Computational Geomechanics. Manuel Pastor

has been included in the force term of the computer code SWANDYNE‐II (Chan 1995) although it is neglected in the left‐hand side of the final algebraic equation when the symmetric solution procedure is used.

      The above set defines the complete equation system for solution of the problem defined providing the necessary boundary conditions have been specified as in (2.18) and (2.19), i.e.

(3.18)

and

      Assuming isotropic permeability, the above equation becomes

      where is the influx, i.e. having an opposite sign to the outflow .

      The total boundary Γ is the union of its components, i.e.

      

      3.2.2 Discretization of the Governing Equation in Space

      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)

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