Runge-Kutta Approach Research Articles

Nonlinear dynamic analysis of the functionally graded graphene platelets reinforced porous plate under moving mass

Recent studies demonstrated that porous materials could gain satisfying improvements in some mechanical properties by adding graphene platelet (GPL) reinforcements. Following this result, the present work exhibits a semi-analytical method to investigate the nonlinear dynamic characteristics of the functionally graded GPLs reinforced porous (FG-GPLRP) plate under a moving mass. Two types of boundary conditions, i.e., simply supported (SSSS) and clamped (CCCC) edges, are incorporated in the study. Based on the refined sinusoidal shear deformation theory (RSSDT) and von Kármán nonlinearity, the governing equations are transformed into a group of ordinary differential equations for the deflection of the plate. Then, the dynamic behaviours of the plate can be investigated by operating the fourth-order Runge–Kutta approach. After verification, several numerical examples are displayed to illustrate the effects of porosity coefficient, GPLs content, Winkler–Pasternak foundation, damping, initial imperfection, and compression stress on the moving-load-bearing capability of the plate. The obtained results demonstrate that, without harming its moving load capacity, it is possible to decrease the mass of the FG-GPLRP plate to a satisfying extent by altering the porosity and GPLs content.

Dynamic analysis of an axially moving underwater pipe conveying pulsating fluid

In this paper, both linear and non-linear dynamics of a slender and uniform pipe conveying pulsating fluid, which is axially moving in an incompressible fluid, are comprehensively studied. The vibration equations of the system are established by considering various factors, including a coordinate conversion system, an “axial added mass coefficient” describing the additional inertia forces caused by the external fluid, the Kelvin–Voigt viscoelastic damping, a kind of non-linear additional axial tension, and the pulsating internal fluid. The vibration equations are discretized by the Galerkin procedure and solved by the Runge–Kutta approach, and the validity of the solution procedure is carefully checked. After that, the linear and non-linear responses of the system are studied when the internal flow velocity and the axially moving speed of the pipe are small. For linear responses, the Kelvin–Voigt viscoelastic damping has great influences on the second and third modes of the system. For the non-linear dynamic, the results are rich and changeful, including the first and second principal parametric resonances, the secondary resonance, the combination resonance, period-1 motion, quasi-periodic motion, and chaotic motion. Finally, the influence of several key system parameters on the non-linear responses is analyzed.

Stability analysis of inclined MHD and Joule effects in Ti-alloy and MWCNT/water hybrid nanofluid: Flow separation

This research article explains the Joule heating and inclined magnetic field effects on Ti6Al4V-MWCNT/[Formula: see text] hybrid nanofluids toward an exponentially shrinking surface. The system of flow governing equations, which is a coupled system of non-linear PDE’s, are transformed into a system of non-dimensional ODE’s by using suitable similarity transformations. The shooting method along with a fourth order Runge-Kutta approach is used to solve the system of resultant ODE’s. As a result, the dual solutions are identified for this problem. By utilizing the least eigenvalue approach, the first solution is found to be realistic and stable whereas the second solution is not stable. In order to verify the accuracy of outcomes, it is compared with the previously published results, and it is found that the present analysis is accurate upto four decimal places. Further, the streamline profiles are also provided in addition to the least eigenvalue profiles to understand the flow patterns. Finally, the flow separation point is identified. The velocity profiles improves in the case of first solution but it declines in the case of second solution with a rise in the shrinking strength parameter, however the temperature profiles behave in opposite manner. The skin friction coefficient and heat transfer rate of Ti6Al4V-MWCNT/[Formula: see text] based hybrid nanofluids are found to be better with higher values of inclined MHD [Formula: see text] and incline angle [Formula: see text].

