A NUMERICAL METHOD FOR ONE-SPEED SLAB-GEOMETRY ADJOINT DISCRETE ORDINATES PROBLEMS WITH NO SPATIAL TRUNCATION ERROR

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Leonor Gorjão
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1 A UMERICAL METHOD FOR OE-SPEED SLAB-GEOMETRY ADJOIT DISCRETE ORDIATES PROBLEMS WITH O SPATIAL TRUCATIO ERROR DAMIAO S. MILITÃO HERMES ALVES FILHO RICARDO C. BARROS UIVERSIDADE DO ESTADO DO RIO DE JAEIRO BRAZIL 22 ICTT PORTLAD SEPT 2011

2 Outie Motivatio It is we kow that the adoit trasport operator pays a very iportat roe i deteriistic ad stochastic partice trasport cacuatios both i o-utipyig ad i utipyig edia e.g. ü Mote Caro siuatio for source-detector probes ü Perturbatio theory We describe today a appicatio of the oe-speed spectra Gree s fuctio SGF oda ethod for adoit discrete ordiates S probes i sab geoetry. The adoit SGF ethod is free fro spatia trucatio errors; i.e. it geerates uerica vaues for the ode-edge ad ode-average adoit aguar fuxes that agree with the aaytic soutio of the adoit S equatios apart fro coputatioa fiite arithetic cosideratios.

3 Outie Spectra Aaysis We first describe a spectra aaysis to the adoit S equatios withi a hoogeeous regio of the sab to deterie a expressio for the oca geera soutio of the adoit S equatios. The adoit SGF oda equatios The adoit SGF oda equatios are coposed of the stadard spatia baace S equatios ad the o-stadard adoit SGF auxiiary equatios that have paraeters which preserve the oca aaytic geera soutio. The adoit BI iterative schee We sove the adoit SGF oda equatios iterativey usig the oeode bock iversio BI iterative schee.

4 Outie The spatia recostructio agorith Oe egative feature of coarse-esh ethods is that they geerate uerica soutio that geeray does ot yied detaied profie of the soutio. Oe aterative way to go aroud this drawback is to use fie-esh uerica ethods or proceed to recostructig the coarse-esh uerica soutio. uerica resuts Cocudig rearks

5 Spectra Aaysis Let s cosider a discretizatio spatia grid ow we write the adoit S equatios for ode Γ Q x x dx d S T 1: ω σ σ µ

6 Spectra Aaysis The aaytic oca geera soutio i Γ ca be writte as x x + h p x The adoit particuar soutio for costat Q appears as p σ T Q σ So

7 Spectra Aaysis To deterie a expressio for the hoogeeous copoet of the oca geera soutio i Γ we cosider the asatz σ x/ x a e T x Γ h We substitute this expressio ito the hoogeeous adoit S equatios ad cosider the oraizatio coditio 1 a ω 1..

8 Spectra Aaysis The resut is the adoit dispersio reatio whose roots are rea eigevaues that appears syetricay about the origi. The expressio for the hoogeeous copoet of the geera soutio ca thus be writte as where the copoets of the eigevectors are give by 1: / 1 x e a x x h T Γ σ β c µ ω. 1: 1: 2 0 c a + µ

9 Spectra Aaysis Therefore the aaytic oca geera soutio of the adoit S equatios i Γ is give by At this poit we reark that the set of eigevaues is the sae as the set of eigevaues υ of the forward S equatios i Γ. However the correspodig eigevectors are ot the sae as a υ for the forward S equatios.. 1: 0 / 1 x Q e a x S T x T Γ + σ σ β σ a

10 The adoit SGF oda equatios ow we itegrate the adoit S equatios withi ode Г ad divide the resut by h to obtai the covetioa discretized spatia baace adoit S equatios µ h σ S 0 Q 1: 1/ 2 + σ 1/ 2 T ω which together with the offered SGF adoit auxiiary equatios Λ + Λ 1/ 2 + H Q + 1/ 2 µ < 0 µ > 0 for the adoit SGF oda equatios.

11 The adoit SGF oda equatios I the adoit SGF auxiiary equatios we deterie the 2 paraeters Λ to preserve the hoogeeous copoet of the oca geera soutio ad the expressio for H Q to preserve the particuar soutio copoet. Λ The 2 etries of the square atrix Λ are the soutios of the iear systes oe for each fixed vaue of for 1 : 2. a h σ T sih σ h T / 2 e σ T h / 2 µ < 0 Λ a + e σ h T / 2 µ > 0 Λ a.

12 The adoit SGF oda equatios The expressio for H Q H Q is give by 1 1 T Λ σ σ S Q. At this poit we reark that the atrix θ for the forward S equatios is equa to the preset atrix Λ for the adoit S equatios. Matrix θ reates the ode-average aguar fuxes i a discrete aguar directios to a icidet ode-edge aguar fuxes. Λ This is i cotrast to atrix which reates the ode-average adoit aguar fuxes i a discrete aguar directios to a exitig ode-edge adoit aguar fuxes.

13 The adoit BI iterative schee We use the vaues of Λ ad H Q i the SGF adoit auxiiary equatios; the we substitute the resut ito the coisio ad scatterig ters of the discretized spatia baace adoit S equatios to write the adoit oe-ode bock iversio BI atrix equatios I 1/2 OUT OUT G+ + 1/2 G 1/2 + S Q µ > 0 I OUT OUT + 1/2 G+ 1/2 G + 1/2 + S Q µ < 0.

14 The adoit BI iterative schee For µ > 0 we sweep fro eft to right x -1/2 x +1/2 For µ < 0 we sweep fro right to eft x -1/2 x +1/2

15 The spatia recostructio agorith As I said a egative feature of coarse-esh ethods is that they geeray do ot geerate detaied profie of the soutio. Sice the adoit SGF ethod is free fro spatia trucatio errors we substitute the coverged adoit SGF uerica soutio ito the geera soutio x 1 β a e σ x/ T + σ x Γ ad sove the resutig syste for the expasio coefficiets β 1 :. With these we are abe to recostruct the adoit soutio at ay poit withi each spatia ode Г 1 : J. T Q σ S 0 1:

16 uerica resuts Ø Mutiayer sab 30 c thick Ø Regio 1: 10 c σ T 1.0 c -1 σ S 0.9 c -1 Ø Regio 2: 5 c σ T 1.0 c -1 σ S 0.9 c -1 Ø Regio 3: 15 c σ T 0.9 c -1 σ S 0.8 c -1 Ø S 128 Guass-Legedre aguar quadrature set Ø By storig 1g gaa rays eitter i regio 1 of the sab we estiate a easureet of a detector respose 5 c σ a 0.5 c -1 that we pace i regio 3 25 < x < 30 to evauate the gaa ray eakage at the oet of the storage oe year ad five years ater whe the source has eaked to regio 2. Ø We use refexive B.C. at x 0 for the forward ad adoit cacuatios ad vacuu B.C. at x 30 for the forward cacuatios ad o-eakage B.C. for the adoit cacuatios.

17 uerica resuts

18 Cocudig rearks The adoit SGF ethod is very efficiet at estiatig easureets of detector resposes due to ucharged radiatio sources. It geerates uerica soutios to adoit S probes that are free fro spatia trucatio errors. Therefore the spatia recostructio schee geerates the exact adoit fux profie withi the sab. As future work we propose a geeraizatio of the preset adoit SGF ethod for eergy utigroup adoit S probes. Thak you!

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