Time Step Criteria Research Articles

Validation and time step determination of discrete element modeling of flexible fibers

In this paper, a flexible fiber model, based on the Discrete Element Method (DEM) and the bonded-particle model by Potyondy and Cundall (International Journal of Rock Mechanics & Mining Sciences 41 (2004) 1329–1364), is comprehensively examined by considering the cases of the bending, stretching, and twisting of a fiber under static loads and dynamic vibrations. The numerical simulation results are in good agreement with the analytical predictions from elastic thin beam theories. In addition, a time step criterion is discussed and proposed in order to ensure numerical stability and obtain correct dynamic behaviors of fibers. It is found that the time step should be less than the time it takes for an axial extensional/compressional wave to travel a single bond length.

Fourier Stability Analysis of Two Finite Element Schemes for Reaction-Diffusion System with Fast Reversible Reaction

In this paper, the stability analysis is performed on two Galerkin finite element schemes for solving reaction-diffusion system with fast reversible reaction. Fourier (Von Neumann) method is implemented to propose time-step criteria for the consistent and the lumped schemes with four popular choices for . We have found that the two schemes are unconditionally stable when , while the consistent scheme is stable under the conditions and when . Also, the lumped scheme is stable under the conditions and when .

N-body integrators with individual time steps from Hierarchical splitting

We review the implementation of individual particle time-stepping for N-body dynamics. We present a class of integrators derived from second order Hamiltonian splitting. In contrast to the usual implementation of individual time-stepping, these integrators are momentum conserving and show excellent energy conservation in conjunction with a symmetrized time step criterion. We use an explicit but approximate formula for the time symmetrization that is compatible with the use of individual time steps. No iterative scheme is necessary. We implement these ideas in the HUAYNO1Available online at www.amusecode.org.1 code and present tests of the integrators and show that the presented integration schemes shows good energy conservation, with little or no systematic drift, while conserving momentum and angular momentum to machine precision for long term integrations.

Accuracy and efficiency of raytracing photoionisation algorithms

Three non-equilibrium photoionisation algorithms for hydrodynamical grid-based simulation codes are compared in terms of accuracy, timestepping criteria, and parallel scaling. Explicit methods with first-order time accuracy for photon conservation must use very restrictive timestep criteria to accurately track R-type ionisation fronts. A second-order accurate algorithm is described which, although it requires more work per step, allows much longer timesteps and is consequently more efficient. Implicit methods allow ionisation fronts to cross many grid cells per timestep while maintaining photon conservation accuracy. It is shown, however, that errors are much larger for multi-frequency radiation than for monochromatic radiation with the implicit algorithm used here, and large errors accrue when an ionisation front crosses many optical depths in a single step. The accuracy and convergence rates of the different algorithms are tested with a large number of timestepping criteria to identify the best criterion for each algorithm. With these criteria selected, the second-order explicit algorithm is the most efficient of the three, and its parallel scaling is significantly better than that of the implicit algorithm. The upgrade from first- to second-order accuracy in explicit algorithms could be made very simply to fixed-grid and adaptive mesh-refinement codes which currently use a first-order method.

A Finite-Element Variable Time-Stepping Algorithm for Solving the Electromagnetic Diffusion Equation

This paper presents a new methodology for applying the Backward Differentiation Formula (BDF) and the Theta algorithm for the variable time-stepping finite-element discretization. The developed BDF-Theta formulation was implemented for solving the Electromagnetic Diffusion equation. An initial-guess prediction algorithm was adopted for convergence acceleration. An algorithm for the selection of the order and time-step size is also included. A minimum time step criterion is adopted. The proposed methodology was validated against an analytical solution and good results were obtained. The adaptive time-step (BDF) requires less computation time than using the fixed time-step implicit Euler.

Calculation of tidal-evolution constants for close binary systems

Tidal torque constants are calculated for each component of 112 close eclipsing binaries of the DMS type (detached components located within the main sequence) from the catalogue by Svechnikov and Perevozkina (1999) on the basis of the evolutionary models by Claret (2004), which are the first to present constants for investigation of tidal effects in close binaries. This study applies conversion of a grid of model tracks into that of isochrones with an arbitrarily small time step based on optimal nonlinear interpolation. The small time-step criterion guarantees application of the linear interpolation procedure to estimate the age of the DMS components and find an individual set of tidal-evolution constants for the close binary under study that refers to the estimated time point. The theoretical values of the apsidal motion period calculated from the individual internal structure constants for each close binary are well consistent with the observations. This confirms the validity of the evolutionary stellar models by Claret and the numerical algorithm proposed here.

