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Article

Malika Remaki

A new finite volume scheme to solve Maxwell’s equations is presented. The approach is based on a leapfrog time scheme and a centered flux formula. This method is well…

Abstract

A new finite volume scheme to solve Maxwell’s equations is presented. The approach is based on a leapfrog time scheme and a centered flux formula. This method is well suited for handling complex geometries, and therefore we can use unstructured grids. It is also able to capture the discontinuities of the electromagnetic fields through different media, without producing spurious oscillations. Owing to these properties, we can treat difficult problems, such a computing a scattered wave across complex objects. An analysis of the scheme is presented and numerical experiments are performed.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 3
Type: Research Article
ISSN: 0332-1649

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Article

Di Yang and Zhiming Gao

A finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl.…

Abstract

Purpose

A finite volume scheme for diffusion equations on non-rectangular meshes is proposed in [Deyuan Li, Hongshou Shui, Minjun Tang, J. Numer. Meth. Comput. Appl., 1(4)(1980)217–224 (in Chinese)], which is the so-called nine point scheme on structured quadrilateral meshes. The scheme has both cell-centered unknowns and vertex unknowns which are usually expressed as a linear weighted interpolation of the cell-centered unknowns. The critical factor to obtain the optimal accuracy for the scheme is the reconstruction of vertex unknowns. However, when the mesh deformation is severe or the diffusion tensor is discontinuous, the accuracy of the scheme is not satisfactory, and the author hope to improve this scheme.

Design/methodology/approach

The authors propose an explicit weighted vertex interpolation algorithm which allows arbitrary diffusion tensors and does not depend on the location of discontinuity. Both the derivation of the scheme and that of vertex reconstruction algorithm satisfy the linearity preserving criterion which requires that a discretization scheme should be exact on linear solutions. The vertex interpolation algorithm can be easily extended to 3 D case.

Findings

Numerical results show that it maintain optimal convergence rates for the solution and flux on 2 D and 3 D meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous.

Originality/value

This paper proposes a linearity preserving and explicit weighted vertex interpolation algorithm for cell-centered finite volume approximations of diffusion equations on general grids. The proposed finite volume scheme with the new interpolation algorithm allows arbitrary continuous or discontinuous diffusion tensors; the final scheme is applicable to arbitrary polygonal grids, which may have concave cells or degenerate ones with hanging nodes. The final scheme has second-order convergence rate for the approximate solution and higher than first-order accuracy for the flux on 2 D and 3 D meshes. The explicit weighted interpolation algorithm is easy to implement in three dimensions in case that the diffusion tensor is continuous or discontinuous.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 3
Type: Research Article
ISSN: 0961-5539

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Article

J.M.F. Trindade and J.C.F. Pereira

This paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured…

Abstract

Purpose

This paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured uniform grids. The main purpose of the paper is to assess the accuracy enhancement with the inclusion of a high‐order reconstruction of the point‐wise velocity field on a fourth‐order accurate numerical scheme for the solution of the unsteady incompressible Navier‐Stokes equations.

Design/methodology/approach

The present finite volume method uses a fractional time‐step for decoupling velocity and pressure. A Runge‐Kutta integration scheme is implemented for integrating the momentum equation along with a polynomial interpolation and Simpson formula for space‐integration. The formulation is based on step‐by‐step de‐averaging process applied to the velocity field.

Findings

The reconstruction of the point‐wise velocity field on a higher‐order basis is essential to obtain solutions that effectively stand for a fourth‐order approximation of the point‐wise one. Results are provided for the Taylor vortex decay problem and for co‐ and counter‐rotating vortices to assess the increase in accuracy promoted by the inclusion of the high‐order de‐averaging procedure.

Research limitations/implications

High‐order reconstruction of the point‐wise velocity field should be considered in high‐order finite volume methods for the solution of the unsteady incompressible form of the Navier‐Stokes equations on structured grids.

Practical implications

The inclusion of a high‐order reconstruction of the point‐wise velocity field is a simple and effective method of enhancing the accuracy of a finite volume code for the computational fluid dynamics analysis.

Originality/value

The paper develops an improved version of a fourth‐order accurate finite volume projection method with the inclusion of a high‐order reconstruction step.

Details

Engineering Computations, vol. 24 no. 8
Type: Research Article
ISSN: 0264-4401

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Article

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

Gang Peng, Zhiming Gao, Wenjing Yan and Xinlong Feng

This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository.

Abstract

Purpose

This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository.

Design/methodology/approach

The nonlinear two-point flux approximation is used to discretize the diffusion flux and has a fixed stencil. The cell-vertex unknowns are applied to define the auxiliary unknowns and can be interpolated by the cell-centered unknowns. The approximation of convection flux is based on the second-order upwind method with a slope limiter.

