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Article
Publication date: 25 May 2012

Mehdi Dehghan and Masoud Hajarian

Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve…

Abstract

Purpose

Solving the non‐linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non‐linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose two new modified Newton's method for solving non‐linear equations. Convergence results show that the order of convergence of the proposed iterative methods for a simple root is four. The iterative methods are free from second derivative and can be used for solving non‐linear equations without computing the second derivative. Finally, several numerical examples are given to illustrate that proposed iterative algorithms are effective.

Design/methodology/approach

In this paper, first the authors introduce two new approximations for the definite integral arising from Newton's theorem. Then by considering these approximations, two new iterative methods are provided with fourth‐order convergence which can be used for solving non‐linear equations without computing second derivatives.

Findings

In this paper, the authors propose two new iterative methods without second derivatives for solving the non‐linear equation f(x)=0. From numerical results, it is observed that the new methods are comparable with various iterative methods. Also numerical results corroborate the theoretical analysis.

Originality/value

The best property of these schemes is that they are second derivative free. Also from numerical results, it is observed that the new methods are comparable with various iterative methods. The numerical results corroborate the theoretical analysis.

Details

Engineering Computations, vol. 29 no. 4
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 November 1999

Scott A. Burns and Keith M. Mueller

The analysis of certain structures must be performed with due consideration to non‐linear behavior, such as material and geometric non‐linearities. The existing methods

Abstract

The analysis of certain structures must be performed with due consideration to non‐linear behavior, such as material and geometric non‐linearities. The existing methods for treating non‐linear structural behavior generally make use of repeated linearization, such as load increment methods. This paper demonstrates that there is an alternative type of linearization that appears to have significant advantages when applied to the analysis of non‐linear structural systems. Briefly stated, this alternative linearization can be thought of as a “monomialization”. This monomial (single‐termed power function) approximation more faithfully models the power function behavior inherent in typical structural systems. Conveniently, it becomes a linear form when transformed into log space. Thus, computational tools based on linear algebra remain useful and effective. Preliminary results indicate that the monomial approximation provides a higher quality approximation to non‐linear phenomena exhibited in structural applications. Consequently, incremental and iterative methods become more effective because larger steps can be taken. The net result is an increase in reliability of the solution process and a significant reduction in computational effort. Two examples are presented to demonstrate the method.

Details

Engineering Computations, vol. 16 no. 7
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 April 1987

CONOR J. FITZSIMONS

This paper examines several modifications to Newton's method for the numerical solution of the nonlinear Poisson equation which describes the electrostatic potential…

Abstract

This paper examines several modifications to Newton's method for the numerical solution of the nonlinear Poisson equation which describes the electrostatic potential distribution in a semiconductor device. Two methods for a more efficient solution of the equation when the device is a Metal‐Oxide‐Semiconductor Field Effect Transistor are proposed. Their extension to the solution of the fully coupled system of equations is also discussed. The modifications to Newton's method are also compared numerically.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 6 no. 4
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 10 August 2010

Alok K. Majumdar and S.S. Ravindran

The purpose of this paper is to present a fast nonlinear solver for the prediction of transients in network flows.

Abstract

Purpose

The purpose of this paper is to present a fast nonlinear solver for the prediction of transients in network flows.

Design/methodology/approach

Broyden method‐based nonlinear solvers are developed to solve the system of conservation equation for fluids by judiciously exploiting physical coupling among the equations.

Findings

To demonstrate the feasibility and robustness of the solvers, two test cases of practical engineering interest were solved. The results obtained by the solvers were verified against analytical results for a simplified case. The performance of the solvers was found to be comparable or better than existing solvers.

Originality/value

The proposed solver enables predictions of fluid and thermal transients in complex flow networks feasible in reduced computational time.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 11 October 2011

Zhenggang Zhu and Michael Kaliske

The purpose of this paper is to present a numerical model of coupled heat, moisture transfer and their effects on the mechanical deformations of wood during the drying process.

Abstract

Purpose

The purpose of this paper is to present a numerical model of coupled heat, moisture transfer and their effects on the mechanical deformations of wood during the drying process.

Design/methodology/approach

Coupling among heat, moisture, and mechanical deformations is solved consecutively by use of sparse solver of MATLAB. The weighted residual of the equilibrium equations of drying process of wood, based on finite element method, is investigated. The stress and plastic strain increments can be solved with Newton's method.

Findings

The numerical model is applied to a plain strain problem of a long wood board taken from the outer region of the wood log. Numerical simulation reveals the stress reversal during the drying process. The mechanical deformations and the principle stresses of a three‐dimensional wood board in consideration of the orthotropic properties are presented.

Originality/value

Plane strain and plane stress are analysed. The tangential modulus is derived. The transformation of the stress and strain tensors between the local coordinate system resulting from the cylindrical properties of wood and the global one is evaluated. Selection of element type for temperature, moisture content and displacement is discussed.

Details

Engineering Computations, vol. 28 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 24 October 2019

Grzegorz Tytko and Łukasz Dawidowski

Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are…

Abstract

Purpose

Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function.

