Last active
April 29, 2024 14:32
-
-
Save wawiesel/7232ba34016b79c51b088c4f8468c69f to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| <?xml version="1.0" encoding="UTF-8"?> | |
| <TEI | |
| xmlns="http://www.tei-c.org/ns/1.0" | |
| xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" | |
| xmlns:xlink="http://www.w3.org/1999/xlink" xml:space="preserve" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 https://raw.githubusercontent.com/kermitt2/grobid/master/grobid-home/schemas/xsd/Grobid.xsd"> | |
| <teiHeader xml:lang="en"> | |
| <fileDesc> | |
| <titleStmt> | |
| <title level="a" type="main" coords="1,105.53,75.81,400.94,12.18">The random ray method applied to fixed source transport problems</title> | |
| <funder ref="#_KWyyfrX"> | |
| <orgName type="full">UK EPSRC</orgName> | |
| </funder> | |
| <funder ref="#_5ETuTgz"> | |
| <orgName type="full">U.S. Department of Energy, Office of Science</orgName> | |
| </funder> | |
| </titleStmt> | |
| <publicationStmt> | |
| <publisher/> | |
| <availability status="unknown"> | |
| <licence/> | |
| </availability> | |
| </publicationStmt> | |
| <sourceDesc> | |
| <biblStruct> | |
| <analytic> | |
| <author> | |
| <persName coords="1,238.85,104.66,54.08,9.57"> | |
| <forename type="first">P</forename> | |
| <surname>Cosgrove</surname> | |
| </persName> | |
| <affiliation key="aff0" coords="1,151.66,128.22,313.17,9.68"> | |
| <note type="raw_affiliation"> | |
| <label>1</label> Department of Engineering , University of Cambridge , Cambridge , UK; | |
| </note> | |
| <orgName type="department">Department of Engineering</orgName> | |
| <orgName type="institution">University of Cambridge</orgName> | |
| <address> | |
| <settlement>Cambridge</settlement> | |
| <country>UK;</country> | |
| </address> | |
| </affiliation> | |
| </author> | |
| <author> | |
| <persName coords="1,311.81,104.66,56.87,9.57"> | |
| <forename type="first">J</forename> | |
| <forename type="middle">R</forename> | |
| <surname>Tramm</surname> | |
| </persName> | |
| <affiliation key="aff1" coords="1,222.16,140.82,172.16,9.68"> | |
| <note type="raw_affiliation"> | |
| <label>2</label> Argonne National Laboratory , IL , USA | |
| </note> | |
| <orgName type="institution">Argonne National Laboratory</orgName> | |
| <address> | |
| <region>IL</region> | |
| <country key="US">USA</country> | |
| </address> | |
| </affiliation> | |
| </author> | |
| <title level="a" type="main" coords="1,105.53,75.81,400.94,12.18">The random ray method applied to fixed source transport problems</title> | |
| </analytic> | |
| <monogr> | |
| <imprint> | |
| <date/> | |
| </imprint> | |
| </monogr> | |
| <idno type="MD5">BA6A3356B6DA1F97A3C909DB2E9A017C</idno> | |
| </biblStruct> | |
| </sourceDesc> | |
| </fileDesc> | |
| <encodingDesc> | |
| <appInfo> | |
| <application version="0.8.1-SNAPSHOT" ident="GROBID" when="2024-04-29T14:31+0000"> | |
| <desc>GROBID - A machine learning software for extracting information from scholarly documents</desc> | |
| <ref target="https://github.com/kermitt2/grobid"/> | |
| </application> | |
| </appInfo> | |
| </encodingDesc> | |
| <profileDesc> | |
| <textClass> | |
| <keywords> | |
| <term>Neutron transport</term> | |
| <term>method of characteristics</term> | |
| <term>fixed source</term> | |
| <term>random ray</term> | |
| </keywords> | |
| </textClass> | |
| <abstract> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <p> | |
| <s coords="1,112.12,238.95,391.88,8.80;1,108.00,250.40,79.16,8.80">The random ray method is a recently developed stochastic method for solving neutral particle transport problems.</s> | |
| <s coords="1,195.08,250.40,308.92,8.80;1,108.00,261.86,331.51,8.80">To-date, it has only been applied in the context of eigenvalue problems in neutron transport; this paper demonstrates its application to fixed source problems.</s> | |
| <s coords="1,443.71,261.86,60.29,8.80;1,108.00,273.32,396.00,8.80;1,108.00,284.78,137.31,8.80">The machinery of the method is naturally suited to generating first collision sources for highly localised fixed source problems and this is demonstrated.</s> | |
| <s coords="1,248.81,284.78,255.19,8.80;1,108.00,296.23,396.00,8.80;1,108.00,307.69,396.00,8.80;1,108.00,319.15,193.70,8.80">On both the Kobayashi dogleg and a more complicated shielding problem, the random ray method is demonstrated to be up to several orders of magnitude more efficient than multigroup Monte Carlo while maintaining accuracy when estimating the fluxes in remote regions subject to significant attenuation.</s> | |
| </p> | |
| </div> | |
| </abstract> | |
| </profileDesc> | |
| </teiHeader> | |
| <facsimile> | |
| <surface n="1" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="2" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="3" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="4" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="5" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="6" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="7" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="8" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="9" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| <surface n="10" ulx="0.0" uly="0.0" lrx="612.0" lry="792.0"/> | |
| </facsimile> | |
| <text xml:lang="en"> | |
| <body> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head coords="1,273.86,361.09,95.28,10.44">INTRODUCTION</head> | |
| <p> | |
| <s coords="1,72.00,386.36,468.00,9.68;1,72.00,398.96,310.73,9.68">The random ray method (TRRM) | |
| <ref type="bibr" coords="1,227.23,386.36,12.78,9.68" target="#b0">[1]</ref> is a variant of the method of characteristics (MoC) | |
| <ref type="bibr" coords="1,479.13,386.36,12.78,9.68" target="#b1">[2]</ref> where the quadrature varies stochastically from one transport sweep to the next. | |
| </s> | |
| <s coords="1,389.27,398.96,150.73,9.68;1,72.00,411.56,468.00,9.68;1,72.00,424.17,127.20,9.68">To-date, it has been demonstrated only on eigenvalue neutron transport problems but there is potential for it to be extended to other forms of radiation transport problems.</s> | |
| <s coords="1,72.00,442.75,468.00,9.68;1,72.00,455.35,75.91,9.68">This paper describes its application in fixed source problems, demonstrating its competitiveness with other transport solvers.</s> | |
| <s coords="1,154.44,455.35,385.56,9.68;1,72.00,467.95,468.00,9.68;1,72.00,480.56,468.00,9.68;1,72.00,493.16,35.01,9.68">This is interesting as one of very few uses of an MoC-based method in a fixed source context: the traditional deterministic MoC method is predominantly applied in 2D eigenvalue problems, and where it has been used for fixed source problems it is often in service of lattice calculations for reactor physics.</s> | |
| <s coords="1,111.28,493.16,383.36,9.68">That being said, there has been some work applying MoC to 3D fixed source problems.</s> | |
| <s coords="1,498.90,493.16,41.10,9.68;1,72.00,505.76,468.00,9.68;1,72.00,518.36,132.16,9.68">However, extreme polar angle discretisation (relative to reactor problems with significant axial regularity) may be necessary for good results | |
| <ref type="bibr" coords="1,188.65,518.36,11.64,9.68" target="#b2">[3]</ref>. | |
| </s> | |
| </p> | |
| <p> | |
| <s coords="1,72.00,536.94,398.28,9.68">In this paper, the utility of random ray MoC is demonstrated on 3D fixed source problems.</s> | |
| <s coords="1,474.82,536.94,65.18,9.68;1,72.00,549.55,468.00,9.68;1,72.00,562.15,319.93,9.68">In fixed source problems, random ray MoC brings the advantages of geometric flexibility, extreme memory reduction, avoiding angular discretisation, and the potential for massive parallelism.</s> | |
| <s coords="1,395.90,562.15,144.10,9.68;1,72.00,574.75,468.00,9.68;1,72.00,587.35,237.42,9.68">Random ray can also be straightforwardly modified to support first collided source calculations to handle highly localised sources -these commonly occur in fixed source problems of interest.</s> | |
| <s coords="1,315.01,587.35,224.99,9.68;1,72.00,599.96,362.50,9.68">These advantages are demonstrated by considering the Kobayashi problem 3 and a more complicated problem devised for this paper.</s> | |
| <s coords="1,440.23,599.96,99.77,9.68;1,72.00,612.56,205.26,9.68">All random ray results are compared against multigroup Monte Carlo.</s> | |
| </p> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head n="2." coords="2,264.89,74.96,8.97,10.44;2,295.90,74.96,51.22,10.44">THEORY</head> | |
| <p> | |
| <s coords="2,72.00,100.23,352.94,9.68">The characteristic form of the multigroup neutron transport equation is given by:</s> | |
| </p> | |
| <formula xml:id="formula_0" coords="2,258.38,116.79,277.36,24.50">d d𝑠 𝜓 𝑔 + Σ tr,𝑔 𝜓 𝑔 = 𝑞 𝑔 , ( | |
| <label>1</label> | |
| </formula> | |
| <formula xml:id="formula_1" coords="2,535.74,124.15,4.26,9.68">)</formula> | |
| <p> | |
| <s coords="2,72.00,145.64,370.64,9.96;2,445.72,145.93,94.29,9.68;2,72.00,158.33,440.80,10.86;2,516.38,158.53,23.62,9.68;2,72.00,170.85,296.00,9.96">where is the distance along a characteristic in a given neutron flight direction, , is the neutron angular flux in energy group along the characteristic, Σ tr, is the transport cross section in group , and is the neutron source in group producing neutrons in the direction of .</s> | |
