# Fast boundary elements for the simulation of eddy currents and their heat production and cooling

## Dr. Jens Breuer

**Nov. 23, 2004, 1:45 p.m. T 112**

We consider an industrial electric device which is driven by an alternating current. The amperage and the frequency are given on some contact areas of the device. Starting with the eddy current model for the Maxwell equations one can de rive boundary integral equations for the unknown traces of the electric and the magnetic fields on the boundary. The given amperage on the contacts translates into a given normal component of the electric field inside the device. This yields a special jump condition for the tangential components of the magnetic field. The unknown magnetic trace can then be restricted to its divergence free part. The jump can be computed by solving an auxiliary problem, the Laplace-Beltrami equation for some surface potential. For its discretization we derive a stabilized mixed Galerkin finite element method on the boundary which leads to a sparse system and gives quasioptimal convergence rates. The discretization of the main system is done via Raviart-Thomas elements on the boundary. The divergence constraint on the magnetic trace is incorporated by using Lagrangian multipliers. The case of multiple materials with different conductivity is handled via a Dirichlet domain decomposition approach. The corresponding discrete system contains fully populated boundary element matrices. So the adaptive cross approximation approach is used to approximate the Laplace and Helmholtz kernels which leads to a sparse boundary element method for the eddy current scheme. The electrical field then enters as a source field into the heat conduction inside the device. For the cooling air flow we assume that it is not affected by the heat production, but can be considered as a given velocity field. This can be the result of the stationary Navier-Stokes equations or the coupling of Prandtl boundary layer equations and a far field potential flow. For the temperature outside the device the nonlinear heat transport equation has to be solved. Unique solvability and convergence of the corresponding finite element scheme can be shown. As a simplified model the nonlinear heat conduction equation is considered. Via the Kirchhoff transformation the resulting system is linearized with constant coefficients. The system can then be handled with boundary integral equations. At the end various numerical examples both the electrical and the thermal part will be shown, which are in agreement with the theoretical results.