Value Of Time Step Research Articles

Accurate numerical evaluation of domain integrals in 3D boundary element method for transient heat conduction problem

This paper presents an improved approach for the numerical evaluation of domain integrals that appear in the solution of transient heat conduction problems when using a time-dependent boundary integral equation method. An implementation of this method requires the accurate evaluation of the domain integrals. As the time step value is very small, the integrand in the domain integral is close to singular, thus rendering accurate evaluation of the integral difficult. First a closest point is introduced when the source point is close to, but not on the cell in the present method. Then a coordinate transformation coupled with a cell subdivision technique is proposed considering the position of the source point or the closest point and the relations between the size of the cell and the time step value. With the new method, accurate evaluation of domain integrals can be obtained. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

On problems of time lag and susceptibility during mathematical simulation of thermal operation of roasting furnace for production of artificial graphite

The method of mathematical simulation has been used for examination of the effect of relationship of thermal time lag of the container (with stock material and fill) and susceptibility of the control system on the temperature pattern of stock material for obtaining artificial graphite; also the effect of computational time step value in time (Δτ) on the results of mathematical simulation has been studied. The results of simulation at any fill bed thickness are shown not to depend practically on Δτ for infinitelyfast system for control of roasting furnace, and at finite susceptibility of the control system the temperature oscillations, being generated by the system with decreasing thermal time lag of the container (fill bed depth), penetrate ever deeper and deeper inside it, however you can detect it with help of computational method only at small Δτ – because big time steps show smoothed pattern with monotonous trends that corresponds to the reality only at significant fill bed depth. The degree of convergence of iteration calculation algorithm is high enough if Δτ is significantly more or significantly less than the period of modeled periodic processes; if these values are of the same order the algorithm can run into a cyclic path or the demanded amount of iterations can increase by some orders.

Simulations of a Heat-Wave Event in New York City Using a Multilayer Urban Parameterization

AbstractThe Weather Research and Forecasting mesoscale model coupled to a multilayer urban canopy parameterization was used to evaluate the evolution of a 3-day heat wave in New York City, New York, during the summer of 2010. Results from three simulations with different degrees of urban modeling complexity and one with an absence of urban surfaces are compared with observations. To improve the city morphology representation, building information was assimilated and the land cover land-use classification was modified. The thermal and drag effects of buildings represented in the multilayer urban canopy model improve simulations over urban regions, giving better estimates of the surface temperature and wind speed. The accuracy of the simulation is further assessed against more simplified urban parameterizations models. The nighttime excessive cooling shown by the Building Energy Parameterization is compensated for when the Building Energy Model is activated. The turbulent kinetic energy is vertically distributed when using the multilayer scheme with a maximum at the average building height, whereas turbulence production is confined to a few meters above the surface when using the simplified scheme. Evidence for the existence of horizontal roll vortices is presented, and the impact that the horizontal resolution and the time step value have on their formation is assessed.

Mixed numerical method for bed evolution

The morphological evolution in shallow water flows was investigated in this study. The shallow water equations were solved for the flows using the lattice Boltzmann method with multiple-relaxation time and an improved scheme for the momentum source terms in which the bed elevation gradients were included. The fifth-order Euler–WENO scheme was coupled with the lattice Boltzmann method to solve the Exner equation for the bed evolution caused by bed-load sediment transport. Five typical test cases were simulated by the coupled model and the numerical results agreed well with either the corresponding experimental data or the analytical solutions. This suggests that the present model has great potential to predict morphological change in shallow water flows. Furthermore, the effect of different values of time step for morphodynamic evolution and flow field were studied and no obvious difference was produced for these two kinds of method in the range tested in the present study.

A Numerical Treatment of Fisher Equation

Fisher equation is commonly arises in chemistry, heat and mass transfer, biology and ecology. In mathematics Fisher equation is also known as Kolmogorov Petrovsky-Piscounov equation, KPP equation or Fisher KPP equation. Fisher equation describes the process of interaction between diffusion and reaction. In this paper a semi implicit method is used to solve the Fisher equation. A semi implicit finite difference scheme has been designed for numerical solution of one dimensional nonlinear Fisher equation. The designed scheme accuracy is first order in time and second order in space. Numerical results are calculated for different values of Diffusion coefficient and time steps are matching with exact solution.

