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Article
Publication date: 14 December 2020

Alia Al-Ghosoun, Ashraf S. Osman and Mohammed Seaid

The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear…

Abstract

Purpose

The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity.

Design/methodology/approach

The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics.

Findings

The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods.

Originality/value

Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

Details

Engineering Computations, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 27 March 2008

P.J. Coelho and D. Aelenei

This paper sets out to implement bounded high‐order (HO) resolution schemes in a hybrid finite volume/finite element method for the solution of the radiative transfer equation.

Abstract

Purpose

This paper sets out to implement bounded high‐order (HO) resolution schemes in a hybrid finite volume/finite element method for the solution of the radiative transfer equation.

Design/methodology/approach

The hybrid finite volume/finite element method had formerly been developed using the step scheme, which is only first‐order accurate, for the spatial discretization. Here, several bounded HO resolution schemes, namely the MINMOD, CLAM, MUSCL and SMART schemes, formulated using the normalized variable diagram, were implemented using the deferred correction procedure.

Findings

The results obtained reveal an interaction between spatial and angular discretization errors, and show that the HO resolution schemes yield improved accuracy over the step scheme if the angular discretization error is small.

Research limitations/implications

Although the HO resolution schemes reduce the spatial discretization error, they do not influence the angular discretization error. Therefore, the global error is only reduced if the angular discretization error is also small.

Practical implications

The use of HO resolution schemes is only effective if the angular refinement yields low‐angular discretization errors. Moreover, spatial and angular refinement should be carried out simultaneously.

Originality/value

The paper extends a methodology formerly developed in computational fluid dynamics, and aimed at the improvement of the solution accuracy, to the hybrid finite volume/finite element method for the solution of the radiative transfer equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 18 no. 2
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 6 July 2015

Di Zhao

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special…

Abstract

Purpose

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES.

Design/methodology/approach

In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

Findings

Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers.

Originality/value

Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

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Article
Publication date: 27 March 2008

Sanjay Kumar Khattri

This paper seeks to develop an adaptive finite volume algorithm, and to present an extensive numerical analysis of it.

Abstract

Purpose

This paper seeks to develop an adaptive finite volume algorithm, and to present an extensive numerical analysis of it.

Design/methodology/approach

The effectiveness of the developed algorithm is demonstrated through practical and computationally challenging problems. The algorithm is tested for a wide range of singularities.

Findings

The convergence of the presented algorithm is independent of the regularity of the problems. It is shown that the our algorithm produces more accurate and well conditioned matrix systems.

Research limitations/implications

Though the presented algorithm works for extreme singularities on rectangular meshes, it may not be as efficient if the underlying meshes are distorted, and it may not converge. Further research is under way for including the multi‐point approximation technique into the algorithm.

Practical implications

Almost all reservoir simulators use the two‐point method, and this algorithm is based on this method. The algorithm can be easily incorporated into the reservoir simulators. The results show that such an implementation will greatly improve the computational efficiency of the simulators. The work is useful for computational scientists, and especially for the researchers in oil industries. The paper reports the numerical work with practical applications.

Originality/value

The paper develops an adaptive finite volume algorithm. It is shown that adaptive meshes represent the underlying problem more accurately, and matrix systems associated with adaptive meshes are easier to solve compared with matrix systems associated with uniform meshes.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 18 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 1 February 2003

Jayantha Pasdunkorale A. and Ian W. Turner

An existing two‐dimensional finite volume technique is modified by introducing a correction term to increase the accuracy of the method to second order. It is well known…

Abstract

An existing two‐dimensional finite volume technique is modified by introducing a correction term to increase the accuracy of the method to second order. It is well known that the accuracy of the finite volume method strongly depends on the order of the approximation of the flux term at the control volume (CV) faces. For highly orthotropic and anisotropic media, first order approximations produce inaccurate simulation results, which motivates the need for better estimates of the flux expression. In this article, a new approach to approximate the flux term at the CV face is presented. The discretisation involves a decomposition of the flux and an improved least squares approximation technique to calculate the derivatives of the dependent function on the CV faces for estimating both the cross diffusion term and a correction for the primary flux term. The advantage of this method is that any arbitrary unstructured mesh can be used to implement the technique without considering the shapes of the mesh elements. It was found that the numerical results well matched the available exact solution for a representative transport equation in highly orthotropic media and the benchmark solutions obtained on a fine mesh for anisotropic media. Previously proposed CV techniques are compared with the new method to highlight its accuracy for different unstructured meshes.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 4 April 2008