Design/methodology/approach

The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them.

Findings

The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time.

Originality/value

The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 38 no. 6
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 1 June 2001

Irina Munteanu, Silvia Drobny, Thomas Weiland and Daniel Ioan

This paper presents a hybrid algorithm used, in conjunction with the Finite Integration Technique (FIT), for solving static and quasistatic electromagnetic field problems…

Abstract

This paper presents a hybrid algorithm used, in conjunction with the Finite Integration Technique (FIT), for solving static and quasistatic electromagnetic field problems in nonlinear media. The hybrid technique is based on new theoretical results regarding the similarities between the Picard‐Banach fixed‐point (polarization) method and the Newton method. At each iteration, the solution is obtained as a linear combination of the old solution, and the new Picard‐Banach and Newton solutions. The numerical solutions are calculated through a “triangle” (bidimensional) minimization of the residual or of the energy functional. The goal of this combination is to increase the robustness of the iterative method, without losing the quadratic speed of convergence in the vicinity of the solution. The proposed method generalizes and unifies in a single algorithm the overrelaxed Picard‐Banach and the underrelaxed Newton methods.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 20 no. 2
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 4 February 2019

Nikhil Kalkote, Ashwani Assam and Vinayak Eswaran

The purpose of this paper is to solve unsteady compressible Navier–Stokes equations without the commonly used dual-time loop. The authors would like to use an adaptive…

Abstract

Purpose

The purpose of this paper is to solve unsteady compressible Navier–Stokes equations without the commonly used dual-time loop. The authors would like to use an adaptive time-stepping (ATS)-based local error control instead of CFL-based time-stepping technique. Also, an all-speed flow algorithm is implemented with simple low dissipation AUSM convective scheme, which can be computed without preconditioning which in general destroys the time accuracy.

Design/methodology/approach

In transient flow computations, the time-step is generally determined from the CFL condition. In this paper, the authors demonstrate the usefulness of ATS based on local time-stepping previously used extensively in ordinary differential equations (ODE) integration. This method is implemented in an implicit framework to ensure the numerical domain of dependence always contains the physical domain of dependence.

Findings

In this paper, the authors limit their focus to capture the unsteady physics for three cases: Sod’s shock-tube problem, Stokes’ second problem and a circular cylinder. The use of ATS with local truncation error control enables the solver to use the maximum allowable time-step, for the prescribed tolerance of error. The algorithm is also capable of converging very rapidly to the steady state (if there is any) after the initial transient phase. The authors present here only the first-order time-stepping scheme. An algorithmic comparison is made between the proposed adaptive time-stepping method and the commonly used dual time-stepping approach that indicates the former will be more efficient.

Originality/value

The original method of ATS based on local error control is used extensively in ODE integration, whereas, this method is not so popular in the computational fluid dynamics (CFD) community. In this paper, the authors investigate its use in the unsteady CFD computations. The authors hope that it would provide CFD researchers with an algorithm based on an adaptive time-stepping approach for unsteady calculations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 1 May 2001

H.M. El‐Hawary

In this paper a spline approximation of deficiency 3 and a step of length 3h method is proposed to approximate the solution of the problem and its derivatives. The…

Abstract

In this paper a spline approximation of deficiency 3 and a step of length 3h method is proposed to approximate the solution of the problem and its derivatives. The Falkner‐ Skan equation has been solved through the use of the shooting technique for handling the problem when the conditions imposed are of boundary‐value rather than an initial‐value type for different values of its parameters. Comparisons are made between the data resulting from the proposed method and those obtained by others.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 11 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 15 August 2016

Qilong Yuan, I-Ming Chen and Teguh Santoso Lembono

Taping, covering objects with masking tapes, is a common process before conducting surface treatments such as plasma spraying and painting. Manual taping is tedious and…

Abstract

Purpose

Taping, covering objects with masking tapes, is a common process before conducting surface treatments such as plasma spraying and painting. Manual taping is tedious and takes a lot of effort of the workers. This paper aims to introduce an automatic agile robotic system and corresponding algorithm to do the surface taping.

Design/methodology/approach

The taping process is a special process which requires correct tape orientation and proper allocation of the masking tape for the coverage. This paper discusses on the design of the novel automatic system consisting of a robot manipulator, a rotating platform, a 3D scanner and a specially designed novel taping end-effectors. Meanwhile, the taping path planning to cover the region of interests is introduced.

Findings

Currently, cylindrical and freeform surfaces have been tested. With improvements on new sets of taping tools and more detailed taping method, taping of general surfaces can be conducted using such system in future.

Originality/value

The introduced taping path planning method is a novel method first talking about the mathematical model of the taping process. Such taping solution with the taping tool and the taping methodology can be combined as a very useful and practical taping package to replace the work of human in such tedious and time-consuming works.

Details

Industrial Robot: An International Journal, vol. 43 no. 5
Type: Research Article
ISSN: 0143-991X

Keywords

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