Ali Reza Motamedi, Nima Noormohammadi, Bijan Boroomand

Computers & Mathematics with Applications

Abstract

A novel meshfree local method is developed in the context of Trefftz approaches to solve eigenvalue problems related to buckling and free vibration of arbitrarily shaped thin plates, potentially having complicated cutouts. The formulation is based on Kirchhoff's assumptions for thin plates with C1 continuity of the solution function. The mesh grid is replaced by a set of nodal points containing the degrees of freedom, located over the domain and its boundaries, thus removing the need for heavy mesh generation process. A sub-domain (cloud) centered at each node and containing several adjacent nodes is considered, in which the approximation functions are defined. The bases are made from Exponential Basis Functions (EBFs), so as to automatically satisfy part of the governing PDE and thus, removing the numerical integration procedure and increasing the solution accuracy. Due to the existence of indefinite values as the critical buckling load or the free vibration frequency, Trefftz approaches have difficulties to directly apply for eigenvalue problems. To resolve this, first the homogeneous eigenvalue problem converts into a non-homogeneous problem by moving the parts with indefinite eigenvalues to the right hand side of the equality and keeping the rest on the left. The solution to the whole eigenvalue problem is approximated using a series of EBFs. Then the solution to the homogeneous part is interpolated using another series of EBFs capable of exactly satisfying the homogeneous part of the newly defined PDE, while the particular solution is approximated using the eigenvalue solution series defined at the first step. Numerical results reveal that the proposed method can efficiently produce results with high accuracy.