Source Code Implementation of Meshless MLPG Method for Cantilever Beam under Concentrated Load

The meshless MLPG (Meshless Local Petrov-Galerkin) method is a numerical computation approach based on local Petrov-Galerkin weak formulations, particularly suited for solving mechanical problems under complex boundary conditions such as cantilever beam analysis under concentrated loads. This method eliminates the constraints of traditional finite element meshing by approximating field variables through node distributions and local support domains, making it especially suitable for large deformation or fracture simulation scenarios.

For implementing the MLPG method for cantilever beams under concentrated loads, the core computational steps typically include: First, constructing Moving Least Squares (MLS) shape functions for displacement field approximation, which operates independently of element topology structures. Code implementation would involve defining nodal influence domains and calculating MLS basis functions through matrix operations. Second, establishing weighted residual equations within local subdomains and performing numerical integration using Gaussian quadrature rules - this requires careful selection of integration points and weight functions in the source code. Finally, applying fixed-end displacement boundary conditions for the cantilever support and concentrated force loading at the free end, followed by iterative solution of the global equilibrium equations through linear system solvers.

Compared to finite element methods, this approach eliminates cumbersome mesh generation steps and naturally handles local high-gradient responses induced by concentrated loads. However, attention must be paid to the completeness of shape functions and numerical integration accuracy in the code implementation, as these are critical factors ensuring solution stability. The programming implementation should include proper validation checks for these parameters.