WS, Beatrice Crippa: A mixed-dimensional model of the electrical treeing

Speaker
Beatrice Crippa,Politecnico di Milano

Abstract
The electrical treeing is a degradation phenomenon in insulating materials, originating from the prolonged action of intense electric fields, that trigger Partial Discharges. Consequently, gas–filled fractures are formed inside the dielectric because of the progressive erosion of the polymeric surface. The underlying physical problem can be modeled by a system of Partial Differential Equations [3], which describes the evolution of the electric field and potential in both the dielectric and the fracture, and the charge movement within the gas.
The typically thin, elongated and highly branched structure of the treeing makes the computational cost for 3D simulations prohibitive. Therefore, to reduce the computational complexity, we propose a dimensional reduction of the treeing geometry, approximated as a 1D graph embedded in the 3D dielectric domain, and derive the corresponding mixed–dimensional 3D-1D reduced model. An advantage of this reduction is the possibility of using a coarser 3D grid for the dielectric, independent of the 1D mesh, which provides a lower number of degrees of freedom in the numerical discretization.
This approach results in two coupled problems: drift–diffusion equations describing the charge movement on a 1D graph, and a 3D-1D system for the electric fields and potentials [1]. We solve the electrostatic problem with mixed Finite Elements (FEM) in 3D and 1D FEM on graph, while on the time–dependent charge transport equations we adopt an operator splitting approach to decouple reaction and drift–diffusion. In particular, we solve the drift–diffusion equations on the 1D graph with upwind Finite Volumes and Two-Point Flux Approximation [2], implicitly discretized in time, and the reaction part with a Patankar–Euler scheme, ensuring monotonicity of the solutions.
We assess the performance and accuracy of the reduced models through tests on simplified domains and then apply them to a realistic tree geometry, achieving a significant reduction on the computational cost. Finally, considering independent meshes on the two domains allows simulations in contexts where the generation of a full conforming 3D mesh would be unfeasible.

References.
[1] B. Crippa, A. Scotti, and A. Villa, “A mixed-dimensional model for the electrostatic problem on coupled domains,” Journal of Computational Physics, p. 114 015, 2025.
[2] B. Crippa, A. Scotti, and A. Villa, “A monotone finite volume scheme for linear drift-diffusion and pure drift equations on one-dimensional graphs,” Networks and Heterogeneous Media, vol. 20, no. 2, pp. 670–700, 2025.
[3] A. Villa, L. Barbieri, M. Gondola, A. R. Leon-Garzon, and R. Malgesini, “A PDE-based partial discharge simulator,” Journal of Computational Physics, vol. 345, pp. 687–705, 2017.