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Calculation of Stresses and Consistent Tangent Moduli from Automatic Differentiation of Hyerelastic Strain Energy Functions Through the Use of Hyper Dual Numbers
Author(s) -
Tanaka Masato,
Sasagawa Takashi,
Omote Ryuji,
Fujikawa Masaki,
Balzani Daniel,
Schröder Jörg
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410202
Subject(s) - hyperelastic material , tangent , moduli , tangent modulus , numerical differentiation , finite element method , perturbation (astronomy) , mathematics , partial derivative , strain energy , anisotropy , mathematical analysis , modulus , physics , geometry , quantum mechanics , thermodynamics
Many materials as e.g. engineering rubbers, polymers and soft biological tissues are often described by hyperelastic strain energy functions. For their finite element implementation the stresses and consistent tangent moduli are required and obtained mainly in terms of the first and second derivative of the strain energy function. Depending on its mathematical complexity in particular for anisotropic media the analytic derivatives may be expensive to be calculated or implemented. Then numerical approaches may be a useful alternative reducing the development time. Often‐used classical finite difference schemes are however quite sensitive with respect to perturbation values and they result in a poor accuracy. The complex‐step derivative approximation does never suffer from round‐off errors, cf. [1], [2], but it can only provide first derivatives. A method which also provides higher order derivatives is based on hyper dual numbers [3]. This method is independent on the choice of perturbation values and does thus neither suffer from round‐off errors nor from approximation errors. Therefore, here we make use of hyper dual numbers and propose a numerical scheme for the calculation of stresses and tangent moduli which are almost identical to the analytic ones. Its uncomplicated implementation and accuracy is illustrated by some representative numerical examples. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)