1.1.1 The Finite Element Method

The finite element method, resulting from the matrix techniques of calculation of the discrete or semi-discrete mechanical structures (assembly of beams), is a tool for resolving problems with partial differential equations involved in physics problems. We will thus tackle this method accordingly because it is useful in modeling mechanical, thermal, neutron and electromagnetic problems [ZIE 79], [SIL 83], [DHA 84], [SAB 86], [HOO 89].

The aim of this chapter is to present the principles of this method which have become essential in the panoply of the engineer. For this presentation, we will only deal with electrostatics. Indeed, this field has a familiar formulation in scalar potential, particularly suitable for the presentation of nodal finite elements, which will be the only ones discussed here.

We will develop two examples of increasing complexity which are manageable "by hand", 1D in a first part and 2D in a second. As it is very close to physical considerations, the variational approach will most of the time be favored. However, the more general method of weighted residues will also be presented. In our examples, we will see how to solve the problems at issue, but also how, using the obtained fields, to extract more relevant information.

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In the third and last part, we will present the concept of a reference element and the principles that make it possible to pass from the local coordinates to the domain coordinates. We will see that beyond the possibility of handling...