Computational Geomechanics. Manuel Pastor

is the coupling matrix‐linking equation (3.23) and those describing the fluid conservation, and

      (3.22 \ bis)

The computation of the effective stress proceeds incrementally as already indicated in the usual way and now (3.15) can be written in discrete form:

(3.27)

      where, of course, D is evaluated from appropriate state and history parameters.

Finally, we discretize Equation (3.17) by pre‐multiplying by (N ^p)^T and integrating by parts as necessary. This gives the ordinary differential equation

(3.28)

      where the various matrices are as defined below

(3.29)

(3.30)

(3.31)

      (3.32)

      where Q^* is defined as in (2.30c), i.e.

(3.33)

      and C_S, S_w, C_w and k depend on p_w.

      3.2.3 Discretization in Time

To complete the numerical solution, it is necessary to integrate the ordinary differential Equations (3.23), (3.27), and (3.28) in time by one of the many available schemes. Although there are many multistep methods available (see, e.g., Wood 1990), they are inconvenient as most of them are not self‐starting and it is more difficult to incorporate restart facilities which are required frequently in practical analyses. On the other hand, the single‐step methods handle each step separately and there is no particular change in the algorithm for such restart requirements.

      Two similar, but distinct, families of single‐step methods evolved separately. One is based on the finite element and weighted residual concept in the time domain and the other based on a generalization of the Newmark or finite difference approach. The former is known as the SSpj – Single Step p^th order scheme for j^th order differential equation (p ≥ j). This was introduced by Zienkiewicz et al. (1980b, 1984) and extensively investigated by Wood (1984a, 1984b, 1985a, 1985b). The SSpj scheme has been used successfully in SWANDYNE‐I (Chan, 1988). The later method, which was adopted in SWANDYNE‐II (Chan 1995) was an extension to the original work of Newmark (1959) and is called Beta‐m method by Katona (1985) and renamed the Generalized Newmark (GNpj) method by Katona and Zienkiewicz (1985). Both methods have similar or identical stability characteristics. For the SSpj, no initial condition, e.g. acceleration in dynamical problems, or higher time derivatives are required. On the other hand, however, all quantities in the GNpj method are defined at a discrete time station, thus making transfer of such quantities between the two equations easier to handle. Here we shall use the later (GNpj) method, exclusively, due to its simplicity.

      In all time‐stepping schemes, we shall write a recurrence relation linking a known value ϕ_n (which can either be the displacement or the pore water pressure), and its derivatives , ,… at time station t_n with the values of Φ_n+1, , ,…, which are valid at time t_n + Δt and are the unknowns. Before treating the ordinary differential equation system (3.23),