SECOND-ORDER ADJOINT SENSITIVITY ANALYSIS PROCEDURE (SO-ASAP) FOR COMPUTING EXACTLY AND EFFICIENTLY FIRST- AND SECOND-ORDER SENSITIVITIES IN LARGE-SCALE LINEAR SYSTEMS: II. ILLUSTRATIVE APPLICATION TO A PARADIGM PARTICLE DIFFUSION PROBLEM Dn G. Ccuci Deprtment of Mechnicl Engineering, University of South Crolin E-mil: ccuci@cec.sc.eu Corresponing uthor: Deprtment of Mechnicl Engineering, University of South Crolin 3 Min Street, Columbi, SC 98, USA Emil: ccuci@cec.sc.eu; Phone: (99) 99 964; Submitte to JCP: August 4, 4 ABSTRACT This work presents n illustrtive ppliction of the secon-orer joint sensitivity nlysis proceure (SO-ASAP) to prigm neutron iffusion problem, which is sufficiently simple to mit n exct solution, thereby mking trnsprent the mthemticl erivtions unerlying the SO-ASAP. The generl theory unerlying SO-ASAP inictes tht, for physicl system comprising N α prmeters, the computtion of ll of the first- n seconorer response sensitivities requires ( N α + ) lrge-scle computtions involving corresponingly constructe joint systems, which we clle secon joint sensitivity systems (SASS). Very importntly, however, the illustrtive ppliction presente in this work shows tht the ctul number of joint computtions neee for computing ll of the first- N α + per response. n secon-orer response sensitivities my significntly less thn For this illustrtive problem, four (4) lrge-scle joint computtions suffice for the complete n exct computtions of ll 4 first- n istinct secon-orer erivtives. Furthermore, the construction n solution of the SASS requires very little itionl effort beyon the construction of the joint sensitivity system neee for computing the first-orer sensitivities. Very significntly, only the sources on the right-sies of the iffusion (ifferentil) opertor neee to be moifie; the left-sie of the ifferentil equtions (n hence the solver in lrge-scle prcticl pplictons) remine unchnge.
All of the first-orer reltive response sensitivities to the moel prmeters hve significntly lrge vlues, of orer unity. Also importntly, most of the secon-orer reltive sensitivities re just s lrge, n some even up to twice s lrge s the first-orer sensitivities. We show tht the secon-orer sensitivities hve the following mjor impcts on the compute moments of the response istribution: () they cuse the expecte vlue of the response to iffer from the compute nominl vlue of the response ; (b) their contributions to the response vrinces n covrinces re reltively minor by comprison to the contributions stemming from the first-orer response sensitivities; n (c) they contribute ecisively to cusing symmetries in the response istribution. Inee, neglecting the secon-orer sensitivities woul nullify the thir-orer response correltions, n hence woul nullify the skewness of the response; consequently, ny events occurring in response s long n/or short tils, which re chrcteristic of rre but ecisive events (e.g., mjor ccients, ctstrophes), woul likely be misse. We expect the SO-ASAP to ffect significntly other fiels tht nee efficiently compute secon-orer response sensitivities, e.g., optimiztion, t ssimiltion/justment, moel clibrtion, n preictive moeling. KEYWORDS: secon-orer joint sensitivity nlysis proceure (SO-ASAP); secon-orer functionl erivtives; response covrinces; response skweness; prticle iffusion.
. INTRODUCTION The ccompnying PART I [] of this work hs presente the secon-orer forwr n joint sensitivity nlysis proceures (SO-FSAP n SO-ASAP) for computing exctly n efficiently the secon-orer functionl erivtives of physicl system responses (i.e., system performnce prmeters ) to the system s moel prmeters. The efinition of system prmeters inclues ll computtionl input t, correltions, initil n/or bounry conitions, etc. The SO-ASAP buils on the first-orer joint sensitivity nlysis proceure (ASAP) for nonliner systems introuce in ([], [3]) n further evelope in ([4]-[6]); for recent pplictions, see lso [7] n references therein. For physicl system comprising N α prmeters n ( N / 3 / α Nα ) N r responses, we note in PART I [] tht the SO-FSAP requires totl of + lrge-scle computtions for obtining ll of the first- n secon-orer sensitivities, for ll N r system responses. On the other hn, the SO-ASAP requires totl of ( N α + ) lrge-scle computtions for obtining ll of the first- n secon-orer sensitivities, for one functionl-type system responses. Therefore, the SO-FSAP shoul be use when N r N α, while the SO-ASAP shoul be use when N α Nr. The ltter cse, where the number of system prmeters is significntly greter thn responses, is the sitution most often encountere in prctice. In this work, we present n illustrtive ppliction of the SO-ASAP to prticle iffusion problem. This prigm problem comprises the mjor ingreients neee for highlighting the slient fetures involve in pplying the SO-ASAP, yet is sufficiently simple to mit n exct solution, thereby mking trnsprent the mthemticl erivtions presente in PART I []. Very importntly, this ppliction will show tht the construction n solution of the secon joint sensitivity system (SASS) requires very little itionl effort beyon the construction of the joint sensitivity system neee for computing the first-orer sensitivities, n tht the ctul joint computtions neee for computing ll of the first- n secon-orer response sensitivities re fr less thn N α per response. This pper is structure s follows: fter efining the prticle iffusion problem in Section Section. n. present the ppliction of the SO-ASAP for obtining the exct expressions of both the first- n secon-orer sensitivities of representtive (etector) 3
response to ll of moel s prmeters. In Section.3, the numericl vlues of these sensitivities re use to highlight their funmentl roles in the propgtion of moel prmeter uncertinties to cuse uncertinties in the compute responses. Very importntly, it will be shown tht they contribute ecisively to cusing ssymetries in the response istribution. In other wors, neglecting the secon-orer sensitivities woul nullify the thirorer response correltions n hence woul nullify the response s skewness. Consequently, ny events occurring in response s long n/or short tils, which re chrcteristic of rre but ecisive events (e.g., mjor ccients, ctstrophies), woul likely be misse. Higherorer moments of the response uncertinty istribution impct subsequent uses of the response istribution, prticulrly for risk quntifiction, ecision nlysis, n quntifying confience intervls, to nme few. Finlly, the concluing remrks in Section 3 highlight irections we re currently pursuing for further generlizing the SO-ASAP. 4
