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Numerical differentiation for use in integrating unified constitutive equations
Author(s) -
Henshall G. A.,
Tanaka T. G.,
Miller A. K.,
Lowe T. C.
Publication year - 1989
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620280509
Subject(s) - constitutive equation , finite element method , partial differential equation , mathematics , computation , numerical analysis , computer science , cauchy elastic material , numerical partial differential equations , algorithm , mathematical analysis , structural engineering , engineering
The NONSS method for numerically integrating time‐dependent constitutive equations requires evaluation of the partial derivatives of the constitutive equations. A numerical method for evaluating these partial derivatives has been developed, thus avoiding the cumbersome task of analytically re‐deriving the partial derivatives when improvements to the constitutive model are evaluated. This method is based upon the adaptive finite‐difference algorithm published by Stepleman and Winarsky. Modifications of their algorithm are described that allow its efficient use within an extended version of NONSS, called NONSS‐ND. Using the MATMOD constitutive equations, the capabilities of NONSS‐ND are demonstrated by comparing the results of a wide variety of deformation simulations with those of NONSS, for which the derivatives are computed from analytical solutions. The numerical differentiation within NONSS‐ND is shown to be sufficiently accurate for use in integrating time‐dependent constitutive equations, but results in a 30–100 per cent increase in computation time. Multi‐element structural calculations are therefore found to be uneconomical, but NONSS‐ND is well suited for testing alternative constitutive models owing to its accuracy and its general applicability to highly non‐linear systems of differential equations.