Neutral Particle Transport Equation Research Articles

Asymptotic Expansion of the Angular Flux Applied to Discrete-Ordinates Source Iterations in Lattice Depletion Calculations

This work presents an asymptotic method based on angular flux expansion in a Neumann series. The technique is aimed at effective reduction of the memory imprint of numerical methods based on collision probabilities (CPs). The asymptotic method has been implemented in the heterogeneous Cartesian cells of the integro-differential transport solver (IDT). The IDT solves the neutral-particle transport equation by discrete ordinates combined with angular-dependent CP matrices. In lattice depletion calculations, because of the change of isotopic concentration along the burnup, methods based on CP discretization, such as current-coupling CP or the one presented in this paper, would require construction and storage of a set of CP coefficients for any depleted pin cell. When the number of media grows, the performances of the solver are bounded by the memory pressure caused by the growth of coefficients. Application of the asymptotic technique, presented in this paper, transforms by two user’s parameters the memory-bound solver in a compute-bound application, where the principal workload is transferred from coefficients to source iterations. In this work, a theoretical study of the method is presented together with two applications to two-dimensional assembly simulations. The effects on self-shielded and depleted materials are highlighted. Preliminary results show an encouraging reduction of memory occupation by a factor 10 without any significant loss of accuracy.

On a Response Matrix Solver for Slab-Geometry Neutral Particle Transport Problems in the Discrete Ordinates and Energy Multigroup Formulations Considering Non-Uniform Interior Sources

We present in this work an extension of the Response Matrix (RM) method for the numerical solution of slab-geometry neutral particle transport equation in the discrete ordinates (S) and energy multigroup formulations considering non-uniform sources. By using the term non-uniform we mean that the particle source is not spatially uniform inside the regions that compose the domain. The extended RM method, differently from the conventional RM method, is based on the solution of “source-independent” auxiliary problems (Green’s functions). The solution of these auxiliary problems is used in conjunction with the given non-uniform source to generate the sweeping matrices for the extended RM method. Numerical results with respect to both uniform and non-uniform source problems are given to illustrate the efficiency of the offered extended RM method.

The Multi-PN angular discretization method of the neutral-particle transport equation for radiation shielding calculations

The well-known Discrete Ordinates (SN) approximation is little effective in multidimensional radiation shielding calculations. In the presence of geometrically large and low-scattering media, discrete-ordinates solutions are generally affected by the ray effect and an adequate refinement of the angular discretization is sometimes prohibitive in terms of calculation cost. The Multi-PN method (MPN) is an angular discretization technique leading to SN-like transport equations but less affected by ray effect. It relies on a solid angle expansion of the interface fluxes, allowing for a local coupling between different angular degrees of freedom within the same solid angle. This coupling enables to solve the equation by means of a standard sweeping algorithm. Furthermore, flux angular moments are directly integrated without any use of quadrature rules. The MPN measured error convergence rate with respect to the number of angular degrees of freedom is significantly larger than the one measured with SN.

An improved spectral approach for solving the nonclassical neutral particle transport equation

An improved modification of the Spectral Approach (SA) used for approximating the nonclassical neutral particle transport equation is described in this work. The term “Spectral” is used to indicate that the nonclassical angular flux is approximated as an expansion in terms of spectral basis functions. In the SA the basis functions are the Laguerre polynomials. The main focus of the modified SA lies on a slight modification of the nonclassical angular flux representation as a truncated Laguerre series. This leads, in some cases, to a considerable decrease of the Laguerre truncation order required to generate accurate solutions, thus improving efficiency. Numerical results are given to illustrate this offered improvement.

