Design-by-Morphing (DbM) is a novel design methodology that creates a search space for topology optimization. Traditional design techniques often impose geometric constraints and, sometimes, the designer's biases on the design space, which restricts the novelty of the designs and allows for only small local changes. On contrary, we show in this paper that DbM does not impose such restrictions on the design space, thus allowing for a radical and expansive search space with only a few design parameters. We compare DbM with other methods in the case of design space generation for 2D airfoils and found that DbM can reconstruct the entire UIUC database to >99.5% accuracy. Furthermore, using a bi-objective genetic algorithm, we optimize the airfoil designs created by DbM to maximize both the lift-over-drag ratio, $CLD_{max}$, and stall angle tolerance, $\Delta \alpha$, which results in a Pareto-front of innovative airfoils that exhibit substantial improvements in both objectives.

Design-by-Morphing (DbM); Topology Optimization; Airfoil

§ INTRODUCTION

Airfoil shape optimization is a critical stage in the design of aerodynamic components, such as aircraft wings wing_1, wing_2, wing_3, wing_4 and wind-turbine blades wind_turbine_1,wind_turbine_2,wind_turbine_3,wind_turbine_4,wind_turbine_5,wind_turbine_6. The airfoil optimization process typically involves three main components: shape parameterization, airfoil evaluation, and optimization. Among these, the parameterization method defines both the design space and the complexity of the optimization problem. To ensure effectiveness, a desirable parameterization technique must be able to encompass a wide design space using a modest amount of design parameters sobester_barrett_2008,sripawadkul_comparison_2010,masters_review_2015,10.1115/1.4036134. This is particularly important during the initial design phase, where minimum geometric constraints are imposed, and the flexibility to make significant changes during optimization is beneficial.

Shape parameterization methods differ in their fidelity and control ranges masters_review_2015,sobester_barrett_2008, and can be placed on a virtual spectrum according to the geometric scope of each design parameter. At one end of the spectrum, adjusting a single parameter alters a local section of the airfoil, which offers precise shape control but modifies the shape slowly. At the opposite end, each design parameter affects the global contour of the airfoil sobester_barrett_2008.

At the local end of the spectrum is the discrete method discrete_method, where the design parameters are exactly the discrete points that define the airfoil surface. Since the position of each point can be adjusted, the design space is potentially limitless survey_parameterization, and precise local control with high fidelity can be achieved.
However, a substantial number of surface points are needed to accurately describe an airfoil shape, which complicates the optimization problem. Gradient-based optimizers are frequently employed to mitigate the increased complexity, but they are likely to get stuck at a sub-optimal solution during the optimization.

As the geometric scope of each parameter is expanded, there emerge the classical approaches that are based on the curve-fittings of regional features or control points.
For example, the popular parametric section (PARSEC) method PARSEC uses eleven or twelve parameters to represent major sectional features of an airfoil, including leading edge radii and upper and lower crest locations, and constructs the airfoil surface using a $6^{\mathrm{th}}$ order polynomial. Another popular method is the Bézier parameterization bezier, which constructs the upper and the lower surfaces of the airfoil through the Bézier curves defined by pre-chosen control points. Additionally, a hybrid of the two techniques, Bézier-PARSEC parameterization, was introduced by bezier-parametric, which uses the parameters of the PARSEC method to define the Bézier curves that form the shape contours. One main issue with the above methods is their inability or inefficiency to include high-fidelity features; the PARSEC and the Bézier-PARSEC methods both have a fixed number of parameters and limited range of fidelity, while the Bézier parameterization requires higher-degree Bézier curves to describe complex shapes which are inefficient to calculate survey_parameterization.

To consider finer details of airfoils or, equivalently, to represent more complex curves, either B-splines sanaye_multi-objective_2014, han_adaptive_2014 or nonuniform rational B-spline (NURBS) nurbs can be used, which creates curves by connecting low-order Bézier segments defined by control points.
As the number of control points increases, these methods move to the local end of the spectrum and become capable of representing high-fidelity features, but the computing complexity also increases. One way to reduce the number of the design parameters is to group the control points together so that the global transformations such as twisting and thickening can be used as the parameters. This is known as the free-form deformation (FFD) method FFD1,FFD2 and is closer to the global end of the spectrum. A similar method, called radial basis function domain element (RBF) approach RBF1,RBF2,10.1115/1.4046650, also exists and makes use of radial basis function to exert deformation on the airfoil.

Near the global end of the spectrum, we see methods using spectral construction of basis functions to form or deform airfoil shapes. One popular choice of the basis functions is the dominant modes from singular value decomposition (SVD) of an airfoil dataset POD1,POD2,PCA, doi:10.2514/6.2020-2707, doi:10.2514/6.2019-1701. Other choices include sinusoidal functions of the Hicks-Henne approach hicks_wing_1978, which create `bumps' on a reference airfoil surface, and surface functions of the class/shape function transformation (CST) method kulfan_fundamental_2006, akram_cfd_2021, which are in the form of the product of a class function and a shape function generated by a linear combination of Bernstein polynomials. Nonetheless, like many other methods on the spectrum, these methods also suffer from the so-called curse of dimensionality that more basis functions or modes are always required to resemble high-fidelity features.

Efforts have been made to overcome the curse of dimensionality
viswanath_forrester_keane_2011, viswanath_forrester_keane_2014, cinquegrana_iuliano_2018. A recent work by chen_chiu_fuge_2020 applied a generative adversarial network (GAN) to learn the major shape variations of an airfoil database and use those to parameterize the shapes while also preserving the high-fidelity features via an additional noise space. However, like many other dimension reduction methods, this study assumes that the optimum design is not far from the database, which is not always true. To address this limitation, they proposed another GAN-based method that encourages diversity during sample generation 10.1115/1.4048626, but a large dataset is still required to initialize the training. In contrast, our paper is motivated by the optimization problem during the early design stage when few initial designs are available. Therefore, we are interested in a parameterization method that is capable of representing high-fidelity features even when the design parameters and initial airfoil designs are limited.

In this paper, we apply the Design-by-Morphing (DbM) parameterization technique to the airfoil optimization problem. DbM is a novel and universal design strategy that was first introduced by 2018CompM6223O and has been used in recent years for geometry optimization of different problems 2018CompM6223O,2019APS..DFDQ14007S,sheikh_2022,2021APS..DFDA15004S. As a global method, it `morphs' homeomorphic baseline shapes together to create new shapes and is able to interpolate and extrapolate the design space, allowing for both high-fidelity representation of shapes without the curse of dimensionality and radical modifications to the shapes without any implicit geometric constraints 2018CompM6223O,sheikh_2022. This strategy is applicable to a variety of 2D and 3D design problems and we aim to conduct a special case study of DbM for the 2D airfoil shape optimization here. Throughout this paper, we aim to make the following scientific contributions:

* Application of DbM to 2D airfoil shape optimization, showing its accurate reconstruction of the existing airfoil database and radical changes in airfoil shapes while being free from geometric constraints and designers' biases by extrapolation of the design space by applying negative weights.

* Evaluation of airfoil design capacity of the DbM strategy and comparison with other typical 2D airfoil design strategies.

* Sensitivity analysis for the number of baseline shapes, convergence analysis compared to conventional airfoil design strategies is shown and the significance of extrapolation for DbM is also shown.

* Optimization within the 2D airfoil search space generated by DbM using a genetic algorithm and investigation of the optimum Pareto-front.

§ DESIGN-BY-MORPHING

Design-by-Morphing (DbM) works by morphing homeomorphic, i.e. topologically equivalent, shapes to create a continuous and constraint-free design search space.

It comes with several advantages. To begin with, DbM is valid for shapes of any dimensions, and capable of creating exotic shapes because radically different baseline shapes can be morphed together. Furthermore, DbM does not impose any geometric constraints on the design parameters. And the only implicit constraints are the selections of the `baseline shapes' themselves, which are necessary to prescribe the problem to be solved. Lastly, it is able to create an extensive design search space, even when the number of pre-existing designs is small, e.g. sheikh_2022, by both the means of `extrapolation', that is to assign negative weights during morphing, and the inclusion of irregular or uncommon shapes. The details of DbM for airfoil optimization are presented in the subsequent subsections.

§.§ Baseline Shapes and Morphing

The DbM technique requires two or more homeomorphic `baseline shapes', mostly chosen from pre-existing designs in the literature, to create the design space. A one-to-one correspondence between the baseline shapes must first be established through some systematic shape collocation methods in either the functional 2018CompM6223O or the geometric space 2019APS..DFDQ14007S,sheikh_2022. Then the new shapes can be generated by applying weights to the collocation vectors of the baseline shapes and summing them together in a linear manner.

$n^{\text{th}}$ airfoil shape

$y$-coordinate collocation vector

An example of DbM. The coordinates of the baseline shapes are weighted, summed, and normalized to form the coordinates of a morphed shape.

For 2D airfoils, the closed shapes can be collocated in the Euclidean coordinate system. It is noted here that all 2D shapes bounded by a single surface are homeomorphic to one another. Using the leading edge of each airfoil as origin, each shape can be collocated by taking fixed and uniformly spaced points along the $x$-axis
, creating a one-to-one correspondence between the shapes. This collocation strategy is demonstrated in Figure <ref>, and the baseline shapes used in this paper are chosen from various airfoils in the literature, which are detailed later. Morphing is performed by multiplying a specific airfoil shape with a scalar weight, summing the weighted vectors, and then normalizing them. For a collection of $N$ baseline shapes, morphing is given by

\begin{equation}
\vec{P}(\vec{x}) = \dfrac{1}{\sum%\limits
_{m=1}^{N}w_m}\sum\limits_{n=1}^{N} w_n \vec{S}_n(\vec{x}) \, .
\label{eq:1}
\end{equation}

Application of DbM to 2D airfoils. Column 1 shows the baseline shapes. Column 2 depicts the elements of the collocation vectors of the baseline shapes plotted as a function of the index $i$ of the collocation vector. Column 3 shows the weighted elements of the collocation vector plotted as a function of the index $i$ of the collocation vector. Column 4 shows the resultant collocation vector of the morphed shape and the morphed shape itself.

Here $\vec{S}_n(\vec{x})$ is the $y$-coordinate collocation vector of the $n^{\text{th}}$ baseline shape, collocated at $\vec{x} = \left[ x_0, \cdots , x_F \right]$ where the $i^\text{th}$ $x$-coordinate $x_i = \left| 1-2i/F \right|$ and $F$ is the number of collocation points. The first half of the elements of $\vec{S}_n$ represents the top surface of the airfoil, and the second half of the elements of $\vec{S}_n$ represents the bottom. $w_n \in [-1,1]$ is the morphing weight applied to the $y$-coordinate vector of the $n^{\text{th}}$ baseline shape, and negative $w_n$ values imply extrapolation. A visual demonstration of the strategy is presented in Figure <ref>.

§.§ Intersection Control

For smooth baseline shapes, applying positive weights, i.e. interpolation, will always create smooth shapes. However, applying negative weights, i.e. extrapolation, may produce non-physical geometries, such as self-intersections, which have `zero-area' regions, as shown in Figure <ref>. One may discard the morphed airfoil shapes with self-intersections during the optimization, but that diminishes the size of our design space. Instead, we recover new shapes by removing the intersections.

