Subdivision of 2D electrodes into segments.

Dealt with elsewhere:

The straight and circular sections are subdivided by the program into straight parts. After rotation about the axis these parts therefore create 'hoops' that have the form of short lengths of cylinders, short lengths of cones or annuli of discs. The charge density is assumed to be uniform over the surface of each hoop.

The number N of subdivisions is entered in the databuilder. Another number p can also be entered using /Advanced options/Concentrate segments at one end/.’

An electrode can be cancelled (that is, ignored) by putting N = 0 (but see cancelling electrode data).

There is an option in the View drop-down menu to see individual segments by creating artificial gaps between them. We encourage the user to check the subdivision of the electrodes with this option.

There is also an option, in /contours/potentials/segments, to output detailed information on segment coordinates and charges.

Important: It is generally true that the best accuracy is obtained when all the segments carry approximately the same charge. This is particularly important for the critical parts of a system, such as a cathode and its surrounding electrodes. It is less important for electrodes that are far from a beam or are not ‘seen’ by the beam. Please look at section 3.4 of the Users Guide or the general advice on segmentation.

Advanced option to concentrate segments at one end.

p denotes the type of subdivision.

It is usually 1, and then the subdivisions are uniformly spaced.

When p is in the range 2 to 4 the subdivisions are more closely spaced at the first end of the electrode.

When p is in the range –4 to -2 they are more closely spaced at the second end.

For example, take zi = 0, zf = 1 and n = 5. Then the following choices of p give the following values of z:

3: 0.00 0.008 0.064 0.216 0.512 1.00

2: 0.00 0.04 0.16 0.36 0.64 1.00

1: 0.00 0.20 0.40 0.60 0.80 1.00

-1: 0.00 0.20 0.40 0.60 0.80 1.00

-2: 0.00 0.36 0.64 0.84 0.96 1.00

-3: 0.00 0.488 0.784 0.936 0.992 1.00

Values of p from -1 to -0.1 and 0.1 to 1 can also be used although the above pattern is changed.

For example, taking zi = 0, zf = 1 and n = 5 again, then the following choices of p give the following values of z:

.5: 0.00 0.447 0.632 0.775 0.894 1.00

-.5: 0.00 0.106 0.225 0.368 0.553 1.00

These are useful for subdividing a disc into equal areas, see below.

Values of p in the range 102 to 104 trigger the option to concentrate the segments at BOTH ends of the electrode. 100 is subtracted by the program and the remainder is applied to the 2 halves of the electrode.

To see the individual subdivisions, click on ‘create gaps to see individual elements’ in the view menu.

The formula used for subdividing an electrode is as follows. Suppose that the electrode extends from z = zi to z = zf. Then the N subdivisions are bounded by z(j), where j = 0 to N, and where the z(j) are given by

p > 0: zi+(zf-zi)*(j/n)**p

p < 0: zf+(zi-zf)*((n-j)/n)**abs(p)

(and where * signifies 'times' and ** signifies 'to the power of').

For p > 0, the length of the first is (1/n)**p and the length of the last can be expanded as p/n - 0.5*p*(p-1)/n**2 + ...

These formulas also apply to values of r (in cylindrical symmetry), x (in planar symmetry) and theta (when a circular arc is being subdivided, in either symmetry).

Choice of p for a free end of an electrode:

The reason for using p = 2 or -2 is that the charge density near the end of a thin electrode is approximately proportional to 1/squareroot(z), where z is the distance from the end (for an example see the file xmpl2d01.dat). For the best accuracy in the Boundary Element Method it is advisable for all the segments to carry approximately the same charge, and so near a free end the segments should become shorter as the end is approached. The formulas are designed to give approximately the same charge on all the segments near the end. Use p = 2 when the first end, zi, is the free end, and p = -2 when the second end, zf, is the free end.

(More generally, if the charge density is proportional to z**a, the charges on the segments are equalised when p = 1/(1-a). )

When the print level 'all' is requested, messages will appear on the screen if an electrode with a free end does not have p = 2 or -2.

Matching segments lengths of touching electrodes:

To help in doing this we give here the lengths of the end segments when p = 2 or -2:

For an electrode of length s and N segments, the shortest and longest segment lengths are s/(n*n) and s*(2n - 1)/(n*n) respectively.

Choice of p for subdividing a disc into equal areas:

The reason for using p = 0.5 or -0.5 is primarily to give segments of uniform area under some conditions. In particular, for a disc centred on the axis, in cylindrical symmetry, these values of p will give annuli of equal area. Use p = 0.5 when the first end, ri, is on the axis (that is, ri = 0), and p = -0.5 when the second end is on the axis (which is different from the rule given above for the sign of p).