. A PARADIGM PARTICLE DIFFUSION PROBLEM Consier the iffusion of neutrons in spent fuel pool like those use by utilities to cool spent fuel rector elements. For simplicity, the pool is consiere to hve one irection (sy, the length) to be much lrger tht the other two, so tht it coul be consiere to be slb of wter. The fuel elements re consiere to be uniformly istribute within the pool, n re consiere to emit 3 Q neutrons cm secon. The neutrons re consiere to be monoenergetic, ll possesing the verge therml energy corresponing to the pool s wter temperture. Mthemticlly, this simple physicl system cn be moele by using the liner neutron iffusion eqution ϕ D Σ, ϕ + Q= x (, ), ϕ ( ± ) =, () where ϕ ( x) enotes the neutron flux, D enotes the iffusion coefficient, Σ enotes the mcroscopic bsorption cross section, Q enotes the istribute source term, n the pool is consiere to be slb of extrpolte thickness. In view of the problem s symmetry, the origin x = cn be conveniently chosen t the mile (center) of the slb. For simplicity, the bounry conitions chosen for Eq. () re tht the neutron flux must vnish t the extrpolte istnce x =. The liner ifferentil eqution in Eq. () cn be solve reily to obtin the generl solution Q cosh ( xk ),, x (, ) Σ cosh ( k ) ϕ x = k Σ D. () where k Σ D enotes the reciprocl iffusion length. The bove expression for the neutron flux ϕ ( x) hols for ny physicl vlues of D, Σ n Q. For prcticl problems, the flux cnnot be obtine nlyticlly s bove, but woul nee to be compute numericlly, using the nominl prmeter vlues Σ, computtion woul yiel the nominl vlue of the flux, ϕ ( x) D, n Q. Such numericl, nmely 5
ϕ Q ( xk ) ( k ) cosh,, ( x) = k Σ D Σ cosh (, ) x. (3) A generic response, R( x ), for the prigm neutron iffusion problem moele by Eq. () is the reing of etector within the slb, for exmple, t istnce b from the slb s miline t x =. The reing of such etector response woul inicte the neutrons rection rte within the etectors rective mteril, n woul be mthemticlly represente in the form Σ ϕ, = (, ), = ( Σ,,, Σ ) R e b e u α α D Q, (4) where Σ represents the etector s equivlent rection cross section, ssume to be constnt, for simplicity. The system prmeters for this problem re thus the positive sclrs Σ, D, Q, n Σ. Referring to the mthemticl nottion use in PART I [], we note the corresponences ( D Q ) ϕ x ( ϕ ) α= Σ,,, Σ, ux, e= e, α, (5) where the gger ( ) enotes trnsposition. The nominl vlues of the stte function (the neutron flux in this illustrtive problem) n moel prmeters will be enote s ( D Q ) ϕ x ( ϕ ) α = Σ,,, Σ, u ( x), e, α. (6) Using Eq. (3) in Eq. (4) gives the following form for the nominl response vlue, R ( e ): ( bk ) ( k ) Q Σ cosh R( e ) =Σ ϕ ( b) =, ( ϕ, ), x (, ). e α (7) Σ cosh 6
For fixe vlue of b, the response R ( e ) is functionl (i.e., sclr-vlue opertor) tht cts linerly on ϕ, s evience by in Eq. (7). On the other hn, even though Eq. () is liner in ϕ, the solution ϕ ( x) epens nonlinerly on α, s highlighte by Eq. (3). The sme is true of the response R ( e ). Even though R ( e ) is liner seprtely in ϕ n in α, s shown in Eq. (4), R ( e ) is not simultneously liner in ϕ n α ; this fct les to nonliner epenence of R ( e) on α. This fct is explicit higlighte by the exct expression of R ( e ) given in Eq. (7)... SO-ASAP Appliction to Compute Exctly n Efficiently the First-Orer Response Sensitivities to Moel Prmeters The prmeters in this problem re etermine from experiments fflicte by uncertinties; in prticulr, the uncertinties for the bsic (microscopic) neutron cross sections re provie in centrlly eposite covrince files. Therefore, the system prmeters α cn n o vry (becuse of uncertinties, externl influences, etc.) from their nominl vlues enote here s α by mounts ( D Q ) h Σ,,, Σ, (8) For esy reference, the nottion in Eq. (8) correspons to the nottion use in PART I []. In prctice, the vritions Σ, D, Q, Σ, usully correspon to the stnr evitions quntifying the uncertinties in the respective moel prmeters. As shown in PART I [], the sensitivities of the response R ( e ) to the vritions h α re given by the (first-orer) G- ifferentil R( ; ) δ e h of R e t (,ϕ ) e α, which is efine s { } ϕ α R( ε ; h) R( ε + εh), with h h, h. (9) ε ε = Applying the bove efinition to the response efine by Eq. (3) gives 7
{ } R( ε ; h) ε( ) ϕ ( x) ε hϕ ( x) α α Rϕ hϕ, ε Σ + Σ + = R ε h + ε () ε = where the irect-effect term R α e h is efine s α ( ) ( b) D Q ( b) R e h ϕ,,, Σ,,, Σ = Σ ϕ, () while the inirect-effect term R h is efine s ϕ ϕ R e h Σ h x. () ϕ ϕ ϕ As Eq. () inictes, the opertor R ( ; ) δ e h is liner in h ; in prticulr, R h is liner opertor on h ϕ. This liner epenence on h is unerscore by writing henceforth DR ( e ; h ) inste of R ( ; ) R e t δ e h to enote the sensitivity of The irect-effect term R α Eq. (), to obtin α e to vritions h. e h cn be evlute t this stge by replcing Eq. (6) into ϕ ϕ ( xk ) ( k ) Q cosh R ( e ) h = ( Σ ), x (, ). (3) Σ cosh However, the inirect-effect term, R h not yet vilble. As escribe in PART I [], h ( x), cnnot be evlute t this stge, since h ( x) ϕ ϕ ϕ is ϕ is relte to h α, n this reltionship is given by the forwr sensitivity equtions (FSE) obtine by computing the G-ifferentils of Eq. (), together with the corresponing bounry conitions. Performing this opertion gives the eqution L α hϕ + Lα ϕ α h α =, (4) together with the bounry conitions 8
. h ± = (5) ϕ In Eq. (4), the opertor L( α ) is efine s L( α ) D Σ, (6) while the opertor L α ( α ) ϕ h α is efine below: Thus, the opertor L α ( α ) of ( Lϕ ) t ϕ L ϕ α h ( D) ( Σ ) ϕ + ( Q), (7) ϕ h α, which mthemticlly represents the prtil G-ifferentil α with respect to α, contins ll of the first-orer prmeter vritions h α. Eqution (4) together with the bounry conitions in Eq. (5) constitute the forwr sensitivity system. To obtin its solution, h ( x) ϕ, for every possible vrition h α, this system of equtions woul nee to be solve repetely (ech such computtion represents lrgescle computtion). Thus, the forwr sensitivity system is vntgeous to use only if the number of responses excees the number of moel prmeters, which is selom the cse in prctice. Nevertheless, Eqs. (4) n (5) provie n inepenent wy to verify the results tht will be obtine lter in this Section by pplying the SO-ASAP. Therefore, we provie below the expression of the solution of Eqs. (4) n (5): = cosh cosh + C x ( xk ) ( k ) ( k ) ( xk ) x ( ) h x C ϕ xk k sinh cosh sinh cosh,,. (8) where the constnts C n C re efine, respectively, s 9