Dual state-parameter optimal estimation of one-dimensional open channel model using ensemble Kalman filter

In this paper, both state variables and parameters of one-dimensional open channel model are estimated using a framework of the Ensemble Kalman Filter (EnKF). Compared with observation, the predicted accuracy of water level and discharge are improved while the parameters of the model are identified simultaneously. With the principles of the EnKF, a state-space description of the Saint-Venant equation is constructed by perturbing the measurements with Gaussian error distribution. At the same time, the roughness, one of the key parameters in one-dimensional open channel, is also considered as a state variable to identify its value dynamically. The updated state variables and the parameters are then used as the initial values of the next time step to continue the assimilation process. The usefulness and the capability of the dual EnKF are demonstrated in the lower Yellow River during the water-sediment regulation in 2009. In the optimization process, the errors between the prediction and the observation are analyzed, and the rationale of inverse roughness is discussed. It is believed that (1) the flexible approach of the dual EnKF can improve the accuracy of predicting water level and discharge, (2) it provides a probabilistic way to identify the model error which is feasible to implement but hard to handle in other filter systems, and (3) it is practicable for river engineering and management.

A simple and effective Verlet-type algorithm for simulating Langevin dynamics

We present a revision to the well known Störmer–Verlet algorithm for simulating second order differential equations. The revision addresses the inclusion of linear friction with associated stochastic noise, and we analytically demonstrate that the new algorithm correctly reproduces diffusive behaviour of a particle in a flat potential. For a harmonic oscillator, our algorithm provides the exact Boltzmann distribution for any value of damping, frequency and time step for both underdamped and overdamped behaviour within the usual stability limit of the Verlet algorithm. Given the structure and simplicity of the method, we conclude that this approach can trivially be adapted for contemporary applications, including molecular dynamics with extensions such as molecular constraints.

Thermally driven classical Heisenberg model in one dimension

We study thermal transport in a classical one-dimensional Heisenberg model employing a discrete-time odd-even precessional update scheme. This dynamics equilibrates a spin chain for any arbitrary temperature and finite value of the integration time step $\ensuremath{\Delta}t$. We rigorously show that in presence of driving, the system attains local thermal equilibrium, which is a strict requirement of Fourier law. In the thermodynamic limit, heat current for such a system obeys Fourier law for all temperatures, as has been recently shown [A. V. Savin, G. P. Tsironis, and X. Zotos, Phys. Rev. B 72, 140402(R) (2005)]. Finite systems, however, show an apparent ballistic transport which crosses over to a diffusive one as the system size is increased. We provide exact results for current and energy profiles in zero- and infinite-temperature limits.

Balance errors generated by numerical diffusion in the solution of non-linear open channel flow equations

► The dissipative numerical methods for solution of nonlinear equations are considered. ► We examine the numerical solution of 1D equations in open channel hydraulics. ► The numerical diffusion generated by the method of solution produces a balance error. ► The balance error is connected with an inadequate conservative form of equation. The paper concerns the untypical aspect of application of the dissipative numerical methods to solve non-linear hyperbolic partial differential equations used in open channel hydraulics. It is shown that in some cases the numerical diffusion generated by the applied method of solution produces not only inaccurate solution but as well as a balance error. This error may occur even for an equation written in the conservative form not only in the solution containing a shock wave but also for the smooth solution with strong gradients. In order to explain this problem 1-D kinematic wave equation and 1-D system of shallow water equations, both solved by the finite-difference implicit box scheme, are considered. Accuracy analysis carried out using the modified equation approach revealed why the numerical solutions of these equations written in the conservative forms contain the balance errors. It was shown that the balance error is directly connected with the additional source term generated by the numerical diffusion during the solution of equation written in an inadequate conservative form. Consequently the balance error depends on the values of time step, spatial interval and the weighting parameters involved in the applied method of solution.

Accelerating incompressible flow calculations using a quasi-implicit scheme: local and dual time stepping approaches

In this paper, a local time stepping approach is proposed for incompressible flow calculations using a quasi-implicit time discretization. Since the non-linear convection term is treated explicitly by this discretization, the convective part of the time step limit still applies. To effectively manage the convection stability limit, the locally calculated time step values are employed. These local time step values are calculated for elements, rather than on nodes to avoid unsymmetric matrices. To avoid any adverse impact of local time stepping on the transient solution, a dual time stepping approach is employed for time dependent problems. The results clearly show that a speed up of more than five times is possible using the local time stepping approach. Although the proposed local time stepping approach is employed for a finite element based spatial discretization, it is applicable to any other form of spatial discretization.

Melting of Argon Cluster: Dependence of Caloric Curves on MD Simulation Parameters

We report on molecular dynamics simulations performed using microcanonical ensemble to predict the melting of argon particles in nanometer size range 10 nm and to investigate the effect of the time step integration. We use the Lennard- Jones potential functions to describe the interatomic interactions, and the results are evaluated by using caloric curves of the melting phenomenon. Thermodynamic properties, including the total energy, Lindemann parameter, kinetic and potential distribution’s functions, are used to characterize the melting process. The data shows bimodal behavior only in a certain interval of integration time step Δt, while the internal energy increases monotonically with the temperature. For the other time step values, the back bending disappears. We claim that negative specific heat is related to a possible decrease of entropy in an isolated system; this can be interpreted as a result of the internal interactions, especially attractive process and specific relaxation time.

