PERTURBATION FINITE ELEMENT METHOD FOR SOLVING GEOMETRICALLY NONLINEAR PROBLEMS OF AXISYMMETRICAL SHELL

In analysing the geometrically nonlinear problem of an axisymmetrical thin-walled shell, the paper combines the perturbation method with the finite element method by introducing the former into the variational equation to obtain a series of linear equations of different orders and then solving the equations with the latter. It is well-known that the finite element method can be used to deal with difficult problems as in the case of structures with complicated shapes or boundary conditions, and the perturbation method can change the nonlinear problems into linear ones. Evidently the combination of the two methods will give an efficient solution to many difficult nonlinear problems and clear away some obstacles resulted from using any of the two methods solely. The paper derives all the formulas concerning an axisym-metric shell of large deformation by means of the perturbation finite element method and gives two numerical examples,the results of which show good convergence characteristics.     

在材料非线性问题中的摄动有限元法

摄动法是解决非线性连续介质力学问题的一种有效方法.这种方法是建立在该问题的线性解析解的基础上的,因此,若得不到一个简单的解析解,应用这种方法去解决一些复杂的非线性问题将遇到困难.有限元法对解非线性问题也是一种十分有用的工具,然而一般来说,它需要相当长的计算时间. 本文介绍摄动有限元法.这种方法吸取上述两种方法的优点,能够解决更复杂的非线性问题,而且也能大量节省计算机的计算时间. 本文讨论了比例加载下的弹塑性力学问题,并提出一个带孔拉板的数值解.     

The perturbation finite element method for solving problems with nonlinear materials

The perturbation method is one of the effective methods for so-lving problems in nonlinear continuum mechanics.It has been de-veloped on the basis of the linear analytical solutions for the o-riginal problems.If a simple analytical solution cannot be ob-tained.we would encounter difficulties in applying this method tosolving certain complicated nonlinear problems.The finite ele-ment method appears to be in its turn a very useful means for sol-ving nonlinear problems,but generally it takes too much time incomputation.In the present paper a mixed approach,namely,theperturbation finite element method,is introduced,which incorpo-rates the advantages of the two above-mentioned methods and enablesus to solve more complicated nonlinear problems with great savingin computing time.Problems in the elastoplastic region have been discussed anda numerical solution for a plate with a central hole under tensionis given in this paper.     

摄动有限元法解一般轴对称壳几何非线性问题

本文在处理几何非线性问题时,利用在变分方程中引入振动过程,得到各级变分摄动方程,并通过有限元法求解.由于有限元法能成功地处理各种复杂边界条件、几何形状的力学问题,摄动法又可将非线性问题转化为线性问题求解.若结合这两种方法的优点,将能够解决大量复杂的非线性力学问题.并能够消除单独使用有限元法或摄动法求解复杂非线性问题所出现的困难. 本文应用摄动有限元法求解了一般轴对称壳的几何非线性问题.