Automobile shape formation and simulation by a computer-aided systematic method

Abstract

A systematic method which combines the shape curvature entropy and computational fluid dynamics analyses was proposed in this study for creating automobile shapes. Various automobile shapes were evaluated by the proposed method and the parameters including the curvature entropy and the drag coefficient of each configuration were calculated so as to determine the optimal shape design. First, shapes of commercially available car models were collected and their curvature entropy values were determined by the proposed equation. Meanwhile, the drag coefficient of each car model was determined by computational fluid dynamics simulation. Both values were compared and the results indicated a positive correlation between an automobile shape’s curvature entropy value and its drag coefficient. Therefore, the curvature entropy is proven to be a suitable indicator for estimating an automobile shape’s aerodynamic performance. For a new automobile shape design, the required number of numerical analyses can be reduced by utilizing the proposed curvature entropy calculation so that the time spent on designing an automobile shape can be greatly reduced.

Keywords

automobile flow analysis shape curvature entropy computer-aided design body surface modeling

Introduction

The sketch model corrections and clay model making and modifications during the conventional procedure of developing automobile shapes usually involve a great deal of manpower and time. Designers’ thought-forming process is often interrupted by this obstacle as compared to the concept of real-time product design by Tovey (1989). The study of Kušar et al. (2014) proposed an approach of concurrent realization for car components by integrating quality standards into the product realization process so that the development costs and time can be reduced.

When designing automobile shapes, consumer requirements must be determined by market research and analyses and further translated into design information. After that, the design concepts must be applied to real vehicles in order to really meet consumers’ needs. In addition to the use of automatic design tools when processing vehicle information, designers also need to flexibly link the intuitive thoughts to their correction intention by Tovey (1992). When defining and translating consumers’ requirements, it is important to clarify and quantize uncertainties so as to expand the design process further and make it transparent by Jones (1992). Nomaguchi et al. (2012) proposed an approach of design process planning based on a growth curve model with a fuzzy number managing uncertain design progress so as to realize the optimization of process planning.

Since the automobile shape design also has safety and power concerns in addition to the aesthetic and symbolic functions, the practical functionality is sometimes more critical. Catalano et al. (2007) proposed a way of building three-dimensional (3D) automobile shapes in a semantic way and retrieving aesthetic elements via a knowledge database. Structurally overt and covert elements can be acquired so as to define and evaluate the beauty of automobile shapes. Setchi et al. (2011) proposed the approach of semantic information retrieval to help with concept design. They used semantic annotations so that designers can easily look for inspiration. The information which is diverse, vague, and uncertain to a certain degree can be acquired by these innovative tools so as to create new designs.

Conventional wind tunnel tests are gradually replaced by simulation software with the rapid development of computer technology in recent years. The software has enabled a numeric computation mode for hydrodynamics by creating a simulative flow field so as to reduce the number of tests required and shorten the development time (Rawnsley and Glynn, 1986; Shaw, 1987). Han et al. (1996) simulated three simplified automobile shapes numerically and compared the results with the experimental values. The results of their study indicated that the turbulent flow model could be the key factor. The results of a simulation analysis by a $κ - ε$ turbulent flow model came closest to the experimental value. By experimenting with different grid points, the results also verified that grid points affect the calculation accuracy. The drag coefficient obviously changed based on a further investigation of the automobile shapes and this resulted in different flow field phenomena. The study of Grun (1996) adopted a zonal approach for the automobile shapes, while Ramnefors et al. (1996), Okumura and Kuriyama (1997), Uchida and Okumura (1997), and Yamane et al. (1997) all adopted the finite volume method to perform systematic flow field simulation. To reduce the air drag, Kim et al. (2002) proposed adding a rectifier device in the rear of large automobiles. Eiamsa-ard et al. (2007) studied relevant aerodynamic features of small automobiles with air rectifier hoods in the rear (i.e. the rear roof). The study indicated that the pressure and resistance can be reduced by changing the vortex model in the rear of the automobile so as to increase the rear pressure. Makowski (2001) used numerical software to help quickly change the design of the inlet of the shuffle box in a Ferrari F-1 racing car. However, the principal concern for automobile shape designers is on the integration of automatic shape-building and analysis software into future engineering applications.

