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Article
Publication date: 1 August 2001

Jaroslav Mackerle

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography…

1635

Abstract

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.

Details

Engineering Computations, vol. 18 no. 5/6
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 1 January 2005

S. D'Heedene, K. Amaratunga and J. Castrillón‐Candás

This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.

Abstract

Purpose

This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.

Design/methodology/approach

Second‐generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities.

Findings

For the 1D Poisson problem, we propose – for any given order of approximation – a compact closed‐form wavelet basis that block‐diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher‐order (n>1) HB do not exhibit this property. We also achieve full scale‐decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D.

Research limitations/implications

Similar techniques may be applied to scale‐decouple the multiresolution finite element (FE) matrices associated with more general elliptic PDEs.

Practical implications

By decoupling scales in the FE matrix, the wavelet formulation lends itself particularly well to adaptive refinement schemes.

Originality/value

The paper explains second‐generation wavelet construction in a Lagrangian FE context. For 1D higher‐order and 2D first‐order bases, we propose a particular choice of wavelet, customized to the Poisson problem. The approach generalizes to other elliptic PDE problems.

Details

Engineering Computations, vol. 22 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 8 June 2012

Syed Tauseef Mohyud‐Din, Elham Negahdary and Muhammad Usman

The purpose of this paper is to present a numerical solution of a family of generalized fifth‐order Korteweg‐de Vries equations using a meshless method of lines. This…

Abstract

Purpose

The purpose of this paper is to present a numerical solution of a family of generalized fifth‐order Korteweg‐de Vries equations using a meshless method of lines. This method uses radial basis functions for spatial derivatives and Runge‐Kutta method as a time integrator and exhibits high accuracy as seen from the comparison with the exact solutions.

Design/methodology/approach

The study uses a meshless method of lines. This method uses radial basis functions for spatial derivatives and Runge‐Kutta method as a time integrator.

Findings

The paper reveals that this method exhibits high accuracy as seen from the comparison with the exact solutions.

Originality/value

This method is efficient method as it is easy to implement for the numerical solutions of PDEs.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 31 December 2021

Alexander Idesman and Bikash Dey

The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world…

Abstract

Purpose

The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry).

Design/methodology/approach

This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes.

Findings

The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry).

Originality/value

It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 20 February 2007

N. Mai‐Duy and R.I. Tanner

To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.

Abstract

Purpose

To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.

Design/methodology/approach

The proposed method is based on a Cartesian grid and a 1D integrated‐radial‐basis‐function scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.

Findings

The proposed method is verified by considering several test problems governed by second‐ and fourth‐order PDEs; very accurate solutions are achieved using relatively coarse grids.

Research limitations/implications

Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further, development is needed for the case of non‐rectangular domains.

Originality/value

The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 17 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 17 June 2019

Muhammad Raees Ul Haq, Hang Xu and Liang Zhao

The purpose of this study is to obtain the numerical as well as regularity results for the nonlinear elliptic set of equations arising in the study of fluid flow in…

Abstract

Purpose

The purpose of this study is to obtain the numerical as well as regularity results for the nonlinear elliptic set of equations arising in the study of fluid flow in microchannel induced by the pressure in the presence of interfacial electrokinetic effects.

Design/methodology/approach

For the numerical study, the authors implemented traditional FDM approach, and for the regularity results they used the classical energy estimates. The interfacial electrokinetic effects result in an additional source term in classical momentum equation, hence affecting the characteristics of the flow and heat transfer. The sinusoidal temperature variation is assumed on side walls.

Findings

The results were obtained for various combinations of physical parameters appearing in the governing equations. This study concludes that in the presence of electric double layer, the average heat transfer rate reduces along with larger values of Reynolds number. It is observed that the heat transfer increases with the increase in amplitude ratio and phase deviation. The flow behavior and heat transfer rate inside the microchannel are also strongly affected by the presence of κ (kappa).

Originality/value

To the best of the authors’ knowledge, the problem of heat transfer through microchannel in combination with sinusoidal temperature variation at boundary with electric double layer effects has not been considered previously. Hence, this paper focuses on the influence of the sinusoidal boundary temperature distributions on both sidewalls of a rectangular microchannel through parallel plates with electrokinetic effects on the pressure-driven laminar flow. In addition, a detailed mathematical analysis is also to be carried out to verify the regularity of this model with the proposed boundary conditions. The study used the classical energy method to get the regularity results.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 10
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 May 2001

Eduardo N. Dvorkin

Engineers have developed robust and efficient incompressible finite element formulations using tools such as the Patch Test and the counting of constraints/variables, the…

Abstract

Engineers have developed robust and efficient incompressible finite element formulations using tools such as the Patch Test and the counting of constraints/variables, the first one aimed at the development of consistent elements and the second one aimed at the development of non‐locking and stable elements. The mentioned tools are rooted in the physics of the continuum mechanics problem. Mathematicians, on the other side, developed complex and powerful tools to examine the convergence of finite element formulations, such as the inf‐sup condition, these methods are based on the properties of the elliptical PDEs that constitute the mathematical model of the continuum mechanics problem. In this paper we intend to understand the inf‐sup condition from an engineering perspective, so as to be able to incorporate it into the package of tools used in the development of finite element formulations.

Details

Engineering Computations, vol. 18 no. 3/4
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 17 November 2021

Mehdi Dehghan, Baharak Hooshyarfarzin and Mostafa Abbaszadeh

This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations…

Abstract

Purpose

This study aims to use the polynomial approximation method based on the Pascal polynomial basis for obtaining the numerical solutions of partial differential equations. Moreover, this method does not require establishing grids in the computational domain.

Design/methodology/approach

In this study, the authors present a meshfree method based on Pascal polynomial expansion for the numerical solution of the Sobolev equation. In general, Sobolev-type equations have several applications in physics and mechanical engineering.

Findings

The authors use the Crank-Nicolson scheme to discrete the time variable and the Pascal polynomial-based (PPB) method for discretizing the spatial variables. But it is clear that increasing the value of the final time or number of time steps, will bear a lot of costs during numerical simulations. An important purpose of this paper is to reduce the execution time for applying the PPB method. To reach this aim, the proper orthogonal decomposition technique has been combined with the PPB method.

Originality/value

The developed procedure is tested on various examples of one-dimensional, two-dimensional and three-dimensional versions of the governed equation on the rectangular and irregular domains to check its accuracy and validity.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 11 May 2022

Yanfei Lu, Futian Weng and Hongli Sun

This paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of…

Abstract

Purpose

This paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).

Design/methodology/approach

The proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.

Findings

To demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.

Originality/value

The H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

Details

Engineering Computations, vol. 39 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 1 March 2001

Bassam A/K Abu‐Hijleh

The problem of laminar cross‐flow forced convection heat transfer from a horizontal cylinder covered with an orthotropic porous layer was investigated numerically. The…

Abstract

The problem of laminar cross‐flow forced convection heat transfer from a horizontal cylinder covered with an orthotropic porous layer was investigated numerically. The effects of porous layer thickness, radial resistance, tangential resistance, and incoming flow Reynolds number on the average Nusselt number were studied in detail. There was up to 40 per cent reduction in the average Nusselt number at high values of Reynolds number. The tangential resistance effect on the Nusselt number was dominant over that of the radial resistance. The effectiveness of the porous layer increased at high values of porous layer thickness as well as at high values of Reynolds number.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 11 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

1 – 10 of 112