Uncertainty Quantification and Robust Optimization in Engineering

The application and use of engineering components such as engines, wings, or complete airplanes are all subject to uncertainties, either of operational nature (variations in speed, angle of attack, pressure, etc) or of geometrical nature (manufacturing tolerances or uncertainties due to wearing). These uncertainties can have an important effect on the performance (output) of these components. The effect of these uncertain parameters should be quantified and included in the final solution to assure and improve the quality of the results. Polynomial chaos is a recent methodology to account for uncertainties that can be described by a distribution function. The method allows to obtain the distribution of the output for given input distributions. Over the last decade, with increasing computational resources and hardware power, design optimization is receiving more and more interest in aeronautical applications. Due to the uncertainties in a design process, the objective is also uncertain. Robust optimization is an extension of conventional optimization where uncertainties are also included in the design procedure. Using polynomial chaos expansion, the uncertain objective can be characterized by its mean and its variance. Therefore, it becomes a multi-objective problem and gradient based optimization requires the gradient of both quantities. These gradients can be obtained from the polynomial chaos expansion of the gradient of the objective. In this chapter, first, a brief introduction to polynomial chaos approach for uncertainty quantification is provided. Further its formulation with adjoint methods is described for gradient based robust optimization. The approach is applied to the optimal shape design of a transonic airfoil under uncertainties.