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E. HINTON, T. ROCK AND 0. C. ZIENKIEWICZ

case of multinoded e

lements, as obviously infinite possibilities of deriving such functions exist, Figure 1? These are questions of considerable importance in many practical problems where it is well established that higher order elements (such as isoparametric quadrilaterals or bricks of parabolic type) are desirable to represent economicallyterms of the P(a) type occurring in equation (1). In this context, it should be mentioned that Key and Beisinger' have presented a method for deriving a diagonal mass matrix from the standard consistent mass matrix for elements with linear or cubic displacement functions.

i

1

2 2

A3

3 3

' 7 Figure 1. Possible functions si for linear and parabolic elements

If equations (I) are derived from the stationarity of a scalar functional, in particular if the term Md derives from the stationarity of a scalar term iT Mfi (5) it is obviously sufficient to show that for convergence the above term will be convergent with the new approximate basis functions. This simply requires that

and allows f l i to be a simple discontinuous function. Thus for convergence the original criteria postulated by Clough4 are adequate. What of the order of convergence? Here, Tong et aLBand Odene study this. The investigations of the former concentrate on the problem of linear vibration and conclude that the order of convergence will be reduced from that of the original basis function N if

p+ 1 -f i< 2(6+ 1- lit) where

(7)

B is order of polynomial in N lit is order of derivatives in P(a)

p is order of polynomial in R in lumped approximation 2 is order of derivative in M

Thus if we seek to lump the mass using discontinuous functions i with f i= 0 and p= 0 in a problem for v which f i= 1 and in which parabolic elements are used, i.e.@= 2, we conclude that by equation (7) the order of convergence will be reduced but nevertheless convergence will occur. In a more general problem in which non-linearity occurs and in which (like in convective diffusion equations) no variational principle exists even these assertions have not yet been proved. It is the object of this note to outline a particular lumping process and to show that good accuracy can be obtained in a linear and non-linear dynamic problem using isoparametric parabolic elements.