Two-dimensional Multigroup Research Articles

A Nodal-iterative Technique for Criticality Calculations in Multigroup Neutron Diffusion Models

In this work, a nodal and iterative technique to evaluate the effective multiplication factor as well as the neutron flux, in multigroup diffusion problems, is presented. An iterative scheme, similar to the source iteration method, is implemented to decouple the system of differential equations which is the fundamental mathematical model. Then, analytical solutions are derived for the one-dimensional transverse integrated equations, of each energy group, resulting from a nodal approach. Constant approximations are assumed for the unknown transverse leakage terms in the contours of the nodes. In addition, constant and linear representations are investigated to express the fluxes in the source term to be updated in the iterative process. Numerical results for the effective multiplication factor were obtained for a series of two-dimensional multigroup problems with up-scattering and down-scattering. The procedure is simple, fast, the analysis of the results indicated a satisfactory agreement with results available in the literature and the use of different approximations to the source term seems to be a good alternative, instead of using higher-order approximations on the contour of the nodes, to improve accuracy.

Three higher order analytical nodal methods for multigroup neutron diffusion equations

This work presents three efficient higher order analytical nodal methods for the numerical solution of a two-dimensional multigroup neutron diffusion equation in Cartesian geometry based on the use of the successive polynomial-weighted transverse integrations technique to convert a one-group diffusion equation to a system of coupled one-dimensional ordinary differential equations. These equations are then solved analytically over each homogenized cell after adequate approximations of the resulting effective sources after transversal integrations. Coupling between the approximate transverse flux-moments is achieved by imposing uniqueness constraint on their moments values. Adjacent elements are coupled by enforcing continuity conditions on the flux and current moments at interfaces cells. The weighted cell-balance equations and current-continuity conditions are then used to derive the discrete equations. These methods are applied for solving numerically various 2D benchmark problems and theirs performances discussed. Numerical results demonstrates more efficiency for the third higher order analytical nodal method for which the alone unknowns considered are the transverse flux moments on the interfaces of the homogenized elements.

COARSE MESH FINITE DIFFERENCE ACCELERATION OF DISCRETE ORDINATE NEUTRON TRANSPORT CALCULATION EMPLOYING DISCONTINUOUS FINITE ELEMENT METHOD

The coarse mesh finite difference (CMFD) method is applied to the discontinuous finite element method based discrete ordinate calculation for source convergence acceleration. The three-dimensional (3-D) DFEM-Sn code FEDONA is developed for general geometry applications as a framework for the CMFD implementation. Detailed methods for applying the CMFD acceleration are established, such as the method to acquire the coarse mesh flux and current by combining unstructured tetrahedron elements to rectangular coarse mesh geometry, and the alternating calculation method to exchange the updated flux information between the CMFD and DFEM-Sn. The partial current based CMFD (p-CMFD) is also implemented for comparison of the acceleration performance. The modified p-CMFD method is proposed to correct the weakness of the original p-CMFD formulation. The performance of CMFD acceleration is examined first for simple two-dimensional multigroup problems to investigate the effect of the problem and coarse mesh sizes. It is shown that smaller coarse meshes are more effective in the CMFD acceleration and the modified p-CMFD has similar effectiveness as the standard CMFD. The effectiveness of CMFD acceleration is then assessed for three-dimensional benchmark problems such as the IAEA (International Atomic Energy Agency) and C5G7MOX problems. It is demonstrated that a sufficiently converged solution is obtained within 7 outer iterations which would require 175 iterations with the normal DFEM-Sn calculations for the IAEA problem. It is claimed that the CMFD accelerated DFEM-Sn method can be effectively used in the practical eigenvalue calculations involving general geometries.

