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Mohamed<EMAIL_ADDRESS>Angela Dai, Faez Ahmed

1Massachusetts Institute of Technology, Cambridge, MA 2Technical University of
Munich, Munich, Germany

# DrivAerNet: A Parametric Car Dataset for Data-Driven Aerodynamic Design and
Graph-Based Drag Prediction

###### Abstract

This study introduces DrivAerNet, a large-scale high-fidelity CFD dataset of
3D industry-standard car shapes, and RegDGCNN, a dynamic graph convolutional
neural network model, both aimed at aerodynamic car design through machine
learning. DrivAerNet, with its 4000 detailed 3D car meshes using 0.5 million
surface mesh faces and comprehensive aerodynamic performance data comprising
of full 3D pressure, velocity fields, and wall-shear stresses, addresses the
critical need for extensive datasets to train deep learning models in
engineering applications. It is 60% larger than the previously available
largest public dataset of cars, and is the only open-source dataset that also
models wheels and underbody. RegDGCNN leverages this large-scale dataset to
provide high-precision drag estimates directly from 3D meshes, bypassing
traditional limitations such as the need for 2D image rendering or Signed
Distance Fields (SDF). By enabling fast drag estimation in seconds, RegDGCNN
facilitates rapid aerodynamic assessments, offering a substantial leap towards
integrating data-driven methods in automotive design. Together, DrivAerNet and
RegDGCNN promise to accelerate the car design process and contribute to the
development of more efficient vehicles. To lay the groundwork for future
innovations in the field, the dataset and code used in our study are publicly
accessible at https://github.com/Mohamedelrefaie/DrivAerNet111Data will be
uploaded on paper acceptance.

###### keywords:

Parametric Design, Computational Fluid Dynamics, Car Aerodynamics, Surrogate
Modeling, Graph Neural Networks

Dataset | Size | Aerodynamics Data | Wheels/Underbody | Parametric | Design Parameters | Open-source
---|---|---|---|---|---|---
$C_{d}$ | $C_{l}$ | u | $p$ | Modeling
Baque el al. 2018 [7] | 2,000 | ✔ | ✗ | ✗ | ✗ | ✗ | ✔ | 21 (3D) | ✗
Umetani et. al 2018 [40] | 889 | ✔ | ✗ | ✔ | ✔ | ✗ | ✗ | - | ✔
Gunpinar et. al 2019 [16] | 1,000 | ✔ | ✗ | ✗ | ✗ | ✗ | ✔ | 21 (2D) | ✗
Remelli et. al 2020 [28] | 1,400 | ✗ | ✗ | ✗ | ✔ | ✗ | ✗ | - | ✗
Jacob et. al 2021 [19] | 1,000 | ✔ | ✔ | ✔ | ✗ | ✔ | ✔ | 15 (3D) | ✗
Usama et. al 2021 [41] | 500 | ✔ | ✗ | ✗ | ✗ | ✗ | ✔ | 40 (2D) | ✗
Rios et. al 2021 [31] | 600 | ✔ | ✔ | ✗ | ✗ | ✗ | ✗ | - | ✗
Song et. al 2023 [36] | 2,474 | ✔ | ✗ | ✗ | ✗ | ✗ | ✗ | - | ✔
Li et. al 2023 [22] | 551 | ✔ | ✗ | ✗ | ✔ | ✗ | ✔ | 6 (3D) | ✗
611 | ✔ | ✗ | ✗ | ✔ | ✗ | ✗ | - | ✗
Trinh et. al 2024 [39] | 1,121 | ✗ | ✗ | ✔ | ✔ | ✗ | ✗ | - | ✗
DrivAerNet (Ours) | 4,000 | ✔ | ✔ | ✔ | ✔ | ✔ | ✔ | 50 (3D) | ✔
Table 1: A comparative analysis of various aerodynamics datasets, highlighting
key aspects such as number of designs in the dataset (size), the inclusion of
aerodynamic coefficients (drag coefficient $C_{d}$ and lift coefficient
$C_{l}$), the inclusion of velocity (u) and pressure ($p$) fields, the
presence of wheels/underbody modeling, the capacity for conducting parametric
studies, the number of design parameters, and open-source availability.

