结构动力方程的样条精细积分法

Spline precise time-integration of structural dynamic analysis

结合精细积分法和样条函数拟合技术的优点,提出了求解结构动力方程的一种有效方法。首先对非齐次项用三次正规化B样条函数进行拟合,然后利用正规化B样条函数形状相同、仅相差一个平移量的特点,构造了一个高效的特解求解方法。按此方法只需求出一个标准B样条项所对应的特解,然后通过时间坐标的平移并结合叠加原理,即可求出任意时刻的特解值。由于特解计算中采用数值积分的方法,避免了矩阵求逆,因而本方法具有较大的适用范围。算例结果证明了该方法的有效性。

The precise time-integration method proposed for linear steady dynamic system can give numerical results approaching the exact solution at the integration points. However, it is more or less difficult when the algorithm is used to the non-homogeneous dynamic systems. The spline precise time-integration method which is based on precise integration and spline function fitting is proposed in this paper. Firstly, the non-homogeneous term is fitted via a set of cubic B-spline functions. The graphs of B- spline functions are exactly the same in shape and only the starting point is different. A highy efficient numerical method for calculating the particular solution to the dynamic system is put forward. With this method, simply solve a dynamic equation of which the non-homogeneous term is a standard B-spline function, then by using the superpose principle, the particular solution is obtained. Together with the general solution which is calculated by the time-integration precise method, the solution satisfying the initial conditions is obtained. This new method avoids the inverse matrix calculations as the particular solution is calculated by numerical quadrature, therefore is valid in general cases. Numerical examples are given to demonstrate the validity and efficiency of the algorithm.