Keywords

Introduction

Aerodynamic shape optimization by means of automatic tools is an industrial relevant field that has to breast several challenges. Some of these challenges are: how to handle deformations in certain regions (such as intersections between wing and fuselage or pylon/nacelle), how to reduce the number of CFD runs required for performing aerodynamic design optimization or how to tackle integrated components. Furthermore, surrogate-based optimization methods require several barriers to be broken when applied to complex configurations, such as the called “curse of dimensionality”, the ability of surrogates to handle a high number of design parameters, efficient constraints handling (Parr et al. 2010), and the proper exploration and exploitation of the whole design space.

In the case of surrogate-based optimization (SBO) methods, the surrogate prediction is also highly influenced by training set size. A huge training set with a proper design space distribution ensures reaching a global optimum, but requires a vast computational cost to be built. On the other hand, a small training set is fast to be built but the accuracy is not enough for optimization purposes. A solution to this issue must be found for the suitable implementation of this method in the aeronautical industry.

In this work, Support Vector Machines (SVM) combined with Evolutionary Algorithms (EAs) and an adaptive sampling method, called Intelligent Estimation Search with Sequential Learning (IES-SL), is proposed. The approach is applied to the multipoint optimization of one typical test case, i.e., the transonic RAE 2822 airfoil. The aim of this work is to provide an analysis of the training set size influence in the behavior of the IES-SL approach proposed.

This paper is structured as follows. In Section “Literature Review”, a review of the recent research efforts in SBO applied to aircraft design is presented. Section “Surrogate-Based Optimization Strategy” presents the applied SBO strategy and Section “Numerical Results” collects the study results. Finally, the conclusions extracted from the results are summarized in Section “Conclusions”.

Literature Review

Recent Research Efforts in SBO Applied to Aircraft Design

Some recent efforts in SBO for aerodynamic shape design includes, e.g., a physics-based surrogates applied to the drag minimization of NACA 0012 and RAE 2822 airfoils in transonic flow conditions (Leifsson et al. 2014). In this work, the geometries were parameterized using PARSEC involving 5–10 design parameters. SBO strategies were applied for the drag minimization of the NLF0416 airfoil using 10 design variables (Li et al. 2001). Variable-fidelity computational fluid dynamics (CFD) combined with shape optimization strategy was applied to the optimization of a transonic airfoil parameterized by the NACA 4-digit definition with three design variables (Koziel and Leifsson 2013).

A surrogate based on proper orthogonal decomposition (POD) applied to the aerodynamic shape optimization of an airfoil is presented by Iuliano (Iuliano and Quagliarella 2013). The geometry was parameterized with 16 design variables defined with the CST method. An approach based on a combination of a genetic algorithm and an artificial neural network is presented by Jahangirian and Shahrokhi (2011). This approach was applied to the shape optimization of an airfoil, which was parameterized by a modified PARSEC involving 10 design variables.

Most of the SBO applications in aerodynamic shape optimization involve two-dimensional configurations, where the number of design variables is usually limited. Nevertheless, some applications to three-dimensional configurations can be found in literature. An investigation about SBO applied to a wing parameterized with 11 design variables was undertaken by Keane (2003). A multi-fidelity surrogate model applied to a three-dimensional wing optimization was addressed by Likeng and Zhenghong (2012). In this case, the design parameters were a combination of 12 variables using the CST method for three wing sections (root, hink and wing tip). Lukaczyk et al. (2014), proposed a method based on an active subspace for effectively searching the whole design space. The method is applied to the optimization of the ONERA M6 transonic wing, which was parameterized with 50 FFD design variables. The aim was to discover a low-dimensional linear subspace of the input space that explained the majority of the variability in the drag and lift coefficients. An SBO application to the aerodynamic shape design of a wing parameterized with volumetric non-uniform rational B-splines (NURBS) was presented by current authors (Andrés-Pérez and Iuliano 2015). Also, in (González-Juárez et al. 2015; Andrés-Pérez et al. 2016) current authors present an application study about the influence of number and location of the design parameters in the behaviour of the IES-SL method applied to the aerodynamic shape optimization. The selected geometries, RAE 2822 airfoil and DPW-w1 wing, were parameterized with volumetric NURBS.

This work is within the aerodynamic shape design and optimization research line of INTA’s Fluid Dynamics Branch.

Surrogate-Based Optimization Strategy

This section introduces each of the components of the SBO approach applied in this study: geometry parameterization through volumetric NURBS, Evolutionary Algorithms (EAs), Support Vector Machines for Regression (SVR) and the Intelligent Estimation Search with Sequential Learning (IES-SL) as the strategy for adaptive sampling focused on optimization.

Geometry Parameterization

Parameterization is a crucial step in an aerodynamic design optimization problem. NURBS have demonstrated to be able to accurately represent a large family of geometries. In aerodynamic design, NURBS provide smooth surfaces while maintaining some deformation locality (Mousavi et al. 2007). In addition, the optimized surface at the end of the optimization process has the correct format to feed directly the CAD and grid generation applications. However, the use of surface NURBS can be impractical, because very frequently requires the additional effort to develop a surface representation that fits the original geometry, with an appropriated arrange of control points for the optimization. An alternative approach is to envelop the geometry in a volumetric NURBS (Martin et al. 2013), which maintain the deformation properties of a conventional 2-dimensional surface, but with the advantage that control points can be set up arbitrarily.

