Discrete simulation parameters

For parameter estimation purposes, simulated annealing can be implemented by discretizing the parameter space. Alternatively, we can specify minimum and maximum values for each unknown parameter, and by using a random number uniformly distributed in the range [0,1], we can specify randomly the potential parameter values as... 

The algorithms discussed earlier for time averaging and local time stepping apply also to velocity, composition PDF codes. A detailed discussion on the effect of simulation parameters on spatial discretization and bias error can be found in Muradoglu et al. (2001). These authors apply a hybrid FV-PDF code for the joint PDF of velocity fluctuations, turbulence frequency, and composition to a piloted-jet flame, and show that the proposed correction algorithms virtually eliminate the bias error in mean quantities. The same code... 

The sampling rate determines the accuracy of the discrete simulation. Changing the OVERSAMP parameter in the code above will reveal that... 

The laminar flov s occurring in the inking process were simulated with a standard finite dement moddlir discretization of the Navier—Stokes equations using a commercial code (COMSOL, FEMLAB GmbH), in which the evaporation was calculated timultaneously to produce boundary conditions for the flow into the meniscus. (See Supplementary Information for details and results for a tar e of simulation parameters). 

In the following, discretization methods for the random parameter field are illustrated, and a mathematical theory for the approximate solution of stochastic elliptic boundary value problems involving a discretized random parameter field that is represented as a superposition of independent random variables is outlined. In the random domain, global and local polynomial chaos expansions are employed. The relation between local approximations of the solution and Monte Carlo simulation is considered, and reliability assessment is briefly discussed. Finally, an example serves to illustrate the different solution procedures. 

In die simulation model, die FCC system was subdivided into discrete elements and suitable subsystems. This model provided all die process parameters such as pressures, flowrates, and temperatures. Figure 6-44 shows die corresponding block diagram. (The model for die expander, piping systems, and vessels is based on a gas turbine model described by GHH Borsig in a paper by W. Blotenberg.)... 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

The simulation module simulates the basic operation(s) which are processed by a combination of a vessel and a station using a discrete event simulator. All necessary data (basic operation(s), equipment parameters, recipe scaling percentage, etc.) is provided by the scheduling-module. The simulator calculates the processing times and the state changes of the contents of the vessels (mass, temperature, concentrations, etc.) that are relevant for logistic considerations. 

Discrete time controllers will not normally be stand-alone units but will be simulated within the software of a digital computer. The capacity of the computer can be used if necessary to produce more complex forms of feedback control than those provided by the standard algorithms of the classical fixed parameter controller. 

The evolution of a system described by an equation related to eqn 2.26 was studied using cell dynamics simulations by Oono and co-workers (Bahiana and Oono 1990 Puri and Oono 1988). In the CDS method the continuous order parameter is discretized on a lattice and at time t is denoted where n labels... 

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

The first term on the right hand side of Eq. (1) gives the amount of particles of / th size at the moment of time tn which neither moved forwards nor broke during one time step of simulation. The second term represents those particles of the same size that moved backwards from the second section. The third term expresses the amount of particles that remained in the first section and have broken from some size higher than x, to that subinterval of size. Term ai(tn) denotes the amount of particles of the / th size that freshly enters the system. Finally, the last term ( Vf - Vb) f(xi-( 1/2)) p(yj, Xi, tn-d) is the fraction of particles of the /th size that left the mill at the moment of time tn-d and was classified by the classifier to recycle into the mill for further grinding. Here, function ///,. 0 < x/j, = t (x,) < 1, describes the operation of the classifier, whilst parameter d denotes the discrete time delay in the recirculation line. 

The aforementioned controllers were implemented on the full-order 2006-dimensional discretization of the original distributed-parameter model, and their performance was tested through simulations. The relevant Matlab codes are presented in Appendix C. 

Different solvation methods can be obtained depending on the way the (Vs(r p)) xj tern1 is calculated. So, for instance, in dielectric continuum models ( Vs(r p)) x is a function of the solvent dielectric constant and of the geometric parameters that define the molecular cavity where the solute molecule is placed. In ASEP/MD, the information necessary to calculate Vs(r, p))[Xj is obtained from molecular dynamics calculations. In this way (Vs(r p))[Xj incorporates information about the microscopic structure of the solvent around the solute, furthermore, specific solute-solvent interactions can be properly accounted for. For computational convenience, the potential Vs(r p)) X is discretized and represented by a set of point charges, that simulate the electrostatic potential generated by the solvent distribution. The set of charges, is obtained in three steps [26] ... 

---