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对于渗流问题的研究在很多领域有着广泛的应用,工程中通常采用有限元法对其进行数值求解,这往往需要耗费较大的计算资源从而限制了其计算规模。弱形式求积元法是一种简单和高效的数值方法,该方法基于问题弱形式描述,可对全域进行高阶近似,具有较快的收敛性,在结构分析领域已有广泛的应用。将弱形式求积元法应用于渗流问题的求解,分析了二维及三维渗流问题,包括承压渗流和无压渗流;对于无压渗流,采用变网格方法,使用积分点位置的多项式插值来近似表示自由面。求解了数值算例并得到了与解析解或者文献解一致的结果。结果表明:与有限元法相比,弱形式求积元法使用较少的自由度就可以得到收敛的结果,显示了弱形式求积元法在渗流分析中的有效性。
The research of seepage problem has been widely used in many fields. The finite element method is often used in engineering to solve the problem, which often requires large computational resources and thus limits its computational scale. Weak formal quadrature method is a simple and efficient numerical method. Based on the weak form of the problem description, this method can perform high-order approximation of the whole region, has faster convergence and has been widely used in the field of structural analysis. The weak form of the quadrature method is applied to solve the seepage problem, and two-dimensional and three-dimensional seepage problems are analyzed, including confined seepage and unpressurized seepage. In the case of unpressurized seepage, the variable grid method is adopted and polynomial interpolation To approximate the free surface. Numerical examples are given and the results are consistent with analytic solutions or literature solutions. The results show that, compared with the finite element method, the weak form quadrature method can obtain the convergence result with less degree of freedom, and shows the validity of weak form quadrature method in seepage analysis.