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我们会通过消息、邮箱等方式尽快将举报结果通知您。基于非结构网格的中子输运方程的球谐函数解法及时空动力学计算研究/Research on the Spherical Harmonics Method and S
基于非结构网格的中子输运方程的球谐函数解法及时空动力学计算研究/Research on the Spherical Harmonics Method and S
本论文首先将球谐函数（PN）方法应用于一种新的与传统一阶和奇偶二阶中子输运方程相比具有多种独特数值优点的二阶自共轭形式的中子输运方程(SAAF: Self-Adjoint Angular Flux)，导出了一组对于任意展开阶数通用的球谐函数微分方程组。由于中子角通量密度的各个球谐矩是相互紧密耦合在一起的，为了便于数值求解，提出了一种有效的迭代求解方法，使得各个微分方程可以依次逐一求解。为了适应任意不规则几何区域的求解，应用有限元方法对空间变量进行离散, 应用有限元分析软件ANSYS的网格剖分功能进行非结构网格剖分；同时为了提高计算速度，应用不完全Cholesky分解预处理共轭梯度方法求解有限元方程组。根据所提出的理论模型研制了多维多群稳态中子输运方程的球谐函数有限元方法计算程序TEPFEM，一系列二维和三维基准题的数值计算结果表明，所建立的理论模型具有很高的计算精度，提出的迭代求解方法具有很好的收敛性。
针对球谐函数方法处理真空边界的不确定性，本文将Marshak边界条件与传统的一阶输运方程相结合，导出了一种适合于SAAF方程的边界条件表达式；然后把双球谐函数（DPN）方法应用于真空边界处理的模型，即在真空区域和非真空区域分别进行球谐函数展开，克服了中子角通量密度在真空直线交界面处关于角度不连续的问题。通过对一系列的数值计算结果的分析，研究了不同真空边界处理方法的精度、特性及适用范围，得出了一些有用的结论。
应用所研制的程序TEPFEM研究了球谐函数方法在求解复杂环境下（“射线效应”问题、任意几何形状、强非均匀介质、强烈各向异性散射、内含真空区域以及深穿透问题等）的中子输运方程的适应性。数值结果表明，该程序具有很高的精度，能够很好地克服“射线效应”，可以应用于任意几何的非结构网格，并具有较高的收敛速度。
然后，针对球谐函数方法计算量大、不适应瞬态中子输运方程求解的缺点，本文将简化球谐函数（SPN）方法应用于SAAF方程中，将球谐函数矩的个数由原来的 个降为 个，大大降低了计算量。数值计算结果表明，SPN方法在保证相应计算精度的条件下，具有比PN方法快得多的计算速度，而且对于维数越高、网格数越多、展开阶数越高的问题，节省时间的效果就越好。
最后，研究了应用SPN方法求解中子输运方程的瞬态时空动力学问题。对时空动力学方程组中中子通量密度方程采用全隐向后差分格式来离散时间变量，缓发中子先驱核浓度方程采用时间积分方法进行解析求解，得到带有固定源的稳态中子输运方程，应用SPN方法求解该固定源问题以提高计算速度。根据所提出的模型研制了多维时空动力学输运计算程序TSPN-T，对几个基准题的计算结果表明， TSPN-T程序具有与国际上同类程序相当的计算精度，计算效率与DOT4-T相当，可以应用于复杂环境下的输运时空动力学计算中。
本课题受国家自然科学基金（）、国防科工委“计算物理”国家重点实验室基金( JW0802)资助。
The spherical harmonics (PN) method is applied to a newly raised second order neutron transport equation named SAAF (Self-Adjoint Angular Flux), which has many special numerical advantages in comparison with the traditional first order transport equation, odd-parity and even-parity second order transport equation. A general set of PN differential equations available to any expanding order N is derived. As the various moments of angular flux are coupled each other, an efficient iterative method is raised to solve them one by one. In order to meet the requirement of irregular geometry, the spatial variables of neutron transport equation are discretized by means of finite element method (FEM). The mesh generation module of the finite element analysis software named ANSYS is used to generate the unstructured-meshes. The incomplete Cholesky decomposition preconditional conjugate gradient method is used to solve the finite element equations. Based on the model, a multi-group neutron transport equation calculation code named TEPFEM is developed. The numerical results of many 2D and 3D benchmark problems demonstrate that this method can provide high precision results and converges rapidly.
As it is impossible to deal with the vacuum boundary condition exactly in PN method, an approximate scheme is applied to derive a boundary expression suitable for SAAF by coupling the Marshak boundary condition with the traditional first order transport equation on the vacuum boundary. And also, the DPN method is utilized to deal with the vacuum boundary. In this method, the angular flux is expanded separately in both the vacuum and non-vacuum regions into different sets of polynomials respectively to simulate the discontinuity between the fluxes in the outward and inward angular regions for a straight free boundary. The precision of various methods to deal with the vacuum boundary is studied through some numerical results.
The program TEPFEM is applied to calculate the transport problems under complex environment, such as ray effect problems, irregular geometry, strongly heterogeneous media, strong anisotropic scattering, void region, deep penetrating problems and so on. The numerical results demonstrate that TEPFEM possesses high accuracy, causes no ray effect, is suitable for unstructured-meshes of any irregular geometry and converges rapidly.
Afterwards, as the PN method has a disadvantage of large amount of calculation and unsuitability for transient transport calculation, the Simplified PN (SPN) method is applied to solve the SAAF. With SPN, the amount of flux moments decreases from (N+1)2 to (N+1) and therefore the computation time decreases greatly. Numerical results of several test problems show that SPN method can obtain satisfactory accuracy with a significantly higher computational speed than PN method, and can save more time when dealing with problems with higher dimension, larger amount of meshes, and larger expanding order N.
Finally, the use of the SPN method to calculate the transient space-time kinetic problem of neutron transport equation is studied. An efficient fully implicit scheme was applied for time discretization of neutron flux equations in combination with direct analytical time integration method of the delayed neutron precursor equations. The resultant fixed-source neutron transport problem was solved by the SPN method in order to increase the calculating speed and meet the requirement of transient calculation. Based on this model the program of multi-dimension space-time neutron kinetic transport equations TSPN-T has been developed and tested by several benchmark problems. The numerical results demonstrated that TSPN-T can provide comparable accuracy with some international similar program at competitive computational efficiency with DOT4-T, and can be used to the transport space-time kinetic calculation of complex environment.
The project was financially supported by the fund of National Nature Science Foundation of China () and fund of China state key laboratory of Computational Physics ( JW0802).没有更多推荐了，
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