球床模块式高温气冷堆核电站是一个复杂的非线性耦合系统，具有多种耦合机制。中子物理-热工水力构成多物理耦合，球床-燃料球-颗粒构成多尺度耦合，堆芯-蒸汽发生器构成多部件耦合，蒸汽发生器连接的一回路-二回路构成多回路耦合，多堆带一机构成多模块耦合。稳定、精确、高效地求解完整的高温气冷堆耦合系统既是发展趋势，也极具挑战性。目前，高温气冷堆耦合计算程序基本采用算符分裂半隐式和Picard迭代等方法，分别求解各个子物理场，然后通过传递耦合参数进行迭代。这些方法存在稳定性差、收敛速度慢、计算精度低等问题。Newton-Krylov方法是一种将所有物理场联立求解的全耦合方法，具有更高的稳定性、效率和精度。DJNK（Divided difference Jacobian Newton-Krylov）和JFNK（Jacobian-free Newton-Krylov）是两种比较有前途的Newton-Krylov方法，DJNK的稳定性更高，预处理潜力更大，但目前的计算效率很低。本文针对高温气冷堆的特点，进行DJNK方法研究，此外，对JFNK方法同步开发并用作性能对比。首先，对于堆芯堆多物理耦合，针对有效增殖因子导致Jacobian矩阵产生稠密行，提出了基于变量代换的稠密行分解技术，实现了高效的矩阵计算；针对不同物理场性质差异大导致耦合矩阵条件数大，提出了降阶最优Scaling技术，保证了矩阵优化的效果和速度。其次，对于多尺度耦合，考虑多批次燃料球和非局部释热，研究了球床-燃料球-颗粒三级多尺度耦合，针对耦合变量数多、大量弱耦合项影响Jacobian矩阵计算速度的难题，提出了广义多层级非线性消去技术，显著提高了求解效率。然后，对于直流蒸汽发生器，针对可动边界模型下复杂的耦合行为，提出了满矩阵差分技术，首次实现了直流蒸汽发生器的DJNK全耦合求解。最后，对于控制棒位搜索，通过建立耦合方程并应用稠密行分解技术，实现了棒位与多物理场的全耦合联立求解，计算效率显著高于传统的外部迭代搜索法。在方法研究基础上，建立了适用于高温气冷堆多种耦合机制的DJNK求解框架，开发了基于DJNK的耦合求解程序。首次实现了一回路中子物理、堆芯热工、多尺度燃料球、蒸汽发生器等模块的DJNK高效求解。计算结果表明，DJNK的效率和稳定性显著高于JFNK，并具有扩展到完整电站全耦合系统高效求解潜力。
The High-Temperature Gas-cooled Pebble-bed Reactor Module is a complex nonlinear coupled system with multiple coupling mechanisms. There are multi-physics coupling, multi-scale coupling, multi-part coupling, multi-loop coupling, and multi-module coupling. Stable, accurate and efficient solution of the complete high-temperature gas-cooled reactor coupling system is both a development trend and extremely challenging.At present, the high-temperature gas-cooled reactor coupling calculation procedures usually use methods such as operator split semi-implicit and Piacrd iteration to solve each sub-physical field separately, and then iterate through transfering the coupling parameters. The Newton-Krylov method is a fully coupled method that solves all physical fields simultaneously, with higher stability, efficiency and accuracy. DJNK and JFNK are two promising Newton-Krylov methods. DJNK has higher stability and greater preconditioning potential, but the current computational efficiency is very low. In this paper, the DJNK method is studied with the characteristics of the high-temperature gas-cooled reactor.Firstly, for the multi-physics coupling of the core, the dense row decomposition technique based on variable substitution is proposed for the effective multiplication factor leading to the generation of dense rows in the Jacobian matrix, which achieves an efficient matrix calculation; for the large difference in the nature of different physical fields leading to the large condition number in the coupling matrix, the reduced-order Scaling technique is proposed to ensure the effect and speed of matrix optimization. Secondly, for multi-scale coupling, considering multi-batch fuel spheres and non-local heat release, the fine pebble bed-fuel sphere-particle multi-scale coupling is studied, and for the problem that the number of coupling variables is large and a large number of weak coupling terms affect the speed of Jacobian matrix calculation, the generalized multi-level nonlinear elimination technique is proposed, which significantly improves the solution efficiency. Then, for the steam generator, the full matrix difference technique is proposed for the complex coupling behavior under the movable boundary model, and the DJNK with full coupling solution of the steam generator is achieved for the first time. Finally, for the control rod position, by establishing the coupling equations and applying the dense row decomposition technique, the fully coupled solution of the rod position and the multi-physics field is realized, and the computational efficiency is significantly higher than that of the traditional external iterative search method.Based on the method study, a DJNK solution framework applicable to multiple coupling mechanisms of the high-temperature gas-cooled reactor was established, and a DJNK-based coupling solution program was developed. For the first time, the efficient DJNK solution for any combination of neutronics, thermal hydraulics, multiscale fuel sphere, steam generator and other modules of the first circuit is realized, with the capability of steady-state and transient simulation of the first circuit. The computational results show that, with the series of methods proposed in this paper, the computational efficiency of DJNK is significantly higher than that of JFNK for the complex one-loop coupled system of the high-temperature gas-cooled reactor, with the potential to be further extended to the stable and efficient solution of power plant-wide coupled systems.
