IMPLEMENTATION OF CHORD LENGTH SAMPLING FOR TRANSPORT THROUGH A BINARY STOCHASTIC MIXTURE

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1 Nuclear Mathematical and Computational Science: A Century in Review, A Century Anew Gatlinburg, Tenneee, April 6-, 003, on CD-ROM, American Nuclear Society, LaGrange Park, IL (003) IMPLEMENTATION OF CHORD LENGTH SAMPLING FOR TRANSPORT THROUGH A BINARY STOCHASTIC MIXTURE T.J. Donovan and T.M. Sutton Lockheed Martin Corp. Schenectady, NY Y. Danon Mechanical, Aeropace, and Nuclear Engineering Department Renelaer Polytechnic Intitute Troy, NY danon@rpi.edu ABSTRACT Neutron tranport through a pecial cae tochatic mixture i examined, in which phere of contant radiu are uniformly mixed in a matrix material. A Monte Carlo algorithm previouly propoed and examined in -D ha been implemented in a tet verion of MCNP. The Limited Chord Length Sampling (LCLS) technique provide a mean for modeling a binary tochatic mixture a a cell in MCNP. When inide a matrix cell, LCLS ue chord-length ampling to ample the ditance to the next tochatic phere. After a urface croing into a tochatic phere, tranport i treated explicitly until the particle exit or i killed. Reult were computed for a imple model with two different fixed neutron ource ditribution and three et of material number denitie. Stochatic phere were modeled a black aborber and varying degree of cattering were introduced in the matrix material. Tallie were computed uing the LCLS capability and by averaging reult obtained from multiple realization of the random geometry. Reult were compared for accuracy and figure of merit were compared to indicate the efficiency gain of the LCLS method over the benchmark method. Reult how that LCLS provide very good accuracy if the cattering optical thickne of the matrix i mall ( ). Comparion of figure of merit how an advantage to LCLS varying between factor of 4 and 5. LCLS efficiency and accuracy relative to the benchmark both decreae a cattering i increaed in the matrix. Key Word: tochatic mixture, tranport, Monte Carlo, MCNP. INTRODUCTION A binary tochatic mixture i defined a a random arrangement of two immicible material. Zimmerman and Adam () propoed the ue of chord length ampling in Monte Carlo to imulate particle tranport through uch a mixture. Donovan and Danon () examined chord length ampling algorithm in -D for the pecial cae of dik of contant radiu mixed in a matrix material and propoed a Limited Chord Length Sampling (LCLS) algorithm that ample chord in the matrix material and treat tranport through the phere explicitly. Thi work extend the pecial cae into a 3-D problem of phere of contant radiu mixed in a matrix material.

2 T.J. Donovan, Y. Danon, T. Sutton Incorporation of a tochatic material treatment uch a LCLS into a production Monte Carlo code uch a MCNP (3) could find a wide audience. Application include pebble bed reactor analye, or heterogeneou hield or target deign. Alo, the explicit treatment of the tochatic phere allow for the incluion of detail uch a multiple hell of varying material, uch a i found in the fuel particle of the MIT High Temperature Ga Pebble Bed Reactor concept (4). Thi work modified MCNP to incorporate the LCLS algorithm for fixed ource problem. Tet model are analyzed uing LCLS a well a a benchmark method. Tally reult are compared and the figure of merit () ratio for both method i calculated.. DESCRIPTION OF WORK.. LCLS Algorithm The LCLS algorithm combine chord length ampling with limited explicit geometric repreentation of the tochatic mixture. The algorithm take advantage of the cloed, imple geometry of the tochatic phere and treat thee region explicitly. When determining the next event, chord length ampling i ued to ample the ditance to a tochatic interface when a particle i within the matrix (material M). Chord length ampling i ued again to determine the poition of the tochatic phere. If the ditance to the tochatic interface i le than the ditance to colliion and the ditance to the nearet real urface, then the particle i moved forward to the urface of the tochatic phere. Particle tranport through the phere i then carried out in the normal