Coarse-mesh Calculations Research Articles

Validation results of the BIPR-8A code, the new module of the software package KASKAD

Abstract The BIPR-8A executable module is a main component of the BIPR 2007 code. This module is intended for three-dimensional coarse mesh calculation of the VVER reactor cores. This paper presents the results of validation of the BIPR-8A executable module. Validation is based on comparison of the calculated results with the operating data of power plants with the VVER-440, VVER-1000 and VVER-1200 reactors, and with the results of calculations obtained by using Monte Carlo codes. The results of comparison of the following values at HZP (hot zero power) power level and operational power level are presented: critical concentration of boric acid, temperature reactivity coefficient, efficiency of the control rods system, relative power distribution of fuel assemblies and 3D power distribution, effective multiplication factor.

An analytical nodal method for energy multi-group discrete ordinates transport calculations in two-dimensional rectangular geometry

A spectral nodal method for energy multi-group X, Y-geometry, discrete ordinates (SN) problems in non-multiplying medium is developed. This analytical coarse-mesh method is referred to as the multi-group spectral Green's function - constant nodal (SGF-CN) method. The SGF-CN method uses the multi-group SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. As the energy-group transfer scattering source terms are treated analytically in the offered method, the only approximations occur in the group transverse leakage terms. Numerical results are given to illustrate the method's accuracy for coarse-mesh calculations.

A hybrid spectral nodal method for multigroup X,Y-geometry discrete ordinates eigenvalue problems

A hybrid spectral nodal method is developed for the solution of multigroup discrete ordinates (SN) eigenvalue problems with isotropic scattering in X,Y geometry. In this method, the spectral diamond - spectral Green's function (SD-SGF) hybrid scheme, originally developed for numerically solving multigroup SN eigenvalue problem in slab geometry with no spatial truncation error, is generalized to solve the energy multigroup transverse-integrated SN nodal equations with constant approximation for the group transverse leakage terms. The resulting SD-SGF-constant nodal (SD-SGF-CN) method is more accurate than conventional nodal methods for criticality calculations because it treats both the scattering and fission source terms exactly; the only approximation involves the transverse leakage terms. This is in contrast to the classic multigroup SN nodal methods, wherein the group transverse leakage terms, the scattering source and the fission source are all approximated, hence decoupling angular directions and energy groups in the transverse integrated SN nodal equations. We solve the SD-SGF-CN equations using the partial one-node block inversion (NBI) iterative scheme (inner iterations) for each estimate of the dominant eigenvalue in the conventional power method (outer iterations). We show numerical results to illustrate the accuracy of the present multigroup SD-SGF-CN method for coarse-mesh calculations.

Finite Element Method with Spectral Green's Function in Slab Geometry for Neutron Diffusion in Multiplying Media and One Energy Group

The physical phenomenon of neutrons transport associated with eigenvalue problems appears in the criticality calculations of nuclear reactors and can be treated as a diffusion process. This paper presents a new method to solve eigenvalue problems of neutron diffusion in slab geometry and one energy group. This formulation combines the Finite Element Method, considered an intermediate mesh method, with the Spectral Green's Function Method, which is free of truncation errors, and it is considered a coarse mesh method. The novelty of this formulation is to approach the spatial moments of the neutron flux distribution by the first-order polynomials obtained from the spectral analysis of diffusion equation. The approximations provided by the new formulation allow obtaining accurate results in coarse mesh calculations. To validate the method, we compare the results obtained with the methods described in the literature, specifically the Diamond Difference method. The accuracy and the computational performance of the proposed formulation were characterized by solving benchmarks problems with a high degree of heterogeneity.

