A New Implementation of the Meshless Finite Volume Method, Through the MLPG "Mixed" Approach

Abstract

The Meshless Finite Volume Method (MFVM) is developed for solving elasto-static problems, through a new Meshless Local Petrov-Galerkin (MLPG) approach. In this MLPG mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through lo- cal meshless interpolation schemes such as the mov- ing least squares(MLS) or radial basis functions(RBF). The nodal values of strains are expressed in terms of the independently interpolated nodal values of displace- ments, by simply enforcing the strain-displacement rela- tionships directly by collocation at the nodal points. The MLPG local weak form is then written for the equilib- rium equations over the local sub-domains, by using the nodal strains as the independent variables. By taking the Heaviside function as the test function, the local domain integration is avoided; this leads to a Meshless Finite Vol- ume Method, which is a counterpart to the mesh-based finite volume method that is popular in computational fluid dynamics. The present approach eliminates the ex- pensive process of directly differentiating the MLS inter- polations for displacements in the entire domain, to find the strains, especially in 3D cases. Numerical examples are included to demonstrate the advantages of the present methods: (i) lower-order polynomial basis can be used in the MLS interpolations; (ii) smaller support sizes can be used in the MLPG approach; and (iii) higher accuracies and computational efficiencies are obtained. keyword: Meshless Local Petrov-Galerkin approach (MLPG), Finite Volume Methods, Mixed Methods, Ra- dial Basis Functions (RBF), and Moving Least Squares (MLS).