Haut PDF MINARET, A DETERMINISTIC NEUTRON TRANSPORT SOLVER FOR NUCLEAR CORE CALCULATIONS

MINARET, A DETERMINISTIC NEUTRON TRANSPORT SOLVER FOR NUCLEAR CORE CALCULATIONS

MINARET, A DETERMINISTIC NEUTRON TRANSPORT SOLVER FOR NUCLEAR CORE CALCULATIONS

CEA - Centre de Saclay DEN/DM2S/SERMA/LLPR,91191 Gif sur Yvette cedex jean-yves.moller@cea.fr; jean-jacques.lautard@cea.fr ABSTRACT We present here MINARET a deterministic transport solver for nuclear core calculations to solve the steady state Boltzmann equation. The code follows the multi-group formalism to discretize the energy variable. It uses discrete ordinate method to deal with the angular variable and a DGFEM to solve spatially the Boltzmann equation. The mesh is unstructured in 2D and semi-unstructured in 3D (cylindrical). Curved triangles can be used to fit the exact geometry. For the curved elements, two different sets of basis functions can be used. Transport solver is accelerated with a DSA method. Diffusion and SPN calculations are made possible by skipping the transport sweep in the source iteration. The transport calculations are parallelized with respect to the angular directions. Numerical results are presented for simple geometries and for the C5G7 Benchmark, JHR reactor and the ESFR (in 2D and 3D). Straight and curved finite element results are compared.
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MINARET or the quest towards the use of time-dependent neutron transport solvers for nuclear core calculations on a regular basis

MINARET or the quest towards the use of time-dependent neutron transport solvers for nuclear core calculations on a regular basis

TIME-DEPENDENT NEUTRON TRANSPORT SOLVERS FOR NUCLEAR CORE CALCULATIONS ON A REGULAR BASIS J.-J. LAUTARD † ‡ , Y. MADAY § ¶k‡ , AND O. MULA † ‡ § Abstract. The present paper deals with the resolution of the time-dependent neutron transport equation that is involved in the field of nuclear safety studies. Through the presentation of the newly implemented kinetic module in the MINARET solver [24] (developed at CEA in the framework of the APOLLO3 R project), we aim first of all at presenting a brief and comprehensive overview of the most widespread resolution techniques employed nowadays in neutron transport industrial codes. Given that the main obstacle in the use of this type of accurate solver on a regular basis relies in the long computing times, MINARET has been used in the present work as a support to rigorously quantify the efficiency of the most common sequential and parallel acceleration techniques that are currently used in this field. An important part of the paper will be devoted to study the performances of an acceleration method that has never been considered before in the resolution of this equation, which is the parallelization of the time variable. In this regard, the parareal in time algorithm (a domain decomposition method for the time variable, [20]) has been implemented to explore its potentialities in this particular application.
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Development of a Hybrid Deterministic-Stochastic Method for Full Core Neutronics

Development of a Hybrid Deterministic-Stochastic Method for Full Core Neutronics

The hybrid method is then introduced. The approach commences by a stochastic simulation to produce neutron flux tallies in few regions or coarse mesh tallies in addition to neutron cross sections. The cross sections are employed in a deterministic lattice calculation to obtain the dominant modes of the transport equation. Once the dominant modes are calculated, they are combined with the tallies scored by the MC solver in order to estimate the amplitudes of the modes and perform flux synthesis and obtain a complete flux distribution. A feasibility study is included to evaluate the accuracy of reconstructing a MC solution using the dominant modes of the transport equation and flux mapping. In the feasibility study, the possibility of reconstructing the neutron flux distribution from tallies in few regions is evaluated. Based on the results produced, the study confirms that the hybrid approach is capable of reconstructing a MC flux distribution from flux tallies in few region. Hence, a considerable reduction in the computational expense is achievable. For better performance, the hybrid method is developed to reconstruct a fine mesh solution starting with a coarse mesh MC solution using the dominant modes of the transport equation. The method is applied to a number of 2D and 3D problems and the accuracy and computational performance are evaluated. Results confirm that in a full core simulation, the hybrid approach could be up to 90% faster than conventional MC while maintaining a comparable accuracy. Solutions produced by the hybrid method for the studied examples are compared to MC reference. In most cases, the relative error between the hybrid solution and the reference is below 2% while errors up to 5% are observed in few cases. Finally, the effects of fuel burnup where evaluated and it is concluded that the method can cope well with dynamic simulations provided the dominant modes are recalculated at the end of each burnup step.
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Development and validation of uncertainty neutron transport calculations at an industrial scale

