Stefan Görtz

Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

The efficiency of building a surrogate model for the output of a computer code can be dramatically improved via variable-fidelity surrogate modeling techniques. In this article, a hierarchical kriging model is proposed and used for variable-fidelity surrogate modeling problems. Here, hierarchical kriging refers to a surrogate model of a highfidelity function that uses a kriging model of a sampled lower-fidelity function as a model trend. As a consequence, the variation in the lower-fidelity data is mapped to the high-fidelity data, and a more accurate surrogate model for the high-fidelity function is obtained. A self-contained derivation of the hierarchical kriging model is presented. The proposed method is demonstrated with an analytical example and used for modeling the aerodynamic data of an RAE 2822 airfoil and an industrial transport aircraft configuration. The numerical examples show that it is efficient, accurate, and robust. It is also observed that hierarchical kriging provides a more reasonable mean-squared-error estimation than traditional cokriging. It can be applied to the efficient aerodynamic analysis and shape optimization of aircraft or any other research areas where computer codes of varying fidelity are in use.

Alternative Cokriging Method for Variable-Fidelity Surrogate Modeling

Surrogate modeling plays an increasingly important role in different areas of aerospace engineering, such as erodynamic shape optimization, aerodynamic data production, structural design, and multidisciplinary design optimization of aircraft or spacecraft. Cokriging provides an attractive alternative approach to conventional kriging to improve the efficiency of building a surrogate model. It was initially proposed and applied in the geostatistics community for the enhanced prediction of less intensively sampled primary variables of interest with the assistance of intensively sampled auxiliary variables. As the underlying theory of cokriging is that of two-variable or multivariable kriging, it can be regarded as a general extension of (one-variable) kriging to a model that is assisted by auxiliary variables or secondary information. In an attempt to apply cokriging to the surrogate modeling problems associated with deterministic computer experiments, this article is motivated by the development of an alternative cokriging method to address the challenge related to the construction of the covariance matrix of cokriging [7]. Earlier work done by other authors related to this study can be found in the statistical community. For example, Kennedy and O’Hagan (KOH) proposed an autoregressive model to calculate the covariances and crosscovariances in the covariance matrix and developed a Bayesian approach to predict the output from an expensive high-fidelity simulation code with the assistance of lower-fidelity simulation codes. This Bayesian approach is identical to a form of cokriging suitable for computer experiments. Later, Qian andWu proposed a similar method, in which a random function (Gaussian process model) was used to replace the constant multiplicative factor of KOH’s method to account for the nonlinear scale change. KOH’s method was applied to multifidelity analysis and design optimization in the context of aerospace engineering by Forrester et al. and Kuya et al. More recently, Zimmerman and Han proposed a cokriging method with simplified cross-correlation estimation. In this article, we propose an alternative approach for the construction of the cokriging covariance matrix and develop a more practical cokriging method in the context of surrogate-based analysis and optimization. The developed cokriging method is validated against an analytical problem and applied to construct global approximation models of the aerodynamic coefficients as well as the drag polar of an RAE 2822 airfoil.

A Variable-Fidelity Modeling Method for Aero-Loads Prediction

A Variable-Fidelity Modeling (VFM) method has been developed as an efficient and accurate aerodynamic data modeling strategy. In this approach, a set of CFD methods with varying degrees of fidelity and computational expense is exercised to reduce the number of expensive high-fidelity computations. Kriging-based bridge functions are constructed to match the low- and high fidelity CFD data. The method is demonstrated by constructing a global approximation model of the aerodynamic coefficients of an RAE 2822 airfoil based on sampled data. The model is adaptively refined by inserting additional samples. It is shown that the method is promising for efficiently generating accurate aerodynamic models that can be used for the rapid prediction of aerodynamic data across the flight envelope.

Non-linear reduced order models for steady aerodynamics

Abstract A reduced order modelling approach for predicting steady aerodynamic flows and loads data based on Computational Fluid Dynamics (CFD) and global Proper Orthogonal Decomposition (POD), that is, POD for multiple different variables of interest simultaneously, is presented. A suitable data transformation for obtaining problemadapted global basis modes is introduced. Model order reduction is achieved by parameter space sampling, reduced solution space representation via global POD and restriction of a CFD flow solver to the reduced POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimization problem. Methods for obtaining feasible starting solutions for the optimization procedure are discussed. The method is demonstrated by computing reduced-order solutions to the compressible Euler equations for the NACA 0012 airfoil based on two different snapshot sets; one in the subsonic and one in the transonic flow regime, where shocks occur. Results are compared with those obtained by POD-based interpolation using Kriging and the Thin Plate Spline method (TPS).

