Runge-Kutta Procedure Research Articles

Semi-analytical consistent zigzag-elasticity formulations with implicit layerwise shear correction factors for dynamic stress analysis of sandwich circular plates with FGM layers

In the present paper, a consistent power series solution is proposed for dynamic behaviors analysis of circular sandwich plates with functionally graded face sheets/cores, for the first time. The proposed solution is a combination of the finite Taylor’s transform and the fourth-order Runge–Kutta procedure. The extension-bending coupling and the higher-order inertias are considered in the present research. The formulation guarantees continuity of the transverse stress components at the interfaces between layers. Another novelty is proposing a zigzag formulation whose results are corrected on the basis of the dynamic three-dimensional theory of elasticity. An interesting discussion for defining explicit/implicit layerwise shear correction factors is also presented. Furthermore, results reveal that using functionally graded face sheets or cores with transition variations of the material properties may lead to very smooth through-the-thickness stress distributions. Consequently, some of the failure modes may be prevented at the interfaces between the layers.

Numerical simulation of microparticles transport in a concentric annulus by Lattice Boltzmann Method

Dispersion and removal of microaerosol particles are investigated numerically in a horizontal concentric annulus by Lattice Boltzmann Method and Lagrangian Runge–Kutta procedure with the assumption of one-way coupling. Drag, buoyancy, gravity, shear lift, Brownian motion and thermophoretic are forces that are included in particle equation of motion. All simulations were performed at Rayleigh number of 104 and particles specific density of 1000. The effect of aspect ratio and particles diameter were determined on particles behavior such as removal and dispersion. Results show that recirculation power increases by decreasing of cylinders gap. Particles move in a thinner quasi-equilibrium region by increasing of their diameter and decreasing of cylinders gap. Brownian motion is dominant removal mechanism in particle with diameter of 1μm.

A well-balanced high-resolution shape-preserving central scheme to solve one-dimensional sediment transport equations

We present a well-balanced high-resolution non-oscillatory central finite volume scheme for solving the shallow water equations with a non-flat bottom topology. Time integration is obtained following a Runge-Kutta procedure, coupled with its natural continuous extension. We use a central scheme with a point value reconstruction algorithm based on average or flux values, which satisfies the monotonicity preserving property. We apply a special treatment for the source term spatial integration, which preserves the time and space accuracy and it results in a well-balanced scheme. Several one-dimensional test cases are used to verify the behaviour and non-oscillatory properties of our scheme.

A transport modeling of the carbon-nitrogen cycle at Igapó I Lake - Londrina, Paraná State, Brazil

This work is a contribution to a better understanding of the effect that domestic sewage discharges may cause in a water body, specifically at Igapó I Lake, in Londrina, Paraná State, Brazil. The simulation of the dynamics of pollutant concentrations throughout the water body was conducted by means of structured discretization of the geometry of Igapó I Lake, together with the finite differences and the finite elements methods. Firstly, the hydrodynamic flow (without the pollutants), modeled by Navier-Stokes and pressure equations, was numerically resolved by the finite differences method, and associated with the fourth order Runge-Kutta procedure. After that, by using the hydrodynamic field velocity, the flow of the reactive species (pollutants) was described through a reaction transport model, restricted to the carbon-nitrogen cycle. The reaction transport model was numerically resolved by the stabilized finite elements method, by means of a semi-discrete formulation. A qualitative analysis of the numerical simulations provided a better understanding of the dynamics of the processes involved in the flow of the reactive species, such as the dynamics of the nitrification process, of the biochemical demand of oxygen and of the level of oxygen dissolved in the water body at Igapó I Lake.

A flexible model for the pricing of perishable assets

We present a flexible and versatile model which addresses the problem of assigning optimal prices to assets whose value becomes zero after a fixed expiry date. (Such assets include the important example of seats on airline flights.) Our model is broad in scope, in particular encompassing the ability to deal with arrivals of customers in groups. It is highly adaptable and can be adjusted to deal with a very extensive set of circumstances.Our approach to the problem is based on elementary and intuitively appealing ideas. We model the arrival of customers (or groups of customers) according to an inhomogeneous Poisson process. We incorporate into the model time dependent price sensitivity (which may also be described as “time dependent elasticity of demand”). In this setting the solution to the asset pricing problem is achieved by setting up coupled systems of differential equations which are readily amenable to numerical solution via (for instance) a vectorised Runge–Kutta procedure. An attractive feature of our approach is that it unifies the treatment of discrete and continuous prices for the assets.