| <s coords="2,371.98,171.13,168.02,9.68;2,72.00,183.74,85.99,9.68">For simplicity, we assume an isotropic source in this work.</s> | |
| <s coords="2,161.95,183.74,347.70,9.68">Further extension to allow anisotropic scattering will be examined in future | |
| <ref type="bibr" coords="2,494.14,183.74,11.64,9.68" target="#b3">[4]</ref>. | |
| </s> | |
| <s coords="2,513.60,183.74,26.40,9.68;2,72.00,196.34,242.03,9.68">Given common approximations, Eq. ( | |
| <ref type="formula" coords="2,208.29,196.34,4.26,9.68" target="#formula_0">1</ref>) can be solved to give: | |
| </s> | |
| </p> | |
| <formula xml:id="formula_2" coords="2,205.90,212.59,334.10,23.89">𝜓 𝑔 (𝑠) = 𝜓 𝑔 (0) 𝑒 -Σ tr,𝑔 𝑠 + 𝑞 𝑔 Σ tr,𝑔 1 -𝑒 -Σ tr,𝑔 𝑠 , | |
| <label>(2)</label> | |
| </formula> | |
| <p> | |
| <s coords="2,72.00,243.48,42.34,9.68;2,115.54,243.05,376.76,10.10">where (0) is the angular flux at the beginning of a characteristic track in a mesh element.</s> | |
| <s coords="2,499.91,243.48,40.09,9.68;2,72.00,256.08,468.00,9.68;2,72.00,268.68,385.44,9.68">During a transport sweep in MoC, when a characteristic/ray is traversed, Eq. ( | |
| <ref type="formula" coords="2,385.76,256.08,4.26,9.68" target="#formula_2">2</ref>) can be rearranged to produce a quantity with which to increment an estimate of the scalar flux in a given mesh element: | |
| </s> | |
| </p> | |
| <formula xml:id="formula_3" coords="2,221.60,287.97,318.40,23.89">𝜙 𝑔 += 𝜓 𝑔 (0) - 𝑞 𝑔 Σ tr,𝑔 1 -𝑒 -Σ tr,𝑔 𝑠 . | |
| <label>(3)</label> | |
| </formula> | |
| <p> | |
| <s coords="2,72.00,318.68,468.00,9.68;2,72.00,331.29,327.34,9.68">In standard MoC, each track would have a unique weight associated with it which would scale this increment, but that is not the case with TRRM, and so this weight is not included here.</s> | |
| <s coords="2,403.23,331.29,136.77,9.68;2,72.00,343.89,468.00,9.68;2,72.00,356.49,467.00,9.68">At the conclusion of the sweep, the final scalar fluxes in each mesh element are obtained by normalising the accumulated fluxes appropriately by the total active ray length traversed that iteration, , and adding contributions from the neutron sources:</s> | |
| </p> | |
| <formula xml:id="formula_4" coords="2,254.62,372.91,285.38,26.36">𝜙 𝑔 = 4𝜋𝜙 𝑔 Σ tr,𝑔 𝑉 𝐿 + 4𝜋𝑞 𝑔 Σ tr,𝑔 , | |
| <label>(4)</label> | |
| </formula> | |
| <p> | |
| <s coords="2,72.00,406.10,229.60,9.68">where is the relative volume of the mesh element.</s> | |
| <s coords="2,306.26,406.10,233.73,9.68;2,72.00,418.70,160.78,9.68">This scalar flux is then used to update the fission and scattering sources across the domain.</s> | |
| <s coords="2,236.55,418.70,303.45,9.68;2,72.00,431.30,86.66,10.66">In eigenvalue problems, the global fission rates are then used to update the estimate of eff .</s> | |
| <s coords="2,164.23,431.30,375.77,9.68;2,72.00,443.91,468.00,9.68;2,72.00,456.51,163.06,9.68">In fixed source problems, provided any multiplication is subcritical, this is no longer necessary, and the power iteration is replaced by a source iteration to converge on the scattering and fission source without further normalisation.</s> | |
| <s coords="2,238.96,456.51,301.04,9.68;2,72.00,469.11,365.25,9.68">Fixed source problems can in principle be performed on supercritical problems provided the fission source is somehow scaled, e.g., by a user-specified .</s> | |
| </p> | |
| <p> | |
| <s coords="2,72.00,487.69,468.00,9.68;2,72.00,500.29,468.00,9.68;2,72.00,512.90,468.00,9.68;2,72.00,525.50,468.00,9.68;2,72.00,538.10,468.00,9.68;2,72.00,550.71,468.00,9.68;2,72.00,563.31,433.03,9.68">With respect to the eigenvalue version of the random ray method, this results in a simplification of the algorithm, with no other changes necessary: ray positions and directions are still sampled from a uniform random distribution across space and angle and then traced on-the-fly; their initial angular flux is still guessed based on the local source and the bias is attenuated by specifying a 'dead length' during which no results are accumulated; all rays traverse the same distance given by a 'termination length'; rays are reflected at both vacuum and reflective boundaries with appropriate modification to their angular flux; active and inactive iterations are performed to accumulate final results estimates and to converge sources, respectively.</s> | |
| </p> | |
| <p> | |
| <s coords="2,72.00,581.89,468.00,9.68;2,72.00,594.49,72.91,9.68">A slight modification to random ray may be desirable in fixed source calculations where low density materials can be common.</s> | |
| <s coords="2,150.03,594.30,389.97,10.86;2,72.00,606.90,242.24,10.86">This modification removes the division by Σ tr, in Eq. (3), which is done by multiplying and dividing by the optical distance, Σ tr, = , to give:</s> | |
| </p> | |
| <formula xml:id="formula_5" coords="2,228.63,623.35,311.37,24.56">𝜙 𝑔 += 𝜏𝜓 𝑔 (0) -𝑠𝑞 𝑔 1 -𝑒 -𝜏 𝜏 , | |
| <label>(5)</label> | |
| </formula> | |
| <p> | |
| <s coords="2,72.00,657.62,434.69,9.68">where, in the limit of small , the exponential term approaches 1 for improved numerical stability.</s> | |
| <s coords="2,511.88,657.62,28.12,9.68;2,72.00,669.78,211.02,10.12;2,283.71,667.91,10.06,6.85;2,294.70,669.78,245.29,10.12;2,72.00,682.82,50.45,9.68">In this work we use a rational approximation of (1 - - ) / which has been found to be numerically robust for small | |
| <ref type="bibr" coords="2,106.93,682.82,11.64,9.68" target="#b4">[5]</ref>. | |
| </s> | |
| </p> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head n="2.1." coords="3,72.00,74.96,17.93,10.44;3,106.47,74.96,137.68,10.44">Uncollided Flux Treatment</head> | |
| <p> | |
| <s coords="3,72.00,100.23,468.00,9.68;3,72.00,112.83,224.10,9.68">In deterministic approaches to fixed source problems, localised (or point) sources are often challenging or impossible to model without specialised treatment.</s> | |
| <s coords="3,300.81,112.83,239.19,9.68;3,72.00,125.43,84.55,9.68">This often consists of an 'uncollided flux/first collided source calculation'.</s> | |
| <s coords="3,160.62,125.43,379.38,9.68;3,72.00,138.04,360.55,9.68">These calculations are often performed to mitigate ray effects (as the uncollided flux is usually the most anisotropic) but are essential where the source is a singular point.</s> | |
| </p> | |
| <p> | |
| <s coords="3,72.00,156.62,468.00,9.68;3,72.00,169.22,468.00,9.68;3,72.00,181.82,413.93,9.68">The essence of these calculations is that they serve as a pre-processing step to a full transport solution by using methods in integral transport to transport the uncollided flux associated with a highly localised source, generating a global first collided source which is distributed throughout the problem domain.</s> | |
| <s coords="3,491.76,181.82,48.24,9.68;3,72.00,194.43,468.00,9.68;3,72.00,207.03,259.59,9.68">A standard transport solve can be used to obtain the further collided fluxes and, at the conclusion of the calculation, the uncollided flux is added to the solution to give the full flux.</s> | |
| </p> | |
| <p> | |
| <s coords="3,72.00,225.61,468.00,9.68;3,72.00,238.21,468.00,9.68">The mechanics of uncollided flux treatments often take the form of some sort of ray tracing procedure, where the flux from the source is deposited along these tracks corresponding to some choice of quadrature | |
| <ref type="bibr" coords="3,524.48,238.21,11.64,9.68" target="#b5">[6]</ref>. | |
| </s> | |
| <s coords="3,72.00,250.81,229.17,9.68;3,301.33,256.25,4.52,3.56;3,309.46,250.81,18.26,9.68;3,327.87,256.25,4.52,3.56;3,336.00,250.81,204.00,9.68;3,72.00,263.42,468.00,9.68">While this may be an involved addition to, say, an S or P solver, robust and flexible ray tracing is a vital component of TRRM: most of the components necessary for such a calculation would be present already.</s> | |
| <s coords="3,72.00,276.02,468.00,9.68;3,72.00,288.62,468.00,9.68">Furthermore, a stochastic ray tracing method for generating uncollided fluxes has already been demonstrated by | |
| <ref type="bibr" coords="3,85.24,288.62,12.78,9.68" target="#b6">[7]</ref> and | |
| <ref type="bibr" coords="3,119.05,288.62,12.78,9.68" target="#b7">[8]</ref> (where it is referred to as MC-MOC), which naturally fits the geometric flexibility of TRRM. | |
| </s> | |
| </p> | |
| <p> | |
| <s coords="3,72.00,301.23,310.50,9.68">The implementation made in this work follows that given by Baker | |
| <ref type="bibr" coords="3,366.98,301.23,11.64,9.68" target="#b6">[7]</ref>. | |
| </s> | |
| </p> | |
| <p> | |
| <s coords="3,72.00,319.81,323.82,9.68">In TRRM, one must uniformly sample phase space to avoid inducing bias.</s> | |
| <s coords="3,399.76,319.81,140.24,9.68;3,72.00,332.41,392.50,9.68">Sampling rays in portions of the geometry where there is no source, however, might be viewed as a waste of computation.</s> | |
| <s coords="3,469.24,332.41,70.76,9.68;3,72.00,345.01,468.00,9.68;3,72.00,357.61,319.50,9.68">This may be the case where sources are highly localised or at least during early source iterations where the scattering source may not have migrated across the problem from a localised fixed source.</s> | |
