Fourth-order Method Research Articles

Modeling and analysis of multi-functional self-healing material using Runge-Kutta Method for investigation of aircraft wing structure

ABSTRACT This paper presents the deployment of Runge-Kutta method to overcome the main challenge in analysis of failure of aircraft wing structure subjected to wind pressure and point loading. Failure phenomenon of any structure is time dependents and is typically referred as dynamic in engineering mechanics and is fairly a complex to investigate. In this context, the dynamic analysis concept has successfully implemented by using computer program in SCILAB software. The numerical technique is adopted as Runge-Kutta fourth order (RK4) method for performing dynamic behaviour of wind structure. The demonstration of failure mode of wing structure is based on function of time. The numerical approach is deemed to provide a detailed description of these phenomena affecting the overall dynamic of failure envelope of wing structure. The parametric study is presented; likewise the effect on failure of wing structure by changing different wind pressure, length and moment, respectively. The wing structure is analytically validated against available literature. Finally, others important failure results obtained from this analysis has discussed in detail.

Influence of inclined magnetic field and cross diffusion on transient forced convective heat and mass transfer flow along a porous wedge with variable Prandtl number

The unsteady 2-D forced convective mass and heat transfer flow of an incompressible, viscous and electrically conducting fluid along a porous wedge is examined in this article using numerical analysis to survey the effects of a magnetic field and thermophoresis. This is done in the existence of temperature-dependent thermal conductivity, a variable Prandtl number, and the effect of cross diffusion. The governing nonlinear PDEs have been converted into nonlinear ODEs by using a similarity transformation. The modified ODEs are solved for similar solutions using the shooting technique and the Runge–Kutta fourth-order method. The temperature, velocity, concentration, thermophoretic velocity, and thermophoretic particle deposition velocity results for various parameters are provided. The Dufour number’s effect enhances the temperature profile and rate of heat transfer. Moreover, it is noted that the rate of mass transfer is meaningfully influenced by the variation of the thermophoresis parameter and Soret number. Thus, in any physical model where the thermal conductivity of the fluid is temperature dependent, the Prandtl number within the BL have to be treated as variable rather than constant.

A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L-stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A-stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented.

L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

Computational study for the Caputo sub-diffusive and Riesz super-diffusive processes with a fractional order reaction–diffusion equation

A Numerical solution of the Caputo-time and Riesz-space fractional reaction–diffusion model is considered in this paper. Based on finite difference schemes, we formulate both second-order and fourth-order numerical methods for the approximation of the Riesz space fractional reaction–diffusion-like equation of Fisher type. In the experiment, it was observed that the fourth-order scheme has better accuracy than the second-order method when applied to solve the fractional diffusion equation. It should be mentioned that the lower order scheme computes rapidly and save more computational time as displayed in the table of results. Finally, some simulation results are presented to justify the effectiveness and applicability of the numerical methods. The one- two- and three-dimensional results obtained for some instances of fractional order (α,β) depict some amazing complex and spatiotemporal patterns which are applicable in applied sciences and engineering.

Composite cylindrical shells with auxetic core on elastic foundation: A nonlinear dynamic analysis

This paper investigated vibration analyses of a composite cylindrical shell consisting of a core with negative Poisson's ratios resting on elastic foundation subjected to pre-loaded axial compression and external excitation force. Governing equations are driven based on Donnell's improved shell theory ignoring the shallowness of cylindrical shells and, with the help of Galerkin's method, Volmir's assumption, and Runge-Kutta's fourth-order method, the dynamical analysis of shells is studied. The purpose of this article is to discuss both the linear and nonlinear aspects of the structure's vibration. Furthermore, several parametric examples are presented and discussed to show how different parameters affect the three-layer cylindrical shell oscillation. To ensure the accuracy and the precision of the numerical results, the proposed method is compared with other literature attempts.

Mathematical Model for Malaria Disease Transmission

Malaria is one of the fatal diseases caused by plasmodium parasites and transmitted to humans through biting of the female of {\it Anopheles} mosquitoes. We proposed a deterministic mathematical model for simulating Malaria disease transmission between humans and mosquitoes. The basic reproduction number $\mathcal{R}_{0}$ is determined by using the next-generation matrix approach. Stability conditions for the model equilibrium points with respect to $\mathcal{R}_{0}$ derived and we show that the forward bifurcation occurred. When $\mathcal{R}_{0} <1$ or $\mathcal{R}_{0} >1$ the Malaria disease dies out or spreads, respectively. The sensitivity analysis for the basic reproduction number $\mathcal{R}_{0}$ fulfilled locally and globally. The model simulation was found by using Runge--Kutta fourth order method in MATLAB. Furthermore, The effects of the important parameters were investigated, and the obtained results were presented in graphical forms. Also, we obtained that the simulation results agree with the stability analysis for $E_{def}$. We discussed the impacts of the Malaria disease control interventions on the important parameter for Malaria disease transmission. Recommendation for control and eradicating Malaria disease transmission provided.

