Einladung zu einem Vortrag von:
Dipl.-Math. Jens Breuer
Universität Stuttgart
Dienstag, 23. November 2004
13:45 Uhr, T 112
Fast boundary elements for the simulation of
eddy currents and their heat production and
cooling
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 elds on the boundary. The given amperage on the contacts translates
into a given normal component of the electric eld inside the device. This yields a
special jump condition for the tangential components of the magnetic eld. 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 nite 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 di
erent conductivity is handled via a Dirich
let 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 eld then enters as a source eld into the heat conduction inside
the device. For the cooling air
ow we assume that it is not a
ected by the heat
production, but can be considered as a given velocity eld. This can be the result
of the stationary Navier{Stokes equations or the coupling of Prandtl boundary
layer equations and a far eld potential
ow. For the temperature outside the
device the nonlinear heat transport equation has to be solved. Unique solvability
and convergence of the corresponding nite element scheme can be shown. As
a simpli ed 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.
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