multi-physics and multi-scale methods used in nuclear reactor analysis

Annals of Nuclear Energy 72 (2014) 104–119

Contents lists available at ScienceDirect

Annals of Nuclear Energy

journal homepage: www.elsevier .com/locate /anucene

Review

Multi-physics and multi-scale methods used in nuclear reactor analysis

http://dx.doi.org/10.1016/j.anucene.2014.05.0020306-4549/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: National Centre for Scientific Research ‘‘Demokritos’’, Institute of Nuclear & Radiological Sciences & Technology, Energy & SafetyResearch Reactor Laboratory, 15310 Aghia Paraskevi, Attiki, Greece. Tel.: +30 210 6503725; fax: +30 210 6503766.

E-mail address: [email protected] (A.G. Mylonakis).

A.G. Mylonakis a,b,⇑, M. Varvayanni a, N. Catsaros a, P. Savva a, D.G.E. Grigoriadis c

a National Centre for Scientific Research ‘‘Demokritos’’, Institute of Nuclear & Radiological Sciences & Technology, Energy & Safety, Nuclear Research Reactor Laboratory, 15310 AghiaParaskevi, Attiki, Greeceb Aristotle University of Thessaloniki, Faculty of Engineering, School of Electrical and Computer Engineering, Nuclear Technology Laboratory, 54124 Thessaloniki, Greecec University of Cyprus, Department of Mechanical and Manufacturing Engineering, Computational Science Laboratory UCY-CompSci, 75 Kallipoleos str., 1678 Nicosia, Cyprus

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 November 2013Received in revised form 30 April 2014Accepted 2 May 2014

Keywords:Neutronic/thermal–hydraulic couplingOperator-Splitting in reactor analysisJacobian-Free Newton Krylov methodPseudo-materials methodPin-power reconstructionMulti-physics explicit/implicit coupling

In an operating nuclear reactor core, various physical phenomena of different nature are interrelated.Multi-physics calculations that account for the interrelated nature of the neutronic and thermal–hydraulicphenomena are of major importance in reactor safety and design and as a result a special effort is devel-oped within the nuclear engineering scientific community to improve their efficiency and accuracy. Inaddition, the strongly heterogeneous nature of reactor cores involves phenomena of different scales.The interaction between different scales is a specificity of these systems, since a local perturbation mightinfluence the behavior of the whole core, or a global perturbation can influence the properties of the mediaon all scales. As a consequence, multi-scale calculations are required in order to take the reactor coremulti-scale nature into account. It should be mentioned that the multi-physics nature of a nuclear reactorcannot be separated from the multi-scale one in the framework of computational nuclear engineering asreactor design and safety require computational tools which are able to examine globally the complicatednature of a nuclear reactor in various scales. In this work a global overview of the current status of two-physics (neutronic/thermal–hydraulic) and multi-scale neutronic calculations techniques is presentedwith reference to their applications in different nuclear reactor concepts. Finally an effort to extract themain remaining challenges in the field of multi-physics and multi-scale calculations is made.

� 2014 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052. Neutronic/thermal–hydraulic coupling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.1. Operator Splitting (OS) methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.2. Jacobian-Free Newton Krylov methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.2.1. Newton iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.2.2. Krylov methods – GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.2.3. A Krylov solver (GMRES) within the framework of JFNK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.2.4. Preconditioning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.2.5. Globalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

2.3. An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.4. Current status of research in coupling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102.5. Cross-section treatment in neutronic/T–H calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3. Multi-scale calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.1. Current status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4. Reactor modeling using coupled neutronic/T–H calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.1. Light Water Reactors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2. Sodium Fast Reactor (SFR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3. Molten Salt Reactors (MSR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
, Nuclear

http://crossmark.crossref.org/dialog/?doi=10.1016/j.anucene.2014.05.002&domain=pdf

http://dx.doi.org/10.1016/j.anucene.2014.05.002

mailto:[email protected]

http://dx.doi.org/10.1016/j.anucene.2014.05.002

http://www.sciencedirect.com/science/journal/03064549

http://www.elsevier.com/locate/anucene

A.G. Mylonakis et al. / Annals of Nuclear Energy 72 (2014) 104–119 105

4.4. Lead cooled Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5. Conclusions – Challenges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

1. Introduction

Nuclear reactors are strongly heterogeneous systems involving,during their operation, various physical phenomena at differentscales. The reactor core is the part of the system where this heter-ogeneity is best illustrated because in the case of a Light WaterReactor (LWR) for instance, it is composed by a great number ofFuel Assemblies (FAs). Furthermore each of these FAs consists offuel pins which contain the nuclear fuel, possibly at various levelsof enrichment. The interaction between different scales is a speci-ficity of these systems, since a local perturbation might influencethe behavior of the whole plant, or a global perturbation maychange the properties of the media on all scales. This special fea-ture induces the need for multi-scale reactor core analysis or anal-ysis which takes phenomena of different scale and their interactioninto account. As has been mentioned above, in a nuclear reactorcore various physical phenomena of different nature are interre-lated, i.e. coupled to each other. For example, neutronic and Ther-mal–Hydraulic (T–H) phenomena are strongly bonded; e.g.microscopic neutron cross-sections of the core materials aredependent on temperature. This interrelation induces the needfor multi-physics calculations, i.e. calculations which take theseinteractions into consideration. Consequently multi-physics calcu-lations capable of considering both the neutronic and T–H phe-nomena are of special interest. The need for sophisticated codescapable of performing both multi-physics and multi-scale analysisbecomes of higher importance for the analysis of the GENIV reac-tors, proposed in the Generation IV International Forum, whichare currently under investigation, since they are characterized byspecial features and high complexity.

The purpose of this work is to summarize the most indicativecoupled neutronic/T–H, multi-scale schemes used in nuclear reac-tor analysis calculations and the main perspectives in this field.The main motivation of this work is that a strong interest forimproving the efficiency and accuracy of multi-physics/multi-scalesimulations exists and lies on the fact that safety analysis shouldmeet more and more strict limits. Furthermore, the continuouslyincreasing computer power in combination with the new moreadvanced numerical methods allows the performance of moredetailed and hence more ‘‘burdensome’’ calculations. In this pointit should be mentioned that the presentation of details concerningthe solvers and techniques in the neutronic calculation on the oneside, and the T–H calculation on the other side, is out of the scopeof this work. For this kind of information, a very interesting paperwritten by Demazière (2013), is highly recommended to be read. Inaddition several references for better understanding of the meth-odologies which are followed by the neutronic and the T–H solversare given in this paper.

After the introduction, the second chapter summarizes some ofthe techniques used in coupling a neutronic and a T–H code. Thethird one presents the current status of multi-scale neutronic cal-culations and the perspectives in this field. The fourth chapter triesto summarize the most indicative neutronic/T–H tools which areimplemented in different reactor concepts. Finally in the conclu-sion part an attempt is made to define the main remaining chal-lenges in the field of multi-physics/multi-scale neutronic/T–Hcalculations.

2. Neutronic/thermal–hydraulic coupling techniques

For decades, reactor analysis was divided into several distinctdomains, such as neutronic and T–H. However as the solution ofone problem is strongly intertwined on the solution of the other,various existing codes have been coupled together in order to takethese interactions into consideration and provide more accuratesolutions of the global reactor analysis problem.

Traditional multi-physics neutronic/T–H tools for reactor analy-sis problems employ extensively validated and verified efficientmono-disciplinary codes coupled together where the output ofthe one code serves as the input of the other. This technique canbe described mathematically by the Operator Splitting (OS) meth-odology. According to OS, the action of the governing equations onthe variables is decomposed into a separate, uncoupled physicaldescription for each part, leading to an inconsistent treatment ofthe nonlinear terms. This technique of coupling standard industrialcodes has the main advantage of avoiding man years of develop-ment and testing. However it is characterized by some importantdrawbacks which are discussed later. Apart from OS techniquesthere are modern multi-physics algorithms like Jacobian-FreeNewton Krylov (JFNK) methods, discussed later in this document,which can be numerically more efficient and accurate than theOS techniques.

