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Citation: Li, K.; Li, H.; Li, S.; Chen, Z.
Fully Convolutional Neural Network
Prediction Method for Aerostatic
Performance of Bluff Bodies Based on
Consistent Shape Description. Appl.
Sci. 2022,12, 3147. https://doi.org/
10.3390/app12063147
Academic Editors: Wenli Chen,
Zifeng Yang, Gang Hu, Haiquan Jing
and Junlei Wang
Received: 21 February 2022
Accepted: 17 March 2022
Published: 19 March 2022
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4.0/).
applied
sciences
Article
Fully Convolutional Neural Network Prediction Method for
Aerostatic Performance of Bluff Bodies Based on Consistent
Shape Description
Ke Li 1,2 , Hai Li 2, Shaopeng Li 1, 2, * and Zengshun Chen 1,2
1Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University,
Ministry of Education, Chongqing 400045, China; keli-bridge@cqu.edu.cn (K.L.);
zengshunchen@cqu.edu.cn (Z.C.)
2School of Civil Engineering, Chongqing University, Chongqing 400045, China; lhtm_study@cqu.edu.cn
*Correspondence: lisp0314@cqu.edu.cn
Abstract:
The shape of a bluff body section is of high importance to its aerostatic performance.
Obtaining the aerostatic performance of a specific shape based on wind tunnel tests and CFD
simulations takes a lot of time, which affects evaluation efficiency. This paper proposes a novel
fully convolutional neural network model that enables rapid prediction from shape to aerostatic
performance. Its main innovations are: (1) The proposal of a new shape description method in
which the shape is described by the combination of the wall distance field and the space coordinate
field, which can efficiently express the influencing factors of the shape on the aerostatic performance.
(2) A step-by-step strategy in which the pressure field is used as the model output and then the
calculation of the aerostatic coefficient is proposed. Compared with the simple direct prediction of
the aerostatic coefficient, the logical connection between input and output can be enhanced and the
prediction accuracy can be improved. It is found that the model proposed in this paper has good
prediction accuracy, and its average relative error is 9.42% compared with the CFD calculation results.
Compared with the direct use of the shape as the model input, the accuracy is improved by 13.25%;
compared with the direct use of the drag coefficient as the model output, the accuracy is improved by
10%. Compared with traditional CFD calculations and wind tunnel experiments, this method can be
used as a fast auxiliary screening method for the optimization of the aerodynamic shapes of bluff
body sections.
Keywords:
deep learning; prediction; aerostatic performance; shape; convolutional neural networks
1. Introduction
Slender structures, such as bridges, high-rise buildings, and others, are prevalent in the
field of civil engineering. These types of structures have low rigidity, and their aerostatic
performance is directly related to the safety of the overall structure under the action of
strong wind. Taking the main beam of a bridge as an example, excessive resistance and
lifting moments will cause the structure to produce excessive lateral displacement and
torsion and, in severe cases, even wind-induced aerostatic instability, which will affect the
safety of the structure.
Slender structures have typical two-dimensional characteristics, and their aerostatic
performance can be determined by the shape of their cross-section. The aerostatic per-
formance of the bluff body section is usually expressed by the three-component force
coefficient, and the aerodynamic shape of the bluff body section is particularly important
among the influencing factors. The section form and aspect ratio are different, and the
corresponding three-component force coefficient and variation law are also different. Kazu-
tosh et al. [
1
] conducted experimental studies and numerical simulations on bridge girder
sections with three different shapes, considering the influence of the degree of central
Appl. Sci. 2022,12, 3147. https://doi.org/10.3390/app12063147 https://www.mdpi.com/journal/applsci
Appl. Sci. 2022,12, 3147 2 of 21
slotting on the three-component force coefficient. Simiu et al. [
2
] explained that the drag
coefficient of a rectangular section shows a trend of first increasing and then decreasing
when the aspect ratio of the rectangular section is between 0 and 2. When the aspect ratio
changes from 2 to 4, the drag coefficient has a certain discreteness. When the ratio is greater
than 4, the drag coefficient remains basically unchanged. Shimada et al. [
3
] studied the
variation of the drag and lift coefficients with the aspect ratio of a rectangular section as
the parameter and found that the drag coefficient reached the maximum value when the
aspect ratio was 0.6.
To sum up, there is an extremely complex nonlinear relationship between the cross-
sectional shape of a bluff body and its aerostatic performance. This relationship does not
have a certain regularity, making it difficult to describe in the form of a specific mathematical
formula. It cannot be quickly passed. The aerodynamic shape obtains the aerostatic three-
component force coefficient. Therefore, it is not possible to quickly obtain the aerostatic
three-component force coefficient using the aerodynamic shape.
At present, there are two common methods for carrying out aerodynamic studies.
The first method involves putting the segment model into a wind tunnel for testing [
4
–
7
],
and the other method involves computational fluid dynamics (CFD) [
8
–
10
]. Although
these two methods are widely used at present, they also have their own shortcomings.
Investing in wind tunnel test equipment is expensive, and significant manpower and
material resources are required for the tests. CFD simulation calculation consumes a lot
of computing resources, which greatly restricts the efficiency of wind resistance design.
Due to its excellent data fitting ability and efficient large data application ability, deep
learning can discover input–output mapping relationships that humans cannot perceive
in a large amount of data and can improve the aerostatic performance efficiency of bluff
body sections.
