WIAS Preprint No. 521, (1999)

Optimal Shape Design of Inductor Coils for Surface Hardening


  • Hömberg, Dietmar
    ORCID: 0000-0001-9460-5729
  • Sokołowski, Jan

2010 Mathematics Subject Classification

  • 49J20 35K55 49J20 49K20 78K60


  • Shape optimization, Maxwell's equations, induction heating, necessary optimality conditions




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.

Appeared in

  • SIAM J. Control Optim., 42 (2003) pp. 1087--1117.

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