Peer-Reviewed Article
Optimization Algorithm for Locating Computational Nodal Points in the Method of Fundamental Solutions to Improve Computational Accuracy in Geosciences Modeling
Authors
Demoes, Noah J., Bann, G.T, Wilkins, B.D, Grubaugh, K.E., Boucher, R., Hromadka II, T.V. AS-0020
Abstract
Using the Complex Variable Boundary Element Method (“CVBEM”) to model ideal fluid flow, a new algorithm is applied to an approximation method that reduces computational requirements while increasing matrix solution demands. Ideal fluid flow is examined by use of the algorithm with the CVBEM as a case study. Traditionally, the modeling nodes are placed on or close to the problem geometry boundary in a somewhat regular pattern. In the current paper, an algorithm is developed and demonstrated that optimizes node locations by examining the possible locations for nodes exterior of the problem domain and then measuring the computational accuracy of the corresponding approximation function with of the analysis approach to other similar problems in Geosciences is straight forward. A three-dimensional application towards modeling groundwater flow about a building foundation is examined as a case study. The methodology is gaining value within the Geosciences toolbox as experience with complex computational techniques continues to advance. The computational Method of Fundamental Solutions is also investigated with similar success.
Keywords: Method of Fundamental Solutions, Optimization Algorithm, Complex Variable Boundary Element Method Introduction
In this paper, the well-known three-dimensional source basis functions are selected for an approximation function. The three-dimensional potential function approximation (real variable) examined is
where the cj’s are constant real-valued coefficients determined by collocation of the approximation to candidate collocation
j’s are the usual non-zero radial distance measures between the nodal
point locations (Pj’s) and arbitrary point P(x,y,z). Other funda- mental basis functions may be used that satisfy the governing
partial differential equation (PDE) which in the current case, is the elliptic Laplace equation. Although variations on the PDE and additional sophistication may be readily included in the approximation, we only carry forward the basic formulation of the above equation. The focus of this paper is the description of the proposed nodal position optimization algorithm. As a case study, a three-dimensional brick-geometry problem domain is examined, representative of a high-rise building foundation element that is located in the midst of a highly urbanized area such as Los Angeles, California. The relevant soils are expansive clays and soil-water is abundant. At issue are the
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Both of the above potential functions are entire functions - ously on the test problem boundary.
Literature Review of MFS and BEM Nodal Point Positioning Techniques
The paper by C. S. Chen (2016) discusses a brief history of the Method of Fundamental Solutions (MFS) and the simplicity associated with this method that makes the method appealing
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solid-water pressures for purposes of designing dewatering system elements and protection against soil-water leakage in subterranean structures such as parking garages. In the usual three-dimensional coordinate system. The three dimen- the problem setting.
The 3D geometry under detailed analysis has dimensions (x, y, z) = (8, 4, 2). Several such elements are deployed in the building foundation, but only one such element is examined in this paper. For demonstration purposes, two three-dimension- al (3D) potential functions are examined as case situations.
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