Analysis of Corner Balance Nonlinear Diffusion Acceleration and Moment-Based, Spatially Linear Transport Discretizations

The recently developed High-Order, Low-Order scheme for the solution of thermal radiative transfer problems is applied as an acceleration method to the neutral particle transport equation. The resulting Corner Balance Nonlinear Diffusion Acceleration (CB-NDA) is derived, and a stability analysis is performed in conjunction with moment-based, spatially linear discretizations. These spatial discretizations correspond to the lumped Linear Discontinuous (LD) and Linear Characteristic (LC) schemes, which possess the thick diffusion limit. The lumped LD and LC schemes satisfy corner balance equations, which in turn are used to derive the CB-NDA. Two variants of the CB-NDA include the net current and partial current formulations. Numerical results are presented that verify the theoretical predictions and implementation. Theoretical spectral radius from the analysis is verified by comparison to values from the numerical solution of a one-dimensional transport problem. Results indicate similar stability between the CB-NDA–accelerated lumped LD and LC schemes. The net current–based CB-NDA is found to be unstable whereas the partial current formulation remains stable over the range of scattering ratios and optical thicknesses.

High-Order Linear Discontinuous and Diamond Differencing Schemes Along Cyclic Characteristics

We are investigating a new class of linear characteristics schemes along cyclic tracks for solving the transport equation for neutral particles with scattering anisotropy. These algorithms rely on linear discontinuous exact integration and diamond differencing, as implemented with the method of discrete ordinates. These schemes are based on linear discontinuous coefficients that are derived through the application of approximations describing the mesh-averaged spatial flux moments in terms of spatial source moments and of the beginning-of-segment and end-of-segment flux values. The linear discontinuous characteristics (LDC) and quadratic-order diamond differencing (DD1) schemes are inherently conservative. In this technical note, we intend to continue the development of the LDC and DD1 schemes by extending their application to cyclic trackings. This extension will make possible the representation of reflective or general albedo boundary conditions. We will present an improved and much shorter derivation of the LDC and DD1 schemes, compared to a previous presentation. Finally, we will implement the new schemes as Matlab scripts for solving a one-dimensional slab benchmark and in the DRAGON5 lattice code for solving a more representative two-dimensional eight-symmetry pressurized water reactor assembly mock-up.

A solution of the monoenergetic neutral particle transport equation for adjacent half-spaces with anisotropic scattering

We present highly accurate solutions to the neutral particle transport equation in a half-space. While our initial motivation was in response to a recently published solution based on Chandrasekhar's H-function, the presentation to follow has taken on a more comprehensive tone. The solution by H-functions certainly did achieved high accuracy but was limited to isotropic scattering and emission from spatially uniform and linear sources. Moreover, the overly complicated nature of the H-function approach strongly suggests that its extension to anisotropic scattering and general sources is not at all practical. For this reason, an all encompassing theory for the determination of highly precise benchmarks, including anisotropic scattering for a variety of spatial source distributions, is presented for particle transport in a half-space. We illustrate the approach via a collection of cases including tables of 7-place flux benchmarks to guide transport methods developers. The solution presented can be applied to a considerable number of one and two half-space transport problems with variable sources and represents a state-of-the-art benchmark solution.

Analytical adjoint discrete ordinates formulation for monoenergetic slab-geometry source-detector calculations

The adjoint to the neutral-particle transport equation in discrete ordinates formulation is solved by the Analytical Discrete Ordinates (ADO) method. A concise and accurate solution is derived for anisotropic scattering problems in multilayer slab geometry with generalised boundary conditions. The derived adjoint discrete ordinates solution is used to compute the absorption rate of neutral particles for source-detector problems. Numerical results for two typical model problems are given to illustrate the efficiency and accuracy of the offered method.

Least-Squares Finite Element Discretization of the Neutron Transport Equation in Spherical Geometry

