A one-dimensional reduced plasma model for the electrical treeing

TL;DR

This work develops a rigorous $1D$ reduction of a 3D drift–diffusion–reaction plasma model for electrical treeing, collapsing slender branches onto a skeleton and using cross-section averages for volume and surface charges. It couples these charges to dipole-moment dynamics via $a_i$, and computes the transverse electric field through a superposition of six basis effects, avoiding costly cross-sectional 2D Poisson solves. A conservative, positivity-preserving numerical scheme—implicit Euler–FV with upwind fluxes and Patankar time stepping for chemistry—ensures physical properties are maintained, while yielding fast simulations on complex branched geometries. The model is validated on a straight line, a branched domain, and a realistic electrical-tree skeleton, capturing avalanche dynamics under a fixed longitudinal field and providing a framework for future coupling with full mixed-dimensional electrostatics. Together, these contributions enable scalable, physics-consistent PD simulations and set the stage for accurate, faster analyses of dielectric breakdown processes in complex insulator geometries.

Abstract

Plasma models, consisting of advection-diffusion Partial Differential Equations coupled with chemical reactions, are widely adopted to describe corona, streamers and dielectric barrier discharges. However, the complex geometry of the electrical treeing represents an obstacle for numerical simulations. We develop a reduced one-dimensional formulation of a plasma model for the electrical treeing, describing the evolution of charge concentrations under the effect of an electric field. The reduced system consists of weakly coupled advection-diffusion-reaction equations for charge concentrations inside the treeing and on the dielectric surface, coupled with production-destruction Ordinary Differential Equations for the dipole moment. A numerical scheme based on Finite Volumes and Patankar-type methods allows efficient simulations, while preserving key physical properties. The model is tested on increasingly complex geometries, from a straight line to a realistic electrical treeing.

Figures (14)

• Figure 1: Simplified domain $\Omega$, given by two coaxial cylinders, corresponding to the dielectric and gas subdomains, $\Omega_s$ and $\Omega_g$, respectively. We call the portion of axis inside the gas domain $\Lambda$ and the interface between the two subdomains $\Gamma$.
• Figure 2: Cylindrical portion $\mathcal{P}$, with lateral surface $\Sigma$, of the gas domain $\Omega_g$, on which we integrate the equation in the gas.
• Figure 3: Section of the adimensionalized domain $\tilde{\Omega}$, on which we define equation \ref{['eq:reduction1d:electric_field:3d:adim']}. The section $\tilde{\mathcal{D}}_g$ is a circle of radius 1, $\tilde{\mathcal{D}}$ is circle of radius $\tilde{R}_s$ and $\tilde{\mathcal{D}}_s = \tilde{\mathcal{D}}\setminus \tilde{\mathcal{D}}_g$ denotes the region between the two.
• Figure 4: Different components of the electric field given by the considered effects: the radial component in Figure \ref{['figure:reduction1d:soe:radial']} is given by the effect of the constant charge concentrations ($\mathbf{\Psi}_5$, $\mathbf{\Psi}_6$), while the components along the $x$ and $y$ axes are given by the respective components of the external electric field and of the dipole moment ($\mathbf{\Psi}_1$ and $\mathbf{\Psi}_3$ along $x$, and $\mathbf{\Psi}_2$ and $\mathbf{\Psi}_4$ along $y$).
• Figure 5: TC1 - Volume concentrations of charged species on $\Lambda=[0,10^{-4}]\ m$, for $\Delta t = 10^{-6}\ \mu s$.

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Lemma 1
• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 2
• Theorem 3
• proof