Development and validation of uncertainty neutron transport calculations at an industrial scale

Abstract. Evaluating uncertainties on nuclear parameters such as reactivity is a major issue for conception of nuclear reactors. These uncertainties mainly come from the lack of knowledge on nuclear and technological data. Today, the common method used to propagate nuclear data uncertainties is Total Monte Carlo [ 1 ] but this method suffers from a long time calculation. Moreover, it requires as many calculations as uncertainties sought. An other method for the propagation of the nuclear data uncertainties consists in using the standard perturbation theory (SPT) to calculate reactivity sensitivity to the desire nuclear data. In such a method, sensitivities are combined with a priori nuclear data covariance matrices such as the COMAC set developed by CEA. The goal of this work is to calculate sensitivites by SPT with the full core diffusion code CRONOS2 for propagation uncertainties at the core level. In this study, COMAC nuclear data uncertainties have been propagated on the BEAVRS benchmark using a two-step APOLLO2/CRONOS2 scheme, where APOLLO2 is the lattice code used to resolve Boltzmann equation within assemblies using a high number of energy groups, and CRONOS2 is the code resolving the 3D full core diffusion equation using only four energy groups. A module implementing the SPT already exists in the APOLLO2 code but computational cost would be too expensive in 3D on the whole core. Consequently, an equivalent procedure has been created in CRONOS2 code to allow full-core uncertainty propagation. The main interest of this procedure is to compute sensitivities on reactivity within a reduced turnaround time for a 3D modeled core, even after fuel depletion. In addition, it allows access to all sensitivites by isotope, reaction and energy group in a single calculation. Reactivity sensitivities calculated by this procedure with four energy groups are compared to reference sensitivities calculated by the iterated fission probability (IFP) method in Monte Carlo code. For the purpose of the tests, dedicated covariance matrix have been created by condensation from 49 to 4 groups of the COMAC matrix. In conclusion, sensitivities calculated by CRONOS2 agree with the sensitivities calculated by the IFP method, which validates the calculation procedure, allowing analysis to be done quickly. In addition, reactivity uncertainty calculated by this method is close to values found for this type of reactor.
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Development and validation of sensitivity and uncertainty calculations using the 3D neutron transport code CRONOS2 at an industrial scale

Development and validation of sensitivity and uncertainty calculations using the 3D neutron transport code CRONOS2 at an industrial scale

between the two codes are also calculated. The second step of this calculation scheme is at core scale using CRONOS2. It uses stored cross sections computed for each assembly by APOLLO2 and it solves flux calculation in diffusion theory with 4 energy groups thanks to MINOS solver. Then, the CRONOS2 procedure developed for sensitivities calculation can be called. Firstly, adjoint flux is calculated. Secondly, CRONOS2 structures which contain cross sections of selected isotopes by reaction are created. They allow calculating production and disappearances operators. Consequently, the scalar product with adjoint flux can be calculated to have production term . In addition, expression of sensitivity can be implemented and have to be breakdown by isotope, by reaction and by energy group, that is equations (32), (33), (34) and (35). Then, the split by energy group is made thanks to unitary sources which are equal to 1 in the wanted energy group and 0 in the other groups. Finally, the numerator scalar product of equations (32), (33), (34) and (35) can be calculated and also sensitivity. Results of this procedure are shown in paragraph 5.3.
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A First Finite Element Solver Shared By Neutron Diffusion, Heat Transfer And Mechanics

A First Finite Element Solver Shared By Neutron Diffusion, Heat Transfer And Mechanics

cyril.patricot@cea.fr, allaire@cmap.polytechnique.fr, olivier.fandeur@cea.fr Abstract - Nuclear reactor core simulations involve several physics, especially in accidental transients. Neutron transport is needed to compute the power distribution. Thermal-hydraulics drives the cooling of the core. Fuel mechanics and heat transfer describe fuel state (including temperature). Mechanics allows to take into account deformations of the core. A lot of works have been done on coupling techniques between these physics, but are usually based on separated codes or solvers. In this paper, we present an alternative approach: the development of a shared solver for the coupled physics. A multiphysics solver is proposed for a time-dependent coupling between neutron di ffusion, heat transfer and linear mechanics. It is based on the finite element method and the Newton algorithm. A very simple application is given and shows the rightness of the developments and the relevance of the solver.
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Condensation and homogenization of cross sections for the deterministic transport codes with Monte Carlo method : Application to the GEN IV fast neutron reactors