FREQUENCY OF NON-LINEAR DYNAMIC RESPONSE OF A POROUS FUNCTIONALLY GRADED CYLINDRICAL PANELS

In this article, a nonlinear dynamical investigation of porose functionally graded cylindrical panels using a proposed analytical model is carried out. The material's properties are considered to be porosity-dependent and graded in the thickness direction, corresponding to a power-law distribution. The classical shell theory, with the geometrical shape of nonlinear in von Karman–Donnell, is employed to get the Lagrange motion equations. By applying the Galerkin procedure, the system of nonlinear dynamic vibration equations is found. The natural frequencies and dynamic amplitude vibrations are obtained by using the fourth-order Runge–Kutta approach. In numerical analyses, the effects of porosity factor, power-law index, porous FGM thickness, frequency–amplitude relation, and excitation force on the dynamic response of thin functionally graded porous cylindrical panels are investigated. Through the obtained results, it is discovered that the porosity coefficients have important effects on the natural frequencies and amplitude of the nonlinear dynamic response of the FG structures. It leads to a reduction in natural frequencies by 5.74 % at 10% pores.

Accelerating Real-Time Coupled Cluster Methods with Single-Precision Arithmetic and Adaptive Numerical Integration.

We explore the framework of a real-time coupled cluster method with a focus on improving its computational efficiency. Propagation of the wave function via the time-dependent Schrödinger equation places high demands on computing resources, particularly for high level theories such as coupled cluster with polynomial scaling. Similar to earlier investigations of coupled cluster properties, we demonstrate that the use of single-precision arithmetic reduces both the storage and multiplicative costs of the real-time simulation by approximately a factor of 2 with no significant impact on the resulting UV/vis absorption spectrum computed via the Fourier transform of the time-dependent dipole moment. Additional speedups─of up to a factor of 14 in test simulations of water clusters─are obtained via a straightforward GPU-based implementation as compared to conventional CPU calculations. We also find that further performance optimization is accessible through sagacious selection of numerical integration algorithms, and the adaptive methods, such as the Cash-Karp integrator, provide an effective balance between computing costs and numerical stability. Finally, we demonstrate that a simple mixed-step integrator based on the conventional fourth-order Runge-Kutta approach is capable of stable propagations even for strong external fields, provided the time step is appropriately adapted to the duration of the laser pulse with only minimal computational overhead.

Pulsating Hydromagnetic Flow of Chemically Reactive Oldroyd-B Nanofluid in a Channel with Brownian Motion, Thermophoresis, and Joule Heating

In the present study, the magnetohydrodynamic pulsating flow of chemically reacting Oldroyd-B nanoliquid via channel with the impressions of Ohmic heating, radiative heat and viscous dissipation is studied. The ruling PDEs (partial differential equations) are changed into ODEs (Ordinary differential equations) by utilizing the perturbation procedure and numerically deciphered by adopting the 4th order Runge-Kutta approach with the aid of the shooting process. The novelty of the current work is to inspect the pulsating flow of Oldroyd-B nanoliquid via channel in the occurrence of an applied magnetic field by deploying the Buongiorno nanofluid model. The application of the proposed physical model is energy production, heating and cooling processes, thermoelectric devices, bio-medical applications like brain tumours, cancer treatment, drug targeting. Detailed analysis on the impacts of several pertinent parameters for velocity, temperature, nanoparticles concentration, rates of heat and mass transfer is done. The outcomes predict that the velocity of nanoliquid is improved with augmenting frequency parameter while it is reduced with acceleration in Hartmann number. The temperature rises with an improvement of thermophoresis, viscous dissipation, and Brownian motion while it falls for a given rise in Hartmann number and thermal radiation. Further, the nanoparticle concentration rises with an increasing Brownian motion while it falls over rising chemical reaction, thermophoresis, and Lewis number.

An efficient symplectic stereo-modeling method for seismic inversion by using deep learning technique

AbstractThis paper proposes an efficient symplectic stereo-modeling (SSTEM) method for full waveform inversion (FWI) by using a deep learning technique. To solve the 2D acoustic equation, the SSTEM method uses a third-order optimal symplectic partitioned Runge–Kutta approach as a time-stepping method. An eighth-order stereo-modeling operator is used for spatial discretization. The SSTEM method is then expressed with a recurrent neural network (RNN). This is realized mainly because the time advancing format of the SSTEM method is similar to that of RNN, and they both use the information from the previous time step to obtain information from the current time step. With SSTEM as the forward modeling method, FWI is implemented using Tensorflow. The well-known adaptive moment estimation (Adam) optimizer and Nesterov adaptive moment estimation (Nadam) optimizer with mini-batch are used. The applicability of the developed code is also verified on GPUs. The numerical results show that the SSTEM method is more efficient and produces less numerical dispersion than the conventional finite-difference (FD) method when the same sampling rate in a wavelength is used. We compare several loss functions. The mean square (MSE) error and absolute (ABS) error loss functions are first tested. Another loss function that adds a physical differential operator to the original loss function is then considered. The FWI results show that this loss function has some improvements. Finally, we implement FWI on the complex Marmousi and SEG/EAGE models, and the inversion results demonstrate that the proposed method is suitable for seismic imaging in complex media.