With the continuous development of neural network technology, artificial intelligence
prediction technology has been gradually applied in the field of wind engineering in recent
years [
11
]. Lute et al. [
12
] applied a support vector machine (SVM) to predict the flutter
derivative of a cable-stayed bridge and to estimate the flutter critical wind speed. June
et al. [
13
] built an artificial neural network model to identify the flutter derivatives of various
main beam cross-section forms, and the prediction results were excellent. Liao et al. [
14
]
accurately predicted the flutter critical wind speed of different streamlined box girders by
comparing four machine learning methods. Hu et al. [
15
,
16
] accurately predicted the wind
pressure and pressure coefficient on a cylindrical surface by comparing multiple machine
learning methods. Guo et al. [
17
] utilized convolutional neural networks (CNN) to predict
velocity fields for several geometries, with an accuracy rate of about 98%. Miyanawala
et al. [
18
] proposed the use of a Euler distance field to express the aerodynamic shape to
predict the lift and drag coefficients and innovatively generated the section shape in the
form of pictures, which greatly reduced section shape expression time. Jin et al. [
19
] used
convolutional neural networks (CNNs) to establish a model of the relationship between
the pressure field and the velocity field of the cylindrical structure and predicted the wake
velocity field based on the surface pressure distribution on the bluff body. Lee and You [
20
]
introduced the flow field continuity condition into flow field prediction and analyzed the
ability of multi-scale convolutional neural networks and adversarial neural networks to
maintain mass conservation and momentum conservation in turbulence prediction.
In summary, scholars have begun to explore the successful application of deep learning
to related fields of wind engineering, which also provides the basis for this study. When
designing the neural network architecture, this paper focuses on the input design and
output design of the network. In terms of the input, since simply using the shape as the
input of the neural network as a parameter cannot express any complex cross-sections, it is
not universal. Therefore, this paper draws on Miyanawala’s idea of using the Euler distance
to describe the aerodynamic shape and proposes to describe the shape as a combination
of the wall distance field and the space coordinate field, which can distinguish the flow
field characteristics at different positions in space and efficiently express the influencing
Appl. Sci. 2022,12, 3147 3 of 21
factors related to aerostatic performance. In terms of the output, the logic of directly
predicting the three-component force coefficient from the aerodynamic shape is too jumpy
to be conducive to the training and testing of the neural network. Therefore, this paper
proposes to divide the aerostatic performance prediction into two steps. First, the pressure
field is predicted, and then, the aerostatic three-component force coefficient is obtained by
traditional integral calculation. A fully convolutional network, with some improvements, is
selected as the neural network. The optimization of the fully convolutional neural network
suitable for classification problems is extended to the pixel regression problem, retaining
the advantages of the efficient spatial feature extraction of the fully convolutional network,
the high-dimensional mapping relationship from shape to pressure field, and the aerostatic
performance prediction of bluff body cross-sectional shapes. In addition, considering that
CFD numerical simulation can provide a large amount of flow field information as the
input and output of a deep learning model, this paper uses CFD simulation to obtain the
required data samples for model training and testing. Ribeiro et al. [
21
] built a DeepCFD
model to predict
Ux
,
Uy
, and
p
by inputting the signed distance function (SDF) and flow
region channel, and the prediction effect was excellent. This has some similarities with the
research content of this paper. However, this paper pays more attention to the aerostatic
performance of the bluff body section. Therefore, a weighted value is added to the wall
distance field in the design input to ensure the prediction accuracy of the wall pressure. The
space coordinate field is proposed to highlight the characteristics of the flow field. When
designing the model, a weighted value is also used for the L2 loss function, and the loss
value closer to the aerodynamic shape adopts a weighted value closer to 1.
Based on the above overall concept, the second and third chapters of this paper
introduce the ideas and implementation details of input and output design. The fourth
chapter introduces the structure of the fully convolutional neural network. The fifth
chapter shows how the adopted CFD setting details and example design are used for the
training and testing of the neural network model; the sixth chapter introduces the model
hyperparameter optimization results; the seventh chapter analyzes the accuracy of the
results and the comparison with other models.
2. Input Design
The input of the neural network model needs to be able to express the aerodynamic
shape information of the bluff body section, and the information should have a consistent
expression structure. For the expression of the aerodynamic shape of the main beam, the
traditional method of using coordinate points will associate the structure of the input data
with the type of shape, causing a data heterogeneity problem. With this problem in mind,
this paper proposes an aerodynamic shape description method in the form of images,
which can be effectively combined with CNN, achieve consistent representation of input
data, and improve information transfer efficiency. Based on this method, the aerodynamic
shape is described by two types of fields, namely, the wall distance field and the space
coordinate field.
2.1. Wall Distance Field
In order to avoid the problem of data heterogeneity, this paper draws on Chen’s [
22
]
idea to express the shape in the form of images, that is, assigning different values to the
internal and external spatial points of the shape. In Chen’s original design, the inside of
the shape is expressed by 1, and the outside of the shape is expressed by 0. Although this
design can express any shape, the efficiency of conveying shape information is not high,
and it cannot provide the distance information of spatial points and walls. Therefore, this
paper proposes the wall distance field as a description of the aerodynamic shape, which can
provide the relative positional relationship between the spatial point and the wall for the
limited receptive field of the CNN and directly introduce the shape pair into the input of
the neural network, influencing the factors of aerostatic performance. In the wall distance
Appl. Sci. 2022,12, 3147 4 of 21
field
IDist
, which is one of the model inputs (
I
), the assignment of the i spatial point is
shown in Equation (1):
IDist,i=β×e−1×m in(Ri,Γ)
B(1)
In the equation,
Ri,Γ
represents the set of distances between the i spatial point and the
set of aerodynamic shape boundary points
Γ
, and
min (Ri,Γ)
refers to the final minimum
value at all distances. For the consideration of dimensionless shapes, B is introduced to
represent the characteristic length of the shape. Because the space near the wall has a
greater impact on the pressure field, we use a negative exponential form to increase the
weight near the boundary layer while limiting the input to 1 to reduce the hidden danger
of gradient explosion.
