改进的移动最小二乘法

近年来发展的无网格方法大多采用移动员小二乘法来构造试函数，而应用移动最小二乘法形成的方程组有时会是病态的甚至奇异的，从而限制了它的发展和应用。本文采用带权正交函数作为基函数对移动最小二乘法做了改进，避免出现病态方程组，且在计算过程中不需要进行短阵求逆运算，提高了计算速度。之后，借鉴牛顿法、平衡法和摄动法对由移动最小二乘法得到的非线性代数方程组提出了新的求解方法。

The Improved Moving Least-Square Approximation

Meshless methods for solving boundary value problems have been extensively popularized in literature in recent years. Moving least-square approximation (MLS) is one of the methods to form trial functions of meshless methods. But sometimes the algebra equations system obtained from moving least-square approximation is ill-conditioned, even with the singular phenomenon. In this paper, the weighted orthogonal functions are used as basis ones, and the improved moving least-square approximation the algebra e-quations system is presented. In the improved moving least-square approximation, the algebra equations system is not ill-conditioned, and the system can be solved without obtaining the inverse matrix, and then the computing speed is higher. And then an new iterative method is presented to solve the nonlinear equations system obtained from moving least-square approximation.