A general, highly simplified but illustrative multi-physicsnuclear engineering problem consists of the time-dependent heatconduction equation (Eq. (1)) coupled to a one group diffusionmodel (Eq. (2)) of neutron kinetics, ignoring delayed neutrons(Gaston et al., 2009). In a Cartesian frame of reference, this problemis described by the following equations:

qCp@T@t�rðKðTÞrTÞ ¼ wRf ðTÞu ð1Þ

1t@u@t�rðDðTÞruÞ þ RaðTÞu ¼ mRf ðTÞu ð2Þ

where q is the material density, CP is the material heat capacity, T isthe material temperature, K is the material thermal conductivity, wthe ‘‘useful’’ energy released per fission, Rf the macroscopic fissioncross-section, Ra the macroscopic absorption cross-section, u thescalar neutron flux, t the neutron velocity, D the neutron diffusioncoefficient and m is the number of neutrons produced per fissionevent. In this model D, K, Rf and Ra are all functions of T. In thishighly simplified homogeneous multi-physics problem, nuclear fis-sion generates heat, which is transferred by conduction. This simplemodel problem, used here for illustrative purposes, is a nonlinearPartial Differential Equation (PDE) system with multiple time andspace scales. For example, the time scale of heat conductive trans-mission lies within a range of 1–10 s. The neutron flux time scaleis in the order of 1 ls since the delayed neutrons are ignored in thissimplified scheme. When coupling existing codes in an OS method-ology, an explicit dependence of variables coming from the othercode is required. Such a first-order in time discretization schemeis the following (Gaston et al., 2009):

qCpTnþ1 � Tn

Dt�rðKðTnÞrTnþ1Þ ¼ wRf ðTnÞun ð3Þ

1t

unþ1 �un

Dt�rDðTnÞunþ1Þ þ RaðTnÞunþ1 ¼ mRf ðTnÞunþ1 ð4Þ

106 A.G. Mylonakis et al. / Annals of Nuclear Energy 72 (2014) 104–119

where the exponents denote time level. On the other hand, when afully implicit first-order in time discretization is used the problemtakes the following form:

qCpTnþ1 � Tn

Dt�rðKðTnþ1ÞrTnþ1Þ ¼wRf ðTnþ1Þunþ1 ð5Þ

1t

unþ1 �un

Dt�rDðTnþ1Þunþ1Þ þRaðTnþ1Þunþ1 ¼ mRf ðTnþ1Þunþ1 ð6Þ

It should be noted that this scheme (Eqs. (5) and (6)) cannot be usedwhen serial code coupling (OS) has been adopted, simply because D,K, Rf and Ra have been evaluated at the previous time step. Theweakness of treating implicitly the involved equations is an impor-tant constraint of the OS methodology. At this point it should bementioned that the inherent time step size limitation and potentialerror of the explicit time integration technique is the primary impe-tus for including fully implicit time integration methods in reactoranalysis codes. On the other hand, a major obstacle impeding thewidespread implementation of fully implicit methods is the ineffi-ciencies associated with solving all of the nonlinear equation setssimultaneously.

2.1. Operator Splitting (OS) methods

An OS technique is actually a tool employed to solve numeri-cally PDEs which describe physical phenomena of different nature.OS techniques are generally used in methods where the differentialoperator is split into several parts, each part representing a partic-ular physical phenomenon, such as convection, diffusion, etc. Thecorresponding numerical method is defined as a sequence of solu-tions of each of the split problems. In most OS techniques, an expli-cit coupling of the physics codes is performed by exchanging thesolution from each physics’ field at every time step as only bound-ary condition to the other physics. This can lead to very efficientmethods, since one can treat each part of the original operatorindependently. In addition, as mono-disciplinary codes, alreadydeveloped and validated, can be employed to solve each field sep-arately, only slight modifications are required in the employedcodes and as a result, the effort which is required is significantlylower. Such methods have been proved to be first-order accuratein time, requiring considerably smaller time steps to obtain anaccurate solution. The general scheme of this approach can be for-mulated as following (Farago, 2008):

1. A small time step (s) is selected which divides the whole timeinterval into subintervals of equal duration.

2. On each subinterval the time-dependent problems are consecu-tively solved, each of which involves only one physical process.

3. The next time subinterval then follows.

It should be mentioned that the distinct problems involved inthe multi-physics calculation are connected via the initial

Fig. 1. A general staggered OS scheme used in a

conditions. As the purpose of this work is not to analyze OS meth-ods in depth, it will be assumed that there are only two operators,i.e., a Cauchy problem of the following form will be used:

dwðtÞdt¼ ðAþ BÞwðtÞ; t 2 ð0; TÞ ð7Þ

wð0Þ ¼ w0 ð8Þ

The sequential splitting, which is actually the mathematicalrepresentation of mono-disciplinary codes coupling, is a very sim-ple algorithm: The problem is firstly solved for the first operatorand the original initial condition, and next for the second operator,using the solution of the first problem at the end-point of the time-subinterval as initial condition. The algorithm is the following(Farago, 2008):

dwn1

dtðtÞ ¼ Awn

1ðtÞ ð9Þ

wn1ððn� 1ÞsÞ ¼ wN

spððn� 1ÞsÞ ð10Þdwn

2

dtðtÞ ¼ Bwn

2ðtÞ ð11Þ

wn2ððn� 1ÞsÞ ¼ wn

1ðnsÞ ð12Þ

for n = 1, 2, . . .,N. Then the split solution at the mesh-points t = ns isdefined as wN

spðnsÞ ¼ wn2ðnsÞ. Here wN

spð0Þ ¼ w0 is given from the ini-tial condition. An important drawback of these conventional cou-pling techniques is that they employ inconsistent treatment of thenonlinear terms due to the explicit treatment of the coupling termsas described below. An OS technique employed in solving a neu-tronic/T–H problem could be illustrated by the following sketch(Fig. 1). This type of coupling is usually referred as ‘‘weak’’ due tothe fact that the solution of the one mono-disciplinary codebecomes the input of the other one without converging the nonlin-ear terms over the time step, inducing inaccuracy (Gan et al., 2003).In order to eliminate this inaccuracy, either smaller time steps areused increasing the computational cost, or ‘‘improved’’ lineariza-tions of the nonlinear terms can be attempted (Ragusa andMahadevan, 2009).

2.2. Jacobian-Free Newton Krylov methods

The JFNK method is a modern fully implicit technique of solvingnonlinear systems of equations, which can achieve a tight conver-gence tolerance without splitting or linearization error. Thismethod employs the Newton and Krylov methods to solve a givenset of nonlinear equations effectively and accurately. Each time-step of the JFNK method is composed by three main parts: the‘‘external’’ Newton iteration (Section 2.2.1), the ‘‘internal’’ Kryloviteration (Section 2.2.2) and the preconditioning part (Section2.2.4). According to Knoll and Keyes (2004) an extra globalizationmethod is often used outside of the Newton part. Recently, dueto the advantages of this method, nuclear technology has been

coupled neutronics/T–H (Gan et al., 2003).

Fig. 2. Generating an orthonormal basis with Gram–Schmidt method.


oriented in using the JFNK method in multi-physics calculations,i.e. in coupled neutronic/T–H calculations.

2.2.1. Newton iterationA system of nonlinear equations can be expressed in the follow-

ing form:

FðuÞ ¼ 0 ð13Þ

where u is the vector of the unknown parameters. By using the Tay-lor expansion, the Newton iteration about the current point can beformed as:

Fðukþ1Þ ¼ FðukÞ þ F 0ðukÞ þ F 0ðukÞðukþ1 � ukÞþ higher order terms ð14Þ

If the right side of the equation is considered equal to zero, Newtonmethod yields to the following linearized system (Knoll and Keyes,2004):

JðukÞduk ¼ �FðukÞ ð15Þukþ1 ¼ uk þ duk for k ¼ 0;1; . . . ð16Þ

where u0 is given, F(uk) is the vector of the nonlinear residuals, J = F0

is its associated Jacobian matrix, u is the vector of the unknownparameters to be found and duk is the Newton correction. Each ele-ment (ith row, jth column) of the Jacobian matrix is given by:

Jij ¼@FiðuÞ@uj

ð17Þ

The iteration continues until the convergence of the nonlinearresidual is satisfied, following:

kFðukÞk2

kFðu0Þk2< eNL ð18Þ

and/or a sufficient small Newton update:

duk

uk

��

2< eupdate ð19Þ

The major drawback of a general Newton iterative method isthat the calculation of each element of the Jacobian matrix by usinganalytic techniques or discrete derivatives is a computationallyintensive procedure which can also introduce errors.

2.2.2. Krylov methods – GMRESSolving a large system of linear equations by using Gauss elim-

ination can be very computationally expensive since a great num-ber of operations are needed. In such cases Krylov methods arehighly recommended as they can significantly reduce the compu-tational burden because they do not require formation of the Jaco-bian matrix.

Krylov methods solve large linear systems of the followingform:

Ax ¼ b ð20Þ

by searching for the solution x within a Krylov subspace:

KkðA; cÞ ¼ spanðc;Ac;A2c; . . . ;Ak�1cÞ ð21Þ

The sequence of the vectors (c,Ac,A2c,A3c, . . .,Ak�1c) whichcomposes the Krylov subspace is called Krylov sequence. Beforegiving a more detailed presentation of a Krylov method, it shouldbe understood why the solution of a linear system is sought withina Krylov subspace. This fact can be easily demonstrated if theinverse of matrix A, the matrix A�1, is written in terms of powersof A as explained in Ipsen and Meyer (1998). They use the minimalpolynomial q(A) of the matrix A to reach this relation between A�1

and A:

0 ¼ qðAÞ ¼ a0I þ a1Aþ . . . amAm ð22Þ

where m is the index of each of the eigenvalues of the matrix A. Ifa0–0:

