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1 – 10 of 10The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline…
Abstract
Purpose
The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations.
Design/methodology/approach
The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points.
Findings
Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain.
Originality/value
First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.
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This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion…
Abstract
Purpose
This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.
Design/methodology/approach
A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.
Findings
Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.
Originality/value
ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.
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Ramesh Chand Mittal, Sudhir Kumar and Ram Jiwari
The purpose of this study is to extend the cubic B-spline quasi-interpolation (CBSQI) method via Kronecker product for solving 2D unsteady advection-diffusion equation. The CBSQI…
Abstract
Purpose
The purpose of this study is to extend the cubic B-spline quasi-interpolation (CBSQI) method via Kronecker product for solving 2D unsteady advection-diffusion equation. The CBSQI method has been used for solving 1D problems in literature so far. This study seeks to use the idea of a Kronecker product to extend the method for 2D problems.
Design/methodology/approach
In this work, a CBSQI is used to approximate the spatial partial derivatives of the dependent variable. The idea of the Kronecker product is used to extend the method for 2D problems. This produces the system of ordinary differential equations (ODE) with initial conditions. The obtained system of ODE is solved by strong stability preserving the Runge–Kutta method (SSP-RK-43).
Findings
It is found that solutions obtained by the proposed method are in good agreement with the analytical solution. Further, the results are also compared with available numerical results in the literature, and a reasonable degree of compliance is observed.
Originality/value
To the best of the authors’ knowledge, the CBSQI method is used for the first time for solving 2D problems and can be extended for higher-dimensional problems.
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Ramesh Chand Mittal and Sumita Dahiya
In this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by…
Abstract
Purpose
In this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations.
Design/methodology/approach
Two schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples.
Findings
A comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method.
Originality/value
For the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.
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Neeraj Dhiman, M.J. Huntul and Mohammad Tamsir
The purpose of this paper is to present a stable and efficient numerical technique based on modified trigonometric cubic B-spline functions for solving the time-fractional…
Abstract
Purpose
The purpose of this paper is to present a stable and efficient numerical technique based on modified trigonometric cubic B-spline functions for solving the time-fractional diffusion equation (TFDE). The TFDE has numerous applications to model many real objects and processes.
Design/methodology/approach
The time-fractional derivative is used in the Caputo sense. A modification is made in trigonometric cubic B-spline (TCB) functions for handling the Dirichlet boundary conditions. The modified TCB functions have been used to discretize the space derivatives. The stability of the technique is also discussed.
Findings
The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. The stability analysis shows that the method is unconditionally stable. Furthermore, this technique is efficient and requires less storage.
Originality/value
The current work is novel for solving TFDE. This technique is unconditionally stable and gives better results than existing results (Ford et al., 2011; Sayevand et al., 2016; Ghanbari and Atangana, 2020).
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N.P. Weatherill, E.A. Turner‐Smith, J. Jones, K. Morgan and O. Hassan
As computer simulation increasingly supports engineering design and manufacture, the requirement for a computer software environment providing an integration platform for…
Abstract
As computer simulation increasingly supports engineering design and manufacture, the requirement for a computer software environment providing an integration platform for computational engineering software increases. The potential benefits to industry are considerable. As a first step in the long‐term development of such a system, a computer software environment has been developed for pre‐ and post‐processing for unstructured grid‐based computational simulation. Arbitrary computer application software can be integrated into the environment to provide a multi‐disciplinary engineering analysis capability within one unified computational framework. Recognising the computational demands of many application areas, the environment includes a set of parallel tools to help the user maximise the potential of high performance computers and networks. The paper will present details of the environment and include an example of, and discussion about, the integration of application software.
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This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.
Abstract
Purpose
This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.
Design/methodology/approach
The parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.
Findings
The IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.
Originality/value
This study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.