##  1 Introduction

Reducing fuel consumption and CO2 emissions through advanced aerodynamic
design is crucial to the automobile industry. This can help with faster
transition towards electric cars, complementing the 2035 ban on internal
combustion engine cars and aligning with the ambitious goal of achieving
carbon neutrality by 2050 to combat global warming [25, 9, 23]. In aerodynamic
design, navigating through intricate design choices involves a detailed
examination of aerodynamic performance and design constraints, which is often
slowed down by the time-consuming nature of high-fidelity CFD simulations and
experimental wind tunnel tests. High-fidelity CFD simulations can take days to
weeks per design [6], while wind tunnel testing, despite its accuracy, is
limited to examining only a handful of designs due to time and cost
constraints. Data-driven approaches can alleviate this bottleneck by
leveraging existing datasets to navigate through design and performance
spaces, thereby speeding up the design exploration process and efficiently
assessing aerodynamic designs.

Although recent advancements in data-driven approaches for aerodynamic design
are promising, these typically concentrate on simpler 2D cases [14, 38] or
lower-fidelity CFD simulations [36, 22, 7] and overlook the complexities
inherent in real-world 3D designs and the challenges posed by high-fidelity
CFD simulations. According to [17], simplifying car designs by excluding
components like wheels and mirrors, and not modeling the underbody, leads to a
significant underestimation of aerodynamic drag. Taking these elements into
account increased the drag by more than 1.4 times, highlighting the importance
of detailed modeling for accurate aerodynamic analysis. Additionally, there is
a dearth of publicly available high-fidelity car simulation datasets,
potentially slowing down research in data-driven method development, as each
researcher may need extensive computational resources to create their own data
and test their methods on them.

Responding to this challenge, our paper introduces DrivAerNet, a comprehensive
dataset that features full 3D flow field information across 4000 high-fidelity
car CFD simulations. It is made publicly available to serve as a benchmark for
training deep learning models in aerodynamic assessment, generative design,
and other machine learning applications.

To demonstrate the importance of large-scale datasets, we also develop a
surrogate model for aerodynamic drag prediction based on Dynamic Graph
Convolutional Neural Networks [42]. Our model, RegDGCNN, operates directly on
extremely large 3D meshes, eliminating the necessity for 2D image rendering or
Signed Distance Fields (SDF) generation. RegDGCNN’s ability to swiftly
identify aerodynamic improvements opens new avenues for creating more
efficient vehicles by streamlining the evaluation of design adjustments. It
marks a significant step towards optimizing car designs more efficiently.

Overall, the contributions of this paper are:

  * •

The release of DrivAerNet, an extensive high-fidelity dataset featuring 4000
car designs, complete with detailed 3D models with 0.5 million surface mesh
faces each, full 3D flow fields, and aerodynamic performance coefficients. The
dataset is 60% larger than the previously available largest public dataset of
cars, and is the only open-source dataset that also models wheels and
underbody, allowing accurate estimation of drag.

  * •

The introduction of a surrogate model, named RegDGCNN, based on Dynamic Graph
Convolutional Neural Networks for prediction of aerodynamic drag. RegDGCNN
outperforms state-of-the-art attention-based models [3, 36] for drag
prediction by 3.57$\%$ on ShapeNet benchmark dataset while using 1000$\times$
fewer parameters, and achieves an $R^{2}$ score of 0.9 on the DrivAerNet
dataset.

In addition, the large size of our dataset is also justified by our analysis
in Section 5.3, which reveals that expanding the training dataset from 560 to
2800 car designs from DrivAerNet resulted in a 75$\%$ decrease in error,
illustrating the direct correlation between dataset size and model
performance. A similar trend is observed with the dataset from [36], where
enlarging the number of training samples from 1270 to 6352 entries yielded a
56$\%$ error reduction, further validating our model’s efficacy and the
inherent value of large datasets in driving advancements in surrogate
modeling.

The structure of our paper is as follows: Section 2 provides an overview of
related work. Section 3 presents the DrivAerNet dataset, detailing the
numerical simulation methods, CFD results, geometric feasibility analysis, and
dataset characteristics. In Section 4, we introduce our RegDGCNN approach
using the Dynamic Graph Convolutional Neural Network for regression tasks.
Section 5 examines the application of the RegDGCNN model for surrogate
modeling of aerodynamic drag, comparing its performance on the DrivAerNet and
ShapeNet datasets and underscoring the benefits of scaling the training
dataset. The paper highlights the limitations of our study and suggests
avenues for future research, followed by a conclusion that summarizes the
results and key findings.

Figure 1: The parametric DrivAer model is depicted with morphing boxes applied
for geometry transformation in ANSA® software using a total of 50 geometric
parameters and 32 morphable entities. The morphing boxes are color-coded to
highlight the areas susceptible to parametric modifications, facilitating the
creation of ‘DrivAerNet’ dataset. Utilizing this morphing technique, we
generated 4,000 unique car designs.

