III. Definitions and Examples of Some Terms

A. Regions

A region is a section of an axis with a beginning coordinate, an end coordinate, a number of divisions , and a number that indicates the type of spacing (equal geometric, logarithmic small to large, or logarithmic large to small). If any region in an axis has already been defined, the beginning border of the next region is automatically set equal to the end border of the previous region. Example 1 shows a set of coordinates on the x-axis with four different regions of the same size, each with ten divisions. The first consists of equall spaced nodes; the second of geometrically spaced nodes with a geometric factor of two; the third of logarithmically spaced nodes, small to large; and the fourth of logarithmically spaced nodes, large to small. This is actually a two-dimensional example, where the y-axis has only one region with one division in order to make the x divisions easier to visualize.

B. Zones

When there are multiple regions on two or more axes, the region boundaries cross, forming areas with a certain type of spacing in the x dimension, another type of spacing in the y dimension, and (if it is three-dimensional) another type of spacing in the z dimension. for example, in a two-dimensional grid, the region boundaries of the y-axis cross the region boundaries of the x-axis, dividing each region in the x-axis into areas with the same spacing in the x dimension, but each with a different type of spacing in the y dimension. These areas are referred to as zones. Example 2a has four regions on the x-axis, similar to example 1, but now there are two regions on the y-axis. The first has ten divisions, and the second has twenty divisions. Both have equally spaced nodes. The resulting cross of region borers creates eight zones. Example 2b is the same grid as example 2a, but it sohows the zones separated for easier visualization.

In AVS output, the zone number is used as the naterial number. If the user defines no properties for these material numbers, AVS will simply show each zone as a different color. Otherwise, the user can define these material properties to his or her preference, making each zone have different properties.

C. Equal Spacing

The user may pick one of four kinds of nodal spacing for each region. The simplest of these is referred to as equal spacing. Equal spacing of nodes simply means that the region is divided equally into the dgiven number of divisions. Each division is the same length. The formula for calculateing the coordinates of equally spaced nodes can be found in Appendix 1.

Example 3 shows a grid with four regions on the x-axis, each with equal spacing. The first two zones are both ten units long (last coordinate of zone munus first coordinate of zone=10) but the first has only three divisions while the second has seventeen. The third zone is only five units long but has ten divisions, and the fourth zone is twenty units long and also has ten divisions.

D. Geomtric Spacing

In a geometrically spaced region, the distance between two consecutive nodes changes by a factor given by the user. TWhen geometric spacing is chosen, the user will be prompted for a "geometric factor" that determines this rate of change. For example, with a geometric factor of two, the largest division is twice the length of the previous division, which is twice the length of the division before that, etc. The coordinates become members of a geometric series. The actual formulas used to calculate the coordinates in a geometrically spaced series can be found in Appendix 1.

Example 4 shows some sample geometrically spaced regions. Each region is ten units long and has 10 divisions. The first has a geometric factor of 1.2, the second has a factor of 4, the third ahs a factor of 1.6, and the fourth has a factor of 0.625. If a user wants the distance between nodes to shrink by a given factor as opposed to growing by that factor, he or she can simply input the value of 1 divided by that factor (1/f) when prompted for a geometric factor. Zones 3 and 4 in Example 4 demonstrate this relationship. Zone 3 has a geometric factor of 1.6, and zone 4 has a geometric factor of 0.625 = 1/1.6. They have the same rate of change of distance between nodes, but in opposite directions (distances in zone 3 are growing from left to right while distances in zone 3 are shrinking from left to right.

A geometric factor of 1 would result in the equivalent of equal spacing.

Also notice that the smallest divisions becomes very small before the geometric factor gets very large at all. As this length decreses, accuracy decreases. I therefore recommend avoidance of large geometric factors as well as avoidance of very large numbers of divisions within a small region.

E. Logarithmic Spacing

In a region with logarithmic spacing, coordinates of nodes are determined as a function of a base 10 logarithm. Since by nature, logarithmic spacing results in smaller divisions growing to larger divisions as we move from left to right across the region, I have included a separate type of spacing that allws the user to assign "backwards" logarithmic spacing, where, from left to right, the divisions are larger and become smaller. Details of how logarithmic spacing is calculated can be found in Appendix 1.

