Mathematics of element formulation

Abstract

The finite element method seeks to offer solutions of displacements arising when loads are applied on a structure. It can also determine the forces that results when a displacement loading condition is imposed on the structure. Finite element meshes play an important role in the derivation of the displacements for a structure. Element formulation deals with derivations needed to deduce displacements acting on a body when discretized by finite elements. Different formulations apply for different types of elements. Derivation of such displacements is dependent on shape functions. This chapter presents the mathematics that drive the element behaviour. This communication is based on simple element types ranging from one-dimensional linear elements, to high-order three-dimensional elements. This is an important chapter as it helps the reader understand how displacements, nodal forces, element stresses and strains are derived for different element types. Readers will also gain the understanding of the limitations of one element type from another, based on the mathematics of their formulation. The content here can be considered for an advanced finite element course, and if one is new to the FEM process, one can skip this chapter.