Boundary Element Methods for Inductive Hardening

This study deals with the simulation of inductive hardening of conducting workpieces made of steel. The aim is to calculate the propagation of heat in the workpiece. Based on this knowledge, the hardened zone can be predicted with sufficient precision. Since the simulation is to be applied in industry, workpieces and inductors are supposed to have a complex three dimensional shape. The electromagnetic calculations are based on the quasi-static approximation of Maxwell’s equations in frequency domain, and the non-linear heat conduction equation is used to evaluate the temperature distribution. The focus of this treatise is on the computation of the electromagnetic fields, especially on the boundary element methods (BEM) applied in order to master the unbounded exterior of the conductors. In the interior of the conductors, the skin effect plays an important role and the electromagnetic fields show a rapid decay. If the numerical solution is to resolve this effect, the mesh must be very fine at the surface, whereas this is not necessary elsewhere. To save storage, the mesh is refined adaptively in the interior, with the aid of a residual based error estimator. The equations for the conducting region are solved using a finite element method (FEM). A hierarchical system of three models is presented for the coupling of the BEM equations for the exterior with the FEM equations for the interior. The eddy current approach is the model with the most convenient properties. The FEM/BEM coupling is strong and symmetric, the equations have a unique solution, and the convergence of an iterative solver can be guaranteed. There is also a quasi-optimal a priori error etimate for a conforming Garlerkin discretization based on edge elements and Raviart-Thomas elements. However in terms of implementation the eddy current approach is also the most complicated one. The impedance model can be used as an approximation. It is based on the same equations for the two regions but in this model the coupling is realized only weakly by imposing so-called impedance boundary conditions on the surface of the conductors. The weak coupling has the advantage that the BEM and FEM parts can be solved independently. In order to get a first rough estimate of the electromagnetic fields, the magnetostatic approach is developed. As far as the BEM computations are concerned it assumes the negligible penetration depth of a perfect conductor and the FEM/BEM parts are coupled only uni-directionally. A kind of scalar magnetic potential is used in all three models, and in regions with nontrivial topology they are multivalued. In that case, the jumps of the magnetic potentials at suitable cutting surfaces or cutting cycles are associated with the total currents in the conductors, these surfaces or cycles must be added to the meshes. For this purpose, an algorithm for the automatic construction and classification of generators of H1(Γh, Z ) for triangulated surfaces is introduced. Unlike the FEM matrices, the BEM matrices are dense and cannot be stored completely. A H-Matrix Approximation is applied on the four utilized kernels of elaborate structure. Analytical solutions are developed to verify the electromagnetic computations. The non-linear heat problem is solved with an implicit Euler method. Measurements of the surface temperature during the process are made for the validation of these calculations. Comparisons of the predicted hardened zone in the simulation with real hardened items are most important for the program’s verification.