Potentials, fields and contours near to boundaries

In regions very near to an electrode (more precisely, at points nearer to a segment than approximately twice the width of the segment) there is a tendency for the inaccuracy to be worse than the requested value. The program uses a technique (see note on faster computation near electrodes) to improve the potentials in these circumstances, but this requires extra computing time. This technique for improvement can be disabled (and time saved) by selecting the appropriate option in the contours menu.

The reason for the apparent weakness is easy to understand. In the BEM the boundary voltages are used to obtain the boundary charges, and then the source of the field becomes these charges, not the original voltages. Therefore in the present form of the BEM the potentials at the centres of the segments are exact but the potentials calculated on other parts of the segments are not exact. Similarly in the FDM and FEM the potentials at the mesh points are exact but the potentials between those points are not exact. In potential plots in the FDM and FEM the mesh points are usually used for the plotting and so the potential distribution appears to be exact over the whole boundary, which it is not! On the other hand in potential plots in the BEM the plotting points usually do not coincide with the centres of the segments and so the potential distribution looks less exact than in the in the FDM and FEM, which it is not! And note also that in the BEM the potential distribution is much more accurate in the interior region, where the particle fly, than in the boundary region.

The largest inaccuracies occur at sharp corners or free edges, unless corrected.

If there is a critical region that is near one or more electrodes it is important to ensure that there is a large number of segments surrounding the region. It is also important not to have other much larger segments in the same vicinity because these might create a less accurate potential that extends over about twice their size.

Inaccuracy of a potential along an electrode

The grid points might not lie on the surfaces of the electrodes. The electrodes might not then appear as exact equipotential surfaces in the contour plots. Using more grid points or a smaller field of view will always improve the picture, as will arranging for grid points to lie on electrode surfaces where this is possible.

Despite appearances the electrodes are in fact almost exact equipotentials in the sense that in the Boundary Element Method (see chapter 1) the potentials at the mid-points of the segments are exactly (that is, within the requested inaccuracy) the potentials applied to the segments.

Inaccuracy at or near a corner

The potential at a boundary region that is not between the centres of neighbouring segments, such as the potential in the interior region of a corner of a square or cube will be less accurate.

The 3rd 2D test can be used to illustrate this feature of the BEM. A cylindrical box of length 2 and radius 1 has voltages -1 and +1 applied to its two ends. The length is divided into 40 segments (after the symmetry reflection) and the radius into 20 segments. It is found that the potentials in the box are consistent with the requested inaccuracy of 0.0001 except at the corner and within a distance of 0.1 from the corner

There are two easy ways of reducing this error:

(1) Subdivide each electrode unevenly, as explained in the relevant note, so that the segments become shorter as the corner is approached (in the present example the subdivision type is made -2 for the first electrode and +2 for the second). The error in the potential at the corner is then reduced to 0.0049 (without increasing the total number of segments).

(2) Add two extra short electrodes that touch the corner but are outside the box, in the radial and axial directions. If each extra electrode has the length 0.1 and is divided into 4 segments the error is reduced to 0.0039. An example is shown in figure 2.4 in the Users Guide.

By combining these two methods the error is entirely eliminated (within the inaccuracy that the user has requested). Yet another method (in practice usually less effective) is of course to significantly increase the total number of segments.

Inaccuracy at or near the edge of a thin electrode

Of more significance are regions outside sharp edges of electrodes, because the field is then strongly non-uniform and so interpolation between the grid points of the Finite Difference and Finite Element methods usually leads to significant inaccuracies. The Boundary Element Method is well suited to this type of simulation, but care must still be taken.

As with the potential at a corner, the error is reduced more effectively by a careful choice of the distribution of segments than by a brute-force increase in the total number of segments. An example is shown in figure 2.5 in the Users Guide.

The CPO programs can of course automatically distribute the segments in the required way, by using the iterative subdivision option.

This note has concerned possible inaccuracies of potentials on or near electrodes. The potentials (and fields) elsewhere in a system are of course calculated accurately, easily and quickly, as illustrated in the many test files.

See also notes on contour details and choosing shapes of 3D triangles and rectangles).