A Reduced-Order Neutron Diffusion Model Separated in Space and Energy via Proper Generalized Decomposition

Kurt A. Dominesey and Wei Ji

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NYdomink@rpi.edu, jiw2@rpi.edu

INTRODUCTION

As radiation transport problems are inherently high-dimensional (posed in space r = (x, y, z), angle Ω = (θ, ϕ),energy E, and time t) the associated computational cost scalesrapidly as O(NrNΩNE Nt). This scaling law can render sim-ulations of even moderate refinement costly or intractable,often necessitating restriction of the problem domain or phys-ical simplifications. By contrast, Reduced-Order Models, orROMs, offer a model which retains the physics of the original,but at a reduced computational effort, disrupting this scalinglaw.

In particular, Proper Generalized Decomposition (PGD)is uniquely useful as an a priori ROM strategy. This is incontrast to a posteriori ROM methods, which rely on multiplerealizations of the full-order model to construct the ROM. Asthis “offline” step can be omitted in PGD, one may circumventthe potentially prohibitive requirement of repeatedly solvingthe full-order model to compute the ROM.

Application of PGD to neutron transport remains prelim-inary but with growing interest in recent years. In specific,Gonzàlez-Pintor et. al. applies PGD to spatially decomposeone-group, two-dimensional neutron diffusion and computethe multiplication factor of a nuclear reactor [1]. This ideais furthered by Senecal and Ji who demonstrate a similarspatially-separated neutron diffusion model with two energygroups [2]. For transients, Alberti and Palmer seek a space-time separated PGD method for one-dimensional diffusion,most recently demonstrated in [3]. Finally, Dominesey andJi use PGD to separate space and angle in one-dimensionalneutron transport and revisit familiar discussions of sourceiteration [4]. However, application of PGD to separate the en-ergetic dimension of neutron transport has yet to be addressed.

As such, we herein present a method for one-dimensionalneutron diffusion separated in space and energy, with anisotropic continuous slowing-down scattering kernel. Numeri-cal convergence with increasing grid refinement and solutionenrichment is demonstrated using the Method of ManufacturedSolutions (MMS). Lastly, the method is applied to a practi-cal problem regarding the slowing down of fission neutrons,where monotonic convergence with enrichment is observed byglobal and local relative numerical indicators.

THEORY

The fundamental axiom of PGD is that a given solution uadmits an approximate separated representation as the finitesum of M products of D mode functions, Xd

m, where eachsuperscript d denotes a unique dimension of the solution

u(x1, x2, . . . xD) ≈M∑

m=1

D∏d=1

Xdm(xd). (1)

For the problem at hand, we seek the scalar flux φ(x, u) interms of spatial modes Xm(x) and lethargy modesUm(u)

φ(x, u) ≈M∑

m=1

Xm(x)Um(u) (2)

where the lethargy u is a dimensionless variable related tothe neutron energy E by u = ln(E∗/E) and E∗ is an arbitraryenergy, often (and presently) chosen as 10 MeV = 107 eV.

Application to Neutron Transport

We begin with the steady-state, energy-dependent neutrondiffusion equation in slab geometry with isotropic scatteringand an isotropic source1

−∇ · D(x, u)∇φ(x, u) + Σt(x, u)φ(x, u)

=

∫ umax

umin

Σs(x, u′ → u)φ(x, u′)du′ + Q(x, u)(3)

and decompose the total and differential scattering cross-sections Σt(x, u) and Σs(x, u′ → u) as

Σt(x, u) =

J∑j=1

N j(x)σt, j(u)

Σs(x, u′ → u) =

J∑j=1

N j(x)σs, j(u′)P j(u′ → u)

(4)

for J unique nuclides with number densities N j and micro-scopic total and scattering cross-sections σt, j and σs, j. More-over, P j(u′ → u) is the probability a neutron scattering offnuclide j will be transferred from lethargy u′ to u. Assum-ing, without loss of generality, the target is at rest (possessingnegligible thermal motion) relative to the neutron we have

P j(u′ → u) =

eu′−u

1−α j, u − ln(1/α j) ≤ u′ ≤ u,

0, otherwise(5)

with

α j =

(A j − 1A j + 1

)2

(6)

where A j is the mass number of nuclide j.Note that we could also resort to a multigroup-collapsed

formulation by dividing our lethargy scale into “groups” g and

1While we employ the diffusion equation for simplicity, our separationin lethargy would proceed almost identically for any other transport method,such as S N , Method of Characteristics, or higher-order PN methods.

