Some Unreliable Finite Element Solutions for Nonlinear Dynamics

This paper on "Some Unreliable Finite Element Solutions for Nonlinear Dynamics" was presented at the 5th International Conference on Reliability of Finite Element Methods for Engineering Applications - 10-12 May 1995, Amsterdam, The Netherlands.

Summary

The average acceleration method or trapezoidal rule is a member of the Newmark family (with β=1/4 and γ=1/2) which is frequently used for both linear and nonlinear dynamics. The technique is 'unconditionally' stable (irrespective of the time step) for linear problems but it will be demonstrated in the present paper that this stability does not extend to nonlinear systems so that often, absurdly small time steps have to be used if the solutions are not to 'blow up' or 'lock'. The latter involves a sudden transfer from kinetic to high-frequency strain energy and may be of such a form that is not obvious that unreliable solutions have been obtained.
As well as highlighting these difficulties, the paper will describe a possible solution by way of a new form of energy conserving time-integration procedure which is linked to the element technology.