Robust Numerical Schemes for an Efficient Implementation of Tangent Matrices: Application to Hyperelasticity, Inelastic Standard Dissipative Materials and Thermo-Mechanics at Finite Strains

Abstract

In this contribution robust numerical schemes for an efficient implementation of tangent matrices in finite strain problems are presented and their performance is investigated through the analysis of hyperelastic materials, inelastic standard dissipative materials in the context of incremental variational formulations, and thermo-mechanics. The schemes are based on highly accurate and robust numerical differentiation approaches which use non-real numbers, i.e., complex variables and hyper-dual numbers. The main advantage of these approaches are that, contrary to the classical finite difference scheme, no round-off errors in the perturbations due to floating-point arithmetics exist within the calculation of the tangent matrices. This results in a method which is independent of perturbation values (in case of complex step derivative approximations if sufficiently small perturbations are chosen). An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of hyperelastic, finite strain elastoplastic, and thermo-elastoplastic boundary value problems, the performance of the proposed approaches is analyzed.