| <s coords="3,396.11,357.61,143.89,9.68;3,72.00,370.22,468.00,9.68;3,72.00,382.82,468.00,9.68;3,72.00,395.42,74.50,9.68">In the context of random ray, the use of an uncollided flux treatment as a preprocessing step is useful in that it allows conversion of highly localised or point sources into a volumetric first collision source term that is defined throughout the entire problem domain.</s> | |
| <s coords="3,150.33,395.42,389.67,9.68;3,72.00,408.02,225.39,9.68">Therefore, uncollided flux treatment can serve to improve the efficiency of TRRM in fixed source problems featuring highly localized sources.</s> | |
| </p> | |
| <p> | |
| <s coords="3,72.00,426.61,269.98,9.68">Uncollided flux treatments also present some complications.</s> | |
| <s coords="3,348.84,426.61,191.16,9.68;3,72.00,439.21,468.00,9.68;3,72.00,451.81,93.52,9.68">First, they necessitate knowing the volume of each mesh element which is traversed -this is in order to calculate the volume-averaged uncollided flux in a given mesh.</s> | |
| <s coords="3,171.18,451.81,368.82,9.68;3,72.00,464.41,241.29,9.68">Where the mesh is a finite volume or finite element mesh, this information is often available already and so this would not be a concern.</s> | |
| <s coords="3,320.64,464.41,219.36,9.68;3,72.00,477.02,468.00,9.68;3,72.00,489.71,28.32,9.45">On the other hand, if the mesh is, say, produced using constructive solid geometry (as in the present work) then these volumes are typically not known a priori.</s> | |
| <s coords="3,104.55,489.62,435.45,9.68;3,72.00,502.22,280.83,9.68">Random ray also needs to know these volumes, but is capable of efficiently computing them during transport due to the uniform ray sampling across the geometry.</s> | |
| <s coords="3,358.86,502.22,181.14,9.68;3,72.00,514.82,468.00,9.68;3,72.00,527.43,43.25,9.68">This computation cannot be done during uncollided flux calculations as rays are sampled non-uniformly, originating from a small portion of the geometry.</s> | |
| <s coords="3,119.10,527.43,343.46,9.68">Hence, in the present work, volumes must be estimated as a preprocessing step.</s> | |
| <s coords="3,466.41,527.43,73.59,9.68;3,72.00,540.03,353.40,9.68">This can be done stochastically, but must be done with sufficient statistics to avoid inducing a bias.</s> | |
| </p> | |
| <p> | |
| <s coords="3,72.00,558.61,468.00,9.68;3,72.00,571.21,130.36,9.68">A second complication is that, for a given source volume, the benefit of an uncollided flux calculation is hard to determine in advance.</s> | |
| <s coords="3,206.66,571.21,333.34,9.68">If the source is a point source then an uncollided calculation is unavoidable.</s> | |
| <s coords="3,72.00,583.82,468.00,9.68;3,72.00,596.42,468.00,9.68;3,72.00,609.02,468.00,9.68;3,72.00,621.62,468.00,9.68;3,72.00,634.23,87.96,9.68">However, if the source is finite then it is ambiguous which will be more efficient: performing an uncollided flux calculation (where volume must be calculated in advance but all rays originate from the source) followed by a potentially cheaper TRRM solve or performing a random ray calculations as normal (where most rays may miss the source during a given sweep, but the source will still be distributed without needing volumes to be precomputed).</s> | |
| <s coords="3,163.91,634.23,374.33,9.68">This trade-off will not be investigated presently but will be considered in future work.</s> | |
| </p> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head coords="4,263.70,74.96,8.97,10.44">3.</head> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head coords="4,294.71,74.96,53.58,10.44">RESULTS</head> | |
| <p> | |
| <s coords="4,72.00,100.23,468.00,9.68;4,72.00,112.83,346.59,9.68">The k-eigenvalue form of TRRM has previously been implemented in the advanced random ray code (ARRC) | |
| <ref type="bibr" coords="4,110.96,112.83,11.64,9.68" target="#b8">[9]</ref>, as well as the Monte Carlo solvers SCONE | |
| <ref type="bibr" coords="4,315.95,112.83,18.26,9.68" target="#b9">[10]</ref> and OpenMC | |
| <ref type="bibr" coords="4,397.59,112.83,16.80,9.68" target="#b10">[11]</ref>. | |
| </s> | |
| <s coords="4,422.38,112.83,117.62,9.68;4,72.00,125.43,468.00,9.68;4,72.00,138.04,200.64,9.68">To demonstrate the method for fixed source problems, the random ray k-eigenvalue implementation in SCONE was modified to include fixed source capabilities, as described above.</s> | |
| <s coords="4,279.23,138.04,260.77,9.68;4,72.00,150.64,468.00,9.68;4,72.00,163.24,468.00,9.68;4,72.00,175.85,113.79,9.68">Although a very simple problem and not demonstrative of TRRM's strengths (such as direct handling of 3D curvilinear geometries and vectorisation over neutron energy groups), the Kobayashi 'dog leg' problem | |
| <ref type="bibr" coords="4,286.58,163.24,18.26,9.68" target="#b11">[12]</ref> is presented as a first demonstration of TRRM applied to fixed source problems. | |
| </s> | |
| <s coords="4,193.06,175.85,346.94,9.68;4,72.00,188.45,122.19,9.68">Subsequently, a more challenging shielding problem which does demonstrate these strengths is presented.</s> | |
| </p> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head n="3.1." coords="4,72.00,213.44,17.93,10.44;4,106.47,213.44,92.47,10.44">Kobayashi dog leg</head> | |
| <p> | |
| <s coords="4,72.00,238.28,468.00,10.10;4,72.00,251.31,316.83,9.68">The dog leg problem consists of a 60 cm × 100 cm × 60 cm rectangular domain where the left/back/bottom boundaries are reflective and the right/front/top boundaries are vacuum.</s> | |
| <s coords="4,393.71,251.31,146.29,9.68;4,72.00,263.92,328.57,9.68">In one corner there is a source of neutrons adjacent to a 'dog leg' duct featuring very low absorbing material.</s> | |
| <s coords="4,404.47,263.92,135.53,9.68;4,72.00,276.52,173.84,9.68">The remainder of the geometry consists of strongly absorbing material.</s> | |
| <s coords="4,251.02,276.52,288.98,9.68;4,72.00,289.12,58.14,9.68">Monoenergetic cross sections are also provided in the benchmark specification.</s> | |
| <s coords="4,135.80,289.12,404.20,9.68;4,72.00,301.72,152.79,9.68">Two sets of cross sections for the problem exist -one without scattering (where ray effects are more significant) and one with.</s> | |
| </p> | |
| <p> | |
| <s coords="4,72.00,320.30,468.00,9.68;4,72.00,332.91,468.00,9.68;4,72.00,345.51,330.02,9.68">The no-scattering problem is 'easier' in the sense that no source iterations are required, but it is also unfavourable to TRRM in that rays may be sampled which never cross the source region and thus perform essentially no useful work (other than contributing to estimating volume).</s> | |
| <s coords="4,408.55,345.51,131.45,9.68;4,72.00,358.11,468.00,9.68;4,72.00,370.71,337.82,9.68">It is also unfavourable in that there is no advantage to using a coarse track laydown to converge the (non-existent) fission/scattering source faster -one of TRRM's most significant advantages in eigenvalue problems.</s> | |
| <s coords="4,414.97,370.71,125.03,9.68;4,72.00,383.32,378.44,9.68">The optimal settings in such a problem to maximise speed would use relatively few rays traversing large distances.</s> | |
| <s coords="4,455.66,383.32,84.34,9.68;4,72.00,395.92,468.00,9.68;4,72.00,408.52,62.11,9.68">This is because the dead length is a pure cost for each unique ray simulated -simulating fewer cycles and rays per cycle will minimise this.</s> | |
| <s coords="4,138.07,408.52,382.66,9.68">Additionally, as there are no sources to converge, no inactive cycles need be performed.</s> | |
| </p> | |
| <p> | |
| <s coords="4,72.00,427.10,468.00,9.68;4,72.00,439.71,239.84,9.68">In reactor calculations, the dead length is often chosen to be several mean free paths long so as to attenuate the majority of the error in the ray's starting condition.</s> | |
| <s coords="4,315.96,439.71,224.04,9.68;4,72.00,452.31,468.00,9.68;4,72.00,464.91,59.50,9.68">In such calculations, after traversal of several mean free paths, the ray's angular flux spectrum is typically dominated by the accumulated scattering and fission source terms.</s> | |
| <s coords="4,137.97,464.91,402.03,9.68;4,72.00,477.51,112.04,9.68">In the dog leg problem, for the void material, a few mean free paths would correspond to 100's of metres in length.</s> | |
| <s coords="4,188.16,477.51,351.84,9.68;4,72.00,490.12,468.00,9.68;4,72.00,502.72,468.00,9.68;4,72.00,515.32,62.70,9.68">This number can be reduced first by considering that the maximum distance to a boundary in this problem is 262 cm (from one vacuum boundary corner, reflecting in the opposite corner, and returning to the vacuum boundary), after which the angular flux estimate will certainly be unbiased (and equal to zero).</s> | |
| <s coords="4,138.62,515.32,401.38,9.68;4,72.00,527.93,200.20,9.68">This can be further reduced given that the largest contiguous distance that a ray can travel in a void region in this problem is about 62 cm.</s> | |
| <s coords="4,277.46,527.93,262.54,9.68;4,72.00,540.53,80.80,9.68">The mean free path in the absorber region in the rest of the problem is 10 cm.</s> | |
| <s coords="4,158.40,540.53,381.60,9.68;4,72.00,553.13,142.43,9.68">Allowing for a few mean free paths to be traversed in this region, 120 cm should be a conservative dead length choice.</s> | |
| </p> | |
| <p> | |
| <s coords="4,72.00,571.71,468.00,9.68;4,72.00,584.31,432.01,9.68">After some experimentation to minimise the runtime for a given degree of uncertainty, 2100 active cycles and 10,000 rays per cycle were used for the absorbing case with a termination length of 1,220 cm.</s> | |
