An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners

Abstract

The Boundary Node Method (BNM) [Int. J. Numer. Meth. Engng 40 (1997) 797] is a boundary-only meshfree method that combines the Moving Least squares (MLS) interpolation scheme with the standard Boundary Integral Equations (BIEs) for a boundary value problem. Curvilinear boundary co-ordinates were originally proposed and used in this method—for both two- [Int. J. Numer. Meth. Engng 40 (1997) 797] and three-dimensional [Boundary methods—elements contours and nodes, 2004] (2D and 3D) problems. Li and Aluru [Comput. Meth. Appl. Mech. Engng 191 (2002) 2337; Engng Anal. Bound. Elem. 27 (2003) 57] have recently proposed an elegant improvement to the BNM (called the Boundary Cloud Method, BCM) that allows the use of Cartesian co-ordinates. Their novel variable basis BCM [Engng Anal. Bound. Elem. 27 (2003) 57] has several advantages relative to the original BCM. It does, however, have a drawback in that continuous approximants are used for all boundary variables, even across corners. It is well known, for example, that the normal derivative of the potential function in potential theory, or the traction in linear elasticity, often suffers jump discontinuities across corners in 2D and across edges and corners in 3D problems. The present paper describes a further improvement to the BNM and the variable basis BCM. This new approach is called the Extended Boundary Node Method (EBNM). This method employs Cartesian co-ordinates with variable bases, together with appropriate approximants for the normal derivative across edges and corners that can model discontinuities in this variable. This new formulation is presented below in detail for 2D problems in potential theory. Numerical results from the EBNM, for 2D potential theory, are compared with those from the variable basis BCM for selected examples.