Numerical Implementation of Meshless Methods for Beam Problems

Abstract

For solving a partial different equation by a numerical method, a possible alternative may be either to use a mesh method or a meshless method. A flexible computational procedure for solving 1D linear elastic beam problems is presented that currently uses two forms of approximation function (moving least squares and kernel approximation functions) and two types of formulations, namely the weak form and collocation technique, respectively. The approximations functions constructed in continuous or in discrete way are used as approximations of the strong forms of partial differential equations (PDEs), and those serving as approximations of the weak forms of PDEs to set up a linear system of equations to reproduce Element Free Galerkin (EFG) and Smooth Particle Hydrodynamics (SPH) meshless methods. To approximate the strong form of a PDE, the partial differential equation is usually discretized by specific collocation technique. The SPH is a representative method for the strong form collocation approach. To approximate the weak form of a PDE, Galerkin weak formulation is used.

Keywords

• numerical methods
• meshless formulation
• EFG
• SPH
• beam discretisation