| <s coords="4,508.65,584.31,31.35,9.68;4,72.00,596.92,468.00,9.68;4,72.00,609.52,29.74,9.68">For the scattering case, the settings are identical except 100 inactive cycles are performed, followed by 2000 active cycles.</s> | |
| <s coords="4,109.00,609.52,431.00,9.68">The mesh consisted of cubes, each with side length 11/10 cm, giving roughly 263k mesh cells.</s> | |
| <s coords="4,72.00,622.12,468.00,9.68;4,72.00,634.72,105.96,9.68">Note that results are estimated across three different lines across the geometry and consist of the fluxes at points along these lines.</s> | |
| <s coords="4,182.55,634.72,357.45,9.68;4,72.00,647.33,164.27,9.68">These lines will be referred to as A, B, and C and can be found in the benchmark definition and are illustrated in Fig. | |
| <ref type="figure" coords="4,228.05,647.33,4.11,9.68" target="#fig_0">1</ref>. | |
| </s> | |
| <s coords="4,240.18,647.33,299.82,9.68;4,72.00,659.93,165.90,9.68">The results used are the mesh-averaged fluxes produced by TRRM at corresponding points in the geometry.</s> | |
| </p> | |
| <p> | |
| <s coords="4,72.00,678.51,468.00,9.68;5,72.00,305.34,38.55,9.68">For all simulations in this paper, multigroup Monte Carlo (MGMC) simulations were also performed with SCONE.</s> | |
| <s coords="5,113.01,303.36,215.95,11.65;5,331.90,305.34,13.39,9.68;5,345.30,303.36,3.99,7.04;5,352.23,305.34,187.77,9.68;5,72.00,317.94,328.51,9.68">The 'mesh' consisted of a uniform lattice of (10) 3 cm 3 boxes, which is simply the most convenient way to represent the benchmark geometry in constructive solid geometry.</s> | |
| <s coords="5,407.11,317.94,132.89,9.68;5,72.00,330.54,468.00,9.68;5,72.00,343.14,102.47,9.68">10M neutrons were simulated over a number of cycles chosen to produce a similar runtime to TRRM (265 for the absorbing case, 230 for the scattering case).</s> | |
| <s coords="5,178.86,343.14,361.14,9.68;5,72.00,355.75,167.70,9.68">Transport was performed using hybrid delta/surface tracking | |
| <ref type="bibr" coords="5,447.68,343.14,18.26,9.68" target="#b12">[13]</ref> and results were scored with a collision estimator | |
| <ref type="bibr" coords="5,218.71,355.75,16.80,9.68" target="#b13">[14]</ref>. | |
| </s> | |
| <s coords="5,244.67,355.75,295.33,9.68;5,72.00,366.38,310.33,11.65;5,385.37,368.35,13.39,9.68;5,398.76,366.38,3.99,7.04;5,405.79,368.35,27.59,9.68">This involved an approximation: the benchmark provides 'flux at a point' results while the collision estimator scored average fluxes in (2) 3 cm 3 boxes.</s> | |
| <s coords="5,437.27,368.35,102.73,9.68;5,72.00,380.95,456.59,9.68">This will result in some bias relative to the benchmark results, but only to a small degree (TRRM will experience a similar bias).</s> | |
| </p> | |
| <p> | |
| <s coords="5,72.00,399.53,468.00,9.68;5,72.00,412.14,34.39,9.68">All simulation were performed on an AMD Ryzen Threadripper node with OpenMP parallelism on 128 threads.</s> | |
| <s coords="5,112.55,412.14,427.45,9.68;5,72.00,424.74,323.48,9.68">The results for the absorbing case are shown in Fig. | |
| <ref type="figure" coords="5,348.10,412.14,5.48,9.68">2</ref> and the results for the scattering case are shown in Fig. | |
| <ref type="figure" coords="5,136.37,424.74,4.11,9.68">3</ref>, with the error bars representing one standard deviation. | |
| </s> | |
| <s coords="5,401.64,424.74,138.36,9.68;5,72.00,437.34,177.00,9.68">For the absorbing case, TRRM took 1m 33s and MGMC took 1m 32s.</s> | |
| <s coords="5,256.01,437.34,259.93,9.68">For the scattering case, TRRM and MGMC took 1m 38s.</s> | |
| <s coords="5,522.96,437.34,17.04,9.68;5,72.00,449.94,468.00,9.68;5,72.00,462.55,441.71,9.68">The parallel 'grind time' for TRRM was about 2.6 ns/integration (with an integration consisting of a ray trace, exponential evaluation, and flux update) and the simulation had a peak memory usage of 48 MB.</s> | |
| <s coords="5,517.63,462.55,22.37,9.68;5,72.00,475.15,468.00,9.68;5,72.00,487.75,265.93,9.68">Most TRRM results are within 2 of the correct results, and where this is not the case the bias is small and likely attributable to averaging the flux over a mesh volume.</s> | |
| <s coords="5,342.96,487.75,197.04,9.68;5,72.00,500.36,468.00,9.68;5,72.00,512.96,85.50,9.68">As opposed to other results published on the benchmark, this implies that little to no bias is present due to ray effects as TRRM samples angular phase space continuously.</s> | |
| <s coords="5,161.91,512.96,378.09,9.68;5,72.00,525.14,424.50,10.10;5,496.50,517.30,7.67,9.37;5,504.22,525.56,35.78,9.68;5,72.06,538.16,455.35,9.68">Furthermore, for results farther from the source (along portions of line B and much of line C), TRRM has a significantly lower standard deviation () than MGMC; note that ∝ 1/ √ where is the number of particles/rays, and so reducing uncertainty by 2 requires 4× as many particles/rays.</s> | |
| <s coords="5,533.30,538.16,6.70,9.68;5,72.00,550.77,468.00,9.68;5,72.00,563.37,468.00,9.68;5,72.00,575.97,237.74,9.68">It should be mentioned that some of the extreme variances MGMC reports might be reduced by, e.g., tallies to estimate flux at a point or variance reduction, although these would also increase runtime to some extent and involve choosing some associated free parameters.</s> | |
| <s coords="5,313.62,575.97,226.38,9.68;5,72.00,588.57,468.00,9.68;5,72.00,601.18,401.90,9.68">Overall, although at positions close to the source (in terms of mean free paths) MGMC produces lower variance results, TRRM obtains a much lower variance at positions far removed from the source which are of interest in deep penetration problems.</s> | |
| <s coords="5,72.00,619.76,468.00,9.68;5,72.00,632.16,107.05,9.87;5,179.46,630.39,3.99,7.04;5,188.23,632.36,136.78,9.68;5,325.41,630.39,3.99,7.04;5,333.47,632.36,156.30,9.68">Fig. | |
| <ref type="figure" coords="5,92.23,619.76,5.48,9.68" target="#fig_2">4</ref> shows the relative, spatially-dependent Figures of Merit (FoMs) of TRRM and MC, where the FoM is defined as = 1/ 2 , where is the runtime and 2 is the variance of a given estimate. | |
| </s> | |
| <s coords="5,496.23,632.36,43.77,9.68;5,72.00,644.96,468.00,9.68;5,72.00,657.57,273.07,9.68">MC tends to do relatively well along line A, where the source is present at position 1 and many source particles are deposited in the absorber at the end of the duct in regions 7-10.</s> | |
| <s coords="5,348.93,657.57,191.07,9.68;5,72.00,670.17,468.00,9.68;5,72.00,682.77,296.40,9.68">Otherwise, relatively few particles reach the deeper portions of the problem to score along line C or the tail of line B. We anticipate that this performance advantage would improve with the optical depth of the desired tally.</s> | |
| </p> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head n="3.2." coords="6,72.00,431.36,17.93,10.44;6,106.47,431.36,130.43,10.44">Larger shielding problem</head> | |
| <p> | |
| <s coords="6,72.00,456.62,468.00,9.68;6,72.00,469.23,195.23,9.68">A more challenging shielding problem was devised to demonstrate TRRM's performance against multigroup Monte Carlo subject to more realistic physics.</s> | |
| <s coords="6,270.88,469.23,269.12,9.68;6,72.00,481.83,468.00,9.68;6,72.00,494.43,213.07,9.68">The problem consists of a cylindrical concrete cask, containing a cube of neutron emitting material contained on one side of a cuboid cavity, a 120 cm thick wall, and a cylinder of water on the other side of the wall.</s> | |
| <s coords="6,292.78,494.43,98.61,9.68">Everything else is air.</s> | |
| <s coords="6,399.10,494.43,140.90,9.68;6,72.00,507.03,245.76,9.68">All surrounding boundaries are vacuum to represent, e.g., a strongly shielded container.</s> | |
| <s coords="6,322.65,507.03,217.35,9.68;6,72.00,519.64,253.27,9.68">The shield also contains a 5 cm diameter circular duct running along the -direction, along the centre-line.</s> | |
| <s coords="6,329.08,519.64,210.91,9.68;6,72.00,532.24,39.27,9.68">Fig. | |
| <ref type="figure" coords="6,348.59,519.64,5.48,9.68" target="#fig_4">5</ref> shows the geometry and dimensions of the problem. | |
| </s> | |
| <s coords="6,115.15,532.24,424.85,9.68;6,72.00,544.84,161.59,9.68">The multigroup data for each material was produced by a very simplified Monte Carlo calculation subject to a 14 MeV neutron source.</s> | |
| <s coords="6,239.22,544.84,300.78,9.68;6,72.00,557.44,190.90,9.68">Only three unique materials are used (concrete, air, and water) with their data produced in a 32 group structure.</s> | |
| <s coords="6,267.47,557.44,272.53,9.68;6,72.00,570.05,82.04,9.68">The source is isotropic and monochromatic, exclusively in the first energy group.</s> | |
| <s coords="6,160.41,570.05,379.59,9.68;6,72.00,582.65,468.00,9.68;6,72.00,595.25,296.93,9.68">The simulations to follow are not aiming for accuracy with respect to the continuous energy calculation, but only to demonstrate the relative performances of TRRM and MGMC on problems with significant attenuation and more sophisticated neutron physics.</s> | |
| </p> | |
| <p> | |
| <s coords="6,72.00,613.83,208.58,9.68">Two variations on this problem are performed.</s> | |
| <s coords="6,287.20,613.83,252.80,9.68">The first has the source as a volume as described above.</s> | |
| </p> | |
| <p> | |
