Fine-mesh Calculations Research Articles

TIME-DEPENDENT HOMOGENIZATION FOR PRESSURIZED HEAVY-WATER REACTORS

A new time-dependent homogenization approach that accounts for inter-assembly leakage has recently been proposed. The new technique extends Generalized Equivalence Theory (GET) to transient simulations through the use of time-dependent, leakage-corrected discontinuity factors that are calculated at each time step by means of a global-local iterative approach to account for the effect of neighbouring nodes so that highly heterogeneous cores are more accurately modelled than when employing single-node, zero-node-boundary-current Assembly Discontinuity Factors (ADFs). The technique has been previously tested for a one-dimensional, two-energy-group, BWR-like benchmark. The present work expands the analysis to a one-dimensional, two-energy-group, Pressurized Heavy-Water Reactor (PHWR) configuration. The PHWR configuration consists of 22 fuel nodes bounded on either side by two nodes of heavy-water (D2O) reflector. Each fuel node spans 28.575 cm and is a one-dimensional stylized representation of a 37-element, natural uranium fuel bundle with D2O coolant residing in a pressure tube that in turn resides in a calandria tube surrounded by D2O moderator. A simple transient induced by instantaneous half-core voiding of the D2O coolant is studied. Three types of calculations are performed: A reference, heterogeneous-node, fine-mesh calculation, a standardly-homogenized-node calculation and a GET-homogenized-node (using ADFs) calculation. The root-mean-square percent errors introduced by standard homogenization and ADF-based homogenization for kinetics calculations in PHWR cores are found to be 4% and 5%, respectively, after 0.5 s. This suggests that the use of a time-dependent homogenization method is desirable, and its use is shown to reduce the RMS errors to a maximum of 0.003% over the course of the transient. The conclusion is that although PHWR cores are not extremely heterogeneous, the accuracy of transient modelling for PHWRs is improved when using time-dependent homogenization over conventional ADFs and that the newly-developed time-dependent homogenization method promises to offer substantial improvements in accuracy for transient results with particular relevance to safety analyses.

A coupled ALE–AMR method for shock hydrodynamics

We present a numerical method combining adaptive mesh refinement (AMR) with arbitrary Lagrangian–Eulerian (ALE) mesh motion for the simulation of shock hydrodynamics on unstructured grids. The primary goal of the coupled method is to use AMR to reduce numerical error in ALE simulations at reduced computational expense relative to uniform fine mesh calculations, in the same manner that AMR has been used in Eulerian simulations. We also identify deficiencies with ALE methods that AMR is able to mitigate, and discuss the unique coupling challenges. The coupled method is demonstrated using three-dimensional unstructured meshes of up to O(107) tetrahedral cells. Convergence of ALE–AMR solutions towards both uniform fine mesh ALE results and analytic solutions is demonstrated. Speed-ups of 5–10 × for a given level of error are observed relative to uniform fine mesh calculations.

Acceleration of within group iteration for pin-by-pin calculations

In thermal reactor physics fine mesh neutron diffusion calculations in which homogenization is only at the pin cell level are of considerable interest. However, the slow convergence of iterations on fine-mesh spatial grids limits their applicability to practical problems. We present here an efficient way to accelerate the within-group iterations for pin-by-pin calculations based on a two-dimensional heterogeneous variational nodal method (VNM). Response matrix (RM) equations are formulated that incorporate multiple pins within each node. Within the nodes, finite elements in the x-y plane are employed to describe the piecewise constant heterogeneous geometry. On the nodal interfaces orthogonal polynomials are employed to approximate current distributions. The RM equations are solved using the Red-Black Gauss-Seidel (RBGS) iteration. Investigations on the coarse nodes acceleration (CNA) by combining homogenized pin cell nodes into larger heterogeneous nodes are performed. A series of meshing schemes are examined with a 2D small modular reactor core problem. With sufficient interface expansion order, CNA do not cause significant errors to either eigenvalues or fission rate distributions compared with fine mesh calculations. It is demonstrated that CNA accelerate the RBGS iteration significantly and achieves favorable accuracy-efficiency trade-off.

