A point cloud deep neural network metamodel method for aerodynamic prediction

Fenfen XIONG, Li ZHANG, Xio HU, Chengkun REN

aSchool of Aerospace Engineering,BeijingInstituteof Technology, Beijing100081,China

bDepartmentofMechanicalEngineering,ImperialCollege London, LondonSW72AZ,UK

KEYWORDSAerodynamic prediction;Deep neural network;Metamodel;Point clouds;Robust shape optimization

AbstractAiming to reduce the high expense of 3-Dimensional(3D)aerodynamics numerical simulations and overcome the limitations of the traditional parametric learning methods,a point cloud deep learning non-parametric metamodel method is proposed in this paper.The 3D geometric data,corresponding to the object boundaries,are chosen as point clouds and a deep learning neural network metamodel fed by the point clouds is further established based on the PointNet architecture.This network can learn an end-to-end mapping between spatial positions of the object surface and CFD numerical quantities.With the proposed aerodynamic metamodel approach,the point clouds are constructed by collecting the coordinates of grid vertices on the object surface in a CFD domain, which can maintain the boundary smoothness and allow the network to detect small changes between geometries.Moreover, the point clouds are easily accessible from 3D sensors.The point cloud deep learning neural network, which employs re-sampling technique, the spatial transformer network and the fully connected layer, is developed to predict the aerodynamic characteristics of 3D geometry.The effectiveness of the proposed metamodel method is further verified by aerodynamic prediction and robust shape optimization of the ONERA M6 wing.The results show that the proposed method can achieve more satisfactory agreement with the experimental measurements compared to the parametric-learning-based deep neural network.

1.Introduction

In the past two decades, aerodynamic optimization based on high-fidelity Computational Fluid Dynamics(CFD)numerical simulation has been of practical interest to aircraft designers.Considering uncertainties widely existing in the aircraft design,manufacturing, and flight environment have inevitable effects on aircraft performances, robust optimization and reliabilitybased optimization have been introduced into the aerodynamic optimization.1–3However, the high-fidelity CFD simulations in practical problems are always computationally expensive,and the introduction of uncertainties also increases the computational cost dramatically.

In order to reduce the high computational expense,the metamodel technique,which can construct an approximate model to replace expensive CFD simulations,has been extensively studied.4–6Weinmeister et al.applied the polynomial chaos and kriging metamodel methods to uncertainty quantification in aerodynamics.7Zhang et al.constructed a double-stage metamodel method by integrating interpolation and regression techniques, and applied it to aerodynamic analysis and design optimizations.8In terms of the complex aerodynamic geometries,not only the nonlinearity of CFD simulation increases significantly, but the shape parametrization always requires numerous geometric variables.For example, only airfoilrelated geometric variables include taper ratio, sweep angle,aspect ratio, wing twist, dihedral angle, wingtip details, etc.The significant nonlinearity and numerous geometric variables greatly increase the number of required sample points and training duration, posing a challenge to the traditional metamodel techniques.

Different from the traditional metamodel methods, the deep neural network method employing a more complicated model structure is dimensionally insensitive and thus exhibits a great potential in solving high-dimensional and nonlinear regression problems.Sun et al.proposed a novel inverse design optimization framework with a two-step deep learning approach and applied it to the design of wind turbine wings.9Thuerey et al.investigated the accuracy of deep learning models for the inference of Reynolds-averaged Navier-Stokes solutions of airfoil flows.10Li et al.proposed a new sampling method for aerodynamic shape optimization based on a deep convolutional generative adversarial network, which was able to generate notably more realistic sample airfoils.11

Seized with an inspiration, the pastor was the high bidder9 at $6.50. His idea was to hang the orange cloth behind the altar to cover the ragged10 hole in the wall.

