基于边界元法对稳态渗流问题的研究

发布时间：2018-07-01 14:53

本文选题：边界元方法 + 渗流　；参考：《哈尔滨工业大学》2014年硕士论文

【摘要】：大部分的工程问题都可归结为偏微分方程边值问题，但由于边界条件复杂、物体性质不均匀、几何形状不规则等原因，一般都求不出问题的解析解，因此人们开始关注求解问题的近似解法——数值分析方法。边界元方法作为数值分析方法的一种，虽然它的出现晚于有限元法，但以其独特的优势在近五十年内得到了快速的发展。如今，，边界元引用的领域也越来越广泛，如位势问题、弹性问题、弹塑性问题、动力学问题等。本论文采用直接边界元法对四种渗流问题进行了研究。根据渗流介质性质的不同，即渗透系数的不同，四种渗流问题分别为：均质各向同性渗流问题（k x ky，且都为常数）、正交各向异性渗流问题（k x ky，但都为常数）、非饱和土渗流问题（k k h se）、分层土的渗流问题。本文除了完成四种渗流问题的解决过程外，每类问题也都编写了matlab程序，并用算例进行验证。具体的研究内容如下：（1）各类渗流问题的边界积分方程的推导。在各类渗流问题的控制方程的基础上，采用加权余量法及格林公式将偏微分方程转化为边界积分方程。其中非饱和土渗流问题比较特殊，因为它的控制方程是非线性的，所以积分方程中还有域内积分项。（2）边界离散方法的研究。在前两个渗流问题中，都采用了常量元与线性元对边界进行离散，并对两种离散方法最终的计算结果精度进行对比，判断优劣。（3）角点处法向流量不连续问题的研究。当采用线性元离散边界，并用单元的两端点作结点时会产生角点问题，本文采用重结点法及重结点单未知量法解决，并针对不同渗流问题得出两种方法的优缺点及适用情况。（4）奇异积分的处理。对于边界积分式中存在的奇异性一般为一阶奇异性，本文采用解析算法处理。对于域内积分式存在的奇异性，本文选择先将积分进行坐标变换，即由直角坐标积分转化为为极坐标积分，再采用半解析半数值法处理。
[Abstract]:Most engineering problems can be attributed to the boundary value problems of partial differential equations. However, due to the complex boundary conditions, uneven properties of objects and irregular geometry, the analytical solution of the problem can not be obtained. Therefore, people began to pay attention to the approximate solution-numerical analysis method. Boundary element method (BEM) is one of the numerical analysis methods. Although it appears later than the finite element method, it has been developed rapidly in the past 50 years with its unique advantages. Nowadays, boundary element references are more and more widely used, such as potential problem, elastic problem, elastoplastic problem, dynamic problem and so on. In this paper, the direct boundary element method is used to study four seepage problems. According to the different properties of percolation medium, that is, the permeability coefficient is different, The four seepage problems are: homogeneous isotropic seepage problem (k x ky,), orthotropic seepage problem (k x ky,), and unsaturated soil seepage problem (k k h se), layered soil seepage problem). In addition to the solution of four kinds of seepage problems, matlab programs are written for each kind of problems and verified by numerical examples. The main contents are as follows: (1) the derivation of boundary integral equations for various seepage problems. On the basis of governing equations of various seepage problems, partial differential equations are transformed into boundary integral equations by using weighted residual method and Green's formula. The problem of unsaturated soil seepage is quite special, because its governing equation is nonlinear, so there are integral terms in the integral equation. (2) the study of boundary discretization method. In the first two seepage problems, the constant element and the linear element are used to discretize the boundary, and the accuracy of the final calculation results of the two discrete methods are compared to judge the merits and demerits. (3) the study of the normal flow discontinuity problem at the corner point. When the boundary is discretized by linear element and the two ends of the element are used as the nodes, the corner problem will arise. In this paper, the double node method and the double node single unknown method are used to solve the problem. The advantages and disadvantages of the two methods and their application are obtained for different seepage problems. (4) the treatment of singular integrals. For the singularity in the boundary integral is generally the first order singularity, the analytic algorithm is used in this paper. For the singularity of the integral expression in the domain, this paper chooses to transform the integral into polar coordinate by means of coordinate transformation, and then use the semi-analytic semi-numerical method to deal with it.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TU43;TV139.1

【相似文献】