Calculation of the effective delayed-neutron fraction and prompt-neutron lifetime in IBR-2M

The main parameters of IBR-2M are presented: the effective delayed-neutron fraction βeff and the promptneutron lifetime τ, calculated using the DORT two-dimensional multigroup neutron-transport compute code and the SCALE4 code with a system of multigroup nuclear constants. For a regular IBR-2M regime βeff = 0.00216 ± 0.00007, τ = (6.5 ± 0.5)·10–8 sec, the delay-neutron value γ = 0.980, the prompt-neutron decay constant in the critical state α = 3.5·104 sec–1. The calculations showed that the effective delayed-neutron fraction for IBR-2M is identical, within the error limits, to the measured value for IBR-2, the prompt-neutron lifetime is approximately 5% longer (βeff = 0.00216, τ = (6.2 ± 0.2)·10–8 sec). It is shown that βeff and τ increase somewhat as the IBR-2 core size increases in the radial direction.

Nodal methods for a two-dimensional static multigroup diffusion calculation of nuclear reactors with hexagonal assemblies

This paper presents three variants of a hexagonal nodal method for determining neutron flux distribution in a nuclear reactor and its critical number. Their common framework constitutes the few-group, coarse-mesh finite-difference diffusion equations for all nodes inside the core domain. To achieve better accuracy without sacrificing efficiency, they are corrected by higher order nodal calculations formulated on pairs of nodes using the transverse integration procedure. This leads to three 1D equations coupled through neutron currents at transverse nodal boundaries. Approximation of the coupling has significant impact on the accuracy of transverse-integration based nodal methods and as such receives considerable attention. Three possibilities are discussed that define the variants of the developed nodal method. Once the transverse coupling term is suitably approximated, solution of nodal equations is computed semi-analytically by using an efficient node-by-node iterative procedure. The three implementations are compared on an example configuration of a VVER-1000 core.

The Proto-Neutron Star Phase of the Collapsar Model and the Route to Long-Soft Gamma-Ray Bursts and Hypernovae

Recent stellar evolutionary calculations of massive low-metallicity fast-rotating main-sequence stars yield iron cores at collapse that are endowed with high angular momentum. It is thought that high angular momentum and black hole formation are critical ingredients for the collapsar model of long-soft γ-ray bursts (GRBs). We present two-dimensional multigroup, flux-limited-diffusion MHD simulations of the collapse, bounce, and immediate postbounce phases of a 35 M☉ collapsar-candidate model of Woosley & Heger. Provided that the magnetorotational instability (MRI) operates in the differentially rotating surface layers of the millisecond-period neutron star, we find that a magnetically driven explosion occurs during the proto-neutron star phase, in the form of a baryon-loaded nonrelativistic jet, and that a black hole, which is central to the collapsar model, does not form. Paradoxically, although much uncertainty surrounds stellar mass loss, angular momentum transport, magnetic fields, and the MRI, current models of chemically homogeneous evolution at low metallicity yield massive stars with iron cores that may have too much angular momentum to avoid a magnetically driven, hypernova-like explosion in the immediate postbounce phase. We surmise that fast rotation in the iron core may inhibit, rather than enable, collapsar formation, which requires a large angular momentum above the core but not in it. Variations in the angular momentum distribution of massive stars at core collapse might explain both the diversity of Type Ic supernovae/hypernovae and their possible association with a GRB. A corollary might be that, through its effect on magnetic field amplification, the angular momentum distribution, rather than the progenitor mass, is the distinguishing characteristic of these outcomes.

Simulations investigating the effect of a deuterium-tritium-ice coating on the motion of the gold cone surface in a re-entrant cone-guided fast ignition inertial confinement fusion capsule

One- and two-dimensional multigroup radiation hydrodynamics simulations have been performed to investigate the motion of the gold plasma generated at the surface of the embedded gold cone in a re-entrant cone-guided inertial confinement fusion capsule. The effect of deuterium-tritium (DT) ice layers, and other possible tampers, of varying thickness, upon the motion of the gold cone plasma has been investigated. The effect of the x-ray drive spectrum incident upon the ice layer is also explored. Ice is shown to tamp the expansion of the gold cone, and whilst denser materials are shown to be more effective in this role, ice does not pollute the ignition region with intermediate-Z ions, which, though preferable to gold contamination, also tend to inhibit the attainment of high fuel-ion temperatures.