Nonlinear dynamic instability of the perovskite solar cell under biaxial mechanical impacts

The perovskite solar cell (PSC) is one of the most promising game-changers in the electro-optic market. This paper examines the nonlinear dynamic performance and stability of the PSC on elastic foundations under different support conditions subjected to various biaxial impacts. By incorporating the third-order shear deformation theory (TSDT) and von-Kármán geometric nonlinearity, the governing equations of the nonlinear dynamic analysis are deduced by employing the Galerkin method. By using the fourth-order Runge-Kutta approach, the nonlinear governing equations can be solved and subsequently, the dynamic buckling behaviour of the structure can be assessed. Multifaceted effects of the velocity impacts, boundary conditions, dynamic buckling criteria, initial imperfections, damping, elastic foundations, plate theories, and impulse loadings on mechanical performance and dynamic stability of the novel PSC are investigated in the numerical study. These numerical investigations inspire the holistic design of practical energy-harvesting devices with strengthened structure capacity against external impulses.

Metode Runge-Kutta Orde 4 Dalam Penyelesaian Persamaan Gelombang 1D Syarat Batas Dirichlet

The problem studied in this study was the movement of deviations in the equation of 1D waves on the rope given the initial deviation value. In this study, the equation of 1D waves with dirichlet boundary problems will be solved analytically using variable and numerical separation using the runge-kutta method approach of order 4. This research begins with examining equation models, solving analytical solutions, applying schemes to the final results of simulations. The decrease in equation model fissile is done by reviewing the working force on a piece of rope, solving an analytical solution with a variable separation method, and numerical completion resulting in the best simulation result with parameter c= 0.5 with time steps ∆x= 0.01 and time interval of 0 ≤ t≤ 1 resulting in a numerical approach close to its analytical solution up to t= 1 s and the influence of parameter c on the movement of wave deviations resulted in that the influence of parameter c affects the magnitude of the deviation of the wave, so the greater the value of parameter c, the smaller deviation of the wave and the faster the speed of the direction. Thus, it can be concluded that the runge-kutta method of order 4 in this study can be said to be one of the numerical approaches of the problem of 1D wave equations with dirichlet boundaries.

Chemical reaction impact on MHD dissipative Casson-Williamson nanofluid flow over a slippery stretching sheet through porous medium

In this study, we investigated the heat and mass transport properties of a non-Newtonian Casson-Williamson nanofluid flow. We explore the effect of viscous dissipation and the velocity slip boundary condition on the mechanism of heat and mass transfer due to a stretching sheet which embedded in a porous medium with heat generation under the influence of both thermal radiation and a uniform magnetic field. All physicochemical characteristics of Casson-Williamson nanofluid are considered to be constant. The nanofluid concentration is investigated under chemical repercussions as a result of the movement of the nanofluid particles. This study assumes that there is no suction (solid wall). A set of nonlinear partial differential equations with boundary conditions are used to mathematically model this physical problem. The numerical solution for the differential equations with the related boundary conditions was illuminated using the Runge–Kutta approach in conjunction with the shooting technique. The numerical examination is then pictorial displayed to show the impact of various governing factors on velocity, temperature, and concentration. The non-Newtonian nanofluid has a faster velocity in the absence of a magnetic field than in the presence of it, although the temperature field has the opposite trend. Further, the skin-friction coefficient increased as the porosity parameter increased, whereas the rate of heat transfer dropped.