A�1 ¼ � 1a0

Xm�1

j¼0

ajþ1Aj ð23Þ

It should be noted that even if a0 = 0 the solution of the linear sys-tem can rely on a Krylov subspace as explained in Ipsen and Meyer(1998). Consequently Eq. (23) shows that the solution of the linearsystem:

x ¼ A�1b ¼ � 1a0

Xm�1

j¼0

ajþ1Aj

" #b ð24Þ

has the form of Krylov subspace elements.The main feature of Krylov methods which makes them suitable

for use in the JFNK methods is that they need only Jacobianmatrix–vector products, therefore the expensive creation of theJacobian matrix is not required (as can be realized by consideringEq. (33), presented later in the text). Although more than one Kry-lov methods exist, this work focuses on the Generalized MinimalRESidual method (GMRES) (Saad and Schultz, 1986) which gener-ates an orthogonal chain of vectors and then combines them toconstruct the new approximate solution. GMRES aims to minimizethe Euclidean norm of the linear residual r:

r ¼ b� Ax ð25Þ

over a Krylov subspace. The value of x which minimizes the residualis sought in a least square sense, within a Krylov subspace by con-structing an orthonormal basis for Kk(A,b) using the Arnoldi method(Arnoldi, 1951) of orthogonalization. These Arnoldi vectors form thetrial subspace from which the solution is constructed. More specif-ically, the Arnoldi method is actually a version of the Gram–Schmidt method of orthogonalization, oriented to Krylov spaces(Ipsen and Meyer, 1998). Gram–Schmidt is a method which gener-ates orthogonal bases; given an arbitrary basis A = {a1,a2, . . .an} ofRn, it produces a new orthonormal basis {v1,v2, . . .vn}. This is doneby the following algorithm:

wi ¼ ai �Xi�1

j¼1

hai;wjikwjk2

2

wj ð26Þ

where w1 = a1, and i lies from 2 to n. The terms hai ;wjikwjk2

2wj represent the

projection of vector a in the direction of w. This procedure isillustrated in Fig. 2. It can be easily seen that a basis consisting ofvectors w is orthogonal but not normal; in order to acquire anorthonormal basis, normalization is required. In this step for eachi from 1 to n the orthonormal basis is created setting:

v i ¼wi

kwik2ð27Þ

Fig. 3. Simplified flowchart of the JFNK method.


As was mentioned above, GMRES uses the Arnoldi method togenerate an orthonormal basis. In Arnoldi, the vector v1 is initiallychosen and for j = 1,2 . . .

wjþ1 ¼ Av j �Xj

i¼1

hijv i ð28Þ

where hij = (Avj,vi) is the scalar product of these two vectors. Afterperforming normalization, the sought orthonormal basis of Krylovsubspace is acquired. This new orthonormal basis, which is gener-ated by the Arnoldi method, composes the vector Vj = (v1, . . .,vj).The auxiliary quantities hij compose the Hessemberg matrix Hj

(j � j). In this framework, the matrix A can be decomposed in thefollowing form:

AVj ¼ Vjþ1Hj ð29Þ

which is called Hessemberg decomposition (where Hj is a (j + 1) � jmatrix). By taking advantage of this decomposition the minimiza-tion of the residual can be transformed to:

minz2KjðA;bÞkb� Azk2 ¼ minykkr0k2e1 � Hkyk2 ð30Þ

where e1 is the first column of the identity matrix, z is the value of xwhich minimize r (Eq. (25)) and y minimizes the least square prob-lem Eq. (30). This least square problem (Eq. (30)) is often solvedwith QR decomposition of HK using plane rotations (Saad andSchultz, 1986), where Q is an orthogonal and R an upper triangularmatrix and the vector y is calculated. Ideally, GMRES method is ter-minated when it produces a zero linear residual. In practice it stopswhen it generates an acceptable small residual vector. The previousalgorithm needs at most n steps (Ipsen and Meyer, 1998) in order toconverge where n � n is the dimension of the matrix A. This meansthat n vectors should be available and consequently stored in mem-ory. However, as the dimension of the problem increases, the mem-ory which is required to store these vectors increases. In order tooverride this problem, the Restarted GMRES method is used. In

the Restarted GMRES method, the solution is constructed by a sig-nificantly smaller (problem dependent) number of vectors (m)and as a result, an iterative scheme is required. The solution canbe formulated as:

xi ¼ xi�1 þ Vmym ð31Þ

where i is the iteration index.

2.2.3. A Krylov solver (GMRES) within the framework of JFNKIn the JFNK approach, a Krylov method is implemented to solve

the linear system of Eq. (15). An initial residual is defined as:

r0 ¼ �FðuÞ � Jdu0 ð32Þ

where du0 is the initial guess. GMRES method aims to minimize theresidual r within each Newton iteration. As each GMRES iteration isperformed within a specific Newton iteration, the k Newton itera-tion index (Eqs. (15) and (16)) is removed and j is used as the Kryloviteration index. duj is picked from the subspace spanned by the Kry-lov vectors, {r0,Jr0,J2r0, . . . Jj�1r0} (Knoll and Keyes, 2004), and can bewritten as:

duj ¼ du0 þXj�1

i¼0

biðJÞir0 ð33Þ

GMRES attempts to minimize the residual:

r ¼ kJduj þ FðuÞk2 ð34Þ

in a least square sense and eventually provide the coefficients bi

which minimize the residual. The convergence criterion for the lin-ear iteration is the following (Knoll and Keyes, 2004):

kJkduk þ FðukÞk2 < eLkFðukÞk2 ð35Þ

In this point it should be mentioned that a balance between thenonlinear (eNL, Eq. (18)) and linear (eL) tolerances should be deter-mined. If the user selects a too large value of eL, more nonlinear iter-


ations will be needed, whereas a too small value would increase thenumber of linear iterations.

Within a GMRES (Krylov) iteration, duj arises as a combinationof the orthonormal Arnoldi vectors. As already mentioned above,the main advantage of Krylov method is that the Jacobian matrixdoes not have to be stored as it can be approximated by matrix–vector products. This matrix–vector product can be approximatedby the following relation:

Jv � Fðuþ evÞ � FðuÞe

ð36Þ

where v is a vector and e is a small perturbation parameter. Someuseful information about the magnitude of e and some approacheswhich are used to well determine e can be found in Knoll and Keyes(2004). A simplified flowchart of GMRES method is shown in Fig. 4;a general simplified global flowchart of JFNK method is depicted inFig. 3.

2.2.4. PreconditioningIn order to improve the numerical efficiency of the JFNK

method, preconditioning, a technique aiming to reduce the numberof Krylov iterations, is usually implemented. Preconditioning is aprocess which approximates the inverse of the Jacobian matrix(JP�1 � I). So instead of solving:

Jdu ¼ �FðuÞ ð37Þ

the following system is solved:

JP�1Pdu ¼ �FðuÞ ð38Þ

As has been shown by Campbell et al. (1996) a good preconditionerP should lead to eigenvalues of JP�1 which are sufficiently clustered.This leads to the minimization of GMRES (Krylov) iterations neededuntil convergence is achieved. Effective preconditioning aims tominimize the storage of Krylov vectors and to reduce physical

Fig. 4. Simplified flowchar

memory computational cost. If the Jacobian matrix is known, theinverse Jacobian could be used as the ideal preconditioner. Howeveras the main aim is to avoid the construction of the Jacobian matrix,other approaches are implemented in different problems in order toapproximate the inverse of the Jacobian matrix in an effective prob-lem dependent way. Physics-based preconditioning is a commonlyused example of such approaches (Knoll and Keyes, 2004).

2.2.5. GlobalizationThe main aim of globalization techniques is to overcome the

possible convergence problems which can arise in the Newton iter-ative technique (Knoll and Keyes, 2004). Since it is out of purposeof this work to present all the globalization techniques, only theline search method is presented here to provide a general idea onthis issue. In order to assure that the solution vector u is movingtowards the correct direction with the correct speed, a parameters can be induced as following:

ukþ1 ¼ uk þ sduk ð39Þ

The selection of the appropriate value of s is performed by requiringa decrease of the nonlinear residual (Knoll and Keyes, 2004):

Fðuk þ sdukÞ < FðukÞ ð40Þ

2.3. An illustrative example

The highly simplified problem (Eqs. (41) and (42)) offers anindicative example of how a steady-state coupled problem canbe solved in an explicit and JFNK context respectively. This simpli-fied coupled steady-state problem is described by the followingequations:

rðDðTÞruÞ � RaðTÞuþ mRf ðTÞu ¼ 0 ð41ÞrðKðTÞrTÞ þwRf ðTÞu ¼ 0 ð42Þ

t of restarted GMRES.


where the first equation is the steady-state neutron diffusion equa-tion and the second one is the steady-state heat conduction equa-tion. For simplicity this nonlinear system of equations is writtenin the following form:

F1ðu; TÞ ¼ 0 ð43ÞF2ðu; TÞ ¼ 0 ð44Þ

or F(u) = 0 where the vector of the unknowns of the system can bewritten:

u ¼uT

� �ð45Þ

In a simplified iterative context this problem would be solved inthe way represented in Fig. 5 where k is the number of iterations. Itshould be noticed that the iterations will stop when a convergencecriterion is satisfied. For example, the fuel temperature (T) can bechecked for convergence according to the following criterion:

Tk � Tk�1

Tk�1< e ð46Þ

where e is a problem-dependent convergence parameter. In asteady-state calculation the approximation order of the time dis-cretization is irrelevant. Such a case is described by Vasquez et al.(2012) (see Section 4.2), where a neutronic stochastic transportcode is coupled with a T–H code in order to perform steady-statecalculations. As it can be understood, the disadvantages of an expli-cit method appear when a transient calculation is performed.