From a mathematical point of view, NURBS surfaces are defined as the tensor product of three NURBS curves, defining a volumetric region, where the deformation is governed by the movement of control points:

$$ S(\xi ,\eta ,\mu ) = \frac{{\sum\nolimits_{i}^{I} {\sum\nolimits_{j}^{J} {\sum\nolimits_{k}^{K} {U_{i,n} (\xi )V_{i,n} (\eta )W(\mu )C_{ijk} } } } }}{{\sum\nolimits_{i}^{I} {\sum\nolimits_{j}^{J} {\sum\nolimits_{k}^{K} {U_{i,n} (\xi )V_{i,n} (\eta )W(\mu )} } } }} $$
(1)

where C are the control points, \( \xi \), \( \eta \), and \( \mu \) are the parametric coordinates, and U, V, and W are the basis functions which are calculated using the following expression:

$$ \begin{aligned} & U_{i,1} (\xi ) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{if}}\,u_{i} \le \xi < u_{i + 1} } \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. \\ & U_{i,k} (\xi ) = \frac{{(\xi - u_{i} )U_{i,k - 1} (\xi )}}{{u_{i + k - 1} - u_{i} }} + \frac{{(u_{i + k} - \xi )U_{i + 1,k - 1} (\xi )}}{{u_{i + k} - u_{i + 1} }} \\ \end{aligned} $$
(2)

The basis coefficients are calculated from the knot vectors \( \bar{U} \), \( \bar{V} \) and \( \bar{W} \), and, which are a sequence of real numbers. Basis functions are equal to zero everywhere except for an interval delimited by the order of the NURBS, defining the area of influence of each control point (Piegl and Tiller 1997). The most common implementation of the control box is to employ uniform basis, which can be obtained with a knot sequence as:

$$ \left\{ {\underbrace {0, \ldots ,0}_{p + 1},} \right.\frac{1}{N} \ldots ,\frac{i}{N}, \ldots , \frac{N - 1}{N},\left. {\underbrace {1, \ldots ,1}_{p + 1}} \right\} $$
(3)

First order is equivalent to a linear interpolation, while second and third orders provide derivative and curvature continuity, respectively.

In this work, the airfoil is parameterized with third order volumetric NURBS, also called control box, and the design variables will be the vertical displacements (z axis) of the 14 control points. Figure 1 depicts the selected parameterization.

Fig. 1
figure 1

RAE 2822 control box parameterization

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To clarify, there are additional control points at the trailing and leading edge that are kept fixed, in order to maintain the angle of attack; so these control points are not considered as design variables.

Evolutionary Algorithm

Evolutionary algorithms (EAs) are bio-inspired methods that clone the behaviour of natural evolution to solve complex optimization problems. The basic elements of an EA are the solution coding, the selection operator and the crossover and mutation operator.

In the design application to be considered in this work, each coding vector is composed by a given parameterization of a geometry, i.e., \( z = \left[ {cp1,cp2,cp3, \ldots ,cpN} \right] \), where cp is the vertical coordinates of each control point.

More details about the EA applied in this paper can be found in a previous work from the authors (Andrés et al. 2012).

Objective Function Approximation Using Support Vector Machines (SVMs)

Support vector machines acts as a meta-model to predict the objective function to be optimized, which in this case is given by the aerodynamic performance of de airfoil.

Support Vector machines for Regression are a powerful tool used on the machine learning field, and a modelling tool for a large amount of regression problems on engineering. The SVR can be solved as a convex optimization problem using kernel theory to face nonlinear problems. The SVR consider not only the prediction error but also the generalization of the model. To obtain the best performance, a search of the most suitable combination of the kernel parameters must be carried on, usually by using cross validation techniques over the training set. To reduce the computational time of this process, different methods have been proposed in the literature to reduce the search space related to these parameters. In this case, it has been applied the one developed by Ortiz-García et al. (2009). Which has proven to require pretty short search times.

More details about the EA applied in this paper can be found in a previous work from the authors (Andrés et al. 2012)

Flowchart of the Proposed Approach

In this article, The Intelligent Estimation Search with Sequential Learning (IES-SL) method is applied. This method allows performing an efficient adaptive sampling guiding the optimization algorithm towards the most promising regions of the design space. The flowchart of the proposed approach is depicted in Fig. 2. First, an initial set of randomly generated (including the baseline) geometries are selected and evaluated with CFD tool (DLR Tau code in this work). With this set, a first surrogate is built and linked within an evolutionary algorithm. The latter will search for the minimum of the surrogate in each of the optimization iterations, and the returned optima will be again evaluated using the high-fidelity CFD solver, and then incorporated to the surrogate model, which is rebuilt and more precise on each iteration. The process will end when a certain number of CFDs budget is reached.

