Table Of ContentDeterministic Numerical Methods
for Unstructured-Mesh Neutron
Transport Calculation
Woodhead Publishing Series in Energy
Deterministic Numerical
Methods for
Unstructured-Mesh
Neutron Transport
Calculation
Edited by
Liangzhi Cao
Professor, School of Nuclear Science and Technology, Xi’an
Jiaotong University, Xi’an, People’s Republic of China
Hongchun Wu
Professor, School of Nuclear Science and Technology, Xi’an
Jiaotong University, Xi’an, People’s Republic of China
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Contributors
Liangzhi Cao Xi’an Jiaotong University,Xi’an, People’s Republic ofChina
Chao Fang Xi’an Jiaotong University,Xi’an, People’s Republic ofChina
Qingming He Xi’an Jiaotong University,Xi’an, People’s Republic of China
Yunzhao Li Xi’an Jiaotong University,Xi’an,People’s Republicof China
Zhouyu Liu Xi’an Jiaotong University,Xi’an, People’s Republic of China
Hongchun Wu Xi’an Jiaotong University, Xi’an, People’s Republic ofChina
HaochunZhangHarbinInstituteofTechnology,Harbin,People’sRepublicofChina
Yining Zhang Harbin Instituteof Technology, Harbin, People’s Republic of China
Youqi Zheng Xi’an Jiaotong University,Xi’an, People’s Republic of China
Foreword
The design, analysis, optimization, and licensing of nuclear reactor plants require
accuratecomputationaltoolsforpredictingthecomplexinteractionsofneutronswith
nuclearfuelandstructuralmaterialswithinreactorcores.Inrecentyears,theincreas-
ingcomplexityofnuclearfueldesignsandthedevelopmentofadvancedreactorcon-
ceptshavecombinedtochallengethecapabilitiesoftraditionalanalysistools.Across
theglobe,researchersarepushingthefrontiersinsolvingthefundamentalBoltzmann
neutron transport equation (NTE) by utilizing massively parallel computers and
advanced methods that minimize approximations in geometrical representations
and energy resolution. While stochastic (Monte Carlo) methods are systematically
maturingintheirabilitytoproducehigh-fidelitysolutionstofull-corereactormodels
forpseudo-steady-stateproblems,theintractabilityofMonteCarloforsolvingtime-
dependent reactor neutron transport problems has left the multigroup deterministic
methodsasthemostpromisingoftechniquesforaccuratelysolvingthebroadestclass
of challenging reactor neutron transportproblems.
Deterministic methods for solving the reactor NTE have been in existence for at
least50years,anddozensoffundamentallydifferentcomputationalapproacheshave
been developed during this time. Until very recently, all reactor applications relied
onlowerdimensionalitymodels(e.g.,1-Dand2-D),compromisedgeometricalrep-
resentations (e.g., Cartesian meshes), and low-resolution angular representations
(e.g.,diffusiontheory).Today,researchersstrivetosolvedirectlytheheterogeneous,
three-dimensional,neutrontransportequationforprecisegeometricalrepresentations
ofnuclearreactorcores.Asaconsequence,eachofthetraditionalapproachesforsolv-
ingtheNTEhasbeenreexaminedforitssuitabilityforextensiontotheseextremely
challengingreactorproblems.Studentsofnuclearengineeringoftenstruggletounder-
standtheimportant differencesbetweenthesemethods,andstudentsmustindividu-
allyassemblenumerousproceedingsoftopicalconferencesandvarioustextbooksin
ordertostudythemyriadoftechniquesthathavebeendevelopedtodate—beforethey
can commence development or extension of modern tools for solving the most
challenging of reactor neutron transportproblems.
OneofthelargereffortstodevelopnewmethodsandtoolsforsolvingtheNTEhas
occurred in the universities of the Peoples Republic of China. In particular, the
NuclearEngineeringComputationalPhysicsLaboratoryatXi’anJiaotongUniversity,
underProf.LiangzhiCaoandProf.HongchunWu,hasproducedasuccessionofstu-
dentsthathavesystematicallystudiedanddevelopedcodes/toolsforeachofthetra-
ditional methods of solving the NTE—with a particular focus on general geometry
extensionsandsolutionsonmoderncomputationalplatforms.Profs.CaoandWuhave
coalesced their student contributions to produce this comprehensive textbook of
x Foreword
techniques for solving the NTE. Students will find this textbook particularly useful
because it compiles, into a single document, the wealth of techniques for solving
the NTE—savingstudentsatremendousamountofeffort inacquiring the materials
needed tounderstand the differences andapplicabilityofthesediverse techniques.
