Dielectric breakdown is the formation of conducting paths through an insulating material in the presence of an extremely strong electric field. High-voltage transformers contain oil as their insulating dielectric. When a critical electric field is exceeded, conduction paths grow at microsecond speeds through the oil, in the form of branched trees, called streamers. These can lead to destructive breakdown.
The intense ionization occurring at the branch tips is high-speed and sub-microscopic in size, so that it connot be observed directly. Overall shape, growth pattern, and timing of the streamer trees can be recorded. We simulate these features by a detailed probability model, which provides three-dimensional graphical output suitable for comparison against high-speed shadow photographs obtained in experiment.
We have parallelized our dielectric breakdown model because the detailed electric-field distribution around the growing tree must be repeatedly calculated at each step of the growth process. This requires a high-resolution three-dimensional treatment. The streamer tree ia a self-avoiding, self-screening fractal structure. Laying the model out on a large rectangular grid (128 X 128 X 128) displays these characteristics well.
A simplified simulation method minimizes programming difficulty. The large cube domain is broken into regular rectangular blocks, which communicate at their interface planes. This creates processes of reasonable size, which operate in parallel like small copies of the original code. The instructions are in Fortran 90; DPARLIB, a NIST-developed extension library, handles the interface communications between processors invisibly.
The program makes extensive use of Fortran 90 array-directed commands, with physical quantities such as voltage field treated as large arrays. For example, neighbor sites to the tree are selected through a set of c-shift instructions. By this method the local model, which closely describes events surrounding one lattice site, is enlarged with great detail across the full domain of over two million sites.
Multiple parallel algorithms were implemented to speed the runs.
Spatial decomposition through block decomposition required each processor to track only its part of the space.
Parallel breakdown was implemented using a randomized red-black algorithm.
Laplace's equation was solved in parallel using SOR.
Time compression was used to reduce the empty (no breakdown) steps for periods of low breakdown probability.
Depending on the stringency of the growth-triggering instructions, the code runs on one to 30 hours on 9 nodes of the IBM SP2, and at similar rates on an analogous SGI multiprocessor.. This makes for convenient turnaround time in the development cycle of results - a task which would greatly exceed the practical capability of a single processor.
Various forms of streamer-tree growth have been approximated by different power-law responses to the electric field, and by the threshold-voltage setting. A linear filter produces dense, multibranched trees; such streamers are seen in experiment at voltages just moderately above threshold. A fourth-power filter gives rise to sparse, forward-directed trees, of a type seen experimentally at high overvoltages. Thus, the method has shown a range and capability to closely simulate some features of experiment.