Hsiao (1994) built a database for basic automobile shape consultations so as to assist designers in creating automobile shapes which meet consumers’ requirements. With an aim to make automobiles maintainable and to find the best location for automobile parts so as to prevent automobile rollovers in the event of accidents, Miao et al. (2008) applied the genetic algorithm (GA) and a new encoding method to automobile configuration design. The GA approach is developed into a multi-objective genetic algorithm (MOGA) which can offer designers appropriate solutions. Ji and Yu (2008) utilized the industrial design approach to display the content and features of light electric vehicles. The purpose of their study was to generate innovative designs by analyzing the design errors of the current products in the market. The parameterized Pareto set was used by Malak and Paredis (2010) to evaluate concept designs and to determine systematic design problems during the component design stage. The study of Helo et al. (2010) indicated that large-scale automobile configuration designs must respond to consumer orders quickly by considering consumer requirements, product features, production flow, and logistics network altogether. They also proposed an integrated vehicle configuration system (IVCS), which uses configuration rules to deal with relevant construction design and production problems so as to meet consumer requirements.

The aim of this study is to determine the correlation between an automobile shape’s curvature entropy and its aerodynamic performance. Drag coefficients of various automobile shapes were determined by numerical simulation and then compared with the curvature entropy values obtained by the proposed equation. It is expected to use the curvature entropy as an indicator of an automobile shape’s aerodynamic performance so that the required number of wind tunnel tests or numerical analyses can be reduced.

Theoretical background

Styling entropy

The information theory, which was proposed by Shannon (1948), belongs to a revolutionary mathematical field. This theory supports quantization of general information and a description of the message, especially a message on the architecture of probability. The information quantity is equal to the entropy value. The unorganized or disorder level of a system is represented by entropy, which is a common word in statistical thermodynamics. The entropy of a system increases when it becomes increasingly unorganized. In other words, entropy measures the complexity of a system. An increase in the entropy value means an increase in the components of the system, and hence an increase in information quantity. The information theory has also been applied to different fields such as communication, astronomy, and crystallography by some researchers. For the complexity of different shapes of satellite images, a type of segmentation arithmetic was developed by Oddo (1992) for acquiring a picture of an airplane so as to build a range in sphere entropy. Cesar and Costa (1997) proposed to measure a shape by multidimensional scaling and to develop class nerve morphology. A directly associated triangle grid data set was proposed by King and Rossignac (1999). Toussaint (1991) resolved the shape of a two-dimensional (2D) polygon and he measured and calculated the complexity of a shape based on the polygon. He also proposed a replacement of the polygon edge, in turn, to generate spiral directions. Chazelle and Incerpi (1984) proposed an algorithm which runs in time as a shape-complexity computation model. Young et al. (1974) used curving power to measure a shape and the figure characteristics of creature objects can be described. The definition of curving power was further expanded by Van Vliet and Verbeek (1993) into a 3D data set. The information theory was utilized by these key methods for the description of shapes. The gray entropy was further expanded by Pal and Pal (1989) into the two-level Shannon’s entropy, which used one L × L gray level co-occurrence matrix (GLCM) to calculate the two-level entropy value.

In the study of Page et al. (2003) on the form complexity, Shannon’s entropy was utilized to measure the complexity of a 2D graph and a 3D structure. In studies related to visual cognition of forms or artificial vision, the thermal function “entropy” was often introduced into the form complexity and was called the “curvature entropy” of the figure. A shape’s complexity is judged by the value of the curvature.

Therefore, curvature entropy represents the amount of curvature data information included in forms and figures and it affects the form complexity. The greater the dissimilar curvature value, the greater the curvature entropy value, and the more complex the graph becomes; vice versa, the smaller the dissimilar curvature value, the smaller the curvature entropy value, and the less complex the graph becomes.

