Grid, computational intervals

The number and shape of the grid blocks in the model depend upon the objectives of the simulation. A 100 grid block model may be sufficient to confirm rate dependent processes described in the previous section, but a full field simulation to be used to optimise well locations and perforation intervals for a large field may contain up to 100,000 grid blocks. The larger the model, the more time consuming to build, and slower to run on the computer. 

Numerical calculations for 7 = 5/3 (a = 1.5) showed that the itera.-tions within the framework of Newton s method converge even if the steps r are so large that the shock wave runs over two-three intervals of the grid W/j in one step r. Of course, such a large step is impossible from a computational point of view in connection with accuracy losses. Thus, the restrictions imposed on the step r are stipulated by the desired accuracy rather than by convergence of iterations. 

The procedure for the selection of the most appropriate time interval requires the integration of the state and sensitivity equations and the computation of lj(t), j=l,...,p at each grid point of the operability region. Next by plotting I/t), j=l,...,p versus time (preferably on a log scale) the time interval [t, tNp] where the information indices are excited and become large in magnitude is determined. This is the time period over which measurements of the output vector should be obtained. 

To follow to actually carry out a TSA simulation a three-dimensional grid, with grid interval of about 0.2 A (5 -106 equispaced points in (132)) is built and the Helmholtz energies at all grid points are computed. Before this can be done in practice, a value for (A2) must be found. Then, local minima and the crest surfaces must be found, using the procedures given in (130,132,165). To study the dynamics of the penetrant molecules on the network of sites a Monte-Carlo procedure is employed, which is presented is some detail in (97). 

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

The expected local error of computation is of second order in time and material properties. In practice an error of about five percent was observed in runs using values of 0.1 for the nondimensional spatial intervals and a factor of 1000 in variation of material parameters across one grid interval. 

The electron density in a crystal, p (xyz), is a continuous function, and it can be evaluated at any point x,y,z in the unit cell by use of the Fourier series in Equations 9.1 and 9.2. It is convenient (because of the amount of computing that would otherwise be required) to confine the calculation of electron density to points on a regularly spaced three-dimensional grid, as shown in Figure 9.3, rather than try to express the entire continuous three-dimensional electron-density function. The electron-density map resulting from such a calculation consists of numbers, one at each of a series of grid points. In order to reproduce the electron density properly, these grid points should sample the unit cell at intervals of approximately one third of the resolution of the diffraction data. They are therefore typically 0.3 A apart in three dimensions for the crystal structures of small molecules where the resolution is 0.8 A. 

While the LBNL team calculated the temperature field using their fully coupled THM model, CEA used measured in situ temperatures for their analysis. The temperature field was imported at regular time intervals into their model grid for a TM analysis. The mid-section y = 21 m was selected for the two-dimensional TM analysis. Calculations were performed using CEA s computer code Castem2000. (Verpeaux et al (1989)). 

First, we compute the integrals in the above equation numerically over the fixed a-interval [0,1]. The interval is split into N subintervals of equal length using N + 1) equi-spaced grid points... 

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