3.1  Finite Element Method

The Finite Element Method (FEM) is a numerical technique used to approximate solutions of PDEs [56]. The technique has surged in the mids 60s and it was intended for solving problems which emerged from elastic theory and structural analysis, for instance, to calculate stress in dams, buildings, and airplanes [57]. In the beginning, FEM was shaped by structural analysts. Thus, the method presentation and nomenclature was particularly suitable for structural mechanics problems. However, the successful progress of the FEM and the rise of the electronic computing industry in the 70s attracted a lot of interest from the scientific community. In the same decade, several mathematicians worked through the details of the FEM, and they could relate the theory behind it to the works of Galerkin, Ritz, and Rayleigh from the late 19th century [57][58]. This development has led to the generalization of the FEM and enabled its use outside the structural mechanics field.

The main idea behind FEM is to approximate the solution of a PDE by a linear combination of functions. Those functions are defined in discrete portions of the PDE domain, the so called finite elements. In principle, FEM can be applied to solve every kind of PDE, but in practice some types of equations are more numerically challenging than others, and FEM would not be a suitable choice, or a modification on the traditional approach is required. The method presented here will be restricted to elliptic PDEs [56], since solid mechanics problems can be represented by one of those.

3.1.1 Variational Form
3.1.2 Galerkin’s Method
3.1.3 Discretization
3.1.4 Basis Functions and Domain Partitioning
 Considerations for 2D and 3D Cases
3.1.5 Geometrical Interpretation of FEM
3.1.6 Final Remarks on FEM