In particular the multi-resolution features of the wavelet analysis, make this approach attractive for the analysis of complex material properties, correlation with modern high density experimental measurements, variability and uncertainty modelling.

The specific properties of wavelet families: compact support, orthogonality, vanishing moments are transferred to the field of finite element analysis, and the wavelets replacing the conventional polynomial interpolation used for shape functions. Previous results in the published literature, showed superior results for the finite element analysis in regions where the solutions have fast variation, while concurrently, good accuracy is being obtained with a reduced mesh density.

The Daubechies wavelet family is used in this paper for the construction of beam finite elements, and the study of the dynamic response under moving load is carried out. The modelling difficulties and the advantages and disadvantages of using WFEM are discussed within the frame of this application, with an emphasis on the multi-resolution capabilities of the scaling functions used for the finite element formulation. The derivatives of the Daubechies scaling functions are highly oscillatory, hence a specific algorithm is required to determine the integral products of the scaling functions and, or their derivatives. These connection coefficients have to be calculated in the wavelet space and then transformed into the physical space and the mapping between the orthogonal base in the fundamental space and the displacement distribution in the physical space has to be preserved.

Numerical simulations using a D10 Daubechies wavelet finite element and the Bernoulli-Euler formulation are presented and two cases are studied: a constant and a harmonic moving load on a simply supported beam which simulates the response of a railway bridge under the moving load of a locomotive. In both the eigenvalue analysis of the simply supported beam and the time integration analysis, a reduced mesh density leads to superior numerical accuracy for the WFEM formulation for the displacements, velocity and acceleration fields in comparison with the standard Bernoulli-Euler formulation.

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