饱和—非饱和土壤渗流过程中Richards方程的分析与计算

发布时间：2018-03-16 23:15

本文选题：Richards方程　切入点：半解析解　出处：《兰州大学》2016年博士论文　论文类型：学位论文

【摘要】：Richards方程是描述土壤渗流过程的基本控制方程.土壤介质的非均质性、土壤导水率的高度非线性性以及区域和初、边值条件的复杂性等诸多因素,为Richards方程数值求解带来很多困难.传统的有限差分方法或者有限元方法会导致计算结果出现非物理的数值震荡或者弥散.寻求有效的Richards方程数值解是土壤水动力学研究的重点内容.本文主要在以下几个方面进行了相关工作:在分层土壤渗流问题中,分层界面处土壤本构关系发生变化,导水率及含水量出现跳跃,对Richards方程数值计算的稳定性带来挑战.我们针对分层土壤下一类Gardener-Basha型Richards方程,分析分层界面处本构关系,得到形式上的解析解,然后离散时间层,通过迭代的方式,得到方程半解析解,避免了分层界面处的数值震荡.进一步,本文对上述方法设计并行计算格式.数值实验显示,并行加速比实验值达到4.392,并行方式为完全可扩展的.非饱和渗流变量(如:压力水头)在较小的空间和较短的时间范围内快速变化,是造成Richards方程数值求解困难的原因之一.提高数值格式精度、提高解在整个区域上的光滑性以及自适应网格剖分是我们在解决这一问题时的基本策略.本文对空间1维采用3次B样条基有限元,对空间2维采用5次Hermit型插值有限元,保证了解u在?上的整体光滑性,缓解非物理震荡现象.进一步,结合多重网格技术,给出自适应计算格式.数值实验显示,基于上述方法得到的结果,对数值震荡的控制优于线性基有限元方法得到的结果.Richards方程刻画的问题多为长时间问题.时间层采用具有保结构性质的数值方法可以有效提高数值稳定性,保证解具备长时间良好数值性态.本文对(1+1)维、(1+2)维Richards方程时间层用s级2s阶全隐辛Runge-Kutta方法进行数值离散,以便在积分过程中长时间保持系统的固有特性.数值实验显示,相对于一般Runge-Kutta方法,该方法对Richards方程刻画长时间问题表现出更高的数值稳定性.考虑与Richards方程相关的一类冻土耦合模型水热耦合模型.该模型是一个科研项目中关注的模型.目前对这类耦合模型的数值求解多采用有限差分方法.本文对耦合模型的中的两个基本方程,分析并整理本构关系,给出基于差分方法的数值计算格式,在此基础上,抽象模型方程,给出基于有限元方法的计算格式.Richards方程的实际应用往往对应海量计算,这使我们必须考虑如何提高并行计算效率.本文对使用广泛的Gauss消去法,设计了一种列行调整双向流水线并行算法,从通信时间、并行度、可扩展性方面分析该算法的性能,并进行了数值实验.数值实验显示,由于在通信阶段数据执行的并发度提高,其并行加速比可达到3.461,优于传统的行列划分并行方式.
[Abstract]:The Richards equation is the basic governing equation for describing the soil seepage process, the heterogeneity of soil media, the high nonlinearity of soil water conductivity, and the complexity of regional and initial boundary conditions, and so on. The traditional finite difference method or finite element method will lead to non-physical numerical oscillation or dispersion of the results. Seeking effective numerical solution of Richards equation is soil hydrodynamic. The main contents of this paper are as follows: in the problem of layered soil seepage, The change of soil constitutive relation at the stratified interface, the jump of water conductivity and moisture content, brings a challenge to the stability of numerical calculation of Richards equation. We analyze the constitutive relation at the stratified interface for a class of Gardener-Basha type Richards equation under stratified soil. The formal analytical solution is obtained, and then the discrete time layer is discretized, and the semi-analytical solution of the equation is obtained by iterative method, which avoids the numerical oscillation at the layered interface. Furthermore, the parallel computing scheme is designed for the above methods. The numerical experiments show that, The parallel speedup ratio is 4.392, and the parallel mode is completely extensible. The unsaturated seepage variables (such as pressure head) change rapidly in smaller space and shorter time range. It is one of the reasons that the Richards equation is difficult to solve numerically. To improve the smoothness of the solution in the whole region and the adaptive mesh generation are the basic strategies for solving this problem. In this paper, we adopt the 3-order B-spline finite element method for the first dimension and the Hermit interpolation finite element method for the second dimension. Make sure you know you're here? Furthermore, combining with the technique of multi-grid, an adaptive computing scheme is given. Numerical experiments show that, based on the results obtained by the above method, The control of numerical oscillation is better than that of the results obtained by linear basis finite element method. The problems described by Richards equation are mostly long time problems. The numerical method with conserved structure property in time layer can effectively improve the numerical stability. It is guaranteed that the solution has good numerical behavior for a long time. In this paper, the time layer of the Richards equation is discretized by the Runge-Kutta method of order 2 s in order to preserve the inherent characteristics of the system for a long time during the integral process. Compared to the general Runge-Kutta method, This method has higher numerical stability for Richards equation to depict long time problems. A kind of coupled model of permafrost related to Richards equation is considered in this paper. The model is a model concerned in scientific research projects. The finite difference method is used to solve this kind of coupling model. In this paper, two basic equations in the coupled model are discussed. The constitutive relation is analyzed and arranged, and the numerical calculation scheme based on the difference method is given. On this basis, the model equation is abstracted, and the practical application of the Richards equation based on the finite element method is given. Therefore, we must consider how to improve the efficiency of parallel computing. In this paper, we design a column and row adjusted bidirectional pipeline parallel algorithm to analyze the performance of the algorithm in terms of communication time, parallelism and extensibility. The numerical experiments show that the parallel speedup ratio can reach 3.461 because of the increase of the concurrency of the data execution in the communication stage, which is superior to the traditional parallel method of column and column partition.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

【相似文献】