An improved moving-boundary heat exchanger model using the variable time step method

Moving-boundary heat exchanger model is a fast low order model. In this paper, an improved moving-boundary heat exchanger model is developed to provide a stable fast model for the research on control strategy of refrigeration system. In this model, the treatment of each iterative time step is added. Before the calculation of the state variables of the next step, the maximum time step of the next iterative is determined by the change state variable values in per unit time of the current iterate. Then, the time step of the next iterative is determined by the maximum time step and the setting time step. Meanwhile, in order to meet the need of the research on control strategy, the independent variable is employed with temperature variable instead of pressure and enthalpy variable in the calculation of refrigerant state parameters. The simulation results shows that the present model can well predict the transient behaviours of heat exchangers using any setting value of time step, and the computing speed of the present model is faster than the speed of original model.

Three-dimensional numerical modeling of vertical ground heat exchangers: Domain decomposition and state model reduction

Modeling of vertical ground heat exchangers is relatively complex because of the three-dimensional transient nature of the problem inside the borehole and in the surrounding ground. Furthermore, the system is characterized by various time scales with rapid changes inside the borehole and slow variations of ground temperature far away from the borehole. Most existing numerical models require important computational resources to adequately represent the short time-scale heat transfer occurring in the immediate vicinity of the borehole, which warrant their use for annual energy simulations. In this article, a three-dimensional reduced model (3D-RM), based on domain decomposition and state model reduction techniques, is proposed to reduce computation time and computer memory. Domain decomposition is used to sub-structure the domain and to vary the time-step values in each sub-domain, and state model reduction is applied to each resulting sub-zone. A comparison with a complete three-dimensional dynamic model indicates that the proposed 3D-RM model reduces computational time by a factor of about 30 without loss of accuracy. A comparison with experimental results shows that the relatively fast transients occurring in the borehole are well predicted by the 3D-RM model not only for the outlet fluid temperature but also for the tube wall temperatures at different depths.

Creep Behavior of Pressurized Tank Composed of Functionally Graded Materials

The creep behavior of a pressurized tank, which is assumed to be made of functionally graded materials, is studied in this paper. The elastic response under the internal and external pressures is first obtained when Young’s modulus obeys a power function along with the wall thickness. If the creep exponent remains constant and the creep coefficient varies with the radial coordinate, a closed-form solution can be derived for the time-dependent behavior of the spherical tank. The effects of material gradients on the creep stress and strain are investigated in detail. The results show that the stress level under the steady creep state is determined by the distribution of the creep properties. However, the magnitude of the creep strain is influenced by the elastic modulus distribution, as well as the creep property distribution inside the functionally graded materials. Compared with the finite element analysis results, the optimum time step value is also investigated. Some fundamental knowledge of the materials distribution is achieved to reduce the maximum creep stress/strain and to uniformize the stress level inside the functionally graded materials tank.

Discretization errors and free surface stabilization in the finite difference and marker-in-cell method for applied geodynamics: A numerical study

The finite difference–marker-in-cell (FD-MIC) method is a popular method in thermomechanical modeling in geodynamics. Although no systematic study has investigated the numerical properties of the method, numerous applications have shown its robustness and flexibility for the study of large viscous deformations. The model setups used in geodynamics often involve large smooth variations of viscosity (e.g., temperature-dependent viscosity) as well large discontinuous variations in material properties (e.g., material interfaces). Establishing the numerical properties of the FD-MIC and showing that the scheme is convergent adds relevance to the applications studies that employ this method. In this study, we numerically investigate the discretization errors and order of accuracy of the velocity and pressure solution obtained from the FD-MIC scheme using two-dimensional analytic solutions. We show that, depending on which type of boundary condition is used, the FD-MIC scheme is a second-order accurate in space as long as the viscosity field is constant or smooth (i.e., continuous). With the introduction of a discontinuous viscosity field characterized by a viscosity jump (η*) within the control volume, the scheme becomes first-order accurate. We observed that the transition from second-order to first-order accuracy will occur with only a small increase in the viscosity contrast (η* ≈ 5). We have employed two methods for projecting the material properties from the Lagrangian markers onto the Eulerian nodes. The methods are based on the size of the interpolation volume (4-cell, 1-cell). The use of a more local interpolation scheme (1-cell) decreases the absolute velocity and pressure discretization errors. We also introduce a stabilization algorithm that damps the potential oscillations that may arise from quasi free surface calculations in numerical codes that employ the strong form of the Stokes equations. This correction term is of particular interest for topographic modeling, since the surface of the Earth is generally represented by a free surface. Including the stabilization enables physically meaningful solutions to be obtained from our simulations, even in cases where the time step value exceeds the isostatic relaxation time. We show that including the stabilization algorithm in our FD stencil does not affect the convergence properties of our scheme. In order to verify our approach, we performed time-dependent simulations of free surface Rayleigh-Taylor instability.

