Computational aspects of dispersive computational continua for elastic heterogeneous media

The present manuscript focusses on computational aspects of dispersive computational continua ($$C^2$$C2) formulation previously introduced by the authors. The dispersive $$C^2$$C2 formulation is a multiscale approach that showed strikingly accurate dispersion curves. However, the seemingly theoretical advantage may be inconsequential due to tremendous computational cost involved. Unlike classical dispersive methods pioneered more than a half a century ago where the unit cell is quasi-static and provides effective mechanical and dispersive properties to the coarse-scale problem, the dispersive $$C^2$$C2 gives rise to transient problems at all scales and for all microphases involved. An efficient block time-integration scheme is proposed that takes advantage of the fact that the transient unit cell problems are not coupled to each other, but rather to a single coarse-scale finite element they are positioned in. We show that the computational cost of the method is comparable to the classical dispersive methods for short load durations.