边界元法在计算地下水稳定水位和流量中的应用

边界元法是一种新的数值计算方法。该法易于处理无限区域的地下水流问题,并且计算流量也较其他方法准确。本文介绍在二维稳定流的情况下如何计算地下水的水头和流量。承压含水层中的稳定流动,水头H满足拉普拉斯方程。利用格林第二公式,可以得到边界积分方程,即边界元的基本公式。可以用数值方法计算这一边界积分。为此,在边界上选取有限个点,称为节点,两节点间的线段称为单元。本文中选用线性单元和线性插值。引进局部坐标系,可以得到表示H和( H/ n)关系的方程。我们可以选一个节点作为固定的基点,其他节点为动点,对于每一选择都可得到一个方程。依次把每一节点作为基点,可得到N个方程,构成一个线性代数方程组。根据边界条件,每一节点中的H或( H/ n)有一个是已知的,解方程组可求出另一个。解出边界上的全部H和( H/ n)以后,可算出内部的水头和流量。对于非均质问题可划分为几个区域来处理。分界线上要满足相容性方程。对于( H/ n)的不连续点,可用“节点多值法”处理。

APPLICATION OF BOUNDARY ELEMENT METHOD TO CALCULATION OF GROUNDWATER HEAD AND DISCHARGE FOR STEADY FLOW

The Boundary element method is a new numerical ealculation method. The method is easy to use in treating infinite domain premieres of groundwater flow and more accurate in calculating discharge, than the finite elemet and finite difference method. This paper discusses how to use the boundary element method to compute groundwater head and discharge for two-dimensional steady flow. In the case of steady flow in a confined aquifer, groundwater head H sa- tisfies Laplace's equation. Using Green's second identity we obtain the bouudary integral equation, The equation can be solved by choosing a finite number of points on the boundary and numerically performing the integration. Those points are called nodes and the line segments between a pair of nodes are cal- led elements. Linear elements and linear interpolation functions are used in this paper. Employing the local co-ordinate system we can get an equation which relates H ana. We can choose a node as a fixed 'base point' and the others as variable 'field points'; there is one equation for each choice. We can make each node, in turn, a base point which will yield N equations. A system of linear algebraic equations for H and (j=1, 2,...., N) can be obtained by gathering those equations. Accrding to the boundary conditions either H or is known. We can solve the system of equations for the unknown H or on the boundary. Then the head H and discharge in the interior of the region can be compared. We can divide a domain into several different zones to deal with unhomo geneous problems. The head H and must satisfy the compatiblity equations across he dividing line. In order to deal with the lack of continuity of at some nodes, a "Multivalue-node Method" is proposed.