THE SUBCYCLED NEWMARK ALGORITHM

The popular Newmark algorithm, used for implicit direct integration of structural dynamics, is extended by means of a nodal partition to per mit use of different timesteps in different regions of a structural mo del. The algorithm developed has as a special case an explicit-explici t subcycling algorithm previously reported by Belytschko, Yen and Mull en. That algorithm has been shown, in the absence of damping or other energy dissipation, to exhibit instability over narrow timestep ranges that become narrower as the number of degrees of freedom increases, m aking them unlikely to be encountered in practice. The present algorit hm avoids such instabilities in the case of a one to two timestep rati o (two subcycles), achieving unconditional stability in an exponential sense for a linear problem. However, with three or more subcycles, th e trapezoidal rule exhibits stability that becomes conditional, fallin g towards that of the central difference method as the number of subcy cles increases. Instabilities over narrow timestep ranges, that become narrower as the model size increases, also appear with three or more subcycles. However by moving the partition between timesteps one row o f elements into the region suitable for integration with the larger ti mestep these the unstable timestep ranges become extremely narrow, eve n in simple systems with a few degrees of freedom. As well, accuracy i s improved. Use of a version of the Newmark algorithm that dissipates high frequencies minimises or eliminates these narrow bands of instabi lity. Viscous damping is also shown to remove these instabilities, at the expense of having more effect on the low frequency response.