Time step, relaxation

So-called false-time-step relaxation is used to achieve stationarity. The semi-implicit method, which considers the pressure-Hnk of the pressure correction equation and the Reynolds equations, is the SIMPLEST algorithm. The sets of algebraic equations for each variable are solved iteratively by means of the ADI technique. An example of the simulated flow field is illustrated in Fig. 3. Good agreement can then be achieved between measured flow details and the simulation results for vessels and impellers of different geometry [1]. 

Wisdom, J. The Origin of the Kirkwood Gaps A Mapping for Asteroidal Motion Near the 3/1 Commensurability. Astron. J. 87 (1982) 577-593 Tuckerman, M., Martyna, G. J., Berne, J. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys. 97 (1992) 1990-2001 Tuckerman, M., Berne, J. Vibrational Relaxation in Simple Fluids Comparison of Theory and Simulation. J. Chem. Phys. 98 (1993) 7301-7318 Humphreys, D. D., Friesner, R. A., Berne, B. J. A Multiple-Time Step Molecular Dynamics Algorithm for Macromolecules. J. Chem. Phys. 98 (1994) 6885-6892... 

T(f) corresponds to the actual temperature at the time t, At is the integration time step, and the relaxation time represents the strength of the coupling (smaller values mean stronger coupling to the bafli). If the coupling is too strong (r smaller... 

In a molecular dynamics calculation, you can add a term to adjust the velocities, keeping the molecular system near a desired temperature. During a constant temperature simulation, velocities are scaled at each time step. This couples the system to a simulated heat bath at Tq, with a temperature relaxation time of "r. The velocities arc scaled bv a factor X. where... 

The simplest method that keeps the temperature of a system constant during an MD simulation is to rescale the velocities at each time step by a factor of (To/T) -, where T is the current instantaneous temperature [defined in Eq. (24)] and Tq is the desired temperamre. This method is commonly used in the equilibration phase of many MD simulations and has also been suggested as a means of performing constant temperature molecular dynamics [22]. A further refinement of the velocity-rescaling approach was proposed by Berendsen et al. [24], who used velocity rescaling to couple the system to a heat bath at a temperature Tq. Since heat coupling has a characteristic relaxation time, each velocity V is scaled by a factor X, defined as... 

In this expression. Ait is the size of the integration time step, Xj is a characteristic relaxation time, and T is the instantaneous temperature. In the simulation of water, they found a relaxation time of Xj = 0.4 ps to be appropriate. However, this method does not correspond exactly to the canonical ensemble. 

Here, At is the size of the time step, Tp is a characteristic relaxation time, and Pg is the pressure of the external constant-pressure bath. The instantaneous pressure can be calculated as follows ... 

In such a process, the water molecule fonned in the elimination step is captured primarily fiom the fixmt side, leading to net retention of configuration for the alcohol. For the ester, the extent of retention and inversion is more balanced, although it vari among individual systems. It is clear om die data in Table 5.18 that the two pairs of stereoisomeric amines do not form the same intermediate, even though a simple mechanistic interpretation would sugg that both would fmm the 2-decalyl cation. The coUap of the ions to product is pvidoitly so rapid that diere is not time for relaxation of the initially formed intermediates to reach a common stnicture. 

We now discuss the translation of the MC time-step into physical time units. It is desirable to map the mobility of the lattice model (due to jumps of the effective monomers) onto the average jump rate of the torsional degrees of freedom, since these motions dominate the relaxation of the overall configuration of the chain. This means that we must allow for a temperature-depen-dent time unit tmc(T) which one attempted MCS per monomer corresponds to, via the formula ... 

In the time constant (relaxation) method, the waveform of P is a negative step which produces a relaxation of the sample temperature from TB + ST to TB. The measure of P(T) may be critical when the power P is comparable with the spurious power or when the thermal conductance G is steeply variable with the temperature (i.e. G oc T3 in the case of contact conductances). 

For the simulation of surface relaxation, KMC has two advantages over Metropolis. Firstly, since an atom moves at every iteration regardless of temperature, lower temperatures can be studied. Secondly, the dynamic (vs. thermodynamic) nature of the algorithm yields a proper time step, whereas is debated whether or not Metropolis does so . 

Calculations were performed with both rigid and flexible lattices. Because of increased computational demands of allowing the framework to relax, flexible lattice simulations were done with a smaller system, together with a shorter time step relative to the rigid framework simulations. The adsorbate atoms were distributed evenly throughout the channels at the start of each simulation, and a nominal temperature of 300 K was selected. The size... 

For case studies that compare model results with in situ observations we apply a nudging technique to ECHAM. At each time step ECHAM is relaxed towards ECMWF analyzed distributions of surface pressure, vorticity, divergence and temperature [35]. This enables the simulation of realistic meteorological situations so that simulated distributions of chemical species can be directly compared with measurement data for a specific time and place, thus... 

The first step, relaxation, involved achieving not only deep physical relaxation but also deep mental relaxation. The induction of such a relaxed state often required a considerable length of time, especially in the beginning it was not unusual for percipients to spend five to ten minutes on this step alone, even after they had learned it well. Modem systematic techniques for relaxing the body, such as Jacobson s progressive relaxation [38], autogenic training [119], or biofeedback procedures would probably work well. Comfortable meditative postures would probably also be effective. 

