basis set is built up (i.e. the L (O) multi-index set defining the basisvectors), otherwise the old and candidate PC basis sets are initialized with onlythe constant term defined by the multi-index 0 in them: GO (O) = GC (O) = 0.

First the level 1 grid corresponding to the multi-index 1 has to be calculated,i.e. the model has to be run with the nominal value of all input parameters(ξ = 0). The old grid set is initialized to O = 1 and the first estimate of the

PC coefficients is calculated as rk = 1h2k

∆(N)1 rk (obviously in case of using

basis adaptivity only the constant term r0 has to be evaluated). The set ofprocessed grids is also initialized as I = 1.

In the second step the level 2 grids are dealt with, these are the lev = 2sparse grids in the F1 forward neighbourhood of 1, simply corresponding tointegration with a level 2 quadrature in all N directions. The active grid setis initialized to A = F1 and the model is run with all input realizations ξ(i)

corresponding to the grids in F1 to obtain the R(ξ(i))response values. Next

the grids are processed by taking the following actions for each l′ ∈ F1 grid:

• The contribution of the grid to the PC basis vectors in the old gridset is calculated, i.e. the coefficients of basis vectors characterized bymulti-indices γ

k∈ GO (O) are updated according to rk = rk + 1

h2k

∆(N)l′

rk,

for all k ≤ PGO .

• (Optional) The set of processed grids is updated as I = I ∪ l′, togetherwith the candidate basis set (gC

(γ)criterion and GC (O) multi-index

set).

• (Optional) Interpolation is used to identify the important basis vectors.The GI (O) set of multi-indices characterizing the interpolation vectors ischosen and the points belonging to the BT

l′ (I) total backward neighbour-hood of l′ are collected. In the one-step approach a single fit is done andthe vectors are separated according to the g+

O

(γ)and g+

A

(γ)criteria.

In the two-step approach first card (ΓGN (O)) interpolations are done

99


with the vector sets ΓGP (O) ∪Ψ(ξ)for each Ψ

(ξ)∈ ΓGN (O), criteria

g+P

(γ)and g+

A

(γ)are used to enrich ΓGP (O) and to put unimportant

basis vectors into ΓGA (O) respectively. In the backward step a finalinterpolation is done with basis vector set ΓGP (O) and criteria g+

O

(γ)

and g+A

(γ)are used to put vectors into the ΓGO (O) old and ΓGA (O)

active basis sets.

• (Optional) If new basis vectors are added to the old basis set, theircoefficient is calculated as rk′ = 1

h2k′

∑l′′∈I

∆(N)l′′

rk′ .

During processing the grids (i.e. updating the values of the PC coefficients)the optional steps are only necessary if basis adaptivity is used, and in thiscase the next step is checking the total contribution of the unimportant basisvectors in ΓGA (O). Until their combined contribution is too high, i.e. until

PGC∑k=PGO +1

(rIk

)2h2k > (1− β1) εσ2 σ2

R the following actions are repeated:

• The basis vector with highest contribution to the variance, i.e.Ψk

(ξ)∈

ΓGA (O) :(rIk

)2h2k = max

k′

(rIk′

)2h2k′ is chosen.

• The old and active basis sets are updated as ΓGO (O) = ΓGO (O)∪Ψk

(ξ)

and ΓGA (O) = ΓGA (O) \Ψk

(ξ).

• The coefficient of the new basis vector is calculated as rk = 1h2k

∑l∈I

∆(N)l rk.

After all grids have been processed and the basis sets have been updated anestimation of the error of the PCE is made:

• For all l ∈ I (≡ O ∪A) grids either the original Gerstner error indicator

is calculated as gOGl = max

α∣∣∣∣∣∣∆(N)l µR

∆(N)1 µR

∣∣∣∣∣∣ , (1− α)n1nl

or the improved

100

4.2. Theory

Gerstner error indicator as gIGl

=

∣∣∣∣∣∣∆(N)l µR

µR

∣∣∣∣∣∣ ,∣∣∣∣∣∣δ

(N)l σ2

R

σ2R

∣∣∣∣∣∣.

• Based on the grid inclusion method the original, the improved or therelaxed global error estimator is calculated as ηOG =

∑l∈A

gOGl , ηIG =

∑l∈A

gIGl

or ηRG =

∑l∈A

gIGl,1 ,maxl∈A

gIGl,2

respectively.

With the known error estimates a decision is made. Depending on the gridinclusion method, if ηOG > εµ or

[ηIG1 > εµ

]∨[ηIG2 > β1εσ2

]or[ηRG1 > εµ

]∨[

ηRG2 > β1εσ2

]is true the algorithm continues, otherwise convergence is detec-

ted and it stops with the most up to date value of the PCE coefficients (∨ isused as a shorthand notation for “logical or”). If the convergence check fails aloop starts involving the following steps:

• The grid with the highest error indicator in the active set is chosen, i.e.l ∈ A : gOGl = max

l′∈AgOGl′ or l ∈ A : gIGl = max

l′∈AgIGl′ . The old and active

grid sets are updated as O = O ∪ l and A = A\l and the new grids areset either as Fl = FAl or as Fl = FA,Rl

• The Fl new grids are calculated and the active set is updated as A =A ∪ Fl.

• The Fl new grids are processed, for each l′ ∈ Fl the same procedure isrepeated as explained for the level 2 grids in F1.

• (Optional) The combined contribution of the unimportant basis vectorsin ΓGA (O) is checked. Same as after processing level 2 grids.

• Error estimates are updated following the same procedure as after level2 grids.

• Convergence is checked with the new estimates for the mean and thevariance using the updated global error estimators.

The loop continues until convergence is reached or until the active set getsempty and there are no more grids to add to the old set (for technical reasons

101


there needs to be a maximum degree of grids, hence if no convergence isachieved at late stages of the algorithm no more grids are added to the activeset).

4.3 ApplicationIn this section a short overview of three applications is given. Each of themwas chosen to test and demonstrate a certain aspect of the developed adaptivePC techniques. The implementation of the problems was done using Matlab,since the polynomial chaos script themselves were also coded in Matlab, partlybased on scripts originally developed by Gilli for his new grid adaptive method(Gilli et al., 2012, 2013b).

4.3.1 Rosenbrock Function

The Rosenbrock function is a popular analytical test problem for differentoptimization algorithms (see Rosenbrock, 1960). The two-dimensional versionused in this chapter reads

R (θ) = R(ξ)

= 100(ξ2 − ξ2

1

)2+ (1− ξ1)2 ,

where both input parameters are chosen to be normally distributed (i.e. fol-lowing Gaussian distribution with zero mean and unity variance). As a resultprobabilists’ Hermite polynomials provide an optimal choice for the PC basisand the exact value of the PC coefficients is known:

R(ξ)

= 402− 2He1 (ξ1)− 200He1 (ξ2) + 601He2 (ξ1) + 100He2 (ξ2)−

−200He2 (ξ1)He1 (ξ2) + 100He4 (ξ1) .

This function is a good candidate for understanding the advantages of usingan adaptive basis set and getting insight into how it can help reducing thenumber of needed function evaluations.

4.3.2 One Effective Delayed Group Point Kinetics withReactivity Insertion

Our second example concerns the time-dependent behaviour of nuclear reactors.Since it is clearly desirable to keep the core in a controlled state at all times it

102

4.3. Application

Initialization

Adaptivebasis?

Generate fullPC basis (L (O)multi-index set)

Initialize PC basis(set GC (O) = 1)

Level 1 gridSet O = 1, calculate and process grid 1.

Level 2 gridsSet A = F1, calculate and process l ∈ A grids.

Adaptivebasis?

Check unimport-ant PC basiscontribution

Calculate error indicatorsand global error estimator

Converged?ExitFind new grids FlSet O = O ∪ l,A = A\l

Calculatenew grids Fl

Set A = A ∪ Fl

Process newgrids Fl

Adaptivebasis?

Check unimportantPC basis con-tribution

Calculate error in-dicators and globalerror estimator

NoYes

Yes

No

YesNo

Yes

No

Figure 4.1: The Fully Adaptive Non-Intrusive Spectral Projection algorithm

103


is vital to be able to predict both long and short term changes. The formerones typically include variations of the power and temperature distributionsdue to the spatially uneven depletion of fissile isotopes and the build-up ofstrongly absorbing fission products, while the latter ones are mostly associatedwith events initiated by the reactor operators and accidental situations.

To study such rapid transients the most widely used and perhaps easiestapproach is the point kinetics approximation (Duderstadt and Hamilton, 1976).Its simplest form is the one effective delayed group case, where the differentprecursors having various fission yields and decay times are combined into asingle group. The equations describing the problem read:

dP (t)dt = ρ (t)− β

Λ P (t) + λC (t) (4.7)

dC (t)dt = −λC (t) + β

ΛP (t) , (4.8)

where P (t), C (t) and ρ (t) are the power, precursor concentration and thereactivity during the transient, while β, λ and Λ are the effective delayedneutron fraction, the averaged decay constant and the neutron generationtime respectively (Duderstadt and Hamilton, 1976). The meaning of Equa-tions 4.7-4.8 is rather straightforward: changes in the reactor power (which isequivalent to the neutron flux) are driven by the neutron multiplication and thedecay of the precursors, while the precursor concentration decreases via decayand increases due to the fission of the fuel. The initial conditions correspondto a steady state at power P0 and precursor concentration C0 = β

ΛλP0, andthe transient is caused by a constant reactivity insertion over a certain timeperiod:

ρ (t) =

0 if t < 0ρ0 if 0 ≤ t ≤ T.

Such a situation could for example correspond to accidental control rod ejectionfrom the core or inadvertent fuel-assembly drop into it. The analytical solutionof the problem is well known, hence supposing that at time T the reactor isscrammed (i.e. a very large negative reactivity is inserted essentially stoppingthe chain reaction) our response of interest, the maximum power during thetransient is given by

R(ξ)

= P (T ) = P0

[ω1 + λ

λ

ω2ω2 − ω1

exp (ω1T ) + ω2 + λ

λ

ω1ω1 − ω2

exp (ω2T )],

104

4.3. Application

where the two modes are

ω1,2 =− (λΛ− ρ0 + β)±

√(λΛ− ρ0 + β)2 + 4Λρ0λ

2Λ .

The values of the input parameters and their distribution are summarized inTable 4.1. The kinetic parameters are representative of a fast reactor, both βand Λ are relatively low. The distributions of the reactivity insertion and thedelayed neutron fraction were intentionally chosen in such a way that promptcriticality (i.e. when ρ0 ≥ β) is just excluded from the possible scenarios(which would not be possible by using the slightly more realistic Gaussiandistributions). Moreover the reactor shutdown time corresponds to roughlythe end of the prompt jump, the very fast increase of power immediately afterthe reactivity insertion, hence the whole transient is very fast.

Table 4.1: The value and distribution of the input parameters of the oneeffective group point kinetics model. The half interval is given in case ofuniform, the relative standard deviation in case of Gaussian variables.

Parameter Distribution Mean Rel. st. dev./half int. [%]

Starting power (P0 [W]) Uniform 1 5Averaged decay c.

(λ[s−1]) Gaussian 0.0944 5

Neutron gen. time (Λ [µs]) Gaussian 0.478 5Effective delayed neutron Uniform 403 5

fraction (β [pcm])Reactivity insertion (ρ0 [pcm]) Uniform 0.90476β 5

Shutdown time (T [s]) Gaussian 0.006 5

Though this example may seem rudimentary it is rather instructive, as theresponse is highly nonlinear in some of the input parameters, therefore itsdistribution is non-Gaussian. As will be seen later this causes on one handtraditional low order full PCE to fail in reconstructing the response PDF, onthe other hand high order full PCE to be computationally expensive due tothe high number of basis vectors. In contrast the basis adaptive PCE is ableto tackle the problem with significantly lower computational cost than MonteCarlo sampling, and the built sparse PCE effectively reconstructs the PDF.

105


4.3.3 Coupled Criticality Problem with Two-Group Diffusionand Heat Conduction

As a last application we return to our coupled criticality problem presentedin Section 2.4.1 with only the thermal coupling being present in the one-dimensional slab. Since it has an analytical solution many responses of interesthave a closed form, moreover the advantage of PCE with respect to the abilityto reconstruct nonlinear dependence can be demonstrated in contrast to adjointmethods for example. The quantities looked at in this chapter are the following:

• steady state power P = 2TB2+ha;

• fast and thermal group fluxes φ1 =TB3

+ha

Q

[Σf

1 + Σf2

Σ1→2D2B2

+ + Σ2

] and

φ2 =TB3

+ha

Q

[Σf

2 + Σf1D2B

2+ + Σ2

Σ1→2

] ;

• and finally the resonance escape probability p = Σtr1→2

Σt1 − Σtr

1→1and fast

fission factor ε = 1 + ν1Σf1

ν2Σf2

Σt2 − Σtr

2→2 +D2B2+

Σtr1→2

.

The input parameters of the problem are the same as in Section 2.4.1 andthey are listed in Table 2.1. 15 of them are chosen to be uncertain with thedistributions shown in Table 4.2.

4.4 Results

In this section the results of applying the different adaptive PC methods to thetest problems detailed in Section 4.3 are discussed by comparing the momentsand the PDFs of chosen responses to either their known analytical value or tobrute force Monte Carlo sampling.

106

4.4. Results

Table 4.2: Uncertainties of the input parameters of the coupled slabproblem

Gaussian St. dev. Gaussian St. dev. Uniform Halfparam. [%] param. [%] param. int. [%]

Σt,11 [cm−1] 5 Σt

2 [cm−1] 5 h[Wcm−1K−1] 10Σt,2

1 [cm−1] 5 a[cm] 5Σf

1 [cm−1] 5 Σf2 [cm−1] 5

D1[cm] 5 D2[cm] 5 Tref [K] 10Σtr

1→1[cm−1] 2 ν1 5Σtr

1→2[cm−1] 2 Σtr2→2[cm−1] 2 ν2 5

4.4.1 Rosenbrock function - Why basis adaptivity helps

As the analytical form of the Rosenbrock function is known it is a goodcandidate to test the basis adaptive techniques, both in terms of convergencewith respect to the number of needed function evaluations and the constructedΓGO (O) PC basis set. Table 4.3 summarizes the results achieved by using theoriginal and improved Gerstner methods with and without basis adaptivity. Forthe test tolerances were set to εµ = εσ2 = 10−5 and the maximum polynomialorder was chosen as O = 4.

Not surprisingly the quickest convergence is achieved with the originalGerstner method, since in this case only the mean is checked, but not thevariance. As a result the algorithm provides the correct mean value, but whenno basis adaptivity is used the estimated variance is more than 8% off from theanalytical value of 1.102406 · 106. In contrast when the PC basis is adaptivelyconstructed the 7 basis vectors are correctly reproduced and the providedvariance is accurate (despite the fact that no explicit check is made on itsconvergence).

When using the improved Gerstner method the results always coincide withthe analytical values, however more function evaluations are used. One can alsonotice that using basis adaptivity decreases the computational cost roughly by33%. Having a deeper look into the progress of the non basis adaptive andthe adaptive runs it becomes apparent that the reason of this is the He4 (ξ2)basis vector present in the full PC basis set Γ (O), but not in the Rosenbrockfunction itself. Hence using the full PC basis set the zero expansion coefficient

107


of He4 (ξ2) only gets properly calculated by grid (1, 3), causing this grid to havea significant error indicator due to the high change in the variance. Thereforethe grid is added to the old set and grids (1, 4) and (2, 3) are added to theactive set causing the 14 extra calculations. In contrast when basis adaptivityis used during processing grid (1, 3) no new basis vectors are added and theerror indicator of the grid is zero, since it causes no change in the coefficientsof the basis vectors (only second order polynomials are present in ξ2, thecoefficients of which are already calculated correctly by grid (1, 2)). He4 (ξ2) isjust an explicit example of the advantage of basis adaptivity that was alreadydiscussed in Section 4.2.5, namely that not including PC basis vectors withnear zero coefficient in the expansion makes it unnecessary to calculate certaingrids. These are grids which would seem important when using the full PCbasis due to the fact that one of their backward neighbours would have a higherror indicator since it would accurately integrate the zero expansion coefficientof one or more basis vectors and cause a significant change in the variance.

Table 4.3: Results of using the FANISP algorithm for the Rosenbrockfunction. Grid inclusion methods OG and IG correspond to the originaland the improved Gerstner methods. The errors of the mean and the

variance are compared to their analytical values.Grid inclusion method OG OG IG IG IG

Basis adaptivity No One-step No One-step Two-stepError of mean [%] 0 0 0 0 0

Error of variance [%] 8.16 0 0 0 0No. of function evaluations 17 17 45 31 31

No. of basis vectors 15 7 15 7 7Index of sparsity 1 ≈ 0.467 1 0.467 0.467

4.4.2 One effective group point kinetics with reactivityinsertion

For comparison the point kinetics problem was first run with 105 realizationsof the 6 input parameters obtained by standard Monte Carlo sampling. Thereference value of the mean and the variance of the maximum power during thetransient was determined based on these MC results, as well as the histogramapproximating the probability density function of the response (generated byusing 200 bins of equal width). Next the FANISP algorithm was used with

108

4.4. Results

different grid inclusion options, basis adaptivity and maximum polynomialorder, in each case setting tolerances to εµ = εσ2 = 10−4. The obtained meanand variance values were compared to their direct MC estimates, whereas theresponse PDF was generated for each case by using 106 samples of the builtPCE and the same 200 bins as for the MC results.

Let us first take a look at the probability density function of the response inFigure 4.2, where both the MC reference and the results of full PCE of differentmaximum orders are shown. The PCE was built in each case with the improvedGerstner method, in order to ensure the proper convergence of the variance aswell. The distribution of the maximum power is highly non-Gaussian, having along tail, hence it is not surprising that a very high polynomial order is neededto accurately reconstruct it. Even an order O = 5 full PCE has slight deviationsfrom the MC reference highlighted by the arrows in Figure 4.2, which becomeeven more apparent in the difference graph where systematic deviations canbe seen spreading over several bins. In contrast the order O = 7 full PCEexhibits only smaller and noisier differences. The underlying reason why sucha high order polynomial expansion is needed is that the maximum power ishighly nonlinear in two input parameters: the effective delayed neutron fractionand the reactivity insertion. As the reactivity insertion is increased and thedelayed neutron fraction is decreased we approach prompt-criticality, causingthe power to have a steep peak (see in Figure 4.3). This dependence can onlybe properly described by high order mixed polynomials, therefore a ratherhigh order is needed despite the simplicity of the problem. As a result, whenusing non-adaptive full PCE the high order polynomials of input parametersupon which the response only depends weakly are also present, causing extra(and unnecessary) computational time to calculate their near zero expansioncoefficients. This makes it a good case for our adaptive PC techniques.

In Table 4.4 the results of using the different grid inclusion methods andpolynomial orders are summarized in case of full, non basis adaptive PCE.The reference value and the standard deviation of the mean and the varianceestimates are µR = 12.3507 ± 0.0195 and σ2

R = 38.168 ± 0.319, based onthe MC results. As expected, the full PCE reproduces both moments withacceptable accuracy once the polynomial order is high enough (at least O = 3),and the more basis vectors there are the more model evaluations are neededto reach convergence. One can also conclude that the computational costscan significantly be reduced by using the simplified and the relaxed Gerstnermethods, without sacrificing too much accuracy. Indeed 20-30% of the model

109


evaluations can be saved by only calculating the reduced admissible forward gridset and relaxing the stopping criterion (SG and RG), compared to calculatingthe full set (IG). Meanwhile the relative change of the mean estimates staysbelow the defined tolerance level (10−4 in this case), and the relative changeof the variance estimates is only slightly higher (roughly 2 · 10−4). Mostimportantly however the estimates of both quantities agree with the MCresults within their respective standard deviations.

5 10 15 20 25 30 35 40−2

−1

0

1

2x 10

−3

5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

Maximum power (P(T)[W])Probability density function [1/W]

Monte Carlo Reference

Full 2nd order PCE

Full 3rd order PCE

Full 5th order PCE

Full 7th order PCE

Error of PDF [1/W]

Figure 4.2: The probability density function of the maximum power duringthe transient obtained by direct MC techniques and using full PCE. Due tothe nonlinearity of the response in 2 input variables a high polynomial orderis needed to properly reconstruct its distribution. Even an O = 5 PCE hassystematic deviations highlighted by the red arrows.

In Table 4.5 the results of the basis adaptive PCE are detailed for the originaland the improved Gerstner methods (as in this case the simplified and therelaxed methods provide the same solution as the improved method, they areneglected from the table). Comparing the values to the ones in Table 4.4 forthe IG method it can be concluded that the sparse PCE retains a similaraccuracy as the full PCE. Just as was the case when using full PCE the meanestimates are always accurate, and the variance is also properly reproduced ifa high enough polynomial order is used. Though it may seem that the original

110

4.4. Results

10

15

20

25

30

35

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

5

10

15

20

25

30

35

40

Reacti

vity in

sert

ion

E"ective delayed neutron fraction

Ma

xim

um

po

we

r [W

]

Figure 4.3: The maximum power as a function of the effective delayed neutronfraction and the reactivity insertion. On the labels the values of the uniformrandom variables are displayed instead of β and ρ0. Black circles represent themodel evaluation points, whereas the surface plot was produced by interpolatingon these values.

