Search Results (49)

This paper studies the reliability and availability of a k-out-of-(M+S) retrial machine repair system with two-way communication, consisting of M primary components and S warm standby components. The system incorporates the retrial behavior of failed components. When the repairman becomes idle, he initiates outgoing calls after a random period either to failed components in the orbit for repair or to components outside the orbit for preventive maintenance. The main contribution of this study is the incorporation of proactive repairman behavior, which more realistically captures operational practices in certain engineering systems. By employing the matrix analytic method together with a recursive approach, the steady-state probabilities of the system are obtained, and several important performance measures are derived. Furthermore, the Runge–Kutta method is used to evaluate the system reliability and the mean time to failure. A sensitivity analysis is conducted to investigate the effects of key system parameters, supported by numerical experiments and graphical illustrations. Finally, a cost–benefit model is formulated, and a genetic algorithm is implemented to determine the optimal values of the decision variables that minimize the cost–benefit ratio. Full article

Vessel vibrations have serious safety risks and must be effectively mitigated. This study investigates the reduction in ship pitch–roll vibrations modeled as a two degrees of freedom of nonlinear spring–pendulum system subjected to multi-parametric excitation, using proportional–derivative controller. The main objective is to develop a rapid and efficient analytical approach to nonlinear vibration analysis. A non-perturbative approach is employed to transform weakly nonlinear oscillators of ordinary differential equations into equivalent linear ones without using Taylor expansions. He’s frequency formula plays a central role in this transformation. The resulting parametric solutions are validated using Mathematica Software (v13) and show a strong agreement with the original nonlinear model. The effects of various parameters on stability are examined. Theoretical analysis is conducted using the multiple time scales method to identify worst resonance conditions and derive frequency response equations. Stability near simultaneous sub-harmonic resonance is assessed using Routh–Hurwitz criterion. Numerical simulations based on the fourth-order Runge–Kutta method confirm the effectiveness of proportional–derivative control. Excellent agreement between analytical and numerical results demonstrates the accuracy, efficiency, and practical applicability of the proposed method. Full article

This study introduces an efficient and accurate two-stage explicit computational scheme for solving partial differential equations (PDEs) containing first-order time derivatives. The suggested method is a modification of the classical Runge–Kutta scheme that introduces a new first-stage formulation. This minimizes numerical error with moderate step sizes while preserving the stability region of the classical method. Spatial discretization is performed using a sixth-order compact finite-difference scheme to obtain high-resolution solutions. The analysis of stability and convergence is strictly determined for both scalar and system forms of convection–diffusion-type equations. To illustrate the suitability of the method, a dimensionless mathematical model of the unsteady, incompressible, laminar flow of a Prandtl-type non-Newtonian nanofluid over a Riga plate is considered, accounting for viscous dissipation, thermophoresis, Brownian motion, and a magnetic field. Here, the Prandtl ternary nanofluid is defined as a non-Newtonian nanofluid that follows the Prandtl rheological model, and it exhibits three critical transport phenomena: heat conduction, viscous dissipation, and nanoparticle diffusion. Representative values of the Prandtl number $P_r=3$ and Reynolds number $R_e=5$ are used to perform the simulation, and other parameters, including but not limited to the Hartmann number $H_a$, Williamson number $W_e$, thermophoresis $N_t$ and Brownian motion $N_b$, are varied to evaluate the flow behavior. Moreover, an artificial neural network (ANN)-developed surrogate model is used to calculate the skin friction coefficient and the local Sherwood number, using five input parameters: the Reynolds number, Prandtl number, Schmidt number, Brownian motion parameter, and thermophoresis parameter. The governing partial differential equations yield high-fidelity numerical data used to train the surrogate model. The data is split into 80% for training, 10% for validation, and 10% for testing. The ANN is tested using regression analysis and error histograms, which demonstrate high accuracy and generalization capacity. Numerical simulation combined with AI-based prediction is a cost-efficient method for real-time estimation of complex non-Newtonian nanofluid systems. Full article

