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
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
Spectral Analysis
Let s consider a discretization spatial grid
Now we write the adjoint SN equations for node Γj
N m Q x x dx d j N n n n j S m Tj m m , 1: 2 ) ( ) ( † 1 † 0 † † + = + = −
∑
= ω ψ σ ψ σ ψ µSpectral Analysis
The analytic local general solution in Γj can be written as
The adjoint particular solution for constant Qj† appears as
)
(
)
(
)
(
† † , , †x
x
x
p m h m mψ
ψ
ψ
=
+
)
(
† † Soj Tj j pQ
σ
σ
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 x m h mx
a
e
x
Tj∈
Γ
=
ς
−σ ςψ
.
1
)
(
1 †∑
==
N n n na
ς
ω
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 † , x a e x j m N x l m N l l h m l Tj ∈Γ = = − =
∑
β ς σ ς ψ 1 ) ( 2 1 0 = +∑
= N n n n j cµ
ς
ω
ς
. : 1 , : 1 , ) ( 2 ) ( 0 † c m N l N a n l l j l m = = + =µ
ς
ς
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 † x a e Q x m N j j S Tj j x l m N l l m l Tj ∈Γ = − + = − =
∑
β ς σ σ ψ σ ς ) ( † l m a ςThe 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 , Q m N h N n j n j n j S j m Tj j m j m j m − + = + = −∑
= − + ψ ω σ ψ σ ψ ψ µ ) ( † † 2 / 1 , 0 , † 2 / 1 , 0 , † ,j m n n j m n n j j m H Q n n + Λ + Λ = + > − <∑
∑
ψ ψ ψ µ µThe 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 n m,
Λ
) ( † j Q H n m,Λ
∑
∑
> − < Λ + Λ = 0 † , 2 / 0 † , 2 / † ). ( ) ( ) 2 / sinh( ) ( . 2 n l j Tj n l j Tj l n n m h l n n m h l j Tj Tj j l l m h e a e a h a µ ς σ µ ς σ ς ς ς σ σ ς ςΛ
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 .
) ( † j Q H . ) 1 ( ) ( † 1 , † Sj Tj j N n n m j Q Q H σ σ − 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 ) (Q†j H
0
,
† † † † † † †=
−
+
>
− − + + − jµ
m OUT 1/2 j OUT 1/2 j IN 1/2 jG
ψ
G
ψ
S
Q
ψ
.
0
,
† † † † † † †=
−
+
<
+ − − + + jµ
m OUT 1/2 j OUT 1/2 j IN 1/2 jG
ψ
G
ψ
S
Q
ψ
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+1/2The 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. N m x Q e a x j j S Tj j x l m N l l m l Tj , , 1: ) ( ) ( ) ( 0 † / † 1 † ∈Γ = − + = − =
∑
β ς σ σ ψ σ ςNumerical 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.
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.