In a JFNK implicit context this problem could be solved in thefollowing way: using Taylor expansion the system of the nonlinearequations can take the form of Eqs. (15) and (16), where

J ¼F1u F1T

F2u F2T

� �ð47Þ

is the Jacobian matrix and the term F(u) is the nonlinear residualwhich can alternatively be written as:

FðuÞ ¼F1

F2

� �ð48Þ

So, in matrix form the problem can be written in the following way(Gaston et al., 2009):

F1uðu; TÞk F1Tðu; TÞk

F2uðu; TÞk F2Tðu; TÞk

" #duk

dTk

" #¼ F1ðu; TÞk

F2ðu; TÞk

" #ð49Þ

ukþ1

Tkþ1

" #¼

uk

Tk

" #þ

duk

dTk

" #for k ¼ 0;1; . . . ð50Þ

where k refers to the current iteration. The solution of the problemusing JFNK method is represented in Fig. 6.

Fig. 5. Solving the problem

2.4. Current status of research in coupling techniques

Mousseau (2004) has showed that the two-phase flow equa-tions coupled with the nonlinear heat conduction are solved moreefficiently using a consistent and accurate numerical scheme basedon JFNK framework, than using traditional OS methods. Howeverin order to ensure this high performance of JFNK method, efficientpreconditioning should be used. Hence optimum preconditioningremains a challenge as also discussed in Knoll and Keyes (2004).A tool which makes use of JFNK method, is the Multiphysics ObjectOriented Simulation Environment (MOOSE) (Gaston et al., 2009).This tool uses the JFNK method combined with physics-based pre-conditioning with aim to solve the system of equations whichdescribe the user-specified problem. In case of existing mono-dis-ciplinary codes, the implementation of JFNK methodology mayrequire extensive modifications to the structure of the codes inorder to obtain access to the residual vectors. Gan et al. (2003) dis-cuss the coupling of the neutronic code PARCS (Joo et al., 1998)with the T–H code TRAC (Spore et al., 1992) using a modified ver-sion of the JFNK method which does not require modifications ofthe codes. It should be mentioned that few attempts to combinewell-known standard sub-system solvers, in nuclear engineeringproblems, using the JFNK methods are met in bibliography. Thispractice might have the advantage of connecting the well testedand reliable solutions of each of the sub-systems in a more effec-tive and accurate numerical framework than the one of the OSmethods.

In the field of OS techniques some research on methods whichcan be applied to restore consistency in the coupling of the nonlin-ear terms is presented in Ragusa and Mahadevan (2009). Generally,the improvement of the OS techniques remains always a challengesince they are steadily used to couple sophisticate mono-disciplin-ary codes in order to avoid re-developing tools from scratch.

2.5. Cross-section treatment in neutronic/T–H calculations

In a multi-physics methodology, after each iteration of a neu-tronic/T–H coupled calculation, new temperatures are providedto the neutronic code as feedback. This means that new nucleardata libraries, which correspond to the new temperatures, shouldbe used. This can be achieved either by using pre-generatedlibraries or by generating them at the exact temperatures duringthe multi-physics calculation. The first procedure requires a suc-cessful guess of the temperature field of the model. In addition, itrequires the pre-generation of a large number of libraries permaterial, using a temperature step. Hence this procedure is quitetime-consuming and it could introduce inaccuracies if the selectedtemperature step is not as small as the problem requires, or if thelibraries have not been generated following the proper procedure.On the other hand, the latter methodology eliminates some of theproblems of the previous approach, but it might significantlyincrease the computational time of the calculation.

An innovative method of treating cross-sections is the one ofthe pseudo-materials which is described and used in Vasquez

(Eqs. (41) and (42)).

Fig. 6. Solving the problem (Eqs. (41) and (42)) using the JFNK method.


et al. (2012) and Kotlyar et al. (2011), and has been originally intro-duced in Bernnat et al. (2000). The method of pseudo-materialsapproximates the cross-sections of a nuclide in a temperature ofinterest (T), using a weighted average of the cross-sections of thenuclide evaluated at a lower temperature Tlow and a higher temper-ature Thigh. The linear version of the pseudo-material which is usedin Vasquez et al. (2012) can be formulated by the followingequation:

RðTÞ ¼ wlowRðTlowÞ þwhighRðThighÞ ð51Þ

where R is the macroscopic cross-section, T is the actual tempera-ture, Tlow, Thigh are the lower and the higher temperatures and wis the mixing coefficient (Vasquez et al., 2012):

whigh ¼T � Tlow

Thigh � Tlowð52Þ

wlow ¼ 1�whigh ð53Þ

The major advantage of this technique is that, if it is used properly,it can eliminate most of the problems of the other methods whichhave been referred above.

Another new advanced technique which accounts for Dopplerbroadening is the On-The-Fly (OTF) Doppler broadening techniquedeveloped by Yesilyurt et al. (2011). Doppler broadened cross-sec-tions are retrieved during the random walk, in a Monte-Carlo code,from a regression model at the exact region temperature and neu-tron energy (Martin, 2012). As Martin refers in Martin (2012), theOTF method is based on Taylor series expansion and asymptoticseries expansions. He also refers that the comparison of the OTFbroadened cross-sections with the ones generated using NJOY(MacFarlane and Muir, 1999) is excellent.

3. Multi-scale calculations

The prediction of the local clad temperature in a specific fuel pinin the core of a nuclear reactor requires the knowledge of the dis-tribution of both the power generation and the water temperaturewithin the core with a fine spatial resolution in which each fuel pinis represented (Monti et al., 2011). Development of a suitable anal-ysis procedure capable to predict the local operating clad temper-atures by implementing a multi-step approach is usually required.In the first step the operative condition should be determined withcoupled neutronic/T–H calculations. The main characteristics cal-culated by this multi-physics calculation concern a coarse repre-sentation of the full core, in which the FA heterogeneity is notresolved in detail. In this way, FA-wise average quantities can bepredicted. In the second step, the obtained FA-wise averaged con-dition should be investigated at sub-channel spatial resolution inorder to provide local, i.e. pin-wise, clad and fuel temperature dis-tribution. In other words, the FA averaged power should be redis-tributed within the fuel pins taking into account the effects of bothneutron flux spatial gradient and local heterogeneity of the FA lat-tice. For this purpose the Pin-Power Reconstruction Technique(PPRT) is introduced in order to calculate the local values of thecritical parameters which are necessary to specify the currentdesign and to suggest improvements.

According to studies, such as Boer and Finnemann (1992), thelocal PPRT make use of a fundamental assumption: detailed pin-by-pin distributions within an assembly can be estimated as theproduct of a global intra-cell (intra-nodal) distribution and a localheterogeneous form function (Dahmani et al., 2011). The formfunction is employed in order to take into account the assemblyheterogeneities and is generated for each fuel assembly type by alattice-physics code at the same time that the homogenized


cross-sections are generated. The pin-by-pin flux is then obtainedby combining these two functions, as said in Dahmani et al. (2011):

Uðx; yÞpin-by-pin ¼ Uðx; yÞhom �Wðx; yÞhet ð54Þ

where U(x,y)hom is the ‘‘homogeneous’’ intra-cell flux at core leveland W(x,y)het is the ‘‘heterogeneous’’ form function evaluated at lat-tice level which accounts for core heterogeneities caused by variousreasons. A single-cell model or multi-assembly model is used inorder to obtain W(x,y)het. This factorization approximation is possi-ble since the neutron mean free path is very small compared to theassembly dimensions so that the lattice and global effects can be‘‘decoupled’’ as has been shown by Boer and Finnemann (1992).

3.1. Current status

Some attempts to modify this technique in order to be used intypes of reactors other than LWRs have been made. For High Per-formance LWR (HPLWR) multi-scale analysis, a multi-scale method(Monti et al., 2011) using a developed PPRT based on the combina-tion of the homogeneous intranodal flux distribution and localform functions is used. The homogeneous intranodal flux is calcu-lated by ERANOS (Rimpault et al., 2002) and the local form func-tions are obtained with MCNP5 (Brown et al., 2002). Anotherreconstruction technique elaborated on CANDU reactors based onthe traditional PPRT presented in Dahmani et al. (2011) and thecalculated reconstructed fluxes are in good agreement with theones calculated using a transport solution. Tohjoh et al. (2006)present a 3D PPRT oriented on Boiling Water Reactors (BWRs),making use of the Monte-Carlo ability to perform detailedcalculations.

4. Reactor modeling using coupled neutronic/T–H calculations

4.1. Light Water Reactors

As LWRs have been developed since some decades, manyattempts in coupling neutronic/T–H codes in order to performmore accurate LWR analysis have been made. Since it is out of pur-pose of this work to present all of these attempts, some indicativeexamples are presented.

