numerical methods in geomechanics. Proceedings of the NATO Advanced Study Institute, University of Minho, Braga, Portugal, held at Vimeiro (24 August–4 September 1981). Boston: D. Reidel.64 Zienkiewicz, O. C. and Shiomi, T. (1984). Dynamic behaviour of saturated porous media: the generalized Biot formulation and its numerical solution, Int. J. Num. Anal. Geomech., 8, 71–96.
65 Zienkiewicz, O. C., Chang, C. T. and Bettess, P. (1980). Drained, undrained, consolidating and dynamic behaviour assumptions in soils, Géotechnique, 30, 4, 385–395.
66 Zienkiewicz, O. C., Chan, A. H. C., Pastor M., Paul, D. K. and Shiomi, T. (1990a). Static and dynamic behaviour of geomaterials – a rational approach to quantitative solutions, Part I: Fully saturated problems, Proc. Roy. Soc. London, A429, 285–309.
67 Zienkiewicz, O. C., Xie, Y. M., Schrefler, B. A., Ledesma, A. and Bicanic, N. (1990b). Static and dynamic behaviour of soils: a rational approach to quantitative solutions, Part II: Semi‐ saturated problems, Proc. Roy. Soc. London, A429, 310–323.
68 Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z. (2005). The Finite Element Method Set (Sixth Edition) Its Basis and Fundamentals, Elsevier.
3 Finite Element Discretization and Solution of the Governing Equations
3.1 The Procedure of Discretization by the Finite Element Method
The general procedures of the Finite Element discretization of equations are described in detail in various texts. Here we shall use throughout the notation and methodology introduced by Zienkiewicz et al. (2013) which is the seventh edition of the first text for the finite element method published in 1967.
In the application to the problems governed by the equations of the previous chapter, we shall typically be solving partial differential equations which can be written as
(3.1)
where A and B are matrices of constants and L is an operator involving spatial differentials such as ∂/∂x, ∂/∂y, etc., which can be, and frequently are, nonlinear.
The dot notation implies time differentiation so that
(3.2)
In all of the above, Φ is a vector of dependent variables (say representing the displacements u and the pressure p).
The finite element solution of the problem will always proceed in the following pattern:
1 The unknown functions Φ are “discretized” or approximated by a finite set of parameters and shape function Nk which are specified in spatial dimensions. Thus we shall write(3.3) where the shape functions are specified in terms of the spatial coordinates, i.e.(3.4a) or (3.4b) are usually the values of the unknown function at some discrete spatial points called nodes which remain as variables in time.
2 Inserting the value of the approximating function into the differential equations, we obtain a residual which is not identically equal to zero but for which we can write a set of weighted residual equations in the form(3.5) which on integration will always reduce to a form(3.6) where M, C, and P are matrices or vectors corresponding in size to the full set of numerical parameters . A very suitable choice for the weighting function Wj is to take them being the same as the shape function Nj:(3.7) Indeed, this choice is optimal for accuracy in the so‐called self‐adjoint equations as shown in the basic texts and is known as the Galerkin process.
If time variation occurs, i.e. if the parameters
Usually, the parameters
Typical finite elements involving linear and quadratic interpolation are shown in Figure 3.1.
Figure 3.1 Some typical two‐dimensional elements for linear and quadratic interpolations from nodal values.
In Section 3.2, we shall consider solution of the approximation based on the u–p form in which the dependent variables are the displacement of the soil matrix and the pore pressure characterized by a single fluid, i.e. water. However, we shall allow incomplete saturation to exist assuming that the air pressure is zero.
The formulation thus embraces all the features of the u–p approximation of Sections 2.2.2, 2.3.1, and 2.3.2 and is the basis of a code capable of solving all low‐frequency dynamic problems, consolidation problems, and static drained or undrained problems of soil mechanics. Only two dimensions will be considered here and in the examples which follow, but extension to three dimensions is obvious.
The code based on the formulation contained in this part of the chapter is named SWANDYNE (indicating its Swansea origin) and its outline was presented in literature by Zienkiewicz et al. (1990a, b). The implicit form used allows long periods to be studied. Indeed, with suitable accuracy control, such codes can be used both for earthquake phenomena limited to hundreds of seconds or consolidation problem with a duration of hundreds of days.
3.2 u‐p Discretization for a General Geomechanics’ Finite Element Code
3.2.1 Summary of the General Governing Equations
We will report here the basic governing equation derived in the previous chapter. However, we shall limit
