多重网格数值求解不可压流体的局部Fourier分析

发布时间：2018-02-24 20:18

  本文关键词： 多重网格方法 局部Fourier分析 光滑性质 聚松弛 分布松弛 渐进收敛因子 波形松弛　出处：《昆明理工大学》2016年博士论文　论文类型：学位论文

【摘要】：多重网格算法是偏微分方程数值求解的一种快速算法。主要针对离散微分方程后所得的代数方程组进行数值求解,在椭圆型偏微分方程的数值解中已被证明是最优的数值算法,其收敛性与网格尺度的大小无关,且计算成本与问题的规模成正比。由于多重网格算法的优越性,使得它成为计算流体力学中一种高效的数值方法而受到广泛关注和研究。本文依托高等学校博士学科点专项科研基金(优先发展领域)(20135314130002)项目、国家自然科学基金面上项目(51279071),研究多重网格法在水力机械内部流数值模拟方面的理论和应用,重点是多重网格光滑理论中的局部Fourier分析方法,对数值求解不可压缩流体控制方程的多重网格方法进行收敛性分析。主要研究内容和创新如下：(1)结合水力机械流道湍流的流动特点,提出了多重网格算法及其误差迭代的格式。基于局部Fourier分析理论,分别定义了离散算子和松弛迭代算子的椭圆率和光滑因子,并利用不同粗、细网格层Fourier组分之间的关系,定义了新的不变子空间,分析了不同粗化方式下网格转化算子的Fourier表述方式,研究了多重网格算法渐进收敛因子的理论计算方法,创新了两色松弛在两种不同的Fourier模态函数不变子空间中的光滑分析方法,得到了基于多色松弛矩阵的Fourier分析的理论表示,并以泊松方程为例给出了相应的分析结果。研究表明,基于多色松弛的多重网格光滑分析过程具有一般的迭代格式,所得结果具有代表性和应用前景。(2)基于交错网格和非交错网格提出了求解Stokes流的离散格式,并对该离散系统实施两种不同多重网格的松弛算法进行了光滑分析：即聚松弛和分布松弛光滑分析。在交错网格的离散系统中实现了多重网格分布松弛,发现该离散系统的光滑性取决于Laplace算子,并得到了相应的光滑因子。其次,在非交错网格离散系统中,分别实施了多重网格分布松弛和聚松弛,在两色松弛的Fourier谐波空间中,讨论了这两种松弛的光滑性质,得出光滑因子关于附加人工压力项参数的表达式。结果表明：松弛方法的收敛性与网格尺度无关,而依赖于附加人工压力项参数。(3)基于最优红黑Jacobi逐点松弛方法,从理论上分析了Possion方程两层网格算法的收敛性。给出了对流扩散方程的一阶上迎风离散格式,分析了对流占优参数和扩散参数对该离散格式的椭圆率影响,探索了对流扩散方程各参数对多重网格光滑性和两层网格收敛性的影响。在提出的理论方法基础上,利用Riemann解的通量差分分裂法-Godunov方法处理Oseen流控制方程的离散,得到了基于一阶上迎风格式的离散方程,并分析了使用多重网格方法求解该离散方程的V-循环算法和W-循环算法的收敛性,并通过局部Fourier分析方法,对获得的离散方程的聚对称交替线Gauss-Seidel松弛的光滑性质进行了系统研究。结果表明：使用多重网格的两层网格及三层网格算法求解具有不同Reynolds数的Oseen流,即便是在较高Reynolds数情况下,聚对称交替线Gauss-Seidel松弛仍然具有很好的光滑性质,且W-循环算法收敛性比V-循环算法好。(4)首次对基于非定常不可压缩流体的NS方程进行基于交错网格离散系统实施多重网格分布松弛。通过局部Fourier分析,发现该离散系统的光滑性质由时间依赖的对流扩散算子决定,并对两种处理时间依赖问题的多重网格松弛,时空松弛和波形松弛进行了系统研究。在交错网格上,提出了非定常不可压缩流体NS方程仅对空间变量进行离散的半离散格式,并对该离散系统实施分布松弛,使得离散系统多重网格松弛的光滑性质仅取决于时间依赖的对流扩散算子。通过局部Fourier分析,对时间依赖的对流扩散问题所使用的时空多重网格方法和波形多重网格方法进行了光滑性分析。另一方面,在时空多重网格方法的光滑分析中,采用了时空离散格式,其中时间离散采用一阶Euler向后格式,而空间离散采用一阶上迎风格式。提出了多重网格的粗化仅对空间粗化的半粗化方法以及与时空多重网格对应的各种松弛的局部Fourier分析方法。而在波形多重网格方法中,首先利用Laplace变换将时间依赖问题转化为带有复参数的定常问题,然后对应用于波形多重网格方法的各种松弛进行局部Fourier光滑分析。通过提出的两种多重网格方法的光滑分析,研究了对流占优参数和雷诺数对各种松弛算子光滑性的影响,给出了相应的最优光滑因子和最佳松弛参数的选取方法。提出的理论和方法部分用于了由导师负责的国家基金面上项目“水轮机旋转湍流全欧拉并行多层网格模拟研究”等项目的算法设计和代码开发应用中,并获得成功。
[Abstract]:The multigrid algorithm is a fast algorithm for solving the partial differential equations for discrete differential equation. The algebraic equations are solved numerically, in the numerical solution of elliptic partial differential equations has been proved to be the optimal numerical algorithm, its convergence and grid scale size, and the computing scale is proportional to the the cost and problems. Due to the superiority of the multigrid method, making it an efficient numerical method in computational fluid dynamics has received widespread attention and research. On the basis of Higher Education Research Fund for the doctoral program (priority areas) (20135314130002) project, the National Natural Science Foundation of China (51279071). Study on the theory and Application of multigrid method in numerical simulation of the internal flow in hydraulic machinery, especially the local Fourier multigrid smooth theory analysis method in the numerical Solution of multigrid method for incompressible fluid control equations of convergence analysis. The main research content and innovation are as follows: (1) according to the flow characteristics of turbulent flow in hydraulic machinery, proposed an iterative multigrid algorithm and its error format. Local Fourier analysis based on the theory of discrete elliptic operator and operator relaxation rate and smooth factor the definition of the use of different coarse and fine mesh, the relationship between Fourier components, the new definition of invariant subspace, analyzed the expression of Fourier grid transformation operator different coarsening method, calculation method of multigrid algorithm convergence factor theory, innovation and relaxation in two different the Fourier mode function invariant subspace smooth analysis method, obtained the relaxation matrix of Fourier color analysis based on the theory of representation, and with the Poisson equation is given. The corresponding analysis results. The research results show that the iterative multigrid relaxation process based on polychromatic smooth analysis with general, the results are representative and applications. (2) staggered and non staggered grid is proposed for discrete format based on Stokes stream, and the implementation of two