However, all of these investigations were conducted with the parametric learning method.If the number of geometry design parameters increases, this method is not effective in the establishment of the aerodynamic prediction model.Moreover,if the parametric modeling and design method change or a refined design is required, the geometry parameters involved will change, and the established model will fail to predict the aerodynamic performance accurately.To address this issue,the non-parametric deep learning metamodel method adopting images as input to establish a Convolutional Neural Network(CNN) has been proposed.Han et al.designed a novel hybrid deep neural network architecture based on the datasets of the flow around the cylinder at different Reynolds numbers to achieve fast and accurate prediction of unsteady flow fields.12Liao et al.constructed a CNN model of the pressure distribution at the leading edge of a mixed airfoil by taking the airfoil picture as the input layer,which significantly reduced the computational time compared with CFD.13Duru et al.proposed an encoder-decoder convolutional neural network-based approach for estimating the pressure field around an airfoil,in which the airfoil shape represented with the distance field map was used as the input of the network.14Sekar et al.developed an inverse design approach of airfoils using deep convolutional neural networks and obtained the mapping of the pressure coefficient to the airfoil shape.15Compared to the high-fidelity CFD simulations, the 2-Dimensional (2D)image-based CNN metamodel methods studied in these works can significantly reduce the computational cost.Moreover, as more aerodynamic features can be explored, the prediction accuracy gets a significant improvement compared to the traditional parametric learning based methods.16

Numerous 3D geometry models in a certain category of aerodynamic profiles are firstly collected, after which the point cloud models are established.Generally,the point cloud models for these geometries can be directly produced by 3D sensors.On the other hand, for the proposed metamodel method, CFD simulation should be carried out for each aerodynamic geometry to obtain aerodynamic characteristics,where the flow field has been grided.Therefore, the grid vertices on a geometry surface can also be directly used to establish the point cloud model.The point cloud model inherits the grid feature in terms of spatial grid distribution(i.e.,fine grids near the object and coarse grids in far fields),which can maintain boundary smoothness and allow the network to detect small difference between two geometries.Thus, the ability to extract the feature of the flow field can be enhanced with the point cloud neural network, which will further improve the accuracy of the aerodynamic prediction.

Clearly,these 2D image-based CNN methods can provide a good aerodynamic prediction for 2D geometries(e.g.,airfoils).However, for 3D geometries that widely exist in practice, the aerodynamic features cannot be represented only by one single image.If multiple 2D images from different perspectives are collected to extract aerodynamic features, the dislocation and the loss of spatial relationships between different features may occur.Moreover,different regions in the domain have different effects on the aerodynamic prediction.For instance,the pressure fields near the wing surface in its wake region are more important than those in other regions.However, when using the CNN based metamodel method, the distribution of CNN pixels is uniform everywhere in the domain, which definitely affect the prediction accuracy.

20. White: White symbolizes light, innocence78 and purity (Matthews 1986). White is also associated with faith and peace. It is a recurring79 color in this version of the tale and is frequently mentioned. With the exception of the witch s red eyes, it is the only color mentioned.Return to place in story.

Taking the ONERA M6 wing as an example, Fig.2 shows the grid vertices on the wing surface, of which the 3D coordinates are stored as a point cloud model.For a clear illustration, part of the data points (N = 700) in a point cloud model of the ONERA M6 wing is shown in Fig.3,where lines represent the geometry boundaries of the 3D wing, and the points are the centers of 700 grid vertices used in CFD simulation on the wing surface.

The rest of this paper is organized as follows.In Section 2,the proposed metamodel method is introduced in detail,including 3D geometry representation via point clouds, data generation for deep neural network training and neural network construction.In Section 3,the proposed method is applied to 3D aerodynamic prediction and robust shape optimization to demonstrate its effectiveness.The conclusion is drawn at last.

2.Proposed aerodynamic metamodel method

The proposed aerodynamic metamodel method of 3D geometry is summarized in Fig.1.First, many 3D aerodynamic geometries are collected, and the point cloud model fed to the deep learning neural network is constructed for each geometry.This is followed by labeling each point cloud model with the corresponding aerodynamic characteristics obtained from CFD simulations or experiments.Finally,a deep learning neural network model is trained as a metamodel to predict the aerodynamic characteristics.

