Keywords: wavelet analysis, multiresolutional techniques, composites materials, finite element method, vibration analysis, transient heat transfer.

Multiresolutional analysis based on the wavelet functions makes it possible to model heterogeneous solids with different macro-, meso-, micro- and even smaller scales. Such functions like Gabor, Morlet, Haar and Daubechies wavelets and the Mexican hat or harmonic functions [6] frequently applied in such analysis can be efficiently used in all those engineering disciplines, where signal processing is necessary (spatial distribution of a wind or fluid pressure, structural dynamic external excitation, geometry of surfaces and interfaces etc.). Multiresolutional methods can find their application also in bioengineering and financial markets prognosis. Considering the opportunity of observation of heterogeneous solids using contrastively different zooms, the wavelet-based techniques perfectly meet the needs of composite materials modelling [2,3,4].

As it is known, wavelet techniques can be divided into continuous wavelet transforms (CWT), where continuous functions may be described using partially constant ones. On the contrary, discrete wavelet transforms (DWT) may find their application, which are based on matrix calculus and which perfectly reflect the needs of discrete numerical methods like the Finite Element Method (FEM), for instance.

The main goal is to employ all the features of wavelet analysis to model interface defects in unidirectional periodic composite materials. Spatial multiresolutional distribution of material properties is given in Figure  in the example of the coefficient of heat conductivity for a composite with (left) and without (right) the interface defects. Some algebraic combination of the Haar wavelet (in macro scale), Mexican hat (interface boundary) and harmonic wavelets (microtructure non-homogeneities) is introduced to describe analytically this coefficient. Uniaxial spatial distribution of composite heat capacity, Young modulus and mass density are characterized quite similarly. It should be mentioned that the interface defects are simulated here as a remarkable decrease of any material property in the neighborhood of multimaterial boundary observed in a material with smaller modulus or coefficient.

The composites with and without interface defects with material characteristics so defined are compared with one another using some basic FEM tests of transient heat transfer [2] and the free vibrations problems [1,3]. The multiresolutional discretization process is completed thanks to various meshes (from zeroth to fifth order), where one, two, four and etc. one-dimensional two-noded linear elastic or three-noded temperature-independent finite elements are introduced into a single Representative Volume Element (RVE) shown above. Necessary numerical data preprocessing is carried out using symbolic computations package MAPLE, which was tailored before for determination of the effective characteristics for analogous composites [2]. Further computational studies using combined symbolic-FEM methodology can be conducted to determine the influence of the interface defects on the overall effective material characteristics [4] of such composites as well as other ones with both two- and three-dimensional distribution of reinforcement [5].

1

J. Argyris, H.P. Mlejnek, Dynamics of Structures, North-Holland, 1991.

2

M. Kaminski, "Multiresolutional homogenization technique in transient heat transfer for unidirectional composites", Proceedings of the 3rd Int. Conf. Engrg. Computational Technology, B.H.V. Topping and Z. Bittnar, edrs., Civil-Comp Press, 2002. doi:10.4203/ccp.75.138

3

M. Kaminski, "Wavelet-based finite element elastodynamic analysis of composite beams", World Congress WCCM V, H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner, Edrs., Vienna 2002.

4

M. Kaminski, "Wavelet-based homogenization of unidirectional multiscale composites", Computational Materials Science, 27, 446-460, 2003. doi:10.1016/S0927-0256(03)00046-6

5

G.W. Milton, "The Theory of Composites", Cambridge University Press, Cambridge, 2002.

6

R.K. Young, "Wavelet Theory and Its Applications", Kluwer, 1993.

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