Collision Source Method Research Articles

A First collision source method using spherical harmonics method for arbitrary shape and order finite element mesh

The discrete ordinates (SN) method is a highly effective and accurate angular discretization method for the neutron transport equation, but suffers from the inherent ray effect in low-scattering regions. The first collision source (FCS) method employing the methods without ray effect to obtain the uncollided flux is a common ray effect mitigation technique. In this paper, a novel FCS method using the spherical harmonics (PN) method to calculate the uncollided flux is proposed. The PN for the first collision source (PN-FCS) method possesses distinctive advantages in several aspects: (1) it can be applied in the problem containing reflective boundary conditions that are difficult to handle with traditional FCS method, (2) it uses the discontinuous Galerkin method to perform spatial discretization, and thus can be easily applied in arbitrary shape and order finite element meshes, (3) it can be effectively calculated in parallel on distributed meshes, (4) the accuracy and computational cost can be flexibly adjusted. Besides, the filter technique is adopted to improve the performance of the PN method in stream-dominated regions. The PN-FCS method has been implemented in two- and three-dimensional neutron transport code and tested by various testing problems. Numerical results show that the PN-FCS method can effectively mitigate the ray effect.

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L2-stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

Coupling of first collision source method with Lattice Boltzmann Method for the solution of neutron transport equation

Despite significant progress in the solution of Neutron Transport Equation (NTE) using Lattice Boltzmann Method (LBM), it suffers from the ray effect a problem similar to SN methods. The ray effect is prominent in the problems where source is localized and medium has low scattering cross-section. Due to ray effect spatial distortion in the scalar flux and angular flux is observed. This effect arises because NTE is solved in discrete directions, allowing neutrons to travel only in a few directions. Therefore, first collision source (FCS) method is applied for SN method. Presently, FCS method is applied to LBM and result are obtained for two-dimensional source problems. It is observed that by applying FCS to LBM, ray effect is mitigated. Compared to SN method, LBM approach is relatively simple in implementation and can be effectively used. Therefore, it has potential to provide easy and convenient solution of neutron transport equation.

Ray effects mitigation using the Monte Carlo first collision source method and application to the Kobayashi benchmark problems

The problem of ray effects, arising in the calculation of local source problems with low scattering media, have been perplexing the discrete ordinates (SN) method. Conventionally, the first collision source technique based on deterministic methods is used to remedy ray effects. In this paper, we investigate a technique based on Monte Carlo method to calculate first collision source distributions and consequently mitigate the ray effects. The Monte Carlo first collision source (MC-FCS) method has been implemented into the three-dimensional (3D) parallel SN transport code Marvin. The 3D Kobayashi benchmark problems were used to systematically verify effectiveness of the MC-FCS solver and to evaluate its computational efficiency. Numerical results show that: the MC-FCS solver has been correctly implemented; the MC-FCS method is highly effective in remedying ray effects; and computational efficiency of MC-FCS is acceptable, especially considering the great improvement in solution accuracy it can bring.

Coupling of First Collision Source Method with Lattice Boltzmann Method for Solution of Neutron Transport Equation

Coupling of First Collision Source Method with Lattice Boltzmann Method for Solution of Neutron Transport Equation

AETIUS solutions for Kobayashi 3D benchmarks with the first collision source method on the volume source and unstructured tetrahedral mesh

The discrete ordinates method (also called the SN method) has been widely used in radiation transport calculations. However, the ray effect and complicated geometry modeling are major shortcomings of the SN calculation compared to the Monte Carlo method.AETIUS is an in-house 3D neutron/photon transport code. It uses the first collision source method on a point or volume source and an unstructured tetrahedral mesh to overcome the two shortcomings mentioned above.A three-dimensional un-collided flux and first collision source code, FNSUNCL3, was previously developed for TORT in three-dimensional X-Y-Z geometry and its benchmark results on Kobayashi problems showed good agreement with the reference. We adapted the methodology used in FNSUNCL3 by modifying the volume source to several pseudo-point sources and summing the contributions from all pseudo-point sources.However, the AETIUS code uses unstructured tetrahedral mesh such that we can apply the first collision source method to any shape of volume sources.The AETIUS results for the original Kobayashi Benchmark Problems and for the Modified Kobayashi Benchmark Problem 2, are shown here to demonstrate the efficacy of using unstructured tetrahedral mesh. These new results show good agreement with reference results.

The adaptive collision source method for discrete ordinates radiation transport

A novel collision source method has been developed to solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. The angular adaptation method is unique in that the flux from each scattering source iteration is obtained, with potentially a different quadrature order used for each. Traditionally, the flux from every iteration is combined, with the same quadrature applied to the combined flux. Since the scattering process tends to distribute the radiation more evenly over angles (i.e., make it more isotropic), the quadrature requirements generally decrease with each iteration. This method allows for an optimal use of processing power, by using a high order quadrature for the first iterations that need it, before shifting to lower order quadratures for the remaining iterations. This is essentially an extension of the first collision source method, and is referred to as the adaptive collision source (ACS) method. The ACS methodology has been implemented in the 3-D, parallel, multigroup discrete ordinates code TITAN. This code was tested on a several simple and complex fixed-source problems. The ACS implementation in TITAN has shown a reduction in computation time by a factor of 1.5–4 on the fixed-source test problems, for the same desired level of accuracy, as compared to the standard TITAN code.

