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'' 


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  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.