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This book covers cutting-edge findings related to uncertainty quantification and optimization under uncertainties (i.e. robust and reliable optimization), with a special emphasis on aeronautics and turbomachinery, although not limited to these fields. It describes new methods for uncertainty quantification, such as non-intrusive polynomial chaos, collocation methods, perturbation methods, as well as adjoint based and multi-level Monte Carlo methods. It includes methods for characterization of most influential uncertainties, as well as formulations for robust and reliable design optimization. A distinctive element of the book is the unique collection of test cases with prescribed uncertainties, which are representative of the current engineering practice of the industrial consortium partners involved in UMRIDA, a level 1 collaborative project within the European Commission's Seventh Framework Programme (FP7). All developed methods are benchmarked against these industrial challenges. Moreover, the book includes a section dedicated to Best Practice Guidelines for uncertainty quantification and robust design optimization, summarizing the findings obtained by the consortium members within the UMRIDA project. All in all, the book offers a authoritative guide to cutting-edge methodologies for uncertainty management in engineering design, covers a wide range of applications and discusses new ideas for future research and interdisciplinary collaborations.

### Vision, Objectives and Research Activities

Virtual prototyping (VP) is a key technology for environment-friendly and cost-effective design in the aircraft industry. However, the underlying analysis and simulation tools are currently applied with a unique set of input data and model variables, although realistic operating conditions are a superposition of numerous uncertainties under which the industrial products operate (uncertainties on operational conditions, on geometries resulting from manufacturing tolerances, numerical error sources and uncertain physical model parameters). Major new developments in this new scientific area of Uncertainty Management and Quantification (UM and UQ) and Robust Design Methods (RDMs) are needed to bridge the gap towards industrial readiness. The UMRIDA project, which stands for Uncertainty Management for Robust Industrial Design in Aeronautics, addresses these objectives by performing major research in both UQ and RDM and developing methods to handle a large number of simultaneous uncertainties including generalized geometrical uncertainties within a quantifiable objective of a turn-around time acceptable for industrial readiness. To assess the quantifiable objective, the developed methods are applied to a unique database with prescribed uncertainties build from industrial challenges provided by the project partners.

Charles Hirsch, Dirk Wunsch

### UMRIDA Test Case Database with Prescribed Uncertainties

The UMRIDA Database for Uncertainty Quantification and Robust Design Methods comprises different test cases with prescribed uncertainties which have been compiled using a common way of describing the different test cases.

Sönke Klostermann

### Uncertainties in Compressor and Aircraft Design

Uncertainties are always present due to limited manufacturing precision and variable operating conditions. Integrating these uncertainties into the design process of compressors and aircraft is a key element to ensure safe and economic operation. This chapter focuses on revealing the actual sources of these uncertainties and proposes how they can be quantified and modelled such that they can be integrated into an uncertainty quantification or a robust design optimization.

Dirk Büche, Sönke Klostermann, G. Rogé, X. Loyatho

### Estimation of Model Error Using Bayesian Model-Scenario Averaging with Maximum a Posterori-Estimates

The lack of an universal modelling approach for turbulence in Reynolds-Averaged Navier–Stokes simulations creates the need for quantifying the modelling error without additional validation data. Bayesian Model-Scenario Averaging (BMSA), which exploits the variability on model closure coefficients across several flow scenarios and multiple models, gives a stochastic, a posteriori estimate of a quantity of interest. The full BMSA requires the propagation of the posterior probability distribution of the closure coefficients through a CFD code, which makes the approach infeasible for industrial relevant flow cases. By using maximum a posteriori (MAP) estimates on the posterior distribution, we drastically reduce the computational costs. The approach is applied to turbulent flow in a pipe at $$Re=$$Re= 44,000 over 2D periodic hills at $$Re_H=5600$$ReH=5600, and finally over a generic falcon jet test case (Industrial challenge IC-03 of the UMRIDA project).

Martin Schmelzer, Richard P. Dwight, Wouter Edeling, Paola Cinnella

### Uncertainties for Thermoacoustics: A First Analysis

An Uncertainty Quantification analysis of a swirled stabilized combustor experiment is performed. The objective is to estimate the modal risk factor of the system, i.e. the probability of a thermoacoustic mode to be unstable, which may facilitate the development and optimization of suitable control methods. To propagate uncertainties, a Monte Carlo method is initially used based on 4000 Helmholtz-based thermoacoustic simulations with random perturbations on the flame input parameters. The analysis of the Monte Carlo database suggests that a reduced two-step Uncertainty Quantification strategy may be efficient to deal with thermoacoustic systems. First, three bilinear surrogate models are tuned from a moderate number of Helmholtz solutions (a few tens). Then, these algebraic models are used to perform a Monte Carlo analysis at reduced cost and approximate the risk factor of the mode. Good agreements are obtained when comparing the risk factor from the full Monte Carlo database and the risk factor from surrogate models.

A. Ndiaye, F. Nicoud

### Numerical Uncertainties Estimation and Mitigation by Mesh Adaptation

This chapter gives an unified formalism that encompasses the two most common mesh adaptation strategies: Hessian-based and goal-oriented. The first one is based on the control of the interpolation error of a solution field. The second one relies on the control of the approximation error of a scalar-output functional. Both of them have been widely used in aeronautics and derived in an anisotropic context by using a metric-based approach. If Hessian-based mesh adaptation is completely generic, it does not account for discretization error of the PDE at hand, contrary to the goal-oriented approach. The scope of this chapter is to extend metric-based mesh adaptation to control a norm of the approximation error. This allows a more accurate output and in particular to control simultaneously the error on multiple functionals of interest as lift, drag, moment, without the need to solve multiple adjoint states. The procedure is based on the derivation of a corrector term that is then used as a source term for adjoint-based mesh adaptation. The estimate is derived within the continuous mesh framework, yielding naturally a fully anisotropic estimate.

