Simulation models, deterministic

Molecular Dynamics and Monte Carlo Simulations. At the heart of the method of molecular dynamics is a simulation model consisting of potential energy functions, or force fields. Molecular dynamics calculations represent a deterministic method, ie, one based on the assumption that atoms move according to laws of Newtonian mechanics. Molecular dynamics simulations can be performed for short time-periods, eg, 50—100 picoseconds, to examine localized very high frequency motions, such as bond length distortions, or, over much longer periods of time, eg, 500—2000 ps, in order to derive equiUbrium properties. It is worthwhile to summarize what properties researchers can expect to evaluate by performing molecular simulations ... 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

In this chapter, we explain the technique of sequential bifurcation and add some new results for random (as opposed to deterministic) simulations. In a detailed case study, we apply the resulting method to a simulation model developed for Ericsson in Sweden. In Sections 1.1 to 1.3, we give our definition of screening, discuss our view of simulation versus real-world experiments, and give a brief indication of various screening procedures. 

Myers and Montgomery (2002). We, however, focus on experiments with computer or simulation models which may be either deterministic or stochastic also see Kleijnen et al. (2002). An introduction to simulation modelling is provided by Law and Kelton (2000). 

The technique of sequential bifurcation is an important and useful method for identifying important factors in experiments with simulation models that involve a large number of factors. We have demonstrated the steps of this technique through a case study on three supply chain configurations in the Swedish mobile communications industry. We have formalized the assumptions of the technique and found that, in practice, these assumptions may not be too restrictive, as our case study illustrates. We have extended the technique of sequential bifurcation to random (as opposed to deterministic) simulations. 

Whatever the geologic causes, there are several purely statistical inferences to be drawn from Figure 16 which bear directly on the issue of reservoir simulation. The size of grid four may be a natural choice for the grid block size in a deterministic simulation model. Such a selection would minimize the variation between blocks and may, in fact, make stochastic assignments of secondary importance (thus, reducing the differences between realizations). The variation of the fifth scale would be incorporated as pseudo functions or megascopic dispersivity into individual blocks. 

The developed optimization is solved with genetic algorithm as the previous study based on deterministic optimization techniques showed that it is often trapped in local optima, due to highly non-linear nature of formulations in the model. The simulation model and genetic algorithm is interacted to produce high quality optimal solution(s), although computational time is relatively expensive. 

Equations 15 to 25 demonstrate a versatile approach to the phenomenological modeling. Ohen and Civan (74) present, additional, auxiliary relationships that make this model deterministic. In all there are 12 parameters (kl9 k2,. . . , (directly measured by experiment, some are evaluated through sensibility tests, and the rest are determined by history matching experimental results with simulations. 

See Law (2007) or Carson (2004) for a more detailed discussion about these prerequisites. The sub-class of deterministic simulation is typically referred to as computer experiments, see Santner et al. (2003) for more details. In this work the focus is on stochastic simulation models. 

In microfluid mechanics, the direct simulation Monte Carlo (DSMC) method has been applied to study gas flows in microdevices [2]. DSMC is a simple form of the Monte Carlo method. Bird [3] first applied DSMC to simulate homogeneous gas relaxation problem. The fundamental idea is to track thousands or millions of randomly selected, statistically representative particles and to use their motions and interactions to modify their positions and states appropriately in time. Each simulated particle represents a number of real molecules. Collision pairs of molecule in a small computational cell in physical space are randomly selected based on a probability distribution after each computation time step. In essence, particle motions are modeled deterministically, while collisions are treated statistically. The backbone of DSMC follows directly the classical kinetic theory, and hence the applications of this method are subject to the same limitations as kinetic theory. 

It may be useful to know that simulation of a one-year-long period of this really complex phenomenon takes 2-3 minutes on an IBM 3031 computer, only three times as much as to simulate the deterministic model. 

Beamon [51] reviewed the relevant research on supply chain design and analysis, and proposed some future research directions. Supply chain model is divided into deterministic models, stochastic analytical models, economic models, and simulation models. The future directions of the research include supply chain performance measurement methods, the establishment of the decision-making model and developing standards and technology of supply chain design and analysis. 

Execution of the method requires the physical domain to be divided into a distribution of conqiutational cells. The cells provide geometric boundaries and volumes, which are used to sample macroscopic properties. Also, only molecules located within the same cell, at a given time, are allowed to collide. The DSMC simulation proceeds from a set of prescribed initial condition. The molecules randomly populate the computational domain. These simulated molecules are assigned random velocities, usually based on the equilibrium distribution. The simulated representative particles move for a certain time step. This molecule motion is modeled deterministically. This process enforces the boundary conditions. With the simulated particles being appropriately indexed, the molecular collision process can be performed. The collision process is modeled statistically, which is different from deterministic simulation methods such as the molecular dynamics methods. In general, only particles within the same computational cell are considered to be possible collision partners. Mthin each cell, collision pairs are selected randomly and a representative set of collisions is performed. The post-collision velocities are determined. There are several... 

To carry out the above mentioned, appear diverse types of models that set up methodologies to represent the system (Bause Kritzinger, 2002 Buzacott Shanthikumar, 1993 Fuqua, 2003 Schryver et al, 2012 Zio Pedroni, 2010). Some of these models are mathematical models, stochastic models, deterministic models, simulation models for discrete events, Markov chains, among others. Each of these models, achieve different representation grades of the system, so its correct selection is relevant to accomplish with the desired objectives. On the other hand, each model possesses different requirements of information and development times, since many times is not possible to apply any model to a specific system. 

Structural reliability methods have been applied to estimate runway-overrun probabilities and were shown to be suitable for this purpose. In particular, the calculated sensitivities and FORM importance measures support the interpretation and further development of the model. With the help of the FORM importance measures, one can simplify the model by modeling the quantities of minor importance deterministically. We have found that the parameters Temperature , Pressure , Reverser deployment , Spoiler deployment and End breaking may be modeled deterministically without inducing a large error. This does, however, not imply that the influence of these quantities is small. It can only be concluded that the uncertainty associated with the quantity is not significant. The parameter sensitivities were in this paper calculated based on the samples obtained with subset simulation. These parameter sensitivities describe the effect of a change of the mean or standard deviation on the probability of failure. We presented the parameter sensitivities in the form of elasticities, which are typically easier to interpret. From the results it can be seen that the variables, which are of main importance according to the FORM importance measures, are... 

Poole, D. and Raftery, A.E. 2000. Inference for deterministic simulation models the Bayesian melding approach. Journal of the American Statistical Association, 95, 1244-1255. 

In the first chapter several traditional types of physical models were discussed. These models rely on the physical concepts of energies and forces to guide the actions of molecules or other species, and are customarily expressed mathematically in terms of coupled sets of ordinary or partial differential equations. Most traditional models are deterministic in nature— that is, the results of simulations based on these models are completely determined by the force fields employed and the initial conditions of the simulations. In this chapter a very different approach is introduced, one in which the behaviors of the species under investigation are governed not by forces and energies, but by rules. The rules, as we shall see, can be either deterministic or probabilistic, the latter leading to important new insights and possibilities. This new approach relies on the use of cellular automata. 

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