• I wanted to get a sense of the baseline electrical field magnitude in different regions and to check the sensitivity to changes in the dielectric properties of surrounding structures.
• I could use the DXF import capability of the Mesh drawing editor to create a set of outlines for the complex turnings. After testing, I moved them directly to the MetaMesh script.
• I wanted benchmark values to check that the 3D solution was set up correctly.

Because 2D solutions run quickly, I incorporated the entire injector assembly in the mesh. Extending this solution volume to three dimensions would have resulted in a huge mesh and many hours of run time. I feel that if a run takes more than one hour, there is probably a better way to solve the problem.

Since the flaws were relatively localized, it would be possible to get a good idea of their effect by limiting the axial extent of the solution to a space near the insulator. In this case, I could define a fixed-potential region on the solution volume boundaries in -z and +z and assign a spatial variation using data from the EStat solution. The existing version of HiPhi supported the definition of potential variations from mathematical functions using the command:

POTENTIAL NoReg > f(x,y,z)

Here, f(x,y,z) is any algebraic function of the Cartesian coordinates. In order to use the feature, it would have been necessary to take a radial scan of potential and then to fit the results with a power series  in ?[x^2 + y^2]. This approach seemed like it would be a lot of work, and I didn’t feel like doing it.

Instead, I took advantage of my unique position as deity of the HiPhi source code and added a new program feature. It is ideally suited to using a 2D solution as the basis of a 3D microscopic solution. Here is the corresponding script command

POTENTIAL NoReg TABLE [x,y,z,r] TabName

The string TabName is the name of a text file defining the potential variation along the specified direction (the variable r is interpreted as ?[x^2+y^2]). It consists of a set of data lines:

r[n]    phi(r[n])

For the project I prepared tables for the downstream and upstream boundaries directly from EStat scans, using ConText to remove unwanted columns. The process took only a few minutes.

The figure below shows a test solution for the potential inside a grounded cylinder. The bottom boundary in z is grounded and the potential on the top follows a tabular variation in r (a partial cosine function). The top picture shows equipotential lines in the plane x = 0.0 and the bottom shows lines in a plane normal to z near the defined fixed-potential boundary.

I modified the HiPhi instruction manual, describing how to use the new command forms. While I was at it, I decided to ex[and the existing commands for relative dielectric constant, conductivity and space-charge density to handle table input.

For more information on HiPhi, please use this link: http://www.fieldp.com/hiphi.html.