Table 4.4: Results of using the FANISP algorithm without basisadaptivity for the one effective group point kinetics model. Grid

inclusion methods IG, SG and RG correspond to the improved, thesimplified and the relaxed Gerstner methods respectively. Mean and

variance values are compared to the MC reference.Grid inclusion PC Order (O) 2 3 5 7

IG

Error of mean [%] 0.0309 0.0306 0.0301 0.0301Error of variance [%] 3.326 0.137 0.410 0.422No. of model eval. 237 461 1367 3293No. of basis vectors 28 84 462 1716

SG


RG


111


Gerstner method provides more accurate variance estimates (with less than0.01% error) than the improved Gerstner method (with 0.3% error), one shouldkeep in mind that both are relative to the MC reference, which is only to beconsidered accurate up to its respective standard deviation of roughly 0.8%.The most important result is however that the number of needed polynomialbasis vectors is much smaller when basis adaptivity is used than the cardinalityof the fixed PC basis sets, roughly 50 to 95% of the basis vectors can beexcluded from the expansion without affecting the results. This also causesthe number of needed model evaluations to be dramatically, up to 95% smallerthan in case of full PCE, e.g. 189 vs. 3293 in case of 7th order polynomials andthe improved Gerstner method. It can also be concluded that the two differentbasis adaptive methods essentially build up the same sparse PCE in all cases.

Finally, Figure 4.4 provides a convincing example of the usefulness of thedeveloped grid and basis adaptive PCE algorithm compared to standard MonteCarlo methods. In Figure 4.4a the probability density functions obtained theby FANISP algorithm are compared to the PDF constructed based on the MCsamples. Building the sparse PCE needs only 121 (OG method) and 189 (IGmethod) model evaluations respectively in contrast to the 105 used for the MCmethod, the constructed PDFs however agree perfectly well. The advantagesof the FANISP algorithm become even more apparent on Figure 4.4b, wherethe relative error of the mean and the variance is depicted as a function of thenumber of model evaluations, with the reference values being the MC estimatesbased on all 105 samples. The mean converges very fast, after roughly 20-30evaluations there is no significant change in its value, in contrast to the MCestimate, which even after 2400 runs has errors of 1%. The same can be saidabout the variance, the adaptive PCE provides a good estimate with less than200 model evaluations, whereas the MC value is more than 2% off even aftermore than 37000 calculations.

Though the presented problem is simple and in no way representative forany real reactor transient, it brings two important messages. First, a handfulof input parameters (upon which the dependence of the response of interestis highly nonlinear) may make it necessary to use high polynomial orders tocorrectly reconstruct PDFs. As a result traditional full PCE will contain alarge number of basis vectors and will need a high number of model evaluations,which in case of a realistic problem (where each model run can take hours oreven days) can make PCE prohibitive. In contrast when using basis adaptivitythe nonlinear dependence on a few of the input parameters only increases the

112

4.4. Results

Table 4.5: Results of using the FANISP algorithm with basis adaptivityfor the one effective group point kinetics model. Grid inclusion methods(Grid IM) OG and IG correspond to the original and the improved

Gerstner methods.Grid Basis PC Order (O) 2 3 5 7IM adapt.

OG One-step

Error of mean [%] 0.0459 0.0459 0.0459 0.0459Error of variance [%] 3.575 0.471 0.007 0.004No. of model eval. 121 121 121 121No. of basis vectors 15 25 39 43Index of sparsity 0.536 0.298 0.084 0.026

OG Two-step


IG One-step


IG Two-step


113


number of basis vectors and model evaluations with the necessary amount.Second, the example is also useful to demonstrate that the developed basis andgrid adaptive PCE can tackle problems with higher efficiency than standardMC sampling, i.e. providing similar or higher accuracy with a significantlylower computational cost.

5 10 15 20 25 30 35 40 45 50 550

50.02

0.04

0.06

0.08

0.1

0.12

0.14

Maximum power (P(T)[W])

Pro

ba

bil

ity

de

nsi

ty f

un

ctio

n [

1/W

]

6 7 8 9 10 11 12 13 140.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

MC reference

FANISP (O=7, OG method)

FANISP (O=7, IG method)

(a) The probability density function ofthe maximum power.

10 100 1000 10000 100000

−15

−10

−5

0

5

10

15

Number of model evaluations

Error in mean/variance [%]

MC mean

MC variance

FANISP mean (O=7, OG method)

FANISP variance (O=7, OG method)

FANISP mean (O=7, IG method)

FANISP variance (O=7, IG method)

37660 model

evaluations

Convergence

at 121 model

evaluations (OG)

Convergence at 189

model evaluations (IG)

2531 model

evaluations

2400 model

evaluations

(b) The convergence of the mean and thevariance of the maximum power.

Figure 4.4: Results obtained by the FANISP algorithm for the one effectivegroup point kinetics model

4.4.3 Coupled criticality problem

Just like in the previous example, first the coupled criticality problem wasrun with 105 MC samples of the input parameters to obtain the referencevalue of the responses and their PDFs (the PDFs were generated using 100bins of equal widths). Next the FANISP algorithm was used with tolerancesεµ = εσ2 = 10−3, fifth order polynomials (O = 5) and general level indexlev = 20. Since the one- and two-step approaches gave very similar results forall studied responses, only the performance of the former is presented.

As can be seen in Table 4.6 the different grid inclusion methods providerather consistent estimates of both the mean and the variance of the responseswith low relative errors. The odd one out is naturally the original Gerstnermethod, providing the least accurate values for the variance as it only checksthe convergence of the mean. It is however computationally very cheap, itneeds more than 80% less model evaluations than the improved method (IG).

114

4.4. Results

It is also reconfirmed that using the simplified and the relaxed methods leadto significant savings as well without degrading the accuracy too much (thevariance changes by roughly 0.8% in the worst case scenario, the fast flux).Moreover all results agree well with the Monte Carlo reference, both the meanand the variance estimates of the PC methods are within the 2σ intervals oftheir MC counterpart. For certain responses it may seem that the relaxedmethod is even more accurate than the simplified and the improved methods,however one should keep in mind that all results are compared to the MCvalues having their own errors.

The very low values of the index of sparsity again provide a good example forthe use of basis adaptivity, since it turns out that even the most complicatedresponses can be sufficiently reconstructed by using a relatively low number ofbasis vectors, the highest number of needed basis vectors is 91 (for the fastflux), compared to the 15504 cardinality of the full fifth order basis set. It isalso worth mentioning that running the algorithm with 10th order maximumallowed polynomials (O = 10) for the fast flux results in the same 91 basisvectors and 891 model evaluations (using the improved Gerstner method),hence prescribing a high polynomial order does not increase the number ofmodel evaluations significantly, provided that the response does not need highorder polynomials to be reproduced.

For another compelling proof, using a full 4th order PCE with 3876 basisvectors for the fast fission factor leads to 7379 model evaluations and thesame PDF as the basis adaptive methods as can be seen Figure 4.5 (the fullfifth order expansion simply run out of memory on the desktop PC used forthe tests). Also note that though the original Gerstner method provides thequickest convergence (with only 35 model evaluations) the slight error in thevariance (and presumably higher errors in the higher moments) causes a slightshift in the resulting PDF, which can be well seen on the difference graph inFigure 4.5, there are both systematic over- and underestimations of the PDF.The other basis adaptive methods as well as the full PCE however only exhibitrandom deviations from the PDF obtained by the MC reference.

Figure 4.6 is another powerful demonstration of the effectiveness of thedeveloped grid and basis adaptive PC methods. Figure 4.6a and Figure 4.6bdepict the convergence of the mean and the variance of the fast flux respectively.The reference values of both moments are based on the 105 MC samples and the1σ errors of the MC estimates are also shown. Again, the PC estimates of the

115


Table 4.6: Results of using the FANISP algorithm with basis adaptivityfor the coupled criticality problem. Grid inclusions OG, IG, SG and RGcorrespond to the original, the improved, the simplified and the relaxedGerstner method respectively, MC to the Monte Carlo reference. All

PC results correspond to the one-step basis adaptive approach.Grid Response Power Fast Thermal Resonance FastIM (P) flux flux escape fission

(φ1) (φ2) prob. (p) factor (ε)

OG

Mean e. [%] 0.0703 0.0600 0.0519 0.0092 0.0039Var. e. [%] 0.7894 1.0823 1.0690 0.2833 1.2423No. eval. 35 91 91 55 35No. basis 16 25 27 19 12Sparsity 0.0010 0.0016 0.0017 0.0012 0.0008

IG


SG


RG


MCMean e. [%] 0.1492 0.1432 0.1503 0.0174 0.0138Var. e. [%] 0.4602 0.4640 0.4695 0.4721 0.4658No. eval. 105 105 105 105 105

116

4.5. Summary

mean converge very fast, essentially after the initial couple of calculations thereis no significant change in its value. In contrast the MC estimate oscillates andeven after 32860 model evaluations has a relative error of 0.268%, exceedingthat of all PC methods. Figure 4.6b tells a similar story, once the PC methodsconverged their estimates of the variance are much more accurate than their MCcounterpart of orders of magnitude higher number of model evaluations. Forexample after 7371 and 35440 runs the MC variance estimates are still 3.58%and 1.28% off respectively, both being even higher than the 1.08% deviationof the result of the original Gerstner method using only 91 evaluations. Notsurprisingly the convergence of the variance is slower than that of the mean andis again rather uneven in case of the PC methods, this is caused by the fact thatthe variance estimates are usually significantly changed each time importantbasis vectors are added to the old basis set. One can also see that compared tothe improved Gerstner method convergence is reached with successively lowercomputational cost when using the simplified, the relaxed and the originalmethods, whereas the results degrade only slightly in case of the last techniquedue to having no explicit check on the variance.

Finally, in Figure 4.7 an example of the advantage of PCE is given comparedto other methods, namely the ability to reconstruct the nonlinear behaviourof the response on the input parameters. This is in sharp contrast withmany deterministic and stochastic approaches, such as first order perturbationbased methods (like the forward and adjoint sensitivity analysis procedures)or linear regression. Moreover due to the non-intrusive nature of the methodno modification is necessary to the original code solving the problem, unlikethe otherwise very effective adjoint methods (see Section 2.5.1 for the resultsobtained using adjoints and compare with Figure 2.1). The price paid forthis simplicity is of course the computational overhead of having to solve theoriginal problem multiple times, instead of solving one adjoint problem once.

4.5 Summary

In this chapter novel polynomial chaos techniques were presented using adaptivesparse grids and low order interpolations for efficiently constructing sparsepolynomial representations of responses of interest. The proposed methods arebased on the concept of non-intrusive spectral projection and were implementedin Matlab as the Fully Adaptive Non-Intrusive Spectral Projection (FANISP)algorithm.

117


1.5 1.6 1.7 1.8 1.9 2 2.10

1

2

3

4

5

6

Fast !ssion factor (ε)

1. 5 1. 6 1.7 1. 8 1. 9 2 2.1−0.5

−0.25

0

0.25

0.5

MC reference

FANISP (IG)

FANISP (SG)

FANISP (RG)

FANISP (OG)

Full PCE (IG)

Pro

ba

bili

ty d

en

sity

fu

nct

ion

Err

or

of

PD

F

Figure 4.5: The PDF of the fast fission factor obtained by MC sampling andthe FANISP algorithm. 5th order polynomials were used to build the sparsePCEs (labeled with FANISP) and 4th for the full PCE. Only the originalGerstner method results in a slight systematic difference from the Monte Carloreference highlighted by the red arrows.

Relying on Gerstner’s technique for calculating multidimensional integralswith adaptive sparse grids four options were introduced for the estimationof the PCE coefficients. The original Gerstner method purely takes intoaccount the convergence of the mean, while the improved Gerstner methodchecks the convergence of both of the first two moments by using an improvederror indicator. The simplified and relaxed methods can significantly reducecomputational costs by excluding certain grids and relaxing the stoppingcriterion. The rationale behind the latter two options is the recognition thatat late stages of the integration all grids have a very small contribution tothe value of the integral, hence in most cases their forward neighbours can beexcluded from the integration without significant loss of accuracy.

Two methods for adaptively constructing the PC basis and obtaining a sparsePCE of responses of interest were also introduced. They are based on low orderinterpolations using the cubature points of the integration as the experimentaldesign and the contribution of the individual PC basis vectors to the variance as

118

4.5. Summary

10 100 1000 10000 100000

−20

−15

−10

−5

0

5

10

15

20


Error of mean [%]

MC reference

Lower 1σ error

Upper 1σ error

FANISP (OG)

FANISP (IG)

FANISP (SG)

FANISP (RG)

Convergence at 91

model evaluations (OG)

Conv. at 295

model eval. (RG)

Conv. at 531

model eval. (SG)

Conv. at 891

model eval. (IG)

32860 model

evaluations

(a) The convergence of the mean of thefast flux

10 100 1000 10000 100000

−20

−10

0

10

20

30

40

50

60


Error of variance [%]

MC reference

Lower 1σ error

Upper 1σ error

FANISP (OG)

FANISP (IG)

FANISP (SG)

FANISP (RG)

Conv. at 91

model eval. (OG)

Conv. at 295

model eval. (RG)

Conv. at 531

model eval. (SG)

Conv. at 891

model eval. (IG)

7371 model

evaluations

35440 model

evaluations

(b) The convergence of the variance ofthe fast flux

Figure 4.6: Results obtained by the FANISP algorithm for the coupled criticalityproblem. All PC methods provide accurate estimates for both the mean andthe variance with much lower computational cost than Monte Carlo sampling,moreover the simplified, the relaxed and the original Gerstner methods convergesuccessively faster than the improved method while retaining a similar accuracy.

−10 −8 −6 −4 −2 0 2 4 6 8 10−40

−30

−20

−10

0

10

20

30

40

50

Σ2

t,1 perturbation [%]

Po

we

r p

ert

urb

ati

on

[%

]

Exact

Adjoint based

Adaptive PCE

Figure 4.7: Comparison of the change of steady state power in the coupledcriticality problem calculated by direct model evaluation, adjoint methodsand adaptive polynomial chaos expansion. PC is capable of reconstructingthe nonlinear behaviour, while the adjoint approach only provides first orderaccuracy.

119


a criteria to distinguish between important and unimportant basis vectors. Bynot using a full PCE, grids that would seem to have a significant contributionto the variance due to accurately integrating the near zero coefficients ofunimportant basis vectors will be correctly deemed as unimportant, hencetheir forward neighbours can be excluded from the calculation. Therefore byusing basis adaptivity both the number of model evaluations and the numberof basis vectors to be sampled during post-processing can be decreased.

The developed methods were tested on three simple yet instructive problems.The results for the two-dimensional Rosenbrock function showed that theFANISP algorithm can correctly reconstruct the analytical form of the functionand gave insight why using basis adaptivity reduces the number of neededmodel evaluations. The point-kinetics problem highlighted how even elementaryproblems may need very high polynomial orders in order to correctly reproduceresponse PDFs, making traditional full PCE impractical or even impossibleto apply. Moreover it also served as a good example of the effectiveness ofusing basis adaptivity, both compared to full PCE and standard Monte Carlosampling, in terms of both the number of needed model evaluations and thenumber of basis vectors in the sparse PCE of the response.

Finally, the larger scale applicability of the FANISP algorithm was demon-strated using the same example as in Section 2.4. This 15 dimensional coupledcriticality problem indicated that in line with the sparsity of effects principlein realistic computational models responses usually only depend strongly on ahandful of the many input parameters, which makes the basis adaptive PCEapplicable.

Due to the non-intrusive nature of the developed algorithm it is ordersof magnitude easier to apply it for the sensitivity and uncertainty analysisof large scale problems than other deterministic approaches (such as adjointtechniques). In the next chapter such an application is presented where theS&U analysis of a transient is carried out with a fully detailed system model ofa Gas Cooled Fast Reactor. Furthermore dimensionality reduction techniquesare presented to further decrease the computational costs and the issue ofparallelization is examined.

120

Chapter 5

S&U Analysis of a GasCooled Fast Reactor

Transient Using AdaptivePC Techniques

5.1 Introduction

Applying traditional PCE for the S&U analysis of large scale problems needinga significant computer time to be solved and containing a lot of variables iscostly. As was shown in Chapter 4 the required computational effort can begreatly decreased by using adaptive sparse grids and basis adaptivity, but itis desirable to find further techniques to reduce costs. Two such methods areproposed and discussed in this chapter: a reduction of the dimensionality ofproblems based on the level 2 grids and an adaptive increase of the polynomialorder O, which can eliminate spurious high order basis vectors if a lower orderrepresentation of the response is already sufficient. The issue of parallelizationis addressed, as well as post-processing the PC expansion. The chapter isconcluded with a demonstration of the large scale applicability of the developedadaptive PC methods by performing the S&U analysis of a Gas Cooled FastReactor (GFR) Unprotected Loss of Flow (ULOF) transient with a detailed

121

5. Large Scale Application of Adaptive PCE

thermal-hydraulic system model of the GFR2400 reactor concept, containing42 uncertain input variables.

5.2 Computational Cost Reduction Techniques

5.2.1 Dimensionality Reduction

Adaptive sparse grids significantly reduce the computational cost associated tobuilding the PCE of responses, but they are based on cubatures, hence theystill suffer from “the curse of dimensionality”. The lower bound on the numberof model evaluations is given by the 2N calculations needed for all level 2

grids, and the 4(N

2

)+ 4N = 2N (N + 1) number needed for all level 3 grids.

For problems with a high number of input parameters (in the thousands) thelevel 2 grids might already be prohibitive, whereas the level 3 grids present acomputational challenge even for smaller problems (for N = 100 variables thereare more than 20000 points belonging to the level 3 grids). Clearly a reductionof the dimensionality of problems can significantly increase the applicabilityand decrease the cost of the developed adaptive PC methods.

The approach discussed here is very similar to the one presented by Alekseevet al. (2011), where the input variables are separated based on their contributionto the variance. These contributions are calculated by first determining thegradient of the response (i.e. the ∇jR = ∂R

(ξ)/∂ξj

∣∣∣0derivatives with

respect to all input variables) then combining it with the individual inputuncertainties (the σαj standard deviations). The gradient is obtained withan adjoint calculation and variables with small contribution are taken intoaccount by the linear approximation, while for the large contributors properPolynomial Chaos Expansion is used.

For codes which can be run in adjoint mode the above technique can providea quasi a-priori dimensionality reduction. For codes which are not adjointcapable one-at-a-time (OAT, see Saltelli et al. (2000)) perturbation of theindividual parameters is an approximate solution for calculating the neededgradients. The level 2 grids calculated by the FANISP algorithm (and anyother spectral projection technique) after the initial computation of the level 1grid correspond exactly to such individual perturbations, hence this approachcan easily be incorporated into PC schemes. The method implemented in

122

5.2. Computational Cost Reduction Techniques

the FANISP algorithm is basically an equivalent alternative, where afterthe calculation of the level 2 grids the contribution of each direction to thevariance is checked without calculating the actual gradient. The less importantparameters are identified according to∑

k:D(γk+1)≡j

r2kh

2k

σ2R

< εDR,

where εDR is a predefined tolerance for the dimensionality reduction, σ2R is the

most up to date value of the variance and the numerator is the contributionof direction j, with D (l) = j ∈ [1, ..., N ] : lj > 1 defining the directions ofmulti-indices. The sum always contains one or two terms only, depending onthe linearity of the response with respect to the parameter, whereas rk are thevalues of the PC coefficients of the basis vectors depending purely on parameterj. For the less important parameters their corresponding [1, 1, ..., 1, 2, 1, ..., 1]multi-index is deleted from the A active set and therefore this direction is alsodeleted from the numerical integration scheme, furthermore only the lineardependence of response is kept.

5.2.2 Incremental Polynomial Order

The use of basis adaptivity avoids adding basis functions to the PCE whichare not important to describe the dependence of the response on the inputparameters. Some further improvement can be achieved by incrementallyincreasing the maximum allowed order of the expansion, thus first trying tobuild a low order expansion then subsequently increasing the global polynomialorder until the change between orders o and o + 1 is below the predefinedtolerance level of the variance (εσ2). Though the savings in computationalcost are smaller than with dimensionality reduction (mainly due to the use ofbasis adaptivity which already discards the vast majority of not needed basisvectors) the method can provide some smaller enhancements.