The main focus of this paper is the numerical solution of the steady-state, mono-energetic Boltzmann transport equation for neutral particles through mixed material media in a spherically symmetric geometry. Standard solution strategies, like the discrete ordinates method, may lead to nonphysical approximate solutions. In particular, a point source at the center of the sphere yields undesirable ray effects. Posing the problem in spherical coordinates avoids ray effects and other nonphysical numerical artifacts in the simulation process, at the cost of coupling all angles in the PDE setting. In addition, traditional finite element or finite difference techniques for spherical coordinates often yield incorrect scalar flux at the center of the sphere, known as flux dip, and oscillations near steep gradients. In this paper, a least-squares finite element method with adaptive mesh refinement is used to approximate solutions to the nonscattering, one-dimensional neutron transport equation in spherically symmetric geometry. It is shown that the resulting numerical approximations avoid flux dip and oscillations. The least-squares discretization yields a symmetric, positive definite, linear system which shares many characteristics with systems obtained from Galerkin finite element discretization of totally anisotropic elliptic PDEs. In general, standard algebraic multigrid techniques fail to scale on non-grid-aligned anisotropies. In this paper, a new variation of smoothed aggregation is employed and shown to be essentially scalable. The effectiveness of the method is demonstrated on several mixed-media model problems.

Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

A method for applying goal-based adaptive methods to the angular resolution of the neutral particle transport equation is presented. The methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution. The angular resolution is adapted across both the spatial and energy dimensions. The spatial domain is discretised using an inner-element sub-grid scale finite element method. The goal-based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution. The goal-based error estimators require the solution of an adjoint system to determine the importance to the specified functional. The error estimators and the novel methods to calculate them are described. Several examples are presented to demonstrate the effectiveness of the methods. It is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error. The novelty of the work is the use of goal-based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation.

Even- and odd-parity kinetic equations of particle transport. 1: Algebraic and centered forms of the scattering integral

We consider an equivalent formulation of the linear kinetic transport equation for neutral particles (neutrons, photons) as a system of two equations for even and odd parts of the distribution function. The particle scattering integral of even- and odd-parity transport equations is converted into a non-linear algebraic form and into a centered form. In the algebraic form of the integral we clearly identify the net result of two opposite processes, i.e., particle scattering from a beam and into the beam. In the centered form of the integral the principal terms of scattering processes are canceled out. An iterative method is proposed for the solution of the system of even- and odd-parity equations with these forms of the scattering integral. Convergence of iterations is studied for a one-dimensional plane problem.

Goal-based angular adaptivity applied to the spherical harmonics discretisation of the neutral particle transport equation

A variable order spherical harmonics scheme has been described and employed for the solution of the neutral particle transport equation. The scheme is specifically described with application within the inner-element sub-grid scale finite element spatial discretisation. The angular resolution is variable across both the spatial and energy dimensions. That is, the order of the spherical harmonic expansion may differ at each node of the mesh for each energy group. The variable order scheme has been used to develop adaptive methods for the angular resolution of the particle transport phase-space. Two types of adaptive method have been developed and applied to examples. The first is regular adaptivity, in which the error in the solution over the entire domain is minimised. The second is goal-based adaptivity, in which the error in a specified functional is minimised. The methods were applied to fixed source and eigenvalue examples. Both methods demonstrate an improved accuracy for a given number of degrees of freedom in the angular discretisation.

High-Order Diamond Differencing along Finite Characteristics

We are investigating a new class of linear characteristics schemes along finite-length tracks for solving the transport equation for neutral particles with scattering anisotropy. These algorithms are based on diamond differencing, as implemented with the method of discrete ordinates. The quadratic-order diamond-differencing (DD1) scheme is based on linear discontinuous coefficients that are derived through the application of approximations describing the mesh-averaged spatial flux moments in terms of spatial source moments and of the beginning- and end-of-segment flux values. This DD1 linear characteristics scheme is inherently conservative. This approach is an improvement relative to other linear characteristics schemes because no information needs to be collected on internal surfaces. Consequently, the DD1 scheme is compatible with existing tracking files for the collision-probability method. The proposed scheme is verified in one-dimensional slab geometry where it is found to be equivalent to a discrete ordinates solution and on simple two-dimensional benchmarks made of regular squares or hexagons.