β
expresses the coefficient inside and outside the shape. When the i
spatial point is located on the wall or inside the shape, it is set to 0, and otherwise, it is set
to 1. The wall distance field
IDist
of an example ellipse with an aspect ratio of 4 is shown in
Figure 1.
Figure 1. Wall distance field of an ellipse with AR = 4.
2.2. Space Coordinate Field
Simply using the wall distance field will result in the loss of some shape information.
To illustrate this problem, the shape expressed in Figure 2can be taken as an example.
The red frame line is the receptive field of the convolutional neural network in a hidden
layer. When it moves to the two positions shown in the figure, it will receive the same wall
distance field information. However, these two positions are upstream and downstream of
the flow field and have different effects on the pressure field and aerostatic performance.
In order to solve this problem, we also need to highlight the flow field characteristics of the
aerodynamic profile of the bluff body section.
Figure 2. Relationship between receptive field and shape.
Appl. Sci. 2022,12, 3147 5 of 21
To this end, we propose to use the coordinate field describing the upstream–downstream
relationship and the coordinate field describing the crosswind position as complementary
information to the wall distance field as the other two inputs to the neural network.
For the coordinate field
IX
, describing the upstream-downstream relationship, the
assignment of the ispatial point is shown in Equation (2):
IX,i=β×1
1+e−Xi
B
(2)
where
Xi
represents the downwind x coordinate of the
i
spatial point. The origin of the
x-coordinate is at the center of the section. The settings of the exponential part can achieve
both the size difference between the upstream and downstream data and the normalization
of the input.
Similarly, for the coordinate field
IY
, describing the position of the crosswind direction,
the assignment of the ispatial point is shown in Equation (3):
IY,i=β×e−|Yi|
B(3)
where
Yi
represents the y coordinate of the crosswind direction of the
i
spatial point. The
origin of the y-coordinate is located at the midpoint of the windward end: for circles and
hexagons, it is the only windward end point, and for rectangles, it is the midpoint of the
windward surface. The settings of the exponential part can achieve both the difference
between the distance of the crosswind and the distance as well as the normalization of
the input.
Figures 3and 4show the values of
IX
and
IY
in the spatial coordinate field for an
example ellipse with an aspect ratio of 4.
Figure 3. IXfield of an ellipse with AR = 4.
Appl. Sci. 2022,12, 3147 6 of 21
Figure 4. IYfield of an ellipse with AR = 4.
3. Output Design
In terms of output design, the simplest and most direct approach is to use the aerostatic
three-component force coefficient as the output of the prediction model. However, the
mapping logic is very complicated, and it directly skips multiple data processing links such
as local sampling of the pressure field and surface pressure integration. Although deep
learning networks have the ability to implement the mappings, this is bound to increase
the burden on the network, requiring more hidden layers and more complex structures. On
the other hand, the pressure field can not only provide information on the aerostatic three-
component force coefficient but also on the surrounding flow field structure, providing
support for other scientific research scenarios such as flow control and heat source analysis.
Therefore, this paper adopts a two-step strategy to obtain the aerostatic performance of
bluff body sections. First, the neural network is used to output the pressure field
Opressure
of the steady flow field, and then, the surface pressure is integrated to obtain the aerostatic
three-component force coefficient. Figure 5shows the output of the neural network for an
example rectangular section with an aspect ratio of 4.
Figure 5. Pressure field Opres sure of a rectangular section with AR = 4.
4. Neural Network Framework Design
Convolutional neural networks have been proposed since 2012 and have achieved
many application results in the fields of image classification and image detection [
23
,
24
].
Appl. Sci. 2022,12, 3147 7 of 21
The multi-layer convolutional structure of CNNs can automatically learn features. Shal-
lower convolutional layers can learn some local region features, while deeper convolutional
layers have larger receptive fields [
25
–
27
]. These properties allow CNNs to learn more
abstract features, and, therefore, it is widely used in image processing problems [
28
]. The
fully convolutional network developed on the basis of CNN is very suitable for solving the
problem of mapping from image to image [
29
]. FCN replaces the last fully connected layer
of CNN with a convolutional layer and uses deconvolution to the last convolutional layer.
The output result is unsampled, and the size of the output result is gradually restored to
the same size as the input image so that a prediction value can be generated for each pixel
point. The spatial information of the original input image is also preserved.
FCN is good at pixel-level classification problems, and this paper predicts the pressure
field or continuous pressure value at each pixel position, which is a regression problem.
Although pixel-level regression problems have higher requirements from neural network
models than classification problems, they generally require training using generative neural
networks. Considering that the pressure field involved in this paper has a gentle gradient
and continuous smoothness, the fully convolutional network can be extended from solving
pixel-level classification problems to solving pixel-level regression problems. The UNet
network proposed by Ronneberger [
30
] in 2015 can complete end-to-end training with
very simple architecture, and the network inference speed is very fast, so a modified
UNet network (Figure 6) is adopted to solve the pixel-level regression problem in this
paper. Figure 6shows the image width, height, and number of channels after convolution,
pooling, and feature fusion. D represents the number of channels, S represents the number
of convolutions, and N represents the number of poolings. These parameters will be studied
in Chapter 6. Meanwhile, the deep learning network structure will be determined. In order
to ensure that the pressure data in the aerodynamic shape is 0, a mask layer consisting of 0
and 1, with a bluff body section shape, is passed when the prediction result is output. Next,
the innovative design of the UNet structure and loss value is explained.
Figure 6. Modified UNet network structure.
