Robuste Optimierung mit Quantilmaßen auf globalen Metamodellen

Die robuste Optimierung bezieht Unsicherheiten der Eingangsvariablen in den Optimierungsprozess ein und bestimmt Lösungen, die unempfindlich gegenüber diesen Schwankungen sind. Sie besitzt einen hohen Anwendungsbezug, da streuende Parameter, wie etwa Fertigungstoleranzen oder Schwankungen in den Materialeigenschaften, in vielen industriellen Anwendungen eine Rolle spielen. In dieser Dissertation wird eine Methodik für die robuste Optimierung mit Quantilmaßen auf globalen Metamodellen entwickelt. Sie ist zugeschnitten auf die Situation, dass nur wenige Funktionsauswertungen zur Verfügung stehen. In der Praxis ist dies immer dann der Fall, wenn ein komplexes System- oder Prozessverhalten nur über aufwändige Computersimulationen oder Experimente erfasst werden kann.

Robust optimization integrates the input variables' uncertainty in the optimization process and determines solutions, which are insensitive to such variations. It has a high practical orientation, because scattering parameters like production tolerances or variations in the material properties play a role in many industrial applications. In this thesis a methodology for robust optimization with quantile measures on global metamodels is developed. It is tailored to the situation, that only a few function evaluations are available. In practical applications this is the case, whenever a complex system or process behaviour can only be modeled by expensive computer simulations or experiments. The focus of the thesis is on characterizing the system's robustness in an appropriate way and on calculating it efficiently. It is demonstrated, that quantile measures meet the industry's requirements to robustness measures in a better way than the standard measures used up to now. They find more appropriate robust areas and give more accurate lower and upper limits for a certain percentage of the centred output distribution, which can be fixed according to the needs of the user. The methodology approximates the system's behaviour with global metamodels. The robustness behaviour is modeled with metamodels for quantiles of the output distributions. The efficient quantile measure computation is done with a new algorithm, which combines Halton sequence and Harrell-Davis estimator and contains an internal error estimation. A numerical study demonstrates its efficiency. A measure for the tolerance of the quantile metamodels is developed, which derives its limits from the original metamodel. Finally a new iterative method supports the user in selecting the robust optimum. The methodology can be accelerated with fast multipole procedures. This way is described and the effort estimated. The validity and efficiency of the methodology is demonstrated with several real application examples from automotive industry.