A two-mesh adaptive mesh refinement technique for [formula omitted] neutral-particle transport using a higher-order DGFEM

We present a two-mesh Adaptive Mesh Refinement technique for the S N neutral-particle transport equation in 2D. A high-order (up to order 4) Discontinuous Galerkin (DG) finite element discretization is employed with standard upwinding. A suitable Diffusion Synthetic Acceleration (DSA) scheme is derived to precondition the transport solves. The DSA scheme, also based on a DG discretization, is derived directly from the discretized high-order DG S N transport equation. Results are presented to demonstrate the superiority of AMR for DSA-accelerated S N transport solves.

Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique

Much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation (SDE). The Random Variable Transformation technique (RVT) is a powerful technique to get the complete solution for the SDE represented by the probability-density function of the solution process. In this paper, the RVT technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one-speed transport equation for neutral particles in a semi-infinite stochastic medium with linear anisotropic scattering. The extinction function of the medium (input stochastic process) is assumed to be a continuous random function of position. The probability-density function and hence, the higher order statistical moments of the solution process are presented. Numerical results are given for different distributions of the input stochastic process.

Multigroup caseology in 1D via the Fourier transform

An alternative view of the Fourier transform inversion provides a new solution to the age-old problem of multigroup neutral particle transport equation in one-dimensional plane geometry. Through analytical continuation, the inversion contour shifts from the real line to accommodate pole contributions as discrete matrix eigenfunctions with continuum contributions appear from the branch cut. We recast the solution in terms of continuum singular matrix eigenfunctions to derive a singular eigenfunction expansion. Closure is an immediate consequence of the Fourier transform inversion. Finally, based on knowledge of the matrix weight factors and closure, we construct a singular eigenfunction expansion, incorporating orthogonality. The significance of this work is that a concise and consistent multigroup eigenfunction expansion emerges for anisotropic scattering that is as easily applied as in the one-group case.

TU‐EE‐A1‐01: Validation of a Prototype Deterministic Solver for Photon Beam Dose Calculations On Acquired CT Data in the Presence of Narrow Beams and Heterogeneities

Purpose: To evaluate a deterministic method for solving the neutral and charged particle transport equations in cases where electron disequilibrium is significant. Methods and Materials: A prototype deterministic solver, Acuros®, has been developed and validated for photon beam radiotherapy. Acuros is based on the Attila® radiation transport code, which solves the differential form of the linear Boltzmann transport equation for neutral particles, and the linear Boltzmann‐Fokker‐Plank transport equation for charged particles. The angular and energy dependent photon and electron flux is solved at every spatial unknown in the computational domain, and quantities such as dose‐to‐medium (DM) are obtained by multiplying the energy dependent particle flux by the energy dependent energy deposition cross section for the associated image pixel material. Comparisons were made with the Monte Carlo code EGSnrc (DOSXYZnrc) for a head‐and‐neck case having eight 1.5×1.5 cm2 photon beams with a 6 MV photon spectrum. The Acuros calculation used 123,000 elements. Since four spatial unknowns were solved in each element, this equated to 492,000 spatial degrees of freedom.The DOSXYZnrc calculation used 2.5×2.5×2.5 mm3 voxels. Results: Results were compared at 95,183 image pixels where dose > 5% of Dmax. 99.02% of pixels satisfied the 3%/3mm criteria (94,251 out of 95,183 pixels). Computational times for the Acuros and DOSXYZnrc calculations were approximately 19.6 CPU minutes (2.2 GHz Opteron processor)and 2,890 CPU minutes (0.4% avg. uncertainty in voxels > Dmax/2), respectively. Through a rewritten, radiotherapy specific solver, Acuros computational times were further reduced to 7.45 CPU minutes without affecting accuracy. When only doses in regions greater than 20% of Dmax are of interest, computational times are further reduced to 3.08 CPU minutes. Conclusions: A clinically viable combination of dose calculation speed and accuracy has been achieved using a radiotherapy specific deterministic solver.Research funded in part by NIH grant 1R43 CA105806‐01A1.

