Predicting three-dimensional fractal dimensions of electrical discharges using two-dimensional projections

Abstract

Three dimensional electrical discharge patterns are simulated to compare the three dimensional structure with the two-dimensional projections of the same. The discharge patterns are simulated using a stochastic dielectric breakdown model. Fractal techniques are used to characterize the morphological structures of the simulated discharge patterns. The discharge patterns are simulated on a 50x50x50 cubic lattice and the fractal dimensions in both three-dimensions and two-dimensions are estimated using the sand box fractal dimension estimation method. The complexity of the simulated electrical tree patterns is strongly dependent on the exponent of the cell potential 'rj'. When the value of the exponent was increased, the growth patterns effectively lost their fractal structure and became a curve with dimension 1. The fractal dimension of the three-dimensional growth patterns and two-dimensional projections when the exponent is close to unity (?1&1) are 1.84 and 1.53 respectively. A strong linear correlation was found between the simulated three-dimensional structures and their two-dimensional projections when the dimension is less than 2. This relation can be used to estimate the three-dimensional fractal dimension from two-dimensional views of complex electrical discharges.