research article unstructured grids and the multigroup ...ning the mesh in the locations around said...

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2013, Article ID 641863, 26 pageshttp://dx.doi.org/10.1155/2013/641863

Research ArticleUnstructured Grids and the Multigroup NeutronDiffusion Equation

German Theler

TECNA Estudios y Proyectos de Ingenieŕıa S.A., Encarnación Ezcurra 365, C1107CLA Buenos Aires, Argentina

Correspondence should be addressed to GermanTheler; [email protected]

Received 22 May 2013; Revised 20 July 2013; Accepted 20 July 2013

Academic Editor: Arkady Serikov

Copyright © 2013 GermanTheler. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The neutron diffusion equation is often used to perform core-level neutronic calculations. It consists of a set of second-orderpartial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved bydiscretizing the neutron leakage term using a structured grid. This work introduces the alternatives that unstructured grids canprovide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities theyoffer. It is by understanding the basic mathematics that lie beneath the equations that model real physical systems; better technicaldecisions can be made. It is in this spirit that this paper is written, giving a first introduction to the basic concepts which can beincorporated into core-level neutron flux computations. A simple two-dimensional homogeneous circular reactor is solved usinga coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements method.Also, the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries, meshing algorithms, basicgeometric entities, discretization schemes, and characteristic grid lengths, giving even more insight into the peculiarities that arisewhen solving the neutron diffusion equation using unstructured grids.

1. Introduction

The better we engineers are able to solve the equationsthat model the real physical plants we design and build,the better services we can provide to our customers, andthus, general people can be benefited with better nuclearfacilities and installations. The Boltzmann neutron transportequation describes how neutrons move and interact withmatter. It involves continuous energy and space-dependentmacroscopic cross-sections that should be knownbeforehandand gives an integrodifferential equation for the vectorial fluxas a function of seven independent scalar variables, namely,three spatial coordinates, two angular directions, energy, andtime. It represents a balance that holds at every point in spaceand at every instant in time. Such an equationmay be tackledusing a variety of approaches; one of them is a simplificationthat leads to the so-called neutron diffusion approxima-tion that states that the neutron current is proportional tothe gradient of the neutron flux by means of a diffusion

coefficient, which is a function of the macroscopic transportcross-section.When this approximation—which is analogousto Fick’s law in species diffusion and to the Fourier expressionof the heat flux—is replaced into the transport equation, apartial differential equation of second order on the spatialcoordinates is obtained. Formally, the neutron diffusionequation may be derived from the transport equation byexpanding the angular dependance of the vectorial neutronflux in a spherical harmonics series and retaining both thezero and one-moment terms, neglecting the contributions ofhigher moments [1, 2]. As crude as it may seem, this diffusionequation gives fairly accurate results when applied under theconditions in which thermal nuclear reactors usually operate.Indeed, it can be shown that in a homogeneous bare criticalreactor, the neutron current is proportional to the neutrongradient for every neutron energy [3].

The energy domain is usually divided into a finite numberof groups, thus transforming one partial differential equationover space and energy into several coupled equations—one

2 Science and Technology of Nuclear Installations

for each group—containing differential operators appliedonly over the spatial coordinates.This resulting set of second-order PDEs is known as the multigroup neutron diffusionequation and is usually used to model, design, and analyzenuclear reactor cores by the so-called core-level calculationcodes.These programs take homogenizedmacroscopic cross-sections (which may depend on the spatial coordinatesthrough changes of fuel burnup, materials temperature orother properties) computed by lattice-level codes as an inputand solve the diffusion equation to obtain the flux (and itsrelated quantities such as power, xenon, etc.) distributionwithin the core.

Given a certain spatial distribution of materials and itsproperties inside a reactor core, chances are that the resultingreactor will not be critical. That is to say, in general, therate of absorptions and leakages will not exactly overcomethe neutrons born by fissions sources, and some kind offeedback—either through an external control system or bymeans of an inherent stability mechanism of the core [4]—is needed to keep the reactor power within a certain interval.Mathematically, this means that the transport—and thus thediffusion—equation does not have a steady-state solution. Inpractice, given a certain reactor configuration, its steady-stateflux distribution is computed by setting all the time deriva-tives to zero as usual but also by dividing the fission sources bya positive value 𝑘eff called the effective multiplication factor,which becomes also one unknown and turns the formulationinto a eigenvalue problem.The largest (or smallest, dependingon the formulation) eigenvalue is therefore the effectivemultiplication factor. If 𝑘eff < 1, the original configurationwas subcritical, and conversely. As successive configurationsmake 𝑘eff → 1, the associated eigenfunctions approach thesteady-state critical flux distribution [5].

Core-level codes traditionally use regular grids to dis-cretize the differential operators over the space. Dependingon the characteristics and symmetry of the reactor core,either squares or hexagons are used as the basic shapeof the mesh. Usual discretization schemes involve cell-centered finite differences or two-step coarse-mesh/couplingcoefficient methods [6–8], which are accurate and efficientenough formost applications.There are, nevertheless, certainlimitations that are not inherently related to structured gridsbut to the way their coarseness is utilized and how they arecomputed and applied to the reactor geometry. These issuescan be overcome by discretizing the spatial operators usinga scheme which could be applied to nonstructured grids,namely, finite volumes or finite elements.

This way, unstructured grids may be used to study,analyze, and understand the numerical errors introduced bythe discretization of the leakage operator with a difference-based scheme over a coarse structured grid by successivelyrefining the mesh whilst comparing the solutions with thestructured one, as depicted in Figure 1. Another example ofapplication may be the improvement of the response matrixof boundary conditions over cylindrical surfaces, which isthe case for most of the nuclear power plants cores. Whena coarse structured mesh is applied to a curved geometry,there appears a geometric condition known as the staircaseeffect. Even though arbitrary shapes cannot be perfectly

tessellated with unstructured grids, for the same number ofunknowns, they better represent the geometry than struc-tured meshes, as Figure 2 illustrates. Again, by refining themesh and comparing solutions, the matrix responses usedto set the boundary conditions on the coarse meshes canbe optimized. Finally, other complex geometries such asslanted control rods (Figure 3) or irradiation chambers can bedirectly taken into account by unstructured grids, possibly byrefining the mesh in the locations around said complexities.

2. The Steady-State Multigroup NeutronDiffusion Equation

In the present work, we take the steady-state multigroupneutron diffusion equation for granted. That is to say, wefocus on the mathematical aspects of the eigenvalue problemand make no further reference to its derivation from thetransport equation nor to the validity of its applicationto reactor problems, as these subjects that are extensivelydiscussed in the classic literature [1, 5, 11].

The differential formulation of the steady-state multi-group neutron diffusion equation over an 𝑚-dimensionaldomain 𝑈 with 𝐺 groups of energy is the set of 𝐺 coupleddifferential equations:

div [𝐷𝑔 (x) ⋅ grad 𝜙𝑔 (x)] − Σ𝑎𝑔 (x) ⋅ 𝜙𝑔 (x)

+

𝐺

∑

𝑔 ̸= 𝑔

Σ𝑠𝑔→𝑔 (x) ⋅ 𝜙𝑔 (x)

+ 𝜒𝑔 ⋅

𝐺

∑

𝑔=1

]Σ𝑓𝑔 (x)𝑘eff

⋅ 𝜙𝑔 (x) = 0,

(1)

where 𝜙𝑔 are the 𝑔 = 1, . . . , 𝐺 unknown flux distributionsand the Σs and the 𝐷s are the macroscopic cross-sectionsand diffusion coefficients, which ought to be computed by alattice-level code and taken as an input to core-level codes.However, for the purposes of fulfilling the objectives of thiswork, we take the macroscopic cross-sections as functions ofthe spatial coordinate x ∈ R𝑚 which is known beforehand.The coefficient 𝜒𝑔 represents the fission spectrum and is thefraction of the fission neutrons that are born into group 𝑔. Asstated above, the ]-fissions are divided by a positive effectivemultiplication factor 𝑘eff and all the time derivatives, that is,the right hand of (1) is set to zero. We note that, in order todefine a consistent nomenclature, we use the absorption crosssection Σ𝑎𝑔 of the group 𝑔, which is equal to the total crosssection Σ𝑡𝑔 minus the self-scattering cross section Σ𝑠𝑔→𝑔.Therefore, the sum over the scattering sources excludes theterm corresponding to 𝑔 = 𝑔. If we had used the totalcross section in the second term of (1), we would have hadto include the self-scattering term as a source.

