Development of a New Algorithm for Solving the Initial Value Problem in Finite Element Elasto-Plastic Analysis
by
Zhongwen Ding
Department of Engineering, Faculty of Engineering and Information Technology, The Australian National University, Canberra, ACT 0200, Australia
Coauthors: S. Kalyanasundaram (Department of Engineering, The Australian National University), M. Cardew-Hall (Department of Engineering, The Australian National University), L. Grosz (School of Mathematical Sciences, The Australian National University), S. Roberts (School of Mathematical Sciences, The Australian National University)

Keywords: Finite element method, initial value problem, Runge-Kutta algorithm, substepping scheme.

The integration of stress-strain relations in elsto-plastic problems can be regarded as a initial value problem. The crudest method for this problem is the Euler algorithm. Since the Euler method is accurate only for very small substeps, conventionally, the whole integration process is broken up into a number of smaller substeps of equal size. Such integration will generally result in the tress change departing from the yield surface and some forms of stress correction are frequently used to ensure that the computed stress remain on the yield surface at any time. Although this method has been used widely in the finite element codes, it have following disadvantages:

1. If the correction-step is applied after each substep, the computational time will increase drastically. However, if it is done at the end of integration process, it can not significantly affect the accuracy.
2. Since the number of substeps is usually determined by an empirical rule which is formulated by trial and error, the inappropriate choose of the number of the substeps usually lead to lose of either accuracy or efficiency.

This paper will present a new algorithm for integrating strain-stress relations. It is based on the third and the fourth order Runge-Kutta method. This substepping scheme controls the error in the integration process by permitting the size of each substep to vary in accordance with the behaviour of the constitutive law. The results indicate that, comparing with the conventional method, the combination of the substepping scheme with stress correction make the solution more accurate and efficient.

References

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S. W. Sloan, "Substepping Schemes for the Numerical Integration of Elastoplastic Stress-strain Relations", Int. J. Numer. Methods Eng., Vol.24, pp.893-911 (1987)

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M. L. James, G. M. Smith and J. C. Wolford, Applied Numerical Methods for Digital Computation, Harper & Row, Publisher, New York, 1985

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R. England, Ërror Extimates for Runge-Kutta Type Solution to Systems of Ordinary Differential Equations", Computer Journal, 12, 166-170, 1969

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B. P. Sommeijer, Parallelism in the numerical integration of initial value problems, Stichting Mathematisch Centrum, Amsterdam, 1993

Date received: July 30, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-79.