First, the deep learning model needs to match the input and output data sizes. The
input involved in this paper is the wall distance field (
IDist
) and the spatial coordinate field
(
IX
,
IY
), and the number of channels is 3. Considering the information content and training
time contained in the image, the image pixels are set to 64
×
64. The output is a pressure
field with the same size as the input image, which is a single channel picture. Since the
Appl. Sci. 2022,12, 3147 8 of 21
regression problem of continuous variables needs to be predicted, it is necessary to modify
the output layer of the UNet and remove the Softmax layer.
After modifying the input and output layers, the hidden layer details of the network
structure are also required. The network architecture used in this paper is based on the
transformation of the UNet and also consists of three parts: downsampling, upsampling,
and skipping structure. The downsampling part includes the two basic components of the
convolution layer and pooling layer, and the feature extraction of the input is realized by
repeating the convolution-pooling process many times. The convolutional layer uses a 3
×
3 small convolution kernel, and the padding size is 1. Using max pooling makes the image
twice as small after each pooling. After multiple downsamplings, the shape information
of the bluff body section contained in the wall distance field and the spatial coordinate
field is highly abstracted into multiple information channels, and the upsampling part is
responsible for restoring the pressure field image based on this information. In order to
retain more detailed information in the downsampling process, the new feature map in the
upsampling process of each level and the feature map at the corresponding downsampling
position are fused on the channel, which is the skip structure. In order to extract more
detailed features before the final output, a 1
×
1 convolution is used to open up all informa-
tion channels to achieve the regression prediction of each pixel. The convolution-pooling
process is performed N times in total, and there are S convolutions each time. Both N and S
will be determined in the model optimization in Chapter 6.
In the model training process, due to the different measurement standards of the
classification model and the regression model, it is necessary to change the loss function
used to deal with the classification problem in the original UNet network to an L2 loss
function to deal with the regression error. At the same time, considering that the prediction
results in this paper pay more attention to the accuracy of the flow field data around the
aerodynamic shape, the L2 loss function is multiplied by the weight value. The loss value
closer to the aerodynamic shape is used, with a weight value closer to 1, and for the loss
value farther away from the aerodynamic shape, the weighted value is closer to 0. The
weighted value of the loss function lossweight is generated according to Equation (4):
losswei ght =IDi st,i(4)
5. Test Design
5.1. Computational Case Design
This chapter mainly shows how to obtain the data for training and verifying the
model. A certain number of aerodynamic shapes are designed as an example. The shape
information and the corresponding pressure field calculation results are used as the input
and output of the neural network. According to different basic shapes, the calculation
examples are divided into four types: regular hexagon, circle, rectangle, and regular
rhombus. Each basic shape is subdivided into different shapes according to the aspect ratio
and corner position, for a total of 105 shapes (Table 1). In order to facilitate the use of a
dimensionless form to determine the shape, the horizontal length of all shapes is fixed as 1
and the aerodynamic shape changes with the vertical height.
Appl. Sci. 2022,12, 3147 9 of 21
Table 1. List of the shape conditions.
Basic shape
Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 21
Table 1. List of the shape conditions.
Basic shape
Distance R
from upper
corner to
X = 0
[0.1,0.2,0.3,0.4] — — —
B 1
AR=B/H [0.5,0.56,0.625,0.71,0.83,1,2,3,4,5,6,7,8,9,10]
Quantity 60 15 15 15
5.2. Data Preparation
In this paper, CFD numerical simulation is used to obtain the steady-state pressure
flow field of the bluff body section. The computational domain was rectangular with a
blockage ratio of 0.9%. It is shown in Figure 7, where B is the length of the bluff body
section. In order to keep the mesh from changing too fast, three computational regions
were defined [31,32]. The rigid zone kept the mesh shape constant, outside of which the
deforming zone underwent smoothing and remeshing. The outermost fixed zone was to
satisfy the blockage ratio requirement. The details of the mesh are illustrated in Figure 8.
Figure 7. Sketch of the two-dimensional (2D) computational domain.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 21
Table 1. List of the shape conditions.
Basic shape
Distance R
from upper
corner to
X = 0
[0.1,0.2,0.3,0.4] — — —
B 1
AR=B/H [0.5,0.56,0.625,0.71,0.83,1,2,3,4,5,6,7,8,9,10]
Quantity 60 15 15 15
5.2. Data Preparation
In this paper, CFD numerical simulation is used to obtain the steady-state pressure
flow field of the bluff body section. The computational domain was rectangular with a
blockage ratio of 0.9%. It is shown in Figure 7, where B is the length of the bluff body
section. In order to keep the mesh from changing too fast, three computational regions
were defined [31,32]. The rigid zone kept the mesh shape constant, outside of which the
deforming zone underwent smoothing and remeshing. The outermost fixed zone was to
satisfy the blockage ratio requirement. The details of the mesh are illustrated in Figure 8.
Figure 7. Sketch of the two-dimensional (2D) computational domain.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 21
Table 1. List of the shape conditions.
Basic shape
Distance R
from upper
corner to
X = 0
[0.1,0.2,0.3,0.4] — — —
B 1
AR=B/H [0.5,0.56,0.625,0.71,0.83,1,2,3,4,5,6,7,8,9,10]
Quantity 60 15 15 15
5.2. Data Preparation
In this paper, CFD numerical simulation is used to obtain the steady-state pressure
flow field of the bluff body section. The computational domain was rectangular with a
blockage ratio of 0.9%. It is shown in Figure 7, where B is the length of the bluff body
section. In order to keep the mesh from changing too fast, three computational regions
were defined [31,32]. The rigid zone kept the mesh shape constant, outside of which the
deforming zone underwent smoothing and remeshing. The outermost fixed zone was to
satisfy the blockage ratio requirement. The details of the mesh are illustrated in Figure 8.
Figure 7. Sketch of the two-dimensional (2D) computational domain.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 21
Table 1. List of the shape conditions.
