Fractal Analysis of 3-Dimensional Simulation Tree Utilizing a Stochastic Model

Abstract

Many complex patterns in nature such as surface discharge and lightning are known as fractal objects. It is well known that the electrical trees in polymeric insulating materials are also fractal objects. However, it is difficult to analyze the real electrical trees in a 3-dimensional space. With the development of a computer system, the stochastic models for electrical trees have been presented and the fractal analysis of their simulation trees has been performed.In this paper, we carried out the 3-dimensional simulation utilizing the stochastic model under a Laplace field, taking into account the growth probability, local field, critical threshold field and voltage drop along tree channelsUsing this numerical simulation model, we obtained the various shapes of simulation trees, such as branch-type, bush-type and densely bush-type trees, and estimated the fractal dimensions of their simulation trees.The results show that the obtained simulation tree patterns have a fractal character and the parameters in an adopted stochastic model influence the fractal character of simulation trees.

Three‐dimensional tree simulation utilizing a three‐layer model

“…The following simulation rules were set up to generate tree patterns [14,15]. (1) In the 3D grid, the Laplace equation ∇ 2 φ = 0 (φ: potential) was solved by the SOR (Successive Over Relaxation) method.…”

“…On the other hand, the case of η = 1 agrees with the DLA (Diffusion-Limited Aggregation) model [17], producing fractal shapes reminiscent of the Lichtenberg figures observed in creeping discharge. In this study, we used η = 1 in the simulations [13,14]. (4) The grid is applied on the tip of the developing tree so that the potential inside the tree discharge portion is decreased by V d .…”

“…In addition to the growth probability, we also considered the critical electric field for tree occurrence and potential drop along a tree discharge path as the most appropriate measures of the actual physical conditions of tree propagation [14]. We also reported a 3D simulation tree based on DBM involving a composite insulator interface and the growth probability [15].…”

Temporal propagation characteristics of simulation tree considering growth probability

“…In order to describe the branch structure in this electric discharge pattern, the simulation of the stochastic model using a computer is proposed. We have performed the three-dimensional tree simulation in a needle-to-plane electrode system based on the simulation with consideration of the growth probability which Niemeyer et al and others proposed [1]- [7], and we have been also attempted the description of various tree patterns from branch-type to bush-type trees. Moreover, we have examined that the physical meaning of three-dimensional simulation trees considering the growth probability, the critical field and potential drop during the tree propagation is closely related to that of real electrical trees.…”