##  2 Related Work

This section starts with an overview of aerodynamics datasets, and then
transitions to discussing recent progress in 3D learning for aerodynamics.

###  2.1 Aerodynamics Datasets

Data-driven aerodynamic design is a methodology that leverages computational
models and machine learning algorithms to optimize car shapes based on large
volumes of aerodynamic performance data, aiming to improve efficiency and
performance. A common type of data-driven aerodynamic design is surrogate
modeling, which uses simplified models to approximate the behavior of complex
aerodynamic phenomena, enabling faster simulations and iterations in the
design process. It is particularly useful for preliminary design phases where
quick evaluations are necessary. However, a significant portion of existing
research on data-driven aerodynamic design is concentrated on simplified 2D
scenarios such as airfoils/2D geometries [8, 38, 14, 41, 20, 16] or simplified
3D models [7, 28, 22, 40, 31, 36, 39]. While these studies are instrumental in
understanding the fundamental physics, they often fall short in terms of
applicability to complex 3D real-world problems. This gap is further widened
by the absence of proprietary data from the industry, posing challenges in
replicating and validating research findings. Moreover, the absence of a
standardized benchmark dataset in the field hampers the ability to
consistently evaluate and compare the efficacy of various machine learning
methodologies. This stands in contrast to fields such as image processing or
3D shape analysis, where benchmark datasets like ImageNet [13] or ShapeNet
[10] have accelerated significant advancements in deep learning by providing a
common ground for methods comparison. Although benchmarks for CFD simulations
and turbulence modeling [4, 5, 6] do exist, their limited size often renders
them inadequate for the demands of deep learning techniques, underscoring the
necessity for a broader and more comprehensive dataset.

Table 1 provides an overview of existing datasets in the literature for data-
driven aerodynamic design. It compares their size, the inclusion of
aerodynamic information such as drag coefficient ($C_{d}$), lift coefficient
($C_{l}$), velocity field (u), pressure ($p$), whether they are parametric,
the number of design parameters, and their availability as open-source. In
addition, we consider the modeling of the rotating wheels and underbody for
evaluation. Our dataset stands out with the largest size of 4,000 samples,
comprehensive aerodynamic information, parametric details, modeling of the
wheels and underbody, and open-source accessibility.

Below, we discuss recent advancements in 3D learning for aerodynamics.

(a) Fastback configuration of the DrivAer baseline model with detailed
underbody, mirrors, and wheels. (b) Computational grid in the $y$-normal
symmetry plane displays four distinct regions of refinement. The addition of
layers around the car body is also evident, all designed to accurately capture
both the wake dynamics and the boundary layer development.

Figure 2: DrivAer Fastback model, complete with detailed features, alongside
its computational mesh, illustrating the mesh refinement zones and additional
layers aimed at accurately modeling aerodynamic phenomena.

###  2.2 Advancements in 3D Learning for Aerodynamics

The study from [39] presented a super-resolution model aimed at refining the
estimated, yet coarsely resolved, flow fields around vehicles from deep
learning predictions to a higher resolution, crucial for aerodynamic vehicle
design. By incorporating a residual-in-residual dense block (RRDB) within the
generator structure and employing a relativistic discriminator for enhanced
detail capture, coupled with a novel distance-weighted loss and physics-
informed loss to ensure physical accuracy, the methodology demonstrated a
marked improvement in flow field enhancement around vehicles, outperforming
previous approaches in this domain. Jacob et. al [19] demonstrated that deep
learning, particularly a modified U-Net architecture using SDF, can accurately
predict drag coefficients for a specific car design without the need for
explicit parameterization.

In recent advancements in computational design, many studies have leveraged
the ShapeNet [10] dataset for shape optimization and surrogate modeling in
aerodynamics. ShapeNet is a dataset consisting of millions of 3D models
spanning 55 common object categories, designed to support research in computer
vision, robotics, and geometric deep learning. Using ShapeNet, [28] enhanced
shape optimization by employing MeshSDF, offering a more flexible alternative
to traditional hand-crafted parameterizations. Song et. al [36] introduced a
novel 2D representation for 3D shapes, coupled with a surrogate model for
aerodynamic drag, showcasing the potential for AI-driven design optimizations.
Rios et. al [31] compared various design representation methods, including
PCA, kernel-PCA, and a 3D point cloud autoencoder, highlighting the
autoencoder’s capability for localized shape modifications in aerodynamic
optimizations. The fourth study introduced the geometry-informed neural
operator (GINO) [22], leveraging SDF, point clouds, and neural operators for
efficient large-scale simulations, achieving significant speed-ups and error
rate reductions in predicting surface pressure and drag coefficients on
different car geometries.