Basic shape
Distance R
from upper
corner to
X = 0
[0.1,0.2,0.3,0.4] — — —
B 1
AR=B/H [0.5,0.56,0.625,0.71,0.83,1,2,3,4,5,6,7,8,9,10]
Quantity 60 15 15 15
5.2. Data Preparation
In this paper, CFD numerical simulation is used to obtain the steady-state pressure
flow field of the bluff body section. The computational domain was rectangular with a
blockage ratio of 0.9%. It is shown in Figure 7, where B is the length of the bluff body
section. In order to keep the mesh from changing too fast, three computational regions
were defined [31,32]. The rigid zone kept the mesh shape constant, outside of which the
deforming zone underwent smoothing and remeshing. The outermost fixed zone was to
satisfy the blockage ratio requirement. The details of the mesh are illustrated in Figure 8.
Figure 7. Sketch of the two-dimensional (2D) computational domain.
Distance R from
upper corner to
X=0
[0.1, 0.2, 0.3, 0.4] — — —
B 1
AR = B/H [0.5, 0.56, 0.625, 0.71, 0.83, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Quantity 60 15 15 15
5.2. Data Preparation
In this paper, CFD numerical simulation is used to obtain the steady-state pressure
flow field of the bluff body section. The computational domain was rectangular with a
blockage ratio of 0.9%. It is shown in Figure 7, where B is the length of the bluff body
section. In order to keep the mesh from changing too fast, three computational regions
were defined [
31
,
32
]. The rigid zone kept the mesh shape constant, outside of which the
deforming zone underwent smoothing and remeshing. The outermost fixed zone was to
satisfy the blockage ratio requirement. The details of the mesh are illustrated in Figure 8.
Figure 7. Sketch of the two-dimensional (2D) computational domain.
Appl. Sci. 2022,12, 3147 10 of 21
Figure 8. Detailed demonstration of the computational grid within the rigid zone.
In order to reduce the calculation time while ensuring the calculation accuracy, the tur-
bulence model uses the Reynolds-averaged Navier–Stokes equations (RANS). A blending
k–
ω
SST turbulence model was used to reduce sensitivity to the inlet boundary condition.
The governing momentum equation of incompressible flow is given as
∂ui
∂t+uj
∂ui
∂xj
=−∂p
ρ∂xi
+ν∂2ui
∂xj∂xj
−∂u0
iu0
j
∂xj
(5)
where
i
,
j=
1, 2 is the two-dimensional analysis,
p
denotes the pressure, and
u0
iu0
j
is the
Reynolds stress component, in which the superscript refers to the fluctuating parts of the
velocity p. With the k–
ω
SST turbulence model, two additional equations are introduced
to obtain the Reynolds stress component, where k is the turbulent kinetic energy and ωis
its rate of dissipation (Equations (6) and (7)). The details of the involved parameters are
explained by Menter [33]:
∂(ρk)
∂t+∂ρujk
∂xj
=P−β∗ρωk+∂
∂xj"(µ+σkµt)∂k
∂xj#(6)
∂(ρω)
∂t+∂(ρujω)
∂xj
=γ
νtP−βρω2+∂
∂xjh(µ+σωµt)∂ω
∂xji
+2(1−F1)ρσω2
ω∂k
∂xj
∂ω
∂xj
(7)
The simulations were conducted using Ansys Fluent. The semi-implicit method
pressure-linked equations consistent (SIMPLEC) algorithm was used to solve the velocity–
pressure coupling problem. To obtain the appropriate numerical accuracy and stability, a
second-order scheme was used for the pressure equation, and the second-order upwind
scheme was used for the momentum equation, turbulent kinetic energy equation, and
specific dissipation rate equation. The inlet boundary was set as a velocity inlet, the
incoming turbulence intensity was set as 4%, and the viscosity ratio was set as two. The
outlet boundary was set as a pressure outlet. Slip conditions were applied to the top and
bottom boundaries for all variables. No-slip conditions were applied to the surfaces of the
bluff body section, which means that the flow state at the object’s surface was equal to the
Appl. Sci. 2022,12, 3147 11 of 21
motion of the bluff body section. Considering that the section of the bluff body with a small
aspect ratio in this paper is difficult to achieve the pressure field under the steady flow field
when the Reynolds number is large, the Re was maintained as 4000 (using the reference
length B) in all cases. The calculation results are shown in Figure 9.
Figure 9. CFD calculation of pressure cloud map.
The pressure field data is acquired once the flow field has stabilized after several
iterations. Each cycle saves 10 instantaneous pressure field data, reads 5 cycles, and
averages 50 instantaneous pressure field data to obtain the pressure field of the steady-
state flow field. The pressure field after data processing is resampled to obtain a 64
×
64
single-channel image. The X and Y coordinates of the sampling range are [
−
1.26, 1.26], and
the sampling size interval is 0.04. It is worth mentioning that this article mainly focuses
on the construction method, so there is no need to strictly verify the accuracy of the CFD
calculation results. The model training and testing are based on different data: 80% are
randomly selected as the training data set, 10% are used as the validation data set, and the
remaining 10% are used as the test data set. The dropout rate for the trained model is 0.2 to
prevent overfitting.
6. Model Performance Evaluation and Optimization
The hyperparameters of a deep learning model directly affect the performance of the
model. Taking model depth as an example, an insufficient number of hidden layers will
make the model less complex and unable to capture the nonlinear mapping between input
and output. However, too many hidden layers will cause unstable gradient calculations. In
order to quantitatively evaluate the performance of the model, we optimize the number
of channels (D), the number of convolutions (S), and the number of pooling times (N)
in Figure 6based on the mean absolute error MAE (Equation (8)) of the extracted drag
coefficient based on the predicted pressure field.