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

In this article, a very simple modified form of the harmonic balance method is used to solve a strongly nonlinear oscillator with cubic nonlinearity and harmonic restoring force. Taylor series expansion up to third term is considered for the harmonic restoring force. The first approximate solutions of the present method pleasantly agree with the numerical solution obtained by Runge–Kutta fourth order method. Accuracy and simplicity of the present method solution is established when compared with the other method solutions. The present method can be utilized to other nonlinear oscillators.

Thermal convection and entropy generation analysis of hybrid nanofluid slip flow over a horizontal poignant thin needle with an inclined magnetic field: A numerical study

Hybrid nanofluids have emerged as a promising field of research in recent years, finding applications in various industries and sectors. The entropy generation and impact of an inclined magnetic field on a water-Ethylene Glycol (50:50) mixture-based nanofluid over a poignant thin needle has been investigated under slip conditions. The flow is swotted using thermal and velocity slip boundary conditions. The numerical effects of an inclined magnetic field on the flow of ferrous and aluminum oxide nanoparticles in a hybrid nanoliquid with as base fluids have been examined. The non-dimensional ODEs along with boundary conditions are solved by numerical technique based on the Runge–Kutta fourth-order method. Velocity profile has been analyzed for the magnetic parameter, inclined angle and volume fractions. The Bejan number provides insights into the balance between heat transfer and entropy generation in a system. From the numerical values of skin friction coefficient and Nusselt number, it is analyzed that [Formula: see text]O3 possesses 1.01% and 1.06% of high heat transfer rate and high surface drag force than [Formula: see text]O4. Heat transfer profile has been analyzed for entropy generation and the Bejan number. The applications of hybrid nanofluids are vast, and one specific area where they can be utilized is in horizontal needle systems.

Free vibration and nonlinear dynamic response of sandwich plates with auxetic honeycomb core and piezoelectric face sheets

This article analyses the nonlinear dynamic behavior and the free vibration of a piezoelectric auxetic honeycomb sandwich plate resting on a Pasternak elastic substrate. The plate includes two face sheets of piezoelectric material and an auxetic honeycomb core with a negative Poisson ratio. The first-order shear deformation plate theory, together with the von Kármán type nonlinear strain-displacement relationship, is used to establish the nonlinear equations of motion. Linear electric potential change with piezoelectric layer thickness is assumed. Two types of boundary conditions, namely immovable and movable simply supported, are considered. The displacement-time and acceleration-displacement curves under dynamic loads are determined by using the Galerkin and Runge-Kutta fourth-order methods. Verification examples have been performed and confirmed the accuracy of the model. Finally, several novel investigations have been conducted to evaluate the effect of material, geometrical parameters, and elastic substrate coefficients on the frequency and nonlinear dynamic characteristics of piezoelectric auxetic honeycomb sandwich plates for both open-circuit and closed-circuit operation.

Computational analysis of nonsimilar electromagnetohydrodynamic radiative nanofluid flow across Darcy–Forchheimer–Brinkman porous media

This study develops nonsimilar analysis for the electrically conducting nanofluid flow over a vertically placed stretching surface adjacent to porous medium with transverse magnetic field assumption. The features of thermal radiations, viscous dissipation, and heat source factors are also incorporated to scrutinize the thermal prospective. The Brinkman–Forchheimer who extended Darcy model is considered for the porous medium and a Rosseland approximation is employed for thermal radiations. Titanium oxide and silica are considered nanoparticles, while kerosine oil and silicon ceramic serving as the base fluids in porous medium, respectively. Tiwari–Dass approach is used to derive the governing system. Appropriate transformations are deployed to render the governing partial differential equations into a dimensionless system. To address the transformed system, the local nonsimilarity technique in conjunction with the bvp4c (fourth-order method boundary value problem) is used. Variations in velocity and thermal distributions of nanofluids have been investigated in relation to developing dimensionless parameters such as nanoparticle concentration, Darcy porosity parameter, Darcy number, electric parameter, Hartmann number, radiation parameter, Richardson number, heat source/sink parameter, and Eckert number. Numerical findings are found to interpret the role of the emerging parameters on physical quantities. The obtained findings reveals that the surge in the estimations of Richardson number enhances the nanofluid velocity while it diminishes the temperature profile. The concentration of nanoparticles improves the thermal profile of both nanofluids under consideration, whereas the enhancement in Darcy porosity parameter reduces velocity and improves temperature distributions. Results found to be in excellent agreement with published results on a special case of the considered problem.