2.1.Geometry representation via point clouds

The mother took them from her, and, instead of behaving better to poor Little Two-eyes, as they ought to have done, they were jealous that she only could reach the fruit and behaved still more unkindly to her

It is assumed that there are N points in a point cloud model.The 3D coordinates(x,y,z)of all points on object boundaries written in a matrix of size N×3 are collected for a point cloud model.Actually, the order in which these points are stored in the matrix is arbitrary, and thus a point cloud model can be represented by many different matrices.

In this work,a new non-parametric metamodel approach is developed for the 3D geometry aerodynamic prediction, in which a point cloud deep neural network is designed.The point cloud model of a 3D aerodynamic geometry is first constructed using the meshed vertices on the object boundaries in CFD simulations,and then the deep learning technology is used to extract the features of a point cloud model.To implement the aerodynamic prediction under different flight conditions,environmental variables (e.g., angle of attack and Mach number) are considered in the convolution layer of the neural network.Based on the constructed aerodynamic metamodel, the aerodynamic robust shape optimization is further implemented to verify the effectiveness of the proposed metamodel method.

His teacher appeared beside me. “He hasn’t put that book down since you gave it to him. He wears it like a shirt, close to his heart. Did you know that’s the first book he’s ever actually owned?”

2.2.Data generation

Fig.2 ONERA M6 wing and surface grid vertices.

Fig.3 Some data points in a point cloud model of ONERA M6 wing.

For each aerodynamic geometry, CFD simulations under different flight conditions are conducted to obtain the corresponding aerodynamic characteristics, such as the lift coefficient, the drag coefficient and the pressure coefficient.The Reynolds-averaged Navier-Stokes (RANS) model compromising well between the computational accuracy and cost is used in our CFD simulations.As many different aerodynamic geometries of a certain category are considered,the simulation results should be validated first on a certain baseline geometry that has the high-fidelity data from wind tunnel tests or direct numerical simulations.Numerical results from different grid refinements are compared to the experimental data.Meanwhile,the calibration of closure coefficients of the turbulence model or the selection of the turbulence model is implemented.Finally, for the turbulence model, its closure coefficients and the computational grid configuration are determined to match this category of geometry and flow condition.The flowchart of CFD simulation validation is shown in Fig.4.

As I was coming into Middle School, grade six, I was really excited because my friend Jennifer was going to be in the same Middle School as me! I was convinced2 that we would be the bestest buds（，）. At first things were great, she introduced me to her friend Amy and we had lots of fun together. None of the girls from my elementary school were in my classes, but I wasn’t worried. I had Amy and Jennifer, and I could make some new friends. Then things started to change. Jennifer was very controlling: I couldn t make new friends, because if I hung out（，） with different people, she would decide that I was mad at her and Amy. So I didn t make new friends, and pretty soon being Jennifer s friend was a struggle.

Fig.1 Principle of proposed aerodynamic metamodel method.

Robust shape optimization is carried out based on the established metamodel.The optimization process based on PC-DNN is shown in Fig.16.The optimization algorithm used here is the genetic algorithm toolbox in python.To ensure the fairness of comparison of optimization results between the two methods,the 50 control points on the five sections of the wing surface are taken as design variables for both methods.The difference is that the design point at the current optimization iteration is directly fed to the network of the P-DNN to predict aerodynamic coefficients, while for PC-DDN, the point cloud model is reconstructed at the current design point.To ensure the physical meaning and smoothness of geometry shape, the wing surface is reconstructed by the B-spline surface method at the current design point, which is then automatically meshed.The 3D coordinates of all the grids on the reconstructed wing surface are recorded as a point cloud model.Then, the resample technique is applied to randomly select n points from the reconstructed point cloud model, which are fed to the network of PC-DNN for aerodynamic prediction.The resample and prediction process is repeated for many times, in case that the selected points cluster in certain region.The mean values are calculated as the predicted aerodynamic coefficients for optimization.

