A mathematical model for induction hardening including mechanical effects

In most structural components in mechanical engineering, there are surface parts, which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect. The mathematical model accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We formulate a consistent model which takes care of effects like transformation strain and transformation plasticity induced by the phase transitions and allows for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument

Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions

Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions

Optimal control of laser hardening

We present a mathematical model for the laser surface hardening of steel. It consists of a nonlinear heat equation coupled with a system of five ordinary differential equations to describe the volume fractions of the occuring phases. Existence, regularity and stability results are discussed. Since the resulting hardness can be estimated by the volume fraction of martens­ite, we formulate the problem of surface hardening in terms of an optimal control problem. To avoid surface melting, which would decrease the workpiece's quality, state constraints for the temperature are included. We prove differentiability of the solution operator and derive necessary conditions for optimality

Sensitivity analysis of an eddy current problem related to induction heating

We study a mathematical model for the inductive heating of steel.It consists of a vector potential formulation of Maxwells equations coupled with a heat equation and an evolution equation for the volume fraction of high temperature phase in steel. An important task for practical applications of induction heating it to find the optimal coupling distance between inductor and workpiece. To this end, we employ the speed method to investigate the sensitivity of solutions to the state equations with respect to perturbations of the inductor coil. We show the existence of strong material derivatives for the state variables and apply the structure theorem to characterize the Eulerian derivative of the cost functional

PID-control of laser surface hardening of steel

We discuss control strategies for the laser surface hardening of steel. The goal is to acchieve a prescribed hardening depth avoiding surface melting. Our mathematical model consists of a system of ODEs for the phase volume fractions coupled with the heat equation. The system is solved semi-implicitely using the finite element method. To obtain a uniform hardening depth the first attempt is to use PID control to achieve a constant temperature in the hot spot of the laser beam on the surface. However, the numerical results prove that this is not sufficient. We show that the best strategy is to control the temperature close to the lower boundary of the hardening zone. Then one can compute the optimal temperature in the hot spot of the beam and use it as the set-point for the pyrometer control of the real process

Suboptimal control of laser surface hardening using proper orthogonal decomposition

Laser surface hardening of steel is formulated in terms of an optimal control problem, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the high temperature phase. The optimal control problem is analyzed and first-order necessary optimality conditions are derived. An error estimate for POD (proper orthogonal decomposition) Galerkin methods for the state system is proved. Finally a strategy to obtain suboptimal controls using POD is developed and validated by computing some numerical examples

Exact controllability on a curve for a semilinear parabolic equation

Motivated by the growing number of industrially important laser material treatments we investigate the controllability on a curve for a semilinear parabolic equation. We prove the local exact controllability and a global stability result in the twodimensional setting. As an application we consider the control of laser surface hardening. We show that our theory applies to this situation and present numerical simulations for a PID control of laser hardening. Moreover, the result of an industrial case study is presented confirming that the numerically derived temperature in the hot-spot of the laser can indeed be used as set-point for the machine-based process control

Optimal Shape Design of Inductor Coils for Surface Hardening

A shape optimization problem is considered related to the design of induction hardening facilities. The mathematical model consists of a vector potential formulation for Maxwell's equations coupled with the energy balance and an ODE to describe the solid-solid phase transition in steel during heating. Depending on the shape of the coil we control the volume fraction of the high temperature phase. The coil is modeled as a tube and is defined by a unit-speed curve. The shape optimiza- tion problem is formulated over the set of admissible curves. The existence of an optimal control is proved. To obtain the form of the shape gradient of the cost functional, the material derivative method is applied. Finally, the first order necessary optimality conditions are estabished for an optimal tube