A three-dimensional nodal method with Channel-wise Intrinsic Axial Mesh Adaptation

In a conventional coarse mesh nodal method the more accurate treatment of intra-nodal axial heterogeneity requires iterative axial node re-homogenization using axial flux profiles either reconstructed from core-wise coarse mesh solution or obtained from channel-wise axial fine mesh calculation. In this paper a new nodal method formulation, using Channel-wise Intrinsic Axial Mesh Adaptation (CIAMA), is proposed to solve this problem in a more fundamental way. For a given transverse (radial) leakage, along each axial channel a rigorous sub-node heterogeneous calculation is performed with the explicit axial heterogeneity within each coarse axial node. However, the transverse leakage between the axial channels is still calculated on the basis of coarse axial nodes, using the axially averaged radial current in each coarse axial node. Since the coupling between the axial channels is through the coarse axial nodes, it is not necessary to match the boundaries of the axial sub-nodes of neighboring axial channels in order to incorporate the axial sub-node calculation as an intrinsic part of the whole core global calculation. Therefore in the CIAMA nodal method, each axial channel is allowed to have its own sub-nodes adapting to its own axial heterogeneity variation. The CIAMA method has been implemented in the commercial code EGRET, which is used to qualify CIAMA. Excellent results of modeling fuel grid and control rod movement are presented. Application of CIAMA to three-dimensional pin-by-pin core calculation is also discussed and demonstrated to work well.

Spatial Homogenization of Transport Discretization Schemes

Neutron transport problems for whole reactor core calculations result in a very large amount of unknowns. Some difficulties related to dimensionality of this kind of problem can be resolved by solving a well-posed homogenized transport problem on a coarse grid. We apply homogenization methodology to develop discretization schemes for solving k-eigenvalue problems on coarse meshes in 1D slab geometry. The step characteristic (SC) method is used for fine-mesh transport calculations. We develop spatially homogenized transport schemes on a basis of two different transport discretization methods: SC and linear discontinuous schemes. The basic approach is to formulate a discretization that is spatially consistent with the given fine-mesh transport discretization. The presented numerical results demonstrate features of the developed methods in preserving grid functions of the angular flux computed with the SC method on fine spatial meshes. We study sensitivity of the proposed schemes to various types of perturbations in spatially averaged cross sections and other homogenization parameters.

Applicability of the Diffusion and Simplified P3 Theories for Pin-by-Pin Geometry of BWR

The pin-by-pin fine-mesh core calculation method is considered as a candidate next-generation core calculation method for BWR. In this study, the diffusion and simplified P3 (SP3) theories are applied to the BWR pin-by-pin fine-mesh calculation. The performances of the diffusion and SP3 theories for cell-homogeneous pin-by-pin fine-mesh calculation for BWR are evaluated through comparison with a cell-heterogeneous detailed transport calculation by the method of characteristics (MOC). Two-dimensional, 2 × 2 multi-assemblies geometry is used to compare the prediction accuracies of the diffusion and SP3 theories. The 2 × 2 multi-assemblies geometry consists of 9 × 9 UO2 fuel assemblies that have two different enrichment splittings. To minimize the cell-homogenization error, the SPH method is applied for the pin-by-pin fine-mesh calculation. The SPH method is a technique that reproduces a result of heterogeneous calculation using that of homogeneous calculation. The calculation results indicated that the diffusion theory shows a discrepancy larger than that of the SP3 theory on the pin-wise fission rate distribution. In contrast to the diffusion theory, the SP3 theory shows a much better accuracy on the pin-wise fission rate distribution. The computation time using the SP3 theory is about 1.5 times longer than that using the diffusion theory. The BWR core analysis consists of various calculations, e.g., the cross section interpolation, neutron flux calculation, thermal hydraulic calculation, and burn-up calculation. The function of the calculation time for the neutron flux calculation is usually less than half in the typical BWR core analysis. Therefore, the difference in the calculation time between the diffusion and SP3 theories would have no significant impact on the calculation time of the BWR core analysis. For these reasons, the SP3 theory is more suitable than the diffusion theory and is expected to have sufficient accuracy for the 2 × 2 multi-assemblies geometry used in this study, which simulates a typical situation of the actual BWR core.

