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Article
Publication date: 2 March 2015

Mas Irfan Purbawanto Hidayat, Bambang Ariwahjoedi and Setyamartana Parman

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction…

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Abstract

Purpose

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction problems.

Design/methodology/approach

In the present method, any governing equations are discretized by B-spline approximation which is implemented in the spirit of FD technique using a local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighbouring nodal values based on B-spline interpolants. The set of neighbouring nodes are allowed to be randomly distributed thus enhanced flexibility in the numerical simulation can be obtained. The method requires no mesh connectivity at all for either field variable approximation or integration. Time integration is performed by using the Crank-Nicolson implicit time stepping technique.

Findings

Several heat conduction problems in complex domains which represent for extended surfaces in industrial applications are examined to demonstrate the effectiveness of the present approach. Comparison of the obtained results with solutions from other numerical method available in literature is given. Excellent agreement with reference numerical method has been found.

Research limitations/implications

The method is presented for 2D problems. Nevertheless, it would be also applicable for 3D problems.

Practical implications

A transient two dimensional heat conduction in complex domains which represent for extended surfaces in industrial applications is presented.

Originality/value

The presented new meshless local method is simple and accurate, while it is also suitable for analysis in domains of arbitrary geometries.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 2
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 23 November 2018

Neeraj Dhiman and Mohammad Tamsir

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In…

Abstract

Purpose

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this method, re-modified quintic B-spline functions and the Crank–Nicolson formulation is used for space and time integration, respectively. Five examples are considered to test out the efficiency and accuracy of the method. The main objective is to develop a method which gives more accurate results and reduces the computational cost so that the authors require less memory storage.

Design/methodology/approach

A new collocation technique is developed to solve the KS type equations. In this technique, quintic B-spline basis functions are re-modified and used to integrate the space derivatives while time derivative is discretized by using Crank–Nicolson formulation. The discretization yields systems of linear equations, which are solved by using Gauss elimination method with partial pivoting.

Findings

Five examples are considered to test out the efficiency and accuracy of the method. Finally, the present study summarizes the following outcomes: first, the computational cost of the proposed method is the less than quintic B-spline collocation method. Second, the present method produces better results than those obtained by Lattice Boltzmann method (Lai and Ma, 2009), quintic B-spline collocation method (Mittal and Arora, 2010), quintic B-spline differential quadrature method (DQM) (Mittal and Dahiya, 2017), extended modified cubic B-spline DQM (Tamsir et al., 2016) and modified cubic B-splines collocation method (Mittal and Jain, 2012).

Originality/value

The method presented in this paper is new to best of the authors’ knowledge. This work is the original work of authors and the manuscript is not submitted anywhere else for publication.

Details

Multidiscipline Modeling in Materials and Structures, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 1573-6105

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Article
Publication date: 11 February 2021

Ghassem Faezian, Ahmad Darabi and Nader Sargolzaei

This study aims to design the rotor geometry of switched reluctance motor (SRM) in a completely flexible way. In the proposed method, there is no default geometry for the…

Abstract

Purpose

This study aims to design the rotor geometry of switched reluctance motor (SRM) in a completely flexible way. In the proposed method, there is no default geometry for the rotor. The initial geometry of the rotor can start from a circle or any other shape and depending on the required performance takes the final shape during the optimal design. In this way, the best performance, possible with geometric design, can be achieved.

Design/methodology/approach

The rotor boundary of a 4/2 SRM is defined by a few B-splines. Some control points are located around the rotor and changing their locations causes customized changes in the rotor boundary. Locations of these points are defined as design variables. A 2-D finite element analysis using MATLAB/PDE is applied to the SRM model and sensitivity analysis is used to optimization design by means of minimizing of objective function.

Findings

The proposed method has many more capabilities for matching different objective functions. For the suggested objective function, while the conventional rotor torque profile difference with the desired torque profile reaches 40%, this difference for B-spline rotor is about 17%. Experimental results from a prototype motor have a close agreement with analysis results.

