The extended boundary node method for three-dimensional potential theory

The boundary node method (BNM) [Mukherjee YX, Mukherjee S. The boundary node method for potential problems. Int J Numer Methods Eng 1997;40:797-815] is a boundary-only mesh-free method that combines the moving least-squares (MLS) interpolation scheme with the standard boundary integral equations (BIEs). Curvilinear boundary co-ordinates were originally proposed and used in this method-for both two [Mukherjee YX, Mukherjee S. The boundary node method for potential problems. Int J Numer Methods Eng 1997;40:797-815] and three-dimensional [Mukherjee S, Mukherjee YX. Boundary methods-elements, contours and nodes. Boca Raton, FL: CRC Press, in press] problems in potential theory and in linear elasticity. Li and Aluru [Li G, Aluru NR. Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis. Comput Methods Appl Mech Eng 2002;191:2337-70; Li G. Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Eng Anal Boundary Elem 2003;27:57-71] 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 [Li G. Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Eng Anal Boundary Elem 2003;27:57-71] 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 two-dimensional (2-D) and across edges and corners in three-dimensional (3-D) problems. The present authors [Telukunta S, Mukherjee S. An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners. Eng Anal Boundary Elem 2004;28:1099-110] have recently proposed a further improvement to the BNM and the variable basis BCM. This new approach is called the extended BNM (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. Two-dimensional problems in potential theory are presented in [Telukunta S, Mukherjee S. An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners. Eng Anal Boundary Elem 2004;28:1099-110]. The present paper is concerned with far more challenging problems-3-D problems in potential theory.