The goal of this paper is a mathematical investigation of dilatometer experiments. These are used to detect the kinetics
of solid-solid phase transitions in steel upon cooling from the high temperature phase. Usually, the data are only used for measuring
the start and end temperature of the phase transition. In the case of several coexisting product phases, expensive microscopic
investigations have to be performed to obtain the resulting fractions of the different phases. In contrast, it is shown in this paper that in the case of at most two product phases the complete phase transition kinetics including the final phase fractions are uniquely determined by the dilatometer data. Numerical results confirm the theoretical result.
The goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal
control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including
mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with
machine-based control.
A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to
render the steel austenitic, followed by quenching to form
a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled
with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions.
We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.
This paper is concerned with the state-constrained optimal control of the
two-dimensional thermistor problem, a quasi-linear coupled system
of a parabolic and elliptic PDE with mixed boundary conditions.
This system models the heating of a conducting material by means of direct current.
Existence, uniqueness and continuity for the state system are derived by employing
maximal elliptic and parabolic regularity. By similar arguments the
linearized state system is discussed, while the adjoint system involving measures
is investigated using a duality argument. These results allow to derive
first-order necessary conditions for the optimal control problem.
We study a mathematical model for laser-induced thermotherapy,
a minimally invasive cancer treatment. The model consists of a
diffusion approximation of the radiation transport equation
coupled to a bio-heat equation and a model to describe the
evolution of the coagulated zone. Special emphasis is laid on
a refined model of the applicator device, accounting for the
effect of coolant flow inside. Comparisons between experiment
and simulations show that the model is able to predict the
experimentally achieved temperatures reasonably well.
We present a thermodynamically consistent model to describe the austenite-ferrite phase transition in steel. We consider the influence of the mechanical displacement field due to eigenstrains caused by volumetric expansions. The model equations are derived in a systematical framework. They are based on the conservation laws for mass and momentum and the second law of thermodynamics. By means of numerical computations for a simplified interface-controlled model, we examine the influence of the mechanical contributions to the transformation kinetics and the equilibrium states.
An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented.
The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is
solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set
strategy is added to the resolution technique.
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissue is destroyed by coagulation. For the dosage planning numerical simulation plays an important role. To this end a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a non-convex cost functional. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient.
We investigate a thermomechanical model of phase transitions in steel. The strain is assumed to be additively decomposed into an
elastic and a thermal part as well as a contribution from transformation induced plasticity. The resulting model can be viewed
as an extension of quasistatic linear thermoelasticity. We prove existence of a unique solution and conclude with some numerical simulations.