Multiscale modeling of beam and plates using customized second-generation wavelets

We have designed bicubic Hermite‐type finite‐element wavelets that decouple the multiresolution stiffness matrix obtained from the discretization of the biharmonic equation. The scale decoupling basis makes the stiffness matrix block diagonal and hence eliminates the coupling between scales. The scale‐decoupled system leads to an incremental procedure for systematic enrichment of the solution without the need for costly remeshing of the whole domain and recalculation of the solution. The solution is obtained by injection of finer‐scale wavelets at locations with high detail coefficients. We conducted some numerical experiments to demonstrate the customized wavelet‐based finite‐element method for the problem of bending of Euler’s beam and Kirchhoff’s plates; we also demonstrate the role of wavelets in resolving localized phenomena.