Simulation reduced parameters

Thus, the z = 0 surface (and equivalently the z = 1 surface) is generated from the original oxidized and reduced simulations. Additional simulations were performed to sample values of the polarization coordinate away from the minima, using parameters scaled according to Eq. (19). 

Figure 6 shows the behavior of the reduced monomer density p z)Rp/Np at increasing anchoring density. The stretching of the chains with increasing surface coverage, which is due to the repulsion between monomers, is evident. This plot has to be compared with Fig. 3b, where the same type of rescaling has been used. However, note that at this point, direct and quantitative comparison is not possible, since it is a priori not clear which value of the interaction parameter /3 in the self-consistent calculation corresponds to which set of simulation parameters ct, N, pa. 

In order to validate the reduced model (uncoupled population of Izhikevich neurons) we chose to perform comparisons with a direct simulation model. In this last model the internal state of each neuron is computed at each time step (with a forth-order Runge-Kutta method) using the equations of the Izhikevich model so we have complete access to individual information as opposed to the population density formalism where only the states distribution can be computed. The simulation parameters were 50 000 Izhikevich neurons in tonic spiking mode (a = 0.02, b = 0.2, c = —65, d = 6 as provided in [29]), a gaussian form of p(v, u, t) at t =0 and a constant input current of / =60 qA was applied to all neurons. The firing rate and the mean membrane potential (M M P) were computed at each time for the two methods during 15 ms. 

Fig. 11.1 (a) Theoretical phase diagrams of temperature versus polymer volume fraction in polymer solutions with the chain length 32 monomers, (b) Molecular simulation results under parallel conditions. The reduced interaction parameter sets are labeled as (Ep/Ec, B/E and the liquid-solid coexistence curve T EpIEc, B/Ec) (Hu et al. 2003a) (Reprinted with permission)... 

Fig. 11.3 Comparisons of phase diagrams of symmetric polymer blends (same chain lengths 32 monomers, only one component crystallizable) obtained from simulations (data point with the same labeled sequences as the solid lines) and from mean-field theory (solid lines for binodals, and dashed lines for liquid-solid coexistence lines) in the cubic lattice space 32. The x-axis is the volume fraction of crystallizable comptment, and the y-axis is the reduced temperature. The data labeled near the solid lines are the reduced eneigy parameter B/Ec, and all the cinves have EpIEc = 1 (Ma et al. 2007) (Reprinted with permission)...

Simulations using free volume theories were performed where the gravitational parameter ag was arbitrarily varied. Lower ag represents reduced gravity environment. Figure 2 shows the volume relaxation of poly (vinyl acetate) after sudden cooling from an initial Ti = 313 K to final Ti = 303 K for three different values of ag, where the curves A, B and C correspond to ag = 0.5 (A), 5.0 (B) and 15.0 (C) respectively and Vf = is the final volume at Tf = 303 K for ag = 0. We note that simulation results are qualitatively in agreement with experiments. An exact comparaison is difficult at this time because the simulation parameters for DRl and PMMA are not available in the literature and must be measured. 

An attractive feature of the developed hybrid model is that the simulation time can be significantly reduced. The parameters of the model can be determined from the simulation of the initial phase of the collision. In the initial phase, the lamella does not appear. Therefore, the simulation can be performed with coarse mesh. With coarse mesh resolution, the CPU-time of simulatirm reduces by a factor of 300 compared to the fully resolved simulatirm of the complete collision. 

Fig. 29 Dynamical phase diagram of a vesicle in shear flow for reduced volume V = 0.59. Symbols correspond to simulated parameter values, and indicate tank-treading discocyte and tank-treading prolate (circles), tank-treading prolate and unstable discocyte (triangles), tank-treading discocyte and tumbling (transient) prolate (open squares), tumbling with shape oscillation (diamonds), unstable stomatocyte (pluses), stable stomatocyte (crosses), and near transition (filled squares). The dashed lines are guides to the eye. From [180]...

In the application described here, a simulation study was performed to develop the ideal control strategy. Additionally, a good model of the system enables the controller parameters to be optimized during the initial engineering phase. This, in turn, means that commissioning time can be substantially reduced. 

Fig. 1.15. Translational and angular velocity correlation functions for nitrogen. MD simulation data from [70], T = 122 K, densities are indicated in the figure. Reduced units for time t = (e/cr2), for density p" = p

A series of simulations were performed to study the effect of variables such as initiator concentration, initiator half-life and activation energy on the optimum temperature and optimum time. It was assumed that initially the polymerization mixture contained S volume percent monomer, the rest of the mixture being solvent and polymer formed earlier. It was required to reduce the monomer concentration from S volume percent to 0.S volume percent in the minimum possible time. The kinetic and tbeimodyamnic parameters used are similar to those of free radical polymerization of MMA. The parameter values are given in Appendix B. 

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

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