fahion.... Matrix Chord Length Ditribution Homogeneou mixing of the tochatic phere in the matrix region i aumed. Excluding boundary effect, Torquato (5), howed that the chord length (λ ) ditribution between phere for uch a mixture can be repreented by the exponential form λ p ( λ ) = exp( ), () λ λ where λ i the average chord length in the matrix material. Although the mixture boundarie may produce edge effect that influence how the tochatic phere may be ditributed, the LCLS method aume that equation till adequately repreent the chord length ditribution. Although ome effort have been made to produce analytical approximation for λ (5, 6), thee approximation are not conidered accurate enough to adequately tet the LCLS method. Therefore λ i determined empirically for a given mixture geometry uing a Monte Carlo technique. Detail on how λ i determined are given in the decription of the model, Section.4. The probability ditribution of equation i ued to ample the ditance to a tochatic interface, D I. If D I i le than the ditance to the nearet real boundary croing, D B, then additional work American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 /

3 Monte Carlo Chord Length Sampling Through a Binary Stochatic Mixture i performed to ample a tochatic phere and tet whether it can fit without overlapping the matrix cell boundarie.... Stochatic Sphere Chord Length Ditribution The tochatic phere coordinate are ampled by firt ampling a chord (λ 0 ) through the phere uing the chord length ditribution for a phere (6) 0 p( λ 0) = λ, () R where R i the phere radiu. Figure illutrate the λ 0 chord and it relationhip to the tochatic phere coordinate. (x,y,z ) (x R Ι,y Ι,z Ι ) R (x λ,y λ,z λ ) (u,v,w) λ 0 / λ 0 / Figure : Sampling Stochatic Sphere Coordinate. In Figure, (x I,y I,z I ) are the coordinate of the point where the particle will cro from the matrix material into the tochatic phere, and (x λ,y λ,z λ ) are the coordinate of the λ 0 chord midpoint. The chord midpoint coordinate are found by moving from point (x I,y I,z I ) a ditance λ 0 / in the direction of particle motion decribed by the direction coine (u,v,w). The center of the tochatic phere i (x,y,z ), which lie on a circle in pace decribed by ( x x ) + ( y y ) + ( z y ) = R λ λ λ, (3) where R R λ 0 =. (4) The circle of equation 3 lie on the x y plane of a newly defined coordinate ytem (x,y,z ), where the z axi i in the direction of particle flight. A coordinate et i choen from the circle by ampling an angle, θ, from (0,π) in the x y plane. Thi angle define a direction vector ( u, v, w ) where u = co(θ), v = in(θ), and w = 0. Thi direction vector point from the chord midpoint to the ampled coordinate point, and the ditance R S eparate the two point. By performing a coordinate tranformation, the ampled direction vector leading to the phere coordinate i tranformed back to the (x,y,z) coordinate frame according to American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 3/

4 T.J. Donovan, Y. Danon, T. Sutton u v = w v u + v u u + v 0 u w u + v v w u + v u + v u u v v. (5) w w The ampled phere coordinate are then x u x y = R v + y z w z λ λ λ. (6) An additional tet i required to check that the ampled tochatic phere doe not interect any urface of the matrix cell. If an interection doe occur, then the ampled phere coordinate are rejected and the ampled ditance to interface D I i reet to a huge value. If a tochatic phere doe not interect the matrix cell urface, then the ampled value of D I i included in the comparion of next-event ditance and the minimum ditance determine the next event. If D I i the minimum next-event ditance, then a urface croing with a tochatic phere take place. The particle i moved to the interection point (x I,y I,z I ), it cell location i changed from the matrix cell to the tochatic phere, and the tranport continue... Incorporation of LCLS into MCNP... Modification to MCNP MCNP verion 4C wa modified to incorporate the LCLS method a decribed in Section.. The only ubroutine that wa modified wa hitory, which control particle tranport. The ubroutine wa modified to provide the following capabilitie: Calculate D I, the ditance to tochatic interface Sample tochatic phere coordinate (x,y,z ) and tet for overlap of the tochatic phere with urface of the matrix cell Include D I in the tet for minimum ditance to determine next event Tranlate particle into and out of the tochatic phere a neceary In addition to the modification made to hitory, three new ubroutine were created. Thee ubroutine are called by hitory to perform the overlap tet, and tranlate the particle in and out of the tochatic phere. American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 4/

