A Meshless Local Petrov-Galerkin Method for Solving the Bending Problem of a Thin Plate

Meshless methods have been extensively popularized in literature in recent years, due to their flexibility in solving boundary value problems. The meshless local Petrov-Galerkin(MLPG) method for solving the bending problem of the thin plate is presented and discussed in the present paper. The method uses the moving least-squares approximation to interpolate the solution variables, and employs a local symmetric weak form. The present method is a truly meshless one as it does not need a mesh, either for the purpose of interpolation of the solution or for the integration of the energy. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions are enforced by the penalty method. Several numerical examples are presented to illustrate the implementation and performance of the present method. The numerical examples presented in the paper show that high accuracy can be achieved for arbitrary nodal distributions for clamped and simply-supported edge conditions. No post processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is of C2 type.