D A M I A N O S . M I L I T Ã O , H E R M E S A L V E S F I L H O , R I C A R D O C . B A R R O S

U N I V E R S I D A D E D O E S T A D O D O R I O D E J A N E I R O

B R A Z I L

2 2 I C T T – P O R T L A N D – 1 1 - 1 6 S E P T 2 0 1 1

A NUMERICAL METHOD FOR ONE-SPEED SLAB-GEOMETRY ADJOINT DISCRETE

ORDINATES PROBLEMS WITH NO SPATIAL TRUNCATION ERROR

Outline

�  Motivation

It is well known that the adjoint transport operator plays a very important role in deterministic and stochastic particle transport calculations both in non-multiplying and in multiplying media, e.g.,

ü  Monte Carlo simulation for “source-detector” problems ü  Perturbation theory

We describe today an application of the one-speed spectral Green’s function (SGF) nodal method for adjoint discrete ordinates (SN) problems in slab geometry.

The adjoint SGF method is free from spatial truncation errors; i.e., it generates numerical values for the node-edge and node-average adjoint angular fluxes that agree with the analytic solution of the adjoint SN equations, apart from computational finite arithmetic considerations.

Outline

●  Spectral Analysis

We first describe a spectral analysis to the adjoint SN equations within a homogeneous region of the slab to determine an expression for the local general solution of the adjoint SN equations.

●  The adjoint SGF nodal equations

The adjoint SGF nodal equations are composed of the standard spatial balance SN equations and the non-standard adjoint SGF auxiliary equations, that have parameters which preserve the local analytic general solution.

●  The adjoint NBI iterative scheme

We solve the adjoint SGF nodal equations iteratively using the “one-node block inversion” (NBI) iterative scheme.

●  The spatial reconstruction algorithm

One negative feature of coarse-mesh methods is that they generate numerical solution that generally does not yield detailed profile of the solution. One alternative way to go around this drawback is to use fine-mesh numerical methods, or proceed to reconstructing the coarse-mesh numerical solution.

●  Numerical results

●  Concluding remarks

Outline

Spectral Analysis

Let’s consider a discretization spatial grid Now we write the adjoint SN equations for node Γj

NmQxxdxd

j

N

nnn

jSmTjmm :1,

2)()( †

1

†0†† =+=+− ∑=

ωψσ

ψσψµ

Spectral Analysis

The analytic local general solution in Γj can be written as

The adjoint particular solution for constant Qj

† appears as

)()()(††

,,

†xxx

pmhmm ψψψ +=

)(

††

SojTj

jp

Qσσ

ψ−

=

Spectral Analysis

To determine an expression for the homogeneous component of the local general solution in Γj we consider the ansatz

We substitute this expression into the homogeneous

adjoint SN equations and consider the normalization condition

.,)()( /††, j

xmhm xeax Tj Γ∈= − ςσςψ

.1)(1

†∑=

=N

nnna ως

Spectral Analysis

The result is the adjoint dispersion relation whose roots are N real eigenvalues that appears symmetrically about the

origin. The expression for the homogeneous component of the general solution

can thus be written as where the components of the N eigenvectors are given by

,:1,,)()( /†

1

†, Nmxeax j

xlm

N

llhm

lTj =Γ∈= −

=∑ ςσςβψ

1)(2 1

0 =+∑

=

N

n n

njcµς

ως

.:1,:1,)(2

)( 0† NlNmc

anl

ljlm ==

+=

µςς

ς

Spectral Analysis

Therefore, the analytic local general solution of the adjoint SN equations in Γj is given by

At this point we remark that the set of N eigenvalues ςl is

the same as the set of N eigenvalues υl of the forward SN equations in Γj. However, the corresponding eigenvectors are not the same as am(υl) for the forward SN equations.

.:1,,)(

)()(0

†/†

1

† NmxQ

eax jjSTj

jxlm

N

llm

lTj =Γ∈−

+= −

=∑ σσ

ςβψ ςσ

)(† lma ς

The adjoint SGF nodal equations

Now we integrate the adjoint SN equations within node Гj and divide the result by hj to obtain the conventional discretized spatial balance adjoint SN equations

which together with the offered SGF adjoint auxiliary

equations form the adjoint SGF nodal equations.

