Please cite this article in press as: R. Ahlfeld et al., SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos, J. Comput. Phys. (2016), http://dx.doi.org/10.1016/j. jcp.2016.05.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. SAMBA AbstractA 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

The new possibilities offered by additive manufacturing (AM) can be exploited in gas turbines to produce a new generation of complex and efficient internal coolant systems. The flexibility offered by this new manufacturing method needs a paradigm shift in the design approach, and a possible solution is offered by topology optimization. The overall goal of this work is to propose an innovative method to design internal channels in gas turbines that fully exploit AM capabilities. The present work contains a new application of a fluid topology sedimentation method to optimize the internal coolant geometries with minimal pressure losses while maximizing the heat exchange. The domain is considered as a porous medium with variable porosity: the solution is represented by the final solid distribution that constitutes the optimized structure. In this work, the governing equations for an incompressible flow in a porous medium are considered together with a conjugate heat transfer equation that includes porosity-dependent thermal diffusivity. An adjoint optimization approach with steepest descent method is used to build the optimization algorithm. The simulations are carried out on three different geometries: a U-bend, a straight duct, and a rectangular box. For the U-bend, a series of splitter is automatically generated by the code, minimizing the stagnation pressure losses. In the straight duct and in the rectangular box, the impact of different choices of the weights and of the definition of the porosity-dependent thermal diffusivity is analyzed. The results show the formation of splitters and bifurcations in the box and “riblike” structures in the straight duct, which enhance the heat transfer.

Uncertainty quantification (UQ) has recently become an important part of the design process of countless engineering applications. However, up to now in computational fluid dynamics (CFD) the errors introduced by the turbulent viscosity models in Reynolds-Averaged Navier Stokes (RANS) models have often been neglected in UQ studies. Although Direct Numerical Simulations (DNS) are physically correct, obtaining a large enough set of DNS data for UQ studies is currently computationally intractable.UQ based only on RANS simulations or on DNS often leads to physical and statistical inaccuracies in the output probability distribution functions (PDF). Therefore, three hybrid methods combining both RANS simulations and DNS to perform non-intrusive UQ are suggested in this work. Low-fidelity RANS simulations and high-fidelity DNS are combined to give an approximation of an output PDF using the advantages of both data sets: the physical accuracy via the DNS and the statistical accuracy via the RANS simulations. The hybrid methods are applied to the flow over 2D periodically arranged hills. It is shown that the Gaussian CoKriging (GCK) method is the best hybrid method and that a non-intrusive hybrid UQ approach combining both DNS and RANS simulations is possible, with both physically more accurate and statistically better PDF.

This work newly proposes an uncertainty quantification (UQ) method named sparse approximation of moment-based arbitrary polynomial chaos (SAMBA PC) that offers a single solution to many current problems in turbomachinery applications. At the moment, every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal probability density functions (PDFs) (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive, and having scarce information is the norm. A third critical limitation is that if using nonintrusive polynomial chaos (NIPC) methods, the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called “curse of dimensionality.” It is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the computational fluid dynamics (CFD) simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results, it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. It is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for ten input random variables are required). It is shown in this paper that SAMBA's algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this, the code needs a very small number of simulations and preserve a 5% error margin. SAMBA's flexibility to handle different forms of input distributions and a high number of input variables is shown on a low-pressure turbine (LPT) blade-based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

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..

The new possibilities offered by Additive Manufacturing (AM) can be exploited in gas turbines to produce a new generation of complex and efficient internal coolant systems. The flexibility offered by this new manufacturing method needs a paradigm shift in the design approach and a possible solution is offered by Topology Optimization. The overall goal of this work is to propose an innovative method to design internal channels in gas turbines that that fully exploit AM capabilities. The present work contains a new application of a Fluid Topology sedimentation method to optimize the internal coolant geometries with minimal pressure losses while maximizing the heat exchange. The domain is considered as a porous medium with variable porosity: the solution is represented by the final solid distribution that constitutes the optimised structure. In this work the governing equations for an incompressible flow in a porous medium are considered together with a conjugate heat transfer equation that includes porosity dependent thermal diffusivity. An adjoint optimisation approach with steepest descent method is used to build the optimisation algorithm. The simulations are carried out on two different geometries: a U-Bend and a straight duct. For the U-Bend a series of splitter is automatically generated by the code, minimizing the stagnation pressure losses. In the straight duct the impact of different porosity dependant thermal diffusivity is analysed to achieve high thermal exchange. The results show “rib like” structures that enhance the heat transfer.

This work newly proposes an uncertainty quantification method named SAMBA PC (Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) that offers a single solution to many current problems in turbomachinery applications. At the moment every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal PDFs (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive and having scarce information is the norm. A third critical limitation is that if using Non-Intrusive Polynomial Chaos methods the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called ‘curse of dimensionality’. In this work it is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the CFD simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. SAMBA is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for 10 input random variables are required). It is shown that SAMBA’s algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this the code needs a very small number of simulations and preserves a 5% error margin. SAMBA’s flexibility to handle different forms of input distributions and a high number of input variables is shown on a low pressure turbine blade based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

This work introduces a new uncertainty quantification method to better deal with scarce data and long simulation run times in Formula 1 design. Race cars are produced in low quantities and for maximum performance. Thus, their designing process is characterised by manufacturing data shortage and complex Computational Fluid Dynamics simulations with long run times. Their car aerodynamics is subject to many random variables that introduce uncertainty into the down-force and drag performance, such as variations in ride height, front wing direction and pitch angle. To accurately predict the car performance during a race, it is important to study the effect of these random variables. This assessment cannot be performed with the standard deterministic Computational Fluid Dynamics approaches used in Formula 1. Even with regard to stochastic approaches, no efficient method has so far been suggested that addresses the problem of data scarcity. The reason for this is that most efficient uncertainty quantification methods fit probability distributions to the scarce data. It is shown in this work that probability distribution fitting can create a significant error using a simple two-dimensional diffuser example. Subsequently, the use of a new data-driven Polynomial Chaos method and its sparse multi-dimensional extension is suggested and demonstrated for Formula 1 to reduce such errors. This method allows to avoid distribution fitting because it is based on pure data. SAMBA’s general formulation also makes it easier to combine any possible inputs within a sparse description for problems with many variables. SAMBA is applied to two realistic car three-dimensional Computational Fluid Dynamics simulations: a NACA 0012 tip wing and the front part of a Formula 1 car. The probabilistic variations of the lift and drag of these two configurations are calculated using SAMBA and shown to be significant.