If the cross sections depend only in an explicit wayon the spatial coordinate x, then (1) is linear. If, as is thegeneral case, the cross sections depend on x through the flux𝜙𝑔(x) itself—such as by means of the xenon concentrationor by local temperature distributions—then (1) is nonlinear.Nevertheless, this general case can be solved by successive

Science and Technology of Nuclear Installations 3

(a) Coarse structured grid commonly used in diffu-sion codes such as in [9]. Each lattice cell is divided intoa 2 × 2 grid for solving the neutron diffusion equation

(b) Discretization of the lattice cell using a fineunstructured grid as proposed in this work

(c) Further refinement of the unstructured grid

Figure 1: Discretization of a homogenized PHWR array of fuel channels for a core-level diffusion computation. Each square is a lattice-levelcell comprising one fuel channel and the surrounding moderator.

linear iterations so the basic problem can be regarded as beingpurely linear.

It should be noted that, if at least one of the diffu-sion coefficients 𝐷𝑔(x) is discontinuous over space, thenthe divergence operator is not defined at the discontinuitypoints. Therefore, the differential formulation—also knownas the strong formulation—is not complete when there existmaterial discontinuities that involve the diffusion coefficients.At thesematerial interfaces, the differential equation has to bereplaced by a neutron current conservation condition:

𝐷+

𝑔(x) ⋅ grad 𝜙+

𝑔(x) = 𝐷−

𝑔(x) ⋅ grad 𝜙−

𝑔(x) , (2)

where the plus and minus sign denote both sides of the inter-face. As the diffusion coefficients are different, the resultingflux distribution𝜙𝑔(x) ought to have a discontinuous gradientat the interface in order to conserve the current.

When transforming the strong formulation into a weakformulation—not just into an integral formulation—both (1)and (2) can be taken into account by a single expression. Ineffect, let 𝜑𝑔(x) be arbitrary functions of x for 𝑔 = 1, . . . , 𝐺.Multiplying each of the 𝐺 equations (1) by 𝜙𝑔(x), integrating

over the domain 𝑈 ∈ R𝑚, and applying Green’s formula [12]to the leakage term, we obtain

∫𝑈

𝐷𝑔 (x) ⋅ [grad 𝜑𝑔 (x) ⋅ grad 𝜙𝑔 (x)] 𝑑𝑚x

+ ∫𝜕𝑈

𝜑𝑔 (x) ⋅ 𝐷𝑔 (x) ⋅ [grad 𝜙𝑔 (x) ⋅ n̂] 𝑑𝑆

+ ∫𝑈

𝜑𝑔 (x) ⋅ Σ𝑎𝑔 (x) ⋅ 𝜙𝑔 (x) 𝑑𝑚x

+ ∫𝑈

𝜑𝑔 (x) ⋅𝐺

∑

𝑔 ̸= 𝑔

Σ𝑠𝑔→𝑔 (x) ⋅ 𝜙𝑔 (x) 𝑑𝑚x

+ 𝜒𝑔 ⋅ ∫𝑈

𝜑𝑔 (x) ⋅𝐺

∑

𝑔=1

]Σ𝑓𝑔 (x)𝑘eff

⋅ 𝜙𝑔 (x) 𝑑𝑚x.

(3)

These 𝐺 coupled equations should hold for any arbitraryset of functions𝜑𝑔(x).Making an analogy between (3) and theprinciple of virtual work for structural problems, we call thesefunctions virtual fluxes. This formulation does not involveany differential operator over the diffusion coefficient and yet,


(a) Continuous two-dimensional domain (b) Structured grid

(c) Unstructured grid

Figure 2: When a continuous domain (a) is meshed with an unstructured grid, there appears a geometric condition known as the staircaseeffect (b). For the same number of nodes, unstructured grids reproduce the original geometry better (c).

Figure 3: Cross section of a hypothetical reactor in which thecontrol rods enter into the core from above with a certain attackangle with respect to the vertical direction.

at the same time, can be shown to be equivalent to the strongformulation given by (1).

In any case, both formulations involve the computationof 𝐺 unknown functions of x and one unknown real value𝑘eff. Except for homogeneous cross sections in canonicaldomains 𝑈, the multigroup diffusion equation needs to besolved numerically. As discussed below, any nodal-baseddiscretization scheme replaces a continuous unknown func-tion 𝜙𝑔(x) by 𝑁 discrete values 𝜙𝑔(𝑖) for 𝑖 = 1, . . . , 𝑁.

Figure 4: The finite volumes method computes the unknown fluxin the cell centers (squares), whilst the finite elements methodcomputes the fluxes at the nodes (circles).

If we arrange these unknowns into a vector 𝜙 ∈ R𝑁𝐺such as

𝜙 =

[[[[[[[[[[[[[[[[

[

𝜙1 (1)

𝜙2 (1)

...𝜙𝐺 (1)

𝜙1 (2)

...𝜙𝑔 (𝑖)

...𝜙𝐺 (𝑁)

]]]]]]]]]]]]]]]]

]

, (4)


(a) Fast flux distribution

−100−50 0 50 100−100

−500

50100

0

0.6

xy

(b) Fast flux unknowns

(c) Matrix 𝑅 structure (d) Matrix 𝐹 structure

Figure 5: A bare homogeneous circle solved with finite volumes (184 unknowns).

then the continuous eigenvalue problem—in either for-mulation—can be transformed into a generalized matrixeigenvalue/eigenvector problem casted in either of the follow-ing forms:

𝑅 ⋅ 𝜙 =1

𝑘eff⋅ 𝐹 ⋅ 𝜙,

𝐹 ⋅ 𝜙 = 𝑘eff ⋅ 𝑅 ⋅ 𝜙,

𝑅−1⋅ 𝐹 ⋅ 𝜙 = 𝑘eff ⋅ 𝜙,

(5)

where 𝑅 and 𝐹 are square 𝑁𝐺 × 𝑁𝐺 matrices. We call𝐹 the fission matrix, as it contains all the ]-fission termswhich are the ones that we artificially divided by 𝑘eff. Therest of the terms are grouped into the removal or transportmatrix 𝑅, which includes the rest of the neutron-matterinteraction mechanisms. It can be shown [11] that, for anyreal set of cross sections, 𝑅−1 exists and that the𝑁𝐺 pairs ofeigenvalue/eigenvector solutions of (5) satisfy that

(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue,

(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive,

(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother.

2.1. Boundary Conditions. Being a differential equation overspace, the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem. These can be imposed flux (Dirichlet), imposed current(Neumann), or a linear combination (Robin). However, dueto the fact that in the linear problem in absence of externalsources—such as (1) or (3)—the problem is homogeneous;if 𝜙𝑔(x) is a solution, then any multiple 𝛼𝜙𝑔(x) is also asolution. That is to say, the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power. Thus, the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constant.Therefore, the allowedDirichlet conditions are zero flux at the boundary; theNeumann conditions should prescribe that the derivative in


(a) Fast flux distribution

−100−50 0 50 100−100

−500

50100

0

0.6

xy

(b) Fast flux unknowns

(c) Matrix 𝑅 structure (d) Matrix 𝐹 structure

Figure 6: A bare homogeneous circle solved with finite elements (218 unknowns).

the normal direction should be zero (i.e., symmetry condi-tion), and the Robin conditions are restricted to the followingform:

grad 𝜙𝑔 (x) ⋅ n̂ + 𝑎𝑔 (x) ⋅ 𝜙𝑔 (x) = 0, (6)

n̂ being the outward unit normal to the boundary 𝜕𝑈 of thedomain 𝑈.