So the young man departed, and went away, away, away, as well as his horse would take him, begging his living as he rode along, and, somehow or other, at last he got to the land of the sun

2.3.Network architecture

Before constructing the point cloud deep neural network using the 3D point cloud model as input, the randomness and rotation problems of point clouds have to be solved.The randomness issue is firstly solved to ensure that the order of the data points stored in a point cloud model will not impact the accuracy of the feature representation for 3D geometry.This is the premise for the construction of a point cloud deep neural network.Each data point in a point cloud model is only represented by its three coordinate values (x, y, z).Geometrically,the order of points should not affect its representation of the overall geometric shape in space.However, actually, the same point cloud model can be represented by two completely different matrices.As shown in Fig.6,two point cloud models share the same position information.However, if these points are stored in the corresponding matrix via the marked number,the corresponding matrices are different.If the two matrices are fed into the general deep neural network for training, the trained network parameters are completely different.

With the rapid development of 3D sensors,such as panoramic camera systems,depth cameras and laser radars,3D point clouds are easily accessible.Compared with 2D images acquired by cameras,the 3D point clouds can provide more spatial information.The points in a point cloud model of 3D geometry are correlated as they come from a distance metric-based space,and this model can capture the local structure and interactions from nearby points.The deep learning technology with a strong data feature extraction capability has been applied to 3D point cloud data analysis by constructing a point cloud deep neural network.At present,the point cloud deep neural network using 3D data as input has been successfully applied to synchronous localization,173D reconstruction,18–22target detection,23–25classification26–28and segmentation.29–31The network structure for the classification of point clouds can be easily converted for numerical prediction by introducing a fully connected structure as the output layer.Therefore,it is necessary to explore the effectiveness of the point cloud deep neural network method in the 3D high-performance aerodynamic prediction, and simultaneously overcoming the limitations of traditional parametric learning based metamodel methods.

The resampling strategy32is usually used to solve the randomness issue.The number of resampled points during the training of each point cloud model is set as n, and n is usually determined according to the number of points in each point cloud model(N,n

When he gathered beach plums for the yearly batch4 of jam, day after day large pots of the warlock’s brew5 simmered on the stove, and the kitchen looked like the playpen of a child who had spilled purple paint

The training process of the point cloud deep neural network is shown in Fig.7.n points （p1;p2;...;pn）are randomly selected from a point cloud model as the input of the network.h represents the feature extraction layer, and g is the symmetric method, such as max-pooling or average-pooling.The maximum pooling strategy is adopted in this work.γ represents higher dimensional feature extraction, which is the Multi-Layer Perception (MLP) based on the fully connected layer after maximum pooling.Finally, to implement numerical prediction, this layer is connected to the output.The process shown in Fig.7 is conducted many times by randomly selecting n points from a point cloud model.In addition,the point cloud models of different aerodynamic geometries generally contain different numbers of points (different N).The employment of a resampling strategy can also solve this issue and realize unified training of all the point cloud models corresponding to different 3D geometries.

After the randomness of the point cloud is solved,the deep neural network can be constructed using the PointNet architecture.Here, the rotation problem of point clouds should be solved to ensure that no matter what orientation of the point cloud model is represented, the network must identify correctly.The point cloud deep neural network structure for aerodynamic prediction is shown in Fig.8.

For the general deep neural network structure, the same point cloud model presented in different orientations can yield different training results.Therefore, the Spatial Transformer Network (STN) is introduced to transform the input and features.STN contains a network for rotation alignment (named as T-Net) that is multiplied by the matrix obtained from the input and transformation network.The size of the first STN is 3 × 3, and the second is 64 × 64.The transformation of the input adjusts the orientation of a point cloud model in space (such as turning the object to the front), which is the most helpful for feature extraction.The second feature transformation aims to aligning the extracted 64D features,i.e.,performing a transformation on the point cloud at the feature level.The coefficients of the two STNs are iteratively adjusted by minimizing the loss of network training.Through using the two STNs, the point cloud model is rotated to the orientation that is most suitable for feature extraction and prediction.Therefore, the influence of the orientation of input 3D model on the network prediction performance is avoided.This will be verified in Section 3.2.The MLP before the maximum pooling operation is realized by convolution with shared weights.After two spatial transformation networks and two MLPs,the 1024D features are extracted from each point in a sample,which are converted into 1024 × 1 global features by maximum pooling.