Frédéric Alauzet, Alain Dervieux, Loïc Frazza, Adrien Loseille

### General Introduction to Polynomial Chaos and Collocation Methods

In this chapter, the basic principles of two methodologies for uncertainty quantification (UQ) are discussed, namely the polynomial chaos method and the collocation method. UQ deals with the propagation of uncertainties through complex numerical models, and in the present context of the UMRIDA project, mostly computational fluid dynamics (CFD) codes. The focus is on non-intrusive methods implying that the model does not require any changes and can be used as a black box.

Chris Lacor, Éric Savin

### Generalized Polynomial Chaos for Non-intrusive Uncertainty Quantification in Computational Fluid Dynamics

This chapter is concerned with the construction of polynomial surrogates of complex configurations arising in computational fluid dynamics for the purpose of propagating uncertainties pertaining to geometrical and/or operational parameters. Generalized homogeneous chaos expansions are considered and different techniques for the non-intrusive reconstruction of the polynomial expansion coefficients are outlined. A sparsity-based reconstruction approach is more particularly emphasized since it benefits from the “sparsity-of-effects” trend commonly observed on global quantities of interest such as the aerodynamic coefficients of a profile. The overall framework is illustrated on a two-dimensional transonic turbulent flow around a RAE 2822 airfoil subjected to a variable free-stream Mach number, angle of attack, and relative thickness of the profile.

Vincent Couaillier, Éric Savin

### Non-intrusive Probabilistic Collocation Method for Operational, Geometrical, and Manufacturing Uncertainties in Engineering Practice

An industry-ready uncertainty quantification tool chain is developed and successfully applied to both simultaneous operational and geometrical uncertainties and uncertainties resulting from manufacturing variability, which are characterized by correlations of the measured coordinates. The non-intrusive probabilistic collocation method is combined with a sparse grid approach to drastically reduce the computational cost. This is one of the key features that make UQ in industrial applications feasible. A second required element is the automatization of the entire simulation chain, from uncertainty definition, simulation setup, post-processing and in case of geometrical uncertainties, geometry modification, and re-meshing. This process is fully automated including the post-processing of the UQ simulations, which consists of output PDF reconstruction and the calculation of scaled sensitivity derivatives. This tool chain is applied to the rotor 37 configuration with imposed uncertainties, demonstrating its capability of handling many simultaneous operational and geometrical or correlated manufacturing uncertainties in turnaround times significantly below the UMRIDA quantitative objectives of less than 1000CPUh for 10 simultaneous uncertainties. It is found that a level 1 sparse grid approach is sufficient if the mean and variance of output quantities are needed and a level 2 sparse grid is sufficient for the reconstructed PDF shape for most engineering applications. For manufacturing uncertainties, it is shown that a level 1 sparse grid can be used for the propagation of manufacturing uncertainties and that a surface reconstruction accuracy of 99% seems necessary for the purpose of UQ studies on manufacturing variability.

Dirk Wunsch, Rémy Nigro, Grégory Coussement, Charles Hirsch

### Non-intrusive Uncertainty Quantification by Combination of Reduced Basis Method and Regression-based Polynomial Chaos Expansion

In this study, an efficient non-intrusive model reduction scheme is developed for uncertainty quantification using proper orthogonal decomposition and the regression-based non-intrusive polynomial chaos approach. The key idea is to retrieve the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine-scale analysis. The reduced basis approach was applied to a highly nonlinear analytical test function, the 2D RAE2822 airfoil with geometrical uncertainties and the NASA rotor 37 with combined geometrical and operational uncertainties. The numerical results show that the developed model is able to produce acceptable results for the statistical quantities of interest. The computation time of the reduced basis method was found to be much lower than that of the classical polynomial chaos expansion method.

Mehrdad Raisee, Dinesh Kumar, Chris Lacor

### Screening Analysis and Adaptive Sparse Collocation Method

In order to treat efficiently UQ problems of industrial relevance, characterized by a large number of uncertainties, we propose in this chapter several techniques for screening analysis (SS-ANOVA, PCA, and Morris) that can be used in order to reduce the number of uncertain parameters to the most significant ones in the problem, and an adaptive version of the Polynomial Chaos Expansion (PCE) that can be used to reduce significantly the number of sampling points needed for an accurate UQ.

Alberto Clarich, Rosario Russo

### General Introduction to Surrogate Model-Based Approaches to UQ

This chapter introduces two popular surrogate modeling methods which can be used to quantify uncertainties such as statistics of the aerodynamic coefficients from scattered data obtained by computational fluid dynamics (CFD) simulations. One is Kriging, which is able not only to interpolate predicted data but also to provide statistical information at unsampled locations in the parameter space based on Bayesian statistics. The other one is the radial basis function (RBF) method. The RBF method is also a powerful nonlinear interpolation method which exactly interpolates the samples, and its various radial basis function types support the interpolated values locally or globally when appropriately selected. Both methods can make use of gradient information, if available, to improve the model accuracy.