Vibration is a critical issue in aerospace structures, where lightweight design, high flexibility, and complex operational environments often lead to pronounced nonlinear dynamic responses. Excessive vibrations induced by harmonic excitations, aerodynamic loads, or onboard equipment can significantly degrade structural integrity, control accuracy, and service life. Consequently, advanced passive vibration suppression techniques with strong robustness and broadband effectiveness are of great importance in aerospace engineering applications. The bifurcation boundary and vibration suppression performance of Piezoelectric–Monostable Nonlinear Energy Sink (PMNES) are crucial for evaluating its effectiveness on the main structure. To simplify the analysis of flexible aerospace structures, a reduced-order model is derived by modal truncation in the low-frequency range, which is then treated as a two-degree-of-freedom main structure. To focus on the underlying nonlinear dynamic mechanisms, an equivalent two-degree-of-freedom lumped-parameter system is adopted as a generic representation of the dominant low-frequency dynamics of flexible aerospace structures. In this work, the electromechanical coupling control equations of the system of a two-degree-of-freedom main structure coupled with PNES are derived through the application of Newton’s second law and Kirchhoff’s voltage law. The methods of complexification-averaging (CX-A) and Runge–Kutta (RK) are employed to assess the vibration suppression performance and stability characteristics of the system under harmonic excitation. The approximate solution is validated through numerical solutions. The approximate solutions of the system are employed to derive the Saddle Node (SN) bifurcation and codimension-two cusp bifurcation points, while the enhanced algorithm is employed to ascertain the most unfavorable amplitude at each external excitation circular frequency and to determine whether the mark represents a Hopf Bifurcation (HB) point. The generalized transmissibility is utilized to assess the efficacy of vibration suppression. The various vibration suppression efficiency regions are created by superimposing the vibration suppression efficiency maps and bifurcation maps. The influence of PNES parameters on the vibration suppression region is investigated. The results indicate that this method can effectively evaluate the bifurcation boundary and vibration suppression performance of PMNES. Full article

We present the ContEvol (continuous evolution) formalism, a family of implicit numerical methods which only need to solve linear equations and are almost symplectic. Combining values and derivatives of functions, ContEvol outputs allow users to recover full history and render full distributions. Using the classic harmonic oscillator as a prototype case, we show that ContEvol methods lead to lower-order errors than two commonly used Runge–Kutta methods. Applying first-order ContEvol to simple celestial mechanics problems, we demonstrate that deviation from equation(s) of motion of ContEvol tracks is still 𝒪(h⁵) (h is the step length) by our definition. Numerical experiments with an eccentric elliptical orbit indicate that first-order ContEvol is a viable alternative to classic Runge–Kutta or the symplectic leapfrog integrator. Solving the stationary Schrödinger equation in quantum mechanics, we manifest ability of ContEvol to handle boundary value or eigenvalue problems. Important directions for future work, including mathematical foundations, higher dimensions, and technical improvements, are discussed at the end of this article. Full article

Predicting wildland fire spread requires numerical schemes that can resolve sharp gradients at the fireline while remaining stable and efficient on practical grids. We develop a compact high-order finite-difference scheme for Hamilton–Jacobi level-set formulations of wildfire propagation, based on the anisotropic spread law of Mallet and co-authors. The spatial discretization employs a compact finite-difference derivative scheme to achieve spectral-like resolution with narrow stencils, improving accuracy and boundary robustness compared with wide-stencil ENO/WENO reconstructions. To control high-frequency artifacts intrinsic to non-dissipative compact schemes, an implicit high-order low-pass filter is incorporated and activated after each Runge–Kutta stage. Convergence is verified on the eikonal expanding-circle benchmark, where the method attains the expected high-order spatial accuracy as the grid is refined. The proposed scheme is then applied to wind-driven wildfire simulations governed by Mallet’s non-convex Hamiltonian, including a single ignition under moderate and strong wind. A complex topology test case is also considered, involving two ignitions that merge into a single front with the evolution of an internal unburnt island. The results demonstrate that the proposed method accurately reproduces fireline evolution even on coarse grids, achieving accuracy comparable to fifth-order WENO while maintaining superior fidelity in complex fireline topologies, where it better resolves multi-front interactions and topological changes in the fireline. This makes the method an efficient, accurate alternative for level-set wildfire modeling and readily integrable into existing frameworks. Full article

This paper presents the development and analysis of novel explicit special two-derivative Runge–Kutta (STDRK) pairs for the numerical integration of ordinary differential equations (ODEs), with a focus on achieving seventh-order accuracy and embedded fifth-order error estimation. The proposed schemes utilize both the first and second derivatives of the solution, leveraging the identity $y^{\prime \prime }=f^{\prime }\left(y\right)f\left(y\right)$, to attain high-order accuracy while minimizing the number of evaluations of the primary function f. A notable feature of the constructed methods is that they require only a single evaluation of f per step, along with five evaluations of $g=f^{\prime }f$, resulting in a significant reduction in computational cost compared to classical Runge–Kutta methods. The necessary order conditions are derived via an algebraic framework based on compositions with parts not exceeding 2. A supporting Mathematica package facilitates the construction of methods of arbitrary order. A new STDRK pair of orders seven and five is derived. Numerical experiments on standard benchmark problems, including the Prothero–Robinson, Kaps, and Kepler systems, highlight the efficiency and competitive performance of the proposed schemes relative to established Runge–Kutta pairs. Full article

