An Unsteady Two-Dimensional Complex Variable Boundary Element Method

Wilkins, Bryce D.; Greenberg, Joshua; Redmond, Brittany; Baily, Alan I.; Flowerday, Nicholas; Kratch, Adam; Hromadka, T.V.; Boucher, Randy; McInvale, Howard; Horton, Steve

doi:10.4236/am.2017.86069

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…As previously mentioned, the steady-state problem is governed by the Laplace Recent research related to the CVBEM has extended its use to modeling problems in three and higher spatial dimensions [11] as well as problems related to the diffusion equation [12]. Current efforts have been dedicated to improving the efficiency of the three-dimensional CVBEM as well as to extending the CVBEM to model applications of the time-dependent three-dimensional diffusion equation.…”

Section: Steady-state Component and The Complex Variable Boundary Elementioning

confidence: 99%

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

Wilkins¹,

Hromadka²,

Boucher³

2017

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In this work, a conceptual numerical solution of the two-dimensional wave partial differential equation (PDE) is developed by coupling the Complex Variable Boundary Element Method (CVBEM) and a generalized Fourier series. The technique described in this work is suitable for modeling initial-boundary value problems governed by the wave equation on a rectangular domain with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The new numerical scheme is based on the standard approach of decomposing the global initial-boundary value problem into a steady-state component and a time-dependent component. The steady-state component is governed by the Laplace PDE and is modeled with the CVBEM. The time-dependent component is governed by the wave PDE and is modeled using a generalized Fourier series. The approximate global solution is the sum of the CVBEM and generalized Fourier series approximations. The boundary conditions of the steady-state component are specified as the boundary conditions from the global BVP. The boundary conditions of the time-dependent component are specified to be identically zero. The initial condition of the time-dependent component is calculated as the difference between the global initial condition and the CVBEM approximation of the steady-state solution. Additionally, the generalized Fourier series approximation of the time-dependent component is fitted so as to approximately satisfy the derivative of the initial condition. It is shown that the strong formulation of the wave PDE is satisfied by the superposed approximate solutions of the time-dependent and steady-state components.

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Section: Steady-state Component and The Complex Variable Boundary Elementioning

confidence: 99%

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

Wilkins¹,

Hromadka²,

Boucher³

2017

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“…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%

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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“…In this work, NPAs 1 and 2 are coupled with the Complex Variable Boundary Element Method (CVBEM), which is a numerical solver for the Laplace equation and related PDEs [9][10][11]. The coupled CVBEM and NPA programs will be used to solve two Dirichlet BVPs of the Laplace equation, which have been selected due to the computational difficulty of modeling potential flow in areas where there is extreme curvature in the flow regime.…”

Section: Introductionmentioning

confidence: 99%

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

Wilkins¹,

Hromadka²,

McInvale³

2020

Int. J. CMEM

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In this paper, we introduce a new node positioning algorithm (NPA) for determining suitable locations of the computational nodes that are a typical feature of mesh reduction numerical methods for partial differential equations-specifically, the Complex Variable Boundary Element Method (CVBEM). The novelty of the introduced NPA is a 'position refinement' procedure, which facilitates the relocation of nodes that are already being used in the current CVBEM model when such relocation reduces the maximum error of the associated CVBEM model. The results of the new NPA (referred to as NPA2) are compared to the results obtained using the recent NPA described in [1] (referred to as NPA1). We compare NPA1 and NPA2 by modeling two example Dirichlet boundary value problems that have been selected due to having regions of extreme curvature in the analytic flow regime that are difficult to model computationally. Consequently, these problems serve as good benchmark problems for testing the efficacy of the current and future NPAs. Our empirical findings suggest that the use of NPA2 can reduce the maximum error of the associated CVBEM model by several orders of magnitude compared to the corresponding result obtained using NPA1.

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An Unsteady Two-Dimensional Complex Variable Boundary Element Method

Cited by 5 publications

References 16 publications

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

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

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

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