有限元法中大单元的构造

梁国平; 傅子智

有限元法中大单元的构造

梁国平¹,
傅子智²

1.
中国科学院数学研究所;
2.
北京石油设计院

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出版历程
- 收稿日期: 1982-02-27
- 刊出日期: 1983-06-15

The Construction of Large Elements in Finite Element Method

Liang Guo-ping¹,
Fu Zi-zhi²

1.
Institute of Mathematics, Academic, Sinica, Beijing;
2.
Beijing Petroleum Design Institute, Beijing

摘要

摘要: 在通常的有限元法中,单元内的插值多项式的阶数固定不变,通过加密剖分网格来提高精度.大单元法则剖分的网格固定不变而通过增加单元内逼近级数的项数来提高精度. 本文提出采用两套变量的办法来构造大单元,即单元内采用一套变量,单元的边界上采用另一套变量,然后用杂交罚函数法把两者联系起来.这种方法能适用于任何椭圆型方程,任意几何形状区域以及任何复杂的边界条件.本文用严密的数学方法证明了:在一般情况下,这种方法的精度比通常的有限元法和文[7]的大单元法高得多.即在达到相同的精度时,本文方法所需要的自由度(即未知数数目)比上述两种方法少得多.

Abstract: In the usual finite element method, the order of the interpolation in an element is kept unchanged, and the accuracy is raised by subdividing the grid denser and denser. Alternatively, in the large element method, the grid is kept unchanged, and the terms of approximate series in the element are increased to raise the accuracy.In this paper, a method for constructing large elements is presented. When using this method,two sets of variables, one set defined inside the element, and the other defined on the boundary of the element, are adopted. Then, these two sets of variables are combined by the hybrid-penalty function method.This method can be applied to any elliptic equations in a domain with arbitrary shape and arbitrary complex boundary condition.It is proved with strict mathematical method in this paper, that in general cases,the accuracy of this method is much higher than that of the usual element and the large element method presented in [7].Therefore,the degrees of freedom needed in this method are much fewer than those in the two methods if the same accuracy is preserved.

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参考文献(10)

[1]	梁国平,分层解法——有限元的一个新算法,应用数学学报,2,1(1979).
[2]	梁国平,半解析法及其误差估计,计算数学,2, 3(1980)
[3]	Zienkiewicz,D.W.Kelly and P.Bettess,The coupling of the finite element method and boundary solution procedure,Int.J.Num.Method.Engng.,11,2,(1977).
[4]	梁国平、顾永耕,平面弹性断裂问题的半解析法,计算数学,1,4(1978).
[5]	Tong,P.and.T.H.H.Pian,A hybrid element approach to crack problems in plane elasticity,Int.J.Numer.Method.Eng.7,(1973),297-308.
[6]	李子才、粱国平,椭圆型方程边值问题的结合法,中国科学,(1981)
[7]	Babuska,I.,B.A.Szabo and L.N.Katz,The-version of the finite element method,SLAM J.Numer.Anal.,18:3,(1981).
[8]	Oden,J.T.and J.N.Reddy,Variational methods in theoretical mechanics,Springer-Verlag,Heidelberg,(1976).
[9]	Oden,J.T.and J.N.Reddy,Mathematical theory of finite elements,John Wiley & Sons,New York,(1976).
[10]	Treves,F.,Basic linear partial differential equations,Academic Press,New York,San Francisco,(1975).