Intersection removal is accomplished by first locating the intersection within the morphed coordinate vector and restructuring the vector by `flipping' it between the intersection points, as shown in Figure <ref>. The vector is then `stiffened' to remove the `zero-area' between the intersections by removing the points in their neighborhoods and linearly interpolating between the broken coordinate vectors. As seen in Figure <ref>, this removes the `zero-area' space and adds some physical area to the shape at the original intersection point. This process is repeated until all intersections are removed, e.g. both intersections in Figure <ref> are successfully removed, and finally, a moving-average smoothing filter is applied to smooth out any sharp edges.

Conditioning for intersection removal. (fig:DbM-intersect_a) Intersections are detected; (fig:DbM-intersect_b) A blown-up image of one intersection, with the shape coordinates direction depicted by arrows; (fig:DbM-intersect_c) Intersection removed by flipping the vector between the intersection; (fig:DbM-intersect_d) The `zero-area' removed by linear interpolation and then smoothed over, as shown by hatted y-coordinates.

§.§ Selection of Baseline Shapes

The selection of baseline shapes is a crucial component of the DbM strategy and ultimately determines the size and novelty of our search space. Metaphorically, the selection of the baseline airfoil shapes serves as the gene pool for the morphed airfoils and its diversity is important for creating a large design space.

One way of selecting the baseline shapes is by performing singular value decomposition (SVD) or principle component analysis (PCA) on a set of shapes and then using the dominant modes as the baseline shapes. Methods such as parametric model embedding <cit.> can help reduce the dimensionality of the problem as well. Although these methods would help in quantitatively choosing baselines, these methods, however, require an existing dataset which might not be available in many shape optimization problems 2018CompM6223O, doi:10.1177/0309524X17709732, sheikh_2022. Therefore, while techniques like SVD and PCA can be easily applied to airfoil shape optimization problems, and provide arguably better baselines, we choose the baseline shapes qualitatively instead to demonstrate the universality of DbM even for engineering problems with few existing designs. In other words, for research purposes we assume that the airfoil database is not a priori knowledge at the selection stage, except for those chosen as baselines.

An additional benefit of directly morphing existing designs is that, from a human designer's perspective, it can be more intuitively informative than handling PCA modes. For example, vertical-axis wind turbines are broadly categorized into drag, lift and hybrid categories, so the weights associated with each type are more informative to a human designer than the weights of the dominant modes. On the other hand, choosing actual shapes as the baseline shapes has the advantage that non-conforming designs can be easily added, as is the case for baseline #19 (mirrored Selig airfoil). Conventional techniques may have much more difficulty adding radical features into the design space, and the significance of such radical baseline shapes is demonstrated in the Results section.

figureTwenty-five baseline airfoil shapes chosen from the UIUC database uiuc_database. See Appendix <ref> for further details.

We selected 25 shapes (see Figure <ref>) from the UIUC airfoil coordinates database uiuc_database as our baseline shapes. They were picked to ensure diversity and to introduce radical features into the design space. Our selection of baseline shape included airfoils that are either known for high lift-to-drag ratio or good stall performance, which are commonly used in the literature and the industry. We also included airfoils with poor aerodynamic performances, as well as airfoils with irregular shapes, to provide novelty to the design space. It is worth noting that, unlike in the conventional airfoil optimization processes koroglu_ozkol_2019, we deliberately included the bad performers so that our optimization could suppress these features by assigning them negative weights, which will be demonstrated in greater detail in our later results. The model names and characteristics of the baseline shapes can be found in Appendix <ref>. To express shapes as collocation vectors, each airfoil shape is represented by 4,001 coordinates that span counterclockwise from the upper surface trailing edge around the leading edge to the lower surface trailing edge, with equally distributed $x$-coordinates parallel to the airfoil chord line of unit length (i.e. $F=4,000$ in Figure <ref>).

§.§ Airfoil Design Capacity Test

As a benchmark, we reconstruct the entire UIUC airfoil database uiuc_database using DbM to test the robustness of our method and the representation capacity of the generated design space. Having noted that one of the key features of DbM is to permit shape extrapolation, we compare our reconstruction results against the results of an interpolation-only DbM (DbM-I) where all DbM weights are non-negative. In addition, we performed the same test on 3 conventional 2D airfoil shape parameterization methods: PARSEC PARSEC, NURBS nurbs and the Hicks-Henne approach hicks_wing_1978. These tests were meant to answer the following questions:

* How much does the extrapolation expand the design space?

Extrapolation is undoubtedly better at creating a wider design search space. However, we shall focus on how quantitatively the search space is broadened by extrapolation, so as to confirm whether this feature genuinely distinguishes DbM from other generic approaches.

* Is DbM comparable to conventional airfoil shape parameterization methods in terms of shape reconstruction?

It should be noted that we selected the baseline airfoil shapes for DbM solely based on the qualitative principle of ensuring diversity and intentionally avoided the use of a selection method that requires a known, rich design database in advance. We shall demonstrate that the answer to the above question is still positive, even though DbM is not a design method specifically for airfoils, unlike the methods that the DbM is compared with.

figureGeometric demonstration of mean absolute error (MAE) between target and reconstructed airfoil surfaces.

For all of the 1,620 airfoils in the UIUC database uiuc_database, we obtained the closest representation of each target shape by running a global optimization of the input design parameters that minimizes mean absolute error (MAE) between the target and the reconstructed airfoil surfaces.

A geometric demonstration of MAE is provided in Figure <ref>. Using the functional expression of an airfoil $y(x_i)= y_i$ ($0\le i \le F$), as introduced in Figure  <ref>,

\begin{equation}
\text{MAE}(target,~reconstructed) = \frac{2}{F} \int_0^F{y_i^{{error}} di} ,
\label{eq:xor}
\end{equation}

where $y^{{error}}_i \equiv | y^{{reconstructed}}_i - y^{{target}}_i|$.
When we express the error in percentage terms, i.e. (MAE $\times$ 100) %, we emphasize that the error is described as a proportion of the area difference to the square of the chord length, as all airfoil shapes are normalized to maintain a unit chord length. The factor of $2$ in Eq. (<ref>) is present for this reason.

Airfoil shape parameterization methods for comparison

[c]Method     [c]Design Variables (DVs)     [c]# of DVs      [c]Remark
PARSEC     $\begin{aligned}  r_{le}^{up/lo}:~& \text{Leading edge radii} \\[-0.5ex] x^{up/lo},~ y^{up/lo}:~& \text{Crest coordinates}\\[-0.5ex] y_{xx}^{up/lo}:~& \text{Crest curvatures}\\[-0.5ex] y_{te},~ t_{te}:~& \text{Trailing edge mid-position and thickness}\\[-0.5ex] \alpha_{te},~ \beta_{te}:~& \text{Trailing edge direction and wedge}  \end{aligned}$     12     [c]Fixed # of parameters

NURBS      $\begin{aligned} x_{ctrl,i}^{up/lo}, y_{ctrl,i}^{up/lo}:~& \text{Control point coordinates}~(i=1,\cdots,~4) \\[-0.5ex] w_{i}^{up/lo}:~& \text{Curve weights}~(i=1,\cdots,~4) \\[-0.5ex] y_{te}^{up/lo}:~& \text{Trailing edge positions} \end{aligned}$      26     [c]Third-order B-spline
Evenly distributed knots

Hicks-Henne     $\begin{aligned} w_{i}^{up/lo}:~& \text{Bump widths}~(i=1,\cdots,~6) \\[-0.5ex] m_{i}^{up/lo}:~& \text{Bump magnitudes}~(i=1,\cdots,~6) \end{aligned}$     24     [c]Base profile: NACA 0012
Cosine-distributed bump points

DbM (Present)     $\begin{aligned} w_{i}:~& \text{Morphing weights}~(i=1,\cdots,~25) \end{aligned}$     25     [c]See Figure <ref> for the baselines

To obtain the closest representation, we utilized a MATLAB-based single-objective genetic algorithm (GA): . The population size is set to 100, and the maximum number of generations is set to 500. The lower bound for MAE was set to $1.44 \times 10^{-3}$ (or 0.144 %) from  Eq. (<ref>) for a chord length of 1, in accordance with the lower limit of Kulfan's typical wind-tunnel tolerance kulfan_fundamental_2006,Masters2017. To ensure a fair comparison, all the airfoil parameterization methods tested underwent the same optimization scheme for shape reconstruction, with similar numbers of design parameters (e.g. 25 design parameters for the DbM) except for PARSEC, which has fixed design parameters. In general, the fidelity of these design methods improves as the number of design variables increases Masters2017.

It is important to note that the objective of our reconstruction tests is to examine both the efficiency and the accuracy of a given parameterization method in the context of shape generation during the design process, which must be distinguished from the accuracy in surface fitting. NURBS, for instance, can achieve arbitrary accuracy for shape fitting if a good initial guess of the parameters is provided, but it may not be ideal for shape generation as the shape it constructs varies slowly during the optimization process. Accordingly, all the reconstructions tests are initialized in a consistent manner to provide a meaningful comparison. In particular, the initial population is set to contain a single parameter set that represents the profile of NACA 0012 with the remaining sets randomly distributed. The results of the reconstruction tests can then be understood as the ability of a method to create various shapes, including the common designs that have been collected in the UIUC database, precisely within a certain number of optimization generations. And the progressive improvement in the design space reconstruction can be observed as a function of number of GA generations.

Percentage of airfoils reconstructed within the mean absolute error tolerance of 0.5 % using DbM, compared to those from DbM only with interpolation (DbM-I) and three airfoil shape parameterization methods (see Table <ref>).

Applying DbM for the reconstruction of the UIUC database, we found that 1,618 of the 1,620 airfoils of the entire UIUC database, were reconstructed with a MAE error < 1%. Even for the two airfoils with the highest error, the DbM reconstruction still resulted in a MAE error of less than 1.5 %. Figure <ref> displays the percentage of airfoils that were reconstructed within the tolerance of 0.5 % MAE error, with respect to the total number of GA generations. A comparison between DbM and DbM-I reveals that the extrapolation feature of DbM significantly contributes to the improved performance of the method, suggesting that the extrapolation feature is indispensable for DbM. On the other hand, at the maximum GA generation, the total percentage of reconstructed airfoils with a MAE error < 0.5% increases from 60 % (DbM-I) to 98 % (DbM). As a result, DbM is converges faster than any other conventional approaches tested here.

GA-based reconstruction of pre-existing airfoil shapes using different design parameterization methods, after 500 GA generations in total.

