中文摘要：

在三维非恒定渗流有限元计算中，不可避免地需计算自由面上的边界积分项。建议一种高精度数值积分的方法求取自由面边界积分项。其基本思路是，基于8节点空间等参单元，根据压力为零的边界条件确定自由面满足的曲面方程，将自由面边界积分转化为轫平面上的二重积分；然后再计算自由面与三维等参单元的交点，将交点投影到局部坐标平面轫上，并根据点与线的相对关系，确定投影点所围成的局部坐标积分区域；再将总积分区域划分成若干个三角形子区域，并利用变步长Simpson方法计算各三角形子区域上的二重积分，从而实现了高精度的自由面边界积分。该方法避免了单元中自由面为平面的假设，可提高计算精度，特别对于单元内自由面变化剧烈情况，更为显著。将该方法应用于砂槽模型和土坝的渗流分析中，计算结果与实际情况较为接近。

英文摘要：

In Getechnical engineering, such as slope, underground tunnel, dam and dike, and soil foundation etc., 3-D unsteady seepage problem often occurs. In performing the numerical analysis of unsteady seepage problem, it is necessary to cal- culate the integral on free surface boundary. In the paper, a high precision numerical integral method is presented. Based on the 8-node isoparametric element, the equation of the free surface boundary in an element is obtained firstly according to zero water pressure condition, and then the integral on the free surface boundary is translated into the integral on the plane Eη/of local reference. And the integral region is calculated according to the projections of the intersecting points where free surface and the boundaries of the dement cross. Finally the whole integral region is divided into several triangle sub-regions, and the Simpson method with changing steps is performed to calculate the integral on each triangle sub - region. The high precision nu- merical integral method can avoid the assumption that the free surface is a plane within an element. It can improve the calculation accuracy, especially for the case that the free surface within an element varies very sharply. The applications in both the sand tank model and an example of homogeneous earth dam are given, it shows that the method is practical.