Numerical Simulation Procedure

NUMERICAL SIMULATION PROCEDURE 3.1. Introduction of finite elements 

For the simulation of the described NMR experiments on dispersed nanoparticles, one needs to account for a number of key variables the time, the orientation of a given particle in space and its location with respect to a local reference system. In addition, in case of sample spinning experiments, the time-dependent orientation of the rotor has to be considered. For numerical calculation, aU these parameters are approximated in a finite element scheme  

Consequently, the full continuum of all possible orientations in space is approximated by a total number of discrete sites. 

The population of spatial positions depends on the shape of the sample 

Time development caused by local variations of the Larmor frequency 

---

Using this approach, the hopping transport was modeled as a quasi-Marcovian process. The details of the analytical formulas forming the basis of the modeling and the numerical simulation procedure are given elsewhere.62 The values of parameters included in the hopping transport model are listed in Table 7. 

Another area where advanced experimental methods typically have to be associated with efficient numerical simulation procedures is when solid-state NMR is used to extract information about molecular motion. Over the years, a large variety of different... 

In this equation, A represents the number of crystals, p is the length of the three-phase junction (i.e., the perimeter of the electrode/crystal interface), and Ax, Az denote the size of the discrete boxes in which the crystal is divided for numerical simulation procedures. 

Tang et al. (1996) present the results of a detailed numerical simulation procedure to model the effects of a bursting spherical vessel. They numerically solved the nonsteady, nonlinear, one-dimensional flow equations. This resulted in a more detailed figure to replace Figure 3.15. 

It should be observed that every element except the powder system in the recovery system is chosen for favorable shock properties which can be confidently simulated numerically. The precise sample assembly procedures assure that the conditions calculated in the numerical simulations are actually achieved in the experiments. The influence of various powder compacts in influencing the shock pressure and mean-bulk temperature must be determined in computer experiments in which various material descriptions are used. Fortunately, the large porosity (densities from 35% to 75% of solid density) leads to a great simplification in that the various porous samples respond in the same manner due to the radial loading introduced from the porous inclusion in the copper capsule. 

Baker et al. (1978a) developed a method which can predict blast pressures in the near field. This method is based on results of numerical simulations (see Section 6.3.1.1) and replaces Step 5 of the basic method (Figure 6.20). The refined method s procedure is shown in Figure 6.25. 

Process validation is the procedure that allows one to establish the critical operating parameters of a manufacturing process. Hence, the constraints imposed by the FDA as part of process control and validation of an SMB process. The total industrial SMB system, as described, is a continuous closed-loop chromatographic process, from the chromatographic to recycling unit and, with the use of numerical simulation software allows the pharmaceutical manufacturer rapidly to design and develop worst-case studies. 

Chemical vapor deposition (CVD) of carbon from propane is the main reaction in the fabrication of the C/C composites [1,2] and the C-SiC functionally graded material [3,4,5]. The carbon deposition rate from propane is high compared with those from other aliphatic hydrocarbons [4]. Propane is rapidly decomposed in the gas phase and various hydrocarbons are formed independently of the film growth in the CVD reactor. The propane concentration distribution is determined by the gas-phase kinetics. The gas-phase reaction model, in addition to the film growth reaction model, is required for the numerical simulation of the CVD reactor for designing and controlling purposes. Therefore, a compact gas-phase reaction model is preferred. The authors proposed the procedure to reduce an elementary reaction model consisting of hundreds of reactions to a compact model objectively [6]. In this study, the procedure is applied to propane pyrolysis for carbon CVD and a compact gas-phase reaction model is built by the proposed procedure and the kinetic parameters are determined from the experimental results. 

Depending on whether or not stochastic features are introduced in the simulation procedure, simulation methods are sometimes classified as stochastic or deterministic. Although the second term is usually applied to methods related to the numerical solution of Newton s equations, the first term is applied to a wide variety of simulation metfiods. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

Without a solution, formulated mathematical systems (models) are of little value. Four solution procedures are mainly followed the analytical, the numerical (e.g., finite different, finite element), the statistical, and the iterative. Numerical techniques have been standard practice in soil quality modeling. Analytical techniques are usually employed for simplified and idealized situations. Statistical techniques have academic respect, and iterative solutions are developed for specialized cases. Both the simulation and the analytic models can employ numerical solution procedures for their equations. Although the above terminology is not standard in the literature, it has been used here as a means of outlining some of the concepts of modeling. 

Wienke D, Lucasius C, Kateman G (1992) Multicriteria target vector optimization of analytical procedures using a genetic algorithm. Part I. Theory, numerical simulations and application to atomic emission spectroscopy. Anal Chim Acta 265 211... 

Generally, the results obtained through the numerical simulation showed good agreement with the experimental data leading to the conclusion that CFD techniques can be effectively used in consequence assessment procedures concerning toxic/flammable dispersion scenarios in real terrains, where box models have limited capabilities. 

This procedure is used for evaluation of the borders and morphology of the pore. The pores are considered as the linked parts of the whole porous space accessible for a probing sphere with a size r=srz where rz is a chosen size [100], For numerous simulations, it is convenient to determine the windows between the cavities as the borders and to assume that the whole porous space is located in the cavities, and windows do not have volume and are only the 2D cross sections between the cavities. In this sense, the windows only determine accessibility of the cavities. This approach allows consideration of the porous space as a network of sites (cavities) and bonds (windows). 

Finally, FDTD may be used to model the coupling of the focal field into the PhC-waveguide, potentially with the presence of an air or glue gap. Even such a simulation procedure with adapted numerical methods for each part of the propagation requires a considerable computation time. To speed up the simulation process for system optimisation remarkably, the FDTD-simulation can be replaced by a formula for the coupling efficiency to a conventional high-index or a PhC-waveguide, ... 

Numerical Tests. How do we measure the heat distribution (Eq. (215)) in numerical simulations of NEAS A powerful procedure that uses the concept of inherent structures goes as follows. The heat exchanged during the time interval (t > t ) has to be averaged over many... 

A systematic study of the validity of such a procedure was performed in collaboration with ETH-Ziirich [15], The validation of the procedure was based on numerical simulations of dynamic experiments and adiabatic runaway curves. These simulations were carried out using different rate equations nth-order, consecutive, branched, and autocatalytic reactions. Moreover, the results were compared to experimental results obtained with over 180 samples of single technical chemical compounds, reactions masses, and distillation residues [17] (Figure 11.8). Thus, they are representative for industrial applications. The line corresponding to this rule (Equation 11.5) is also represented (full line) in Figure 11.8. All experimental points lie above the line and the safety margin remains reasonable. Thus, the method is conservative, but delivers a reasonable safety margin. 

---