To provide better insight, we plot the reconstruction results of 10 airfoil shapes in Figure <ref>, which represent the less successful reconstruction attempts. In particular, we ranked the results based on the unweighted average of all MAE errors from the five tested methods for each airfoil case and made the selections at every percentile bin from the worst. These 10 shapes are depicted in row-major order, from one in the $90^{\mathrm{th}}$ percentile (HOR 20, average MAE 0.35 %) to one in the $99^{\mathrm{th}}$ percentile (FX 79-W-660A, average MAE 1.1 %). Even these less successful results appear to reasonably reconstruct the target airfoil shapes. It is worth paying attention to the worst case, FX 79-W-660A, which is designed for use on a thick rotor blade of a wind turbine and far from the typical streamlined airfoil shapes. DbM-I encountered a notable failure in this case because none of our chosen baseline airfoil shapes for DbM were as thick as the target shape. As a result, the reconstruction just ended up with the thickest baseline airfoil shape, #23 (see Figure <ref>). This specific example underscores the significance of the extrapolation feature of DbM, which provides the opportunity to explore extraordinary designs, such as much thicker airfoils in this case. We also observed that NURBS occasionally produced thorn-like local structures (e.g. at the leading edge of GOE 511 in Figure <ref>), which resulted from the locally deforming nature of NURBS. These artifacts are normally removed by fitting softwares, such as  in Rhinoceros 3D, or manual handling of the control points by the designer.

Another way DbM explicitly introduces novelty is by using novel shapes directly as baselines. Generally speaking, novel designs that contain unconventional features can be challenging to construct. For example, the `mirrored' airfoil in our DbM baseline shapes (#19) is considered off-design by conventional airfoil parameterization methods that prescribe fixed edge geometries for the airfoil, such as relatively `blunt' and `sharp' edges at $x=0$ and $x=1$, respectively. Figure <ref> displays the results of GA-based reconstruction of the mirrored airfoil using the tested methods. PARSEC and the Hicks-Henne approach, which implicitly define edge geometries for airfoils, clearly struggle in reconstructing the mirrored edges. At $x=1$, these methods still exhibit hints of the sharp edge in their reconstructed shapes. On the other hand, NURBS performs well as it is more flexible in handling curvatures through weight parameters. However, although not considered in this study, if one wishes to introduce a tentative higher-order feature, such as a stepped wing lumsdaine_investigation_1974, NURBS may require a larger number of design parameters (i.e. more control points and weights). On the contrary, DbM would only require one additional design parameter (i.e. by adding it as a new baseline shape) to introduce novelty regardless of the complexity of the new design.

For the current study, we note here that we used 25 baselines based on the computational budget available and our study shows that the number of baseline shapes was sufficient. However, a smaller number of baseline shapes might have proven to be enough as well. to this end, Figure <ref> presents a sensitivity study of DbM in relation to the number of design variables (baseline shapes) used, where the convergence trend confirms that 20-25 baseline shapes are sufficient. Note that the current sensitivity study is restricted to the same baseline shape set in Figure <ref>. All tests were done five times with five random subsets for each case to consider sensitivity to the choice of baseline shapes within the subset. Future efforts will be directed towards conducting additional sensitivity analyses of DbM by varying the selection of the initial 25 baseline shapes themselves.

Overall, we have shown that DbM is competitive against conventional local parameterization methods despite being on the global end of the spectrum. In addition, DbM's ability to generate extraordinary designs through the extrapolation feature enhances the chances of finding novel solutions that deviate from the inputs (baseline shapes), which is important because the aerodynamic performance of an airfoil can be non-intuitively correlated with geometric features of the airfoil. More importantly, DbM is not just a method for design parameterization, but rather a universal design strategy for broader design search. While we have compared DbM to airfoil shape parameterization methods in the context of airfoil optimization, DbM can be useful for any type of problem that aims to introduce more novelty in design search.

figureReconstruction of a deliberately “mirrored" airfoil shape. In contrast to convention, the blunt edge is at $x=1$ and the sharp edge is at $x=0$. PARSEC and the Hicks-Henne method, which implicitly define edge geometries, face challenges in reconstructing the mirrored edges. NURBS looks to perform well due to its better flexibility in adjusting curvatures through weights. DbM has no problem because it can take such non-conforming shapes as baselines as needed.

figureRelative percentage of airfoils reconstructed within the mean absolute error tolerance of 0.5 % by DbM using a subset of the chosen baseline shapes (see Figure <ref>) to those using all twenty-five baseline shapes at the maximum GA generation. All tests were done five times with five random subsets for each case. Error bars indicate the standard deviation of the five test results.

§ OPTIMIZATION METHODOLOGY

Our airfoil optimization methodology is built around the DbM technique introduced in Sec. <ref>. As shown by the flowchart in Figure <ref>, the optimization starts with the selection of the baseline shapes and then evaluates and optimizes the airfoils formed by morphing these baseline shapes using DbM. Our methodology does not rely on a specific airfoil evaluation tool or a specific optimizer, and discussions on their choices are provided in Sec. <ref> and Sec. <ref>, respectively.

figureGeneral flowchart of airfoil optimization via DbM.

§.§ Airfoil Evaluation

Our optimization methodology is not limited to a specific airfoil performance analyzer. Any reliable CFD or experimental methods can be used. For optimizing airfoil shapes using CFD-based solvers, the evaluation of the objective functions (i.e. aerodynamic properties) is typically divided into two categories: the full Reynolds-averaged Navier-Stokes (RANS) based approach and the interacted viscous/inviscid zonal approach. The RANS-based approach is computationally expensive and demands a highly efficient optimizer. To accommodate a large number of design variables, which is common in the aerodynamic designs, a gradient-based optimizer coupled with adjoint methods for computing derivatives is deemed most feasible rans_piotrowski_2022,rans_he_2019,rans_kenway_2016. On the other hand, the viscous/inviscid zonal approach, which combines separated solutions for inviscid external flow and viscous shear layer flow in an iterative manner to form a continuous profile, is faster and less expensive.

Among a number of inviscid/viscous zonal airfoil analysis codes, XFOIL drela_1989 has been the most dominant and widely adopted program xfoil_use_ronsten_1992,xfoil_use_selig_1998,xfoil_use_jones_2000,xfoil_use_mueller_2003,xfoil_use_johnson_2005,xfoil_use_batten_2006,xfoil_use_lafountain_2010,xfoil_use_ramanujam_2017,10.1115/1.4048626. It combines a vorticity panel method for exterior flow with an integral boundary-layer method for viscous boundary layers and uses an $e^9$-type amplification formulation to determine the transition point drela_1989. Its suitability for airfoil designs has been demonstrated in the past literature, where its predictions of aerodynamic properties are found to be in good agreement with both wind-tunnel experiment data xfoil_McGhee_1988,xfoil_selig_1995 and the RANS-based simulation results xfoil_morgado_2016.

While our choice of the evaluation tool is flexible, for this work, we opt for XFOIL due to its acceptable accuracy under our flow condition and its low computation cost. Its widespread usage also allows for a quick reproduction of our optimization results. It is used in a black-box manner so that any other commercial or in-house airfoil analysis tools can be incorporated into our optimization framework if necessary. Our detailed airfoil evaluation setup is given in Appendix <ref>.

§.§ Optimization

When a set of solutions is given, the most optimal solution within the set can be determined without difficulty for single-objective optimization problems, which is the case for most of the previous airfoil optimization studies  10.1115/1.4046650, 10.1115/1.4054631,10.1115/1.4048626. However, for multi-objective optimization, multiple and potentially conflicting objectives must be considered simultaneously to determine the optimal answer in the solution set Kais99, doi:10.1080/23311916.2018.1502242. If the designer has a quantitative ranking of the objectives, these objectives can be combined together to formulate a single-objective problem, but when no such ranking exists, constructing a Pareto-front is the most common methodology barron, das, mgp, which has applications in the design of architected materials doi:10.1142/S1758825120501161, doi:10.1126/sciadv.abk2218 and turbo-machinery inproceedings, 10.3389/fenrg.2021.708230, 2016, RODRIGUES2016587,2011CMAME.200..883W,10.1115/1.2779899, process engineering Nguyen_2021, HuangGAO85, 10.1093/bioinformatics/btz544, shape design LI2020525,app11020729,CIARDIELLO2020115984, and structural engineering 10.3389/fmars.2019.00017, article.

We pose the multi-objective optimization problem as

\begin{equation}
\vec{w}_{opt}= \argmax_{\vec{w} \in \mathcal{W}} (\vec{f}(\vec{w})),
\label{eq:multiobj}
\end{equation}

where $\vec{f}(\vec{w})=[f_1(\vec{w}),f_2(\vec{w}), \cdots ,f_{\text{\textit{K}}}(\vec{w})]$. Here $f_1, \cdots ,f_{\text{\textit{K}}}$ are the $K$ objectives to be maximized, and $\vec{w}$ is the design variable vector. Generally, $\vec{w}$ is a $d$-dimensional vector defined over a bounded set $\mathcal{W}\subset \mathbb{R}^d$ representing $d$ continuous variables. $\{\vec{w}_{opt}\}$ is a set of Pareto-optimal solution vectors, i.e. vectors that are not Pareto-dominated by any other vectors. For the reader's convenience, it is noted that a design variable vector $\vec{\hat{w}}$ is Pareto-dominated by another design variable vector $\vec{\tilde{w}}$ if $f_{k}(\vec{\hat{w}}) \leq f_{k}(\vec{\tilde{w}})$ for all $k \in \left\{ 1, \cdots,K \right\}$. To obtain the Pareto-front, especially when objectives cannot be weighted or when a non-convex black-box function is considered,
evolutionary or genetic algorithms are a natural choice 10.1093/bioinformatics/btz544, 860052. In fact, these algorithms have been commonly implemented in many previous aerodynamic optimization studies due to their gradient-free nature and wide search domain akram_cfd_2021, skinner_state—art_2018, rahmad_single-_2020, nsga_zhao_2014. An alternate choice is Bayesian optimization method, which has been proven to be efficient when the cost functions are expensive to compute (e.g. when using experiments or CFD as an evaluation tool) mixmobo.

Our study considers a bi-objective ($K=2$) two-dimensional airfoil shape optimization. In particular, we optimize the shape of a subsonic airfoil operating in an incompressible flow with $\mathrm{Re} \equiv {U c}/{\nu}$ of $1\times 10^6$, where $U$ and $\nu$ are the free-stream flow speed and fluid kinematic viscosity, respectively, and $c$ is the airfoil chord length. The parameter to be optimized is the morphing weight vector for the DbM technique, defined as:

\begin{equation}
\vec{w} \equiv (w_1, \cdots , w_{25}) \in \mathcal{D}^{25},
\end{equation}

where $\mathcal{D} = [-1,1]\subset \mathbb{R}$ and $w_i$ ($i = 1, 2, \cdots, 25$) is the weight applied to the $i^{\mathrm{th}}$ baseline shape. The design objectives are the maximum lift-drag ratio over all possible angles of attack $\alpha$, i.e. $f_1(\vec{w})=CLD_{max}(\vec{w})$,
and the difference between the stall angle $\alpha_s$ and the angle where the maximum lift-drag ratio occurs, i.e. $f_2(\vec{w})=\Delta\alpha(\vec{w})$, often called the stall angle tolerance. This particular combination of design objectives has applications in the design of vertical-axis wind turbines 2019APS..DFDQ14007S, and the precise definitions of these design objectives are explained in Appendix <ref>. Both objectives are evaluated using XFOIL, which is efficient enough to be used with the GA.

