Baseline o n normal upper Gamma equals normal upper Gamma Subscript w EndLayout"/> Assuming isotropic permeability, the above equation becomes
where
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 pw is achieved by suitable shape (or basis) functions, writing
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 Γu and Γp 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
and the “load vector” f (1), equal in number of components to that of vector
(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
The discrete, ordinary differential equation now becomes
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:
