A highly accurate 1st- and 2nd-order differentiation scheme for hyperelastic material models based on hyper-dual numbers

In this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models, which are derived in terms of the first and second derivatives of a strain energy function. This numerical scheme provides a compact model-independent framework, which means that once the framework is implemented, any other hyperelastic material model can be incorporated by solely modifying the energy function. The method is based on the numerical calculation of strain energy derivatives using hyper-dual numbers and thus referred to as hyper-dual step derivative (HDSD). The HDSD does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values. Furthermore, it enables the calculation of derivatives of arbitrary order. This is a great advantage compared to other numerical approaches as, e.g., the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values. Another alternative, the complex-step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values, but it only provides first derivatives and is thus not able to calculate stresses and moduli at once. In this paper, representative numerical examples using an anisotropic model are provided showing the performance of the proposed method. In detail, an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme. Furthermore, examples demonstrate the robustness and simple implementation of the HDSD scheme in finite element software. It turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time.