Robust Shape Optimization under Uncertainties in Device Materials, Geometry and Boundary Conditions

Abstract

We address the shape optimization problem of electronic and electric devices under geometrical and material uncertainties. Thereby, we aim at reducing undesirable phenomena of these devices such as hot-spots or torque fluctuations. The underlying minimization is based on the computation of a direct problem with random input data. To investigate the propagation of uncertainties through two- and three-dimensional, spatial models the stochastic collocation method (SCM) has been used in our work. In particular, uncertainties, which result from imperfections of an industrial production, are modelled by random variables with known probability distributions. Then, the polynomial chaos expansion (PCE) is used to construct a suitable response surface model, which can be effectively incorporated into the robust optimization framework. Correspondingly, the gradient directions of a cost functional, comprised of the expectation and the variance value, are calculated using the continuum design shape sensitivity and the PCE in conjunction with the SCM. Finally, the optimization results for the relevant electronics/electrical engineering problems demonstrate that the proposed method is robust and efficient. Overall, this work demonstrates, how recent techniques from shape and topology optimization can be combined with uncertainty quantification to solve complicated real-life problems.