This textbook systematically progresses through many of the classic solution
methods in a sequence of chapters covering: Collision Probabilities, Transmission
Probabilities, Current-Coupled Collision Probabilities, Method of Characteristics,
SphericalHarmonics/FiniteElements,DiscreteOrdinates,Mesh-FreeDiffusionThe-
ory, as well as an introduction to Wavelets, Variational Nodal, and Hybrid Monte
Carlo/MOC methods.Ineach chapter,systematicmathematicalderivationsandfor-
mulationsarepresentedforeachmethod,andnumerousnumericalexamplesareused
toprovidethereaderwithanunderstandingoftherequireddiscretization/accuracyof
eachmethod.Manyoftheexamplesareforcomplexgeometricalrepresentationson
unstructuredmeshes—whichare seldom treated inothertextbooks.
Students and nuclear professionals will find this textbook to be a valuable asset
when starting research into modern methods for solving the NTE and their applica-
tions for the analysis of complex nuclear fuels and reactor designs that are rapidly
evolving inthe world of advanced nuclear reactor engineering.
KordSmith
ProfessorofthePracticeofNuclearScienceandEngineering,MassachusettsInstitute
of Technology,Cambridge, MA, UnitedStates
Preface
Neutrontransportequation,derivedfromtheBoltzmannequation,isthebasicequa-
tiontodescribeneutronbehaviorsduringtheirtransportindifferentmedia.Numerical
solutiontotheneutrontransportequationisofgreatimportanceinthenuclearareas
thusattractingmoreandmoreattentionsinthecommunityofnuclearengineeringas
well as nuclear technology applications. During the past decades, many effective
numericalmethodshavebeenproposedanddevelopedtosolvetheneutrontransport
equation.Theycanbeclassifiedintotwocategories:MonteCarlomethodsanddeter-
ministicmethods.BecausethebasicideaofMonteCarlomethodsistotallythenum-
ber ofparticlesbytrackingtheneutron inthe realgeometricstructures,itoffersthe
MonteCarlomethods highgeometricadaptability bynature, although thecomputa-
tional cost is high. For the deterministic methods, however, the angular and spatial
variablesarealwaysdiscretizedbymeshes,sothegeometricadaptabilityrelieshighly
onthediscretizationmethod.Traditionaldeterministicmethodsaremainlydeveloped
basedonthestructuredmeshes,whichmeanstheycanonlybeusedforregulargeo-
metric problems, or the real engineering problems have to be converted to regular
geometrywithapproximations.Inordertoreducethoseapproximationsandimprove
the accuracy of the solution, deterministic methods should be developed based on
unstructured meshes.
Inrecentyears,thedemand forhigh-fidelitynumericalsolutiontoneutrontrans-
portequationisincreasingduetotherapiddevelopmentoftheadvancednuclearreac-
torconceptsaswellasthehigh-performancecomputationaltechnologies.However,
eventhoughtherearemanyjournalandconferencepapersdiscussingtherecentpro-
gressesinthedeterministicneutrontransportcalculationmethods,itisstillverydif-
ficulttofindacomprehensivebookfocusingonthistopicuptonow.Thereareafew
booksdiscussingtheFEMsolutionstoNTEandsomeotherbooksontraditionalSN
andPNmethods,butnotfocusingontheunstructuredmeshes.Actually,themethodof
characteristic (MOC), for example, has been well developed for more than 20years
and has been widely employed in the industrial codes like CASMO5, APOLLO3,
etc. Even for the traditional collision probability method (CPM) and transmission
probabilitymethod(TPM),recentstudyhasextendedthemtotriangularunstructured
meshes.Thoseprogresseshavenotyetbeenwellsummarized.So,theprimarymoti-
vationofthisbookistoputtogetherthosenewlydevelopedmethodsforthesolutionto
NTE with unstructured meshes.
The Nuclear Engineering Computational Physics (NECP) Lab at Xi’an Jiaotong
University,createdandledbytheauthorssince2004,hasbeenfocusingonthedevel-
opmentofdeterministicnumericalmethodsforunstructured-meshneutrontransport
calculation from the very beginning of its establishment. Over the past 16years,
dozens of graduate students have worked on this topic by developing different
xii Preface
methodsandtestingthemwithnumericalsimulation.Manyin-housecodeshavebeen
writtentoverifythosemethodsagainstsomebenchmarkproblems.Amongthem,we
selected some relatively mature ones to composite thisbook.
