Boundary element modeling with variable nodal and collocation point locations

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: a b s t r a c t In both the real variable and Complex Variable Boundary Element Methods (CVBEM), nodal points are typically located on the problem boundary and then various techniques are used to fit boundary condition values at the nodal point locations such as collocation (equating approximation function to boundary condition values at a discrete set of locations on the boundary) or least squares minimization on the boundary, among others. In this paper, the CVBEM is used to examine the significant improvement in approximation accuracy achieved by using as additional approximation variables the actual nodal point locations (both on the problem boundary as well as exterior of the problem domain union boundary), and to also use as additional approximation variables the locations where boundary conditions are fitted (i.e. collocation points). The developed concepts also apply directly to the more commonly used real variable boundary element technique. Our results show that significant improvement in modeling accuracy is achieved by including the nodal point coordinates and also the collocation point coordinates as additional variables to be optimized. The real variable Boundary Element Method (BEM) and also the Complex Variable Boundary Element Method (CVBEM) are well documented and described in the literature (for example, for the CVBEM see Hromadka and Whitley [1] and Hromadka [2], and for the BEM see Brebbia [3]), and so the reader is referred to those publications, among others, for detailed background information into these numerical methods. In the current work, the CVBEM is focused upon assessing the advantages achieved by using a new approach towards improving approximation accuracy. This new approach, described for the CVBEM, would also apply to the well-known real variable BEM. Therefore, discussions regarding the CVBEM applications would similarly apply to the case of the BEM. The new approach being presented is an extension of Dean and Hromadka [4]. In Dean and Hromadka [4], nodal point locations used in the CVBEM approximation function are included as variables to be optimized in minimizing approximation error in matching problem boundary conditions at collocation points that are located on the problem boundary. In other words, the general approach in boundary element methods is to fix …