A mixed-dimensional model for the electrostatic problem on coupled domains

TL;DR

This paper introduces a mixed-dimensional 3D-1D model for electrostatic problems involving thin inclusions, enabling efficient computation of electric fields and potentials in complex geometries with validation against analytical solutions.

Contribution

It presents a novel mixed-dimensional formulation combining 3D and 1D models for electrostatics, reducing computational costs for thin inclusions in larger domains.

Findings

Validated approach against analytical solutions

Compared mixed-dimensional model with full 3D solutions

Applied model to electrical treeing geometry

Abstract

We derive a mixed-dimensional 3D-1D formulation of the electrostatic equation in two domains with different dielectric constants to compute, with an affordable computational cost, the electric field and potential in the relevant case of thin inclusions in a larger 3D domain. The numerical solution is obtained by Mixed Finite Elements for the 3D problem and Finite Elements on the 1D domain. We analyze some test cases with simple geometries to validate the proposed approach against analytical solutions, and perform comparisons with the fully resolved 3D problem. We treat the case where ramifications are present in the one-dimensional domain and show some results on the geometry of an electrical treeing, a ramified structure that propagates in insulators causing their failure.