A continuum model for describing pseudo-turbulent flows of a dispersed phase is developed using a statistical approach based on the kinetic equation for the probability density of particle velocity and temperature. The introduction of the probability density function enables a statistical description of the particle ensemble through equations for the first and second moments, replacing the dynamic description of individual particles derived from Langevin-type equations of motion and heat transfer. The lack of detailed dynamic information on individual particle behavior is compensated by a richer statistical characterization of the motion and heat transfer within the particle continuum. A numerical simulation of the unsteady flow of a gas–particle suspension generated by the interaction of a shock wave with a particle cloud is performed using an interpenetrating continua model and equations for the first and second moments of both gas and particles. Numerical methods for solving the two-phase gas dynamics equations—formulated using a two-velocity and two-temperature model—are discussed. Each phase is governed by conservation equations for mass, momentum, and energy, written in a conservative hyperbolic form. These equations are solved using a high-order Godunov-type numerical method, with time discretization performed by a third-order Runge–Kutta scheme. The study analyzes the influence of two-dimensional effects on the formation of shock-wave flow structures and explores the spatial and temporal evolution of particle concentration and other flow parameters. The results enable an estimation of shock wave attenuation by a granular backfill. The extended pressure relaxation region is observed behind the cloud of particles. Full article

In this work, explicit Two-Derivative Runge–Kutta methods of the general case (that use several evaluations of the right-hand side function and its derivative per step) are considered. In order to address problems of orbital or oscillatory character, we develop methods with frequency-dependent coefficients. We construct three exponentially/trigonometrically fitted methods following two approaches suggested by Vanden Berghe and Simos. Also, we construct a phase-fitted and amplification-fitted method. The efficiency of the new modified methods is demonstrated by numerical examples. Full article

In this paper, general Runge–Kutta–Nyström (GRKN) methods are developed and analyzed, tailored for second-order initial value problems of the form $y^{″}=L{y}^{\prime }+My+g\left(t\right)$, where $L,M\in {\mathbb{R}}^{n\times n}$ are constant matrices with $n\ge 1$. The construction of embedded pairs of orders $6\left(4\right)$ and $7\left(5\right)$, suitable for adaptive integration strategies, is emphasized. By utilizing rooted tree theory and recent simplifications for linear inhomogeneous systems, symbolic order conditions are derived, and efficient schemes are designed through algebraic and evolutionary techniques. Numerical tests verify the superiority of our new derived pairs. In particular, this work introduces novel embedded GRKN pairs with reduced-order conditions that exploit the linearity and structure of the underlying system, enabling the construction of low-stage, high-accuracy integrators. The methods incorporate FSAL (First Same As Last) formulations, making them computationally efficient. They are tested on representative physical systems in one, two, and three dimensions, demonstrating notable improvements in efficiency and accuracy over existing high-order RKN methods. Full article

We propose a new strategy for constructing Runge–Kutta (RK) methods using evolutionary computation techniques, with the goal of directly minimising global error rather than relying on traditional local properties. This approach is general and applicable to a wide range of differential equations. To highlight its effectiveness, we apply it to two benchmark problems with oscillatory behaviour: the (2+1)-dimensional nonlinear Schrödinger equation and the N-Body problem (the latter over a long interval), which are central in quantum physics and astronomy, respectively. The method optimises four free coefficients of a sixth-order, eight-stage parametric RK scheme using a novel objective function that compares global error against a benchmark method over a range of step lengths. It overcomes challenges such as local minima in the free coefficient search space and the absence of derivative information of the objective function. Notably, the optimisation relaxes standard RK node bounds ($c_i\in [0,1]$), leading to improved local stability, lower truncation error, and superior global accuracy. The results also reveal structural patterns in coefficient values when targeting high eccentricity and non-sinusoidal problems, offering insight for future RK method design. Full article

