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Nuclear Engineering and Design 252 (2012) 108– 120

Contents lists available at SciVerse ScienceDirect

Nuclear Engineering and Design

j ourna l ho me pag e: www.elsev ier .com/ locate /nucengdes

he AMP (Advanced MultiPhysics) Nuclear Fuel Performance code

evin T. Clarnoa,∗, Bobby Philipa, William K. Cochrana, Rahul S. Sampatha, Srikanth Allua, Pallab Baraia,rdjan Simunovica, Mark A. Berrill a, Larry J. Otta, Sreekanth Pannalaa, Gary A. Diltsb,ogdan Mihailab, Gokhan Yesilyurtc, Jung Ho Leec, James E. Banfieldd

Oak Ridge National Laboratory, United StatesLos Alamos National Laboratory, United StatesArgonne National Laboratory, United StatesUniversity of Tennessee-Knoxville, United States

i g h l i g h t s

New, three-dimensional, parallel, multi-physics code to simulate fuel behavior in nominal operation.Fully-coupled thermomechanics for nominal operation and operation during transients.Isotopic depletion using Scale/ORIGEN-S within a fuel performance code.Leveraging of existing, validated material models from existing fuel performance codes.Initial validation evaluation of an advanced modeling and simulation code for fuel performance.

r t i c l e i n f o

rticle history:eceived 13 December 2011eceived in revised form 9 July 2012ccepted 20 July 2012

a b s t r a c t

The AMP (Advanced MultiPhysics) Nuclear Fuel Performance code is a new, three-dimensional, multi-physics tool that uses state-of-the-art solution methods and validated nuclear fuel modelsto simulate the nominal operation and anticipated operational transients of nuclear fuel. TheAMP Nuclear Fuel Performance code leverages existing validated material models from traditional fuelperformance codes and the Scale/ORIGEN-S spent-fuel characterization code to provide an initial capa-bility that is shown to be sufficiently accurate for a single benchmark problem and anticipated to be

accurate for a broad range of problems. The thermomechanics foundation can be solved in a time-dependent or quasi-static approach with any variation of operator-split or fully-coupled solutions ateach time step through interoperable interfaces to leading computational mathematics tools, includ-ing PETSc, Trilinos, and SUNDIALS. A baseline validation of the AMP Nuclear Fuel Performance code hasbeen performed through the modeling of an experiment in the Halden Reactor Project (IFA-432) thatdemonstrates the integrated capability and provides a baseline of the initial accuracy of the software.

. Introduction

Developing new fuels and qualifying them for large-scaleeployment in power reactors is a lengthy and expensive process,ypically spanning a period of two decades from concept to licens-ng. In recent years, fuel performance capabilities based on firstrinciples have been playing more of a role in what has tradition-lly been an empirically dominated process. Nonetheless, nuclearuel behavior is based on the interaction of multiple complex phe-

omena, and recent evolutionary approaches are being appliedore on a phenomenon-by-phenomenon basis, targeting localized

∗ Corresponding author. Tel.: +1 865 241 1894; fax: +1 865 574 9619.

029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.nucengdes.2012.07.018

© 2012 Elsevier B.V. All rights reserved.

problems, as opposed to a systematic approach based on a funda-mental understanding of all interacting parameters.

Advanced nuclear fuels are generally more complex and lesswell understood than the traditional fuels used in existing reactors(e.g. ceramic UO2 with burnable poisons and other minor addi-tives). The added challenges are primarily caused by a less completeempirical database and, in the case of recycled fuel, the inherentvariability in fuel compositions. It is clear that it would be challeng-ing, if not impossible, to use the traditional approach to develop andqualify fuels over the entire range of variables pertinent to the U.S.Department of Energy (DOE) Office of Nuclear Energy on a timely

basis with available funds.

As a result the Advanced Modeling and Simulation Office(AMSO) in the DOE Office of Nuclear Energy has launched theNuclear Energy Advanced Modeling and Simulation (NEAMS)

ering a

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K.T. Clarno et al. / Nuclear Engine

pproach to revolutionize fuel development. This new approach isredicated upon transferring the recent advances in computationalcience and computer technologies into the fuel development pro-ram. The effort will couple computational science with recentdvances in the fundamental understanding of physical phenom-na through ab initio modeling and targeted phenomenologicalesting to leapfrog many fuel-development activities. Realizinghe full benefits of this approach will likely take some time.owever, it is important that the developmental activities for mod-ling and simulation be tightly coupled with the experimentalctivities to maximize feedback effects and accelerate both thexperimental and the analytical elements of the program toward aommon objective. As current-practice modeling approaches (suchs FRAPCON, Berna et al., 1997 used to license fuel by the U.S.uclear Regulatory Commission) use many empirical models, these

ools have been shown to predict, within conservative margins,ight-water reactor (LWR) fuel performance. Therefore, incorpo-ating models from these existing codes for legacy equivalence isital to maintaining experimental accuracy. The close integrationf modeling and simulation with experimentation is essential toeveloping a useful fuel performance simulation capability, pro-iding a validated design and analysis tool, and understanding thencertainties within the models and design process.

The long-term vision for the Fuels Integrated Performance andafety Code (IPSC) within the NEAMS program is to deliver auel performance capability that can predict, with uncertaintystimates, the failure of fuel pins during nominal operation andnticipated operational transients for a wide variety of fuel andeactor forms. These include, but are not limited to, UO2 or mixedxide (MOX) in an LWR and metal or oxide transmutation fuel in aodium-cooled fast reactor (SFR). This will require, at minimum,oupling a continuum-scale nuclear fuel performance capabilityith an atomistically-informed grain-scale physics code that canrovide material property data at the continuum, and coupling with

reactor simulator that can provide the global source-terms andoundary conditions (flow and neutronics) during an anticipatedperational transient.