Transactions of the American Nuclear Society, Vol. 120, Minneapolis, Minnesota, June 9–13, 2019

457Computational Methods: General

writing

Σs(x, u′ → u) =

H∑h=1

Πh(x)Σs,h,g′→g,ug′−1 ≤ u′ ≤ ug′

ug−1 ≤ u ≤ ug

0, otherwise

Πh(x) =

1, x ∈ xh

0, otherwise

(7)

for H unique materials h each located on set xh. (A similarform would likewise be assumed for the total cross-section,Σt.) However, this requires a priori knowledge of the flux φ tocompute flux-averaged multigroup cross-sections Σs,h,g′→g.

As the continuous slowing-down model (Equation 4) re-quires finer lethargic resolution (and no prerequisite flux distri-bution), we find it a more compelling initial application of ourROM. However, any ROM developed for continuous-slowingdown should be readily applicable to multigroup-collapsedcalculations (which simplify the cross-sections and scatteringkernel to piecewise-constant over each group g = 0, 1 . . .G).

In a similar fashion to our cross-sections, we now take aseparable form of our source,

Q(x, u) =

N∑n=1

Q(n)x (x)Q(n)

u (u). (8)

Furthermore, let us assume a homogeneous medium of a sin-gle nuclide j with unit number density, such that Equation 3(omitting the subscript j = 1) simplifies to

−D(u)∂2

∂x2 φ(x, u) + σt(u)φ(x, u)

=

∫ u

ulim

σs(u′)eu′−u

1 − αφ(x, u′)du′ +

N∑n=1

Q(n)x (x)Q(n)

u (u)

(9)

whereulim = max(u − ln(1/α), umin). (10)

Multiplying by an arbitrary test function ϕ∗(x, u) and integrat-ing over all space x ∈ Ωx and lethargy u ∈ Ωu we achieve thefollowing weak form. For brevity, let us introduce the notation(

•, •)Ωa×Ωb...

=

∫Ωa×Ωb...

• • da db . . . (11)

such that we find(−D

∂2

dx2 φ+σtφ−

∫ u

ulim

σseu′−u

1 − αφdu′, ϕ∗

)Ωx×Ωu

=

( N∑n=1

Q(n)x Q(n)

u , ϕ∗)Ωx×Ωu

(12)In keeping with PGD methodology we assume our test

function to be of the form

ϕ∗(x, u) = X∗(x)UM(u) + XM(x)U∗(u) (13)

whereX∗ andU∗ are arbitrary functions. As Equation 12 mustbe satisfied for any choice of X∗ andU∗, our weak form canbe distributed to form two separate equations. Subsequently

substituting in our PGD approximation of the flux (Equation2) we obtain for the first term of the test function, X∗UM

−

M∑m=1

( d2

dx2Xm,X∗

)Ωx

(DUm,UM

)Ωu

+

M∑m=1

(Xm,X

∗

)Ωx

(σtUm,UM

)Ωu

−

M∑m=1

(Xm,X

∗

)Ωx

( ∫ u

ulim

σseu′−u

1 − αUm(u′)du′,UM(u)

)Ωu

=

N∑n=1

(Q(n)

x ,X∗)Ωx

(Q(n)

u ,UM

)Ωu

(14)

and the second,2 XMU∗

−

M∑m=1

( d2

dx2Xm,XM

)Ωx

(DUm,U

∗

)Ωu

+

M∑m=1

(Xm,XM

)Ωx

(σtUm,U

∗

)Ωu

−

M∑m=1

(Xm,XM

)Ωx

( ∫ u

ulim

σseu′−u

1 − αUm(u′)du′,U∗(u)

)Ωu

=

N∑n=1

(Q(n)

x ,XM

)Ωx

(Q(n)

u ,U∗)Ωu

.

(15)

As each equation contains products of unknowns (XM andUM) owing to the underlined terms, our problem is thereforenonlinear. However, we may solve this system for our latestmodes M by employing an iterative linearization strategy.

Picard Iteration

We here select Picard, or “fixed-point”, iteration for theresolution of our nonlinear system. Specifically, we guessU(0)

Mas a uniform distribution (in lethargy), then solve Equation14 for X(1)

M , treating UM := U(0)M as a known. This alternat-

ing iteration is repeated until convergence, as measured by asuitable indicator, here taken to be a relative L2 metric3

ε(k) =||X

(k)MU

(k)M − X

(k−1)M U

(k−1)M ||2

||X(k−1)M U

(k−1)M ||2

. (16)

Additionally, as the solution of our nonlinear system istruly some product of the constituent modes XMUM , we findit prudent to normalize one mode within each Picard iteration

U(k)M := U(k)

M /||U(k)M ||2 (17)

to avoid “drifting” of modes between iterations, where onemode shrinks while the other grows. We summarize our PGDalgorithm, separated in space and lethargy, in Algorithm 1.

2These equations would be identical in the multigroup-collapsed

case, save for the substitutions of∫ u

ulim

σseu′−u

1−α Um(u′)du′ and σt with∑Gg′=0

∫ ug′

ug′−1Σs,g′→gUm(u′)du′ and Σt,g forUm(u) in each group g = 0, . . . ,G.

