Jacobian-free Newton Krylov coarse mesh finite difference algorithm based on high-order nodal expansion method for three-dimensional nuclear reactor pin-by-pin multiphysics coupled models

Abstract

The nuclear reactor core is a large-scale, complicated and tight-coupled nonlinear multiphysics coupled system including neutron transport, heat conduction of nuclear fuel pins, coolant flow, heat transfer and so on. To meet the demands of the advanced nuclear reactor analysis and design, there is an urgent need to reduce the computational costs and improve the convergence rate for the three-dimensional (3D) nuclear reactor core multiphysics coupled models, especially for a high-fidelity pin-by-pin (at the level of fuel pin by fuel pin) full core simulation. In this work, we develop CNJ, an efficient Jacobian free Newton Krylov (JFNK) coarse mesh finite difference (CMFD) algorithm based on the nodal expansion method (NEM) to simultaneously solve the complicated pin-by-pin neutronics-thermal hydraulic (N-TH) coupled models. The high-order NEM can improve the numerical accuracy of the CMFD method on a coarse mesh and the JFNK method can ensure the rapid convergence and high efficiency on large-scale and nonlinear coupled discrete systems. By combining the CMFD, NEM and JFNK methods and making full use of their respective advantages, the CNJ algorithm can obtain high accuracy and efficiency even on a coarse mesh to greatly reduce the number of solution variables and the computational cost of the 3D coupled models. Finally, two representative and complicated 3D pin-by-pin N-TH coupled models are analyzed to evaluate the numerical accuracy and efficiency advantage of CNJ in detailed.

Introduction

The nuclear reactor has been applied to carbon-free electricity, space or marine nuclear power, medical isotope production and military security due to its high power density, long endurance and efficiency in harsh environments. The reactor core consists of a great number of fuel assemblies (FAs) and each of the FAs is composed of many different fuel pins (fuel rod, plate or sphere) at various levels of enrichments [1], [2]. The energy is released in the nuclear fuel pin by nuclear fission reactions between neutrons and fission fuel and then conducted through the fuel pins to the coolant in order to extract the heat and transport it to other equipment [3]. Therefore, the neutron distribution, the heat conduction of nuclear fuel pins, the coolant flow and heat transfer are highly coupled with each other. For example, the temperature of the fuel pins and the density distribution of the coolant, which depend on the fission power and the coolant conditions, affect the microscopic neutron cross-section or the probability of the certain nuclear reactions and the fission power.

To meet the demands of the advanced nuclear reactor analysis and improve its economics and safety, there is an urgent need to reduce the computational costs and improve the convergence rate for three-dimensional (3D) nuclear reactor core multiphysics coupled models, especially for high-fidelity pin-by-pin (at the level of fuel pin by fuel pin) full core simulation. However, good numerical convergence and efficiency are of major importance and difficult for the large-scale nonlinear coupled models. The traditional multiphysics coupled tools employ extensively operator Splitting (OS) or Picard iterative methods to decompose the coupled systems into some separate and uncoupled physical problems. The solutions of each individual or split problems are solved separately and then interact just as only boundary conditions to other physics processes, which leads to an inconsistent treatment of nonlinear effects and poor convergence of the coupled systems [1]. Recently, Jacobian-free Newton Krylov (JFNK) methods have been gradually developed to simultaneously solve the reactor neutron transport or diffusion problems [4], thermal-hydraulic coupled models [5] and the neutronics-thermal hydraulic (N-TH) coupled problems [6] due to the good convergence rate of the Newton and Krylov methods and the tight coupled form without introducing errors due to splitting [1], [7], [8]. Some popular multiphysics coupled platforms, such as LIME [8], MOOSE [9] and VERA [10] which use JFNK methods based on the finite difference/volume/element method (FDM/FVM/FEM), have also been developed to solve complicated nuclear reactor multiphysics coupled models.

In addition, coarse mesh nodal expansion methods (NEM) [11] have high accuracy and efficiency possible with high-order polynomial expansion, which has been widely adopted in reactor physics calculation [11] and even extended to solve the thermal hydraulic models [12] and the coupled models [13]. Compared with FDM and FVM, NEM can use a coarser mesh size for the same accuracy as FDM/FVM and greatly reduce the number of discrete solution variables and the computational cost. However, this introduces additional variables such as the transverse-integrated interface currents or fluxes and nodal expansion coefficients. To take advantage of the simple discrete form of the FDM and the high-order accuracy of the NEM, the coarse mesh finite difference (CMFD) scheme based on NEM has been developed to further improve the computational efficiency of the NEM or the accuracy of the CMFD on a coarse mesh [14].

Recently, JFNK methods have been successfully developed to solve discrete systems of the NEM for 3D and multigroup neutron diffusion problems to further improve the computational efficiency of the NEM and the critical technology and strategy of combining the NEM with the JFNK method are also analyzed in detailed [15]. Although some local elimination strategies are proposed to eliminate some expansion coefficients and intermediate variables of the NEM to reduce the number of JFNK solution variables, transverse-integrated interface currents or fluxes, nodal expansion coefficients, nodal average fluxes and the effective multiplication factor of the nuclear fission chain reaction still need to be chosen as the solution variables. In contrast, the CMFD method only evaluates the residual functions of nodal average fluxes and the effective multiplication factor.

Therefore, to further reduce the number of solution variables of the JFNK method based on NEM, an efficient multiphysics coupled algorithm CNJ is developed here to simultaneously solve the complicated pin-by-pin N-TH coupled models in the tight-coupled form by combining the CMFD, NEM and JFNK methods and making full use of their respective advantages. In CNJ, only the nodal average fluxes and the effective multiplication factor of the nuclear fission chain reaction need to be solution variables of the residual systems. High-order NEM can improve the numerical accuracy of the CMFD method on a coarse mesh size and the JFNK method can ensure the rapid convergence rate of large-scale and nonlinear multiphysics coupled discrete systems so that the developed CNJ can obtain high accuracy and efficiency even on a coarse mesh and greatly reduce the number of solution variables and the computational cost of 3D coupled models. In addition, the residual systems with the physics-based left preconditioners are obtained by making full use of the original Picard iterative coupled framework. Both the scaling and the Modified Incomplete LU (MILU) method are proposed as a right preconditioner to improve the convergence rate.

The paper is organized as follows: the formulation, derivation, preconditioner and the iterative strategy of CNJ are presented in Section 2. Two representative pin-by-pin N-TH coupled models with eight-group (in 8 energy ranges) macroscopic cross-sections of nuclear reactions are analyzed in Section 3 to evaluate CNJ numerical accuracy and computational efficiency. The conclusions are discussed in Section 4.