The AMP Nuclear Fuel Performance code (AMP) is a new three-imensional (3D) multi-physics tool that uses state-of-the-artolution methods and validated nuclear fuel models to simu-ate nominal operation and anticipated operational transients ofuclear fuel (Turner et al., 2009; Clarno et al., 2011; Dilts et al.,010). AMP leverages existing validated material models fromraditional fuel performance codes and the Scale/ORIGEN-S codeSCALE, 2005; Gauld et al., 2005) to provide an initial fuel per-ormance capability. There are many existing efforts to develop aarallel, 3D fuel performance code, including PLEIADES/ALCYONEThouyenin et al., 2006, 2007), BISON (Newman et al., 2009), andACO (Marino et al., 2007). However, AMP Nuclear Fuel Performanceode is unique in several respects:

It is designed to easily incorporate external packages with veryminor modifications allowing access to existing, validated mate-rial models and codes, as demonstrated through the use of simpleinterfaces to Scale/ORIGEN-S and (some routines of) MATPRO(Berna et al., 1997).It is designed to be easily incorporated as a tightly-coupled com-ponent in an external code system, such as a reactor simulator orlower-length-scale code suite, which will allow others to lever-age its continuum-scale fuel performance capabilities (as will bedemonstrated in a future publication).

It incorporates the isotopic depletion, decay, and transmuta-tion capabilities of Scale/ORIGEN-S to provide comprehensive,validated time-variation of more than 1900 nuclides for manyconceivable fuel concepts (Section 3.2).

nd Design 252 (2012) 108– 120 109

This paper is written to serve as an overview of themethods and capabilities that make up the core of theAMP Nuclear Fuel Performance code. However, because theAMP Nuclear Fuel Performance code is under continuous devel-opment, this paper represents a “snapshot in time” of the version0.9 of AMP Nuclear Fuel Performance code, as distributed by theRadiation Safety Information Computational Center (RSICC) inSeptember 2010. For that reason, the focus is on the fundamentalmethods used in the code and a demonstration of its use. Addition-ally, where existing methods and mathematics are well understoodor commonplace, references are given in order to save space forthe more novel aspects of the AMP Nuclear Fuel Performance code.In Section 2, the overall objectives and current capabilities ofthe AMP Nuclear Fuel Performance code are presented, which leadto the long-term requirements for the code. Section 3 providesan overview of the fundamental components that make up theAMP Nuclear Fuel Performance code. In Section 4, the initial valida-tion problem and results are presented, with an analysis of boththeir significance and their insignificance. Section 5 concludes thepaper with a summary of the findings.

2. The AMP Nuclear Fuel Performance code

2.1. Objectives

AMP is a new code that was built from scratch with majorleveraging of existing off-the-shelf (OTS) codes to meet interimobjectives to (1) deliver a useful capability to the user community;(2) enhance understanding of the software and user requirements;(3) demonstrate an understanding of the coupled physics simula-tion process with best-of-class software; and (4) gain experiencedeveloping software as a multi-institutional team with a singleset of coding conventions, standards, and tools. To meet theseprimary objectives, the AMP Nuclear Fuel Performance code projectembarked on a path to develop

• a tightly-coupled, 3D thermo-mechanical solver• approximate models for the material properties, depletion, heat

generation, and convective heat transfer similar to those foundin FRAPCON (Berna et al., 1997) and SCALE (SCALE, 2005)

• a simple user interface to set up, simulate, and understand theperformance of LWR oxides

• a compiled version that executes in parallel on a cluster at OakRidge National Laboratory

However, the initial release (0.9) of theAMP Nuclear Fuel Performance code was rapidly designed anddeveloped without a focus on extensibility or software qualityengineering (especially within the OTS components). A full fuel pincapability is under development and testing, with a public releaseof the software expected in 2012.

2.2. Overview of capabilities

AMP computes the 3D thermal, mechanical, and stoichiomet-ric state of traditional nuclear fuel (UO2) in an LWR. Nuclear fuelin an LWR is composed of hundreds of individual ceramic UO2pellets stacked inside a protective metal cladding tube, whichis surrounded by flowing water to cool the system. The nuclearheat produced in the materials is transported through the fuel,across a gap, and through the cladding and then removed by

the coolant. The materials respond mechanically to the thermalstresses, and the stoichiometric state changes as a result of ther-mal gradients. Because the temperature distribution is dependentupon both the mechanical and stoichiometric state, the physics are

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lways nonlinearly coupled. Nuclear irradiation changes the iso-opic and elemental composition of the materials, which changeshe material properties of the materials and, along with thermalradients, imposes slowly varying stresses (densification, creep,nd swelling).

To model these physical processes theMP Nuclear Fuel Performance code includes

a neutron-flux-derived fission and gamma heat sourceincremental elastic–plastic mechanics within solid bodies, butneglecting mechanical stresses between solid bodiesmechanics models for thermal expansion, irradiation-induceddensification and swelling, relocation, and temperature-,irradiation-, and stress-induced creeptime-dependent nonlinear thermal diffusion within solid bodiesand approximate heat transfer between solid bodiessteady-state, one-dimensional, single-phase coolant heatremovalnonlinear material properties for UO2, MOX, U–Zr, U–Pu–Zr, Na,H2O, and Henonlinear and linear solversimplicit and explicit time integratorsa general computational backplane

However, several significant types of science are not modeled,ncluding, but not limited to, chemistry, mechanical contact andracture, multidimensional flow and neutronics, and grain-levelhysics. A primary purpose of the AMP Nuclear Fuel Performanceode is to serve as an exploratory tool for understanding the soft-are requirements associated with incorporating these physics in

he planning for a future, predictive nuclear fuel performance code.herefore, the AMP Nuclear Fuel Performance code is designed tonable the incremental incorporation of additional physics for rapidrototyping. Each physics capability that may be modeled will have

different degree of coupling with the other physics and requireesolution on different time scales.

