Test3d06, 6th 'benchmark test' data file for CPO3D

Capacitance of a cube

This benchmark test calculates the capacitance of a unit cube. All faces of the cube are set to voltage +1, and the capacitance is deduced from the total charge on the faces. The capacity is obtained in a few sec with an error of 0.4%, and by extrapolating to an infinite number of segments it is obtained with an estimated inaccuracy of 0.008%.

The number of segments used in the present example is small enough for the example to be run with the ‘demo’ version of CPO3D. Higher accuracy could of course be obtained with more segments, using the standard or full versions of CPO3D.

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.

A cube is set up as in file test3d03.dat. All 4 planes of reflection symmetry are used, so the end face starts in the minimum octant as a single triangle, while a side face starts in the minimum quadrant as a single square. The technique of iterative subdivision (also called adaptive segmentation) has been used, in 4 stages. All faces of the cube are at voltage +1.

The printing level has been put at 'a', for 'all', so that the charges are shown in the output data file tmp4a.dat. The cumulative charge shown in the output data file is 1.83056E-14.  Multiplying this by 8 ( = 2**3), to allow for the 3 reflections, and multiplying it also by 500 (to scale the side of the cube from 2mm to 1m), gives the capacity of a unit cube as 7.32224E-11 Farad.  Using 300 segments gives 7.33816E-11 Farad, and extrapolating to infinity, as explained in the User's Guide, gives the result 7.3498(6)E-11 Farad.  These three results differ by 0.39%, 0.18% and 0.017% respectively from the value 7.351040(7) given by F H Read, J. Computational Physics 133, 1-5 (1997), and obtained with a more careful extrapolation.  This in turn differs by 0.0006% from the previous most accurate value, given by Goto et al, J. Computational Physics, 100, pp 105-115, 1992, who claim (with interesting reasoning) an error of 0.00008% !

Later work (2004):

For detailed information see the paper 'Capacitances and singularities of the unit triangle, square, tetrehadron and cube' by F H Read, COMPEL, VOL 23, 572-578 (2004) and the footnotes of the following data files:

test3d06: Capacitance of a cube.

test3d07: Capacitance of a sphere.

test3d15: Capacitance of a circular disc.

test3d17: Capacitance and singularities of a tetrahedron.

test3d18: Capacitance of a unit square.

test3d19: Capacitance of a unit triangle, and summary of capacitance tests.

The capacitances of a sphere and disc are found essentially exactly, within the given number of digits -see F. H. Read, J. Computational Physics 133, 1-5 (1997).

Summary of the capacitances of a square, tetrahedron and cube:

Capacitance                                 exponent

along                      across

 Present Goto et al mid-line edge edge Triangle 0.25084(12) -0.759(12) -0.255(6) -0.5 Square 0.3667896(8) 0.3667892(11) -0.700(4) -0.204(4) -0.5 Tetrahedron 0.35651(4) -0.677(4) -0.295(8) -0.378 Cube 0.6606767(4) 0.6606747(5) -0.536(5) -0.210(4) -0.333