Daigo Maruyama, Stefan Görtz, Dishi Liu

### Comparing Surrogates for Estimating Aerodynamic Uncertainties of Airfoils

Different surrogate models are compared in terms of their efficiency in estimating statistics of aerodynamic coefficients of the RAE2822 airfoil due to geometric input uncertainties. A comparison with direct integration and polynomial chaos methods is also performed. The aerodynamic coefficients and their partial gradients with respect to the uncertain input parameters are computed with a CFD solver and its adjoint counterpart. Reference statistics are computed in order to quantify the error of the different methods. The efficiency of the different methods is discussed in terms of the error in estimating a statistical quantity as a function of the number of CFD (including adjoint) computations used to construct the surrogate model. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods for the same computational cost. Sampling techniques are discussed in the context of estimating stochastic quantities used for risk management. While the mean and standard Deviation (used for mean-risk approach) can be efficiently computed by distributing the samples in the input parameter space with its probability density function, the maximum or minimum value (used for worst-case scenario) can be led more accurately by an expected improvement based adaptive sampling technique. This fact indicates that advanced sampling techniques are required for evaluating both the mean risk and worst-case risk at the same time.

Daigo Maruyama, Dishi Liu, Stefan Görtz

### Ordinary Kriging Surrogates in Aerodynamics

This chapter describes the methodology used to construct Kriging-based surrogate models and their possible application to uncertainty quantification and robust shape optimization. More particularly, a two-dimensional RAE 2822 airfoil at transonic speed is considered for which the shape of the baseline profile is altered by localized bumps of small amplitudes. The flow around the airfoil is then subjected to important changes compared to the baseline configuration. The aim of the surrogate is to assess their influence on the aerodynamic performance of the profile as quantified by its lift-to-drag ratio. An optimization analysis is subsequently carried out in order to extract the local extrema of this performance measure. Assigning some uncertainty to the bump amplitudes, it also reveals that the global maximum identified by a high-quality surrogate is not necessarily the most robust one. This example constitutes an interesting benchmark for testing uncertainty quantification and robust optimization strategies.

Antoine Dumont, Jean-Luc Hantrais-Gervois, Pierre-Yves Passaggia, Jacques Peter, Itham Salah el Din, Éric Savin

### Surrogates for Combustion Instabilities in Annular Combustors

While the computational power is still increasing, thus arousing the interest for high-fidelity simulations, the need of low-order models is also felt to both predict and understand combustion instabilities at low costs. Historically applied to simple systems like longitudinal Rijke tubes to unveil the driven mechanisms leading to instability, they have recently been adapted to more complex configurations such as annular combustors. A network model is presented here to predict thermo-acoustic modes in an annular combustion chamber fed by burners connected to an annular plenum, typical of modern combustor designs. Explicit expressions of the growth rate are derived in several cases showing key parameters controlling the stability. In more general situations, no explicit solution can be obtained. Nevertheless, such an analytical model can be solved numerically at low cost compared with 3D acoustic tools and high-fidelity simulations. In this framework, efficient sensitivity techniques and UQ methods can be developed to tackle the UQ problem: “How can we assess the risk of instability in industrial combustors at the predesign stage?".

M. Bauerheim, A. Ndiaye, F. Nicoud

### General Introduction to Monte Carlo and Multi-level Monte Carlo Methods

In this chapter, we present a general introduction to Monte Carlo (MC)-based methods, sampling methodologies, stratification methods, and variance reduction techniques. In the first part, we will discuss the theoretical basis and the convergence proprieties of MC methods. The next part is devoted to pseudorandom and quasi-random number generation, the generation of random variables and the application of stratification. It is followed by techniques for correlation and discrepancy control. The third part presents the concept of Latin Hypercube Sampling (LHS). The last part introduces the concept of Multi-Level Monte Carlo (MLMC).

Robin Schmidt, Matthias Voigt, Michele Pisaroni, Fabio Nobile, Penelope Leyland, Jordi Pons-Prats, Gabriel Bugeda

### Latin Hypercube Sampling-Based Monte Carlo Simulation: Extension of the Sample Size and Correlation Control

Latin hypercube sampling (LHS) is frequently used in Monte Carlo-type simulations for the probabilistic analysis of systems due to its variance reducing properties compared with random sampling. LHS allows for an extension of the sample size by doubling them or adding an even multiple of the sample size depending on the selection of the sample values. This can become a drawback of LHS compared to random sampling especially in the presence of a large sample size. In this chapter, a new approach to the multiple extension of a Latin hypercube samples is presented. The objective is to extend the sample size but to keep the increase in the number of realizations constant. It is of particular importance that the present approach is able to maintain the correlations between the input variables in the probabilistic analysis.

Robin Schmidt, Matthias Voigt, Ronald Mailach

### Multi-level Monte Carlo Method

Uncertainty quantification has gained interest during the recent years. Two clear examples are NODESIM-CFD and, the just finished, UMRIDA projects.

Jordi Pons-Prats, G. Bugeda

### Continuation Multi-level Monte Carlo

In this chapter, we describe the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed in Collier et al. [1] and apply it to efficiently propagate operating and geometric uncertainties in internal and external aerodynamic simulations. The key idea of MLMC, presented in the previous chapter, is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigation on a hierarchy of nested computational grids (levels). In the continuation algorithm (C-MLMC) the parameters that prescribe the number of levels and simulations per level are computed on the fly to further reduce the overall computational cost.

Michele Pisaroni, Fabio Nobile, Penelope Leyland

### Introduction to Intrusive Perturbation Methods

Perturbation methods based on derivatives can be applied to uncertainties in two particular contexts. First, surrogate models based on Taylor expansion can be built after methods for evaluation of first and second derivatives have been developed. Second, PC-based probabilistic functionals can be minimized after a gradient is computed. Both contexts use derivations which are intrusive. This chapter, after a definition of these approaches, concentrates on Taylor surrogate models or Moment methods.

Alain Dervieux

### Algorithmic Differentiation for Second Derivatives

This chapter gives the derivatives of an equation-constrained functional in order to help the derivation of a Taylor-based surrogate model. Two approaches are described, the Tangent-on-Tangent approach and the Tangent-on-Reverse approach according to the two composite differentiation modes.

