仿样有限条U变换逼近法的精确解及其收敛性

Convergence and Exact Solutions of the Spline Finite Strip Method Using a Unitary Transformation Approach

仿样有限条法(spline finite strip method)是分析等截面结构最流行的数值方法之一.在以往的研究中,与一些基准问题的解析结果相比较,论证了该方法数值结果的有效性和收敛性,但至今未对该方法的精确解和显式误差项进行过数学推导,解析地论证过其收敛性.该文在对平板的分析中,使用酉变换(简称U变换)逼近法,导出了仿样有限条法精确的数学解,这是首次在公开文献中给出的精确解.和常规的仿样有限条法相比较,总矩阵方程的集成及其数值解都不同,U变换法的总矩阵方程,减少为仅含有2个未知量的方程,然后导出仿样有限条法显式的精确解.精确解按Taylor级数展开,导出误差项和收敛率,并和其他数值方法直接比较.在这一点上可以发现,仿样有限条法收敛速度和非协调有限元相同时,包含的未知量少得多,收敛率比常规的有限差分法快得多.

The spline finite strip method was one of the most popular numerical methods for analyzing prismatic structures. Efficacy and convergence of the method had been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems. To date, no mathematical exact solutions of the method or its explicit forms of error terms had been derived to demonstrate analytically its convergence. As such, mathematical exact solutions of spline finite strips in plate analysis were derived using a unitary transformation approach （abbreviated as the U-transformation method herein）. These exact solutions were presented for the fwst time in open literature. Unlike the conventional spline finite strip method which involves assembly of the global system of matrix equation and its numerical solution, the U-transformation method decoupled the global matrix equation into one involving only two unknowns, thus rendering exact solutions of the spline finite strip to be derived explicitly. By taking Taylor＇ s series expansion of the exact solution, error terms and convergence rates were also derived explicitly and compared directly with other numerical methods. In this regard, it was found that the spline finite strip method converged at the same rate as a non-cow forming fmite element, yet involving smaller number of unknowns compared to the latter. The convergence rate was also found superior to the conventional finite difference method.