The framework for the generation of a computationally efficient approximation of aero- dynamic loads determined from expensive high-fidelity calculations consists of the fol- lowing steps:
1. The independent variables and their range are specified, and initial sampling is used to begin the procedure and to provide a quick overview of aerodynamic data throughout the parameter space. Aerodynamic data are calculated at preset initial sample points using aerodynamic models appropriate for the given task. 2. A surrogate model based on kriging interpolation fitting data in the form of
input/output combinations is generated.
3. The parameter space is iteratively refined by adding sample points at untried locations to improve the accuracy and verify the robustness of the surrogate model.
4. If aerodynamic data have been obtained using different fidelity models, data fusion is then used to combine the low- and high-fidelity predictions in one single dataset of forces and moments.
2.3.1 Initial Sampling
The task of the initial sampling is to provide an informative picture of the function at minimum cost [88]. When confronted with a deterministic computer simulation as opposed to wind-tunnel or flight tests, a given set of input parameters always produces the same aerodynamic data. Without information on the function, a design optimal distribution method can only apply space-filling sampling, in the sense that all areas of the parameter space are sampled. Several space-filling methods requiring only infor- mation on the domain are available in the literature, such as Monte Carlo [89], latin hypercube sampling [90] and Sobol [91]. A major disadvantage of space-filling methods is that samples are randomly selected, can cluster together and each high-fidelity sim- ulation may not provide significant additional information. To overcome the potential lack of uniformity, optimal latin hypercube sampling [92] ensures a more uniform design of experiment obtained by optimizing a spreading criteria, e.g. minimum distance or correlation, of the sample points. However, sampling methods which include informa- tion on the full-order function in sample distribution are preferable. These methods are named a posteriori sequential sampling methods, as opposed to a priori sampling used for the space-filling methods. The set of sample points is iteratively refined and an additional high-fidelity simulation is run for a combination of input parameters in which the model exhibits maximum error. The drawback is that the high-fidelity sim- ulations are launched one at a time, whereas they would be launched in parallel with an a priori method.
The a priori approach is first considered to initialize the procedure. Each three- parameter sub-table defines a three-dimensional domain. An initial set of ten sample points is generated as follows. Eight samples are placed at the vertices of the parameter space, and two sample points are located within the parameter space, typically at the highest value of the angle of attack and for a given Mach number. This choice is sound because it avoids the need for extrapolation, and high-fidelity simulations are located in regions where aerodynamic loads are likely to exhibit non-linearities due to vortical flow developments, compressible effects and their mutual interaction. The simulations at the ten sample points provide an initial view of the behaviour of the aerodynamic data. A sequential sampling approach is then considered to iteratively refine the parameter space and to verify the accuracy of the approximation model when including additional data.
2.3.2 Kriging Interpolation
Kriging interpolation is used to approximate non-linear multi-dimensional deterministic functions by interpolating available sampled data, typically, from a full-order simula- tion. It is a parametric approach which presumes that the global functional form of the relationship between the response and the design variables is known.Once constructed,
kriging is a computationally cheap model to be used in place of the expensive full- order simulation for prediction of the function at untried points. Kriging has been successfully applied in different areas and, in particular, to problems involving fixed- and rotary-wing aerodynamics [17, 18].
Here, the mathematical formulation of the aerodynamic coefficients in Eqs. (2.2) and (2.3) represents the function to be approximated with kriging interpolation. With suitable assumptions, the 4 +Nc aerodynamic terms have been expressed as a function
of three input parameters. To illustrate, consider the baseline table with nα·nM·nβ
entries and the requirement for a high-fidelity representation of the static aerodynamic force and moment coefficients, Ci0 fori=L, D, m, Y, l and n. From a small ensemble
of sample points, kriging interpolation is used to predict the aerodynamic data at the nα·nM·nβflight conditions. For convenience, letybe a single scalar function represen-
tative of each aerodynamic coefficient in turn,y=Ci. Assume a given set ofnspnumer-
ical samples, [x1, . . . ,xnsp]T, where xi = [αi, Mi, βi]T is a vector of input parameters,
and the corresponding full-order aerodynamic coefficients, yx = [y(x1), . . . , y(xnsp)]T.
