Application of Meshless Local Petrov-Galerkin (MLPG) to Problems with Singularities, and Material Discontinuities, in 3-D Elasticity

In this paper, a truly meshless method, the Meshless Local Petrov-Galerkin (MLPG) Method, is developed for three-dimensional elasto-statics. The two simplest members of MLPG family of methods, the MLPG type 5 and MLPG type 2, are combined, in order to reduce the computational requirements and to obtain high efficiency. The MLPG5 method is applied at the nodes inside the 3-D domain, so that any domain integration is eliminated altogether, if no body forces are involved. The MLPG 2 method is applied at the nodes that are on the boundaries, and on the interfaces of material discontinuities, so that the boundary conditions, and material discontinuities, are satisfied without any further tedious integration. Two classical three-dimensional elasto-statics problems, viz., the Boussinesq problem, and the Eshelby’s inclusion problem, are analyzed with the current method. The MLPG results agree excellently with the analytical solutions, and demonstrate their superiority over the results obtained from the traditional displacement-finite-element method. The MLPG method is straightforward, easy to implement, efficient, and accurate; therefore it holds a great promise to replace the finite element method in three-dimensional analysis in the near future. keyword: meshless method, Meshless Local PetrovGalerkin (MLPG) methods, local symmetric weak form, Moving Least Squares (MLS) interpolation