Richard Ahlfeld

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyaks algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBAs efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

Uncertainty Quantification of Leakages in a Multistage Simulation and Comparison With Experiments

April 10-15, 2016 Abstract The present paper presents a numerical study of the impact of tip gap uncertainties in a multistage turbine. It is well known that the rotor gap can change the gas turbine efficiency but the impact of the random variation of the clearance height has not been investigated before. In this paper the radial seals clearance of a datum shroud geometry, representative of steam turbine industrial practice, was systematically varied and numerically tested. By using a Non-Intrusive Uncertainty Quantification simulation based on a Sparse Arbitrary Moment Based Approach, it is possible to predict the radial distribution of uncertainty in stagnation pressure and yaw angle at the exit of the turbine blades. This work shows that the impact of gap uncertainties propagates radially from the tip towards the hub of the turbine and the complete span is affected by a variation of the rotor tip gap. This amplification of the uncertainty is mainly due to the low aspect ratio of the turbine and a similar behavior is expected in high pressure turbines. .

Stochastic Variation of the Aero-Thermal Flow Field in a Cooled High-Pressure Transonic Vane Configuration

In transonic high-pressure turbine stages, oblique shocks originated from vane trailing edges impact the rear suction side of each adjacent vane. High-pressure vanes are usually cooled to tolerate the combustor exit temperature levels, which would reduce dramatically the residual life of a solid vane. Then, it is highly probable that shock impingement will occur in proximity of one of the coolant rows. It has already been observed that the presence of an adverse pressure gradient generates non-negligible effects on heat load due to the increase in boundary layer thickness and turbulence level, with a detrimental impact on the local adiabatic effectiveness values. Furthermore, the generation of a tornado-like vortex has been recently observed that could further decrease the efficacy of the cooling system by moving cold flow far from the vane wall. It must be also underlined that manufacturing deviations and in-service degradation are responsible for the stochastic variation of geometrical parameters. This latter phenomenon greatly alters the unsteady location of the shock impingement and the time-dependent thermal load on the vane. Present work starts from what is shown in literature and provides a highly-detailed description of the aero-thermal field that occurs on a model that represents the flow conditions occurring on the rear suction side of a cooled vane. The numerical model is initially validated against the experimental data obtained by the University of Karlsruhe during TATEF2 EU project, and then an uncertainty quantification methodology based on the probabilistic collocation method and on Pades polynomials is used to consider the probability distribution of the geometrical parameters. The choice of aleatory unknowns allows to consider the mutual effects between shock-waves, trailing edge thickness and hole diameter. Turbulence is modelled by using the Reynolds Stress Model already implemented in ANSYS® Fluent®. Special attention is paid to the description of the flow field in the shock/boundary layer interaction region, where the presence of a secondary effects will completely change the local adiabatic effectiveness values.

Autonomous Uncertainty Quantification for Discontinuous Models Using Multivariate Padé Approximations

Problems in turbomachinery computational fluid dynamics (CFD) are often characterized by nonlinear and discontinuous responses. Ensuring the reliability of uncertainty quantification (UQ) codes in such conditions, in an autonomous way, is challenging. In this work, we suggest a new approach that combines three state-of-the-art methods: multivariate Pade approximations, optimal quadrature subsampling (OQS), and statistical learning. Its main component is the generalized least-squares multivariate Pade- Legendre (PL) approximation. PL approximations are globally fitted rational functions that can accurately describe discontinuous nonlinear behavior. They need fewer model evaluations than local or adaptive methods and do not cause the Gibbs phenomenon like continuous polynomial chaos methods. A series of modifications of the Pade algorithm allows us to apply it to arbitrary input points instead of optimal quadrature locations. This property is particularly useful for industrial applications, where a database of CFD runs is already available, but not in optimal parameter locations. One drawback of the PL approximation is that it is nontrivial to ensure reliability. To improve stability, we suggest to couple it with OQS. Our reasoning is that least-squares errors, caused by an ill-conditioned design matrix, are the main source of error. Finally, we use statistical learning methods to check smoothness and convergence. The resulting method is shown to efficiently and correctly fit thousands of partly discontinuous response surfaces for an industrial film cooling and shock interaction problem using only nine CFD simulations.

Uncertainty quantification for fat-tailed probability distributions in aircraft engine simulations

Rare event simulation is vital for industrial design because some events, so-called black swans, can have fatal consequences despite their low probability of occurrence. Finding low-probability events far off the mean design is a challenging task for realistic engineering models because they are characterized by high computational demands, many input variables, and often insufficient statistical information to build parametric probability distributions. Therefore, an adaptive and arbitrary polynomial chaos method, called sparse approximation of moment-based arbitrary polynomial chaos, is suggested in this work. Sparse approximation of moment-based arbitrary polynomial chaos creates custom polynomial basis functions and grids based on statistical moments to avoid incorrect statistical assumptions. The contribution of this work is that it is derived how rare event simulation can conveniently be integrated into adaptive sparse grid methods by calculating polynomial chaos expansions based on the statistical m...