2.2. Grids and Schemes. One way of solving the neutrondiffusion equation—and in general any partial differentialequation over space—is by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid. Given an 𝑚-dimensional domain, a grid definesa partition composed of a finite number of simple geometricentities. In structured grids, these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structure.On the other hand, the geometric entities that composean unstructured grid are arranged in an irregular pattern,and the identification of the elementary entities needs to

be separately specified, for example, by means of a sortedlist. For instance, Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a). Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions. In the case of Figure 2(c)that shows an unstructured grid, there is no way to system-atically make a reference to a particular quadrangle withoutany further information.

Almost all of the grid-based schemes—which are knownas nodal schemes, which are to be differentiated from modalschemes based on series expansions—are based in either thefinite differences, volumes, or elements method. Finite differ-ences schemes provide themost simple and basic approach toreplace differential (i.e., continuous) operators by difference(i.e., discrete) approximations. However, they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in space,which is the case for any reactor core composed of atleast two different materials. Moreover, boundary conditionsare hard to incorporate and usually give rise to incorrectresults.


20 40 600.995

0.9975

1

Analytical solutionFinite volumesFinite elements

keff

a/c

Figure 7: The effective multiplication factor of a two-group barecircular reactor of radius 𝑎 as a function of the quotient 𝑎/ℓ

𝑐between

the radius and the characteristic length of the cell/element.

20 40 60

Finite volumesFinite elements

Erro

r ink

eff

10−3

10−4

10−5

a/c

Figure 8: Absolute value of the error committed in the computationof 𝑘eff versus 𝑎/ℓ𝑐.

Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity, applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entity’s surface using informationcontained in its neighbors. In this context, each elementaryentity is called a cell, and finite volumes methods give themean value of each of the group fluxes 𝜙𝑔(𝑖) at the 𝑖th cell,which may be associated to the value of 𝜙𝑔(x𝑖) where x𝑖 isthe location of the cell barycenter. Being an integral-basedmethod, spatial-discontinuous parameters are handled moreefficiently than in finite differences, and boundary conditionscan be easily incorporated as forced cell fluxes. Nonetheless,even though these methods may be applied to unstructuredgrids, the estimation of the fluxes on the cells’ surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened.

10

70

90

130

150

170

10 30 50 70 90 110 130 150 170

4

1 38

36 37353

3 32 33 34312

26 27 28 29 3025

19 20 21 22 23 2418

10 11 12 13 14 15 16 17

2 331 4 5 6 7 8 9

x (cm)

y (cm

)

Figure 9: The 2D IAEA PWR benchmark geometry.

Moreover, the simple integral approach cannot take intoaccount discontinuous diffusion coefficients, so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2).

Finally, finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions. The method isbased on shape functions that are local to each elementaryentity—now called element, defined by nodes as corners—and on finding a set of nodal values such that a certaincondition is met, which is usually that the residual of thesolution has to be orthogonal to each of the shape functions.This condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions. Thesemathematical properties make finite elements schemes veryattractive. However, these features depend on a large numberof integrations that ought to be performed numerically, soa computational effort/desired accuracy tradeoff has to beconsidered. Not only does the finite elements method givethe values of the flux 𝜙𝑔(x𝑖) at the 𝑖th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain𝑈. Boundary conditions are divided into essentialand natural.Thefirst group comprises theDirichlet boundaryconditions which are satisfied exactly—within the precisionof the eigenvalue problem solver—by the obtained solution.The latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values.

The same unstructured grid may be used either for thefinite volumes or for the finite elements method. In the first


Largest eigenvalue 1.029690 (2883.36 pcm)11.35 @ (30.66, 30.79)13.06 @ (51.71, 130.76)

Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 0.74 seconds

Max 𝜙2(x, y) @coreMax 𝜙2(x, y) @reflector

2.483 × 10−8

1.223 × 10−8

4.03 × 10−9

keff

quarter-symmetry core meshed using Delaunay (triangles, c = 3) solved with finite volumesMilonga’s 2D LWR IAEA benchmark problem case no. 002

(a)

123456789

1011121314151617181920212223242526272829303132333435363738

k Pk 𝜙1k 𝜙2k0.75 32.82 5.551.33 42.42 9.841.47 46.45 10.901.23 39.21 9.110.61 26.77 4.530.94 29.95 6.940.93 29.27 6.890.74 20.11 5.49— 3.22 7.561.46 45.97 10.781.50 47.30 11.101.33 41.99 9.851.07 34.05 7.891.04 32.71 7.670.95 29.71 7.000.72 19.56 5.34— 3.06 7.331.49 46.95 11.021.36 42.89 10.071.18 37.33 8.761.07 33.67 7.920.97 28.92 7.180.66 16.28 4.91— 2.34 5.491.20 37.97 8.910.97 30.96 7.180.90 28.42 6.690.84 22.21 6.19— 5.68 12.15— 0.67 2.570.47 20.42 3.480.68 20.55 5.040.58 14.14 4.30— 2.22 5.620.57 13.74 4.25— 3.88 8.20— 0.53 2.03— 0.60 2.29

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)

Figure 10: (a) Mesh and thermal flux distribution. (b) Power and fluxes. (c) Flux distribution 𝜙𝑔(𝑥, 0) along the 𝑥-axis. (d) Flux distribution

𝜙𝑔(𝑥, 𝑥) along the diagonal.

case, the unknowns are themean value of the fluxes over eachcell, whilst in the latter the unknowns are the fluxes evaluatedat each node. Therefore, the number of unknowns 𝑁𝐺 isdifferent for each method, even when using the same grid.Figure 4 shows this situation, and in Section 3.1, we further

illustrate these differences. A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13].





1.394 × 10−8

6.865 × 10−9

3.425 × 10−9

keff

quarter-symmetry core meshed using Delaunay (quads, c = 2) solved with finite volumesMilonga’s 2D LWR IAEA benchmark problem case no. 013

(a)

123456789

1011121314151617181920212223242526272829303132333435363738

k Pk 𝜙1k 𝜙2k0.74 32.44 5.491.33 42.30 9.821.47 46.51 10.921.24 39.46 9.170.60 26.45 4.430.94 29.97 6.960.93 29.38 6.910.74 20.05 5.45— 3.01 7.461.45 45.88 10.761.50 47.20 11.081.33 41.98 9.841.09 34.65 8.051.04 32.80 7.690.95 29.86 7.040.71 19.62 5.28— 2.93 7.151.48 46.80 10.981.36 42.88 10.071.19 37.55 8.811.07 33.85 7.960.97 29.05 7.210.67 16.53 4.93— 2.20 5.561.21 38.08 8.930.98 31.15 7.240.92 28.89 6.800.83 22.47 6.15— 5.43 12.41— 0.67 2.600.45 19.77 3.300.69 20.76 5.090.58 14.60 4.28— 2.26 5.740.56 13.64 4.13— 3.71 8.21— 0.52 2.00— 0.60 2.38

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



3. Results

We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron

diffusion problem. The examples are two-dimensional prob-lems, as they contain some of the complexities a real three-dimensional reactor posse, yet the reported results are notso complicated as to be easily understood and analyzed. Inparticular, we state some basic differences between the finite


Largest eigenvalue 1.029695(2883.88 pcm)11.12 @ (30.76, 30.32)8.38 @ (50.00, 130.00)



1.056 × 10−8

5.202 × 10−9

5.122 × 10−9

keff

quarter-symmetry core meshed using Delaunay (quads, c = 2) solved with finite volumesMilonga’s 2D LWR IAEA benchmark problem case no. 018

(a)

123456789

1011121314151617181920212223242526272829303132333435363738

k Pk 𝜙1k 𝜙2k0.74 32.20 5.491.30 41.52 9.611.44 46.51 10.681.20 38.43 8.900.61 26.48 4.520.94 29.96 6.930.94 29.59 6.960.72 20.61 5.69— 3.58 8.621.42 44.95 10.541.47 46.34 10.881.31 41.27 9.681.07 34.09 7.891.04 32.75 7.680.96 30.03 7.080.71 20.14 5.54— 3.42 8.201.46 46.06 10.811.34 42.22 9.911.18 37.11 8.711.07 33.75 7.940.98 29.29 7.270.62 16.92 5.22— 2.58 6.321.19 37.52 8.800.96 30.84 7.150.91 28.58 6.730.80 22.80 6.33— 6.11 12.77— 0.80 3.180.47 20.43 3.500.69 20.88 5.100.54 14.63 4.50— 2.55 6.480.51 14.18 4.39— 4.10 8.54— 0.64 2.52— 0.71 2.85

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid, although a rather coarseone, so the differences can be observed directly into theresulting figures.We then solve the classical two-dimensionalLWR problem, also known as the 2D IAEA PWR benchmark.

Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithms,basic shapes, and characteristic lengths of the mesh.

To solve the two examples shown below, we employed themilonga code, which was written from scratch by the author





5.408 × 10−8

2.665 × 10−8

8.424 × 10−9

keff

Milonga’s 2D LWR IAEA benchmark problem case no. 031quarter-symmetry core meshed using Delquad (quads, c = 4) solved with finite volumes

(a)

123456789

1011121314151617181920212223242526272829303132333435363738

k Pk 𝜙1k 𝜙2k0.77 34.22 5.701.40 44.68 10.401.53 48.36 11.351.28 40.70 9.480.60 27.04 4.480.95 30.17 7.020.91 28.61 6.730.69 19.18 5.08— 2.64 6.891.51 47.85 11.221.55 48.97 11.491.37 43.34 10.171.11 35.35 8.231.04 32.72 7.680.92 29.08 6.850.67 18.76 4.96— 2.54 6.581.54 48.54 11.391.39 43.92 10.311.20 37.92 8.901.07 33.60 7.900.95 28.31 7.020.61 15.74 4.51— 1.97 5.151.23 38.68 9.070.99 31.32 7.300.90 28.18 6.630.78 21.68 5.79— 4.81 11.91— 0.55 2.170.44 19.84 3.280.67 20.27 4.980.52 13.60 3.88— 1.90 5.200.52 13.29 3.83— 3.21 7.92— 0.43 1.68— 0.48 1.95

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



of this paper and is currently still under development withinhis ongoing PhD thesis. There exists a first public release[14] under the terms of the GNU General Public License—that is, it is a free software—that can only handle structuredgrids. There is a second release being prepared, whose

main relevance is that it can work with nonstructured gridsas well, which is the main feature of the code. It uses ageneral mathematical framework—coded from scratch aswell—which provides input file parsing, algebraic expres-sions evaluation, one- and multidimensional interpolation of





2.296 × 10−12

1.131 × 10−12

6.877 × 10−13

keff

Milonga’s 2D LWR IAEA benchmark problem case no. 043eighth-symmetry core meshed using Delaunaya (triangs,c = 2) solved with finite volumes

(a)

1 0.74 32.41 5.482 1.31 41.88 9.713 1.45 45.82 10.764 1.22 38.77 9.005 0.60 26.44 4.456 0.93 29.86 6.927 0.93 29.41 6.928 0.75 20.30 5.529 — 3.22 7.79

10 1.43 45.23 10.6111 1.48 46.67 10.9512 1.31 41.45 9.7213 1.07 34.23 7.9414 1.03 32.69 7.6715 0.95 29.93 7.0616 0.73 19.88 5.3917 — 3.12 7.5118 1.47 46.37 10.8819 1.34 42.43 9.9620 1.18 37.21 8.7321 1.07 33.70 7.9322 0.98 29.14 7.2323 0.67 16.51 4.9924 — 2.35 5.7125 1.19 37.63 8.8226 0.96 30.77 7.1527 0.91 28.47 6.7128 0.83 22.55 6.1729 — 5.76 12.2530 — 0.68 2.6631 0.46 20.16 3.4132 0.68 20.58 5.0533 0.59 14.43 4.3534 — 2.33 5.8935 0.57 13.81 4.2036 — 3.81 8.2337 — 0.54 2.0838 — 0.62 2.44

k Pk 𝜙1k 𝜙2k

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



scattered data, shared-memory access, numerical integrationfacilities, and so forth. The milonga code was designed withfour design basis vectors in mind—which are thoroughlydiscussed in the documentation [15]—which includes thetype of problems it can handle, the code scalability, which

features are expected, and what to do with the obtainedresults.

It works by first reading an input file that, using plain-text English keywords and arguments, defines the number𝑚 of spatial dimensions, the number 𝐺 of group energies,





2.019 × 10−8

9.946 × 10−9

7.138 × 10−9

keff

Milonga’s 2D LWR IAEA benchmark problem case no. 047eighth-symmetry core meshed using Delaunay (triangles, c = 3) solved with finite volumes

(a)

1 0.74 31.94 5.452 1.29 41.17 9.533 1.43 45.15 10.604 1.19 38.14 8.835 0.61 26.34 4.506 0.93 29.84 6.907 0.94 29.54 6.958 0.71 20.64 5.719 — 3.62 8.65

10 1.41 44.58 10.4511 1.46 45.99 10.7912 1.30 40.97 9.6113 1.06 33.88 7.8414 1.03 32.63 7.6515 0.95 29.98 7.0716 0.68 20.15 5.5717 — 3.46 8.2318 1.45 45.72 10.7319 1.33 41.94 9.8420 1.17 36.91 8.6621 1.07 33.63 7.9222 0.98 29.25 7.2623 0.59 16.96 5.2824 — 2.63 6.3525 1.18 37.28 8.7426 0.96 30.68 7.1127 0.91 28.50 6.7228 0.79 22.80 6.3529 — 6.19 12.7630 — 0.81 3.2131 0.47 20.36 3.4932 0.68 20.85 5.0933 0.51 14.65 4.5434 — 2.58 6.5035 0.49 14.19 4.4336 — 4.16 8.5437 — 0.65 2.5438 — 0.71 2.88

k Pk 𝜙1k 𝜙2k

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



and optionally a mesh file. Currently, only grids generatedwith the code gmsh [16] are only supported, mainly becauseit is also a free software and it suits perfectly well milonga’sdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters. Afterward,the physical entities defined in the grid are mapped into

materials with macroscopic cross sections, which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributions,which in turn may be defined by algebraic expressions, byinterpolating data located in files, by reading shared-memoryobjects or by a combination of them. In the same sense,



Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error


2.028 × 10−12

9.993 × 10−13

1.012 × 10−12

keff

Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1.256 seconds

9.993 × 10

1.012 × 10−12

Milonga’s 2D LWR IAEA benchmark problem case no. 058eighth-symmetry core meshed using Delaunay (quads, c = 2) solved with finite volumes

(a)

1 0.74 32.07 5.472 1.29 41.36 9.573 1.44 45.34 10.644 1.20 38.28 8.875 0.61 26.38 4.506 0.93 29.85 6.907 0.94 29.48 6.948 0.72 20.53 5.669 — 3.57 8.59

10 1.42 44.78 10.5011 1.46 46.17 10.8412 1.30 41.11 9.6413 1.06 33.96 7.8614 1.03 32.63 7.6515 0.95 29.91 7.0616 0.71 20.05 5.5217 — 3.40 8.1618 1.45 45.88 10.7719 1.33 42.06 9.8720 1.17 36.98 8.6821 1.07 33.62 7.9122 0.98 29.18 7.2523 0.62 16.86 5.2024 — 2.58 6.2925 1.18 37.37 8.7626 0.96 30.73 7.1227 0.91 28.48 6.7128 0.80 22.72 6.3129 — 6.09 12.7230 — 0.80 3.1731 0.47 20.35 3.4832 0.69 20.81 5.0933 0.54 14.58 4.4834 — 2.55 6.4635 0.52 14.13 4.3736 — 4.09 8.5137 — 0.64 2.5138 — 0.71 2.84

k Pk 𝜙1k 𝜙2k

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



boundary conditions are applied to appropriate physicalentities of dimension𝑚−1.The problemmatrices𝑅 and𝐹 arethen built and stored in an appropriate sparse format usingthe free PETSc [17, 18] library, and the eigenvalue problemis solved using the free SLEPc [19] library. The results arestored into milonga’s variables and functions, which may be

evaluated, integrated—usually using the free GNU ScientificLibrary [20] routines—and of course written into appropriateoutputs. Milonga can also solve problems parametrically andbe used to solve optimization problems. As stated above, thecode is a free software so corrections and contributions aremore than welcome.



Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti


2.6 × 10−8

1.281 × 10−8

4.996 × 10−9

keff

Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0.604 seconds

1.281 × 10 8

4.996 × 10−9


(a)

1 0.75 32.97 5.552 1.34 42.68 9.923 1.48 46.60 10.944 1.23 39.26 9.145 0.59 26.45 4.406 0.94 29.97 6.977 0.93 29.14 6.858 0.72 20.02 5.349 — 3.01 7.73

10 1.46 46.06 10.8011 1.50 47.35 11.1112 1.33 42.12 9.8813 1.08 34.48 8.0214 1.04 32.71 7.6715 0.94 29.54 6.9616 0.70 19.54 5.2117 — 2.88 7.3518 1.49 46.91 11.0119 1.36 42.89 10.0720 1.19 37.48 8.8021 1.07 33.59 7.9022 0.96 28.80 7.1423 0.64 16.36 4.7524 — 2.17 5.6325 1.21 38.10 8.9426 0.98 31.10 7.2427 0.90 28.29 6.6628 0.81 22.27 5.9929 — 5.36 12.5730 — 0.62 2.5031 0.45 20.02 3.3632 0.68 20.40 5.0033 0.55 14.14 4.0934 — 2.15 5.7935 0.55 13.59 4.0836 — 3.59 8.3437 — 0.50 1.9938 — 0.61 2.36

k Pk 𝜙1k 𝜙2k

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



3.1. A Coarse Bare Homogeneous Circle. Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 𝑎 over an unstructured grid whichis deliberately coarse, using the finite volumes method andthe finite elements method, respectively. Two group energies

were used and a null-flux boundary conditionwas fixed at theexternal surface. Figures 5(a) and 6(a) compare the obtainedfast flux distributions. In the first case, the numerical solutionprovidesmean values for each neutron flux group in each cell,while in the latter, the solution is computed at the nodes, and



Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm


5.66 × 10−12

2.789 × 10−12

1.595 × 10−12

keff

Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 0.9921 seconds

5.66 × 10

2.789 × 10−12

1.595 × 10−12


(a)

1 0.73 31.51 5.402 1.27 40.59 9.393 1.41 44.57 10.464 1.18 37.68 8.715 0.61 26.15 4.506 0.93 29.79 6.887 0.94 29.71 6.998 0.67 20.92 5.829 — 3.75 8.85

10 1.39 43.97 10.3111 1.44 45.41 10.6612 1.28 40.52 9.5013 1.05 33.65 7.7814 1.03 32.59 7.6415 0.96 30.16 7.1216 0.66 20.45 5.6817 — 3.59 8.4118 1.43 45.20 10.6119 1.32 41.55 9.7520 1.16 36.72 8.6221 1.07 33.66 7.9222 0.98 29.51 7.3323 0.54 17.28 5.4324 — 2.73 6.5125 1.17 37.00 8.6826 0.95 30.57 7.0827 0.91 28.60 6.7428 0.74 23.11 6.4829 — 6.42 13.0030 — 0.84 3.3231 0.48 20.41 3.5332 0.69 21.03 5.1333 0.46 14.93 4.6734 — 2.69 6.6735 0.46 14.44 4.5536 — 4.32 8.7137 — 0.67 2.6338 — 0.74 2.97

k Pk 𝜙1k 𝜙2k

(b)

0 40 80 120 1600

15

30

45

𝜙2(x, 0)𝜙1(x, 0)

(c)

0 40 80 120 1600

15

30

45

𝜙2(x, x)𝜙1(x, x)

(d)



continuous functions are evaluated by means of the shapefunctions used in the formulation. Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace. It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases. Finally, the structure ofthe sparse eigenvalue problemmatrices is shown for each case

with blue, red, and cyan representing positive, negative, andexplicitly inserted zero values. In the finite volume case, thereare 184 unknowns (92 cells × 2 groups), while in the secondcase there are 218 unknowns (109 nodes × 2 groups). Thevolumes’ fissionmatrix is almost diagonal: it has a bandwidthequal to the number of energy groups, which in this case is


500 1000 2000 5000 10000 20000 50000Number of unknowns

2840

2860

2880

2900

2920

ORNLRisøKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes

Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes

Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements

𝜌(p

cm)

105 2 × 105 5 × 105

Figure 19: Static reactivity versus number of unknowns. The four original solutions as published in 1977 [10] are included as reference.

two.The off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 𝜙 ∈ R184. Other orderings may be used,but the rate of convergence of the eigenvalue problem can bedeteriorated. On the other hand, the elements’ fission matrixhas a nontrivial structure, because the grid is unstructured,and the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generated.This effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21].The diagonal block of element’s 𝑅 matrix and the null blockin 𝐹 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface. Other types of boundary conditions lead to differentkinds of structures within the problem matrices.

In case a part of the domain contained a nonmultiplicativematerial such as a reflector, then there would appear sectionsof the fission matrix with null values in both methods,rendering 𝐹 singular in both methods. Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution. It can be seen in the elements’ matricesa particular structure that implements the boundary condi-tions at the external surfaces. This structure does not appearin the volumes’ matrices because the boundary conditions

appear as flux terms which are summed up over all thesurfaces of each cell, so they aremasked inside the volumetricdiscretization of the divergence and gradient operators.

As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution, it is adequate tocompare how the two proposed numerical schemes relate toit. In the studied problem, we ignored upscattering and fastfissions. Then, the analytical effective multiplication factor is

𝑘eff =]Σ𝑓2 ⋅ Σ𝑠1→2

[Σ𝑎1 + Σ𝑠1→2 + 𝐷1(]0/𝑎)2] [Σ𝑎2 + 𝐷2(]0/𝑎)

2]

(7)

being, ]0 = 2.4048 . . ., the smallest root of Bessel’s first-kindfunction of order zero 𝐽0(𝑟).

Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement, indicated bythe quotient 𝑎/ℓ𝑐 between the radius of the circle and thecharacteristic length of the cell/elements. We can see thatthe 𝑘eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution. Indeed,it can be proven that for a bare one-dimensional slab thisis always the case [13]. However, this result does not hold


2000 3000 6000 10000 20000 30000 60000

0.1

0.03

0.01

0.3

1

3

10

30

100

300To

tal w

all t

ime (

seco

nds)

Number of unknowns

105 2 × 105 3 × 105

Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements

Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements

Figure 20: Total wall time versus number of unknowns.

for problems with nonuniform cross sections, even in simpleone-dimensional reflected reactors.

We may draw two other conclusions from Figure 7. First,that the finite element method seems to provide a bettersolution than the volumes-based scheme and, second, thateven though the error committed tends to decrease with finergrids, its behavior is not monotonic for finite volumes. Infact, Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scale,where both conclusions are even more evident. We deferthe discussion of such differences between the discretizationscheme until the next section. It is worth to note, however,that the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases. Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions. Nevertheless, this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids.