Condensation and homogenization of cross sections for the deterministic transport codes with Monte Carlo method : Application to the GEN IV fast neutron reactors

reaction rates. This direction-X current will be applied to the fuel/reflector configuration which has been mentioned previously as a challenging topic for neutronic physicists. 8.3 Application to fuel-reflector calculation Within the framework of the CIRANO program[38] which was carried out in the nuclear experimental facility, MASURCA, there was a core configuration named ZONA2B. This one is a typical fuel-reflector core example. Its fuel sub-assembly is composed of MOX and sodium rodlet, while its reflector sub-assembly consists of steel and sodium rodlet. Previous analysis on ZONA2B core revealed remarkable discrepancies between the deter- ministic calculation results and the Monte-Carlo reference results. In order to figure out this problem, a preliminary research was conducted during Jacquet’s Ph.D. work[77]. A simplified core geometry, made up of a homogeneous fuel medium and a homogeneous reflector medium, was used by Jacquet. In this section,we shall apply the direction-X cur- rent involved IGSC method on the same geometry to verify the applicability of our method. Figure 8.11 illustrates the fuel-reflector geometry, especially the spatial refinement required in deterministic calculations, in particular for ECCO code. The boundary con- ditions are reflective for the fuel side and void for the reflector side. And homogenized multi-group constants are produced for each distinguished medium as shown in Figure 8.11. It was pointed out in Jacquet’s thesis that this explicit geometry modeling should be able to account for the fuel-reflector interface effect. However, unsatisfactory results persisted in the ECCO/BISTRO calculations which are quoted in Table 8.8. A brief re- sum´ e Jacquet’s analysis procedure follows.
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A Deterministic against Monte-Carlo Depletion Calculation Benchmark for JHR Core Configurations

A Deterministic against Monte-Carlo Depletion Calculation Benchmark for JHR Core Configurations

obtaining standard deviations on the effective multiplication factor of less than 90 pcm which could be acceptable in this context because of the calculation time limitations. Deterministic 3D calculations are performed on a hexagonal spatial meshing with the CRONOS2 diffusion code using 6 energy groups [2]. Transport-diffusion equivalence factors are defined and used in order to correct the impact of the diffusion approximation and the impact due to the following simplifications: collapsing, homogenization, and approximations in the CRONOS2 calculations [2]. The CRONOS2 depletion calculations are performed using one energy group and the depletion chain only considers a restricted number of isotopes. In CRONOS2, the heavy nuclide content in each cell is estimated by data interpolation from an APOLLO2 infinite lattice calculation for the corresponding burnup value of the considered mesh cell. Only the 10 B, 135 Xe and the 149 Sm isotopic contents are calculated by CRONOS2. As for the TRIPOLI-4 ® D calculation, nuclear data come from the European nuclear data library JEFF3.1.1 [5].
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Validation of an Isotope Evolution Model for Apollo3 Calculations in SFR Core

Validation of an Isotope Evolution Model for Apollo3 Calculations in SFR Core

The core step performs full core calculations with a coarser energetic mesh and spatial dis- cretization. Proper techniques are used to deal with 3D reactor geometries and apply ther- modynamical and kinetic models in order to simulate operational and accidental conditions. As it is possible to note, core calculations are the result of a synergistic effort of more than one code. Along with lattice and core codes, it is necessary to take into account all the codes which elaborate the nuclear data. All these codes, together, compose a platform. In France, the platform used by CEA, EDF and AREVA is composed by: a code that treats nuclear data and estimates uncertainties: CONRAD [7]; a code that evaluates proper input data libraries for deterministic and stochastic codes: GALILEE [8]; a reference Monte Carlo code: TRIPOLI4 [9]; a deterministic lattice code used principally for thermal and epithermal reac- tors, APOLLO2, and its core counterpart CRONOS2 ; a deterministic lattice code used for fast reactors, ECCO, and its core counterpart ERANOS [10]. As it will be discussed later, during the fuel cycle, inside and outside the core, material compositions change, because of neutron induced reaction and radioactive decay. A depletion code which describes such evolution is consequently required to complete the platform. DARWIN3 is currently used by the French organization presented above [11].
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A 2D/1D algorithm for effective cross sections generation in fast reactor neutronic transport calculations