High-order multirate explicit time-stepping schemes for the baroclinic-barotropic split dynamics in primitive equations

In order to treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. Based on the framework of strong stability-preserving Runge-Kutta approach, we propose two high-order multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) for the resulting split system in this paper. The proposed schemes allow for a large time step to be used for the three-dimensional baroclinic (slow) mode and a small time step for the two-dimensional barotropic (fast) mode, in which each of the two mode solves just need to satisfy their respective CFL conditions for numerical stability. Specifically, at each time step, the baroclinic velocity is first computed by advancing the baroclinic mode and fluid thickness of the system with the large time-step and the assistance of some intermediate approximations of the barotropic mode obtained by substepping with the small time step; then the barotropic velocity is corrected by using the small time step to re-advance the barotropic mode under an improved barotropic forcing produced by interpolation of the forcing terms from the preceding baroclinic mode solves; lastly, the fluid thickness is updated by coupling the baroclinic and barotropic velocities. Additionally, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are relieved via a reconciliation process with carefully calculated flux deficits. Two benchmark tests from the “MPAS-Ocean” platform are carried out to numerically demonstrate the performance and parallel scalability of the proposed SSPRK-SE schemes.

Impact of nonlinear radiation on magnetohydrodynamic flow of hybrid nanofluid with heat source effect

This investigation is performed to deliberate the novel significance of nonlinear thermal radiation effect on magnetohydrodynamic flow of Casson hybrid nanoliquid caused by a curved stretching sheet. The flow and thermal transport nature of three different hybrid nanoliquids in the stimulus of viscous dissipation, nonlinear radiation and magnetic force is discussed numerically. The novelty of this work is to obtain the simultaneous solutions for three different types of hybrid nanoliquids, they are obtained by suspending titanium alloy (Ti6Al4vs.) is composed of titanium, aluminum and vanadium in the ratio of 90:6:4 respectively and aluminium alloy (AA7075) is compossed of Aluminum, Zinc, Magnesium and Copper nanoparticles in the ratio of 90:6:3:1 respectives with added metals Silicon, Ferrous and Magnesium. in three various working liquids viz. methanol, ethylene glycol and engine oil. The flow governing PDE's are transmuted into ODE's with suitable similarity transformation and solved by using Runge-Kutta and Newton's approach. Numerical outcomes of flow and temperature profiles are presented via graphical trends, also skin friction coefficient and rate of thermal transfer are illustrated via tabulated values. Major outcomes reveals that, inclusion of hybrid nanometer sized particles in pedestal liquid lead to an large hike in thermal transfer performance. Importantly, thermal transport rate and temperature profiles of the hybrid nanoliquid rises with improving nonlinear radiation values and these results are significant in nonlinear radiation case as compared with linear radiation case.

Computational Stochastic Investigations for the Socio-Ecological Dynamics with Reef Ecosystems

The motive of this work is to present a computational design using the stochastic scaled conjugate gradient (SCG) neural networks (NNs) called as SCGNNs for the socio-ecological dynamics (SED) with reef ecosystems and conservation estimation. The mathematical descriptions of the SED model are provided that is dependent upon five categories, macroalgae M(v), breathing coral C(v), algal turf T(v), the density of parrotfish P(v) and the opinion of human opinion X(v). The stochastic SCGNNs process is applied to formulate the SED model based on the sample statistics, testing, accreditation and training. Three different variations of the SED have been provided to authenticate the stochastic SCGNNs performance through the statics for training, accreditation, and testing are 77%, 12% and 11%, respectively. The obtained numerical performances have been compared with the Runge-Kutta approach to solve the SED model. The reduction of mean square error (MSE) is used to investigate the numerical measures through the SCGNNs for solving the SED model. The precision of the SCGNNs is validated through the comparison of the results and the absolute error performances. The reliability of the SCGNNs is performed by using the correlation values, state transitions (STs), error histograms (EHs), MSE measures and regression analysis.

Optimal Prevention and Treatment Control on SVEIR Type Model Spread of COVID-19

COVID-19 pandemic has disrupted the world's health and economy and has resulted in many deaths since the first case occurred in China at the end of 2019. Moreover, The COVID-19 disease spread throughout the world, including Indonesia on March 2, 2020. Coronavirus quickly spreads through droplets of phlegm through the throat to the lungs. Researchers in the medical field and the exact science are jointly examined the spread, prevention, and optimal control of COVID-19 disease. Due to the prevention of COVID-19, a vaccine has been found in early 2021, which at the time, the vaccination process was carried out worldwide against COVID-19. This paper examines the spread model of SVEIR-type COVID-19 by considering the vaccination subpopulation. In this study, control of prevention efforts ( and ) and healing efforts are given and being analyzed with the fourth-order Runge-Kutta approach. Based on numerical simulations, it can be seen that using the controls and can reduce the number of infected individuals in the subpopulation compared to those without control. The control can increase the number of recovered individual subpopulations.Keywords: COVID-19; SVEIR model; optimal control; treatment; vaccination.

