Improved Quality Assessment of Probabilistic Simulations and Application to Turbomachinery
Authors: Prots Andriy , Högner Lars , Voigt Matthias , Mailach Ronald , Danner Florian 
Published: January 2021

Abstract
Probabilistic methods are gaining in importance in aerospace engineering due to their ability to describe the behavior of the system in the presence of input value variance. A frequently employed probabilistic method is the Monte Carlo Simulation (MCS). There, a sample of random representative realizations is evaluated deterministically and their results are afterwards analyzed with statistical methods. Possible statistical results are mean, standard deviation, quantile values and correlation coefficients. Since the sample is generated randomly, the result of a MCS will differ for each repetition. Therefore, it can be regarded as a random variable. Confidence Intervals (CIs) are commonly used to quantify this variance. To gain the true CI, many repetitions of the MCS have to be conducted, which is not desirable due to limitations in time and computational power. Hence, analytical formulations or bootstrapping is used to estimate the CI. In order to reduce the variance of the result of a MCS, sampling techniques with variance reduction properties like Latin Hypercube Sampling (LHS) are commonly used. But the known methods to determine the CI do not consider this variance reduction and tend to overestimate it instead. Furthermore, it is difficult to predict the change of the CI size with increasing size of the sample.
In the present work, new methods to calculate the CI are introduced. They allow a more precise CI estimation when LHS is used for a MCS. For this purpose, the system is approximated by means of a meta model. The distribution of the result value is now approximated by repeating the MCS many times. The time consuming deterministic calculations of a MCS are thus replaced with an evaluation on the meta model. These so called virtual MCS can therefore be performed in a short amount of time. The estimated distribution of the result value can be used to estimate the CI. It is, however, not sufficient to use only the meta model. The error ε, defined as the difference between the true value y and the approximated value y, must be considered as well. The generated meta model can also be used to predict the size of the CI at different sample sizes. The suggested methods were applied to two test cases. The first test case examines a structural mechanics application of a bending beam, which features low computational cost. This allows to show that the predicted sizes of the CI are sufficiently precise. The second test case covers the aerodynamic application. Therefore, an aerodynamic Computational Fluid Dynamics (CFD) analysis accounting for geometrical variations of NASA’s Rotor 37 is conducted. For this, the blade is parametrized with the in-house tool Blade2Parameter. For different sample sizes, blades are generated using this parametrization. Their geometrical variance is based on experience values. CFD calculations for these blades are performed with the commercial software NUMECA. Afterwards, the CIs for result values of interest like mechanical efficiency are evaluated with the presented methods. The suggested methods predict a narrower and thus less conservative CI.