n C C ( δσ ) Q Σ ( δq), Σ ( cosh k ) ( δ ) ( δ ) D D Σ Σ Q ( k ) D Σ cosh. (9) () Hence, the expression of the inirect-effect term is obtine by replcing Eq. (8) in Eq. () to obtin: ( e ) cosh cosh C x ( xk ) ( k ) ( k ) ( xk ) x ( ) R ϕ hϕ x =Σ C xk k +Σ sinh cosh sinh cosh,,. () As ws generlly shown in [], n reclling the nottions introuce in PART I [], the funmentl prerequisite for pplying the SO-ASAP is the introuction of Hilbert spce, H u, pproprite to the problem t hn. For our illustrtive exmple, it is pproprite to chose H to be the rel Hilbert spce H L ( Ω ), with Ω (, ) u inner prouct u, equippe with the Hu L () ( Ω) Ω ( ) f x, g x f x g x, for f, g,,. In view of Eq. (), the inirect-effect term Rϕ form s follows: e h cn be expresse in inner prouct ϕ R e h Σ h b = Σ h x x b = Σ x b, h. (3) ϕ ϕ ϕ ϕ ϕ The next step unerlying the SO-ASAP is the construction of the opertor L + ( α ) tht is formlly joint to L( α ). For this purpose, Eq. (4) is multiplie by squre-integrble, but
t this stge otherwise still rbitrry, function ψ ( x) = ( Ω) integrte over x to obtin H L, n is subsequently Q h ϕ ϕ ϕ Σ = ( Σ ) +. (4) ψ x D h x ψ x D ϕ Q Integrting the left-sie of Eq. (4) by prts twice, to trnsfer ll opertions on h ( x) opertions on ψ ( x), les to ϕ to h ϕ ψ x ψ ( x) D Σ hϕ ( x) = D ψ ( x) hϕ ( x) Σ h ϕ + D ψ h ϕ ψ. (5) The right-sie of the bove equtions shows tht the forml joint of L( α ) is the opertor α D (6) * Σ. L Note tht the function ψ ( x) is still rbitrry t this stge, except for the requirement tht Q ψ H = L Ω. The next step in the construction of the joint system is the ientifiction of the source term, which is chieve by requiring the first terms on the right-sies of Eqs. (3) n (5) to represent the sme functionl. Imposing this requirement yiels the eqution * ψ L ( α ) ψ D Σ ψ ( x) =Σ ( x b), (7) The bounry conitions for ψ ( x) cn now be selecte by requiring tht unknown vlues of } h ϕ, such s the erivtives { hϕ, woul be eliminte from Eq. (5). Since h is ϕ known t x = ± from Eq. (5), the elimintion of unknown vlues of h ϕ cn be ccomplishe by choosing the bounry conitions
ψ ( ± ) =. (8) The bove selection of the joint bounry conitions completes the construction of the joint sensitivity system, consisting of Eqs. (7) n (8). Since the bounry conitions in Eq. (8) for the joint function ψ ( x) re the sme s the bounry conitions for h ( x) φ in * Eq. (5), it follows from this n from Eqs. (6) n (6) tht the opertors L ( α ) n L( α ) re not just formlly, but bon-fie self-joint. Using Eqs.(4) n (7) in Eq.(3) les to the following expression for the inirect-effect term Rϕ e h in terms of the joint function ψ ( x) : ϕ ϕ R ϕ ( e ) hϕ = ψ ( x) ( D) ( Σ ) ϕ ( x) + ( Q). (9) As expecte, the joint sensitivity system, cf., Eqs. (7) n (8), is inepenent of prmeter vritions h α, so it nees to be solve only once to obtin the joint function ψ ( x). Very importnt, too, is the fct (chrcteristic of liner systems) tht the joint system is inepenent of the originl solution ( x) ϕ n cn therefore be solve inepenently (n without ny knowlege) of the forwr neutron flux ϕ ( x). Of course, the joint system epens on the response, which provies the source term s shown in Eq. (7). Solving the joint system for our illustrtive exmple yiels the following expression for the joint function ψ ( x) : ( b ) Σ sinh k ψ ( x) = ( x ) k H( x b) ( x b) k sinh ( k + + ΣD ) sinh sinh, (3) where H( x b) is the Hevisie-step functionl efine s
H x, for x < =., for x (3) Replcing the joint function ψ ( x) from Eq. (3) in Eq. (9) n crrying out the respective integrtion over x yiels the first-orer sensitivity DR ( e ; h ) of R ( e ) t in the system prmeters: e to vritions h α where R R R R DR ( ; ) ( ) ( D) ( Q), D Q e h = Σ + + + Σ (3) Σ Σ R S ( ) = ψ ( x) ϕ ( x) S = QS S Ak + Bk S D S (33) R ϕ S Q S = x = x x + x ( ) ψ ψ ϕ ψ D D D Q S = D S kbk (34) R ( ) ( S ) ( 3 3 = ψ x =S S A k ) Q (35) R ( ) ( S ) ( 4 = ϕ x x b = Q S A k ) S (36) with the qutities Ak n B k efine, respectively, s Ak ( bk ) ( k ) cosh, (37.) cosh 3
A ( ) ( B k = cosh k ) sinh ( k ) cosh ( bk ) bsinh ( bk ) cosh ( k ). k k (37.b) One of the min uses of sensitivities is for rnking the reltive importnce of prmeter vritions in influencing vritions in responses. Reltive sensitivities re use for this purpose, since they re imensionless numbers; the reltive sensitivity of response R ( e ) to the i th -prmeter, rel α i, is efine s Si ( R αi) αi R( e e ). It is evient from Eqs. (7), (35) n (36) tht the reltive sensitivities of the etector response R ( e ) to the source ( α3 3 Q) n to the etector s rection cross section ( α4 Σ ) rel rel S3 3 ( R Q) Q R( e ) = n S4 ( R S) S R( e ) = position ( x b) e e re unity, i.e.,, regrless of the = of the loction of the etector or the mteril properties of the meium. This mens tht % increse/ecrese in either Q or Σ will inuce % increse/ecrese in nominl vlue of the response R ( e ). These re rther lrge sensitivities, inee! The reltive influence of response s sensitivities to the meium s mcroscopic bsorption cross section, Σ, n to the meium s iffusion coefficient, D, cn best be illustrte by consiering, specificlly, tht = 5 cm, n the istribute neutron sources emit nominlly 7 3 S = neutrons cm s. Furthermore, the mteril properties escribing the bsorption n iffusion of therml neutron in wter hve the following nominl vlues: Σ =.97 cm, D =,6 cm. Consier tht mesurements re performe with n ielize, infinitely thin, etector hving n inium-like nominl etector cross section Σ = 7.438cm. Consier now six symmetric responses, t the following etector loctions: (i) b = ± cm, locte close to the wter pool s center; (ii) b = ± 4cm, locte close to the pool s bounries; n (iii) b = ± 49.5cm, locte very close to the ege, where the neutron flux grient becomes very steep, n the iffusion coefficient mrkely influences the etector s response. We esignte the etector s responses t these loctions s R ( cm), R (4 cm), R 3 (49.5 cm), R 4 (- cm), R 5 (-4 cm), n R 6 (-49.5 cm). These symmetric loctions, t which Eq. (7) clerly inictes tht R( x) = R( x), were elibertely chosen in orer to 4