MAE =1
m
m
∑
i=1
|ti−yi|(8)
Pytorch performs very well in scientific research, which is mainly reflected in the very
Pythonic style of Pytorch. Pytorch reduces the difficulty of getting started and can be built
layer by layer when building a deep learning model, which is convenient for real-time
modification. Therefore, this paper uses Pytorch to build the deep learning model.
Appl. Sci. 2022,12, 3147 12 of 21
6.1. Optimization of the Number of Hidden Layers
The number of hidden layers has a greater impact on the model performance than
the size of the hidden layers. Therefore, this paper determines the suitable model depth
for the prediction of the aerodynamic performance of the bluff body section [
34
,
35
]. The
convolutional layer, pooling layer, and ReLU layer are considered to be downsampling
components, and the number of these components (that is, the number of pooling times
(N) in Figure 6) is optimized. Considering that the size of the data image is 64
×
64, after
6 iterations of max pooling, the input information is converted into multiple channels of
1
×
1, and, therefore, the maximum number of downsampling components is set to 6.
When naming the model, UNet_2-3 represents two downsampling components, and each
downsampling component layer contains three convolution operations. Using the same
training and validation sets, the comparison of different models is shown in Table 2.
Table 2. Model performance comparison—number of downsampling components.
Model Settings UNet_2-2 UNet_3-2 UNet_4-2 UNet_5-2 UNet_6-2
MAE 0.2313 0.2038 0.1942 0.2190 0.1593
Parameter quantity 1.016 ×1064.262 ×1061.724 ×1076.915 ×1072.768 ×108
From the results in Table 2, it can be seen that as the number of model layers increases,
MAE first decreases, then increases, and finally decreases to its lowest value. The model
obtains the smallest drag coefficient MAE when using six downsampling components.
However, the data in Table 2are based on two convolution operations. Therefore, it
is necessary to continue to observe the effect of the number of convolution operations
(Table 3).
Table 3. Model performance comparison—number of convolutions.
Model Settings UNet_2-3 UNet_3-3 UNet_4-3 UNet_5-3 UNet_6-3
MAE 0.1801 0.1175 0.1696 0.1456 0.1234
Parameter quantity 1.533 ×1066.401 ×1062.587 ×1071.037 ×1084.152 ×108
Combining the observations in Tables 2and 3, it can be seen that when three convolu-
tions are used, the MAE is significantly lower than when two convolutions are used. At
the same time, the UNet_3-3 model has the lowest MAE and fewer parameters than the
other models, and its computational cost is lower. Considering that increasing the number
of convolution operations can significantly improve the model performance, in order to
determine the optimal number of convolution operations for the UNet_3 model, UNet_3-4
is added for comparison. The calculation shows that the prediction result of UNet_3-4
is
MAE =
0.1675 and the parameter quantity is 8.540
×
10
6
. It can be seen that when
the number of downsampling components is 3, the MAE does not decrease but increases
after the number of convolution operations is increased to 4. In summary, considering
the performance and the amount of computation, this paper chooses 3 as the number of
downsampling components and uses the UNet network model with 3 convolutions.
6.2. Hidden Layer Size Optimization
After determining the depth of the hidden layer, it is necessary to further optimize its
size, that is, the number of channels (D) in Figure 6. The number of hidden layer channels
of the first downsampling component in the selected UNet network for comparison is set
to 16, 32, and 128. The number of hidden layer channels in the subsequent downsampling
component is twice the model width of the previous layer. The same training set and
validation set are used, and the comparison of different models is shown in Table 4. When
naming the model, UNet_3-3-16 represents that the number of hidden layer channels of the
first downsampling component is 16.
Appl. Sci. 2022,12, 3147 13 of 21
Table 4. Model performance comparison—number of convolutions.
Model Settings UNet_3-3-16 UNet_3-3-32 UNet_3-3-64 UNet_3-3-128
MAE 0.1531 0.0942 0.1175 0.1700
Parameter quantity 4.010 ×1051.601 ×1066.401 ×1062.559 ×107
It can be seen from the results in Table 5that as the number of hidden layer channels
increases, the MAE first decreases and then increases. When the initial hidden layer channel
number D is 32, the MAE is the smallest and the number of model parameters is moderate.
Therefore, UNet_3-3-32 is selected for research in subsequent chapters.
Table 5. Average relative error of predicted drag coefficient after changing input.
Shape R = 0.1
Hexagon
R = 0.2
Hexagon
R = 0.3
Hexagon
R = 0.4
Hexagon Circle Rectangle Diamond
Average relative error (%) 7.690 22.20 45.33 26.72 17.26 21.37 18.13
7. Pressure Field Prediction Effect
The pressure field prediction accuracy directly affects the subsequent extraction of the
drag coefficient, so it is necessary to check the prediction results of the flow field. After the
model training is completed, the wall distance field and spatial coordinate field of the test
data set are input into the model to obtain the predicted value of the pressure field. It is
then converted into a visualized image and compared with the CFD calculation results, as
shown in Figure 10. Due to space limitations, representative shapes of pressure fields are
selected for presentation.
Figure 10. Cont.
Appl. Sci. 2022,12, 3147 14 of 21
Figure 10.
Comparison of pressure field prediction effects of four typical shapes. The first column is
the predicted value. (
a1
) The ellipse with AR = 0.71. (
b1
) The hexagon with AR = 0.56 R = 0.3. (
c1
)
The rectangle with AR = 0.71. (
d1
) The diamond with AR = 0.71. The second column is the true value.
(
a2
) The ellipse with AR = 0.71. (
b2
) The hexagon with AR = 0.56 R = 0.3. (
c2
) The rectangle with AR
= 0.71. (d2) The diamond with AR = 0.71.
