Computational Geomechanics. Manuel Pastor

(3.27), and (3.28), we shall illustrate the time‐stepping scheme on the simple example of (3.6) by adding a forcing term:

(3.34)

      From the initial conditions, we have the known values of Φ_n, . We assume that the above equation has to be satisfied at each discrete time, i.e. t_n and t_n+1. We can thus write:

(3.35)

      and

(3.36)

      From the first equation, the value of the acceleration at time t_n can be found and this solution is required if the initial conditions are different from zero.

The link between the successive values is provided by a truncated series expansion taken in the simplest case as GN22 as Equation (3.34) is a second‐order differential equation j and the minimum order of the scheme required is then two: as (p ≥ j)

      (3.37)

      Alternatively, a higher order scheme can be chosen such as GN32 and we shall have:

      (3.38)

In this case, an extra set of equations is required to obtain the value of the highest time derivatives. This is provided by differentiating (3.35) and (3.36).

      (3.39)

      and

      (3.40)

      In the above equations, the only unknown is the incremental value of the highest derivative and this can be readily solved for.

      Returning to the set of ordinary differential equations we are considering here, i.e. (3.23), (3.27), and (3.28) and writing (3.23) and (3.28) at the time station t_n+1, we have:

(3.41)

(3.42)

      assuming that (3.27) is satisfied.

      Using GN22 for the displacement parameters and GN11 for the pore pressure parameter ^w, we write:

      (3.43a)

      and

      (3.43b)