Tensor Analysis for Finite Elements

In this chapter, the fundamentals of tensor analysis are compiled for use within the finite element method. Similar to the classical approaches such as [Klingbeil 1966], the differential geometry of the spatial bodies is treated first, followed by the mechanical fundamentals for the deformation of elastic continua. The tensor representations are then specialized to the finite element approximations for the modeling of the geometry and the description of the displacement fields. These explanations follow the classical compendia, such as [Zienkiewicz/Taylor 1989] or even the textbook [Meissner/Maurial 2000]. Here, however, they are consistently presented in the preferred index notation for matrices and tensors. Derived exemplary for numerical purposes are the matrix relations for the approximate spanning of the geometry of individual elements and for their stiffness relations under external loading as well as for the calculation of internal forces. Since the focus is primarily on the principle of matrix methodology, the plane continua of the plate theory are selected as a relatively transparent field of application. This is sufficiently complex, but nevertheless illustrative, in order to demonstrate the advantages of the new matrix methods with typical examples.