4 Finite Element Method

Abstract

In this chapter, the fundamentals of the displacement-based finite element method are reiterated in a concise form, details can be found in a number of textbooks and the references therein. For larger finite element systems, model reduction techniques are commonly applied in order to save computational efforts, so that in this chapter two most commonly applied model reduction techniques – the mode displacement method and the mode acceleration method – are reviewed. Various formulations of the finite element method have been developed in the past years, e.g. the displacement-based finite element formulation and several mixed finite element formulations. However, the most frequently used formulation of the finite element method is the displacement-based formulation, i.e. the displacements are the unknown variables which have to satisfy certain boundary conditions. Once the displacements have been determined, strains and stresses can be calculated. The displacement-based finite element analysis can be derived by the principal of virtual displacements (or work), which in turn is equivalent to setting the first variation of the total potential energy equal to zero.