kms 대한수학회

3월 23일, 3월 24일 4시 30분에 예정되었던 본 세미나는 취소 되었습니다.

혼란을 드려 죄송합니다.

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NIMS 초청 세미나 및 산업문제 세미나

연사: Dietmar Homberg (TU Berlin & WIAS)
일시: 3월 23일(월) 16:30~17:30 과 3월 24일(화) 16:30~17:30
장소: 국가수리과학연구소 수학원리응용센터 중형세미나실
제목: Modelling, simulation and control of surface heat treatments (3/23)
The production of modern multi-phase steels -- a challenge for industrial mathematics (3/24)
초록

- Abstract -
3월 23일 (월)
Modelling, simulation and control of surface heat treatments

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. Depending on the respective heat source one can distinguish between different surface hardening procedures, the most important ones being induction hardening and radiation treatments like laser and electron beam hardening.

In the introductory part of my talk I will briefly present the different hardening techniques and a mathematical model which allows to describe the phase transitions during heating and cooling, which are responsible for the changing microstructure and the desired surface hardness. In the second part I will discuss beam hardening. Here, the model consists of a semi-linear heat equation and a rate law for the micro-structural changes. I will present an adaptive finite element simulation of the process. An important technical problem is to avoid surface melting, especially above cavities in the work-piece. Here, I will show that best results can be achieved by combining open-loop optimal control with a machine-based feedback control.

The last part of the talk is devoted to multi-frequency induction hardening. Here, a well directed heating by electromagnetic waves and subsequent quenching of the work piece increases the hardness of the surface layer.
The process is very fast and energy efficient and plays an important role in modern manufacturing facilities in many industrial application areas. Although the original process is quite old, recent years have seen an important progress due to a new technology which allows to work simultaneously with several frequencies in one induction coil. For the first time this technology allows for the contour close hardening of complicated components such as gears in one induction coil. However, the process control especially the adjustment of the frequency fractions is quite delicate and requires costly experiments. Hence there is a high demand for simulation and optimal control of multifrequency hardening.

I will present some results of a collaboration between two industrial partners and four scientific partners on this topic funded by the German Ministry of Education and Research. The model for multifrequency induction hardening of steel parts consists of a system of partial differential equations including Maxwell's equations and the heat equation. We show that the coupled system admits a unique weak solution. Then I will discuss the numerical approximation of the problem. It turns out to be quite intricate since one has to cope with different time scales for heat diffusion and the Maxwell system. Moreover, owing to the skin effect only the boundary layers of the component are heated by induced eddy currents, hence we also have to consider different spatial scales.
We present a numerical algorithm based on adaptive edge-finite elements for the Maxwell system, which allows to treat these difficulties. We show some 3D simulations and conclude with results of an experimental validation in an industrial setting.

3월 24일 (화)
The production of modern multi-phase steels -- a challenge for industrial mathematics

Despite the development of sophisticated composite materials in recent years, steel is still the basic construction material for industrial societies. Steel is also a modern material, e.g., 80% of the steel grades sold by the German steel producer Thyssen Krupp have been developed in the last 15 years. In particular, this development has been triggered by the the demands of automotive industry. In 1999, a consortium of 33 international steel producers formed the ULSAB-AVC (UltraLight Steel Auto Body-Advanced Vehicle Concepts) consortium to pursue a steel-intensive family car, fit for the 21st century, that would be safe, affordable and fuel-efficient.

Besides the development of structural components such as tailored blanks and tubes the main part of the innovation came from a consequential employment of multi-phase steels. Approximately 75% of the ULSAB-AVC bodystructure design uses dual-phase (DP) steels. These steels have shown high potential for automotive applications due to their remarkable property combination with high strength and good formability. The standard process route for the production of DP steel is by hot rolling and subsequent controlled cooling. It provides good microstructure homogeneity with acceptable surface quality for many applications.

In the first part of the talk I will review models suitable to describe the ferrite growth in DP steels. We revisit the classical Johnson-Mehl-Avrami-Kolmogorov nucleation and growth models and show how to use this approach to account also for soft impingement effects. In the second part we will ask how one can detect these phase transitions experimentally. This leads us to the consideration of dilatometers, measuring the length and temperature changes in a specimen during controlled cooling. We show how mathematics, in this case the solution of an inverse problem, can help to detect the phase transition kinetics.

The macroscopic Avrami models give no information about grain size. In the third part of the talk we discuss a possibility to incorporate grain size information into our modelling approach. To this end we study a Fokker-Planck type evolution equation for the grain-size distribution. In the last part of the talk we finally consider the production of DP steel. To simulate a real run-out table (ROT) we consider the pilot hot-rolling mill at IMF Freiberg, Germany. In a first step another inverse problem has to be solved to identify the cooling conditions on ROT. Then, we set up an optimal control problem for the production of DP steel and conclude with some numerical results using a sequential quadratic programming approach and an experimental validation.