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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.

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.

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.

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

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

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

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.

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.

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).

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.

The purpose of this paper is to propose a smooth double-spline interpolation method for six-degree-of-freedom rotational robot manipulators, achieving the global C2 continuity of the robot trajectory.

This paper presents a smooth double-spline interpolation method, achieving the global C2 continuity of the robot trajectory. The tool center positions and quaternion orientations are first fitted by a cubic B-spline curve and a quartic-polynomial-based quaternion spline curve, respectively. Then, a parameter synchronization model is proposed to realize the synchronous and smooth movement of the robot along the double spline curves. Finally, an extra u-s function is used to record the relationship between the B-spline parameter and its arc length parameter, which may reduce the feed rate fluctuation in interpolation. The seven segments jerk-limited feed rate profile is used to generate motion commands for algorithm validation.

The simulation and experimental results demonstrate that the proposed method is effective and can generate the global C2-continuity robot trajectory.

The main contributions of this paper are as follows: guarantee the C2 continuity of the position path and quaternion orientation path simultaneously; provide a parameter synchronization model to realize the synchronous and smooth movement of the robot along the double spline curves; and add an extra u-s function to realize arc length parameterization of the B-spline path, which may reduce the feed rate fluctuation in interpolation.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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

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

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.

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.

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.

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).

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.

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.

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.

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.

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

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.

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.

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.

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

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.

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

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