In Ellis et al. (2013), PARCS (Joo et al., 1998) 3D reactor coresimulator which solves the steady-state and time-dependent,multi-group neutron diffusion and transport equations in orthogo-nal and non-orthogonal geometries, is coupled explicitly, through ageneral passing interface, to the T–H code TRACE (TRACE, 2008),which is able to analyze large/small break Loss Of Coolant Acci-dents (LOCAs) and system transients in both Pressurized WaterReactors (PWRs) and BWRs. TRACE is executed within a time stepand provides the temperature and flow field information. After-wards the cross-section libraries are updated and provided toPARCS, as a feedback, in order to solve the steady state diffusionequation. The null-transient calculation is terminated when asteady-state condition is reached or after the number of time stepsspecified by the user is reached. An attempt to couple these twocodes to solve an implicit transient problem by implementing animplicit steady-state solution in it, is also described in Ellis et al.(2013). The first results show that this coupling scheme mayachieve lower computational time since larger time steps can beused. In this scheme the T–H equations, the heat transfer equationsand the steady state neutron diffusion equations are solved simul-taneously using the Newton’s iterative scheme. It should be notedthat in this case the Jacobian matrix is formed using analytic tech-niques in contrast with the JFNK implicit method where the com-putationally expensive Jacobian matrix formation is eschewed.Finally, the resulting linear system is solved using a direct linear

solver. This method produces the correct steady-state solutionfor varying time step sizes for each mesh (Ellis et al., 2013). Thecomputational time still needs optimization since the implicitnumerical scheme leads to more time-consuming calculations thanthe traditional explicit one.

ATHLET/DYN3D (Kliem et al., 2007) is another attempt of cou-pling a neutronic and a T–H code for analyzing complex transients.DYN3D (Grundmann et al., 2000), a 3D code for the calculation ofsteady states and transients in LWRs, is coupled with the ATHLET(ATHLET, 2003), a thermo-fluid-dynamic system code for a widerange of applications. More specifically, DYN3D has been developedfor the analysis of Reactivity Initiated Accidents (RIA) in thermalnuclear reactors. The power distributions are calculated with thehelp of 3D nodal expansion methods for hexagonal and Cartesiangeometry. DYN3D also includes a fuel rod model and a two phaseT-H one, integrated in the module FLOCAL, which provides fueltemperatures, coolant temperatures and densities as well as boronconcentrations for the calculation of feedback effects on the basis ofcross-section libraries generated by cell codes. This DYN3D schemeis suitable for the calculation of RIA initiated by moved control rodsand/or perturbations of the coolant flow. In FLOCAL the reactor coreis modeled by parallel cooling channels which can describe one ormore fuel elements. The parallel channels are coupled hydraulicallyby the condition of equal pressure drop over all core channels. Formore complex transients DYN3D is coupled with ATHLET. ATHLETcan be applied to the whole spectrum of operational and accidenttransients, small and intermediate leaks up to large breaks of cool-ant loops or steam lines at PWRs and BWRs. According to Kliemet al. (2007), three different ways of coupling these two codes areoffered; the external, the internal and the parallel. The three waysof coupling between ATHLET and DYN3D are illustrated in Fig. 7(Helmholtz-Zentrum Dresden-Rossendorf). The symbols used inFig. 7 are explained in Table 1. When the external coupling is imple-mented, the mass flow rates at the core inlet/outlet and the coreoutlet coolant enthalpy pass from DYN3D to ATHLET, whereas thecore inlet/outlet pressure and the enthalpy and boron concentra-tion at the core inlet pass from ATHLET to DYN3D. When the inter-nal coupling option is used, the core T–H is performed by ATHLETand DYN3D calculates only the neutron kinetics part. Finally inthe parallel mode, the core T–H calculation is performed by the cou-pled code in the same iteration loop of the code system (Kliem et al.,2007). ATHLET calculates the core inlet/outlet boundary conditionsand send them to DYN3D. Afterwards, DYN3D recalculates the coreT–H features and uses these results to upgrade the power distribu-tion. In this point it should be mentioned that as multi-physics/multi-scale calculations are of major importance for reactor safety,the simulation of Nuclear Power Plant accident conditions requires3D modeling of the reactor core to ensure a realistic description ofthe physical phenomena and hence evaluate safety margins in amore accurate way. Challenges in this domain are presented inIvanov and Avramova (2007). Ivanov and Avramova also discussthe extension of the modeling capabilities of the coupled semi-implicit neutronic/T–H scheme TRAC-PF1/NEM by including a pinpower reconstruction scheme coupled to a sub-channel model, inorder to obtain an improved transient fuel rod response. Amongthe others, they also discuss the adaptive high-order table lookupmethod (Watson and Ivanov, 2002), a new sophisticated uniquecross-section representation methodology for 3D coupled transientcalculations.

Ravnik et al. (2008) discuss CORD-2 (Trkov and Ravnik, 1994), acore design system for PWRs with multi-physics capabilities. Theneutronic part consists of WIMS (Askew et al., 1966), a well-knownand widely used lattice code, and GNOMER (Trkov, 2008), a 3D dif-fusion code. The neutronic part provides a solution for the wholecore by solving the diffusion equation using a coarse mesh nodalmethod. The T–H part employs a simple module from CTEMP

Fig. 7. ATHLET/DYN3D coupling options (Helmholtz-Zentrum Dresden-Rossendorf).

Table 1Explanation of the symbols appeared in Fig. 7.

Symbol Explanation

Tf(r), TM(r) Fuel, moderator temperatureqM(r) Moderator densityCB(r) Boron concentrationP(r) Power distributionGin, Gout Mass flow rate in the inlet and the outlet of the corehin, hout Coolant Enthalpy in the inlet and the outlet of the corepin, pout Coolant Pressure in the inlet and the outlet of the core


(Kromar et al., 1991) T–H code, in order to provide a T–H feedbackto the neutronic part; the cross-sections are upgraded according tothe temperature distribution and the boron concentration. An iter-ative serial coupling scheme between the neutronic and the T–Hpart is used.

In Zare et al. (2010), WIMS lattice code, CITATION (Fowler,1999) deterministic neutronic code for core analysis calculations

and COBRA-EN (Wheeler et al., 1976) T–H code are coupled inorder to perform a complete neutronic/T–H steady-state and tran-sient analysis of reactor cores. A module which controls the exter-nal data transfer between the two codes has been developed.Coolant density, coolant temperature and fuel temperature areupdated in each iteration and pass to CITATION which performsthe core neutronic analysis (Fig. 8) using the new updated cross-section values. Subsequently, the power distribution, calculatedby CITATION, passes to COBRA-EN. The iterations continue untilthe temperature distribution converges suitably. In addition WIMSand CITATION are used to study the effect of the fuel BurnUp (BU)on the excess reactivity, neutron flux and power distribution of thecritical core of VVR-S (Kozlov and Zemlyanskii, 1963) researchreactor. The results for the VVR-S obtained by this method are ingood agreement with the experimental and computational resultsaccording to the operational history of the reactor, as referred inZare et al. (2010).

In Kotlyar et al. (2011), MCNP (MCNP) stochastic (Monte-Carlo)neutronic code and SARAF (Kotlyar et al., 2011), a BU and decay

Fig. 8. CITATION/COBRA-EN coupling scheme (Zare et al., 2010).


module, are coupled using serial coupling, with THERMO(Shaposhnik et al., 2010), a simplified T–H feedback module forsteady-state calculations, creating the BGCore system (Fig. 9). Inthis point it should be mentioned that the use of a Monte-Carloneutronic code in multi-physics calculations has both advantagesand disadvantages (Martin, 2012). Some of the advantages are thatthe continuous energy Monte-Carlo codes are capable of analyzingreactor configurations with arbitrary geometrical complexity, lim-ited by the ability of the code to represent arbitrary shapes in acomputational model; and arbitrary physics complexity which islimited by the provided cross-sections that describe the physicalphenomena being modeled. In addition to the flexibility of

Monte-Carlo methodology to simulate the most complex geome-try, the continuous energy Monte-Carlo methodology treatsthe neutron energy dependence correctly with essentially noapproximations. On the other hand, one of the most importantdisadvantages is the extreme computational burden to carry outa full-core Monte-Carlo simulation, including prohibitive computa-tional time and excessive memory demand. BGCore’s main aim isto perform full core level BU calculations with T–H feedback inassembly sub-channel level and not in pin level sub-channel reso-lution of T–H parameters. Kotlyar et al. (2011) refer that as hasbeen shown in Shaposhnik et al. (2010), a detailed description ofthe radial void distribution within a single fuel pin sub-channel

Fig. 9. BGCore schematic representation (Kotlyar et al., 2011).


of a BWR assembly has negligible effect on the neutronic charac-teristics of the whole system. In addition since in PWRs operatingconditions the water density variations across a fuel assemblysub-channel is much smaller compared to a BWR, a negligibleneutronic effect is also expected. The results are compared withthe ones calculated by DYN3D and it can be said that the differencefor full PWR analysis is not so important; there is good agreementin the results given by the two codes.

In Aghaie et al. (2012), WIMS and CITATION neutronic modulesand a reactor kinetics equation solver along with the RELAP5/3.2(RELAP5/MOD3.3, 2001) T–H module have been combined, usingexplicit coupling, creating IRTRAN, a package used in steady stateand transient analysis of PWRs (Fig. 10). To ensure that IRTRANresults are accurate, the drop of a control rod transient case is sim-ulated. Furthermore, the reactivity coefficients of BoushehrNuclear Power Plant which is a VVER-1000 are calculated. Theobtained transient results are in good agreement with the pub-lished data.