verify the preservtion of symmetry uring n fter the computtion of the respective responses sensitivities to the moel prmeters. In units of, the nominl 3 [ neutrons cm s ] compute response vlues obtine using Eq. (7) for these etector loctions were s follows: 9 9 ( ) = ( ) = 3.77, R ( cm) R ( cm) R cm R cm 4 8 ( 49.5 ) = ( 49. 5 ) = 6. 6 R cm R cm 3 6 7. 4 = 4 = 3.66, n 4 Note lso tht R( x) = R( x), so tht only three (rther thn six) lrge-scle joint sensitivity computtions [i.e., soving Eqs. (7) n (8)] suffice to obtin ll of results presente in Tble (bsolute sensitivities, in the respective units, which re, for brevity omitte from the tble) n Tble (reltive sensitivities). Tble. Absolute sensitivities for six etector responses First-orer Absolute Sensitivities R ( cm) R 4 (- cm) R (4 cm) R 5 (-4 cm) R 3 (49.5 cm) R 6 (-49.5 cm) S (SIG) -.97x -758x -.673x S (D) -.33x 5 -.39x 9 -.737x 9 S 3 (Q) 3.776x 3.663x 6.76x S 4 (SIG) 5.76x 8 4.94x 8 8.68x 7 Tble. Reltive sensitivities for six etector responses First-orer Reltive Sensitivities R ( cm) R 4 (- cm) R (4 cm) R 5 (-4 cm) R 3 (49.5 cm) R 6 (-49.5 cm) S (SIG)_rel -. -.95 -.54 S (D)_rel -5.64x -6 -.5 -.46 S 3 (Q)_rel... S4 (SIG)_rel... The bsolute sensitivities will be use in Section.3 to compute response uncertinties rising from prmeter uncertinties. They re ifficult to use, however, for rnking the importnce of prmeters in influencing the respective responses. By contrst, the reltive sensitivities presente in Tble reily inicte the importnce of the moel prmeters in influencing the etector responses. Of course, these prmeter importnces reflect the reltive importnce of the physicl processes ssocite with the respective prmeters in influencing rel rel the etector s response. As lrey mentione, S 3 = S 4 =, n these sensitivities turn out to be the most importnt ones for the etector response, regrless of the etector s position. This result is physiclly ue to the fct tht the neutron source strength, on the one hn, n 5
the etector s interction cross section, shoul be very importnt. Note, in prticulr, tht the rel sensitivity S 4 escribes irect effect term specific to the etector s properties (in this rel illustrtive prigm exmple, S 4 is ctully inepenent of the meium s mteril properties). The other two reltive sensitivities, nmely S rel ( R S) S R( e ) ( ) ( e ) e n rel S R D D R e escribe opposing physicl processes n, hence, influences on rel the etector responses.thus, S is very importnt in the center of the slb, where it hs rel lrge negtive vlue of comprble bsolute mgnitue to S 3. Towrs the ege of the slb, rel S ecreses in bsolute mgnitue but remins negtive throughout the meium. rel Physiclly, the behvior of S inictes tht the bsorption process of neutrons cross section rel is loss mechnism (hence the negtive sign of S throughout the meium), n its importnce is lmost s lrge s the prouction by the neutron sources (s inicte by the rel positive sign n mgnitue of S 3 ), throughout most of the meium. Very close to the physicl bounry of the meium, where the neutron flux must vnish becuse of the chosen bounry conitions, the bsorption mechnism iminishes rpily in importnce. The flux grient is very steep over smll istnce towrs the bounry, inicting bounrylyer-like behvior of the bsorption process, which obviously must cese t the meium s extrpolte bounry, where the flux must mthemticlly vnish ue to the impose bounry conitions. On the other hn, the behvior of S rel ( R D) D R( e ) e inictes tht the iffusion coefficient n consequently the etils of the iffusion process, ply very minor role in rel influencing the etector response, except close to the bounries of the slb. The sign of S is negtive, inicting loss of neutrons in contributing to the etector s response (i.e., incresing the iffusion coefficient, n hence the importnce of the iffusion process, cuses fewer neutrons to interct with the etector n hence contribute to its response). Towrs the extrpolte bounry of the meium, nmely t.5 cm from the respective extrpolte rel rel bounries, the mgnitue of S becomes comprble to tht of S, both being negtive, inicting neutron losses from the etector s response. This result is expecte since the 6
neutron flux ws (mthemticlly) require to vnish t the extrpolte bounries, n the physicl mechnisms tht reuce the neutron flux cn only stem from bsoption combine with iffusion... SO-ASAP Appliction to Compute Exctly n Efficiently the Secon-Orer Response Sensitivities to Moel Prmeters As shown in the generl theory presente in Section 3. of PART I [], the funmentl philosophicl n strting point for computing the secon-orer erivtives (sensitivities) S α R α α = R α α ws the consiertion of the first-orer sensitivities to ij i j j i be functionls of the form S i ϕψ,, α, i =,,3,4. Bse on this funmentl consiertion, the SO-ASAP procees by computing the first-orer G-ifferentil, S i (,, ) the functionls (,, ) Si u αψ, t the point (, ) e ψ, from the efinition δ e ψ g, of (ny of) (,, ;,, ) (,, Si ϕ ψ α hϕ hψ hα Si ϕ + εhϕ ψ + εhψ + ε α) α h (38) ε ε = for n rbitrry sclr ε F, n ll (i.e., rbitrry) vectors ( h, h, ) ( x) ( x) N h H Ω H Ω H. For our illustrtive exmple, there re only ϕ ψ α ϕ ψ α ( N ) α α + = istinct secon-orer erivtives, ue to the symmetry property R αi α j = R α j αi. It is convenient to compute the sensitivities ij S α by strting with the simplest ones n progressing to the more ifficult ones. An exmintion of Eqs. (33) through (36) reily revels tht S4 ( α ) n 3 while S α hve the simplest expressions, S α hs the most complicte expression. Therefore, it is computtionlly vntgeous to compute first the four secon-orer erivtives ( α ) αi S4 R Q, =,,3,4, followe by the three secon-orer erivtives i ( α ) R S α ( = ), since 34 S3 i i,,,3 S α woul lrey be vilble. The computtions woul then procee by etermining S ( ) R i S i, =, n, 7
lstly, sub-sections. S R D α. These computtions will be performe explicitly in the following... Applying the SO-ASAP to Compute the Secon-Orer Response Sensitivities ( S ) S R α 4i i Applying the efinition shown in Eq. (38) to Eq. (36) yiels the corresponing G-ifferentil, 4 DS α, of the first-orer sensitivity S 4 α, s DS4 h x b h x x b ( ) = ( ϕ + ε ϕ) ( ) = ϕ ( ), (39) ε ε = where the function h ϕ is the solution of Eqs. (4) n (5). Note tht the entire contribution to 4 DS α comes from the inirect-effect term; there is no irect-effect term contribution to 4 DS α. Applying the generl theoreticl consiertion presente in PART I [] for the SO-ASAP, we note tht the system joint to Eqs. (4) n (5), but corresponing to the response DS4 α, cn be obtine by following the proceure outline in Section., bove, when eriving the (first) joint sensitivity system for the joint function ψ ( x), cf., Eqs. (7) n (8). We will esignte the system joint to Eqs. (4) n (5), but corresponing to the response DS4 ( α ) s the secon joint sensitivity system, n its solution will be esignte s λ 4 ( x). Thus, following the logic scribes in Section., bove, we reily obtin the following secon joint sensitivity system for the joint function λ 4 ( x) : λ4 D Σ λ4 ( x) = ( x b), (4) 8
