Rational Chebyshev Approximation Research Articles

Probabilistic approach of solving burnup problems

This paper presents a new approach to solve nuclear burnup problems using a probabilistic method. Unlike the traditional methods that rely on complex matrix exponential calculations, the proposed method tracks the transformation of nuclei over a period of time, enabling the estimation of nuclide concentrations. The method is implemented in a C++ program CNUCTRAN and verified against the Chebyshev Rational Approximation Method (CRAM). Three sample calculations are presented, showing a detailed comparison of results obtained using CNUCTRAN and CRAM48. The relative error analysis emphasizes the accuracy of the probabilistic method, with the most significant error observed below 0.001%. The computational efficiency is discussed in terms of CPU time, indicating that CNUCTRAN requires more CPU time than CRAM but with improved accuracy as the time step decreases. The numerical results for various burnup problems demonstrate the accuracy and efficiency of the probabilistic method, making it a promising alternative for simulating realistic nuclear transmutations.

Application of the Gauss-Seidel Method to the Chebyshev Rational Approximation Method for Solving Nuclear Fuel Depletion Systems

The Chebyshev Rational Approximation Method (CRAM) has become one of the dominant methods for solving the Bateman equations for nuclear fuel depletion analysis. Since its introduction over a decade ago, several improvements have been made to CRAM improving its accuracy and reducing its run time. We analyzed its run time using two previously published methods for solving the CRAM system of equations, direct matrix inversion (DMI) and sparse Gaussian elimination (SGE), for depletion systems of varying numbers of nuclides to see how the two methods perform relative to one another. In addition to these two methods, we introduced the Gauss-Seidel (GS) method for solving the CRAM system of equations and compared its performance relative to DMI and SGE for depletion systems with varying numbers of nuclides. We demonstrated that for practical purposes, GS is faster than SGE and DMI and achieves a practical level of accuracy. All testing was performed using the CRAM implementation in the Griffin reactor physics analysis application.

Error analyses and evaluation for solving burnup equations with MMPA method

The Mini-Max Polynomial Approximation (MMPA) is a new method to solve burnup equations, which has attracted much attention due to its advantages of good computational accuracy and efficiency. Understanding its error mechanism is significant for its application in practical burnup calculations. In this research, a mathematical model is proposed to clarify the error mechanism of the MMPA method for the burnup calculation. Then the coefficients of the MMPA method with different orders are given by the Remez algorithm. The errors of the MMPA method with different orders in burnup calculation are compared and analyzed, especially in single-step burnup calculation and infinite-step burnup calculation. Three cases are selected to verify the correctness of the mathematical error model, including fresh fuel burnup, depleted fuel burnup and decay calculation. The results show that when the order is greater than 36, the calculation accuracy of the MMPA method is less than the numerical noise of double precision calculation. As the order increases, the step length sensitivity of the MMPA method decreases. Finally, the computational accuracy and computational efficiency of the MMPA method with different orders are compared with those of the 16th order Chebyshev Rational Approximation Method (CRAM). The accuracy of the 16th order CRAM is between the 32nd and 36th order MMPA method. And the computation time of MMPA methods below order 76 is less than that of the 16th order CRAM. The theoretical analysis results of the mathematical error model are consistent with the numerical results. Then several best practices for solving burnup equations using the MMPA method are presented: (1) It is recommended to use the MMPA method with orders above 36 and below 76, whose accuracy and efficiency are better than 16th order CRAM in different degrees. (2) In multi-physics coupled computation which may require a strategy of large step length calculation, increasing the order of the MMPA method can not only improve the computational accuracy but also improve the computational efficiency.

Practical methods for GPU-based whole-core Monte Carlo depletion calculation

Several practical methods for accelerating the depletion calculation in a GPU-based Monte Carlo (MC) code PRAGMA are presented including the multilevel spectral collapse method and the vectorized Chebyshev rational approximation method (CRAM). Since the generation of microscopic reaction rates for each nuclide needed for the construction of the depletion matrix of the Bateman equation requires either enormous memory access or tremendous physical memory, both of which are quite burdensome on GPUs, a new method called multilevel spectral collapse is proposed which combines two types of spectra to generate microscopic reaction rates: an ultrafine spectrum for an entire fuel pin and coarser spectra for each depletion region. Errors in reaction rates introduced by this method are mitigated by a hybrid usage of direct online reaction rate tallies for several important fissile nuclides. The linear system to appear in the solution process adopting the CRAM is solved by the Gauss-Seidel method which can be easily vectorized on GPUs. With the accelerated depletion methods, only about 10% of MC calculation time is consumed for depletion, so an accurate full core cycle depletion calculation for a commercial power reactor (BEAVRS) can be done in 16 h with 24 consumer-grade GPUs.