Appl. Sci. 2022,12, 3147 15 of 21
The analysis and prediction results show that the larger part of the prediction error
for the ellipse, hexagon, and diamond generally exists in the negative pressure area that
deviates far from the shape, and the predicted pressure is larger than the pressure field
calculated by CFD. The larger part of the error of the rectangular section exists in the
positive pressure area that deviates far from the shape, and the predicted pressure is
smaller than that calculated by CFD. The authors believe that this phenomenon is mainly
caused by two factors. First, when designing the loss function, the aerostatic performance is
considered to be more closely related to the pressure field near the wall. Therefore, in order
to predict the wall pressure value more accurately, the loss function gives more weight
to the prediction accuracy close to the wall and has more relaxed requirements for the
prediction value of the grid points farther away from the wall of the bluff body section.
Secondly, the improved UNet model is still too simple, and it is necessary to use a more
complex generative neural network in subsequent research. However, the improved Unet
network can still achieve rapid prediction of the wall pressure and third-component force
coefficient to a certain extent, as will be discussed in subsequent chapters.
8. Wall Pressure Prediction Results
Wall pressure is of great significance in the evaluation of aerostatic performance
and can also be used to extract third-component force coefficients. Figure 11 shows the
comparison of the wall pressure prediction results and the CFD calculation results. Due to
space limitations, only four typical shapes and two for each shape are shown. Since the
shapes used in this paper are all symmetrical up and down, only the wall pressure in the
upper part is shown.
Figure 11. Cont.
Appl. Sci. 2022,12, 3147 16 of 21
Figure 11. Comparison of the prediction effects of the wall pressure field of the four shapes. (a) The
ellipse with AR = 0.71. (
b
) The ellipse with AR = 4. (
c
) The hexagon with AR = 0.56 R = 0.3. (
d
) The
hexagon with AR = 7 R = 0.1. (
e
) The rectangle with AR = 0.71. (
f
) The rectangle with AR = 4. (
g
) The
diamond with AR = 0.71. (h) The diamond with AR = 4.
In general, the analysis of the wall pressure prediction results shows that the basic
trend of the wall pressure distribution can be simulated based on the model proposed in
this paper, but there are certain errors. The prediction results of the positive pressure area
are generally better than those of the negative pressure area. Among them, the area with
the largest error is mainly concentrated in the negative pressure area generated by flow
separation and the negative pressure area near the wake. The absolute error value does
not exceed 0.2 at the most, and the range is small. Meanwhile, the prediction errors of the
rectangle and rhombus are generally larger than those of the ellipse and hexagon. The
author believes that this is mainly caused by the uncomplicated UNet model. The original
UNet model is mainly used to deal with classification problems. Although this paper has
modified it, the modified UNet model still has a lot of room for improvement when dealing
with regression problems.
9. Drag Coefficient Prediction
The drag coefficient under the shape can be obtained through the wall pressure of
each aerodynamic shape, and the aerostatic wind load in the wind resistance design can
be obtained from the drag coefficient. Therefore, the prediction of the drag coefficient is
very important, and the input and output of the deep learning model also directly affect
the prediction accuracy of the drag coefficient. In this paper, the three distance fields (
IDist
,
IX
and
IY
) are used as the input of the model, and the output is the pressure field. By
Appl. Sci. 2022,12, 3147 17 of 21
comparing the accuracy of the drag coefficient, it is shown that the innovative input and
output proposed in this paper can effectively improve the prediction accuracy.
9.1. Compared to Using Shape as Input
Donglin Chen sets the internal aerodynamic shape to 1 and the external to 0 to express
the aerodynamic shape, then uses it as the model input for training. Since the sampling area
in this paper is large to ensure the normal training of the model, the internal aerodynamic
shape is set to 0 and the external is set to 1 to express the aerodynamic shape as the model
input. In this paper, three distance fields (
IDist
,
IX
, and
IY
) are innovatively proposed to
express the aerodynamic shape and flow field information. This paper uses the UNet model
determined in Chapter 6 and exactly the same parameters for training and compares the
accuracy of the drag coefficient to illustrate that the three distance fields (
IDist
,
IX
, and
IY
) proposed in this paper can effectively improve the prediction accuracy of the model.
Taking an ellipse with an aspect ratio of 4 as an example, the 0–1 distance field is shown in
Figure 12. The average relative error of the drag coefficient of each basic shape predicted
by the model is shown in Table 5.
Figure 12. Partial 0–1 distance field.
It can be seen from Table 6that the average relative error of the R = 0.3 hexagon
reaches 45.33%, and the average relative errors of the R = 0.4 hexagon and rectangle also
exceed 20%, making the total average relative error reach 22.67%. In contrast, the average
relative error of the three distance fields (
IDist
,
IX
, and
IY
) acting as the model input is
9.42%, which is 13.25% higher than the average relative error of the drag coefficient when
the 0–1 distance field is used as the model input. It can be seen that the wall distance field
and spatial coordinate field can make the model learn and predict more efficiently.
Table 6. Average relative error of predicted drag coefficient after changing output.
Shape R = 0.1
Hexagon
R = 0.2
Hexagon
R = 0.3
Hexagon
R = 0.4
Hexagon Circle Rectangle Diamond
Average relative error (%) 3.450 7.740 12.14 5.010 9.110 13.08 15.42
The average relative error when using the 0–1 distance field as the model input is much
higher than the average relative error of the wall distance field and the spatial coordinate
field proposed in this paper. The main reason for this is that the 0–1 distance field is too
simple to express the aerodynamic shape and cannot provide effective distance information
and flow field information, while the drag coefficient is closely related to the distance
information and flow field information. It is, therefore, difficult for deep learning models
to establish a relationship between the aerodynamic shapes and the drag coefficient.