3.2. The 2D IAEA PWR Benchmark Problem. This is aclassical two-group neutron diffusion problem, first designedin the early 1970s and taken as a reference benchmark forcomputational codes. A number of codes were used to solveeither this problem or its three-dimensional formulation[22, 23], including milonga using a structured grid [24]. Theoriginal formulation can be found in the reference [10], andthere is a reproduction that may be easily found online inreference [24]. The geometry consists of a one-quarter ofa PWR core depicted in Figure 9, and the homogeneousmacroscopic cross sections are listed in Table 1. An axialbuckling term 𝐵2

𝑧,𝑔= 0.8×10

−4 should be taken into account.The external surface should be subject to a zero incomingcurrent condition, which may be written as

𝜕𝜙𝑔

𝜕𝑛= −

0.4692

𝐷𝑔

⋅ 𝜙𝑔. (8)

The expected results are as follows.

(1) Maximum eigenvalue.(2) Fundamental flux distributions:

(a) Radial flux traverses 𝜙𝑔(𝑥, 0) and 𝜙𝑔(𝑥, 𝑥).



105 2 × 105 3 × 105



0.1

0.03

0.01

0.3

1

3

10

30

100

300Ti

me c

onsu

med

to m

esh

the g

eom

etry

(sec

onds

)

Figure 21: Time needed to mesh the geometry versus number of unknowns.

Table 1: Macroscopic cross-sections (units are not stated in the original reference, but they are assumed to be in cm−1 or cm as appropriate).

𝐷1

𝐷2

Σ1→2

Σ𝑎1

Σ𝑎2

]Σ𝑓2

Material1 1.5 0.4 0.02 0.01 0.080 0.135 Fuel 12 1.5 0.4 0.02 0.01 0.085 0.135 Fuel 23 1.5 0.4 0.02 0.01 0.130 0.135 Fuel 2 + rod4 2.0 0.3 0.04 0 0.010 0 Reflector

Note: the fluxes shall be normalized such that

1

𝑉core∫𝑉core

∑

𝑔

]Σ𝑓𝑔 ⋅ 𝜙𝑔𝑑𝑉 = 1. (9)

(b) Value and location of maximum power density.This corresponds to maximum of 𝜙2 in thecore. It is recommended that the maximumvalues of 𝜙2 both in the inner core and at thecore/reflector interface be given.

(3) Average subassembly powers 𝑃𝑘

𝑃𝑘 =1

𝑉𝑘

∫𝑉𝑘

∑

𝑔

]Σ𝑓𝑔 ⋅ 𝜙𝑔𝑑𝑉, (10)

where 𝑉𝑘 is the volume of the 𝑘th subassembly and𝑘 designates the fuel subassemblies as shown inFigure 9.

(4) Number of unknowns in the problem, number ofiterations, and total and outer.

(5) Total computing time, iteration time, IO-time, andcomputer used.

(6) Type and numerical values of convergence criteria.(7) Table of average group fluxes for a square mesh grid

of 20 × 20 cm.(8) Dependence of results on mesh spacing.

Even though the original problem is based on a quarter-core situation, the problem has an eighth-core symmetry


Quarter Delaunay triangles elements

Tim

e con

sum

ed to

read

the m

esh

(sec

onds

)


105 2 × 105 3 × 105

Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes

Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements


0.1

0.03

0.01

0.3

1

3

10

30

100

300

Figure 22: Time needed to read the mesh versus number of unknowns.

which cannot be taken into account by structured gridswhich are the main target of the benchmark. However,nonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four. Answers to items (1)–(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at http://dx.doi.org/10.1155/2013/641863). As asked initem (8), how results depend not only on the mesh spacingbut also on the meshing algorithm, on the grid’s elementarygeometric shape, and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves.

Taking advantage of milonga’s capability of reading andparsing command-line arguments, the selection of the coregeometry (quarter or eighth), the meshing algorithm (delau-nay [16] or delquad [25]), the shape of the elementaryentities (triangles or quadrangles), the discretization scheme(volumes or elements), and the characteristic length ℓ𝑐 ofthe mesh can be provided at run time. Fixing five values forℓ𝑐 = 4, 3, 2, 1, 0.5 gives 2 × 2 × 2 × 2 × 5 = 80 possiblecombinations, which we solve with successive invocations to

milonga with the same input file but with different argumentsfrom a simple script. Figures 10, 11, 12, 13, 14, 15, 16, 17, and18 show the results corresponding to items (1)–(7) for someillustrative cases. The complete set of figures and the codeused to generate themmay be provided upon request. Table 2compiles the answers to the problem for every case studied.

As the milonga code is still under development, itsnumerical routines are not yet fully optimized nor designedfor parallel computation. Therefore, the reported times areonly rough estimates and should be taken with care. Thesolution comprises five steps:

(1) generate the grid with the requested geometry, mesh-ing algorithm, basic shape, and characteristic lengthby calling to gmsh;

(2) read the generated mesh;(3) build the matrices;(4) solve the eigenvalue problem;(5) compute the requested results.

The CPU time reported in Figures 10–18 is thus thesum of all of these five steps, but not the time needed togenerate the figures—which in some caseswith coarsemeshes


Table2

:Resultsofthe2

DIAEA

PWRBe

nchm

arkp

roblem

obtained

with

them

ilong

acod

efor

thee

ightyp

ropo

sedcombinatio

nsofsymmetry,m

eshing

algorithm

,basicshape,discretization

scheme,andcharacteris

ticelem

ent/c

elllength.Th

ereference

solutio

nis𝜌=2874.9×10−5,w

hich

correspo

ndsto𝑘eff=1.0296.

Case

no.

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ𝑐

(cm)

Δ𝜌

(pcm

)max𝜙2

(—)

Total

unkn

owns

Outer

iter.

Linear

iter.

Inner

iter.

