Simulations methods, discrete

To perform simulations of relatively large systems for relatively long times, it is essential to optimize the computational strategy of discrete particle simulations. Obviously, the larger the time step 5t, the more efficient the simulation method. For the soft-sphere model, the maximum value for 5t is dictated by the duration of a contact. Since there are two different spring-dashpot systems in our current model, it is essential to assume that tcontact>n — tcontacUU so that... 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

IR, Raman and related phenomena) to describe with a static approach the salient aspects of phenomena, which are essentially of a dynamical nature [1], This regime was later shown to be essential for a correct description of the photophysical phenomena. It introduces in the QM formalism aspects that are not present in the standard formulation, particularly, that the excited states activated by the excitation process are not orthogonal to the fundamental one (a similar effect is present in the emission process). The orthogonality among states is a basic tenet of the standard formulation, and the selection rules are based on this property. The description obtained with this model is more realistic than the standard one, when the chromophore is immersed into a responsive medium. Discrete solvent simulation methods could hardly describe these effects. 

Turner, J.S., Discrete simulation methods for chemical kinetics, J. Phys. Chem., 1977, 81, 2379-2408. 

Figure 14 shows the resulting grid of points. At each drawn point, there is a value of c. The digital simulation method now consists of developing rows of c values along x, (usually) one f-step at a time. Let us focus on the three filled-circle points Cj t, c.j and ei+1 at time tj. One of the various techniques to be described will compute from these three known points a new concentration value e = c (f = (J + 1 )St) (empty circle) at for the next time value fJ+1, by expressing (1.1) in discrete form ... 

In microporous materials where Knudsen diffusion prevails, De cannot be calculated by solving Fick s law. The use of a discrete particle simulation method such as dynamic MC is appropriate in such cases (Coppens and Malek, 2003 Zalc et al., 2003, 2004). In the Knudsen regime, relatively few gas molecules collide with each other compared with the number of collisions between molecules and pore walls. One of the fundamental assumptions of the Knudsen diffusion is that the direction in which a molecule rebounds from a pore wall is independent of the direction in which it approaches the wall, and is governed by the cosine law the probability d.v that a molecule leaves the surface in the solid angle dm forming an angle 0 with the normal to the surface is... 

Simulation methods for problems with free surfaces governed by Navi-er Stokes equations were reviewed by Scardovelli and Zaleski (1999). The specific problems of these simulations are the location of the interface and the choice of the spatial discretization ... 

The development of the discrete solution simulation methods and their application to other systems (the pNA/MNA/dioxane solutions have been used here as the most exhaustively studied exemplar, but similar problems are inherent in all studies of organic molecules of a comparable or greater size in solution) is well described in recent publications which can be located, in particular, via the recent series by Reiss and collaborators. Here attention is drawn to a few more recent publications that are relevant to the subject. 

An important factor to be considered is the computational cost. Continuum methods are noticeably less expensive than simulation methods based on discrete models. On the other hand, simple properties, as the solvation energy AGso of small and medium-size solutes are computed equally well with both continuum and discrete methods, reaching chemical accuracy (Orozco et al., 1992 Tomasi, 1994 Cramer and Truhlar, 1995a). There is large numerical evidence ensuring that the same conclusion holds for other continuum and discrete methods as well. The evaluation of A(jso at TS has given almost identical results in several cases, but here numerical evidence is not sufficient to draw definitive conclusions. 

Modifications and enhancements in the basic finite volume method, necessary for simulations of complex multiphase or reactive flows, are discussed in this chapter. Approximations invoked in linearization of source terms and interpolation practices need to be examined carefully in light of their implications on convergence and accuracy. For most of the multiphase flow simulation methods, suitable modifications need to be incorporated in the discretized equations to avoid non-physical results. Some such modifications are discussed in this chapter. Complex geometry of the... 

Shirvanyants, D., Ding, F., Tsao, D., Ramachandran, S., Dokholyan, N.V. Discrete molecular dynamics an efficient and versatile simulation method for fine protein characterization. J. Phys. Chem. B 116, 8375-8382 (2012)... 

The lattice Boltzmann method is a mesoscopic simulation method for complex fluid systems. The fluid is modeled as fictitious particles, and they propagate and coUide over a discrete lattice domain at discrete time steps. Macroscopic continuum equations can be obtained from this propagation-colhsion dynamics through a mathematical analysis. The particulate nature and local d3mamics also provide advantages for complex boundaries, multiphase/multicomponent flows, and parallel computation. 

Advanced discrete simulation methods applied to repairable multistate systems... 

Huseby, A. B., Eide, K. A., Isaksen, S. L., Natvig B. Gasemyr, J. 2008. Advanced discrete event simulation methods with application to importance measure estimation. Safety, Reliability and Risk Analysis. Theory, Methods and Applications, volume 3. London CRC Press. 1747-1753. 

Figure 1 provides a simple example of the visual representations of a Finite State Diagram (a sub-set of Harel Statecharts) and a Petri Net system for a simple repairable failure mode which is revealed or non-revealed. The example is too simple to demonstrate the modeling power of either Harel Statechart or Petri Net models, but it shows the general principles the system is modeled as a time sequence of states and transitions for which transitions rules are employed and pre- and post conditions may apply. The systems are analysed using Discrete Event Simulation methods providing the required results, i.e. the PFD and other interesting quantities. 

Kozicki, J. et al. 2008. A new open-source software developed for numerical simulations using discrete modeling methods. Computer Methods in Applied Mechanics and Engineering 197 4429-4443. 

A hierarchy of computational models is available to simulate dispersed gas-liquid-solid flows in three-phase slurry and fluidized bed reactors [84] continuum (Euler-Euler) method, discrete particle/bubble (Euler-Lagrange) method, or front tracking/capturing methods. While every method has its own... 

The discrete phase simulation method described in Secs. 4.1 through 4.4 is capable of predicting the flow behavior in gas-liquid-solid three-phase flows. In this section, several simulation examples are given to demonstrate the capability of the computational model. First, the behavior of a bubble rising in a liquid-solid suspension at ambient pressure is simulated and compared to experimental observations. Then the effect of pressure on the bubble rise behavior is discussed, along with the bubble-particle interaction. Finally, a more complicated case, that is, multibubble formation dynamics with bubble bubble interactions, is illustrated. 

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