.3. Software design

AMP was developed with a design that was defined by the initialequirements of the software and developers; the primary objec-ives include the following:

Provide the software infrastructure to perform end-to-end cal-culations of time-dependent, coupled nonlinear multi-physicsproblems.Leverage existing investments in software by DOE and others fortime integrators, nonlinear solvers, linear solvers, discretizations,and meshing.Couple only loosely to external software packages, ensuring thatAMP is not tied to the development path or bottlenecks of externalpackages for time integrators, nonlinear solvers, linear solvers,discretizations, meshing, and materials databases.Leverage the strengths of multiple packages simultaneously; forexample, use SUNDIALS (Hindmarsh et al., 2005), PETSc (Balayet al., 2010), and Trilinos (Heroux et al., 2005) at the same time totake advantage of their strengths.Provide extensibility to allow developers to explore new algo-rithms for any of the areas mentioned above and to use consistentinterfaces to test their ideas with other parts of the frameworkseamlessly.

Allow for incremental development in multi-physics problemswithout a need to rewrite code as new physics is introduced.Allow the development of intrusive and nonintrusive uncertaintyquantification techniques.

nd Design 252 (2012) 108– 120

Since AMP is only loosely coupled to external packages, its long-term stability and viability are not restricted or compromised bythe existence or nonexistence of any external package.

3. Software components

An AMP Nuclear Fuel Performance Technical Specificationsdocument is distributed with the software. However, a brief intro-duction to the physics and mathematics modeled is included in thissection. In addition, these capabilities have varying degrees of ver-ification, integration with other components, and completeness, asexpected in a rapid prototyping effort.

3.1. Power

The power component within the AMP Nuclear Fuel Performancecode establishes the spatial distribution of the heat source duringthe irradiation. While traditional fuel performance codes allow theuser to define the average linear heat generation rate and axialshape functions, as a function of time, and the burnup-dependentradial distribution within the fuel is computed internally, AMP pro-vides the user with additional flexibility to provide the power (orneutron flux), as a function of time, in three-dimensions.

Eq. (1) is the general function that describes the power (or flux)distribution P(r, �, z) as a function of the radius (r) from the cen-ter of the pellet, height (z), and azimuthal-angle (�) about the axialdimension (z), which is used to compute the heat-source as a func-tion of time at every Gauss-point in the problem. The user-definedpower definition allows for a simple definition of the axial shapefunctions, the radial rim effect, and azimuthal variations in thepower due to assembly features. The radial power shape includesthe option to use a model that coincides with the empirically-derived TUBRNP model (Eq. (2)) in TRANSURANUS (Lassmann,1992; Lassmann et al., 1994; Schubert et al., 2008) (also includedin FRAPCON).

P(r, �, z) = 1 + a F(r) + b� sin(�) +∑k>0

ckPk(z); (1)

F(r) = 1 + 3.45 exp[−3(R − r)0.45]. (2)

In Eq. (1), the user-defined coefficients (a, b� , and ck) define themagnitude of spatial variation in each dimension, where Pk(z) areLegendre polynomials (Weisstein, 2011) about the axial dimension,and F(r) is the TUBRNP model that is based on the radius and theouter radius of the fuel pellet (R), and � is the azimuthal variationaround the axial dimension, which is used to represent effects ofassembly features, such as guide-tubes or control rods. At present,the TUBRNP model is used to define the power distribution, andthe Scale/ORIGEN-S depletion is performed with a constant powerapproximation. Figs. 1 and 2 display representative power and bur-nup distributions (at a mid-cycle time step of the initial validationproblem described in Section 4).

3.2. Depletion

The governing transmutation equation (Eq. (3)) describes theradioactive decay of one isotope into another, the depletion of fuelthrough fission, and the transmutation of one isotope into anotheras a result of a neutron absorption. The transmutation equation isnondimensional in space and can apply to a single location, cell, oraverage for a material.

dN̂

dt= �SN̂ + LN̂ + b̂, (3)

where N̂ is the concentration of all isotopes being tracked (upto 2000), � is the neutron flux (neutrons per square centimeter

K.T. Clarno et al. / Nuclear Engineering a

pdracti(rstiic

pS

Fig. 1. Representative power distribution within a fuel pellet.

er second), S is the matrix representing the transmutation andepletion of isotopes due to neutron absorptions, L is the matrixepresenting the radioactive decay constants (decays per second),nd b̂ is the vector of external rates of isotope production, whichould be due to spatial redistribution through fission gas or speciesransport. The decay matrix (L) is inherent to the isotopes andndependent of the problem being solved. The reaction rate matrix�S) is problem-dependent, because the probability of a particulareaction can depend on the neutron velocity distribution (energypectra). Therefore, it is properly defined only with a known solu-ion to the energy-dependent Boltzmann transport equation, whichs beyond the scope of a fuel performance code but is commonn many lattice physics codes or continuous-energy Monte Carlo

odes.

The externally defined power, or flux, distribution isrovided to the isotopic decay and transmutation codecale/ORIGEN-S, which was modularized and embedded within

Fig. 2. Representative burnup distribution within a fuel pellet.

nd Design 252 (2012) 108– 120 111

the AMP Nuclear Fuel Performance code to provide a well-validateddepletion capability for a wide range of fuel types. Included withthe distribution of AMP are cross-section libraries representativeof specific fuel types in a given reactor environment, including UO2in a pressurized water reactor (PWR), boiling water reactor (BWR),and CANDU; MOX in a PWR and BWR; uranium oxycarbide (UCO)in a high-temperature gas reactor; and U-TRU-Zr in an SFR. Eachof these libraries was developed and is distributed with the Scalenuclear analysis code suite, which has been well validated for eachof these fuel/reactor configurations. The depletion capability canuse any of the cross section libraries that are distributed with theORIGEN-ARP sequence in Scale.

The isotopes and transmutation rates and distributions may beplotted, compressed to provide the elemental concentrations andtransmutation rates (such as oxygen), and compressed to providerequired “reduced quantities,” such as metal and fission gas concen-trations and transmutation rates. The individual chemical elementsare provided to the chemical species component as an input, andthe fission gas generation rate will be provided to the gas releasemodels. There are many challenges and nuances associated with theuse of Scale/ORIGEN-S within a fuel performance code, which willbe discussed in a later publication. For this work, the depletion ismerely used as a means of accurately computing all isotopic distri-butions for the given fuel pellet, regardless of fuel type, throughoutthe fuel cycle and does not directly affect the thermo-mechanicssimulation.

