PN Method Research Articles

A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes

This document presents a new numerical scheme dealing with the Boltzmann transport equation. This scheme is based on the expansion of the angular flux in a truncated spherical harmonics function and the discontinuous finite element method for the spatial variable. The advantage of this scheme lies in the fact that we can deal with unstructured, non-conformal and curved meshes. Indeed, it is possible to deal with distorted regions whose boundary is constituted by edges that can be either line segments or circular arcs or circles. In this document, we detail the derivation of the method for 2D geometries. However, the generalization to 2D extruded geometries is trivial.

A comparison of transport methods for the solution of a problem with shadowing effects in spherical geometry

The PN method, the post-processed PN (ppPN) method, the discrete-ordinates (SN) method and the Monte Carlo method are used to solve the problem of the exterior of a sphere subject to a known incoming angular flux at the surface. This geometrical configuration introduces shadowing effects in the solution that show up as position-dependent angular-flux discontinuities in the angular variable. A brief description of the methods is given, including a prescription on how to normalize the angular flux, the scalar flux and the current in the SN and Monte Carlo codes used (ANISN and MCNP6), in order to allow for a proper comparison with the results from the other methods. Numerical results reported in graphical form for two test cases demonstrate that only the ppPN and Monte Carlo methods are able to reproduce well the profiles of the angular flux and the magnitudes of the angular-flux discontinuities. In addition, numerical results for the scalar flux and current are provided in tabular form for the test cases studied.

The eigenvalues of one-speed neutrons in a slab with forward and backward scattering

The eigenvalue spectrum is studied for one-speed neutrons in a slab with forward and backward scattering. First, the transport equation describing the interaction of neutrons in a system with general geometry is given. Then, the scattering function in transport equation is chosen as the forward-backward-isotropic (FBI) scattering model. The resultant transport equation is solved using the Legendre polynomials expansion (PN method) and the Chebyshev polynomials of second kind expansion (UN method) in neutron angular flux. Then, the PN and UN moments of the equations are obtained using the properties of the Legendre and the Chebyshev polynomials of the second kind. Finally, the eigenvalues for various values of the collision and scattering parameters are calculated using different orders of the presented methods and they are given in the tables for comparison.

On SPN theory

We have generalized Pomraning’s variational derivation of the Simplified PN (SPN) method and apply it to the first-order transport equation with anisotropic scattering and sources without restriction on the parity of N. The result is a new system of coupled linear integral equations for N+1 scalar functions, which are shown to be equivalent to the traditional first-order and second-order (even or odd) SPN equations. The integral operator in these equations is the square root of the Laplacian. We view these integral equations as a natural paradigm for SPN in that they do not introduce the artificial current vectors which add two artificial degrees of freedom (per current) to the traditional first-order SPN equations. Interface conditions are derived by requiring the solution to behave smoothly across the interface between homogeneous regions. Surprisingly, these conditions are identical to the traditional ones obtained assuming a slab-like behavior for the angular flux. In an infinite homogeneous medium the integral SPN equations give the exact PN solution and here we join the work of Ackroyd et al. and Chao. We have also derived a simple recurrence relation to obtain the well-known PDE for the scalar flux. By virtue of the equivalence SPN-PN in an infinite homogeneous medium, this technique gives the scalar-flux PDE for PN, which was obtained for N=1,2,3 by Davison in a much more involved way. We include also a comparative review of most of the derivations of the SPN equations in the literature, including the recent work by Chao, which shows the progress in the understanding of these equations which culminates with Chao’s work. Our integral equations are fully equivalent to those of Chao, in that they predict the same equations and associated SPN angular flux.

Meta-analysis of brain functional images under N-back and DMTS paradigm in human working memory

Objective To study the similarities and differences in brain activation under N-back and DMTS paradigms of working memory. Methods Through searching of BrainMap function database by the combination keywords, a total of 122 articles with 122 experiments (N-back: 67 experiments, 937 subjects, 900 foci; Delayed Matched to Sample: 55 experiments, 806 subjects, 872 foci). Maps were analyzed using the FDR PN (P<0.0001) method based coordinates of the Montreal Neurological Institute (MNI) space. Results In the combine of the N-back and DMTS paradigms, these regions were activated including the bilateral inferior parietal lobe (Left: -36, -52, 44; Right: 38, -52, 48) and insula(Left: -34, 22, 0; Right: 34, 24, 2), the superior frontal gyrus (2, 16, 50) and inferior frontal gyrus (-44, 8, 30)of the left brain, and the middle frontal (46, 36, 24) and sub-gyral (30, 4, 56)of the right brain. The N-back paradigm was similar to the above results, whereas the DMTS paradigm only activated the precentral gyrus (-50, 8, 34) of the left brain and the right insula (34, 24, 4). Conclusion The neural circuit of working memory is the frontal-parietal networks. N-back paradigm with the brain activation consists to the neural circuit of working memory.However, DMTS paradigm activates less brain regions and is inconsistent to the neurons of working memory. Key words: Health controls; Working memory; N-back paradigm; Delayed matched to sample paradigm; Meta-analysis; Brain functional images

