Chebyshev Rational Approximation Method 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.

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.

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.

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.

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.

Study on algorithms for solving depletion sparse matrix based on DESCAR module

Depletion solver with Chebyshev rational approximation method (DESCAR) was developed by Nuclear Power Institute of China (NPIC) to solve depletion equation. In this paper, the algorithms for solving depletion sparse matrix are studied, including symbolic Lower-Triangle and Upper-Triangle (LU) decomposition using elimination directed acyclic graphs (EDAGS), symbolic LU decomposition using symmetric reduction (FSNZ), symbolic LU decomposition using partial path-symmetric reduction (FPNZ), symbolic decomposition using elimination tree (ETREE), the direct sparse solver using pardiso routine in Intel math kernel library (PDMKL). An optimized direct factorization algorithm (ODFZ) is proposed to improve the speed of solving sparse matrix. The above methods are developed and tested in DESCAR module using a 17 × 17 pressurized water reactor (PWR) assembly containing Gd2O3-UO2 fuel rods. Results show that ODFZ method has highest calculation efficiency, the calculation speed is about 3.5 times that of PDMKL method.

ONIX: An open-source depletion code

Open Source software enables innovative, community-based software development. ONIX brings this concept to the field of depletion calculations. It is an open-source depletion software to be used for nuclear reactor simulations, for fissile material production analysis as well as for nuclear arms control applications. ONIX provides a module to solve the depletion equation using a Chebyshev Rational Approximation Method. For the generation of one-group cross sections, it includes a coupling interface for the open-source neutron transport code, OpenMC, as well as a module to read pre-computed values in a stand-alone mode. ONIX has special features to optimize nuclear data libraries, to update isomeric branching ratio during burnup, and to support automation of simulations for nuclear archaeology. ONIX has been validated against results from numerical and experimental benchmarks, and its results agree with other methods within expected error ranges.

Global error analysis of the Chebyshev rational approximation method

The Chebyshev rational approximation method (CRAM) has become a widely adopted method for solving nuclear depletion problems. Therefore, understanding CRAM’s accuracy is important for the safe operation of nuclear power plants. This article performs a global error analysis of CRAM and finds that, as the length of the time step approaches zero, the relative error measured between the exact and CRAM solutions at a fixed end time approaches one and infinity for even and odd orders, respectively; for intermediate time step sizes, a minimum in relative error is observed. We show that the reason for CRAM’s behavior is that the method is inconsistent. Two best practices for using CRAM, derived from these results, are: (1) use CRAM order 16 or higher, (2) if necessary, increase the CRAM order when multiphysics coupling requires smaller time steps.

Solving Burnup Equations by Numerical Inversion of the Laplace Transform Using Padé Rational Approximation

Burnup calculations play a very important role in nuclear reactor design and analysis, and solving burnup equations is an essential topic in burnup calculations. In the last decade, several high-accuracy methods, mainly including the Chebyshev rational approximation method (CRAM), the quadrature-based rational approximation method, the Laguerre polynomial approximation method, and the mini-max polynomial approximation method, have been proposed to solve the burnup equations. Although these methods have been demonstrated to be quite successful in the burnup calculations, limitations still exist in some cases, one of which is that the accuracy becomes compromised when treating the time-dependent polynomial external feed rate. In this work, a new method called the Padé rational approximation method (PRAM) is proposed. Without directly approximating the matrix exponential, this new method is derived by using the Padé rational function to approximate the scalar exponential function in the formula of the inverse Laplace transform of burnup equations. Several test cases are carried out to verify the proposed new method. The high accuracy of the PRAM is validated by comparing the numerical results with the high-precision reference solutions. Against CRAM, PRAM is significantly superior in handling the burnup equations with time-dependent polynomial external feed rates and is much more efficient in improving the accuracy by using substeps, which demonstrates that PRAM is the attractive method for burnup calculations.

Development and validation of Burn-up Calculation Code IMPC-Burnup2.0 for accelerator-driven sub-critical system

Fuel depletion analysis is important for the evolutionary behavior of nuclear materials in the reactor. For the accelerator-driven sub-critical system (ADS), the main research focuses on the transmutation of minor actinides (MA) and long-lived fission products (LLFP). To perform nuclide inventory calculation of ADS, a new nuclide point-depletion code IMPC-Depletion has been developed by utilizing Chebyshev Rational Approximation Method (CRAM) and Transmutation Trajectory Analysis method (TTA) to solve the Bateman equation. Besides, a new burnup code system named IMPC-Burnup2.0 which couples the OpenMC neutron transport code and IMPC-Depletion is also developed to perform burnup behavior analysis of ADS sub-critical system. Finally, the decay of Np237 and fixed neutron flux irradiation of 3.4% UO2 are calculated and compared with other reference results to verify IMPC-Depletion. Moreover, IAEA-ADS and OECD/NEA ADS Minor Actinide Burner benchmark are calculated using IMPC-Burnup2.0 to testing the capability for burnup analysis for ADS sub-critical core. Numerical results show that IMPC-Burnup2.0 is reliable when compared with reference values and it can be used to solve burnup problem of ADS.