Increasing the critical time step: micro-inertia, inertia penalties and mass scaling

Explicit time integration is a popular method to simulate the dynamical behaviour of a system. Unfortunately, explicit time integration is only conditionally stable: the time step must be chosen not larger than the so-called "critical time step", otherwise the numerical solution may become unstable. To reduce the CPU time needed to carry out simulations, it is desirable to explore methods that increase the critical time step, which is the main objective of our paper. To do this, first we discuss and compare three approaches to increase the critical time step: micro-inertia formulations from continuum mechanics, inertia penalties which are used in computational mechanics, and mass scaling techniques that are mainly used in structural dynamics. As it turns out, the similarities between these methods are significant, and in fact they are identical in 1D if linear finite elements are used. This facilitates interpretation of the additional parameters in the various methods. Next, we derive, for a few simple finite element types, closed-form expressions for the critical time step with micro-structural magnification factors. Finally, we discuss computational overheads and some implementational details.