Analysis and simulations of multifrequency induction hardening

TL;DR

This paper develops a mathematical model for induction hardening of steel, proving existence, uniqueness, and stability of solutions, and supports findings with finite element simulations.

Contribution

It introduces a coupled differential system for induction hardening, providing rigorous mathematical analysis and simulation results.

Findings

Existence and uniqueness of solutions established.

Stability estimates derived for the model.

Finite element simulations demonstrate the model's applicability.

Abstract

We study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwell's equations coupled with an internal energy balance and an ODE for the volume fraction of {\sl austenite}, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system.