非饱和带水分运移的有限分析数值模拟理论方法

发布时间：2018-11-07 18:48

【摘要】：包气带是连接大气降水、地表水与地下水的重要纽带,并且对于保护干旱、半干旱地区的生态环境具有重大意义。由于描述非饱和带水分运移的动力学参数具有高度的非线性,导致了非饱和带水分运移的Richard方程同样也具有高度的非线性,因此求解该类方程极其困难。通常在相对简单的定解条件下,可以获得描述包气带水分运移问题的解析解,然而在研究较为复杂条件下的非饱和带水流问题时,当前最为有效的方法还是采用数值模拟方法。Richard方程在空间上是一个二阶椭圆型方程,但是对于时间是一阶的,具有抛物线型方程的性质。所以在使用有限差分法及有限元法等传统的数值模拟方法求解时,容易造成数值弥散、数值振荡及质量不守恒等一系列问题。上世纪80年代陈景仁提出了一种新的数值计算方法——有限分析法。有限分析法的基本思想是:将控制方程的局部解析解组成整体数值解,这样得到的解可以比较好的保持原有问题的物理特性,格式通过自动调节有限分析系数来反映对流与扩散效应,可以获得单调无震荡性解,数值稳定性高。因此,本文应用有限分析法求解Richard方程,探讨包气带水分运移问题。本文提出了四种形式包气带在均质条件下水分运移的有限分析计算格式及一种非均质条件下的有限分析计算格式。通过定性定量化的研究,可以获得以下几个方面的成果及结论:1.在数学上严格证明了有限分析计算格式的稳定性及收敛性,即在理论上说明了有限分析法是无条件稳定及收敛的。2.均质条件下四种形式的有限分析计算格式中,除了负压型的有限分析计算格式外,其它三种形式的有限分析计算格式都可以获得稳定性高、收敛性好及精度高的数值解,并且能够较好的控制质量平衡误差。通过与解析解进行比较,混合型Richard方程的有限分析计算格式可以获得计算精度最高的数值解并且能够最好的控制质量平衡误差;含水量型Richard方程及Kirchhoff变换的Richard方程的有限分析计算格式次之,负压型Richard方程的有限分析计算格式最差。因此不推荐使用负压型Richard方程的有限分析计算格式。3.混合型Richard方程、含水量型Richard方程及Kirchhoff变换的Richard方程等三种有限分析计算格式在相对粗糙的空间步长条件下仍然可以获得精度较高的数值解。4.首次应用有限分析法推导了基于Kirchhoff变换后负压型Richard方程的有限分析分析计算格式,用来求解包气带非均质条件水分运移问题。通过和解析解对比可知,本文推导的算法可以很好的解决Kirchhoff变换后,变量在非均质界面不连续的问题,并且该算法在较为宽松步长条件下也能够获得较高精度的数值解。然而使用有限差分法(应用几何平均计算内部节点的渗透系数)求解包气带非均质条件水流问题时,对空间步长的要求较高,在非均质界面上会引起巨大的局部单点误差,并且其还会影响整个计算域的计算精度。5.应用有限分析法模拟了一个室内实验。有限分析法的模拟结果与实测值拟合度较高,说明了有限分析法能够应用于求解相对复杂的实际问题。
[Abstract]:The package gas zone is an important link between the atmospheric precipitation, the surface water and the ground water, and is of great significance to the protection of the ecological environment in the arid and semi-arid areas. Because the dynamic parameters describing the water migration of unsaturated zone have a high degree of non-linearity, the Richard's equation, which results in the water migration of unsaturated zone, also has a high degree of non-linearity, so it is extremely difficult to solve this kind of equation. In this paper, the analytical solution to the problem of water migration in the air zone of the bag can be obtained under relatively simple fixed conditions. However, in the case of the water flow in unsaturated zone under the complicated conditions, the most effective method is to adopt the numerical simulation method. Richard's equation is a second-order elliptic equation in space, but it has a parabolic equation for time. Therefore, a series of problems such as numerical dispersion, numerical oscillation and non-conservation of quality can be easily caused when using the traditional numerical simulation method such as the finite difference method and the finite element method. In the 1980s, Chen Jingren proposed a new numerical method of numerical calculation _ finite analytic method. The basic idea of the finite analytic method is to form the whole numerical solution by the local analytic solution of the control equation, so that the obtained solution can maintain the physical property of the original problem, and the format can reflect the convection and the diffusion effect by automatically adjusting the limited analysis coefficient, and the numerical stability is high. In this paper, the finite analytic method is applied to solve the problem of water migration in the package gas zone. In this paper, the finite analysis and calculation scheme of the water migration under the homogeneous condition and the finite analytic calculation method under the non-homogeneous condition are presented in this paper. Through the qualitative and quantitative study, the following results and conclusions can be obtained: 1. In this paper, the stability and the convergence of the finite analytic calculation form are strictly proved, that is, the finite analytic method is proved to be unconditionally stable and convergent. In addition to the limited analysis and calculation of the negative pressure type, the finite analysis and calculation form of the four forms of homogeneous condition can obtain the numerical solution with high stability, good convergence and high precision, and can control the quality balance error better. By comparison with the analytical solution, the finite analysis and calculation form of the mixed-type Richard equation can obtain the numerical solution with the highest calculation accuracy and the best control quality balance error; the water content type Richard equation and the Kirchhoff-transformed Richard equation are the second in the finite analysis and calculation format, The finite analysis of the negative pressure type Richard's equation is the worst. Therefore, the limited analysis and calculation format of the negative pressure type Richard equation is not recommended. A numerical solution with high accuracy can still be obtained under the condition of relatively rough space steps, such as the mixed-type Richard's equation, the water-content-type Richard equation and the Kirchhoff-transformed Richard's equation. The finite analysis and analysis of the negative pressure type Richard's equation based on Kirchhoff's transformation are derived for the first time. The method is used to solve the problem of moisture migration in the non-homogeneous condition of the package gas zone. Compared with the analytical solution, the proposed algorithm can solve the problem that the variable is not continuous in the non-homogeneous interface after the Kirchhoff transformation, and the algorithm can obtain high-precision numerical solution under the condition of looser step. However, using the finite difference method (using the geometric mean to calculate the permeability coefficient of the internal node), the requirement of the space step size is high, and a great local single-point error can be caused on the non-homogeneous interface. and it also affects the calculation accuracy of the whole computing domain. The application of finite analytic method to the simulation of an indoor experiment. The simulation results of the finite analytic method are higher than that of the measured values, and the finite analytic method can be used to solve the relative complex practical problems.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：P641.2

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