Fig.4 Flowchart of CFD validation.

Fig.5 Flowchart of CFD data generation.

Fig.6 Point clouds with different orders of data points.

Fig.7 Schematic diagram of deep neural network training.

Considering that the prediction of aerodynamic characteristics generally involves different flight conditions (e.g., inflow Mach number and angle of attack), the data related to environmental variables are introduced to the network after the three-layer convolution of point cloud data and the maximum pooling operation.Thus,after three layers of convolution,the size of the network becomes 1026 × 1.Finally, the aerodynamic characteristics is predicted through an MLP (512, 256,1) fully connected layer.

The Mean Absolute Error (MAE) function, which is the mean of the discrepancy between the predicted value f(x) of the network and the CFD simulation result y, is considered as the loss during network training.

In practice,aerodynamic prediction often has to be done on many different categories of geometry.Based on the neural networkmodel trained for a certaincategoryof aerodynamic geometry, the transfer learning technique33 can be employed to fast adjust the network parameters and reduce the number of samples of the new category of geometry.

From Table 1, it can be observed that the CFD simulation result begins to converge in Case 4,and the corresponding liftto-drag ratio (15.64) is very close to the excremental value(15.52).Therefore, this grid is selected for the subsequent CFD simulations.

3.Application to aerodynamic prediction and robust shape optimization

The proposed point cloud deep neural network metamodel method is applied to the aerodynamic characteristic prediction and robust shape optimization design of a 3D wing.The ONERA M6 wing34is mainly investigated here whose geometry can be found in Fig.2.For the M6 wing,it takes approximately 116 min to implement one CFD simulation on a modern personal computer.As a large number of CFD simulations are required during robust optimization, the proposed point cloud deep neural network method is used to construct metamodels of the lift and drag coefficients to reduce the computational cost of the high-fidelity CFD simulation.Based on the metamodels, robust aerodynamic shape optimization is conducted considering the uncertainties from flight conditions.At each design point, the corresponding point cloud model is reconstructed and fed to the point cloud deep neural network to predict the aerodynamic characteristics.

3.1.Validation of CFD results

The grids are generated using the CFD preprocessing software gambit, and Fluent 19.0 is used as the CFD solver.For this problem, the number of the baseline grid in CFD is about 590000.The method introduced in Section 2.2 is used to generate CFD data on a personal computer with 16 G memory,an NVIDIA 1660 graphics card and an Intel(R) Core(TM)I5-10400F CPU.Table 1 shows the results of the grid convergence test,in which five cases with different grid configurations are considered.

She went straight to the place from which the noise came, and, to her great surprise, beheld3 the same lion stretched on the ground with a deep wound across his face

Fig.8 Point cloud neural network structure of aerodynamic prediction.

Table 1 Results of grid convergence test.

Fig.9 shows the pressure coefficients (Cp) of four sections generated by CFD and experiments.35These sections are illustrated in Fig.10,which respectively lies at y/b=0.2,0.4,0.65,0.8 of the wing along the wingspan starting from the wing root.Clearly,the pressure coefficients of the four sections produced by CFD simulations show good agreements with the experimental data, demonstrating its effectiveness.

3.2.Training of point cloud deep neural networks

Fig.9 Pressure coefficients produced by CFD simulations.

Fig.10 Wing geometry with sections.

Fig.11 Design boundaries and baseline boundaries of wing root surface.

