Airfoil GAN: Encoding and Synthesizing Airfoils forAerodynamic-aware Shape Optimization
AAirfoil GAN: Encoding and Synthesizing Airfoils forAerodynamic-aware Shape Optimization
Yuyang Wang, Kenji Shimada and Amir Barati Farimani
Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213
The current design of aerodynamic shapes, like airfoils, involves computationally intensivesimulations to explore the possible design space. Usually, such design relies on the prior defi-nition of design parameters and places restrictions on synthesizing novel shapes. In this work,we propose a data-driven shape encoding and generating method, which automatically learnsrepresentations from existing airfoils and uses the learned representations to generate newairfoils. The representations are then used in the optimization of synthesized airfoil shapesbased on their aerodynamic performance. Our model is built upon VAEGAN, a neural networkthat combines Variational Autoencoder with Generative Adversarial Network and is trainedby the gradient-based technique. Our model can (1) encode the existing airfoil into a latentvector and reconstruct the airfoil from that, (2) generate novel airfoils by randomly samplingthe latent vectors and mapping the vectors to the airfoil coordinate domain, and (3) synthesizeairfoils with desired aerodynamic properties by optimizing learned features via a genetic algo-rithm. Our experiments show that the learned features encode shape information thoroughlyand comprehensively without predefined design parameters. By interpolating/extrapolatingfeature vectors or sampling from Gaussian noises, the model can automatically synthesizenovel airfoil shapes, some of which possess competitive or even better aerodynamic propertiescomparing with training airfoils. By optimizing shape on learned features via a genetic algo-rithm, synthesized airfoils can evolve to have specific aerodynamic properties, which can guidedesigning aerodynamic products effectively and efficiently.
Nomenclature 𝑧 = latent feature vectors of airfoilsˆ 𝑧 = random noise sampled from Gaussian distribution 𝑥 = airfoils˜ 𝑥 = airfoils reconstructed from 𝑧 ˆ 𝑥 = airfoils synthesized from ˆ 𝑧 L 𝑝𝑟𝑖𝑜𝑟 = prior loss a r X i v : . [ c s . C E ] J a n 𝑟𝑒𝑐𝑜𝑛 = reconstruction loss L 𝑙𝑎𝑦𝑒𝑟 = layer like loss L 𝐺𝐴𝑁 = adversarial GAN loss 𝐶 𝑙 = lift coefficient 𝐶 𝑑 = drag coefficient I. Introduction R ecent years have witnessed the success of deep learning [1] in many fields like computer vision [2], naturallanguage process [3] and robotics [4] [5]. Such data-driven methods can automatically learn compact andcomprehensive representations from samples. However, most of the prevalent deep learning models are based onsupervised learning, meaning the samples are paired with manually tagged labels. Such supervision makes the modelhard to generalize since both the amount of labeled data and the information contained in the label are limited. Hence,self-supervised learning is proposed to learn the features directly from data. Variational Autoencoder (VAE) [6] [7]follows the insight via an encoder-decoder structure, where the encoder down-samples high-dimensional input into alatent feature domain while the decoder reconstructs the sample from learned low-dimensional feature. By minimizingthe difference between the reconstructed sample and the original one, VAE automatically learns features without anylabels. Generative Adversarial Network (GAN) [8] pushes learning from self-supervision even further via a min-maxgame between a generator and a discriminator. The discriminator works as a classifier to determine real samples fromsynthesized fake ones; meanwhile, the generator which synthesizes samples from random noise is intended to cheat thediscriminator. By jointly training the two components, GAN can generate super high-quality realistic samples [9], whichVAE fails to achieve. Through learning to reconstruct or synthesizing samples, the self-supervised models automaticallyencode high-dimensional input into informative features, which can be generalized to different tasks without restrictionsfrom human labels.Such a self-supervised learning manner can help the parameterization of various geometries using the learnedrepresentations. Geometry parameterization plays an important role in shape design [10–12], and geometry heavilyinfluences the performance, especially in the design of aerodynamic products like airfoils. A practical and effective airfoildesign must meet certain aerodynamic requirements, like lift, drag, pitching moment, and critical-speed characteristics[13]. However, due to the curse of dimensionality, design optimization based on CFD simulations can be hard or eveninfeasible on the airfoil geometry domain, especially for gradient-free methods [14]. Therefore, parameterization ordimension-reduction is required to define the design domain before any design optimization is applied. Traditionalparameterization or dimension-reduction techniques rely on manually selected design parameters like control pointsof Bézier curves [15] or B-splines [16], which places restrictions on the generalization to various shapes as well as2ynthesizing novel geometries. Implementation of self-supervised deep learning methods on shape parameterization,like VAE and GAN, can overcome the limitations of traditional techniques and synthesize shapes with great novelty,which can provide insights for future geometric design.In this work, VAEGAN [17], which is a combination of Variational Autoencoder (VAE) and Generative AdversarialNetwork (GAN), is utilized to learn feature vectors and generate novel airfoils without any human prior. VAEGANtakes advantage of both VAE and GAN. With the encoder-decoder architecture from VAE, the model learns to explicitlyencode an existing airfoil shape into a low-dimensional feature domain and reconstruct the shape with little error. Withthe discriminator from GAN, our model can generate a large number of high-quality novel airfoils from random noisewith no manually designed parameters.Our experiments show that generated airfoils are smooth even without any smoothing post-process. K-meansclustering in the learned feature domain demonstrates that feature vectors encode essential shape information in away that each cluster represents various shape patterns. Further test on learned latent features illustrates that differentgeometry information is encoded in each dimension of the representation. The performance of the model in synthesizingnovel airfoils is examined as well. By either interpolation or extrapolation of feature vectors, a synthesized airfoilinherits features of parent samples, while generated airfoils from sampled Gaussian noise show great novelty in a waythat it is not a simple combination of two existing airfoils. A further experiment on the aerodynamic properties ofsynthesized airfoils, either by interpolating, extrapolating, or sampling, indicates the synthesized airfoils can possesscompetitive aerodynamic properties, and some even surpass the existing ones. With a genetic algorithm [18], airfoilgeometries can be optimized on the feature domain and evolve to possess specific aerodynamic properties. Our modelproves its ability to parameterize the existing airfoil shape as well as generating novel and practical airfoils; both lead todesigning the new generation of airfoils more intelligently and efficiently without intensely relying on experimentalexperience or manually design parameters. II. Related Works
We propose to use a generative model to synthesize novel airfoils without any predefined design parameters. Throughtraining via the gradient-based method, our deep learning model also automatically learns to parameterize airfoilsinto latent feature vectors. To better address the insight of our work, this section reviews previous work on shapeparameterization, especially on aerodynamic geometries and the implementation of deep learning on geometry designand synthesis.
