Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation
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This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.
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- 3.Shabana, A. A. and Yakoub, R. Y., ‘Three dimensional absolute nodal coordinate formulation for beam elements: Theory’, ASME Journal of Mechanical Design 123, 2001, 606–613.Google Scholar
- 4.Yakoub, R. Y. and Shabana, A. A., ‘Three dimensional absolute nodal coordinate formulation for beam elements: Implementation and application’, ASME Journal of Mechanical Design 123, 2001, 614–621.Google Scholar
- 6.Shabana, A. A. and Mikkola, A. M., ‘Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity’, ASME Journal of Mechanical Design 125, 2003, 342–350.Google Scholar
- 7.Brenan, K. E., Campbell, S. L., and Petzold, L. R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, Pennsylvania, 1996.Google Scholar
- 8.Ascher, U. M. and Petzold, L. R., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, Pennsylvania, 1998.Google Scholar
- 9.Stoer, J. and Bulirsch, R., Introduction to Numerical Analysis, Springer-Verlag, New York, 1993.Google Scholar
- 10.Gurtin, M. E., An Introduction to Continuum Mechanics, Academic Press, San Diego, California, 1981.Google Scholar
- 11.Zienkiewich, O. C. and Taylor, R. L., The Finite Element Method, Vol. 1, McGraw-Hill, London, 1994.Google Scholar
- 13.Sopanen, J. T. and Mikkola, A. M., ‘Studies on the stiffness properties of the absolute nodal coordinate formulation for three-dimensional beams’, in Proceedings of the ASME 2003 Design Engineering Technical Conferences, Chicago, Illinois, September 2–6, 2003.Google Scholar
- 14.García de Jalón, J. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems—The Real-Time Challenge, Springer-Verlag, New York, 1993.Google Scholar