A highly efficient class of optimal fourth-order methods for solving nonlinear systems

A highly efficient class of optimal fourth-order methods for solving nonlinear systems

Magnetohydrodynamic double-diffusive peristaltic flow of radiating fourth-grade nanofluid through a porous medium with viscous dissipation and heat generation/absorption

This article focuses on determining how to double diffusion affects the non-Newtonian fourth-grade nanofluids peristaltic motion within a symmetrical vertical elastic channel supported by a suitable porous medium as well as, concentrating on the impact of a few significant actual peculiarities on the development of the peristaltic liquid, such as rotation, initial pressure, non-linear thermal radiation, heat generation/absorption in the presence of viscous dissipation and joule heating with noting that the fluid inside the channel is subject to an externally induced magnetic field, giving it electromagnetic properties. Moreover, the constraints of the long-wavelength approximation and neglecting the wave number along with the low Reynolds number have been used to transform the nonlinear partial differential equations in two dimensions into a system of nonlinear ordinary differential equations in one dimension, which serve as the basic governing equations for fluid motion. The suitable numerical method for solving the new system of ordinary differential equations is the Runge–Kutta fourth-order numerical method with the shooting technique using the code MATLAB program. Using this code, a 2D and 3D graphical analysis was done to show how each physical parameter affected the distributions of axial velocity, temperature, nanoparticle volume fraction, solutal concentration, pressure gradients, induced magnetic field, magnetic forces, and finally the trapping phenomenon. Under the influence of rotation Omega, heat Grashof number {Gr}_{t}, solutal Grashof number {Gr}_{t}, and initial stress {P}^{*}, the axial velocity distribution u changes from increasing to decreasing, according to some of the study’s findings. On the other hand, increasing values of nonlinear thermal radiation R and temperature ratio {theta }_{w} have a negative impact on the temperature distribution theta but a positive impact on the distributions of nanoparticle volume fraction Phi and solutal concentration gamma. Darcy number Da and mean fluid rate F also had a positive effect on the distribution of pressure gradients, making it an increasing function.

Modeling fracture in brittle materials by higher-order phase field method using C1 non-Sibsonian interpolants

PurposeThis study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C1 basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.Design/methodology/approachThe weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C0 non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C1) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.FindingsIt is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.Originality/valueFourth-order PFM is implemented using C1 non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

High-resolution frequency-difference beamforming for a short linear array.

Conventional beamforming (CBF) is a commonly employed approach for detecting and estimating the direction-of-arrival (DOA) of acoustic signals in underwater environments. However, CBF becomes ambiguous due to spatial aliasing when the received signal's half wavelength is smaller than the array spacing. Frequency-difference beamforming (FDB) allows for processing data in the lower frequency Δf without encountering spatial aliasing by utilizing the product of array data at frequency f with its complex conjugate at frequency f+Δf. However, lower frequency results in a wider mainlobe, which can lead to poorer DOA performance for short arrays. In this paper, a fourth-order cumulants FDB method and a conjugate augmented FDB method are proposed to extend an M-element uniform linear array to 2M-1 and 4M-3 elements. The proposed methods generate narrower beams and lower sidelobe levels than the original FDB for short arrays with large spacing. And by setting the signal subspace dimension reasonably, the proposed methods can improve the weak target detection ability under strong interference compared with FDB. Last, we verify the excellent performance of the proposed methods through simulations and experimental data.

HORIZONTAL ACCELERATION OF FLOOR-MOUNTED NONSTRUCTURAL COMPONENTS UNDER ROCKING MOTION

Floor-mounted nonstructural components (NSCs) that are attached through various types of angles are susceptible to seismic damage as they experience severe rocking motion resulting from their slender geometric configuration. However, current design codes do not reflect such complex dynamic behavior and prescribe design factors based on simplified single-degree-of-freedom (SDOF) models, which might cause an under-estimation of seismic design forces. Therefore, to properly control their seismic damage, the complex rocking dynamic behavior needs to be thoroughly investigated. This paper analyzed the rocking behavior of the floor-mounted NSCs and their effects on the horizontal acceleration response. Floor-mounted NSCs and their mounting attachments were idealized as rigid blocks restrained by linear vertical springs at the corner of the block. The nonlinear equations for the rocking behavior of rigid block were developed, and the analysis was conducted based on the fourth-order Runge-Kutta integration method. First, it is shown that the resonance period of rocking NSCs is mainly affected by the NSC slenderness and the location of the center of gravity of the component. The horizontal acceleration tends to increase as the slenderness and the height of the center of gravity of NSCs increase. Based on the observations, some useful design recommendations were suggested for designing attachments of floor-mounted NSCs.