A. Geometry Parameterization and Dimension Reduction
A lot of work has been done in parameterizing complex shapes and reducing geometric dimensions. Some researchfocuses on analytically expressing the curves. In [19], it is introduced that by adding analytic shape functions to the3aseline shape, a compact formulation for parameterization can be obtained. The design variables, in this case, are thecoefficients correlated with shape functions. By this means, the analytical function of the airfoil curve is formulated.Also, [20] follows the same formulation strategy but with different shape functions. Another common and efficientmethod is PARSEC [21], which defines eleven geometric parameters to thoroughly express the airfoil shape, includingupper and lower curvature, thickness, leading-edge radius, etc. With the defined parameters, a linear combination ofshape functions is introduced to describe the airfoil shape as well. Such methods work well for a specific set of curvesbut may fail to express complex geometries since they rely on manually designed parameters.Polynomial and spline are also utilized to help dimension reduction. With different orders of polynomials as thebasis, the airfoil shape can be described as a linear combination of the basis [22, 23]. However, high-order terms canoverfit to high-frequency noise, especially when coefficients are of different magnitudes. Besides polynomial, Béziercurve [15], which is built upon the Bernstein polynomials, is another mathematical formulation of curves. In detail, 𝑛 + 𝑛 -degree Bézier curve. Although Bézier and polynomial curves aremathematically equivalent, Bézier usually perform better in controlling a curve since control points are closely related tothe curve position and shape. To mitigate the rounding error, De Casteljau [24], a recursive algorithm, is introduced tocompute the Bernstein polynomials numerically. Besides, Bézier-PARSEC [16], which combines Bézier and PARSEC,uses PARSEC parameters to define Bézier curves. The B-spline curve with B-spline basis functions is also utilized todescribe the airfoil shape. Yet B-spline formulation fails to represent implicit conic sections accurately. That is whyNon-uniform rational B-spline (NURBS) is introduced [25]. NURBS can accurately represent both standard geometricobjects like lines, circles, ellipses, and cones, as well as free-form geometry, which are prevalent in industrial design.Another popular technique in shape parameterization is free-form deformation (FFD), which utilizes high-levelshape deformation instead of lower-level geometric entities to represent a shape. Based on this insight, [26] presents atechnique which can apply deformation either locally or globally based on trivariate Bernstein polynomials. Deformationis manipulated by control points of trivariate Bézier volumes. In [27], an extended free-form deformation (EFFD)method is presented, which allows arbitrarily shaped deformations by using non-parallelepiped lattices. Researchpresented in [28] incorporates FFD and sensitivity analysis where geometry changes and structural responses arecorrelated, and the shape satisfying deformation or stress constraints can be found easily.Other methods like leveraging camber and thickness mode shapes derived from existing airfoils are also usedto parameterize the airfoil shapes [29]. Also, the linear reduction method like the SVD is utilized to extract airfoilrepresentations and optimize shape design [30]. The conventional dimension reduction or parameterization techniqueshave been implemented in different scenarios and successfully represented the existing airfoil shapes. However, thesemethods usually require pre-defining the design space as well as the boundary of design space, like the design parametersin NACA, shape functions, etc., which can degrade the synthesis of novel/new airfoils and optimization towards thedesired design.” 4 . Deep Learning in Geometry Design In recent years, deep learning has been a great success in extracting informative features from data [1]. Especially in asupervised visual learning manner, with the invention of convolutional neural network (CNN) [31, 32], deep learninghas been general solutions in many fields, like image classification [33] [34], object detection [35, 36], and segmentation[37, 38]. Also, with the introduction to GAN [8], deep learning in a self-supervised manner has been widely used insynthesizing realistic samples, and some sophisticated GAN-based model can generate high-resolution images whichare even hard to distinguish by humans [9]. With the ability to extract representative features and synthesize realisticsamples, deep learning has been widely used to make geometric design more systematic and efficient. This section willintroduce some work of deep learning in geometric design, especially in aerodynamic shape design.In the work of [39], a multi-layer perceptron (MLP), which takes the angle of attack and flap setting as input, istrained to predict multiple aerodynamic coefficients, including lift coefficient, drag coefficient, and moment of inertia.Further, [40] utilizes CNN with airfoil images as input to learn the lift coefficients of different airfoils in multiple flowMach numbers, Reynolds numbers, and diverse angles of attack. Similarly, in [41], CNN is implemented to predictthe pressure coefficient value at the test point. Moreover, [42] presents a CNN-based method to learn the correlationbetween airfoil geometry and pressure distribution. The model can also conduct an inverse airfoil design given thepressure.On the other hand, the GAN model [8], as a self-supervised learning framework, has been a prevalent and greatsuccess in generating realistic samples. GAN, with a discriminator and a generator, is designed to learn features andgenerate samples without manually tagged labels. The discriminator is a classifier telling the true input from thesynthesized fake one, while the generator is intended to generate plausible samples to cheat the discriminator. Sincefirst introduced [8], many pieces of research have been dedicated to pushing the edge of GAN. In Wasserstein GAN[43], by introducing Wasserstein distance, the quality of generated samples can be well measured during training.ConditionalGAN [44] and InfoGAN [45] extend GAN to generate a sample from various categories within one model.Also, DCGAN [46] and StyleGAN [9] can generate high-quality realistic samples with sophisticated architectures.With the ability to generate plausible samples, GAN provides a powerful architecture to learn representations that canhelp shape design as well. Based on this insight, [47, 48] introduce BézierGAN, which uses a GAN to generate Béziercurve control points and then uses the control points to formulate the boundary of airfoils. Such pipeline guaranteesgenerated airfoils to be smooth, and further shape optimization can be conducted on the feature domain [48]. Such amethod is, however, restricted to Bézier curves and fails to encode existing airfoil shapes explicitly. Further, GAN can beimplemented in generating three-dimensional samples, as shown in [49–51]. [49] proposes to use a three-dimensionalconvolutional layer to generate volumetric objects. While [50] trains a generative model of three-dimensional shapesurfaces, which directly encodes surface geometry and shape structure, [52] further proposes a model to representprobabilistic relationships between properties of shape components and relates them to learned underlying causes of5tructural variability within the domain.In our work, we propose to use a VAEGAN-based model [17] to extract features of airfoil shapes and synthesize newairfoil designs. The learned latent features encode the airfoil shape and can be utilized to synthesize new designs throughinterpolation or extrapolation. In comparison to conventional parameterization methods, our model can generate awider variety of new airfoils, and some show promising aerodynamic properties. By applying the genetic algorithm, theVAEGAN synthesized airfoils can be optimized to desired aerodynamic properties.
