一种基于单元局部坐标系求解二维孔隙渗流问题的数值方法

›› 2014, Vol. 299 ›› Issue (2): 584-590.

一种基于单元局部坐标系求解二维孔隙渗流问题的数值方法

冯春，李世海，王理想

中国科学院力学研究所，北京 100190

收稿日期:2012-11-25 出版日期:2014-02-11 发布日期:2014-02-18
作者简介:冯春，男，1982年生，助理研究员，博士研究生，主要从事岩土力学领域数值计算方法的研究工作。
基金资助:
国家重点基础研究发展计划（973）项目资助（No. 2010CB731506）；国家科技支撑计划资助（No. 2012BAK10B00）；国家自然科学基金青年基金项目资助（No. 11002146）

A numerical method to solve pore seepage problems based on element local coordinate system

FENG Chun, LI Shi-hai, WANG Li-xiang

Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China

Received:2012-11-25 Online:2014-02-11 Published:2014-02-18

摘要/Abstract

摘要： 将数值计算区域用三角形单元进行离散，并为每个单元构建局部坐标系。局部坐标系的X轴为三角形单元某一条边的方向，局部坐标系的原点为该边的其中一个端点。在局部坐标系下，基于“格林公式”及达西定律推导了单元压力梯度及单元流速的解析表达式，给出了流经单元各棱及各节点的流量计算方法。形成了类似固体弹簧系统的渗流管道网络，建立了管道压差与流量的函数关系。将各单元局部坐标系下求得的流速及流量转换至整体坐标系，并在节点上进行凝聚。通过引入流体体积模量实现了节点渗透压力的显式求解，通过引入节点饱和度实现了非饱和问题的求解。基于局部坐标系的方法具有物理意义明确、求解过程简单等特点。通过在局部坐标系下构建管道压差与管道流量的对应关系，将有限元的渗透刚度矩阵简化为两个管道的渗透刚度值，从而节省了内存，提高了计算效率。4个数值算例的计算结果与理论解基本一致，表明了该方法在求解稳态、非稳态、饱和、非饱和渗流问题时的精度。

关键词: 孔隙渗流, 数值方法, 局部坐标系, 体积模量, 饱和度

Abstract: Numerical domain is discretized by triangle elements; and local coordinate system is set up for each one. X-axis of local coordinate system is along one edge of the element; and the origin point is at the endpoint of the edge. In local coordinate system, based on Green formula and Darcy law, analytical expression of pressure gradient and flow velocity of element, discharges of each edge and each node are given. The seepage “pipeline” network similar to solid spring system is formed; and the relationship between pressure difference and discharge of pipeline is built. Flow velocity and discharge of each element in element local coordinate system should be transformed to global coordinate system, and should be accumulated in each node. By introducing fluid bulk modulus and node saturation, pore pressure of each node could be calculated explicitly and the unsaturated problems could be simulated well. The method based on local coordinate system is clear in physical meaning and simple in solution process. According to constructing the relationship between pressure difference and discharge of pipeline in local coordinate system, the seepage stiffness matrix of FEM is simplified to two seepage stiffness values; so the memory is saved and the efficiency is improved. The results of 4 numerical cases almost coincide with analytical solutions so as to demonstrate the solution precision when simulating steady-state, non-steady-state, saturated, unsaturated seepage problems.

Key words: pore seepage, numerical method, local coordinate system, bulk modulus, saturation

中图分类号:

O 241

冯春，李世海，王理想. 一种基于单元局部坐标系求解二维孔隙渗流问题的数值方法[J]. , 2014, 299(2): 584-590.

FENG Chun, LI Shi-hai, WANG Li-xiang . A numerical method to solve pore seepage problems based on element local coordinate system[J]. , 2014, 299(2): 584-590.

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