P.S.B. Zdanski, M. Vaz and G.R. Inácio

Numerical simulation of polymer injection processes has become increasingly common in mould design. In industry, such a task is accomplished mainly by using commercial…

Abstract

Purpose

Numerical simulation of polymer injection processes has become increasingly common in mould design. In industry, such a task is accomplished mainly by using commercial packages. Owing to the complexities inherent of this class of problems, most commercial codes attempt to combine realistic rheological descriptions with simplified numerical models. In spite of the apparent success, such approaches are not able to capture important aspects of the flow topology. The present work aims to describe a more elaborate mathematical model based on finite volumes which is able to provide both accurate solutions and further insights on the physics of the polymer flow.

Design/methodology/approach

The mathematical model comprises the momentum and energy equations and a Poisson equation for pressure to impose the incompressibility constraint. The governing equations are discretized using the finite volume method based on central, second‐order accurate formulas for both convection and diffusion terms. Artificial dissipation terms are added externally in order to control the odd‐even decoupling problem.

Findings

The numerical model was conceived within the framework of a generalized Newtonian formulation. The capability of the numerical scheme is illustrated by simulations using three distinct constitutive relations to approach the non‐Newtonian behaviour of the polymer melt: isothermal power‐law, modified Arrhenius power‐law and cross models.

Originality/value

This paper extends the computational strategies previously developed to Newtonian fluids to account for more complex constitutive relations. The velocity and temperature coupled solution for polymer melts using only second‐order accurate formulas constitute also a relevant contribution.

Details

Engineering Computations, vol. 25 no. 3
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 July 1996

V. PENNATI, M. MARELLI and L.M. DE BIASE

In this paper new cubic v‐splines monotonic one‐dimensional profiles are presented, for the finite volume solution of convection‐diffusion problems. By studying the…

Abstract

In this paper new cubic v‐splines monotonic one‐dimensional profiles are presented, for the finite volume solution of convection‐diffusion problems. By studying the profile in normalized variables, some weight functions have been determined for the profile. Being free of the requirement that the volumes be equal, the volume size can be reduced where needed. Numerical properties of the proposed method were formally analysed and are confirmed by numerical examples included here.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 6 no. 7
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 1 June 1999

P.J. Coelho and J. Gonçalves

The finite volume method for radiative heat transfer calculations has been parallelized using two strategies, the angular domain decomposition and the spatial domain…

Abstract

The finite volume method for radiative heat transfer calculations has been parallelized using two strategies, the angular domain decomposition and the spatial domain decomposition. In the first case each processor performs the calculations for the whole domain and for a subset of control angles, while in the second case each processor deals with all the control angles but only treats a spatial subdomain. The method is applied to three‐dimensional rectangular enclosures containing a grey emitting‐absorbing medium. The results obtained show that the number of iterations required to achieve convergence is independent of the number of processors in the angular decomposition strategy, but increases with the number of processors in the domain decomposition method. As a consequence, higher parallel efficiencies are obtained in the first case. The influence of the angular discretization, grid size and absorption coefficient of the medium on the parallel performance is also investigated.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 9 no. 4
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 11 February 2021

J.N. Reddy, Matthew Martinez and Praneeth Nampally

The purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of…

Abstract

Purpose

The purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of linear differential equations to the solution of nonlinear heat transfer and like problems in one and two dimensions.

Design/methodology/approach

In the DMCDM, a mesh of finite elements is used for the approximation of the variables and another mesh of control domains for the satisfaction of the governing equation. Both meshes fully cover the domain but the nodes of the finite element mesh are inside the mesh of control domains. The salient feature of the DMCDM is that the concept of duality (i.e. cause and effect) is used to impose boundary conditions. The method possesses some desirable attributes of the finite element method (FEM) and the finite volume method (FVM).

Findings

Numerical results show that he DMCDM is more accurate than the FVM for the same meshes used. Also, the DMCDM does not require the use of any ad hoc approaches that are routinely used in the FVM.

Originality/value

To the best of the authors’ knowledge, the idea presented in this work is original and novel that exploits the best features of the best competing methods (FEM and FVM). The concept of duality is used to apply gradient and mixed boundary conditions that FVM and its variant do not.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 6
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 5 May 2015

Martin Joseph Guillot and Steve C McCool

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume

Abstract

Purpose

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

Design/methodology/approach

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

Findings

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

Practical implications

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

Originality/value

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

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