different multigrid relaxation algorithm for the discrete the system of smooth Analysis: Poly relaxation and smooth distribution of relaxation analysis. In the discrete staggered grid system is implemented in the distributed relaxation multigrid, the discrete system depends on the smoothness of the Laplace operator, and the smooth factor accordingly. Secondly, on a non staggered grid discrete system, multi grid distribution respectively. Relaxation and relaxation in Fourier poly implementation, and relaxation in harmonic space, discusses the two kinds of relaxation of the smooth nature, the smooth factor on additional artificial pressure parameters The expression. The results show that the convergence and grid scale relaxation method to rely on additional artificial pressure parameters. (3) the optimal Jacobi point relaxation method based on theoretical analysis of the convergence of the two grid algorithm Possion equation. First order upwind discretization scheme is presented for convection diffusion equation and analyzed the influence of convection and diffusion parameters of the elliptic discrete format rate, exploring the various parameters of the convection diffusion equation of multi grid smoothness and two grid convergence effect. Based on the proposed method, the use of Riemann solution of the discrete flux difference splitting method -Godunov method Oseen flow control the obtained equation, discrete equations of first order upwind scheme based on the analysis, and the convergence of the use of multigrid method for solving the discrete equations of the V- cycle and W- cycle algorithm algorithm, and through the Bureau Fourier analysis method, symmetric alternating poly line Gauss-Seidel relaxation of smooth properties of discrete equations obtained were studied. The results show that using multi grid two grid and the three grid algorithm with different Reynolds number Oseen flow, even at high Reynolds number, poly symmetric alternating line Gauss-Seidel relaxation smooth still has good properties, and W- cycle convergence than V- cycle algorithm. (4) for the first time on the NS equation based on the unsteady incompressible fluid of staggered grid discrete system using multiple grid distribution. Based on relaxation through local Fourier analysis, found that the smooth nature of the convection of discrete systems by time the dependence of the diffusion operator, multigrid relaxation and dependence on two kinds of treatment time, temporal relaxation and waveform relaxation were studied. On staggered grid is proposed. The unsteady incompressible NS equations only the spatial variables for semi discrete scheme, implementation and distribution of relaxation of the discrete system, the smooth nature of the discrete system multigrid relaxation only depends on the time dependent convection diffusion operator. Through the local Fourier analysis, space-time multigrid method and multigrid method of diffusion wave the problem of time-dependent convection using the smoothness analysis. On the other hand, in the smooth temporal and spatial analysis of multigrid method, the temporal discrete format, in which time the first order discrete backward Euler format, and the space is discretized using first order upwind scheme is proposed. The semi coarsening method of multiple coarsening only the space grid coarsening and local Fourier relaxation and corresponding space-time multigrid method. The waveform in the multigrid method, we use Laplace Transform time dependent problem into a constant problem with complex parameters, and then the corresponding waveform relaxation for a variety of multigrid methods for local Fourier smooth analysis. Through the analysis of two kinds of smooth multigrid method proposed, studied the convection parameters and Reynolds number on various relaxation effects of smoothness operator selection method is given, the corresponding optimal smoothing factor and the optimum relaxation parameter. Some theories and methods for state funds by the tutor on the project of "rotating turbulent turbine Euler parallel multi grid simulation research" project, algorithm design and code design application, and achieved success.

【学位授予单位】：昆明理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82
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本文编号：1531573