III. Proposed Method
A. Data Pre-processing
The UIUC Coordinates Database [53], which contains more than 1,600 2-dimensional airfoils, is used to train thegenerative model. Each airfoil in the database is represented by varying numbers of points with x, y coordinates. Suchvariation restricts data to be fed directly into a neural network model, which requires a homogeneous input. To deal withit, we first scale the x coordinates of all airfoils to [ , ] . Then all airfoils are interpolated by splines, and 𝑁 points areselected with 𝑥 given in Eq. 1, where 𝑖 represents the index of each point. 𝜃 𝑖 = 𝜋 ( 𝑖 − ) 𝑁𝑥 𝑖 = − cos ( 𝜃 𝑖 ) (1)In our case, 𝑁 is set to be 200 with 100 points from the upper boundary and the other 100 from the lower boundary. Bythis means, all the interpolated airfoils share the same x coordinates. Therefore, only the y coordinates of each airfoilare fed into the model, which reduces the dimensionality of the data. Finally, all y coordinates are scaled to [− , ] bymultiplying a normalization coefficient. As illustrated in Fig 1, the first row shows the original airfoils from the UIUCCoordinate Database, while the second row shows the corresponding processed airfoils. Fig. 1 Data pre-processing: the first row shows the origin airfoil coordinates from the UIUC database, thesecond row shows the corresponding processed airfoilsB. VAEGAN
Our model is based upon VAEGAN [17], which takes advantage of both VAE [6] [7] and GAN [8]. VAE contains twocomponents: an encoder and a decoder. The former encodes a high-dimensional sample, 𝑥 , into a low-dimensional6atent representation, 𝑧 . While the decoder takes as input the latent vector, 𝑧 , and upsamples from the representationdomain to the original data domain, ˜ 𝑥 . The encoder and decoder are given as: 𝑧 ∼ Enc ( 𝑥 ) = 𝑞 ( 𝑧 | 𝑥 ) , ˜ 𝑥 ∼ Dec ( 𝑧 ) = 𝑝 ( ˜ 𝑥 | 𝑧 ) . (2)To regularize the encoder, VAE takes into consideration a prior distribution of the latent vector, 𝑝 ( 𝑧 ) . Here it is assumedthat 𝑧 ∼ N ( , I ) , which follows an isotropic Gaussian distribution. The loss function for VAE to minimize is given by: L 𝑉 𝐴𝐸 = L 𝑟𝑒𝑐𝑜𝑛 + L 𝑝𝑟𝑖𝑜𝑟 , (3)with L 𝑟𝑒𝑐𝑜𝑛 = || ˜ 𝑥 − 𝑥 || , and L 𝑝𝑟𝑖𝑜𝑟 = 𝐷 𝐾 𝐿 ( 𝑞 ( 𝑧 | 𝑥 )|| 𝑝 ( 𝑧 )) , (4)where L 𝑟𝑒𝑐𝑜𝑛 measures how well the reconstructed data, ˜ 𝑥 , is comparing to the original 𝑥 by the mean square error(MSE), and L 𝑝𝑟𝑖𝑜𝑟 is the Kullback Leibler divergence (KL divergence), which measures the difference between encodedrepresentation vectors and Gaussian distribution.VAE learns the representation of samples and can reconstruct them from 𝑧 with the encoder-decoder architecture.However, it suffers from poor performance in generating novel samples, which have not been seen before [8] [17]. Tothis end, a generative adversarial network (GAN) [8] is introduced, which contains a discriminator, D , and a generator, G , competing with each other in a self-supervised manner. G tries to generate plausible samples to fool D , while D keeps sharpening its decision boundary to determine synthesized fake samples from real ones. In detail, the generator, G ,is fed with random noise ˆ 𝑧 ∼ 𝑝 ( ˆ 𝑧 ) and maps the noise to the data sample domain to generate fake ˆ 𝑥 . The discriminatortakes both 𝑥 and ˆ 𝑥 to predict whether the input is from the real dataset or generated by G . The cross-entropy lossfunction for the min-max game is given as:min 𝐺 max 𝐷 L 𝐺𝐴𝑁 = log D ( 𝑥 ) + log ( − D (G( ˆ 𝑧 ))) . (5)As one may expect, GAN generates samples purely from random noise, making it hard to obtain an explicit mappingfrom the data domain to the feature domain. Therefore, we build our model upon VAEGAN, which combines VAEand GAN. Namely, the generator is replaced by an encoder-decoder structure from VAE, as shown in Figure 2. Noticethat in VAEGAN, the model generates a reconstructed sample, ˜ 𝑥 , given a real sample, 𝑥 , and meanwhile generates afake sample, ˆ 𝑥 , directly from noise, ˆ 𝑧 . Both ˜ 𝑥 and ˆ 𝑥 should be classified as fake by the discriminator, D , and only 𝑥 is7 ig. 2 VAEGAN combines the encoder-decoder structure from VAE and discriminator, D , from GAN recognized as the real sample. Hence GAN loss function L 𝐺𝐴𝑁 from Eq. 5 is modified to: L 𝐺𝐴𝑁 = log (D ( 𝑥 )) + log ( − D ( Dec ( 𝑧 ))) + log ( − D ( Dec ( Enc ( 𝑥 )))) , (6)which takes into consideration the real sample, 𝑥 , reconstructed sample, ˜ 𝑥 , and fake sample, ˆ 𝑥 . Besides, to stabilize thetraining process and sharpen the decision boundary of D , another loss function, L 𝑙𝑎𝑦𝑒𝑟 , is introduced when trainingthe encoder and decoder. L 𝑙𝑎𝑦𝑒𝑟 , as given in Eq. 7, measures the 𝑙 distance between the values of the neurons in oneparticular layer of D when fake samples are fed and the values when real samples are fed. L 𝑙𝑎𝑦𝑒𝑟 = ||D 𝑙 ( 𝑥 ) − D 𝑙 ( Dec ( ˆ 𝑧 ))|| . (7)The complete loss function for VAEGAN is a weighted combination of all the loss terms given by: L = 𝜆 L 𝑝𝑟𝑖𝑜𝑟 + 𝜆 L 𝑟𝑒𝑐𝑜𝑛 + 𝜆 L 𝑙𝑎𝑦𝑒𝑟 + 𝜆 L 𝐺𝐴𝑁 . (8)The three components, encoder, decoder and discriminator are trained jointly, and each term of the loss function isassigned with different weights 𝜆 when training each component. C. Airfoil Synthesis
