A novel weak-form meshless method based on Lagrange interpolation for mechanical analysis of complex thin plates

Abstract

In this paper, a novel weak-form meshless method, Galerkin Free Element Collocation Method (GFECM), is proposed for the mechanical analysis of thin plates. This method assimilates the benefits of establishing spatial partial derivatives by isoparametric elements and forming coefficient matrices node by node, which makes the calculation more convenient and stable. The pivotal aspect of GFECM is that the surrounding nodes can be freely chosen as collocation elements, which can adapt to irregular node distribution and suitable for complex models. Meanwhile, each collocation element is used as a Lagrange isoparametric element individually, which can easily construct high-order elements and improve the calculation accuracy, especially for high-order partial differential equations such as the Kirchhoff plate bending problem. In order to obtain the weak-form of the governing equation, the Galerkin form of the governing equation is constructed based on the virtual work principle and variational method. In addition, due to the Lagrange polynomials possessing the Kronecker delta property as shape functions, it can accurately impose boundary conditions compared with traditional meshless methods that use rational functions. Several numerical examples are proposed to verify the correctness and effectiveness of the proposed method in thin plate bending problems.