Mesh Finite Difference Research Articles

3D Real-Time Simulation of Reactor Core based on Discontinuity Factor Ratio

During the development of full scope simulator (FSS) in nuclear power plant, it is difficult to satisfy with high precision and efficiency of the 3D distribution calculation of reactor power. The accuracy of conventional coarse mesh finite difference method (CMFD) cannot meet the requirement if the nodal size is large. According to the generalized equivalence theory (GET), the finite difference solution of diffusion equation based on discontinuity factor correction was deduced, the calculation method of the discontinuity factor and boundary albedo are presented through the process of assembly homogenization. Through the calculation of typical LMW benchmark problem, the simulation results showed that using CMFD with discontinuity factor ratio (CMFD-DFR) to solve the time-dependent neutron kinetic equation, can not only improve the simulation precision, and can keep the calculation speed, and can meet the requirement of reactor core real-time model of FSS.

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes.

Finite-element analysis of top-casing electric source method for imaging hydraulically active fracture zones

Imaging hydraulically active fracture zones (HAFZ) is of paramount importance to subsurface resource extraction, geologic storage, and hazardous waste disposal. We have developed advanced 3D finite-element (FE) electrical imaging algorithms for HAFZ in the presence of a steel-cased well. The algorithms use tetrahedral FE meshes in the simulation domain and coarse rectangular finite-difference meshes in the imaging domain. This heterogeneous dual-mesh approach is well suited to modeling the multiscale earth model due to steel-cased wells. We find that the algorithms accurately and efficiently simulate surface electric field measurements over a 3D HAFZ at depth when one end point of a surface electric source is connected to a wellhead. For brevity, this configuration is called the top-casing electric source method. By replacing a hollow cased well with a solid prism, we improve our computational efficiency without affecting the solution accuracy. The sensitivity of the top-casing source method to HAFZ highly depends on the continuity of a steel-cased well because it makes currents preferentially flow to HAFZ. The sensitivity also depends on conductivity structures around the well because they control current leaking from the steel-cased well. We find that the method can image a localized HAFZ and detect changes in its width and height. The imaging results are improved when a volume of the imaging domain is constrained from geomechanical perspectives. A primary advantage of the method is the fact that the sources and receivers are placed on the surface, thus not interrupting the well operation.

Stabilization of multi-group neutron transport with transport-corrected cross-sections

Many deterministic neutron transport solvers rely on the source iteration method to solve the multi-group neutron transport equation. Often, these solvers rely on transport corrected cross-sections for accurate prediction of the neutron flux distribution. Transport corrected within-group scattering cross-sections can become negative, and if these negative cross-sections are large in magnitude, source iteration can become unstable and fail to converge for certain cases. In this study, we present evidence of this convergence issue for the method of characteristics (MOC) on full-core PWR problems with common transport correction schemes. A theoretical discussion is presented to illustrate the reason for the convergence issues. Previously established stabilization methods are compared with a newly proposed stabilization method. Results show that the new stabilization method allows for faster convergence than previous techniques. In addition, the effect of Coarse Mesh Finite Difference (CMFD) acceleration on stability is analyzed, showing that CMFD acceleration with full-group structure can overcome the convergence issues, but a stabilization technique is necessary for convergence when a condensed group structure is used.

A robust fitted numerical method for singularly perturbed turning point problems whose solution exhibits an interior layer

In this paper we considerer singularly perturbed convection-diffusion-reaction problems with a turning point whose solution exhibits an interior layer. We establish bounds on the solution to these problems and their derivatives. We construct a fitted mesh finite difference method (FMFDM). The method consists of an upwind scheme on an appropriately designed piecewise uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. A rigorous error analysis shows that the developed method is uniformly convergent of order almost one. In order to improve the accuracy of the proposed FMFDM, we apply Richardson extrapolation. Two numerical examples are considered to illustrate the theoretical findings. Results show that this method is reliable and competitive.