By training on the UIUC Database, the VAEGAN model automatically learns to encode airfoils into latent features andreconstruct airfoils from the feature domain. The learned latent features can be directly utilized for dimension reductionand shape parameterization. Moreover, the VAEGAN model is intended to synthesize novel airfoils which are differentfrom samples in the training dataset. To this end, we propose three synthesis methods: interpolation, extrapolation, andsampling, all of which are conducted on the latent feature domain.8ore specifically, in interpolation or extrapolation, two airfoils from the UIUC Database are first mapped to latentfeature vectors, 𝑧 and 𝑧 , via a well-trained encoder. A new feature vector ¯ 𝑧 , which is an affine combination of 𝑧 and 𝑧 , is calculated as given in Eq. 9: ¯ 𝑧 = 𝜈𝑧 + ( − 𝜈 ) 𝑧 , (9)where 𝜈 is the coefficient controlling the weight between 𝑧 and 𝑧 . When 0 ≤ 𝜈 ≤
1, ¯ 𝑧 is an interpolated feature vector,else it is an extrapolation between 𝑧 and 𝑧 . The interpolated/extrapolated feature vector, ¯ 𝑧 , is then fed into the decoderto synthesize an airfoil. Also, such interpolation/extrapolation between two airfoils can be directly extended to a tripletcase. Given 𝑧 , 𝑧 , and 𝑧 are three feature vectors mapped from three different airfoils via the encoder, the expressionof triplet interpolation/extrapolation is shown in Eq. 10:¯ 𝑧 = 𝛼𝑧 + 𝛽𝑧 + 𝛾𝑧 , where 𝛼 + 𝛽 + 𝛾 = . (10)Similarly, when 0 ≤ 𝛼, 𝛽, 𝛾 ≤
0, ¯ 𝑧 is an interpolation of the three feature vectors, and an extrapolation otherwise.Besides interpolation and extrapolation, sampling is another method to synthesize novel airfoils. Unlike interpolationor extrapolation, which relies on feature vectors from existing airfoils, sampling generates airfoils directly from randomnoise. A feature vector ˆ 𝑧 is randomly sampled from an isotropic Gaussian distribution N ( , I ) and then mapped to anairfoil via the decoder. By this means, synthesized airfoils from sampling are less restricted since sampled latent vectorsare not constrained by features extracted from airfoils in the UIUC Database and are more likely to introduce novelty tothe synthesized shapes. D. Aerodynamic-aware Shape Optimization
So far, how the VAEGAN model is built and used to generate novel airfoils has been introduced. However, the noveltyin shape does not guarantee a better airfoil design. To design engineering effective airfoils, aerodynamic propertiesare supposed to be considered. To this end, we propose to use a genetic algorithm (GA) [54] [18] to optimize airfoilshapes by controlling feature vectors learned from the VAEGAN model so that the airfoils can evolve to have the desiredaerodynamic properties. Specifically, lift coefficient, 𝐶 𝑙 , and drag coefficient, 𝐶 𝑑 , which measure the aerodynamic forceperpendicular and horizontal to the direction of motion, are considered to evaluate the aerodynamic performance of thesynthesized airfoils.As a non-gradient optimization technique, GA is inspired by natural selection and is intended to force individuals togradually evolve to the optimal. Assume the GA has 𝑁 generations in total and 𝑀 individuals in each generation. In ourcase, individuals are feature vectors. We use 𝑧 𝑖 to represent all individuals in the 𝑖 th generation, and 𝑧 𝑖, 𝑗 for the 𝑗 th individual in the 𝑖 th generation; also the airfoil decoded from 𝑧 𝑖, 𝑗 is annotated as 𝑎 𝑖, 𝑗 . Similarly, 𝐶 𝑖, 𝑗𝑙 and 𝐶 𝑖, 𝑗𝑑 representslift and drag coefficients of 𝑎 𝑖, 𝑗 , respectively. The fitness score, 𝑠 𝑖, 𝑗 , is used to measure the aerodynamic performance of9he individual, 𝑧 𝑖, 𝑗 , as shown in Eq. 11: 𝑠 𝑖, 𝑗 = −( 𝐶 𝑖, 𝑗𝑙 − 𝐶 𝑡𝑙 𝐶 𝑡𝑙 ) − ( 𝐶 𝑖, 𝑗𝑑 − 𝐶 𝑡𝑑 𝐶 𝑡𝑑 ) , (11)where the square of the difference between the target and current aerodynamic coefficients is calculated and normalizedby the squared target 𝐶 𝑡𝑙 and 𝐶 𝑡𝑑 . The fitness score is supposed to approach zero as individuals evolve on each generation.As shown in Algorithm 1, the initial generation, 𝑧 , is randomly sampled from an isotropic Gaussian distribution, N ( , I ) . The GA starts with the selection from the initial generation by randomly picking two individuals and comparingtheir fitness scores. The one with a higher fitness score wins the tournament and becomes one of the parents. 𝑝 𝑖 and 𝑝 𝑖 denote all the parents 1 and parents 2 in the 𝑖 th generation respectively. Single-point crossover is then implemented togenerate offspring from parents 1 and 2. Namely, a crossover point on the parent vector is randomly selected, and allelements after that point are swapped between the two parents. Mutation in the natural selection process is also imitatedwith additive Gaussian noises. Algorithm 1
Aerodynamic-aware Shape Optimization via GA procedure GA(
𝑁, 𝑀, 𝐶 𝑡𝑙 , 𝐶 𝑡𝑑 , 𝑝 ) Initialize 𝑖 : = Sample first generation 𝑧 𝑗 ∼ N ( , I ) , for 0 < 𝑗 < 𝑀 while 𝑖 < 𝑁 do Synthesize airfoils 𝑎 𝑖, 𝑗 from 𝑧 𝑖, 𝑗 via decoder Compute 𝐶 𝑖, 𝑗𝑙 , 𝐶 𝑖, 𝑗𝑑 and fitness score 𝑠 𝑖, 𝑗 = −( 𝐶 𝑖, 𝑗𝑙 − 𝐶 𝑡𝑙 𝐶 𝑡𝑙 ) − ( 𝐶 𝑖, 𝑗𝑑 − 𝐶 𝑡𝑑 𝐶 𝑡𝑑 ) Select 𝑀 parent 1, 𝑝 𝑖 , and 𝑀 parent 2, 𝑝 𝑖 , from 𝑧 𝑖 by tournament Generate next generation, 𝑧 𝑖 + , through single-point crossover With probability 𝑝 , 𝑧 𝑖 + , 𝑗 will add a Gaussian noise N ( , I ) 𝑖 : = 𝑖 + return The individual with the highest score from 𝑧 𝑖 IV. Experiments