##### Limitations of Existing Studies and the Impact of Comprehensive
Modeling:

Despite the novel methodologies employed in these studies, they faced
limitations stemming from the inherent drawbacks of the ShapeNet dataset, such
as lower mesh resolution, lack of watertight geometries, small dataset size,
and oversimplifications like modeling cars as single-bodied entities without
detailed considerations for components like wheels, underbody, and side
mirrors, which can significantly impact real-world aerodynamic performance.
This oversimplification can significantly impact real-world aerodynamic
performance; according to [17], including these details in the DrivAer
fastback model, increased the drag value from 0.115 to 0.278 in CFD
simulations and from 0.125 to 0.275 in wind tunnel experiments. These
increases represent a substantial increase in drag by approximately 142$\%$
and 120$\%$ respectively, underscoring the critical role of comprehensive
modeling in achieving accurate aerodynamic assessments. Another common hurdle
in both surrogate modeling and design optimization is the scarcity of data,
which complicates efforts to replicate results or benchmark various models and
approaches. Addressing this challenge, our contribution introduces DrivAerNet,
a comprehensive benchmark dataset tailored for data-driven aerodynamic design,
aiming to facilitate comparison and validation of future methodologies.

##  3 DrivAerNet Dataset

##### Introduction to DrivAerNet Dataset and Model Background:

DrivAer model [18] is a well-established conventional car reference model
developed by researchers at the Technical University of Munich (TUM). It is a
combination of the BMW series 3 and Audi A4 car designs in order to be
representative of most conventional cars. The DrivAer model was developed to
bridge the gap between open-source oversimplified models like the Ahmed [2]
and SAE [11] bodies and the complex designs of the manufacturing companies,
which are not publicly available. To accurately assess real-world aerodynamic
designs, we selected the fastback configuration with a detailed underbody,
wheels, and mirrors (FDwWwM) as our baseline model, as shown in Figure 2(a).
This choice of the FDwWwM model was driven by the substantial impact of
wheels, mirrors, and underbody geometry on aerodynamic drag, a conclusion
supported by the findings in [17]. Specifically, the detailed underbody
geometry adds 32-34 counts to the drag, the inclusion of mirrors introduces an
additional 14-16 counts, and the presence of wheels elevates the total drag
coefficient by 102 counts 222In aerodynamic evaluations, “drag counts” are
used as a unit of measurement to denote small changes in the drag coefficient
($C_{d}$). One drag count is defined as a $0.0001$ increment in $C_{d}$..

##### Selecting the Baseline Parametric Model:

In order to create a comprehensive dataset for training deep learning models
for surrogate modeling and design optimization, we first created a parametric
model of the DrivAer model. This approach was necessitated by the limitations
of the original model, which was provided as a single, non-parametric .stl
file. To adequately capture the geometric variations and design modifications
relevant to real-world automotive design challenges, we developed a version of
the DrivAer model that is defined by 50 geometric parameters and includes 32
morphable entities (see Figure 1) using the commercial software ANSA®. This
parametric model allows for a more detailed exploration of the design space,
facilitating the generation of 4,000 unique design variants through the
application of the Optimal Latin Hypercube sampling method, specifically
employing the Enhanced Stochastic Evolutionary Algorithm (ESE) as outlined by
[12].

Figure 3: Mesh resolution comparison across various datasets; first row
features the Ahmed body mesh from Li et al. (2023) [22], demonstrating a
coarse resolution. Second row shows medium-resolution mesh from the ShapeNet
dataset, as utilized by Song et al. (2023) [36]. Final row presents our high-
resolution mesh, providing greater detail for in-depth aerodynamic design.
Figure 4: Car models from the DrivAerNet dataset, illustrating a range of
aerodynamic designs. The far left model exhibits the maximum volume within the
dataset, while the far right model represents the design with the smallest
volume, highlighting the diversity and scope of aerodynamic profiles studied.

##### Techniques for Generating Diverse Car Designs:

The parametric model, along with the constraints and the bounds applied during
the Design of Experiments (DoE), significantly enriches the dataset, making it
a robust foundation for the development and training of advanced deep learning
models aimed at surrogate modeling and design optimization tasks. We provide
access to the parametric model with the incorporated morphing features for
further reference and utilization333The parametric model is available at
https://github.com/Mohamedelrefaie/DrivAerNet/ParametricModel.