| <s coords="6,72.00,626.44,468.00,9.68;6,72.00,639.04,144.15,9.68">The second considers the source to be a point, instead of a cube; this is done to investigate the use of the uncollided calculation in TRRM.</s> | |
| <s coords="6,218.64,639.04,308.36,9.68">Together, this produces four calculation cases to run with each method.</s> | |
| <s coords="6,530.87,639.04,9.13,9.68;6,72.00,651.64,468.00,9.68;6,72.00,664.24,241.03,9.68">In each calculation, we are concerned with the flux integrated over space and energy in both the water cylinder and the cube volume which defines the original source.</s> | |
| </p> | |
| <p> | |
| <s coords="6,72.00,682.82,468.00,9.68;7,72.00,303.27,214.29,9.68">Fig. | |
| <ref type="figure" coords="6,92.57,682.82,5.48,9.68" target="#fig_5">6</ref> shows the discretisation of the problem used by TRRM: the air was coarsely discretised, while the water, wall, and source regions are relatively fine. | |
| </s> | |
| <s coords="7,290.12,303.27,202.63,9.68">Just over 600k cells were used in the geometry.</s> | |
| <s coords="7,496.57,303.27,43.43,9.68;7,72.00,315.87,468.00,9.68;7,72.00,328.47,45.65,9.68;7,117.66,326.50,3.99,7.04;7,124.88,328.47,90.40,9.68;7,215.28,326.50,3.99,7.04;7,222.50,328.47,13.39,9.68;7,235.89,326.50,3.99,7.04;7,243.12,328.47,41.70,9.68">A second, finer discretisation is also used for TRRM, where the mesh in the source and container region is refined from 1 cm 3 elements to (0.5715) 3 cm 3 elements.</s> | |
| </p> | |
| <p> | |
| <s coords="7,72.00,347.05,468.00,9.68;7,72.00,359.66,19.18,9.68">Each solver was run with two sets of settings: one expending more computational effort, and the other less.</s> | |
| <s coords="7,95.42,359.66,444.58,9.68;7,72.00,372.26,295.98,9.68">Coarser simulation settings for TRRM were 80k rays per iteration, a dead length of 350 cm, an active length of 850 cm, 300 inactive iterations, and 200 active iterations.</s> | |
| <s coords="7,372.96,372.26,167.04,9.68;7,72.00,384.86,468.00,9.68;7,72.00,397.46,468.00,9.68;7,72.00,410.07,209.01,9.68">The dead length was chosen such that it guarantees a ray strikes a vacuum boundary at some point during the length, while the population and active length were chosen to ensure a sufficient ray coverage during a given iteration, i.e., over 99% of cells are hit by an active ray during a given iteration.</s> | |
| <s coords="7,285.64,410.07,254.36,9.68;7,72.00,422.67,374.78,9.68">The inactive iterations were chosen by experimentation to ensure convergence, though this could be automated using, e.g., Shannon entropy | |
| <ref type="bibr" coords="7,431.27,422.67,11.64,9.68" target="#b8">[9]</ref>. | |
| </s> | |
| <s coords="7,450.84,422.67,89.16,9.68;7,72.00,435.27,407.77,9.68">The active iterations were chosen somewhat arbitrarily to ensure that good statistics on final results were achieved.</s> | |
| <s coords="7,483.71,435.27,56.29,9.68;7,72.00,447.88,468.00,9.68;7,72.00,460.48,120.70,9.68">For the point source calculations, additional simulation settings had to be chosen for the volume precomputation and the uncollided flux calculation.</s> | |
| <s coords="7,198.26,460.48,341.74,9.68">The volume calculation used 50M rays, each travelling a distance of 800 cm.</s> | |
| <s coords="7,72.00,473.08,468.00,9.68;7,72.00,485.68,197.36,9.68">The uncollided flux calculation used 200k rays and 20 batches for statistics, with a distance sufficient for each ray to strike the boundary and terminate.</s> | |
| <s coords="7,273.16,485.68,266.84,9.68;7,72.00,498.29,257.13,9.68">The finer settings used the finer mesh as described previously, while increasing the number of rays per iteration to 100k.</s> | |
| <s coords="7,334.99,498.29,205.01,9.68">The uncollided calculation also used 5M rays.</s> | |
| <s coords="7,72.00,510.89,468.00,9.68;7,72.00,523.49,390.03,9.68">The coarser MGMC calculation used 50M particles, simulated over 200 batches, while the finer calculation used the same number of particles per batch but increased the number of batches by 10×.</s> | |
| </p> | |
| <p> | |
| <s coords="7,72.00,542.07,227.92,9.68">Results are shown in Table | |
| <ref type="table" coords="7,189.80,542.07,3.65,9.68" target="#tab_0">I</ref> for all calculation cases. | |
| </s> | |
| <s coords="7,303.74,542.07,236.26,9.68;7,72.00,554.68,401.54,9.68">Uncertainties are included for all flux results, but these are small for TRRM and for the MGMC source integrals where all MC particles are born.</s> | |
| <s coords="7,479.22,554.68,60.78,9.68;7,72.00,567.28,468.00,9.68;7,72.00,579.88,446.68,9.68">In the coarser cases, TRRM only slightly under-predicts MGMC in the source region (about 1% difference) and overlaps within 2 standard deviations in the water region -the finer calculation cases agree within about 0.5%.</s> | |
| <s coords="7,522.96,579.88,17.04,9.68;7,72.00,592.48,446.52,9.68">The runtimes for the point source TRRM result include times for uncollided flux and volume calculations.</s> | |
| <s coords="7,522.96,592.48,17.04,9.68;7,72.00,605.09,468.00,9.68;7,72.00,617.69,72.91,9.68">The volume calculation took about 23s, while the uncollided calculation took 1s for the coarse case and 65s for the fine case.</s> | |
| <s coords="7,149.78,617.69,390.22,9.68;7,72.00,630.29,468.00,9.68;7,72.00,642.89,59.33,9.68">Given results which agree well with MGMC, this implies that overheads associated with uncollided flux calculations are not substantial: less than 10% of runtime for the coarse case and 14% for the fine case.</s> | |
| <s coords="7,138.92,642.89,401.08,9.68;7,72.00,655.50,215.07,9.68">Notably, even to achieve the quite high standard deviation in the water region required a significantly longer MGMC runtime than TRRM.</s> | |
| <s coords="7,289.42,655.50,250.58,9.68;7,72.00,668.10,327.34,9.68;7,399.74,666.13,3.99,7.04;7,404.18,668.10,49.89,9.68;7,454.47,666.13,3.99,7.04;7,462.76,668.10,77.24,9.68;7,72.00,680.70,174.86,9.68">This can be quantified using Figures of Merit with respect to the uncertainty of the integral fluxes in water (defined as FoM= 1/ 2 where 2 is the normalised variance of the flux and is the runtime).</s> | |
| <s coords="7,250.63,680.70,289.37,9.68;9,72.00,384.85,468.00,9.68;9,72.00,397.45,468.00,9.68">For this particular problem and result, TRRM outperforms MGMC However, definitively stating that TRRM is accurate at estimating the flux in the water region is difficult to say due to the very high MGMC uncertainties associated with these results, despite the significant runtimes.</s> | |
| <s coords="9,72.00,410.05,468.00,9.68;9,72.00,422.66,425.97,9.68">Both sets of MGMC results agree with the TRRM results within uncertainty, although further resolving the effects of discretisation error in a practical manner would require variance reduction calculations.</s> | |
| </p> | |
| </div> | |
| <div | |
| xmlns="http://www.tei-c.org/ns/1.0"> | |
| <head n="4." coords="9,247.04,449.64,8.97,10.44;9,278.05,449.64,86.90,10.44">CONCLUSIONS</head> | |
| <p> | |
| <s coords="9,72.00,474.91,291.91,9.68">A fixed-source random ray method was implemented and tested.</s> | |
| <s coords="9,371.72,474.91,168.28,9.68;9,72.00,487.51,468.00,9.68;9,72.00,500.12,147.37,9.68">On the Kobayashi dogleg problem, it produced accurate results at all points and was up to 100× more efficient than MGMC at estimating fluxes in deeper regions of the problem.</s> | |
| <s coords="9,224.47,500.12,315.53,9.68;9,72.00,512.72,468.00,9.68;9,72.00,525.32,43.53,9.68">On the other hand, for results in regions adjacent to the particle source, MC had up to a 100× FoM advantage, although these quantities are typically of lesser interest in shielding problems.</s> | |
| <s coords="9,122.10,525.32,417.90,9.68;9,72.00,537.92,468.00,9.68">This performance was similar in a more realistic shielding problem where, for results subject to significant particle attenuation, TRRM was 8 to 9 orders of magnitude more efficient than MGMC.</s> | |
| <s coords="9,72.00,550.53,468.00,9.68;9,72.00,563.13,375.65,9.68">Additionally, TRRM's compatability with uncollided flux calculations was demonstrated, showing that it is a minor (up to 15%) overhead for point source calculations in realistic problems.</s> | |
| <s coords="9,454.61,563.13,85.39,9.68;9,72.00,575.73,468.00,9.68;9,72.00,588.33,468.00,9.68;9,72.00,600.94,223.46,9.68">The conclusions of this study might be affected by using variance reduction techniques with MGMC, but these would entail additional complication versus TRRM which produces accurate results in acceptable runtimes without requiring any acceleration techniques, e.g., CMFD.</s> | |
| <s coords="9,298.06,600.94,241.94,9.68;9,72.00,613.54,391.08,9.68">This also suggests that TRRM will be performant when used to aid MC variance reduction, as considered in a sister paper at this conference | |
| <ref type="bibr" coords="9,442.08,613.54,16.80,9.68" target="#b14">[15]</ref>. | |
| </s> | |
| </p> | |
| <p> | |
| <s coords="9,72.00,632.12,468.00,9.68;9,72.00,644.72,299.46,9.68">Further improvements might be obtained by using linear sources, allowing for a coarser representation of geometries and thus reducing the computational cost of a solution | |
| <ref type="bibr" coords="9,355.95,644.72,11.64,9.68" target="#b3">[4]</ref>. | |
| </s> | |