3.3. Heat transfer

Heat transfer in AMP models the various physics with appro-priate models, including nonlinear thermal diffusion in the pelletand clad, axial coolant flow in the channel, and gap conductancebetween the pellet and cladding.

The thermal diffusion component of AMP uses standard finiteelement technology for solving the time-dependent (Eq. (4)), orstatic (�Cp(∂T/∂t) = 0), diffusion equation and derives materialmodels from FRAPCON and the literature. It treats the general casein which all the material-dependent coefficients can be fairly com-plicated nonlinear functions. The thermal diffusion is solved on theinitial mesh and does not account for the effects of displacementsdue to mechanics.

�Cp∂T

∂t= ∇ · [k∇T] + Q̇ . (4)

In Eq. (4), � is the density, Cp is the heat capacity at constantpressure, T is the temperature, t is the time, k is the conductivity,and Q̇ is the power.

A simple uniaxial coolant flow model (Eq. (5)) evaluates the bulkcoolant temperature (Tb) as a function of fuel height (z), whichremoves heat from the pellet surface in steady-state operation,based upon the mass flow rate (G), effective diameter (De), and heatcapacity at constant pressure (Cp), due to a surface heat flux (q′′) as afunction of cladding height (z). This is defined with a mapping oper-ation of the two-dimensional clad surface temperature and heatflux distributions to the azimuthally-smeared one-dimensional (z)temperature and heat flux distributions. It has been verified totransfer the magnitude and a linear slope with exact numericalprecision.

Tb(z) = Tin +∫ z

0

4q′′(z)CpGDe

dz (5)

The thermal heat source (Q̇ ) is derived from the neutronics pack-age, which can provide user-defined space–time distributions. Atthis time, only piecewise constant variations in time have beenimplemented, leading to rapid thermal transients. A representative

112 K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120

F

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Fig. 4. Representative radial displacement distribution within a fuel pellet andshape of the radial surface (pink line). (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of the article.)

Table 1Material properties in the AMP Nuclear Fuel Performance code.

Thermal conductivity (k) Thermal expansion coefficient (˛axial , ˛radial)Heat capacity at constant

pressure (Cp)Youngs modulus (E)

Density (�) Poissons ratio ()Elastic yield stress (�0) Linear strain hardening (H)Viscosity () Strain hardening exponent (n)

advantageous to include the derivatives of some of each of theseproperties with respect to the condition (e.g. temperature or bur-nup). At this time, AMP has libraries of properties for the materialslisted in Tables 2–4 for fuel, cladding, and coolant, respectively.

Table 2Fuel materials in the AMP Nuclear Fuel Performance code.

Fuel Sources

ig. 3. Representative temperature distribution within a fuel pellet and cladding.

uel, clad, and coolant temperature distribution is shown in Fig. 3at a mid-cycle time step of the initial validation problem describedn Section 4).

.4. Mechanics

The mechanics component of AMP approximates the govern-ng equation that describes a body, �, undergoing an infinitesimaltrain and displacement. It satisfies Eq. (6) at static mechanicalquilibrium.

� + �b = 0 within � (6)

Here, �, � and b are the symmetric Cauchy stress tensor, den-ity, and (internal and external) body forces, respectively. � is aunction of the displacements, u; the precise form of this functionepends on the material used in the body. This is solved with atandard finite-element technology; however, certain types of ele-ents, such as the 8-noded hexahedral element, exhibit a behavior

alled volumetric locking, in which the element behaves more stifflynder some conditions; the reduced integration scheme is used toeduce this locking.

The mechanics material models provide the correct estimate oftress (�ij) at each Gauss point subjected to some external strain�ij), based on the deformation mechanisms (e.g. thermal expan-ion, swelling, densification) causing the displacement of the fuelellet and the clad. These material models also return the constitu-ive matrix (Cijkl) at each Gauss point, which relates the strain withhe stress through the equation (Simo and Hughes, 1998),

ij = Cijkl�kl. (7)

mall strain conditions are assumed in developing the materialodels. The total strain (�tot

ij) can be written in additive form,

totij = �e

ij + �pij

+ �thij + �sw

ij + �deij + �c

ij + �reij , (8)

here �eij

is the elastic strain; �pij

is the plastic strain; �thij

is the

train due to thermal expansion; �swij

is the swelling strain; �deij

ishe densification strain; �c

ijdenotes the strain due to temperature,

rradiation and stress-induced creep; and �reij

signifies the reloca-

ion strain. All strains can be computed either implicitly (default)r explicitly, and all properties are functions of various parameters,ncluding temperature and burnup. The fuel and cladding lever-ge the identical infrastructure for solving Eqs. (6)–(8), but not all

Fuel swelling Creep terms (linear, nonlinear, transition)Yield stress Shear modulus (G)Fuel densification Explicit and implicit creep

strains are included in both materials (e.g. cladding has no densifi-cation, swelling, or relocation terms). Fig. 4 shows a representativeradial displacement within a fuel pellet (at a mid-cycle time step ofthe initial validation problem described in Section 4), in which thepink line represents the shape of the fuel pellet surface.

3.5. Material properties

In AMP, a material library is defined by a collection of properties(e.g. thermal conductivity or density), which can each be definedby a model that is used to evaluate the property at a given condi-tion (e.g. temperature, linear heat generation rate, oxygen-to-metalratio, burnup, etc.), where the properties and models originatedfrom a common source or methodology (UO2 in MATPRO). Somematerial models (such as the thermal diffusion coefficient or creep)are library-independent, but require properties (e.g. creep terms)from a library. The list of material properties, with traditional vari-able notation in parenthesis, that are available to use by the physicsoperators are given in Table 1.