Applicability of the Diffusion and Simplified P3 Theories for Pin-by-Pin Geometry of BWR

The pin-by-pin fine-mesh core calculation method is considered as a candidate next-generation core calculation method for BWR. In this study, the diffusion and simplified P3 (SP3) theories are applied to the BWR pin-by-pin fine-mesh calculation. The performances of the diffusion and SP3 theories for cell-homogeneous pin-by-pin fine-mesh calculation for BWR are evaluated through comparison with a cell-heterogeneous detailed transport calculation by the method of characteristics (MOC). Two-dimensional, 2 × 2 multi-assemblies geometry is used to compare the prediction accuracies of the diffusion and SP3 theories. The 2 × 2 multi-assemblies geometry consists of 9 × 9 UO2 fuel assemblies that have two different enrichment splittings. To minimize the cell-homogenization error, the SPH method is applied for the pin-by-pin fine-mesh calculation. The SPH method is a technique that reproduces a result of heterogeneous calculation using that of homogeneous calculation. The calculation results indicated that the diff...

Impact of Pin-by-Pin Thermal-Hydraulic Feedback Modeling on Steady-State Core Characteristics

In this paper, the effect of a pin-by-pin thermal-hydraulic feedback treatment on the core characteristics at a steady-state condition is investigated using a three-dimensional fine-mesh core calculation code. Currently, advanced nodal codes treat the inside of an assembly as homogeneous, and the temperature distribution inside a node is usually ignored. Namely, the fuel temperature is estimated from the assembly average power density, and the moderator temperature is calculated from the nodewise closed-channel model. However, the validity of a flat temperature distribution inside a node has not yet been investigated, because a three-dimensional pin-by-pin whole-core calculation must be done for comparison. A three-dimensional pin-by-pin nodal-transport code for a pressurized water reactor (PWR) core analysis, SCOPE2, was used in this study since it can directly treat the pin-by-pin feedback effect. A whole-core subchannel analysis code was developed to enhance the thermal-hydraulic capability of SCOPE2. The pin-by-pin feedback models for fuel and moderator temperature were established, and their impact on the core characteristics was investigated in a 3 × 3 multiassembly and the whole PWR core geometries. The calculations showed that modeling of the pin-by-pin temperature distribution revealed a negligible effect on core reactivity and only a slight impact on the radial peaking factor. The difference in the radial peaking factor that is exposed by the pin-by-pin temperature modeling is less than 0.005 in the test calculations.

A Variational Nodal Expansion Method for the Solution of Multigroup Neutron Diffusion Equations with Heterogeneous Nodes

A new nodal diffusion method for the neutronics analysis of light water reactor cores has been developed. The method is based on an expansion of neutron fluxes within a node into a series of functions that are numerically obtained from single-assembly calculations without the process of assembly homogenization. The assembly heterogeneity effect can be taken into account in whole-core calculations in a consistent way with the heterogeneous single-assembly calculations, providing highly accurate results including intranodal pin-power distributions. The expansion coefficients are determined by a classical Ritz procedure in such a way that the solution becomes the most accurate—in the least squares sense—approximation to the exact solution. The present method was implemented in a two-dimensional nodal diffusion code and tested for benchmark cases both for boiling water reactors and pressurized water reactors. The root-mean-square errors of both node average powers and nodal maximum pin powers were observed to be <1%, with computing time of less than a few percent of the reference, fine-mesh calculation. It was also observed that the accuracy of the present method could be improved to almost any desired degree only by increasing the order of expansion polynomials.

Parallel computations of finite element output bounds for conjugate heat transfer

This paper investigates the a posteriori finite element bound method applied to a heat transfer problem in a multi-material electronic components array. The temperature field is obtained by solving Poisson equations and convection–diffusion equations in different regions of the computational domain. The bound method calculates very sharp lower and upper bounds of the temperature of the hottest component which is assumed to be the engineering output of interest. This paper shows that for this two-dimensional problem the bound method can yield more than an 80-fold reduction in simulation time over a fine mesh calculation (330,050 d.o.f.) while still maintaining quantitative control over the accuracy of the engineering output of interest. Parallel implementation on a Beowulf cluster is also reported.

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.