Two‐dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation

SUMMARYIn this article, an extension to the total variation diminishing finite volume formulation of the lattice Boltzmann equation method on unstructured meshes was presented. The quadratic least squares procedure is used for the estimation of first‐order and second‐order spatial gradients of the particle distribution functions. The distribution functions were extrapolated quadratically to the virtual upwind node. The time integration was performed using the fourth‐order Runge–Kutta procedure. A grid convergence study was performed in order to demonstrate the order of accuracy of the present scheme. The formulation was validated for the benchmark two‐dimensional, laminar, and unsteady flow past a single circular cylinder. These computations were then investigated for the low Mach number simulations. Further validation was performed for flow past two circular cylinders arranged in tandem and side‐by‐side. Results of these simulations were extensively compared with the previous numerical data. Copyright © 2011 John Wiley & Sons, Ltd.

A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain

The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.

Basic features of a charged particle dynamics in a laser beam with static axial magnetic field

AbstractIn this paper, the trajectory and kinetic energy of a charged particle, subjected to interaction from a laser beam containing an additionally applied external static axial magnetic field, have been analyzed. We give the rigorous analytical solutions of the dynamic equations. The obtained analytical solutions have been verified by performing calculations using the derived solutions and the well known Runge-Kutta procedure for solving original dynamic equations. Both methods gave the same results. The simulation results have been obtained and presented in graphical form using the derived solutions. Apart from the laser beam, we show the results for a maser beam. The obtained analytical solutions enabled us to perform a quantitative illustration, in a graphical form of the impact of many parameters on the shape, dimensions and the motion direction along a trajectory. The kinetic energy of electrons has also been studied and the energy oscillations in time with a period equal to the one of a particle rotation have been found. We show the appearance of, so-called, stationary trajectories (hypocycloid or epicycloid) which are the projections of the real trajectory onto the (x, y) plane. Increase in laser or maser beam intensity results in the increase in particle’s trajectory dimension which was found to be proportional to the amplitude of the electric field of the electromagnetic wave. However, external magnetic field increases the results in shrinking of the trajectories. Performed studies show that not only amplitude of the electric field but also the static axial magnetic field plays a crucial role in the acceleration process of a charged particle.At the authors of this paper best knowledge, the precise analytical solutions and theoretical analysis of the trajectories and energy gains by the charged particles accelerated in the laser beam and magnetic field are lacking in up to date publications. The authors have an intention to clarify partly some important aspects connected with this process. The presented theoretical studies apply for arbitrary charged particle and the attached figures-for electrons only.

동적효과를 고려한 저류함수모형의 최적 매개변수 결정

강우로부터 유출현상은 고유적으로 비선형성이다. 더욱이 실제적으로 이와 같은 비선형성의 해석은 많은 어려움을 내포하고 있다. 또한, 부정류효과의 동적작용을 고려한 저류개념은 매개변수의 유역특성상 추정하기가 상당히 복잡하기 때문에 피해오고 있는 실정이다. 본 연구에서는 이와 같은 동적효과를 고려한 비선형의 저류함수에 대한 매개변수의 최적치를 얻고자 시도한다. 이를 위한 수치해법은 금강의 보청천유역의 관측치와 계산치의 오차를 최소로 하는 최소자승법에 의거 준선형화, Runge-Kutta 및 pattern-search 법들을 적용한다. 본 연구의 동적효과를 고려한 비선형 개념적모형의 적용성은 비선형성만을 고려한 저류함수모형 및 기존의 Nash 모형과 비교하여 검토하였다. 그 결과 2계모형이 l계모형보다 강우로부터 유출예측치를 보다 더 잘 재현하는 것을 알 수 있었으며, Nash 모형과는 대등함을 보여주었다. 여기서 획득된 매개변수들은 물리적 의미뿐만 아니라 본 모형의 국내 적용성도 제공한다. The basin response to storm is regarded as nonlinearity inherently. In addition, the consistent nonlinearity of hydrologic system response to rainfall has been very tough and cumbersome to be treated analytically. The thing is that such nonlinear models have been avoided because of computational difficulties in identifying the model parameters from recorded data. The parameters of nonlinear system considered as dynamic effects in the conceptual model are optimized as the sum of errors between the observed and computed runoff is minimized. For obtaining the optimal parameters of functions, the historical data for the Bocheong watershed in the Geum river basin were tested by applying the numerical methods, such as quasi-linearization technique, Runge-Kutta procedure, and pattern-search method. The estimated runoff carried through from the storage function with dynamic effects was compared with the one of 1st-order differential equation model expressing just nonlinearity, and also done with Nash model. It was found that the 2nd-order model yields a better prediction of the hydrograph from each storm than the 1st-order model. However, the 2nd-order model was shown to be equivalent to Nash model when it comes to results. As a result, the parameters of nonlinear 2nd-order differential equation model performed from the present study provided not only a considerable physical meaning but also a applicability to Korean watersheds.