λ ± =. (4) 4 DS α oes not epen on hψ ( x), the secon joint function [sy, θ Since 4 4 x ], which woul hve correspone to hψ ( x), is ienticlly zero, since the joint system for θ woul hve been liner homogeneous eqution with zero source n zero bounry conitions. In terms of the joint function 4 x becomes: 4 x λ, the expression of the ifferentil DS4 ( α ) ϕ DS4( ) = λ4( x) ( D) + ( S ) ϕ ( x) ( Q) (4) = S S + S D + S Q + S S ( ) ( ) ( ) ( ) 4 4 43 44, where R 4 = λ4 ϕ, S S S x x (43) R ϕ 4 = λ4, S D S x (44) S R 43 = λ4 ( x), S Q 3 (45) S R (46) S S 44. Note tht 44 S, since the expression of 4 DS α, nmely Eq.(39), oes not contin the quntity ( Σ ). Note lso tht the joint system stisfie by λ 4 ( x) n (4), is the sme s woul be obtine by setting by the joint function ψ ( x), cf. Eqs. (7) n (8). Therefore, iviing the result in Eq. (3) by Σ or, equivlently, setting, comprising Eqs.(4) Σ into the joint system stisfie λ is simply obtine by 4 x Σ in Eq. (3), to obtin: 9
λ ( x) = { ψ ( x) } 4 Σ = ( b ) sinh k sinh ( x ) k H( x b) sinh ( x b) k. sinh ( k + + ΣD ) (47) { } Even more, since λ ( x) ψ ( x) =, it is not even necessry to solve Eqs. (4) n (4) to 4 Σ compute λ 4 ( x), n it is not necessry to introuce Eq. (47) into Eqs. (43) through (45), n to perform the respective integrtions. In fct, one simply replces Σ in Eqs. (33) through (35), or, equivlently, ivies these equtions through Σ, to obtin: ( ) R S S4( ) = = Q ( S ) Ak + Bk, (48) S S S S D S S S ( ) R S Q, (49) 4( ) = = kbk S D S D S ( ) 3 43 = = S S Q S ( ) Ak 3 R S. (5)... Applying the SO-ASAP to Compute the Secon-Orer Response Sensitivities ( ) S3i R Q αi 3. The G-ifferentil, 3 DS α, of the first-orer sensitivity S efinition given in Eq. (38) to Eq. (35) to obtin 3 α is compute by pplying the DS3 h h x ( ) = ( ψ + ε ψ ) = ψ, (5) ε ε =
where the function h ψ is the solution of the G-ifferentite (first) joint sensitivity system, cf., Eqs. (7) n (8). Thus, pplying the efinition of the G-ifferentil [cf., Eq. (38)] to Eqs. (7) n (8) yiels: h ψ ψ D Σ hψ ( x) = ( D) + ( Σ ) ψ + ( Σ) ( x b), (5) h ± =. (53) ψ Once gin, the entire contribution to 3 is no irect-effect term contribution to 3 DS α comes from the inirect-effect term; there DS α. The system joint to Eqs. (5) n (53), n lso corresponing to the response DS3 ( α ) is constructe by pplying the sme proceure s lrey use in the previous Sections, to obtin: D θ Σ = (54) 3 θ3 ( x), θ ± =. (55) 3 DS α oes not epen on hϕ ( x), the secon joint function [sy λ 3 ( x) Since 3 ], which woul hve correspone to h ϕ, is ienticlly zero, since the joint system for λ 3 ( x) woul hve been liner homogeneous eqution with zero source n zero bounry conitions. In terms of the joint function 3 x θ, the expression of the ifferentil 3 DS α becomes: ψ DS3( ) = θ3( x) ( D) ( S) ψ ( x) ( S) ( x b) (56) = S S + S D + S Q + S S ( ) ( ) ( ) ( ) 3 3 33 34, where R 3 = θ3 ψ, Q S S 3 x x (57)
R ψ S 3 = θ x 3 3 Q D ( ) θ ψ ( ) θ = S D x x + S D x x b 3 3, (58) S R 3 = (59) Q Q 33, R S 3 = θ x x b = θ b (6) 34 3 3. Q S Note tht the joint system comprising Eqs. (54) n (55) oes not even nee to be solve, since simple comprison of this system to the originl iffusion Eq. () for the neutron flux ϕ ( x) shows tht θ ϕ (54) n (55) to obtin x = x Q. This fct cn be reily verifie by ctully solving Eqs. 3 / θ ( x) ( xk ) ( k ) cosh =. 3 Σ cosh (6) Introucing the bove expression for θ 3 ( x) into Eqs.(57) through (6), n performing the respective integrtions les to the following results for the secon-orer sensitivities S 3i : R S 3 S S Ak Bk (6) 3( ) = + Q S S D S ( S ) S 3 R S k (63) 3( ) = B k Q D D S S 3 R Ak (64) 34( ) = θ3 ( b) = Q Σ Σ
Note tht the computtion of S 34 vi Eq. (6) gives the sme result s hs lrey been obtine from the computtion of S 43 vi Eq. (5). This importnt consiertion provies n inepenent verifiction tht the joint functions θ 3 ( x) n, respectively, λ 4 ( x) inee been compute correctly., hve..3. Applying the SO-ASAP to Compute the Secon-Orer Response Sensitivities ( S ) S R. i i The G-ifferentil, DS α, of the first-orer sensitivity S efinition given in Eq. (38) to Eq. (33), to obtin α is compute by pplying the DS h h h h ( ) = ( ψ + ε ψ )( ϕ + ε ϕ) = ( ϕψ + ψϕ ) (65) ε ε = The function h ϕ is the solution of Eqs. (4) n (5), while h ψ is the solution of Eqs. (5) n (53). Note tht lthough the entire contribution to DS α comes from the inirect-effect term [the irect-effect term contribution in Eq. (65) is ienticlly zero], DS α oes epen on both h ϕ n h ψ. Consequently, s shown in the generl theory presente in Prt I, corresponing to functions, λ ( x) n θ ( x) DS α there will be two non-zero joint systems n two joint, which will be the solutions of the systems joint to those corresponing to the systems stisfie by h ϕ n, respectively, h ψ. Thus, the joint system λ, corresponing to Eqs. (4) n (5) s well s to the response for x s DS α is obtine D λ Σ = (66) λ ( x) ψ, 3
λ ± =, (67) while the joint system corresponing to Eqs. (5) n (53) s well s to the response DS α is obtine s D θ Σ = (68) θ ( x) ϕ, θ ± =. (69) In view of Eqs. (65) (69), the G-ifferentil joint functions λ ( x) n ( x) θ in the form DS α cn be expresse in terms of the ϕ DS( ) = λ( x) ( D) + ( S ) ϕ ( x) ( Q) + + S + S = S S + S D + S Q + S S ψ θ ( x) ( D) ( ) ψ ( x) ( ) ( x b) (7) ( ) ( ) ( ) ( ) 3 4, where R = λ ϕ + θ ψ S S S x x x x, (7) R ϕ ψ S( ) = λ ( x) + θ ( x) D S ( ) λ ϕ θ ψ λ = S D x x + x x + Q D x ( ) θ ( ) S D x x b, (7) S R 3 = Q S 3 λ x, (73) 4
R 4 = θ = θ S S S x x b b. (74) The solution λ ( x) of joint system shown in Eqs. (66) n (67) is ( k ) λp λp λp + λp λ( x) = λp ( x) sinh ( x) cosh ( x), (75) sinh cosh ( k ) with λ Σ ( bk k ) ( k ) p ( x) = D Σ cosh Σ sinh x cosh + + cosh + sinh cosh 4k Σ D ( xk k ) ( xk k ) ( xk ) ( xk ) ( b) H x x cosh + cosh sinh cosh 4k ( xk bk ) ( xk bk ) ( xk ) ( xk ) (76) Similrly, solving the joint system consisting of Eqs.(68) n (69) yiels θ ( x) Q = Σ Σ sinh cosh sinh cosh cosh ( k ) ( xk ). ( k ) D Q cosh + ( Σ ) cosh k xk x xk k (77) The secon-orer sensitivity S α cn now be compute from Eqs. (77) n (74) to obtin ( ) 4 the sme expression s ws obtine in Eq. (48) for S ( α ), nmely: 4 S ( ) = θ b = S ( ) = Q S Ak + Bk ( ) 4 4 S D S. (48) 5