The curvature entropy can be calculated by the equation as follows

H = - \sum_{i = 1}^{n} p_{i} \cdot \log_{2} p_{i}

(1)

where H is the complexity, n is the number of occurrences of a styling figure on the line curvature, and p_i is the possibility of occurrence of the ith type of figure on the line curvature.

Equations for flow field analysis around an automobile

The history of a Chevrolet Lumina took part in the NASCAR Winston Cup from 1989 to 1994 was studied by Laise and Bayless (1994). They investigated the way how the automobile achieved the best aerodynamic performance during races. The study of Wickern and Lindener (2000) indicated that Audi’s wind tunnel equipment (including a ground simulation system driven by five belts) can be placed right at the front of the automobile. This formed a projection area up to 3 m² and a wind speed of up to 300 km/h. The external flow field of the Opel Astra was simulated by Kleber (2001) in the Fluent Software and the results offered a general introduction of the whole simulation process, including preprocessing, calculation, and post-processing.

As shown in Figure 1, the boundary of the numerical model is formed by the inlet, outlet, block, and boundary wall. The domain height is five times the vehicle height, the flow field in front of the automobile is five times the vehicle length, and the flow field behind the automobile is six times the vehicle length.

The fluid properties are as follows:

Air density: 1.189 kg/m³

Coefficient of dynamic viscosity: 1.544 ×10⁻⁵ m²/s

Pressure coefficient : C_{p} = \frac{Δ p}{\frac{1}{2} ρ V_{\infty}^{2}}

(2)

Coefficient of frictional resistance : C_{r} = \frac{τ_{w}}{\frac{1}{2} ρ V_{\infty}^{2}}

(3)

Drag coefficient : C_{d} = \frac{F_{p} + F_{f}}{\frac{1}{2} ρ A_{t} V_{\infty}^{2}}

(4)

where $C_{p}$ is the pressure coefficient, $C_{r}$ is the coefficient of frictional resistance, $C_{d}$ is the drag coefficient, $Δ p$ is the relative pressure difference, $ρ$ is the air density, $V_{\infty}$ is the traveling speed of the vehicle, $τ_{w}$ is the shearing strength on the wall, $F_{p}$ is the profile drag, $F_{f}$ is the pressure resistance, and $A_{t}$ is the area of the section in the forward direction.

The 2D grid data of the vehicle body were first entered into the Fluent software so as to analyze the flow field around the automobile. The analysis results can generate several simulation charts, including the $C_{p}$ value curve chart, pressure distribution chart, $C_{d}$ value, and the airflow field on the surface of the vehicle. These served as reference factors for evaluating the automobile shapes and performance by calculating the wind drag and the wind pressure around the vehicle. The results can be used for the shape correction afterward and this serves as the criteria for automobile shape evaluation. Computational fluid dynamics (CFD) analyses have been widely used for improving vehicle performance. For example, Tsai et al. (2009) proposed an effective numerical model to investigate the flow field around a passenger car with wing-type rear spoiler by using the Fluent software as well. Their study also analyzed the aerodynamics and aero-acoustics of the most suitable spoiler design. Similarly, Sharma and Bansal (2009) proposed a numerical model to obtain the flow structure around a passenger car with tail plates and they recommended using add-on devices to reduce the drag force and enhance fuel economy of a vehicle. Khan and Umale (2014) also investigated the effects of the radius of Ahmed body and the ground clearance on the drag coefficient by CFD analyses and compared the results with actual wind tunnel data.