ARES: A Parallel Discrete Ordinates Transport Code for Radiation Shielding Applications and Reactor Physics Analysis

ARES is a multidimensional parallel discrete ordinates particle transport code with arbitrary order anisotropic scattering. It can be applied to a wide variety of radiation shielding calculations and reactor physics analysis. ARES uses state-of-the-art solution methods to obtain accurate solutions to the linear Boltzmann transport equation. A multigroup discretization is applied in energy. The code allows multiple spatial discretization schemes and solution methodologies. ARES currently provides diamond difference with or without linear-zero flux fixup, theta weighted, directional theta weighted, exponential directional weighted, and linear discontinuous finite element spatial differencing schemes. Discrete ordinates differencing in angle and spherical harmonics expansion of the scattering source are adopted. First collision source method is used to eliminate or mitigate the ray effects. Traditional source iteration and Krylov iterative method preconditioned with diffusion synthetic acceleration are applied to solve the linear system of equations. ARES uses the Koch-Baker-Alcouffe parallel sweep algorithm to obtain high parallel efficiency. Verification and validation for the ARES transport code system have been done by lots of benchmarks. In this paper, ARES solutions to the HBR-2 benchmark and C5G7 benchmarks are in excellent agreement with published results. Numerical results are presented which demonstrate the accuracy and efficiency of these methods.

Application of the first collision source method to CSNS target station shielding calculation**Supported by Major National S&T Specific Program of Large Advanced Pressurized Water Reactor Nuclear Power Plant (2011ZX06004-007), National Natural Science Foundation of China (11505059, 11575061), and the Fundamental Research Funds for the Central Universities (13QN34).

Ray effects are an inherent problem of the discrete ordinates method. RAY3D, a functional module of ARES, which is a discrete ordinates code system, employs a semi-analytic first collision source method to mitigate ray effects. This method decomposes the flux into uncollided and collided components, and then calculates them with an analytical method and discrete ordinates method respectively. In this article, RAY3D is validated by the Kobayashi benchmarks and applied to the neutron beamline shielding problem of China Spallation Neutron Source (CSNS) target station. The numerical results of the Kobayashi benchmarks indicate that the solutions of DONTRAN3D with RAY3D agree well with the Monte Carlo solutions. The dose rate at the end of the neutron beamline is less than 10.83 μSv/h in the CSNS target station neutron beamline shutter model. RAY3D can effectively mitigate the ray effects and obtain relatively reasonable results.

ACS Algorithm in Discrete Ordinates for Pressure Vessel Dosimetry

The Adaptive Collision Source (ACS) method can solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. This is similar to, and essentially an extension of, the first collision source method. Previously, the ACS methodology has been implemented into the TITAN discrete ordinates code, and has shown speedups of 2-4 on a simple test problem, with very little loss of accuracy (within a provided adaptive tolerance). This work examines the use of the ACS method for a more realistic problem: pressure vessel dosimetry with the VENUS-2 MOX-fuelled reactor dosimetry benchmark. The ACS method proved to be able to obtain accurate results while being approximately twice as efficient as using a constant quadrature in a standard source iteration scheme.

MultiTrans SP3 code in coupled photon–electron transport problems

Detailed modeling of photon and electron transport is advantageous in radiotherapy and dosimetry. MultiTrans radiation transport code is based on tree multigrid technique and SP3 approximation. The standard SP3 approximation has been extended into a form that takes into account anisotropic group-to-group scattering. A first collision source method has been implemented. Applicability range of standard and extended SP3 approximations in coupled photon–electron transport problems is examined by comparing calculated doses with deterministic MultiTrans and stochastic EGS4 radiation transport codes.

Partisn calculations of 3D radiation transport benchmarks for simple geometries with void regions

We have solved the 3D radiation transport benachmark problems proposed by Prof Kobayashi using the transport code PARTISN. We have used different angular discretizations represented by the Sn method of various orders of n to investigate the resultant sensitivity of the solution. In addition, we have employed our first collision source method as incorporated in the code to mitigate the substantial ray effects present. We also include the computation time needed for these problems on our SGI Origin/2000 platform.

An Extended First Collision Source Method for Electron Beam Source Problems

Beam source problems are difficult to treat numerically because of the associated singularities in angle and space. For electrons, conventional first collision source techniques offer little help because the cross sections are so large and anisotropic that the first collision source and original source are not very different. By extending the definition of the uncollided flux to include particles that have not deviated significantly from the original beam direction, an extended first collision source is obtained that is smooth enough for use in S[sub N] codes. Through the use of effective cross sections, the extended first collision source is determined using standard first collision source techniques. The effective cross sections model electron transport with a reduced number of collisions, but larger deflections per collision. These cross sections are generated using a brute-force S[sub N] solution of the space-independent Spencer-Lewis equation on a restricted cone of directions, centered about the beam direction. Several sample calculations are given in this paper.

A First-Collision Source Method That Satisfies DiscreteSnTransport Balance

The standard first collision source method is modified to include the angle-integrated, mesh cell balance equation. This modification produces a method that is fully consistent with our normal S/sub N/ balance which is essential for diagnostic purposes. The method is also second order in mesh refinements. Illustrative results are also presented in the paper. 3 refs., 5 figs., 1 tab.