We use a MATLAB-based variant of the popular NSGA-II algorithm  996017, which is a controlled, elitist genetic algorithm. Its practical employment can be found in keane_robust_2020 for the purpose of airfoil design optimization, as in the case with ours. Our initial population consists of the single-objective optimums of each design target and random samples in the design space. A population size of 372 is used with a total of 3,000 GA generations. Within each generation, solutions are actively ranked to maintain diversity and to prevent over-crowding in the Pareto-optimal solution set. Our setup was tested using the commonly used set of `$ZDT$' benchmark problems for multi-objective problems, as suggested by 10.1162/106365600568202. The test problems and validation results are detailed in Appendix <ref>.

The Pareto-front consisting of the optimal airfoil shapes, resulting from the NSGA-II 3,000 generation runs. The gray points are the whole evaluation outcomes of the UIUC reference database uiuc_database, with the DbM baseline cases in the present study highlighted as red hollow circles with respective indices. See Appendix <ref> for the clustering. $\mathrm{Re_{chord}} = 1\times 10^6$.

§ RESULTS

A Pareto-front on the $\Delta \alpha$ - $CLD_{max}$ objective plane resulting from a total of 3,000 generations of the GA runs is depicted in Figure <ref>. See Appendix <ref> for how we validated the maximum generation number. The convergence of the front is confirmed by the large number of generations with the population size of 372, involving around 1.1 million XFOIL evaluations of $CLD_{max}$ and $\Delta \alpha$. Without duplicates, a set of 80 Pareto-optimal airfoil shapes was obtained via DbM from the 25 chosen baseline shapes. For comparison, the whole UIUC database uiuc_database, as well as the baseline cases, are evaluated and plotted in Figure <ref> together. It is noted that baseline #19 has zero $CLD_{max}$ and $\Delta \alpha$ because it is intentionally mirrored, and XFOIL failed to evaluate its aerodynamic performance. We assigned zero values to failing cases like this because they represented airfoil geometries found to be aerodynamically unviable in the XFOIL space. The GA optimization successfully developed the Pareto-front, with two ends at $\left(CLD_{max}, \Delta \alpha \right) = \left( 30.63, 40^{\circ}\right)$ and $\left(CLD_{max}, \Delta \alpha \right) = \left( 264.17, 11^{\circ}\right)$. Even in the largest maximum lift-drag ratio case, the angle of attack gap between the stall and design point is $11^{\circ}$, providing the airfoil with a decent tolerant range for off-design operations.

The Pareto-front is divided into three different clusters, each constitutes a segment of the front that does not overlap with the others. It is worth noting that the non-overlapping division of the front is a result of clustering through Principal Component Analysis (PCA), rather than being manually assigned. The details of the clustering are provided in Appendix <ref>.

Figure <ref> depicts nine representative optimal airfoil shapes on the Pareto-front, arranged in ascending order of $CLD_{max}$. From each cluster, three airfoil shapes with distinct objective function values have been selected for representation. Also, note that Figure <ref>fig:pareto_shape_a shows the extreme case of the smallest $CLD_{max}$ and largest $\Delta \alpha$, while Figure <ref>fig:pareto_shape_i depicts the opposite extreme of the largest $CLD_{max}$ and smallest $\Delta \alpha$. It can be seen that within each cluster the overall shape remains unchanged, with only a gradual decrease in airfoil thickness as $CLD_{max}$ increases. Since thin airfoils such as bird-like airfoils ananda2018design, e.g. #13 and #14 of the baseline shapes, are known for their high $CLD$ performance, the trend of airfoil thickness observed in the Pareto-front appears to be reasonable.

Nine representative Pareto-optimal airfoil shapes. (fig:pareto_shape_a)-(fig:pareto_shape_c) are in cluster 1, (fig:pareto_shape_d)-(fig:pareto_shape_f) are in cluster 2 and (fig:pareto_shape_g)-(fig:pareto_shape_i) are in cluster 3. See Appendix <ref> for the clustering.

Cluster 1, made up by 48 optimal airfoil shapes, resembles the total mean of the Pareto-front, which is the average of all airfoil shapes on the Pareto-front (see Figure <ref>fig:pca_shape_a). This makes sense as they make up the majority of airfoil shapes located on the front. Moreover, this cluster is located near the origin in the PCA-projected weight space (see Figure <ref> in Appendix <ref>), indicating that no radical morphing of the airfoil shape took place from the mean shape.

Next, cluster 2 contains 24 optimal airfoil shapes. Compared to those in cluster 1, the most distinguishing feature is their narrow trailing edge regions, which are typically favorable for increasing lift. However, these airfoils are not greatly different from the origin in the PCA-projected weight space and are close to the total mean Pareto-front.

Finally, 8 optimal airfoil shapes are found in cluster 3 from the optimization. This cluster includes the airfoil shapes experiencing more drastic morphing than the other clusters. This is manifested by the fact that they are the thinnest airfoils where the leading edge region's thickness also becomes narrow.

The mean weight distributions with respect to 25 original baseline shapes are shown in Figure <ref>. Overall, the weight distributions of the three clusters conform to the weight distribution of the total mean. It turned out that baseline shape #13 (model name: AS6097) was commonly the most significant for morphing. Since this baseline shape has the best in $CLD_{max}$ and the second best in $\Delta \alpha$ among the 25 baseline shapes (see Figure <ref>), it was likely to persist in the GA runs over generations against the selection pressure that only sorts out dominant individuals in terms of both $CLD_{max}$ and $\Delta \alpha$. However, excellence in the objectives of an individual baseline shape does not necessarily guarantee its survival, which is the case for the globally best baseline shape #6 (model name: AH 79-100C). An individual's superior `phenotype' may be no longer revealed or even suppressed after the morphing is done and all `genes' are mixed up. In the same sense, inferiority in the objectives of an individual does not necessarily result in elimination, as demonstrated by the `mirrored' baseline #19.

Mean weight distributions of the Pareto-optimal airfoil shapes with respect to twenty-five baseline airfoil shapes.

As discussed from the examination of the morphed airfoil shapes, both cluster 1 and 2 show no significant differences from the total mean Pareto-front. Through small shape variations in shape from the total mean Pareto-front, as seen in Figure <ref>fig:pca_shape_a, reaching these optima would be relatively easily. In contrast, cluster 3 has a number of weights that are quite different from the mean (e.g., #6 and #11) and substantial morphing would be required if one starts with the total mean airfoil shape.

In the context of the present study, each axis obtained by PCA can be considered as a unique morphed airfoil shape because the 25 PCA coefficient vectors defined in the weight space $\mathcal{D}^{25}$ are orthogonal to each other. These 25 new morphed airfoils span the entire design space and therefore can serve as alternative baseline shapes in lieu of the original ones. More importantly, the dominance of the first two PCA axes with respect to the data point variance, accounting for 95% of total variance explained, suggests that the major geometric features of the 208 airfoil shapes we found through optimization are virtually generated by morphing of these two new airfoils. A small variance of a PCA axis indicates that the data points are not significantly deviated from their mean on the axis. In other words, the baseline shape corresponding to this PCA axis has an marginal impact on morphing the airfoil shape for optimization.

Once we pick two baseline shapes from the first two dominant PCA axes, whose associated collocation vectors are denoted as $\vec{P}_1$ and $\vec{P}_2$ for example, and use them to morph the airfoil shape obtained from the total mean of the Pareto-optimal weight vector set, which corresponds to the mean collocation vector $\vec{P}_{mean}$, we gain a better understanding of how the morphing, especially along each PCA axis, influences major geometric changes in the optimal airfoil shapes. These airfoil shapes are depicted in Figure <ref>, where the black and red surfaces are distinguished to emphasize that they represent the first and second halves of the collocation points, respectively. For example, we note that the orientation of the two surface of $\vec{P}_1$ is mirrored in comparison to that of $\vec{P}_{mean}$, meaning that the stronger the weight of PCA axis 1 in the positive direction is, the narrower the morphed airfoil shape gets.

Total mean

PCA axis 1

PCA axis 2

Morphed airfoil shapes generated by the optimal weight vectors, representing (fig:pca_shape_a) the total mean of all optimal airfoils' weights, (fig:pca_shape_b) the coefficients of the PCA axis having the most variance and (fig:pca_shape_c) the coefficients of the PCA axis having the second-most variance. The black and red surfaces correspond to the first and second half of the collocation points, respectively.

§ DISCUSSION

Most shape parameterization methods rely upon the careful selection of geometric constraints and parameters, which determines the likelihood of success. The fidelity offered by such methods largely depends on the number of the parameters chosen. Moreover, these designs are limited by the parametric constraints and the implicit designer bias, making it difficult to perform extrapolation or make radical global changes. Data-driven methods typically rely on the assumption that the optimum solutions are not far from the training data-set, which again limits the ability to make radical shape changes.

DbM, on the other hand, creates a design space that is not inhibited by geometric constraints, enables extrapolation from the design space, which is particularly useful for airfoil design, and is applicable to a wide range of engineering design problems. It does not suffer from the curse of dimensionality when parameterizing airfoils by control points and allows for high-fidelity representation of airfoils without increasing the number of independent parameters in the problem. Using only 25 baseline shapes from the UIUC database, we were able to recreate 99.87% of UIUC database with an MAE error < 1%. We also showed that extraordinary and broad searches are possible using DbM. By applying it to the bi-objective shape optimization with the objectives of maximizing $CLD_{max}$ and $\Delta \alpha$, we could achieve significant results compared to our baseline shapes. We posit that for the design parameterization of airfoils as well as other 2D/3D shapes, DbM should be the preferred method for creating an unconstrained, unbiased and non-database-driven design space that allows for radical modifications, which can often result in non-conforming shapes.

In this paper, our qualitative selection of 25 baseline shapes adequately spanned the design space with tolerable error. However, it would be possible that even a smaller number of baseline shapes than 25 could successfully construct the design space if some of the current baseline shapes were redundant. To further understand DbM, our future work will focus on performing sensitivity analysis of DbM on the baseline shape selection and applying DbM for design of turbo-machinery.

§ CONCLUSION

The DbM design strategy creates a design space that contains novel and radical 2D airfoils that are not constrained by geometric parameters or designer bias. Optimization within the design space created, for the dual objectives of $CLD_{max}$ and $\Delta \alpha$, resulted in remarkable improvements in both objectives and provided a Pareto-front of optimal airfoil designs. The final airfoils showed significant advancements compared to the input baseline shapes.

Overall, in our optimization study with respect to the 2D airfoil optimization problem, DbM is a suitable method for design space creation. In addition, our methodology is highly adaptable and can be utilized for shape optimization of other fluid machinery. Our ongoing work includes the applications of DbM in conjunction with Bayesian optimization to 3D airfoil optimization and vertical-axis wind turbine optimization problems.

§ ACKNOWLEDGMENT

The authors would like to thank Prof. Ömer Savasş, affiliated with University of California at Berkeley, for providing insightful discussions regarding airfoils and aerodynamics. The authors acknowledge the use of the Extreme Science and Engineering Discovery Environment (XSEDE), supported by National Science Foundation grant number ACI-1548562 through allocation TG-CTS190047.