An adaptive p-refinement strategy applied to nodal expansion method in 3D Cartesian geometry

The aim of this work is to develop a coarse mesh treatment strategy using adaptive polynomial, p, refinement approach for average current nodal expansion method in order to solve the neutron diffusion equation. For performing the adaptive solution process, a posteriori error estimation scheme, i.e. flux gradient has been utilized for finding the probable numerical errors. The high net leakage in a node represents flux gradient existence between neighbor nodes and it may indicate the source of errors for the coarse mesh calculation. Therefore, the relative Cartesian directional net leakage of nodes is considered as an assessment criterion for mesh refinement in a sub-domain. In our proposed approach, the zeroth order nodal expansion solution is used along coarse meshes as large as fuel assemblies to treat neutron populations. Coarse nodes with high directional net leakage may be chosen for implementing higher order polynomial expansion in the corresponding direction, i.e. X and/or Y and/or Z Cartesian directions. Using this strategy, the computational cost and time are reduced relative to uniform high order polynomial solution. In order to demonstrate the efficiency of this approach, a computer program, APNEC, Adaptive P-refinement Nodal Expansion Code, has been developed for solving the neutron diffusion equation using various orders of average current nodal expansion method in 3D rectangular geometry. Some well-known benchmarks are investigated to compare the uniform and adaptive solutions. Results demonstrate the superiority of our proposed strategy in enhancing the accuracy of solution without using uniform high order solution throughout the domain and increasing the number of unknowns.

Spectral nodal method for numerically solving two-energy group X,Y geometry neutron diffusion eigenvalue problems

We describe in this paper the recent advances in spectral nodal methods applied to diffusion problems in Cartesian geometry for neutron multiplying systems. In particular, we present a constant spectral nodal method for two-energy group X,Y geometry applied to neutron diffusion eigenvalue problems. We consider an arbitrary rectangular spatial grid defined on a two-dimensional rectangular domain and we use a transverse integration procedure to transform the two-dimensional problem into two 'one-dimensional' problems wherein the transverse leakage terms are approximated by constants. As a result, we obtain the transverse-integrated constant nodal diffusion equations that we discretise using the spectral nodal method. The discretised balance diffusion equations together with appropriate auxiliary equations, continuity and boundary conditions form the two-energy group X,Y geometry spectral constant nodal diffusion equations. The auxiliary equations have parameters that are to be determined such that the analytical general solutions of the transverse-integrated constant nodal diffusion equations are preserved. We show numerical results to illustrate the method's accuracy for coarse-mesh calculations in homogeneous and heterogeneous domains.

Simultaneous Optimization of Tallies in Difficult Shielding Problems

Monte Carlo is quite useful for calculating specific quantities in complex transport problems. Many variance reduction strategies have been developed that accelerate Monte Carlo calculations for specific tallies. However, when trying to calculate multiple tallies or a mesh tally, users have had to accept different levels of relative uncertainty among the tallies or run separate calculations optimized for each individual tally. To address this limitation, an extension of the Consistent Adjoint Driven Importance Sampling (CADIS) method, which is used for difficult source/detector problems, has been developed to optimize several tallies or the cells of a mesh tally simultaneously. The basis for this method is the development of an importance function that represents the importance of particles to the objective of uniform Monte Carlo particle density in the desired tally regions. This method utilizes the results of a forward discrete ordinates solution, which may be based on a quick coarse-mesh calculation, to develop a forward-weighted source for the adjoint calculation. The importance map and the biased source computed from the adjoint flux are then used in the forward Monte Carlo calculation to obtain approximately uniform relative uncertainties for the desired tallies. This extension is called forward-weighted CADIS, or FW-CADIS.

The spectral Green’s function linear-nodal method for one-speed X, Y-geometry discrete ordinates deep penetration problems

A new spectral nodal method is developed for the solution of one-speed discrete ordinates ( S N) problems with isotropic scattering in X, Y geometry. In this method, the spectral Green’s function (SGF) scheme, originally developed for solving S N problems in slab-geometry with no spatial truncation error, is generalized to solve the one-dimensional transverse-integrated S N linear-nodal equations with linear polynomial approximation for the transverse leakage terms. The resulting SGF-linear-nodal (SGF-LN) equations are solved with the full-node inversion (FNI) iterative scheme, which uses the best available estimates for the node-entering quantities to evaluate the node angular quantities in all the exiting directions as the equations are swept across the system. We give numerical results that illustrate the accuracy of the SGF-LN method for coarse-mesh calculations.

