A time-dependent massively-parallel Monte Carlo particle transport calculational module (ParticleMC) for inertial confinement fusion (ICF) applications is described. The ParticleMC package is designed with the long-term goal of transporting neutrons, charged particles, and gamma rays created during the simulation of ICF targets and surrounding materials, although currently the package treats neutrons and gamma rays. Neutrons created during thermonuclear burn provide a source of neutrons to the ParticleMC package. Other user-defined sources of particles are also available. The module is used within the context of a hydrodynamics client code, and the particle tracking is performed on the same computational mesh as used in the broader simulation. The module uses domain-decomposition and the MPI message passing interface to achieve parallel scaling for large numbers of computational cells. The Doppler effects of bulk hydrodynamic motion and the thermal effects due to the high temperatures encountered in ICF plasmas are directly included in the simulation. Numerical results for a three-dimensional benchmark test problem are presented in 3D XYZ geometry as a verification of the basic transport capability. In the full paper, additional numerical results including a prototype ICF simulation will be presented

Brantley, P. S.

Stuart, L. M.

UNT Digital Library

UCRL-CONF-226076Monte Carlo Particle Transport Capabilityfor Inertial Confinement FusionApplicationsPatrick S. Brantley, Linda M. StuartNovember 13, 2006M&C + SNA 2007Monterey, CA, United StatesApril 15, 2006 through April 19, 2006Disclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.  Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications (M&C + SNA 2007)Monterey, California, April 15-19, 2007, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2007)MONTE CARLO PARTICLE TRANSPORT CAPABILITY FOR INERTIALCONFINEMENT FUSION APPLICATIONSPatrick S. Brantley and Linda M. StuartLawrence Livermore National LaboratoryP.O. Box 808, L-023Livermore, CA 94551 USAbrantley1@llnl.govABSTRACTA time-dependent massively-parallel Monte Carlo particle transport calculational module (ParticleMC)for inertial confinement fusion (ICF) applications is described. The ParticleMC package is designedwith the long-term goal of transporting neutrons, charged particles, and gamma rays created during thesimulation of ICF targets and surrounding materials, although currently the package treats neutrons andgamma rays. Neutrons created during thermonuclear burn provide a source of neutrons to theParticleMC package. Other user-defined sources of particles are also available. The module is usedwithin the context of a hydrodynamics client code, and the particle tracking is performed on the samecomputational mesh as used in the broader simulation. The module uses domain-decomposition and theMPI message passing interface to achieve parallel scaling for large numbers of computational cells. TheDoppler effects of bulk hydrodynamic motion and the thermal effects due to the high temperaturesencountered in ICF plasmas are directly included in the simulation. Numerical results for athree-dimensional benchmark test problem are presented in 3D XYZ geometry as a verification of thebasic transport capability. In the full paper, additional numerical results including a prototype ICFsimulation will be presented.Key Words: Inertial confinement fusion, Monte Carlo particle transport.1. INTRODUCTIONThe Monte Carlo method has been used for decades to simulate particle transport. The accurate simulationof the transport of neutrons, charged particles, and gamma rays in inertial confinement fusion (ICF) targetsand surrounding materials can aid a correct understanding of target performance as well as design ofdiagnostics. In this paper, we briefly describe a time-dependent massively-parallel Monte Carlo particletransport calculational module (ParticleMC) that has been developed for inertial confinement fusion (ICF)applications.The remainder of this paper is organized as follows. In Section 2, we briefly summarize the numericalalgorithms in the ParticleMC package. We present numerical results for a three-dimensional benchmarktest problem in Section 3. We conclude with a brief discussion in Section 4.2. Monte Carlo AlgorithmsThe ParticleMC package is written using object-oriented techniques in the standard C++ programminglanguage. Extensive use of the concepts of inheritance and