§ DATA AND MATERIALS AVAILABILITY

The data needed to evaluate the conclusions are present in the paper and Appendices. The data files and optimization setup will be posted in a public repository upon publication of the paper.

[A]$c$Airfoil chord length ($\mathrm{m}$)
[A]$d$Drag force of an airfoil per unit span ($\mathrm{N~m^{-1}}$)
[A]$l$Lift force of an airfoil per unit span ($\mathrm{N~m^{-1}}$)
[A]$\vec{P}$$y$-coordinate collocation vector of a morphed airfoil
[A]$\vec{S}$$y$-coordinate collocation vector of a baseline airfoil
[A]$U$Free-stream flow speed ($\mathrm{m~s^{-1}}$)
[A]$w$Design-by-Morphing weight factor

[B]$\alpha$Airfoil angle of attack ($\mathrm{^{\circ}}$)
[B]$\alpha_s$Airfoil stall angle ($\mathrm{^{\circ}}$)
[B]$\Delta \alpha$Stall angle tolerance, the range of $\alpha$ between the stall point and the maximum lift-drag ratio point ($\mathrm{^{\circ}}$)
[B]$\nu$Fluid kinematic viscosity ($\mathrm{m^2~s^{-1}}$)
[B]$\rho$Fluid density ($\mathrm{kg~m^{-3}}$)

[C]$C_d$Drag coefficient of an airfoil per unit span, $2l/(\rho U^2 c)$
[C]$C_l$Lift coefficient of an airfoil per unit span, $2d/(\rho U^2 c)$
[C]$CLD$Lift-drag ratio of an airfoil, $C_l / C_d$
[C]$CLD_{max}$Maximum lift-drag ratio of an airfoil, $\max_{\alpha} CLD (\alpha)$
[C]$\mathrm{Re}$Reynolds number based on airfoil chord length, $U c/\nu$

§ AERODYNAMIC OPTIMIZATION OBJECTIVES

Airfoil optimization has become a common practice in aerodynamic design problems that involve maximization of one or more performance parameters of airfoils. We mainly consider the following two performance parameters: the lift-drag ratio and stall angle. Given the flow speed $U$, fluid density $\rho$ and airfoil chord length $c$, the lift and drag coefficients of an airfoil per unit span at an angle of attack $\alpha$, $C_l$ and $C_d$, are expressed as:

\begin{equation}
C_l(\alpha) \equiv \frac{l(\alpha)}{\frac{1}{2}\rho U^2 c}
C_d(\alpha) \equiv \frac{d(\alpha)}{\frac{1}{2}\rho U^2 c}.
\label{eq:clcd}
\end{equation}

where $l$ and $d$ are lift and drag force per unit span, respectively, both of which change with respect to $\alpha$. In this paper, these parameters are predicted using XFOIL drela_1989, a program for analyzing a subsonic 2D airfoil, with varying $\alpha$ and then used for optimization. Based on $C_l$ and $C_d$, the lift-drag ratio $CLD$ is calculated as:

\begin{equation}
CLD(\alpha) = \frac{C_l \,(\alpha)}{C_d \,(\alpha)}.
\label{eq:l/d}
\end{equation}

On the other hand, we define the stall angle $\alpha_s$ as an angle of attack where $C_l$ reaches its first local maximum as the angle increases from 0${}^{\circ}$, or:

\begin{equation}
\begin{gathered}
\alpha_s \equiv \min_{\alpha \ge 0} \alpha ~~ \text{where}~~ {}^{\exists} \delta > 0 ~~ \text{such that}\\ C_l\,(\alpha) \ge C_l\,(x) ~~ {}^{\forall} x \in [\alpha - \delta ,~ \alpha + \delta] .
% \alpha_s \equiv \argmax_{\alpha \in \mathbb{R}}{C_l (\alpha)} .
\end{gathered}
\label{eq:stall}
\end{equation}

Note that this definition is more conservative than the typical definition of stalling, where the flow at the rear region begins to fully separate and $C_l$ is globally maximized. $\alpha_s$ is occasionally smaller than the global maximum of $C_l$. Nonetheless, this approach helps avoid overestimation of the stall angle, which is expected to occur in XFOIL due to the nature of its flow solver having a limited accuracy in stall and post-stall conditions.

$CLD$ and $\alpha_s$ have been typically considered to be significant in characterizing airfoil performance. For example, when it comes to lift-type wind turbines, the point where $CLD$ is maximized is often chosen as the design point. However, since wind turbines cannot always operate under design conditions, $\alpha_s$ needs to be additionally considered to evaluate how far they run under increasing lift conditions. For well-designed airfoils, $\alpha_s$ generally occurs later than the design point, which yields operational tolerance beyond the design point. Consequently, the stall angle tolerance $\Delta \alpha$, i.e. the range between these two angles of attack, expressed as

\begin{equation}
\Delta \alpha \equiv \max \left(0,~ \alpha_s - \argmax_{\alpha \in \mathbb{R}}{CLD (\alpha) } \right) ,
\label{eq:deltaalpha}
\end{equation}

can be a proper choice to evaluate the off-design performance  li_method_2013. Figure <ref> depicts a schematic diagram of how $CLD$ and $\Delta \alpha$ are determined on airfoil performance curves.

Schematic diagram of airfoil performance curves.

§ BASELINE AIRFOIL SHAPES AND EVALUATION

The model names, features, shape outlines, and XFOIL evaluation results of the 25 baseline shapes used by DbM in this paper. The coordinates of the baseline shapes are obtained from the UIUC airfoil coordinates database uiuc_database. The airfoil evaluation results are obtained for an incompressible flow with a chord Reynold number of $1\times10^6$. The reference evaluation results are interpolated from the online XFOIL database airfoiltools; N/A indicates that no data is available.

2*Model Name
2*Series (Features)
1c2*Airfoil Shape

$CLD_{max}$     $\Delta\alpha$     $CLD_{max}$     $\Delta\alpha$

1      NACA 0012                  NACA (4-digit)                   75.6         8.50       69.3         6.75

2      NACA 2412                  NACA (4-digit)                   101.4        12.00      99.5         12.00

3      NACA 4412                  NACA (4-digit)                   129.4        1.75       126.2        11.50

4      E 205                      Eppler                           128.3        8.50       124.4        10.50

5      AH 81-K-144 W-F Klappe     Althaus                          89.7         2.00       91.6         2.00

6      AH 79-100 C                Althaus                          183.0        14.75      170.6        15.50

7      AH 79-K-143/18             Althaus                          110.9        1.50       107.0        1.50

8      AH 94-W-301                Althaus                          103.0        4.00       101.4        2.75

9      NACA 23112                 NACA (5-digit)                   98.6         6.75       96.9         8.00

10     NACA 64(2)-415             NACA (6-digit)                   120.6        12.50      113.8        13.00

11     NACA 747(A)-315            NACA (7-digit)                   111.5        12.00      105.8        13.00

12     Griffith 30% Suction      Griffith (Suction)               17.3         0.00       17.9         0.00

13     AS 6097                    Selig (Bird-like)                N/A          N/A        171.2        14.00

14     E 379                      Eppler (Bird-like)               N/A          N/A        160.0        2.00

15     Clark YS                   Clark                            85.7         5.25       82.3         5.75

16     Clark W                    Clark                            116.1        11.00      114.8        11.00

17     Clark Y                    Clark                            114.8        11.75      113.7        12.75

18     Chen                       Chen                             125.4        0.00       126.7        0.00

19     S2027 Mirrored              Selig (Mirrored)                  N/A          N/A        0.00          0.00

20     GOE 417A                   Gottingen (Thin plate)           86.7         5.25       90.4         5.25

21     GOE 611                    Gottingen (Flat bottom)          125.6        9.00       129.7        9.00

22     Dragonfly Canard           Dragonfly                        144.6        2.50       147.5        3.00

23     FX 79-W-470A               Wortmann (Fat)                   N/A          N/A        23.9         9.25

24     Sikorsky DBLN-526          Sikorsky (Fat)                   53.3         4.75       51.5         4.25

25     FX 82-512                  Wortmann                         99.1         14.75      98.7         13.00

Our optimization methodology does not rely on one specific airfoil evaluation tool. To compare our results with previous literature and help future researchers quickly reproduce our results, we used XFOIL drela_1989 in the present study. The two design objectives, $CLD_{max}$ and $\Delta\alpha$, are obtained from the $C_l$ and $C_d$ data calculated by the XFOIL at different angle of attacks (see Figure <ref>).

For improved efficiency and consistency, we used XFOIL to generate performance data and did not rely on any of its built-in paneling features. The conditioning and re-paneling of the morphed airfoil coordinates are custom-built at the end of our DbM algorithm, transforming the coordinates into 200 or 250 vortex panels with a relatively higher concentration where the curvature is high. To reduce evaluation time, we first performed a rough scan with an $\alpha$ increment of $1^{\circ}$ to estimate the range determining $\Delta \alpha$, and then finer scans for $CLD_{max}$ and $\Delta \alpha$ separately with an $\alpha$ increment of $0.25^{\circ}$ within and around the estimated range of $\Delta \alpha$ from the initial rough scan.

It is worth noting that XFOIL uses a global Newton's method drela_1989 to solve the boundary layer and transition equations simultaneously and uses the solution from the previous angle of attack as a starting guess. As a result, ill-conditioned airfoil coordinates and the occurrence of flow separation can both lead to non-convergence of the XFOIL evaluation. To ensure the robustness and correctness of our airfoil evaluation, our XFOIL wrapper attempts to reach convergence by restarting the root-finding with a fresh starting guess and gradually increasing the number of panels. If both attempts fail, the wrapper will check convergence at neighboring points, which will indicate whether flow separation occurs or not. Besides non-convergence, we further verify the correctness of the Newton's method by comparing the calculated viscous and inviscid drag coefficients. The later is determined purely by the potential flow theorem and have to be smaller than its viscous counterpart due to its neglect of the friction (viscous effect). Any angle with incorrect result will undergo the same treatment as non-converging ones, hence ensuring the correctness of our airfoil performance evaluation. A comparison between our XFOIL evaluation and an existing database of the same airfoils under the same flow conditions is provided in Table <ref>.

§ OPTIMIZATION TEST FUNCTIONS AND VALIDATION

Multi-objective optimization of benchmark test functions using GA.

We used the multi-objective problems suggested by 10.1162/106365600568202 to test our GA setup. The details of the test functions are given in Table <ref>. All the test functions are aimed to be minimized with 25 variables in the given design space.

Benchmark Test Functions. All of the test functions are bi-objective with extended to n-dimensional constrained search space.