Alain Dervieux

### Second-Order Derivatives for Geometrical Uncertainties

The paper presents a method for handling geometrical uncertainties. In this case, discretization of continuous uncertainty field leads to a large set of correlated uncertainties/random variables. In order to reduce dimensionality of the problem, the authors propose a method that takes advantage of both the probabilistic information (covariance) and the local behavior of the objective (up to a second-order derivatives). The proposed method is verified for the UMRIDA BC-03 test case (UMRIDA Consortium, Test case description innovative database for UQ and RDM, 2014). The method is shown to outperform the Karhunen-Loeve decomposition and the analysis based purely on the Hessian matrix. The method allows to keep the same level of accuracy with a significant reduction of the number of uncertainties.

Marcin Wyrozębski, Łukasz Łaniewski-Wołłk, Jacek Rokicki

### Application of Uncertainty Quantification Methodologies to Falcon

In this section, we give both the description of the IC-03 Falcon test case (Industrial Challenge [1]) and comparison of results (Delft UMRIDA workshop, [2] and other partner’s inputs). A database incorporating modelization (turbulence), numerical error (mesh), operational (AoA), and geometrical (wing spanwise twist distribution) uncertainties have been created. Six Partners are involved in IC-03: Alenia Aermacchi/Leonardo (Italy), INRIA (France), NUMECA (Belgium), TU Delft (Netherlands), VUB (Belgium), Dassault Aviation (France).

G. Rogé, X. Loyatho

### Application of UQ to Combustor Design

The present chapter investigates an uncertainty quantification (UQ) approach for the simulations of combustion instabilities (CI). They stem from the interaction between acoustic waves and heat release fluctuations. CI are harmful to gas turbines engines if they are not mastered at the design level. The targeted test case within the UMRIDA project is a realistic full annular helicopter combustor from Safran Helicopter Engines. This combustor is equipped with several burners and flames, each of them described by two uncertain input parameters. Therefore, we are facing a “curse of dimensionality” as around 10–20 independent uncertain parameters are generally involved in real combustors. In order to break the curse, active subspace methods (Constantine et al, J Sci Comput, 2013) and efficient surrogate techniques are used to assess the risk factor of the system, i.e., the probability of an acoustic mode to be unstable. For such high-dimensional complex systems, active subspace methods based on gradient correlations are known to provide dimension reduction of the necessary input parameters. Efficiency of the proposed method was shown effective on practical examples (Bauerheim et al. Combust Flame 161(5):1374–1389, 2014). In the scope of our overall approach, an Analytical Tool to Analyze and Control Azimuthal Mode in Annular Chambers named ATACAMAC (Bauerheim et al. Combust Flame 161(5):1374–1389, 2014) is used to deal with the complexity of the combustor features. Besides, it is hardly conceivable having recourse to 3D LES or even 3D Helmholtz solvers to deal with uncertainties in the high-dimensional spaces within a reasonable computational timeframe. To avoid expensive Helmholtz simulations, the quasi-analytical tool ATACAMAC is first applied to generate a reference Monte Carlo database to obtain the statistical benchmark database for the risk factor of the acoustic mode. Thereafter, the dimension of the system is drastically reduced to much less than 20–40 parameters using the active subspace methodology. Linear and quadratic surrogate models are introduced based on moderate active variables previously determined. Such models proved satisfactory in cheaply and accurately estimating the risk factor of the mode (Ndiaye et al, ASME Turbo Expo, 2015). The surrogate models are then fitted with the thousands of ATACAMAC simulations performed within the Monte Carlo analysis. These low-order models are then reused 100 000 times to determine the PDF of the growth rate of the acoustic disturbances, a necessary quantity to estimate the risk factor of the mode. A discussion ensued to evaluate the uncertainty quantification approach adopted.

S. Richard, J. Lamouroux, A. Ndiaye, F. Nicoud

### Manufacturing Uncertainties for Acoustic Liners

The present chapter investigates an uncertainty quantification (UQ) approach for the simulations of aircraft noise (overall sound pressure level—OASPL) attenuation performed by acoustic liners with respect to manufacturing tolerances. They are devices typically installed inside engine nacelles of turbofan airliners to mitigate the noise produced by the engines. Aircraft noise pollution is harmful to people onboard but mostly to those on ground near airports and is constantly addressed by stringent ICAO standards updates. The targeted test case within the UMRIDA project is the mathematical representation of a real regional jet engine nacelle from Leonardo Finmeccanica production. An acoustic liner in the simplest form consists of a sandwich panel with a top perforated sheet, an interior honeycomb structure, and a rigid back plate, and at a minimum, it can be characterized by four main independent geometrical uncertainties due to manufacturing tolerances. The methodology proposed consists first of all in the execution of experimental tests, needed for the determination of a database of measurements. The database is then used to quantify the geometrical uncertainties of the acoustic panel through the application of a dedicated tool of modeFRONTIER software developed by ESTECO: the distribution fitting tool to find the statistical distribution which better fits the experimental data. Subsequently, in order to quantify accurately the performance distributions, a numerical model of the liner, provided by FNM, is integrated into modeFRONTIER, allowing the automatic execution of a series of computational acoustic simulations with MSC ACTRAN software developed by Free Field Technology. The results are therefore interpreted by modeFRONTIER, accordingly to the tools developed during UMRIDA project, to obtain the UQ of the acoustic performance following two criteria: highest accuracy and lowest number of sampling points. Since the number of uncertainties considered is not large, the use of a SS-ANOVA screening to determine the most important ones is not necessary. We will therefore apply a modified version of regression analysis, called adaptive sparse polynomial chaos methodology (Blatman and Sudret, Adaptive Sparse Polynomial Chaos Expansion Based on Least Angle Regession, 2010) [1], which aims not to reduce the number of uncertainties to apply polynomial chaos expansion to, but rather to reduce the number of the terms of the same polynomial chaos expansion to avoid overfitting. Thus, it is possible to perform an accurate UQ by a reduced number of sampling points (making the subsequent RDO more feasible), without completely discarding any of the uncertain parameters.