5.2.3 Parallelization

For any computationally expensive calculation parallelization is a key issueand S&U studies are no different. For stochastic methods the parallelization isstraightforward, each sample can be run in parallel since they are independentof each other and one does not need any information about the others. Adaptive

123


integration techniques are however different since the grids to be calculated ineach step are chosen based on the values of the error indicator, which have tobe recalculated after the addition of each new sparse grid and hence clearlydepend on the previous calculations. Naturally the cubature points belongingto a grid can always be calculated in parallel, however there are further options.Instead of adding only the grid with the highest error indicator in the activeset to the old set, i.e. only adding one grid in each integration step, we canadd multiple grids, thereby decreasing the number of steps the algorithmneeds. A logical criterion for grids to be added is gl > min εµ, εσ2, since iftheir individual contribution to the mean and the variance is higher than thetolerance their combined contribution is bound to be higher as well. If theerror indicator of all grids in the active set is individually smaller than thepredefined tolerance but the global error indicator is still above it, we canswitch to adding the first

Nparallel =

η1 − εµmaxlgl,1

or Nparallel = max

η1 − εµmaxlgl,1

,η2 − εσ2

maxlgl,2

number of grids in each step in decreasing order of their error indicators,depending on whether the original or the relaxed, or the improved or simplifiedGerstner methods are used. dxe stand for the ceiling function, i.e. the firstnatural number bigger or equal to x and the reasoning behind is that withthe current estimates of the error indicators at least Nparallel number of gridshave to be added to the old set for the global error estimator to fall belowthe tolerance. As will be seen the parallelized FANISP algorithm is usuallyless efficient than the serial version in terms of the total number of functionevaluations, however it reaches convergence in a significantly smaller number ofsteps. Hence for expensive calculations where at each step long time is neededfor running the computer code with the realizations of the inputs correspondingto the cubature points of the newly added grids parallelization can offer savingsin computational time at the expense of more actual model runs.

5.3 Post-Processing PCE - Sensitivity Analysisand Variance Decomposition

Once the PCE of a response has been obtained several important quantities canbe readily calculated from it in a fast and easy way. The estimates of the first

124

5.3. Post-Processing PCE

two moments (mean and variance) are given by µR = r0 and σ2R =

∑PGk=1 r

2kh

2k.

It is possible to derive analytical expressions for the higher moments as well,however they are significantly more complicated. Local sensitivity analysis canbe performed in a straightforward way, for standard variables (for which theirαj(ξ)

= [a0]j +[aj

]jξj PCE contains only two terms ) the

SPCER,αj =PG∑k=0

rk∂Ψk

(ξ)

∂ξj

∣∣∣∣∣∣0

1[aj

]j

absolute and the

SPCER,αj =

α0j

R0

PG∑k=0

rk∂Ψk

(ξ)

∂ξj

∣∣∣∣∣∣0

1[aj

]j

relative sensitivity coefficients are easy to calculate and their meaning is clear.For constrained variables the needed sensitivities are given by

SPCER,αj =∑i

∂R(ξ)

∂ξi

∣∣∣∣∣∣0

∂ξi∂αj

=PG∑k=0

rk∑i

∂Ψk

(ξ)

∂ξi

∣∣∣∣∣∣0

∂ξi∂αj

∣∣∣∣∣0,

where ∂ξi/∂αj |0 is the Jacobian of the inverse of the PCE of the inputs. In thecase of linearly constrained Gaussian variables these can be represented as α =α0 + UT ξ (where C

α= U

TU is a decomposition of their singular C

αcovariance

matrix), hence the derivatives are given by ∂ξi/∂αj |0 =[(UT)−1

left

]i,j. To

convert the derivatives into appropriate constrained sensitivities (similar to thecase of the fission spectrum) the SC

αf

α,αinput sensitivity matrix (shown later in

Section 5.5.1) is needed to calculate ∂ξi/∂αj |Cαf0 =

∑l

[(UT)−1

left

]i,l

[SC

αf

α,α

]l,j

(for more details see Chapter C and Perkó et al. (2014a)). The final constrainedsensitivities are obtained as

SPCE,CαfR,αj

=α0j

R0

∑i

∂R(ξ)

∂ξi

∑l

[(UT)−1

left

]i,l

[(SC

αf

α,α

)T ]l,j

=

α0j

R0

PG∑k=0

rk∑i

∂Ψk

(ξ)

∂ξi

∑l

[(UT)−1

left

]i,l

[(SC

αf

α,α

)T ]l,j.

125


Variance decomposition can also be readily performed using the PCE ofresponses. The traditional total Sobol sensitivity indices (Saltelli et al., 2000;Dimov and Georgieva, 2010) can be calculated as

STj = 1σ2R

∑γ∈GO∧j∈D(γ+1)

r2γh

2γ ,

and similar expressions can be derived for all other indices. The underlyingreason is that the PCE provides exactly the dependence of the response on theindividual parameters and the interactions of multiple parameters (via basisvectors depending on two, three, etc. parameters).

Since the PCE of a response is basically a meta-model of the problem it canalso be used to determine the full PDF of responses or failure probabilities.For this one can employ a traditional brute force Monte-Carlo approach, sincethe sampling of polynomials is orders of magnitude faster than that of theoriginal problem for any realistic situation, hence even millions of input-outputpairs can be generated within a matter of seconds on a desktop PC.

5.4 Application to a Gas Cooled Fast ReactorTransient

As a demonstration of the large scale applicability of the presented adaptivePolynomial Chaos techniques the sensitivity and uncertainty analysis of anunprotected loss of flow (ULOF) transient was performed in a Gas CooledFast Reactor. The investigated design is the so-called GFR2400 concept thatwas developed within the European FP6 GoFastR project (Stainsby et al.,2014; Zabiego et al., 2013; Perkó et al., 2014b). This section provides a briefoverview of the most relevant details of the reactor, the Cathare model thatwas used for calculating the transient (Geffraye et al., 2011; Darona, 2011) aswell as the input uncertainties that were taken into account.

5.4.1 The European GFR2400 Gas Cooled Fast Reactor

The reference Gas Cooled Fast Reactor (GFR) design that was studied inthe GoFastR project is the GFR2400 seen in Figure 5.1a, featuring 2400 MWthermal power, He primary coolant and ceramic UPuC fuel and SiC structuralmaterials withstanding the high temperatures needed for high thermal efficiency.

126

5.4. Application to a Gas Cooled Fast Reactor Transient

The core has a 60 rotational symmetry and consists of 516 hexagonal fuelassemblies arranged in an inner region with lower, and an outer region withhigher Pu enrichment (see Figure 5.1b). Each assembly contains 217 fuel pinsin a hexagonal lattice and a surrounding hexagonal SiC tube. Thin metallicliners made from rhenium and a tungsten-rhenium alloy are envisioned on theinner side of the cladding tubes to ensure fission product confinement, sincethe SiC cladding itself is not expected to be entirely leak-proof.

Core

Lower plenum

Downcomer

Upper plenum

PCSDHR loop

DHR loop

(a) Conceptual design of the GFR2400primary circuit. Three power conversionsystems (PCS) and three decay heat re-moval loops (DHR loop) are connectedto the vessel.

2 2

2

22

2

22

2

33

33

3

33

33

3

3

3

3

33

3

44

44

44

44

44

44

44

44

44

4

1

66

66

6

66

66

66

66

66

66

666

55

55

55

55

55

55

55

5

55

555

5

Inner region FA

Outer region FA

Control rod

Safety rod

Re!ector

(b) Core layout of GFR2400. Only halfof the core is depicted and the numberssignal which thermal-hydraulic channelthe fuel assemblies belong to in theCathare model.

Figure 5.1: Primary circuit and core layout of the GFR2400 Gas Cooled FastReactor Design. Pictures adopted from (Mikityuk, 2012).

The most relevant thermal-hydraulics parameters characterizing the reactor

127


are summarized in Table 5.1. The primary loop mass flow of m0 = 1216 kg/sis provided by three primary blowers, each with an inertia of I0

PB = 130 kg m2,while the bypass flow is expected to be 5% of the nominal flow, i.e. m0

Bypass =0.05m0. Superscript 0 denotes the nominal, “unperturbed” value of thestochastic inputs throughout this chapter. The coolant inlet temperatureis Tin = 400 C, the outlet temperature after mixing with the bypass flow is780 C and the nominal pressure of the primary circuit is P 0

primary = 7 MPa.The secondary coolant is a helium nitrogen mixture and the plant efficiency isestimated to be around 45%. The relatively small pressure drop in the coremakes it possible to remove the decay heat by natural circulation under pres-surized conditions, while for depressurized accidents three dedicated decay heatremoval loops and six gas reservoirs are available (offering triple redundancy).For a more detailed description of the reactor design see (Stainsby et al., 2014;Perkó et al., 2014b).

As in this work an unprotected 50% loss of flow transient was simulated (byreducing the mass flow provided by the primary blowers to 50% of the nominalvalue in 4 seconds) the specifications of the elaborate safety systems (controlrods, decay heat removal loops, accumulators, etc.) are not included here.

Table 5.1: Thermal-Hydraulic Parameters of GFR2400Parameter Value Parameter Value

Thermal power [MW] 2400 Primary coolant HeInlet pressure [MPa] 7 Pressure drop in core [MPa] 0.143Mass flow rate [kg/s] 1216 Bypass flow rate [kg/s] 60.8Inlet temperature [K] 673 Coolant specific heat [J/kg/K] 5195

5.4.2 The Cathare Model of GFR2400

For the simulation of the chosen transient the Cathare 2 code system was used(version V25_2 mod7.1 (Geffraye et al., 2011)). Within the GoFastR projectan appropriately detailed Cathare input deck was developed (consisting ofmore than 6000 lines of Cathare code language) encompassing all relevantparts of the reactor including the primary, secondary and tertiary circuits, heatexchangers, decay heat removal loops, gas reservoirs, compressors, turbines,etc. This model was made available for the current research and here only themost important data are given.

128


Cathare (Lavialle, 2006) uses the point-kinetics equations to describe thetime evolution of the neutron flux and the power of the reactor as:

dPfis (t)dt = βeff

Λ (ρ(t)− 1)Pfis (t) +ND∑i=1

λiCi (t) + Sext (t)

dCi (t)dt = βi

Λ Pfis (t) − λiCi (t) i = 1, ..., ND.

Pfis is the instantaneous fission power (directly proportional to the neutronflux), Sext is an arbitrary external power source, while ρ(t) is the reactivity inunits of $. The rest of the notation is as usual. 8 groups of delayed neutronsare included (ND = 8) with delayed neutron fractions and decay constantsshown later in Table 5.4. The prompt neutron generation time is Λ0 = 0.63 µsand the effective delayed neutron fraction is βeff,0 = 4.03 · 10−3 = 403 pcm.

Since in this work an unprotected transient was simulated only two reactivityeffects were taken into account: feedback due to fuel temperature (Doppler-effect) and due to coolant density variation (void-effect). The Doppler-effect isgiven by

δρD(t) = αD · (ln (Tfuel (t) [K])− ln (Tfuel (0) [K])) ,

where αD is the Doppler constant defined as

αD = (ρper − ρref ) /βeff

ln (Tper[K]/Tref [K]) = ∆ρDβeff

,

with ρref and ρper being the reactivities of reactor cores having reference andperturbed fuel temperatures. T [K] denotes that the temperature is expressedusing units of Kelvin, similarly later T [C] is used to signal using units ofCelsius. The void-effect is given by

δρV (t) = αV ·(ρdensitycoolant (t)− ρdensitycoolant (0)

),

where αV is the void constant defined as

αV = (ρper − ρref ) /βeff

ρdensityper − ρdensityref

= ∆ρV∆ρdensityβeff ,

for some reference and perturbed coolant densities ρdensityref and ρdensityper . Sinceno external reactivity insertion was supposed the total reactivity in the point-kinetics is given by ρ(t) = δρD (t) + δρV (t). For the current design the values

129


of ∆ρ0D = 1060 pcm and ∆ρ0

V = 330 pcm were taken, the latter having beencalculated for complete depressurization of the core (Perkó et al., 2014b).

Most of the Pfis instantaneous fission power represents an immediate energyrelease, this power is the nominal power given by

Pnom (t) =

1−NH∑j=1

Ej

Pfis (t) .

The remaining Etot =∑NHj=1Ej fraction of the power represents a delayed heat

source through the decay of actinides and fission-products, this power is thePres (t) residual power and at any time the effective power heating the coreis the sum of the nominal and residual powers: Peff (t) = Pnom (t) + Pres (t).The relation between the fission and residual powers is given by the decay-heatequations:

dHj (t)dt = −λHj Hj (t) + EjPfis (t) j = 1, ..., NH ,

where Hj (t) is the energy associated to decay-heat group j, while λHj and Ejare the decay constants and effective energy fractions of the groups. Thusthe residual power is given by Pres (t) =

∑NHj=1 λ

Hj Hj (t). Cathare uses 11

decay-heat groups with decay constants and the effective energy fractionspresented in Table 5.2. In total E0

tot = 6.636% of the heat is emitted as decay-heat, while the effective power of the core at the beginning of the transient isP 0eff (0) = 2400 MW.

The active zone of the reactor was modelled with seven parallel channels(axial elements): six channels account for 6 groups of fuel assemblies withsimilar power densities (Pelloni, 2011) and one for the bypass flow. The numberof fuel assemblies in the different groups, their total power and the derivedradial power peaking factors are listed Table 5.3. The peak power productionoccurs in group 1 (at the border of the inner and outer regions, see Figure 5.1b),while the lowest power density is experienced in group 6 (at the edge of the corenear the reflector ring). In the reference case the mass flows in channels 2-6were set by adjusting the singular pressure drops simulating the flow resistanceat the entrance of the fuel assemblies in a way that the exit temperatures atthe top are equal to 800 C, while for the hottest channel it was set to thesame value as the surrounding assemblies in the higher enriched outer zone(channel 5).

130


Table 5.2: Effective energy fractions and decay constants of the 11decay-heat groups

Group Effective energy fraction(E0j

)Decay constant

(λHj [s−1]

)1 5.620 · 10−3 6.855 · 10−1

2 1.103 · 10−2 1.571 · 10−1

3 1.061 · 10−2 2.793 · 10−2

4 7.860 · 10−3 6.671 · 10−3

5 9.620 · 10−3 9.642 · 10−4

6 8.600 · 10−3 2.454 · 10−4

7 3.790 · 10−3 3.083 · 10−5

8 4.060 · 10−3 4.049 · 10−6

9 1.870 · 10−3 6.834 · 10−7

10 2.370 · 10−3 6.619 · 10−8

11 9.300 · 10−4 8.403 · 10−9

Total 6.636 · 10−2 -

Table 5.3: The power peaking factor and the number of fuel assembliesin the six channels of the core model

Channel Number Total power Power per Radialof FAs (Wi) (P 0

eff,i[MW]) FA (P 0i [MW]) PPF (P 0

i /P0)

1 6 34.25 5.708 5.708/4.6512 54 233.78 4.329 4.329/4.6513 96 428.40 4.462 4.462/4.6514 114 531.82 4.665 4.665/4.6515 126 663.09 5.263 5.263/4.6516 120 508.66 4.239 4.239/4.651

Total 516 2400 4.651 -

131


The calculation of the temperature distribution along the pins (i.e. insidethe pellets and in the cladding) requires further data, namely the axial powerdistribution, the thermal properties of the UPuC fuel and the SiC cladding,plus the heat transfer in the gap (the thin metallic liners covering the innerside of the cladding were neglected in the Cathare model, since they representnegligible heat resistance). All of these were taken from proprietary GoFastRdocuments (Mikityuk, 2012; Zabiego et al., 2011; Somers et al., 2006). Thethermal conductivity and specific heat of the fuel are

k0UPuC [W/mK] = 11.044− 0.00107 ·T [K] + 1.32 · 10−6 ·T [K]2

C0UPuC [J/kgK] = 200.8 + 3.84 · 10−2 ·T [K],

where T [K] is the temperature of the UPuC in units of Kelvin. The temperaturedrop in the gap between fuel and cladding is modelled with a heat resistancegiven by

h0Gap

[W/m2K

]= 1918.9 + 2.0691 ·T [C] + 2 · 10−4 ·T [C]2,

where T [C] is the cladding inner wall temperature in units of C. Finally, thethermal conductivity and specific heat of the SiC cladding are:

k0SiC [W/mK] = 9.95−4.51 · 10−3 ·T [K]+7.23 · 10−7 ·T [K]2−3.60 · 103 ·T [K]−2

C0SiC [J/kgK] = 608.2 + 0.5918 ·T [C].

The axial power profile along the pins was based on calculation done by theERANOS code (Rimpault et al., 2002) and is shown later in Figure 5.2. Theprofile was supposed to be identical in all 6 channels and no radial dependencewas assumed inside the fuel pins.

All other parts of the primary loop, as well as the secondary, tertiary and thedecay heat removal loops were modelled as usual in system codes (upper andlower plenums with 0D volumes, hot and cold ducts, the downcomer, all pipeswith appropriate axial elements, heat exchangers with specific heat exchangermodules, etc.).

5.5 Sources of UncertaintiesAll the values presented in Section 5.4.2 correspond to the means of the inputparameters. This section lists the associated uncertainties coming from several

132

5.5. Sources of Uncertainties

different sources. The neutronic uncertainties are mostly based on ERANOScalculations done with the JEFF 3.1 cross section library and the 15 groupBOLNA covariance library. They include the kinetic parameters (effectivedelayed neutron fraction, prompt neutron generation time), power distribution(radial and axial power peaking factors), and reactivity effects (Perkó et al.,2014b). For certain thermal-hydraulic parameters uncertainties where based on(Marqués et al., 2011), such as initial effective power, residual power fraction,primary circuit pressure, primary blower inertia, etc. For parameters where nouncertainty data was available an estimate of 5% or 20% was made.

5.5.1 Neutronic Uncertainties

Table 5.4 gives an overview of the uncertainties of the neutronics parameters.The 5.4% overall uncertainty of the effective delayed neutron fraction wasdistributed among the 8 delayed neutron groups supposing that all follow aGaussian distribution with the same relative standard deviation, and these 8effective delayed neutron fractions were considered as the independent inputparameters instead of βeff . No uncertainty was available for the neutrongeneration time and the decay constants of the delayed neutron groups, hencea Gaussian distribution was supposed for all 9 parameters with a relativestandard deviation of 5%. For the reactivity effects and the global powerpeaking factor (as well as the effective delayed neutron fraction) the uncertaintywas derived with generalized perturbation theory and they were also assumedto be normally distributed.

The initial effective power of the reactor was chosen to be uniformly dis-tributed with a half interval of 2% (Marqués et al., 2011), i.e. Peff ∼U(P 0eff − σP , P 0

eff + σP)with σP = 0.02 ·P 0

eff . According to two-dimensionaltransport calculations (Perkó et al., 2014b) the uncertainties in the powerpeaking factor and the power lower depression factor are 2.13% and 2.11%respectively. Hence the ˜ppf i = Pi/P radial power peaking factors in thethermal-hydraulic channels were supposed to be normally distributed witha 2.12% uncertainty, i.e. σ ˜ppf i

= 0.0212 · ppf0i and ˜ppf i ∼ N

(ppf0

i , σ ˜ppf i

).

These values are constrained and cannot all be chosen independently, sincethey have to satisfy the constraint of

5∑i=0

˜ppf i ·Wi = Wtot = 516. (5.1)

133


Table 5.4: Neutronic Parameters of GFR2400 and their UncertaintiesGaussian parameters Value Uncertainty

β-effective(βeff,0 [pcm]

)403 5.4%

Effective DNF of the 5.96, 64.33, 22.64,of the 8 groups 57.82, 126.7, 54.67, 12.6%(βeff,0i [pcm]

)46.16, 24.74

Decay constant 0.0125, 0.0283, 0.0425,of the 8 groups 0.133, 0.292, 0.666, 5%(

λ0i

[s−1]) 1.63, 3.55

Neutron generation time (Λ [µs]) 0.63 5%Doppler effect (∆ρD [pcm]) -1060 4.86%Void effect (∆ρV [pcm]) 330 8.12%

Radial power 1.23, 0.93, 0.96, 2.12%peaking factors 1.00, 1.13, 0.91

The condition given by Equation 5.1 yields two difficulties: the constrainedfuel assembly powers in the different channels have to be sampled correctlyand the individual effects of the changes of the power peaking factors have tobe evaluated. This problem is very similar to the sensitivities and uncertaintiesassociated with the fission spectrum in pure neutron transport problems. Thesolution used in this thesis is detailed in Chapter C, here only the results aresummarized.

First a proper C ˜ppf covariance matrix is produced for the power peakingfactors by iteratively normalizing their covariance matrix (analogous to thefission spectrum covariance matrix normalization) as

(SC

αf

α,α

)TCαSC

αf

α,αand

resetting the variances to their calculated values until the changes between twoiterations are below a tolerance limit (see Section C.2). The covariance matrixobtained this way is shown in Equation 5.2 and it is singular since the powerpeaking factors are linearly constrained (the SC

αf

α,αinput sensitivity matrix

is displayed in Equation 5.3 and was obtained using full normalization, seeEquation C.5 in Section C.1). The C ˜ppf = UTTU eigenvalue decompositionof the gained covariance matrix reveals that one of the eigenvalues is zero.Deleting it from T and the corresponding eigenvector from U the original

134


covariance matrix can be reconstructed as C ˜ppf = UTU , where U is shown in

Equation 5.4. For the calculation of constrained sensitivities (similar to theconstrained sensitivities to the fission spectrum in neutron transport problems)the left inverse of UT is also needed, which is shown in Equation 5.5.

