An algorithm for optimizing CVBEM and BEM nodal point locations

The Complex Variable Boundary Element Method or CVBEM is a numerical technique for approximating particular partial differential equations such as the Laplace or Poisson equations (which frequently occur in physics and engineering problems, among many other fields of study). The advantage in using the CVBEM over traditional domain methods such as finite difference or finite element based methods includes the properties that the resulting CVBEM approximation is a function: (i) defined throughout the entire plane, (ii) that is analytic throughout the problem domain and almost everywhere on the problem boundary and exterior of the problem domain union boundary; (iii) is composed of conjugate two-dimensional real variable functions that are both solutions to the Laplace equation and are orthogonal such as to provide the “flow net” of potential and stream functions, among many other features. In this paper, a procedure is advanced that locates CVBEM nodal point locations on and exterior of the problem boundary such that error in matching problem boundary conditions is reduced. That is, locating the nodal points is part of modeling optimization process, where nodes are not restricted to be located on the problem boundary (as is the typical case) but instead locations are optimized throughout the exterior of the problem domain as part of the modeling procedure. The presented procedure results in nodal locations that achieve considerable error reduction over the usual methods of placing nodes on the problem boundary such as at equally spaced locations or other such procedures. Because of the significant error reduction observed, the number of nodes needed in the model is significantly reduced. It is noted that similar results occur with the real variable boundary element method (or BEM).The CVBEM and relevant nodal location optimization algorithm is programmed to run on program Mathematica, which provides extensive internal modeling and output graphing capabilities, and considerable levels of computational accuracy. The Mathematica source code is provided.