基于隐式重启Arnoldi方法的中子扩散本征值问题求解及其降阶研究

Apply implicitly restarted Arnoldi method to solving eigenvalue problem and reducing dimensionality in neutron diffusion

中子扩散方程高阶谐波可用于重构堆芯中子注量率分布,但传统源迭代与源修正迭代法求解时的收敛速度慢,计算耗时长。采用隐式重启Arnoldi方法(Implicitly Restarted Arnoldi Method,IRAM)求解本征值问题的中子扩散方程获得谐波数据,通过本征正交分解(Proper Orthogonal Decomposition,POD)与伽辽金(Galerkin)投影相结合的方法构建POD-Galerkin低阶模型,并重构二维稳态TWIGL基准题中子注量率分布。研究结果表明:IRAM方法在求解中子扩散方程的高阶本征值和谐波问题上具有较高的精度;基于POD-Galerkin低阶模型重构中子注量率分布具有较高的保真性与计算效率,有效增值系数与参考解的误差为8.7×10^(-5),对角线上快群和热群中子注量率最大相对误差为2.56%,且低阶模型计算用时仅为全阶模型的10.18%。本研究为堆芯中子注量率重构提供了一种可靠且高效的方法,该方法不仅可用于重构稳态时堆芯中子注量率分布,还具有在瞬态情况下预测中子注量率分布的潜力,有望在未来的应用中进一步拓展。

[Background]High-order harmonics of neutron diffusion equations can be used to reconstruct the neutron flux distribution in a reactor core,but traditional source iteration methods or modified source iteration methods have low solving efficiency.[Purpose]This study aims to provide a reliable and efficient method for reconstructing the neutron flux distribution in reactor cores.[Methods]Firstly,the neutron diffusion equation was discretized using the finite difference method.Then,the implicitly restarted Arnoldi method(IRAM)was employed to solve the eigenvalue problem of the neutron diffusion equation and obtain high-order harmonic samples for different macroscopic cross-section states.Subsequently,a low-order model for the neutron diffusion equation was constructed by using these samples and a combination of proper orthogonal decomposition(POD)and Galerkin projection,and an error model was developed to characterize the accuracy of eigenvalue and harmonic calculations.Finally,relevant programs were developed to reconstruct the neutron flux distribution in the two-dimensional steadystate TWIGL benchmark problem and validate the accuracy of the model.[Results]The computation results show that the IRAM exhibits high accuracy in solving the high-order eigenvalues and harmonic problems of the neutron diffusion equation,with an error on the order of 10−14.The reconstruction of the neutron flux distribution based on the POD-Galerkin low-order model also maintains a high level of accuracy.The solution error increases with the order of the eigenvalues,with an error magnitude less than or equal to 10−12.The reconstructed neutron flux distribution closely matches the reference solution in the reactor core,and the error in the effective multiplication factor is only 8.7×10^(−5).Additionally,the computation time for the low-order model is only 10.18%of the full-order model.[Conclusions]This study provides a reliable and efficient method for reconstructing the neutron flux distribution in reactor cores.The method can be used not only to reconstruct the steady-state neutron flux distribution but also has the potential to predict the transient neutron flux distribution,which is expected to be further expanded in future applications.