Shape and topology optimization of a permanent-magnet machine under uncertainties

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.

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