Meshless Local Petrov-Galerkin (MLPG) approaches for solving 3D Problems in elasto-statics

Three different truly Meshless Local Petrov-Galerkin (MLPG) methods are developed for solving 3D elasto-static problems. Using the general MLPG concept, these methods are derived through the local weak forms of the equilibrium equations, by using different test functions, namely, the Heaviside function, the Dirac delta function, and the fundamental solutions. The one with the use of the fundamental solutions is based on the local unsymmetric weak form (LUSWF), which is equivalent to the local boundary integral equations (LBIE) of the elasto-statics. Simple formulations are derived for the LBIEs in which only weakly-singular integrals are included for a simple numerical implementation. A novel definition of the local 3D sub-domain is presented, which enables the numerical integrations to be performed in an accurate and efficient way, based on a truly meshless implementation. The augmented radial basis functions (RBF) and the moving least squares (MLS) are chosen to construct the shape functions, for the three MLPG methods. Numerical examples are included to demonstrate that the present methods are very promising for solving the elastic problems, as compared to the traditional Galerkin Finite Element Method. keyword: Meshless Local Petrov-Galerkin approach (MLPG), Local Boundary Integral Equations (LBIE), Radial Basis Functions (RBF), Moving Least Squares (MLS).