A 2D/1D algorithm for effective cross sections generation in fast reactor neutronic transport calculations

Abstract Fast resolution of the Boltzmann transport equation over a nuclear reactor core presup- poses the definition of homogenized and energy collapsed cross sections. In modern sodium fast reactors that rely on heterogeneous core designs, anisotropy in the neutrons propagation cannot be neglected so three-dimensional models should be preferred to compute those effec- tive cross sections. In this paper, the 2D/1D approximation is used to avoid computationally expensive 3D calculations while preserving consistent angular representations of the neutron flux. An iterative procedure is defined to solve the 2D/1D equations and produce coarse group homogenized cross sections that account for 3D transport effects. Accuracy of the algorithm is tested on a realistic model of the ASTRID core showing very good results against Monte Carlo simulations for all neutronic parameters (eigenvalue, sodium void worth and fission map distribution).
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A First Finite Element Solver Shared by Neutron Diffusion,Heat Transfer and Mechanics

A First Finite Element Solver Shared by Neutron Diffusion,Heat Transfer and Mechanics

cyril.patricot@cea.fr, allaire@cmap.polytechnique.fr, olivier.fandeur@cea.fr INTRODUCTION Nuclear reactor cores can be deformed by thermal expan- sion, irradiation effects or during particular accidental tran- sients. These deformations are likely to impact neutron trans- port in all reactor types. Fast neutron reactors are nevertheless particularly sensible to these e ffects, because of their thermal features (large temperature gradients, potentially strong tem- perature variations in case of accident) and the way neutrons evolve within the core (important role played by leakages in the neutron balance, weak fraction of delayed neutrons). As a consequence, in the context of the development of the fourth generation, methodologies to take into account core distortions in deterministic neutron transport codes have been developed. These tools can aim at providing linear feedback coe ffi- cients [1, 2], or to compute the neutron flux. In this last case, the mesh used can itself be deformed [3, 4] or not [5, 6] (a geometry projection method is then needed).
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A Dynamic Homogenization Method for Nuclear Reactor Core Calculations

A Dynamic Homogenization Method for Nuclear Reactor Core Calculations

The third class of methods is based on a very different philosophy, because it does not require any off-line calculation. A lattice solver is used as an “on-the-fly” cross- section generator for the core solver, where the boundary conditions are calculated by the low-order operator and updated at each iteration. This approach is definitely more expensive but it can be easily parallelized. The method was firstly called Dy- namic Homogenization (DH), proposed by Mondot and Sanchez in 2003 [79] and tested for simple cases in 1D problems. In their work, the fine information on the environ- ment came directly from the neighboring assemblies and from the core currents and eigenvalue. Each assembly has then different boundary conditions that account for its position in the core and for the isotopic content of the neighboring assemblies: both sources of gradients in a reactor. In 2008 Takeda et al. [80] applied a similar iterative process but, to redefine the RHP at each iteration, they adopted albedo boundary con- ditions determined with the core currents instead of using the outgoing angular flux of the neighbor. It ensues that each lattice calculation is independent of the others because it only depends on core quantities, which entails that the spectrum of the neighboring assemblies cannot be properly taken into account. In 2014 Colameco et al. [81] extended the DH method to a 2D configuration obtaining good agreement against reference transport calculation for a 2x2 cluster of UOx and MOx assemblies with re- flection boundary conditions. The method was also adopted for Pebble Bed Reactors by Grimod and Sanchez in 2015 [82] where the pebbles were depleted with their own fine-group fluxes.
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A domain decomposition method in APOLLO3$^R$ solver, MINARET