Problem     Bounds     Objective Functions     Optima     Note
$ZDT1$     $\begin{aligned} w_i & \in [0,1], \\[-.5ex] i &= 1,\dots,n \end{aligned}$       $\begin{aligned}  f_1(\vec{w})&=w_1 \\[-.5ex] f_2(\vec{w})&=g(\vec{w})\left[1-(f_1(\vec{w})/g(\vec{w}))^{1/2}\right]\\[-.5ex] g(\vec{w})&=1+9\left(\sum_{i=2}^nw_i\right)/(n-1) \end{aligned}$       $\begin{aligned} w_1 & \in [0,1]\\[-.5ex] w_i &=0, \\[-.5ex] i &= 2,\dots,n \end{aligned}$      convex

$ZDT2$     $\begin{aligned}  w_i & \in [0,1], \\[-.5ex] i &= 1,\dots,n \end{aligned}$      $\begin{aligned}  f_1(\vec{w})&=w_1 \\[-.5ex] f_2(\vec{w})&=g(\vec{w})\left[1-(f_1(\vec{w})/g(\vec{w}))^{2}\right]\\[-.5ex] g(\vec{w})&=1+9\left(\sum_{i=2}^nw_i\right)/(n-1) \end{aligned}$       $\begin{aligned} w_1 & \in [0,1]\\[-.5ex] w_i &=0, \\[-.5ex] i &= 2,\dots,n \end{aligned}$      non-convex

$ZDT4$     $\begin{aligned} w_1 & \in [0,1]\\[-.5ex] w_i & \in [-5,5], \\[-.5ex] i &= 2,\dots,n \end{aligned}$       $\begin{aligned}  f_1(\vec{w})&=w_1 \\[-.5ex] f_2(\vec{w})&=g(\vec{w})\left[1-(f_1(\vec{w})/g(\vec{w}))^{1/2}\right]\\[-.5ex] g(\vec{w})&=10n+\sum_{i=2}^n\left(w_i^2-10\cos(4\pi w_i)\right)-9 \end{aligned}$       $\begin{aligned} w_1 & \in [0,1]\\[-.5ex] w_i &=0,  \\[-.5ex] i &= 2,\dots,n \end{aligned}$      non-convex

$ZDT6$     $\begin{aligned}  w_i & \in [0,1], \\[-.5ex] i &= 1,\dots,n \end{aligned}$      $\begin{aligned}  f_1(\vec{w})&=1-\exp(-4w_1)\sin^6(6 \pi w_1) \\[-.5ex] f_2(\vec{w})&=g(\vec{w})\left[1-(f_1(\vec{w})/g(\vec{w}))^{2}\right]\\[-.5ex] g(\vec{w})&=1+9\left[\left(\sum_{i=2}^nw_i\right)/(n-1)\right]^{1/4} \end{aligned}$      $\begin{aligned} w_1 & \in [0,1]\\[-.5ex] w_i &=0,  \\[-.5ex] i &= 2,\dots,n \end{aligned}$     $\begin{aligned} &\text{non-convex,} \\[-.5ex] &\text{non-uniform} \end{aligned}$

MATLAB's NSGA-II genetic algorithm, a fast sorting and elitist multi-objective genetic algorithm, was used for practical implementation. Initialization was performed through single objective optimization for each objective and random sampling. A population size of 372 was used, with a total of 3,000 maximum generations. The `phenotype' crowding distance metric was used. This setup was validated on the test functions described above. All the problems were benchmarked with $25$ variables ($d = 25$) and two objective functions ($K = 2$), as with the present airfoil optimization problem. The results of our setup on these four benchmark problems are shown in Figure <ref>. The algorithm could accurately capture $ZDT1$, $ZDT2$, and $ZDT4$ and predict $ZDT6$, which is the most complicated due to its non-convex and non-uniform properties, reasonably well.

Projection of the 25-dimensional optimal weight vectors to the 2-dimensional subspace spanned by 2 PCA axes of the dominant variance. $k$-means clustering with the cluster size of 3 is used to identify the clusters.

§ AIRFOIL SHAPE CLUSTERING

To analyze characteristics of the optimized airfoil shapes in detail, the airfoil shapes on the Pareto-front were classified into three clusters using $k$-means clustering based on the Euclidean distance with $k = 3$. The clustering was performed in the design variable space, or weight space, of $\mathcal{D}^{25}$ rather than in the objective plane because the purpose of clustering was to identify common geometric features over different airfoil shapes as a result of the optimization. The selection of the cluster size was based on the PCA of the optimal weight vector set.

It should be noted here that the baseline shapes chosen might be linearly dependent. The distances in the PCA weight space, thus, might not be rigorous as a morphed shape on the Pareto-front may be represented by another set of weights. However, this PCA analysis was used only to identify if qualitative classes within the Pareto-front could be found and clustered together and to glean some additional insights of our Pareto-front results.

Figure <ref> shows the projection of the 25-dimensional weight vector set to the 2-dimensional subspace spanned by the 2 PCA axes having the first- and second-most variance. The explained variance ratios of PCA axes 1 and 2 are 80.7% and 14.0%, respectively. On the other hand, the PCA axis of the third-most variance only accounts for 1.7% of the variance, affirming that the 2-dimensional projection in Figure <ref> adequately scatters the clusters. Based on this observation, $k = 3$ was chosen to be the most appropriate cluster size.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleApplied Soft Computing
shorttitleConstrained multi-objective optimization algorithms
titleConstrained multi-objective optimization algorithms: Review and comparison with application in reinforced concrete structures
abstractAirfoil (aerofoil) tools and applications. Search for airfoil coordinates and dat files. Plot and comapare airfoil shapes
titleAirfoil Tools

abstractThis paper presents the parameterization and optimization of two well-known airfoils. The aerodynamic shape optimization investigation includes the subsonic (NREL S-821) and transonic airfoils (RAE-2822). The class shape transformation is employed for parametrization while the genetic algorithm is used for optimization purposes. The absolute scheme of the optimization process is carried out for the minimization of the drag coefficient and maximization of lift to drag ratio. In-house MATLAB code is incorporated with a genetic algorithm to calculate the drag coefficient and lift to drag ratio of the resulting optimized airfoil. The panel method is utilized in genetic algorithm optimization code to calculate pressure distribution, lift coefficient, and lift to drag ratio for optimized airfoil shapes and validates with XFOIL and NREL experimental data. Furthermore, CFD analysis is conducted for both the original (NREL S-821) and optimized airfoil obtained. The present method shows that the optimized airfoil achieved an improvement in lift to drag ratio by 7.4% and 15.9% of S-821 and RAE-2822 airfoil, respectively, by the panel technique method and provides high design desirable stability parameters. These features significantly improve the overall aerodynamic performance of the newly optimized airfoils. Finally, the improved aerodynamics results are reported for the design of turbulence modeling and NREL phase II, Phase III, and Phase VI HAWT blades.
journaltitleApplied Sciences
titleCFD Analysis and Shape Optimization of Airfoils Using Class Shape Transformation and Genetic Algorithm—Part I

titleDesign and performance analysis of an airborne wind turbine for high-altitude energy harvesting

booktitle2018 AIAA Aerospace Sciences Meeting
titleDesign of bird-like airfoils

abstractEngineering problems that are modeled using sophisticated mathematical methods or are characterized by expensive-to-conduct tests or experiments are encumbered with limited budget or finite computational resources. Moreover, practical scenarios in the industry, impose restrictions, based on logistics and preference, on the manner in which the experiments can be conducted. For example, material supply may enable only a handful of experiments in a single-shot or in the case of computational models one may face significant wait-time based on shared computational resources. In such scenarios, one usually resorts to performing experiments in a manner that allows for maximizing one’s state-of-knowledge while satisfying the above-mentioned practical constraints. Sequential design of experiments (SDOE) is a popular suite of methods that have yielded promising results in recent years across different engineering and practical problems. A common strategy that leverages Bayesian formalism is the Bayesian SDOE, which usually works best in the one-step-ahead or myopic scenario of selecting a single experiment at each step of a sequence of experiments. In this work, we aim to extend the SDOE strategy, to query the experiment or computer code at a batch of inputs. To this end, we leverage deep reinforcement learning (RL)-based policy gradient methods, to propose batches of queries that are selected taking into account the entire budget in hand. The algorithm retains the sequential nature, inherent in the SDOE while incorporating elements of reward based on task from the domain of deep RL. A unique capability of the proposed methodology is its ability to be applied to multiple tasks, for example, optimization of a function, once its trained. We demonstrate the performance of the proposed algorithm on a synthetic problem and a challenging high-dimensional engineering problem.
journaltitleJournal of Mechanical Design
titleReinforcement Learning-Based Sequential Batch-Sampling for Bayesian Optimal Experimental Design

abstractThree published approximation formulae for selecting the best multiattribute alternative based on rank-ordered weights are evaluated. All formulae are surprisingly efficacious in determining the best multiattribute alternative. Rank order centroid (ROC) weights are more accurate than the other rank-based formulae; furthermore, the ROC formula generalizes to incorporate both other forms of partial information about attribute weights and partial rank order information as well. Because a ROC-based analysis is so straightforward and efficacious, it provides an appropriate implementation tool.
issn0025-1909, 1526-5501
journaltitleManagement Science
titleDecision Quality Using Ranked Attribute Weights

journaltitleRenewable Energy
titleHydrodynamics of marine current turbines

American Institute of Aeronautics

booktitle36th AIAA Aerospace Sciences Meeting and Exhibit
titleTwo-dimensional aircraft high lift system design and optimization

Cambridge ; New York

Cambridge University Press

seriesCambridge monographs on applied and computational mathematics
shorttitleRadial basis functions
titleRadial basis functions: theory and implementations
Radial basis functions

journaltitleApplied Mathematical Modelling
titleRevised multi-choice goal programming

journaltitleApplied Energy
titleReview of performance optimization techniques applied to wind turbines

booktitleWind Turbines - Design, Control and Applications
shorttitleWind Turbine Design
titleWind Turbine Design: Multi‐Objective Optimization

abstractDeep generative models are proven to be a useful tool for automatic design synthesis and design space exploration. When applied in engineering design, existing generative models face three challenges: (1) generated designs lack diversity and do not cover all areas of the design space, (2) it is difficult to explicitly improve the overall performance or quality of generated designs, and (3) existing models generally do not generate novel designs, outside the domain of the training data. In this article, we simultaneously address these challenges by proposing a new determinantal point process-based loss function for probabilistic modeling of diversity and quality. With this new loss function, we develop a variant of the generative adversarial network, named “performance augmented diverse generative adversarial network” (PaDGAN), which can generate novel high-quality designs with good coverage of the design space. By using three synthetic examples and one real-world airfoil design example, we demonstrate that PaDGAN can generate diverse and high-quality designs. In comparison to a vanilla generative adversarial network, on average, it generates samples with a 28% higher mean quality score with larger diversity and without the mode collapse issue. Unlike typical generative models that usually generate new designs by interpolating within the boundary of training data, we show that PaDGAN expands the design space boundary outside the training data towards high-quality regions. The proposed method is broadly applicable to many tasks including design space exploration, design optimization, and creative solution recommendation.
journaltitleJournal of Mechanical Design
titlePaDGAN: Learning to Generate High-Quality Novel Designs