Improved Extrapolation of Steady Turbulent Aerodynamics using a Non-Linear POD-based Reduced Order Model

A reduced-order modelling (ROM) approach for predicting steady, turbulent aerodynamic flows based on computational fluid dynamics (CFD) and proper orthogonal decomposition (POD) is presented. Model-order reduction is achieved by parameter space sampling, solution space representation via POD and restriction of a CFD solver to the POD subspace. Solving the governing equations of fluid dynamics is replaced by solving a non-linear least-squares optimisation problem. The method will be referred to as LSQ-ROM method. Two approaches of extracting POD basis information from CFD snapshot data are discussed: POD of the full state vector (global POD) and POD of each of the partial states separately (variable-by-variable POD). The method at hand is demonstrated for a 2D aerofoil (NACA 64A010) as well as for a complete industrial aircraft configuration (NASA Common Research Model) in the transonic flow regime by computing ROMs of the compressible Reynolds-averaged Navier-Stokes equations, pursuing both the global and the variable-by-variable POD approach. The LSQ-ROM approach is tried for extrapolatory flow conditions. Results are juxtaposed with those obtained by POD-based extrapolation using Kriging and the radial basis functions spline method. As a reference, the full-order CFD solutions are considered. For the industrial aircraft configuration, the cost of computing the reduced-order solution is shown to be two orders of magnitude lower than that of computing the reference CFD solution.

Interpolation-based reduced-order modelling for steady transonic flows via manifold learning

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inflow conditions to obtain an accurate prediction of shocks. The reduced-order representation of the data is derived using the Isomap ML method, which is applied to a set of sampled computational fluid dynamics (CFD) data. In order to develop a ROM that has the ability to predict approximate CFD solutions at untried parameter combinations, Isomap is coupled with an interpolation method to capture the variations in parameters like the angle of attack or the Mach number. Furthermore, an approximate local inverse mapping from the reduced-order representation to the full CFD solution space is introduced. The proposed ROM, called Isomap+I, is applied to the two-dimensional NACA 64A010 airfoil and to the 3D LANN wing. The results are compared to those obtained by proper orthogonal decomposition plus interpolation (POD+I) and to the full-order CFD model.

Reduced-Order Modeling of Steady Flows Subject to Aerodynamic Constraints

A novel reduced-order modeling method based on proper orthogonal decomposition for predicting steady, turbulent flows subject to aerodynamic constraints is introduced. Model-order reduction is achieved by replacing the governing equations of computational fluid dynamics with a nonlinear weighted least-squares optimization problem, which aims at finding the flow solution restricted to the low-order proper orthogonal decomposition subspace that features the smallest possible computational fluid dynamics residual. As a second and new ingredient, aerodynamic constraints are added to the nonlinear least-squares problem. It is demonstrated that the constrained nonlinear least-squares problem can be solved almost as efficiently as its unconstrained counterpart and outperforms all alternative approaches known to the authors. The method is applied to data fusion, seeking to combine the use of computational fluid dynamics with wind-tunnel or flight testing to improve the prediction of aerodynamic loads. It is also ...

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.

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.

Multi-Level MDO of a Long-Range Transport Aircraft Using a Distributed Analysis Framework

DLRs work on developing a distributed collaborative MDO environment is presented. A multi-level Approach combining high-fidelity MDA for aerodynamics and structures with conceptual aircraft design methods is employed. Configuration-specific sizing loads are evaluated and used for sizing the structure. A gradient-free optimization algorithm is used to optimize the fuel burn of a generic long-range wide-body transport aircraft configuration with 9 shape parameters. The results show a truly multidisciplinary improvement of the modified design. The result of a gradient-free high-fidelity MDO with preselected load cases and five shape parameters is also presented, comparing a full mission analysis with results for the Breguet range equation.