## 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, to reproduce Element Free Galerkin (EFG) and Smooth Particle Hydrodynamics (SPH) meshless methods. The numerical implementation for beam problems of these two formulations is discussed and numerical tests are presented to illustrate the difference between the formulations.

## Extended 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.

Numerical applications for beam problems are obtained by implementing various quadrature techniques to perform the integrations of the system equations in EFG method or for discretize the continuous form of the displacement of the SPH. Numerical comparison between these two formulations performed in the aim to illustrate the difference between the formulations.

## Comments (0)