基于节块展开法的Jacobian-Free Newton Krylov联立求解物理-热工耦合问题

Jacobian-Free Newton-Krylov based on nodal expansion method for neutronic-thermal hydraulic coupling problem

目前反应堆物理热工耦合程序通常采用固定点迭代思路,这可能导致部分工况收敛速度慢,甚至出现不收敛的现象,严重影响了计算效率.基于此,本文将高效的粗网节块展开法(NEM)与Jacobian-Free Newton-Krylov(JFNK)方法结合,成功地开发出了一套新方法 NEM_JFNK,实现了联立求解物理热工耦合问题.首先将NEM推广到热工问题的求解,之后使用NEM来离散物理-热工耦合问题的所有控制方程,使得所有变量都能在粗网格下进行离散,从而大大减小求解问题的规模;其次将NEM离散后的方程经过某些特殊的处理,成功地嵌入JFNK的计算框架,最终开发出了基于线性预处理的NEM_JFNK,即LP_NEM_JFNK.此外,为了充分利用原有的迭代程序,避免JFNK残差方程的重新建立,本文还开发了无需重构残差方程的NEM_JFNK,即NRC_NEM_JFNK,并实现"黑箱"耦合.文中以一维中子-热工模型为例,给出LP_NEM_JFNK和NRC_NEM_JFNK数学模型,并对计算结果进行分析.结果表明:新方法无论是收敛速度还是计算效率都具有明显优势.

The traditional fixed-point iteration method is typically used for neutronics/thermal-hydralics coupling problems in most nuclear safety analysis codes.But the fixed-point iteration method has a tendency to fail to be used in computing the coupling problems due to slow convergence rate in some cases and even no convergence,and thus resulting in a limited efficiency,especially for the tight-coupling and fast-transient problems.In addition,for the reactor thermal-hydraulic calculation,the traditional finite difference or volume method（FDM or FVM） is used.However,both FDM and FVM require fine mesh size to achieve the desired precision and thus also result in a limited efficiency for the large scale problems.In this paper,to ensure the accuracy,efficiency and convergence for large-scale and complicated coupling problems,the new methods-NEM_JFNK are successfully developed to simultaneously solve the neutronics /thermalhydralics coupling problems by combining the advantage of the efficient coarse nodal expansion method（NEM） and Jacobian-Free Newton-Krylov method（JFNK）.The NEM has been widely used in the reactor physics analysis due to its high efficiency and accuracy in the reactor physics analysis,and it has proved to be superior to FDM and FVM.To improve the efficiency and accuracy for the large scale problems,the NEM is first extended to thermal hydraulic problems from the reactor physics calculation.Then all the governing equations of the neutronics/thermal-hydralics coupling problems can be discretized by the NEM and all the variables can be solved on the coarse meshes so that the size of coupling problems is greatly reduced.To ensure the high accuracy for the coupling problems on the coarse meshes,the high-order coefficients in NEM are successfully transferred between the coupling terms by our research.After that,to ensure the convergence of complicated coupling problems,JFNK based on the NEMs needs to be developed.However,the researches of JFNK based on the NEM in reactor analysis are less and the existing JFNK methods are mostly based on FVM or FDM or the finite element method.In this paper,the NEM discretization equations are successfully integrated into the framework of JFNK through the special treatment and the NEM_JFNK with linear-based preconditioner named LP_NEM_JFNK is also successfully developed.In addition,to take advantage of the existing code and avoid the construction of residual formulations,the non-residual construction NEM_JFNK named NRC_NEM_JFNK is presented and the ＂black-box＂ coupling method is achieved by NRC_NEM_JFNK so that the existing codes only need the simple modification to achieve the combination of the NEM and JFNK.Numerical results show LP_NEM_JFNK and NRC_NEM_JFNK outperform traditional fixed-point iteration method in the sense of convergence rate and efficiency.Further studies are needed to extend the NEM_JFNK method to the multi-dimensional neutronic/thermal hydraulic coupling problems in the high temperature gas-cooled reactor.