The Boundary Node Method for Potential Problems

Abstract

SUMMARYThe Element-Free Galerkin (EFG) method allows one to use a nodal data structure (usually with an underlying cell structure) within the domain of a body of arbitrary shape. The usual EFG combines Moving Least-Squares (MLS) interpolants with a variational principle (weak form) and has been used to solve two-dimensional (2-D) boundary value problems in mechanics such as in potential theory, elasticity and fracture. This paper proposes a combination of MLS interpolants with Boundary Integral Equations (BIE) in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter! This new method, called the Boundary Node Method (BNM), only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself. An underlying cell structure is again used for numerical integration. In principle, the BNM, for 3-D problems, should be extremely powerful since one would only need to put nodes (points) on the surface of a solid model for an object.Numerical results are presented in this paper for the solution of Laplace's equation in 2-D. Dirichlet, Neumann and mixed problems have been solved, some on bodies with piecewise straight and others with curved boundaries. Results from these numerical examples are extremely encouraging.

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.…”

New Implementation Method for Essential Boundary Condition to Extended Element-Free Galerkin Method: Application to Nonlinear Problem

“…In order to resolve the above difficulty, many meshless approaches have been proposed [1][2][3]. In the approaches, elements of a geometrical structure are no longer necessary and, hence, the preparation of input data is extremely simplified.…”

Application of Two Dimensional Extended Boundary Node Method to Potential Problem

“…In order to resolve the above demerit, Mukherjee et al have proposed the boundary-node method (BNM) [1]. Since the BNM is one of the meshless approaches, a preparation of input data can be extremely simplified.…”