Computational Geomechanics. Manuel Pastor

Computational Geomechanics - Manuel Pastor


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      (3.22 \ bis)bold f Superscript left-parenthesis 1 right-parenthesis Baseline equals integral Underscript normal upper Omega Endscripts left-parenthesis bold upper N Superscript u Baseline right-parenthesis Superscript normal upper T Baseline rho bold b d upper Omega plus integral Underscript normal upper Gamma Subscript t Baseline Endscripts left-parenthesis bold upper N Superscript u Baseline right-parenthesis Superscript normal upper T Baseline bold t overbar d upper Gamma

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

      where the various matrices are as defined below

      (3.32)bold f Superscript left-parenthesis 2 right-parenthesis Baseline equals minus integral Underscript normal upper Omega Endscripts left-parenthesis bold upper N Superscript p Baseline right-parenthesis Superscript normal upper T Baseline bold nabla Superscript normal upper T Baseline left-parenthesis bold k upper S Subscript normal w Baseline rho Subscript normal f Baseline bold b right-parenthesis d upper Omega plus integral Underscript normal upper Gamma Subscript normal w Baseline Endscripts left-parenthesis bold upper N Superscript p Baseline right-parenthesis Superscript normal upper T Baseline bold q overbar d upper Gamma

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

      and CS, Sw, Cw and k depend on pw.

      3.2.3 Discretization in Time

      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 pth order scheme for jth order differential equation (pj). 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 ϕn (which can either be the displacement or the pore water pressure), and its derivatives ModifyingAbove bold upper Phi With ampersand c period dotab semicolon Subscript n, ModifyingAbove bold upper Phi With two-dots Subscript n,… at time station tn with the values of Φn+1, ModifyingAbove bold upper Phi With ampersand c period dotab semicolon Subscript n plus 1, ModifyingAbove bold upper Phi With two-dots Subscript n plus 1,…, which are valid at time tn + Δt and are the unknowns. Before treating the ordinary differential equation system (3.23),


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