Test2d21, 21st 'benchmark test' data file for CPO2D

Field at the centre of a double cylinder lens.

The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.

The geometry is as described by D W O Heddle (paper submitted for publication, 1999):

There are two thick-walled cylinders, their inner and outer radii being 0.5 and 0.75 respectively.

The cylinders are terminated at z = +/- 5

The potential difference between the cylinders is 1

The gap G between the cylinders is variable.

Heddle has shown analytically that when G = 0 the field at the centre of the system is 1.3262275051

The above data file simulates this system with G = 0.0025

The value obtained for the field is 1.326223, in a computing time of 52 seconds, using a 450Mhz computer.

Using a modified version of the commercial program, to allow the inaccuracies to be set at 0.0000001 and to give the output values to 8 significant figures, the value becomes 1.32622117 (in 60 secs).

The values obtained for this and other values of G are as follows, where the first field value is obtained with the commercially available program and the second with the modified program:

G field

0.1000 1.316317

0.0750 1.320615

0.0500 1.323718

0.0250 1.325599

0.0200 1.325825

0.0150 1.326002

0.0100 1.326128

0.0050 1.326204

0.0025 1.326223 1.32622117

0.0020 1.32622345

0.0015 1.32622522

0.0010 1.32622649

0.0005 1.32622725

Extrapolation to G = 0 gives the values 1.326229 and 1.32622751 for the lower and higher accuracy results respectively. The first is in error by 0.0000015 and the second is exact to within 0.00000001.

A second type of extrapolation is to an infinite number of segments. This results in a decrease in the last significant figure (of the more accurate results above) by approximately 1. (This procedure also requires a slight increase in the computing time, but as recommended in Help the extrapolation is achieved by using a series of smaller values of the number of segments, and so the increase in computing time is less than a factor of 2.)

The most accurate value obtained by Heddle using a Finite Difference Method and extrapolating to a zero gap is 1.3262350, which is in error by 0.0000075, in a computing time of approximately 350 sec (using the same speed of computer). Applying a second type of extrapolation, to a zero mesh size, gives 1.3262280, which is in error by 0.0000005 (and which also requires a longer computing time).

In summary, the approximate inaccuracies and approximate computing times for the zero-gap results are as follows:

(the times are for single calculations and so do not include times to obtain sets of results for extrapolations)

Program |
inaccuracy |
time (secs) |

CPO2D (commercial version) |
0.0000015 |
52 |

CPO2D (special version) |
0.00000001 |
60 |

CPO2D (special) extrap to infinite segs |
<0.00000001 |
60 |

FDM (Heddle) |
0.0000075 |
350 |

FDM (Heddle) extrap to zero mesh |
0.0000005 |
350 |