The Finite Element
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
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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 Rd . • A polynomial function space P on K. • A set of n D dim.P / linear functionals Li ./, i D 1; 2; : : : ; n, defining the socalled 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 fSj gnj D1 . The basis functions Sj are generally called shape functions.
M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__8, © Springer-Verlag Berlin Heidelberg 2013
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8 The Finite Element
A finite element is said to be unisolvent if the functionals can uniquely determine the shape functions. Unisolvency can be thought of as a necessary compatibility condition for Li ./, P , and K. By definition, it is equivalent to Li .v/ D 0 v D 0, for all v 2 P and all i . Even though it can be a bit hard to establish unisolvency, the actual calculation of the shape functions is easy as it simply amounts to solving a linear system. Indeed, the shape functions Sj are determined from the n linear algebraic equations Li .Sj / D ıij ;
i; j D 1; 2; : : : ; n
(8.1)
By taking a linear combination of shape functions and coefficients we get a polynomial or finite element function in P on each polygon K. Besides specifying the shape functions on each polygon K, the functionals also specify the behavior of the these functions between adjacent polygons. For instance, if we want finite element functions that are continuous on the whole mesh K, then we must take care in choosing functionals, so that the resulting shape functions become continuous, especially across the polygon boundary @K. Indeed, the functionals determine both the local and the global properties of the finite element space Vh . The particular choice of functionals Li ./ give rise to groups or families of finite
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