N. Magnino

### Manufacturing Uncertainties in High-Pressure Compressors

A method to deal with correlated manufacturing uncertainties based on the non-intrusive probabilistic collocation method and the principal component analysis is applied to a 1.5-stage high-pressure compressor. The uncertainties are defined based on a set of optical measurements, leading to realistic deformations. The parametric model, which is built based on the optical measurements and used to represent the blade geometry, allows the representation of the uncertainties by 15 correlated parameters. The results of the UQ computation are analysed in terms of computational cost and sensitivity of the quantities of interest with respect to the input uncertain parameters. It is shown that a level 1 sparse grid is sufficient to have a convergence of the two first statistical moments, which are the mean and the standard deviation and thus sufficient for the treatment of manufacturing uncertainties. Moreover, the sensitivity of the quantities of interest with respect to the input uncertainties are computed and compared with a Monte Carlo simulation found in the literature on the same test case. It is shown that the NIPColM coupled with the PCA allows reducing the computational cost by a factor 16 in comparison with the Monte Carlo simulation.

Rémy Nigro, Dirk Wunsch, Grégory Coussement, Charles Hirsch

### Formulations for Robust Design and Inverse Robust Design

In order to apply Robust Design Optimization to problems of industrial relevance, characterized by large number of variables and high computational effort, it is important to define the best strategy to solve every kind of problem. Different approaches are here presented, including multi-objective optimization and reliability optimization, based on exploitation of Polynomial Chaos Expansion for the quantification of percentiles. In addition, Tolerance Design or Inverse Robust Design methodology is presented, as an efficient approach to reduce excessive warranty costs in manufacturing process while keeping the expected quality level, by minimizing the standard deviation of the uncertain parameters.

Alberto Clarich, Rosario Russo

### Robust Design of Initial Boundary Value Problems

We study hyperbolic and incompletely parabolic systems with stochastic boundary and initial data. Estimates of the variance of the solution are presented both analytically and numerically. It is shown that one can reduce the variance for a given input, with a specific choice of boundary condition. The technique is applied to the Maxwell, Euler, and Navier–Stokes equations. Numerical calculations corroborate the theoretical conclusions.

Jan Nordström, Markus Wahlsten

### Robust Optimization with Gaussian Process Models

In this chapter, the application of the Gaussian regression models in the robust design and uncertainty quantification is demonstrated. The computationally effective approach based on the Kriging method and relative expected improvement concept is described in detail. The new sampling criterion is proposed which leads to localization of the robust optimum in a limited number of steps. The methodology is employed to the optimal design process of the intake channel of the small turboprop engine.

Krzysztof Marchlewski, Łukasz Łaniewski-Wołłk, Sławomir Kubacki, Jacek Szumbarski

A strategy for robust design optimization (RDO) is proposed, i.e., optimization under uncertainties reducing the variability of the system output with respect to the input uncertainties. This strategy relies on the non-intrusive probabilistic collocation method for the uncertainty propagation and a surrogate-assisted optimization strategy. In order to allow for RDO within reasonable turnaround times, a mixed Design of Experiments (DoE) is built, which comprises design variables and uncertainties as individual dimensions. This reduces the cost by one order of magnitude compared to an approach where each point in the DoE is run with a UQ simulation. The robust design optimization problem is formulated as a simultaneous maximization of the mean efficiency and minimization of standard deviations of efficiency and of other global output quantities at the example of the Rotor 37. Three designs on the chosen four-dimensional Pareto front are compared with the deterministic design. The reconstruction of PDFs of global output quantities visualizes their reduced standard deviation. Scaled sensitivity derivatives allow in a direct way to identify the uncertainties, which are responsible for an increase or decrease in sensitivity of output quantities, and they prove to be a very useful tool for the understanding of system dependencies. Full performance curves are run for the selected designs, and the optimal robust designs are discussed. The computational overhead of the presented robust design optimization varies between 1.4 and 1.9 times the computational cost of a deterministic optimization.

Rémy Nigro, Dirk Wunsch, Grégory Coussement, Charles Hirsch

### Robust Design Measures for Airfoil Shape Optimization

Two kinds of robustness measures are introduced and applied to design optimization of the UMRIDA BC-02 transonic airfoil test case under uncertainty. Robust design optimization (RDO) aims at minimizing the mean and standard deviation of the drag coefficient. Reliability-based design optimization (RBDO) targets minimizing the maximum drag coefficient. Both robustness measures are efficiently evaluated by using efficient sampling techniques assisted by a gradient-enhanced Kriging model. The airfoil is parameterized with 10 deterministic design variables, which are optimized by a gradient-free Subplex algorithm. The nominal airfoil geometry is assumed to be perturbed by a Gaussian random field which is parameterized by 10 independent variables through a truncated Karhunen–Loève expansion. Two operational parameters are also considered uncertain. The airfoil obtained by optimizing the two robustness measures has similar geometrical features and shows better performance in terms of the robustness measures than the initial and the deterministically designed airfoils. The strength and location of the shock wave of the robustly designed airfoils are shown to be less sensitive to random geometrical perturbations than the initial and deterministically designed airfoils.

