Due to the very general nature of the Meshless Local Petrov-Galerkin (MLPG) method, it is very easy and natural to introduce the upwinding concept (even in multidimensional cases) in the MLPG method, in order to deal with convection-dominated flows. In this paper, several upwinding schemes are proposed, and applied to solve steady convectiondiffusion problems, in one and two dimensions. Even fo...

Two meshless local Petrov-Galerkin (MLPG) methods based on two different trial functions but that use a simple linear test function were developed for beam and column problems. These methods used generalized moving least squares (GMLS) and radial basis (RB) interpolation functions as trial functions. These two methods were tested on various patch test problems. Both methods passed the patch tes...

The truly Meshless Local Petrov-Galerkin (MLPG) method is extended to solve the incompressible Navier-Stokes equations. The local weak form is modified in a very careful way so as to ovecome the so-called Babus̃ka-Brezzi conditions. In addition, The upwinding scheme as developed in Lin and Atluri (2000a) and Lin and Atluri (2000b) is used to stabilize the convection operator in the streamline di...

Discussed is here the multipoint meshless finite difference method (MMFDM) following the original Collatz [2] multipoint concept, and the essential ideas of the meshless FDM [3]. The Collatz approach was based on interpolation, regular meshes and the local formulation of b.v. problems. On the other hand in the MFDM we deal with the moving weighted least squares (MWLS) approximation, arbitrarily...

The general Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement & traction boundary integral equations are presented, for solids undergoing small deformations. Using the directly derived non-hyper singular integral equations for displacement gradients, simple and straight-forward derivations of weakly singular traction BIE's for solids undergoing small deformations are als...

This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions. In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary conditio...

The Rajendran-Grove (RG) ceramic damage model is a three-dimensional internal variable based constitutive model for ceramic materials, with the considerations of micro-crack extension and void collapse. In the present paper, the RG ceramic model is implemented into the newly developed computational framework based on the Meshless Local Petrov-Galerkin (MLPG) method, for solving high-speed impac...

This paper presents the implementation of a three-dimensional dynamic code, for contact, impact, and penetration mechanics, based on the Meshless Local Petrov-Galerkin (MLPG) approach. In the current implementation, both velocities and velocity-gradients are interpolated independently, and their compatibility is enforced only at nodal points. As a result, the time consuming differentiations of ...