5 Monte Carlo Chord Length Sampling Through a Binary Stochatic Mixture... MCNP Input to Define a Stochatic Mixture Cell The matrix cell i defined in the normal way and i filled with the matrix material. A repreentative tochatic phere i created in a void region of zero importance in the model. Creating the tochatic phere in uch a region pecifie the geometry and material of the tochatic phere but place it in a region of the model that particle do not tranport through. Particle tranport through the tochatic phere can only occur if LCLS information i included in the input deck. The tochatic phere may have multiple hell (each it own cell) built with concentric phere. The MCNP idum and rdum input array were ued to pecify LCLS information. The following information i pecified: idum() Matrix cell number idum() i, the number of concentric phere making up the tochatic phere. idum(+ to +i) The cell number of the tochatic phere idum(3+i) The urface number of the outermot urface of the tochatic phere idum(4+i) The number of the cell on the poitive ene of urface idum(3+i) rdum() rdum() rdum(3) rdum(4) rdum(5) The average chord length in the matrix material The radiu of the tochatic phere X coordinate of the repreentative tochatic phere Y coordinate of the repreentative tochatic phere Z coordinate of the repreentative tochatic phere The only other input required to pecify the tochatic mixture cell are the volume of the matrix cell and the tochatic cell. Thee volume mut be entered uing the VOL card o thoe tallie that ue the cell volume will be correctly normalized. The volume that MCNP calculate automatically for thee cell are not the true volume of the tochatic mixture region. The true matrix cell volume i equal to the empty matrix cell minu the volume of the tochatic phere that in reality are dipered within it. Similarly, becaue the ingle repreentative tochatic phere repreent all of the tochatic phere in the matrix, it true volume i equal to the volume of the ingle repreentative phere multiplied by the number of phere that it repreent. If the tochatic phere i modeled a a number of concentric pherical hell, then the repreentative volume of each hell cell mut be pecified..3. Optimization of Benchmark Thi ection dicue how benchmark tally reult were obtained for comparion to the LCLS reult. The benchmark tally reult i obtained by running NITER MCNP job, each with a different random realization of the tochatic mixture, NPS particle per job, and a different random number eed. The average of the et of y i reult i the benchmark tally reult y, or yi i y = (7) NITER American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 5/

6 T.J. Donovan, Y. Danon, T. Sutton If the enemble averaged tally reult y were the only quantity of interet, the combination of NITER and NPS would not be of much concern. Both value would be et high enough to enure that the error etimate on the mean reult would be acceptably low and that the random effect due to the different poible mixture realization are captured. However, the econd objective of thi tudy wa to examine the difference in the for the LCLS and Benchmark tallie. Thi comparion i only ignificant if the benchmark tally ha been optimized. It can be hown that the variance in the population of y i reult may be etimated a the um of the geometric variance and the pooled tochatic variance of each of the y i reult, S y i S p = S g + (8) NPS where S y i i the variance of the y i reult and S g i the geometric variance. The geometric variance i the variance in the NITER reult due olely to the random effect of the placement of the tochatic phere within the matrix cell. S i the pooled, or