# Modelling, Simulation and Optimization of Multifrequency Induction Hardening

Induction hardening is a modern method for the heat treatment of workpieces made of steel. A well directed heating by electromagnetic waves and subsequent quenching of the workpiece increases the hardness of the surface layer. With the multi-frequency concept, medium and high frequency power are supplied simultaneous on a single induction coil, which can result in a uniform contour hardening of gears.

#### Background

A given current density in the induction coil induces eddy currents inside the workpiece. Because of the Joule effect, these eddy currents lead to an increase in temperature in the boundary layers of the workpiece. Then the current is switched off, and the workpiece is quenched by spray-water cooling producing the desired hard, martensitic microstructure in the boundary layer. Due to the skin effect, the eddy currents tend to distribute in a small surface layer. The penetration depth of these eddy currents depends on the material and essentially on the frequency. Therefore, it is difficult to obtain a uniform contour hardened zone for complex workpiece geometries such as gears using a current with only one frequency.

The effect of medium-, high- and multi-frequency induction hardening. MF (left): only the root of the gear is hardened, HF (middle): only the tip of the gear is hardened, MF+HF (right): contour hardening can be achieved

#### Modelling

The aim of the simulation is the determination of the temperature, the phase distribution and the magnetic field as the source of the heating. The model consists of the vector potential equation of Maxwell’s equations, coupled to the energy balance and a rate law to determine the phase distribution in the workpiece. Since the characteristic material parameters depend on the temperature and the phase fraction itself, there results a nonlinear system of partial differential equations.

#### Simulation

The numerical realization is based on an adaptive finite element method using edge elements to discretize the magnetic vector potential. The system of partial differential equations is solved using the finite element toolbox WIAS-pdelib.

#### Example

Within the project, simulations of multifrequency induction hardening of gears have been performed and compared to experimental results. In general, a very good correspondence between numerical and experimental results could be achieved. The picture shows a comparison of the numerically and experimentally obtained hardening profile for a gear.

Simulated and experimental hardening profile using the multi-frequency approach

#### Framework of the project

The project was funded as a joint research network by the  German Federal Ministry of Education and Research within the funding program “Mathematik für Innovationen in Industrie und Dienstleistungen” as a cooperation between

#### Contact

Dr. Thomas Petzold
Weierstrass Institute for Applied Analysis and Stochastics, Berlin
E-mail: thomas.petzold@wias-berlin.de

Prof. Dr. Dietmar Hömberg
Weierstrass Institute for Applied Analysis and Stochastics, Berlin
E-mail: dietmar.hoemberg@wias-berlin.de