The Search for Superconvergence in Spherical Harmonics Approximations

The occurrence of superconvergence in various first-order spherical harmonics approximations of the neutral particle transport equation is being investigated. Superconvergence refers to the added accuracy gained in evaluating the solution of the transport equation at optimally chosen base points of the finite element trial functions. It has been observed that this phenomenon is happening when primal and dual discretizations in space and angle lead to the same numerical result, a property also referred as primal-dual agreement. A systematic search is presented for primal-dual agreement on one-dimensional slab, tube, and spherical geometries and on Cartesian two-dimensional geometries based on complete and simplified Pn approximations. Primal-dual agreement was successfully obtained in all Cartesian geometries but not in tube and spherical geometries, due to the angular redistribution term.

TH‐E‐224C‐02: Performance Assessment of a Deterministic Method Incorporating Coupled Photon‐Electron Transport for Photon Beam Dose Calculations On Acquired CT Data

Purpose: To evaluate the performance of a neutral and charged particle deterministic solution method for external photon beam dose calculations using acquired CT data. Methods and Materials: The Attila® radiation transport code, which solves the differential form of the linear Boltzmann transport equation for neutral particles and the Boltzmann‐Fokker‐Planck transport equation for charged particles, has been adapted for calculating dose distributions from acquired CT images. Comparisons were made with the Monte Carlo code EGSnrc (DOSXYZnrc) for a sample prostate treatment consisting of 8 10×10 cm2 open beams with a realistic 6 MV photon spectrum. The Attila computational mesh consisted of 125,000 arbitrary tetrahedral elements, of approximately uniform size, which encompassed an imaged torso region. CT numbers were mapped to the Attila tetrahedral elements using one of four materials: air, tissue, adipose tissue, and bone, each having four discrete densities. Attila dose was determined at the image resolution by extracting the energy dependent particle flux at each 1×1×2.5 mm3 pixel from Attila's calculated linear discontinuous finite element spatial representation, and multiplying by the energy dependent dose deposition response for that pixel material and density.The DOSXYZnrc calculation used 2.5×2.5×2.5 mm3 voxels, and was run until an average statistical uncertainty of 0.4% was achieved for voxels in the target region. Results: Computational times Attila and DOSXYZnrc were 36 CPU minutes (2.4 GHz Opteron processor) and approximately 8,500 CPU minutes, respectively. Employing a deterministic electron cut‐off below 300 keV reduced the Attila computational time to 22 CPU minutes. Agreement between both codes was excellent in both high and low dose regions, with differences less than 2%/2mm for greater than 95% of points on a 2‐D plane through the entire torso. Conclusions: A general purpose deterministic solver was successfully applied for dose calculations using CT image data.Research funded by NIH grant 1R43 CA105806‐01A1.

A Diffusion Synthetic Acceleration Method for Block Adaptive Mesh Refinement

A multidimensional block-based adaptive mesh refinement (BAMR) method for the neutral particle transport equation with diamond and linear discontinuous spatial differencing was developed several years ago. This method was implemented in the PARallel TIme-dependent SN (PARTISN) deterministic transport code. However, the only source acceleration method available with BAMR was transport synthetic acceleration. Although the block-based adaptive mesh is orthogonal, the individual mesh cells may not be simply connected. Because of this lack of simple connectivity, development of a fully consistent diffusion synthetic acceleration (DSA) method has not been possible. This paper describes the development of a DSA method based upon an additive correction to the scalar flux iterate after a transport sweep. This DSA equation is differenced using a vertex-centered diffusion discretization that is diamond-like and may be characterized as “partially” consistent. It does not appear algebraically possible to derive a diffusion discretization that is fully consistent with diamond transport differencing on AMR meshes. The diffusion matrix is symmetric positive definite, and the DSA method is effective for most applications. This BAMR-DSA solver has been implemented and tested in two dimensions for rectangular (X-Y) and cylindrical (R-Z) geometries. As expected, results confirm that a partially consistent BAMR-DSA method will introduce instabilities for extreme cases (e.g., scattering ratios approaching 1.0 with optically thick cells), but for most realistic problems, e.g., the iron-water shielding problem, the BAMR-DSA method provides an effective acceleration method.