Under cyclic moving load action, tensile-dominant structures are prone to crack initiation due to cumulative damage effects. The presence of cracks leads to structural stiffness degradation and nonlinear redistribution of dynamic characteristics, thereby compromising str18uctural integrity and service performance. The current research on the dynamic behavior of cracked structures predominantly focuses on transient analysis through high-fidelity finite element models. However, the existing methodologies encounter two critical limitations: computational inefficiency and a trade-off between model fidelity and practicality. Thus, this study presents an innovative analytical framework to investigate the dynamic response of cracked simply supported beams subjected to moving loads. The proposed methodology conceptualizes the cracked beam as a system composed of multiple interconnected sub-beams, each governed by the Euler–Bernoulli beam theory. At crack locations, massless rotational springs are employed to accurately capture the local flexibility induced by these defects. The transfer matrix method is utilized to derive explicit eigenfunctions for the cracked beam system, thereby facilitating the formulation of coupled vehicle–bridge vibration equations through modal superposition. Subsequently, dynamic response analysis is conducted using the Runge–Kutta numerical integration scheme. Extensive numerical simulations reveal the influence of critical parameters—particularly crack depth and location—on the coupled dynamic behavior of the structure subjected to moving loads. The results indicate that at a constant speed, neither crack depth nor position alters the shape of the beam’s vibration curve. The maximum deflection of beams with a 30% crack in the middle span increases by 14.96% compared to those without cracks. Furthermore, crack migration toward the mid-span results in increased mid-span displacement without changing vibration curve topology. For a constant crack depth ratio (γ_i = 0.3), the progressive migration of the crack position from 0.05 L to 0.5 L leads to a 26.4% increase in the mid-span displacement (from 5.3 mm to 6.7 mm). These findings highlight the efficacy of the proposed method in capturing the complex interactions between moving loads and cracked concrete structures, offering valuable insights for structural health monitoring and assessment. Full article

A time-consistent upwind difference scheme with a preconditioned numerical flux for unsteady gas-liquid multiphase flows is presented and applied to the analysis of cavitating flows. The fundamental equations were formulated in general curvilinear coordinates to apply to diverse flow fields. The preconditioning technique was applied specifically to the numerical dissipation terms in the upwinding process without changing the time derivative terms to maintain time consistency. This approach enhances numerical stability in unsteady multiphase flow computations, consistently delivering time-accurate solutions compared to conventional preconditioning methods. A homogeneous gas-liquid two-phase flow model, third-order Runge-Kutta method, and the flux difference splitting upwind scheme coupled with a third-order MUSCL TVD scheme were employed. Numerical tests of two-dimensional gas-liquid single- and two-phase flows over backward-facing step with different step height and flow conditions successfully demonstrated the capability of the present scheme. The calculations remained stable even for flows with a very low Mach number of 0.001, typically considered incompressible flows, and the results were in good agreement with the experimental data. In addition, we analyzed unsteady cavitating flows at high Reynolds numbers and confirmed the effectiveness and applicability of the present scheme for calculating unsteady gas-liquid two-phase flows. Full article

To solve the nonlinear vibration problems of second- and third-order nonlinear oscillators, a modified harmonic balance method (HBM) is developed in this paper. In the linearized technique, we decompose the nonlinear terms of the governing equation on two sides via a constant weight factor; then, they are linearized with respect to a fundamental periodic function satisfying the specified initial conditions. The periodicity of nonlinear oscillation is reflected in the Mathieu-type ordinary differential equation (ODE) with periodic forcing terms appeared on the right-hand side. In each iteration of the linearized harmonic balance method (LHBM), we simply solve a small-size linear system to determine the Fourier coefficients and the vibration frequency. Because the algebraic manipulations required for the LHBM are quite saving, it converges fast with a few iterations. For the Duffing oscillator, a frequency–amplitude formula is derived in closed form, which improves the accuracy of frequency by about three orders compared to that obtained by the Hamiltonian-based frequency–amplitude formula. To reduce the computational cost of analytically solving the third-order nonlinear jerk equations, the LHBM invoking a linearization technique results in the Mathieu-type ODE again, of which the harmonic balance equations are easily deduced and solved. The LHBM can achieve quite accurate periodic solutions, whose accuracy is assessed by using the fourth-order Runge–Kutta numerical integration method. The optimal value of weight factor is chosen such that the absolute error of the periodic solution is minimized. Full article

Effects of the road, such as speed bumps, can significantly affect a car’s stability. This study focuses on how a quarter-car model is affected by a basic harmonic speed hump and how Cubic Negative Velocity Control (CNVC) is used to control the amplitude of disturbances. This study differs from earlier research in considering various control and force kinds that impact the system. The external forces in this context are a component of a non-linear dynamic system. Two-degree-of-freedom (2DOF) differential coupled equations describe the system’s equation. Numerous numerical experiments have been conducted, including proportional derivative (PD), negative derivative feedback (NDF), positive position feedback (PPF), linear negative velocity control (LNVC), and CNVC; the results show that when the hump is represented as a simple harmonic hump, CNVC has the best effect and can regulate vibrations more precisely than the other approaches on this system. Subsequently, the vibration value of the system was numerically analyzed both before and after the control was implemented. Using the frequency response equation and phase plane approaches in conjunction with the Runge–Kutta fourth order method (RK-4) in the context of resonance situation analysis, the stability of the numerical solution has been evaluated. Full article