Findings

Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

Originality/value

A new positivity-preserving nonlinear finite volume scheme is proposed to simulate the far-field model used in the geological radioactive waste repository. Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 2
Type: Research Article
ISSN: 0961-5539

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Article

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

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Article

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

G.K. Despotis and S. Tsangaris

The extrudate swell phenomenon is analysed by solving, simultaneously,the Navier‐Stokes equations along with the continuity equation bymeans of a finite volume method. In…

Abstract

The extrudate swell phenomenon is analysed by solving, simultaneously, the Navier‐Stokes equations along with the continuity equation by means of a finite volume method. In this work, the planar jet flows of incompressible viscous Newtonian and power‐law fluids for Reynolds numbers as high as 75 are simulated. The method uses the velocity components and pressure as the primitive variables and employs an unstructured triangular grid and triangular or polygonal control volume for each separate variable. The numerical results show good agreement with previously reported experimental and numerical results. Shear thickening results in an increase in swelling ratio, while the introduction of surface tension results in a describes in swelling ratio.

Details

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

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Article

Ajay Vadakkepatt, Sanjay R. Mathur and Jayathi Y. Murthy

Topology optimization is a method used for developing optimized geometric designs by distributing material pixels in a given design space that maximizes a chosen quantity…

Abstract

Purpose

Topology optimization is a method used for developing optimized geometric designs by distributing material pixels in a given design space that maximizes a chosen quantity of interest (QoI) subject to constraints. The purpose of this study is to develop a problem-agnostic automatic differentiation (AD) framework to compute sensitivities of the QoI required for density distribution-based topology optimization in an unstructured co-located cell-centered finite volume framework. Using this AD framework, the authors develop and demonstrate the topology optimization procedure for multi-dimensional steady-state heat conduction problems.

Design/methodology/approach

Topology optimization is performed using the well-established solid isotropic material with penalization approach. The method of moving asymptotes, a gradient-based optimization algorithm, is used to perform the optimization. The sensitivities of the QoI with respect to design variables, required for optimization algorithm, are computed using a discrete adjoint method with a novel AD library named residual automatic partial differentiator (Rapid).

Findings

Topologies that maximize or minimize relevant quantities of interest in heat conduction applications are presented. The efficacy of the technique is demonstrated using a variety of realistic heat transfer applications in both two and three dimensions, in conjugate heat transfer problems with finite conductivity ratios and in non-rectangular/non-cuboidal domains.

Originality/value

In contrast to most published work which has either used finite element methods or Cartesian finite volume methods for transport applications, the topology optimization procedure is developed in a general unstructured finite volume framework. This permits topology optimization for flow and heat transfer applications in complex design domains such as those encountered in industry. In addition, the Rapid library is designed to provide a problem-agnostic pathway to automatically compute all required derivatives to machine accuracy. This obviates the necessity to write new code for finding sensitivities when new physics are added or new cost functions are considered and permits general-purpose implementations of topology optimization for complex industrial applications.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 28 no. 2
Type: Research Article
ISSN: 0961-5539

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Article

Paramjeet Singh, Santosh Kumar and Mehmet Emir Koksal

The purpose of this paper is to develop and apply a high-order numerical method based on finite volume approximation for quadratic integrate-and-fire (QIF) neuron model…

Abstract

Purpose

The purpose of this paper is to develop and apply a high-order numerical method based on finite volume approximation for quadratic integrate-and-fire (QIF) neuron model with the help of population density approach.

Design/methodology/approach

The authors apply the population density approach for the QIF neuron model to derive the governing equation. The resulting mathematical model cannot be solved with existing analytical or numerical techniques owing to the presence of delay and advance. The numerical scheme is based along the lines of approximation: spatial discretization is performed by weighted essentially non-oscillatory (WENO) finite volume approximation (FVM) and temporal discretization are performed by strong stability-preserving explicit Runge–Kutta (SSPERK) method. Compared with existing schemes of orders 2 and 3 from the literature, the proposed scheme is found to be more efficient and it produces accurate solutions with few grid cells. In addition to this, discontinuity is added in the application of the model equation to illustrate the high performance of the proposed scheme.

Findings

The developed scheme works nicely for the simulation of the resulting model equation. The authors discussed the role of inhibitory and excitatory parts in variation of neuronal firing. The validation of the designed scheme is measured by its comparison with existing schemes in the literature. The efficiency of the designed scheme is demonstrated via numerical simulations.

Practical implications

It is expected that the present study will be a useful tool to tackle the complex neuron model and related studies.

Originality/value

The novel aspect of this paper is the application of the numerical methods to study the modified version of leaky integrate-and-fire neuron based on a QIF neuron. The model of the current study is inspired from the base model given in Stein (1965) and modified version in Kadalbajoo and Sharma (2005) and Wang and Zhang (2014). The applicability was confirmed by taking some numerical examples.

Details

Engineering Computations, vol. 36 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

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