Deterministic modeling techniques

This chapter provides an overview of the most frequently applied numerical methods for the simulation of polymerization processes, that is, die calculation of the polymer microstructure as a function of monomer conversion and process conditions such as the temperature and initial concentrations. It is important to note that such simulations allow one to optimize the macroscopic polymer properties and to influence the polymer processability and final polymer product application range. Both deterministic and stochastic modeling techniques are discussed. In deterministic modeling techniques, time variation is seen as a continuous and predictable process, whereas in stochastic modeling techniques, a random-walk process is assumed instead. 

Within the deterministic modeling techniques, a first distinction can be made between numerical methods enabling the simulation of only averages of the CLD as a function of monomer conversion, that is, the method of moments, and methods enabling the simulation of the full CLD, that is, the/w// CLD methods. In what follows, the main aspects of these methods are discussed in detail. For illustration purposes, it is assumed that only two populations of macrospecies, namely the living and dead polymer molecules (Fig. 10.1), exist and that only a limited number of chain-growth polymerization reactions can take place. 

Once the seismic source and Earth crustal model have been adequately described, nearfault ground motion simulations in the low-frequency range (e.g., below 1 Hz) can be performed using deterministic modeling techniques [e.g., discrete wavenumber method (DWN), finite difference method (FDM), finite element method (FEM), boundary element... 

In the present study, the problem is written as a nonlinear programming problem and is solved with SQP technique. Two process models are evaluated when the process is optimized using the SQP technique. The first one is a deterministic model with the kinetic parameters determined by Atala et al. (1), and the second one is a statistical model obtained using the factorial design technique combined with simulation. 

When the optimization problem is solved using the SQP technique and the deterministic model, the values of temperature and concentrations in the fermentor have to be considered as optimization variables, so the number of the optimization variables is higher than using RSM. The optimization variables are concentrations of viable cells (Xv), dead cells (Xd), substrate (S), product (P), temperature (T) as well as the variables used by Costa et al. (5) S0, R, tr, and r. 

B. Solving Deterministic, Continuum Differential Equation Models Techniques and Status 8... 

Deterministic models are used to estimate chronic, or long-term, dietary exposures. Although research is currently being conducted on the use of distributional techniques for chronic dietary risk assessment, the author is not aware of any working model at this time. 

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. 

Numerical alternating direction techniques required 10 hours of computation time on the IBM-7094 to achieve solutions with 80% convergence to the true steady state (4). Computation time was reduced to about 30 min per solution for triple the model time length, using Monte Carlo methods (5) to solve the deterministic model on an IBM-360 model 40 (the 7094 is about 10 times faster than the model 40). A digital computer solution of a three-component model (02, C02, and glucose)... 

Jain, A. Prasad Indurthy, S. K. V. (2003) Comparative Analysis of Event-based Rainfall-runoff Modeling Techniques-Deterministic, Statistical, and Artificial Neural Networks, Journal of Hydrologic Engineering, 8, p. 93-98. 

In many ways modeling the repair process is difficult because the repair process is quite different from the failure process. Random failures are due to a stochastic process and most of our modeling techniques were created for these stochastic processes. Certain aspects of the repair process are deterministic. Other aspects of the repair process are stochastic. Fortunately, we can approximate the repair process more accurately with Markov models than most other techniques. 

The aim of this chapter is to describe the basis and the advantage of a molecular conception of emulsion polymerization, by means of molecular modeUng techniques. Indeed, a molecular picture of the processes should provide a deeper understanding than might be achieved with time- and space-averaged deterministic models. 

Although there are several techniques for estimating the reaction rate constants based upon the deterministic model, these methods are usually rather complicated, and the results cannot be statistically characterised. That is why from time to time estimates based upon one or another stochastic model are suggested. Such a suggestion has earlier been described under the name fluctuation-dissipation theorem , and similar methods have been presented by Mulloolly (1971, 1972, 1973), Hilden (1974), and Matis and Hartley (1971). 

This section is an overview of the most important deterministic and stochastic modeling techniques to obtain the polymer microstracture as a function of monomer conversion and polymerization conditions at the microscale. It is assumed that, for this scale, the bulk concentrations and temperature are known. The simplest case is the simulation of a batch polymerization reactor on laboratory scale with perfect macromixing and isothermicity implying a reactor with spatial homogeneity of the bulk concentrations and temperature. 

A main distinction has been made between deterministic and stochastic modeling techniques. A further distinction has been proposed based on the scale for which the mathematical model must be derived (eg, micro-, meso-, and/or macroscale). Notably, the complexity of the model approach depends on the desired model output. Detailed microstractural information is only accessible using advanced modeling tools but these are associated with an increase high in computational cost. The advanced models allow one to directly relate macroscopic properties to the polymer synthesis procedure and, thus, to broaden the application market for polymer products, based on a fundamental understanding of the polymerization kinetics and their link with polymer processing. 

Several systematic test design techniques and testing approaches have been developed to assist in systematically exploring the available choices and selecting a representative sample (Beer and Heindl 2007). Criteria for selecting such a sample in safety critical systems are typically based on requirements, system models, control flow, data flow and an operational profile. Deterministic or non-deterministic models of different categories and formality are used. Relevant questions are ... 

If problems of this kind are to be avoided, potential users will have to be trained to an exceptionally high level in the modelling technique used in the old methods and the overhead in doing this will be high. The Monte Carlo method could be the remedy. Due to its inherent slowness relative to the deterministic methods it cannot be used at present for all routine calculations. However, it can provide reference values which are transparent to physicists and engineers. 

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