Computational Investigation of the Effect of Reinforcement Schemes and Angles on the Stress-Strain State and Strength of Composite Cylinders Under Axisymmetric Internal Explosion. Part 1. Effect of the Discretization Spacings of the Computational Domain on the Accuracy of Determination of Stress-Strain State and Strength

The effect of two-dimensional regular finite-difference mesh spacings on the accuracy of calculation of the dynamic axisymmetric stress-strain state and strength of hollow composite cylinders of finite length, fixed overall dimensions and thickness has been numerically determined by the Wilkins method modified for helical orthotropy and implemented in an application package, created earlier by the authors. The cylinders were made by a ribbon consisting of VMPS glass fibers, impregnated with an EDT-10 epoxy binder, on a technological mandrel. Loading is done by exploding a spherical explosive charge in the center of symmetry of a cylinder in an air atmosphere. The obtained results allow one to choose mesh spacings along the radial and axial coordinates, which ensure an acceptable accuracy of determination of the maximum values of hoop stresses and strains, as well as strength functions by maximum stress and strain criteria and by the generalized von Mises criterion.

Performance evaluation of CMFD on inter-cycle correlation reduction of Monte Carlo simulation

This paper presents a performance examination of the Coarse Mesh Finite Difference (CMFD) method for a reduction of inter-cycle correlations in Monte Carlo (MC) active cycle simulations. Sensitivity tests using the UNIST in-house MC code ‘MCS’ reveal the characteristics of CMFD effectiveness depending on problem sizes, dimensions, and cross-section dependency on neutron energy (multi-group vs. continuous). Also, by applying this method to the three-dimensional BEAVRS whole core problem, it is successfully demonstrated that the CMFD can reduce the inter-cycle correlations of practical, large size, and high dominance ratio problems, and that the figure of merit of efficiency for a practical reactor problem increases by a factor of six.

3D whole-core neutron transport simulation using 2D/1D method via multi-level generalized equivalence theory based CMFD acceleration

A new 2D/1D method with multi-level generalized equivalence theory (GET) based coarse mesh finite difference (CMFD) acceleration was proposed for the whole core transport calculation in this paper. Fouressential features of this new 2D/1D methods are as follows: (1) Two new defined factors, nodal discontinuity factor (NDF) and modified diffusion coefficient factor (MDF), were defined in this new GET based CMFD method to achieve equivalence between CMFD and the transport solutions and especially insure that only positive coupling coefficients would occur in the CMFD linear system; (2) For accelerating the convergence of the CMFD, a multi-level acceleration scheme was implemented and an innovative self-developed RSILU preconditioned GMRES solver was developed to solve the CMFD linear system; (3) Within the new CMFD framework, 2D MOC and 1D two-node nodal expansion solvers were developed for 3D whole core transport calculation and the sub-plane technique was applied to minimize the nodal error successfully while maintain a reasonable computing time; (4) For implementation, transverse leakage splitting technique was applied to avoid the total source to become negative, and domain decomposition method based on MPI was implemented to take advantages of high performance clusters. The accuracy of this 2D/1D method via multi-level GET based CMFD and the effectiveness and acceleration performance of the multi-level CMFD method were examined for the well-known C5G7 benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level acceleration schemeis efficient and enhances the converge speed of both the GET based CMFD acceleration and the MOC calculation.

Functionalization of the Discontinuity Factor in the Albedo-Corrected Parameterized Equivalence Constants (APEC) Method

The Albedo-corrected Parameterized Equivalence Constants (APEC) method, a new leakage correction method for two-group nodal analysis of light water reactors, has been extended to discontinuity factor (DF) correction. First, the error of nodal calculations induced by an inaccurate assembly discontinuity factor (ADF) is evaluated using the reference two-group cross section (XS) and DF calculated from heterogeneous core transport calculations. Functionalization of DF is performed by finding relationships between surfacewise current-to-flux ratio and change of DF from ADF. The least-squares method is used to fit several candidate functions to various core calculation results. The coefficients of APEC XS and DF correction functions are determined considering several color-set models. In this work, the two-dimensional method of characteristics–based lattice code DeCART2D is used for reference core calculations and lattice calculations. The extended APEC method is implemented in an in-house NEM nodal code using the partial-current coarse mesh finite difference acceleration. A small modular reactor (SMR) initial core benchmark is analyzed to evaluate the performance of the extended APEC method. In addition, the extended APEC method is applied to several variants of the SMR core and large variants to assess its general applicability.

