弹性力学的一种边界无单元法

A BOUNDARY ELEMENT-FREE METHOD FOR LINEAR ELASTICITY

首先对移动最小二乘逼近法进行了研究,针对其容易形成病态方程组的缺点,提出了以带权的正交函数作为基函数的方法——改进的移动最小二乘逼近法.改进的移动最小二乘逼近法比原方法计算量小、精度高,且不会形成病态方程组.然后,将弹性力学的边界积分方程方法与改进的移动最小二乘逼近法结合,提出了弹性力学的一种边界无单元法.这种边界无单元法是边界积分方程的无网格方法,与原有的边界积分方程的无网格方法相比,该方法直接采用节点变量的真实解为基本未知量,是边界积分方程无网格方法的直接解法,更容易引入边界条件,且具有更高的精度.最后给出了弹性力学的边界无单元法的数值算例,并与原有的边界积分方程的无网格方法进行了较为详细的比较和讨论.

In this paper, moving least-square approximation method is discussed first. Sometimes the method forms an ill-conditioned system of equations so that the solution can not be obtained correctly. And so the modified moving least-square approximation method is presented. In the modified method the orthogonal function system mixed a weight function is used as the basis functions. The modified method has higher computational efficiency and precision than the old method, and does not form an ill-conditioned system of equations. Then combining the boundary integral equation method and the modified moving least-square approximation method, a meshless method of boundary integral equation for linear elasticity, which is called boundary element-free method (BEFM), is presented. The corresponding formulae of BEFM are obtained. A numerical example of BEFM is given. And the differences between BEFM and the old meshless methods of boundary integral equation, such as boundary node method (BNM) and local boundary integral equation method (LBIE), are discussed in detail. Comparing with BNM and LBIE, BEFM need not increase new boundary, and need not any evaluation point. In BEFM, the basic unknown quantity is the real solution of the nodal variables; but in BNM and LBIE, the basic unknown quantity is the approximations of the values of the nodal variables. In BEFM, the numerical solutions of the real nodal variables can be obtained; but in BNM and LBIE, the numerical solutions of the approximations of the real nodal variables can be obtained only. In BEFM, the boundary conditions can be applied directly and easily; but in BNM and LBIE, the boundary conditions are applied after they are transformed into their approximations on boundary nodes with moving least-squares approximation. And from the formulae of moving least-squares approximation, the numerical solutions of the variables on any boundary point, which is not a boundary node, must be obtained from the real solutions on boundary nodes, but from the approximations of the real solutions. BEFM is a direct numerical method of the meshless methods of boundary integral equation, but BNM and LBIE are indirect ones. So BEFM has higher computational precision. BEFM can be applied easily to other problems which can be solved with boundary element method.