averaged, tochatic variance S p p of each of the NITER y i reult. The term i an etimate of the variance in the NITER NPS reult due to the tochatic nature of the olution method. The variance on y i then The time to compute the average benchmark tally i S yi S y =. (9) NITER T ( + D NPS) = NITER C, (0) where C i an overhead time per realization and D i a time per hitory. The i typically defined a the invere of the product of the relative uncertainty quared and the time required to compute the tally. =. () S y y In a typical Monte Carlo calculation, a ingle job i run with NPS particle. Since the relative uncertainty decreae a the invere of the quare root of NPS and the time i proportional to NPS, the i expected to be a contant (i.e., independent of NPS) for a given tally. However, for the cae in quetion, where the tally i an average of NITER reult each obtained with NPS particle, the i not independent of NPS. Subtituting equation 8, 9, and 0 into equation, the reulting expreion i T American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 6/

7 Monte Carlo Chord Length Sampling Through a Binary Stochatic Mixture y S S g p + NITER C NITER NITER NPS = ( + D NPS). () NITER drop from equation, leaving a a function of the variance parameter and NPS. The value of NPS that produce the maximum i determined by differentiating Equation with repect to NPS, etting the reulting expreion to zero, and olving for NPS. The optimum value of NPS i found to be NPS opt C S =. (3) D S p g To compute a benchmark tally for a given model, the variance and timing parameter are etimated by running a erie of tet job with varying combination of NITER and NPS. The benchmark tally i then calculated again uing the optimum NPS, and NITER i choen to achieve the deired relative error or total number of hitorie. It i important to note that in order to determine the optimum NPS for a given benchmark tally, the benchmark tally mut be computed to a high degree of accuracy. Since the tally (and it ) i known, it i omewhat unrealitic to recalculate the tally imply to enure that the lat calculation wa the mot efficient. However, thi approach i the mot conervative and allow the comparion of LCLS to the mot ideal benchmark. For the model to be examined, the optimum NPS wa determined for a ingle tally, but three tallie were actually computed. Becaue the remaining tallie were not optimized for their, the LCLS to benchmark ratio are not given for thee tallie..4. Stochatic Mixture Model.4.. Model Decription The model conit of a phere of radiu 5cm, containing 78 randomly arranged phere of cm radiu. The number and ize of the tochatic phere wa choen to obtain a 5% volume fraction of phere in the matrix region. The tochatic phere do not poe a hell ubtructure for thi model. Becaue phere were placed in the allowable volume equentially, thi method of creating a realization of the random mixture i known a Random Sequential Addition, or RSA (5). Randomly placed phere were not permitted to overlap one another or the urface of the matrix phere. In addition, the tochatic phere were not permitted to fall on the origin located at the center of the matrix phere. Thi retriction allow a point ource to be pecified in the matrix cell at the origin for all benchmark realization. Thi non-phyical retriction on the mixing of the tochatic phere for the benchmark model i neceary becaue the LCLS modification to MCNP do not currently permit ource particle to originate from the tochatic phere. Starting a point ource at the origin in the LCLS model reult in ource particle originating in the matrix material. Becaue the ource particle i born in the matrix American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 7/

