3 Electrostatic application: building the mesh Figure 9: Alternative ways to model coaxial cylinders with a 2D code. In the previous chapter, we followed a prepared example without going into the details of how the input files were generated and how the mesh parameters were chosen. Now, we’re ready to build a complete solution and to learn about how the geometry of the mesh affects the accuracy of the solution. The best approach to make comparisons is to model a system that has an analytic solution. A good choice is a set of coaxial cylindrers with an applied voltage where there are spatial variations of potential and electric field.
There are two ways to model coaxial cylinders with a 2D code (Fig. 9 ). The first is to use cylindrical coordinates z - r and to model the infinite length in z with Neumann conditions 1 at the upper and lower z boundaries. This approach, where the electrode boundaries are simply straight lines, would not exercise the conformal mesh capability. Instead, we will use planar coordinates (Fig. 9 , lower) where the cross section is in the x - y plane and the system extends infinitely in z (out of the page). For specific parameters, take r i = 5 . 0 cm, r o = 15 . 0 cm and V i = 100 . 0 V. The space between the cylinders is filled with polyethylene ( ǫ r = 2 . 7) and the outer electrode is grounded.
Using formulas available in introductory electromagnetism texts, 1 The Neumann condition specifies that the normal component of electric field, E z , is zero at the boundaries. 14