Magnetohydrodynamic Mixed Convection Heat and Mass Transfer of Nanofluid Flow Over a Stretching Wedge-Shaped Surface with the Effect of Thermophoresis and Brownian Motion

The present study has been investigated to the consequence of the magnetic parameter, Grashof number, modified Grashof number, Prandtl number, thermal radiation parameter, Brownian motion parameter, thermophoresis parameter, heat generation parameter, Schmidt number, Biot number, stretching parameter, Lewis number, and chemical reaction parameter, respectively, over a stretching wedge of the magnetohydrodynamic (MHD) BL nanofluid flow. The main goal of this paper is to numerically investigate the nature of the MHD BL nanofluid flow along a stretching wedge-shaped surface with radiation, heat source, and chemical reaction parameters. The fundamental equations has been transformed into ordinary differential equations (ODEs) by the usual transformation. The numerical solutions are found by employing Runge-Kutta fourth-order method by exploiting symbolic software MATLAB via the shooting method. The novelty of the current study is implicated in the area of fluid dynamics to solve nonlinear differential equations numerically and is an important contribution to the field of nanofluids flow. Numerical solutions reveal that the concerned physical parameters lead to progress in the skin friction factor, rate of change of heat transfer as well as the rate of change of concentration. Brownian motion and thermophoresis parameters play a crucial role in the variation of temperature and concentration profiles and also in the development of thermal and concentration boundary layers.

Efficient Fourth-Order Scheme for Multiple Zeros: Applications and Convergence Analysis in Real-Life and Academic Problems

High-order iterative techniques without derivatives for multiple roots have wide-ranging applications in the following: optimization tasks, where the objective function lacks explicit derivatives or is computationally expensive to evaluate; engineering; design finance; data science; and computational physics. The versatility and robustness of derivative-free fourth-order methods make them a valuable tool for tackling complex real-world optimization challenges. An optimal extension of the Traub–Steffensen technique for finding multiple roots is presented in this work. In contrast to past studies, the new expanded technique effectively handles functions with multiple zeros. In addition, a theorem is presented to analyze the convergence order of the proposed technique. We also examine the convergence analysis for four real-life problems, namely, Planck’s law radiation, Van der Waals, the Manning equation for isentropic supersonic flow, the blood rheology model, and two well-known academic problems. The efficiency of the approach and its convergence behavior are studied, providing valuable insights for practical and academic applications.

A Parallel Solution of Timing Synchronization in High-Speed Remote Sensing Data Transmission

Considering the problem that the timing synchronization calculation in high-speed remote sensing signal reception is complex and it is difficult for it to be parallel, this paper deduces and designs a parallel timing error estimation and correction scheme. This paper presents the design of polyphase DFT filter banks with non-maximum decimation. The feedforward timing error estimation and correction method is then improved to enhance synchronization performance. Finally, an implementation scheme for parallel timing error estimation and correction is proposed using the polyphase filter bank time domain decomposition method and the filter polyphase model. In the estimation module, the parallel implementation structure of the joint second-order and fourth-order cyclic statistics methods is designed, which improves the estimation accuracy. In the correction module, a fractional delay filtering method with higher accuracy is adopted in order to improve the calibration accuracy and reduce the computational complexity. The timing synchronization of a high-speed remote sensing signal with timing error is simulated and verified. The experimental results show that the parallel method proposed in this paper greatly reduces the processing speed of subband data, and has a good synchronization performance, which is close to the theoretical limit in the demodulation error rate. This paper utilizes a multi-phase DFT filter bank architecture to achieve parallel timing synchronization, which presents a novel approach for the future parallel reception of high-speed remote sensing signals.

Efficient exponential methods for genetic regulatory systems

In this paper, we construct and analyze a new second-order exponential method for simulating genetic regulatory systems. For such systems involving highly nonlinearities, a novel splitting technique is proposed to improve the stability and thus further increase accuracy and efficiency of the method. This idea can be also used with high-order exponential Runge–Kutta methods. Another possibility is to use exponential Rosenbrock methods which are derived based on dynamic linearization of the system. In particular, we additionally suggest a second- and fourth-order exponential Rosenbrock methods for genetic regulatory systems depending on their structure and properties. The linear stability analysis of the proposed methods is also presented. Our numerical experiments on a 2-gene model, 3-gene repressilator and a 13-gene budding yeast cell cycle model show the effectiveness of the proposed methods and technique over several widely used methods for genetic regulatory systems in the literature.