103 ·C ˜ppf =

0.6770 0.0964 0.0401 −0.0057 −0.1044 0.00570.0964 0.3894 −0.0001 −0.0362 −0.1151 −0.02470.0401 −0.0001 0.4137 −0.0858 −0.1733 −0.0695−0.0057 −0.0362 −0.0858 0.4521 −0.2251 −0.1080−0.1044 −0.1151 −0.1733 −0.2251 0.5754 −0.1946

0.0057 −0.0247 −0.0695 −0.1080 −0.1946 0.3733

(5.2)

SCαf

α,α=

0.9857 −0.0108 −0.0112 −0.0117 −0.0132 −0.0106−0.1284 0.9026 −0.1004 −0.1050 −0.1184 −0.0954−0.2283 −0.1732 0.8215 −0.1866 −0.2105 −0.1696−0.2711 −0.2056 −0.2120 0.7784 −0.2500 −0.2013−0.2997 −0.2273 −0.2343 −0.2449 0.7237 −0.2225−0.2854 −0.2165 −0.2231 −0.2333 −0.2631 0.7081

(5.3)

U =

0.0052 −0.0178 0.0039 0.0014 0.0003 0.00300.0029 −0.0005 −0.0166 0.0001 −0.0001 0.0133−0.0066 0.0008 0.0086 −0.0155 −0.0042 0.0122

0.0188 0.0044 0.0008 −0.0124 0.0105 −0.0028−0.0157 −0.0072 −0.0070 −0.0075 0.0211 −0.0054

(5.4)

(UT)−1

l=

13.936 −48.102 10.571 3.7043 0.8018 8.13126.3787 −1.1426 −35.883 0.2113 −0.1692 28.879−12.598 1.4840 16.365 −29.546 −8.0053 23.345

29.100 6.7613 1.2588 −19.235 16.313 −4.3596−17.809 −8.2081 −7.9862 −8.5825 24.035 −6.1107

(5.5)

With these matrices the constrained power peaking factors can be sampledand represented as ˜ppf = ppf0 + U

Tξ, where ξ ∼ N (0, 1)5. These factors will

have exactly the prescribed mean values and covariance matrix, furthermorethey are normally distributed as desired, since the linear combination ofnormally distributed variables is also normally distributed.

The second problem to tackle is the accurate computation of the effects ofthe power peaking factors. Since in Cathare the fractional power depositions

135


have to be given in the different channels, which are always normalized, tocalculate unconstrained sensitivities the total power and all the power peakingfactors have to be adjusted simultaneously. The total power is given by

Peff = Peff +5∑i=0

(˜ppf i − ppf0

i

)PWi = Peff +

5∑i=0

(˜ppf i − ppf0

i

) PeffWtot

Wi,

since the Peff perturbed power has to be adjusted according to the plus powerwe wish to deposit in the channels as a result of the changing power distribution.The final power which is the input to Cathare is therefore given as

Peff = Peff

1−5∑i=0

(˜ppf i − ppf0

i

) Wi

Wtot

.

The last step is the correct choice of the power peaking factors, such that allthe extra power when calculating the unconstrained sensitivities goes to thespecific channels, i.e. such that

PeffWtot

˜ppf iWi = ppfiPeffWtot

Wi.

The power peaking factors satisfying the above condition and given as inputsto Cathare are

ppfi = ˜ppf iPeffPeff

= ˜ppf i

(1−

5∑i=0

(˜ppf i − ppf0

i

) Wi

Wtot

).

Remembering that Peff and ˜ppf i were chosen to be normally distributed itmay seem at first glance that the effective power Peff and the power peakingfactors ppfi calculated as above do not follow a Gaussian distribution. Thisis not true however, since due to the specific sampling procedure the sum

5∑i=0

(˜ppf i − ppf0

i

)Wi/Wtot always adds up to zero. This is only a technicality

needed to be able to produce unconstrained sensitivities.

To simulate the uncertainty of the axial power distribution first it wasapproximated by a quadratic fit of the form of:(

P pini

P pin

)0

calc

≈(P pini

P pin

)0

fit

= A0i2 +B0i+ C0.

136


(P pini /P pin

)0

calcand

(P pini /P pin

)0

fitare the calculated and the fitted power

peaking factors along the 22 axial sections of the pins (i = 1, ..., 22), andthe coefficients are A0 = −0.006894, B0 = 0.1463 and C0 = 0.5063. Theuncertainty was generated by using this second order polynomial fit andassigning a uniform distribution for the quadratic and linear coefficients with 5%half intervals, i.e. A ∼ U

[A0 − σA, A0 + σA

]and B ∼ U

[B0 − σB, B0 + σB

]with σA = 0.05A0 and σB = 0.05B0, while adjusting the constant term so thatthe normalization is still valid. Hence the ∆

(P pini /P pin

)fit

deviations weregenerated as(

P pini

P pin

)fit

−(P pini

P pin

)0

fit

=(A−A0

)i2 +

(B −B0

)i+

(C − C0

),

and the constant term was adjusted such that:

0 =22∑i=1

∆(P pini

P pin

)fit

=22∑i=1

[(A−A0

)i2 +

(B −B0

)i+

(C − C0

)].

Using the coefficients generated this way the realizations of the power profileare gained as (

P pini

P pin

)calc

=(P pini

P pin

)0

calc

+ ∆(P pini

P pin

)fit

.

5.5.2 Thermal-Hydraulic Uncertainties

No uncertainty data was available for the thermal properties of the fuel pins,hence a constant (i.e. temperature independent) standard deviation wasassumed for all parameters. This constant deviation was chosen to correspondto a 5% relative standard deviation at a characteristic temperature (seen inTable 5.5). The implementation of these uncertainties was done by samplingthe constant first terms in the temperature laws of the thermal properties withthe standard deviations listed in Table 5.5 according to a Gaussian distribution.

The uncertainties on the initial effective and residual powers, as well as onthe inertia of the primary blower and the bypass flow were taken from Marquéset al. (2011). For these parameters uniform distribution was assumed with the

137


0 5 10 15 200.2

0.4

0.6

0.8

1

1.2

Axial position

Ax

ial

po

we

r p

ea

kin

g f

ac

tor

y = − 0.006894*x2+ 0.1463*x + 0.5063

Original data

Quadratic "t

Realization 1

Realization 2

Figure 5.2: The axial power peaking factor along the pin. A quadratic fitis a good approximation for the original data, hence it was used to generatedifferent realizations of the power profile.

Table 5.5: Standard deviation of the uncertainties in thethermal-hydraulic properties of the fuel pins

Property Characteristic Characteristic Standardtemperature value deviation

Fuel th. cond. (kUPuC) 1000 C 11.82 W/mK 0.591 W/mKFuel sp. heat (CUPuC) 1000 C 249.68 J/kgK 12.48 J/kgK

Gap cond. (hGap) 900 C 3943 W/m2K 197.1 W/m2KClad th. cond. (kSiC) 800 C 5.942 W/mK 0.297 W/mKClad sp. heat (CSiC) 800 C 1082 J/kgK 54.08 J/kgK

lower and upper boundaries seen in Table 5.6. For the pressure drop coefficientsmodelling flow resistance at the entrance of the thermal-hydraulic channelsGaussian distribution was chosen with 20% relative standard deviation.

All together there are N = 42 uncertain input parameters, for a summarysee Tables 5.7-5.8. This makes the investigated problem 42 dimensional,significantly higher than what is usual in Polynomial Chaos studies.

138


Table 5.6: Uncertainties on power and flow characteristicsParameter Nominal Minimum Maximum

value value valueInitial effective power (Peff [MW]) 2400 0.98 ·P 0

eff 1.02 ·P 0eff

Residual power fraction (Etot) 0.06636 0.9 ·E0tot 1.1 ·E0

tot

Bypass flow (mBypass [kg/s]) 60.8 0.025 · m0 0.075 · m0

Primary blower inertia(IPB

[kg m2]) 130 0.75 · I0

PB 1.25 · I0PB

Table 5.7: Summary of Thermal-Hydraulic Input UncertaintiesParameter Notation Reference Distribution Relative std./

value Half-intervalFuel cond. kUPuC 11.82 W/mK Gaussian 5%

Fuel sp. heat CUPuC 249.68 J/kgK Gaussian 5%Gap cond. hGap 3943 W/m2K Gaussian 5%Clad cond. kSiC 5.942 W/mK Gaussian 5%

Clad sp. heat CSiC 1082 J/kgK Gaussian 5%Init. eff. power Peff 2400 MW Uniform 2%Res. power fr. Etot 0.06636 Uniform 10%Bypass flow mBypass 60.8 kg/s Uniform 50%

Flowdistribution(singular

pressure dropcoefficients inchannels)

K1 2.392 Gaussian 20%K2 8.664 Gaussian 20%K3 7.516 Gaussian 20%K4 5.959 Gaussian 20%K5 2.393 Gaussian 20%K6 9.507 Gaussian 20%

Pr. pressure Pprimary 7 MPa Uniform 20%Pr. blower in. IPB 130 kgm2 Uniform 25%

139


Table 5.8: Summary of Neutronic Input UncertaintiesParameter Notation Reference Distribution Relative std./

value Half-inteval

Effectivedelayedneutronfractions

βeff1 5.96 pcm Gaussian 12.6%βeff2 64.33 pcm Gaussian 12.6%βeff3 22.64 pcm Gaussian 12.6%βeff4 57.82 pcm Gaussian 12.6%βeff5 126.7 pcm Gaussian 12.6%βeff6 54.67 pcm Gaussian 12.6%βeff7 46.16 pcm Gaussian 12.6%βeff8 24.74 pcm Gaussian 12.6%

Decayconstants ofdelayedneutronfamilies

λ1 0.0125 s−1 Gaussian 5%λ2 0.0283 s−1 Gaussian 5%λ3 0.0425 s−1 Gaussian 5%λ4 0.133 s−1 Gaussian 5%λ5 0.292 s−1 Gaussian 5%λ6 0.666 s−1 Gaussian 5%λ7 1.63 s−1 Gaussian 5%λ8 3.55 s−1 Gaussian 5%

Generation time Λ 0.63 µs Gaussian 5%Doppler effect ∆ρD −1060 pcm Gaussian 4.86%Void effect ∆ρV 330 pcm Gaussian 8.12%

Radial powerdistribution(peakingfactors inthermal-hydraulicchannels)

ppf1 1.23 Gaussian 2.12%ppf2 0.93 Gaussian 2.12%ppf3 0.96 Gaussian 2.12%ppf4 1.00 Gaussian 2.12%ppf5 1.13 Gaussian 2.12%ppf6 0.91 Gaussian 2.12%

Axial powerdistribution

A -0.006894 Uniform 5%B 0.1463 Uniform 5%

140

5.6. Results

5.6 ResultsAs responses several output parameters were investigated, including the max-imum fuel, cladding and outlet temperatures in the different channels duringthe transient, the maximum effective power, as well as the maximum temper-ature in the upper plenum. The Polynomial Chaos results were compared toestimates obtained by standard Monte Carlo sampling using 10000 samples.Means, variances, probability density functions and sensitivities were calcu-lated, moreover the convergence and accuracy of the different approaches wereinvestigated.

5.6.1 Response Moments and Probability Denstity Functions

First let us study the first two moments of the different responses in Table 5.9.The reference values of the mean and the variance were determined based onthe 10000 Monte Carlo samples and standard error propagation rules wereapplied to calculate the errors of these estimates. The Polynomial Chaosresults were obtained with the original Gerstner method using basis adaptivity,a polynomial order of O = 2, a maximum grid order (i.e. integration level)of GO = 10 and a tolerance of εµ = εσ2 = 1%. For almost all responses thePC estimate of the mean (µR) is within the 2σ interval around the µR MCreference (i.e. µR ∈ [µR − 2σµR , µR + 2σµR ]), and even for two exceptions itis only slightly outside it. However the computational costs are drasticallydifferent, the PC estimates need only 85 model evaluations, less than 1% of the10000 MC runs. For many of the responses the PC estimate of the varianceis also accurate, despite the fact that with the original Gerstner method noexplicit check is made on its convergence. This is a result of using basisadaptivity, with which only those PC basis vectors are added to the PCE forwhich the quadrature can be supposed to be relatively accurate.

In Figure 5.3 the probability density functions of several different responsesare depicted. The reference PDFs were constructed by using the 10000 MonteCarlo samples and 100 bins of equal width in each case, whereas the PCestimates were obtained by using the PCE given by the original Gerstnermethod (using only 85 model evaluations and O = 2 second order polynomials)and 106 samples. As can be seen in general there is an almost perfect overlapbetween the results, with the notable exception of the bypass temperature(Figure 5.3e). This is not surprising, since the variance estimate for this responsewas already off 11.5% (see Table 5.9). The cause of the discrepancy is that

141


Table 5.9: Mean and variance estimates of responses of interest withMonte Carlo sampling and the original Gerstner Method with O = 2.Although only the convergence of the mean is checked with the PCtechnique, almost all variance values are acceptable. Monte Carlo

estimates are shown with their respective standard deviations, while PCones with their deviations from the MC reference.

Resp. MC mean PC mean MC variance PC varianceTmaxFuel,1 1549.2± 0.024% 1548.9 (0.019%) 1380.1± 1.44% 1379.7 (0.03%)TmaxFuel,2 1348.8± 0.026% 1348.8 (0.001%) 1174.3± 1.43% 1171.5 (0.24%)TmaxFuel,3 1364.4± 0.024% 1364.0 (0.033%) 1097.3± 1.41% 1131.2 (3.09%)TmaxFuel,4 1386.8± 0.024% 1387.0 (0.017%) 1077.0± 1.42% 1079.4 (0.22%)TmaxFuel,5 1456.3± 0.023% 1456.1 (0.013%) 1070.1± 1.43% 1125.1 (5.19%)TmaxFuel,6 1338.2± 0.025% 1338.6 (0.026%) 1100.6± 1.42% 1152.1 (4.68%)TmaxClad,1 1292.8± 0.027% 1292.9 (0.004%) 1178.2± 1.42% 1292.0 (9.66%)TmaxClad,2 1152.5± 0.030% 1152.1 (0.034%) 1202.9± 1.41% 1226.9 (1.99%)TmaxClad,3 1161.0± 0.028% 1160.4 (0.050%) 1057.0± 1.42% 1112.3 (5.23%)TmaxClad,4 1172.7± 0.027% 1173.2 (0.047%) 989.95± 1.42% 1048.5 (5.91%)TmaxClad,5 1211.7± 0.024% 1211.6 (0.007%) 865.05± 1.44% 981.71 (13.5%)TmaxClad,6 1146.8± 0.029% 1146.4 (0.027%) 1104.5± 1.44% 1131.6 (2.45%)Pmaxeff 2430.8± 0.012% 2431.0 (0.009%) 854.18± 1.02% 876.20 (2.58%)TmaxOut,1 1093.9± 0.026% 1093.8 (0.009%) 790.76± 1.43% 795.67 (0.62%)TmaxOut,2 996.25± 0.030% 996.03 (0.022%) 883.96± 1.40% 868.94 (1.70%)TmaxOut,3 1001.2± 0.028% 1000.6 (0.060%) 765.01± 1.43% 766.06 (0.14%)TmaxOut,4 1007.5± 0.026% 1007.7 (0.021%) 695.72± 1.42% 681.14 (2.10%)TmaxOut,5 1030.1± 0.023% 1029.9 (0.017%) 574.34± 1.43% 594.77 (3.56%)TmaxOut,6 993.03± 0.029% 992.94 (0.009%) 814.24± 1.43% 814.51 (0.03%)TmaxOut,B 399.94± 0.013% 399.83 (0.027%) 27.073± 0.99% 30.185 (11.5%)TmaxUP 971.62± 0.012% 971.47 (0.016%) 131.76± 1.35% 132.37 (0.46%)

142

5.6. Results

the PDF of the bypass temperature is more complicated than the other PDFs,hence the used PCE is not accurate enough (only a low polynomial order wasused and no check was made on the convergence of the variance). Figure 5.3is nevertheless a forceful demonstration of the usefulness of the developedadaptive PC techniques: they enable an almost perfect reconstruction of theresponse PDFs and satisfactory estimates for the first two moments with only85 model runs for many responses of interest.

To properly represent more complicated responses such as the bypass tem-perature by their PCE these have to be produced with a method that alsotakes into account the convergence of the higher moments, not only the mean.The relaxed Gerstner method is the cheapest of these options, the obtainedresults are shown in Figure 5.4. As can be seen already the second orderPCE sufficiently represents the PDFs of the bypass temperature and the twocladding temperatures (the ones with a relative high error in their variancefrom the original Gerstner method), and the higher order (O = 5) repres-entation provides some further small improvement for the former. Naturallythe computational cost of building these PC expansions is higher than theoriginal Gerstner method, for the 3 responses the second order expansionsneed 689, 621 and 161, whereas the 5th order ones 845, 697 and 169 modelevaluations respectively. For all other responses building the PCE with therelaxed Gerstner method using O = 2 needs 145 to 761 model evaluations,whereas using O = 5 requires 145 to 845 runs.

Figures 5.3-5.4 already highlighted a big advantage of PC methods, namelythat they enable the construction of smoother PDFs than MC methods, sincesampling the PCE of a response is orders of magnitude faster than running theoriginal code. This becomes especially important in cases where the probabilitydensity functions have long tails which are difficult to sample or for problemswhere only a limited number of runs are affordable. To get an insight into theconvergence of the PDFs as a function of the computational costs in Figure 5.5probability density functions of two responses are depicted produced by usingdifferent numbers of MC samples as well as their PCE (produced with theoriginal Gerstner method and O = 2). For a similar computational cost thePDFs produced by PC are clearly superior to the MC ones, which hardly giveany information about the actual shape of the distributions. Moreover eventhe 10 and 100 times more expensive MC estimates are significantly noisierthan their PC counterpart.

143


1200 1300 1400 1500 1600 17000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Maximum fuel temperature [°C]

Pro

ba

bili

ty d

en

sity

fu

nct

ion

[1

/°C

]

TFuel,1

Max

TFuel,2

Max

TFuel,3

Max

TFuel,4

Max

TFuel,5

Max

TFuel,6

Max

(a) Maximum fuel temperatures in thesix thermal-hydraulic channels

1000 1100 1200 1300 1400 15000

0.005

0.01

0.015

Maximum cladding temperature [°C]

Pro

ba

bili

ty d

en

sity

fu

nct

ion

[1

/°C

]

TClad,1

Max

TClad,2

Max

TClad,3

Max

TClad,4

Max

TClad,5

Max

TClad,6

Max

(b) Maximum cladding temperaturesthe six thermal-hydraulic channels

2.35 2.4 2.45 2.50

2

4

6

8

10

12

14

Maximum power [GW]

Pro

ba

bil

ity

de

nsi

ty f

un

ctio

n [

1/G

W]

Pe!

max

(c) Maximum effective power

800 900 1000 1100 12000

0.005

0.01

0.015

0.02

Maximum outlet temperature [°C]

Pro

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[1

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]

TOutlet,1

Max

TOutlet,2

Max

TOutlet,3

Max

TOutlet,4

Max

TOutlet,5

Max

TOutlet,6

Max

(d) Maximum outlet temperatures

380 390 400 410 4200

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TOut,B

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(e) Maximum bypass temperature

940 960 980 1000 10200

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Maximum upper plenum temperature [°C]

Pro

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[1

/°C

]

TUP

max

(f) Maximum upper plenum temp.

Figure 5.3: The probability density functions of different responses obtainedby Monte Carlo sampling (continuous lines) and the FANISP algorithm usingthe original Gerstner method (dashed lines).

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5.6. Results

1200 1250 1300

0.008

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[1

/°C

]

TClad,5

Max − MC T

Clad,1

Max − MC

TClad,1

Max − OG, O=2T

Clad,5

Max − OG, O=2

TClad,1

Max − RG, O=2T

Clad,5

Max − RG, O=2

TClad,1

Max − RG, O=5T

Clad,5

Max − RG, O=5

(a) PDF of maximum cladding temper-atures in the 1st and the 5th channels ob-tained by the original and relaxed Gerstnermethods with different polynomial orders

380 390 400 410 4200

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Pro

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]

Maximum bypass temperature [°C]

Monte Carlo

Original Gerstner

Relaxed G., O=2

Relaxed G., O=5

(b) PDF of maximum bypass temperatureobtained by the original and relaxed Ger-stner methods with different polynomialorders

Figure 5.4: The probability density functions of different responses obtainedby Monte Carlo sampling and the FANISP algorithm using different options.Abbreviations OG and RG stand for the original and relaxed Gerstner methodsrespectively.

1150 1200 1250 1300 1350 1400 14500

0.005

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0.015

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0.025

0.03


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C]

MC − 100 runs

MC − 1000 runs

MC − 10000 runs

PC − 85 runs

(a) PDF of the maximum cladding temper-ature in the hottest channel (1st)

940 960 980 1000 10200

0.01

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Maximum upper plenum temperature [°C]

Pro

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1/°

C]

MC − 100 runs

MC − 1000 runs

MC − 10000 runs

PC − 85 runs

(b) PDF of the maximum upper plenumtemperature

Figure 5.5: The convergence of the probability density functions of differentresponses

145


5.6.2 Convergence and Accuracy of Monte Carlo and PCEstimates

To have a quantitative comparison between the accuracy of the developedadaptive PC methods and Monte Carlo sampling the convergence of the meanand the variance estimates were investigated. As can be seen in Figure 5.6the means always converge very fast with the PC estimates, basically afterthe calculation of the level 2 grids we have a satisfactory result. In contrastthe MC results are much less accurate even after significantly higher numberof calculations. The same can be said about the variance estimates, once thePC technique converged it provides a value that is significantly closer to thereference value (i.e. the result of the 10000 MC runs) than MC estimatesof much higher computational cost. For example the deviation of the PCestimate of the variance for the maximum fuel temperature in the hottestchannel (Figure 5.7a) after 845 calculation is 1.195% from the MC reference,whereas that of the MC estimate after 7780 calculations is 1.203%. For thecladding in the same channel (Figure 5.7b) there is 2.042% deviation in the MCvariance after 6195 model evaluations whereas the PC estimate is only 2.037%off after 689 runs. We could also look at the errors at the same computationalcost for the maximum upper plenum temperature as an example (Figure 5.7f),where the deviations of the PC estimates after 213 and 267 runs are 1.386%and 1.628%, whereas those of the MC values at the same number of runs are8.696% and 8.142% respectively.