Evaluating Quantities of Interest Other Than Nuclide Densities in the Bateman Equations

Traditionally, analysts solve the Bateman depletion equations to calculate the nuclide number density (NND) of each nuclide since these densities impact other reactor parameters, such as reactivity, as they change. Many quantities of interest, such as radiation damage, are calculated using simple integration methods, assuming that the NNDs are constant over a given depletion interval. However, the NNDs are time dependent, which can be accurately represented only by the Bateman depletion equations. We propose that these quantities can be calculated simultaneously with the NNDs within the Bateman depletion equations, preserving the coupled nature of these quantities to the time-dependent NNDs. We implemented this functionality in Griffin, demonstrating that only minor code modifications were necessary in order to accommodate an evaluation of these quantities in the Bateman depletion equations. The Chebyshev Rational Approximation Method was used to successfully solve for these additional quantities in the Bateman depletion equations. For radiation damage, the results calculated by Griffin were very accurate, differing by less than 2.5% from an analytical benchmark. For other quantities, the discrepancy between quantities calculated by the Bateman depletion equations versus those calculated by the Forward Euler method exceeded 10% for decay energy and 2% for fissions per initial heavy metal atom and kinetic energy released per unit mass when few depletion intervals were used. As the number of depletion intervals increased, both methods began to converge as expected.

RATIONAL AND GENERALISED RATIONAL CHEBYSHEV APPROXIMATION PROBLEMS AND THEIR APPLICATIONS

RATIONAL AND GENERALISED RATIONAL CHEBYSHEV APPROXIMATION PROBLEMS AND THEIR APPLICATIONS

Numerical solution of fractional Volterra integral equations based on rational Chebyshev approximation

Numerical solution of fractional Volterra integral equations based on rational Chebyshev approximation

Methods and performance of a GPU-based pinwise two-step nodal code VANGARD

The methods and performance of a GPU-based pinwise two-step nodal core calculation code VANGARD are presented which is targeted for use in the commercial nuclear design analyses. In order to realize high-speed core cycle depletion with a single consumer-grade GPU mounted on ordinary personal computers, various elaborated computational methods are introduced as follows. The computationally intense pinwise nodal calculation is optimized by adaptive employment of low-order expansion. The two-level surface current correction method is used for updating pinwise incoming partial currents within the assembly-level coarse mesh finite difference framework. A cross section data compression technique is introduced to fit the enormous pinwise group constant data into the limited GPU memory. A massively parallelized depletion scheme based on the Chebyshev Rational Approximation Method is implemented. The neighbor-informed burnup correction method is developed to ensure the accuracy of the pinwise depletion for gadolinia-bearing pins. Performance examinations are performed for the realistic APR1400 and AP1000 core problems to demonstrate that three-dimensional pinwise cycle depletion can be completed within 3 minutes with high accuracy and resolution.

Introduction of the Adding and Doubling Method for Solving Bateman Equations for Nuclear Fuel Depletion

This paper introduces and evaluates the Adding and Doubling Method (ADM) for solving the Bateman equations for depletion systems with varying numbers of nuclides and compares it to the Chebyshev Rational Approximation Method (CRAM), both implemented in the reactor physics analysis application Griffin. ADM, when applied to the Crank-Nicolson Finite Difference method, can produce results comparable in accuracy and precision to CRAM with comparable run times for systems with 35 or 297 nuclides. For systems with more than 300 nuclides, the matrix-matrix operations required by ADM are significantly more costly than the matrix-vector operations required by CRAM, making CRAM the more efficient method for systems with large numbers of nuclides. ADM is an accurate method that maintains other advantages over CRAM in that it does not depend on pre-generated coefficients or require complex number operations. ADM also manages to outperform CRAM by a factor of more than 250 in terms of run time for depletion systems that require multiple Bateman solves while the depletion matrix and time step size remain constant over all depletion intervals.