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleAirfoil Design Parameterization and Optimization Using Bézier Generative Adversarial Networks

abstractThis paper shows how to measure the intrinsic complexity and dimensionality of a design space. It assumes that high-dimensional design parameters actually lie in a much lower-dimensional space that represents semantic attributes—a design manifold. Past work has shown how to embed designs using techniques like autoencoders; in contrast, the method proposed in this paper first captures the inherent properties of a design space and then chooses appropriate embeddings based on the captured properties. We demonstrate this with both synthetic shapes of controllable complexity (using a generalization of the ellipse called the superformula) and real-world designs (glassware and airfoils). We evaluate multiple embeddings by measuring shape reconstruction error, pairwise distance preservation, and captured semantic attributes. By generating fundamental knowledge about the inherent complexity of a design space and how designs differ from one another, our approach allows us to improve design optimization, consumer preference learning, geometric modeling, and other design applications that rely on navigating complex design spaces. Ultimately, this deepens our understanding of design complexity in general.
journaltitleJournal of Mechanical Design
titleDesign Manifolds Capture the Intrinsic Complexity and Dimension of Design Spaces

journaltitleApplied Energy
titleMulti-objective approach to the optimization of shape and envelope in building energy design

journaltitleComputers & Fluids
titleInvestigation of adaptive design variables bounds in dimensionality reduction for aerodynamic shape optimization

abstractThis work involves the design optimization of metal–ceramic through the thickness of functionally graded material (FGM) plates subjected to thermomechanical loadings. Constrained optimization was performed for minimum mass and minimum material cost of the FGM plates. The design process of FGM plate structures requires a good choice of metal and ceramic materials and the adequate definition of the components volume fractions through the thickness direction in order to accomplish a certain structural behavior, while optimizing the material costs and/or the plate mass. Here, the optimization problems are solved with the simulated annealing (SA) algorithm, not requiring the calculation of the derivatives of the objective or constraint functions. Constrained single objective optimization cases are studied, and validated with alternative solutions, considering the p-index and the FGM plate thickness as design variables. New optimization cases, involving additionally the metal and ceramic materials as design variables, are presented both for benchmark purposes and to demonstrate the suitability of the SA algorithm to solve those optimization problems.
journaltitleApplied Sciences
titleOptimization of Metal–Ceramic Functionally Graded Plates Using the Simulated Annealing Algorithm

issn0934-4373, 1615-1488
journaltitleStructural Optimization
titleA closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems

journaltitleIEEE Transactions on Evolutionary Computation
shorttitleA fast and elitist multiobjective genetic algorithm
titleA fast and elitist multiobjective genetic algorithm: NSGA-II

journaltitleLecture Notes in Engineering
titleXFOIL: An Analysis and Design System for Low Reynolds Number Airfoils

WORLD SCIENTIFIC

booktitleFrontiers of Computational Fluid Dynamics 1998
titlePros & Cons of Airfoil Optimization

journaltitleAerospace Science and Technology
titleWinglet multi-objective shape optimization

booktitleCurves and Surfaces for Computer-Aided Geometric Design
titleThe Bernstein Form of a Bézier Curve

journaltitleFrontiers in Marine Science
titleAn Efficient Multi-Objective Optimization Method for Use in the Design of Marine Protected Area Networks

issn2095-0233, 2095-0241
journaltitleFrontiers of Mechanical Engineering
titleIntelligent methods for the process parameter determination of plastic injection molding

Hefei, China

booktitleProceedings of the 3rd World Congress on Intelligent Control and Automation (Cat. No.00EX393)
titleStudy on multi-objective genetic algorithm

abstractMetamaterials (MMs) are composites that are artificially engineered to have unconventional mechanical properties that stem from their microstructural geometry rather than from their chemical composition. Several studies have shown the effectiveness of viscoelastic MMs in vibration attenuation due to their inherent vibration dissipation properties and the Bragg scattering effect. This study presents a multiobjective optimization based on genetic algorithms (GA) that aims to find a viscoelastic MM crystal with the highest vibration attenuation in a chosen low-frequency range. A multiobjective optimization allows considering the attenuation due to the MM inertia versus the Bragg scattering effect resulting from the periodicity of the MM. The investigated parameters that influence wave transmission in a one-dimensional (1D) MM crystal included the lattice constant, the number of cells and the layers’ thickness. Experimental testing and finite element analysis were used to support the optimization procedure. An electrodynamic shaker was used to measure the vibration transmission of the three control specimens and the optimal specimen in the frequency range 1–1200[Formula: see text]Hz. The test results demonstrated that the optimized specimen provides better vibration attenuation than the control specimens by both having a band-gap starting at a lower frequency and having less transmission at its passband.
issn1758-8251, 1758-826X
journaltitleInternational Journal of Applied Mechanics
titleOptimization of Viscoelastic Metamaterials for Vibration Attenuation Properties

Honolulu, Hawaii

American Institute of Aeronautics

booktitle53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR>20th AIAA/ASME/AHS Adaptive Structures Conference<BR>14th AIAA
titleA POD-based Reduced Order Design Scheme for Shape Optimization of Air Vehicles

journaltitleJournal of Solar Energy Engineering
titleNew Airfoils for Small Horizontal Axis Wind Turbines

Honolulu, Hawaii

American Institute of Aeronautics

booktitle53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference
titleHybrid Optimization for Wind Turbine Thick Airfoils

journaltitleCogent Engineering
shorttitleA review of multi-objective optimization
titleA review of multi-objective optimization: Methods and its applications

issn1389-4420, 1573-2924
journaltitleOptimization and Engineering
titleAn adaptive geometry parametrization for aerodynamic shape optimization

journaltitleAerospace Science and Technology
titleRobust aerodynamic shape optimization—From a circle to an airfoil

issn0021-8669, 1533-3868
journaltitleJournal of Aircraft
titleWing Design by Numerical Optimization

issn0885-7474, 1573-7691
journaltitleJournal of Scientific Computing
titleAerodynamic design via control theory

journaltitleComputers & Fluids
titleThirty years of development and application of CFD at Boeing Commercial Airplanes, Seattle

journaltitleJournal of Aircraft
titleAerodynamic and Aeroacoustic Optimization of Rotorcraft Airfoils via a Parallel Genetic Algorithm

abstractModern wind turbine airfoil designs are increasingly emphasizing low sensitivity to the leading edge roughness in addition to good aerodynamic performances under varying wind conditions. In this study, a multi-point robust design optimization method has been systematically established for the wind turbine airfoil. The objective is to not only maximize the lift-to-drag ratio and lift coefficient at and near the design point, but also minimize their sensitivity to the leading edge roughness associated with the geometry profile uncertainty. The geometry parameterization of the airfoil is conducted by the Bezier curves. In the robust optimization, the multi-objective genetic algorithm, Monte Carlo simulation technique, and artificial neural network model are used. The results show that both the robust Pareto optimal (RPO) and deterministic Pareto optimal (DPO) airfoils have higher lift-to-drag ratio and lift coefficient than the original design FX 63-137 at and near the design point. A smaller maximum camber and larger radii near the leading edge help the RPO airfoil outperform both the DPO and original ones in terms of low sensitivity to the leading edge roughness. This study may be useful to the future development of the wind turbine airfoils with higher efficiency and reliability.
issn0957-6509, 2041-2967
journaltitleProceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy
titleMulti-point robust design optimization of wind turbine airfoil under geometric uncertainty

abstractAbstract The use of surrogate models (response surface models, curve fits) of various types (radial basis functions, Gaussian process models, neural networks, support vector machines, etc.) is now an accepted way for speeding up design search and optimization in many fields of engineering that require the use of expensive computer simulations, including problems with multiple goals and multiple domains. Surrogates are also widely used in dealing with uncertainty quantification of expensive black-box codes where there are strict limits on the number of function evaluations that can be afforded in estimating the statistical properties of derived performance quantities. Here, we tackle the problem of robust design optimization from the direction of Gaussian process models (Kriging). We contrast two previously studied models, co-Kriging and combined Kriging (sometimes called level 1 Kriging), and propose a new combined approach called combined co-Kriging that attempts to make best use of the key ideas present in these methods.
journaltitleJournal of Computational Design and Engineering
titleRobust design optimization using surrogate models

VIRTUAL EVENT

American Institute of Aeronautics

booktitleAIAA AVIATION 2020 FORUM
titleGeneric Modal Design Variables for Aerodynamic Shape Optimisation

journaltitleAIAA Journal
titleMultipoint Aerodynamic Shape Optimization Investigations of the Common Research Model Wing

journaltitle2019 IEEE 10th International Conference on Mechanical and Aerospace Engineering (ICMAE)
titleoptimization of an Airfoil Characteristics to Minimize the Turn Radius of a Small Unmanned Aerial Vehicle

Portsmouth, Virginia

American Institute of Aeronautics

booktitle11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
title"Fundamental" Parameteric Geometry Representations for Aircraft Component Shapes

journaltitleComputer Applications in Engineering Education
titleUse of XFOIL in design of camber-controlled morphing UAVs

journaltitleIEEE Computer Graphics and Applications
titleNURBS-based free-form deformations

abstractAn improved aerodynamic optimization technique for two-dimensional wind turbine airfoils is presented in this article. On the basis of the combination of response surface method and uniform experimental design, a space filling design of experiment, this technique optimizes the lift-to-drag ratio of an airfoil at the design angle of attack. In order to reduce the number of design variables, the upper and lower surfaces of a reference airfoil are, respectively, fitted by using two B-spline curves with only four control points each. By using the ordinates of the eight control points as design variables, the ranges of the design variables are set and the airfoil is parameterized. Then, according to the uniform design tables, the values of each design variable at different levels are determined and various airfoil shapes are generated by using the specified combinations of the design variables. Finally, the optimal ordinates of the eight control points are obtained using a quadratic polynomial expression that is the regression of the computational results of flows around the airfoils. The computational results of flows around the optimized shapes of four airfoils reveal that the proposed technique, compared with some other optimization methods in the literature, is a time-saving and effective method to seek better aerodynamic performance for wind turbine airfoils.
issn0957-6509, 2041-2967
journaltitleProceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy
titleAerodynamic optimization of wind turbine airfoils using response surface techniques

journaltitleJournal of Renewable and Sustainable Energy
titleA method to evaluate the overall performance of the CAS-W1 airfoils for wind turbines

journaltitleRenewable Energy
titleImproving wind turbine blade based on multi-objective particle swarm optimization

titleInvestigation of the Kline-Fogleman airfoil section for rotor blade applications

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleGeometric Comparison of Aerofoil Shape Parameterization Methods

Kissimmee, Florida

American Institute of Aeronautics

booktitle53rd AIAA Aerospace Sciences Meeting
titleReview of Aerofoil Parameterisation Methods for Aerodynamic Shape Optimisation

abstractExperimental results were obtained for an Eppler 387 airfoil in the Langley Low Turbulence Pressure Tunnel. The tests were conducted over a Mach number range from 0.03 to 0.13 and a chord Reynolds number range for 60,000 to 460,000. Lift and pitching moment data were obtained from airfoil surface pressure measurements and drag data for wake surveys. Oil flow visualization was used to determine laminar separation and turbulent reattachment locations. Comparisons of these results with data on the Eppler 387 airfoil from two other facilities as well as the Eppler airfoil code are included.
titleExperimental results for the Eppler 387 airfoil at low Reynolds numbers in the Langley low-turbulence pressure tunnel