Fig. 2
figure 2

Flowchart of the proposed approach

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The aim of this work is to study the influence of the initial training size in the precision of the surrogate and the convergence of the proposed approach.

Numerical Results

Baseline Geometry

The selected geometry for this study was the well-known RAE2822 airfoil features described in Table 1. The airfoil is a rear-loaded, sub-critical geometry, designed to exhibit a roof-top type pressure distribution at design conditions (Mach = 0.66, Cl = 0.56 ESDU 1973). It has been tested in the RAE wind tunnel in 11 different flow conditions in the range of Mach numbers from 0.676 to 0.750 and at several Reynolds numbers (Cook et al. 1979).

A 56 k points unstructured grid was generated for this study.

Table 1 Baseline airfoil features
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Test Case Definition

The proposed approach is applied to 5 optimizations cases with 4, 8, 16, 32 and 64 initial random training points respectively. The multipoint optimization problem of the RAE 2822 is selected. The flow conditions for both design points 1 & 2 are:

 

DP1

DP2

Mach

0.734

0.754

Re

6.5 M

6.2 M

Turb. Model

SA

κω TNT

The objective function selected was \( Min \left( {\frac{{C_{D} }}{{C_{L} }}} \right) \) with some considerations. These are:

  • Aerodynamics constraints and penalties:

    1. 1.

      Prescribed minimum lift coefficient: \( \left. {C_{l}^{0} } \right|_{k} :\left. {C_{l} } \right|_{k} \ge \left. {C_{l}^{0} } \right|_{k} \).

    2. 2.

      Prescribed minimum pitching coefficient: \( \left. { C_{m}^{0} } \right|_{k} : \left. {C_{m} } \right|_{k} \ge \left. {C_{m}^{0} } \right|_{k} \).

    3. 3.

      Drag penalty: if constraint on minimum pitching moment is not satisfied, the penalty will be 1 drag count per 0.01 in \( \Delta C_{m} \).

  • Geometric constraints

    1. 1.

      Limit: ±20% of the initial control points’ values.

    2. 2.

      Prescribed maximum thickness ratio \( ( {\text{t}}/{\text{c)}}_{ \hbox{max} } :{\text{max(t}}/{\text{c)}} = ( {\text{t}}/{\text{c)}}_{ \hbox{max} } \).

    3. 3.

      Prescribed minimum thickness ratio \( ( {\text{t}}/{\text{c)}}_{ \hbox{min} }^{80} \) at \( {\text{x}} = 0.8{\text{c}} \): \( ( {\text{t}}/{\text{c)}}^{80} \ge ( {\text{t}}/{\text{c)}}_{ \hbox{min} }^{80} \).

    4. 4.

      Prescribed minimum leading edge nose radius \( {\text{R}}_{ \hbox{min} }^{\text{le}} : {\text{R}}^{\text{le}} \ge {\text{R}}_{ \hbox{min} }^{\text{le}}. \)

Sensitivity Study Results

In this section, the results of the present study are presented. Three issues are analysed. First, the influence of the initial training size in the convergence of the method. Next, the influence in the method precision of the initial data set. Finally, the value of the objective function reached in each case.

Regarding the first analysis, Fig. 3 shows the convergence of the IES-SL for each test case. As can be seen, the five test cases have a huge oscillation during the “training period”. This is the expected behaviour since the points in this data set are generated randomly. A lower size of initial training means the optimizer requires more iterations to reach the “optimum region”. The reason is that the initial surrogate is more intelligent with a huge initial data set, but it requires more time to be built. At last, the five test cases reach the same optimum region (see Table 2).

Fig. 3
figure 3figure 3

SBGO convergence versus initial data set size

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Table 2 OF evolution respect the initial training size
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Figure 4 illustrates the accuracy of the method with respect the initial training size. As expected, an initial surrogate with a vast number of points has an initial accuracy higher than one with a small set of points. This is in the same line that the convergence. Nevertheless, it requires more time to start the optimum seek.

Fig. 4
figure 4figure 4

Approach accuracy for each initial training size

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Last, but not least, Table 2 summarizes the value of the OF reached in each case. It can be seen that there is no influence of the initial training size in the final value of the OF (with a reasonable budget of iterations). This is the main advantage of the IES-SL proposed.

Conclusions

The aim of this work was to provide an analysis about how the initial training size of the surrogate affects the behaviour of the proposed IES-LS method. The following conclusions have been extracted from the solutions:

  • The optimum region reached is the same independently the training set size. A model with higher initial data set size requires less iterations to reach de optimum region, but it requires more computational time to be built, which is not feasible from the industry point of view.

  • In the same trend, the initial accuracy of the surrogate increases with the number of training samples, but the drawback is the same which is exposed in the previous point.

  • As summarized in Table 2, the training set size has no influence in the OF reached by the proposed IES-SL approach.

In summary, the main advantage of the proposed SBO method is that it can reach the global optimum with a small number of initial samples. This is feasible due to sequential learning allows the surrogate to become accurate each iteration. So, there’s an important reduction of the initial computational cost that requires a standard offline SBO.