10·î21Æü(ÌÚ) 15:30 -- 17:00 Dietmar Hoemberg (Weierstrass Inst. for Appl. Anal. and Stochastics) ``The induction hardening of steel - modeling, analysis and optimal design of inductor coils'' ¡ã¸¦µæÈ¯É½Í׻ݡä 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 first part of my talk I will present a mathematical model for surface heat treatments. It consists of a system of ODEs to describe the occuring phase transitions coupled with a nonlinear heat equation and a model for the respective heat source. In the case of induction hardening we use a vector potential formulation of Maxwell's equations. Unfortunately, in most cases the geometry of the region to be hardened does not allow to have a simple annular inductor shape. And even when the principal topology of the inductor is already fixed, the coupling distance between inductor and workpiece and the spacing of the coil turns have to be adjusted carefully in order to obtain the desired heating or hardening pattern. This problem is addressed in the second part of my talk. A major issue is to find a decent mathematical formulation of the design problem. We show that induction coils can conveniently be described as tubes, constructed from space curves.To investigate the shape sensitivity with respect to perturbations of the coil we employ the speed method for an admissible velocity field.