Complementary to the already mentioned multi-physics toolsfor neutronic/T–H analysis, the NURESIM, an indicative representa-tive of simulation platforms is presented (Fig.11). A EuropeanReference Simulation Platform for Nuclear Reactors, is being

developed since 2005 (Chauliac et al., 2011).Within this project aplatform; i.e. a tool able to provide a complete, accurate represen-tation of the physical phenomena occurring in a reactor core, isdeveloped. Emphasis was given in the implementation of the latestadvances in core physics, two-phase T–H and fuel modeling. Itshould be noted that the completed NURESIM project (2005–2008) and its continuation the NURISP (2009–2011) and NURE-SAFE (2013–2015) projects focus on GEN-II and GEN-III reactors.One of the main aims of NURESIM was to create a common frame-work to facilitate multi-physics and multi-scale calculations withspecial focus on neutronic/thermal hydraulic calculations. Throughthis framework a pre-, pro-processing open source platform SAL-OME was created. Last but not least, validation of the individualmodels, solvers, codes and the platform was one of the main tasksthrough the project (Chauliac et al., 2011). As mentioned in Zerkaket al. (2010), the coupling scheme developed during the NURESIMproject is based on the OS methodology, mainly due to the lowimplementation effort. In addition Calleja et al. (2014) describe,the coupling between the 3D neutron diffusion code COBAYA3(Jiménez et al., 2010) and the sub-channel code SUBCHANFLOW(Imke and Sanchez, 2012) in an OS context, aiming to solve tran-sient problems, within the NURESIM platform.

Fig. 10. IRTRAN flowchart (Aghaie et al., 2012).

Fig. 11. The NURESAFE platform (NURESAFE EXCOM, 2013). Code references: (Santamaret al., 2003; Toumi et al., 2000; Bartosiewicz et al., 2008; Farvaque, 1991; Repetto et al.


4.2. Sodium Fast Reactor (SFR)

SIMMER-III (Maschek et al., 2005), originally developed forSFRs, is a 2D multi-velocity-field, multi-phase, multi-component,Eulerian, fluid dynamics code coupled with a space, time andenergy-dependent neutron transport theory model. The neutronicmodule obtains the transient neutron flux distribution with theimproved quasi-static method. The equations of the involvedsub-codes are solved on three geometrical mesh levels (fluiddynamics, pin, neutronics) with three time-step hierarchies, i.e.the neutronic flux shape, the reactivity plus the heat transfer andthe nuclear power plus the fluid dynamics.

In Vasquez et al. (2012), the MCNP neutronic code is coupled,using serial methodology, with the sub-channel T–H codeCOBRA-IV with purpose to achieve a simultaneous analysis of thesteady-state neutronic/T–H characteristics of a nuclear reactor.The code is applied in a SFR concept at both FA and full core scale.The temperature dependence of the fuel cross-section libraries ishandled with the pseudo-material approach. In this work, aftereach iteration, updated values of fuel, clad and coolant tempera-tures as well as coolant densities are provided in MCNP as T–Hfeedback (Fig. 12). The convergence criterion requires the temper-ature difference between the previous and the actual step to besmaller than e. It should be noted that the convergence is checkedat each axial level and in each rod.

4.3. Molten Salt Reactors (MSR)

In MSRs, the unusual characteristic of the fluid fuelled core showsitself in the form of a strong nonlinear coupling between the fuelmotion and neutron dynamics, because delayed neutron precursorscreated in the core can decay in a different position of the primaryloop affecting the overall neutron balance. These characteristics dif-ferentiate MSR analysis from the one of the conventional solid fueledreactors; this analysis requires special techniques and tools.

Yamamoto et al. (2005, 2006), discuss a multi-physics model forsteady-state and transient analysis for the Small Molten Salt

ina et al., 2009; Both et al., 1994; Jiménez et al., 2010; Baudron et al., 1999; Bieder, 2009).

Fig. 12. MCNP/COBRA coupling (Vasquez et al., 2012).


Reactor (SMSR) comprising continuity and momentum conserva-tion equations for fuel salt flow, two-group neutron diffusionequations for fast and thermal neutron fluxes, transport equationsfor six-group delayed neutron precursors, and energy conservationequations for fuel salt and graphite moderators in order toinvestigate the case when the reactor operates in steady-stateand the case when a duct blockage accident occurs.

In Wang et al. (2006), SIMMER-III, mentioned in Section 4.2, ismodified in order to be used for the analysis of the neutronic/T–H characteristics of MSRs. Among the modifications, a point kinet-ics model which considers the flow effects of the fuel salt is devel-oped and established. The founded models and developed code areapplied to analyze the safety characteristics of the molten salt acti-nide recycler and transmuter system (MOSART). The results con-firm the importance of the T–H performance of this type ofreactor which affects its operation and safety.

In Krepel et al. (2007), DYN3D-MSR, a modified version of theLWR code DYN3D, for MSRs analysis is discussed. The neutronicand the T–H part of DYN3D have been modified to take intoaccount the MSRs’ special features like the drift of delayed neutronprecursors. The neutronic part uses the diffusion approach. Morespecifically the time-dependent diffusion equation for two energygroups is solved using the nodal expansion method. An implicitfinite difference scheme together with the exponential transforma-tion techniques is used for the integration of the two-group diffu-sion equation and the equation for the volumetric concentration ofthe delayed neutron precursors. The T–H part uses a modified ver-sion of the DYN3D T–H module FLOCAL in order to solve the energybalance equation together with the mass and momentum equa-tions. DYN3D-MSR has been validated on experimental resultsfrom the MSR Experiment (MSRE) performed in ORNL.

In Gammi et al. (2011), a modeling approach to the dynamics ofMSRs is proposed. This modeling approach does not follow thegeneral approaches used in coupling neutronic and T–H codes

which are well evaluated only for conventional nuclear reactorconcepts. Significant modifications may be required in MSR mod-eling due to their special features like the unusual distribution ofthe delayed neutron precursors. As a result, a set of nonlinear par-tial differential coupled equations which describe the physicalphenomena occurring in a nuclear reactor is used in Gammiet al. (2011). The neutronic field is represented by the 2-group dif-fusion theory where the velocity field of the fuel is taken intoaccount in the balance equations of the six families of the delayedneutron precursors, by introducing a convection term. The T–Hfield is described by the Reynolds Averaged Navier–Stokes equa-tions and the field of heat transfer by the energy balance equationsfor fuel salt and graphite. This set of equations is solved by usingthe finite element software COMSOL Multiphysics (COMSOL,2010). The solver which is employed in this work is based onthe finite elements method using quadratic Lagrangian elements.In addition, the single-fluid MSR is chosen as reference configura-tion and a single-channel of this reactor has been analyzed. Asregards the transient analysis, it is performed for pump-drivenreactivity transients and in presence of periodic perturbations.The evaluation of the results showed that the approach used isrepresentative and well descriptive of the Molten Salt BreederReactor core behavior.

In Guo et al. (2013), a Multiple-channel Analysis Code (MAC) isdeveloped and coupled with MCNP4c (MCNP) neutronic code toanalyze the neutronic/T–H behavior of MSR experiment. MAC cal-culates the T–H parameters, such as temperature distribution, flowdistribution and pressure drop and then it provides them toMCNP4c which calculates the effective multiplication factor, neu-tron flux and power distribution. Coupling is achieved by a linkagecode developed to exchange data between MAC and MCNP4c andimplement the coupling iteration process. The results of thesteady-state analysis of an MSRE are in reasonable agreement withthe analytic solutions given from the ORNL. Only 1/8 of the MSREcore, divided in 20 control volumes along the axial direction ineach fuel channel, as well as the graphite moderator bar, is simu-lated. Temperature and density distribution used in MCNP4c areupdated at the beginning of each iteration. The iterative processis carried out until the power convergence is obtained. The cou-pling procedure takes into account the fuel salt temperature,graphite temperature, fuel salt density effect and graphite densityeffect. The pseudo-materials technique is used for the cross-sectionupdating.

4.4. Lead cooled Reactors

In Aufiero et al. (2011), an extension of the Multi-Physics Mod-eling implicit approach for nuclear reactor analysis is presented,with reference to ELSY, the European lead-cooled fast reactor sys-tem. This tool obtains the solution of the coupled neutronic/T–Hproblem by solving simultaneously the 6-group diffusion equationalong with the balance equations for six groups of precursors andthe Reynolds-averaged Navier–Stokes equations. As regards theheat transfer the energy balance equations are used. In addition,aiming to take into account the thermal expansion effects in thefuel and cladding, the equations of linear elasticity are also insertedin the system of equations which describes this multi-physicsproblem. This set of partial differential equations is solved byimplementing the general-purpose finite element software COM-SOL Multiphysics. This methodology is applied in a single-channelrepresentative of the active core average conditions of the ELSYreactor, at the beginning of life. The results show that thisapproach is able to simulate effectively the reactor behavior insteady state and transient condition in reasonable computationaltime.


In Bonifetto et al. (2013) the capabilities of FRENETIC, a multi-physics simulation tool for lead-cooled fast reactors, are discussed.FRENETIC has been developed for the quasi-3D analysis of a lead-cooled fast reactor core characterized by hexagonal fuel elements.The neutronic module consists of a 2D and 1D full-core multi-group diffusion solver developed based on a coarse-mesh nodalscheme. Cross-section generation is performed by ECCO, a moduleof the transport code ERANOS. The T–H module solves implicitly intime the transient 1D mass, momentum and energy balances ineach fuel assembly. The coupling between the neutronic and T–Hmodules is performed explicitly by using the commercial platformTISC (TLK-Thermo GmbH). More specifically, the distribution of thepower calculated by the neutronic module is transferred to the T–Hmodule at each T–H time step. According to Bonifetto et al. (2013),one FRENETIC’s innovation is that it is the only existing tool able toperform coupled neutronic/T–H calculations in hexagonal geome-try at the full-core level. The model assumes a uniform distributionof the different relevant parameters at the hexagon level.