Numerical simulation for the investment casting process of a large-size titanium alloy thin-wall casing

To optimize the investment casting process when producing high quality large-size titanium alloy thin-wall components is a time-consuming job due to the complicated metallurgical process. Numerical simulation is a high-efficiency method compared with trial and error, and therefore is introduced to the investment casting process optimization to shorten the new product development cycle and reduce the production cost. In this study, weakly compressible model (WCM) and ununiformed finite difference mesh (UFDM) was developed to reduce the memory consumption and ensure the simulation efficiency. The precision of the WCM and UFDM were verified by numerical simulation of cavity heat convection in a square cavity and hydraulics simulation of centrifugal filling in a transparent cavity. The numerical simulation of the investment casting process of a titanium alloy thin-wall casing under different process conditions was accomplished using a self-developed software, and the distribution characteristics of potential shrinkage defects were predicted. It was found that the predicted defects in the titanium alloy casing matched well with the actual X-ray experimental results. For the components investigated in this paper, more numerical simulation results show that the centrifugal casting process with respect to gravity casting had no obvious improvements in the concentrated shrinkage defects, and the gravity casting process can be more reasonable from the engineering point of view.

Unstructured coarse mesh finite difference method to accelerate k-eigenvalue and fixed source neutron transport calculations

Unstructured coarse mesh finite difference (CMFD) method was implemented to accelerate the k-eigenvalue and fixed source neutron transport calculations in the nodal SN transport code DNTR, which was based on arbitrary triangular-z meshes. The same triangular-z meshes were adopted in the CMFD calculation to accommodate the arbitrary problem boundaries and simplify the coarse mesh generation and mapping processes. Two-level CMFD formulations were derived for the k-eigenvalue problems, and the few-group CMFD calculation was used to speed up the convergence of the multigroup CMFD calculation. For the fixed source problems in the neutronics analyses of subcritical reactors or transient neutron transport calculations, a special ks iteration method by introducing the source multiplication factor ks was employed in the SN transport calculation to remove the dependence of the convergence rate on the subcriticality. Multigroup CMFD formulations were established to accelerate the convergence of the ks iteration procedure in the fixed source neutron transport calculation. The acceleration effects of CMFD on the k-eigenvalue and fixed source calculations were verified using four representative neutron transport problems, i.e. the small fast breeder reactor with Cartesian boundaries, the BN-600 fast reactor with hexagonal boundaries, the space nuclear power reactor with cylindrical boundaries, and the accelerator driven subcritical reactor with hexagonal boundaries. The numerical results showed that for the first three eigenvalue problems, the multigroup CMFD achieved a speedup of total computation time by a factor of 2–5, and the two-level CMFD reduced the computation time by a factor of 3–7. It was also shown that the multigroup CMFD was able to accelerate the ks iteration procedure in the fixed source problems by a factor of about 6. The implementation of the unstructured CMFD method turns the DNTR code into an efficient neutron transport solver with flexible geometry treatment capability for both k-eigenvalue and fixed source problems.

A modified predictor-corrector quasi-static method in NECP-X for reactor transient analysis based on the 2D/1D transport method

The 2D/1D transport method is a widely applied method for the whole-core heterogeneous transport calculation. The transient simulation based on the pin-resolved 2D/1D method is implemented in the high-fidelity code NECP-X. The 3D transient fixed source problem (TFSP) is transformed into 2D TFSP and 1D TFSP coupled by leakage, and the pin-based coarse mesh finite difference method (CMFD) is used to accelerate convergence of the 2D/1D TFSP. In this paper, a modified predictor-corrector quasi-static (mPCQS) method is developed to provide better reactivity prediction for PK calculations. Furthermore, the simplified source scaling method (SSSM) is used to accelerate the convergence of the CMFD TFSP iterations. The results of three problems are presented including the 2D C5G7-TD benchmark, a MiniCore-3D rod-ejection problem and a 3D heterogeneous rod-ejection problem. Numerical results reveal that the SSSM method can significantly reduce the number of CMFD iterations without increasing the computational complexity, and the mPCQS method can provide better results than the standard PCQS for step perturbation transient event and rod ejection transient event no matter in the homogeneous case or in the heterogeneous case.

The Design of 3D-Printed Lattice-Reinforced Thickness-Varying Shell Molds for Castings.