( ) ,:1,2 1

††,

0†,

†2/1,

†2/1, NmQ

h

N

njnjn

jSjmTjjmjm

j

m =+=+−− ∑=

−+ ωψσ

ψσψψµ

)( ††2/1,

0,

†2/1,

0,

†, jjnnmjnnmjm QH

nn

+Λ+Λ= +>

−<

∑∑ ψψψµµ

The adjoint SGF nodal equations

In the adjoint SGF auxiliary equations we determine the N2 parameters to preserve the homogeneous component of the local general solution and the expression for to preserve the particular solution component.

The N2 entries of the square matrix are the

solutions of the N linear systems, one for each fixed value of m for l = 1 : N

nm,Λ

)( †jQH

nm,Λ

∑∑>

−

<

Λ+Λ=0

†,

2/

0

†,

2/†

).()()2/sinh()(.2

n

ljTj

n

ljTjlnnm

hlnnm

hljTj

Tjj

llm aeaehha

µ

ςσ

µ

ςσ ςςςσσ

ςς

Λ

The adjoint SGF nodal equations

The expression for is given by At this point we remark that the matrix θ for the forward SN

equations is equal to the present matrix for the adjoint SN equations. Matrix θ relates the node-average angular fluxes in all discrete angular directions to all incident node-edge angular fluxes.

This is in contrast to matrix which relates the node-average

adjoint angular fluxes in all discrete angular directions to all exiting node-edge adjoint angular fluxes .

)( †jQH

.)1(

)(

†

1,

†

SjTj

j

N

nnm

j

QQH

σσ −

Λ−=

∑=

Λ

Λ

The adjoint NBI iterative scheme

We use the values of Λm,n and in the SGF adjoint auxiliary equations; then we substitute the result into the collision and scattering terms of the discretized spatial balance adjoint SN equations to write the adjoint one-node block inversion (NBI†) matrix equations

)( †jQH

0,††††††† >+−= −−++− mµjOUT1/2j

OUT1/2j

IN1/2j QSψGψGψ

.0,††††††† <+−= +−−++ mµjOUT1/2j

OUT1/2j

IN1/2j QSψGψGψ

The adjoint NBI iterative scheme

For µm > 0 we sweep from left to right For µm < 0 we sweep from right to left

xj-1/2

xj-1/2

xj+1/2

xj+1/2

The spatial reconstruction algorithm

●  As I said, a negative feature of coarse-mesh methods is that they generally do not generate detailed profile of the solution. ●  Since the adjoint SGF method is free from spatial

truncation errors, we substitute the converged adjoint SGF numerical solution into the general solution

and solve the resulting system for the expansion coefficients βl , l = 1 : N. With these, we are able to reconstruct the adjoint solution at any point within each spatial node Гj, j = 1 : J.

NmxQ

eax jjSTj

jxlm

N

llm

lTj :1,,)(

)()(0

†/†

1

† =Γ∈−

+= −

=∑ σσ

ςβψ ςσ

Numerical results

Ø Multilayer slab 30 cm thick Ø Region 1: 10 cm, σT = 1.0 cm-1, σS = 0.9 cm-1

Ø Region 2: 5 cm, σT = 1.0 cm-1, σS = 0.9 cm-1

Ø Region 3: 15 cm, σT = 0.9 cm-1, σS = 0.8 cm-1

Ø S128 Guass-Legendre angular quadrature set

Ø By storing 1g (gamma rays emitter) in region 1 of the slab, we estimate a measurement of a detector response (5 cm, σa = 0.5 cm-1) that we place in region 3 (25 < x < 30) to evaluate the gamma ray leakage at the moment of the storage, one year and five years later, when the source has leaked to region 2.

Ø We use reflexive B.C. at x = 0 for the forward and adjoint calculations and vacuum B.C. at x = 30 for the forward calculations and no-leakage B.C. for the adjoint calculations.

Numerical results

Concluding remarks

�  The adjoint SGF method is very efficient at estimating measurements of detector responses due to uncharged radiation sources.

�  It generates numerical solutions to adjoint SN problems that are free

from spatial truncation errors. �  Therefore, the spatial reconstruction scheme generates the exact

adjoint flux profile within the slab. �  As future work, we propose a generalization of the present adjoint

SGF method for energy multigroup adjoint SN problems.

Thank you!