8 T.J. Donovan, Y. Danon, T. Sutton material at the origin, a tochatic phere cannot exit at that location at the beginning of the particle hitory. However, if a ource particle catter back near the origin later on in it hitory, the LCLS method may then ample a tochatic phere that overlap the origin. Figure how plot of a benchmark realization of thi model and the LCLS model of the ame ytem. Each plot i a planar view of the x-y view at z=0. In the LCLS model, the phere outide and to the right of the matrix phere i the repreentative tochatic phere. In the benchmark model, each tochatic phere i a unique cell placed randomly in the matrix phere. Two matrix cell are defined, repreenting the two hemiphere in the poitive and negative x direction. Each matrix cell i the interection of pace inide the matrix phere, in the poitive or negative x pace, and outide the 78 tochatic phere. Figure : Benchmark MCNP model (left) and LCLS model (right). (x-y plane at z=0) The matrix material (M) i Nitrogen-4 and the tochatic phere material (M0) i Boron-0. Thee material were choen for their moothly varying cro ection in the thermal energy range. Alo, the N-4 cro ection i dominated by cattering wherea B-0 i a trong aborber. M0 denity wa pecified to make the tochatic phere black aborber. Three different M denitie were pecified to tudy the effect of increaing catter in the matrix. The three M denitie were: 0.00 gm/cc, 0. gm/cc, and gm/cc. For the thermal energy ource, thee denitie correpond to mean free path length in the matrix of approximately 000cm, 0cm, and cm, repectively. Given that the average chord length in the matrix i approximately 0cm, the three matrix denitie correpond roughly to optical thicknee of 0.0 (tranparent),, and 0 (highly cattering), repectively. The model wa analyzed for a fixed thermal energy neutron ource (.53E-8 MeV) in two different patial ditribution. The firt ditribution wa a point ource located at the center of the matrix phere. The econd ditribution wa uniformly pread throughout the matrix material. With two ource ditribution and three M denitie, there were ix cae analyzed in total. American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 8/

9 Monte Carlo Chord Length Sampling Through a Binary Stochatic Mixture Three tallie were computed for each cae. Thee tallie were: an F urface current tally on the urface of the matrix cell, an F4 cell flux tally in the matrix cell itelf, and an F4 cell flux tally in the tochatic phere. For the benchmark job the cell flux tally for the tochatic phere i the average flux in all the explicitly modeled phere..4.. Determination of the Matrix Mean Chord Length The mean chord length in the matrix for thi geometry wa determined empirically to be.087 cm. Thi value wa determined by firt modeling a realization of the tochatic mixture. The coordinate origin wa defined a the center of the matrix phere. The poitive and negative x, y, and z axe (total of 6 line) were then checked to determine how many did not have tochatic phere overlapping them. Thi wa repeated many time and the total number of intance where the 6 line had no overlapping phere wa tallied. Thi value divided by ix time the number of attempt define the probability that a ource particle originating at (0,0,0) at an arbitrary direction would encounter no tochatic phere along it initial trajectory. Thi probability, referred to a p(0), i related to the average chord length λ in the matrix. The relationhip between p(0) and λ can be expreed mathematically by tarting a particle at (0,0,0) and examining the method by which LCLS ample tochatic phere location. In order to implify the integral that will be developed, the following argument aume that the tochatic phere radiu i much le than the radiu of the matrix phere. From the tarting location (0,0,0), LCLS firt ample a λ chord in the matrix material. If λ i greater than R m -R, where R m i the matrix phere radiu and R i the tochatic phere radiu, then the particle ee no tochatic phere along it flight path. Thi i a contribution to p(0) but i not the only poibility. If λ i between (R m -R) and (R m -R), there i a poibility that the ampled tochatic phere overlap the matrix phere and will be rejected. If o, then particle again ee no tochatic phere along it path. The total probability that a particle at the origin will ee no tochatic phere along it path i p(0), and i equal to the um of the two ampling probabilitie decribed. Thi um can be expreed formally uing the two chord length ditribution of equation and, reulting in Rm R x x p(0) = dx exp( ) + exp( ) λ λ Rm R Rm R λ λ ( Rm R x) dydx Equation 4 how the relationhip between p(0) and the deired quantity λ. Given the integration limit and the value obtained for p(0), thi expreion wa olved for λ. 3. RESULTS American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 9/ R y R. (4) Table I and II contain reult for the point ource and uniform matrix ource, repectively. Each table contain the enemble averaged benchmark reult and the LCLS reult for the three different matrix material denitie. Each tally reult alo how an etimate of the one tandard