3D printing technologies have been used gradually for the fabrication of sand molds and cores for castings, even though these molds and cores are dense structures. In this paper, a generation method for lattice-reinforced thickness-varying shell molds is proposed and presented. The first step is the discretization of the STL (Stereo Lithography) model of a casting into finite difference meshes. After this, a shell is formed by surrounding the casting with varying thickness, which is roughly proportional to the surface temperature distribution of the casting that is acquired by virtually cooling it in the environment. A regular lattice is subsequently constructed to support the shell. The outside surface of the shell and lattice in the cubic mesh format is then converted to STL format to serve as the external surface of the new shell mold. The internal surface of the new mold is the casting’s surface with the normals of all of the triangles in STL format reversed. Experimental verification was performed on an Al alloy wheel hub casting. Its lattice-reinforced thickness-varying shell mold was generated by the proposed method and fabricated by the binder jetting 3D printing. The poured wheel hub casting was sound and of good surface smoothness. The cooling rate of the wheel hub casting was greatly increased due to the shell mold structure. This lattice-reinforced thickness-varying shell mold generation method is of great significance for mold design for castings to achieve cooling control.

Resistivity inversion of transversely isotropic media

In this paper, we have suggested a finite difference algorithm for resistivity forward and inverse modeling with electrically anisotropic media in applied geophysics. Electrical anisotropy affects the surface measurement in a fashion that may make interpretation erroneous. This means that the resistivity section obtained by an inversion method that does not incorporate electrical anisotropy gives the wrong subsurface structure. We used a classical multielectrode dataset of a profile for estimating not only horizontal but vertical resistivity as well. Thus, the electrical anisotropy can be calculated. Finally, the finite difference mesh can be corrected by using the estimated anisotropic coefficient. The result of the developed algorithm was verified with a 2-layered analytic solution. Furthermore, the present method was also tested on a field dataset.

The Numerical Simulation of the Shock Wave of Coal Gas Explosions in Gas Pipe*

For the problem of large deformation and vortex, the method of Euler and Lagrange has both advantage and disadvantage. In this paper we adopt special fuzzy interface method(volume of fluid). Gas satisfies the conditions of conservation equations of mass, momentum, and energy. Based on explosion and three-dimension fluid dynamics theory, using unsteady, compressible, inviscid hydrodynamic equations and state equations, this paper considers pressure gradient’s effects to velocity, mass and energy in Lagrange steps by the finite difference method. To minimize transport errors of material, energy and volume in Finite Difference mesh, it also considers material transport in Euler steps. Programmed with Fortran PowerStation 4.0 and visualized with the software designed independently, we design the numerical simulation of gas explosion with specific pipeline structure, check the key points of the pressure change in the flow field, reproduce the gas explosion in pipeline of shock wave propagation, from the initial development, flame and accelerate the process of shock wave. This offers beneficial reference and experience to coal gas explosion accidents or safety precautions.

Multi-level coarse mesh finite difference acceleration with local two-node nodal expansion method

A computationally efficient and effective multi-level coarse mesh finite difference (CMFD) acceleration is proposed using the local two-node nodal expansion method for solving the three-dimensional multi-group neutron diffusion equation. The multi-level CMFD acceleration method consists of two essential features: (1) a new one-group (1G) CMFD linear system is established by using cross sections, flux and current information from the multi-group (MG) CMFD to accelerate the MG CMFD calculation, (2) an adaptive Wielandt shift method is proposed to accelerate the inverse power iteration of 1G CMFD in order to provide an accurate estimate of the eigenvalue at the beginning of the iteration for both 1G and MG CMFD linear system. Additionally, a nodal discontinuity factor and a diffusion coefficient correction factor are defined to achieve equivalence of the 1G and MG CMFD system. The accuracy and acceleration performance of multi-level CMFD are examined for a variety of well-known multi-group benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level CMFD acceleration method is efficient, particularly for the larger, multi-group systems.