Theory of turbulent flow model

The governing equations for the analysis of a flow field include the following: a continuity equation, a momentum equation, a turbulence kinetic energy equation ( $κ$ ), and a dissipation rate of turbulence energy equation ( $ε$ ). Among them, the turbulent flow model uses the standard $κ - ε$ turbulent flow model of Launder and Spalding for flow field calculations. The standard $κ - ε$ turbulent flow model is a semi-empirical turbulent flow model, and it can be used to derive the transport equation of turbulence kinetic energy ( $κ$ ) and dissipation rate ( $ε$ ). These parameters determine the turbulent flow transportation behavior mainly according to basic physical control equations as follows.

Turbulence kinetic energy equation ( $κ$ )

\frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial x_{i}} (ρ k u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + G_{k} + G_{b} - ρ ε - Y_{M}

(5)

Dissipation rate equation ( $ε$ )

\frac{\partial}{\partial t} (ρ ε) + \frac{\partial}{\partial x_{i}} (ρ ε u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] + C_{1 s} \frac{ε}{k} (G_{k} + C_{3 ε} G_{b}) - C_{2 g ρ} \frac{ε^{2}}{k}

(6)

Turbulence viscosity coefficient $(μ_{t})$

μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}

(7)

where $G_{k}$ represents the turbulence kinetic energy generated by the velocity gradient; $G_{b}$ represents the turbulence kinetic energy generated by buoyancy; $Y_{M}$ refers to the fluctuation due to excessive diffusion of the compressible turbulent flow; $σ_{k}$ and $σ_{ε}$ are the turbulent Prandtl number of the turbulence kinetic energy and the turbulence dissipation, respectively; and $C_{1 ε}$ , $C_{2 ε}$ , and $C_{3 ε}$ are the empirical constants, whose suggested values are shown in Table 1.

Table 1.

Standard values of experimental coefficients in $κ - ε$ turbulence model.

$C_{1 ε}$	$C_{2 ε}$	$C_{μ}$	$C_{k}$	$C_{ε}$
1.44	1.92	0.09	1.0	1.314

The $κ - ε$ model is an equation based on the assumption that the flow field is completely turbulent and the molecular viscosity can be neglected. Therefore, it can obtain better results when calculating the complete turbulent flow field.

Implementation procedure

The implementation procedure is shown in Figure 2.

Step 1: define vehicle shape and construct shape features

The first step in this study was to collect side elevations of sports sedans which were manufactured from the year of 1985–2011. In all, 115 sample images were collected to acquire a diversity of vehicle shapes and to avoid any similar shapes and this range covers various shapes of mass-produced sports sedans and concept sports sedans.

The aerodynamics of an automobile shape might be altered by rectifier devices, rear spoilers, or tail plates, which are difficult to depict the profile and determine the exact curvature entropy values. It is assumed in this study that all the automobile shapes are not installed with any type of rectifier devices so that the curvature entropy values can be correctly determined.

The side profile of a vehicle was divided into three parts in this study: head shape, roof shape, and tail shape. The head shape is defined as the contour which starts from the lower edge of the bumper and to the front edge of the windscreen, the roof shape starts from the front edge of the windscreen to the upper edge of the rear windscreen, and the tail shape starts from the upper edge of the rear windscreen to the lower edge of the rear bumper. To define these contours, the SolidWorks software was used to make radiation lines from the central point of the chassis and obtain intersection points where the radiation lines crossed the contour. In addition, due to the different degrees of line complexity in different areas, a different number of radiation lines were created for different sections. Totally, 32 lines were created for the head, 16 lines for the roof, and 24 lines for the tail, as shown in Figure 3.

The automobile images were converted, according to their proper proportions, into figure point data in the software and stored in Excel so as to build a contour database. The database was then loaded by sections (head shape, roof shape, and tail shape) into the IBM SPSS Statistics Professional software to perform cluster analysis by Ward’s.

According to the cluster analysis results, the head shape can be divided into 10 groups, the roof shape can be divided into 7 groups, and the tail shape can be divided into 11 groups. Representative shapes of all groups were also obtained. Figure 4 shows the representative samples of the head shape groups.