10 T.J. Donovan, Y. Danon, T. Sutton deviation uncertainty on the lat ignificant digit in parenthee. Figure of merit comparion are only made for the F urface current tally on the matrix phere. For both ource ditribution, reult for the firt two matrix denitie how excellent agreement (within %) between the LCLS etimate and the benchmark enemble averaged tally reult. For the third matrix denity cae the accuracy of the LCLS method degrade ignificantly. Increaing cattering in the matrix caue the LCLS reult for leakage to underpredict the benchmark reult. Logically, thi alo caue the LCLS reult for the flux in the tochatic phere to increae. The ratio on the urface current tally vary widely from a factor of 4 for the lowet matrix denity point ource cae, to 4.8 for the medium matrix denity uniform ource cae. Becaue of the overhead cot in creating random benchmark realization, the LCLS model how the mot advantage over the benchmark method for model and tallie that poe the larget geometric variance (i.e., thoe benchmark model that require a larger NITER and maller NPS for the optimized ). The point ource cae with the lowet matrix denity produce very little catter in the matrix. For particle to ecape the matrix phere they mut have a free line of ight from the origin to the matrix urface along their direction of travel. The urface current tally for thi cae i more dependent on the random placement of the tochatic phere than the other cae examined. A cattering increae in the matrix the geometric variance i reduced becaue a particle ha an increaed chance of channeling around tochatic phere and contributing to the urface current tally. Table I. Thermal neutron point ource at (0,0,0). Tally and reult for three et of matrix material denitie. Set: ρ0 = 0 [gm/cc], ρ = 0.00 [gm/cc] Tally Benchmark LCLS Surface Current [n/ource n] () (5) 4 Matrix Cell Flux [n/cm /ource n].40e-4 (8).4E-4 () - Stochatic Cell Flux [n/cm /ource n] 8.6E-8 () 8.64E-8 () - Set : ρ0 = 0 [gm/cc], ρ = 0. [gm/cc] Tally Benchmark LCLS Surface Current [n/ource n] 0.04 (8) (4) 87 Matrix Cell Flux [n/cm /ource n].47e-4 (9).449E-4 () - Stochatic Cell Flux [n/cm /ource n] 9.05E-8 () 9.07E-8 () - Set 3: ρ0 = 0 [gm/cc], ρ = [gm/cc] Tally Benchmark LCLS Surface Current [n/ource n] 7.0E-4 () 6.E-4 () 7 Matrix Cell Flux [n/cm /ource n].359e-4 (9).9E-4 (5) - Stochatic Cell Flux [n/cm /ource n] 5.0E-8 (5) 5.44E-8 (7) - tandard deviation uncertaintie on the lat ignificant digit hown in parenthee. LCLS bench LCLS bench LCLS bench American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 0/

11 Monte Carlo Chord Length Sampling Through a Binary Stochatic Mixture Table II. Thermal neutron ource uniformly ditributed in matrix. Tally and reult for three et of matrix material denitie. Set: ρ0 = 0 [gm/cc], ρ = 0.00 [gm/cc] Tally Benchmark LCLS Surface Current [n/ource n] (4) (3) 5.7 Matrix Cell Flux [n/cm /ource n].88e-4 ().84E-4 (9) - Stochatic Cell Flux [n/cm /ource n] 6.046E-8 (8) 6.07E-8 (7) - Set : ρ0 = 0 [gm/cc], ρ = 0. [gm/cc] Tally Benchmark LCLS Surface Current [n/ource n] (4) (4) 4.8 Matrix Cell Flux [n/cm /ource n].80e-4 ().798E-4 () - Stochatic Cell Flux [n/cm /ource n] 6.6E-8 (8) 6.7E-8 (8) - Set 3: ρ0 = 0 [gm/cc], ρ = [gm/cc] Tally Benchmark LCLS Surface Current [n/ource n] (3) 0.75 () 8.07 Matrix Cell Flux [n/cm /ource n].56e-4 (6).0969E-4 (6) - Stochatic Cell Flux [n/cm /ource n] 3.938E-8 (6) 4.00E-8 (6) - tandard deviation uncertaintie on the lat ignificant digit hown in parenthee. LCLS bench LCLS bench LCLS bench 4. CONCLUSIONS The cae examined tet the effectivene of the LCLS method for treating particle tranport through a pecified binary tochatic mixture. The tochatic phere are modeled