非线性动力方程的精细时空有限元方法  

THE PRECISE SPACE-TIME FINITE ELEMENT METHOD FOR NONLINEAR DYNAMIC EQUATION

作　　者：江小燕[1] 王建国[1] 

机构地区：[1]合肥工业大学土木与水利工程学院,合肥230009

出　　处：《工程力学》2014年第1期23-28,共6页Engineering Mechanics

摘　　要：根据Hamilton变作用定律构造了时空有限元矩阵;并根据传递矩阵原理,利用时间的一维性将时空有限元矩阵变换为时间方向的传递矩阵,将初值问题转化为一般矩阵相乘问题以方便求解。为了保证计算的稳定性,参考了精细积分的思想提出精细时空有限元方法,并给出线性问题在时间级数荷载作用下的计算式。数值分析结果证明该方法在线性问题分析上非常准确并可以推广到非线性动力方程的求解;只需将非线性解看作初始解和增量解的叠加,通过精细时空有限元线性求解方法计算增量解,逐步修正后即可得到非线性解。结果表明该方法是一个有效的求解非线性动力方程的方法。Based on the Hamilton variable action law, a space-time finite element is deduced. According to the principle of a transfer matrix and one-dimensionality of time, a space-time finite element matrix is transformed to a time-transfer matrix and initial value problems are converted to matrix multiplication problems so as to be solved more easily. In order to ensure the stability of the calculation, the precise space-time finite element method is established on the basis of the precise integration idea, and the calculation formula under the action of series loads is presented. The results of numerical analysis on a linear problem show that this method is very accurate and can be extended to nonlinear dynamic problems. Firstly, regarded a nonlinear solution as a superposition of an initial solution and an incremental solution, and then to calculate the incremental solution using a precise space-time finite element linear solution method, to modify solution gradually, the nonlinear solution can be achieved finally. The results of numerical experiments show that this method is a stable and efficient calculation method on solving nonlinear dynamic problems.

关 键 词：非线性动力方程 Hamilton变作用定律 时空有限元 精细方法 传递矩阵 

分 类 号：O241[理学—计算数学]