On the PN Method in Spherical Geometry: A Stable Solution for the Exterior of a Sphere

A numerically stable version of the PN method for solving the one-speed problem of neutron transport in the exterior of a sphere is developed. A transformation is used to obtain a plane-geometry-like neutron transport equation, where the angular derivative part of the streaming term in spherical geometry is considered as a source. In this way, conventional PN solutions developed for plane geometry can be used to solve the problem and avoid having to deal with ill-conditioned solutions expressed in terms of the modified spherical Bessel functions of the third kind.

Even-parity Boltzmann transport equation applied for response (contributon) flux calculation based on the spatial channel theory

An even parity approach for the detection of main stream channels of response flux inside the material is presented. The product of forward and adjoint flux is called the response flux which plays an important role in assessing the performance of shielding materials. Based on two distinct maximum principles, even parity forward and adjoint fluxes (ψ and ψ†) are obtained respectively. Weak and strong points of shielding materials can be well understood using the spatial channel theory and this analysis is performed using the even parity Boltzmann transport equation. The PN method as well as the finite element method are employed to approximate the angular and spatial components of the fluxes, respectively. Also, we extend the concept of spatial channel theory to fissile materials. A number of test cases are provided to evaluate the performance of the proposed approach.

An extended half-range spherical harmonics method for first-order neutron transport equation based on variational treatment

A new variational approach with anisotropic scattering kernel for first order neutron transport equation based on Finite Element Method (FEM) and Double-PN (DPN) approximation has been introduced. In presented variational principle, the angular dependence of the neutron flux has been separated into two sub-ranges of the forward and backward moving particles. The applied boundary conditions for the DPN method are straightforward in comparison to the conditions for the PN method. The method has also been extended for 2D plane geometry by using extended half-range spherical harmonics method. By defining a new variational principle, the discontinuity of angular flux on the boundary surface has been treated. A new computing program has been developed, to calculate the angular flux by this method which has the capability to solve neutron transport equation by arbitrary DPN approximation in 1D anisotropic scattering media. Moreover, by investigating extended half-range spherical harmonics method, the program has been developed for 2D geometry. The efficacy of the method is assessed by comparing the required order of angular expansion which is necessary to achieve fine results in several benchmarks. The results demonstrate the superiority of the method against traditional PN method.

Accurate reflective boundary condition in the PN method

Accurate implementation of reflective Boundary Condition (B.C.) in the PN method for both forward and adjoint analysis in one, two and three dimensional geometries is investigated in detail. Slab, spherical and cylindrical geometries are covered for 1D space, while a comprehensive procedure for general 3D Cartesian geometry is explained which encapsulates 2D (XY) geometry as a special case. It is shown that the reflective B.C. in the PN method ends to a series of moment couplings on the mirror surface which should be treated as an essential B.C. While these constraints are derived for the conventional PN method, they are applicable for the even parity PN approach, as well. Necessary guideline for imposing reflective constraints on the final system of linear equations in eigenvalue search or fixed source problems is also described. We finally verify the proposed approach by several test cases at the end.

An arbitrary geometry finite element method for the adjoint neutron transport equation

In this paper a variational formulation for the adjoint even parity neutron transport equation (NTE) based on the generalized least squares method is adopted. The so-called PN method or expansion via Spherical Harmonics Polynomials (SHPs) is then summoned to treat the angular dependency of the equation while Finite Element Method (FEM) is invoked for the spatial domain. Based on our matrix form approach, it is shown that the scalar adjoint flux, ϕ†(r), can be determined through an efficient approach solely or in parallel with the scalar forward flux, ϕ(r). Eigenvalue problems as well as detector readings are of most useful applications of the adjoint calculation which are also discussed in this survey. To verify the equations we upgraded our Even parity Neutron TRANSport code, ENTRANS, with an adjoint solver which is capable of handling one-, two- and three-dimensional problems of arbitrary geometry. Ability to cover any order of multigroup scattering anisotropy (with or without up-scattering) as well as higher order elements is also embedded in the ENTRANS. Finally, several test cases are examined against reference results to illustrate the accuracy and efficiency of the proposed approach.

