The multiscale wavelet finite element method for structural dynamics

Abstract

The Wavelet Finite Element Method (WFEM) involves combining the versatile wavelet analysis with the classical Finite Element Method (FEM) by utilizing the wavelet scaling functions as interpolating functions; providing an alternative to the conventional polynomial interpolation functions used in classical FEM. Wavelet analysis as a tool applied in WFEM has grown in popularity over the past decade and a half and the WFEM has demonstrated potential prowess to overcome some difficulties and limitations of FEM. This is particular for problems with regions of the solution domain where the gradient of the field variables are expected to vary fast or suddenly, leading to higher computational costs and/or inaccurate results. The properties of some of the various wavelet families such as compact support, multiresolution analysis (MRA), vanishing moments and the “two-scale” relations, make the use of wavelets in WFEM advantageous, particularly in the analysis of problems with strong nonlinearities, singularities and material property variations present. The wavelet based finite elements (WFEs) presented in this study, conceptually based on previous works, are constructed using the Daubechies and B-spline wavelet on the interval (BSWI) wavelet families. These two wavelet families possess the desired properties of multiresolution, compact support, the “two scale” relations and vanishing moments. The rod, beam and planar bar WFEs are used to study structural static and dynamic problems (moving load) via numerical examples. The dynamic analysis of functionally graded materials (FGMs) is further carried out through a new modified wavelet based finite element formulation using the Daubechies and BSWI wavelets, tailored for such classes of composite materials that have their properties varying spatially. Consequently, a modified algorithm of the multiscale Daubechies connection coefficients used in the formulation of the FGM elemental matrices and load vectors in wavelet space is presented and implemented in the formulation of the WFEs. The approach allows for the computation of the integral of the products of the Daubechies functions, and/or their derivatives, for different Daubechies function orders. The effects of varying the material distribution of a functionally graded (FG) beam on the natural frequency and dynamic response when subjected to a moving load for different velocity profiles are analysed. The dynamic responses of a FG beam resting on a viscoelastic foundation are also analysed for different material distributions, velocity and viscous damping profiles. The approximate solutions of the WFEM converge to the exact solution when the order and/or multiresolution scale of the WFE are increased. The results demonstrate that the Daubechies and B-spline based WFE solutions are highly accurate and require less number of elements than FEM due to the multiresolution property of WFEM. Furthermore, the applied moving load velocities and viscous damping influence the effects of varying the material distribution of FG beams on the dynamic response. Additional aspects of WFEM such as, the effect of altering the layout of the WFE and selection of the order of wavelet families to analyse static problems, are also presented in this study.