| <s coords="9,375.19,644.72,164.81,9.68;9,72.00,657.33,468.00,9.68">This is particularly attractive in a fixed source context as these problems tend to feature significantly larger homogeneous regions than reactors.</s> | |
| <s coords="9,72.00,669.93,468.00,9.68;9,72.00,682.53,457.49,9.68">For example, they do not tend to have composition gradients due to depletion/temperature variation which would impose limits on how coarsely discretised a problem might be while maintaining physical fidelity.</s> | |
| </p> | |
| </div> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" xml:id="fig_0" coords="5,74.47,269.98,463.07,9.57;5,272.98,282.58,66.04,9.57;5,227.19,72.00,157.61,187.99"> | |
| <head>Figure 1 .</head> | |
| <label>1</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="5,74.47,269.98,41.90,9.57">Figure 1.</s> | |
| <s coords="5,120.32,269.98,417.22,9.57;5,272.98,282.58,66.04,9.57">A modified image of the Kobayashi geometry, illustrating the lines across which the flux is estimated [12]</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| <graphic coords="5,227.19,72.00,157.61,187.99" type="bitmap"/> | |
| </figure> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" xml:id="fig_1" coords="6,77.48,213.50,457.04,9.57;6,122.84,226.10,366.31,9.57"> | |
| <head>Figure 2 .Figure 3 .</head> | |
| <label>23</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="6,77.48,213.50,457.04,9.57;6,122.84,226.10,133.92,9.57">Figure 2. Computed/exact fluxes in the Kobayashi problem without scattering across lines given in the benchmark specification.</s> | |
| <s coords="6,260.72,226.10,228.44,9.57">The error bars represent one standard deviation.</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| </figure> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" xml:id="fig_2" coords="7,90.61,267.91,430.77,9.57;7,101.67,280.51,408.67,9.57"> | |
| <head>Figure 4 .</head> | |
| <label>4</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="7,90.61,267.91,430.77,9.57;7,101.67,280.51,42.15,9.57">Figure 4. Spatially-dependent relative Figures of Merit of TRRM and MC for the Kobayashi problem.</s> | |
| <s coords="7,147.77,280.51,362.56,9.57">The results are shown for the three lines given in the benchmark specification.</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| </figure> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" xml:id="fig_3" coords="8,224.76,240.38,162.48,8.70;8,242.64,374.84,126.73,8.70"> | |
| <head>( a )</head> | |
| <label>a</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="8,238.86,240.38,148.38,8.70"> view at 25 cm above the ground.</s> | |
| <s coords="8,242.64,374.84,126.73,8.70">(b) view at the centreline.</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| </figure> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" xml:id="fig_4" coords="8,195.35,394.94,221.30,9.57;8,74.34,426.40,463.34,92.67"> | |
| <head>Figure 5 .</head> | |
| <label>5</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="8,195.35,394.94,221.30,9.57">Figure 5. Geometry colored by material region.</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| <graphic coords="8,74.34,426.40,463.34,92.67" type="bitmap"/> | |
| </figure> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" xml:id="fig_5" coords="8,174.76,672.81,262.49,9.57;8,74.34,537.49,463.34,107.89"> | |
| <head>Figure 6 .</head> | |
| <label>6</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="8,174.76,672.81,41.90,9.57">Figure 6.</s> | |
| <s coords="8,220.61,672.81,216.64,9.57">Discretisation of the geometry used by TRRM.</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| <graphic coords="8,74.34,537.49,463.34,107.89" type="bitmap"/> | |
| </figure> | |
| <figure | |
| xmlns="http://www.tei-c.org/ns/1.0" type="table" xml:id="tab_0" coords="9,72.00,73.49,468.00,302.45"> | |
| <head>Table I . Results comparison of TRRM and MGMC on the larger shielding problem including runtimes, integral fluxes and uncertainties and Figures of Merit for scoring in the water region.</head> | |
| <label>I</label> | |
| <figDesc> | |
| <div> | |
| <p> | |
| <s coords="9,104.03,353.66,98.39,9.68">9 orders of magnitude.</s> | |
| <s coords="9,206.29,353.66,333.71,9.68;9,72.00,366.27,437.26,9.68">Within the limits of Monte Carlo uncertainty, it appears that TRRM provides an accurate and efficient solution on reasonably complicated shielding problems relative to MGMC.</s> | |
| </p> | |
| </div> | |
| </figDesc> | |
| <table coords="9,72.00,110.64,453.48,252.70"> | |
| <row> | |
| <cell cols="2">Source Solve</cell> | |
| <cell>Total runtime (s)</cell> | |
| <cell>Integral flux in source (n cm /s)</cell> | |
| <cell>Integral flux in water (n cm /s)</cell> | |
| <cell>FoM in water</cell> | |
| </row> | |
| <row> | |
| <cell cols="2">Volume TRRM</cell> | |
| <cell>363</cell> | |
| <cell cols="3">1.400E+05±4.699E+00 1.979E-03±5.241E-08 4.017E+06</cell> | |
| </row> | |
| <row> | |
| <cell/> | |
| <cell>TRRM finer</cell> | |
| <cell>544</cell> | |
| <cell cols="3">1.407E+05±3.179E+00 1.992E-03±4.729E-08 3.357E+06</cell> | |
| </row> | |
| <row> | |
| <cell/> | |
| <cell>MGMC</cell> | |
| <cell>3915</cell> | |
| <cell cols="3">1.415E+05±1.687E+00 2.161E-03±3.492E-04 9.788E-03</cell> | |
| </row> | |
| <row> | |
| <cell/> | |
| <cell>MGMC high stats</cell> | |
| <cell>38,348</cell> | |
| <cell cols="3">1.415E+05±5.404E-01 1.956E-03±9.893E-05 1.020E-02</cell> | |
| </row> | |
| <row> | |
| <cell>Point</cell> | |
| <cell>TRRM</cell> | |
| <cell>377</cell> | |
| <cell cols="3">2.461E+01±2.118E-03 1.983E-07±4.674E-12 4.770E+06</cell> | |
| </row> | |
| <row> | |
| <cell/> | |
| <cell>TRRM finer</cell> | |
| <cell>626</cell> | |
| <cell cols="3">2.481E+01±2.655E-03 2.034E-07±4.467E-12 3.314E+06</cell> | |
| </row> | |
| <row> | |
| <cell/> | |
| <cell>MGMC</cell> | |
| <cell>4563</cell> | |
| <cell cols="3">2.493E+01±2.504E-04 1.964E-07±3.533E-08 6.775E-03</cell> | |
| </row> | |
| <row> | |
| <cell/> | |
| <cell>MGMC high stats</cell> | |
| <cell>44,927</cell> | |
| <cell cols="3">2.493E+01±7.686E-05 2.022E-07±1.193E-08 6.389E-03</cell> | |
| </row> | |
| <row> | |
| <cell>by 8 to</cell> | |
| <cell/> | |
| <cell/> | |
| <cell/> | |
| <cell/> | |
| </row> | |
| </table> | |
| </figure> | |
| </body> | |
| <back> | |
| <div type="acknowledgement"> | |
| <div> | |
| <head coords="10,236.46,74.96,139.09,10.44">ACKNOWLEDGEMENTS</head> | |
| <p>The submitted manuscript was supported by the | |
| <rs type="funder">U.S. Department of Energy, Office of Science</rs>, under contract No. | |
| <rs type="grantNumber">DE-AC02-06CH11357</rs> and by the | |
| <rs type="funder">UK EPSRC</rs> grant | |
| <rs type="grantNumber">EP/T022159/1</rs>. | |
| <rs type="person">P. Benie</rs> assisted through maintaining the Cambridge Nuclear Energy Group workstations. | |
| </p> | |
| </div> | |
| </div> | |
| <listOrg type="funding"> | |
| <org type="funding" xml:id="_5ETuTgz"> | |
| <idno type="grant-number">DE-AC02-06CH11357</idno> | |
| </org> | |
| <org type="funding" xml:id="_KWyyfrX"> | |
| <idno type="grant-number">EP/T022159/1</idno> | |
| </org> | |
| </listOrg> | |
| <div type="references"> | |
| <listBibl> | |
| <biblStruct coords="10,95.71,177.04,444.28,9.68;10,95.71,188.95,354.09,9.72" xml:id="b0"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,342.49,177.04,197.51,9.68;10,95.71,189.00,38.32,9.68">The Random Ray Method for neutral particle transport</title> | |
| <author> | |
| <persName> | |
| <forename type="first">John</forename> | |
| <forename type="middle">R</forename> | |
| <surname>Tramm</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0002-5397-4402</idno> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Kord</forename> | |
| <forename type="middle">S</forename> | |
| <surname>Smith</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Benoit</forename> | |
| <surname>Forget</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Andrew</forename> | |
| <forename type="middle">R</forename> | |
| <surname>Siegel</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.jcp.2017.04.038</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,146.50,189.08,148.27,9.45">Journal of Computational Physics</title> | |
| <title level="j" type="abbrev">Journal of Computational Physics</title> | |
| <idno type="ISSN">0021-9991</idno> | |
| <imprint> | |
| <biblScope unit="volume">342</biblScope> | |
| <biblScope unit="page" from="229" to="252"/> | |
| <date type="published" when="2017-08">2017</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">J. R. Tramm, K. S. Smith, B. Forget, and A. R. Siegel. "The Random Ray Method for neutral particle transport." Journal of Computational Physics, volume 342, pp. 229-252 (2017).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,208.38,444.28,9.68;10,95.71,220.33,292.24,9.68" xml:id="b1"> | |
| <monogr> | |
| <title level="m" type="main" coords="10,147.24,208.38,392.76,9.68;10,95.71,220.33,19.07,9.68">A Characteristics Formulation of the Neutron Transport Equation in Complicated Geometries</title> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">J</forename> | |
| <surname>Askew</surname> | |
| </persName> | |
| </author> | |
| <imprint> | |
| <date type="published" when="1972">1972</date> | |
| </imprint> | |
| <respStmt> | |