To verify the effectiveness of the proposed Point Cloud Deep Neural Network method (PC-DNN), the traditional Parametric Deep Neural Network method (P-DNN) adopting geometry shape parameters as inputs is also used to construct the metamodels for comparison.To ensure the fairness of comparison, the two approaches (PC-DNN and P-DNN) adopt the same method to generate samples.Five sections with equal intervals along the wingspan are selected on the wing, which respectively lies at y/b=0,0.25,0.5,0.75,1 of the wing along the wingspan starting from the wing root referring to Fig.10.On each section surface,ten control points(five at the top and five at the bottom) are selected.The horizontal positions of these control points are fixed at the chord length (0.1, 0.3,0.5, 0.7, and 0.9 along the chord of the wing, starting from the leading edge), and their vertical positions can be changed.The design space of the vertical positions for each section is specified by expanding and shrinking it by 0.25 with respect to the baseline.Fig.11 shows the design boundaries and baseline boundaries of the wing root surface (y/b = 0).Here the baseline boundaries correspond to the boundaries of ONERA M6 wing.

First,the Latin hypercube sampling strategy is employed to generate samples (sample size is 500) for the 50 control points and flight conditions (Ma and α), at which the lift and drag coefficients are calculated by CFD.For PC-DDN, the wing surface is reconstructed by the B-spline surface method36for each set of 50 control points, and then meshed by gambit.The 3D coordinates of all the grids on the wing surface are recorded and stored as a point cloud model.The number of points in one point cloud model of wing is limited within 5500–6000.The number of points selected from one point cloud model is set as n = 5000 during resampling to achieve unified training of all 500 samples.The final sample size fed to the network of PC-DNN is 1 × 3 × 5000.For the PDNN,the inputs of the deep neural network are the 50 control points, Ma and α.

Fig.12 shows the evolution of loss for the training of the point cloud deep neural network of the proposed method.Clearly,with the increase in the number of samples,the errors of both networks gradually decrease.

Afterwards the teacher decided to do a class project to see what kind of impact recognition would have on a community. She gave each of the students three I more ribbons and instructed them to go out and spread this acknowledgment ceremony. Then they were to follow up on the results, see who honored whom and report back to the class in about a week.

Another 50 samples are generated randomly in the same way as the training samples to check the prediction accuracy of the metamodels constructed by the two methods.The average relative error of the proposed PC-DNN method for the lift(CL)and drag(CD)coefficients is 7.7%and 7.4%,respectively,which is smaller than those of the P-DNN(12.6%and 13.1%).Fig.13 shows the aerodynamic prediction for the baseline wing at different Ma and α for the two methods, where the plots in the left column correspond to α=3.06°,while the right corresponds to Ma =0.8395.It is noticed that with the increase of Ma and α, generally CLand CDbecome larger for this problem, which is consistent with theoretical analysis.Meanwhile,the aerodynamic coefficients produced by PC-DNN are basically closer to those by CFD compared to the existing PDNN method, demonstrating its effectiveness and advantage in improving prediction accuracy.This is because the proposed PC-DNN employs point clouds as input that can more accurately and reasonably capture the flow features; while PDNN directly treats geometry parameters as input.

To investigate the gain in computational cost of PC-DNN over P-DNN, based on the above test, we gradually increase the sample size of P-DNN and update the corresponding metemodels.Fig.14 shows the average relative errors of predicted CLand CDby P-DNN, from which it is found that with the increase of samples, the average relative errors for both CLand CDare quickly reduced.When the sample size is increased to 650, the average relative error with P-DNN is 7.5% (CL)and 7.1% (CD), which is very close to that of PC-DNN(7.7%and 7.4%).It is concluded that PC-DNN can reduce the computational cost by about 23% ((650–500)/650) compared with P-DNN for this problem.

To my dearest wife, by the time you are reading this, I m sure I m no longer around, I bought this policy for you, though the amount is only $100k, I hope it will be able to help me continue my promise that I have made when we got married, I might not be around anymore, I want this amount of money to continue taking care of you, just like the way I will if I could have live longer. I want you to know I will always be around, by your side. I love you.

A paper outlining the discovery and the properties of this new mineral will be published in the July issue of the journal American Mineralogist, and is available online now

Fig.12 Evolution of loss for network training diagram.

Fig.13 Aerodynamic predictions of PC-DNN and P-DNN.