Study on two-dimensional transfer of radiative heating wave

A two-dimensional (2D) multigroup radiation transfer hydrodynamics code LARED-R-1 is used to simulate a supersonic wave experiment performed earlier by the Livermore group. The main result is that, contrary to the conclusion of Back et al. (2000a), the average-atom opacity model is sufficient to explain the obtained experimental results, provided that an adequate description of the radiation transport was used. The simulation results from LARED-R-1 show the spectrum of radiation in foam with radius and length of several optical depths is not in Planckian distribution and the angular intensity distribution is anisotropic.

RTk/SN Solutions of the Two-Dimensional Multigroup Transport Equations in Hexagonal Geometry

This paper describes an extension to the hexagonal geometry of some weakly discontinuous nodal finite element schemes developed by Hennart and del Valle for the two-dimensional discrete ordinates transport equation in quadrangular geometry. The extension is carried out in a way similar to the extension to the hexagonal geometry of nodal element schemes for the diffusion equation using a composite mapping technique suggested by Hennart, Mund, and del Valle. The combination of the weakly discontinuous nodal transport scheme and the composite mapping is new and is detailed in the main section of the paper. The algorithm efficiency is shown numerically through some benchmark calculations on classical problems widely referred to in the literature.

A finite subelement generalization of the variationalnodal method

The variational nodal method is generalized by dividing each spatial node into a number oftriangular finite elements designated as subelements. The finite subelement trial functions allow for explicit geometry representations within each node, thus eliminating the need for nodal homogenization. The method is implemented within the Argonne National Laboratory code VARIANT and applied to two-dimensional multigroup problems. Eigenvalue and pin-power results are presented for a four-assembly DECD/NEA benchmark problem containing enriched UO 2 and MOX fuel pins. Our seven-group model combines spherical or simplified spherical harmonic approximations in angle with isoparametric linear or quadratic subelement basis functions, thus eliminating the need for fuel-coolant homogenization. Comparisons with reference seven-group Monte Carlo solutions indicate that in the absence of pin-cell homogenization, high-order angular approximations are required to obtain accurate eigenvalues, while the results are substantially less sensitive to the refinement of the finite subelement grids.

Two‐dimensional, Time‐dependent, Multigroup, Multiangle Radiation Hydrodynamics Test Simulation in the Core‐Collapse Supernova Context

We have developed a time-dependent, multi-energy group, multiangle (Sn) Boltzmann transport scheme for radiation hydrodynamics simulations in one and two spatial dimensions. The implicit transport is coupled to both one-dimensional (spherically symmetric) and two-dimensional (axially symmetric) versions of the explicit Newtonian hydrodynamics code VULCAN. The two-dimensional variant, VULCAN/2D, can be operated in general structured or unstructured grids, and although the code can address many problems in astrophysics, it was constructed specifically to study the core-collapse supernova problem. Furthermore, VULCAN/2D can simulate the radiation hydrodynamic evolution of differentially rotating bodies. We summarize the equations solved and methods incorporated into the algorithm and present the results of a time-dependent two-dimensional test calculation. A more complete description of the algorithm is postponed to another paper. We highlight a two-dimensional test run that follows the immediate postbounce evolution of a collapsed core for 22 ms. We present the relationship between the anisotropies of the overturning matter field and the distribution of the corresponding flux vectors as a function of energy group. This is the first two-dimensional multigroup, multiangle, time-dependent radiation hydrodynamics calculation ever performed in core-collapse studies. Although the transport module of the code is not gray and does not use flux limiters (however, there is a flux-limited variant of VULCAN/2D), it still does not include energy redistribution and most velocity-dependent terms.