Thefirstchapterisdevotedtothederivationofthetransportequationanditsdif-
ferent forms including first-order and second-order differential forms and integral
forms.TheadjointneutrontransportequationisalsointroducedinChapter1.Then,
inChapters2–4,sometraditionalmethods,CPM,TPM,andCCCP,basedonintegral
transport equation will be expanded into unstructured meshes. Chapter 5 introduces
the most widely used MOC method for complex geometry problems. Chapters 6
and 7 are devoted to two classical deterministic methods, PN and SN method, with
unstructuredmeshesmainlycoupledwithFEM.InChapter8,themesh-freemethod
whichhasbeenappliedtoneutrondiffusionandtransportcalculationinrecentyears
will be briefly introduced. Finally, Chapter 9 includes some other nonclassical
methodswhicharestillunderdevelopment,suchasthewaveletmethod,variational
nodal method, and the Monte Carlo deterministic hybrid method. In most of the
abovementionedmethods,thetheoreticalmodelwillbederivedfirstandsomenumer-
icalresults will follow.
Theauthors wishtoexpresstheirappreciation tomanycontributorstothisbook.
Firstofall,theyacknowledgeformerandcurrentgraduatestudentsofNECPlabwho
developed those methods and codes and contributed greatly to the contents of this
book. They are Haoliang Lu (Chapter 7), Pingping Liu (Chapter 3), Guoming Liu
(Chapter3),QichangChen(Chapter5),QiZheng(Chapter9).Theauthorsalsothank
Dr.WeiShenatCANDUOwnerGroupwhoofferedthoughtfulsuggestionsandcom-
mentstothemanuscript.Duringthepreparationofthisbook,manycurrentgraduate
studentsprovidedalotofhelpstothem.TheyareChaoFang,QiZheng,YifanZhang,
JianxinMiao,XiaoyangZou,etc.Atlast,butnottheleast,theauthorsthankMs.Maria
Convey,Ms.MichelleFisher,andothercolleaguesfromElsevierfortheirkindsup-
port and constant help tomake this bookhappen.
Liangzhi Cao
Hongchun Wu
Xi’an Jiaotong University,Xi’an, People’s Republic ofChina
1
Neutron transport equation
LiangzhiCao
Xi’an JiaotongUniversity, Xi’an, People’sRepublic ofChina
1.1 Introduction
Neutronsplayaveryimportantroleinmanynuclearfacilities,forexample,nuclear
fissionandfusionreactors.Thetheorytodescribeandsimulatethetransportbehavior
ofneutrons iscalledneutrontransporttheory.Therootsoftransporttheory goback
morethanacenturytotheBoltzmannequation,firstformulatedforthestudyofthe
kinetictheoryofgases[1].Withtherapiddevelopmentofhigh-performancecomputer
technology,thenumericalsimulationofneutrontransportbehaviorisbecomingmore
and more important inengineering design andanalysis.
Inrecentyears,thedemand forhigh-fidelitynumericalsolutiontoneutrontrans-
port equation (NTE) has increased due to the rapid development of the advanced
nuclear reactor concepts as well as the high-performance computater technologies
[2, 3].
This chapter introduces the basic concept of neutron transport equation and its
alternativeforms.Somefundamentalsforthenumericalsolutionofneutrontransport
calculation are also briefly introduced inthis chapter.
1.2 Definition of the ordinates and basic elements
In the transport theory, neutrons are considered as point particles, which means the
neutron motion state can be represented by the determined position and velocity.
As shown in Fig. 1.1, the position of neutrons in space can be expressed by r. The
velocityvectorνof the particlesis written interms of itssolid angleas
ν¼υΩ (1.1)
where υ¼jνj is the magnitude of velocity, and its relation to the kinetic energy of
neutrons E is E¼mυ2/2, where m is the mass of neutrons and Ω is the unit vector
ofthedirectionofmotion.Itsmodulusisequalto1.Itsdirectionisexpressedbypolar
coordinate system through polar angle θ and azimuth angleφ.
Therefore,atanymoment,thestateofneutronmotioncanbedescribedbysixinde-
pendentvariablessuchaspositionvectorr(x,y,z),energyE anddirection ofmotion
,
Ω(θ, φ).Fordifferent coordinate systems, the expressions ofr andΩ are different.
DeterministicNumericalMethodsforUnstructured-MeshNeutronTransportCalculation.
https://doi.org/10.1016/B978-0-12-818221-5.00003-9
Copyright©2021ElsevierLtd.Allrightsreserved.