Because some material libraries come from literature thatexplores a specific physical process (e.g. Mihaila et al., 2009), notall material libraries have all properties defined. In addition, someapplications require extensions of the base material properties toinclude new features. For example, in the process of ”method ofmanufactured solution” testing of the diffusion operator, it was

UO2 in an LWR Berna et al. (1997) and Mihaila et al. (2009)MOX in an LWR Berna et al. (1997)U–Zr in an SFR Karahan (2009)U–Pu–Zr in an SFR Karahan (2009)

K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120 113

Table 3Cladding materials in the AMP Nuclear Fuel Performance code.

Cladding Source

Zircaloy 4 Berna et al. (1997)316 stainless steel Mihaila et al. (2009)HT9 steel Karahan (2009)

Table 4Coolant materials in the AMP Nuclear Fuel Performance code.

Coolant Source

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Table 6Time integration algorithms available in the AMP Nuclear Fuel Performance code.

Explicit Implicit DAE

Euler BDF1-5 BDF1-52nd order Runge-Kutta

Water Incropera and Dewitt (1996)Sodium Fink and Leibowitz (1995)Helium Mihaila et al. (2009)

etails of the particular models within each library, along withriginal references for each model in the library, are provided inhe associated sources of Tables 2–4, as well as the User Manual forhe AMP Nuclear Fuel Performance code (Clarno et al., 2010).

.6. Computational backplane

The mathematical problem addressed by AMP involves solvingarge discretizations of coupled nonlinear time-dependent systemsf differential equations. DOE has invested millions of dollars inhe development of general purpose mathematical solver soft-ares to tackle just such problems on parallel cluster computing

rchitectures. The effort has taken place at several national lab-ratories over approximately two decades through the Office ofcience Advanced Scientific Computing Research and SciDAC pro-rams and the National Nuclear Security Administration Advancedimulation and Computing program. AMP heavily leverages manyf these tools, including the third-party libraries listed in Table 5.hese in turn depend on standard libraries like MPI, HDF5, andAPACK, which often must be site-built along with the others. Theseibraries have been integrated into multiphysics simulation toolsMahadevan et al., 2012), but are primarily based on the use of aingle mesh for all science domains (Gaston et al., 2009).

Establishing a single mesh database and solver library couldave been made at the outset of the project; but doing so wouldave restricted a major high-level requirement: to provide a fuelapability that can be embedded in an externally-developed reactorimulation tool, and a general software development philosophyf allowing developers to choose the best tool available for a givenpplication rather than restricting them to the available capabili-ies in a single package. However, each library of Table 5 has its ownay of representing vectors and matrices for use in its solvers. Using

bject-oriented technology, the AMP Nuclear Fuel Performance codeomputational backplane provides a high-level uniform inter-ace that frees application developers working with AMP from

he technical details of interfacing among the various packages.his interface makes it relatively painless, for example, to use aUNDIALS time integrator with a PETSc inexact Newton–Krylovonlinear solver and a Trilinos multi-level preconditioner. It also

able 5xternal libraries used in the AMP Nuclear Fuel Performance code.

Library Capabilities Reference

PETSc Vectors, matrices, linear andnonlinear solvers

Balay et al. (2010)

Trilinos Vectors, matrices, linear andnonlinear solvers

Heroux et al. (2005)

SUNDIALS Time integrators withnonlinear solvers

Hindmarsh et al. (2005)

Libmesh Meshes, finite element tools Kirk et al. (2006)HYPRE Linear solvers Falgout et al. (2005)

4th order Runge-KuttaBogacki-Shampine

makes it straightforward to add new packages later. This feature isone of the great strengths of AMP.

3.7. Solvers

Performing the simulations that AMP is targeting requires solv-ing a number of static and time-dependent partial differentialequations that describe diverse physical processes. The coefficientsin the differential equations representing material properties maybe extremely nonlinear and depend on many different variables.It is necessary to solve a large system of highly-coupled nonlin-ear algebraic equations of considerable complexity for each timestep. It is often nearly impossible, expensive, or time consumingto construct an explicit tangent stiffness matrix for such a system.A common numerical tool of choice for meeting such a challengeis the Jacobian-Free Newton Krylov (JFNK) method. JFNK can beused to solve coupled multi-physics problems without the need tocompute expensive explicit Jacobian matrices.

The user is able to define the collection of tools from the exter-nal packages (Table 5) or internal to AMP that are most appropriatefor the particular problem. In addition to using a JFNK method (viaPETSc, or within SUNDIALS), AMP also provides internal solvers,such as the Nonlinear Krylov Accelerated Inexact Newton’s method.The most commonly used suite, and default for the fuel designerinterface, are the extensively-developed Scalable Nonlinear Equa-tions Solvers (SNES) nonlinear inexact Newton, and KSP linearKrylov solvers from Argonne National Laboratory’s PETSc library.Iterative Krylov solvers require a user-selected preconditioner or anapproximation to the matrix of interest with an easier-to-computesolution for efficiency. The AMP code uses the block-diagonalmatrix built from the linear finite element stiffness matrix for eachequation and, in most cases, the ML multi-level linear solver fromthe Trilinos library. The AMP backplane, however, allows the use ofany user-constructed preconditioner solver.

Development of reliable, scalable parallel solvers is an expensiveand time-consuming process. The availability of PETSc and Trilinoshas thus greatly accelerated the development of AMP.

3.8. Time integrators

A suite of explicit and implicit time-integrators are availablewith AMP that were extracted and reused from the SAMRSolverscode. Because the mechanics component is solved in a quasi-static mode, and other physics, such as heat transfer, may betime-dependent, the resulting problem is a differential-algebraicequation (DAE) system. The integration method used by the IDAtime integrator, within the SUNDIALS code (Hindmarsh et al.,2005), is variable-order, variable-coefficient backward differentia-tion formula (BDF), in fixed-leading-coefficient form. A summary ofthe time-integration capabilities in AMP Nuclear Fuel Performancecode is shown in Table 6.

