The Iterated Fission Matrix

ANS Winter Meeting Las Vegas, NV, Nov. 610, 2016

The Iterated Fission Matrix Manuele Auero, Yishu Qiu, Massimiliano Fratoni University of California, Berkeley

Introduction The Iterated Fission Matrix Conclusions and ongoing work

Outline

Introduction:

The Fission Matrix method & Èigenmode' perturbation theory

How we obtain higher eigentriplets via Monte Carlo transport (...and why we need them)

The Iterated Fission Matrix

Combining Fission Matrix and Iterated Fission Probability

Asymptotically exact eigenvalues w/ nite mesh size (...yet another trick with latent generations)

Conclusions and ongoing work:

A `matrix-free' approach to the Fission Matrix method

(Petrov-)Galerkin eigenvalue method on Krylov subspaces (...it's all about projection) Manuele Auero UC Berkeley


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The Fission Matrix method Èigenmode' Perturbation Theory Accuracy of the Fission Matrix estimates for ki

The Fission Matrix method Eigenvalue problem for a multiplying system discretized into N cells:

1 F · Sn kn

Sn = Sn,I =

N 1 X FI ←J · Sn,J kn J=1

k0 and

S0 : fundamental eigenpair

FI ←J : average eective number of ssion neutrons produced in cell I from a neutron born in cell J

Manuele Auero UC Berkeley


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The Fission Matrix method

Two main sources of error: Statistical uncertainty (not discussed here)

Spatial discretization (studied in this work)



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The Fission Matrix method Estimates for discretized F are usually obtained in criticality source Monte Carlo simulations:

(0) FI ←J

=

R

R

r 0 ∈VI

r ∈VJ

F(r 0 ← r ) · S0 (r ) · dr dr 0 R

S0 (r ) · dr

r ∈VJ

S0 (r ) is the (continuous) fundamental ssion source distribution estimated via Monte Carlo (0)

F represents a consistent discretization for the solution of: S0 = Manuele Auero UC Berkeley

1 F · S0 k0 The Iterated Fission Matrix

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The Fission Matrix method Higher eigenmodes Consistent discretization of F(r 0 ← r ) for higher eigenmodes:

R (n) FI ←J

=

R

r 0 ∈VI r ∈VJ

F(r 0 ← r ) · Sn (r ) · dr dr 0 R

Sn (r ) · dr

r ∈VJ

Unfortunately, Because to use

(0)

Sn (r ) is not available!

lim

VI ,VJ →zero

(n) FI ←J

→

(0) FI ←J ,

it is common practice

F instead of (n) F , to obtain higher eigenpairs.



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The Fission Matrix method Higher eigenmodes Case study: UAM GENIII PWR MOX core (2D)



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The Fission Matrix method Higher eigenmodes



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k1 /k0 ' 0.995



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k2 /k0 ' 0.995



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k3 /k0 ' 0.985



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k4 /k0 ' 0.984



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k5 /k0 ' 0.972



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Èigenmode' Perturbation Theory The eect of a perturbation x on the ssion source distribution S0 is projected onto the set of eigenmodes S = S1 , S2 , S3 · · ·

d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki



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d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki



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d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki



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d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki

dF can be eciently calculated via Monte Carlo (X)GPT for dx several perturbed parameters x in a single Serpent run (collisionhistory approach)



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d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki

Continuous XGPT estimators for S†i


T

dF S0 dx


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d S0 = dx

∞ X

Si

†

Si ·

i=1

Td

F

S0

dx k0 − ki

Today's problem



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Èigenmode' Perturbation Theory



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Èigenmode' Perturbation Theory Perturbation of

239

Pu resonance parameters -- Γf @0.2956 eV

Cross sections sensitivities (3% Γf relative perturbation)

XS sensitivity [-]

0.5 0.0 -0.5

MT102 MT18

-1.0 4

Reference XS [b]

10

MT102 MT18

3

10

2

10

1

10

0

10

-7

-6

10

10

-5

10

Energy [MeV]



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d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki



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d S0 = dx

∞ X i=1

Si

†

Si ·

Td

F

S0

dx k0 − ki

1 · 10−3 error on ki → 10% 20% error on the projection coecient Very accurate estimates for higher eigenvalues ki are required (target accuracy . 10−4 )



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Accuracy of the Fission Matrix estimates for k1

k1 /k0 ' 0.995



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k1 /k0 ' 0.995



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k3 /k0 ' 0.985



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k3 /k0 ' 0.985



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k5 /k0 ' 0.972



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k5 /k0 ' 0.972



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The Iterated Fission Matrix Spatial convergence of the Iterated Fission Matrix Implementation in SerpentOpenFOAM

The Iterated Fission Matrix (IFM) If kn is a real, positive eigenvalue,

Sn =

1 · kn

F Sn is equivalent to Sn =

1

kn

`

·

`

F Sn

The main idea:

Replace the ssion kernel F(r 0 ← r ) with the `th iterated ssion kernel (`) F(r 0 ← r ), to produce the `th IFM New MC estimators based on

Iterated Fission Probability

` = number of latent generations (as in IFP)



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Ìterated' ssion kernel (self-convolution of the ssion kernel)

F(r 0 ← r )

P FI ←J =

current gen.

wnew · δ(rnew ∈I ) · δ(rprogenitor ∈J)

P prev. gen.

wprogenitor · δ(rprogenitor ∈J)



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Ìterated' ssion kernel (self-convolution of the ssion kernel) (2)

F(r 0 ← r ) = (F ∗ F) (r 0 ← r ) ≡ F(r 0 ← r 00 ) · F(r 00 ← r ) · dr 00 R

V

P (2)

FI ←J =

current gen.


P gen. −2




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Ìterated' ssion kernel (self-convolution of the ssion kernel) (3)

F(r 0 ← r ) ≡

s

F(r 0 ← r 00 ) · F(r 000 ← r 00 ) · F(r 00 ← r ) · dr 00 dr 000

V

P (3)

FI ←J =

current gen.


P gen. −3




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Calculating the Iterated Fission Matrix The Iterated Fission Matrix

R (`) (0) FI ←J

=

r 0 ∈V

R I

(`) (0) (`)

F can be derived as:

F(r 0 ← r ) · S0 (r ) · dr dr 0

r ∈VJ

R

S0 (r ) · dr

r ∈VJ

Analog estimator for the numerator in criticality simulation: number of ssion neutrons in cell I with the `th -generation progenitor tagged with cell-label J Propagation of information to descendants as in IFP



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Discretization errors Because of discretization errors, (`) FI ←J VI ,VJ →zero (n)

lim

(`) (0)

F

6=

→

(`) (0) FI ←J

→

(0) FI ←J

(0)

F

`

converges faster than lim

VI ,VJ →zero

(n) FI ←J



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Convergence of the Iterated Fission Matrix for k1

k1 /k0 ' 0.995 1.000 0.995

Dominance ratio k 1/k 0

0.990 0.985 0.980 0.975 0.970

l =1 l =2 l =6 l =11 l =16

0.965 0.960 0.955 0.950 10 0

10 1

10 2

10 3

10 4

Mesh size



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k1 /k0 ' 0.995 10-1 l =1 l =2 l =6 l =11 l =16

Error

10-2

10-3

10-4 0 10

10

1

10

2

10

3

10

4

Mesh size



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k1 /k0 ' 0.972 k5/k0

−2

Error

10

−3

10

g=0 g=1 g=5 g=10 g=15

−4

10

1

10

2

3

10

10 mesh size



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Convergence of the Iterated Fission Matrix...

WHY?

The main assumption behind the Fission Matrix Method is that any neutron born in the cell J has the same probability of producing a neutron in the cell I,

at the next generation

The main assumption behind the Iterated Fission Matrix is that any neutron born in the cell J has the same probability of producing a neutron in the cell I, after ` generations



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Implementation in SerpentOpenFOAM OpenFOAM: C++ Finite-Volume multiphysics toolkit SerpentOpenFOAM internal coupling (OF as library)

OpenFOAM classes for unstructured mesh + multiphysics coupling (CFD/Porous Media CFD)

lapacke C wrapper for lapack FORTRAN linear algebra package arpack++ C++ wrapper for arpack FORTRAN library for large sparse (generalized) eigenvalue problems via ARNOLDI method



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Conclusions Current applications (Petrov-)Galerkin method for the continuous ssion kernel

Summary The problem

The calculation of higher eigenpairs is often performed adopting the Fission Matrix Method Some application (e.g., perturbation theory) requires very accurate estimates for the eigenvalues ki (∼tens of pcm) FM needs a very ne spatial discretization to achieve accurate ki estimates (might be prohibitive in 3D cases)



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Summary Our solution The Iterated Fission Matrix (IFM) is a combination of the Fission Matrix and the Iterated Fission Probability The implementation∗ requires the propagation of cell indexes to descendant ∗

straightforward implementation for MC codes w/ Fission Matrix and


IFM showed to converge much faster (in the considered case) → smaller meshes



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Summary Our solution The Iterated Fission Matrix (IFM) is a combination of the Fission Matrix and the Iterated Fission Probability The implementation∗ requires the propagation of cell indexes to descendant ∗

straightforward implementation for MC codes w/ Fission Matrix and


IFM showed to converge much faster (in the considered case) → smaller meshes Behavior of statistical errors should be investigated! Manuele Auero UC Berkeley