Heat transfer process simulation by finite differences for online control of ladle furnaces

A numerical method for solving the two-dimensional transient heat conduction equation is presented to determine temperature profiles of ladle furnaces during the casting of special steels. The method is based on an explicit formulation of the heat transfer process in finite differences. In order to improve the efficiency of the method, two auxiliary algorithms are added, one for correcting finite difference results from coarse mesh calculations and the other to perform time extrapolation of the temperature behaviour between furnace batches. The calculated steel temperatures are found to be in good agreement with experimental data, obtained at a Brazilian thin steel casting facility. Currently, about 80% of the ladle batches at this facility come out within their commercial specifications, against 45% previously observed without the use of such a model. With average processing times of the order of 100 s in an AMD K6II 300 MHz computer, the method is suitable for real time thermal tracking of ladles and steel.

A posteriori finite element output bounds with adaptive mesh refinement: application to a heat transfer problem in a three-dimensional rectangular duct

Numerical simulations based on an a posteriori finite element bound method with adaptive mesh refinement are presented for the three-dimensional convection–diffusion equation. The bound method provides relevant, quantitative, inexpensive, and rigorous lower and upper bounds to the output on a very fine discretization (“truth” discretization) at a cost close to the coarse mesh calculation (“working” discretization). To achieve a desired bound gap (i.e., difference between upper and lower bounds) at the lowest cost, an adaptive mesh refinement technique is used to refine the mesh only where needed. An optimal stabilization parameter is also applied to improve the sharpness of the bound gap. In this paper, the output of a heat transfer problem in a rectangular duct with a given velocity field is investigated. The average temperature at one section of the duct is bounded for a given inlet temperature and heat flux. For this problem, the adaptive mesh refinement strategy provides the same bound gap with only half the number of elements required by an uniform mesh refinement strategy. The hybrid flux calculation on the coarse mesh introduced for the domain decomposition approach is compared with the hybrid flux calculation on the fine mesh to analyze the contribution of the hybrid flux to the bound gap.

A coarse-mesh numerical method for one-speed neutron transport eigenvalue problems in two-dimensional Cartesian geometry

We describe a hybrid spectral nodal method applied to one-speed S N eigenvalue problems in x, y-geometry for nuclear reactor global calculations. To solve the transverse-integrated S N nodal equations, we generalize the spectral diamond (SD) method, that we developed for numerically solving slab-geometry S N eigenvalue problems with no spatial truncation error. In the present generalization, we approximate the transverse leakage through the edges of each spatial node by constants, so we call our method the SD-constant nodal (SD-CN) method that we use in the fuel regions of the nuclear reactor core. In the non-multiplying regions, e.g., reflector and baffle, we use the spectral Green's function-constant nodal (SGF-CN) method; hence the hybrid characteristic of our method. In order to converge the numerical solution for each S N “fixed source” problem (inner iterations) in each outer iteration (power method), we use the one-node block inversion (NBI) scheme. We show some numerical results to a typical model problem to illustrate the method's accuracy in coarse-mesh calculations.

A posteriori bounds for linear functional outputs of Crouzeix-Raviart finite element discretizations of the incompressible Stokes problem

We present a finite element technique for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions; the finite element discretization is effected by Crouzeix-Raviart elements, the discontinuous pressure approximation of which is central to our approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic energy reformulation of the desired output; the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the intersubdomain continuity conditions on velocity. Appeal to the dual max-min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization; the Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires solution only of local subdomain Stokes problems on the fine-mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flowrate past, and the lift force on, a body immersed in a channel.

A posteriori bounds for linear functional outputs of Crouzeix–Raviart finite element discretizations of the incompressible Stokes problem

A finite element technique is presented for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions. The finite element discretization is effected by Crouzeix–Raviart elements, the discontinuous pressure approximation of which is central to this approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic ‘energy’ reformulation of the desired output, the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the inter-sub-domain continuity conditions on velocity. Appealing to the dual max–min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization. The Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires the solution of only local sub-domain Stokes problems on the fine mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flow rate past and the lift force on a body immersed in a channel. Copyright © 2000 John Wiley & Sons, Ltd.

Accounting for Strong Localized Heterogeneities and Local Transport Effect in Core Calculation

Two methods based on the variational nodal transport method have been developed to account for localized heterogeneities and local transport effects in full core calculations. A local mesh refinement technique relies on using the projected partial ingoing surface currents produced during coarse-mesh iterations as boundary conditions for fine-mesh calculations embedded within the coarse-mesh calculations. The outgoing fine-mesh partial currents are averaged to serve in the coarse-mesh iterations. Then, a mixed transport-diffusion method using two levels of angular approximations for the surface partial currents depending on the node considered has been implemented to account for local transport effects in full core diffusion calculations. These methods have been tested for a model of the Superphenix complementary shutdown rods.