3This nonlinear tolerance is set at 10−6 in all numerical examples shown.

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458 Computational Methods: General

Algorithm 1: Space-Lethargy PGD

φ← 0;for m← 1, 2 do Boundary Conditions

Prescribe Xm,Um to satisfy boundary conditions;φ← φ + Xm ⊗Um;// Omitted for homogenous Dirichlet// m = 2 omitted if φleft(u) ∝ φright(u)

for m← 3, 4 . . . M do Enrichment LoopInitializeU(0)

m as guess; // Here uniformfor k ← 1, 2 . . . do Picard LoopX

(k)m ← solution of Equation 14 givenU(k−1)

m ;U

(k)m ← solution of Equation 15 given X(k)

m ;U

(k)m ←U

(k)m /||U(k)

m ||2; // Normalizationε(k) ← compute by Equation 16;if ε(k) < tolerance then // Here 10−6

Xm ←X(k)m ;

Um ←U(k)m ;

breakφ← φ + Xm ⊗Um;

NUMERICAL RESULTS

Through Picard iteration, the underlined terms of Equa-tions 14 and 15 become iterative constants. Reverting tothe strong form of each equation, we denote each constantD,T,S,QX/Um

(where the specific definition can be deter-mined by inspection) and write for the spatial mode at M = 1

−DU1

d2

dx2X1 +(TU1 − SU1

)X1 =

N∑n=1

Q(n)U1

Q(n)x (18)

which is a diffusion-reaction equation, akin to traditional one-group neutron diffusion. At present, we discretize this equationusing centered finite-differences. Likewise forU1 we find(−DX1 D + TX1σt

)U1 − SX1

∫ u

ulim

σseu′−u

1 − αU1du′ =

N∑n=1

Q(n)X1

Q(n)u ,

(19)or a Volterra integral equation of the second kind, similar to theneutron slowing-down equation in infinite media.4 Discretiza-tion proceeds by a trapezoidal expansion method, expandingUM and σs as nodal values interpolated by first-order La-grange polynomials. Note for M > 1, our equations becomecumbersome, but no more difficult to solve, as all summandsof m < M are known, effectively becoming source terms.

Method of Manufactured Solutions

We begin by applying the Method of Manufactured Solu-tions (MMS) for an exactly separable function

φ(x, u) = σs(u)−1 sin (πx) sin(

3π (10 ln(1/α) − u)10 ln(1/α)

)(20)

4This Volterra nature is specific to the assumption of target-at-rest kine-matics; upscattering renders this equation Fredholm. Moreover, steady-statefission would introduce a nonlinear (eigenvalue) Fredholm operator.

which can be represented in a single mode (that is, with M =1). Selecting our cross-sections and diffusion coefficient

σt = 1/E = eu/E∗, σs = σt, D = σs/(3σ2t ) (21)

we solve for a slab Ωx = [0, 1] with A = 12 over lethargyinterval u ∈ Ωu = [0, 10 ln(1/α)] given homogenous Dirichletboundary conditions (that is, φ = 0 on ∂Ωx). We find our PGDimplementation converges to the numerical solution withinone mode, such that all subsequent modes are of negligiblemagnitude. Comparing the numerical error of our methodwith increasing grid refinement, we achieve Figure 1 wherethe L2 error is defined as

L2 Error =||φ − φexact||2

||φexact||2(22)

which suggests we have achieved second-order, or O(h2), ac-curacy (where h is conventionally the “mesh width” and isinversely proportional to N − 1).

102 103

Nodes N

10−6

10−5

10−4

L2

Err

orO(h2)

Fig. 1. Numerical convergence with grid refinement.

Satisfied with the numerical accuracy of our method, wenext move to characterize a manufactured solution which re-quires infinitely many modes to represent exactly

φ(x, u) = sin(π(1 − x2

) (1 − u2

))(23)

and show the error with increasing number of modes M inFigure 2. In solving this problem we again take homogeneousDirichlet boundary conditions on Ωx = [−1, 1], with

σt = E = E∗e−u, σs = σt, D = σs/(3σ2t ) (24)

over lethargy Ωu = [−1, 1] where A = 2 and N = 500. As themanufactured source will also be of infinite rank, we decom-pose the discrete source via Singular Value Decomposition(SVD) to obtain a form compatible with Equation 8.

From Figure 2, we observe near optimal separation up tothe first three modes, almost matching the ideal decompositionof the exact solution given by SVD. However, the error thenstagnates, indicating we have reached the discretization errorincurred by numerical evaluation of our differential (spatialdiffusion) and integral (lethargic scattering) operators.

