Numerical approximation of a transient thermo-electromagnetic problem in axisymmetric geometries

TL;DR

This paper develops a finite element method for transient thermo-electromagnetic problems in axisymmetric geometries, incorporating nonlinear coupling and providing error estimates validated by simulations.

Contribution

It introduces a variational formulation and fixed-point proof for existence and uniqueness, along with a finite element discretization and error analysis for the coupled problem.

Findings

Finite element method effectively models transient thermo-electromagnetic phenomena.

Error estimates are derived and validated through numerical experiments.

Simulations demonstrate practical industrial applications.

Abstract

This paper analyzes a transient thermo-electromagnetic problem arising in the modeling of induction heating processes. Unlike previous studies that focused on steady-state scenarios, we consider a time-dependent thermal problem coupled with a nonlinear time-harmonic electromagnetic problem through temperature-dependent electrical conductivity and Joule effect. Exploiting cylindrical symmetry and assuming a purely azimuthal current density, we formulate the problem on a two-dimensional meridional section. We introduce a variational formulation in appropriately weighted Sobolev spaces and prove existence of a solution by a fixed-point argument. Under reasonable assumptions on the physical parameters, we also prove uniqueness. A finite element discretization combined with implicit time stepping is used to compute the numerical solution. To evaluate the accuracy of the approximation, a priori error estimates are derived and validated by numerical experiments. Finally, numerical simulations illustrate the effectiveness of the proposed approach in an industrially relevant configuration.