A domain decomposition method in APOLLO3$^R$ solver, MINARET

IDT [9] of the APOLLO3 R code. This paper presents the implementation of Domain Decomposition inside the core solver MINARET. It is motivated by an assessment: the "traditional" deterministic calculation scheme in two steps is built on ”severe” approximations, thought to offer a satisfying compromise between precision and calculation time. Yet, nowadays, these approximations become less and less accurate with the growing complexity of reactor core concepts. Bypassing that standard source of approx- imations would both increase the precision and bring deterministic schemes closer to reference calculations. The most natural way to do so consists in computing a full core calculation in fine transport. In this context, the use of Domain Decomposition is essential to overcome the increase in calculation time which goes along with the increase of precision.
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Adjoint-based deviational Monte Carlo methods for phonon transport calculations

Adjoint-based deviational Monte Carlo methods for phonon transport calculations

This paper focuses on the benefits that can be derived by exploiting the observation that many phonon transport problems of scientific and practical interest involve relatively small driving forces (temperature differences) and thus may be treated by the linearized Boltzmann equation. Recent work [ 4 ] has shown that using the linearized Boltzmann equation in the presence of temperature differences of magnitude smaller than 10% of the reference temperature (e.g., temperature differences on the order of ±30 K for a reference temperature of 300 K) results in errors on the order of a few percent. Examples of problems featuring small temperature differences include calculations of the effective thermal conductivity of nanostructures (where, in fact, small temperature differences are required to prevent a nonlinear response), simulation of transient thermoreflectance experiments [ 5 ], and simulations of the thermal behavior of thin films, nanowires, and su- perlattices. This observation was already used in a previous paper [ 5 ] in which a kinetic Monte Carlo type scheme for simulating the Boltzmann equation was developed. In that work, linearity enabled the decoupling of deviational particle trajectories leading to an algorithm that was significantly faster and easier to code while exhibiting no time-step error [ 4 , 5 ].
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Classic nuclear localization signals and a novel nuclear localization motif are required for nuclear transport of porcine parvovirus capsid proteins

Classic nuclear localization signals and a novel nuclear localization motif are required for nuclear transport of porcine parvovirus capsid proteins

ABSTRACT Nuclear targeting of capsid proteins (VPs) is important for genome delivery and precedes assembly in the replication cycle of porcine parvovirus (PPV). Clusters of basic amino acids, corresponding to potential nuclear localization signals (NLS), were found only in the unique region of VP1 (VP1up, for VP1 unique part). Of the five identified basic regions (BR), three were im- portant for nuclear localization of VP1up: BR1 was a classic Pat7 NLS, and the combination of BR4 and BR5 was a classic bipar- tite NLS. These NLS were essential for viral replication. VP2, the major capsid protein, lacked these NLS and contained no region with more than two basic amino acids in proximity. However, three regions of basic clusters were identified in the folded pro- tein, assembled into a trimeric structure. Mutagenesis experiments showed that only one of these three regions was involved in VP2 transport to the nucleus. This structural NLS, termed the nuclear localization motif (NLM), is located inside the assembled capsid and thus can be used to transport trimers to the nucleus in late steps of infection but not for virions in initial infection steps. The two NLS of VP1up are located in the N-terminal part of the protein, externalized from the capsid during endosomal transit, exposing them for nuclear targeting during early steps of infection. Globally, the determinants of nuclear transport of structural proteins of PPV were different from those of closely related parvoviruses.
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Azimuthal anisotropies as stringent test for nuclear transport models

Azimuthal anisotropies as stringent test for nuclear transport models

the detector biases on the extraction of the in-plane anisotropy ratio. Finally, we confront the data to the model calculations for different parametrisations of the nuclear EoS. A. The IQMD model The Quantum Molecular Dynamics (QMD) model [13,17,43,55] is a n-body theory which calculates the time evolution of intermediate energy heavy ion reactions on an event by event basis. Nucleons are represented as gaussian wave functions whose centroids are propagated according to the classical equations of motion and their width parameter L is kept fixed. These wave packets are submitted to two and three body local Skyrme forces plus Yukawa and long range Coulomb potentials. An additional term describing the momentum depen- dence of the nuclear interaction (MDI) is included. The latter, which is computed from the momentum dependence of the optical p-nucleus potential, produces an additional trans- verse deflection of nucleons early in the reaction. The parameters of the density dependent Skyrme and MDI potentials are adjusted to reproduce the ground state properties of the infinite nuclear matter and to fix the choice of the incompressibility K of the EoS. The numerical simulation of a complete event includes the initialisation procedure of projectile and target (see below), the propagation of nucleons through the previous potentials and the nucleon-nucleon interactions via a stochastic scattering term. For each binary collision, the phase space distributions in the final stage of the scattering partners are checked to obey the Pauli principle, otherwise the collision is blocked. At the final stage of the reaction (t = 200f m/c), composite particles are formed if the centroid distances are lower than 3f m. The IQMD (Isospin Quantum Molecular Dynamics) [43] is a QMD version which takes into account isospin degrees of freedom for the cross sections and the Coulomb interaction. This version uses a gaussian width fixed to 4L = 8.66f m 2 for Au nuclei. The initialisation
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Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

