The Finite Element

Abstract

In this chapter we study the concept of a finite element in some more detail. We begin with the classical definition of a finite element as the triplet of a polygon, a polynomial space, and a set of functionals. We then show how to derive shape functions for the most common Lagrange elements. The isoparametric mapping is introduced as a tool to allow for elements with curved boundaries, and to simplify the computation of the element stiffness matrix and load vector. We finish by presenting some more exotic elements. 8.1 Different Types of Finite Elements 8.1.1 Formal Definition of a Finite Element Formally, a finite element consists of the triplet: • A polygon K R d. • A polynomial function space P on K. • A set of n D dim.P / linear functionals L i ./, i D 1; 2; : : : ; n, defining the so-called degrees of freedom. The polygon K is of different type depending on if the space dimension d is 1, 2, or 3. The most common types of polygons in use are lines, triangles, quadrilaterals, tetrahedrons, and bricks. Occasionally, prisms are used. Each polygon stems from a mesh K D fKg of the computational domain ˝. Triangle and tetrahedron meshes are able to easily represent domains with curved boundaries, while quadrilateral and brick meshes are easy to implement in a computer. Prisms are primarily used for domains with cylindrical symmetries, such as pipes, for instance. Let us equip P with a basis fS j g n j D1. The basis functions S j are generally called shape functions.