Aerodynamics-guided machine learning for design optimization of electric vehicles

Automobile aerodynamic analysis

Our dataset spans a wide range of industry-quality automobile geometries, which exhibit vastly different unsteady wake behavior and aerodynamic performance, as shown by the voxelized geometries in Fig. 2a. The baseline car designs considered in this study come from different production model car geometries including SUVs, hatchbacks, sedans, and box cars, with multiple vehicle models for each car type considered. Additionally, the features of the baseline designs for each model are parametrically modified to produce a variety of geometries. The flow is computed with a large-eddy simulation utilizing a moving mesh that has been validated with industry wind-tunnel experiments. Further details of the aerodynamic analysis are given in the Methods section.

a Representative voxelized vehicle geometries for different car types. b Example 3D and 2D (center plane) isocontours of normalized total pressure coefficient, C_p,t, for different car types. C_p,t = 0 isocontour is shown, indicating separated flow. c Probability density plot, p(C_D), of normalized drag coefficient, C_D, with respect to vehicle type.

The time-averaged flow around automobiles shares several salient features that are useful in estimating the aerodynamic drag^{14,28,29,30,31}. For even the most simplified automobile geometries, three-dimensional separated flow regions are generated in the wake behind the vehicle. The flow past most automobiles exhibits two recirculation regions that are formed when the flow separates at the top and bottom edges of the back of the vehicle. The size of the recirculation regions depends on how the shape of the rear geometry directs the trailing wake flow. A pair of trailing vortices are formed when the fluid boundary layer rolls around the sides of the vertical supports at the rear, referred to as the “C-pillars”^14,15,16. These wake structures create a region of low pressure behind the vehicle and contribute to the induced drag. Conversely, a region of high pressure exists at the front of the vehicle where the flow is stopped by the front geometry. The difference in pressure over the surface of the vehicle results in the pressure drag, which is the primary source of drag force for automobiles¹⁴. Figure 2b depicts the total pressure fields for the different car types in the dataset. The zero total pressure isocontour is shown (C_p,t = 0), which highlights where the flow may separate from the vehicle. We observe that in addition to the rear wake, we also see flow separation at other regions, including the wheels, the front roof edge, the front windshield edges (“A-pillar”), and the front bumper edges. We note that the use of the moving mesh LES solver captures physical phenomena not seen in previous analyses. One example is the wake generated from the outflow from the rotating front wheel, which produces appreciable differences in the drag estimate but is not captured with a static mesh¹³. With the variation of geometric features, the relative sizes of the aforementioned structures and the resulting aerodynamic performance can vary drastically from case to case. For example, if the transition between the front windshield and the top roof panel is abrupt, the flow may separate from the front of the roof (Box). A different flow pattern can appear in the rear where, depending on the rear windshield angle, the flow may separate over the rear spoiler only to reattach lower on the bottom of the windshield (Sedan). Figure 2c shows a probability density plot of the normalized drag coefficients C_D, which demonstrates that the different geometries in our dataset span vastly different ranges of drag coefficients.

Although we can qualitatively analyze how fluid structures and aerodynamic performance relate to the vehicle geometry, the wide range of vehicle parameters and the nonlinear nature of fluid flow make it challenging to formulate aerodynamic prediction models, especially when comparing visually similar geometries. For example, the box car exhibits much larger separated flow regions at the front geometry, as well as a larger low-pressure wake behind the vehicle compared to the other geometries. However, we can see that in Fig. 2c some of the box cars in the current dataset can exhibit a lower drag coefficient than some SUV and hatchback designs, possibly due to differences in the front geometry. Other complexities arise in other flow interactions, such as ground effects in the wake and underbody regions, which can be profoundly altered by minor geometric changes, and require careful aerodynamic analysis. As such, the nonlinear behavior of the flow confounded with a large number of geometric parameters makes a reasonable prediction of the drag performance from geometries difficult to infer. However, we expect that there exists some nonlinear functional dependency between the vehicle geometry and its drag coefficient that can be captured through data-driven approaches. We aim to capture this relation utilizing a small number of latent variables that be used as both a low-dimensional representation of a vehicle geometry as well as the estimated drag coefficient. For this objective, we use an observable-augmented autoencoder assisted by principle component analysis (PCA)^{32,33,34,35,36} that reconstructs the high-dimensional car vehicle information while estimating the vehicle drag coefficient from the latent space, as illustrated in Fig. 3.