a black aborber, and the effect of increaed cattering in the matrix material are examined. The three matrix denitie correpond to optical thicknee of roughly 0.0,, and 0. For both ource ditribution, LCLS and benchmark reult compare cloely for the firt two matrix denitie. A cattering in the matrix increae, LCLS accuracy degrade. The aumption of an exponential chord length ditribution and the value choen for the mean chord length may contribute to the lo in accuracy becaue ampling from equation i performed more and more frequently a cattering increae. However, it i believed that the lo of accuracy i due principally to the aumption of Markovian tranport made by the LCLS method. Markovian tranport implie that a particle future i independent of it pat. In a homogenou medium, tranport i Markovian. However, particle tranport through a tochatic mixture i not Markovian becaue the ditance to the urrounding material interface depend on the pat trajectory of the particle. In the LCLS method, the ampled location of a tochatic phere i dicarded after a particle exit the tochatic phere. Upon a cattering event in the matrix, a particle could interect a phere where there had previouly been none. For thi reaon, a cattering in the matrix material increae the LCLS method i expected to overpredict the flux in the tochatic phere. American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 /

12 T.J. Donovan, Y. Danon, T. Sutton One of the intereting finding of thi work lie in the optimization of the for an enemble averaged Monte Carlo reult. The combination of geometric and tochatic randomne in the averaged benchmark reult allow for the optimization of the number of particle run per realization. Becaue of the overhead cot to build a complex random cell geometry, the benchmark method i leat efficient when the optimized run trategy require many realization with a relatively mall number of particle per realization. Converely, if the variance in the enemble averaged reult i not trongly affected by the randomne of the geometry, then an accurate reult can be computed with only a few realization, each running a large number of particle. The point ource cae with the lowet matrix denity how a ratio of 4 in favor of LCLS. The benchmark for thi cae required a large number of realization with relatively few particle per realization. In addition, becaue the matrix i o tranparent, the tranport time per particle for both LCLS and benchmark i low. Therefore for the benchmark the total time i dominated by the time required to build all of the realization. Thi i the reaon that the ratio i the highet for thi cae. For the uniform matrix ource and the lowet matrix denity, the ratio drop to 5.7. The geometric variability i greatly reduced by ditributing the ource through the matrix. Thi caue the efficiency of the benchmark to increae. The advantage in the LCLS method in thi cae i driven by the reduced tranport time per particle for LCLS. Tranport time per particle for LCLS i le becaue of the reduced number of cell and the impler matrix cell a compared with the benchmark model. The geometric implicity of LCLS i obtained at the expene of the additional chord length ampling and aociated calculation. REFERENCES. G.B. Zimmerman, M.L. Adam, Algorithm for Monte Carlo Particle Tranport in Binary Statitical Mixture, Tranaction of the American Nuclear Society, 64, pp , (99).. T.J. Donovan, Y. Danon, Application of Monte Carlo Chord-Length Sampling Algorithm to Tranport Through a Two-Dimenional Binary Stochatic Mixture, Nuclear Science & Engineering, 43, pp. -4, (003). 3. J.F. Briemeiter, editor, MCNP A General Monte Carlo N-Particle Tranport Code, LA-3709-M, April J.R. Johnon, J.R. Lebenhaft, M.J. Dricoll, Burnup Reactivity and Iotopic of an HTGR Fuel Pebble, Tranaction of the American Nuclear Society, 5 pp , (00). 5. S. Torquato, Random Heterogeneou Material: Microtructure and Macrocopic Propertie, Springer-Verlag New York, (00). 6. B. Su, G.C. Pomraning, A Stochatic Decription of a Broken Cloud Field, Tranaction of the American Nuclear Society, 5, pp , (994). American Nuclear Society Topical Meeting in Mathematic & Computation, Gatlinburg, TN, 003 /