templating are utilized throughout the package.The ParticleMC package has, since inception, been designed for massively-parallel three-dimensionalPatrick S. Brantley and Linda M. Stuartsimulations with large numbers of spatial zones in the computational mesh. As a result, the package canoperate on computational meshes that are spatially domain-decomposed. Parallel communication isperformed using the Message Passing Interface (MPI) [1]. Templating on the computational mesh typeresulted in significant code reuse between 3D XYZ and 2D RZ geometry versions of the ParticleMCpackage.The ParticleMC package tracks Monte Carlo particles (currently neutrons and gamma rays, and eventuallylight charged particles) through a computational mesh. Source particles can be generated either by a MonteCarlo simulation of thermonuclear processes [2] or by user-specified external sources. Each Monte Carloparticle maintains a three-dimensional Cartesian representation of its position, for both 3D XYZ meshesand 2D RZ meshes, as well as direction cosines with respect to the Cartesian coordinate system. EachMonte Carlo particle also maintains a type identifier (currently neutron or gamma ray), a spatial zoneidentifier corresponding to its position, its current kinetic energy, an energy group index, a weight valuecorresponding to the number of physical particles the Monte Carlo particle represents, and its currentsimulation time. Finally, each Monte Carlo particle carries an individual random number generator state toensure that it encounters the same random number stream regardless of the number of processors on whichthe simulation is run. The ParticleMC package currently uses the RNG random number generatorlibrary [3] developed at LLNL, and the state size for this random number generator is small. Becausefloating point arithmetic is not commutative, the simulation results are not yet reproducible on differentnumbers of processors.The Monte Carlo particles are tracked through the computational mesh in the laboratory frame. Since thepackage is designed for use in simulations with hydrodynamics, particles are transformed to the fluid frameto access nuclear cross sections and to perform collision kinematics. The ParticleMC package is typicallyused for time-dependent simulations, so Monte Carlo particles must be tracked from the beginning of thetimestep to the end of the timestep (assuming they are not captured or escape the problem boundary withinthe timestep). Monte Carlo particles that reach the end of the timestep are stored in a census list ofparticles. At the beginning of the next timestep, the census list is processed through a census comb toselect, via splitting and Russian Roulette, the desired number of Monte Carlo particles to be followedduring the timestep. The Monte Carlo particle population can decrease due to capture or escape, canincrease over simulation time if few particles are captured or escape during a timestep (e.g. for smalltimestep sizes or for highly-scattering media), and can significantly increase through the application ofvariance reduction techniques. Additional functionality maintains the desired apportioning of the MonteCarlo particles to be tracked between census particles and particles from external sources. At the beginningof a timestep, the position of a particle and the zone identifier it maintains may be inconsistent as a result ofhydrodynamic motion or mesh relaxation. To make these two quantities consistent, we have implementedan idea suggested by Zimmerman [4] that tracks the particle from the center of the zone corresponding toits stored zone index to the position of the particle. This algorithm is efficient and naturally handles anycommunication required as a result of crossing a spatial domain boundary using the existing Monte Carloparticle tracking functionality.Isotopic species sampling, nuclear reaction type sampling, collision kinematics calculations, and outgoingparticle characteristics sampling are performed using the Monte Carlo All Particle Method (MCAPM)library [5]. This library utilizes a continuous value for the neutron kinetic energy and performs continuouscollision kinematics but uses a multigroup representation of the neutron cross sections with typically 175energy groups. Gamma ray cross sections are stored as pointwise data with typically 176 energy points.We have implemented three neutron thermalization models to account for the effects of target thermalJoint