An efficient iterative method for solving the governing equation of water infiltration problems in unsaturated soils

One of the significant problems in geo-environmental and geotechnical engineering is the unsaturated flow of soil in unsaturated soils. The model of this phenomenon in porous media is governed by the Richards equation. In this paper a new, efficient, iterative method is used to handle the Richards equation. This new technique is obtained from the variational iteration method by a simple reconstruction that is the Laplace iteration method (LIM). In order to evaluate the efficiency and accuracy of the solutions obtained by the proposed method, two representative examples were investigated. The obtained results show that the Laplace iteration method is a very effective method, simplifies the difficulty of classical techniques and is quite accurate for systems of partial differential equations.     

加速型改进迭代法在非饱和土渗流中的应用研究

Richards方程常用于非饱和土渗流问题,并且应用广泛。在数值求解中,对Richards方程线性化,进而采用有限差分法进行数值离散以及迭代计算。其中传统的迭代法比如Jacobi迭代、Gauss-Seidel迭代法（GS）和连续超松驰迭代法（successive over-relaxation method,简称SOR）迭代收敛率较慢,尤其在离散空间步长较小以及离散时间步长较大时。因此,采用整体校正法以及多步预处理法对传统迭代法进行改进,提出一种基于整体校正法的多步预处理Gauss-Seidel迭代法（improved Gauss-Seidel iterative method with multistep preconditioner based on the integral correction method,简称ICMP(m)-GS）求解Richards方程导出的线性方程组。通过非饱和渗流算例,并与传统迭代法和解析解对比,对改进算法的收敛率和加速效果进行了验证。结果表明,提出的ICMP(m)-GS可以很大程度地改善线性方程组的病态性,相较于常规方法GS,SOR以及单一改进方法,ICMP(m)-GS具有更快的收敛率,更高的计算效率和计算精度。该方法可以为非饱和土渗流的数值模拟提供一定参考。     

Application of an improved P(m)-SOR iteration method for flow in partially saturated soils

This paper studies the potential of using the successive over-relaxation iteration method with polynomial preconditioner (P(m)-SOR) to solve variably saturated flow problems described by the linearized Richards’ equation. The finite difference method is employed to numerically discretize and produce a system of linear equations. Generally, the traditional Picard method needs to re-evaluate the iterative matrix in each iteration, so it is time-consuming. And under unfavorable conditions such as infiltration into extremely dry soil, the Picard method suffers from numerical non-convergence. For linear iterative methods, the traditional Gauss-Seidel iteration method (GS) has a slow convergence rate, and it is difficult to determine the optimum value of the relaxation factor w in the successive over-relaxation iteration method (SOR). Thus, the approximate optimum value of w is obtained based on the minimum spectral radius of the iterative matrix, and the P(m)-SOR method is extended to model underground water flow in unsaturated soils. The improved method is verified using three test examples. Compared with conventional Picard iteration, GS and SOR methods, numerical results demonstrate that the P(m)-SOR has faster convergence rate, less computation cost, and good error stability. Besides, the results reveal that the convergence rate of the P(m)-SOR method is positively correlated with the parameter m. This method can serve as a reference for numerical simulation of unsaturated flow.

     

一种新改进的DPIV方法

数字粒子图像测速技术DPIV(Digital Particle Image Velocimetry)是一种快速提取流场速度的方法,但测速精度较低。已有的改进方法虽然使精度得到了提高,却丧失了DPIV快速处理的优点。本文综合已有方法的优点,提出了一种兼有快速和精度较高的处理方法。该方法以Willert提出的DPIV方法为基础,吸收了Huang提出的改进方法中的迭代思想,既保留了DPIV快速处理的优点,又在一定程度上提高了测速的精度,非常适合于多工况和非恒定流的测量。     

对迭代法位场向下延拓方法的剖析

位场向下延拓迭代法的实质内容是"向上延拓而不是向下延拓"和以"迭代的结果趋近于观测值"为标准的操作过程。根据数据操作流程剖析了位场向下延拓迭代法的运行机制,得到了迭代数据在空间域的变化规律,即用迭代法将观测高度的位场向下延拓一个深度h。这实际上是通过不同高度的向上延拓来实现的。也就是说,迭代次数增加一次,涉及的上延平面就增大一个h的高度。一般地,迭代次数n与上延高度h的关系为n~(n+1)h。在空间域中,初值、上延结果、差以及每一次校正后的结果都能用满足莱布尼兹定理的交错级数表示,从而得出了迭代法能够收敛的结论;或者,以"观测高度上的实测值与计算值的差值小到可以忽略"为标准,从数学上也能证明迭代法能够收敛。数学推论和模型试验结果说明了迭代的位场初值可以任意给定。在实际操作中,迭代误差标准的影响和由于迭代误差标准不恰当可能出现不能达到迭代标准的情况,需引起注意,也值得进一步研究。     

Performance of a flux splitting scheme when solving the single-phase pressure equation discretized by MPFA

In this article we present a series of tests to study how well suited the TPFA coefficient matrix is as a preconditioner for the MPFA discrete system of equations in an iterative solver, using a flux splitting method. These tests have been conducted for single-phase flow for a wide range of anisotropy, heterogeneity, and grid skewness (mainly parallelogram grids). We use the K-orthogonal part of the MPFA transmissibilities for a parallelogram grid to govern the TPFA transmissibilities. The convergence of the flux splitting method is for each test case measured by the spectral radius of the iteration matrix.     

非饱和土中溶质迁移参数反演的HISR方法

以非饱和土中溶质迁移参数反演问题为背景, 依据正则化方法的思路, 以Itakura Saito距离作为同伦函数中的平凡问题, 将同伦方法引入非线性参数反演问题的求解, 进而提出一种求解非线性参数反演问题的大范围收敛(HomotoyItakura SaitoRegularization, HISR) 方法.为保证迭代稳定性, 并同时削弱观测噪声的影响, 同伦参数的修正采用了连续化修正方法.本文将HISR方法应用于求解带有平衡及非平衡吸附效应的一维非饱和土中溶质迁移参数反演问题, 计算结果表明HISR方法具有大范围收敛性及计算稳健性, 同时有较强的抵抗观测噪声的能力.      

重力界面反演改进算法的比较与应用

为得到更接近地下真实分布的反演界面,笔者比较分析Parker-Oldenburg反演和改进迭代反演,发现改进迭代反演方法不仅可以避免计算放大项,而且能够得到更好结果。通过理论模型,对经典Parker-Oldenburg方法和改进迭代反演方法的计算速度、迭代收敛性和结果精确性进行比较,结果表明改进迭代反演方法在保证计算速度和迭代收敛的情况下,能获得更精确的地下界面结构。通过实际数据验证了改进迭代反演的有效性和实用性。     

大型地下洞室群围岩应力-损伤-渗流耦合分析

以渗透体力来考虑渗流场的力学效应,建立了应力-损伤-渗透系数关系方程来考虑应力和损伤对渗流场的影响,结合岩体结构的三维弹塑性损伤有限元分析,建立了大型地下洞室开挖围岩应力-损伤-渗流耦合的计算模型。该计算模型求解的难度主要体现在岩体材料弹塑性、损伤、渗流自由面边界、渗流溢出边界、应力-损伤-渗流相互影响关系等。提出分步迭代法对以上因素进行归纳后按一定顺序分别进行迭代求解,取得了良好的计算效果。将该方法应用于某水电站大型地下洞室群分析,得出了一系列有意义的结论。