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Ongoing work Applications

Applications involving IFM & MC Perturbation Theory: Nuclear data uncertainty propagation for power distribution in large and loosely-coupled reactors Convergence acceleration of coupled Monte CarloCFD multiphysics simulations via Newton iterations (submitted to M&C2017)



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ARNOLDI iteration for the eigenvalue problem The adoption of ARNOLDI iteration in the SerpentOpenFOAM implementation ensures fast and cheap (CPU & memory) solution of the eigenvalue problem The discretized ssion matrix F is sparse and large If F is N × N , only m N eigenmodes are of interest Full decomposition of F is expensive (CPU & memory) Arnoldi method operates on a Krylov subspace of F 2 3 k K = span v, Fv, F v, F v , · · · , F v Only matrix-vector multiplications, it never operates on F



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Matrix-free approach to the Fission Matrix Method Question Why should we spend resources building a huge matrix F if we never use it? Answer uhhh... maybe we shouldn't! Idea Let's use a method that doesn't need to access elements of F and let it operate directly on the ssion kernel F(r 0 ← r ) What? (Petrov-)Galerkin projection and Krylov subspace expansion on random ssion source distributions



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Fission kernel Petrov-Galerkin projection

Trial subspace: Φ = {φ1 (r ), φ2 (r ), φ3 (r ), · · · , φm (r )} Test subspace: Ψ = {ψ1 (r ), ψ2 (r ), ψ3 (r ), · · · , ψm (r )}



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Trial subspace: Φ = {φ1 (r ), φ2 (r ), φ3 (r ), · · · , φm (r )} Test subspace: Ψ = {ψ1 (r ), ψ2 (r ), ψ3 (r ), · · · , ψm (r )} Z

Z

aij = F (ψi , φj ) =

ψi (r 0 ) F(r 0 ← r ) φj (r ) · dr dr 0

r 0 ∈V r ∈V



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Z


ψi (r 0 ) F(r 0 ← r ) φj (r ) · dr dr 0

r 0 ∈V r ∈V

Analog estimator for all the aij in a single MC run:∗ P aij =

current gen.

P prev. gen.

∗

wnew · ψi (rnew ) · φj (rprogenitor )

wprogenitor · φj (rprogenitor ) · φj (rprogenitor )

Correction for S0 (r ) required



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Z


ψi (r 0 ) F(r 0 ← r ) φj (r ) · dr dr 0

r 0 ∈V r ∈V

Z bij = hψi , φj i =

ψi (r ) φj (r ) · dr r ∈V



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Z


ψi (r 0 ) F(r 0 ← r ) φj (r ) · dr dr 0

r 0 ∈V r ∈V



A = F (Φ, Ψ) ;


B = hΦ, Ψi


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Z


ψi (r 0 ) F(r 0 ← r ) φj (r ) · dr dr 0

r 0 ∈V r ∈V



A = F (Φ, Ψ) ;

B = hΦ, Ψi

Reduced (m × m) generalized eigenvalue problem: A y = λB y Manuele Auero UC Berkeley


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Reduced Eigenvalue problem If λn and

ny

are the nth eigenpair of

A y = λB y,

eigenvalues and eigenmodes of the full problem

Sn = k1n F Sn

can be approximated as:

kn ' λn ;

Sn (r ) ' Φ n y =

m X

yn,h · φh (r )

h=1



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Reduced Eigenvalue problem If λn and

ny

are the nth eigenpair of

A y = λB y,

eigenvalues and eigenmodes of the full problem

Sn = k1n F Sn

can be approximated as:

kn ' λn ;

Sn (r ) ' Φ n y =

m X

yn,h · φh (r )

h=1

If the subspaces Φ and Ψ are not good enough...

Krylov subspace expansion via IFP (in the same run): Φ = {Φ, F Φ, (F ∗ F) Φ, (F ∗ F ∗ F) Φ} Ψ = Ψ, F† Ψ, F† ∗ F† Ψ, F† ∗ F† ∗ F† Ψ Manuele Auero UC Berkeley


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First test: 2D homogeneous molten salt mixture

50 orthogonal random vectors (red noise):



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First test: 2D homogeneous molten salt mixture

50 orthogonal random vectors (red noise):



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Results

10th forward eigenmode 1-G diusion Galerkin



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Results




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Results

23th forward eigenmode 1-G diusion



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Results




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Results

23th forward eigenmode Krylov-Galerk. 1-G diusion Galerkin



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Thank you for the attention!

Questions? Suggestions?



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Èigenmode' Perturbation Theory Sn =

1 · F Sn kn

S†n =

and

T 1 · F S†n kn

The eigenmodes are bi-orthogonal and normalized such that: T

S† S = I      

S†0 S†1 .. .

S†n





   S0 S1 · · ·  


 1 0 ···    0 1 · · · Sn  . . .. = .   .. .. 0 0 ···


 0 0  ..  .

1

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