Nonlinear 3D numerical field calculation of a magnetic recording head

Methods for calculating the field of a magnetic recording head with nonlinear B-H characteristics in three dimensions are described. The mesh-refinement iteration technique finite integration technique, matrix difference-splitting iteration technique, and fine-mesh calculation technique are applied to save storage and iteration time. The field and permeability distributions inside interesting domains are presented. >

Disk-instability model for outbursts of dwarf novae: Fine mesh calculations

We have executed fine mesh calculations which can almost fully resolve the transition front in the accretion disk of dwarf novae. Results show that the effects of thermal diffusion become very important only when the heating wave passes by. But it is unlikely that they cause drastic changes in the situation of wave propagation. The validity of the localized front approximation is examined. It is found that this approximation is relatively good for heating waves but it is marginal for cooling waves.

A Two-Dimensional Two-Group Albedo Model for Pressurized Water Reactor Reflector

A two-dimensional, two-group albedo model is developed for replacement of a pressurized water reactor reflector. Spatially dependent albedos in the vicinity of an outward concaving corner of the core boundary are derived using the same formulation and technique developed by the authors for the theory of the interface flux nodal method. The model is mathematically simple and has been successfully tested for applications in both fine-mesh finite difference and coarse-mesh nodal calculations, for cases of a reflector with and without a shroud.

The Design of Reload Cores Using Optimal Control Theory

A formal approach for the optimization of the final design of reload cores has been devised and verified. The method is based on applying the calculus of variations (Pontryagin's principle) to the normal flux and depletion system equations. The resulting set of coupled system, Euler-Lagrange (E-L), and optimality equations are solved iteratively. This is done by assuming a loading pattern for the old fuel, first solving the system equations, and then the E-L equations. The pattern is then modified by using the optimality (or Pontryagin) condition, and the process is repeated until no further improvements can be made. A computer program, OPMUV, implementing these procedures has been written and verified. The code can handle two-dimensional, quarter-core symmetric configurations with up to 241 assemblies and 4 nodes per assembly with modified one-group theory. It also has the capability of optimizing over the entire depletion cycle as well as just at the beginning of cycle (BOC). The results show that the procedure does work. In all cases tried, the method led to a reduction in nodal peaks of 1 to 3% over the final designer-obtained loading pattern within a couple of iterations. These savings carry over to comparable reductions in pin peaksmore » when the optimized patterns are used in four-group, fine-mesh calculations. Since the changes on each iteration are limited to ensure convergence, the method is thus well suited for the final fine tuning of the normally obtained patterns to gain an extra few percent in power flattening.« less

The constrained finite element approach to coarse-mesh transport computations

Coarse-mesh computational methods have attracted widespread interest for both diffusion theory and for higher-order transport approximations. Fine-mesh diffusion calculations for the analysis of large, neutronically loosely-coupled cores such as the BWR shown in Fig. 1 are often prohibitively expensive for routine use. Therefore, techniques that reduce the spatial grid points to a manageable number without resulting in an inordinate loss of accuracy are of considerable interest. A variety of nodal methods have been developed for the treatment of PWR cores for this reason. For the BWR core shown in Fig. 1, however, these methods are not nearly as effective, for they require that the fuel assembly cross sections be homogenized before the global calculation is carried out. But such homogenization obliterates the strong coupling of the local spectral effects due to the water gaps and control rods with the global flux variation. Neither are fine-mesh transport methods effective for LWR fuel assembly problems, such as the configuration shown in Fig. 2, without the individual fuel cells first being homogenized. In such problems, containing burnable poisons and control-rod slots, the explicit representation of the many circular fuel-coolant

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.

Effective one-group coarse-mesh method for calculating three-dimensional power distribution in fast reactors

An effective one-group coarse-mesh method for calculating three-dimensional power distribution in fast reactors has been developed. This method uses direction dependent one-group diffusion coefficients in each region so as to preserve neutron leakage rates before and after energy condensation. Furthermore, to correct coarse meshes of one point per hexagonal assembly used in the three-dimensional diffusion calculation, one-group cross-sections are modified by applying Askew's method. Based upon this method an effective three-dimensional diffusion code has been made. Results obtained in test cases on a prototype fast reactor indicate that this method is as accurate as six-group fine-mesh diffusion calculations using six mesh points per assembly in each radial plane and the solution of this method is a factor of 40 faster than the six-group fine-mesh calculations.