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

11/4

Delaunay

Vo

lumes

4.0

−12.0

11.45

8264

332

1033

6𝑒−09

3𝑒−09

1𝑒−09

349470

21/4

Delaunay

Vo

lumes

3.0

8.5

11.35

15740

332

1967

2𝑒−08

1𝑒−08

4𝑒−09

4813401

31/4

Delaunay

Vo

lumes

2.0

19.7

11.24

31188

332

3898

1𝑒−08

6𝑒−09

3𝑒−09

7621932

41/4

Delaunay

Vo

lumes

1.015.3

11.23

127476

332

15934

7𝑒−09

3𝑒−09

3𝑒−09

273

81886

51/4

Delaunay

Vo

lumes

0.5

18.7

11.19

510676

332

63834

2𝑒−09

9𝑒−10

1𝑒−09

1148

352261

61/4

Delaunay

Elem

ents

4.0

17.3

11.00

4308

332

538

2𝑒−08

8𝑒−09

4𝑒−09

318794

71/4

Delaunay

Elem

ents

3.0

11.7

11.07

8108

332

1013

5𝑒−09

2𝑒−09

2𝑒−09

4712946

81/4

Delaunay

Elem

ents

2.0

8.4

11.12

15936

332

1992

6𝑒−09

3𝑒−09

2𝑒−09

7321347

91/4

Delaunay

Elem

ents

1.06.2

11.16

64478

332

8059

6𝑒−09

3𝑒−09

4𝑒−09

262

77105

101/4

Delaunay

Elem

ents

0.5

5.8

11.17

256700

332

32087

1𝑒−09

5𝑒−10

1𝑒−09

1091

331969

111/4

Delaunay

◻Vo

lumes

4.0

5.3

11.49

5042

332

630

2𝑒−08

1𝑒−08

5𝑒−09

288007

121/4

Delaunay

◻Vo

lumes

3.0

34.6

11.33

8560

332

1070

7𝑒−09

3𝑒−09

2𝑒−09

3910895

131/4

Delaunay

◻Vo

lumes

2.0

30.8

11.31

15576

332

1947

1𝑒−08

7𝑒−09

3𝑒−09

541564

514

1/4Delaunay

◻Vo

lumes

1.017.7

11.22

60774

332

7596

8𝑒−09

4𝑒−09

3𝑒−09

167

50115

151/4

Delaunay

◻Vo

lumes

0.5

19.9

11.20

244936

332

30617

4𝑒−09

2𝑒−09

2𝑒−09

698

215678

161/4

Delaunay

◻Elem

ents

4.0

19.6

11.00

5222

332

652

3𝑒−08

1𝑒−08

8𝑒−09

4713098

171/4

Delaunay

◻Elem

ents

3.0

12.3

11.07

8810

332

1101

2𝑒−08

9𝑒−09

7𝑒−09

6918598

181/4

Delaunay

◻Elem

ents

2.0

9.011.12

15922

332

1990

1𝑒−08

5𝑒−09

5𝑒−09

107

30591

191/4

Delaunay

◻Elem

ents

1.06.3

11.16

61456

332

7682

5𝑒−09

2𝑒−09

4𝑒−09

389

113237

201/4

Delaunay

◻Elem

ents

0.5

5.8

11.17

246332

332

30791

8𝑒−10

4𝑒−10

1𝑒−09

1676

498192

211/4

Delq

uad

Vo

lumes

4.0

−4.5

11.44

6228

332

778

4𝑒−08

2𝑒−08

7𝑒−09

308495

221/4

Delq

uad

Vo

lumes

3.0

57.0

11.53

11820

332

1477

3𝑒−08

1𝑒−08

4𝑒−09

4010879

231/4

Delq

uad

Vo

lumes

2.0

45.9

11.03

24100

332

3012

2𝑒−08

1𝑒−08

5𝑒−09

6217715

241/4

Delq

uad

Vo

lumes

1.051.2

11.04

9640

03

3212050

2𝑒−08

1𝑒−08

9𝑒−09

207

61714

251/4

Delq

uad

Vo

lumes

0.5

52.8

11.06

385600

332

48200

2𝑒−09

1𝑒−09

2𝑒−09

838

255735

261/4

Delqu

ad

Elem

ents

4.0

22.0

10.93

3288

332

411

3𝑒−08

2𝑒−08

6𝑒−09

308495

271/4

Delq

uad

Elem

ents

3.0

14.0

11.04

6148

332

768

4𝑒−08

2𝑒−08

8𝑒−09

39110

0028

1/4Delq

uad

Elem

ents

2.0

8.2

11.10

12392

332

1549

2𝑒−08

9𝑒−09

6𝑒−09

6117067

291/4

Delqu

ad

Elem

ents

1.06.3

11.15

48882

332

6110

2𝑒−09

1𝑒−09

1𝑒−09

197

5804

630

1/4Delq

uad

Elem

ents

0.5

5.8

11.16

194162

332

24270

2𝑒−09

9𝑒−10

2𝑒−09

790

238769

311/4

Delq

uad

◻Vo

lumes

4.0

−41.6

11.74

3132

332

391

5𝑒−08

3𝑒−08

8𝑒−09

267322

321/4

Delq

uad

◻Vo

lumes

3.0

−24.8

11.51

5922

332

740

9𝑒−09

4𝑒−09

2𝑒−09

318779

331/4

Delq

uad

◻Vo

lumes

2.0

−13.2

11.39

12050

332

1506

1𝑒−08

7𝑒−09

4𝑒−09

4412239

341/4

Delq

uad

◻Vo

lumes

1.0−4.9

11.26

48200

332

6025

6𝑒−09

3𝑒−09

2𝑒−09

122

36094

351/4

Delq

uad

◻Vo

lumes

0.5

−2.3

11.23

192800

332

24100

5𝑒−09

2𝑒−09

3𝑒−09

464

140228

361/4

Delq

uad

◻Elem

ents

4.0

22.4

10.93

3294

332

411

3𝑒−08

1𝑒−08

7𝑒−09

359852

371/4

Delq

uad

◻Elem

ents

3.0

14.1

11.04

6144

332

768

3𝑒−08

2𝑒−08

9𝑒−09

5114018

381/4

Delq

uad

◻Elem

ents

2.0

9.211.11

12392

332

1549

1𝑒−08

6𝑒−09

5𝑒−09

8122526

391/4

Delq

uad

◻Elem

ents

1.06.5

11.15

48882

332

6110

5𝑒−09

3𝑒−09

4𝑒−09

276

78431


Table2:Con

tinued.

Case

no.

Symm-

etry

Mesh

algorithm

Basic

shape

Solutio

nmetho

dℓ𝑐

(cm)

Δ𝜌

(pcm

)max𝜙2

(—)

Total

unkn

owns

Outer

iter.

Linear

iter.

Inner

iter.

Resid

ual

norm

Relativ

eerror

Error

estim

ate

Mem

ory

(Mb)