Investigation and improvement of the Mini-Max Polynomial Approximation method for solving burnup equations

In this work, a new efficient solution algorithm named Combined Matrix Iterative Algorithm based on the Mini-Max Polynomial Approximation (MMPA) method is proposed for burnup calculations. As a new matrix exponential method in recent years, MMPA method has good numerical stability and computational accuracy, and all calculation are real number operations. As the same with Chebyshev Rational Approximation Method (CRAM) which is widely and successfully used in solving burnup equations, the method can also directly deal with rigid and sparse burnup matrices. Compared with the traditional solution algorithms based on MMPA method, the Combined Matrix Iterative Algorithm (MMPA-CMIA) proposed in this paper can effectively reduce the computational complexity and improve the computational efficiency after equivalently transforming the expression of the MMPA method. Then the MMPA-CMIA is used to theoretically demonstrate and analyze the solution efficiency of random matrices with different orders which has the similar distribution characteristics to burnup matrices. Compared with the 16-order CRAM method and the other two traditional solution algorithms based on MMPA method, the algorithm shows good acceleration effect and computational precision. Finally, the algorithm is used in the actual burnup benchmarks to verify and validate the accuracy and efficiency. The burnup matrices of different orders are constructed through detailed burnup chain and simplified burnup chain. The results show that the algorithm has good accuracy and numerical stability. And comparing with 16-order CRAM, the algorithm can save about 30% of computing time for solving burnup equations. It shows that the algorithm is a potentially powerful method for accelerating the solution of large-scale burnup calculations.

Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation

In this work, we investigate the poroelastic waves by solving the time‐domain Biot‐JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson‐Koplik‐Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time‐domain Biot‐JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square‐root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge‐Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.

A spent nuclear fuel source term calculation code BESNA with a new modified predictor-corrector scheme

This paper introduces a new point depletion-based source term calculation code named BESNA (Bateman Equation Solver for Nuclear Applications), which is aimed to estimate nuclide inventories and source terms from spent nuclear fuels. The BESNA code employs a new modified CE/CM (Constant Extrapolation – Constant Midpoint) predictor-corrector scheme in depletion calculations for improving computational efficiency. In this modified CE/CM scheme, the decay components leading to the large norm of the depletion matrix are excluded in the corrector, and hence the corrector calculation involves only the reaction components, which can be efficiently solved with the Talyor Expansion Method (TEM). The numerical test shows that the new scheme substantially reduces computing time without loss of accuracy in comparison with the conventional scheme using CRAM (Chebyshev Rational Approximation Method), especially when the substep calculations are applied. The depletion calculation and source term estimation capability of BESNA are verified and validated through several problems, where results from BESNA are compared with those calculated by other codes as well as measured data. The analysis results show the computational efficiency of the new modified scheme and the reliability of BESNA in both isotopic predictions and source term estimations.

Validation of the advanced neutron transport lattice code KYLIN-V2.0 with critical and burnup experiments

KYLIN-V2.0 is an advanced lattice code developed in Nuclear Power Institute of China. The subgroup method, MOC method and Chebyshev rational approximation method (CRAM) are applied in the resonance, transport and burnup calculation to provide the macro cross-section library. In this paper, several critical experiments and burnup benchmarks, including B&W critical experiment, KRITZ critical experiment and Takahama-3 burnup benchmark, are applied in the validation of KYLIN-V2.0. KYLIN-V2.0 is validated by 21 cases in B&W-1484, 23 cases in B&W-1810 and 16 samples in Takahama-3. Eigenvalue, pin power distribution and nuclides density results of these cases show the good consistency compared with measured data. The KYLIN-V2.0 code shows good heterogeneous calculation ability for two-dimensional lattices.

Efficient implementation of the exact artificial boundary condition for the exterior problem of the Stokes system in three dimensions

AbstractIn this paper the Stokes system in an unbounded domain is solved by the artificial boundary method. The novelty lies in an operator form of the exact Dirichlet-to-Neumann (DtN) mapping. With the help of the Chebyshev rational approximation of the square root function, we derive a highly accurate approximate DtN mapping, which can be numerically implemented without resorting to the eigen-decomposition in terms of the vectorial spherical harmonics. In addition, we develop an efficient block preconditioner for the augmented truncated saddle point problem. Numerical experiments demonstrate the effectiveness of the proposed method.

Verification of depletion capability of OpenMC using VERA depletion benchmark

Depletion capability has been implemented in the open-source state-of-the-art Monte Carlo particle transport code, OpenMC, via the Python application programming interface (API) and C API. The Python API provides the main depletion functionality including the depletion matrix solver based on the Chebyshev rational approximation method, and high-fidelity numerical integration algorithms for flux and nuclide densities updates. OpenMC serves as the data provider for problem dependent transmutation reactions needed to form the depletion matrix. The data exchange between the OpenMC solver and the depletion solver is controlled by the C API to facilitate in-memory transfers. This paper firstly demonstrates the accuracy of this implementation by comparing results of VERA depletion benchmark against the well-verified Monte Carlo depletion code, Serpent2, including both multiplication factor and nuclide densities of interest. Additionally, a simplified depletion chain based on that of the CASL project is evaluated by comparing its accuracy and efficiency with a full depletion chain based on ENDF/B-VII.1 library. The benefits of high-order numerical integration algorithms have also been verified on selected benchmarks with and without burnable absorbers. Finally, the impact of the ENDF/B-VIII.0 data library on select benchmarks was also evaluated.