Example 5 shows some sample regions with logarithmic spacing. There are 4 regions on the x-axis, each 10 units long. The first two have 5 divisions each, the first with spacing of small divisions to large and the second with large to small, and the last two have 30 divisions each, the third with small divisions to large and the fourth with large to small.

IV. Limitations

A. Dimensions

This program can generate 1, 2, or 3 dimensional grids, but a 2-dimensional grid will use only the x- and y-axes, and a 1-dimensional grid will use only the x-axis.

B. Limited Number of Nodes and Regions

As the program standds now, there can be no more than 200 nodes in any one axis, and no more than 50 regions. If desired, these limits can be easily changed in the source code. They are constants that are defined at the beginning of the code called MAXNODES and MAXZONES.

C. Accuracy

Coordinates are given with accuracy up to 6 significant digits. If the number of divisions becomes very large with respect to the size of the regionk, or if the geometric factor of a region with geometric spacing gets too large, the smallest divisions can become too small with respect to the other divisions in the grid, and accuracy decays. No more than 6 decimal digits difference in length of divisions can be calculated. I recommend avoidance of these practices that may result in verry small sized divisions.

Happy Gridding!

V. Appendix I

This appendix gives the formulas use to calculate the different kinds of spacing available from this grid generator, including equal spacing, geometric spacing, and logarithmic spacing. For all of the formulas given in this appendix, the following conventions will be used:

dx is the distance between first two consecutive nodes in region

n1 is the first coordinate in the region (This is inputted by the user for the first region. For all subsequent regions, n1 is set equal to the last coordinate of the previous region.

n2 is the second coordinate in the region, etc..

nx is the last corrdinate in the region,

d is the number of divisions in the region

x is the total number of coordinates in region = d + 1.

f is the geometric factor.

L is the length of the region.

px is the percentage associated with node nx.

A. Equal Spacing

Equal spacing of nodes is calculated according to the following formula:

If the region being configured is the first region of the axis, n1 is inputted as the "beginning coordinate of the region" by the user. Ptherwise, n1 is the same coordinate that ended the previous region. The program does not duplicate this coordinate as a separate node in the axis.

B. Geometric Spacing

Geometric spacing is calculated according to the following formula:

C. Logarithmic Spacing

To achieve logarithmic spacing, all cases are determined with respect to an analogous region starting at 1 and ending at 10. The code divides this temporary region equally into the given number of divisions, takes the logarighm of each of those numbers, and treats the resulting numbers as percentages of the size of the actual region. That is the reason for choosing the region from 1 to 10: log(1) = 0, and log(10) = 1. Therefore, the first coordinate of the region is placed at 0% of the way across the actual region (at the beginning), and the last node is placed at 100% of the way across the region (at the end). Let's consider the simplest example:

Let's say the user actually wants a region that starts at 1 and ends at 10, so we need no analogous region., Let's also say the he or she wants 9 divisions. The code divides the region into 9 equally spaced divisions, giving us the coordinates 1, 2, 3, 4 .... 10. It then takes the log of each of these numbers, giving us the following percentages: (the code actually calculates the logarithms to 6 decimal points precision)

log(1) = 0% log(2) = 0.310 = 30.1% log(3) = 0.477 = 47.7% log(4) = 0.602 = 60.2% ... log(10) = 1 = 100% We now calculate the actual coordinates of the nodes by the following formula: nx = n1 + (px * L)

So for example,

n1 = n1 + (p1 * L) = 1 + (0% * 9) = 1

n2 = n1 + (p2 * L) = 1 + (30.1% * 9) = 3.709

n3 = n1 + (p3 * L) = 1 + (47.7% * 9) = 5.29 , ...etc.

n10 = 1 + (p10 * L) = 1 + (100% * 9) = 10

The formula for logarithmic spacing, large to small, is similar: nx = n1 - (px*L), where n1 in this case is the last coordinate of the region, the end border.

Even if the example above had not begun at 1 and ended at 10, we would have followed the same procedure, dividing a region from 1 to 10 into the number of divisions given by the user, taking the logs of those numbers, and using them as percentage distances across the region.

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