差分格式收敛性研究的一种新方法

提出了一种对差分格式收敛性进行研究的新方法．应用U变换法和有限差分法，分析了均质简支梁的静力问题，求出了在均布荷载作用下梁的挠度和弯矩的精确解析表达式，并研究其收敛性，得到了收敛率系数的精确值．

Convergence study of the finite difference method: a new approach

In structural analysis, the U-transformation method is one of the most efficient methods in finding the exact solutions of periodic structures. Performing U-transformation on a periodic structure can uncouple the governing equations. Therefore, exact analytical solutions of the periodic structures can easily be obtained and the amount of computation is then greatly reduced. In the past 20 years the U-transformation method has successfully been applied in the static and dynamic analyses of periodic structures such as beams, plates, trusses, grillwork and so on. And, it has a great potential for further development. The present paper extends the U-transformation method to the convergence study of the finite difference method. Without loss of generality, the exact static analysis of a simply supported beam is conducted as an example. When the beam is divided into the mesh of equal finite elements, the mesh may be treated as a periodic strucutre. Thus the U-transformation method is used to obtain exact analytical solutions of the deflections and bending moments of the beam subjected to uniformly distributed loads. When the number of finite elements approaches to infinity, the exact error expression and the exact convergence rate of the solutions are obtained. They cannot be obtained if other methods are used instead.