Spatial grid

It is readily apparent that finer and finer structures get resolved as the number of spatial grids is increased. Statistical quantities, such as average slip velocity between the gas and particle phases, obtained by averaging over the whole domain, were found to depend on the grid resolution employed in the simulations and they became nearly grid-size independent only when grid sizes of the order of a few ( 10) particle diameters were used. Thus, if one sets out to solve microscopic TFM equations, grid sizes of the order of few particle... 

Using the spatial grid kernel function, (6.207), the estimated histogram is given by... 

A special case is the application domain of discrete functions (e.g., measurements on some spatial grid). The Fourier transform of a discrete function can be computed quite efficiently by a special algorithm (Fast Fourier Transform) at discrete points in Fourier space [132]. 

At this point the pros and cons of the two approaches seem to roughly counterbalance. This apparent equity extends to most accuracy criteria as well. If a timescale is not resolved, neither solution method can give detailed profiles of phenomena occurring on that scale. Similarily, to compute spatial gradients accurately they must be resolved with enough spatial grid points in either type of calculation. 

The diffusion-reaction equations are solved using finite-difference techniques employing a multilayer spatial grid to account for the corrosion product deposits present on the fuel and carbon steel surfaces, Fig. 21. For further details the reader is referred to more extensive discussions published elsewhere (6,23). 

Equation (7) can most readily be solved by an explicit finite difference scheme which steps forward in time across the spatial grid. The value of G is updated at each spatial grid point in turn. When all of the spatial grid has been updated a solution at that point in time has been calculated for the problem considered. 

Figure 11.17. Video and Raman image of an integrated circuit, obtained with line scanning and stage translation on a Dilor XY spectrometer. Area in white box was observed with a 28 x 28 spatial grid, and the Raman image was reconstructed from 28 CCD exposures. White regions in the Raman image correspond to higher Raman intensity in the 501 to 536 cm Raman shift range.

Figure 11.24. Alternative representations of the data in Rgure 11.23A. Image A is the raw 28 X 28 spatial grid, with white representing the most intense 858 cm band. Image B is an axonometric plot showing intensity as both a height above the x-y plane and as a false color scale, shown on the right. (See color plates.)...

Finer spatial grid resolution and model downscaling ... 

Thus, for both the ionic and the dipolar systems, the actual use of the rigorously derived Ewald summation for slab systems loads to a substantial increase in computer time. One way of dealing with this problem would be to employ precalculated tables [252] for potential energies (and forces) on a three-dimensional spatial grid amended by a suitable interpolation scheme. Another strategy is to employ approximate methods such as the one presented in the subsequent Section 6.3.2. 

Dissinger, G. R. GRD1 - A New Implicit Integration Code for the Numerical Solution of Partial Differential Equations on Either Fixed or Adaptive Spatial Grids, Doctoral Dissertation, Lehigh Univ. Bethlehem, PA, 1983. 

PAA can form associates even in much diluted solutions because of interaction between polar groups of different molecular chains which results in appearance of spatial grid and formation of a gel. 

In systems of high symmetry such as atoms, it is possible to represent the orbitals (i.e., the spatial part of the spin orbitals) numerically on a spatial grid. For polyatomic systems, however, it is more common to represent the spin orbitals as linear expansions of a set of N simple, analytical one-electron basis functions Xk(x), mostly centered on the atoms in the system. These linear expansions may then be written as ... 

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