Chapter 1Finite Elements and Shape Functions
There is a wide range of existing literature on finite elements, both on theoretical aspects (for example [Oden, Reddy-1977], [Ciarlet-1978], [Hughes-1987], [Ciarlet-1991]) and on practical aspects ([Zienkiewicz, Taylor-1989], [Bathe-1996], [Dhatt et al. 2007]). The purpose of this chapter, therefore, is not to provide yet another description of this method, but rather to introduce a point of view that is strongly oriented toward the underlying geometric aspects. Indeed, classically, these are all the aspects of approximations of functions (polynomial space or others, convergence, convergence rate, etc.) that are examined. We will, thus, only review basic definitions related to finite elements1, as well as their shape functions. The classic case of finite elements whose degrees of freedom are nodal values of the considered functions (in other words, like Lagrange elements) is described for complete elements, reduced elements as well as rational elements. The less common finite elements such as Hermite elements, for example, where nodal or other derivatives are involved are not explicitly considered2.
1.1. Basic concepts
The finite elements method allows us to calculate an approximate solution to a problem formulated in terms of a system of partial derivatives over a continuum Ω across two related approximations: a spatial approximation and an approximation for calculated solutions. The physical problem under study is modeled by ...