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Jiawei Feng, Jianzhong Fu, Zhiwei Lin, Ce Shang and Bin Li
T-spline is the latest powerful modeling tool in the field of computer-aided design. It has all the merits of non-uniform rational B-spline (NURBS) whilst resolving some flaws in…
Abstract
Purpose
T-spline is the latest powerful modeling tool in the field of computer-aided design. It has all the merits of non-uniform rational B-spline (NURBS) whilst resolving some flaws in it. This work applies T-spline surfaces to additive manufacturing (AM). Most current AM products are based on Stereolithograph models. It is a kind of discrete polyhedron model with huge amounts of data and some inherent defects. T-spline offers a better choice for the design and manufacture of complex models.
Design/methodology/approach
In this paper, a direct slicing algorithm of T-spline surfaces for AM is proposed. Initially, a T-spline surface is designed in commercial software and saved as a T-spline mesh file. Then, a numerical method is used to directly calculate all the slicing points on the surface. To achieve higher manufacturing efficiency, an adaptive slicing algorithm is applied according to the geometrical properties of the T-spline surface.
Findings
Experimental results indicate that this algorithm is effective and reliable. The quality of AM can be enhanced at both the designing and slicing stages.
Originality/value
The T-spline and direct slicing algorithm discussed here will be a powerful supplement to current technologies in AM.
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E. HINTON, N.V.R. RAO and J. SIENZ
This paper deals with structural shape and thickness optimization of axisymmetric shell structures loaded symmetrically. In the finite element stress analysis use is made of newly…
Abstract
This paper deals with structural shape and thickness optimization of axisymmetric shell structures loaded symmetrically. In the finite element stress analysis use is made of newly developed linear, quadratic, and cubic, variable thickness, C(0) elements based on axisymmetric Mindlin‐Reissner shell theory. An integrated approach is used to carry out the whole shape optimization process in a fully automatic manner. A robust, versatile and flexible mesh generator is incorporated with facilities for generating either uniform or graded meshes, with constant, linear, or cubic variation of thickness, pressure etc. The midsurface geometry and thickness variations of the axisymmetric shell structure are defined using cubic splines passing through certain key points. The design variables are chosen as the coordinates and/or the thickness at the key points. Variable linking procedures are also included. Sensitivity analysis is carried out using either a semi‐analytical method or a global finite difference method. The objective of the optimization is the weight minimization of the structure. Several examples are presented illustrating optimal shapes and thickness distributions for various shells. The changes in the bending, membrane and shear strain energies during the optimization process are also monitored.
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Sreekanth M.P., Rajesh Ranganathan and Arivazhagan Pugalendhi
Laparoscopic surgeons suffer because of discomfited body posture while performing surgery and experience discomfort owing to lack of customized surgical instruments. Accordingly…
Abstract
Purpose
Laparoscopic surgeons suffer because of discomfited body posture while performing surgery and experience discomfort owing to lack of customized surgical instruments. Accordingly, this paper aims to recommend an individual customization strategy by developing an ergonomically designed laparoscopic forceps handle and thereby increase the comfort of surgeons.
Design/methodology/approach
Hand anthropometric parameters of 282 south Indian male subjects are used to customize the handle. uPrint and Objet260 Connex, which works based on fused deposition modeling and PolyJet, respectively, are used to fabricate the prototype of the handle. Design modifications include a pistol-type grip, the increased contact area between the hand and handle, and neutral wrist posture.
Findings
Ergonomic evaluation parameters such as grip, functionality, comfort and wrist posture using subjective ratings from laparoscopic surgeons were recorded and obtained average values of 4.1, 3.6, 4.1 and 4.1, respectively, based on a five-point ordinal scale. Additionally, stress analysis also confirms the safety of the handle based on von Mises stress criteria.
Research limitations/implications
Anthropometric data are limited to 282 subjects and subjective evaluation is conducted using a prototype, not the end-use product.
Originality/value
Evaluation using subjective rating confirms the ascendancy of a modified handle over the existing handle in terms of assessed parameters. The proposed individual customization strategy can be applied for other industrial hand tools to enhance comfort.
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