Voxelized vehicle geometries are taken as input and output, and the drag coefficient is estimated from the encoded latent representation.

Latent manifold discovery

If we directly attempt to learn a relationship between the vehicle geometries and the drag coefficient for a given dataset, we can learn many possible models that provide similar levels of accuracy. This can make shape optimization difficult, because such models may prove less reliable when generalizing to produce new geometries. With this in mind, we seek low-dimensional coordinates that capture a relationship between input geometries and the drag coefficient, while reconstructing a geometry from the low-dimensional representation. Training both tasks simultaneously helps to mitigate overfitting as well as allows us to observe geometric similarities, which is helpful for identifying relevant features for design optimization.

The original vehicle geometry is first preprocessed to a voxelized representation. This represents the dimensionality reduction step in Fig. 1, as we effectively reduce our high-fidelity input from a mesh represented with a set of points connected by smooth, continuous segments to a regular grid with finite resolution. The voxelized geometries are then used for identifying a low-dimensional latent space. What this amounts to is learning a curvilinear coordinate system that parameterizes a manifold representation of our dataset. We learn such a manifold by using a nonlinear autoencoder, a type of unsupervised machine-learning model³⁷. A basic autoencoder consists of an encoder, which reduces the dimension of input data into a lower-dimensional latent space, and a decoder which reconstructs the input from the encoded representation.

For this work, we leverage an observable-augmented nonlinear autoencoder³² that is simultaneously trained to estimate the drag coefficient from the compressed representation of the vehicle geometries, as illustrated in Fig. 3. Since the drag coefficient needs to be estimated from the latent variables, the identified latent representation holds a relationship between vehicle geometries and their aerodynamic performance in a low-order manner. In other words, the neural network weights are trained such that features relevant to estimations of the drag coefficient are captured. This can also reduce the problem of analyzing qualitative similarities of geometric trends in the original high-dimensional geometries to observing changes in salient features extracted in our low-dimensional space. Moreover, such an aerodynamically relevant manifold provides a desired direction to improve aerodynamic performance in a low-order manner. In other words, we can identify an optimal modification of the vehicle design with reduced computational cost. Further details of this approach are given in the Methods section.

Shown in Fig. 4 is the discovered three-dimensional latent space manifold, (ξ₁, ξ₂, ξ₃), that is learned by the present autoencoder. We note that for our dataset, a three-dimensional latent space is enough to achieve sufficiently reasonable geometry reconstruction as well as accurate drag estimation, as we observed very little gain in accuracy without a substantial increase in the latent space dimension. Through the drag decoder, we can obtain an estimated drag coefficient corresponding to the geometry parameterized by any point in our three-dimensional latent space in Fig. 4. We also note that the geometry reconstruction from the decoder is qualitatively indistinguishable from the original input for a number of cases, depicted in Figs. 3 and 4. The average percent absolute error of estimated drag values lies within 2% of the reference value for the training, validation, and test sets. As seen in Fig. 4, each point in the low-dimensional space corresponds to a vehicle geometry. We also observe that vehicles cluster in the low-order space based on both geometric similarity and the estimated drag coefficient with distinct point clouds corresponding to the different car types (shown by the marker shape). Increased estimated drag approximately correlates with increasing ξ₁ and ξ₃ in the shown manifold.

