A time-discontinuous Galerkin method for the dynamical analysis of porous media

Abstract

SUMMARYWe present a time-discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space-time finite element scheme.

A new hybrid velocity integration method applied to elastic wave propagation

“…Moreover, bothũ n andv n are known quantities, which are either solutions from the previous time slab Q n−1 or the prescribed initial values. So far we have obtained the weak form (25) formulated in the velocity field v. The displacement field u is embedded implicitly as a function of the velocity v, cf. (23); more details concerning the FEM formulation are given in the Appendix.…”

“…According to the time discretization, the so-called time slab Q n is defined analogously by Q n := × I n . For further details concerning the notation and the space-time discretization, we refer to [25]. …”

“…The jump is introduced as 'v( := |v + (t n ) − v − (t n )|, see Figure 1(a). The solution strategy with respect to the first-order equation has been discussed extensively in [25], which will not be repeated here. Hence, we proceed to the treatment of the displacement field u in …”

“…Next, we apply the upwind flux treatment to thev. Applying the upwind flux treatment in time was introduced in [26] and has been successfully applied to the dynamical analysis of porous materials in [25]. The upwind flux ofv reads…”

Generalized EVI‐space‐time Galerkin method for dynamical modeling in porous media

“…In the previous works, a so-called Embedded Velocity Integration (EVI) technique [3] was proposed to solve the governing second-order time-dependent problems without introducing the extra auxiliary equation via the order-reduction technique. The resulting single first-order equation with the unkonwn of the rate term can be solve efficiently by the time-discontinuous Galerkin method [1]. In the current work, a stabilization factor α is introduced into the EVI formulation.…”