渗流转角奇异问题的数值流形单元法

Turning point singularity problem in seepage by numerical manifold method

在水利工程中的渗流计算中,大量存在着转角奇异性问题。一般有限元方法需要构建特殊的奇异形函数和单元属性来处理,存在不便利性,而且广泛使用的等参单元又忽视了奇异性问题。运用数值流形方法,提出了采用构造局部特殊片的方法来解决该奇异性问题。首先,将转角奇异的解析解构建为一个基函数,与常数项基函数一起形成奇异片覆盖函数,在奇异点影响到的星点上采用奇异片覆盖函数,其他星点采用常数片覆盖函数进行求解,计算结果逼近真解,并能很好地反映转角中的强奇异性。其次,给出这类问题的流形元离散列式,将经典正三角形流形元改进为正四边形,并介绍程序的编制步骤,和该特殊物理片的施加方法。再次,3个工程算例分别为坝基不透水帷幕有压渗流解析解对比、经典流速跌落试验对比以及堰下板桩有压渗流与已有计算流网对比,算例表明了该方法的有效性和正确性,为进一步分析奇异性对结构和复杂工程的影响打下基础。该方法也适用于其他基于单位分解插值的数值分析方法,也为其他场变量连续而偏导数不连续问题提供了思路。

In seepage calculation of hydraulic engineering,there are a large number of singularity problems at turning point.It is inconvenience that the general finite element method needs to construct special singularity shape function and element property to deal with the singularity problem,which used to be ignored when the isoparametric element was widely used.In this paper,a new method by constructing a local special patch is proposed to solve this singularity problem by using numerical manifold method.Firstly,the singularity covering function is formed together by the constant term basis function and a new basis function constructed by the analytic solution at turning point.The singularity covering function is used on the star points affected by the singularity point,at the same time,the constant patch covering function is used on the other star points,and the calculation result approximates the real solution and can well reflect the strong singularity in the corner.Secondly,the discrete formula of the numerical manifold method about this kind of problem is given,and the classical regular triangle of manifold element is improved to square,as well as,the procedure of the program and the application method of the special physical patch are introduced.Thirdly,three engineering examples are respectively confined seepage of dam foundation with impervious curtain contrast with the analystic solution,the comparison of the classical flow velocity drop test,and the confined seepage of weir with sheet pile compared with flow net which had been calculated,those show the effectiveness and correctness of the method,and provide a basis for further analysis of the effects of singularity on the structure and complex engineering.Finally,The proposed method is also applicable to other numerical analysis methods based on the partition of unity method of interpolation,and also provides an approach for those problems with continuous field and discontinuous partial derivatives.