5. Conclusions – Challenges

As has been mentioned above, neutronic/T–H multi-scale calcu-lations are of special interest since they can significantly contributeto the improvement of the reactor design and the fulfillment of thecontinuously increasing reactor safety requirements. Especially inthe framework of GENIV, it is obvious that the different reactorconcepts require special computation approaches since they arecharacterized by high complexity.

The main remaining challenge in the field of neutronic/T–H cal-culations is the improvement of coupling techniques in terms ofaccuracy and efficiency of numerical simulations. So far, OS meth-ods have been the main numerical tool used in this field. The mainreason for their popularity is that this type of coupling requires thelowest implementation effort. Due to the inaccuracy that isinserted in the coupled calculations by OS methodology, very smalltime steps are required in time-dependent calculations to over-come this problem. As a result, the computational cost increasessignificantly and sometimes may make their use impractical. Inthe same field, a topic that concentrates the interest of manyresearchers is the use of the JFNK methods instead of the OS meth-ods, since JFNK has been proven to provide more accurate solu-tions. However in order to achieve high accuracy and lowcomputational cost, proper preconditioning in combination withproper selection of the needed parameters, like Newton and Krylovtolerances, is required. The use of JFNK as the coupling methodbetween mono-disciplinary codes seems to be very promising.However it requires quite extensive changes in the codes in orderto acquire access to the needed quantities.

Staying on the field of multi-physics calculations, it could beargued that computational cost and accuracy could be improvedby investigating new more efficient cross-section generation meth-ods or by improving the already existing ones. OTF Doppler broad-ening technique seems to be a step towards this direction.

Last but not least, as the nuclear reactor safety limits concern-ing the temperature even in a small point in the center of thenuclear reactor are becoming continuously stricter, a calculationable to simulate the reactor core characteristics of such a very het-erogeneous system in a very detailed scale is extremely burden-some. As a result, techniques which can predict the behavior ofthe reactor in specific points by using information generated fromhomogenized models is of major importance. Despite PPRT tech-niques are currently extensively used, many improvements canbe done in order to improve the balance between accuracy andcomputational cost, especially for calculations concerning GEN-IVreactor concepts.

Acknowledgments

This work has been co-financed by the European Union (Euro-pean Social Fund – ESF) and Greek national funds through theOperational Program ‘‘Education and Lifelong Learning’’ of theNational Strategic Reference Framework (NSRF) - Research Fund-ing Program: Thales. Investing in knowledge society through theEuropean Social Fund.

References

Aghaie, M., Zolfaghari, A., Minuchehr, A., 2012. Coupled neutronic thermal–hydraulic transient analysis of accidents in PWRs. Ann. Nucl. Energy 50, 158–166.

Arnoldi, W.E., 1951. The principle of minimized iteration in the solution of thematrix eigenvalue problem. Quart. Appl. Math. 9, 17–29.

Askew, J.R., Fayers, F.J., Kemshell, P.B., 1966. A general description of lattice codeWIMSD4. J. Br. Nucl. Energy Soc. 5, 564.

ATHLET Mod 2.0 Cycle A, Input Data Description, GRS-P-1/Vol. 1, Rev. 3,Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) mbH, Germany, 2003.

Aufiero, M., Cammi, A., Fiorina, C., Luzzi, L., 2011. Development of a single-channelmulti-physics model for lead fast reactors. In: 20th Int. Conf. Nuclear Energy forNew Europe, Slovenia.

Bartosiewicz, Y., Laviéville, J., Seynhaeve, J.-M., 2008. A first assessment of theNEPTUNE CFD code: instabilities in a stratified flow, comparison between theVOF method and a two-field approach. Int. J. Heat Fluid Flow 29, 460–478.

Baudron, A.M., Lautard, J.J., Schneider, D., 1999. Mixed dual methods for neutronicreactor core calculations in the CRONOS SYSTEM. In: Mathematical Methodsand Supercomputing in Nuclear Applications, Madrid, Spain.

Bernnat, W., Matters, M., Keinert, J., Conti, A., 2000. Application of the Monte CarloMethod for the Calculation of Safety Related Reactor Physics Parameter forPower Reactor with Consideration of Detailed Burn-up Distributions. Universityof Stuttgart, IKE-6-2000.

Bieder, U., Calvin, Ch., Mutelle, H., 2003. Detailed thermal hydraulic analysis ofinduced break severe accidents using the massively parallel CFD code Trio_U.In: Proceedings of the International Conference on Supercomputing in NuclearApplications, Paris, France.

Boer, R., Finnemann, H., 1992. Fast analytical flux reconstruction method for nodalspace-time nuclear reactor analysis. Ann. Nucl. Energy 19, 617–628.

Bonifetto, R., Dulla, S., Ravetto, P., Savoldi Richald, L., Zavino, R., 2013. A full-corecoupled neutronic/thermal–hydraulic code for the modeling of lead-coolednuclear fast reactors. Nucl. Eng. Des. 261, 85–94.

Both, J.P., Derriennic, M., Morillon, B., Nimal, J.C., 1994. A survey of TRIPOLI-4. In:Proc. of Eighth Conf. on Radiation Shielding – Arlington, USA, vol. I, p. 373.

Brown, F.B., Barrett, R.F., Booth, T.E., Bull, J.S., Cox, L.J., Forster, R.A., Goorley, J.T.,Mosteller, R.D., Post, S.E., Prael, R.E., Selcow, E.C., Sood, A., Sweezy, J., 2002.MCNP Version 5. In: American Nuclear Society Winter Meeting, LA-UR-02-3935,Washington, DC, November 17–21, 2002.

Calleja, M., Sanchez, V., Jimenez, J., Imke, U., Stieglitz, R., Macian, R., 2014. Couplingof COBAYA3/SUBCHANFLOW inside the NURESIM platform and validation usingselected benchmarks. Ann. Nucl. Energy 71, 145–158.

Campbell, S.L., Ipsen, I.C.F., Kelley, C.T., Meyer, C.D., 1996. GMRES and the minimalpolynomial. BIT 36, 664–675.

Chauliac, C., Aragones, J.-M., Bestion, D., Cacuci, D.G., Crouzet, N., Weiss, F.-P.,Zimmermann, M.A., 2011. NURESIM – a European simulation platform fornuclear reactor safety: multi-scale and multi-physics calculations, sensitivityand uncertainty analysis. Nucl. Eng. Des. 241, 3416–3426.

COMSOL Multiphysics 4.1, Users Guide, COMSOL Inc., 2010.Dahmani, M., Phelps, B., Shen, W., 2011. Verification and validation of the flux

reconstruction method for CANDU applications. Ann. Nucl. Energy 38, 2410–2416.

Demazière, C., 2013. Multi-physics modelling of nuclear reactors: current practicesin a nutshell. IJNEST 7 (4), 288–318.

Ellis, M.S., Watson, J., Ivanov, K., 2013. Progress in the development of an implicitsteady state solution in the coupled code TRACE/PARCS. Prog. Nucl. Energ. 66,1–12.

Farago, I., 2008. A modified iterated operator splitting method. Appl. Math. Model.32, 1542–1551.

Farvaque, M., 1991. User’s manual of CATHARE 2- version V1.3E, CEA,GENGSTR?LMK?EM?91-61.

Fowler, T.B., 1999. Citation-LDI2 Nuclear Reactor Core Analysis System, CCC 643,Oak Ridge National Laboratory, Oak Ridge, Tennessee.

Gammi, A., Marcello, V.D., Luzzi, L., Memoli, V., Ricotti, M.E., 2011. A multi-physicsmodelling approach to the dynamics of Molten Salt Reactors. Ann. Nucl. Energy38, 1356–1372.

Gan, J., Xu, Y., Downar, T.J., 2003. A matrix-free newton method for coupledneutronics thermal-hydraulics reactor analysis, Nuclear Mathematical andComputational Sciences: A Century in Review, A Century Anew, Catlinburg,Tennessee, April 6–11, 2003.

Gaston, D., Hansen, G., Kadioglu, S., Knoll, D.A., Newman, C., Park, H., Permann, C.,Taitano, W., 2009. Parallel multiphysics algorithms and software forcomputational nuclear engineering. J. Phys.: Conf. Ser. 180.

http://refhub.elsevier.com/S0306-4549(14)00216-3/h0175









































Grundmann, U., Rohde, U., Mittag, S., 2000. DYN3D – three-dimensional core modelfor steady-state and transient analysis of thermal reactors. In: Proc. PHYSOR2000, Pittsburgh, Pennsylvania, USA.

Guo, Z., Zhou, J., Zhang, D., Chaudri, K., Tian, W., Su, G., Qiu, S., 2013. Coupledneutronics/thermal–hydraulics for analysis of molten salt reactor. Nucl. Eng.Des. 258, 144–156.