Finite element and implicit Runge-Kutta implementation of an acoustics-convection upstream resolution algorithm for the time-dependent two-dimensional Euler equations

The second of a two-paper series, this paper details a solver for the characteristics-bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi-dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi-dimensional independently of the orientation of the reference axes and computational cells. The time-dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non-linearly stable for the spatially-discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non-oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique- and interacting-shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.

Effect of vibration on fatigue crack growth of an edge crack for a rectangular plate

The coupling analysis of vibration and fatigue crack growth for the rectangular plate with an edge crack is presented. The modified Forman's equation is employed to fatigue crack growth, and the plate equations of von Karman's theory are considered in vibration analysis. The equations of vibration are reduced to an ordinary differential equation by assuming mode shapes and Galerkin's procedure. The amplitude and natural frequency are determined by means of the Runge–Kutta scheme. The modified Forman's equation for crack propagation and the Runge–Kutta procedure for vibrating amplitude are simultaneously employed cycle by cycle. The effect of vibration on fatigue crack growth of the cracked rectangular plate is determined and discussed. The unsteady vibration affects the fatigue life evidently in this analysis.

Analysis and implementation of a parallelization strategy on a Navier–Stokes solver for shear flow simulations

A parallel computational solver for the unsteady incompressible three-dimensional Navier–Stokes equations implemented for the numerical simulation of shear flow cases is presented. The computational algorithms include Fourier expansions in the streamwise and spanwise directions, second-order centered finite differences in the direction orthogonal to the solid walls, third-order Runge–Kutta procedure in time in which both convective and diffusive terms are treated explicitly; the fractional step method is used for time marching. Based on the numerical algorithms implemented within the computational solver, three different (MPI based) parallelization strategies are devised. The three schemes are evaluated with particular attention to the impact of the communications onto the whole computational procedure, and one of them is implemented. Computations are executed on two different parallel machines and results are shown in terms of parallel performance. Processes using different number of processors combined with different number of computational grid points are tested.

Heat transfer on A/C humidifier operating by solar energy

A solar humidifier unit used for humidification of an air conditioning system is presented. A Runge–Kutta procedure of fifth order has been used to solve the governing equations. Solar energy data as well as ambient temperature data for one hot summer day were used in calculation the inside temperature of the humidifier as well as other parameters to determine the effective use of the solar humidifier. The results obtained are promising to corporate it with an A/C system. The amount of vapor produced reaches a maximum value at the time when the ambient temperature reaches also a maximum value and the relative humidity reaches a minimum values.

Modeling the anode boundary layer of high-intensity argon arcs

This paper is concerned with numerical studies of the anode region of a high-intensity argon arc. The anode region is divided into three subzones: the anode boundary layer, the presheath and the sheath. The governing equations of the dominating processes with the boundary conditions taken from the solutions of LTE plasmas are solved by applying the Runge–Kutta procedure. One parameter θ, i.e., the ratio of the heavy particle temperatures at the free-fall edge to the anode surface temperatures, is introduced into this study. The results indicate that the parameter θ is an important factor influencing the local plasma properties in the boundary layer, the boundary layer thickness and the sheath potential. An estimate of the value for θ is made. The potential drop in the boundary layer is slightly positive.

Analysis of dynamical behaviour of a rotor with deflection limiters

The dynamical behaviour of a rotor running into contact with a non-annular stator is studied by numerical simulations. The rotor is represented by a 2-dof model moving in one plane. The stator models an arbitrary number of discrete stops by superimposing annular stators with large clearances and high offsets. Friction forces are included in the contact description. The strongly nonlinear differential equations are integrated using a fourth-order Runge-Kutta procedure. Time histories for start-up and run-down sequences are simulated in two numerical examples. Different stop configurations have been evaluated. For a configuration with four symmetrically distributed stops an interesting bouncing phenomenon was discovered. This was not found for a non-symmetrical configuration. When the bouncing phenomenon occurred, the radial displacement of the rotor was larger for the stators with stops than for an annular stator.