Originality/value

The B-splines have been used to design machines and electromagnetic devices. However, this method is used for the first time in design of the whole rotor of a SRM.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 40 no. 3
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 8 February 2021

Ouadie Koubaiti, Said EL Fakkoussi, Jaouad El-Mekkaoui, Hassan Moustachir, Ahmed Elkhalfi and Catalin I. Pruncu

This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous…

Abstract

Purpose

This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution.

Design/methodology/approach

In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory.

Findings

The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity.

Originality/value

The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

Details

Engineering Computations, vol. 38 no. 7
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 26 October 2020

Özlem Ersoy Hepson

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

Abstract

Purpose

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

Design/methodology/approach

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

Findings

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

Originality/value

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

Details

Engineering Computations, vol. 38 no. 5
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 20 March 2017

Jian Gao, Hao Wen, Zhiyuan Lin, Haidong Wu, Si Li, Xin Chen, Yun Chen and Yunbo He

Remanufacturing of worn blades with various defects normally requires processes such as scanning, regenerating a geometrical reference model, additive manufacturing (AM…

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Abstract

Purpose

Remanufacturing of worn blades with various defects normally requires processes such as scanning, regenerating a geometrical reference model, additive manufacturing (AM) through laser cladding, adaptive machining and polishing and quality inspection. Unlike the manufacturing process of a new part, the most difficult problem for remanufacturing such a complex surface part is that the reference model adaptive to the worn part is no longer available or useful. The worn parts may suffer from geometrical deformation, distortion and other defects because of the effects of harsh operating conditions, thereby making their original computer aided design (CAD) models inadequate for the repair process. This paper aims to regenerate the geometric models for the worn parts, which is a key issue for implementing AM to build up the parts and adaptive machining to reform the parts. Unlike straight blades with similar cross sections, the tip geometry of the worn tip of a twist blade needs to be regenerated by a different method.

Design/methodology/approach

This paper proposes a surface extension algorithm for the reconstruction of a twist blade tip through the extremum parameterization of a B-spline basis function. Based on the cross sections of the scanned worn blade model, the given control points and knot vectors are firstly reconstructed into a B-spline curve D. After the extremum of each control point is calculated by extremum parameterization of a B-spline basis function, the unknown control points are calculated by substituting the extremum into the curve D. Once all control points are determined, the B-spline surface of the worn blade tip can be regenerated. Finally, the extension algorithm is implemented and validated with several examples.

Findings

The proposed algorithm was implemented and verified through the exampled blades. Through the extension algorithm, the tip geometry of the worn tip of a twist blade can be regenerated. This method solved a key problem for the repair of a twist blade tip. It provides an appropriate reference model for repairing worn blade tips through AM to build up the blade tip and adaptive machining/polishing processes to reform the blade geometry.

Research limitations/implications

The extension errors for different repair models are compared and analyzed. The authors found that there are several factors affecting the accuracy of the regenerated model. When the cross-section interval and the extension length are set properly, the restoration accuracy for the blade tip can be improved, which is acceptable for the repairing.

Practical implications

The lack of a reference geometric model for worn blades is a significant problem when implementing blade repair through AM and adaptive machining processes. Because the geometric reference model is unavailable for the repair process, reconstruction of the geometry of a worn blade tip is the first crucial step. The authors proposed a surface extension algorithm for the reconstruction of a twist blade tip. Through the implementation of the proposed algorithm, the blade tip model can be regenerated.

Social implications

Remanufacturing of worn blades with various defects is highly demeaned for the aerospace enterprises considering sustainable development. Unlike straight blades, repair of twist blades encountered a very difficult problem because the geometric reference model is unavailable for the repair processes. This paper proposed a different method to generate the reference model for the repair of a twist blade tip. With this model, repair of twist blades can be implemented through AM to build up the blade tip and adaptive machining to subtract the extra material.

Originality/value

The authors proposed a surface extension algorithm to reconstruct the geometric model for repair of twist blades.