The left plot depicts the latent space of sport utility vehicles (SUVs), hatchbacks, sedans, and box geometries. Right parity plots show the comparison of estimated and reference normalized drag coefficients C_D. Comparisons of example vehicle geometries and autoencoder reconstructions are shown and the corresponding points marked in the latent space with a–d. The value underneath each decoded geometry reports the percent L₂ reconstruction error metric, \(\varepsilon =\parallel \bfp-\hat\bfp\parallel _2/\parallel {\bfp}\parallel _2\).

The relative distances between latent points can be taken to be representative of the similarity between vehicle cases. For example, vehicles with very different geometries and aerodynamic performance, such as sedans and box cars, are placed far apart in the latent space. On the other hand, many of the hatchback geometries have an intermediate drag value, which is reflected by their placement in between the low-drag sedans and the high-drag SUV and box cars.

Data-driven shape optimization

With the identified latent space manifold, we aim to modify existing vehicle geometries to improve aerodynamic performance. Since the present latent manifold relates car geometries with drag performance, optimization can be performed directly in the low-dimensional latent space rather than the high-dimensional space of the input data which can accelerate computations of the gradients. Knowing this, we obtain a direction in the latent space corresponding to a decrease in the drag coefficient for a given geometry by computing the gradient of drag at a given latent space point. By modifying the latent space coordinate in the direction of reduced drag and observing changes in the decoded geometry, we can identify regions of the vehicle geometry to modify in order to reduce the drag coefficient. The produced car geometries from this iterative descent process can then be used to generate baseline designs for further validation.

However, it is important that any modified geometries produced during this optimization process should not be “extrapolation” cases. That is to say, we must ensure that the geometry produced by our optimization process still resembles a physically realizable car design. If the optimized design is too dissimilar to any cases in training, this may also put the reliability of the estimated drag into question. Since the distance in the latent space can be considered a measure of similarity, the present data-driven optimization considers a constraint based on the latent space distance between the optimized design and the training data. We consider a soft distance constraint which penalizes the optimization from moving too far from the training data. This distance constraint adds a penalty term to the cost function, which grows as the distance from the training data increases. Further details of optimization formulation and the constraints are given in the Methods section.

We demonstrate our data-driven vehicle shape optimization on the discovered manifold in Fig. 5. We present an example case of geometry optimization starting from a high-drag SUV case (initial normalized drag coefficient, C_D ≈ 0.86). The initial geometry and the geometry after performing the shape optimization with the soft constraint are also visualized in Fig. 5. The model estimates an 11% reduction of the normalized drag (C_D ≈  0.77) for the modified geometry. Along the optimization trajectory, we decode geometries and validate the drag estimate with LES performed on the smoothed voxel data. As seen in Fig. 5 we observe agreement between the trend of the estimated drag and the value corresponding to the CFD simulation for a few of the sampled validation cases.

The latent optimization path with the corresponding normalized drag coefficient is shown. Orange points depict drag values obtained from validation computational fluid dynamics (CFD) simulation. a The given initial car and b optimized geometry are also depicted. The regions highlighted in yellow highlight some features that have been substantially modified.

In Fig. 6, we show the pressure fields around the validation geometry. In the optimized geometry, we observe that the rear geometry is modified. The slope of the roof leading to the rear spoiler is lowered, and the end is shifted backward. This provides a boattail-like effect as this reduces the pressure gradient where the flow separates (at the spoiler), which in turn reduces the pressure drag. Additionally, there is a smaller observed pressure gradient around the C-pillar, and we note that the trailing wake vortices are elongated and tapered, reducing their influence on the induced drag. The edges of the front geometry are smoothed, which reduces the size of the front face, which the high-pressure zone (red) acts on, in addition to allowing the flow to smoothly transition around the front geometry. The validation cases demonstrate that the model has learned a trend between geometric features and aerodynamic performance rather than just memorizing the training data.

(Left) Zero total pressure (C_p,t = 0) isocontour colored by the pressure coefficient, C_p. (Right) Midplane slice of pressure coefficient field, C_p. The pressure difference around the rear spoiler region is reduced.