Our VAEGAN model consists of 3 components: an encoder, a decoder, and a discriminator, which are all built onmulti-layer perceptron (MLP). As illustrated in Fig 2, the encoder encodes 200-dimensional airfoil coordinates into a32-dimensional feature domain while the decoder maps the feature back to the airfoil. The discriminator is a classifierexamining whether the input is a real airfoil from the UIUC Database, or a fake one reconstructed from the decoder,or synthesized from random noises. In detail, the encoder is modeled by a 3-layer MLP with the number of neurons [ , , ] in each layer, and LeakyReLU [55] is implemented as the activation function in each layer. The decoderis also a 3-layer MLP with the number of neurons [ , , ] in each layer. A hyperbolic tangent (Tanh) functionworks as the activation function in the output layer to scale all outputs into [− , ] . Similarly, the discriminator containsthree layers with the number of neurons [ , , ] , and outputs the probability of whether the input is real or fake10hrough a Sigmoid activation function.To automatically learn the latent features and synthesize airfoils, the VAEGAN model is trained on the UIUCDatabase for 5000 epochs, and each epoch goes through all the samples in the dataset. Initial learning rates for all threecomponents: encoder, decoder, and discriminator are set to be 0 . . /
100 of the database size. Adam optimizer [56] is utilized to updateall the parameters in the model. As mentioned in Section III C, different coefficients are assigned to each term in the lossfunction Eq. 8; also different components, namely the encoder, decoder, and discriminator, have different coefficients,respectively. Coefficients of different loss terms and components are shown in Eq. 12: L 𝐸𝑛𝑐 = . L 𝑝𝑟𝑖𝑜𝑟 + . L 𝑙𝑎𝑦𝑒𝑟 + L 𝑟𝑒𝑐𝑜𝑛 , L 𝐷𝑒𝑐 = . L 𝑝𝑟𝑖𝑜𝑟 + . L 𝑙𝑎𝑦𝑒𝑟 + L 𝑟𝑒𝑐𝑜𝑛 + L 𝐺𝐴𝑁 , L D = L 𝐺𝐴𝑁 , (12)where L 𝐸𝑛𝑐 , L 𝐷𝑒𝑐 , and L D represent loss functions for the encoder, decoder, and discriminator, respectively.To better investigate our VAEGAN-based model, we compare the performance with two other paramterizationmethods, principle component analysis (PCA) and variational autoencoder (VAE) [6]. PCA conducts a lineartransformation from the pre-processed airfoil point coordinates into prioritized latent variables. In our case, the top 32dimensions are kept as the feature. The VAE follows the same encoder-decoder architecture as the VAEGAN, but lacksthe discriminator. The latent feature dimension is also set to 32, and the loss function is given in Eq. 13: L 𝐸𝑛𝑐 = . L 𝑝𝑟𝑖𝑜𝑟 + L 𝑟𝑒𝑐𝑜𝑛 . (13) A. Airfoil Reconstruction via Encoder-decoderFig. 3 Reconstructed airfoils: the first row shows the airfoils from UIUC Coordinate Database, the second rowshows the reconstructed airfoils with VAE, and the third row shows the reconstructed airfoils with smoothness
The VAEGAN model can automatically learn feature vectors, namely mapping the high-dimension airfoils into11ow-dimension representations. To estimate whether or not the feature vector fully encodes the geometric information ofthe original airfoil,we first feed airfoils from UIUC Coordinate Database into the encoder to obtain the encoded featurevectors. The decoder then takes the vectors as input and outputs the reconstructed airfoils. Also, the Savitzky-Golayfilter [57], a moving polynomial fitting, is implemented to smoothen the boundary of reconstructed airfoils. In our case,the second-order polynomial is used in the Savitzky-Golay filter, and the length of the moving window is set to be 7.In Fig 3, the first row illustrates samples from the UIUC Database, and the second row shows reconstructed airfoilsfrom corresponding feature vectors, with an MSE, 3.65345 × − , between the reconstructed and original airfoils. Thissmall error indicates the learned features well represent the shape of airfoils. The third row shows reconstructed airfoilswith the Savitzky-Golay filter with an MSE, 3.65054 × − , comparing to the original airfoils. These results furtherdemonstrate that the encoder-decoder can reconstruct airfoils that are smooth and realistic without smoothing filters.Also, our VAEGAN-based model is compared with PCA and VAE as shown in Table 1. PCA reaches the lowest MSEsince it provides a close form solution, whereas VAE and VAEGAN are optimized numerically via the gradient-basedmethod. With the discriminator and adversarial loss from GAN, VAEGAN model performs slightly better than VAE inreconstruction. It should be pointed out that all the three parameterization methods have small reconstruction MSEs ofmagnitude 10 − , meaning all the features extracted well encodes the airfoil shapes from the UIUC database. Table 1 Mean squared error of airfoil reconstruction via different featurization techniques
Featurization PCA VAE VAEGANMSE 1.29208 × − × − × − B. Clustering in Feature Domain