Page

faults

401/4

Delq

uad

◻Elem

ents

0.5

5.9

11.17

194162

332

24270

1𝑒−09

5𝑒−10

1𝑒−09

1104

318967

411/8

Delaunay

Vo

lumes

4.0

−9.9

11.39

4228

332

528

4𝑒−12

2𝑒−12

1𝑒−12

359630

421/8

Delaunay

Vo

lumes

3.0

5.0

11.32

7712

332

964

1𝑒−11

6𝑒−12

2𝑒−12

4211578

431/8

Delaunay

Vo

lumes

2.0

17.7

11.20

15776

332

1972

2𝑒−12

1𝑒−12

7𝑒−13

5515436

441/8

Delaunay

Vo

lumes

1.014.2

11.18

63902

332

7987

3𝑒−12

1𝑒−12

1𝑒−12

160

46691

451/8

Delaunay

Vo

lumes

0.5

17.6

11.15

254612

332

31826

2𝑒−12

8𝑒−13

1𝑒−12

555

168526

461/8

Delaunay

Elem

ents

4.0

18.3

10.96

2252

332

281

6𝑒−12

3𝑒−12

2𝑒−12

359589

471/8

Delaunay

Elem

ents

3.0

11.9

11.04

4040

224

505

2𝑒−08

1𝑒−08

7𝑒−09

41114

3748

1/8Delaunay

Elem

ents

2.0

8.5

11.08

8156

332

1019

2𝑒−12

8𝑒−13

6𝑒−13

5515053

491/8

Delaunay

Elem

ents

1.06.3

11.12

32480

332

4060

1𝑒−12

6𝑒−13

8𝑒−13

151

43671

501/8

Delaunay

Elem

ents

0.5

5.8

11.13

128358

332

1604

47𝑒−13

3𝑒−13

7𝑒−13

527

157581

511/8

Delaunay

◻Vo

lumes

4.0

3.7

11.44

2380

224

297

5𝑒−08

2𝑒−08

1𝑒−08

5314173

521/8

Delaunay

◻Vo

lumes

3.0

21.8

11.20

4194

332

524

7𝑒−12

4𝑒−12

2𝑒−12

3610034

531/8

Delaunay

◻Vo

lumes

2.0

34.5

11.10

7960

332

995

5𝑒−12

3𝑒−12

1𝑒−12

4712879

541/8

Delaunay

◻Vo

lumes

1.016.5

11.17

30652

332

3831

2𝑒−12

1𝑒−12

7𝑒−13

100

28851

551/8

Delaunay

◻Vo

lumes

0.5

19.1

11.17

122066

224

15258

2𝑒−08

8𝑒−09

9𝑒−09

343

103825

561/8

Delaunay

◻Elem

ents

4.0

17.5

10.97

2526

332

315

4𝑒−12

2𝑒−12

1𝑒−12

6016159

571/8

Delaunay

◻Elem

ents

3.0

11.4

11.04

4386

332

548

4𝑒−12

2𝑒−12

2𝑒−12

5013768

581/8

Delaunay

◻Elem

ents

2.0

9.011.08

8234

332

1029

2𝑒−12

1𝑒−12

1𝑒−12

7320094

591/8

Delaunay

◻Elem

ents

1.06.2

11.12

31186

224

3898

2𝑒−08

7𝑒−09

9𝑒−09

208

59589

601/8

Delaunay

◻Elem

ents

0.5

5.8

11.13

123122

224

15390

9𝑒−09

5𝑒−09

1𝑒−08

807

2366

7061

1/8Delq

uad

Vo

lumes

4.0

−21.0

11.68

3224

332

403

1𝑒−11

5𝑒−12

2𝑒−12

43117

8462

1/8Delq

uad

Vo

lumes

3.0

36.9

11.50

6000

332

750

1𝑒−11

6𝑒−12

3𝑒−12

5013502

631/8

Delq

uad

Vo

lumes

2.0

31.2

11.04

12316

332

1539

2𝑒−11

8𝑒−12

5𝑒−12

6016395

641/8

Delq

uad

Vo

lumes

1.044

.010.94

48714

332

6089

9𝑒−12

4𝑒−12

4𝑒−12

1183444

165

1/8Delq

uad

Vo

lumes

0.5

53.4

10.81

193980

332

24247

5𝑒−12

2𝑒−12

4𝑒−12

423

126768

661/8

Delq

uad

Elem

ents

4.0

24.0

10.90

1750

332

218

9𝑒−12

4𝑒−12

2𝑒−12

43117

8667

1/8Delqu

ad

Elem

ents

3.0

15.0

11.00

3184

332

398

1𝑒−11

5𝑒−12

2𝑒−12

5013509

681/8

Delq

uad

Elem

ents

2.0

8.9

11.06

6426

332

803

5𝑒−12

3𝑒−12

2𝑒−12

5916047

691/8

Delq

uad

Elem

ents

1.06.4

11.11

24886

332

3110

2𝑒−12

1𝑒−12

1𝑒−12

113

32788

701/8

Delq

uad

Elem

ents

0.5

5.9

11.13

98052

224

12256

1𝑒−08

5𝑒−09

8𝑒−09

398

118153

711/8

Delq

uad

◻Vo

lumes

4.0

−39.8

11.72

1650

332

206

1𝑒−11

5𝑒−12

2𝑒−12

369907

721/8

Delq

uad

◻Vo

lumes

3.0

−24.9

11.46

3058

332

382

4𝑒−12

2𝑒−12

1𝑒−12

349349

731/8

Delq

uad

◻Vo

lumes

2.0

−13.1

11.36

6228

224

778

3𝑒−08

1𝑒−08

5𝑒−09

4010930

741/8

Delq

uad

◻Vo

lumes

1.0−5.1

11.23

24536

224

3067

3𝑒−08

2𝑒−08

1𝑒−08

7822716

761/8

Delq

uad

◻Elem

ents

4.0

23.7

10.90

1752

332

219

6𝑒−12

3𝑒−12

2𝑒−12

4111213

771/8

Delq

uad

◻Elem

ents

3.0

14.5

11.01

3184

332

398

6𝑒−12

3𝑒−12

2𝑒−12

43118

1178

1/8Delq

uad

◻Elem

ents

2.0

9.411.07

6426

332

803

3𝑒−12

1𝑒−12

1𝑒−12

6016220

791/8

Delqu

ad◻

Elem

ents

1.06.5

11.11

24874

332

3109

1𝑒−12

6𝑒−13

8𝑒−13

156

4400

080

1/8Delq

uad

◻Elem

ents

0.5

5.9

11.13

9804

22

2412255

8𝑒−09

4𝑒−09

8𝑒−09

567

162168


Tim

e con

sum

ed to

bui

ld th

e mat

rices

(sec

onds

)


105 2 × 105 3 × 105


Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements

0.1

0.03

0.01

0.3

10

30

100

300

1

3

Figure 23: Time needed to build the matrices versus number of unknowns.

was considerably larger than the solution time itself. Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm

𝑅 ⋅ 𝜙 − (1/𝑘eff) ⋅ 𝐹 ⋅ 𝜙

‖𝑅‖ + (1/𝑘eff) ⋅ ‖𝐹‖< 10−8. (11)

The reported residual norm and relative error are𝑅 ⋅ 𝜙 −

1

𝑘eff⋅ 𝐹 ⋅ 𝜙

,

𝑅 ⋅ 𝜙 − (1/𝑘eff) ⋅ 𝐹 ⋅ 𝜙

(1/𝑘eff) ⋅ 𝜙

,

(12)

respectively. The fields marked as outer, linear, and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm. The computer used to solve the problem has anIntel i7 920@ 2.67GHz processor with 4Gb of RAM runningDebian GNU/Linux Wheezy.

When using a finite volumes-based scheme over anunstructured mesh, the solver has to gather information

about which cells are neighbors and which are not. Currentlygmsh does not write this kind of lists in its output files,so milonga has to explicitly solve the neighbors problem.Performing a linear search is an 𝑂(𝑁2) task, which isunacceptable for problem sizes 𝑁 of interest. The code usesa search based on a 𝑘-dimensional tree [27], which can inprinciple be performed in𝑂(𝑁) steps. Still, for large values of𝑁, the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution. Thisstep is not needed in finite elements, although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals, which thenhave to be assembled. Again, for large 𝑁, this step (numberthree) takes up most of the time needed to solve the problem.

The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles. If the mesh needs to be based on quadranglesinstead, a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle, whenever is possi-ble. However, for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere, which may be a


Tim

e con

sum

ed to

solv

e the

eige

nval

ue p

robl

em (s

econ

ds)


105 2 × 105 3 × 105



0.1

0.03

0.01

0.3

10

30

100

300

1

3

Figure 24: Time needed to solve the eigenvalue problem versus number of unknowns.

desired property of the resulting grid. For finite elements, thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change. However, the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure. On the other hand,the number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid.

Figure 19 shows how the computed static reactivity (i.e.,1 − 1/𝑘eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme. The four accepted results published in the originalreference [10] are included for reference, although it shouldbe taken into account that said reactivities were computedalmost forty years ago. Figure 20 shows the total wall timeneeded to solve the problem as a function of 𝑁𝐺, whileFigures 21, 22, 23, and 24 show the times needed for eachof the first four steps involved in the solution, maintainingthe same logarithmic scale for both the abscissas and theordinates. Green data represent finite volumes, whilst bluebullets correspond to finite elements. Solid lines indicatequarter-core and dashed lines eighth-symmetry. Fillet bullets

are results obtained by theDelaunay triangulation, and emptybullets were computed with the delquad algorithm. Finally,triangle-shaped data correspond to triangles and squares, anddiamonds denote quadrangles as the basic geometry of thegrid.

It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 𝑘eff than finite volumes with therefinement of the mesh. This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ𝑐. Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements. As expected,eighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknowns.For small problems, finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible. When the problem size grows, thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices. Also,at least for this configuration, it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements.


4. Conclusions

Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores. These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders, and therefore,not only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved. A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry, meshing algorithm,basic shape, and discretization scheme plus any value ofthe grid’s characteristic length by using a single input file.The complete set of input files and code—executable andsource—is available either online or upon request, withcomments, experiences, suggestions, and corrections beingmore than welcome. Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm. Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement. A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem.

Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families. Finite volumesmethods compute cell mean values, whilst finite elementsgive functional values at the grid’s nodes. In general, finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes. Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices. For a large number ofunknowns, the process of finding which cell is neighbor ofwhich—even based on a 𝑘-dimensional tree—overwhelmsthe computation and assembly of elementary matrices, andfinite-element methods perform better.

References

[1] G. Glasstone and S. Bell, Nuclear Reactor Theory, KriegerPublishing Company, 1970.

[2] J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis,John Wiley & Sons, New York, NY, USA, 1976.

[3] C. Gho, Reactor Physics Undergraduate Course Lecture Notes,Instituto Balseiro, 2006.

[4] G. G. Theler and F. J. Bonetto, “On the stability of the pointreactor kinetics equations,”Nuclear Engineering andDesign, vol.240, no. 6, pp. 1443–1449, 2010.

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