Design Methods for Reducing Failure Probabilities with Examples from Electrical Engineering

This thesis addresses the quantification of uncertainty and optimization under uncertainty. We focus on uncertainties in the manufacturing process of devices, e.g. caused by manufacturing imperfections, natural material deviations or environmental influences. These uncertainties may lead to deviations in the geometry or the materials, which may cause deviations in the operation of the device. The term yield refers to the fraction of realizations in a manufacturing process under uncertainty, fulfilling all performance requirements. It is the counterpart of the failure probability (yield = 1 - failure probability) and serves as a measure for (un)certainty. The main goal of this work is to efficiently estimate and to maximize the yield. In this way, we increase the reliability of designs which reduces rejects of devices due to malfunction and hence saves resources, money and time. One main challenge in the field of yield estimation is the reduction of computing effort, maintaining high accuracy. In this work we propose two hybrid yield estimation methods. Both are sampling based and evaluate most of the sample points on a surrogate model, while only a small subset of so-called critical sample points is evaluated on the original high fidelity model. The SC-Hybrid approach is based on stochastic collocation and adjoint error indicators. The non-intrusive GPR-Hybrid approach uses Gaussian process regression and allows surrogate model updates on the fly. For efficient yield optimization we propose the adaptive Newton-Monte-Carlo (Newton-MC) method, where the sample size is adaptively increased. Another topic is the optimization of problems with mixed gradient information, i.e., problems, where the derivatives of the objective function are available with respect to some optimization variables, but not for all. The usage of gradient based solvers like the adaptive Newton-MC would require the costly approximation of the derivatives. We propose two methods for this case: the Hermite least squares and the Hermite BOBYQA optimization. Both are modifications of the originally derivative free BOBYQA (Bound constrained Optimization BY Quadratic Approximation) method, but are able to handle derivative information and use least squares regression instead of interpolation. In addition, an advantage of the Hermite-type approaches is their robustness in case of noisy objective functions. The global convergence of these methods is proven. In the context of yield optimization the case of mixed gradient information is particularly relevant, if - besides Gaussian distributed uncertain optimization variables - there are deterministic or non-Gaussian distributed uncertain optimization variables. The proposed methods can be applied to any design process affected by uncertainties. However, in this work we focus on application to the design of electrotechnical devices. We evaluate the approaches on two benchmark problems, a rectangular waveguide and a permanent magnet synchronous machine (PMSM). Significant savings of computing effort can be observed in yield estimation, and single- and multi-objective yield optimization. This allows the application of design optimization under uncertainty in industry