3.9. User interfaces

There are two primary user interfaces for accessing the capabil-ity within the initial release of the AMP Nuclear Fuel Performancecode (version 0.9): a simple fuel designer interface for executing

114 K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120

gram

ldabte

3

niwiatsdt

ddcupitd

idpTsdm(fpwststaidTNsulTSno

Fig. 5. Chart depicting flow dia

egacy-equivalent traditional nuclear fuel studies and an experteveloper interface that provides access to the full functionalityvailable. Not all available features of the expert interface haveeen rigorously tested. These user interfaces are documented inhe User Manual for the AMP Nuclear Fuel Performance code (Clarnot al., 2010), which is distributed with the software.

.10. Solution strategy

AMP is designed to be extensible to allow developers to exploreew algorithms for any of the areas mentioned and use standard

nterfaces to test their ideas with other parts of the frame-ork seamlessly. This design allows for incremental development

n multi-physics problems without a need to rewrite code asdditional physics is introduced. Therefore, there is no fixed “solu-ion strategy” for all problems. However, the standard solutiontrategy for fully coupled quasi-static problems and implicit time-ependent full-coupled problems includes a preprocessor step, aime-varying multi-physics solve, and a post-processing step.

The preprocessing step includes input processing (from a fuelesigner input to an expert developer database), reading andecomposing data (including meshes), and neutronics prepro-essing. The neutronics preprocessing step includes mapping aser-defined power/flux shape as a function of time to Gauss-oints, depleting the fuel at every Gauss point, plotting the

sotope and elements, creating databases of source-terms for theime-varying multi-physics solve and post-processing steps, andeallocating memory used in the neutronics preprocessing.

The time-varying multi-physics solve includes constructing thendividual operators (mechanics, heat transfer) for every physicalomain (fuel regions, gap, clad, coolant) and creating a single multi-hysics operator, which can be either static or time-dependent.he outer loop in the solution strategy is over user-defined timeteps (to model changes in reactor operating conditions). For time-ependent problems, there is a loop within each outer loop toodel the transient multi-physics problem. For each of these

series of static or time-dependent) steps, the burnup is computedrom an integral of the power shape over the given time step. Theower and burnup are used as a source-term and a componentithin the material models. The single multi-physics operator is

olved with a JFNK approach in which the unknown vector includeshe entire solution vector (temperature, displacement, and non-toichiometry in fuel; temperature and displacement in cladding;emperature in coolant). The nonlinear operator uses a linear oper-tor within the preconditioner, where the standard preconditioners block-diagonal, and each block is a single type of physics (thermaliffusion and mechanics) on a single mesh (pellet, clad, coolant).he IDA time-integrators from SUNDIALS limit the solve to a singleewton iteration per time-step, which quickly reduces the time-

tep size. The explicit time-integration solvers have not yet beensed for fuel performance modeling. For the quasi-static prob-

ems, the multi-physics problems is solved to a tight convergence.

he primary solvers used for quasi-static problems are the PETScNES and the Trilinos ML linear solver within the block diago-al preconditioner. The post-processing step presently includesnly a preliminary species formation capability, which uses the

of a fuel simulation with AMP.

temperature and elemental distributions from the multi-physicssolve and the preprocessing, respectively. This solution strategy isdepicted in Fig. 5.

4. Validation and code-to-code comparison

AMP is designed to incorporate models and capabilities of exist-ing fuel performance codes in three dimensions while extendingto a more predictive capability with additional resolution in timeand space, modeling additional physics, and incorporating modelsthat account for more of the fundamental physics. In this process,AMP will be continuously benchmarked against both experimentsand existing fuel performance codes. The first validation and code-to-code comparison is of the Halden Reactor Project experimentIFA-432.

The IFA-432 experiment includes experimental results for thefuel centerline temperature at five axial locations of three indepen-dent rods. The rod assembly was fabricated at Pacific NorthwestNational Laboratory and irradiated in the test reactor in Halden,Norway, for 759 days. At the conclusion, the rods were inspectedat Harwell, England. The complete description and associated datafor the IFA-432 experiment, and associated results from FRAPCON,are described in the FRAPCON-3 Integral Assessment (Lanning et al.,1997). Because many of the material properties and models usedin AMP originated in FRAPCON, a code-to-code comparison wasperformed to evaluate the proper implementation of the modelsand physics. Because there are differences in dimensionality, andin some models, there should not be an exact agreement betweenthe AMP and FRAPCON results.

The FRAPCON code was used to model the experiment, thoughsome of the data were neglected in the Integral Assessment report(Lanning et al., 1997). The experiment assembly consisted of sixrods, of which only the first three were used in the FRAPCONassessment. Rods 1–3 had normal (9 mil), small (3 mil), and large(15 mil) gaps between pellet and clad. For the initial validationand code-to-code comparison, there was a focus on Rod 1, whichwas the reference rod in the experiment. Rod 1 was designedand demonstrated not to have a mechanical interaction betweenthe pellets and clad. This was a beneficial benchmark because theAMP Nuclear Fuel Performance code was not yet modeling the ther-momechanical contact in version 0.9 of the software, upon whichthis publication is based. For this publication, only Rod 1 was usedfor validation; in the future, the other rods of IFA-432, as well asother benchmarks on the International Fuel Performance Experi-ments (IFPE) database (Turnbull, 1996) will be modeled as well.

4.1. Validation with experiment

This validation evaluation was an extension of the FRAPCON-3 Integral Assessment and leveraged identical inputs. However,this does present a complication: the Integral Assessment reportutilized processed experimental data, including an approximate

power history, shown in Fig. 6. The upper region of Rod 1 was usedas an initial validation case for AMP Nuclear Fuel Performance codein this paper, which included a thermocoupled that failed around 8Wd/MT (170 days). On-going work includes simulation of the other

K.T. Clarno et al. / Nuclear Engineering a

Ft

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4

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F

ig. 6. Comparison of measured (thick line) and input (thin line) power for Rod 1 athe upper thermocouple (Fig. A2.8(a) from Lanning et al., 1997).

ods and a comparison, using both AMP Nuclear Fuel Performanceode and FRAPCON, with the original data from the IFPE database.

