Moving Least Squares Differential Quadrature Based on Radial Basis Functions for the Vibration Analysis of Beams, Plates and Shells

The numerical stability and accuracy of strong form collocation approaches, such as the Differential Quadrature (DQ) method, depend on the choice of the basis functions and point collocations. DQ methods allow to approximate a partial or total derivative as a weighted linear sum of the unknown functional values. The calculation of these weighting coefficients is not trivial, because they can be not accurate when some selections are made. In fact, classic DQ method demonstrated ill-conditioned matrices when a large number of points was considered. Among all the different families of DQ methods, in the present work a Moving Least Squares DQ (MLSDQ) method based on Radial Basis Functions (RBFs) is carried out. In particular, RBFs are used as basis functions for the weighting coefficient calculation. Generally, DQ method needs structured grids to perform well. On the contrary MLSDQ method can achieve good results also using a random point collocation. All the unknown functional values are expressed through a MLS approximation. The proposed method is applied to one- and two-dimensional systems. In particular the accuracy, stability and reliability of this technique is presented for beams, plates and shells with variable radii of curvature. The effect of composite materials with anisotropic behavior will be underlines throughout several numerical applications compared to the open literature. Furthermore, the present implementation will be included into a strong form finite element framework for the investigation of structures when discontinuities and cracks are presented.