| <orgName>UK Atomic Energy Establishment</orgName> | |
| </respStmt> | |
| </monogr> | |
| <note type="report_type">Technical report</note> | |
| <note type="raw_reference">J. Askew. "A Characteristics Formulation of the Neutron Transport Equation in Complicated Geome- tries." Technical report, UK Atomic Energy Establishment (1972).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,239.71,444.28,9.68;10,95.71,251.66,349.29,9.68" xml:id="b2"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,281.98,239.71,258.02,9.68;10,95.71,251.66,87.90,9.68">Recent developments in the GENESIS code based on the Legendre polynomial expansion of angular flux method</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Akio</forename> | |
| <surname>Yamamoto</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Akinori</forename> | |
| <surname>Giho</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Tomohiro</forename> | |
| <surname>Endo</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.net.2017.06.016</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,209.37,251.75,200.94,9.45">Nuclear Engineering and Technology</title> | |
| <title level="j" type="abbrev">Nuclear Engineering and Technology</title> | |
| <idno type="ISSN">1738-5733</idno> | |
| <imprint> | |
| <biblScope unit="volume">49</biblScope> | |
| <biblScope unit="issue">6</biblScope> | |
| <biblScope unit="page" from="1143" to="1156"/> | |
| <date type="published" when="2017-09">2017</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">A. Yamamoto, A. Giho, and T. Endo. "Application of the GENESIS Code to the Kobayashi 3D Benchmark Problem." In Transactions of the American Nuclear Society (2017).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,271.04,444.28,9.68;10,95.71,282.96,346.32,9.72" xml:id="b3"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,243.00,271.04,297.00,9.68;10,95.71,283.00,34.16,9.68">A Linear Source Approximation Scheme for the Method of Characteristics</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Rodolfo</forename> | |
| <forename type="middle">M</forename> | |
| <surname>Ferrer</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Joel</forename> | |
| <forename type="middle">D</forename> | |
| <surname>Rhodes</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.13182/nse15-6</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,141.41,283.09,145.34,9.45">Nuclear Science and Engineering</title> | |
| <title level="j" type="abbrev">Nuclear Science and Engineering</title> | |
| <idno type="ISSN">0029-5639</idno> | |
| <idno type="ISSNe">1943-748X</idno> | |
| <imprint> | |
| <biblScope unit="volume">182</biblScope> | |
| <biblScope unit="issue">2</biblScope> | |
| <biblScope unit="page" from="151" to="165"/> | |
| <date type="published" when="2017">2017</date> | |
| <publisher>Informa UK Limited</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">R. M. Ferrer and J. D. Rhodes. "A Linear Source Approximation Scheme for the Method of Charac- teristics." Nuclear Science and Engineering, volume 182, pp. 151-165 (2017).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,302.38,444.28,9.68;10,95.71,314.33,328.69,9.68" xml:id="b4"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,371.22,302.38,168.78,9.68;10,95.71,314.33,285.71,9.68">Three-Dimensional Full-Core BEAVRS Using OpenMOC with Transport Equivalence</title> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">G</forename> | |
| <surname>Giudicelli</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0001-9714-4382</idno> | |
| </author> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">B</forename> | |
| <surname>Forget</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">K</forename> | |
| <surname>Smith</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1080/00295639.2024.2321661</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j">Nuclear Science and Engineering</title> | |
| <title level="j" type="abbrev">Nuclear Science and Engineering</title> | |
| <idno type="ISSN">0029-5639</idno> | |
| <idno type="ISSNe">1943-748X</idno> | |
| <imprint> | |
| <biblScope unit="page" from="1" to="13"/> | |
| <date type="published" when="2019">2019</date> | |
| <publisher>Informa UK Limited</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">G. Giudicelli, W. Wu, C. Josey, K. Smith, and B. Forget. "Adding a third level of parallelism to OpenMOC, an open source deterministic neutron transport solver." (2019).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,333.71,444.29,9.68;10,95.71,345.62,444.28,9.72;10,95.71,357.62,31.96,9.68" xml:id="b5"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,422.41,333.71,117.59,9.68;10,95.71,345.67,155.26,9.68">Uncollided flux techniques for arbitrary finite element meshes</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Milan</forename> | |
| <surname>Hanuš</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Logan</forename> | |
| <forename type="middle">H</forename> | |
| <surname>Harbour</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Jean</forename> | |
| <forename type="middle">C</forename> | |
| <surname>Ragusa</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Michael</forename> | |
| <forename type="middle">P</forename> | |
| <surname>Adams</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Marvin</forename> | |
| <forename type="middle">L</forename> | |
| <surname>Adams</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.jcp.2019.07.046</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,269.44,345.76,153.43,9.45">Journal of Computational Physics</title> | |
| <title level="j" type="abbrev">Journal of Computational Physics</title> | |
| <idno type="ISSN">0021-9991</idno> | |
| <imprint> | |
| <biblScope unit="volume">398</biblScope> | |
| <biblScope unit="page">108848</biblScope> | |
| <date type="published" when="2019-12">2019</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">M. Hanuš, L. H. Harbour, J. C. Ragusa, M. P. Adams, and M. L. Adams. "Uncollided flux techniques for arbitrary finite element meshes." Journal of Computational Physics, volume 398, p. 108848 (2019).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,377.00,444.28,9.68;10,95.71,388.96,135.77,9.68" xml:id="b6"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,146.60,377.00,209.82,9.68">A stochastic first collision source for TWODANT</title> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">R</forename> | |
| <surname>Baker</surname> | |
| </persName> | |
| </author> | |
| </analytic> | |
| <monogr> | |
| <title level="m" coords="10,386.33,377.09,153.67,9.45;10,95.71,389.05,101.07,9.45">Proc. 8th International Conference on Radiation Shielding</title> | |
| <meeting>8th International Conference on Radiation Shielding</meeting> | |
| <imprint> | |
| <date type="published" when="1994">1994</date> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">R. Baker. "A stochastic first collision source for TWODANT." In Proc. 8th International Conference on Radiation Shielding (1994).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,408.34,444.28,9.68;10,95.71,420.29,444.29,9.68;10,95.71,432.20,136.84,9.72" xml:id="b7"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,452.20,408.34,87.80,9.68;10,95.71,420.29,277.14,9.68">Computation of the uncollided-flux moments with advanced MOC and QP methods</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Matteo</forename> | |
| <surname>Falabino</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0002-5410-0969</idno> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Daniele</forename> | |
| <surname>Sciannandrone</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0002-8821-4278</idno> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Emiliano</forename> | |
| <surname>Masiello</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Jean-François</forename> | |
| <surname>Vidal</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Igor</forename> | |
| <surname>Zmijarevic</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.jcp.2022.111156</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,386.97,420.38,148.53,9.45">Journal of Computational Physics</title> | |
| <title level="j" type="abbrev">Journal of Computational Physics</title> | |
| <idno type="ISSN">0021-9991</idno> | |
| <imprint> | |
| <biblScope unit="volume">460</biblScope> | |
| <biblScope unit="page">111156</biblScope> | |
| <date type="published" when="2022-07">2022</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">M. Falabino, D. Sciannandrone, E. Masiello, J.-F. Vidal, and I. Zmijarevic. "Computation of the uncollided-flux moments with advanced MOC and QP methods." Journal of Computational Physics, volume 460, p. 111156 (2022).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.71,451.63,444.28,9.68;10,95.71,463.54,408.00,9.72" xml:id="b8"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,343.50,451.63,196.50,9.68;10,95.71,463.58,128.14,9.68">ARRC: A random ray neutron transport code for nuclear reactor simulation</title> | |
| <author> | |
| <persName> | |
| <forename type="first">John</forename> | |
| <forename type="middle">R</forename> | |
| <surname>Tramm</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Kord</forename> | |
| <forename type="middle">S</forename> | |