Helmholtz-Zentrum Dresden-Rossendorf home page: www.hzdr.de.Imke, U., Sanchez, V., 2012. Validation of the subchannel code SUBCHANFLOW using

the NUPEC PWR test (PSBT). Sci. Technol. Nucl. Inst..Ipsen, I.C.F., Meyer, C.D., 1998. The idea behind Krylov methods. Am. Math. Mon.

105 (10), 889–899.Ivanov, K., Avramova, M., 2007. Challenges in coupled thermal–hydraulics and

neutronics simulations for LWR safety analysis. Ann. Nucl. Energy 34, 501–513.Jiménez, J., Cuervo, D., Aragonés, J.M., 2010. A domain decomposition methodology

for pin by pin coupled neutronic and thermal–hydraulic analyses in COBAYA3.Nucl. Eng. Des. 240, 313–320.

Joo, H.G., Barber, D., Jiang, G., Downar, T.J., 1998. PARCS, A Multi-dimensional Two-group Reactor Kinetics Code based on the Nonlinear Analytic Nodal Method, PU/NE-98-26, Purdue University.

Kliem, S., Hohne, T., Rohde, U., Kozmenkov, Y., 2007. DYN3D/ATHLET and ANSYSCFX calculations of the OECD VVER-1000 collant transient benchmark. In: TheFifth International Conference: Safety Assurance of NPP with WWER, Russia.

Knoll, D.A., Keyes, D., 2004. Jacobian-free Newton–Krylov methods: a survey ofapproaches and applications. J. Comput. Phys. 153, 357–397.

Kotlyar, D., Shaposhnik, Y., Fridman, E., Shwageraus, E., 2011. Coupled neutronicthermo-hydraulic analysis of full PWR core with Monte-Carlo based BGCoresystem. Nucl. Eng. Des. 241, 3777–3786.

Kozlov, V.F., Zemlyanskii, M.G., 1963. Design details of the VVR-S experimentalnuclear reactor. J. Nucl. Energy. Parts A/B Reactor Sci. Technol. 17, 33–39.

Krepel, J., Rohde, U., Grundmann, U., Weiss, F., 2007. DYN3D-MSR spatial dynamicscode for molten salt reactors. Ann. Nucl. Energy 34, 449–462.

Kromar, M., Slavic, S., Trkov, A., 1991. CTEMP: A Code for ThermohydraulicCalculations. Institute Jozef Stefan, Ljubljana, Slovenia, IJS-DP-6143.

MacFarlane, R.E., Muir, D.W., 1999. NJOY99.0 Code System for Producing Pointwiseand Multigroup Neutron and Photon Cross Sections from ENDF/B Data, PSR-480/NJOY99.00. Los Alamos National Laboratory, Los Alamos.

Martin, W.R., 2012. Challenges and prospects for whole-core Monte Carlo Analysis.Nucl. Eng. Technol. 44 (2).

Maschek, W., Rineiski, A., Suzuki, T., Wang, S., Mori, Mg., Wiegner, E., Wilhelm, D.,Kretzschmar, F., 2005. SIMMER-III and SIMMER-IV Safety Code Development forReactors with Transmutation Capability. In: Mathematics and Computation,Supercomputing, Reactor Physics and Biological Applications, Palais des Papes,Avignon, France, September 12–15, 2005.

MCNP home page: mcnp.lanl.gov.Monti, L., Starflinger, J., Schulenberg, T., 2011. Development of a coupled neutronic/

thermal–hydraulic tool with multi-scale capabilities and applications to HPLWRcore analysis. Nucl. Eng. Des. 241, 1579–1591.

Mousseau, V., 2004. Implicitly balanced solution of the two-phase flow equationscoupled to nonlinear heat conduction. J. Comput. Phys. 200, 104–132.

NURESAFE EXCOM, 2013, D1.11-Project presentation, CEA, NURESAFE-D1.11-version 1.2.

Ragusa, J., Mahadevan, V., 2009. Consistent and accurate schemes for coupledneutronics thermal-hydraulics reactor analysis. Nucl. Eng. Des. 239, 566–579.

Ravnik, M., Trkov, A., Kromar, M., Slavic, S., Mele, I., Zefran, B., Zeleznic, N., 2008.CORD-2 Core Design system for PWR Type reactors, report. Institute JozefStefan, Slovenia.

RELAP5/MOD3.3 CODE MANUAL, Code structure, system models, and solutionmethods’’, Nuclear Safety Analysis Division, Information Systems Laboratories,Inc., volume 1 Rockville, Maryland, Idaho Falls, Idaho, 2001.

Repetto, G., Jacq, F., Lamare, F., Ricaud, J., Barre, F., Nihon Genshiyoku Gakkai, 2009.DRACCAR, a new 3d-thermal mechanical computer code to simulate locatransient on nuclear power plants: status of the development and thevalidation. In: International Congress, Advances in nuclear power plants, Tokyo.

Rimpault, G., Plisson, D., Tommasi, J., Jacmin, R., 2002. The ERANOS code and datasystem for fast reactor neutronic analyses. In: Proc. Int. Conf. PHYSOR 2002,Seoul, Korea.

Saad, Y., Schultz, M., 1986. GRMES: a generalized minimal residual algorithm forsolving non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856.

Santamarina, A., Bernard, D., Blaise, P., Leconte, P., Vaglio, C., Vidal, J.F., 2009.APOLLO2.8: a validated code package for PWR neutronics calculations. In: Proc.Int. Conf. Advances in Nuclear Fuel Management IV (ANFM2009), Hilton HeadIsland, South Carolina, USA, April 12–15, 2009.

Shaposhnik, Y., Shwageraus, E., Elias, E., 2010. The effect of radial distribution ofcoolant density on the multiplication factor. In: Proc. of IYNC, Cape Town, SouthAfrica.

Spore, J.W., Guffee, L.A., Steinke, R.G., Woodruff, S.B., 1992. TRAC-PF1/MOD2.Programmer’s Guide, Los Alamos National Lab., Technical Report, US NuclearRegulatory Commission Report NUREG/CR-5673-V3.

TLK-Thermo GmbH home page: www.tlk-thermo.com.Tohjoh, M., Watanabe, M., Yamamoto, A., 2006. Three-dimensional pin power

reconstruction for the axially heterogeneous region in BWR. Ann. Nucl. Energy33, 242–251.

Toumi, I., Bergeron, A., Gallo, D., Royer, E., Caruge, D., 2000. FLICA-4: a three-dimensional two-phase flow computer code with advanced numerical methodsfor nuclear applications. Nucl. Eng. Des. 200, 139–155.

TRACE V5.0 (patch 01): User’s Manual, US NRC, Office of Nuclear RegulatoryResearch, 2008.

Trkov, A., 2008. GNOMER – Multigroup 3-Dimensional Neutron Diffusion NodalCode with Thermohydraulic Feedbacks. Institute Jozef Stefan, Ljubljana,Slovenia, IJS- DP-6688.

Trkov, A., Ravnik, M., 1994. CORD-2 package for PWR nuclear core designcalculations. In: Proceedings of the International Conference on ReactorPhysics and Reactor Computations, Tel-Aviv, Ben-Gurion University of theNegev Press, 23–26 January, 1994.

Vasquez, M., Tsige-Tamirat, H., Ammirabile, L., Martin-Fuertes, F., 2012. Coupledneutronics thermal–hydraulics analysis using Monte Carlo and sub-channelcodes. Nucl. Eng. Des. 250, 403–411.

Wang, S., Rineiski, A., Maschek, W., 2006. Molten salt related extensions of theSIMMER-III code and its application for a burner reactor. Nucl. Eng. Des. 236,1580–1588.

Watson, J., Ivanov, K., 2002. Improved cross-section modeling methodology forcoupled three-dimensional transient calculations. Ann. Nucl. Energy 29, 937–966.

Wheeler, C., Stewart, C., Cena, R., Rowe, D., Sutey, A., 1976. COBRA-IV-I: An InterimVersion of COBRA for Thermal-Hydraulic Analysis of Rod Bundle Nuclear FuelElements and Cores, BNWL-1962.

Yamamoto, T., Mitachi, K., Suzuki, T., 2005. Steady state analysis of small molten saltreactor. JSME Int. J. B Fluid T. Ser. B 48 (3).

Yamamoto, T., Mitachi, K., Ikeuchi, K., Suzuki, T., 2006. Transient characteristics ofsmall molten salt reactor during blockage accident. Heat Tran. Asian Res. 35,434–450.

Yesilyurt, G., Martin, W.R., Brown, F.B., 2011. On-the-Fly Doppler broadening forMonte Carlo codes. Nucl. Sci. Eng. 171 (3).

Zare, N., Fadaei, A.H., Rahgoshay, M., Fadaei, M.M., 2010. Introducing and validatinga new method for coupling neutronic and thermal-hydraulic calculations. Nucl.Eng. Des. 240, 3727–3739.

Zerkak, O., Manera, A., Gajev, I., Kozlowski, T., 2010. Review of Multi-physicsCoupling Techniques and Suggestions of Improvements in the Context ofNURISP, D-3.2.1.1-Revision 0, PSI, 2010.




http://www.hzdr.de





















http://mcnp.lanl.gov










http://www.tlk-thermo.com























multi-physics and multi-scale methods used in nuclear reactor analysis

Documents