Consider a micellar solution at equilibrium that is subject to a sudden temperature change (T-jump). At the new temperature the equilibrium aggregate size distribution will be somewhat different and a redistribution of micellar sizes will occur. Aniansson and Wall now made the important observation that when scheme (5.1) represents the kinetic elementary step, and when there is a strong minimum in the micelle size distribution as in Fig. 2.23(a) the redistribution of micelle sizes is a two-step process. In the first and faster step relaxation occurs to a quasi-equilibrium state which is formed under the constraint that the total number of micelles remains constant. Thus the fast process involves reactions in scheme (5.1) for aggregates of sizes close to the maximum in the distribution. This process is characterized by an exponential relaxation with a time constant Tj equal to... 

Time dependence The inherent time dependence of fires sets strong requirements for computational efficiency. In RANS models, the radiation field must be updated within the internal iterations of the time step, but the computational cost can be relaxed by solving RTE only every Mh iteration. In SOFIE, for example, it is typical to use N = 10. In FDS, the time accuracy of the radiation field has been relaxed by solving the FVM equations typically every third time step and only part of the directions at a time. 

During the assimilation of climatic shipborne data, the relaxation coefficient featured an inverse dependence on the relative dispersion of the observation errors. Owing to the growth of the latter with depth, this coefficient decreased, which allowed one to smooth the vertical differences in the rates of adaptation recognized in [40]. The climatic temperature and salinity fields were interpolated over the nodes of the calculation grid and were assimilated at each model time step. This way, the degree of the disagreement between the calculated and observed fields at the moment of assimilation was reduced to its minimum. 

In relaxation equations, the total flow rates and -values do not change from one time step to the next. Once the compositions are... 

Any of the global Newton methods can be converted to a relaxation form in Ketchum s method by making both the temperatures and the liquid compositions time dependent and by having the time step increase as the solution is approached. The relaxation technique should be applied to difflcult-to-solve systems and the method of Naphtali and Sandholm (42) is best-suited for nonideal mixtures since both the liquid and vapor compositions are included in the independent variables. Drew and Franks (65) presented a Naphtali-Sandholm method for the dynamic simulation of a reactive distillation column but also stated that this method could be used for finding a steady-state solution. 

The time step, At, is used to switch the method from being a relaxation method to a global Newton method. When the time step is small, e.g., if = 0.1, then the changes in the independent variables are small. The method performs like a damped Newton-Raphson method, where the steps are small but in the direction of the solution and without any oscillation. When the value of At is large, i.e., At = 1000, the method performs like a Newton-Raphson method. The value of At at each column trial determines the speed and stability of the method, The units of the time step are the same as the flows to and from the column. The calculation sequence of the Ketchum method is as follows ... 

Computer simulations of the coagulation of equal-sized particles of unit density in air at 298 K and 1 atm were performed based on the algorithm discussed in the previous section. Since the Hamaker constant for most of the aerosol systems is of the order of 10 12 erg, this value was used for the calculation of the interaction potential between the particles. Computations were performed on a CDC 815 computer. In all the computations, the duration of the time step ts for the random force was taken to be equal to one-tenth of the relaxation time for Brownian motion, i.e., ls = O.lf, so that the condition ts -4 f l is satisfactorily fulfilled. The time step f, for the motion of the fictitious particle in the region of the potential well, i.e., region II, was taken to be 0.05. In other words, the values of the dimensionless times 6 and 0 were taken as... 

Hahn [47] developed a hybrid simulation based on BD and Monte Carlo methods. Incorporation of the statistical techniques of Monte Carlo methods relaxes the constraint that time steps must be sufficiently short such that external force fields can be considered constant, and the BD improves upon the Monte Carlo methods by allowing dynamic information to be collected. Hahn applied the model to the investigation of theoretical deposition by simulating a... 

Obviously, the above algorithms are not suitable when transients of the finer scale model are involved (Raimondeau and Vlachos, 2000), as, for example, during startup, shut down, or at a short time after perturbations in macroscopic variables have occurred. The third coupling algorithm attempts fully dynamic, simultaneous solution of the two models where one passes information back and forth at each time step. This method is computationally more intensive, since it involves continuous calls of the microscopic code but eliminates the need for a priori development of accurate surfaces. As a result, it does not suffer from the problem of accuracy as this is taken care of on-the-fly. In dynamic simulation, one could take advantage of the fast relaxation of a finer (microscopic) model. What the separation of time scales between finer and coarser scale models implies is that in each (macroscopic) time step of the coarse model, one could solve the fine scale model for short (microscopic) time intervals only and pass the information into the coarse model. These ideas have been discussed for model systems in Gear and Kevrekidis (2003), Vanden-Eijnden (2003), and Weinan et al. (2003) but have not been implemented yet in realistic MC simulations. The term projective method was introduced for a specific implementation of this approach (Gear and Kevrekidis, 2003). 

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