Stability of explicit radiation-material coupling in radiative transfer calculations

We show that explicit radiation-material coupling, which is essentially always stable for infrared radiative transfer is conditionally stable in the high energy density regime. A linearized stability analysis is first performed for a simple infinite-medium problem that yields both a criterion for unconditional stability, a time-step restriction that applies for conditional stability, and a time-step criterion that always applies for non-oscillatory solutions. This analysis is then extended to include space dependence with the result that the system is always conditionally stable, but with a time step restriction somewhat different from the infinite-medium case. Nonetheless, the time step restriction for non-oscillatory solutions remains the same. Computations are presented that confirm the predictions of our analysis, and conclusions are given.

Fourier stability analysis of two-dimensional finite element schemes for shallow water equations

This article presents a stability-based analysis of amplification factors obtained using Fourier or von Neumann method for the finite element formulation of shallow water equations. A Galerkin finite element model is developed for two-dimensional shallow water equations to obtain a linearised form of the error equations. Fourier analysis is performed at the element as well as nodal levels to propose time-step criteria for consistent, upwind and lumped methods using explicit, semi-implicit and implicit schemes. The minimum and upper bound Courant numbers are computed for each finite element scheme using the coefficient method. We observed a close agreement between the amplification factors determined using integer and half-integer multiples of Courant numbers of scalar and matrix eigenvalue problems and that of minimum and upper bound Courant numbers of the coefficient method. We demonstrate that ignoring the amplification factor behaviour at the computational grid node may result in an inappropriate representation of numerical stability for certain schemes. The amplitude behaviour as a function of the Courant and wave numbers is analysed for square elements for two-dimensional field solution at each computational grid node. A relationship is derived between element error-squared norm and nodal error-squared norm.

Turbulence modelling of multiphase flow in high-pressure trickle-bed reactors

Computational fluid dynamics (CFD) has been used as a successful tool for single-phase reactors. However, fixed-bed reactors design depends overly in empirical correlations for the prediction of heat and mass transfer phenomena. Therefore, the aim of this work is to present the application of CFD to the simulation of three-dimensional interstitial flow in a multiphase reactor. A case study comprising a high-pressure trickle-bed reactor (30 bar) was modelled by means of an Euler–Euler CFD model. The numerical simulations were evaluated quantitatively by experimental data from the literature. During grid optimization and validation, the effects of mesh size, time step and convergence criteria were evaluated plotting the hydrodynamic predictions as a function of liquid flow rate. Among the discretization methods for the momentum equation, a monotonic upwind scheme for conservation laws was found to give better computed results for either liquid holdup or two-phase pressure drop since it reduces effectively the numerical dispersion in convective terms of transport equation. After the parametric optimization of numerical solution parameters, four RANS multiphase turbulence models were investigated in the whole range of simulated gas and liquid flow rates. During RANS turbulence modelling, standard k– ε dispersed turbulence model gave the better compromise between computer expense and numerical accuracy in comparison with both realizable, renormalization group and Reynolds stress based models. Finally, several computational runs were performed at different temperatures for the evaluation of either axial averaged velocity and turbulent kinetic energy profiles for gas and liquid phases. Flow disequilibrium and strong heterogeneities detected along the packed bed demonstrated liquid distribution issues with slighter impact at high temperatures.

High performance direct gravitational N-body simulations on graphics processing units II: An implementation in CUDA

We present the results of gravitational direct N-body simulations using the graphics processing unit (GPU) on a commercial NVIDIA GeForce 8800GTX designed for gaming computers. The force evaluation of the N-body problem is implemented in “Compute Unified Device Architecture” (CUDA) using the GPU to speedup the calculations. We tested the implementation on three different N-body codes: two direct N-body integration codes, using the 4th order predictor–corrector Hermite integrator with block time-steps, and one Barnes-Hut treecode, which uses a 2nd order leapfrog integration scheme. The integration of the equations of motions for all codes is performed on the host CPU. We find that for N > 512 particles the GPU outperforms the GRAPE-6Af, if some softening in the force calculation is accepted. Without softening and for very small integration time-steps the GRAPE still outperforms the GPU. We conclude that modern GPUs offer an attractive alternative to GRAPE-6Af special purpose hardware. Using the same time-step criterion, the total energy of the N-body system was conserved better than to one in 10 6 on the GPU, only about an order of magnitude worse than obtained with GRAPE-6Af. For N ≳ 10 5 the 8800GTX outperforms the host CPU by a factor of about 100 and runs at about the same speed as the GRAPE-6Af.

CFD modelling of a liquid–solid fluidized bed

A multifluid Eulerian computational fluid dynamics (CFD) model with granular flow extension is used to simulate a liquid–solid fluidized bed. The numerical simulations are evaluated qualitatively by experimental data from the literature and quantitatively by comparison with new experimental data. The effects of mesh size, time step and convergence criteria are investigated. Varying the coefficient of restitution did not alter the results significantly. The Gidaspow drag relationship predicted a higher voidage than the Wen and Yu drag law. Two different liquid distributors (uniform and non-uniform) were simulated and compared, but a better representation of the geometry of the distributor plate did not greatly influence the results. Qualitatively, the simulations show trends similar to experimental trends reported by various authors. The predictions are also compared with new experimental results for 1.13 mm glass spheres at a wide variety of superficial liquid velocities (0.0085–0.110 m/s) and two different temperatures (12 and 33 ∘ C ) significantly affecting the liquid viscosity. The CFD model predictions are within 5% of the steady-state experimental data and show the correct trend with variation in viscosity.

