Grundlagen der Tensorrechnung

Tensor calculus, with its consistent formulation of invariants and transformations, has for years acquired a high significance in engineering. Thus, early works on the mechanics of load-bearing structures, such as [Green/Zerna 1954], which renewed the formulation of the theory, and compendium of mathematics on the tensor calculus in index notation, such as [Duschek/Hochrainer 1968], can serve as a basis at this point. In this chapter, the most important basics of tensor algebra are recapitulated in this context, especially to make the approach of the object-oriented matrix calculus comprehensible and to provide the ability to apply the presented methodology consistently. This is because the index notation allows the arithmetic operations of tensor calculus to be seamlessly transformed into the practical numeric of programming tools using the new object-oriented classes and methods of matrix calculus. This capability is demonstrated with the tensor class BASIS and its application in the numeric function METRIC.