A Finite Subelement Generalization of the Variational Nodal Method

The variational nodal method is generalized by dividing each spatial node into a number of triangular finite elements designated as subelements. The finite subelement trial functions allow for explicit geometry representations within each node, thus eliminating the need for nodal homogenization. The method is implemented within the Argonne National Laboratory code VARIANT and applied to two-dimensional multigroup problems.Eigenvalue and pin-power results are presented for a four-assembly Organization for Economic Cooperation and Development/Nuclear Energy Agency benchmark problem containing enriched UO2 and mixed oxide fuel pins. Our seven-group model combines spherical or simplified spherical harmonic approximations in angle with isoparametric linear or quadratic subelement basis functions, thus eliminating the need for fuel-coolant homogenization. Comparisons with reference seven-group Monte Carlo solutions indicate that in the absence of pin-cell homogenization, high-order angular approximations are required to obtain accurate eigenvalues, while the results are substantially less sensitive to the refinement of the finite subelement grids.

Monte Carlo Simulation of the TRIGA Mark II Benchmark Experiment with Burned Fuel

Monte Carlo calculations of a criticality experiment with burned fuel on the TRIGA Mark II research reactor are presented. The main objective was to incorporate burned fuel composition calculated with the WIMSD4 deterministic code into the MCNP4B Monte Carlo code and compare the calculated keff with the measurements. The criticality experiment was performed in 1998 at the “Jožef Stefan” Institute TRIGA Mark II reactor in Ljubljana, Slovenia, with the same fuel elements and loading pattern as in the TRIGA criticality benchmark experiment with fresh fuel performed in 1991. The only difference was that in 1998, the fuel elements had on average burnup of ~3%, corresponding to 1.3-MWd energy produced in the core in the period between 1991 and 1998. The fuel element burnup accumulated during 1991-1998 was calculated with the TRIGLAV in-house-developed fuel management two-dimensional multigroup diffusion code. The burned fuel isotopic composition was calculated with the WIMSD4 code and compared to the ORIGEN2 calculations. Extensive comparison of burned fuel material composition was performed for both codes for burnups up to 20% burned 235U, and the differences were evaluated in terms of reactivity. The WIMSD4 and ORIGEN2 results agreed well for all isotopes important in reactivity calculations, giving increased confidence in the WIMSD4 calculation of the burned fuel material composition. The keff calculated with the combined WIMSD4 and MCNP4B calculations showed good agreement with the experimental values. This shows that linking of WIMSD4 with MCNP4B for criticality calculations with burned fuel is feasible and gives reliable results.

A spectral-Galerkin coarse-mesh approach to solving the two-dimensional multigroup diffusion equations

Based on the spectral-Galerkin method and domain decomposition method, a novel coarse-mesh model (SGCM) has been developed for solving the two-dimensional multigroup neutron diffusion equations. In this study, the reactor domain is decomposed into subdomains (or course meshes). Within any individual subdomain, the neutron diffusion equations with the Robin boundary condition are first reformulated in a weak (variational) form, which is then solved by means of the Legendre-spectral method. As for interfacial coupling between subdomains, it is relaxed by an interfacial relation of discontinuity between incoming and outgoing partial currents if discontinuity factors are considered on subdomain interfaces. The resulting discretized system with block-sparse-structured matrix is solved by the block successive over-relaxation method (BSOR). The method was examined with two well-known PWR benchmark problems and a realistic reactor core. It turns out that the method can generate not only extremely accurate assemblywise average quantities, but also satisfactory smooth homogenized fluxes, and heterogeneous reconstructed pinwise fluxes, which results from the fact that all the formulations are derived in a variational form in which the admissible solution space consists of higher-order Legendre polynomials.

