Taylor Meshless Method for bending and buckling of thin plates

Abstract

This paper introduces a new meshless method named Taylor Meshless Method (TMM) using Taylor series to deduce the shape functions. Next the problem is discretized by point-collocation only on the boundary and without integration. The discrete boundary problem is solved by least-squares method. In this talk, this method is applied to bending and buckling of isotropic and anisotropic plates. The shape functions are polynomials that coincide with harmonic polynomials in the case of Laplace equation. These polynomials are computed numerically by solving the PDE approximately in the sense of Taylor series. Of course, this leads to an error that decreases asymptotically with the degree. TMM can be considered as a Trefftz Method, but we search approximated solutions in the sense of Taylor series while Trefftz Method is generally based on exact solutions of the PDE. As a counterpart, one is able to build this complete family of approximated solutions, whatever be the studied equation. A strong reduction of the number of degrees of freedom is the main advantage of this class of discretization techniques. The main drawback is the ill-conditioning of the final matrix, what can limit the size of the solved problem, but it was established that TMM is able to solve large scale problems. In the case of nonlinear PDEs as Föppl-Von Karman plate models, one applies first a linearization technique as Newton iterative technique. Here we apply Asymptotic Numerical Method. Next the resulting linear equations have variable coefficients are they are solved by TMM. Several linear and nonlinear numerical results will be presented for isotropic and anisotropic laminated plates.