International Topical Meeting on Mathematics & Computation andSupercomputing in Nuclear Applications (M&C + SNA 2007), Monterey, CA, 20072/5Monte Carlo Capability for ICFmotion encountered in the high-temperature plasmas typical of ICF calculations. We describe these modelsin order of increasing fidelity and computational expense. The first and simplest “32kT floor” model usesroom temperature microscopic neutron cross sections but enforces a minimum neutron energy of 32kT ,where kT is the zonal ion temperature, after a neutron interaction with the background matter:Eoutn = max(Eoutn ,32kT ). Energy is conserved by subtracting the amount of energy required to increasethe neutron to the zone thermal energy from the energy to be deposited from nuclear reactions to thebackground matter. The second neutron thermalization model was recently implemented in the MCAPMlibrary [5] based on the the THERMAL subroutine of Cullen [6] and assumes that temperature-broadenedmicroscopic neutron cross sections are available (the MCAPM library can provide thesetemperature-dependent cross sections). This model resamples elastic scattering collisions usingthermally-broadened cross sections if the incident neutron energy satisfies Einn < 104 × (32kT ). The thirdneutron thermalization model treats the interaction of a neutron with the hot plasma in two stages. The firststage involves determining the correct distance to collision which can only be accomplished using thethermally-broadened total macroscopic cross section [2, 7]. Once it is determined that a collision occurs,the second stage involves sampling the characteristics of the neutron reaction. The isotopic speciesundergoing the collision is sampled using the thermally-broadened cross sections. Given the sampledisotopic species, the thermal velocity of the nucleus must be sampled favoring energy and direction pairswith larger reaction rates (not simply an isotropic Maxwellian distribution) [2]. We accomplish this usingthe rejection technique recommended in Ref. [2]. After the target thermal velocity is sampled, the collisionkinematics are performed using the MCAPM library collision functions in the frame of the thermal target.For time-dependent, domain-decomposed simulations, statistical error estimates are difficult to obtain viatechniques employed in typical Monte Carlo codes in which particle histories are tracked from beginning toend. As a result, we have chosen to use the technique known as batching [2] in which each Monte Carloparticle has a batch identifier. An estimate of the statistical error in a desired tally can then be obtained bycomputing the standard deviation in the mean of the tally as computed from the particles in each batch.We have currently implemented variance reduction via importance regions coupled with splitting andRussian roulette [8]. The importance values can be specified on a zonal basis if desired. An importantaspect of the implementation of this technique for a Monte Carlo method employed in time-dependentsimulations is to correctly account for the importance values during the application of the census comb.3. NUMERICAL RESULTSIn this section, we present numerical results for the semi-analytic three-dimensional searchlight benchmarkproblem proposed by Kornreich and Ganapol [9] as a verification of the basic Monte Carlo transportcapability. This problem models a monoenergetic and monodirectional beam of neutrons incident on thesurface of a semi-infinite isotropically-scattering half-space at the point (x, y, z) = (0, 0, 0) cm. Theincident beam of neutrons is canted with respect to the free face of the half-space, resulting in athree-dimensional scalar flux distribution. The neutron transport is considered to be monoenergetic. Thesemi-analytic benchmark values for the scalar flux are tabulated every 0.5 mean free paths (mfp) into thehalf-space (along the z axis) in Ref. [9] for several values of the scattering ratio c. We have considered thecase of the incident beam canted with a cosine of µ0 = 0.9 and a scattering ratio of c = 0.9. This problem ischallenging for a Monte Carlo transport simulation, as the scalar flux decreases by over a factor of onehundred for the points of comparison. We solved this problem in a spatial domain of 60 × 60 × 30 mfpusing an orthogonal computational mesh with 14,749 zones domain-decomposed into four domains. Thespatial zones along the z axis (in which the zonal scalar flux values were computed