Automatic differentiation for stress and consistent tangent computation

Abstract

In the field of nonlinear finite element analysis, the linearization within the Newton–Raphson and the Multilevel-Newton–Raphson algorithm requires the computation of the stresses and the so-called consistent tangent operator. There are several possibilities for coding the derivatives either by deriving analytical expressions, by using numerical differentiation schemes, or by applying the concept of automatic differentiation. In this article, the three possibilities are compared for three different kinds of constitutive models in order to ascertain particular differences. These models are a finite strain hyperelasticity relation, a more sophisticated finite strain viscoplasticity model, and a small strain von Mises-type thermo-viscoplasticity model. Especially for the last example, it is required to provide a number of tangents to be derived within a monolithic, fully coupled finite element computation. Numerical examples investigate the advantages and disadvantages of analytical, numerical, and automatic differentiation. It is shown that the latter one is a very helpful tool in the developing stage of constitutive modeling.