A two-stage fourth-order gas-kinetic CPR method for the Navier-Stokes equations on triangular meshes

An efficient gas-kinetic scheme with fourth-order accuracy in both space and time is developed for the Navier-Stokes equations on triangular meshes. The scheme combines an efficient correction procedure via reconstruction (CPR) framework with a robust gas-kinetic flux formula, which computes both the flux and its time-derivative. The availability of the flux and its time-derivative makes it straightforward to adopt an efficient two-stage temporal discretization to achieve fourth-order time accuracy. In addition, through the gas-kinetic evolution model, the inviscid and viscous fluxes are coupled and computed uniformly without any separate treatment for the viscous fluxes. As a result, the current scheme is more efficient than traditional explicit CPR methods with a separate treatment for viscous fluxes, and a fourth-order Runge-Kutta approach. Furthermore, a robust and accurate subcell finite volume limiting procedure is extended to the CPR framework for troubled cells, resulting in subcell resolution of flow discontinuities. Numerical tests demonstrate the high accuracy, efficiency and robustness of the current scheme in a wide range of inviscid and viscous flow problems from subsonic to supersonic speeds.

Nonlinear dynamic investigation of the perovskite solar cell with GPLR-FGP stiffeners under blast impact

The perovskite solar cell (PSC), as a potential disruptive space market entrant, has fascinated both the scientific community and aerospace industry due to the high specific power, flexibility, and low fabrication cost. With the aim of reducing structure weight while strengthening the blast-carrying capacity of the PSC, the novel graphene platelets (GPL) reinforced functionally graded porous (GPLR-FGP) stiffeners have been incorporated as enhancements against impact. This research explores the nonlinear dynamic characteristics of the PSC with GPLR-FGP stiffeners under blast load. Integrating the von-Kármán geometric nonlinearity and the first-order shear deformation theory, the governing motion equations are derived by utilizing Airy's stress function and the Galerkin method. The fourth-order Runge-Kutta approach is employed to capture the solutions of dynamic equations effectively. Diverse influences of the stiffener material, boundary condition, plate theory, porosity distribution, GPL dispersion, porosity coefficient, GPL weight fraction, GPL geometry, damping, nonlinear elastic foundation, and initial imperfection are investigated in the numerical study. Besides, the optimal parameters of the PSC with GPLR-FGP stiffeners are discovered, facilitating the following paces of space design and practical implementation in extraterrestrial circumstances.

Comparisons of Safety Factors for Slope in Nonlinear Soils

Slope stability assessment is one of the most important issues for geotechnical engineers. In the framework of a nonlinear failure criterion, four different types of safety factors are presented and their relationships are investigated for soil slope in this study. The variational method is incorporated into the kinematic approach of limit analysis to assess the stability of soil slope with known geometric boundary, and seismic effects are considered. The seismic loads are simplified as external forces acting on the slope. A rotational failure mechanism is used to describe the sliding mode of slopes. The differential equations for the sliding surface and corresponding stress distribution are derived using the variational method and then are employed to generate the sliding surface via the fourth-order Runge-Kutta approach. To avoid the computational complexity, the energy-work balance equation of the kinematic approach, instead of the static equilibrium equation, is used to judge whether the state of the slope is critical. The safety factor is designed as the minimum factor that brings the slope in the limit state. Computational schemes are proposed to calculate the values of safety factors. The validity of the proposed approach is demonstrated through comparison with previous works. Finally, parametric study is conducted to further reveal the relationships among the four types of safety factors.