To show the effectiveness of PC-DNN in dealing with randomness issue, five samples are selected from the validation set.For each sample, the storage locations of the first 50%and the last 50%points in the point cloud model are swapped.The modified five samples are fed to the PC-DNN network,and the predicted coefficients are shown in Table 2.It is noticed that the predicted coefficients before (Original) and after(Modified)location swap are very close.The results indicate that the storage order of points has little impact on prediction, demonstrating that the randomness issue has been effectively solved by PC-DNN.

Fig.14 Average relative errors of predicted coefficients by P-DNN with increased samples.

Table 2 Prediction of five samples.

Moreover, to show the effectiveness in dealing with rotation issue, the orientations of wing before and after training are illustrated in Fig.15.Cleary, more details of the wing can be shown after training, which is more beneficial to network feature extraction.For this wing problem, as the points of all the point cloud models are constructed in the same coordinate system, each original point clouds sample is rotated with the same angle.

3.3.Robust shape optimization

The shape optimization problem considered here aims to minimize the drag coefficient subjected to constraints on the lift coefficient and the maximum thickness of the wing by optimizing the wing geometry.It is assumed that the angle of attack(α)and Mach number(Ma)are uniformly distributed and vary within ±0.2° and ±0.1 with respect to their mean values,respectively.To reduce the sensitivity of lift (CL) and drag(CD) coefficients to uncertainties, robust aerodynamic shape optimization is carried out as follows:

where μ and σ represent the mean and standard deviation of a quality.C0Lis the lift coefficient and t0maxis the maximum thickness of the baseline M6 wing.

Fig.16 Procedure of optimization for proposed PC-DNN.

For other geometries of the same category, based on the grid of the baseline geometry, batch processing is conducted to generate grids automatically.The procedure of generating the CFD data for one certain category of aerodynamic geometry is shown in Fig.5.

Fig.15 Orientations of 3D wing.

Table 3 Optimization results with different methods.

The optimal results with the consideration of the same uncertainties from Ma and α.are shown in Table 3, in which μCD, σCD, μCL, σCL, P(CL≥C0L) are calculated with Monte Carlo Simulations (MCS, 500 runs) by substituting the obtained optimal design variables into the CFD simulation model, P(CL≥C0L) represents the probability of CL≥C0L,and a larger value corresponds to a more reliable design solution.DO denotes the results obtained by the deterministic optimization based on the P-DNN metamodel without considering any uncertainties.It is noticed that the mean values of the drag coefficients (μCD) obtained by optimization are all reduced compared to that of the baseline wing, and it is reduced most for DO.However,these two robust optimizations yield smaller variations in the drag coefficient (σCD) than those of DO and the baseline.As uncertainties are not considered during optimization design for DO, the variations in the drag and lift coefficients are the largest (see σCDand σCL), exhibiting their large sensitivity and weak robustness to uncertainties.Generally, the proposed PC-DNN performs slightly better than PDNN (smaller μCDand σCD, larger P(CL≥C0L)).This is ascribed to the employment of the point cloud DNN with coordinates of surface grid vertices as input that can enhance the flow feature extraction and finally improve aerodynamic prediction accuracy.

Fig.17 Pressure coefficients at four sections by different methods.

Fig.17 shows the pressure coefficients at the four sections for the wing geometry corresponding to the baseline, DO,PC-DNN and P-DNN.Compared with the baseline wing,basically,the pressure coefficient curves by all the optimization methods do not vary drastically as those of baseline,especially for the regions close to the leading edge(x/c=0).This means that the pressure variations on the four sections become smaller via optimization.Therefore, the shock wave is weakened,which reduces the drag.Compared to the robust optimization,the deterministic optimization yields larger fluctuations in pressure coefficients, larger gaps between the upper and lower curves of pressure coefficients, indicating that the variation of drag and lift by DO are larger.These results show great agreement with the results presented above,which is ascribed to the high accuracy of constructed metamodels.These results demonstrate the effectiveness of the proposed PC-DNN.