using a pathlengthJoint International Topical Meeting on Mathematics & Computation andSupercomputing in Nuclear Applications (M&C + SNA 2007), Monterey, CA, 20073/5Patrick S. Brantley and Linda M. Stuartestimator [8]) were 0.1 × 0.1 × 0.1 mfp in size. The simulation used 20 million Monte Carlo neutrons and32 particle batches for statistical error estimation. In Table I, we present the benchmark [9] and computedscalar flux values. We also show the percent relative standard deviation (PRSD) in the Monte Carlo scalarflux values as computed using the batching approach as well as the percent relative error (PRE) in theMonte Carlo values as compared to the benchmark values. Good agreement is obtained between the MonteCarlo results and the benchmark values: 40% and 90% of the points agree to within one and two standarddeviations, respectively.Table I. Benchmark and Computed Scalar Flux ValuesScalar Fluxz Benchmark Computed PRSD PRE0.5 5.9281×10−1 5.9967×10−1 0.28 1.161.0 2.2346×10−1 2.2479×10−1 0.48 0.601.5 1.1243×10−1 1.1201×10−1 0.69 -0.372.0 6.3715×10−2 6.3878×10−2 0.91 0.262.5 3.8547×10−2 3.7996×10−2 1.18 -1.433.0 2.4311×10−2 2.4205×10−2 1.21 -0.433.5 1.5784×10−2 1.5468×10−2 1.58 -2.004.0 1.0470×10−2 1.0652×10−2 1.85 1.744.5 7.0614×10−3 6.7934×10−3 2.64 -3.805.0 4.8256×10−3 4.6568×10−3 3.13 -3.504. CONCLUSIONSIn this paper, we briefly described the capabilities of a time-dependent massively-parallel Monte Carloparticle transport capability designed for inertial confinement fusion applications. We presented numericalresults for a three-dimensional benchmark transport problem as a verification of the basic Monte Carlotransport capability. In the full paper, additional numerical results including a prototype ICF simulationwill be presented.In future work, we plan to implement a Monte Carlo charged particle transport capability. In addition, weplan to pursue stategies to improve the computational efficiency of the package.ACKNOWLEDGEMENTSThis work was performed under the auspices of the U.S. Department of Energy by the University ofCalifornia Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. We would like toacknowledge the Monte Carlo particle tracking and parallel communication capabilities initially developedby Nick Gentile upon which the ParticleMC package is built. We also acknowledge many helpfulconversations and much helpful guidance from George Zimmerman.Joint International Topical Meeting on Mathematics & Computation andSupercomputing in Nuclear Applications (M&C + SNA 2007), Monterey, CA, 20074/5Monte Carlo Capability for ICFREFERENCES[1] W. Gropp, E. Lusk, and A. Skjellum, Using MPI: Portable Parallel Programming with theMessage-Passing Interface, The MIT Press, Cambridge, Massachusetts, USA (1999).[2] G.B. Zimmerman, “Monte Carlo Methods in ICF,” Proceedings of Thirteenth InternationalConference on Laser Interactions and Related Plasma Phenomena, Monterey, CA, April 13-18,1997 (1997). See also Lawrence Livermore National Laboratory internal report UCRL-JC-126845(1997).[3] B.R. Beck and E.D. Brooks III, “The RNG Random Number Library,” Lawrence LivermoreNational Laboratory internal report UCRL-MA-141673 (2000).[4] G.B. Zimmerman, private communication (2002).[5] P.S. Brantley, C.A. Hagmann, and J.A. Rathkopf, “MCAPM-C Generator and Collision RoutineDocumentation (Revision 1.2),” Lawrence Livermore National Laboratory internal reportUCRL-MA-141957 (2003).[6] D.E. Cullen, “THERMAL: A Routine Designed to Calculate Neutron Thermal Scattering,”Lawrence Livermore National Laboratory internal report UCRL-ID-120560 (1995).[7] E.H. Canfield, “Monte Carlo Methods for Neutrons in a Hot Gas,” Lawrence Livermore NationalLaboratory internal report UCIR-694 (1973).[8] E.E. Lewis and W.F. Miller, Computational Methods of Neutron Transport, American NuclearSociety, Lagrange Park, Illinois, USA, Ch. 1 (1993).[9] D.E. Kornreich and B.D. Ganapol, “Numerical Evaluation of the Three-Dimensional SearchlightProblem in a Half-Space,” Nucl. Sci. Engr., 127, pp. 317-337 (1997).Joint International Topical Meeting on Mathematics & Computation andSupercomputing in Nuclear Applications (M&C + SNA 2007), Monterey, CA, 20075/5

Monte Carlo Particle Transport Capability for Inertial Confinement Fusion Applications

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Monte Carlo Particle Transport Capability for Inertial Confinement Fusion Applications

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