One-Dimensional Galerkin Meshless Method MATLAB Implementation

In numerical computation, meshing is a fundamental concept that divides the computational domain into small cells for analysis. However, meshless methods utilize point-based approximations to eliminate traditional meshes, thereby avoiding initial mesh generation and remeshing procedures. These methods not only preserve computational accuracy but also substantially reduce meshing difficulties. The one-dimensional Galerkin meshless MATLAB program represents such a meshless approach that has found extensive applications in numerical computations. The MATLAB implementation typically involves several key components: node generation using linspace or similar functions, shape function formulation through moving least squares (MLS) approximation, and numerical integration via Gaussian quadrature. The code structure generally includes modules for domain discretization, stiffness matrix assembly using Galerkin weak form, and boundary condition enforcement. However, current meshless approximations generally lack analytical expressions and are predominantly based on Galerkin principles, leading to computational demands that exceed traditional finite element methods. Furthermore, since meshless approximations are primarily fitting-based, handling displacement boundaries remains challenging, with Lagrange multiplier methods being commonly employed for boundary enforcement. In the MATLAB code, this often translates to additional constraint equations and expanded system matrices. Although these factors increase computational complexity, the advantages of meshless methods continue to make them worthy of investigation and application, particularly for problems involving large deformations or moving boundaries where traditional meshing becomes problematic.