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Wall distance calculation using the Eikonal/Hamilton‐Jacobi equations on unstructured meshes A finite element approach

Wall distance calculation using the Eikonal/Hamilton‐Jacobi equations on unstructured meshes A... Purpose – The purpose of this paper is to numerically solve Eikonal and Hamilton‐Jacobi equations using the finite element method; to use both explicit Taylor Galerkin (TG) and implicit methods to obtain shortest wall distances; to demonstrate the implemented methods on some realistic problems; and to use iterative generalized minimal residual method (GMRES) method in the solution of the equations. Design/methodology/approach – The finite element method along with both the explicit and implicit time discretisations is employed. Two different forms of governing equations are also employed in the solution. The Eikonal equation in its original form is used in the explicit Taylor Galerkin discretisation to save computational time. For implicit method, however, the convection‐diffusion form in its conservation form is used to maintain spatial stability. Findings – The finite element solution obtained is both accurate and smooth. As expected the implicit method is much faster than the explicit method. Though the proposed finite element solution procedures in serial is slower than the standard search procedure, they are suitable to be used in a parallel environment. Originality/value – The finite element procedure for Eikonal and Hamilton‐Jacobi equations are attempted for the first time. Though the finite volume and finite difference‐based computational fluid dynamics (CFD) solvers have started employing differential equations for wall distance calculations, it is not common for finite element solvers to use such wall distance calculations. The results presented here clearly show that the proposed methods are suitable for unstructured meshes and finite element solvers. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Engineering Computations Emerald Publishing

Wall distance calculation using the Eikonal/Hamilton‐Jacobi equations on unstructured meshes A finite element approach

Engineering Computations , Volume 27 (5): 13 – Jul 20, 2010

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Publisher
Emerald Publishing
Copyright
Copyright © 2010 Emerald Group Publishing Limited. All rights reserved.
ISSN
0264-4401
DOI
10.1108/02644401011050921
Publisher site
See Article on Publisher Site

Abstract

Purpose – The purpose of this paper is to numerically solve Eikonal and Hamilton‐Jacobi equations using the finite element method; to use both explicit Taylor Galerkin (TG) and implicit methods to obtain shortest wall distances; to demonstrate the implemented methods on some realistic problems; and to use iterative generalized minimal residual method (GMRES) method in the solution of the equations. Design/methodology/approach – The finite element method along with both the explicit and implicit time discretisations is employed. Two different forms of governing equations are also employed in the solution. The Eikonal equation in its original form is used in the explicit Taylor Galerkin discretisation to save computational time. For implicit method, however, the convection‐diffusion form in its conservation form is used to maintain spatial stability. Findings – The finite element solution obtained is both accurate and smooth. As expected the implicit method is much faster than the explicit method. Though the proposed finite element solution procedures in serial is slower than the standard search procedure, they are suitable to be used in a parallel environment. Originality/value – The finite element procedure for Eikonal and Hamilton‐Jacobi equations are attempted for the first time. Though the finite volume and finite difference‐based computational fluid dynamics (CFD) solvers have started employing differential equations for wall distance calculations, it is not common for finite element solvers to use such wall distance calculations. The results presented here clearly show that the proposed methods are suitable for unstructured meshes and finite element solvers.

Journal

Engineering ComputationsEmerald Publishing

Published: Jul 20, 2010

Keywords: Finite element analysis; Iterative methods; Differential equations; Meshes

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