The bove result provies n inepenent verifiction tht the joint functions θ ( x) n λ 4 ( x) hve been compute correctly. Next, inserting the expressions of λ n θ into Eqs. (7) through (73), n performing the respective integrtions, yiels the following expressions: x x R Q S 5 S 3 Ak kbk ( k) Ck S (78) ( ) = 3 + 4 4 ( S ) where R Q S( ) S = k B ( k ) kc( k ) D S (79) 4D S ( S ) Bk R S Ak 3 (6) 3( ) = S3( ) = + Q Σ Σ Σ D S B A C k bk k k k 3 = = cosh cosh k k ( + ) 3 + bsinh k sinh bk cosh k b cosh bk cosh k (8) Of course, the expression of S ( α ) = S ( α ) hs lrey been compute in Eq. (6), which 3 3 is the reson why it ws ccoringly lbele bove, so its computtion from Eq. (73) serves only s n itionl inepenent verifiction tht the joint function λ ( x) hs been correctly compute. This verifiction is in the sme spirit s the verifiction tht ws performe on the joint functions θ ( x) when computing S α from Eq. (74) n then ( ) 4 ensuring tht the resulting expression ws ienticl to tht lrey obtine in Eq. (48) for S ( α ). 4 On the other hn, the expressions of S ( α ) n S α obtine in Eqs. (78) n (79), respectively, re in ition to the expressions lrey obtine for the secon-orer sensitivities in Sections.. n... 6
..4. Applying the SO-ASAP to Compute the Secon-Orer Response Sensitivities Si R D αi. The G-ifferentil, DS α, of the first-orer sensitivity S the the efinition given in Eq. (38) to Eq. (34), to obtin α is compute by pplying DS S + e S h h Q Q h ( ) = ( ψ + e ψ )( ϕ + e ϕ) + + e( ) ( ψ + e ψ ) e D + e D e = { DS ( )} DS ( ) = + { }, irect inirect (8) where n { } ( D) S ( S ) D DS x x ψ ϕ irect ( D ) ( ) ( ) D Q D Q + ( D ) ψ x, (8) S Q S { ( )} ϕ ψ + ψ ϕ DS h x x h x x. inirect D D D (83) In Eqs. (8) through (83), the function h ϕ is the solution of Eqs. (4) n (5), while h ψ is the solution of Eqs. (5) n (53), just s ws the cse in the previous sections. Note tht, in contristinction to the G-ifferentils DSi ( α ), the G-ifferentil DS α comprises not 7
only n inirect-effect term, but lso (non-zero) irect-effect term. The irect-effect term cn be evlute immeitely since ll of the quntities in Eq. (8) re lrey vilble. { } On the other hn, the inirect-effect term DS α epens on both h ϕ n h ψ, so inirect tht two non-zero joint functions, lbele here s λ ( x) n θ ( x), re neee in orer to evlute it. These joint functions will be solutions corresponing to the systems joint to those stisfie by h ϕ n h ψ, respectively, but with sources erive from the representtion of { ( )} DS α in Eq. (83). Applying by now the fmilir SO-ASAP yiels the secon inirect joint system for λ ( x) [which correspons to Eqs. (4) n (5), s well s to the response in Eq. (83)], of the form D λ Σ Σ = (84) D λ ( x) ψ, λ ± =, (85) together the secon joint system for θ ( x) [which correspons to Eqs. (5) n (53), s well s to the response in Eq. (83)], of the form D θ Q Σ Σ = (86) D D θ ( x) ϕ, θ ± =. (87) { } The G-ifferentil λ ( x) n ( x) θ s DS α cn now be expresse in terms of the joint functions inirect ϕ { ( )} = λ ( ) ( ) + S ϕ DS x D x Q inirect ψ + θ ( x) ( D) + ( S ) ψ ( x) + ( S ) ( x b). (88) 8
{ } Combinig the bove result with the irect-effect term, yiels: ( ) ( ) ( ) ( ) ( ) 3 4, DS α, efine in Eq. (8), DS = S S + S D + S Q + S S (89) irect where R ( ) = ϕ ψ + λ ϕ D S S D x x x x + θ ( x) ψ x, (9) R S Q S( ) = ϕ ( x) ψ ( x) ϕ ( x) ψ ( x) D D D ϕ ψ λ( x) + θ ( x), (9) R 3 ψ λ D Q = S 3 D x x, (9) R 4 = θ = θ D S S x x b b. (93) To compute the lst integrl on the right-sie of Eq. (9), we use Eqs. () n (7) to obtin ϕ ψ Q λ( x) θ( x ) λ ( x) Σ ϕ ( x) θ ( x) Σ Σ + = + ψ ( x) + ( x b), D D D D n subsequently replce the right-sie of the bove expression in the lst integrl on the right-sie of Eq. (9). Compring Eqs. (84) n (85) with Eqs. (66) n (67) reily revels tht the sources of these two liner systems re proportionl to ech other by the fctor ( Σ D ). Hence, the 9
solutions of these two systems will be proportionl to ech other by the sme fctor. Consequently, Eqs. (84) n (85) nee not even be solve since the solution λ ( x) cn be immeitely written own in terms of the λ ( x) s λ ( x) ( D ) λ ( x) = Σ (94) with the expression of λ ( x) provie in Eq. (75). On the other hn, the solution Eqs. (86) n (87) is obtine by conventionl methos s θ of x θ ( x) ( D ) cosh ( k ) Q xsinh xk cosh k sinh k cosh xk =. (95) k Since the expression of the secon-orer sensitivity S α is lrey vilble from Eq. ( ) 4 (49), it follows tht Eq. (93) provies n exct verifiction of the joint function θ ( x). This is inee the cse, s emonstrte by the fct tht 4 4 S ( α ) = S ( α ) = θ b (96) The expression of S α hs lrey been obtine in Eq. (79), n the expression of ( ) S α hs lrey been obtine in Eq. (63). Therefore, Eqs. (9) through (9) cn serve s ( ) 3 multiple verifictions of the correcteness of the vrious computtions. This is inee the cse: inserting the expressions given in Eqs. (94) n (95) [for λ ( x) n θ ( x) together with the Eqs. (3) n (3) [i.e., the expressions of ϕ ( x) n ( x), respectively] ψ ] into Eqs. (9) through (9), n performing the respective intergrtions, les to the following results: n S S Q S k B k kc k 4D ( ) = ( ) = ( S ), (97) S k S S B k 3( ) = 3( ) = D S. (98) 3
The only new result which is ctully obtine when computing other ifferentils DSi ( α ) DS α, fter ll of the hve lrey been compute, is the expression for the seconorer sensitivity S α. Thus, performing the integrtions in Eq. (9) yiels ( ) R Q S k S( ) 3 = 3 B ( k ) + C ( k ). (99) D 4 S D ( D ) Hving now finishe the computtion of ll of the secon-orer response sensitivities, it is instructive to count the number of joint systems tht neee to be solve, since these computtions woul be the equivlent of the lrge-scle computtions performe if the SO- ASAP were use in prctice. For one functionl-type response, the count is s follows: (i) one joint computtion to etermine the function ψ ( x), which suffices to compute, using just qurtures, ll of the first-orer sensitivities S R α,=,,3,4; n for subsequently etermining (through trivil ivision) ll of the secon- orer sensitivities S4 R Q α,=,,3,4, i i i i (ii) two joint computtions for etermining S R S = ; i i,, (iii) one joint computtion for etermining S R D. To summrize: for ech response, four (4) lrge-scle joint computtions suffice for the complete n exct computtions of ll (4) first- n () istinct secon-orer erivtives, incluing the vrious verifictions of the joint functions n erivtive-symmetry properties. Also, for ll of the joint computtions, only the sources on the right-sie of Eq. () [i.e., the initil iffusion eqution for the neutron flux ϕ ( x) ] neee moifictions. The ctul solver for the iffusion eqution remine unchnge. By comprison, forwr methos, e.g., the SO-FSAP, require 4 lrge-scle forwr computtions (i.e., solutions of the iffusion eqution) for computing ll of the first- n (istinct) secon-orer sensitivities. It is instructive to compute the secon-orer reltive sensitivities using the sme t (n for the sme responses) s use in Section. for the computtions of the first-orer response sensitivities. The corresponing numericl results re presente in Tble 3 (secon-orer 3
