A truly-meshless galerkin method, through the MLPG mixed approach

Abstract

A truly meshless Galerkin method is formulated in the present study, as a special case of the general Meshless Local Petrov-Galerkin (MLPG) Mixed approach. The Galerkin method is implemented as a truly meshless method, for solving elasto-static problems. In the present Galerkin method, the test function is chosen to be the same as the trial function, as a special case of the MLPG approach. However, the MLPG local weak form is written over a local sub-domain which is completely independent from the trial or test functions. Even though in the present Galerkin approach, the trial and test functions are the same, the present MLPG approach (wherein the support sizes of the nodal trial and test function domains, as well as the size of the local subdomain over which the local weak-form is considered, can be arbitrary) may lead to either symmetric or unsymmetric "stiffness" matrices. These matrices are sparse and are well-conditioned. The present MLPG Galerkin Mixed Method does not require any background meshes (or cells) for performing the numerical integration of the local weak-forms, and makes the present method to be truly meshless. In addition, the mixed approach is also used to interpolate the nodal values of strains independently from the nodal values of displacements. The present mixed approach eliminates the expensive process of directly differentiating the interpolations for displacements in the entire domain, to find the derivatives, such as strains and stresses. The present MLPG Galerkin Mixed Method is not plagued by the so-called LBB conditions, which are common in the Galerkin Mixed Finite Element Method. Numerical examples are included to demonstrate the advantages of the present method: i) the truly meshless implementation; ii) the simplicity of the mixed approach wherein lower-order polynomial basis and smaller support sizes can be used; and iii) higher accuracies and computational efficiencies, and iv) no LBB conditions.