Comparisons of two meshfree local point interpolation methods for structural analyses

Abstract

As truly meshless methods, the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are examined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical implementations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formulations are developed for structural analyses of 2-D elastodynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics.

The basis of meshless domain discretization: the meshless local Petrov?Galerkin (MLPG) method

“…If, in addition, we take N = m in local LSQ method, then one can get the so-called local point interpolation [33]. Obviously, this method inherits all the shortcomings of local LSQ scheme due to the same truncated locality, listed in the proceeding paragraph.…”

“…Moreover, for an arbitrarily chosen set of scattered nodes, special techniques should be used to assure a successful computation of shape functions (to avoid the matrix P to be singular). If the basis functions are taken to be globally supported radial basis functions, then one can get the so-called Local Radial Point Interpolation (LRPIM) [33]. This method is stable and flexible compared to LPIM.…”

“…Obviously, this method inherits all the shortcomings of local LSQ scheme due to the same truncated locality, listed in the proceeding paragraph. In [33], the basis functions are taken to be monomial; and thus, as a matter of fact, it is not a new method: it is just a FEM type interpolation. Here, we list the procedure.…”

An implicit RBF meshless approach for time fractional diffusion equations

“…The so-called meshless methods, using various global weak-forms, were proposed about twenty years ago. This category of meshless methods includes the element-free Galerkin (EFG) method [20], the reproducing kernel particle method (RKPM) [21], and the point interpolation method (PIM) [22,23], the meshless local Petrov-Galerkin (MLPG) method [24], the local radial point interpolation method (LRPIM) [25,26], the boundary node method (BNM) [27], and the boundary point interpolation method (BPIM) [28,29].…”

Solución del problema de Kirsch mediante elementos libres de malla, utilizando funciones de interpolación de base radial

“…El método de Free Galerkin (EFG) propuesto por Lu et al en 1994 [16], el método de reproducción de partículas (RKMP) propuesto por Jun et al [17], el método de los puntos de interpolación propuestos por Liu et al [18], el método sin malla de Petrov-Galerkin (MLPG) propuesto por Atluri et al [19], el método de nodos de frontera (BNM) expuestos por Mukherjee [20], el método de interpolación de puntos límites (BPIM) por Liu y Gu [21], [22], MeshFree fuertes y débiles (MWS) propuestos por Liu y Gu [23]. Donde las funciones de aproximación se construyen mediante un conjunto de nodos arbitrarios, y ningún elemento o conectividad de los nodos se hace necesario para la aproximación de dichas funciones.…”