Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis

An extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard FEM software. This method is based on the complex-step derivative approximation (CSDA) approach. The CSDA is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e.g. finite difference approximation. Therefore, the CSDA can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision. Implementation details of the robust numerical approximation of tangent moduli from stress calculations using the CSDA are given and their performance is illustrated through representative examples involving finite deformations. In addition to that, we focus on the determination of material instabilities. Therefore, an accompanying localization analysis is performed, where the acoustic tensor is directly computed from the approximation of the moduli. It is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation. On the other hand, the CSDA approach provides high-accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely.