Seepage analysis based on weak finite element method
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摘要: 在坝、闸和堤防等水工建筑物的设计和运行管理中，渗流计算占有非常重要的地位。目前渗流分析的方法以一般有限元方法为主。在渗流分析时往往希望通过加密网格来追求更高精度的数值结果，但是由于一般有限元法属于协调有限元方法，它不适合处理混合剖分网格，加密网格容易导致计算格式不稳定、计算结果不收敛。而弱有限元方法是一种非协调有限元方法，用弱函数的弱梯度算子替代一般有限元方法变分式中的经典梯度算子，并在变分式中增加稳定子项，保证了计算格式的绝对稳定性，并且它适用于混合网格。对于同一剖分网格，弱有限元法的总体代数方程组的自由度远远比一般有限元方法的要大很多，为了降低弱有限元法的自由度，探讨根据弱有限元方法适用于混合剖分网格的特点，采用有针对性的混合剖分网格，运用弱有限元法求解渗流自由面和进行有防渗帷幕的闸基渗流分析，数值结果表明弱有限元方法能灵活处理混合剖分网格，并且计算结果具有高精度。Abstract: In the design and operation management of hydraulic structures such as dams, gates and embankments, the seepage calculation plays a very important role. The current methods for seepage analysis are mainly based on the general finite element method. It is often hoped, by refining the mesh, to pursue the calculated results with higher-precision in the seepage analysis. However, the general finite element method, which belongs to the coordinated finite element method, cannot handle hybrid grids, and the refined mesh may lead to unstable calculation formats and the divergence of the calculated results. The weak finite element method is a non-coordinated one, which replaces the classical gradient operator in the variational formula for the general finite element method with the weak gradient operator of the weak function, and adds a stable sub-term in the variational formula, to obtain an absolute stability of the calculation format, and it can handle hybrid grids. For the same mesh grid, the degree of freedom of the overall algebraic equations for the weak finite element method is much larger than that of the general finite element method. In order to reduce the degree of freedom of the weak finite element method, the targeted mixed meshing grid is used to establish the weak finite element method which is used to solve the free surface of seepage and to analyze the seepage field of the gate foundation with anti-seepage curtain. The numerical simulation shows that the weak finite element method can be used to process flexibly hybrid grids, and it is of high accuracy.
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Keywords:
- weak finite element method /
- hybrid grid /
- seepage /
- free surface /
- brake base
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图 9 防渗墙深5 m时水头等值线分布图
Figure 9. Distribution of water head contour line at depth of anti-seepage wall of 5 m
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图 10 防渗墙深10 m时水头等值线分布图
Figure 10. Distribution of water head contour line at depth of anti-seepage wall of 10 m
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图 11 防渗墙深15 m时水头等值线分布图
Figure 11. Distribution of water head contour line at depth of anti-seepage wall of 15 m
表 1 剖分单元信息
Table 1 Information of elements
单元编号 单元类型 节点编号 1 3 3 2 4 2 3 3 1 2 3 4 3 4 5 6 4 5 3 6 7 8 1 
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表 2 节点坐标信息
Table 2 Information of node coordinates
节点 1 2 3 4 5 6 7 8 x 1 2 1 2 2 1 0 0 y 0 0 1 1 2 2 2 0
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表 3 计算结果对比Ⅰ
Table 3 Comparison of calculated results Ⅰ
横坐标 解析解 FEM 误差1 WG 误差2 0 10.00 10.00 0.00 10.00 0.00 1 9.59 9.73 0.14 9.62 0.03 2 9.17 9.39 0.22 9.26 0.09 3 8.72 8.99 0.27 8.79 0.07 4 8.25 8.53 0.28 8.30 0.05 5 7.75 8.03 0.28 7.78 0.03 6 7.21 7.46 0.25 7.23 0.02 7 6.63 6.83 0.20 6.63 0.00 8 6.00 6.11 0.11 6.04 0.04 9 5.29 5.21 -0.08 5.34 0.05 10 4.47 4.50 0.03 4.50 0.03
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表 4 计算结果对比Ⅱ
Table 4 Comparison of calculated results Ⅱ
横坐标 解析解 WG 误差 0 10.00 10.00 0.00 1 9.59 9.59 0.00 2 9.17 9.18 0.01 3 8.72 8.77 0.05 4 8.25 8.23 -0.02 5 7.75 7.74 -0.01 6 7.21 7.16 -0.05 7 6.63 6.61 -0.02 8 6.00 6.00 0.00 9 5.29 5.34 0.05 10 4.47 4.50 0.03
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表 5 水力梯度值
Table 5 Values of hydraulic gradient
防渗墙
深/m防渗墙底端处水力梯度 闸基出口处
水力梯度5 0.323 0.332 10 0.333 0.262 15 0.358 0.190
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