The encoded features obtained from the encoder in our method can help better understand the shape of current airfoils.All airfoils from the UIUC Coordinate Database are first mapped to feature vectors, and an unsupervised learningalgorithm, K-Means [58], is used to cluster these airfoils in the feature domain. To visualize the 32-dimensionalfeature domain, we use Parametric t-distributed Stochastic Neighbor Embedding (parametric t-SNE) [59, 60] as avisualization tool. Parametric t-SNE is modeled by MLP, which maps the high dimensional feature vector 𝑧 𝑖 into alow-dimension embedding 𝑦 𝑖 , while keeps the similarity between points. It converts similarities between data points tojoint probabilities and minimizing the KL divergence between the joint probabilities of embedding 𝑦 𝑖 and the originalfeature vector 𝑧 𝑖 . Eq. 14 shows the cost function 𝐶 , which t-SNE is expected to minimize: 𝑝 𝑗 | 𝑖 = exp (−(cid:107) 𝑧 𝑖 − 𝑧 𝑗 (cid:107) / 𝜎 𝑖 ) (cid:205) 𝑘 ≠ 𝑖 exp (−(cid:107) 𝑧 𝑖 − 𝑧 𝑘 (cid:107) / 𝜎 𝑖 ) , 𝑞 𝑗 | 𝑖 = exp (−(cid:107) 𝑦 𝑖 − 𝑦 𝑗 (cid:107) / 𝜎 𝑖 ) (cid:205) 𝑘 ≠ 𝑖 exp (−(cid:107) 𝑦 𝑖 − 𝑦 𝑘 (cid:107) / 𝜎 𝑖 ) ,𝐶 = 𝐾 𝐿 ( 𝑃 || 𝑄 ) = ∑︁ 𝑖 ∑︁ 𝑗 𝑝 𝑗 | 𝑖 log 𝑝 𝑗 | 𝑖 𝑞 𝑗 | 𝑖 , (14)12here 𝜎 𝑖 is calculated by a binary search given a fixed perplexity that is specified by the user [59]. Fig. 4 shows theK-means clustering results visualized with parametric t-SNE, where different colors represent different clusters, and thecentroid of each cluster is also shown. The centroid of each cluster is different from each other in symmetry, height,camber, etc., and each represents the geometric pattern of each cluster. Features from close clusters represent similarairfoil shapes. The distance between feature points intuitively reflects the difference between the two airfoil shapes.This indicates that our VAEGAN-based model learns features that maintain the similarity of input airfoil shapes. Fig. 4 Parametric t-SNE visualization for clustering on feature domainC. What is Encoded in the Feature Domain
Also, experiments are conducted to investigate what geometric features are encoded in each dimension of the learnedrepresentation. A series of manually designed feature vectors are fed into the decoder, where all the elements are set tozero except for one specific dimension. That particular element is changed gradually from −
10 to 10, and the designedfeature vectors are mapped to the airfoil coordinate domain by the decoder. Changes of generated airfoils are illustratedin Fig 5, and the 2D embedding of feature vectors using parametric t-SNE is shown in Fig 7. Here, only four dimensionsare chosen for analysis purposes.The 1 st dimension encodes the height of the upper boundary, as illustrated in Fig 5a. As the 1 st dimension increasingfrom −
10 to 10, the height of the front half airfoil increases while the tail becomes thinner. The 8 th dimension encodesthe camber of both the upper boundary and the lower boundary. It is shown that by tuning the 8 th dimension, the upperboundary of the airfoil changes from a concave curve to a horizontal straight line, while the lower boundary evolves13rom a concave to a convex curve. Interestingly, the 22 nd dimension encodes quite similar representations as the 1 st dimension while in the opposite direction. In other words, generated airfoils from feature vectors whose 22 nd dimensionchange from −
10 to 10 are like those with the 1 st dimension change from 10 to −
10 as illustrated in Fig 7a and Fig 7c.Besides, Fig 5d shows how the last dimension is connected to the camber of the lower boundary. In detail, the curvatureof the lower boundary decreases as the 32 nd dimension increases.In comparison to the features learned by our VAEGAN-based model, Fig. 6 shows the VAE-synthesized airfoils whenchanging only one feature dimension. As shown in 6a, the first dimension of VAE features fails to encode any shaperepresentations. Even in the dimensions where representations are learned as dimension 3 and 31 shown in Fig. 6b andFig. 6c, the feature does not change continuously as we observe in the VAEGAN results. Our VAEGAN-based modellearns more representative and thorough features than the VAE model. Also, the representations in each dimensionare entangled, like the 3 rd dimension encodes both the upper bound and lower bound. These results indicate that,without manually designed parameters, our VAEGAN-based model learns geometrically meaningful features, and eachdimension of the learned feature domain encodes informative and different geometry features. (a)(b)(c)(d) Fig. 5 VAEGAN-generated airfoils by gradually changing only one dimension of the feature domain: (a)changes the 1 st dimension, (b) changes the 8 th dimension, (c) changes the 22 nd dimension, and (d) changes the32 nd dimensionD. Synthesizing Novel Airfoils To make use of the encoder-decoder architecture to synthesize novel airfoils, we conduct experiments on interpolationand extrapolation of feature vectors obtained from the UIUC Database airfoils as well as sampling from random Gaussiannoises following the method introduced in Section III C.As given in Eq. 9, the affine combination of two feature vectors, 𝑧 and 𝑧 , are computed with 𝜈 = .