5.6.3 Response Sensitivities

The PCE of the different responses does not only provide the probabilitydensity function and the moments but it can also be used to derive sensitivitiesby simply taking the partial derivatives with respect to the different inputparameters (as was described in Section 5.3). As a reference again the 10000Monte Carlo samples are used. For each response a multi-linear regression wasperformed and the coefficients were transformed into sensitivity coefficients,whereas for the PC results simply the derivatives were calculated. The MCreference sensitivities are listed in Table 5.10, where only the most importantparameters are displayed to which at least one of the investigated responses ofinterest is sensitive enough. As can be seen the most important parameters arethe initial effective power and the power distribution (power peaking factors).The sensitivity of the maximum power during the transient to the starting

146

5.6. Results

10 100 1000 10000

−1.5

−1

−0.5

0

0.5

1


Error of mean [%]

MC

Lower 1σ interval

Upper 1σ interval

OG, O=2

RG, O=2

RG, O=5

85 model eval. 845 model eval.761 model eval.

(a) Max. fuel temperature in channel 1

−1

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Err

or

of

me

an

[%

]

MC

Lower 1σ interval

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OG, O=2

RG, O=2

RG, O=585 model eval.

689 model eval. 845 model eval.

10 100 1000 10000

(b) Max. clad temperature in channel 1

−0.4

−0.2

0

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Err

or

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Upper 1σ interval

OG, O=2

RG, O=2


145 model eval.

145 model eval.

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Err

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OG, O=2

RG, O=2


497 model eval.

629 model eval.

10 100 1000 10000

(d) Max. outlet temp. in channel 1

−0.8

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Err

or

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OG, O=2

RG, O=2

RG, O=5

85 model eval.


10 100 1000 10000


−0.4

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Err

or

of

me

an

[%

]

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Upper 1σ interval

OG, O=2

RG, O=2


267 model eval.

213 model eval.

10 100 1000 10000


Figure 5.6: Convergence of the mean of different responses obtained by MonteCarlo sampling and the FANISP algorithm using different options.

147


−100

−80

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0

20

40

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or

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]

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Lower 1σ interval

Upper 1σ interval

OG, O=2

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85 model eval.

845 model eval.

761 model eval.

10 100 1000 10000

(a) Max. fuel temperature in channel 1

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−80

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Lower 1σ interval

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OG, O=2

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85 model eval.


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[%

]

10 100 1000 10000

(b) Max. clad temperature in channel 1

−100

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OG, O=2

RG, O=2

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85 model eval.

145 model eval.

145 model eval.

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[%

]

10 100 1000 10000


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85 model eval.

497 model eval.

629 model eval.

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[%

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10 100 1000 10000

(d) Max. outlet temp. in channel 1

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85 model eval.

169 model eval.

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213 model eval.

Err

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[%

]

10 100 1000 10000


Figure 5.7: Convergence of the variance of different responses obtained byMonte Carlo sampling and the FANISP algorithm using different options.

148

5.6. Results

power is essentially 1, and the maximum fuel and cladding temperatures inthe different thermal-hydraulic channels also show strong correlations with it,with sensitivity values ranging between 0.7 and 0.8. The constrained powerpeaking sensitivities can clearly be recognized: each channel has positivesensitivities to its corresponding power peaking factor and negative to all theothers, furthermore the higher number of assemblies these channels containthe more negative the sensitivity gets. Next in line are the sensitivities to thegap conductance and the cladding thermal conduction. These provide negativevalues to each responses, since an increase in their value causes a decrease inthe heat resistance from the fuel to the coolant, resulting in lower temperaturesin both steady state and transient situations.

The PC estimates of the sensitivities are listed in Table 5.11 and for mostparameters they show good agreement with the MC reference (Table 5.10).The differences between the PC and the MC results are within a percent forthe most relevant parameters, to the less important ones (i.e. the parameterswith small sensitivities) the differences are somewhat higher. Furthermore oneneeds to remember that the MC estimates are calculated using multivariatelinear regressions, which is only an approximation if the response dependenceis nonlinear.

For the power peaking factors separate calculations were performed to deriveunconstrained sensitivities and check the constrained sensitivities obtained bythe procedure described in Section 5.3. The results of one-at-a-time perturba-tions to the power peaking factors (and correspondingly the effective power)by one standard deviation are summarized in Table 5.12. As expected theunconstrained sensitivities are significantly different from their constrainedcounterparts, most notably for the sensitivities of responses associated witha certain channel to the power peaking factors of the other channels. Theunconstrained sensitivities are close to zero in such cases (such as the sensitivityof the fuel temperature in channel 1 to the power peaking factors of the otherchannels), whereas the constrained ones are significantly higher and alwaysnegative, since the increase of power in one channel has to be counterbalancedby a decrease in all the other channels to satisfy the constraint. For the samereason the unconstrained sensitivities of the maximum effective power are allpositive and the constrained ones are all near zero, since in the latter case onlythe power distribution changes, not the total starting power. Furthermore theconstrained sensitivities listed in Table 5.12 correspond well to the sensitivitiesshown in Tables 5.10-5.11, there are only small deviations, mainly in the values

149


Table 5.10: Relative sensitivities of various responses to the differentinput parameters obtained with Monte Carlo sampling and linear

regression. Only parameters with a sensitivity of at least 0.01 are listed.Par. TmaxFuel,1 TmaxFuel,2 TmaxClad,1 TmaxClad,2 Pmaxeff TmaxOut,1 TmaxOut,2 TmaxOut,B TmaxUP

∆ρD -0.083 -0.074 -0.080 -0.069 -0.017 -0.070 -0.060 -0.001 -0.061∆ρV 0.041 0.037 0.039 0.034 0.027 0.034 0.029 -0.000 0.030βeff2 0.014 0.012 0.014 0.012 -0.002 0.014 0.012 0.000 0.012βeff5 0.011 0.010 0.010 0.009 -0.004 0.008 0.007 0.000 0.007ppf1 0.762 -0.012 0.806 -0.013 -0.000 0.748 -0.013 0.001 -0.002ppf2 -0.080 0.641 -0.083 0.655 -0.001 -0.078 0.604 -0.001 -0.006ppf3 -0.138 -0.127 -0.145 -0.129 -0.001 -0.134 -0.119 -0.000 -0.000ppf4 -0.176 -0.161 -0.187 -0.164 -0.001 -0.173 -0.151 0.000 -0.003ppf5 -0.213 -0.196 -0.227 -0.202 0.002 -0.211 -0.187 0.000 0.006ppf6 -0.156 -0.145 -0.164 -0.147 0.000 -0.152 -0.135 0.001 0.005kUPuC -0.063 -0.059 -0.011 -0.010 -0.002 -0.010 -0.009 0.000 -0.009CUPuC -0.008 -0.009 -0.012 -0.013 0.013 -0.004 -0.005 -0.000 -0.004hGap -0.135 -0.128 -0.037 -0.033 -0.010 -0.034 -0.030 -0.000 -0.030kSiC -0.124 -0.099 -0.104 -0.078 -0.008 -0.015 -0.014 0.001 -0.015CSiC -0.004 -0.005 -0.004 -0.005 0.004 -0.014 -0.014 0.000 -0.014Peff 0.756 0.704 0.796 0.717 1.003 0.734 0.660 0.082 0.655Etot 0.002 0.002 0.002 0.002 -0.001 0.002 0.002 -0.000 0.002mByp 0.015 0.015 0.023 0.021 0.000 0.024 0.021 0.006 -0.014A -0.091 -0.078 0.050 0.070 -0.004 0.027 0.034 0.003 0.032B 0.004 0.007 0.081 0.091 -0.005 0.024 0.030 0.004 0.029K1 0.031 -0.001 0.046 -0.001 0.000 0.049 -0.001 -0.000 -0.000K2 -0.007 0.071 -0.010 0.103 0.000 -0.011 0.108 -0.003 0.002K3 -0.011 -0.010 -0.016 -0.015 -0.000 -0.017 -0.015 -0.005 0.004K4 -0.012 -0.012 -0.018 -0.016 -0.000 -0.019 -0.017 -0.005 0.003K5 -0.008 -0.007 -0.011 -0.010 -0.000 -0.012 -0.011 -0.004 0.002K6 -0.015 -0.014 -0.022 -0.020 -0.000 -0.023 -0.021 -0.007 0.006Ppr -0.021 -0.014 -0.068 -0.052 0.024 -0.080 -0.060 0.109 -0.054

150

5.6. Results

Table 5.11: Relative sensitivities of various responses to the differentinput parameters obtained with the FANISP algorithm. Only

parameters with a sensitivity of at least 0.01 are listed.Par. TmaxFuel,1 TmaxFuel,2 TmaxClad,1 TmaxClad,2 Pmaxeff TmaxOut,1 TmaxOut,2 TmaxOut,B TmaxUP

∆ρD -0.083 -0.075 -0.081 -0.070 -0.017 -0.071 -0.061 0.000 -0.061∆ρV 0.041 0.037 0.039 0.034 0.027 0.034 0.029 0.000 0.029βeff2 0.013 0.012 0.013 0.012 -0.002 0.013 0.011 0.000 0.011βeff5 0.011 0.010 0.011 0.009 -0.003 0.008 0.007 0.000 0.007ppf1 0.761 -0.012 0.805 -0.012 0.000 0.748 -0.012 0.000 -0.002ppf2 -0.083 0.639 -0.088 0.650 0.000 -0.082 0.601 0.000 -0.008ppf3 -0.137 -0.127 -0.144 -0.129 0.000 -0.134 -0.119 0.000 0.001ppf4 -0.174 -0.160 -0.184 -0.163 0.000 -0.171 -0.151 0.000 0.000ppf5 -0.211 -0.195 -0.225 -0.200 0.000 -0.209 -0.185 -0.001 0.008ppf6 -0.156 -0.144 -0.164 -0.145 0.000 -0.152 -0.134 0.001 0.001kUPuC -0.062 -0.058 -0.011 -0.010 -0.002 -0.010 -0.009 0.000 -0.009CUPuC -0.008 -0.009 -0.012 -0.014 0.012 -0.005 -0.005 0.000 -0.005hGap -0.136 -0.127 -0.038 -0.033 -0.010 -0.035 -0.030 0.000 -0.031kSiC -0.123 -0.097 -0.101 -0.081 -0.007 -0.014 -0.014 0.000 -0.014CSiC -0.003 -0.004 -0.004 -0.004 0.004 -0.014 -0.014 0.000 -0.014Peff 0.743 0.689 0.777 0.695 1.005 0.714 0.637 0.051 0.635Etot 0.003 0.003 0.003 0.000 0.000 0.003 0.000 0.000 0.002mByp 0.015 0.015 0.023 0.021 0.000 0.024 0.022 0.006 -0.014A -0.092 -0.081 0.047 0.074 0.000 0.026 0.034 0.004 0.031B -0.004 0.000 0.083 0.094 0.000 0.025 0.032 0.004 0.030K1 0.031 0.000 0.046 0.000 0.000 0.049 0.000 0.000 0.000K2 -0.007 0.072 -0.010 0.102 0.000 -0.011 0.108 -0.003 0.002K3 -0.011 -0.011 -0.017 -0.015 0.000 -0.018 -0.016 -0.005 0.004K4 -0.012 -0.012 -0.018 -0.016 0.000 -0.019 -0.017 -0.006 0.004K5 -0.008 -0.007 -0.011 -0.010 0.000 -0.012 -0.011 -0.004 0.002K6 -0.015 -0.014 -0.022 -0.020 0.000 -0.023 -0.021 -0.007 0.006Ppr -0.021 -0.014 -0.069 -0.050 0.023 -0.081 -0.059 0.109 -0.054

151


of the near zero sensitivities.

Table 5.12: Unconstrained and constrained sensitivities of variousresponses to the power peaking factors obtained with individual

perturbations.Par. TmaxFuel,1 TmaxFuel,2 TmaxClad,1 TmaxClad,2 Pmaxeff TmaxOut,1 TmaxOut,2 TmaxOut,B TmaxUP

Unconstrained relative sensitivitiesppf1 0.777 -0.003 0.819 -0.003 0.014 0.762 -0.003 0.001 0.007ppf2 -0.012 0.775 -0.012 0.790 0.107 -0.013 0.730 0.005 0.060ppf3 -0.006 -0.005 -0.007 -0.006 0.217 -0.008 -0.006 0.010 0.137ppf4 -0.012 -0.010 -0.014 -0.012 0.285 -0.015 -0.012 0.014 0.178ppf5 -0.007 -0.005 -0.012 -0.010 0.386 -0.014 -0.011 0.016 0.253ppf6 0.003 0.003 0.002 0.003 0.270 0.001 0.002 0.013 0.176

Constrained relative sensitivitiesppf1 0.766 -0.013 0.808 -0.014 -0.004 0.752 -0.013 -0.000 -0.005ppf2 -0.084 0.701 -0.088 0.715 -0.018 -0.082 0.662 -0.001 -0.019ppf3 -0.139 -0.140 -0.146 -0.142 -0.011 -0.135 -0.131 -0.000 -0.008ppf4 -0.177 -0.177 -0.186 -0.181 0.002 -0.173 -0.167 0.000 -0.002ppf5 -0.212 -0.214 -0.226 -0.220 0.033 -0.211 -0.205 -0.000 0.029ppf6 -0.155 -0.157 -0.163 -0.159 -0.001 -0.150 -0.146 0.001 0.005

5.6.4 Response Uncertainties and UncertaintyDecomposition

The final uncertainties of the different responses are listed in Table 5.13,where the relative standard deviation values are displayed obtained by MCsampling and PC methods using the relaxed Gerstner method and two differentpolynomial orders (O = 2 and O = 5). The values are rather similar, theuncertainties of the maximum cladding temperatures are between 2.5% and 3%,for the maximum fuel temperatures they range between 2.2% and 2.5% andfor the helium outlet temperatures between 2.3% and 3%. The uncertaintiesof the maximum power, the upper plenum and the bypass temperature aresomewhat lower, 1.2%, 1.2% and 1.3% respectively.

To identify the main sources of uncertainties the Sobol sensitivity indiceswere also calculated for the responses. Figure 5.8 gives an overview of the

152

5.6. Results

uncertainty decomposition of a selected number of them with only the mostimportant parameters being displayed. Since the power peaking generatingfactors individually do not have a specific meaning, their sensitivity indices werelumped into a single power peaking factor effect. As can be seen this providesthe most important source of uncertainty for the maximum fuel and claddingtemperatures, followed by the initial effective power, the gap conductance,the pressure loss coefficient for the hottest channel (i.e. the flow distributioneffecting this channel) and finally the cladding heat conduction. For the outlettemperature in this channel the picture is similar, except that the pin relatedquantities are replaced by the flow related ones (bypass flow and pressure). Forthe maximum power reached during the transient and the outlet temperaturein the bypass channel almost solely the initial effective power and the primarycircuit pressure are important. Finally for the upper plenum temperature thethree important quantities are the initial effective power, the primary circuitpressure and the bypass flow.

0

0.2

0.4

0.6

0.8

1

Input parameters

To

tal S

ob

ol i

nd

ice

s

∆ρD ppf hGap Pe K1∆ρV kUPuC kSiC mBypass Pprimary

TFuel,1

max

TClad,1

max

Pe!

max

TOut,1

max

TOut,B

max

TUP

max

Figure 5.8: The total Sobol sensitivity indices of different responses calculatedwith PCE

153


Table 5.13: Relative uncertainties of the different responses obtained byMC sampling and the FANISP algorithm. The PC results correspond tothe relaxed Gerstner method with different maximum PC orders and

the corresponding computational cost is also displayed.Respon- Relative unc. (σ/µ [%]) No. of model evaluations

ses MC PC (O = 2) PC (O = 5) PC (O = 2) PC (O = 5)TmaxFuel,1 2.40 2.37 2.38 761 845TmaxFuel,2 2.54 2.54 2.54 689 697TmaxFuel,3 2.43 2.43 2.43 761 769TmaxFuel,4 2.37 2.35 2.35 625 629TmaxFuel,5 2.25 2.28 2.28 685 697TmaxFuel,6 2.48 2.49 2.51 757 773TmaxClad,1 2.66 2.63 2.65 689 845TmaxClad,2 3.01 2.95 2.95 557 565TmaxClad,3 2.80 2.77 2.78 621 769TmaxClad,4 2.68 2.63 2.63 561 697TmaxClad,5 2.43 2.44 2.44 621 697TmaxClad,6 2.90 2.88 2.88 561 565Pmaxeff 1.20 1.22 1.22 145 145TmaxOut,1 2.57 2.54 2.55 497 629TmaxOut,2 2.98 2.95 2.95 389 397TmaxOut,3 2.76 2.75 2.75 441 449TmaxOut,4 2.62 2.57 2.57 393 397TmaxOut,5 2.33 2.33 2.34 441 505TmaxOut,6 2.87 2.87 2.87 389 397TmaxOut,B 1.30 1.30 1.30 161 169TmaxUP 1.18 1.17 1.17 213 267

154

5.7. Computational Costs and Cost Reduction Techniques

5.7 Computational Costs and Cost ReductionTechniques

The presented results show that adaptive PC techniques can provide accurateestimates of response sensitivities and uncertainties at significantly lowercomputational effort than standard sampling based methods. The latterhowever have the advantage of providing estimates of all responses at once,whereas the use of adaptivity can only steer the computation towards theefficient calculation of the chosen responses. Since different input parametersare important for different responses the total cost of building a PCE for allresponses can roughly be as high as the sum of the needed model evaluationsof the different responses. In practice however many of the needed grids arethe same, hence by saving the value of all responses when building the PCE forthe chosen one it is possible to simply reuse these when building the next PCE,resulting in a lower computational cost. For our example it turns out that only1037 and 1095 different calculations are needed for an O = 2 and O = 5 PCEof all responses using the relaxed Gerstner method, which is well comparableto a typical choice of a 1000 runs using Monte Carlo methods. It is also worthmentioning that the use of basis adaptivity makes a significant difference inthe number of needed model evaluations, should we chose to build a full secondorder (O = 2) PCE with P + 1 = 946 basis vectors for each response, all level3 grids (GO = 3) would be needed equalling 3697 calculations.

To evaluate the effectiveness of the dimensionality reduction techniquedetailed in Section 5.2.1 the FANISP algorithm was run for the differentresponses with an O = 2 maximum polynomial order and a tolerance ofεµ = εσ2 = 0.001 using the relaxed Gerstner method, without and with areduction in dimensionality using εDR = 0.0005. The results summarized inTable 5.14 show that a substantial reduction can be achieved in the numberof needed model evaluations without significantly affecting the accuracy ofthe results. As can be seen the effective dimensionality of the problem issignificantly reduced even for a relatively small tolerance for the contributionof the parameters to the variance, meaning that as dictated by the sparsityof effects principle (Montgomery, 2001) each response is influenced by only asubset of the inputs (though this is naturally problem dependent). Typicallydelayed neutron fractions, decay constants and the singular pressure dropcoefficients are identified as the less significant variables. The reduced numberof parameters also results in a smaller number of model evaluations, for the

155


current test up to 24% of the computational costs can be saved. At thesame time the mean and variance estimates do not degrade significantly, thedifference between the mean estimates is below the εµ tolerance level in allcases, whereas the variance estimates change with less than 1% for almost allresponses (there are 3 exceptions). More important is however the fact thatthe results obtained by dimensionality reduction can be considered to have asimilar accuracy compared to the MC reference, keeping in mind that thoseestimates have a standard deviation of 1.4% in most cases (see Table 5.9).

To evaluate the effectiveness of the incremental polynomial order the FAN-ISP algorithm was run for the different responses with an O = 5 maximumpolynomial order and a tolerance of εµ = εσ2 = 0.01 using the relaxed Gerstnermethod. The results are summarized in Table 5.15 and show minor improve-ments for certain responses. For example for the maximum outlet temperaturein the hottest channel roughly 10% of the calculations could be saved, howeverfor most of the outputs the number of needed model runs does not change.Nevertheless in many cases the number of basis vectors in the final PCE issmaller than without the use of incremental order, since all of them convergebefore reaching the defined 5th order maximum.

