Comparison of two algorithms for locating computational nodes in the complex variable boundary element method (CVBEM)

Wilkins, Bryce D.; Hromadka, T.V.; McInvale, Jackson

doi:10.2495/cmem-v8-n4-289-315

Cited by 3 publications

(14 citation statements)

References 19 publications

(29 reference statements)

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“…On the other hand, only a subset of the boundary data are used in the collocation approach. The refinement decision refers to the distinction between NPAs 1 and 2, as defined in [8]. When nodal position refinement is used, the algorithm is referred to as NPA2.…”

Section: The Collocation Approachmentioning

confidence: 99%

“…Additionally, Tikhonov regularization can improve the condition of the matrix A, which may lead to more numerically stable solutions for the coefficients. The improved conditioning is a consequence of pre-pending a scalar multiple of the identity matrix on top of the standard least squares matrix, as indicated in eqn (8). That is, the lower N B rows of the matrix equation in eqn (8) are exactly the same as in eqn ( 5), but the first 2n rows are a scalar multiple of the identity matrix.…”

Section: The Least Squares Adpproachmentioning

confidence: 99%

“…Discussion regarding the selection of basis functions for use with the CVBEM can be found in [5,6]. Algorithms for selecting the computational nodes and collocation points to be used in the CVBEM modeling process have been developed in [7,8]. The fourth implementation decision refers to the selection of a norm for measuring the error of the CVBEM approximation function.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…These terms each correspond to a source point (i.e. computational node) located in the exterior of the problem domain, and the placement of these source points is determined according to the node-positioning algorithms (NPAs) discussed in [7,8]. Furthermore, it is noted that in this implementation of the CVBEM, the candidate nodes are located strictly in the exterior of the problem domain.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Wilkins¹,

Hromadka²

2022

Int. J. CMEM

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Two approaches for formulating a computational Complex Variable Boundary Element Method (CVBEM) model are examined. In particular, this paper considers a collocation approach as well as a least squares approach. Both techniques are used to fit the CVBEM approximation function to given boundary conditions of benchmark boundary value problems (BVPs). Both modeling techniques provide satisfactory computational results, when applied to the demonstration problems, but differ in specific outcomes depending on the number of nodes used and the type of BVP being examined. Historically, the CVBEM has been implemented using the collocation approach. Therefore, the novelty of this work is in formulating the least squares approach and applying the least squares formulation to a Dirichlet BVP as well as a mixed BVP. This work does not claim that one technique should always be used over the other, but rather it seeks to demonstrate the viability of the least squares approach and assert that both techniques for determining the coefficients of the CVBEM approximation function should be considered during the modeling process.

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Section: The Collocation Approachmentioning

confidence: 99%

Section: The Least Squares Adpproachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Wilkins¹,

Hromadka²

2022

Int. J. CMEM

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

Ali,

Hromadka

2023

IJCMEM

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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.

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Stress Distribution in Cantilever Beams with Different Hole Shapes: A Numerical Analysis

Ali,

Najem,

Karash

et al. 2023

IJCMEM

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The main duty of engineers is to guarantee that structures are both erect and adhere to codes, which proves their outstanding functionality and economic viability. In today's elastic materials, the von Mises stress values have to be verified when examining fatigue or failure. In the domains of heavy lifting, robotics, mechanical and offshore engineering, oil and gas engineering, and civil engineering, the Von Mises criteria are among the most often used benchmarks for assessing productivity conditions. In this study, seven I-beams models will be built, the first model without holes and the other six models with holes in various shapes (square, triangular, circular, hexagonal, and rectangular). The ANSYS program will be used to solve it using the finite element method. For the upper surface of these models, equal loads will be applied. The findings demonstrate that the shear stress values for the seven models were less than the shear stress values of the metal, which came to (370MPa), in line with the theory of maximum shear stress. With a value of (62.7MPa), the second-best model was the best. One of the most important conclusions when comparing the values of von Mess stresses with the von Mess theory of stress is that the third model (with rectangular openings) performed better than the other models when compared to the first model because its value was the same in both models (370MPa). The seventh model (hexagonal holes) had the lowest maximum value of stress intensity at 261MPa, per the results. being aware that this model weighs (70Kg) less than the first.

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Comparison of two algorithms for locating computational nodes in the complex variable boundary element method (CVBEM)

Cited by 3 publications

References 19 publications

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

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

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

Stress Distribution in Cantilever Beams with Different Hole Shapes: A Numerical Analysis

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