5. Fig 8 showsthe interpolated airfoils from two different clusters. The labels under each airfoil indicate which two clusters are 𝑧 and 𝑧 come from. As illustrated in Fig 8, the interpolated airfoil inherits features from both clusters. For instance, Cluster 6and Cluster 11 both represent symmetric airfoil but with variant heights. The interpolation between these two clusters14 a)(b)(c) Fig. 6 VAEGAN-generated airfoils by gradually changing only one dimension of the feature domain: (a) changethe 1 st dimension, (b) changes the 3 rd dimension, and (c) changes the 31 st dimension (a) (b)(c) (d) Fig. 7 Path visualization of gradually changed feature on 2D embedded space using parametric t-SNE: (a)changing the 1 st dimension, (b) changing the 8 th dimension, (c) changing the 22 nd dimension, and (d) changingthe 32 nd dimension synthesizes a symmetric airfoil with a medium height, as shown in the last airfoil of Fig 8. Also, Clusters 3 and 4 bothencode thin airfoils. However, the lower boundary is concave in Cluster 3, while Cluster 4 represents a convex lowerboundary making the airfoil symmetric in shape. The interpolation between these two generates a thin airfoil with a flatlower boundary, which is a combination of concave and convex curves. By interpolation, novel airfoils with featuresfrom different clusters can be generated. Extrapolation between airfoils from different clusters is conducted as well.15ollowing Eq. 9, two feature vectors, 𝑧 and 𝑧 , are encoded from two different airfoils, and coefficient 𝜈 is set to be 2.Fig 9 shows the generated results from the extrapolation. Similar to interpolation, extrapolated airfoils inherit featuresfrom 𝑧 and 𝑧 . Fig. 8 Interpolation of airfoils from different clustersFig. 9 Extrapolation of airfoils from different clusters
Besides interpolation and extrapolation, the performance of the sampling synthesis method is also estimated. Insampling, a Gaussian noise, ˆ 𝑧 ∼ N ( , I ) , is directly fed into the decoder to generate novel airfoils. Shown in Fig. 10care airfoils synthesized by our VAEGAN model through sampling. Besides, sampled airfoils using PCA and VAE arealso included in Fig. 10a and Fig. 10b. PCA, as a linear projection technique, fails to synthesize realistic airfoils throughrandom sampling. Both deep-learning-based generative models, VAE and VAEGAN, can synthesize different smoothairfoil shapes. To quantitatively measure the synthesized airfoils from VAE and VAEGAN, we here introduce FréchetInception Distance (FID) [61], which is used to evaluate the quality of samples from deep learning-based generativemodels. FID is calculated by computing the Fréchet distance between two feature representations. Generally, lower FIDindicates higher generative sample quality. In our case, we feed the synthesized airfoils and the UIUC airfoils into thewell-trained discriminator from VAEGAN and extract the second hidden layer as the representation. The FID for VAEand VAEGAN are 1.38788 and 0.65366, respectively, meaning VAEGAN synthesizes more realistic airfoils. Also,airfoils synthesized via VAEGAN possess more novelty while maintains the general geometric pattern of airfoils. Forinstance, in Fig. 10c, the first airfoils in the fourth row and the third one in the second row are different from existingsamples in the UIUC database. Though such novelty does not guarantee better aerodynamic properties, some airfoilsare likely to have negative lift coefficients, which are infeasible in practice. The VAEGAN-based model can synthesizea wide variety of airfoils that serve as candidates for further optimization through CFD simulation as we will investigate16n Section IV F. (a) PCA (b) VAE (c) VAEGAN Fig. 10 Generated airfoils by sampling with different featurization techniques.E. Aerodynamic Properties of Synthesized Airfoil (a) (b)
Fig. 11 Lift coefficient 𝐶 𝑙 v.s. drag coefficient 𝐶 𝑑 of: (a) interpolated/extrapolated airfoils, and (b) sampledairfoils Aerodynamic properties of the synthesized airfoils are also tested since airfoils have to meet certain aerodynamicproperties to make the design feasible and effective. XFoil ∗ is utilized to compute the lift coefficient, 𝐶 𝑙 , and thedrag coefficient, 𝐶 𝑑 . The experiments on XFoil is set for a low speed condition: Reynolds number 𝑅𝑒 = × ,Mach number 𝑀𝑎 = .