that the triangulation of the domain is extracted from a first decomposition of Ω into several horizontal layers of prisms each of them being subsequently cut into three tetrahedrons as explained in Figure 1(left). This enables interpolating the cross sections, as sketched in Figure 1(right) where the immersion of the control rod into the tetrahedron is illustrated. The cross section (represented with color) are taken into account relatively with the volume occupied by control rod. In this case positive values, between 0 and 1, is attributed to the cross section related to the rods, whereas cross section related to fuel are deduced automatically i.e. interpolated using the complement value. It is worthy noting that this procedure is done before finite element matrix assembly. We update cross sections at each time step using interpolation, then we assemble the main matrix and solve the system of (8).
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TRIPOLI-4 and FREYA for Stochastic Analog Neutron Transport. Application to Neutron Multiplicity Counting.

TRIPOLI-4 and FREYA for Stochastic Analog Neutron Transport. Application to Neutron Multiplicity Counting.

Left: multiple 3 He tubes removed for clarity. Right: upper polyethylene plug removed. 2. Developments in TRIPOLI-4 ® for NMC TRIPOLI-4 ® solves the linear Boltzmann equation for neutrons, photons, electrons and positrons with the Monte Carlo method, in any 3-D geometry. The code uses ENDF format continuous-energy cross sections from various international evaluations. It has advanced variance reduction methods to address deep penetration issues and can be run in parallel. TRIPOLI-4 ® is used as a reference code for industrial purposes (fission/fusion) for CEA 1 , EDF 2 and branches of AREVA, as well as an R&D and teaching tool, for radiation protection and shielding, core physics, nuclear criticality safety and nuclear instrumentation.
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A parallel solver for incompressible fuid flows

A parallel solver for incompressible fuid flows

In this paper, we focus on FD discretizations (of second order in time and space) of the three- dimensional NS equations which are solved using the classical prediction-projection method [12, 23, 18, 10]. This method relies on an explicit treatment of the non-linear terms and an implicit treatment of the viscous and pressure terms to ensure stability. As detailed in Section 2, it leads to the solution of 3-D diffusion (velocity) and Poisson (pressure) equations that consti- tute the main computational workload of the simulation. Further, we restrict ourself to spatial discretizations on structured cartesian grids resulting from the tensorization of one-dimensional grid. Although not general, the structured character of these grids greatly facilitates the design and efficient implementation of the algorithms on modern computer architectures, and encom- passes many situations of interest. Even for these simple meshes, 3-D parallel solvers are needed since complex flow simulations and targeted applications commonly require grid resolution corre- sponding to 10 7 − 10 9 unknowns to properly capture the flow dynamics. These large simulations
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Comparison of Monte Carlo methods for adjoint neutron transport

Comparison of Monte Carlo methods for adjoint neutron transport

In several applications emerging in radiation shielding and reactor physics, the adjoint approach might yield faster and more accurate answers than the regular forward Monte Carlo approach [1]. In particular, forward particle transport becomes inefficient when the detector is ‘small’ 1 in the phase space: in the limit case of a point-like detector, almost no particles can contribute to the simulation, the probability that a forward random walk coming from the source will cross the detector phase space being negligibly small. On the contrary, all adjoint random walks will start from the detector, and there will be a higher chance that they will eventually cross the source (unless the source is also very small). The so-called reciprocity theorem ensures that arbitrary observables can be estimated by resorting to either forward or backward Monte Carlo methods, each leading to the same unbiased score, upon inverting the roles of source and detector [3]. The choice of either scheme must be guided by a variance minimization principle, which is problem-dependent. Ideally, zero-variance Monte Carlo schemes might be conceived by weighting the forward random walks by the importance obtained by solving the corresponding adjoint problem [1, 4, 5].
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