bsolute sensitivities, with the corresponing units ommitte) n Tble 4 (secon-orer reltive sensitivities). Tble 3. Secon-orer (bsolute vlues, sns units) sensitivities for six etector responses n Orer Absolute Sensitivities R ( cm) R 4 (- cm) R (4 cm) R 5 (-4 cm) R 3 (49.5 cm) R 6 (-49.5 cm) S.95x 3.67x 3.8x S 5.8x 7.4x 5.8x S 3 -.9x 4 -.76x 4 -.67x 3 S 4 -.58x -.36x -.5x 9 S -4.59x 6 -.97x 9.53x S 3 -.33x - -.4x -.74x S 4 -.79x 4 -.67x 8 -.33x 8 S 33 S 34 5.8x 4.9x 8.6 S 44 Tble 4. Secon-orer (reltive) sensitivities for six etector responses n Orer Reltive Sensitivities R ( cm) R 4 (- cm) R (4 cm) R 5 (-4 cm) S _rel..77.8 S _rel 4.4x -5..7 S 3 _rel -. -.95 -.54 S 4 _rel -. -.95 -.54 S _rel -3.x -5 -.4.65 S 3 _rel -5.64x -6 -.54 -.46 S 4 _rel -5.64x -6 -.54 -.46 S 33 _rel S 34 _rel... S 44 _rel R 3 (49.5 cm) R 6 (-49.5 cm) The vlues of the bsolute secon-orer sensitivities presente in Tble 3 will be use in the following Section to illustrte their essentil role for quntifying non-gussin fetures (e.g., symmetries) of the vrious response istributions. To quntify ssymetries in istribution, t the very lest the thir-orer ( skewness ) response correltions nee to be compute, which require the exct computtion of (t lest) the first- n secon-orer response sensitivities to moel prmeters. Also, we note from Tble 4 tht, conventionl folklore (which is to rgue tht secon-orer sensitivities re sufficiently smll to be neglecte ) potentilly ignores 3
significnt informtion bout the system uner stuy.in rector physics, The results in Tble 4 clerly inicte tht secon-orer reltive sensitivities cn be just s lrge s (or even significntly lrger thn) first-orer sensitivities..3. Illustrtion of the Essentil Role Plye by the Secon-Orer Response Sensitivities for Quntifying Non-Gussin Fetures of the Response Uncertinty Distribution In generl, the moel prmeters re experimentlly erive quntities n re therefore subject to uncertinties. Specificlly, consier tht the moel comprises N α uncertin prmeters α i, which constitute the components of the (column) vector α of moel prmeters, efine s α = ( α α ),..., N α. The usul informtion vilble in prctice comprises the men vlues of the moel prmeters together with uncertinties (stnr evitions n, occsionlly, correltions) compute bout the respective men vlues. The components of vector ( α α ) s α i n efine s α,..., N α of men vlues of the moel prmeters re enote αi αi, f f α p α α,. () where the ngulr brckets enotes integrtion of generic function f ( α ) over the unknown joint probbility istribution, p ( α ), of the prmeters α. The prmeter istribution s secon- ij orer centrl moments, µ α, re efine s ij µ α αi αi αj αj ρσσ ij i j; i, j =,, N α. () 33
ii The centrl moments ij moments µ ( α α ) evition of i µ α vr i re clle the vrince of α i, while the centrl α cov i, j ; i j, re clle the covrinces of α i n ii α is efine s σ µ i α j. The stnr α. For (univrite) istribution of vrite α i, the expecte (or men) vlue α i is mesure of the loction of the respective istribution, ii while the stnr evition σ µ i α provies mesure for the ispersion of the respective istribution roun the men vlue. In the sme sense, the men vlue cn be interprete s the istribution s center of grvity, while the vrincecn be interprete s the istribution s moment of inerti (whoich in mechnics linerly reltes the pplie torque to the inuce ccelertion). When the moel uner consiertion is use to compute vector form s r = ( r r N ),..., r moel s prmeters, i.e., r = r( α ). It follows tht r = N r responses (or results), enote in, ech of these responses will be implicit functions of the r α will be vector-vlue vrite which obeys (generlly intrctble) multivrite istribution in α. For lrge-scle systems, the probbility istribution p ( α ) is not known in prctice n, even if it were known, the inuce istribution in r = r( α ) woul still be intrctble, since p ( α ) coul not be propgte exctly through the lrge-scle moels use in for simulting relistic physicl systems. The time-honore eterministic metho for computing uncertinties in response r ( α ) rising from uncertinties in the prmeters α relies on expning formlly the response r ( α ) in Tylor series roun α, constructing pproprite proucts of such Tylor series, 34
n intergrting formlly the vrious proucts over the unknown prmeter istribution function p ( α ), to obtin response correltions. This metho for constructing response correltions stemming from prmeter correltions is known s the propgtion of errors or propgtion of moments metho. Using this metho, reference [5] provies expressions for response correltions up to fourth-orer, using up to fourth-orer prmeter correltions n response sensitivities. For illustrting the effects of secon-orer response sensitivities for the prigm neutron iffusion problem consiere in this work, it suffices to tke from Ref. [5] response correltions up to thir-orer, for the very simple cse when: (i) the prnmeters re uncorrelte n normlly istribute; n (ii) only the first- n secon-orer response sensitivities re vilble. For these prticulr conitions, the response correltions erive in [5] reuce to the following expressions for the first three response moments: (i) The expecte vlue of response k r, enote here s UG E r k, which rises ue to uncertinties in uncorrelte normlly-istribute moel prmeters (the superscript UG inictes uncorrelte Gussin prmeters), is given by the expression E r Nα k k = k + σ i i= αi UG r r α, () where k r α enotes the compute nominl vlue of the response; 35
