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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 are time‐dependent, Equation (3.6), which is now an ordinary differential equation, requires solution in the time domain. This can be, once again, achieved by discretization in time and use of finite elements there, although many alternative approximations (such as the use of finite differences or other integration schemes) are possible.

      Usually, the parameters represent simply the values of Φ^h at specified points – called nodes and the shape functions are derived on a polynomial basis of interpolating between the nodal values for elements into which the space is assumed divided.

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,

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