The development of fuel management code TRACS for HFETR based on Unstructured-mesh variational nodal method

High Flux Engineering Test Reactor (HFETR) focuses on the irradiation test of reactor fuel elements, thus the accurate determination of power or flux distribution becomes important. Since HFETR contains complex geometry assembly and different types of irradiation channels, the traditional simulation methods such as Monte Carlo method and deterministic method based on rectangle and hexagonal mesh are not applicable in taking into account calculation accuracy and efficiency. In this study, based on “two-step” calculation scheme, a fuel management code named TRACS was developed. Homogenization of few-group cross-sections was performed by KYLIN-II lattice code, and reactor core calculation was performed by unstructured-mesh variational nodal method. Criticality position of control rod searching adopts difference method, and micro-depletion calculation is performed by solving microscopic burnup equation using an open source matrix exponential solver EXPOKIT of Chebyshev rational approximation method. Finally, two benchmarks of HFETR sample model with different irradiation channels were used to verify TRACS code, and the results by Monte Carlo code OpenMC serve as reference. The comparison results show that the keff error is less than 300 pcm for both two cases, and the relative power error of nearly all the fuel assembly and regions in irradiation assembly at both BOL and EOL stay below 5%, and the critical positions of control rods between TRACS and OpenMC during reactor operation are nearly the same. Overall, TRACS agrees well with OpenMC, and it can be used for fuel management calculation of HFETR with complex geometrical irradiation channel.

Stability analysis of higher-order neutronics-depletion coupling schemes and Bateman operators

Previous work has introduced the stability analysis of coupled neutronics-depletion solvers for the standard explicit Euler and predictor-corrector methods. The present work is an extension of this analysis to higher-order schemes that are commonly used, including the LE, LE/LI, LE/QI, and their implicit versions where such a method exists. Substepping, extrapolation, and linear and quadratic interpolation are investigated, and their effects on numerical stability are discussed. A realistic, numerically-stiff depletion system is considered by applying automatic differentiation to the Chebyshev rational approximation method; accounting for initial nonlinear behaviour, the predictions from the stability analysis match the outcomes of simulation.

The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions

In this paper, we demonstrate that a well-known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimization. Although this correspondence is not surprising, it naturally connects rational and generalized rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and, therefore, offers more theoretical and computational tools for solving this problem. The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters that are also the decision variables in the corresponding optimization problems.

Study on an improved burnup algorithm in Kylin-2 code

Kylin-2 is an advanced neutronics lattice code, developed by Nuclear Power Institute of China. Chebyshev Rational Approximation Method (CRAM) is used in Kylin-2 to solve the burnup chains. This method has the characters of high calculation accuracy and good computational stability, which is the most widely used numerical solution nowadays. However, when solving the large-scale matrices, the computing time would sharply increase, so the burnup calculation efficiency of Kylin-2 is not very high. In this study, an improved CRAM based on order reduction with Krylov Subspace Method (KSM), Combined Algorithm, has been used to solve burnup equations. According to the verification and analysis, comparing to CRAM, Kylin-2 code with Combined Algorithm has higher efficiency in burnup calculation while maintaining the accuracy.

RAST-K v2—Three-Dimensional Nodal Diffusion Code for Pressurized Water Reactor Core Analysis

The RAST-K v2, a novel nodal diffusion code, was developed at the Ulsan National Institute of Science and Technology (UNIST) for designing the cores of pressurized water reactors (PWR) and performing analyses with high accuracy and computational performance by adopting state-of-the-art calculation models and various engineering features. It is a three-dimensional multi-group nodal diffusion code developed for the steady and transient states using microscopic cross-sections generated by the STREAM code for 37 isotopes. A depletion chain containing 22 actinides and 15 fission products and burnable absorbers was solved using the Chebyshev rational approximation method. A simplified one-dimensional single-channel thermal-hydraulic calculation was performed with various values for the thermal conductivity. Advanced features such as burnup adaptation and CRUD modeling capabilities are implemented for the multi-cycle analysis of commercial reactor power plants. The performance of RAST-K v2 has been validated with the measured data of PWRs operating in Korea. Furthermore, RAST-K v2 has been coupled with a sub-channel code (CTF), fuel performance code (FRAPCON), and water chemistry code for multiphysics analyses. In this paper, the calculation models and engineering features implemented in RAST-K v2 are described, and then the application status of RAST-K v2 is presented.