“Virtual density” and traditional boundary perturbation theories: Analytic equivalence and numeric comparison

We compare and contrast “virtual density” perturbation theory with the traditional boundary perturbation theory developed by Pomraning, Larsen, and Rahnema in the context of diffusion theory. First, after reviewing that literature, we mathematically prove that virtual density perturbations and traditional boundary perturbations are precisely equivalent for arbitrary 1-D problems, which constitute non-uniform isotropic expansions. We also mathematically prove that these two perturbation theories are equivalent for 2-D boundary shift problems, which constitute non-uniform anisotropic expansions. Extension of this proof to swellings or 3-D problems is straightforward. We compare the two theories numerically for a series of alternating uranium and sodium 1-D slabs in finite difference diffusion, and we show that virtual density theory predicts reactivities much more accurately and efficiently than traditional boundary perturbation theory. Boundary perturbation theory is often very inaccurate on a coarse mesh but converges to the virtual density solution as the mesh becomes finer. We also compare the two theories for axial assembly swelling in an abbreviated FFTF benchmark with a coarse mesh. Here we find that reactivity coefficients obtained via virtual density perturbation theory agree with reference solutions to within 0.1%, while those obtained via boundary perturbation theory exhibit sporadic accuracy – sometimes in the range of 1–5% error, more frequently in the range 5–20% error, and occasionally well over 100% error in control rod assemblies. We conclude that although virtual density perturbation theory and boundary perturbation theory are analytically equivalent, boundary perturbations in diffusion theory are often thwarted in coarse mesh finite difference solutions due to inaccurate flux gradients along mesh cell surfaces in heterogeneous cores.

Acceleration and Real Variance Reduction in Continuous-Energy Monte Carlo Whole-Core Calculation via p-CMFD Feedback

In the three-dimensional (3-D) continuous-energy whole-core reactor analysis, the partial current–based coarse mesh finite difference (p-CMFD) feedback was applied to the Monte Carlo (MC) k-eigenvalue problem simulation for both inactive and active iterations (cycles). To reduce the stochastic errors in the p-CMFD parameters and their biases due to the ratio-type estimators, the first-in-first-out (FIFO) accumulation scheme was introduced in the MC/p-CMFD procedure. The MC/p-CMFD procedure was tested on a typical pressurized water reactor 3-D continuous-energy whole-core problem while varying the FIFO queue lengths and the results were compared with the conventional power iteration. The Shannon entropy analysis showed that MC/p-CMFD accelerates the convergence of the fission source distributions and mitigates the spatial clustering phenomenon. The real variance analysis also showed that MC/p-CMFD reduces the interiteration correlation, leading to the most real variance reduction in the local MC tallies at the optimum queue length (L = 5). It was also shown that a nontrivial bias was introduced by the p-CMFD feedback, especially for the global tally (keff) with L = 1. However, the bias decreased as the tally bin size became smaller and it was effectively reduced by increasing the queue length (L ≥ 5). In conclusion, the MC/p-CMFD procedure showed promising capability for 3-D continuous-energy whole-core reactor analysis by MC simulation.

Space-Dependent Wielandt Shifts for Multigroup Diffusion Eigenvalue Problems

We present a new concept—the space-dependent Wielandt shift (SDWS)—for accelerating the convergence of the power iteration (PI) scheme for multigroup diffusion k-eigenvalue problems. The SDWS improves on standard Wielandt shift (WS) techniques, which are empirical in nature and are typically effective only when the current estimate of the solution is reasonably converged. By accounting for the physics of the problem through SDWS, we are able to improve the acceleration for the initial iterates when the current estimate of the solution is not close to convergence. Numerical results from one-dimensional problems suggest that, compared to standard WS techniques, the new SDWS techniques can provide upward of a 46% reduction in the number of PIs required for convergence and a 40% reduction in the computational time required. This improvement is sensitive to several problem-dependent factors, such as the geometry and energy-dependence of the problem, the spatial discretization, and the initial guess. The reduction in computational time is also dependent on the linear solver in the PI scheme, as it is well known that WSs can significantly worsen the conditioning of the diffusion linear system. In this paper, we provide a detailed study of the impact of WSs on the performance of several iterative linear solvers. Results from our implementation of SDWS in the three-dimensional (3D) code MPACT show that SDWS can provide similar speedups for 3D multigroup diffusion eigenvalue problems. These results also show that moderate speedups can be obtained by applying SDWS to the coarse mesh finite difference (CMFD) solver in a CMFD-accelerated transport scheme. However, the benefit of doing this may be limited because all but the first few CMFD solves are relatively easy to converge, regardless of the WS used.

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Abstract In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.