Variable grid spacing

If the OCFE solution procedure is to be used regularly for system analysis, a careful study of the problem of locating elements is necessary. Finlayson (1980) gives a basis for this study in a cursory examination of variable grid spacing and elements sizes, or adaptive meshes as he calls them. The simplest procedure is based on physical information about the solution profile such as in the present case the point where CO is depleted within the reactor bed. 

The drawings of the mesh in Figures 1 and 2 show that variable grid spacing is considered in the grid design, which will improve the accuracy of the numerical simulation near by the pumping... 

This method is applicable to equations with more than two independent variables, but only two will be considered here. A region over which a solution is sought is divided into a grid with m divisions of variable x and n divisions of variable y, spaced at distances Ax and Ay. A first derivative is approximated by... 

Instead of using the method of separation of variables, a numerical solution to Eq. (4.144) can be easily obtained. Here, a numerical finite-difference solution will be discussed. A series of grid lines in the Z and R directions are introduced as shown in Fig. 4.16, a uniform grid spacing, ARt being used in the radial coordinate direction. 

Figure 3.3. Typical PC 1 vs. PC 2 loading plot from a GRID/ PCA analysis. The points in the plot represent the variables of theX matrix, i.e. the positions in the grid space.

The random phase volume model is defined for any number of composition variables, and the number of phase points is defined by requiring the average spacing between phase points to be = 0.25. To avoid edge effects, additional points are added in a belt of width 2 around the simplex of allowed compositions. The number of phase points for different grid spacing is shown in Table 1. 

For the grid method, the number of samples in the composition space is and the number of samples in the noncomposition space is. The grid spacing of the composition variables is =... 

Asymptotically, hyperspherical coordinates become inadequate since the energetically allowed space contains fewer and fewer grid points. It is therefore necessary to map the wave function onto other coordinates (e.g., Jacobi coordinates). However, in the semiclassical treatment of the problem this is not possible since the wave function is known only in a restricted phase space, i.e., in either (6, ) or (p, 0, < >) space. It is therefore necessary either to carry out the projection in these coordinates by using variable grid methodology or to introduce a mixed Jacobi-hyperspherical coordinate treatment. This latter procedure is possible since we can express the Hamiltonian as... 

D QSAR analyses based on similarity matrices offer a valuable new tool for the quantitative description of structure-activity relationships. Also hydrophobic fields and interaction fields with different probe atoms may be implemented, like in CoMFA studies. It is hoped that the preliminary results [1064, 1065] stimulate active research in this field to achieve further methodological improvements. Several CoMFA-inherent problems apparently do not arise in the molecular similarity matrices approach, e.g. the cut-off selection, a proper grid spacing, and the elimination of variables having low variance. 

Finally, we employed the CGLT of Kierlik et al. [7], which is a simplified DFT (nearest neighbor), to model adsorption in our 5 nm Vycor sample. We used the adsorbate-substrate energy grid that was used in the GCMC simulations [13] with variable lattice spacing all... 

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