Daigo Maruyama, Stefan Görtz, Dishi Liu

### Robust Design with MLMC

In this chapter, we present a robust optimization approach based on the combination of evolutionary algorithms and the Continuation Multi-Level Monte Carlo (C-MLMC) methodology, presented in Chap. https://doi.org/Continuation Multi-level Monte Carlo, to estimate robust designs, without relying on derivatives and meta-models. We present numerical studies for the 2D RAE-2822 transonic airfoil design affected by operating uncertainties. The performance of a robust optimal designs is compared to the deterministic optimal solutions to underline the improvements in robustness that can be achieved.

Michele Pisaroni, Fabio Nobile, Penelope Leyland

### Value-at-Risk and Conditional Value-at-Risk in Optimization Under Uncertainty

This work is related to the use of various risk measures in the context of robust- and reliability-based optimization. We start from the definition of risk measure and its formal setting, and then, we show how different risk functional definitions can lead to different approaches to the problem of optimization under uncertainty. In particular, the application of value-at-risk (VaR) and conditional value-at-risk (CVaR), also called quantiles and superquantiles, is here illustrated. These risk measures originated in the area of financial engineering, but they are very well and naturally suited to reliability-based design optimization problems and they represent a possible alternative to more traditional robust design approaches. We will then discuss the implementation of an efficient risk measure-based optimization algorithm based on the introduction of the weighted empirical cumulative distribution function (WECDF) and on the use of methods for changing the probability measure. Subsequently, we will discuss the problems related to the error in the estimation of the risk function and we will illustrate the “bootstrap” computational statistics technique to get an estimate of the standard error on VaR and CVaR. Finally, we will report some simple application examples of this approach to robust and reliability-based optimization.

Domenico Quagliarella

### Combination of Polynomial Chaos with Adjoint Formulations for Optimization Under Uncertainties

The main idea of this approach is to combine the non-intrusive polynomial chaos-based uncertainty quantification methods with adjoint formulations for optimization under uncertainties. Introducing uncertainties in a design process, the objective is also uncertain. 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 are obtained from the polynomial chaos expansion of the gradient of the objective. The proposed approach is applied to the optimal shape design of the transonic RAE2822 airfoil under uncertainties. The optimization procedure is performed using a Python-based optimizer SciPy which uses the sequential least square programming (SLSQP) algorithm.

Dinesh Kumar, Mehrdad Raisee, Chris Lacor

### Robust Multiphysics Optimization of Fan Blade

Fan blade is a complicated object, and obviously it is subjected to geometrical uncertainties from manufacture tolerances and other production deviations. In spite of all uncertainties, the fan blade should provide stable aerodynamic efficiency and strength properties. That is why it is considered to solve multidimensional and multidisciplinary optimization task (aerodynamics, strength, and flutter sensitivity) in robust statement under geometrical uncertainties. In the proposed test case, geometrical uncertainties from fan blade manufacture tolerances and deviations are considered. The probability density function (pdf) was obtained as a result of statistical operation upon the results of blades coordinate measurements. Approximately, 2500 fan blades were measured by means CMM process to reconstruct pdf for more than 40 geometrical uncertainties (there are blade thicknesses for different airfoil locations in several cross sections). CFD and FEM calculations were carried out in NUMECA FINE/Turbo and ANSYS software correspondingly. A surrogate model technique (the response surface and the Monte Carlo method implemented to RSM results) was applied for the uncertainty quantification and the robust optimization process for the task under consideration. In the present work, Approx software was used for surrogate model construction. The IOSO technology was employed as one of the robust optimization tools. This technology is also based on a widespread application of the response surface technique. As a result, robust optimal solutions (the Pareto set) for all four considered criteria were obtained. Probabilistic criteria were assessed based on the results obtained. Robust optimization results were compared with deterministic optimization results.

K. Vinogradov, G. Kretinin, I. Leshenko

### UQ Sensitivity Analysis and Robust Design Optimization of a Supersonic Natural Laminar Flow Wing-Body

The robust design optimization of a natural laminar flow wing for a supersonic business jet is the objective of the reported research work. In particular, the pursued goal is to obtain a wing shape whose performance is influenced as least as possible by geometrical uncertainties. The starting point is a supersonic business jet wing-body that was already optimized for natural laminar flow using a deterministic approach within the EU funded SUPERTRAC Project. This configuration was firstly analyzed to identify the main dependencies, and interactions of the parameters that describe the uncertainty sources in the robust design problem, and in a second step, a robust design optimization algorithm was used to obtain an optimal solution less sensible to geometrical perturbation with respect to the baseline. The optimization algorithm is an evolutionary one and its principal requirement is the resilience to noise in the objective function values. The objective function that defines the goal of the optimization is based on special risk functions, namely value-at-risk (VaR) and conditional value-at-risk (CVaR), that are widely used in financial engineering community and that offer interesting advantages with respect to more classical approaches based on expectation or variance risk functions. The initial part of the optimization task is based on VaR risk function computed using a very coarse sample set. In a second step, the CVaR function, computed over a finer sample is used to further improve the results. The confidence intervals of VaR and CVaR estimations are computed using the bootstrap computational statistics technique. The results illustrate the feasibility of such a robust optimization approach for the application to industrial class robust design optimization techniques.

Domenico Quagliarella, Emiliano Iuliano

### Robust Compressor Optimization by Evolutionary Algorithms

Evolutionary algorithms are ideal candidates for robust design optimization as they are robust in two different ways. First, their search is robust as the risk of premature convergence is low. Second, their population-based search allows for direct (implicit) extraction of robustness information from the population. This avoids additional computational effort for (explicit) uncertainty quantification. In this work, a standard implicit and a novel approach for robust design optimization based on the CMA-ES are presented. Both approaches are first tested on a test function and then applied to the robust design optimization of a compressor impeller.

