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Keywords: finite difference, fractal approximation, square medium error, target points, stretching parameters.

Summary

In the present work, as an improvement of the previous proposed methods, the authors study the fractal interpolation method [1,2] for the analysis of dynamical time dependent structural systems by means of the generalised finite difference method (GFDM).

The response of a structure subject to dynamical loads is determined by means of the finite element approach [3] which transform the structure into a finite collection of elementary oscillators.

An iterated function system (IFS) transformation of the generic unknown function for the generic oscillator and the Taylor's expansion of the same function gives an error function which is minimised at the target points. By means of the Gauss-Lagrange method an unbounded minimum error problem is solved and the values of the unknown function, of its first and second derivative, are calculated at the fixed points. Then the deformed shape of the structure and the derivative displacement with respect to time are determined. According to minimum least square functional error FDM expression for derivatives are also obtained in closed form. The influence of stretching parameters is analysed for small and large time intervals and an approximation value is given.

The numerical experiments obtained from the response of a simply supported beam to an imposed time dependent set of displacements, show the fast rate of convergence of the results with very low errors for displacements and derivatives. The procedure is utilised for variable space and time discretisation. The proposed method is not influenced by the variability of the time interval both for the displacement and for the first derivative.

References

1: M.Barnsley, Fractals everywhere, Academic Press, Inc.,Boston.
2: S. Basu, E. Foufoula-Georgiu, F. Portè-Agel, "Synthetic turbulence, fractal interpolation, and large-eddy simulation", Physical Review E, 70, 026310. doi:10.1103/PhysRevE.70.026310
3: K.J. Bathe, Finite Element Procedures, New Jersey, Prentice Hall Inc, 1996.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 88 PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and M. Papadrakakis Paper 191 The Fractal Generalized Finite Difference Method in Elastodynamics G.M. Cocchi and P. Tiriaca Department of Architecture, Construction and Structures, Faculty of Engineering, Università Politecnica delle Marche, Ancona, Italy doi:10.4203/ccp.88.191 purchase the full-text of this paper Full Bibliographic Reference for this paper G.M. Cocchi, P. Tiriaca, "The Fractal Generalized Finite Difference Method in Elastodynamics", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 191, 2008. doi:10.4203/ccp.88.191 Keywords: finite difference, fractal approximation, square medium error, target points, stretching parameters. Summary In the present work, as an improvement of the previous proposed methods, the authors study the fractal interpolation method [1,2] for the analysis of dynamical time dependent structural systems by means of the generalised finite difference method (GFDM). The response of a structure subject to dynamical loads is determined by means of the finite element approach [3] which transform the structure into a finite collection of elementary oscillators. An iterated function system (IFS) transformation of the generic unknown function for the generic oscillator and the Taylor's expansion of the same function gives an error function which is minimised at the target points. By means of the Gauss-Lagrange method an unbounded minimum error problem is solved and the values of the unknown function, of its first and second derivative, are calculated at the fixed points. Then the deformed shape of the structure and the derivative displacement with respect to time are determined. According to minimum least square functional error FDM expression for derivatives are also obtained in closed form. The influence of stretching parameters is analysed for small and large time intervals and an approximation value is given. The numerical experiments obtained from the response of a simply supported beam to an imposed time dependent set of displacements, show the fast rate of convergence of the results with very low errors for displacements and derivatives. The procedure is utilised for variable space and time discretisation. The proposed method is not influenced by the variability of the time interval both for the displacement and for the first derivative. References 1 M.Barnsley, Fractals everywhere, Academic Press, Inc.,Boston. 2 S. Basu, E. Foufoula-Georgiu, F. Portè-Agel, "Synthetic turbulence, fractal interpolation, and large-eddy simulation", Physical Review E, 70, 026310. doi:10.1103/PhysRevE.70.026310 3 K.J. Bathe, Finite Element Procedures, New Jersey, Prentice Hall Inc, 1996. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £145 +P&P)
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