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Automatic Implementation of Elasto‐plastic Incremental Formulations at Finite Strains using Hyper‐Dual Numbers
Author(s) -
Tanaka Masato,
Balzani Daniel,
Schröder Jörg
Publication year - 2015
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201510174
Subject(s) - dissipative system , cauchy stress tensor , mathematics , associative property , computer science , automatic differentiation , range (aeronautics) , dual (grammatical number) , finite element method , mathematical optimization , algorithm , computation , mathematical analysis , physics , pure mathematics , structural engineering , engineering , quantum mechanics , art , literature , aerospace engineering
Numerical simulation of standard dissipative materials undergoing finite strains remains an important and challenging topic in computational mechanics. The incremental variational formulation (IVF), firstly proposed by Ortiz et al. [1], provides a general variational framework which is suitable for the implementation of a broad range of constitutive laws for standard dissipative materials. The IVF recasts the inelasticity theory as an equivalent optimization problem where the incremental stress potential is minimized with respect to the internal variables. However, their implementation often requires more effort than classical formulations due to high‐order tensor derivatives. In this contribution, a novel implementation of IVFs is presented to arrive at a fully automatic and robust scheme with computer accuracy using hyper‐dual numbers (HDNs). The HDNs, which are originally developed by Fike [2], derive exact and automatic derivative calculations without any cumbersome choice of perturbation values. Its uncomplicated implementation for associative finite strain elasto‐plasticity and its performance is illustrated by a representative numerical example. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)