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1 – 10 of over 1000
Article
Publication date: 27 March 2019

Ranjan Kumar Mohanty and Gunjan Khurana

This paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D…

Abstract

Purpose

This paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.

Design/methodology/approach

In this method, three grid points for the unknown function w(x,t) and two half-step points for the known variable x in spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.

Findings

The paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.

Research limitations/implications

There are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.

Practical implications

Physical problems with singular and nonsingular coefficients are directly solved by this method.

Originality/value

The paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

Article
Publication date: 10 February 2020

P. Veeresha, D.G. Prakasha and Jagdev Singh

The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM).

Abstract

Purpose

The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM).

Design/methodology/approach

The proposed technique (q-HATM) is the graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana-Baleanu (AB) operator.

Findings

The fixed point hypothesis considered to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional-order model. To illustrate and validate the efficiency of the future technique, the authors analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order.

Originality/value

To illustrate and validate the efficiency of the future technique, we analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. The obtained results elucidate that, the proposed algorithm is easy to implement, highly methodical, as well as accurate and very effective to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

Details

Engineering Computations, vol. 37 no. 6
Type: Research Article
ISSN: 0264-4401

Keywords

Content available

Abstract

Details

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

Article
Publication date: 1 January 2004

M. Inc, Y. Cherruault and K. Abbaoui

Studies an analytic solution and a reliable numerical approximation of linear and non‐linear wave equations by using the Adomian decomposition method. The solution is calculated…

Abstract

Studies an analytic solution and a reliable numerical approximation of linear and non‐linear wave equations by using the Adomian decomposition method. The solution is calculated in the form of a series with easily computable components. The non‐homogeneous problem is quickly solved by observing the self‐cancelling “noise terms” where the sum of the components vanishes in the limit. Several linear or non‐linear partial differential equations are considered and their numerical approximate solutions are compared with its numerical analytic solutions by applying the Adomian decomposition method and using a computer algebraic system (MATLAB). The numerical results show the effectiveness of the method for these types of equations.

Details

Kybernetes, vol. 33 no. 1
Type: Research Article
ISSN: 0368-492X

Keywords

Article
Publication date: 5 March 2018

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.

Details

Engineering Computations, vol. 35 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

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 time-harmonic…

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. 39 no. 1
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 15 November 2022

Sandeep Kundu, Kapil Kumar Kalkal, Monika Sangwan and Devender Sheoran

The purpose of the present paper is to investigate the thermo-mechanical interactions in an initially stressed nonlocal micropolar thermoelastic half-space having void pores under…

152

Abstract

Purpose

The purpose of the present paper is to investigate the thermo-mechanical interactions in an initially stressed nonlocal micropolar thermoelastic half-space having void pores under Lord–Shulman model. A moving thermal shock is applied to the formulation.

Design/methodology/approach

The normal mode technique is adopted to obtain the exact expressions of the physical quantities.

Findings

Numerical computations for stresses, displacement components, temperature field and change in the volume fraction field are performed for suitable material and are depicted graphically. Some comparisons have been shown in figures to estimate the effects of micropolarity, initial stress, voids, nonlocal parameter and time on the resulting quantities.

Originality/value

The exact expressions for the displacement components, stresses, temperature and change in the volume fraction field are obtained in the physical domain. Although numerous investigations do exist to observe the disturbances in a homogeneous, isotropic, initially stressed, micropolar thermoelastic half-space, the work in its current form has not been established by any scholar till now. The originality of the present work lies in the formulation of a fresh research problem to investigate the dependence of different physical fields on nonlocality parameters, micropolarity, initial stress, porosity and time due to the application of a moving thermal shock.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 7 March 2023

M.A. Alosaimi and D. Lesnic

When modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer…

Abstract

Purpose

When modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.

Design/methodology/approach

The non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.

Findings

The reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.

