35 years of advancements with the complex variable boundary element method

Abstract

The Complex Variable Boundary Element Method, or CVBEM, was first published in Journal of Numerical Methods in Engineering in year 1984 by authors Hromadka and Guymon [1]. Since that time, several papers and books have been published that present various aspects of the numerical technique as well as advances in the computational method such as extension to three or higher dimensions for arbitrary geometries, nonhomogeneous domains, extension to use of a Hilbert Space setting as well as collocation methods, inclusion of the time derivative via coupling to generalized Fourier series techniques, examination of various families of basis functions including complex monomials, the product of complex polynomials with complex logarithm functions (i.e., the usual CVBEM basis functions), Laurent series expansions, reciprocal of complex monomials, other complex variable analytic functions including exponential and others, as well as linear combinations of these families. Other topics studied and developed include rotation of complex logarithm branch-cuts for extension of the problem computational domain to the exterior of the problem geometry, depiction of computational error in achieving problem boundary conditions by means of the approximate boundary technique, mixed boundary value problems, flow net development and visualization, display of flow field trajectory vectors in two and three dimensions for use in depicting streamlines and flow paths, among other topics. The CVBEM approach has also been extended to solving partial differential equations such as Laplace's equation, Poisson's equation, unsteady flow equation, and the wave equation, among other formulations that include sources, sinks and combinations of these equations with mixed boundary conditions. In the current paper, a detailed examination is made of the performance between four families of basis functions in order to assess computational efficiency in problem solving of two dimensional potential problems in a high aspect ratio geometric problem domain. Two selected problems are presented as case studies to demonstrate the different levels of success by each of the four families of examined basis functions. All four families involve basis functions that solve the governing partial differential equation, leaving only the goodness of fit in matching boundary conditions of the boundary value problem as the computational optimization goal. The modeling technique is implemented in computer programs Mathematica and MATLAB. Recommendations are made for future research directions and lessons learned from the current study effort.

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

“…where 𝑐 𝑗 ⊂ ℂ is a complex coefficient, 𝑔 𝑗 (𝑧) is a complex variable function that is analytic within Ω and harmonic, and 𝑛 is the number of basis functions, or nodes, used in the approximation function [4,5]. In contrast with real variable methods that utilize real coefficients, the complex coefficients consist of a real and imaginary part.…”

A least squares approach for determining the coefficients of the CVBEM approximation function

“…2 and 9 for problems 1 and 2, respectively. In particular, the black dots are the candidate nodes, whose locations correspond to the singularities of the basis functions defined in eqn (9), which are optimized using the NPA. The blue dots are the boundary data at which the boundary conditions are known.…”

“…In most of the recent works related to the CVBEM, collocation has been used to determine the coefficients of the approximation function [4][5][6][7][8][9][10][11]. Since the collocation approach is well-documented in the aforementioned references, the focus of this paper is on developing the least squares approach.…”