The basic idea of the finite element method is to divide the field to be analysed into sub-domains, the so-called finite elements, of simple shape: e. g. triangles, quadrilaterals with linear, parabolic or cubic sides for two-dimensional analysis. In each finite element, an analytically simple equation is postulated for the variable to be determined, i. e. the coordinate or displacement for solid mechanics, and the fluid pressure, temperature or concentration for diffusion problems. In order to obtain continuity, the unknown variable field has to be continuous at the limit between finite elements. This requirement is obtained thanks to common values of the field at specific points, the so-called nodes, which are linking the finite elements together. The field values at nodal points are the discretised problem unknowns.
For most solid mechanics and diffusion problems, isoparametric finite elements seem to be optimal (Zienkiewicz et al., 1988). The unknown field l (here representing a set of displacements in all directions) may then be written, for solid mechanics cases1 as:
l = Nl g, n)xL L = 1, m (11.14)
where m is the number of nodes in the model. This unknown field, l_, then depends on the nodal unknowns xL (not only referring to the x — direction) and on shape functions Nl, which, themselves, depend on the isoparametric coordinates, f, n, defined on a reference, normalised, space. The strain rate and the spin may then be derived thanks to Eq. 11.5, the stress rate is obtained by Eqs. 11.3 and 11.4 and is time integrated. Eventually, equilibrium (Eq. 11.1) has to be checked.
For scalar diffusion or advection-diffusion problems, the unknown field, p, representing a general pressure (which could be pore pressure, u (the use here), temperature, T, or concentration, C, by appropriately changing the notation) may then be written:
p = Nl d, n) Pl L = 1, m (11.15)
Where p depends on the nodal unknowns, pL, and on the shape functions, Nl. Then Darcy’s fluid velocity and the storage changes may be derived thanks to Eqs. 11.7 and 11.8 (or, respectively, Eqs. 11.9 and 11.10). No time integration is required here. Finally, the balance equation (Eq. 11.6) has to be checked.
The finite element method allows an accurate modelling of the boundary condition, thanks to an easily adapted finite element shape. Internal boundaries of any shape between different geological layers or different solids can be modelled. Specific finite elements for interfacial behaviour or for unilateral boundaries have also been developed (e. g. Charlier & Habraken, 1990). Variations of the finite element
size and density over the mesh are also easy to manage, with the help of modern mesh generators.
