Xmpl3d23, 23rd 'example' data file for CPO3D

Field and potential penetration through a mesh

A flat mesh of square holes formed from crossed thin wires is simulated, and a non-zero field is created on one side of it. The penetration of potential into the field-free region is investigated.

The file was written 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.

One of a set of files:

test2d18.dat: Field penetration through a flat mesh of flat strips.

xmpl2d19.dat: Field penetration through a cylindrical mesh of round wires.

xmpl2d20.dat: Field penetration through a flat mesh of round wires.

xmpl3d23.dat: Field and potential penetration through a flat mesh of crossed round wires.

xmpl3d29.dat: Field and potential penetration through a flat mesh of crossed flat wires.

For further details see the papers:

Short and long range penetration of fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward Nucl. Instr. Meth. A427, 363-367 (1999).

Penetration of electrostatic fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward, Rev. Sci. Instrum. 69, 2000-2006 (1998).

A flat mesh is composed of thin crossed wires that form an array of square holes, and has a transparency of 90 percent.

The objective of this study is to measure the penetration of the field and potential through the mesh. To do this the field on one side of the mesh should be set up as non-zero while the other side is set up as a nominally field-free region. In fact in this example all the planes of reflection symmetry are used, including the plane through the mesh, and so neither side is nominally field-free, but after the computation a constant field can be subtracted from one side to make it nominally field-free.

The mesh is at zero potential and fields are created on both sides of it by means of continuous end plates at a different potential. An extrapolation technique is used to simulate a mesh that is infinite in extent. It is found that the external field penetrates only a short distance through the mesh, but that there is a finite change of the potential in the field region.

In the limit of a vanishingly small ratio r0/s, Read and Bowring show that the potential in the field-free region is

phi0 = (0.25*E*s/pi)*ln(s/(4*pi*r0)) + 0.25*E*r0,

where s is the repeat distance of the mesh, r0 is the radius of the wires and E is the field strength at the surface of the mesh.

The computed value can (and should) be corrected in three different ways:

(1) The number N of segments should be extrapolated to infinity.

(2) The number M of mesh holes should be extrapolated to infinity.

Detailed results are given in the papers referred to above.

Details of setting up the electrodes at the crossing points:

The complete cylinders that form the mesh touch at their ends, leaving a square-section hole between the cylinders. The basic shape that has to be attached to the end of each cylinder to fill this hole has a form similar to that of a Bishop's mitre. Because there is reflection in the z=0 plane, only half of the basic shape is required, consisting of a single curved triangle. This triangle can be cut along the direction of the axis of the cylinder, to form two triangles. Each of these triangles is cut further in the present data file into 4 triangles.

For example, consider the basic mitre shape at the y=0.475 end of the cylinder whose axis extends from (0.5,0,0) to (0.5,0.450,0), and which has a radius of 0.050. The schematic diagram below represents a flattened version of one half of the 'single curved triangle' referred to above:

D

/ | \

C / | \ G

/|\ | /|\

B / | \ | / | \ H

/ \ | \ | / | / \

A /_____\|___\|/___|/_____\ I

F E J

The labels A, B etc are referred to in the data file. The corresponding coordinates are:

A |
0.475 |
0.475 |
0. |

B |
0.47835 |
0.47835 |
0.01250 |

C |
0.4875 |
0.4875 |
0.02165 |

D |
0.5 |
0.5 |
0.050 |

E |
0.5 |
0.475 |
0.050 |

F |
0.4875 |
0.475 |
0.02165 |

G |
0.51250 |
0.4875 |
0.02165 |

H |
0.52165 |
0.47835 |
0.01250 |

I |
0.5250 |
0.475 |
0. |

J |
0.51250 |
0.475 |
0.02165 |

The triangle DEG represents the other half of the 'single curved triangle'.

(The reason for subdividing into 4 triangles is as follows: The points A, B, C and D all lie in the x=y plane, and after further subdivision the points along the line ABCD should in principle continue to lie entirely in the x=y plane. However the line ABCD is a geodesic that does not have the form of an arc of a circle, and so in practice when the triangles are further subdivided points are created along ABCD that do not lie exactly on the geodesic. The use of 4 triangles lessens this simulation. If the 'single curved triangle' were not divided at all, the deviations from the geodesic would be large. The use of segments of the form of 'end cylindrical triangles' does not help here.).