Computational Geomechanics. Manuel Pastor

where and are as yet undetermined quantities. The parameters β₁, β₂, and are usually chosen in the range of 0 to 1. For β₂ = 0 and = 0, we shall have an explicit form if both the mass and damping matrices are diagonal. If the damping matrix is non‐diagonal, an explicit scheme can still be achieved with β₁ = 0, thus eliminating the contribution of the damping matrix. The well‐known central difference scheme is recovered from (3.41) if β₁ = 1/2, β₂ = 0 and this form with an explicit and implicit scheme has been considered in detail by Zienkiewicz et al. (1982) and Leung (1984). However, such schemes are only conditionally stable and for unconditional stability of the recurrence scheme, we require

      The optimal choice of these values is a matter of computational convenience, the discussion of which can be found in literature. In practice, if the higher order accurate “trapezoidal” scheme is chosen with β₂ = β₁ = 1/2 and = 1/2, numerical oscillation may occur if no physical damping is present. Usually, some algorithmic (numerical) damping is introduced by using such values as

      or

      Dewoolkar (1996), using the computer program SWANDYNE II in the modelling of a free‐standing retaining wall, reported that the first set of parameters led to excessive algorithmic damping as compared to the physical centrifuge results. Therefore, the second set was used and gave very good comparisons. However, in cases involving soil, the physical damping (viscous or hysteretic) is much more significant than the algorithmic damping introduced by the time‐stepping parameters and the use of either sets of parameters leads to similar results.

Inserting the relationships (3.43) into Equations (3.41) and (3.42) yields a general nonlinear equation set in which only and remain as unknowns.

      This set can be written as

      (3.44a)

      (3.44b)

      where and can be evaluated explicitly from the information available at time t_n and

      (3.45)

      In this, must be evaluated by integrating (3.27) as the solution proceeds. The values of _n+1 and _n+1 at the time t_n+1 are evaluated by Equation (3.43).

      The equation will generally need to be solved by a convergent, iterative process using some form of Newton–Raphson procedure typically written as

      (3.46a)