Recent developments in the GENESIS code based on the Legendre polynomial expansion of angular flux method

Abstract This paper describes recent development activities of the GENESIS code, which is a transport code for heterogeneous three-dimensional geometry, focusing on applications to reactor core analysis. For the treatment of anisotropic scattering, the concept of the simplified Pn method is introduced in order to reduce storage of flux moments. The accuracy of the present method is verified through a benchmark problem. Next, the iteration stability of the GENESIS code for the highly voided condition, which would appear in a severe accident (e.g., design extension) conditions, is discussed. The efficiencies of the coarse mesh finite difference and generalized coarse mesh rebalance acceleration methods are verified with various stabilization techniques. Use of the effective diffusion coefficient and the artificial grid diffusion coefficients are found to be effective to stabilize the acceleration calculation in highly voided conditions.

A Computationally Viable Version of the PN Method for Spheres

The long-standing problem of implementing the method effectively for spherical geometry is revisited in this work. It is shown that a least-squares approach to the method resolves to a great extent the numerical instability reported for the first time by Aronson in 1984. In the proposed version of the method, a small loss of accuracy is still observed for intermediate orders of the approximation, but in high order (typically ), full accuracy is recovered, and the method can be used with confidence even for extremely high orders of the approximation. Numerical results of benchmark quality are tabulated for the quantities of interest for two basic transport problems in spherical geometry: the albedo problem for a sphere and the critical-sphere problem, both including cases that show the effects of scattering anisotropy described by the binomial law.

A hybrid reconstructed discontinuous Galerkin and continuous Galerkin finite element method for incompressible flows on unstructured grids

A hybrid reconstructed discontinuous Galerkin and continuous Galerkin method based on an incremental pressure projection formulation, termed rDG(PnPm)+CG(Pn) in this paper, is developed for solving the unsteady incompressible Navier–Stokes equations on unstructured grids. In this method, a reconstructed discontinuous Galerkin method (rDG(PnPm)) is used to discretize the velocity and a standard continuous Galerkin method (CG(Pn)) is used to approximate the pressure. The rDG(PnPm)+CG(Pn) method is designed to increase the accuracy of the hybrid DG(Pn)+CG(Pn) method and yet still satisfy Ladyženskaja–Babuška–Brezzi (LBB) condition, thus avoiding the pressure checkerboard instability. An upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field. A number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rDG(PnPm)+CG(Pn) method. The numerical experiments indicate that both rDG(P0P1)+CG(P1) and rDG(P1P2)+CG(P1) methods can attain the designed 2nd order and 3rd order accuracy in space for the velocity respectively. Moreover, the 3rd order rDG(P1P2)+CG(P1) method significantly outperforms its 2nd order rDG(P0P1)+CG(P1) and rDG(P1P1)+CG(P1) counterparts: being able to not only increase the accuracy of the velocity by one order but also improve the accuracy of the pressure.

Application of UN method to neutron transport equation in slab geometry using HG phase function

AbstractThe steady-state one speed neutron transport equation without sources is studied in slab geometry using Henyey-Greenstein scattering function. The critical thicknesses are calculated for various c (mean number of secondary neutrons per collision) and t scattering parameters using UN method and obtained values are compared with PN method. All the numerical results for different collision scattering functions and t parameters of UN and PN methods are presented in the tables.

Advanced Differential Approximation Formulation of the PN Method for Radiative Transfer

The spherical harmonics (PN) method, especially its lowest order, i.e., the P1 or differential approximation, enjoys great popularity because of its relative simplicity and compatibility with standard models for the solution of the (overall) energy equation. Low-order PN approximations perform poorly in the presence of strongly nonisotropic intensity distributions, especially in optically thin situations within nonisothermal enclosures (due to variation in surface radiosities across the enclosure surface, causing rapid change of irradiation over incoming directions). A previous modification of the PN approximation, i.e., the modified differential approximation (MDA), separates wall emission from medium emission to reduce the nonisotropy of intensity. Although successful, the major drawback of this method is that the intensity at the walls is set to zero into outward directions, while incoming intensity is nonzero, resulting in a discontinuity at grazing angles. To alleviate this problem, a new approach, termed here the “advanced differential approximation (ADA),” is developed, in which the directional gradient of the intensity at the wall is minimized. This makes the intensity distribution continuous for the P1 method and mostly continuous for higher-order PN methods. The new method is tested for a 1D slab and concentric spheres and for a 2D medium. Results are compared with the exact analytical solutions for the 1D slab as well as the Monte Carlo-based simulations for 2D media.