In Table 5.16 the results of running the FANISP algorithm in parallel aresummarized for O = 2 and O = 5, with the relaxed Gerstner method and anaccuracy of εµ = εσ2 = 0.01. Both the total number of calculations as well asthe number of integration steps are listed, together with the comparison to theserial calculations (detailed in Table 5.13 and in Table 5.15 for example). Formost of the responses the computational costs are clearly higher in terms ofthe number of needed model evaluations, the parallel calculations need up to84% more code runs than their serial counterpart. The reason for this increaseis that there can be grids in the active set which have an error indicator thatis higher than the tolerance, but is smaller than the highest error indicator.In the parallel algorithm these grids are all added to the old set and all theirforward neighbours are calculated. In contrast, in the serial algorithm onlythe grid with the highest error indicator is put to the old set, only its forwardneighbours are calculated and the error indicators of the remaining grids inthe active set are updated. Hence it can happen that the error indicator of agrid is higher than the tolerance before an integration step, but falls below thelimit after it. These grids are kept in the active set when the serial algorithmis used, but are added to the old set in the parallel version, leading to extramodel evaluations and the increased computational costs.

156

5.7. Computational Costs and Cost Reduction Techniques

Table 5.14: Accuracy and computational cost of PCE with the relaxedGerstner method with and without dimensionality reduction. The PCEsfor the different responses were built using O = 2 with a tolerance ofεµ = εσ2 = 0.001, and results were compared to the MC reference (seen

in Table 5.9).No dimensionality Dimensionality reductionreduction (εDR = 0) with εDR = 0.0005

Para- Mean Variance No. of Mean Variance No. of Eff.meter error [%] error [%] runs error [%] error [%] runs dim.TmaxFuel,1 0.017 2.32 1193 0.018 1.85 1105 24TmaxFuel,2 0.010 0.59 1025 0.013 0.07 941 22TmaxFuel,3 0.051 0.63 1089 0.051 0.83 1033 23TmaxFuel,4 0.025 1.36 1119 0.000 1.35 933 21TmaxFuel,5 0.022 2.77 921 0.023 3.16 845 20TmaxFuel,6 0.006 1.03 1243 0.002 1.70 1143 23TmaxClad,1 0.029 1.53 1379 0.025 0.44 1113 23TmaxClad,2 0.037 2.94 877 0.036 2.14 833 21TmaxClad,3 0.070 1.07 1125 0.068 0.17 989 22TmaxClad,4 0.001 3.64 1147 0.010 2.64 973 22TmaxClad,5 0.030 1.31 1017 0.029 2.44 853 21TmaxClad,6 0.021 0.53 1041 0.021 0.00 965 23Pmaxeff 0.010 3.14 289 0.009 2.97 221 9TmaxOut,1 0.040 1.77 997 0.040 1.15 845 24TmaxOut,2 0.117 2.25 681 0.044 2.08 625 22TmaxOut,3 0.082 0.70 829 0.082 0.46 769 22TmaxOut,4 0.002 3.21 897 0.002 2.86 769 22TmaxOut,5 0.042 0.88 905 0.042 1.56 765 21TmaxOut,6 0.031 0.22 833 0.031 0.07 769 23TmaxOut,B 0.010 0.32 1429 0.002 1.88 1113 18TmaxUP 0.029 1.87 1067 0.032 1.06 933 21

157


Table 5.15: Computational cost and the number of basis vectors of PCEwith the relaxed Gerstner method with incremental polynomial order.The PCEs for the different responses were built incrementally increasingthe polynomial order with a maximum of O = 5, with a tolerance of

εµ = εσ2 = 0.01.Without incremental order With incremental order

Para- No. of No. of basis No. of No. of basis Maximummeter runs vectors runs vectors PC orderTmaxFuel,1 845 73 845 67 3TmaxFuel,2 697 73 697 73 3TmaxFuel,3 769 72 769 70 3TmaxFuel,4 629 63 629 61 3TmaxFuel,5 697 57 697 55 3TmaxFuel,6 773 77 765 74 3TmaxClad,1 845 82 845 76 3TmaxClad,2 565 63 565 61 3TmaxClad,3 769 70 769 67 3TmaxClad,4 697 72 697 67 3TmaxClad,5 697 77 697 69 3TmaxClad,6 565 62 565 60 3Pmaxeff 145 24 145 24 2TmaxOut,1 629 67 565 62 3TmaxOut,2 397 49 397 44 2TmaxOut,3 449 50 449 50 3TmaxOut,4 397 52 397 51 3TmaxOut,5 505 59 449 55 3TmaxOut,6 397 47 397 45 2TmaxOut,B 169 24 169 22 3TmaxUP 267 45 229 43 3

158

5.8. Summary

Although the total number of code runs is generally higher with paralleliza-tion, the number of integration steps is significantly, 50 to 85% less than in theserial case. The number of integration steps is equal to the number of instancesthe external code has to be run, hence assuming infinite computer power (i.e.being able to run the code for all cubature points at each step at once) it is alsothe bottle-neck of the calculation. Hence savings in the integration steps canalso be thought of as the efficiency of parallelization. In case of dealing withproblems which need a long time to run each time and having sufficiently largecomputational capacity, parallelization can therefore substantially decrease thetime that is needed for building the PCE of responses, at the cost of a loweroverall efficiency.

5.8 Summary

This chapter demonstrated the applicability of novel adaptive PolynomialChaos techniques for the sensitivity and uncertainty analysis of large scale,realistic problems. As an example a sensitivity analysis and an uncertaintyquantification of an Unprotected Loss of Flow transient was performed for theEuropean GFR2400 Gas Cooled Fast Reactor design using the Cathare 2 codesystem and the FANISP algorithm. A high number of uncertain sources weretaken into account including both neutronic and thermal-hydraulic paramet-ers, amounting to a total of 42 independent inputs. Several responses wereinvestigated, most importantly the maximum fuel and cladding temperaturesoccurring during the transient as well as the gas outlet temperatures and themaximum power of the reactor.

It was shown that due to the non-intrusive nature of the developed PCtechniques they are relatively easy to apply to realistic problems such as thecomplete system model of a reactor, since from the PC point of view eventhe most complicated problem description is treated as a black box. Theapplied Fully Adaptive Non-Instrusive Spectral Projection algorithm proved tobe efficient in determining response sensitivities, uncertainties, as well as fullPDFs, despite the large number of input parameters. For all the investigatedresponses PCE provided an affordable alternative to standard Monte Carlosampling while offering a significantly higher accuracy. For many responsesthe simplest and fastest original Gerstner method used together with basisadaptivity proved to be sufficient needing only 85 model evaluations, whilefor even the most difficult and complicated ones the relaxed Gerstner method

159


Table 5.16: Computational cost of PCE with the relaxed Gerstnermethod with parallel calculation. In the parentheses the number of

integration steps in the serial calculations are shown (corresponding tothe number of runs listed in Table 5.13), as well as the computational

overhead compared to the serial calculation.O = 2 O = 5

Parameter No. of No. of integ- No. of No. of integ-runs ration steps runs ration steps

TmaxFuel,1 769 (+1.1%) 4 (20) 949 (+12%) 5 (23)TmaxFuel,2 697 (+1.2%) 3 (19) 1049 (+51%) 6 (22)TmaxFuel,3 769 (+1.1%) 4 (20) 909 (+18%) 6 (24)TmaxFuel,4 629 (+0.6%) 4 (18) 761 (+21%) 6 (20)TmaxFuel,5 697 (+1.8%) 4 (19) 777 (+12%) 5 (19)TmaxFuel,6 765 (+1.1%) 4 (20) 1177 (+52%) 6 (23)TmaxClad,1 841 (+22%) 3 (19) 1553 (+84%) 6 (24)TmaxClad,2 565 (+1.4%) 3 (17) 849 (+50%) 6 (17)TmaxClad,3 629 (+1.3%) 3 (18) 1089 (+42%) 5 (20)TmaxClad,4 625 (+11%) 3 (17) 909 (+30%) 5 (19)TmaxClad,5 697 (+12%) 3 (18) 953 (+37%) 6 (21)TmaxClad,6 565 (+0.7%) 4 (17) 753 (+33%) 8 (17)Pmaxeff 145 (+0.0%) 3 (7) 145 (+0.0%) 3 (7)TmaxOut,1 565 (+14%) 3 (16) 661 (+5.1%) 5 (19)TmaxOut,2 397 (+2.1%) 3 (14) 397 (+0.0%) 4 (14)TmaxOut,3 449 (+1.8%) 4 (15) 449 (+0.0%) 4 (15)TmaxOut,4 397 (+1.0%) 4 (14) 397 (+0.0%) 4 (14)TmaxOut,5 449 (+1.8%) 3 (15) 513 (+1.6%) 5 (16)TmaxOut,6 397 (+2.1%) 4 (14) 397 (+0.0%) 4 (14)TmaxOut,B 169 (+5.0%) 4 (12) 169 (+0.0%) 4 (12)TmaxUP 229 (+7.5%) 4 (13) 267 (+0.0%) 5 (15)

160

5.8. Summary

provided an accurate description with less than a thousand runs.

The uncertainty of the maximum fuel and cladding temperatures was foundto be around 2.5% and 3% respectively, whereas for the maximum power andthe maximum upper plenum temperature an uncertainty estimate of 1.2%was made. The sensitivity analysis and the variance decomposition revealedthat the most important parameters are the initial effective power of thereactor and the exact power distribution. Other relevant parameters includethe heat transfer quantities (mainly the gap conductance and the claddingheat conduction), reactivity feedback coefficients, bypass mass flow and theflow distribution in the core.

The presented work greatly expands the applicability of polynomial chaosexpansion methods, to 40-50 parameters in contrast with the majority ofprevious works with typically less than 10 input variables. This opens a lot ofnew opportunities for application where the responses can be expected to berelatively smooth functions of the inputs. Nevertheless for even higher numberof inputs the limitations of quadrature based numerical integration remainand the computational cost become prohibitive. Hence future works shouldmainly include new types of applications rather than further optimization, andpossible extensions of the methods to less smooth responses.

161


162

Chapter 6

Conclusions andRecommendations

In this thesis efficient and accurate methods were developed for the sensitivityand uncertainty (S&U) analysis of coupled problems encountered in the nuclearfield. The main motivation for the research was that due to the increaseduse of computationally expensive multi-physics multi-scale simulations - withorders of magnitude different spatial and temporal scales - it is essential tominimize the costs associated to their S&U analysis, as well as to enhance thecapabilities of these techniques. In the nuclear community this is particularlyrelevant with the spread of Best Estimate Plus Uncertainty methodologies,requiring high fidelity multi-physics solvers and appropriate sensitivity anduncertainty analysis methods to be developed in parallel.

6.1 Conclusions

The main contribution of this thesis to S&U analysis methodologies is in twofields: adjoint techniques and adaptive Polynomial Chaos (PC) methods. Themost important findings can be summarized as follows:

• It was shown that the adjoint methods well known for pure criticalityproblems can be extended to coupled systems as well, where the effects

163

6. Conclusions and Recommendations

of the perturbations can be interpreted in three different ways: eitherby allowing the steady-state power level to adjust to the new conditions(power perturbation), or by constraining the power and allowing theeigenvalue to differ from unity (eigenvalue perturbations), or by sim-ultaneously perturbing a control variable to conserve the criticality ofthe system at the unperturbed power level (control parameter perturba-tion). Each approach has its practical relevance and their novelty comesfrom allowing the investigation of the effects of interdependencies aswell as providing more accurate uncertainty estimations by enabling theuncertainties originating from all subsystems to be taken into account.

• The computational aspects of the coupled adjoint theory were also in-vestigated and an efficient method was proposed for the solution of thecoupled adjoint problem. It is based on the use of Krylov methodswrapped around the neutron transport and augmenting codes, whichuses the individual programs to perform the necessary matrix-vectormultiplications and inversions for preconditioning. The presented blockiteration preconditioner provides very fast convergence, with typicallyless than 10 iterations being enough for sufficiently low relative residuals.It was shown that the large scale applicability of the method involvesonly modest code development effort if the augmenting codes are adjointcapable. The main challenge therefore lies in the mapping of the neut-ronics and augmenting phase-spaces upon each other, which is neededfor any coupled calculations anyway.

• In the second half of this thesis Polynomial Chaos methods were invest-igated. The adaptive sparse grid techniques based on the original ideaof Gerstner for calculating multidimensional integrals were shown to beefficient in significantly reducing the computational cost associated withcalculating the PC coefficients of the Polynomial Chaos Expansion (PCE)of quantities of interest, both compared to traditional full grids as wellas to static Smolyak sparse grids.

• To further reduce costs adaptive basis construction methods were pro-posed in order to build up the PC basis set as needed by the dependenceof the output instead of using a fixed set. The presented one- and two-step approaches are based on using interpolation with the points of thesparse grids to choose between important and unimportant basis vectors.

164

6.2. Recommendations

The two techniques yield very similar results, in most cases building upalmost identical sparse PC representations.

• For large scale applications an adaptive increase of the global polynomialorder and an automatic dimensionality reduction method were introduced.The former yielded modest enhancements, while the latter was shownto be effective in reducing computational costs by eliminating thosevariables early in the calculations which contribute less to the variancethan a predefined tolerance limit.

• To demonstrate the true large scale applicability of the developed FullyAdaptive Non-Intrusive Spectral Projection (FANISP) algorithm thesensitivity and uncertainty analysis of a Gas Cooled Fast Reactor transientwas performed, with 42 input parameters. The results confirmed thatthe FANISP algorithm can be applied even to such high dimensionalproblems, as only a few of the parameters are important for each response,most can be discarded after the first step of the calculation (i.e. afterthe level 2 grids).

6.2 Recommendations

For the use of the coupled adjoint theory, adjoint capable augmenting codesshould be implemented. With advances in programming disciplines (e.g. theuse of automatic differentiation tools) both new and existing solvers should beadapted with adjoint modules. Once such codes reach maturity coupling themwith traditional neutron transport codes could be done relatively easily.

To deal with nonlinearities in coupled criticality problems one could usethe coupled adjoint theory together with the adaptive PC techniques. It wasalready shown that the level 2 grids can effectively be used to decrease thenumber of input parameters significantly, this step could completely be replacedby an adjoint calculation, providing the values of the derivatives with respectto all input parameters at once. The derivative information could be used topropagate uncertainties associated to the large number of unimportant inputparameters, whereas proper PCE could be performed for parameters with largederivatives, where nonlinearities are expected.

To extend the applicability of the presented PC methods to generic problemswith hundreds or more input parameters PCE could be coupled with stochastic

165

6. Conclusions and Recommendations

tools. One could start with traditional sampling (possibly Latin Hypercubeor Stratified Sampling) in order to eliminate the non-important variables andonly use PCE for the remaining relevant ones. A similar, fully PCE basedapproach would be to merge sampling and quadrature based PCE, first usinginterpolations (e.g. least square or least angle regression) and random samplesto choose the most relevant parameters and basis vectors, then switch to thegrid based approach to get the accurate values of the PC coefficients.

In the FANISP algorithm, the importance of the sparse grids is based ontheir contribution to the first two moments of the response. One could devisean error indicator in order to take into account higher moments as well, or tosteer the calculation in a different way (e.g. first by the contributions to thefirst moment, then to the second, etc.). The algorithm could also be extendedto better deal with multiple responses. The present approach is to save thevalues of all outputs at the already calculated quadrature points, then reloadthem when FANISP is run with a different response, thus only the new pointshave to be calculated again. Instead of such looping over the outputs, onecould devise an error indicator that takes into account all responses at thesame time and therefore steers the calculations to be globally accurate.

A very interesting extension of the presented PC methods would be toconsider discontinuous responses or responses with discontinuous derivatives.For such cases the basis vectors would have to be changed, instead of globalpolynomials one could employ polynomials with a local support when needed(just like in Multi-Element gPC) and adapt the sparse grids accordingly.Noisy responses (e.g. when using Monte Carlo transport) represent anotherchallenging case, with possibly high practical relevance. In such a case a lot ofthe sparse grids would seem to be important due to slightly changing the valuesof the mean and the variance. Such oscillatory behaviour could be identifiedby looking into the history of the error indicator or the PC coefficients.

The adjoint and PC methods presented in this thesis represent very attractivetechniques to tackle the S&U analysis of the two most common types of coupledcalculations encountered in the nuclear field. As adjoint techniques are uniquelyapplicable to handle all input parameters at once, whereas Polynomial ChaosExpansion is well suited to handle nonlinearities, their combined use has evenhigher potentials. This thesis provides a strong basis both for the practicalapplications of the developed methods as well as for further research to expandtheir range of applicability and enhance their effectiveness.

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178

Appendix A

Notation

In this thesis the following notation is used:

• Single and double underlines are used to denote vectors v and matricesM .

• [v]i and vi both denote the i-th element of a vector v, whereas[M]i,j

and Mi,j both signal the element of matrix M in the i-th row and thej-th column.

• α denotes the vector of N input variables and their unperturbed value isdenoted by α0.

• O denotes an operator and(O)∗

its adjoint.

• 〈 , 〉φ and 〈 , 〉T are used to signal inner products in the neutronicand augmenting phase spaces, respectively. 〈 , 〉 without subscript isused to signal the inner product of L2 (θ,P) in Chapters 4-5.

• L and F are the traditional loss and fission operators, M is the transportoperator and N is a generic augmenting operator. Furthermore the

179

A. Notation

following shorthand notations are used in Chapter 2:

L0 = L(α0, T 0); ∂L

∂T

∣∣∣∣∣0

= ∂L(α, T )∂T

∣∣∣∣∣α0,T 0

; ∂L

∂α

∣∣∣∣∣0

= ∂L(α, T )∂α

∣∣∣∣∣α0,T 0

;

F 0 = F (α0, T 0); ∂F

∂T

∣∣∣∣∣0

= ∂F (α, T )∂T

∣∣∣∣∣α0,T 0

; ∂F

∂α

∣∣∣∣∣0

= ∂F (α, T )∂α

∣∣∣∣∣α0,T 0

;

∂L

∂αc

∣∣∣∣∣0

= ∂L(α, T )∂αc

∣∣∣∣∣α0,T 0

; ∂F

∂αc

∣∣∣∣∣0

= ∂F (α, T )∂αc

∣∣∣∣∣α0,T 0

;

M0 = L0 − λ0F 0; ∂M

∂T

∣∣∣∣∣0

= ∂L

∂T

∣∣∣∣∣0− λ0 ∂F

∂T

∣∣∣∣∣0

;

∂M

∂α

∣∣∣∣∣0

= ∂L

∂α

∣∣∣∣∣0− λ0 ∂F

∂α

∣∣∣∣∣0

; ∂M

∂αc

∣∣∣∣∣0

= ∂L

∂αc

∣∣∣∣∣0− λ0 ∂F

∂αc

∣∣∣∣∣0

;

N0 = N(α0, T 0, φ0); ∂N

∂α

∣∣∣∣∣0

= ∂N(α, T, φ)∂α

∣∣∣∣∣α0,T 0,φ0

;

∂N

∂T

∣∣∣∣∣0

= ∂N(α, T, φ)∂T

∣∣∣∣∣α0,T 0,φ0

; ∂N

∂φ

∣∣∣∣∣0

= ∂N(α, T, φ)∂φ

∣∣∣∣∣α0,T 0,φ0

;

∂N

∂αc

∣∣∣∣∣0

= ∂N(α, T, φ)∂αc

∣∣∣∣∣α0,T 0,φ0

;

P 0f = Pf (α0, T 0); ∂Pf

∂α

∣∣∣∣0

= ∂Pf (α, T )∂α

∣∣∣∣α0,T 0

;

∂Pf∂T

∣∣∣∣0

= ∂Pf (α, T )∂T

∣∣∣∣α0,T 0

; ∂Pf∂αc

∣∣∣∣0

= ∂Pf (α, T )∂αc

∣∣∣∣α0,T 0

;

• Calligraphic letters L,G, I,F ,B,O,A stand for sets containing multi-indices (either identifying sparse grids or PC basis vectors).

• Γ (O) signals different sets of PC basis vectors of orders less than or equalto O.

180

Appendix B

Derivation of PerturbationFormulas and Adjoint

Problems

B.1 Derivation of the Power PerturbationExpressions

For the unperturbed system characterized by α0, the steady-state solution isφ0 and T 0 satisfying

L(α0, T 0)φ0 = F (α0, T 0)φ0 (B.1)N(α0, T 0, φ0) = 0. (B.2)

For the perturbed system described by input parameters α0 + ∆α the steady-state solution is φ0 + ∆φ and T 0 + ∆T , also obeying Equations B.1-B.2. Asusual, the Taylor expansion of operators L, F and N around the unperturbedsolution can be made, after which discarding second order terms and considering

181

B. Derivation of Perturbation Formulas and Adjoint Problems

Equations B.1-B.2 yields the following expressions:(L0 − F 0

)∆φ +

(∂L

∂T

∣∣∣∣∣0

∆T − ∂F

∂T

∣∣∣∣∣0

∆T)φ0

+(∂L

∂α

∣∣∣∣∣0

∆α− ∂F

∂α

∣∣∣∣∣0

∆α)φ0 = 0 (B.3)

∂N

∂φ

∣∣∣∣∣0

∆φ + ∂N

∂T

∣∣∣∣∣0

∆T + ∂N

∂α

∣∣∣∣∣0

∆α = 0. (B.4)

Taking the inner product of Equation B.3 and Equation B.4 with weight

functions w0φ(x) and w0

T (y) respectively, introducing J0∆T = ∂M

∂T

∣∣∣∣∣0

∆Tφ0 =(∂L

∂T

∣∣∣∣∣0

∆T − ∂F

∂T

∣∣∣∣∣0

∆T)φ0 and using the property of the adjoint operators

the following expressions are gained:⟨(M0

)∗w0φ,∆φ

⟩φ

+⟨(J0)∗w0φ,∆T

⟩T

+⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

= 0 (B.5)

⟨(∂N

∂φ

∣∣∣∣∣0

)∗w0T ,∆φ

⟩φ

+⟨(

∂N

∂T

∣∣∣∣∣0

)∗w0T ,∆T

⟩T

+⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

= 0. (B.6)

The relative change in the power (constraint) value corresponding to theperturbation is given by

∆PP 0 = 1

P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

+ 1P 0

⟨(∂Pf∂T

∣∣∣∣0

)∗φ0,∆T

⟩T

+ 1P 0

⟨P 0f ,∆φ

⟩φ.