Step 2: integrate overall shapes

The fluid numerical simulation was used in this study to match the six head feature groups with fixed roof feature groups and nine tail feature groups with the most samples. This results in 21 automobile shapes as shown in Table 2.

Step 3: calculate the shape curvature entropy (SCE)

Table 2.

Summary of 21 automobile shapes.

Index	S1	S2	S3	S4	S5
Shape #
Index	S6	S7	S8	S9	S10
Shape #
Index	S11	S12	S13	S14	S15
Shape #
Index	S16	S17	S18	S19	S20
Shape #
Index	S21
Shape #

The data of the shape sample of the six new automobile shapes was entered into equation (1), the SCE equation, to calculate the curvature entropy of each automobile shape. The resulting values are shown in Table 3.

Step 4: simulation data for the analysis of automobile flow field

Table 3.

Shape curvature entropy values of 21 shapes.

Shape #	Shape 1	Shape 2	Shape 3	Shape 4	Shape 5	Shape 6
SCE	6.327	6.199	6.209	6.800	6.656	6.312
Shape #	Shape 7	Shape 8	Shape 9	Shape 10	Shape 11	Shape 12
SCE	6.931	6.301	6.735	6.703	6.295	6.727
Shape #	Shape 13	Shape 14	Shape 15	Shape 16	Shape 17	Shape 18
SCE	6.689	6.753	6.738	5.891	6.223	6.741
Shape #	Shape 19	Shape 20	Shape 21
SCE	7.058	7.259	7.02

SCE: shape curvature entropy.

The procedure is as follows:

Preprocessing: define the physical models, material properties, and boundary conditions.

The boundary conditions are as follows:

Free boundary condition $\partial φ / \partial n = 0$ , relative pressure P = 0.

Air speed at inlet V = 100 km/h, initial condition KE = 0.

The surface of the body is not porous with porosity = 0; it stands still and is not sliding.

Ground speed V = 0 km/h.

The fluid in the flow field is air and its properties are as follows:

Density is ρ = 1.189 kg/m³.

Laminar kinematic viscosity ν = 1.544 × 10⁻⁵ m²/s.

Specific heat = 1005 J/kg K.

Prandtl number = 0.715.

Obtaining a solution: conduct simulations and calculations and load the defined file from preprocessing into the solver. The settings and calculation procedures are as follows:

Define entrance/exit speed, pressure, 2D automobile shape grid data, and other boundary conditions.

The entire analysis area for the 2D automobile model lies on the plane of x = 1, and the number of grids is 42 × 87 in y- and x-directions, respectively; the convergence criterion of the numerical values of the residuals is set to 10⁻³.

Set the fluid properties as air and the solid property for other materials.

Solving for the velocity field:

Choose the flow field solver and define it as an incompressible and stable k–ε turbulence model.

Define various relaxation coefficients.

Define the number of iterations and relevant monitoring points.

Solve for the flow field.

Post-processing: present the analysis results in different physical quantities, including the following:

Vector or scalar.

Graph or data.

Results and discussions

Curvature entropy

The larger the curvature entropy value, the more complex the shape; vice versa, the smaller the dissimilar curvature value, the smaller the curvature entropy value and the less complex the shape becomes. As mentioned earlier, the side profile of a target vehicle was divided into three sections, which include the head section, roof section, and tail section. To ensure consistent simulation results, the first step is to investigate the most feasible number of grids. As a preliminary verification, each of these three sections was assigned a constant number of 100, 200, or 500 grids. The results obtained with different numbers of grids were further compared and it was found that the difference between the 200-200-200 grids and the 500-500-500 grids was within 1%. Therefore, the simulation of the rest of car models remained a configuration of 500-500-500 grids.

Therefore, by comparing the resulting curvature entropy values of the 21 vehicle shapes in Table 3, we can rank them according to the complexity of their head shapes: S20 > S19 > S21 > S7 > S4 > S14 > S18 > S15 > S9 > S12 > S10 > S13 > S5 > S1 > S6 > S8 > S11 > S17 > S3 > S2 > S16.