Method for evaluating an extended Fault Tree to analyse the dependability of complex systems: Application to a satellite-based railway system

Evaluating dependability of complex systems requires the evolution of the system states over time to be analysed. The problem is to develop modelling approaches that take adequately the evolution of the different operating and failed states of the system components into account. The Fault Tree (FT) is a well-known method that efficiently analyse the failure causes of a system and serves for reliability and availability evaluations. As FT is not adapted to dynamic systems with repairable multi-state components, extensions of FT (eFT) have been developed. However efficient quantitative evaluation processes of eFT are missing. Petri nets have the advantage of allowing such evaluation but their construction is difficult to manage and their simulation performances are unsatisfactory. Therefore, we propose in this paper a new powerful process to analyse quantitatively eFT. This is based on the use of PN method, which relies on the failed states highlighted by the eFT, combined with a new analytical modelling approach for critical events that depend on time duration. The performances of the new process are demonstrated through a theoretical example of eFT and the practical use of the method is shown on a satellite-based railway system.

Solution methods of neutron transport equation in nuclear reactors

A few numerical methods that usually used to solve neutron transport equation in nuclear reactor are SN dan PN method, Monte Carlo, Collision Probability and Methods of Characteristics . First two methods have been developed using diffusion approach, and last three methods suitable are applicated for transport approximation. Those of three methods have important role in the desain of nuclear reactors. In addition to follow the development of advanced reactor designs, the three methods were chosen because they do not use diffusion approach these are more accurate methods, as well as less need considerable computer memory. Of all the existing methods, the CP method has several advantages among the others. Keywords : Neutron transport, SN, PN, CP, MOC, MC

Exact Solutions of the Two-Dimensional Cell Problem

A method has been developed that provides analytic solutions for two-dimensional cell problems for the neutron transport equation. This is made possible by assuming an infinite, repeating lattice of rectangular regions. The solution is effected by means of a finite Fourier transform, the periodicity of which is related to the size of the unit cell. In order to drive the flux, we assume that the cell is composed of two regions: an inner circular region and the remaining exterior part. Different sources are placed in each region thereby leading to a situation rather like the conventional reactor cell problem but with no spatial variation of the cross sections. The method is illustrated by two examples: the Levermore-Pomraning equations and the two-group equations. In the former case, we have obtained the stochastically averaged flux within the cell and also the Pomraning χ-function. In addition, we have calculated the ratio of the spatially averaged flux in the outer region to that in the inner circular region, i.e., the disadvantage factor. Fluxes and disadvantage factors are also obtained for the two-group equations, and the rate of convergence is shown. These results are exact transport theory solutions and are offered as benchmarks for checking transport theory codes. The calculations are also repeated using diffusion theory. The SPN method, which we show to be exact for our problem, is used to demonstrate the rate of convergence of the PN method for two-dimensional cell problems.

Testing shrinkage factors: comparison of methods and correlation with index properties of soils

The paper presents a study on the shrinkage properties of three clay soils from Poland. Shrinkage limit, volumetric shrinkage and relative volumetric shrinkage were determined, tested according to PN-88/B-04481 (1988) and BS1377: Part 2 (1990) and correlated with the index properties of soils. The shrinkage limit was also calculated from Krabbe’s (1958) equation. The results showed that shrinkage limit values obtained by the BS method are lower than those obtained using the PN method, but the values calculated from Krabbe’s equation differ significantly. While no strong correlation was found between shrinkage limit and index soil parameters, linear relationships were obtained between volumetric shrinkage and initial moisture content, shrinkage range, plasticity index and clay content. Following multivariable regression analysis, the relative volumetric shrinkage was expressed as a function of plasticity index and moisture content. This function can be used to predict volumetric changes of the foundations based on the moisture content and material properties of the soil.

The criticality calculations for one-speed neutrons in a reflected slab with anisotropic scattering using the modified UN method

Abstract The reflected critical slab problem is studied using Chebyshev polynomials of second kind in one-speed neutron transport theory. The variation of the critical thickness of the medium is investigated both in the case of linearly anisotropic scattering and reflecting boundary conditions. The critical half-thicknesses of the slab are calculated for selected values of the reflection coefficient and the anisotropy parameter. Numerical results obtained for the critical half-thickness are identical with the ones obtained by the PN method and the ones available in literature.