# An Analysis of the Stability and Convergence Properties of a Crank-Nicholson Algorithm for Nonlinear Elasto-Dynamics Problems

## Summary

The stability of nonlinear step by step computing schemes for solid mechanics problems can be insured by either requiring that the energy of the approximation be conserved or by insisting that the energy be bounded for all time. While techniques exist for analyzing conserving procedures, few analysis methods exist which are capable of establishing the boundedness of the energy of nonconserving computing schemes. In this paper an analysis procedure for demonstrating the stability and convergence of nonconserving step by step solution algorithms is introduced. As a typical example this technique is used to analyze a Crank-Nicholson-Galerkin algorithm for nonlinear elastodynamics problems.

## Keywords

Truncation Error Geneous Boundary Condition Energy Error Solid Mechanic Problem Element Basis Function## Preview

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## References

- 1.Belytschko, T., and Schoeberle, D.F., “On the Unconditional Stability of an Implicit Algorithm for Nonlinear Structural Dynamics”, Journal of Applied Mechanics, 42, 1975, pp. 865–869.ADSCrossRefGoogle Scholar
- 2.Hughes, T.J.R., “Stability, Convergence and Growth and Decay of Energy of the Average Acceleration Method in Nonlinear Structural Dynamics”, Computers and Structures, 6, 1976, pp. 313–324.MathSciNetMATHCrossRefGoogle Scholar
- 3.Hughes, T.J.R., Caughey, T.K., and Liu, W.K., “Finite Element Methods for Elastodynamics which Conserve Energy”, Journal of Applied Mechanics, 45, 1978, pp. 366–370.ADSMATHCrossRefGoogle Scholar
- 4.Wellford, L.C., and Hamdan, S.M., “An Analysis of an Implicit Finite Element Algorithm for Geometrically Nonlinear Problems of Structural Dynamics, Part 1: Stability”, Computer Methods in Applied Mechanics and Engineering, 14, 1978, pp. 377–390.ADSMATHCrossRefGoogle Scholar
- 5.Oden, J.T., and Reddy, J.N., Mathematical Theory of Finite Elements, Wiley Interscience, New York, 1976.MATHGoogle Scholar
- 6.Wellford, L.C., “On the Theoretical Basis of Finite Element Methods for Geometrically Nonlinear Problems of Structural Analysis”, Formulations and Computational Algorithms in Finite Element Analysis, Ed.,K.J. Bathe, et. al., MIT Press, Cambridge, Massachusetts, 1977.Google Scholar