Test2d26, Axial potentials of a 2-tube lens.

The axial potentials are compared with the 'benchmark' values published by David Edwards, Jr, address IJL Research Center, Newark, VT 05871, USA, and obtained by the multiregion FDM method. Private communication, Feb 2007.

Here the anti-symmetry plane z = 0 is used and the voltage applied to the tube at z < 0 is -4.5V. The voltage applied to the tube at z > 0 therefore automatically becomes +4.5V.

The ray output file gives:

Initial total number of segments= 320

Potentials (volts) at 6 equally spaced points between

r, z = 0.0000E+00, -1.0000E+00 and 0.0000E+00, 0.0000E+00

Requested inaccuracy = 1.00E-07

potentialtest2d26, axial potentials 2-tube lens

0.000E+00 -1.000E+00 -4.440641E+00

0.000E+00 -8.000E-01 -4.345198E+00

0.000E+00 -6.000E-01 -4.099613E+00

0.000E+00 -4.000E-01 -3.492394E+00

0.000E+00 -2.000E-01 -2.163024E+00

0.000E+00 -5.551E-17 0.000000E+00

Using a version of CPO2D in which the inaccuracy is reduced from 1E-07 to 1E-10 and the calculated potentials are given with more significant figures, we obtain the following results:

Initial total number of segments= 320

Requested inaccuracy = 1.00E-10

potential

0.000E+00 -1.000E+00 -4.4406413072E+00

0.000E+00 -8.000E-01 -4.3451979438E+00

0.000E+00 -6.000E-01 -4.0996133516E+00

0.000E+00 -4.000E-01 -3.4923940612E+00

0.000E+00 -2.000E-01 -2.1630238517E+00

0.000E+00 -5.551E-17 -3.7171643844E-14

Then increasing the number of segments we obtain the following results:

Initial total number of segments= 640

Requested inaccuracy = 1.00E-10

0.000E+00 -1.000E+00 -4.4406408521E+00

0.000E+00 -8.000E-01 -4.3451975265E+00

0.000E+00 -6.000E-01 -4.0996133845E+00

0.000E+00 -4.000E-01 -3.4923952160E+00

0.000E+00 -2.000E-01 -2.1630258745E+00

0.000E+00 -5.551E-17 -7.4343287687E-15

Initial total number of segments=1280

Requested inaccuracy = 1.00E-10

0.000E+00 -1.000E+00 -4.4406407881E+00

0.000E+00 -8.000E-01 -4.3451974642E+00

0.000E+00 -6.000E-01 -4.0996133839E+00

0.000E+00 -4.000E-01 -3.4923953840E+00

0.000E+00 -2.000E-01 -2.1630261821E+00

0.000E+00 -5.551E-17 -7.4343287687E-15

Initial total number of segments=2560

Requested inaccuracy = 1.00E-10

0.000E+00 -1.000E+00 -4.4406407798E+00

0.000E+00 -8.000E-01 -4.3451974590E+00

0.000E+00 -6.000E-01 -4.0996133917E+00

0.000E+00 -4.000E-01 -3.4923954193E+00

0.000E+00 -2.000E-01 -2.1630262358E+00

0.000E+00 -5.551E-17 -7.4343287687E-15

The results are not changed by using the option to improve the matrix inverse.

For each value of z we now extrapolate to an infinite number of segments, plotting V versus 1/N^p, where N is the number of segments and p = 2.5, an empirical value. (In one case, the value for 1280 segments, z = 0.6, is omitted -we have not yet looked into the reasons for this apparently

incorrect value.)

z         extrap value         +5.5                 Edward values             Difference

1.0 4.44064077601 9.94064077601 9.94064077208 0.00000000393

0.8 4.34519745495 9.84519745495 9.84519746827 -0.00000001332

0.6 4.09961339117 9.59961339117 9.59961340479 -0.00000001362

0.4 3.49239541859 8.99239541859 8.99239542414 -0.00000000555

0.2 2.16302623621 7.66302623621 7.66302624025 -0.00000000404

The largest fractional difference is 1.4E-9