A spectral-finite difference solution of the Navier-Stokes equations in three dimensions

A new computational code for the numerical integration of the three-dimensional Navier–Stokes equations in their non-dimensional velocity–pressure formulation is presented. The system of non-linear partial differential equations governing the time-dependent flow of a viscous incompressible fluid in a channel is managed by means of a mixed spectral–finite difference method, in which different numerical techniques are applied: Fourier decomposition is used along the homogeneous directions, second-order Crank–Nicolson algorithms are employed for the spatial derivatives in the direction orthogonal to the solid walls and a fourth-order Runge–Kutta procedure is implemented for both the calculation of the convective term and the time advancement. The pressure problem, cast in the Helmholtz form, is solved with the use of a cyclic reduction procedure. No-slip boundary conditions are used at the walls of the channel and cyclic conditions are imposed at the other boundaries of the computing domain. Results are provided for different values of the Reynolds number at several time steps of integration and are compared with results obtained by other authors. © 1998 John Wiley & Sons, Ltd.

Kinetic modelling of the stabilizer consumption and of the consecutive products of the stabilizer in a gun propellant

AbstractDiphenylamine is used for the stabilization of solid propellants based on nitrocellulose. It reacts with the autocatalyticly acting decomposition product of nitrocellulose and forms consecutive products, which also stabilize. The kinetics of these stabilizing reactions are described by four reaction models. With respect to DPA and the DPA consecutive products the models have one, two and four steps. The four step model contains the DPA consecutive products up to trinitro‐DPA. By approximations an analytical equation for the stabilizer decrease could be derived, which describes the stabilizer consumption very well. The other equation systems have been integrated numerically by a Runge–Kutta procedure and their reaction rate constants have been obtained by a non‐linear least squares fit parameter calculation. This allows an evaluation of the reaction kinetic models. Relative reactivities of the DPA consecutive products with respect to DPA are calculable with the obtained reaction rate constants and an effective DPA concentration can be established. The different reactivities of mixed stabilizers as MNA and 2‐NO2‐DPA can be described with the presented method of modelling.

High Resolution of Unsteady Shock Waves around Compression Corners Using Unstructured Adaptive Meshes.

Numerical simulation, which can be used to examine flow fields quantitatively and in detail, has become very important recently. However, a numerical method which agrees well with the experimental results has not yet been established. This is due to the lack of resolution in complicated flow patterns containing numerous shocks. In the present work, an adaptive refinement/coarsening algorithm is developed and applied to transient shocked flows over single and double wedges. The MUSCL approach for unstructured adaptive meshes is employed to obtain high-order accuracy for spatial discretization and a four-stage Runge-Kutta procedure. The results demonstrate that the adaptive method gives better results than those obtained using structured meshes and it shows good agreement with experiments.

Transient analysis of bandwidth allocation strategies in Broadband ISDN

We study the transient behavior of B-ISDN bandwidth allocation strategies in this paper. First, we present the united state transition equations and obtain the steady-state probability distribution vector by the matrix-geometric solution method. Second, we treat the problem in a uniform way that establishing the coupled differential equations describing the dynamic flow process of B-ISDN channel with the steady state probability as its initial condition and solve it numerically based on the fourth-order Runge-kutta procedure which allows a good balance between computing time and accuracy. Finally, experiment results and discussion are given to illustrate the effectes of the method presented in this paper and show that the idea and analysis way can be extended to all dynamic traffic controlling and management in B-ISDN systems.

An off-line computer simulation for controlling the repulp section in a plant

An off-line computer simulation program is proposed to control a repulping section of a plant that has previously been controlled solely by an operator. The off-line computer program can be used as a tool by the operator to improve the control of repulping a certain material before leaching commences. Controlling the repulping section is of special importance, especially to processes employing unit operations after the repulping section with fast reaction kinetics. The fast reaction kinetics make it essential to control the conditions of the incoming pulp to the leaching vessel. The simulation uses dynamic mass balance equations to simulate the physical properties of the system. It then solves the set of differential equations with an iteration procedure that uses the fourth order Runge-Kutta(RK) procedure and the Golden Section Search(GSS) optimisation procedure, thus obtaining optimum settings so that the desired conditions in the process could be maintained or reached as quickly as possible. The simulation was tested on a plant and indicated that considerable improvement in the stability of the operation could be achieved. Two calculation methods were also investigated for the mathematical models, i.e. the Runge-Kutta procedure mentioned above and the Laplace transform solution. Comparing the two methods it was evident that the Laplacian solution was not as accurate as the Runge-Kutta solution due to the inherent linearisation step employed. This was especially evident for large deviations from the desired conditions.