Numerical simulation of Richards equation in partially saturated porous media: under-relaxation and mass balance

Numerical simulation of Richards equation in unsaturated soil is known to be difficult because of the highly non-linear material properties involved. An extensive study was conducted to clarify the role of mass balance and under-relaxation on the rate in which the correct solution is approached. Both finite element and finite difference techniques were considered. For the former, the h-based formulation was compared with a mass-conservative mixed form. The conductivity function (K) was under-relaxed in two ways while the capacity function (C) is computed following the standard mass or non-mass conservative schemes recommended in literature. For fairly coarse discretisation, it was found that large errors were produced when K was under-relaxed by evaluating it at the average of heads from current and previous time step (UR1), regardless of the numerical scheme used. Maintaining global mass balance is found to have little impact on the accuracy. All numerical schemes that under-relaxed K by computing it at the average of two most recent iterations in the current time step (UR2) converged quicker to the correct solution with increasing discretisation, although more iterations per time step than UR1 is needed to achieve a stable solution. An important practical ramification is that it appears to be possible to achieve reasonably accurate and oscillation-free results using fairly coarse discretisation by making only minor modifications (namely, using UR2 for conductivity function) to the h-based finite element formulation and applying some minimum time step criteria.

An optimum time-stepping scheme forN-body simulations

We present a new time-stepping criterion for N-body simulations that is based on the true dynamical time of a particle. This allows us to follow the orbits of particles correctly in all environments since it has better adaptivity than previous time-stepping criteria used in N-body simulations. Furthermore, it requires far fewer force evaluations in low-density regions of the simulation and has no dependence on artificial parameters such as, for example, the softening length. This can be orders of magnitude faster than conventional ad hoc methods that employ combinations of acceleration and softening and is ideally suited for hard problems, such as obtaining the correct dynamics in the very central regions of dark matter haloes. We also derive an eccentricity correction for a general leapfrog integration scheme that can follow gravitational scattering events for orbits with eccentricity e→ 1 with high precision. These new approaches allow us to study a range of problems in collisionless and collisional dynamics from few body problems to cosmological structure formation. We present tests of the time-stepping scheme in N-body simulations of two-body orbits with eccentricity e→ 1 (elliptic and hyperbolic), equilibrium haloes and a hierarchical cosmological structure formation run.

Accuracy of Hydrodynamic Modeling of Flood Detention Reservoirs

Hydrodynamic modeling of flood detention reservoirs is an effective tool to evaluate reservoir performance, to plan modifications, and to design new reservoirs. The one-dimensional (1D) Saint Venant equations (SVE) are suitable to simulate the standing waves and other degenerations that occur in the reservoirs. The accuracy of these simulations depends on the Courant numbers used in the simulations. Consequently, the use of an appropriate time step for a specific grid size is crucial for the accuracy of the solution. A series of numerical experiments were performed and compared to observed data to develop time step criteria for solving the 1D SVE using the Abbott-Ionescu scheme. The time step was normalized by the wave celerity to make it problem dependent. The new time step criteria were evaluated on a different reservoir and proved to be adequate for selecting a time step for reservoir modeling. The criteria could readily be integrated in existing models to be used in an adaptive time stepping scheme.

Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations

AbstractThe finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time‐stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so‐called minimum time step criteria are established for the Crank‐Nicolson scheme, the Galerkin‐time scheme, and the backward‐difference scheme used in the temporal discretization. For the forward‐difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one‐dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Stability and accuracy of two-dimensional kinematic wave overland flow modeling

A two-dimensional finite element based overland flow model was developed and used to study the accuracy and stability of three numerical schemes and watershed parameter aggregation error. The conventional consistent finite element scheme results in oscillations for certain time step ranges. The lumped and the upwind finite element schemes are tested as alternatives to the consistent scheme. The upwind scheme did not improve on the stability or the accuracy of the solution, while the lumped scheme provided stable and accurate solutions for time steps twice the size of time steps needed for the consistent scheme. A new accuracy based dynamic time step estimate for the two-dimensional overland flow kinematic wave solution is developed for the lumped scheme. The newly developed dynamic time step estimates are functions of the mesh size, and time of concentration of the watershed hydrograph. Due to lack of analytical solutions, the time step was developed by comparing numerical solutions of various levels of discretization to a reference solution using a very fine mesh and a very small time step. The time step criteria were tested on a different set of problems and proved to be adequate for accurate and stable solutions. A sensitivity analysis for the watershed slope, Manning’s roughness coefficient and excess rainfall rate was conducted in order to test the effect of parameter aggregation on the stability and accuracy of the solution. The results of this analysis show that aggregation of the slope data resulted in the highest error. The roughness coefficient had a smaller effect on the solution while the rainfall intensity did not show any significant effect on the flow rate solution for the range of rainfall intensity used. This work pioneers the challenge of providing guidelines for accurate and stable numerical solutions of the two-dimensional kinematic wave equations for overland flow.