New York

Springer Science+Business Media, LLC

titleNonlinear multiobjective optimization

journaltitleAerospace Science and Technology
titleXFOIL vs CFD performance predictions for high lift low Reynolds number airfoils

journaltitleAnnual Review of Fluid Mechanics
titleAERODYNAMICS OF SMALL VEHICLES

abstractAbstract This work aims to review the proposed trajectory optimization algorithms and their applications in solving flight trajectory planning and optimal control problems. Beside the classification of trajectory optimization algorithms listed in Chai R, Savvaris A, Tsourdos A and Chai S such as the deterministic; the stochastic; and the multi-objective algorithms, this work adds in reviewing the effective and frequently-used approach for solving optimal problems— the hybrid/combined methods. The results of related works show that this hybrid approach is effective and can provide feasible solutions for solving the flight vehicle trajectory optimal design and control problems.
issn1742-6588, 1742-6596
journaltitleJournal of Physics: Conference Series
titleA review of Hybrid/combined methods for trajectory optimization of flight vehicles

issn0178-7675, 1432-0924
journaltitleComputational Mechanics
titleFinding the optimal shape of the leading-and-trailing car of a high-speed train using design-by-morphing

journaltitleAIAA SCITECH 2022 Forum
titleInvestigation of a Smooth Local Correlation-based Transition Model in a Discrete-Adjoint Aerodynamic Shape Optimization Algorithm

San Diego, California

American Institute of Aeronautics

booktitleAIAA Scitech 2019 Forum
titleEfficient Aero-Structural Wing Optimization Using Compact Aerofoil Decomposition

Tangerang Selatan, Indonesia

titleSingle- and multi-objective optimization of a low-speed airfoil using genetic algorithm

journaltitle35th Wind Energy Symposium
titleImproving Airfoil Lift Prediction

abstractAerospace design optimization typically explores the effects of structural performance and aerodynamics on the geometry of a component. This paper presents a methodology to incorporate manufacturing cost and fatigue life models within an integrated system to simultaneously trade off the conflicting objectives of minimum weight and manufacturing cost while satisfying constraints placed by structural performance and fatigue. A case study involving the design of a high pressure turbine disk from an aircraft engine is presented. Manufacturing cost and fatigue life models are developed in DECISIONPRO™, a generic modeling tool, whereas finite element analysis is carried out in the Rolls-Royce PLC proprietary solver SC03. A multiobjective optimization approach based on the nondominated sorting genetic algorithm (NSGA) is used to evaluate the Pareto front for minimum cost and volume designs. A sequential workflow of the different models embedded within a scripting environment developed in MATLAB™ is used for automating the entire process.
journaltitleJournal of Mechanical Design
titleApplying Multiobjective Cost and Weight Optimization to the Initial Design of Turbine Disks

journaltitleApplied Mathematical Modelling
titleAn airfoil optimization technique for wind turbines

journaltitleRenewable and Sustainable Energy Reviews
titleMulti-objective optimization of wind farm layouts – Complexity, constraint handling and scalability

abstractA new parameterization method is developed for airfoils. The method combines the best characteristics of two previous polynomial methods: the Bézier parameterization and the PARSEC parameterization. The new Bézier–PARSEC (BP) parameterization is a second-order continuous combination of four Bézier curves, two each for the camber and thickness distributions. Twelve parameters control the geometry. These are all based on the aerodynamics of the shape, such as leading edge radius and angle, rather than on the Bézier control points themselves. The intent of the new parameterization is to improve the characteristics of the design space in an aerodynamic optimization problem by reducing the nonlinear interactions between design parameters. It is demonstrated that the new method is able to parameterize a broad range of airfoils, and that it has potential for accelerating convergence of evolutionary aerodynamic optimization.
journaltitleCanadian Aeronautics and Space Journal
titleBézier–PARSEC parameterization for airfoil optimization

journaltitleJournal of Wind Engineering and Industrial Aerodynamics
titleStatic pressure measurements on a rotating and a non-rotating 2.375 m wind turbine blade. Comparison with 2D calculations

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleSurvey of Shape Parameterization Techniques for High-Fidelity Multidisciplinary Shape Optimization

journaltitleJournal of Renewable and Sustainable Energy
titleMulti-objective optimization of airfoil shape for efficiency improvement and noise reduction in small wind turbines

titleSTRUCTURAL OPTIMIZATION OF AEROFOILS FOR MANY CRITERIA

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleShape design for thin-walled beam cross sections using rational B splines

abstractA technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved.The scheme is based on trivariate Bernstein polynomials, and provides the designer with an intuitive appreciation for its effects.
journaltitleACM SIGGRAPH Computer Graphics
titleFree-form deformation of solid geometric models

Department of Aeronautical
Astronautical Engineering University of Illinois at Urbana-Champaign

titleUIUC Airfoil Database

Virginia Beach, Va

SoarTech Publications

titleSummary of low speed airfoil data

abstractMethodologies for reducing the design-space dimensionality in shape optimization have been recently developed based on unsupervised machine learning methods. These methods provide reduced dimensionality representations of the design space, capable of maintaining a certain degree of the original design variability. Nevertheless, they usually do not allow to use directly the original parameterization method, representing a limitation to their widespread application in the industrial field, where the design parameters often pertain to well-established parametric models, e.g. CAD (computer aided design) models. This work presents how to embed the parametric-model original parameters in a reduced-dimensionality representation of the design space. The method, which takes advantage from the definition of a newly-introduced generalized feature space, is demonstrated, as a proof of concept, for the reparameterization of 2D Bezier curves and 3D free-form deformation design spaces and the consequent solution of simulation-driven design optimization problems of a subsonic airfoil and a naval destroyer in calm water, respectively.
journaltitleComputer Methods in Applied Mechanics and Engineering
titleParametric model embedding
Dimensionality reduction,Representation learning,Karhunen–Loève expansion,Parametric model embedding,Simulation-driven optimization,Shape optimization

journaltitleAPS Division of Fluid Dynamics Meeting Abstracts
titleShape optimization methodology for fluid flows using mixed variable Bayesian optimization and design-by-morphing

Elsevier B.V.

journaltitleComputer Methods in Applied Mechanics and Engineering
titleOptimization of the Shape of a Hydrokinetic Turbine's Draft Tube and Hub Assembly Using Design-by-Morphing with Bayesian Optimization
bayesian optimization,design-by-morphing,draft-tubes,hydro-kinetic turbines,mixmobo,shape optimization

journaltitleAPS Division of Fluid Dynamics Meeting Abstracts
titleVertical axis wind turbine design using design-by-morphing and Bayesian optimization

journaltitleStructural and Multidisciplinary Optimization
titleBayesian optimization for mixed-variable, multi-objective problems
itected meta-materials,Bayesian optimization,HedgeMO,MixMOBO,Mixed variables,Multi-objective

abstractThis article aims to present a two-dimensional parametric analysis of a modified Savonius wind turbine using computational fluid dynamics. The effects of three independent parameters of the rotor, namely, shape factor, overlap ratio, and tip speed ratio on turbine performance were studied and then optimized for maximum coefficient of performance using response surface methodology. The rotor performance was analyzed over specific domains of the parameters under study, and three-variable Box-Behnken design was used for design of experiment. The specific parametric combinations as per design of experiment were simulated using ANSYS Fluent ® , and the response variable, coefficient of performance (C p ), was calculated. The sliding mesh model was utilized, and the flow was simulated using Shear Stress Transport (SST) k − ω model. The model was validated using past experimental results and found to predict parametric effects accurately. Minitab ® and ReliaSoft DOE++ ® were used to develop regression equation and find the optimum combination of parameters for coefficient of performance over the specified parametric domains using response surface methodology.
issn0309-524X, 2048-402X
journaltitleWind Engineering
titleComputational fluid dynamics analysis of a modified Savonius rotor and optimization using response surface methodology

journaltitleApplied Soft Computing
titleState-of-the-art in aerodynamic shape optimisation methods

journaltitleThe 26th Congress of ICAS and 8th AIAA ATIO
titleQuest for a Truly Parsimonious Airfoil Parameterization Scheme

Vieweg+Teubner Verlag

booktitleRecent Development of Aerodynamic Design Methodologies
titleParametric Airfoils and Wings

Fort Worth, Texas

American Institute of Aeronautics

booktitle13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference
titleA Comparison of Airfoil Shape Parameterization Techniques for Early Design Optimization

abstractTo further reduce the computational expense of metamodel-based design optimization (MBDO), a novel sequential radial basis function (RBF)-based optimization method using virtual sample generation (SRBF-VSG) is proposed. Different from the conventional MBDO methods with pure expensive samples, SRBF-VSG employs the virtual sample generation mechanism to improve the optimization efficiency. In the proposed method, a least squares support vector machine (LS-SVM) classifier is trained based on expensive real samples considering the objective and constraint violation. The classifier is used to determine virtual points without evaluating any expensive simulations. The virtual samples are then generated by combining these virtual points and their Kriging responses. Expensive real samples and cheap virtual samples are used to refine the objective RBF metamodel for efficient space exploration. Several numerical benchmarks are tested to demonstrate the optimization capability of SRBF-VSG. The comparison results indicate that SRBF-VSG generally outperforms the competitive MBDO methods in terms of global convergence, efficiency, and robustness, which illustrates the effectiveness of virtual sample generation. Finally, SRBF-VSG is applied to an airfoil aerodynamic optimization problem and a small Earth observation satellite multidisciplinary design optimization problem to demonstrate its practicality for solving real-world optimization problems.
journaltitleJournal of Mechanical Design
titleSequential Radial Basis Function-Based Optimization Method Using Virtual Sample Generation

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleGeometric Filtration Using Proper Orthogonal Decomposition for Aerodynamic Design Optimization

journaltitleScience Advances
shorttitleStrength through defects
titleStrength through defects: A novel Bayesian approach for the optimization of architected materials

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleAirfoil and Wing Design Through Hybrid Optimization Strategies

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleDimension Reduction for Aerodynamic Design Optimization

issn0001-1452, 1533-385X
journaltitleAIAA Journal
titleConstrained Design Optimization Using Generative Topographic Mapping

issn1367-4803, 1367-4811
titleArtificial intelligence-based multi-objective optimization protocol for protein structure refinement

journaltitleComputer Methods in Applied Mechanics and Engineering
titleMulti-objective optimization of turbomachinery using improved NSGA-II and approximation model

Cambridge, UK ; New York

Cambridge University Press

seriesCambridge monographs on applied and computational mathematics
titleScattered data approximation
Approximation theory,Multivariate analysis

journaltitleFrontiers in Energy Research
titleThe Parametric Modeling and Two-Objective Optimal Design of a Downwind Blade

journaltitleJournal of Mechanical Design
titleA Shape Parameterization Method Using Principal Component Analysis in Applications to Parametric Shape Optimization

journaltitleApplied Mathematics and Mechanics
titleRobust design of natural laminar flow supercritical airfoil by multi-objective evolution method

abstractIn this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Pareto-optimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search.
issn1063-6560, 1530-9304
journaltitleEvolutionary Computation
shorttitleComparison of Multiobjective Evolutionary Algorithms
titleComparison of Multiobjective Evolutionary Algorithms: Empirical Results