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Heterogeneous Multiscale Methods for Mechanical Systems with Vibrations

We are concerned with heterogeneous multiscale methods for the numerical integration of the equations of motion of mechanical systems subjected to fast vibrations. We suggest easily formulated asynchronous algorithms that bypass the need for explicitly determining the relations between macro- and microstates. The new algorithms have good geometric properties and in some cases may incorporate a simplified filtering technique that leads to large savings in computational effort. The problems considered may be modeled either by ordinary differential equations (state space description) or by differential algebraic equations (descriptor form). A SHAKE-SHAKE and a RATTLE-SHAKE algorithm are presented, but higher-order versions exist.