Dirk Büche

### Robust Optimization of Acoustic Liners

The present chapter documents the innovative methodology for the automatic multi-objective robust design optimization of engine nacelle acoustic liners. These devices are typically installed inside nacelles of turbofan aircraft to attenuate the noise produced by the engines as aircraft noise pollution is harmful to people onboard but mostly to those on ground near airports and is regulated by ICAO standards constantly revised. The targeted test case analyzed within the UMRIDA project is the mathematical model of a real regional jet engine nacelle selected from Leonardo Finmeccanica vast production. This liner in the simplest form consists of a sandwich panel with a top perforated sheet, an interior honeycomb structure, and a rigid back plate, and at a minimum, it can be characterized by four main independent geometrical uncertainties due to manufacturing tolerances. The methodology proposed consists first of all in the execution of experimental tests, needed for the determination of a database of measurements, starting-point to quantify the geometrical uncertainties of the acoustic panels, through the application of a dedicated tool of modeFRONTIER software from ESTECO, i.e., the distribution fitting tool. This allows to find the statistical distribution which better fits the experimental data. Then, to accurately quantify the performance distributions, a numerical model of the liner incorporating proprietary semi-empirical impedance model is integrated in modeFRONTIER workflow, allowing the automatic execution of a series of acoustic simulations with MSC ACTRAN software developed by Free Field Technology. The results are processed with a Python script to have the PLTViewer extract the concise performance index overall sound pressure level (OASPL). Since the number of uncertainties considered is not large, using the efficient adaptive sparse Polynomial Chaos methodology, it is possible to perform an accurate UQ keeping the number of sampling points low and increasing accuracy by avoiding overfitting. We based the final robust design optimization on efficient single-objective reliability-based formulation, using two optimization algorithms SIMPLEX and MOGAII to compare against in terms of effectiveness versus computational cost.

N. Magnino

### Application of Robust Design Methodologies to Falcon

In this section, we give the description both of deterministic and robust optimization problems in order to find the “best” wing shape for the IC-03 Falcon (UMRIDA Project: Test Case Description Innovative Database for UQ and RDM: IC-03 [1]). Comparison of results and discussion about impact of robustness conclude this work.

G. Rogé, X. Loyatho

### Uncertainties Identification and Quantification

Uncertainties are present in any engineering task. If the predicted result of a numerical simulation agrees with test results or operational data, then uncertainties are typically ignored or simply not even recognized. In case of disagreement, they often become of key interest. This section describes the different types of uncertainties in a product development process and how to handle them in an uncertainty quantification and robust design optimization.

Dirk Büche, Sönke Klostermann, Martin Schmelzer

### Polynomial Chaos and Collocation Methods and Their Range of Applicability

In this chapter the different polynomial chaos and stochastic collocation methodologies used within the UMRIDA project are compared. Guidelines for their use and applicability are formulated.

Chris Lacor, Éric Savin

### Surrogate Model-Based Approaches to UQ and Their Range of Applicability

Efficient surrogate modeling approaches are presented in the context of robust design. The type of surrogate model, and the number and distribution of the sample points are discussed. The test case is the UMRIDA BC-02 airfoil with two uncertain operational and 10 uncertain geometrical parameters. Statistics of the quantity of interest (QoI) are evaluated based on surrogate models of the QoI. Here, the QoI is lift coefficient or drag coefficient. Both Kriging and gradient-enhanced Kriging (GEK) surrogate models are considered. The surrogate models are generated based on scattered samples of QoI. A Sobol sequence is used to generate samples with a low-discrepancy distribution, for which the QoI and its gradients with respect to the uncertain parameters are evaluated with a Computational Fluid Dynamics (CFD) solver and its adjoint counterpart. The mean and standard deviation of the QoI are efficiently evaluated by using GEK with more than 12 samples for large numbers of uncertainty parameters more than 10. The accuracy of the surrogate models is also investigated in terms of the derived robust design solutions. The error dispersion of the stochastic objective function due to the sample distribution affects the optimal solution. Thirty sample points are necessary to reduce the error dispersion to within one drag count, which is considered to be on the same order of magnitude as the epistemic uncertainty due to CFD errors.

Daigo Maruyama, Dishi Liu, Stefan Görtz

### Monte Carlo-Based and Sampling-Based Methods and Their Range of Applicability

The present section will focus on the applicability issues of Monte Carlo-based methods, as well as those methods based on sampling techniques. Special focus will be put on the Multi-Level Monte Carlo method and the two implementations developed during the UMRIDA project, namely the Continuous MLMC and MLMC. All named methods have been described in the above sections of this book.

Robin Schmidt, Matthias Voigt, Michele Pisaroni, Fabio Nobile, Penelope Leyland, Jordi Pons-Prats, Gabriel Bugeda

### Introduction to Intrusive Perturbation Methods and Their Range of Applicability

Two Taylor-based perturbation methods for generating surrogate models are examined: Tangent-on-Tangent differentiation, Tangent-on-Reverse differentiation. Their range is evaluated in terms of computational cost.

Alain Dervieux

### Use of Open Source UQ Libraries

For this section, we aim to demonstrate the use of open source software for aircraft preliminary design at the example of a wing configuration robust design optimization. On airfoil level, we demonstrate the meta-modelling of the relation between the lift and drag coefficients and the parameters used to describe the airfoil shape. The meta-model is built with the open source library OpenTURNS that is dedicated to the treatment of uncertainty.