Originality/value

Although previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

Details

Engineering Computations, vol. 40 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 20 December 2019

Raju Kumhar, Santimoy Kundu, Manisha Maity and Shishir Gupta

The purpose of this paper is to examine the dependency of dispersion and damping behavior of Love-type waves on wave number in a heterogeneous dry sandy double layer of finite…

Abstract

Purpose

The purpose of this paper is to examine the dependency of dispersion and damping behavior of Love-type waves on wave number in a heterogeneous dry sandy double layer of finite thickness superimposed on heterogeneous viscoelastic substrate under the influence of hydrostatic initial stress.

Design/methodology/approach

The mechanical properties of the material of both the dry sandy layers vary with respect to a certain depth as quadratic and hyperbolic function, while it varies as an exponential function for the viscoelastic semi-infinite medium. The method of the separation of variables is employed to obtain the complex frequency equation.

Findings

The complex frequency equation is separated into real and imaginary components corresponding to dispersion and damping equation. After that, the obtained result coincides with the pre-established classical equation of Love wave, as shown in Section 5. The response of all mechanical parameters such as heterogeneities, sandiness, hydrostatic stress, thickness ratio, attenuation and viscoelasticity on both the phase and damped velocity against real wave number has been discussed with the help of numerical example and graphical demonstrations.

Originality/value

In this work, a comparative study clarifies that the Love wave propagates with higher speed in an isotropic elastic structure as compared to the proposed model. This study may find its applications in the investigation of mechanical behavior and deformation of the sedimentary rock.

Details

Multidiscipline Modeling in Materials and Structures, vol. 16 no. 4
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 16 June 2022

Bhanu Pratap Rajak, Santimoy Kundu, Raju Kumhar and Shishir Gupta

The purpose of this study is stated regarding the impact of the horizontally polarized shear wave vibration on a composite medium in the terms of phase and damped velocity.

Abstract

Purpose

The purpose of this study is stated regarding the impact of the horizontally polarized shear wave vibration on a composite medium in the terms of phase and damped velocity.

Design/methodology/approach

The assumed composite is composed of magneto-elastic fiber-reinforced (MEFR) layer constrained between heterogeneous viscoelastic layer and heterogeneous elastic half-space. The considered heterogeneity is associated with the directional rigidity and mass density in the uppermost layer and half-space of quadratic and trigonometric types, respectively. The coupled field equations related to the respective medium are solved analytically by employing the method of separation of variables.

Findings

The dispersion relation of the stated problem is secured by using the continuity assumptions, imposed at the stress-free surface and the interfaces of the expressed medium. The adopted numerical examples are used to compute the dispersion relation and plot the graphs between phase/damped velocity and wave number. Parametric studies on the phase and damped velocity yield five main conclusions: (1) Phase velocity decreases with increasing value of wave number and damped velocity increases up to a certain number and then starts falling simultaneously with increasing magnitude of wave number while keeping the rest parametric values fixed. (2) The presence of heterogeneity in the upper layer enhances the phase velocity and diminishes the damped velocity, but the presence of heterogeneity in the half-space enhances both the phase and damped velocity. (3) The appearance of reinforced parameters enhances the phase velocity for the considered crystalline graphite material and diminishes the phase velocity for the rest materials (carbon fiber-epoxy resin and steel) of the MEFR layer. Similarly, damped velocity decreases for the assumed crystalline graphite material of the MEFR layer and increases for the rest materials of the MEFR layer. (4) The induced dissipation factor due to viscoelastic property shows reversal decreasing and increasing effect on phase and damped velocity of SH-wave. (5) Ascending values of the angle at which the wave crosses the magnetic field increase the phase velocity and decrease the damped velocity for all the considered MEFR examples.

Originality/value

Till date, the mathematical modeling as well as vibrational analysis of wave propagation through the composite structure consisting of MEFR layer constrained between viscoelastic media and elastic half-space under the effect of different varying properties with depth remains a new challenging issue for the researchers around the globe. The current analysis is an approach to move ahead in the era of wave propagation in different realistic models based on their parametric studies. Also, these studies are very helpful to find their applications in the field of mechanical, construction, aerospace, automobile, biomedical, marine, manufacturing industries and many branches of science and technology where magnetic fields induced in elastic deformation occur.

Details

Engineering Computations, vol. 39 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

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