Stability and accuracy of finite element schemes for the one-dimensional kinematic wave solution

Solving the kinematic wave equations for overland flow using the conventional consistent Galerkin finite element scheme is known to result in numerical oscillations due to the non-symmetric first spatial derivative terms in the kinematic wave equations. In this paper the lumped and the upwind finite element schemes are evaluated as alternatives to the consistent Galerkin finite element scheme. Stability analysis of the upwind scheme shows that the damping effect, that could reduce the oscillations, is small for the high Courant numbers encountered in overland flow problems. The upwind scheme, using upwind factors of 0.1 and 1.0, did not provide any improvement to the stability of the lumped and the consistent schemes. The lumped scheme considerably reduces oscillations without significant reduction in the overall solution accuracy. No analytical guidelines for time-step criteria that will insure stability and accuracy were provided by the stability analysis performed for the three schemes. Problem specific accuracy-based dynamic time-step criteria was developed and evaluated for the lumped scheme. These time-steps were found to be on average, double the size of the dynamic time-steps for the consistent scheme.

Dynamic Time Step for One-Dimensional Overland Flow Kinematic Wave Solution

Kinematic wave theory is widely used in modeling a variety of hydrologic processes. Results of applying the kinematic wave overland flow solution using different time steps showed that the conventionally used stability criterion known as the Courant condition fails to give a time step estimate that ensures stable and accurate numerical solutions. Accordingly, a new accuracy-based dynamic time step estimate for the one dimensional overland flow kinematic wave solution is developed. The newly developed dynamic time step estimates are functions of the mesh size, watershed slope, roughness, excess rainfall, and time of concentration. The new criteria were developed by comparing the consistent formulation of the Galerkin-Crank Nicholson numerical solution of the kinematic wave equation to the characteristic method-based analytical solution, using different time steps and meshes. For each simulation, characterized by boundary and initial conditions and mesh size, an optimal time step that integrates the problem within 5% error was determined. The series of mesh sizes and corresponding optimal time steps were used to develop the dynamic time step. The time step criteria were tested on a variety of problems, including a steady state and time varying rainfall scenarios, and proved to be adequate for accurate and stable results within an efficient computational time. The criteria can be easily integrated in flow routing models to select the optimal time step with minimal user input.

Validation Methods and Results for a Two-Dimensional Ice Accretion Code

A research project is underway at NASA Lewis Research Center (now called JohnH. Glenn Research Center at Lewis Field) to producea library of computercodes thatcanaccurately predict ice growthunderanymeteorological conditions for any aircraft surface. Results are presented from themost recent version of the two-dimensional code LEWICE. This version differs from previous releases due to its robustness and its ability to reproduce results accurately for different point spacing and time step criteria across several computing platforms. It also differs in the extensive amount of effort undertaken to compare the results in a quantiŽ able manner against the database of ice shapes that havebeen generated in the NASA Lewis Icing Research Tunnel. A quantitativemethod is described that has been applied to both the computational and experimental ice shapes. The result of this comparison shows that the difference between the predicted ice shape from LEWICE 2.0 and the average of the experimental data is 7.2%whereas the variability of the experimental data is 2.5%.

Dynamic time step and stability criteria comparison for the heat diffusion equation

Dynamic time step estimates were determined for four single step numerical schemes in one dimension, and three finite element and three finite difference single step schemes in two dimensions. Unlike other stability-based time step criteria, the dynamic time step estimates are functions of grid size, material properties, and boundary conditions. For a certain problem, an initial coarse mesh is required to determine its lowest eigenvalue. The problem needs to be re-meshed and solved with the new mesh to satisfy a desired accuracy. Correlations between the dynamic time step and the stability criteria were conducted. A stability criteria equivalence was defined as the stability criteria that gives a time step estimate equivalent to the dynamic time step estimate. For the range of problems solved and for the central difference time stepping scheme, the stability criteria equivalence ranges from 0.5 to 2.7 for one-dimensional problems and 0.46 to 9.13 for tqo-dimensional problems, depending on the problem boundary conditions. The study shows that, unlike the dynamic time step estimate, the stability criteria used for selecting a time step is not adequate, since it does not change with the problem boundary conditions.