It can be seen by comparing Shape 20 with Shape 16 that the curved shape of the former is more complex. The former has a longer front water tank hood as compared to the conventional head shape of the latter. Shape 5 and Shape 6 belong to current popular small car models, but their curvature entropy did not increase since they have a shorter radiation line. Instead, their curvature entropy is smaller due to their rounder shape and they have more consistent curvature values accordingly. Shape 20 is long and narrow and this leads to a greater and more complex change in the shape curvature while its curvature entropy is larger. It was also found that the curvature entropy values of car shapes before the 1990s are in average larger than those after the 1990s. This can be attributed to the fact that the aerodynamics of cars was greatly improved due to the advances in the numerical simulation.

Analysis results of Cp values of the flow field

By calculating the pressure (P), horizontal velocity (V), vertical velocity (W), KE, and PE of the automobile shapes via iterations of the flow field analysis, the flow field features can be examined and the influences can be investigated via the analysis along the coordinates of the grid points. The external streamline distribution of two representative automobile shapes is shown in Figure 5. For (A) Shape 2, the distribution of streamlines is very smooth without any sudden change. On the contrary, the streamlines of (B) Shape 20 encounter a sudden change around its head shape and the entire flow field in front of the windshield is greatly affected. It is apparent that the air drag of Shape 20 is much higher than that of Shape 2.

The velocity distribution in Figure 6 also indicated that the flow field behind the rear pillar of (B) Shape 20 includes a very large area in which the air velocity is small. This leads to a very large back pressure, and thus the air drag is apparently much higher than (A) Shape 2.

It can be seen from Figure 7 that the surface pressure gradient of the 2D automobile shape is the largest at the head portion and the tail portion and the shape of the grids at these locations changes greatly. This indicates that a larger number of grids are required at these portions to prevent the p-value from being diverged.

The distribution of pressure coefficient values at the head and tail portions can be calculated according to equation (2), which is the equation of pressure coefficient. The distribution of Cp values on the surface of the head and tail portions of Shape 2 is shown in Figure 8 (air pressure coefficient). In general, due to a larger frontal area, the negative value of the Cp value of the head also increases and this leads to uneven pressure distribution on the vehicle body.

From the head to the location between the hood and the windshield, Cp > 0, while behind the windshield, Cp < 0; the Cp values on the upper surface of the tail increase dramatically (Figure 9).

On the locations of an air intake grille and outlet and ventilation system, the distribution of Cp values should also be taken into consideration. The air intake grille must be located on the surface of the vehicle body with a higher pressure, such as the surface of the head or the hood where the Cp value is above 0. Air outlets should be located at locations with a greater bleeding effect and the Cp value is low. Cooling and ventilation systems should be located at positions with pressure coefficient changes while the fans should have less area.

Comparison of drag coefficients for different automobile shapes

The drag coefficients $C_{d}$ of the 21 automobile shapes can be analyzed by the numerical simulation according to equation (4). The resulting values are shown in Table 4, and they are ranked from large to small as follows: S20 > S19 > S21 > S7 > S4 > S14 > S18> S15 > S9 > S12 > S10 > S13 > S5 > S1 > S6 > S8 > S11 > S17 > S3 > S2 > S16. By creating charts from the data obtained, the resulting Figure 10 (drag coefficient from small to large) indicated clearly that the automobile shape with the smallest drag coefficient $C_{d}$ is Shape 16. This result corresponds to the ranking of SCE values. It is obvious from this figure that the greater the curvature entropy, the larger the drag coefficient; vice versa, the smaller the entropy value, the smaller the drag coefficient. A positive correlation exists between these two parameters.

Table 4.

Drag coefficients of the 21 automobile shapes.

