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Article
Publication date: 1 February 2006

A. Rap, L. Elliott, D.B. Ingham, D. Lesnic and X. Wen

To develop a numerical technique for solving the inverse source problem associated with the constant coefficients convection‐diffusion equation.

Abstract

Purpose

To develop a numerical technique for solving the inverse source problem associated with the constant coefficients convection‐diffusion equation.

Design/methodology/approach

The proposed numerical technique is based on the boundary element method (BEM) combined with an iterative sequential quadratic programming (SQP) procedure. The governing convection‐diffusion equation is transformed into a Helmholtz equation and the ill‐conditioned system of equations that arises after the application of the BEM is solved using an iterative technique.

Findings

The iterative BEM presented in this paper is well‐suited for solving inverse source problems for convection‐diffusion equations with constant coefficients. Accurate and stable numerical solutions were obtained for cases when the number of sources is correctly estimated, overestimated, or underestimated, and with both exact and noisy input data.

Research limitations/implications

The proposed numerical method is limited to cases when the Péclet number is smaller than 100. Future approaches should include the application of the BEM directly to the convection‐diffusion equation.

Practical implications

Applications of the results presented in this paper can be of value in practical applications in both heat and fluid flow as they show that locations and strengths for an unknown number of point sources can be accurately found by using boundary measurements only.

Originality/value

The BEM has not as yet been employed for solving inverse source problems related with the convection‐diffusion equation. This study is intended to approach this problem by combining the BEM formulation with an iterative technique based on the SQP method. In this way, the many advantages of the BEM can be applied to inverse source convection‐diffusion problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 16 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 20 August 2021

Salam Adel Al-Bayati and Luiz C. Wrobel

The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed…

Abstract

Purpose

The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.

Design/methodology/approach

The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.

Findings

The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.

Originality/value

Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

Details

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

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Article
Publication date: 29 October 2021

Frédérique Le Louër and María-Luisa Rapún

The purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of…

Abstract

Purpose

The purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).

Design/methodology/approach

The elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.

Findings

As the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.

Originality/value

The authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

Details

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

Keywords

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Article
Publication date: 1 August 2005

M. Inc and Y. Cherruault

To use the modified decomposition method to obtain numerical solutions of fourth‐order boundary value problems.

Abstract

Purpose

To use the modified decomposition method to obtain numerical solutions of fourth‐order boundary value problems.

Design/methodology/approach

The modified form of Adomian decomposition method (ADM) originated by Wazwaz in 1997/1998 is considered and applied. In addition, a comparison of the modified decomposition method and the perturbed collocation method when applied to the solutions of fourth‐order boundary value problems is presented.

Findings

The comparison of these two methods has shown that the modified or standard decomposition methods are reliable, efficient and easy to use for solving higher‐order boundary value problems.

Research limitations/implications

This study provides both an analysis of the method as well as a comparison of the numerical solutions of fourth‐order boundary value problems. Future research in the application of this methodology to other areas should prove productive.

Practical implications

The numerical results, using the modified technique, were obtained from selected fourth‐order boundary value problems which were linear and non‐linear problems.

Originality/value

Assesses, by comparison, methods for the numerical solutions of fourth‐order boundary value problems. Shows through original numerical results the value of the modified or standard decomposition methods in the solution of these problems.

Details

Kybernetes, vol. 34 no. 7/8
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 9 July 2020

Tadeusz Sobczyk and Marcin Jaraczewski

Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using…

Abstract

Purpose

Discrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations.

Design/methodology/approach

The DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images.

Findings

An alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems.

Research limitations/implications

The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes.

Practical implications

Based on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established.

Originality/value

New finite-difference expressions for the first- and second-order derivatives have been applied.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 39 no. 4
Type: Research Article
ISSN: 0332-1649

Keywords

Content available

Abstract

Details

Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 7 June 2013

Michael Dawson, Duncan Borman, Robert B. Hammond, Daniel Lesnic and Dominic Rhodes

The purpose of this paper is to apply the meshless method of fundamental solutions (MFS) to the two‐dimensional time‐dependent heat equation in order to locate an unknown…

Abstract

Purpose

The purpose of this paper is to apply the meshless method of fundamental solutions (MFS) to the two‐dimensional time‐dependent heat equation in order to locate an unknown internal inclusion.

Design/methodology/approach

The problem is formulated as an inverse geometric problem, using non‐invasive Dirichlet and Neumann exterior boundary data to find the internal boundary using a non‐linear least‐squares minimisation approach. The solver will be tested when locating a variety of internal formations.

Findings

The method implemented was proven to be both stable and reasonably accurate when data were contaminated with random noise.

Research limitations/implications

Owing to limited computational time, spatial resolution of internal boundaries may be lower than some similar case investigations.

Practical implications

This research will have practical implications to the modelling and monitoring of crystalline deposit formations within the nuclear industry, allowing development of future designs.

Originality/value

Similar work has been completed in regards to the steady state heat equation, however to the best of the authors' knowledge no previous work has been completed on a time‐dependent inverse inclusion problem relating to the heat equation, using the MFS. Preliminary results presented here will have value for possible future design and monitoring within the nuclear industry

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 5
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 19 April 2013

Lazhar Bougoffa and Randolph C. Rach

The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations…

Abstract

Purpose

The purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Design/methodology/approach

The authors first transform the given nonlocal boundary value problems of first‐ and second‐order differential equations into local boundary value problems of second‐ and third‐order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations.

Findings

The new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

Originality/value

The paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

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Article
Publication date: 17 June 2021

Lucas Fernandez and Ravi Prakash

The purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.

Abstract

Purpose

The purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.

Design/methodology/approach

The method consists in rewriting the inverse reconstruction problem as a topology optimization problem and then to use the concept of topological derivatives to seek a higher-order asymptotic expansion for the topologically perturbed cost functional. Such expansion is truncated and then minimized with respect to the parameters under consideration, which leads to noniterative second-order reconstruction algorithms.

Findings

In this paper, the authors develop two different classes of noniterative second-order reconstruction algorithms that are able to accurately recover the unknown penetrable obstacles from partial measurements of a field generated by incident waves.

Originality/value

The current paper is a pioneer work in developing a reconstruction method entirely based on topological derivatives for solving an inverse scattering problem with penetrable obstacles. Both algorithms proposed here are able to return the number, location and size of multiple hidden and unknown obstacles in just one step. In summary, the main features of these algorithms lie in the fact that they are noniterative and thus, very robust with respect to noisy data as well as independent of initial guesses.

Details

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

Keywords

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Article
Publication date: 20 September 2011

Mehdi Dehghan and Fatemeh Shakeri

Multi‐point boundary value problems have important roles in the modelling of various problems in physics and engineering. This paper aims to present the solution of…

Abstract

Purpose

Multi‐point boundary value problems have important roles in the modelling of various problems in physics and engineering. This paper aims to present the solution of ordinary differential equations with multi‐point boundary value conditions by means of a semi‐numerical approach which is based on the homotopy analysis method.

Design/methodology/approach

The convergence of the obtained solution is expressed and some typical examples are employed to illustrate validity, effectiveness and flexibility of this procedure. This approach, in contrast to perturbation techniques, is valid even for systems without any small/large parameters and therefore it can be applied more widely than perturbation techniques, especially when there do not exist any small/large quantities.

Findings

Unlike other analytic techniques, this approach provides a convenient way to adjust and control the convergence of approximation series. Some applications will be briefly introduced.

Originality/value

The paper shows how an important boundary value problem is solved with a semi‐analytical method.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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