Huemul—a two-dimensional multigroup collision probability code for general geometries

The need for a flexible tool for control rod calculations in research and power reactors resulted in the development of HUEMUL, a 2-D transport code using the collision probability (CP) method in fairly general geometries and able to solve a great variety of problems. HUEMUL has been tested with copper activity measurements performed in the Canadian D20 facility ZED-2 with stainless-steel adjuster rods obtaining a very good agreement (better than 2%). Comparisons with results from the WIMS code for a light water cell with 3% wt enriched UO 2 has also shown a very good agreement in fluxes and multiplication constant (better than 1.5% in fluxes and 0.5 mk in k x ).

The Coarse Space-Energy Mesh Rebalancing Method for Neutron Diffusion Calculations in Fast Reactors

This paper reports on the coarse space-energy mesh rebalancing method studied for the purpose of convergence acceleration on two-dimensional multigroup neutron diffusion calculations with a seven-point finite difference scheme, a uniform triangular mesh, and an arbitrary scattering matrix. The rebalancing method provides convergences without numerical instability for a range of fast reactor problems with varying numbers of neutron energy groups and mesh points. The number of outer iterations is decreased with the rebalancing method by a factor of 2 in comparison to the case when only asymptotic fission source extrapolation and successive overrelaxation acceleration techniques are applied. With the rebalancing method. The HIVER code solves the problems 5 to 20 times faster than the existing reference CITATION code. The relative calculation speed of the reference code increases with the problem size.

A two-dimensional finite element multigroup diffusion theory for neutral atom transport in plasmas

Solution of the energy-dependent diffusion equation in two dimensions is formulated by multigroup approximation of the energy variable and general triangular mesh, finite element discretization of the spatial domain. Finite element formulation is done by Galerkin's method. Based on this formulation, a two-dimensional multigroup finite element diffusion theory code, FENAT, has been developed for the transport of neutral atoms in fusion plasmas. FENAT solves the multigroup diffusion equation in X-Y cartesian and R-Z cylindrical/toroidal geometries. Use of the finite element method allows solution of problems in which the plasma cross section has an arbitrary shape. The accuracy of FENAT has been verified by comparing results to those obtained using the two-dimensional discrete ordinate transport theory code, DOT-4.3. Results of application of FENAT to the transport of limiter-originated neutral atoms in a tokamak fusion machine are presented.

Spherical Harmonics Method for a Two-Dimensional Multigroup Transport Equation Using a Semi-Discrete Ordinates Equation: Part II

Spherical Harmonics Method for a Two-Dimensional Multigroup Transport Equation Using a Semi-Discrete Ordinates Equation: Part II

Spherical Harmonics Method for a Two-Dimensional Multigroup Transport Equation Using a Semi-Discrete Ordinates Equation: Part I

It is shown that, after integrating the transport equation over the azimuthal angle of the polar coordinates, the resulting discrete ordinates equation with respect to the polar angle is equivalent to that of the spherical haromonics method provided that the discrete ordinates were chosen as the roots of the associated Legendre functions. The form of this semi-discrete ordinates equation is independent of the order of the approximation and simpler than those of the usual spherical harmonics method. The present method may be regarded as an extension of the Wick-Chandrasekhar method to multidimensional problems, since the present equation is reduced to the second-order form of the Wick-Chandrasekhar equation in the case of one-dimensional slab geometry.

Discontinuous Finite Element Method for Solving the Two-Dimesional Neutron Transport Equation on Lagrangian Meshes

It is often desirable to solve the two-dimensional multigroup transport equation for (r-z) geometries directly given by hydrodynamic calculations. Usually, only Monte Carlo codes can compute kappa or k eigenvalues on such geometries. Most deterministic codes use an orthogonal mesh or restrict the mesh to a regular triangular grid. Other methods were developed at Commissariat a l'Energie Atomique and Los Alamos National Laboratory but do not solve the problem of sliding between two Lagrangian blocks. Thus, a production code has developed that solves these problems and is able to obtain kappa or k eigenvalues with a good degree of accuracy for such geometries.