| <surname>Smith</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Benoit</forename> | |
| <surname>Forget</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0003-1459-7672</idno> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Andrew</forename> | |
| <forename type="middle">R</forename> | |
| <surname>Siegel</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.anucene.2017.10.015</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,236.78,463.67,111.51,9.45">Annals of Nuclear Energy</title> | |
| <title level="j" type="abbrev">Annals of Nuclear Energy</title> | |
| <idno type="ISSN">0306-4549</idno> | |
| <imprint> | |
| <biblScope unit="volume">112</biblScope> | |
| <biblScope unit="page" from="693" to="714"/> | |
| <date type="published" when="2018-02">2018</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">J. R. Tramm, K. S. Smith, B. Forget, and A. R. Siegel. "ARRC: A random ray neutron transport code for nuclear reactor simulation." Annals of Nuclear Energy, volume 112, pp. 693-714 (2018).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.72,482.96,444.28,9.68;10,95.71,494.87,444.29,9.72;10,95.71,506.87,31.96,9.68" xml:id="b9"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,365.64,482.96,174.36,9.68;10,95.71,494.92,188.92,9.68">SCONE: A Student-Oriented Modifiable Monte Carlo Particle Transport Framework</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Mikolaj</forename> | |
| <forename type="middle">Adam</forename> | |
| <surname>Kowalski</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0003-3940-7767</idno> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Paul</forename> | |
| <surname>Cosgrove</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0003-2829-6299</idno> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Jakob</forename> | |
| <surname>Broman</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Eugene</forename> | |
| <surname>Shwageraus</surname> | |
| </persName> | |
| <idno type="ORCID">0000-0002-7309-4920</idno> | |
| </author> | |
| <idno type="DOI">10.3390/jne2010006</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,300.09,495.00,139.42,9.45">Journal of Nuclear Engineering</title> | |
| <title level="j" type="abbrev">JNE</title> | |
| <idno type="ISSNe">2673-4362</idno> | |
| <imprint> | |
| <biblScope unit="volume">2</biblScope> | |
| <biblScope unit="issue">1</biblScope> | |
| <biblScope unit="page" from="57" to="64"/> | |
| <date type="published" when="2021-03-08">2021</date> | |
| <publisher>MDPI AG</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">M. A. Kowalski, P. Cosgrove, J. Broman, and E. Shwageraus. "SCONE: A Student-Oriented Modifiable Monte Carlo Particle Transport Framework." Journal of Nuclear Engineering, volume 2, pp. 57-64 (2021).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.72,526.25,444.29,9.68;10,95.71,538.16,179.23,9.72" xml:id="b10"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,230.12,526.25,217.23,9.68">The OpenMC Monte Carlo particle transport code</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Paul</forename> | |
| <forename type="middle">K</forename> | |
| <surname>Romano</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Benoit</forename> | |
| <surname>Forget</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.anucene.2012.06.040</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,460.69,526.34,79.31,9.45;10,95.71,538.29,29.29,9.45">Annals of Nuclear Energy</title> | |
| <title level="j" type="abbrev">Annals of Nuclear Energy</title> | |
| <idno type="ISSN">0306-4549</idno> | |
| <imprint> | |
| <biblScope unit="volume">51</biblScope> | |
| <biblScope unit="page" from="274" to="281"/> | |
| <date type="published" when="2013-01">2013</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">P. K. Romano and B. Forget. "The OpenMC Monte Carlo particle transport code." Annals of Nuclear Energy, volume 51, pp. 274-281 (2013).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.72,557.58,444.28,9.68;10,95.71,569.54,352.26,9.68" xml:id="b11"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,292.49,557.58,247.51,9.68;10,95.71,569.54,172.89,9.68">3D radiation transport benchmark problems and results for simple geometries with void region</title> | |
| <author> | |
| <persName> | |
| <forename type="first">Keisuke</forename> | |
| <surname>Kobayashi</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Naoki</forename> | |
| <surname>Sugimura</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName> | |
| <forename type="first">Yasunobu</forename> | |
| <surname>Nagaya</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/s0149-1970(01)00007-5</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j">Progress in Nuclear Energy</title> | |
| <title level="j" type="abbrev">Progress in Nuclear Energy</title> | |
| <idno type="ISSN">0149-1970</idno> | |
| <imprint> | |
| <biblScope unit="volume">39</biblScope> | |
| <biblScope unit="issue">2</biblScope> | |
| <biblScope unit="page" from="119" to="144"/> | |
| <date type="published" when="2000">2000</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| <respStmt> | |
| <orgName>OECD NEA</orgName> | |
| </respStmt> | |
| </monogr> | |
| <note type="report_type">Technical report</note> | |
| <note type="raw_reference">K. Kobayashi, N. Sugimura, and Y. Nagaya. "3-D Radiation Transport benchmark problems and results for simple geometries with void regions." Technical report, OECD NEA (2000).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.72,588.92,444.29,9.68;10,95.71,600.83,444.29,9.72;10,95.71,612.83,73.05,9.68" xml:id="b12"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,159.13,588.92,380.87,9.68;10,95.71,600.87,236.09,9.68">Performance of Woodcock delta-tracking in lattice physics applications using the Serpent Monte Carlo reactor physics burnup calculation code</title> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">Jaakko</forename> | |
| <surname>Leppänen</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.anucene.2010.01.011</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,347.84,600.96,114.51,9.45">Annals of Nuclear Energy</title> | |
| <title level="j" type="abbrev">Annals of Nuclear Energy</title> | |
| <idno type="ISSN">0306-4549</idno> | |
| <imprint> | |
| <biblScope unit="volume">37</biblScope> | |
| <biblScope unit="issue">5</biblScope> | |
| <biblScope unit="page" from="715" to="722"/> | |
| <date type="published" when="2010-05">2010</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">J. Leppänen. "Performance of Woodcock delta-tracking in lattice physics applications using the Serpent Monte Carlo reactor physics burnup calculation code." Annals of Nuclear Energy, volume 37, pp. 715-722 (2010).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.72,632.21,444.28,9.68;10,95.71,644.12,403.90,9.72" xml:id="b13"> | |
| <analytic> | |
| <title level="a" type="main" coords="10,164.71,632.21,375.28,9.68;10,95.71,644.16,123.59,9.68">On the use of delta-tracking and the collision flux estimator in the Serpent 2 Monte Carlo particle transport code</title> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">Jaakko</forename> | |
| <surname>Leppänen</surname> | |
| </persName> | |
| </author> | |
| <idno type="DOI">10.1016/j.anucene.2017.03.006</idno> | |
| </analytic> | |
| <monogr> | |
| <title level="j" coords="10,232.68,644.25,111.51,9.45">Annals of Nuclear Energy</title> | |
| <title level="j" type="abbrev">Annals of Nuclear Energy</title> | |
| <idno type="ISSN">0306-4549</idno> | |
| <imprint> | |
| <biblScope unit="volume">105</biblScope> | |
| <biblScope unit="page" from="161" to="167"/> | |
| <date type="published" when="2017-07">2017</date> | |
| <publisher>Elsevier BV</publisher> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">J. Leppänen. "On the use of delta-tracking and the collision flux estimator in the Serpent 2 Monte Carlo particle transport code." Annals of Nuclear Energy, volume 105, pp. 161-167 (2017).</note> | |
| </biblStruct> | |
| <biblStruct coords="10,95.72,663.54,444.28,9.68;10,95.71,675.50,387.79,9.68" xml:id="b14"> | |
| <monogr> | |
| <title level="m" type="main" coords="10,345.71,663.54,194.28,9.68;10,95.71,675.50,262.20,9.68">Application of the Random Ray Method to Global Variance Reduction in Radiation Shielding Problems</title> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">V</forename> | |
| <surname>Raffuzzi</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">P</forename> | |
| <surname>Cosgrove</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">P</forename> | |
| <surname>Shriwise</surname> | |
| </persName> | |
| </author> | |
| <author> | |
| <persName coords=""> | |
| <forename type="first">J</forename> | |
| <surname>Tramm</surname> | |
| </persName> | |
| </author> | |
| <idno>PHYSOR 2024</idno> | |
| <imprint> | |
| <date type="published" when="2024">2024</date> | |
| </imprint> | |
| </monogr> | |
| <note type="raw_reference">V. Raffuzzi, P. Cosgrove, P. Shriwise, and J. Tramm. "Application of the Random Ray Method to Global Variance Reduction in Radiation Shielding Problems." In PHYSOR 2024 (2024).</note> | |
| </biblStruct> | |
| </listBibl> | |
| </div> | |
| </back> | |
| </text> | |
| </TEI> |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment