There are many books devoted to a general introduction to the finite-element method. It is impossible to list all of them here, but those referenced in the footnotes appear in increasing order of difficulty. Interested readers could select one or two for study, depending on their backgrounds. In this chapter a very simple introduction is given, mainly in view of applications to the numerical modeling of metal-forming processes.

A first introduction is given in Section 2.1, with a comparison with another popular numerical method: the finite-difference method. The selected thermal problem allows a complete “hand calculation” by the two methods, and a comparison of the results with an analytical solution. The aim of this first overview is to summarize very briefly the main aspects (and vocabulary) of the method. It shows that a quasi-intuitive approach is conceivable and could help to demythologize the finite-element method.

A more systematic approach follows in Sections 2.2 to 2.5, in order to separate as clearly as possible the steps that lead to the rational development of the method. In most occasions the basic concepts are first explained in detail by using one-dimensional examples, and the generalization is only mentioned, as more detail will be given in Chapters 3 and 4 for two- and three-dimensional approaches.

The basic idea is to follow four steps:

1. Establish the physical equations of the model (see Chapter 1, or an earlier work on this subject).