Convolution quadrature boundary element method for quasi-static visco- and poroelastic continua

The main difference of convolution quadrature method (CQM)-based boundary element formulations to usual time-stepping BE formulations is the way to solve the convolution integral appearing in most time-dependent integral equations. In the CQM formulation, the convolution integrals are approximated by a quadrature rule whose weights are determined by the Laplace transformed fundamental solutions and a multi-step method. So, there is no need of a time domain fundamental solution. For quasi-static problems in visco- or poroelasticity time-dependent fundamental solutions are available, but these fundamental solutions are highly complicated yielding to very sensitive algorithms. Especially in viscoelasticity, for every rheological model a separate fundamental solution must be deduced. Here, firstly, viscoelastic as well as poroelastic constitutive equations are recalled and, then, the respective integral equations are presented. Applying the usual spatial discretization and using the CQM for the temporal discretization yields the final time-stepping algorithm. The proposed methodology is tested by two simple examples considering creep behavior in viscoelasticity and consolidation processes in poroelasticity. The algorithm shows no stability problems and behaves well over a broad range of time step sizes.