Daubechies wavelet beam and plate finite elements

In the last few years, wavelets analysis application has called the attention of researchers in a wide variety of practical problems, particularly for the numerical solutions of partial differential equations using different methods such as finite differences, semi-discrete techniques or finite element method. In some mathematical models in mechanics of continuous media, the solutions may have local singularities and it is necessary to approximate with interpolatory functions having good properties or capacities to efficiently localize those non-regular zones. Due to their excellent properties of orthogonality and minimum compact support, Daubechies wavelets can be useful and convenient, providing guaranty of convergence and accuracy of the approximation in a wide variety of situations. In this work, we show the feasibility of a hybrid scheme using Daubechies wavelet functions and the finite element method to obtain numerical solutions of some problems in structural mechanics. Following this scheme, the formulations of an Euler-Bernoulli beam element and a Mindlin-Reisner plate element are derived. The accuracy of this approach is investigated in some numerical test cases.