EARTHQUAKE ENGINEERING A N D STRUCTURAL DYNAMICS, VOL. 4, 245-249 (1976)

A NOTE ON MASS LUMPING AND RELATED PROCESSES IN THE FINITE ELEMENT METHOD E. HINTON, T. ROCK AND 0. C. ZIENKIEWICZ Department of Civil Engineering, The University of Wales, Swansea, Wales

SUMMARY

The general problem of mass lumping and related processes in the finite element method are discussed. A mass lumping scheme is presented for parabolic isoparametric elements. Examples are presented to show the good accuracy which can be obtained in linear and non-linear dynamic problems using the scheme. In many problems of structural dynamics, transient heat conduction, etc., the finite element approximation leads to equations of the type (dots referring to time differentiation)

P(a)+Ca+Mi= F after the introduction of basic functions Ni describing the continuous variable as

(1)

(2) In this note we restrict our attention to problems of structural dynamics although much of what is discussed has wider application for problems defined by equation (1). For structural dynamics problems, nodal displacements a, are represented by the displacement vector a. In the equilibrium relationship expressed in equation (l), the terms on the left side represent the inertia force, damping force and elastic (or non-linear) force vectors, respectively, and the right side is the dynamically applied load vector. In equation (l), the term P(a) generally involves h t h order derivatives of Ni while the damping matrix C and the mass matrix M contain derivatives of order 5 (5 frequently being zero, i.e. involving no differentiation). A common structure of the matrix M (or C) is of the form

9= CN,a,

Mij=

R being the integration domain.

sn

N r pNj dR

(3)

In the solution of such equations it would be extremely useful from the computational viewpoint if the matrices M and C could be made diagonal. In the early days the engineer has (apologetically) lumped his masses or forces by physical reasoning alone and appeared glad when a proper discretization procedure introduced him to consistent mass/force approximations of the form given by equation (3).1-3In recent years there has been a partial return to mass lumping as investigators found that the use of consistent masses did not always lead to improved accuracy and always involved additional computational work. Clough? Washizu5 and otherss have demonstrated this point with the use of simple elements for which lumping procedures are physically obvious. Clough4 postulated that such lumping is simply the outcome of using V substitute basis functions i to approximate the terms of equation (3) by

in which N are piecewise constant, non-overlapping functions covering the whole element. Is this approximation always tenable with respect to convergence? What is a good form of such an approximation in the Received 12 December 1974 Revised I0 March 1975 @ 1976 by

John Wiley& Sons, Ltd.

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