Signorini问题的无网格边界点解法

发布时间：2018-03-18 03:03

本文选题：无网格方法　切入点：Signorini问题　出处：《重庆师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：Signorini问题是一类重要的数学物理问题,该问题的Signorini边界条件是由一个等式约束下的两个不等式以互补形式给出的,其边界条件的这种不确定性给求解带来一定的麻烦,因此Signorini问题形成了一类特殊的椭圆边值问题。由于该问题的Signorini条件是定义在求解区域的边界上,因此基于边界积分方程的无网格方法很适合求解这类问题。本文利用投影算法,将移动最小二乘逼近法和边界积分方程相结合,提出了求解Signorini问题的一种无网格边界点投影迭代算法。该方法首先利用一个简单的投影算子将Signorini边界条件转化为与之等价的投影不动点方程,然后分别构造出Signorini问题的隐式和显式投影迭代格式。隐式投影迭代格式是在不动点方程的基础上直接构造的,而显式投影迭代格式则是通过引入了一个与不动点方程等价的残量函数,在残量函数的基础上构造的。这样在每一次的迭代过程中,我们就将一个特殊的椭圆边值问题转化成为一个可求解的一般椭圆边值问题并且采用无网格边界点方法进行数值求解,并且对隐式投影迭代格式收敛且唯一收敛到Signorini问题的唯一解进行了证明。最后,对于隐式和显式投影算法,我们分别以数值算例对其求解效率给予验证,结果表明了本文方法在求解Signorini问题时的可行性和有效性,相对于边界元方法也具有更好的精度和收敛速度。
[Abstract]:The Signorini problem is an important mathematical and physical problem. The Signorini boundary condition of the problem is given in the form of complementarity by two inequalities under an equality constraint. The uncertainty of the boundary condition brings some trouble to the solution of the problem. So the Signorini problem forms a kind of special elliptic boundary value problem. Because the Signorini condition of the problem is defined on the boundary of the solution region, the meshless method based on the boundary integral equation is very suitable for solving this kind of problem. The moving least square approximation method is combined with the boundary integral equation. In this paper, a meshless boundary point projection iterative algorithm for solving Signorini problem is proposed. Firstly, a simple projection operator is used to transform the Signorini boundary condition into an equivalent projective fixed point equation. Then the implicit and explicit projection iterative schemes of Signorini problem are constructed respectively. The implicit projection iterative schemes are constructed directly on the basis of fixed point equations. The explicit projection iterative scheme is constructed on the basis of the residual function by introducing a residue function equivalent to the fixed point equation. We transform a special elliptic boundary value problem into a general elliptic boundary value problem that can be solved and numerically solve it by using the meshless boundary point method. The convergence of implicit projection iterative scheme and its unique convergence to the unique solution of Signorini problem are proved. Finally, the efficiency of implicit and explicit projection algorithms is verified by numerical examples. The results show that the proposed method is feasible and effective in solving Signorini problem and has better accuracy and convergence speed than BEM.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

【相似文献】