Molecular dynamics method for simulation of thermodynamic equilibrium

Simulation of a system of particles consisting of argon atoms using the molecular dynamics method is considered. An original molecular dynamics program was developed to calculate the parameters of gases on the basis of the Lennard-Jones potential. the conservativeness of the system in the process of calculation is provided by optimization of the time-step value. The process of establishing a uniform distribution of atoms in velocity is investigated.

On the analysis of dispersion property and stable time step in meshfree method using the generalized meshfree approximation

This paper studies the dispersion characteristic and stable time step of the meshfree method in explicit dynamic problems using the generalized meshfree (GMF) approximation. In the dispersion analysis, the von Neumann method is applied to analyze the numerical dispersion errors for the spatial semi-discretization of a partial differential equation (PDE) in meshfree method using various approximations. The study emphasizes on the influence from the Kronecker-delta property in the approximation that is constructed using the GMF approximation and the full transformation method. In the stable time step analysis, we derive the analytical value of the critical time step based on the eigenvalue analysis. In addition, special attention is paid on the study of the boundary condition effect in suggesting a stable time step used for the practical analysis. Three full-discretization examples are presented to verify the study in the semi-discretization analysis.

Sensitivity of the Indian summer monsoon rainfall and its interannual variation to model time step

The sensitivity of the Indian summer monsoon rainfall (ISMR) and its interannual variation (IAV) to model time step is investigated using NCAR-Community Atmosphere Model version 3 (CAM3). A set of multiyear numerical experiments is performed using the atmospheric model inter-comparison project (AMIP) protocol with observed sea surface temperature (SST). The default value of time step for 64 × 128 horizontal resolution with semi-Lagrangian dynamical core is 60 min. The model overestimates the mean and underestimates the standard deviation. The mean and standard deviation of ISMR systematically decrease with decrease of time step size. With respect to observations, the mean becomes more reasonable but standard deviation becomes less reasonable. There is a decrease in precipitation over the Saudi Arabia, Maritime Continent, and northwestern Arabian Sea with decrease in time step, while over the Eastern Indian Ocean, Eastern Arabian Sea, and Eastern Bay of Bengal there is an increase in precipitation. The pattern correlation of precipitation with observation systematically increases with decrease of time step. In regard to the IAV of ISMR, simulation with 20 min time step outperforms the other time steps i.e. 60, 40, 30, and 05 min. When it is decreased to 20 min, the model bias in precipitation climatology is reduced and the low-level westerly jet over the Indian peninsular becomes more realistic. There is an overall improvement in the climatology of rainfall and winds in the vicinity of Indian summer monsoon region with 20 min time step.

Coupling a Network HVAC Model to a Computational Fluid Dynamics Model Using Large Eddy Simulation

An HVAC network model was coupled to FDS v5.5. The HVAC model allows a user to specify the topology of an HVAC system along with dampers, fans, and forward/reverse flow loss through ducts and fittings. The model was indirectly coupled with the FDS flow solver. The HVAC model uses prior time step values as its boundary conditions and provides to FDS wall boundary conditions of temperature, velocity, and species for prediction of the next FDS time step. The current implementation does not account for transport times with the HVAC network. This paper describes the governing equations for the HVAC network model which is based upon the MELCOR, a nodal network model used for nuclear power plant containment buildings, solver. The specific numerical implementation of the equations within FDS is described. A series of verification exercises demonstrate that the network model correctly models HVAC flows and that its coupling with FDS maintains mass conservation. A simple and a complex validation exercise show that the combined solvers can accurately predict HVAC flows for a duct network in a complex geometry with fire effects

NONLINEAR STRUCTURAL ANALYSIS USING DYNAMIC RELAXATION METHOD WITH IMPROVED CONVERGENCE RATE

This paper develops the optimal time-step ratio and the critical damping for nonlinear structural analysis, when using dynamic relaxation method (DRM). The convergence rate of the DRM depends on the value of the time step and the critical damping factor. In the present study, the effect of the time-step ratio is separated from the time-step value. Additionally, the effect of the eigenvalue on the error function is investigated. Using the eigenvalue definition, the critical damping is updated in each dynamic relaxation (DR) step to decrease the required computations. Moreover, the optimal time-step ratio is calculated. It is also shown that the value of the constant time step has no effect on the rate of convergence. The proposed procedure provides a simple and accelerated DRM, which can be used in the linear and nonlinear analyses of structures. The efficiency of the new method was verified by its application to a wide range of typical structures. The results show that the suggested scheme accelerates the convergence rate of the DR process.