Investigation of Dynamic Load Sharing Behavior for Herringbone Planetary Gears considering Multicoupling Manufacturing Errors

In this study, based on the lumped-parameter theory and the Lagrange approach, a novel and generalized bending-torsional-axial coupled dynamic model for analyzing the load sharing behavior in the herringbone planetary gear train (HPGT) is presented by taking into account the actual structure of herringbone gears, manufacturing errors, time-dependent meshing stiffness, bearing deflections, and gyroscopic effects. The model can be applied to the analysis of the vibration of the HPGT with any number of planets and different types of manufacturing errors in different floating forms. The HPGT equivalent meshing error is analyzed and derived for the tooth profile errors and manufacturing eccentric errors of all components in the HPGT system. By employing the variable-step Runge–Kutta approach to calculate the system dynamic response, in conjunction with the presented calculation approach of the HPGT load sharing coefficient, the relationships among manufacturing errors, component floating, and load sharing are numerically obtained. The effects of the combined errors and single error on the load sharing are, respectively, discussed. Meanwhile, the effects of the support stiffness of the main components in the HPGT system on load sharing behavior are analyzed. The results indicate that manufacturing errors, floating components, and system support stiffness largely influence the load sharing behavior of the HPGT system. The research has a vital guiding significance for the design of the HPGT system.

Explicit high-order exponential time integrator for discontinuous Galerkin solution of Maxwell's equations

In this paper, exponential propagator (EP) is introduced to the discontinuous Galerkin finite element method (DG-FEM) for solving transient electromagnetic problems. As a kind of explicit time-stepping method, the proposed EP has fourth order accuracy in time and requires only one storage unit for each of the degrees-of-freedom. It provides an attractive alternative to the high-order low-storage explicit Runge-Kutta (LSERK) approach, which is the most popular time-marching scheme currently used in high-fidelity simulation of Maxwell's equations with DG-FEM. When centered numerical flux is utilized, the discretized Maxwell's system in lossless region is symplectic structure-preserving (energy conserving), and thus very suitable for accurately computing the long-time propagation of waves in non-dissipative medium, or simulating the motion of charged particles in electromagnetic fields. Our method can also handle upwind numerical flux at the sacrifice of symplectic-conserving virtue for higher accuracy in spatial discretization and spurious inhibition. A major contribution of this work is that, the application range of this scheme is extended to lossy media such as conductors and particularly unsplit perfectly matched layers in the framework of DG-FEM. This greatly facilitates the treatment of open boundary conditions in scattering and/or radiation problems. Extensive numerical examples, including the analysis of photonic crystals, dielectric coated perfect conducting cylinder, propagation of waves in zero-index materials, and corrugated O-type Cherenkov oscillators etc., are presented to demonstrate its stability, accuracy, and performance. All the numerical experiments show that the presented method is superior to the LSERK scheme when centered numerical flux is adopted.

Nonlinear Static and Dynamic Buckling Analysis of Imperfect Stiffened Double-Layer FGP Shallow Spherical Shells Embedded Within Elastic Foundations

In this work, an analytical method is presented to analyze the nonlinear static and dynamic buckling of imperfect stiffened double-layer functionally graded porous (FGP) shallow shells subjected to external pressure. The shallow spherical shell consists of two layers, where upper layer of shell is nonporous functionally graded (FG) material and the lower layer of shell is FG porous. This spherical shell is resting on elastic foundations, which consist of two-term Winkler–Pasternak’s model. Two FG porous distributions types consisting of symmetric porosity distribution (SPD) and nonsymmetric porosity distribution (NSPD) are investigated. The porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam. Utilizing the classical theory of shells considering geometrical nonlinearity based on von Kármán–Donnell relations and Hooke’s law, the governing equations of stiffened double-layer FG porous spherical shells are extracted. To discretize the partial differential equation of shell and obtaining the system equations of motion, the Galerkin method is applied. Regarding the discretized equation of motion, the explicit expressions for dynamic and static critical buckling load are determined. To investigate the dynamic behavior, the equation of motion is solved via a numerical method named Runge–Kutta approach, and then a presented approach by Budiansky–Roth is employed to obtain a critical load for the nonlinear dynamic buckling. The influence of various porosity distributions and coefficients of the shallow shell, elastic foundations, and geometrical characteristics on the nonlinear static and dynamic buckling analysis of imperfect stiffened double-layer FG porous shallow shells is examined.