Multi-Resolution Finite Wavelet Domain Method for Fast Transient Dynamic Analysis in Homogeneous and Heterogeneous Rods and Beams

Abstract

A Multi-resolution wavelet-based numerical method is developed for the fast prediction of transient response in elastic homogeneous and heterogeneous rods and beams. The method takes advantage of the remarkable mathematical properties of Daubechies wavelet and scaling functions as basis functions for the spatial approximation of state variables. The Multiresolution capability of the Daubechies wavelet family, provides the hierarchical computational framework that incorporates both scaling and wavelet functions. An uncoupled solution system between each resolution is formulated, using an explicit time integration scheme. The first level of analysis provides the coarse solution, while finer approximations are sequentially calculated and superimposed on the coarse solution, until the desired level of accuracy is achieved, without discarding the previous results obtained at coarser resolutions. Additionally, due to the orthogonality and compact support of Daubechies wavelet family, the decoupled mass matrices of each resolution are diagonal, or block diagonal and the stiffness matrices are banded. The proposed method uses a uniform grid which remains practically unchanged when increasing the order of interpolation (p-method), owing to its meshless character. Numerical results for the simulation of high-frequency wave propagation in isotropic and orthotropic rods and beams are presented and compared against confirmed models, demonstrating substantial reduction in computational effort. Furthermore, additional benefits of the proposed method are shown, such as the localization capabilities of the fine solutions, which exhibit high sensitivity in detecting discontinuities resulted from material inhomogeneity.

Enhanced Simulation of Guided Waves and Damage Localization in Composite Strips Using the Multiresolution Finite Wavelet Domain Method

“…The localization and isolation capabilities of the MR-FWD method are initially demonstrated in the simulation of guided waves in a damaged laminated beam modeled with Timoshenko wavelet-based beam elements. The formulation of the wavelet-based Timoshenko beam elements is described in [15]. The laminated beam is 1 m long, with a lay-up of [0/90/0/90] S carbon/epoxy plies, has a cross section of 10 −4 m 2 , is clamped at its left edge and transversely excited at its center by a 5-cycle tone burst with 50 kHz central frequency.…”

“…Also, to the best of our knowledge, none of the previous or the other well-established numerical methods have demonstrated inherent damage detection properties that can assist the effective design of SHM systems. Recently, the authors introduced the Multiresolution finite wavelet domain (MR-FWD) method for the rapid simulation of elastic wave propagation in rods, 2D plane strain solids [14] and Timoshenko beams [15]. The MR-FWD method exhibits critical computational gains by exploiting the non-converged solution to achieve convergence and evinces extra functionalities such as the ability to leave the mesh practically unchanged when increasing the order of interpolation (pmethod) because of its meshless character.…”