Fig. 7 shows a comparison of the centerline temperature fromhe experiment, AMP, and the FRAPCON Integral Assessment onto

consistent piecewise-constant-in-time graph with a consistentxis for Rod 1 of IFA-432. In Fig. 7, the centerline temperature inhe experiment and two codes are shown on the left axis, while theelative error of the AMP solution, with respect to the experimentalata, and the relative difference of the AMP solution, with respect tohe FRAPCON solution, are shown on the right axis. The comparisonf centerline temperature for Rod 1 of IFA-432, for both FRAPCONnd AMP, demonstrates that AMP is capable of computing a cen-erline temperature that is relatively equivalent to FRAPCON andithin 70 K of the experiment for the life of the thermocouple.

he largest discrepancy between AMP (and FRAPCON) and thexperiment is immediately preceding the failure of the thermo-ouple, which is likely due to the imminent failure. In addition, thether discrepancies are between the codes and experiment coulde attributed to a variety of things, including the approximations

n the power history, as shown in Fig. 6.

.2. Code-to-code comparison with FRAPCON

Code-to-code comparisons provide a useful benchmark forgures of merit that are often difficult to measure (on-line tempera-ure distribution) and can supplement in understanding differencesetween the software. However, code-to-code comparisons do not

ig. 7. Centerline temperature, as measured in the experiment, computed with FRAPCO

nd Design 252 (2012) 108– 120 115

serve as validation (which requires comparison with experiment)or verification (which requires ensuring the software is writtencorrectly). A key challenge in any code-to-code comparison isunderstanding the differences in approximations and models andunderstanding trends in the simulations.

There are several important aspects the experiment and differ-ences to note between the FRAPCON and AMP simulations, beforeresults are discussed.

• FRAPCON is one (radial) dimension with axial coupling throughthe gap and plenum models and coolant flow, whereas AMP mod-els the full 3D space.

• The version 0.9 release of the AMP Nuclear Fuel Performance codecan model only a single fuel region (in this case a pellet at thethermocouple location) plus cladding, whereas FRAPCON wasused to model the entire 1.9 ft fuel stack.

• In the AMP Nuclear Fuel Performance code model, the 0.069 in.diameter centerline hole in the fuel pellet was neglected, becausethe experimental data with which the results were comparedused an effective temperature that (approximately) accountedfor this channel in determining the centerline temperature.

• FRAPCON models the time dependence with a quasi-staticapproximation, so the AMP Nuclear Fuel Performance code simu-lation did the same.

• The IFA-432 experiment took place in a heavy BWR, whichis modeled in FRAPCON with a fixed cladding outer surfacetemperature; thus the one-dimensional flow capability of theAMP Nuclear Fuel Performance code was not used.

• In the bottom axial level of the IFA-432 experiment, pellet–cladmechanical interaction did not occur until the end ofthe simulation, as seen in the FRAPCON results, so theAMP Nuclear Fuel Performance code approximation of neglectingthis effect is reasonable.

• The primary full pin effects, fission gas release, is seen in the gapconductance (heat transfer between pellet and cladding) model;the AMP Nuclear Fuel Performance code simulation used the gapheat transfer coefficient from the FRAPCON solution.

• The relocation and densification material models in the

AMP Nuclear Fuel Performance code are from the FALCON code,which are different than FRAPCON.

• The axial deformation from AMP that was used for compari-son purposes with FRAPCON was based on the maximum axial

N, and computed with AMP with the approximate power history shown in Fig. 6.

116 K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120

Fig. 8. FRAPCON calculated temperature distribution.

Fig. 9. AMP calculated temperature distribution.

Fig. 10. Relative difference in temperatures between AMP and FRAPCON.

K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120 117

eratu

•

utr

AtTadst

Fig. 11. Relative difference in temp

displacement in the material, though AMP computes a displace-ment that varies radially.The radial deformation from AMP that was used for comparisonpurposes with FRAPCON was based on the maximum radial dis-placement in the material, though AMP computes a displacementthat varies axially (such as the hour-glassing seen in the pink lineof Fig. 4).

The mesh used in the AMP Nuclear Fuel Performance code sim-lation is shown in Fig. 3, where color represents calculatedemperature at day 759 of the experiment. For this work, a meshefinement study was not performed.

A 3D plot at the final time step in the validation problem of theMP Nuclear Fuel Performance code results for the power, burnup,emperature, and displacement are shown in Figs. 1–4, respectively.he thermal results for FRAPCON and AMP are shown in Figs. 8

nd 9, respectively, for the linear heat generation rate (blue withots), clad inner surface temperature (purple with triangles), fuelurface temperature (green with diamonds), and fuel centerlineemperature (red with squares). The FRAPCON and AMP results

Fig. 12. FRAPCON calculated d

res during the initial power ramp.

trend well together, with the significant changes in temperaturearising directly from changes in the specific power. The reductionof the thermal conductivity due to burnup of the fuel is appar-ent from the relatively flat centerline temperature from 20,000 to35,000 MWd/MTU, despite the gradual decrease in power.

The relative differences between AMP and FRAPCON are shownin Fig. 10. The relative difference between AMP and FRAPCON isless than 5% throughout the lifetime of the fuel, and the initial dif-ferences are less than 0.4% (Fig. 11). There is a bias with burnup inthe AMP results that is attributable to a different treatment for theneutronics as a function of time. In FRAPCON, the radial flux shape(and power magnitude) are defined by the user, and the powershape is allowed to vary as a result of local depletion; for this sim-ulation, the AMP used a constant spatial depletion treatment (forwhich Scale/ORIGEN-S is well validated) and a (constant in time)user-defined power shape. This small difference in the time varia-

tion of the power shape is likely the cause of the small differencein the fuel centerline temperature.

The mechanics results for FRAPCON and AMP are shown in Figs.12 and 13, respectively, for the linear heat generation rate (blue

isplacement distribution.