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Article
Publication date: 14 June 2011

İdris Dağ, Aynur Canivar and Ali Şahin

The purpose of this paper is to provide numerical solutions of the time‐dependent advection‐diffusion problem by using B‐spline finite element methods in which Taylor…

Abstract

Purpose

The purpose of this paper is to provide numerical solutions of the time‐dependent advection‐diffusion problem by using B‐spline finite element methods in which Taylor series expansion is used for the related time discretization.

Design/methodology/approach

The solution domain is partitioned into uniform mesh. The collocation and the Galerkin methods where B‐spline functions are used as base functions are applied to advection‐diffusion equation.

Findings

Given methods are unconditionally stable and the obtained results are comparable with some earlier studies in terms of accuracy.

Originality/value

Quadratic and cubic B‐spline base functions are used with Taylor series expansion for the discretization of the equation.

Details

Kybernetes, vol. 40 no. 5/6
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 7 August 2017

Ram Jiwari and Ali Saleh Alshomrani

The main aim of the paper is to develop a new B-splines collocation algorithm based on modified cubic trigonometric B-spline functions to find approximate solutions of…

Abstract

Purpose

The main aim of the paper is to develop a new B-splines collocation algorithm based on modified cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic Burgers’-type equations with Dirichlet boundary conditions.

Design/methodology/approach

A modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and an algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithm reduced the Burgers’ equations into a system of first-order nonlinear ordinary differential equations in time variable. Then, strong stability preserving Runge-Kutta 3rd order (SSP-RK3) scheme is used to solve the obtained system.

Findings

A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from to the schemes developed in Abbas et al. (2014) and Nazir et al. (2016), and the developed algorithms are free from linearization process and finite difference operators.

Originality/value

To the best knowledge of the authors, this technique is novel for solving nonlinear partial differential equations, and the new proposed technique gives better results than the results discussed in Ozis et al. (2003), Kutluay et al. (1999), Khater et al. (2008), Korkmaz and Dag (2011), Kutluay et al. (2004), Rashidi et al. (2009), Mittal and Jain (2012), Mittal and Jiwari (2012), Mittal and Tripathi (2014), Xie et al. (2008) and Kadalbajoo et al. (2005).

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 8
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 16 August 2013

Youdong Chen, Liang Yan, Hongxing Wei and Tianmiao Wang

This paper aims to present a technique for optimal trajectory planning of industrial robots that applies a new harmony search (HS) algorithm.

Abstract

Purpose

This paper aims to present a technique for optimal trajectory planning of industrial robots that applies a new harmony search (HS) algorithm.

Design/methodology/approach

The new HS optimization algorithm adds one more operation to the original HS algorithm. The objective function to be minimized is the trajectory execution time subject to kinematical and mechanical constraints. The trajectory is built by quintic B‐spline curves and cubic B‐spline curves.

Findings

Simulation experiments have been undertaken using a 6‐DOF robot QH165. The results show that the proposed technique is valid and that the trajectory obtained using quintic B‐spline curves is smoother than the trajectory using cubic B‐spline curves.

Originality/value

The proposed new HS algorithm is more efficient than the sequential quadratic programming method (SQP) and the original HS method. The proposed technique is applicable to any industrial robot and yields smooth and time‐optimal trajectories.

Details

Industrial Robot: An International Journal, vol. 40 no. 5
Type: Research Article
ISSN: 0143-991X

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Article
Publication date: 16 October 2009

Dursun Irk

The purpose of this paper is to investigate the numerical solutions of the Burgers' and modified Burgers' equation using sextic B‐spline collocation method.

Abstract

Purpose

The purpose of this paper is to investigate the numerical solutions of the Burgers' and modified Burgers' equation using sextic B‐spline collocation method.

Design/methodology/approach

Crank‐Nicolson central differencing scheme has been used for the time integration and sextic B‐spline functions have been used for the space integration to the modified and time splitted modified Burgers' equation.

Findings

It has been found that the proposed method is unconditionally stable and obtained results are consistent with some earlier published studies.

Originality/value

Sextic B‐spline collocation method for the Burgers' and modified Burgers' equation is given.

Details

Kybernetes, vol. 38 no. 9
Type: Research Article
ISSN: 0368-492X

Keywords

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