Keywords: finite elements, hybrid-mixed formulations, wavelets on the interval, elasticity, plate problems.

This communication illustrates the use of wavelets defined on the interval as approximation functions in hybrid-mixed stress finite element models applied to the solution of linear elastic plate stretching and plate bending problems. Special attention is given to the algorithms and techniques involved in the generation and manipulation of such functions.

In the hybrid-mixed stress model discussed here [1,2], both the stress and displacement fields are simultaneously and independently approximated in the domain of each element. The displacements on the static boudary, which is considered to include the inter-element boundaries, is also independently modelled. None of the fundamental relations - equilibrium, compatibility and elasticity - is locally enforced. All field equations are imposed in a weighted residual form so designed as to ensure that the discrete numerical model embodies all the relevant properties of the continuum field it represents, static kinematic duality and elastic reciprocity.

Wavelet series defined on the interval are used to approximate both the static and kinematic fields. Wavelet systems are nowadays being used in a wide range of both theoretical and applied fields. The most successfull applications lie in the image and signal processing areas. In many cases, the traditional Fourier analysis has been replaced with remarkable advantages by new techniques based on wavelet systems.

In recent years, some effort has been devoted to the development of effective numerical models for the solution of Solid Mechanics problems using hybrid-mixed stress formulations combined with wavelet approximations [2]. Some interesting results have already been obtained and reported [3]. However, the extraordinary properties of such systems of functions is far from being completely exploited.

In the model presented here, the orthogonal Daubechies [4] wavelet system is used. In order to make possible the definition of the approximation over a closed and limited interval, the domain of an element, the construction introduced by Monasse and Perrier [5] is implemented. This construction follows the ideas originally expressed by Cohen, Daubechies and Vial [6].

In this communication, the basic operations involved in the generation and manipulation of orthogonal wavelets defined on the interval are presented and discussed in some detail. In the context of hybrid-mixed models, special attention is given to the derivation of closed form solutions for the integrations associated with the definition of all structural operators. Due to these analytical definitions, numerical integration schemes can be fully avoided.

A set of numerical examples is presented to illustrate and to validate the use of the hybrid-mixed model. The performance of the model is assessed by comparison with other numerical schemes presented in the literature.

1

JAT Freitas, JP Moitinho de Almeida and EMBR Pereira, "Non-conventional formulations for the finite element method", Computational Mechanics, 23, pp. 488-501, 1999; doi:10.1007/s004660050428

2

LMS Castro, "Wavelets e Séries de Walsh em Elementos Finitos", PhD thesis, Technical University of Lisbon, 1996;

3

LMS Castro and JAT Freitas, "Wavelets in hybrid-mixed stress elements", Computer Methods in Applied Mechanics and Engineering, vol. 190/31, pp. 3977-3998, 2001; doi:10.1016/S0045-7825(00)00313-3

4

I Daubechies, "Orthonormal bases of compactly supported wavelets", Comm. Pure and Applied Math., 41, pp. 909-996, 1988; doi:10.1002/cpa.3160410705

5

P. Monasse and V. Perrier, "Orthonormal wavelet bases adapted for partial differential equations with boundary conditions", SIAM J. Math. Anal., 29(4), pp. 1040-1065, 1998; doi:10.1137/S0036141095295127

6

A Cohen, I Daubechies and P. Vial, "Wavelets on the Interval and fast wavelet transforms", Appl. Comput. Harmon. Anal., 1(1), pp. 54-81, 1993; pp. 909-996, 1988; doi:10.1006/acha.1993.1005

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