Appl. Sci. 2022,12, 3147 18 of 21
9.2. Compared to Using the Drag Coefficient as Output
There are two ways to predict the drag coefficient of the bluff body section through
the deep learning method. One is to directly predict the drag coefficient as the output;
the other is to predict the pressure flow field and then pass it through the data to get the
drag coefficient.
The direct method still uses the three distance fields of
IDist
,
IX
, and
IY
as the input of
the model and the processed drag coefficient as the output. The CNN model consists of
the convolutional layer, pooling layer, ReLU layer, and fully connected layer. The CNN
model, with a model depth of 3 layers and an initial model width of 128, was selected by
comparing the size of the MAE and the number of parameters, as shown in Figure 13.
Figure 13. Determined CNN model.
By directly predicting the drag coefficient through the CNN model, the overall average
relative error of the test set reached 19.64%, the average relative error of the diamond
reached 38.81%, and the average relative errors of the R = 0.2 regular hexagon and the
R = 0.3 regular hexagon were less than 10%.
The author believes there are two main reasons for the large error in the direct pre-
diction of the drag coefficient through the CNN model. The first reason is that there are
very obvious differences between the basic shapes, which makes it difficult to accurately
associate the shape input with the drag coefficient in the deep learning model. Therefore,
we can only fit relatively satisfactory results under different shapes, which leads to large
errors in some shapes. The second reason is that it is too simplistic to directly use the drag
coefficient data as the output and directly use the image as the output. When predicting
a number such as a drag coefficient, it is difficult for a deep learning model to establish
a certain connection between the input and output. The randomness is also large, which
makes the model prediction less stable.
In reference to the first reason, better deep learning model architecture can be selected
by changing the structure of the model. However, the CNN model is the optimal solution
for the direct prediction of the drag coefficient in this paper. Therefore, the output needs
to be redesigned for the second reason. The drag coefficient can be obtained indirectly by
predicting the pressure field of the steady-state flow field. The input of the model is still the
three distance fields of
IDist
,
IX
, and
IY
, and the output is the stable pressure flow field. Due
to the different input and output models, the UNet network is selected and the optimal
model is determined, as shown in Figure 6.
Appl. Sci. 2022,12, 3147 19 of 21
Considering that the drag coefficients of different shapes are quite different, the
average relative error is used to comprehensively evaluate the prediction accuracy of
different shapes, as shown in Table 6. At the same time, the comparison between the
predicted drag coefficient and the CFD-calculated drag coefficient for each shape is shown
in Figure 14.
Figure 14. Drag coefficient prediction results.
From the prediction results given in Table 6, it can be seen that the average relative
errors of the drag coefficients of the R = 0.1 hexagon, R = 0.2 hexagon, and R = 0.4 hexagon
and circle are less than 10%. A direct relationship is maintained between the accuracy
of the wall pressure prediction and the prediction accuracy of the drag coefficient. The
relative errors of the drag coefficients predicted by the R = 0.3 hexagon, rectangle, and
diamond exceed 10% but do not exceed 20%. The average relative error is the largest for
the diamond, reaching 15.42%. The average relative error for all shapes is 9.42%. It can be
seen that the UNet model adopted by the indirect method has better performance than the
CNN model that directly predicts the drag coefficient, and the average relative error of the
drag coefficient is increased by 10.22%. It can be seen that the indirect method of predicting
the drag coefficient strengthens the connection between the input and the output. The
prediction of the flow field can also provide applications for flow field control and other
scenarios, and the practical performance of the model is greatly improved.
10. Conclusions
This paper proposes a novel, fully convolutional neural network model that enables
rapid prediction from shape to aerostatic performance.
(1)
The overall average relative error of the drag coefficient is 9.42%.
(2)
The shape is described by the combination of the wall distance field and the space
coordinate field. This improves the prediction accuracy by 13.25% compared to when
the shape is directly used as the model input.
(3)
A step-by-step strategy in which the pressure field is used as the model output is
proposed. The model prediction accuracy is improved by 10.22% when using the
pressure field as the model input compared to directly predicting the drag coefficient.
Appl. Sci. 2022,12, 3147 20 of 21
Author Contributions:
Conceptualization, K.L.; methodology, K.L.; software, H.L.; validation, H.L.;
formal analysis, H.L.; investigation, K.L.; resources, K.L.; data curation, Z.C.; writing—original draft
preparation, H.L.; writing—review and editing, K.L.; visualization, H.L.; supervision, S.L.; project
administration, S.L.; funding acquisition, K.L. and S.L. All authors have read and agreed to the
published version of the manuscript.
Funding:
This research was funded by the National Science Foundation for Young Scientists of China
grant number 51808075; the National Science Foundation of China grant number 51978108, the 111
project of the Ministry of Education and the Bureau of Foreign Experts of China grant number B18062;
the Natural Science Foundation of Chongqing China grant number cstc2020jcyj-msxmX0773 and
cstc2020jcyj-msxmX0937; the Fundamental Research Funds for the Central Universities grant number
2020CDJ-LHZZ-018, 2021CDJQY-025 and 2020CDJ-LHZZ-016; the Chongqing full-time postdoctoral
exit and stay in the Chongqing Project grant number 2020LY07.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments:
This paper is supported by the National Science Foundation of China (51808075,
51978108), 111 Project (B18062), the Natural Science Foundation of Chongqing, China (cstc2020jcyj-
msxmX0773, cstc2020jcyj-msxmX0937), the Fundamental Research Funds for the Central Universities
(2020CDJ-LHZZ-018, 2021CDJQY-025 and 2020CDJ-LHZZ-016), and the Chongqing full-time post-
doctoral exit and stay in the Chongqing Project (2020LY07).
Conflicts of Interest: The authors declare no conflict of interest.
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