In order to eliminate the indirect terms (the last two terms containing ∆φ and∆T ) the adjoint problem that has to be solved is(

M0)∗w0φ +

(∂N

∂φ

∣∣∣∣∣0

)∗w0T = w0

P

P 0 P0f (B.7)

(J0)∗w0φ +

(∂N

∂T

∣∣∣∣∣0

)∗w0T = w0

P

P 0

(∂Pf∂T

∣∣∣∣0

)∗φ0, (B.8)

182

B.2. Derivation of the Eigenvalue Perturbation Expressions

with w0P being arbitrarily chosen. To see this, one has to take the inner product

of Equations B.7-B.8 with ∆φ and ∆T respectively, add the two equations,then subtract the sum from the sum of Equations B.5-B.6. The result of theseoperations after rearrangement is:

1P 0

⟨(∂Pf∂T

∣∣∣∣0

)∗φ0,∆T

⟩T

+ 1P 0

⟨P 0f ,∆φ

⟩φ

=

− 1w0P

⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

− 1w0P

⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

.

Therefore the relative change in the power level is given by Equation B.9:

∆PP 0 = 1

P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

− 1w0P

⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

− 1w0P

⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

. (B.9)

The derivation of Equation 2.11 is essentially the same as that of Equation B.9,the only difference is that in Equations B.7-B.8 the right hand side is ∂R

∂φ

∣∣∣∣0

and ∂R

∂T

∣∣∣∣0respectively, hence the indirect change in response R will be given

by

∆Rindirect = −⟨wPT ,

∂N

∂α

∣∣∣∣∣0

∆α⟩T

−⟨wPφ ,

∂M

∂α

∣∣∣∣∣0

∆α⟩φ

.

B.2 Derivation of the Eigenvalue PerturbationExpressions

When the reactivity worth of different perturbations is searched the unper-turbed solution satisfies Equations B.10-B.12:

L(α0, T 0)φ0 = λ0F (α0, T 0)φ0 (B.10)N(α0, T 0, φ0) = 0 (B.11)⟨

Pf (α0, T 0), φ⟩φ

P 0 = 1. (B.12)

183


In the perturbed system the changed dependent variables φ0 +∆φ and T 0 +∆Talso satisfy Equations B.10-B.12 with a changed eigenvalue λ0 + ∆λ. Again,making the Taylor expansion of the operators, only keeping first order termsand using Equations B.10-B.12 the following equations are gained:

(L0 − λ0F 0

)∆φ +

(∂L

∂T

∣∣∣∣∣0

∆T − λ0 ∂F

∂T

∣∣∣∣∣0

∆T)φ0

+(∂L

∂α

∣∣∣∣∣0

∆α− λ0 ∂F

∂α

∣∣∣∣∣0

∆α)φ0 =∆λF 0φ0 (B.13)

∂N

∂φ

∣∣∣∣∣0

∆φ + ∂N

∂T

∣∣∣∣∣0

∆T + ∂N

∂α

∣∣∣∣∣0

∆α = 0 (B.14)

1P 0

⟨P 0f ,∆φ

⟩φ

+ 1P 0

⟨∂Pf∂T

∣∣∣∣0

∆T, φ0⟩φ

+ 1P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

= 0. (B.15)

Now taking the inner product of Equation B.13 and Equation B.14 with weightfunctions w0

φ(x) and w0T (y) respectively, while multiplying Equation B.15 with

−w0P , then using the property of the adjoint operators the following is gained:

⟨(M0

)∗w0φ,∆φ

⟩φ

+⟨(J0)∗w0φ,∆T

⟩T

+⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

= ∆λ⟨w0φ, F

0φ0⟩φ

(B.16)⟨(∂N

∂φ

∣∣∣∣∣0

)∗w0T ,∆φ

⟩φ

+⟨(

∂N

∂T

∣∣∣∣∣0

)∗w0T ,∆T

⟩T

+⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

= 0 (B.17)

− w0P

P 0

⟨P 0f ,∆φ

⟩φ− w0

P

P 0

⟨(∂Pf∂T

∣∣∣∣0

)∗φ0,∆T

⟩T

−w0P

P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

= 0. (B.18)

Summing up Equations B.16-B.18 and arranging for ∆λ:

∆λ =[ ⟨(

M0)∗w0φ,∆φ

⟩φ

+⟨(

∂N

∂φ

∣∣∣∣∣0

)∗w0T ,∆φ

⟩φ

− w0P

P 0

⟨P 0f ,∆φ

⟩φ

184

B.2. Derivation of the Eigenvalue Perturbation Expressions

+⟨(J0)∗w0φ,∆T

⟩T

+⟨(

∂N

∂T

∣∣∣∣∣0

)∗w0T ,∆T

⟩T

− w0P

P 0

⟨(∂Pf∂T

∣∣∣∣0

)∗φ0,∆T

⟩T

+⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

+⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

− w0P

P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

]·

1⟨w0φ, F

0φ0⟩φ

.

It is easy to see that the adjoint problem that has to be solved to make the termscontaining ∆φ and ∆T disappear is the same as for the power perturbation(Equations B.7-B.8). Having obtained the necessary adjoint functions w0

φ andw0T the change in the critical eigenvalue can be calculated by Equation B.19:

∆λ =[− w0

P

P 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

+⟨w0T ,∂N

∂α

∣∣∣∣∣0

∆α⟩T

+⟨w0φ,∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

]1⟨

w0φ, F

0φ0⟩φ

. (B.19)

When other response functionals R (α, T, φ) are considered the adjoint problemthat has to be solved is given by Equations B.20-B.21:

(M0

)∗wλφ +

(∂N

∂φ

∣∣∣∣∣0

)∗wλT = wλP

P 0 P0f + ∂R

∂φ

∣∣∣∣0

(B.20)

(J0)∗wλφ +

(∂N

∂T

∣∣∣∣∣0

)∗wλT = wλP

P 0

(∂Pf∂T

∣∣∣∣0

)∗φ0 + ∂R

∂T

∣∣∣∣0. (B.21)

Again, taking the inner product of Equation B.20 with ∆φ and Equation B.21with ∆T , adding the resulting equations, then subtracting the sum of Equa-tions B.16-B.18 with superscripts 0 changed to λ on w0

φ, w0T and w0

P , thefollowing expression is gained for the indirect change of the response:⟨

∂R

∂φ

∣∣∣∣0,∆φ

⟩φ

+⟨∂R

∂T

∣∣∣∣0,∆T

⟩T

= ∆λ⟨wλφ, F

0φ0⟩φ

+ ...

+ wλPP 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

−⟨wλT ,

∂N

∂α

∣∣∣∣∣0

∆α⟩T

−⟨wλφ,

∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

.

185


The term containing ∆λ can be made disappear by choosing the value of wλPsuch that ⟨

wλφ, F0φ0

⟩φ

= 0.

Let us denote this appropriate value of wλP by wλ,aP and assume that thecorresponding adjoint functions are wλ,aφ and wλ,aT , hence:⟨

wλ,aφ , F 0φ0⟩φ

= 0.

Furthermore suppose that the lambda-adjoint (Equations B.7-B.8) was solvedfor w0

P . When the generalized adjoint equations (Equations B.20-B.21) aresolved for a wλP = wλ,aP +a ·w0

P , the corresponding solution is wλφ = wλ,aφ +a ·w0φ

and wλT = wλ,aT + a ·w0T , with which:⟨

wλφ, F0φ0

⟩φ

=⟨wλ,aφ , F 0φ0

⟩φ

+ a⟨w0φ, F

0φ0⟩φ

= a⟨w0φ, F

0φ0⟩φ.

This can be used to update the wλP parameter in the generalized adjointequation during iteration:

wλ,k+1P = wλ,kP − a ·w0

P = wλ,kP −

⟨wλ,kφ , F 0φ0

⟩φ⟨

w0φ, F

0φ0⟩φ

w0P . (B.22)

The final formula for the response variation is given by Equation B.23:

∆R = ∂R

∂α

∣∣∣∣0

∆α+ wλ,aPP 0

⟨∂Pf∂α

∣∣∣∣0

∆α, φ0⟩φ

−⟨wλ,aT ,

∂N

∂α

∣∣∣∣∣0

∆α⟩T

−⟨wλ,aφ ,

∂M

∂α

∣∣∣∣∣0

∆αφ0⟩φ

, (B.23)

with the auxiliary condition that⟨wλ,aφ , F 0φ0

⟩φ

= 0.

186

Appendix C

Constrained Quantities inSensitivity and Uncertainty

Analysis

C.1 Constrained SensitivitiesIn sensitivity analysis we are interested in R (α) responses, which can dependboth explicitly on the α input parameters as well as implicitly through thesolution of the problem in question. As a result the sensitivities around aparticular α0 point in the phase-space of the inputs can be written as

SR,αj = dR (α)dαj

∣∣∣∣∣α0

= dR (α)dαj

∣∣∣∣∣0

= ∂R (α)∂αj

∣∣∣∣∣direct

α0

+ ∂R (α)∂αj

∣∣∣∣∣indirect

α0

, (C.1)

where the first expression represents the usual direct, the second the indirectterm accounting for the implicit dependence through the solution. When theinputs are unconstrained Equation C.1 directly gives the sought sensitivitiesof the response to the different input parameters.

When the input parameters are constrained, only such α values are allowedwhich satisfy the constraint of

Cαf (α) = C, (C.2)

187

C. Constrained Quantities in Sensitivity and Uncertainty Analysis

where Cαf (α) is a scalar valued function of the inputs (the constraint function,not to be confused with the Cf (x) constraint function used in Chapters 2-3).In such a case the absolute sensitivities of the response to parameter αi aredefined as

SCαfR,αj

= dR (α)dαj

∣∣∣∣∣Cαf

0=∑i

dR (α)dαi

∣∣∣∣0

∂αi∂αj

∣∣∣∣∣Cαf

0=∑i

dR (α)dαi

∣∣∣∣0SCαfαi,αj ,

whereas the relative sensitivities are given as

SCαfR,αj

=α0j

R0dR (α)

dαj

∣∣∣∣∣Cαf

0=α0j

R0

∑i

dR (α)dαi

∣∣∣∣0

∂αi∂αj

∣∣∣∣∣Cαf

0=α0j

R0

∑i

dR (α)dαi

∣∣∣∣0SCαfαi,αj ,

with the reference being R0 = R(α0). In the above formulas the dR (α)

dαi

∣∣∣∣0

derivatives on the right hand sides incorporate both the direct and indirectterms according to Equation C.1, and Cαf in the superscript of the derivativesindicates that they should be calculated with the constraint being taken into

account. The computation of the SCαfαi,αj = ∂αi

∂αj

∣∣∣∣∣Cαf

0sensitivity coefficients

is a principle issue when dealing with constrained inputs, since they con-tain the information about how the perturbation to parameter αj should becounterbalanced by parameters αi such that the constraint is still respected.

By differentiating Equation C.2 with respect to a parameter αj the followingequation is gained:∑

i

∂Cαf (α)∂αi

∣∣∣∣∣0SCfαi,αj = 0 j = 1, ..., N. (C.3)

Since Equation C.3 represents only N conditions regarding the N2 numberof[SC

αf

α,α

]j,i

= SCαfαi,αj input sensitivity coefficients, one can see that there are

many ways the latter can be derived. The approach used in this thesis is afull normalization of the input parameters, meaning that an α = α0 + ∆αset of input parameters not satisfying Equation C.2 is transformed into anα = α0 + ∆α set satisfying it according to α = κ (α) α. The normalizationconstant κ (α) is fixed by requiring that the

SCαfαi,αj = ∂αi

∂αj

∣∣∣∣∣Cαf

0= ∂αi∂αj

= lim∆αj→0

∆αi∆αj

(C.4)

188

C.2. Covariance Matrix of Linearly Constrained Quantities

coefficients satisfy Equation C.3, leading to the following input sensitivitycoefficients (Perkó et al., 2014a):

[SFNα,α

]j,i

= SFNαi,αj = lim∆αj→0

∆αi∆αj

= δi,j −

∂Cαf (α)∂αj

∣∣∣∣∣0∑

k

∂Cαf (α)∂αk

∣∣∣∣∣0α0k

α0i . (C.5)

Using the input sensitivity coefficients given by Equation C.5 the SFNR,αj andSFNR,αj constrained sensitivity coefficients of a response R are obtained as

SFNR,αj = SR,αj −

∂Cαf (α)∂αj

∣∣∣∣∣0∑

k

∂Cαf (α)∂αk

∣∣∣∣∣0α0k

∑i

SR,αiα0i (C.6)

and

SFNR,αj = SR,αj − α0

j

∂Cαf (α)∂αj

∣∣∣∣∣0∑

k

∂Cαf (α)∂αk

∣∣∣∣∣0α0k

∑i

SR,αi , (C.7)

where we used that 1R0SR,αiα

0i = SR,αi

C.2 Covariance Matrix of Linearly ConstrainedQuantities

In case of linearly constrained quantities, i.e. when

Cαf (α) =N∑j=1

αjWj = W Tα = C (C.8)

holds, the covariance matrix of the input parameters is singular. To see thislet us consider the inputs as a sum of their α0 = E (α) expected value and a∆α deviation from it as α = α0 + ∆α. Substituting into Equation C.8 yields

W T(α0 + ∆α

)= C,

189


from which after multiplying with ∆αT from the right and taking the expecta-tion value Equation C.9 is gained:

W TCα

= 0. (C.9)

In the above derivation we made use of the fact that E (∆α) = 0 by definition.Equation C.9 means that the weighted column (and row) sum of the covariancematrix of linearly constrained quantities is zero, in analogy with the fissionspectrum terminology this can be called a “generic zero column and row sum”condition (Yang et al., 2008).

Ideally the covariance matrix of any linearly constrained quantities wouldsatisfy the “generic zero column and row sum” condition, however this if oftennot so and can be caused by several factors. For example in case of the fissionspectrum the numerical accuracy of the used computer codes and the singleprecision number representation is known to produce inaccurate covariancematrices (Yang et al., 2008; Nagaya et al., 2009). In other situations only thevariances of the inputs are known together with the constraint they have tosatisfy, but not the proper covariances (Favorite et al., 2013). This is the case inSection 5.5.1 as well where only the uncertainties of the power peaking factorsare known but not their full covariance matrix. As a solution to the fissionspectrum problem the ENDF/B-6 Manual suggests a correction to be appliedto the covariance matrix to satisfy the “zero column and row sum” condition(ENDFB6 Format Manual). It is easy to generalize this procedure, one onlyhas to choose an input transformation procedure, i.e. the SC

αf

α,αinput sensitivity

matrix and remember that the ∆α admissible variation corresponding to anarbitrary ∆α variation around the α0 expectation value of the inputs is givenby ∆α =

(SC

αf

α,α

)T∆α (Equation C.4). As a result the “corrected”, or in other

words normalized covariance matrix of linearly constrained quantities can bewritten as

CCαf

α=(SC

αf

α,α

)TE(∆α∆αT

)SC

αf

α,α=(SC

αf

α,α

)TCαSC

αf

α,α. (C.10)

The normalization procedure given by Equation C.10 can also be usedto generate correlation data for constrained quantities for which only theirvariances are known together with the constraint. To do so one has to normalizethe covariance matrix according to Equation C.10, then set the diagonal valuesof the resulting matrix to the given variance values. The procedure can

190

C.3. Sensitivities with Polynomial Chaos Expansion

iteratively be repeated until the set accuracy is reached, i.e. until during thenormalization the elements of the matrix change with less than a predefinedtolerance. The covariance matrix obtained this way satisfies the “generic zerocolumn and row sum” condition, and has the desired variance values. InSection 5.5.1 this procedure is used to generate the covariance matrix for thepower peaking factors given by Equation 5.2.

C.3 Sensitivities with Polynomial ChaosExpansion

When applying PCE in a problem the first step is to express the α physical inputquantities as Polynomial Chaos Expansions of the ξ independent random vari-ables. Independent physical quantities can easily be represented by their PCE,since it involves only one random variable as αj

(ξ)

=∑k∈Pαj

[ak]j ψj,γk,j (ξj),where Pαj is the set of indices corresponding to the j-th input, ak is thecoefficient vector for the k-th basis vector, and ψj,γk,j (ξj) = Ψk

(ξ)is the

k-th basis vector (which is only uni-variate due to the special selection of k).For usual random variables (e.g. following a normal or uniform distribution)their expansion only contains 2 terms, Pαj = 0, j holds (supposing that the0-th basis vector is the constant term and the following N vectors are thefirst order uni-variate polynomials Ψj

(ξ)

= ψj,1 (ξj) , j = 1, ..., N) and the

expansion is simply αj(ξ)

= [a0]j +[aj

]jψj,1 (ξj), most often reducing to

αj(ξ)

= [a0]j +[aj

]jξj . In this case the derivatives with respect to ξj can

easily be converted into derivatives with respect to the αi physical quantitiesand the sensitivity coefficients can readily be obtained.

When the input variables are correlated or constrained special attentionis needed for their proper representation with PCE. Here we only discussthe case of linearly constrained Gaussian variables needed in Section 5.5.1for the power peaking factors. Since their derived CC

αf

αcovariance matrix is

singular, the usual technique used for correlated Gaussian variables based onthe CC

αf

α= ZTZ Cholesky decomposition of their covariance matrix cannot

be applied, since the decomposition does not exist. However their covariancematrix is symmetric and positive-semidefinite, hence it has an eigenvaluedecomposition of the form of CC

αf

α= UTTU , where T is a diagonal matrix

191


containing the eigenvalues, one (or more) of which is zero. Supposing that thezero eigenvalue is

[T]

1,1= 0 the matrix can be reconstructed as CC

αf

α= U

TU ,

where U ∈ R(N−1)×N and

[U]i,j

=N∑k=2

√[T]i+1,k

[U]k,j,

i.e. we delete the first row of U containing the eigenvector corresponding tothe zero eigenvalue. As a result the linearly constrained Gaussian parameterscan be represented as α

(ξ)

= α0 + UTξ, where ξ ∈ RN−1, meaning that the

N constrained quantities are only dependent on at most N − 1 independentvariables (due to the constraint).

Once the input parameters are properly represented by the independent ξrandom variables the PCE of the R

(ξ)output quantities can be built. Deriv-

ing the SR,ξj =dR

(ξ)

dξj

∣∣∣∣∣∣0

sensitivity coefficients is straightforward, however

when a ξj random variable is used to represent constrained physical variablesthe meaning of these coefficients is not evident. In such a case ξ are thegenerating variables for the α = α0 + U

Tξ constrained physical variables

always satisfying Equation C.2. Therefore ∆α = UTξ always represent ad-

missible perturbations, which correspond to arbitrary ∆α perturbations via∆α =

(SC

αf

α,α

)T∆α according to Equation C.4. The final relationship between

the arbitrary perturbations of the physical quantities and the generating vari-ables is therefore ξ =

(UT)−1

left

(SC

αf

α,α

)T∆α, where

(UT)−1

leftis the left inverse

of the non-square matrix UT . Finally, the change in the response due to anadmissible perturbation induced by a ∆αj arbitrary perturbation is given by

∆R =∑i

∂R(ξ)

∂ξiξi =

∑i

∂R(ξ)

∂ξi

∑l

[(UT)−1

left

]i,l

[(SC

αf

α,α

)T ]l,j

∆αj .

Therefore the constrained sensitivities are obtained as

SPCE,CαfR,αj

=∑i

∂R(ξ)

∂ξi

∑l

[(UT)−1

left

]i,l

[(SC

αf

α,α

)T ]l,j

(C.11)

192

C.3. Sensitivities with Polynomial Chaos Expansion

SPCE,CαfR,αj

=α0j

R0

∑i

∂R(ξ)

∂ξi

∑l

[(UT)−1

left

]i,l

[(SC

αf

α,α

)T ]l,j, (C.12)

or in vector formSPCE,CfR,α = SCf

α,α

[(UT)−1

left

]TSR,ξ

SPCE,CαfR,α = 1

R0α0SC

αf

α,α

[(UT)−1

left

]TSR,ξ.

