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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 88
Edited by: B.H.V. Topping and M. Papadrakakis
Paper 135

The Wavelet-Based Theory of Spatial Naturally Curved and Twisted Linear Beams

E. Zupan1,2, D. Zupan2 and M. Saje2

1Research Department, VEPLAS, Velenje, Slovenia
2Faculty of Civil and Geodetic Engineering, University of Ljubljana, Slovenia

Full Bibliographic Reference for this paper
E. Zupan, D. Zupan, M. Saje, "The Wavelet-Based Theory of Spatial Naturally Curved and Twisted Linear Beams", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 135, 2008. doi:10.4203/ccp.88.135
Keywords: wavelets, shape functions, linear beam theory, analytical solution.

Wavelets have received an increased attention in the last decade in various engineering disciplines. They have proven to provide a suitable mathematical background for signal processing, processing of images, pattern recognition, diagnosing and monitoring the disturbances, and similar problems. The generality of their applicability stands directly on the attractive properties the sets of wavelets have: periodicity, orthogonality and linear independency. For the theoretical foundation of the wavelet theory, the reader is referred to [1].

We would like to point out that there are a large number of types of scaling and wavelet functions, which, unfortunately, could not be expressed explicitly in a general way and do not have proper interpolation properties. That is why the implementation of the wavelets in finite element theories is non-trivial and often demands some additional theoretical work. One of the implementations is the wavelet-Galerkin method, which demands the shape functions to be expressed in a form of a product of wavelet functions and wavelet coefficients. In such an approach, the relation between the wavelet coefficients and the physical quantities is non-trivial, which makes it difficult to treat element boundary conditions. Alternatively, an additive-type, spline-wavelet-based interpolation was proposed by Han, Ren and Huang [2] to resolve this difficulty for conventional displacement-based finite elements.

The present formulation, in contrast to [2], is based on the variant of wavelets presented by Prestin and Quak [3]. The related set of the scaling functions (of any order) and the corresponding set of wavelets not only preserve the properties of the wavelets (they represent the orthonormal base functions), but also possesses two additional properties of an utmost importance in the finite element implementation: they can be expressed explicitly and have interpolation properties.

The wavelet-based discretization is applied to the linearized finite strain beam theory [4] which assumes small displacements, rotations and strains but is capable of considering an arbitrary initial geometry and material behaviour. In the numerical solution algorithm, we base our derivations on the vector of strain measures as the only unknown functions in a finite element.

We point out that the scaling functions and wavelet based decomposition can be a suitable choice of shape functions in finite element formulations. Scaling functions and wavelet based formulations are in contrast to Lagrangian polynomial-based formulations numerically stable for an arbitrary order of straight and curved elements. The stability of the numerical solution regarding the order of interpolation indicates that such a formulation is appropriate when mesh refinement is required.

I. Daubechies, "Ten lectures on wavelets", CBMS - NSF Regional Conference Series in Applied Mathematics, Department of Mathematics, University of Lowell, MA. SIAM: Philadelphia, PA, 1992.
J.G. Han, W.X. Ren, Y. Huang, "A spline wavelet finite-element method in structural mechanics", Int. J. Numer. Methods Engng. 26, 717-730, 1988.
J. Prestin, E. Quak, "Trigonometric interpolation and wavelet decompositions", Numerical Algorithms 9, 293-317, 1995. doi:10.1007/BF02141593
D. Zupan, M. Saje, "The linearized three-dimensional beam theory of naturally curved and twisted beams: The strain vectors formulation", Comput. Methods Appl. Mech. Engrg. 195, 4557-4578, 2006. doi:10.1016/j.cma.2005.10.002

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