Initial samples are selected according to the above guidelines. In kriging, the unknown function of interesty is assumed to be a realization of a stochastic process
b
y(x) = f(x) + Z(x) (2.5) The first term is a low-order regression function (constant, linear or quadratic) and the second term is a stochastic process, assumed to be Gaussian and with variance σ2. The regression model,f(x), realizes a globally valid trend function, and theZ(x) adjusts the prediction for local deviations fromf(x), and guarantees that the kriging predictor by gives the exact value of y at a sample location. The assumption that the system response in Eq. (2.5) is a random process is made because the deviations from the regression model can resemble the realization of a stochastic process [15]. The covariance matrix ofZ(x) is a measure of how strongly correlated two points are, and is equal to the variance σ2 multiplied by the correlation matrix, R. The correlation matrix of the samples has dimensionsnsp × nsp, and each element is given by
Rij(p,xi,xj) = Y k scfpk, x(k)i − x (k) j (2.6)
where scf is a user defined spatial correlation function, and x(k)i denotes the k-th component of the i-th sample point. This matrix is dense symmetric positive definite with diagonal elements equal to one, and becomes ill-conditioned when samples are too close together. The order of the kriging correlation matrix depends only on the number of samples considered, nsp, and not on the dimension of the input vector, in
this case nα·nM·nβ. Several correlation functions are available, such as exponential,
phenomenon being approximated [86]. For a continuously differentiable problem, the spline function should be considered because it has a parabolic behaviour near the origin. To model physical phenomena, which usually have a linear behaviour near the origin, the use of exponential or linear functions was suggested [93]. The exponential is the most common correlation function, which is adapted to a wide range of physical applications. Depending on the correlation function selected and for a large dimension of the input vector, x, the corresponding correlation matrices can be ill-conditioned, which is an issue when using kriging for high-dimensional interpolation. In this study, the correlation function used is the exponential. The regression model is formulated as a low-order function,f(x) =f(x)T β, with the two vectors having dimension equal to the number of basis functions of the selected polynomial,Nbasis. Here, the regression model
was taken as a linear function and, for the example illustrated,f(x) =1, α, M, βT. A matrixF(x), of dimensionnsp×Nbasis, is constructed from the vectorsf(x), where the
i-th row corresponds to the evaluation of the basis functions at thei-th sample input. With the generalized least-squares estimates of β and σ, denoted by βb and σb, the unknown correlation parameterspkare found by maximizing a likelihood function [16], which represents the probability that the stochastic process Z(x) produced the value of the aerodynamic coefficients at the sampled data points. The estimatedpkrepresent
the fitting parameters that are most consistent with the sampled data. The kriging interpolation of the functiony(x) is given as
b
y(x) = f(x)T βb + r(x)T R−1
(yx − Fβb) (2.7)
where the correlation vector, written as
r(x) =R11(p,x1,x), . . . , Rnsp1(p,xnsp,x) T
(2.8) represents the correlation between the provided set of sampled points [x1, . . . ,xnsp]T
and an arbitrary unsampled locationx in the parameter space. The parameters of the kriging model are determined from a small ensemble of expensive numerical simulations, likely obtained from a CFD solver, and the system response is approximated at an arbitrary unsampled location not included in the initial set at the expense of two scalar products on f(x) and r(x), as shown in Eq. (2.7). A Matlab toolbox implements the kriging interpolation [86], and is freely available 1.
2.3.3 Iterative Sampling
The quality of the kriging interpolation depends on the number of sample points and their location, which is case-dependent. For a systematic methodology to be efficient and reduce the uncertainty in the prediction of the full-order function, there is a need
1
to minimize a suitable figure of merit. Two methodologies are currently adopted, based on the root Mean Squared Error (MSE) and Expected Improvement Function (EIF). The goal of both methodologies is to improve the accuracy of the kriging model at untried sample points, adding additional samples until a criterion for termination is fulfilled.
The root Mean Squared Error, ϕ, is referred to as the standard error of the kriging model and is a measure of uncertainty in the prediction. It is evaluated as
ϕ2(x) = σ2 1 − r(x)T R−1
r(x) + uT FTR−1
F−1 u (2.9) with the vector u(x) = FTR−1
r(x) − f(x) and the process variance σ2 = 1 nsp yx − Fβb T R−1 yx − Fβb (2.10) By construction, the MSE is zero at a sample point and increases as the distance between samples increases, with its maximum value being σ. To further improve the kriging interpolation, an additional high-fidelity simulation is run at the untried sample point x where the MSE is maximum. This criterion seeks for a global approximation of the exact function because it is driven by the weighted distance correlation for the error terms.
The Expected Improvement Function can be used to concentrate new sample points around the global minimum or maximum. This is very useful when, for instance, the maximum lift coefficient is of interest. The kriging predictor in Eq. (2.5) can be interpreted as a random variable with mean given by the predictor and variance given by the mean standard error. A probability can then be computed so that the system response at any point will fall below (or above) the current minimum (or maximum). The kriging model is iteratively refined placing an additional high-fidelity simulation at the untried sample point where the EIF has a maximum. This method is suited for searches of the global maxima or minima but is unable to find local maxima or minima. A good practice is the use of a combined global and local search [94].
As a high-fidelity evaluation of the system response is obtained in a new suitable sample point, the procedure is repeated until the maximum error in the prediction is below a threshold. The number of high-fidelity simulations is likely to be limited by constraints on computational time and resources, and typically this requirement is regarded as an additional criterion to stop the sampling procedure.
2.3.4 Data Fusion
The aerodynamic coefficients can be obtained using different aerodynamic sources. Data fusion combines aerodynamic predictions from different sources. Consider that data are available from two sources, which are, respectively, cheap (low-fidelity) and
expensive (high-fidelity) to evaluate. The cheap samples are considered to provide information at least about the trend of the target function, whereas the expensive samples give quantitative information. The cheap estimates are usually available in more locations than the expensive ones. A Kriging function bη is calculated from the samples of the cheap aerodynamic evaluations. This Kriging function is then evaluated at the locations at which expensive predictions are available, ηb(xi). The vector of
the input parameters at the expensive samples, xi, is augmented by the evaluation
of the kriging function for the cheap samples. In the case of the baseline table, the augmented vector would have dimension four and be xaugi = [αi, Mi, βi,bη(xi)]T, with
corresponding aerodynamic coefficients yi = y(xi) for each of the nsp sample points.
A Kriging function is calculated for these augmented samples,ηb(xaugi ), with the extra component bringing information to the correlation calculation from the cheap samples.