Peer-Reviewed Article
Using a Node Positioning Algorithm to Improve Models of Groundwater Flow Based on Meshless Methods
Bryce D. Wilkins1, Theodore V. Hromadka II2, AS-0020, William Nevils3, Benjamin Siegel3, Prarabdha Yonzon3
1 Carnegie Mellon University 2 Distinguished Professor, United States Military Academy 3 Cadet, United States Military Academy
Abstract
Recently, the Computational Engineering Mathematics program at the United States Military Academy has made significant advancements in determining suitable locations of nodal points (or model sources) for use in meshless numerical methods for partial differential equations such as the Complex Variable Boundary Element Method (CVBEM). The latest node positioning algorithm (NPA) to emerge from the group was used to develop high-precision outcomes for several benchmark problems in potential flow. Such high-precision outcomes provide more confidence in the predictions obtained for various geoscience results. For example, groundwater flow models are of high importance to many areas of analysis in which a key point of investiga- tion is determination of the location of a source of contamination given a location of damage downgradient caused by accumulation of the contaminant. The improved computational accuracy afforded by the coupled CVBEM/NPA models buttresses the other forms of evidence typically used in performing such a forensics analysis including geochemical testing and dating of chemical compounds, among other techniques. In this paper, we apply the recent NPA to model a new set of challenging computational problems with the goal of identifying the likely source of a hypothetical contaminant among a set of candidate sources.
Keywords: contamination detection, potential flow, Complex Variable Boundary Element Method (CVBEM), node positioning algorithms (NPAs), mesh reduction methods, applied complex variables
Introduction
The Complex Variable Boundary Element Method (CVBEM) [1, 2] is a numerical procedure for approximating the solution of boundary value problems (BVPs) of the Laplace and related partial differential equations (PDEs) [3, 4]. In the early 2000’s, the CVBEM methodology was extended to modeling BVPs in three and higher spatial dimensions [5, 6]. Recent develop- ments have focused on developing algorithms for determin- ing suitable locations of collocation points and computational nodes for use in CVBEM models [7, 8, 9]. The CVBEM is related to other mesh reduction numerical methods for PDEs including the real variable boundary element methods [10, 11] and the Method of Fundamental Solutions (MFS) [12, 13]. Several studies have focused on the application of the CVBEM to groundwater modeling and contaminant transport, and this work builds upon these studies [14, 15, 16]. The CVBEM
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approximation function is constructed as a linear combination of complex variable basis functions that are each analytic
(1) where denotes a complex coefficient and is ana-
cj is a complex coefficient, it has two parts: a real part and an imaginary part. Thus, for n coefficients,
there are 2n unknown values that need to be determined dur- ing the CVBEM modeling process. These coefficients can be determined in a number of ways including by collocation or least squares minimization. When collocation is used, it is nec- essary to specify 2n collocation points on the problem boundary where the values of the boundary conditions are known and
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