Comparison of Current Complex Variable Boundary Element Method (CVBEM) Capabilities in Basis Functions, Node Positioning Algorithms (NPAs), and Coefficient Determination Methods

Abstract

CVBEM is a numerical method of solving boundary value problems that satisfy Laplace's Equation in two dimensions. Three key parameters that impact the computational error and functionality of CVBEM are the basis function, the positions of the modeling nodes, and the coefficient determination methodology. To demonstrate the importance of these parameters, a case study of 2D ideal fluid flow into a 90-degree bend and over a semicircular hump was conducted comparing models using original CVBEM, complex log, complex pole, and digamma function variants basis functions, using two different NPAs, NPA1 and NPA2, and using collocation and least squares methods to determine coefficients. Results indicate that the combination of the original CVBEM basis function, NPA2, and least squares results in an approximation with the least computational error. Moreover, least squares appear to enable stability in both NPAs regarding reduction of computational error due to taking advantage of all boundary data and more stable condition number growth. By exploring the interaction of the three main CVBEM parameters, this paper clarifies the unique impact they have on the modelling process and explicitly identifies a fourth parameter, collocation point placement, as being impactful on computational error.