Quasi-Monte Carlo mesh-free integration for meshless weak formulations

In meshless methods based on weak formulation, the governing integral equation is obtained from the weak form of elasticity over global or local domains. Many of the so-called meshless methods, such as the element-free Galerkin (EFG) method [Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. Int J Numer Methods Eng 1994;37:229–56], are based on the global weak form over the entire domain. The local weak forms are the basis of the meshless local Petrov–Galerkin (MLPG) [Atluri SN, Shen S. The meshless local Petrov–Galerkin (MLPG) method. Stuttgart: Tech Science Press; 2002] method. The integration of the resulting discrete set of algebraic equations are complex and require strategies using a large number of integration points, and may, therefore, be time-consuming. A possible improvement is the use of the Monte Carlo integration technique. The technique is straightforward and efficient, and is simpler and easier compared with other global or local integration methods. The method is implemented for MLPG and EFG methods. Numerical tests show the mesh-free integration method to be accurate and robust. Various Quasi-Monte Carlo sequences are performed and compared. These sequences are examined for three-dimensional elasticity problems. The numerical results prove the efficiency of the integration techniques for both meshless formulations tested.