康托洛维奇-里茨杂交法及其应用

Kantorovich-Ritz Hybrid Method and Its Applications

本文将Kantorovich法与Ritz法进行适当组合,吸收了二者的主要优点,提出了康托洛维奇法的一种改进方法。以二维问题为例,Kantorovich法在一个方向(例如y方向)的分布完全预先选定,这含有很大的主观任意性,因而限制了近似解的精度,改进法则在y方向仿Ritz法改进为一个含有若干个自由参数的分布函数,由于增加了近似解的自由度,故可改善解的精度。对Kantorovich法的另一改进是在计算高阶近似解时,通过逐项求解待定函数避免了求解更高阶微分方程或含更多方程的方程组,减少了计算量,降低了计算难度。用改进法求解了固体力学里的矩形截面柱体扭转问题和四边固支矩形板的弯曲问题,通过算例充分说明了此方法的特点和优越性。

A new hybrid method was proposed by combining classical Kantorovich and Ritz methods. Taking the two dimensional problem for example, a prescribed series expansion is chosen to approximate the solution along one axis （such as y） in classical Kantorovich method, leaving the other one determined by the specific feature of the problem to solve. So that the accuracy is usually lower in the prescribed direction than that in the post determined direction. To cure this drawback, the free parameters into the prescribed part of approximate function like Ritz method were added. By taking advantage of free parameters, more flexible approximate solutions were employed, which is a significant improvement to classical Kantorovich method. As a result, the accuracy is greatly improved. In traditional Kantorovich approach, a high order differential equation or a more complex differential equation set has to be solved when the approximate function is high-order. In contrast, this is circumvented in this new method through a sequential calculation process. Three numerical examples were carried out, and the results justify the advantage of the new method over traditional Kantorovich method.