193


194

Summary

This thesis presents novel adjoint and spectral methods for the sensitivityand uncertainty (S&U) analysis of multi-physics problems encountered inthe field of reactor physics. The first part focuses on the steady state ofreactors and extends the adjoint sensitivity analysis methods well establishedfor pure neutron transport problems to coupled criticality calculations, wherefeedbacks are present between neutronics and other phenomena (e.g. thermal-hydraulics or fission product poisoning). The second part presents novelspectral techniques, namely grid and basis adaptive Polynomial Chaos (PC)methods for the S&U analysis of generic problems, together with a large scaleapplication of the developed Fully Adaptive Non-Intrusive Spectral Projection(FANISP) algorithm for the sensitivity and uncertainty analysis of a transient.

Following a short introduction in Chapter 1 to some of the most frequentlyused S&U analysis techniques Chapter 2 presents the theory for the adjointbased sensitivity analysis of coupled criticality problems. This enables thecomputation of first order changes in responses of interest due to variations ofboth neutronic input parameters (such as cross sections) and those describingaugmenting phenomena (e.g. thermal-hydraulics). The chapter also presents avery efficient procedure for calculating the necessary neutronics and augmentingadjoint functions that relies on using Krylov algorithms together with theindividual neutron transport and augmenting codes to perform the requiredmatrix-vector multiplications and inversions during preconditioning. As a proofof principle study a one-dimensional slab model is investigated, where two-groupdiffusion theory is coupled with heat-conduction and xenon-poisoning.

In Chapter 3 the larger scale applicability of the coupled adjoint theory isstudied. A deeper look into the exact form of the adjoint operators reveals thatfor the most common cases of coupling neutron transport to thermal-hydraulicsand fission product poisoning the effects of the operators can easily be calculatedby routines present in the adjoint capable neutron transport and augmentingcodes. This enables their reuse with little code modifications, therefore themain challenge lies in the coupling scheme rather than in dedicated codedevelopment (once both codes are already suited for solving the individualadjoint problems). As a more realistic application the S&U analysis of acoupled model of an infinite array of fuel pins was performed employing apurpose made thermal-hydraulics code and a general purpose discrete ordinates

195

Summary

neutron transport solver. The results confirmed that the preconditioned Krylovalgorithm provides excellent performance in calculating the necessary adjointfunctions and these properly provide the first order changes of responses ofinterest due to perturbations in any of the system input parameters.

In Chapter 4 the development of novel adaptive Polynomial Chaos techniquesis detailed aimed at the S&U analysis of generic problems. Two types ofadaptivity is discussed: adaptive sparse grid algorithms relying on Gerstner’soriginal technique for calculating multidimensional integrals and adaptive PCbasis set construction for building up the Polynomial Chaos Expansion (PCE)of responses in an efficient way. The details of the implementation of thedeveloped methods in the FANISP algorithm are also presented and threedemonstrational problems are studied. They all focus on specific merits of theadaptive algorithms and confirm that they are significantly more effective thantraditional spectral techniques as well as brute force Monte Carlo methods.

As a truly large scale, realistic application of the FANISP algorithm thesensitivity and uncertainty analysis of an Unprotected Loss Of Flow transientin the GFR2400 Gas Cooled Fast Reactor is performed in Chapter 5. Fordealing with the 42 considered uncertain input parameters two further costreduction techniques are discussed based on a reduction of dimensionality andan adaptive increase of the global polynomial order. It is shown that due tothe non-intrusive nature of the developed PC methods they are easy to applyand can be far more efficient in determining sensitivities, uncertainties andeven full probability density functions than standard techniques. The mostimportant merit of the chapter is that it greatly expands the usefulness of PCmethods to problems with up to 40-50 input parameters, providing a lot ofnew opportunities for application.

In conclusion, the work presented in this thesis provides attractive methodsfor the sensitivity and uncertainty analysis of the two most common types ofmulti-physics calculations in the nuclear field, namely determining the coupledsteady state and the transient behaviour of reactors. The practical applicabilityof both the developed coupled adjoint method and the FANISP algorithmwas demonstrated, which serves as a strong basis for future use. Since adjointmethods are uniquely capable of taking into account all input parameters atonce, whereas PC techniques can deal with nonlinearities, their combined usehas even higher potentials. As a closure several possible future research topicsare highlighted among the recommendations.

196

Samenvatting

In dit proefschrift worden nieuwe geadjugeerde en spectrale methoden gep-resenteerd voor de gevoeligheids- en onzekerheidsanalyse (S&U-analyse) vangekoppelde fysicaproblemen in de reactorfysica. In het eerste gedeelte wordtde nadruk gelegd op reactoren in stationaire toestand en worden de gead-jugeerde gevoeligheidsanalysemethoden voor neutrontransportproblemen uit-gebreid naar gekoppelde kritieke massa berekeningen, waarbij terugkoppelingentussen neutronica en andere verschijnselen (bijvoorbeeld thermohydraulicaof splijtingsproductenvergiftiging) bestaat. In het tweede gedeelte wordennieuwe spectraaltechnieken gepresenteerd, namelijk rooster- en basisadap-tieve Polynoom Chaos (PC) methoden voor de S&U-analyse van algemeneproblemen, samen met een toepassing op grote schaal van het ontwikkeldeFANISP-algoritme (Fully Adaptive Non-Intrusive Spectral Projection) voor degevoeligheids- en onzekerheidsanalyse van een transiënt.

Na een korte introductie van enkele van de meest gebruikte S&U-analyse-technieken in Hoofdstuk 1, wordt in Hoofdstuk 2 de theorie van de geadjugeerdegevoeligheidsanalyse van gekoppelde kritieke massa problemen gepresenteerd.Dit maakt de berekening van de veranderingen van eerste orde in de responsies,door variaties in zowel nucleaire invoerparameters (zoals reactiedoorsneden)als ook in die versterkende fenomenen beschrijven (zoals thermohydraulica),mogelijk. Dit hoofdstuk beschrijft ook een zeer efficiënte procedure om de gead-jugeerde functies voor de neutronica en de augmenting codes uit te rekenen, dienodig zijn voor het gebruik van Krylov-algoritmes met losse neutrontransport-en augmentingcodes om de benodigde matrixvectorvermenigvuldigingen eninversies gedurende preconditioning uit te voeren. Als werkend voorbeeldwordt een vlakkeplaatmodel bestudeerd, waarbij de tweegroeps-diffusietheoriegekoppeld is aan warmtegeleiding en xenonvergiftiging.

In Hoofdstuk 3 wordt de toepasbaarheid van de gekoppelde geadjugeerdetheorie op grotere schaal bestudeerd. Voor de gevallen waarbij neutrontransportgekoppeld is aan thermohydraulica en splijtingsproductenvergiftiging, blijken deeffecten van de operatoren makkelijk berekend te kunnen worden door routinesdie aanwezig zijn in de geadjugeerde neutronentransport- en augmentingcodes.Dit maakt hergebruik mogelijk met weinig veranderingen aan de code; hetkoppelingsschema is daarom de belangrijkste uitdaging, in plaats van deontwikkeling van de toegepaste code (als beide codes eenmaal geschikt zijn

197

Samenvatting

om individuele geadjugeerde problemen op te lossen). Een S&U-analysevan een realistischere toepassing in de vorm van een oneindig rooster vanbrandstofpinnen is uitgevoerd, waarbij gebruik gemaakt is van een doelgerichtethermohydraulicacode en een breed toepasbare neutrontransportcode op basisvan discrete ordinaten. De resultaten bevestigen dat het ’preconditioned’Krylov-algoritme zeer efficiënt is met betrekking tot de benodigde geadjugeerdefuncties en dat deze functies de eerste-orde veranderingen in de responsies tengevolge van verstoringen in een van de systeemparameters juist bepalen.

In Hoofdstuk 4 wordt de ontwikkeling van nieuwe adaptieve Polynoom Chaostechnieken gedetailleerd beschreven met als doel de gevoeligheid en onzekerheidvan algemene problemen te analyseren. Twee soorten van adaptiviteit wordenbesproken: adaptieve ’sparse grid’-algoritmes, die afhankelijk zijn van Gerstnersoorspronkelijke techniek om multi-dimensionele integralen te berekenen enadaptieve PC-basissetconstructie voor het opbouwen van de Polynoom ChaosExpansie (PCE) van responsies op een efficiënte manier. De details van deimplementatie van de ontwikkelde methodes in het FANISP-algoritme wordenook besproken en drie demonstratieproblemen worden bestudeerd. Alle leggende nadruk op specifieke voordelen van de adaptieve algoritmes en bevestigendat ze significant effectiever zijn dan zowel traditionele spectraaltechnieken alsook Monte Carlo-methoden.

Voor een echt grootschalige en realistische toepassing van het FANISP-algoritme is in Hoofdstuk 5 een gevoeligheids- en onzekerheidsanalyse van een’Unprotected Loss Of Flow’-transiënt in de GFR2400 gasgekoelde snelle reactorgepresenteerd. Om te kunnen omgaan met de 42 beschouwde onzekerheids-invoerparameters, worden nog twee kostenreductietechnieken besproken, diegebaseerd zijn op dimensionaliteitsreductie en een adaptieve toename van deglobale polynoomorde.

Het is aangetoond dat deze technieken makkelijk toepasbaar zijn en datze, door de ’non-intrusive’ natuur van de ontwikkelde PC-methoden, veelefficiënter kunnen zijn in het bepalen van gevoeligheden, onzekerheden en zelfsvolledige kansdichtheidsfuncties dan standaardtechnieken. De belangrijksteverdienste van deze methode is dat het de bruikbaarheid van PC-techniekenvoor problemen met tot wel 40-50 invoerparameters sterk uitbreidt, hetgeenveel mogelijkheden biedt voor de toepassing daarvan.

Kort samengevat biedt het werk in dit proefschrift aantrekkelijke methodenvoor de gevoeligheids- en onzekerheidsanalyse van de twee meest voorkomende

198

typen gekoppelde fysicaberekeningen op nucleair gebied; zowel het bepalenvan de gekoppelde stationaire toestand als ook het tijdsafhankelijke gedragvan reactoren. De praktische toepasbaarheid van zowel de ontwikkelde gekop-pelde geadjugeerde methode als het FANISP-algoritme is aangetoond, hetgeeneen sterke basis vormt voor toekomstig gebruik. Omdat alleen geadjugeerdemethoden in staat zijn om met alle invoerparameters tegelijk rekening tehouden, terwijl PC-technieken overweg kunnen met niet-lineariteiten, heefthet gecombineerde gebruik van beide groete potentie. Als afsluiting worden inde aanbevelingen verscheidene mogelijke toekomstige onderzoeksonderwerpengesuggereerd.

(Dutch translation provided by Jurriaan Peeters, with corrections made byJan Leen Kloosterman, Danny Lathouwers and Aldo Hennink.)

199

Samenvatting

200

Acknowledgements

Without a doubt it was a great pleasure to work in the PNR/NERA groupin the last four and a half years, and there are many people to whom I amthankful for making my time at TU Delft truly memorable. First and foremostI need to thank my supervisor Jan Leen Kloosterman, who offered me theopportunity to do a PhD not only once, but also for a second time after myinitial decline. Jan Leen always supported me and gave me all the freedomI could have wanted to do my research as I see fit. I also have to thank mypromotor Tim van der Hagen, whose canny ability to see beyond the detailsand quickly pinpoint what is important was helpful many times. Last, butcertainly not least among the staff members, I would like to thank DannyLathouwers. Without his constant guidance, help, and his willingness to alwaysfind time for me to discuss whatever issue I faced, this thesis would certainlynot look the way it does now. It has been a pleasure to work with you Danny,and I am sure it will continue to be so in the future as well.

I also want to thank my fellow PhD students and postdocs, who created afriendly, funny and dynamic atmosphere in the office, and with whom I sharedmany great moments outside working hours as well. First and foremost Károlyand József, my fellow Hungarian gang members have to be mentioned. Theirhelp proved invaluable countless times, let it be some linear algebra problem,coding, or the whimsical nature of SCALE; and talking about Hungarianpolitics with them has always been highly enjoyable. I am also truly grateful toLuca, with whom I could discuss my topic down to the very last detail any time,and whose insight into (1,1,2,1) and (1,1,3,1) will remain unsurpassable. Ofcourse I can never forget Stuart, whom I harassed constantly with my questionson the proper use of English, and who was nevertheless always willing to help.I am also thankful to my two Dutch colleagues, Bart and Gert Jan, whoalthough live in the skies, still were fantastic friends for even such a smallEarthly creature as myself. To my fellow last warrior Frank, thank you for ourmany pleasant conversations and your occasional help. Special thanks go toWim for his kind invitation to Gent and the superb time we spent there, andI have to apologize for not becoming his colleague. To Jurriaan elvtárs I amespecially grateful for the wonderful concerts he invited me to. I also have tothank Edith, our adopted colleague for the many enjoyable cooking discussions,as well as for the many actual cakes she baked for us. Two postdocs cannot beleft out from this list: Christophe, whose sarcasm and dark mind are always a

201

Acknowledgements

bright point on any day, and Dimitrios, who, well, what to say. One just hasto meet Dimitrios to fully be able to appreciate the fun he is.

I also have to thank all my bachelor and master students. Thank youCharlotte. I immensely enjoyed our work together, and I am already missingour daily fruit hours with the delightful conversations we had. I also greatlyappreciate all the work Sebastian has done despite the little time he has spentin our group so far, and I am sorry I will not be able to personally assist himduring his whole project. To Mark and Dick, although I was not officially yoursupervisor, I am glad I could help you many times with your research. I amalso grateful to some of the other students who made life at our group pleasant,in random order thank you Hunor, Joran, Lodewijk, Sjoerd and Wilhelm. Thesame goes for Aldo, who was also kind enough to have a final look at the Dutchtranslation of my propositions and summary made by Jurriaan and havingbeen corrected by Jan Leen and Danny.

There are some other members of the PNR/NERA group who made mystay a lot of fun, our coffee breaks entertaining and our lunch conversationsinteresting. Martin, Peter, Dick, Hugo, Eduard, Norbert, you were alwaysgreat to have around, and for your occasional help I am grateful. I cannotthank enough our two secretaries, Ine and Thea. If it was not for you, I amsure I would have been buried by papers at some point in the last years. Andfor the NERA Next Generation, Valentina, Matteo, Denis and Sara, I am gladI got to know you, and I wish you the best of luck. I also have to thank someregulars of the Koepeltje, as well as some people outside the university, so bigthanks to Tekla, Ingrid, Kari, Attila, Kocka, Cseka, Fábi, Geri, Zsolti, Béla,Ákos, Nafi, Sanyi, Máté, Szabolcs, Léon, Mischa, Stefan, and certainly someothers I have forgotten to mention.

Last, but most importantly, I would like to thank my parents, my brother, andmy whole family. Kedves család! Ezúton is szeretném megköszönni mindazt aszeretet, gondoskodást és támogatást amit az elmúlt négy, sőt, lassan az elmúlt30 évben kaptam tőletek. Gyermek nem is kívánhatna magának törődőbbés nagylelkűbb szülőket, megértőbb és jobb fej tesót, valamint kedvesebbrokonságot. Nélkületek biztosan nem sikerült volna ilyen “jelentős” művetalkotnom.

202

List of Publications

Journal papers

Z. Perkó, S. Fehér and J.L. Kloosterman. Minor Actinide Transmutation inGFR600. Nuclear Technology, 177, pp. 83-97 (2012).

Z. Perkó, D. Lathouwers and J.L. Kloosterman. Adjoint-Based SensitivityAnalysis of Coupled Criticality Problems. Nuclear Science and Engineering,173, pp. 118-138 (2013).

Z. Perkó, L. Gilli, D. Lathouwers and J.L. Kloosterman. Grid and Basis Ad-aptive Polynomial Chaos Techniques for Sensitivity and Uncertainty Analysis.Journal of Computational Physics, 260, pp. 54-84 (2014).

Z. Perkó, D. Lathouwers, J.L. Kloosterman and T.H.J.J. van der Hagen. LargeScale Applicability of a Fully Adaptive Non-Intrusive Spectral Projection Tech-nique: Sensitivity and Uncertainty Analysis of a Transient. Annals of NuclearTechnology, 74, pp. 272-292 (2014).

Z. Perkó, S. Pelloni, K. Mikityuk, et al. Core Neutronics Characterization ofthe GFR2400 Gas Cooled Fast Reactor. Progress in Nuclear Energy (2014),http://dx.doi.org/10.1016/j.pnucene.2014.09.016

Z. Perkó, D. Lathouwers, J.L. Kloosterman and T.H.J.J van der Hagen. Con-strained Quantities in Sensitivity and Uncertainty Analysis. In press in NuclearScience and Engineering, (2014).

Conference papers

Z. Perkó, S. Fehér, J.L. Kloosterman and S.A. Christie. Recycling of VVERminor actinides in a Gas-Cooled Fast Reactor. Proceedings of InternationalConference on the Physics of Reactors, PHYSOR 2010, pp. 2003-2015, Pitts-burgh, Ohio, US (2010).

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Z. Perkó, D. Lathouwers and J.L. Kloosterman. Sensitivity Analysis of CoupledCriticality Calculations. Proceedings of International Conference on the Phys-ics of Reactors, PHYSOR 2012, pp. 3067-3080, Knoxville, Tennessee, US(2012).

Z. Perkó, D. Lathouwers, J.L. Kloosterman and T.H.J.J. van der Hagen. Ad-aptive Polynomial Chaos Techniques for Uncertainty Quantification of a GasCooled Fast Reactor Transient. Procedings of International Conference onMathematics and Computational Methods Applied to Nuclear Science and En-gineering, M&C 2013, pp. 1084-1095, Sun Valley, Idaho, US (2013).

K. Mikityuk, Z. Perkó, G. Girardin. Reactivity effect of steam/water ingressin Generation IV Gas-Cooled Fast Reactor Core. Proceedings of InternationalConference on Fast Reactors and Related Fuel Cycles: Safe Technologies andSustainable Scenarios, FR13, Paris, France (2013).

Z. Perkó, D. Lathouwers, J.L. Kloosterman and T.H.J.J. van der Hagen. LargeScale Applicability of a Fully Adaptive Non-Intrusive Spectral ProjectionTechnique: S&U Analysis of a Transient. Technical Track Proceedings ofInternational Youth Nuclear Congress, IYNC 2014, Burgos, Spain (2014)

Z. Perkó, D. Lathouwers, J.L. Kloosterman and T.H.J.J. van der Hagen. Con-strained Quantities in Uncertainty Quantification: Ambiguity and Tips toFollow. Proceedings of International Conference on the Physics of Reactors,PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future,Kyoto, Japan (2014).

Project reports

Z. Perkó, J.L. Kloosterman, R. Stainsby, Student Workshop on Gas CooledFast Reactors - A training course. Deliverable D6.2, GoFastR project, October(2011).

Z. Perkó, J.L. Kloosterman, C. Poette, et al. Contribution to the Final Reporton GFR Core Performance & Uncertainties. Deliverable D1.1-15, GoFastR

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project, September (2012).

Z. Perkó, J.L. Kloosterman, C. Poette, et al. GFR2400 Core Neutronics andThermal-hydraulics Characterization. Deliverable D1.1-11, GoFastR project,March (2012).

Z. Perkó, K. Peers, R. Stainsby, et al. GFR Severe Accident Analysis. Deliver-able D1.3-8, GoFastR project, February (2013).

K. Peers, R. Stainsby, Z. Perkó, et al. GFR Severe Accident Model Develop-ment. Deliverable D1.3-4, GoFastR project, January (2013).

Z. Perkó, J.L. Kloosterman. Contribution of TU Delft to Deliverable D6.1.5-1- Sensitivity and uncertainty analysis of transient/accident scenarios in AL-LEGRO using novel adaptive Polynomial Chaos techniques. Contribution toDeliverable D6.1.5-1, ESNII+ project, October (2014).

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Curriculum Vitae

Zoltán Perkó was born in Keszthely, Hungary on the 4th of July 1985. Hegrew up in his home city of Keszthely where he first attended the CsokonaiVitéz Mihály Általános Művelődési Központ (primary school), then the specialmathematics class of the Vajda János Gimnázium (secondary school). Aftergraduating with a special Vajda Plaquet from secondary school he enrolled inthe engineering-physics five years masters program at the Budapest Universityof Technology of Economics in 2004. During his thesis work he investigated thefuel cycle and transmutational capabilities of Gas Cooled Fast Reactors andhad the opportunity to spend 4 months as a guest researcher at the Physics ofNuclear Reactors (PNR) section of the Department of Radiation, Radionuclides& Reactors (R3) of the Delft University of Technology in the Netherlands. Hedefended his thesis and received his masters’ degree in 2010.

The 4 months he spent in Delft in 2009 turned out to be a cornerstone in hislife, since as a result of his research he was offered a PhD position at Delft. InJuly 2010 he re-joined the PNR section of the Delft University of Technologyas a PhD researcher, working on Gas Cooled Fast Reactors and sensitivityand uncertainty analysis methods within the framework of the FP7 EuropeanGoFastR project. The results of his 4 years of research at PNR - currentlyknown as the Nuclear Energy and Radiation Applications (NERA) section ofthe Department of Radiation Science and Technology (RST) - are summarizedin the current thesis.

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Curriculum Vitae

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Sensitivity and Uncertainty Analysis of Coupled Reactor Physics … · 2017-09-14 · Sensitivity...

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