Introduction to Numerical Electrostatics Using MATLAB

9Refining the Finite Difference Method

In this chapter we’ll refine and expand the finite difference solution technique and its applications. We’ll look at refined grids for better accuracy in high-field regions (regions with rapidly changing voltage), mixed dielectrics, other coordinate systems, multielectrode structures, and magnetic wall boundary conditions, including calculation of symmetric structures. We’ll show how, in many cases, we can calculate both upper and lower bounds on our estimates of C (and for transmission line cross sections L and Z₀). We’ll take a brief look at solving three-dimensional (3d) problems.

9.1 Refined Grids

Regions such as outer corners of electrodes have very high electric fields as compared to other regions in a structure. Saying that the field magnitude is high is equivalent to saying that the voltage in that region changes rapidly with position. In such a situation we want to reduce the value of h to ensure that a significant portion of the voltage variation does not take place within one cell. The brute-force approach to this is to simply increase the value of h throughout by, say, a factor of 100 in both axes. This will, of course, increase the number of nodes by a factor of 10,000 (assuming a simple rectangular box geometry). If we had started out with a 50 × 250 = 12,500-node system, then increasing the number of nodes by a factor of 10,000 would not seem too practical. An alternative approach is to decrease h only where (we think) ...