##### Background on contour plots

When we make a contour plot we set up a grid of x,y,z coordinates and calculate the field at those points. We then do interpolations to sketch out the lines that get plotted and displayed on the screen. The interpolations are simply a series of straight lines that connect points of similar values. There will be cases where points are near corners and that will mean interpolations might actually cut through a material but this is simply a display issue and does not mean you have a bad solution. You can improve this particular problem by choosing to do a contour plot with a higher density (Medium instead of Coarse or Very Fine instead of Fine, etc.) This includes the ability to choose a user defined grid where you can input the number of points in a Row, Column format.

If you are calculating the field near a source of high field value (sharp corner, etc.) then the maximum field value will vary as you change either the size of your contour plot or the density of the plot. This happens because the coordinates where the field is calculated for a contour plot is somewhat arbitrary so the maximum value displayed by the contour plot will depend purely on how close one of the points used for the contour plot comes to the maximum value. The best thing that you can do is set the maximum value to be a bit lower than the value reported by the program and see how much of the contour plot disappears. If nothing noticeable changes then you are probably seeing a singularity (e.g. corner) that is giving a high field value, if a section of the contour plot disappears then you are seeing real values of the field that you would be concerned about in a design.

A better way to look at the field values from geometry is to do a graph along a line from one point in space to another. You can vary the number of points you are creating your graph from and it is much easier to see a continually varying result. It also happens to generate results faster.

##### A Note on FEM in cases such as these:

One of the drawbacks of FEM is that it only calculates the voltage (or in magnetics the magnetic (or vector) potential) at particular points in space and then interpolates everything from those points. This can cause interpolation errors and can also cause rippling in the field plots, as you need to differentiate discrete potential data to get the field. The difference between the interpolation errors in FEM and the ones in BEM are that FEM interpolation errors are real and will affect your solution whereas the ones in BEM are only there for displaying things like contours. If you query a point in space in FEM the program will interpolate from known points and produce an approximate answer based on solved points around the point that you want. In BEM if you query a point in space the value at that exact point will be calculated and no interpolation error would result. It is because of this fact that BEM is so highly valued for charged particle optics packages.

One other result of interpolations done with FEM results is that they tend to artificially suppress field values at corners because the high field values get missed. The “maximum” is thus a feature of the FEM mesh rather than a true physical aspect of the model. In BEM no such artificial limits occur. The user has greater control based on the solution being more correct to theoretical result for the given geometry. Thus, the user can intelligently find the maximum field based on either rounding the geometry, or by ignoring the results at locations unrealistically close to the corner when analyzing. For more information on the modeling issues involved, see:

Benchmark Problem for Simulating Electric Fields Near Sharp Corners and

Benchmark Problem for Simulating Electric Fields Near Triple Junctions