多介质输运问题健壮高效的数值方法研究

发布时间：2018-01-23 12:37

本文关键词： S_N 输运方程多群辐射扩散方程离散纵标方法简单隅角平衡方法 h自适应扭曲网格耦合建模区域分解并行离散格式　出处：《中国工程物理研究院》2016年博士论文　论文类型：学位论文

【摘要】：输运问题在惯性约束聚变和武器物理等领域中有着广泛而重要的应用,它具有多变量、多尺度、多物理等特征,其数值模拟难度居现代科学计算领域的前列。实际应用中输运方程的求解存在如下的问题：1)数值解出现非物理振荡、出负等现象；2)大变形网格下的计算精度较低,有时甚至计算失败；3)数值计算代价昂贵。本文主要针对上述问题,为提高输运方法的健壮性开展相关的离散格式和自适应算法研究,并为了提高求解效率开展多物理耦合模拟及并行算法研究,为实际应用问题提供算法与技术支撑。本文主要成果如下：(1)对一维粒子输运方程的子网格平衡格式开展了理论研究,证明了该格式的稳定性和收敛性；并通过分析格式设计的不足,提出了基于嵌套网格的两种离散格式。数值结果表明,新格式的精度要优于步格式,与菱形格式、子网格平衡格式的精度相当,且能明显抑制菱形格式和子网格平衡格式计算中的非物理振荡。对于强散射问题的计算,新格式的迭代收敛速度较快。(2)对耦合流体计算的二维辐射传输问题中存在的因网格大变形导致精度较低及扫描死锁问题,提出了一种采用三角剖分的改进子网格平衡格式,并设计了基于网格几何品质的h自适应加密输运算法,与原有算法相比,新算法能改善大变形网格上的计算精度,并且解决了实际应用问题中由于凹网格死锁带来的“算不下去”的问题。进一步,在间断有限元的框架下给出了混杂网格上输运算法的稳定性和收敛性证明。(3)对输运问题的不同层次建模,开展了多群扩散与单群扩散、多群扩散与单温热传导两种耦合建模的数值模拟方法研究.基于区域分解的方法求解耦合模型,并分别提出了自洽的耦合界面连接条件,数值模拟结果表明,耦合建模计算精度与单一精细建模的计算精度相当,其计算代价与低层次建模的计算代价相当。(4)对二维扩散方程提出了一种无条件稳定的具有二阶精度的单元中心型守恒并行离散格式,格式的构造不需要预估和校正步,并且满足离散的能量守恒。理论上严格证明了离散数值解在H1范数下的无条件稳定性和二阶收敛性,数值实验验证了理论分析的结果。
[Abstract]:Transport problem has been widely used in inertial confinement fusion and weapon physics. It has the characteristics of multi-variable, multi-scale, multi-physics and so on. The difficulty of numerical simulation is in the forefront of modern scientific calculation. In practical application, there exists the following problems in solving the transport equation: 1) the numerical solution has the phenomena of non-physical oscillation and negative. 2) the calculation accuracy of large deformation mesh is low, and sometimes it even fails; 3) numerical computation is expensive. In this paper, the discrete scheme and adaptive algorithm are studied in order to improve the robustness of transport methods. And in order to improve the efficiency of solving the multi-physical coupling simulation and parallel algorithm research. The main results of this paper are as follows: (1) the subgrid equilibrium scheme of one-dimensional particle transport equation is studied theoretically, and the stability and convergence of the scheme are proved. Two discrete schemes based on nested meshes are proposed by analyzing the shortcomings of the scheme design. The numerical results show that the accuracy of the new scheme is better than that of the step scheme and the precision of the subgrid balance scheme is the same as that of the rhombus scheme. Moreover, the non-physical oscillation in the calculation of rhombus scheme and sub-grid balance scheme can be obviously suppressed, and the strong scattering problem can be calculated. The iterative convergence rate of the new scheme is faster. (2) in the two-dimensional radiative transmission problem of coupled fluid computation, the problem of low precision and scanning deadlock caused by large mesh deformation exists. An improved submesh balance scheme using triangulation is proposed, and an h adaptive encryption transport algorithm based on mesh geometry quality is designed, which is compared with the original algorithm. The new algorithm can improve the computational accuracy on large deformed meshes, and solve the problem of "not being able to calculate" caused by the deadlock of concave meshes in practical applications. In the framework of discontinuous finite element, the stability and convergence proof of the transport algorithm on hybrid meshes is given, and the multi-group diffusion and single-group diffusion are developed. The numerical simulation method of multi-group diffusion and single-temperature heat conduction coupling modeling is studied. The domain decomposition method is used to solve the coupled model and the self-consistent coupling interface connection conditions are proposed respectively. The numerical simulation results show that. The computational accuracy of coupled modeling is equivalent to that of single fine modeling. The computational cost is the same as that of low level modeling.) an unconditionally stable conserved conserved parallel discrete scheme with second-order accuracy is proposed for two-dimensional diffusion equations. The construction of the scheme does not require the prediction and correction steps, and satisfies the discrete energy conservation. In theory, the unconditional stability and the second order convergence of the discrete numerical solutions under the H 1 norm are strictly proved. The results of theoretical analysis are verified by numerical experiments.
【学位授予单位】：中国工程物理研究院
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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