A boundary face method for potential problems in three dimensions

Abstract

SUMMARYThis work presents a new implementation of the boundary node method (BNM) for numerical solution of Laplace's equation. By coupling the boundary integral equations and the moving least-squares (MLS) approximation, the BNM is a boundary-type meshless method. However, it still uses the standard elements for boundary integration and approximation of the geometry, thus loses the advantages of the meshless methods. In our implementation, here called the boundary face method, the boundary integration is performed on boundary faces, which are represented in parametric form exactly as the boundary representation data structure in solid modeling. The integrand quantities, such as the coordinates of Gauss integration points, Jacobian and out normal are calculated directly from the faces rather than from elements. In order to deal with thin structures, a mixed variable interpolation scheme of 1-D MLS and Lagrange Polynomial for long and narrow faces. An adaptive integration scheme for nearly singular integrals has been developed. Numerical examples show that our implementation can provide much more accurate results than the BNM, and keep reasonable accuracy in some extreme cases, such as very irregular distribution of nodes and thin shells.

An adaptive element subdivision technique for evaluating 3D weakly singular boundary integrals

“…There are some advantages of the proposed adaptive element subdivision technique. In this method, the sub-elements including the source point have better shape than that in conventional subdivision method [3], while the remaining subelements also have better shape due to the property of sphere and merging operation. The second is that more Gaussian points are shifted towards the source point and a large number of Gaussian points are avoided compared to the conventional subdivision method which will be mentioned in next section.…”

“…Usually, polar coordinate transformation is used to solve this problem [2]. Zhang proposed a new coordinate transformation denoted as   ,   transformation to deal with weakly singular integrals [3]. However, due to the reason of conventional polar coordinate transformation and   ,   transformation just by directly connecting the singular point with each vertex of element, shape of sub-elements will be poor when the singular point is located near the edge or in the edge etc., which will result in poor calculation accuracy.…”

A General 2D Meshless Interpolating Boundary Node Method Based on the Parameter Space

“…Meshless (mesh-free) methods have been developed in the past decades to reduce the required effort for mesh generation. Many meshless methods have been proposed thus far, including the smoothed particle hydrodynamics (SPH) method [6,7], the reproducing kernel particle methods (RKPM) [8], the hpclouds method [9], the element-free Galerkin method (EFG) [10], the meshless local Petrov-Galerkin (MLPG) approach [11], the boundary node method (BNM) [12][13][14], the boundary element-free method (BEFM) [15], the hybrid boundary node method (hybrid BNM) [16][17][18][19][20], the Galerkin boundary node method (GBNM) [21], the boundary face method (BFM) [22,23], and the boundary point interpolation method [24].…”

Multiple reciprocity boundary face method for transient heat conduction in functionally graded materials

“…To avoid the geometric errors between the geometric model and the analysis model, our method is implemented in the framework of boundary face method (BFM) program [14,15].…”