MATLAB Implementation of Meshless Local Petrov-Galerkin (MLPG) Method

The Meshless Local Petrov-Galerkin (MLPG) method is a meshless numerical computation technique based on local weak forms, particularly suitable for complex geometries and moving boundary problems. Its core innovation lies in eliminating traditional mesh constraints by achieving numerical solutions through node distribution and local integration. The Moving Least Squares (MLS) method serves as the mathematical foundation for constructing shape functions. MLS approximates field functions through local support domains around nodes, offering high-order continuity and arbitrary precision. A critical implementation aspect is the selection of weight functions, typically employing Gaussian or spline-type functions to control the influence range of local approximations. In MATLAB code, this involves defining support domain radii and implementing weight function calculations using vectorized operations for efficiency. Gauss quadrature is employed in MLPG for numerical integration over local domains. Since meshless methods lack explicit grids, shape functions and their derivatives must be recalculated at each integration point. For 2D implementations, square or circular integration domains are commonly used, with Gauss point quantities selected based on accuracy requirements. Code implementation requires careful management of integration point coordinates and corresponding weight factors through nested loops or array operations. The MLPG implementation workflow primarily includes: - Node distribution and support domain definition (using spatial indexing algorithms for neighbor search) - MLS shape function construction (matrix operations for solving least squares problems) - Local integration domain generation - System matrix assembly (sparse matrix techniques for efficient storage) - Boundary condition handling (penalty method or Lagrange multipliers) - Linear system solution (using MATLAB's built-in solvers like backslash operator or iterative methods) This method avoids the mesh generation difficulties inherent in traditional finite element analysis, making it especially suitable for problems involving large deformations and crack propagation. However, it requires substantial computational resources, necessitating optimization of support domain sizes and integration schemes through parameter studies and adaptive algorithms.