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The spatial discretization involving the variables u and p_w is achieved by suitable shape (or basis) functions, writing

(3.19)

Note that the nodal values of the pore pressures are indicated with a superscript.

We assume here that the expansion is such that the strong boundary conditions (3.18) are satisfied on Γ_u and Γ_p automatically by a suitable prescription of the (nodal) parameters. As in most other finite element formulations, the natural boundary condition will be obtained by integrating the weighted equation by parts.

To obtain the first equation discretized in space, we premultiply (3.8) by (N ^u)^T and integrate the first term by parts (see for details Zienkiewicz et al (2013) or other texts) giving:

(3.20)

where the matrix B is given as

(3.21)

and the “load vector” f ⁽¹⁾, equal in number of components to that of vector contains all the effects of body forces, and prescribed boundary tractions, i.e.

(3.22)

At this stage, it is convenient to introduce the effective stress see (3.12) now defined to allow for effects of incomplete saturation as

(3.23)

The discrete, ordinary differential equation now becomes

(3.24)

where

(3.25)

is the MASS MATRIX of the system and

(3.26)

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.