118 K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120

d disp

wafaivflddcrTst(5t

A1ct

Fig. 13. AMP compute

ith dots), average clad radial displacement (orange with pluses),verage clad axial displacement (purple with triangles), averageuel radial displacement (green with diamonds), and average fuelxial displacement (red with squares). The cladding displacements dominated by the thermal expansion, which has a relatively smallariation throughout the irradiation, which leads to a relativelyat displacement profile during the experiment. The fuel radialisplacement is dominated by the relocation model, which areerived from different sources (the AMP Nuclear Fuel Performanceode model is from the FALCON code), but lead to relatively similaresults and are both pure radial models with no axial component.he fuel axial displacement is dominated by the densification andwelling of the fuel (with variations due to thermal expansion,hat is related to the power). The displacement is initially large125 �m) due to the initial power ramp, and shrinks over the next00 MWd/MTU due to densification, before growing throughouthe life of the fuel due to swelling.

The relative differences between the

MP Nuclear Fuel Performance code and FRAPCON are shown in Fig.4. The relative differences between AMP Nuclear Fuel Performanceode and FRAPCON are less than 18% throughout the lifetime ofhe fuel, which is dominated by the initial power ramp (Fig. 15).

Fig. 14. Relative difference in displacements between AMP and FRAPCON.

lacement distribution.

There are likely many differences, including the use of an averagedisplacement for a surface (which is not uniformly displacedin AMP Nuclear Fuel Performance code because it is a 3D code);differences in material models used (relocation and densification);and the neglect of the expansion due to fission gas release in AMP.Despite these differences, the agreement in the mechanics solutionis relatively good. There is no persistent growth (bias) in the differ-ence throughout burnup, and the error remains small throughoutthe process. If thermal expansion, relocation, densification, andswelling are neglected, these results are more than 50% differentat various times. Therefore, AMP Nuclear Fuel Performance codehas the use of these models with a reasonable agreement. Somedifferences can be attributed to the fundamental in nature (3Dversus 1D) and a thorough study of the fundamental differenceswill be the topic of a future publication.

4.3. Computing information

The solution strategy employed in this benchmark is describedas the quasi-static solution strategy from Section 3.10.

This problem had 5226 spatial elements in the fuel and 4488elements in the cladding with 4 degrees of freedom per node

Fig. 15. Relative difference in displacements during the initial power ramp.

K.T. Clarno et al. / Nuclear Engineering and Design 252 (2012) 108– 120 119

the fir

(tepi

stsiWfw

eaILvpoEp

5

pvsindeMd(PpiPcd

Fig. 16. Convergence plots for

3 displacement +temperature). The convergence criteria requiredhat the L2-norm of the final residual be less than 10−5. It wasxecuted with all eight cores of a dual quad-core AMD Opteronrocessor (3.2 GHz) and ran in 170 (wall-clock) minutes, which

ncluded pre- and post-processing time.The convergence plots for the first five quasi-static time steps are

hown in Fig. 16, where the rapid initial reduction is associated withhe temperature and the slower rate with the mechanics. There isubstantial room for improvement in run-time, which was signif-cantly affected by the incorporation of an implicit creep model.

hen implicit creep was turned off, the final time step was reducedrom 107 nonlinear iterations to 39, and the total wall-clock timeas reduced from 170 min to 69.

AMP has been compiled and executed on desktop comput-rs to leadership class hardware, including Jaguar (NCCS, 2011a)nd Frost (NCCS, 2011b) at ORNL, Icestorm (Whiting, 2011) atdaho National Laboratory, and Yellowrail at Los Alamos Nationalaboratory. For single pellet–single clad problem, such as thisalidation case, there is no need to utilize high-performance com-uters. However, AMP Nuclear Fuel Performance code has been runn over 10,000 cores of Jaguar for a full fuel assembly simulation.xtensive scaling studies and analysis will be included in a futureublication.

. Conclusions

This paper has provided an overview of the scientific com-onents, methods, and capabilities that make up the core of theersion (0.9) of the AMP Nuclear Fuel Performance code and demon-trated the integrated capability through modeling an experimentn the IFPE database. Each of the individual science compo-ents, including neutronics, heat transfer, and mechanics has beenescribed and demonstrated with validated material models fromxisting nuclear fuel codes, including FRAPCON, FALCON, andATPRO. The computational backplane has been described andemonstrated through the use of a block-diagonal preconditionedusing Trilinos ML) with a Jacobian-Free Newton Krylov solver (usingETSc SNES and KSP) for each of the quasi-static time-steps in theroblem. The integrated use of Scale/ORIGEN-S for modeling the

sotopic depletion, decay, and transmutation within a Nuclear Fuelerformance code provides a validated capability to evaluate thehanges in the composition of a wide variety of (fuel) materialsuring irradiation.

st five quasi-static time steps.

The demonstration problem has provided a baseline validationof the AMP Nuclear Fuel Performance code through the modeling ofan experiment in the Halden Reactor Project (IFA-432), which isthe first validation problem incorporated in the FRAPCON IntegralAssessment report. The results of the simulation have demon-strated that AMP can model this fuel performance benchmarkwith reasonable accuracy. The differences between the code andexperiment are well within the experimental uncertainty and thedifferences between AMP and FRAPCON are be due to fundamentaldifferences in approximations, models, and the dimensionality ofthe two codes.

Additional development has been focused on integration withradiation transport and thermal-fluid dynamics to accurately incor-porate the affects of assembly-level physics on the source-termsand boundary conditions of the fuel rods.

Acknowledgements

The development of the AMP Nuclear Fuel Performance code wasfunded by the Fuels IPSC element of the NEAMS program ofthe U.S. DOE Office of Nuclear Energy AMSO. Several researcherswere critical in guiding and developing this work, including PhaniNukala (ORNL) in computational mechanics and Cetin Unal (LANL)for overall project guidance, development, and contributions tounderstanding and guiding the overall multiscale fuel perfor-mance simulation efforts within the NEAMS program. Mark Berrillacknowledges support from the Eugene P. Wigner Fellowship atOak Ridge National Laboratory, managed by UT-Battelle, LLC, for theU.S. Department of Energy under Contract DE-AC05-00OR22725.

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