02, and attack angle 𝛼 = ° . As illustrated in Fig 11a, 𝐶 𝑙 and 𝐶 𝑑 are tested on three airfoils:NACA1412, NACA2424 and NACA4415 from the UIUC database. Following Eq. 10, triplet interpolation/extrapolationis conducted with feature vectors, 𝑧 , 𝑧 , and 𝑧 , encoded from the three NACA airfoils. The interpolated airfoils,marked by green dots, possess 𝐶 𝑙 and 𝐶 𝑑 in between the three NACA airfoils. While extrapolated airfoils, markedby black crossings, have significantly different aerodynamic properties from the interpolated airfoils. Some airfoilssynthesized by extrapolation have high 𝐶 𝑙 with relatively low 𝐶 𝑑 , located at the upper part of Fig. 11a. This demonstrates ∗ https://web.mit.edu/drela/Public/web/xfoil/ F. Shape Optimization on Aerodynamic Properties via Genetic Algorithm (a) (b)(c)
Fig. 12 Shape optimization via genetic algorithm: (a) average score of each generation; (b) 𝐶 𝑙 and 𝐶 𝑑 ofsynthesized airfoils for different generations; (c) synthesized airfoil geometry for different generation The VAEGAN model has been proven to be able to parameterize existing airfoils to latent feature vectors and synthesizenovel airfoils automatically. However, whether or not the learned features and synthesized airfoils can be optimized topossess desired aerodynamic properties remains untested. To this end, this section demonstrates that with the VAEGANmodel, airfoil shapes can be optimized to realize the target 𝐶 𝑙 and 𝐶 𝑑 value via a genetic algorithm (GA). The lift anddrag coefficients, 𝐶 𝑙 and 𝐶 𝑑 , calculated are under the same condition as Section IV E. As illustrated in Fig. 12, thetarget lift coefficient is 𝐶 𝑡𝑙 = .
6, and target drag coefficient is 𝐶 𝑡𝑑 = .
06. In our case, the total number of generations, 𝑁 , is set to be 60, and the number of populations on each generation, 𝑀 , to be 25. Fig. 12a and Fig. 12b show howthe average score and 𝐶 𝑙 , 𝐶 𝑑 change with generation, respectively. As Fig. 12c illustrates, the airfoil shape graduallyevolves to the target 𝐶 𝑙 and 𝐶 𝑑 . Also, we compare the performance of airfoil optimization using different featurization18echniques, PCA, and VAE. The genetic algorithm with the same objective function and settings are conducted. Thelift coefficient 𝐶 𝑙 , drag coefficient 𝐶 𝑑 , and fitness score of the last generation in the genetic algorithm is reported inTable 2. The VAEGAN-synthesized airfoils reach an averaged lift coefficient of 0.5857 and an averaged drag coefficientof 0.0061. The coefficients are close to the desired aerodynamic properties. Whereas PCA and VAE fail to synthesizedesired airfoils within the same number of generations and population size. This is because our VAEGAN-based modelgenerates a wider variety of airfoils that serves as potential candidates in design optimization. Such experiments provethat a simple optimization technique like GA and the well-trained VAEGAN model can synthesize airfoils with desiredaerodynamic properties, which can guide designing effective and efficient aerodynamic products. Table 2 Airfoil design optimization results with different featurization techniques 𝐶 𝑙 𝐶 𝑑 Fitness scoreFeaturization mean std mean std mean stdPCA 0.41953 0.012672 0.0069430 6.9231 × − -0.12893 0.046250VAE 0.52746 0.0017073 0.0056662 1.4631 × − -0.017819 0.0014068VAEGAN 0.58570 0.0028946 0.0061030 7.0711 × − -8.5312 × − V. Conclusion
In this work, a data-driven method is proposed to achieve three goals: (1) automatically featuring airfoil geometries fromthe UIUC Database [53] without manually designed parameters, (2) synthesizing novel airfoils by either interpolating orextrapolating the encoded features, as well as generating from random noise, and (3) optimizing the features to synthesizeairfoils with desired aerodynamic properties. Our model is built upon VAEGAN, which combines the encoder-decoderarchitecture from VAE [6] [7] and the discriminator from GAN [8]. With the encoder-decoder structure, our modellearns explicit mappings from airfoil coordinates to latent feature domain as well as from feature vectors to airfoils,while with the discriminator, the model can automatically synthesize realistic samples. Also, our model is trained in aself-supervised manner. Namely, the model learns compact and informative features directly from airfoil shapes withoutmanually tagged labels or designed parameters. Optimized on the learned feature domain via GA, the synthesizedairfoils can evolve to have desired aerodynamic properties.Experiments show that our model learns compact and comprehensive features encoding shape information of airfoilsand can automatically generate novel airfoils. First, airfoils can be reconstructed via decoding the learned features withminor error compared to the origin coordinates. Second, K-Means [58] clustering on the feature domain of the UIUCCoordinate Database further demonstrates the learned representations are meaningful in a way that the centroid of eachcluster represents different shapes. It is also investigated what is encoded in each dimension of the feature domain bygradually changing the feature vector on one specific dimension with all other dimensions fixed. Without human prior,each dimension encodes different geometric information like height, camber, symmetry, and even coupled features.19oreover, novel airfoils are synthesized by interpolating and extrapolating learned features from different airfoils as wellas directly generated from random noise. By interpolating or extrapolating, the synthesized airfoil inherits and blendsfeatures from existing airfoils, which provides insights for designing new airfoils. On the other hand, airfoils generatedfrom Gaussian noise are more aggressive in a way that they follow a less geometrical format of existing airfoils, andmore novelty is introduced to the airfoil design. Finally, the synthesized airfoils can be optimized via GA to possesscompetitive or even better aerodynamic properties in comparison to existing ones, indicating the synthesized geometriesare not only plausible in shape but also practical in aerodynamic performance.
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