Sönke Klostermann, R. Lebrun

### Uncertainty Quantification in an Engineering Design Software System

The application of uncertainty quantification (UQ) techniques in the daily engineering practice requires a toolchain that can be used intuitively by a wide range of design engineers. Ideally, this toolchain does not require detailed knowledge of the underlying UQ methods and is highly automated to ease the design tasks of the engineer using it. Such an automated chain is proposed in FINETM, where the user input is limited to the selection of the input uncertainties and a decision on the needed output. The steps of a fully automated UQ chain from simulation set-up, geometry modification, re-meshing and post-processing are detailed in this chapter.

Dirk Wunsch, Rémy Nigro, Grégory Coussement, Charles Hirsch

### Use of Automatic Differentiation Tools at the Example of TAPENADE

This chapter gives details of the implementation of automatic differentiation for differentiating the software corresponding to an equation-constrained functional. The reference automatic differentiation tool considered is the code-to-code differentiator TAPENADE.

Alain Dervieux

### Formulations for Robust Design and Inverse Robust Design

The classical approach to solve a Robust Design Optimization problem is based on the definition of a multi-objective optimization problem, consisting generally in the optimization of the mean value of the performances and on the minimization of their standard deviation.

Alberto Clarich, Rosario Russo

### Use of RD in Multiphysics Applications

The present chapter aims at providing first guidelines for the application of uncertainty quantification method to robust design in industrial problems exhibiting complex physics. The first part of this chapter is dedicated to the exploitation of UQ-based techniques to predict the risk of occurrence of combustion instabilities in aeronautic engines. Combustion is by nature a multiphysics problem including fluid mechanics, chemistry and heat transfer. Including UQ in such complex applications requires to reduce the problem solve to reduced order models accounting for the main phenomena implying a limited number of parameters. Within the project, reduced order models have been developed at CERFACS for combustion–acoustic interactions, allowing the introduction of UQ in the field of combustion instabilities. This chapter therefore focuses on this specific application and paves the way towards the use of UQ techniques for combustor robust design. A second part is dedicated to efficient UQ and RDO for aero-engine acoustic liners. Preliminary designing of acoustic liners is by nature a multiobjective problem due to different flight conditions prescribed by certification authorities. The number of uncertainties was limited to the geometrical variables that most influence the acoustic impedance according to the proprietary semi-empiric model exploited in the mathematical representation of the nacelle modelled into finite element code MSC Actran, which is used to conduct computational aero-acoustic simulations. The Adaptive Sparse Polynomial Chaos method has been used for an efficient UQ, allowing the reduction of the terms, rather than the reduction of the variables, retaining accuracy by avoiding overfitting. A single-objective reliability-based formulation is successfully applied with SIMPLEX and MOGAII algorithm for efficient RDO.

S. Richard, N. Magnino

### Geometrical Uncertainties—Accuracy of Parametrization and Its Influence on UQ and RDO Results

A large number of geometrical uncertainties of a transonic RAE2822 airfoil are parameterized by a truncated Karhunen–Loève expansion (KLE), and the influence of the truncation on the statistics of aerodynamic quantities is investigated both in terms of efficiency and accuracy. Direct integration of a very large number of quasi-Monte Carlo samples computed with CFD is used to compute the mean and standard deviation for different levels of truncation, i.e., for different numbers of uncertain parameters. We show that a parameterization based on a well-truncated KLE can efficiently reduce the number of geometrical uncertainties while maintaining accuracy. Excessive truncation will not improve the efficiency of surrogate-based statistics integration and will inevitably lead to a loss of accuracy of the estimated statistics. This is attributed to the use of a gradient-enhanced surrogate model that employs an adjoint flow solver to compute the gradient of the aerodynamic coefficients with respect to the uncertain parameters. All partial gradients can be computed at the cost of one adjoint solution; i.e., the cost of computing all partial gradients is independent of the number of uncertain parameters. It is also shown that a loss of accuracy due to an improper truncation may influence the results of robust design optimization.

Dishi Liu, Daigo Maruyama, Stefan Görtz

### Analysis and Interpretation of Probabilistic Simulation Output

Obtaining the input/output cumulative distribution function (CDF) or probability density function (PDF) would be the main goal when dealing with uncertain input parameters.

Alberto Clarich, Rosario Russo

### Summary of UMRIDA Best Practices

Uncertainty quantification (UQ) is becoming a strategic step in the design phase. Robust Design Optimization (RDO) is the following step. The Technological Readiness Level (TRL) of intrusive and non-intrusive methodologies is increasing rapidly, although several limitations remain. Nowadays, UQ is a major trend in research, because there is a lot of room for improvement.

Jordi Pons-Prats, Gabriel Bugeda

### Project Summary and Outlook

The UMRIDA project, on Uncertainty Management for Robust Industrial Design in Aeronautics, addressed major new developments in the new scientific area of uncertainty management and quantification (UM and UQ) and robust design methodologies (RDM), which are needed to bridge the gap toward industrial readiness of these methods. For this reason, research was performed in both UQ and RDM and developing methods to handle a large number of simultaneous uncertainties including generalized geometrical uncertainties within a quantifiable objective of a turnaround time acceptable for industrial readiness. A series of different methodologies was investigated and successfully assessed on the unique database with prescribed uncertainties, which was built from industrial challenges provided by the project partners. The UMRIDA project allowed to raise the Technology Readiness Level for various methodologies for UQ and RDO from a basic level 2–3 to an industrial applied level of 5–6, demonstrated by the application of the developed methods to industrial challenges form the UMRIDA database while respecting the quantifiable object defined. Finally, as an outcome of the project and final workshop, future challenges are identified.

Charles Hirsch, Dirk Wunsch