Numerical simulations of fast crack growth in brittle solids

Dynamic crack growth is analysed numerically for a plane strain block with an initial central crack subject to tensile loading. The continuum is characterized by a material constitutive law that relates stress and strain, and by a relation between the tractions and displacement jumps across a specified set of cohesive surfaces. The material constitutive relation is that of an isotropic hyperelastic solid. The cohesive surface constitutive relation allows for the creation of new free surface and dimensional considerations introduce a characteristic length into the formulation. Full transient analyses are carried out. Crack branching emerges as a natural outcome of the initial-boundary value problem solution, without any ad hoc assumption regarding branching criteria. Coarse mesh calculations are used to explore various qualitative features such as the effect of impact velocity on crack branching, and the effect of an inhomogeneity in strength, as in crack growth along or up to an interface. The effect of cohesive surface orientation on crack path is also explored, and for a range of orientations zigzag crack growth precedes crack branching. Finer mesh calculations are carried out where crack growth is confined to the initial crack plane. The crack accelerates and then grows at a constant speed that, for high impact velocities, can exceed the Rayleigh wave speed. This is due to the finite strength of the cohesive surfaces. A fine mesh calculation is also carried out where the path of crack growth is not constrained. The crack speed reaches about 45% of the Rayleigh wave speed, then the crack speed begins to oscillate and crack branching at an angle of about 29° from the initial crack plane occurs. The numerical results are at least qualitatively in accord with a wide variety of experimental observations on fast crack growth in brittle solids.

A Spectral Nodal Method for One-GroupX, Y-Geometry Discrete Ordinates Problems

This paper reports on a nodal method that is developed for the solution of one-group discrete ordinates (S{sub N}) problems with linearly anisotropic scattering in x, y-geometry. In this method, the spectral Green's function (SGF) scheme, originally developed for solving S{sub N} problems in slab geometry with no spatial truncation error, is generalized to solve the one-dimensional transverse-integrated S{sub N} nodal equations with the constant approximation for the transverse leakage terms. The resulting SGF-constant nodal (SGF-CN) method is more accurate than conventional coarse-mesh methods for deep penetration problems because it treats the scattering source terms implicitly and exactly; the only approximation involves the transverse leakage terms. In conventional S{sub N} nodal methods, the transverse leakage terms and scattering source are both approximated. The authors solve the SGF-CN equations using the one-node block inversion iterative scheme, which uses the best available estimates for the node-entering fluxes to evaluate the node-exiting fluxes in the directions that constitute the incoming fluxes for the adjacent nodes as the equations are swept across the system. Finally, the authors give numerical results that illustrate the accuracy of the SGF-CN method for coarse-mesh calculations.

Application of Improved Coarse Mesh Method to BWR Core Calculations

The improved coarse mesh method, which was originally derived by Askew and applied to fast reactor core calculations by one of the authors, has been applied to boiling water reactor core calculations. Since direct application of the method leads to negative fluxes in the reflector region, a procedure has been developed in which the correction factor for coarse meshes is not utilized in the reflector. The results of the improved coarse mesh calculations were compared with those of fine mesh calculations. The negative fluxes were eliminated and good agreement was obtained for the neutron multiplication factor and power distributions in fuel assemblies near control rods, though the accuracy of power distributions near the reflector was as that of conventional coarse mesh calculations.

Application of improved coarse mesh method to BWR core calculations.

The improved coarse mesh method, which was originally derived by Askew and applied to fast reactor core calculations by one of the authors, has been applied to boiling water reactor core calculations. Since direct application of the method leads to negative fluxes in the reflector region, a procedure has been developed in which the correction factor for coarse meshes is not utilized in the reflector. The results of the improved coarse mesh calculations were compared with those of fine mesh calculations. The negative fluxes were eliminated and good agreement was obtained for the neutron multiplication factor and power distributions in fuel assemblies near control rods, though the accuracy of power distributions near the reflector was as that of conventional coarse mesh calculations.