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BDF Methods for DAEs in Multi-body Dynamics: Shortcomings and Improvements

  • Michael F. Steigerwald
Conference paper
Part of the NATO ASI Series book series (volume 69)

Abstract

If backward differentiation formula (BDF) methods are used to solve sets of differential-algebraic equations (DAEs), problems may arise if the step size (h), becomes very small. The step size may decrease for a variety of reasons, but the most difficult problem to solve involves a sudden change, or (near) discontinuity in any of the system variables.

In general, BDF methods encounter stability problems if used in programs that allow for sudden changes in the system variables. To solve problems such as these, in the general case, the program will decrease the step size to integrate through the discontinuity. As the step size decreases, the Jacobian matrix, if used for Newton iterations, becomes ill-conditioned. As a result, round-off error makes it almost impossible to solve the equations of motion. Details of the problem and solutions are discussed.

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References

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    C. W. Gear, The Automatic Integration of Ordinary Differential Equations, Communications of the ACM, Volume 14, Number 3 (1971), pp. 176–190.MathSciNetzbMATHCrossRefGoogle Scholar
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    L. R. Petzold and P. Lotstedt, Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints II: Practical Implications, SLAM J. Sci. Stat. Comput., Volume 7, Number 3 (1986), pp. 720–733.MathSciNetzbMATHCrossRefGoogle Scholar
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    T. Park and E. Haug, III-conditioned Equations in Kinematics and Dynamics of Machines, International Journal for Numerical Methods in Engineering, Volume 26 (1988), pp. 217–230.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Michael F. Steigerwald
    • 1
  1. 1.Mechanical Dynamics, Inc.Ann ArborUSA

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