cov r r (ii) The covrince, (, ), between two responses r k n r rising from normllyistribute uncorrelte prmeters is given by k Nα Nα UG rk r r k r 4 cov ( rk, r ) = σi + σ i (3) i= αi αi i= αi αi The vrince, vr ( r ) k, of response r k is obtine by setting r k r in the bove expression l to obtin N N UG r k r k 4 k = σi + σ i i= i i= i ( r ) (4) vr ; (iii) The thir-orer response correltion, m ( r r r ) hs the following expression: k l m 3,,, mong three responses ( r k, r n r m ) Nα UG rk rl rm rk rl rm rk rl r m 4 m3 ( rk, rl, rm) = + + σ. i i= αi αi αi αi αi αi αi αi α (5) i In prticulr, the thir-orer centrl moment, ( r ) setting k = l = m in Eq. (5) to obtin µ 3 k, of the response k r is etermine by Nα UG r k rk 4 3 ( rk) = 3 i i= αi α (6) i µ σ The skewness, γ ( r ) k, of response r k cn be compute using the customry efinition ( r ) ( r ) ( r ) 3/ γ k µ 3 k vr k. (7) 36
The skewness of istribution quntifies the eprture of the subject istribution from symmetry. Symmetric univrite istributions (e.g., the Gussin) re chrcterize by γ r k =. A istribution with long right til woul hve positive skewness while γ <, r k istribution with long left til woul hve negtive skewness. In other wors, if then the respective istribution is skewe towrs the left of the men UG lower vlues of k r reltive to E( r ) UG k. On the other hn, if ( r k ) istribution is skewe towrs the right of the men UG reltive to UG E r k. E r k, fvoring γ >, then the respective E r k, fvoring higher vlues of r k As inicte by the expressions in Eqs. () - (7), the secon-orer sensitivities hve the following impcts on the response moments: () They cuse the expecte vlue of the response, UG compute nominl vlue of the response, k r α ; E r k, to iffer from the (b) They contribute to the response vrinces n covrinces; however, since the contributions involving the secon-orer sensitivities re multiplie by the fourth power of the prmeters stnr evitions, the totl of these contributions is expecte to be reltively smller thn the contributions stemming from the first-orer response sensitivities; (c) They contribute ecisively to cusing ssymetries in the response istribution. As Eq. (5) inictes, neglecting the secon-orer sensitivities woul nullify the thir-orer response correltions n hence woul nullify the skewness, γ ( r ) k, of response r k, cf. Eq. (7). Consequently, ny events occurring in response s long n/or short tils, which re chrcteristic of rre but ecisive events (e.g., mjor ccients, ctstrophies), woul likely be misse. 37
To illustrte the bove points numericlly for our prigm neutron iffusion problem, it is convenient to recll here tht: = 5 cm, 7 3 S = neutrons cm s, Σ =.97 cm, D =,6 cm, Σ = 7.438cm. Recll lso tht the compute responses h the following vlues in units of 3 [ neutrons cm s ] R cm = R cm = 3.77, : 9 4 9 8 ( 4 ) = ( 4 ) = 3.66, n R ( 49.5cm) = R ( 49. 5cm) = 6. 6 R cm R cm 5 3 6 7. We will now investigte the effects of prmeter uncertinties by using Eqs. () (7) to quntify the resulting uncertinties inuce in the respecte responses. Severl combintions of prmeters reltive stnr evitions, enote s δσ A (%), δ D (%), δ Q (%) δσ D, n % will be consiere for this purpose; the vrious cses n corresponing reltive stnr evitions re liste in Tble 5. Tble 5. Moel prmeters reltive stnr evitions Cse δσ (%) δ D (%) δ Q (%) δσ (%) 5 5 3 5 4 5 5 The expression given in Eq. (4) hs been use to compute the stnr evitions inuce in responses by the prmeter uncertinties liste in Tble 5. The results of these computtions re shown in Tble 6. Exmintion of cses n inictes tht uncertinties rising solely in Q n Σ, respectively, propgte one-to-one, inucing corresponing uncertinties in ll of the responses; this fct is expecte in view of their first-orer reltive sensitivities of unity, for ll responses (s shown in Tble ). The uncertinties in Σ propgte lmost one-to-one to prouce the response uncertinties in the interior of the slb, but cuse iminishingly smller uncertinties in the etector response towrs the slb s bounries; gin, this behvior is expecte in view of the vlues for the corresponing firstorer sensitivities shown in Tble. The impct of the uncertinties in D on the vrious responses is lso ictte by the behvior of the corresponsing first-orer sensitivities shown in Tble : the uncertinties in D hve negligeble impct on the responses in the interior of the slb, but their impct increses s the etector responses pproch the meium s 38
bounries. Finlly, when ll of the prmeters re uncertin, their combine contribution to the respective response uncertinty is lrger thn ny one prmeter s uncertinty, but is smller thn the the squre-root of the sums of squres, becuse the response sensitivities to Σ n D re not unity. We lso mention tht the results (which re not shown, for brevity) of using Eq. (3) to compute the response correltions (vlue rnges.998 to.) inicte responses re fully (or lmost fully) correlte. In ll of these computtions, the secon-orer sensitivities h negligible impct on the responses stnr evitions n correltions. Tble 6. Response reltive stnr evitions Cse σ ( R ) = σ ( R ) σ ( R ) = σ ( R ) σ ( R ) = σ ( R ) rel rel 4 rel rel 5 rel 3 rel 6.5.5.5.5.5.5 3.5.45.8 4 9.8x -7.8.69 5.74.7.58 As inicte by the results presente in Tble 7 for the skewness of the vrious responses, the secon-orer sensitivities hve ecisive impct on the ssymetries of the inuce response istributions. Since the secon-orer sensitivities of the responses to both Q n Σ re zero, cf. Tble 3, it follows tht uncertinties rising solely in these quntities woul not ffect the symmetry of the resulting response istribution in our illustrtive prigm problem. This expecttion is inee confirme by the results presente uner Cse n Cse in Tble 7. On the other hn, Cse 3 in Tble 7 inictes tht uncertinties stemming solely from the bsoption cross section Σ cuse the resulting response istributions to be significntly skewe to the right of their corresponing expecte vlues, throughout the meium; this symmetry iminishes somewht towrs the meium s bounries. Cse 4 inictes somewht counter-intuitive behvior of the response istributions, in tht uncertinties rising solely from the iffusion coefficient D cuse the response istribution to unergo complex behvior, s follows: (i) the response istribution is skewe significntly to the left of the response s expecte vlue in the meium s interior; (ii) the bove-mentione symmetry iminishes towrs the meium s bounry; 39