It should be pointed out that the tests conducted above aims to verifying the effectiveness and advantage of the proposed PC-DNN method over the existing parametric PDNN.In practical applications,other design variables can also be selected for PC-DNN,and the wing surface does not necessarily need to be constructed during each optimization iteration.For example, some key points of the point cloud model on areas of interest for aerodynamic shape can be directly picked as design variables.In this way,the aerodynamic shape optimization can be implemented very flexibly with the proposed PC-DNN.This will be explored in the further work.

3.4.Perdition considering small geometry variation

To investigate the ability of the proposed metamodel method to detect small changes between geometries which is difficult for the traditional parameterization based method,37another test with a small change in geometry is conducted.For example,the wing may have roughness locally on the surface due to manufacture,and it will affect the smoothness of the wing surface as well as the aerodynamic performance.It is assumed that there is a tiny bump on the upper wing surface that lies between 20%–40% along the chord length starting from the leading edge(i.e., x/c is between 0.2–0.4).Fig.18 shows the region (dotted box)on the upper surface considering the bump.

Meanwhile, the shape of the bump studied here is assumed to be Gaussian stochastic due to manufacturing.And thirteen new samples are generated by directly changing the positions of points within this area in the point cloud model of the baseline wing.And the mean and standard deviation of the perturbation is set as 0.5% and 0.15% of the positions for baseline points, respectively.Then the deep neural network is trained using the meta-learning method38based on ten of these new samples, in which the network parameters are updated.The network updating can quickly get converged using only a small number of samples, as the meta-learning method is employed.

Fig.18 Region on the upper surface considering bump.

Table 4 Predicted results considering tiny bump on wing surface.

The rest three samples are used to test the prediction accuracy of the updated network,of which the results are shown in Table 4.Theoretically,the bump will impact the smoothness of the wing surface, and easily induce a shock wave during the flight,which increases the drag,but has little impact on the lift.Clearly, CDare all increased at the three test samples, exhibiting great agreement to the theoretical analysis.Meanwhile,generally, the average bulge of Sample 1 is the smallest, followed by Sample 2 and Sample 3.Theoretically,the drag coefficient of the wing for Sample 3 is the largest, followed by Sample 2 and Sample 1, and the lift coefficients are close.The results in Table 3 show great agreement with theoretical analysis.The proposed PC-DNN method is demonstrated to be effective for predicting small changes in a geometry shape,which is ascribed to the employment of grid vertices on the object surface in a CFD domain as a point cloud model.However, for the parameterized method, it is complicated and difficult to effectively describe the small geometry change by geometry parameters,as many new parameters should be carefully introduced.

4.Conclusions

In this paper, a point cloud deep learning metamodel method with point clouds of 3D geometry as input in replacement of the computationally expensive CFD simulation is developed.The point clouds of one geometry shape are constructed by collecting the coordinates of grid vertices on the object surface in a CFD domain or data from 3D sensors.A deep learning neural network metamodel fed by the point clouds is established based on the PointNet architecture.The application to the ONERA M6 wing shows that the aerodynamic prediction accuracy of the proposed metamodel method can be improved by about 5% with the same number of CFD evaluations, and the computational cost can be reduced by about 23%with the comparable prediction accuracy, compared to the traditional parametric learning based deep neural network.Moreover,numerical results demonstrate that the proposed metamodel method can effectively detect small changes between geometries that are difficult for the parameterization based method,exhibiting great potential in quantifying aerodynamic characteristic variation of surface roughness or ice accretion that is difficult to describe by the parameterization based approach.Future work will be explored on optimizing the positions of some key points in the point clouds using the proposed point cloud deep learning metamodel, while satisfying physical limitation of geometry by carefully exerting constraints.

Declaration of Competing Interest

8.Circle around her with chalk:A circle is drawn49 to protect a magician from the demon50 that he/she has summoned (Biedermann 70), but according to Jung, the circle can also symbolize the whole self (Nataf 66).Return to place in story.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (No.52175214) and the Basic Research Program of Equipment Development Department (No.514010103-302).