Shape #	Shape 1	Shape 2	Shape 3	Shape 4	Shape 5	Shape 6
Drag coefficient, $C_{d}$	0.499	0.368	0.388	0.691	0.516	0.490
Shape #	Shape 7	Shape 8	Shape 9	Shape 10	Shape 11	Shape 12
Drag coefficient, $C_{d}$	0.698	0.473	0.538	0.535	0.473	0.536
Shape #	Shape 13	Shape 14	Shape 15	Shape 16	Shape 17	Shape 18
Drag coefficient, $C_{d}$	0.527	0.612	0.566	0.356	0.432	0.581
Shape #	Shape 19	Shape 20	Shape 21
Drag coefficient, $C_{d}$	0.714	0.718	6.985

The curvature entropy values of S16, S2, and S3 are the top three lowest. Those of S17, S11, S8, S6, S1, S5, S13, S10, S12, and S9 are between 0.432 and 0.55 to which most of the automobile shapes belong. The curvature entropy values of S20, S19, S21, and S7 are on the higher end. As compared to the study by Hucho (1987), it is apparent that these car models have higher drag coefficients. This result supports the positive correlation between the curvature entropy and the drag coefficient.

The investigations in this study also indicated that the car models before 1990s have a curvature entropy which is larger than 7.0 while their drag coefficients are larger than 0.7. This is followed by the car models between 2000 and 2005, which have a curvature entropy which is larger than 6.7 and a drag coefficient which is larger than 0.5. Therefore, any future car models are recommended to target at a curvature entropy that is smaller than 6.0 and a drag coefficient that is smaller than 0.35.

Conclusion

A systematic method for evaluating the aerodynamic performance of an automobile shape was proposed in this study. Commercially available automobile shapes were collected and their features captured for further analyses. By calculating their curvature entropy values, these automobile shapes were ranked accordingly. Meanwhile, the flow field distribution around an automobile shape was analyzed by CFD analyses so that its shape pressure coefficient and drag coefficient can be determined. The results obtained from the analyses and comparisons indicated that when a shape is more complex, it has a larger curvature entropy value along with a larger drag coefficient. This indicated a positive correlation between an automobile shape’s curvature entropy value and its drag coefficient. As compared to wind tunnel tests which are expensive and time consuming, it is desirable to conduct flow field analyses by the numerical simulations so that the number of wind tunnel tests required can be greatly reduced. With the positive correlation mentioned above, the proposed curvature entropy serves as a suitable indicator for estimating an automobile shape’s aerodynamic performance. As a result, the required number of CFD analyses or wind tunnel tests can be greatly reduced.

For the future scope of work, it is recommended to extend the curvature entropy calculations from 2D to 3D so that the geometrical variations along the additional dimension can be taken into account. Moreover, it is also worth investigating the correlation between the curvature entropy and other vehicle performance indicators such as noise, vibration, or weight allocation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Ministry of Science and Technology of the Republic of China under grant MOST-103-2221-E-029-026.

Author biographies

SHIH-WEN HSIAO is a distinguished professor in the Department of Industrial Design at National Cheng Kung University, Tainan, Taiwan. His major research interests include application of fuzzy set theory on product design, concurrent engineering in product design, computer-aided design, application of the neural network, genetic algorithm, and gray theory in product design, color planning for product design, heat transfer analysis, and application of reverse engineering in product design.

HSIN-HUNG LIN is currently a PhD candidate in the Department of Industrial Design at National Cheng Kung University, Taiwan. His major research interests include design management, innovation design, computer-aided design, concurrent engineering, application of fuzzy set theory on product design, green design, and new product development.

CHI-HUNG LO is an assistant professor in the Department of Industrial Design at Tunghai University, Taichung, Taiwan. His research interests include product aesthetics measures, product family development, concurrent engineering in product designs, and artificial-intelligence-based inference system design.

YA-CHUAN KO is an assistant professor in the Department of Creative Product Design at Asia University, Taichung, Taiwan. She is also an advisor to the Department of Medical Research, China Medical University Hospital, Taichung, Taiwan. Her research interests include product family development, product design, concurrent engineering in product design, and usability evaluation.

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