On Stability of Multitime Step Integration Procedures

Stability of multitime step integration methods for finite-element computations in structural dynamics is analyzed. Multitime step procedures based on the Newmark family of methods are described. The basic idea of multitime step methods is to utilize various time step sizes in different domains of an element mesh. Interpolated nodal values from the large time step domain are used in the computation of displacements, velocities, and accelerations in the small time step domain. An analytical study of the errors introduced by the interpolation is given. The analysis shows instability because of the resonance phenomena and because of the propagation of spurious high frequencies caused by unbalanced forces originating from interpolation errors. Numerical results exhibiting instability problems are presented. To complete the study, examples of solutions stabilized with numerical damping of high frequencies are included in the presentation. The conclusion from the present study is that multitime step methods are unstable in their nature. Numerical damping can stabilize the computations but alters the response of free vibrating undamped systems.