Slowing Down of Fission Neutrons

We move now to a more practical problem. Specifically,let us simulate the flux arising from a source

Q(x, u) = Π(x)χ(u) (25)

Transactions of the American Nuclear Society, Vol. 120, Minneapolis, Minnesota, June 9–13, 2019

459Computational Methods: General

1 2 3 4 5 6Modes M

10−6

10−5

10−4

10−3

10−2

10−1

L2

Err

or

Progressive PGD

SVD

Fig. 2. Numerical convergence with modal enrichment.

where Π represents a uniform source in the center of the slab

Π(x) =

1, −1 cm ≤ x ≤ 1 cm0, otherwise

(26)

and

χ(u) = 0.453e−1.036E∗e−usinh

(√2.29E∗e−u

)(27)

is an expression for the prompt neutron fission spectrum of235U (in MeV) given by [5]. We take our system to be a 10 cmthick slab of a fictitious material with the mass of deuterium(A = 2). Discretizing over N = 201 nodes in both dimensions,we assume our cross-sections and diffusion coefficient to be

σt = min(103eu/E∗, 1), σs = 0.9σt, D = σs/(3σ2t ).(28)

and compute the flux over the energy interval ΩE =[10 MeV, 10−3 eV], or Ωu = [0, 23.03] with E∗ = 10 MeV.Figure 3 displays the first five modes in each dimension.

−4 −2 0 2 4

x [cm]

0.0

0.5

1.0

1.5X1

X2

X3

X4

X5

Q(1)x · ||Q(1)

u ||2

0 2 4 6 8 10

u/ ln(1/α)

0.00

0.25

0.50

0.75

1.00

1.25

10−4

10−3

10−2

10−1

100

σt

[cm

2]

U1 · ||X1||2U2 · ||X2||2U3 · ||X3||2U4 · ||X4||2U5 · ||X5||2Q

(1)u · ||Q(1)

x ||2/||Q(1)u ||2

σt

Fig. 3. Scalar flux induced by prototypical 235U fission source.

In Figure 4 we estimate the convergence of our numericalproblem. Unlike the previous two studies, we have no knownsolution against which to compare, and so instead measure therelative change between iterations. Specifically, we employglobal L2 and local L∞ indicators (||Xm||2 and ||Xm||∞||Um||∞),

normalized to the contribution of the first mode, m = 1. Eachmetric reports monotonic convergence (with agreement be-tween global and local criteria), and is reduced by over sevendecades given M = 15 modes. Moreover, we expect a toler-ance on either measure would provide an appropriate termina-tion criterion for the enrichment process in practice.

2 4 6 8 10 12 14Mode m

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

||Xm||2/||X1||2(||Xm||∞||Um||∞)/(||X1||∞||U1||∞)

Fig. 4. Estimated convergence with modal enrichment for theslowing down of fission neutrons.

CONCLUSIONS

In summary, we have derived and demonstrated a methodfor neutron diffusion separated in space and energy via PGD.Using MMS, we have demonstrated O(h2) numerical accuracyand detailed a test case in which we obtain near optimal de-composition (compared to SVD) before converging upon thediscretization error. Lastly, we characterize convergence withenrichment by global and local error indicators for a prototyp-ical fission slowing-down problem. Future efforts will seekto characterize the method using higher-fidelity models (forspatial transport and energetic scattering) and more realisticproblems (including experimentally-measured cross-sections).

ACKNOWLEDGMENTS

This research was performed under appointment of thefirst author to the Rickover Fellowship Program in NuclearEngineering sponsored by Naval Reactors Division of theNational Nuclear Security Administration.

REFERENCES

1. S. GONZÁLEZ-PINTOR, D. GINESTAR, and G. VERDÚ,“Using Proper Generalized Decomposition to Compute theDominant Mode of a Nuclear Reactor,” Mathematical andComputer Modeling, 57, 1807–1815 (2013).

2. J. P. SENECAL and W. JI, “Characterization of the ProperGeneralized Decomposition Method for Fixed-Source Dif-fusion Problems,” Ann. Nucl. Energy, 126, 68–83 (2019).

3. A. L. ALBERTI and T. S. PALMER, “Reduced Order Mod-eling of Non-Linear Radiation Diffusion Via Proper Gen-eralized Decomposition,” Trans, Am. Nucl. Soc., 119, 691–694 (2018).

4. K. A. DOMINESEY, J. P. SENECAL, and W. JI, “AReduced-Order Neutron Transport Model Separated inSpace and Angle,” Trans. Am. Nucl. Soc., 119, 687–690(2018).

5. J. J. DUDERSTADT and L. J. HAMILTON, Nuclear Reac-tor Analysis, John Wiley & Sons, Inc. (1976).

Transactions of the American Nuclear Society, Vol. 120, Minneapolis, Minnesota, June 9–13, 2019

460 Computational Methods: General