Diverging from the approach taken in [19], we implemented a broader range of
morphing techniques, enabling us to explore a more diverse array of car
designs. This approach aims to enhance the adaptability of deep learning
models, allowing them to generalize across various car designs instead of
being limited to minor geometric modifications within a single design. Figure
3 depicts the variation in mesh quality, ranging from coarse to high-
resolution across various datasets. Our original mesh with 540k mesh faces,
provides a denser and more detailed representation compared to the mesh
resolutions in the studies by [22] and [36], thereby revealing more detailed
geometric and design features.

Additionally, Figure 4 presents a spectrum of car shapes from the DrivAerNet
dataset, illustrating the variability in design dimensions and features. This
range from the largest to the smallest volume model underscores the dataset’s
capacity to cover a comprehensive span of aerodynamic profiles.

###  3.1 Numerical Simulation

####  3.1.1 Domain and Boundary Conditions

The DrivAer fastback model scaled 1:1, was selected for conducting the CFD
simulations. The simulations were carried out using the open-source software
OpenFOAM®, a comprehensive collection of C++ modules for tailoring solvers and
utilities in CFD studies. In this study, the coupling between pressure and
velocity is achieved through the SIMPLE algorithm (Semi-Implicit Method for
Pressure Linked Equations), as implemented in the simpleFoam solver, which is
designed for steady-state, turbulent, and incompressible flow simulations. The
$k$-$\omega$-SST model, based on Menter’s formulation [24], was chosen for the
Reynolds-Averaged Navier-Stokes (RANS) simulations due to its ability to
overcome the limitations of the standard $k$-$\omega$ model, particularly its
dependency on the freestream values of $k$ and $\omega$, and its effectiveness
in predicting flow separation.

The simulations were performed at a flow velocity ($u_{\infty}$) of 30 m/s,
which corresponds to a Reynolds number of roughly $9.39\times 10^{6}$, using
the car length as the characteristic length scale. The computational mesh is
constructed using the SnappyHexMesh (SHM) tool, featuring four distinct
refinement zones. Additional layers have been added around the car body to
precisely represent wake dynamics and boundary layer evolution (see Figure
2(b)). Boundary conditions were defined with a uniform velocity at the inlet
and pressure-based conditions at the outlet. To avoid backflow into the
simulation domain, the velocity boundary condition at the outlet was
configured as an inletOutlet condition. The car surface and ground were
assigned no-slip conditions, while the wheels were modeled to rotate with the
rotatingWallVelocity boundary condition. Slip conditions were applied to the
lateral and top boundaries of the domain.

Near-wall viscosity effects were addressed using the nutUSpaldingWallFunction
wall function approach. The selected wall function for the viscosity term
applies a continuous turbulent viscosity profile near the wall, based on
velocity, following the approach proposed by [37]. For divergence terms, the
default Gauss linear scheme is used, with the velocity convective term
discretized using a bounded Gauss linearUpwindV scheme applied to the gradient
of velocity, ensuring second-order accuracy. Gradient calculations employed
the Gauss linear method complemented by a multi-dimensional limiter to enhance
the stability of the solution. The quantities of interest are the 3D velocity
field, surface pressure, and wall-shear stresses, as well as the aerodynamic
coefficients.

####  3.1.2 Validation of the Numerical Results

Comparison of Mesh Resolutions and Experimental/Simulation Results
---
Mesh name | Cell count | DrivAer face count | CPU-hours | Difference Exp. | Difference Ref. Sim.
RANS coarse | $8\times 10^{6}$ | $5.4\times 10^{5}$ | 88 | 2.81$\%$ | 1.7$\%$
RANS medium | $16\times 10^{6}$ | $5.4\times 10^{5}$ | 205 | 0.81$\%$ | 1.88$\%$
RANS fine | $40\times 10^{6}$ | $8.7\times 10^{5}$ | 512 | 5.7$\%$ | 6.78$\%$
Table 2: A comparison of different mesh resolutions used in our simulations
for the baseline DrivAer model, including cell count and DrivAer face count.
The simulation times are listed alongside the percentage differences when
compared to experimental results and a referenced simulation, as reported in
[17].

The selection of the DrivAer fastback model is justified by the availability
of both computational and experimental references, enabling us to benchmark
our results against established data [17, 43]. Before commencing the
simulations, we conducted a preliminary assessment of how mesh refinement
influences the results. This involved comparing the drag coefficients obtained
from three different mesh resolutions with experimental values and reference
simulation, as detailed in Table 2. The objective was to identify an optimal
balance between simulation accuracy and computational efficiency. This balance
is crucial since our aim is to generate a large-scale dataset for training
deep learning models, necessitating both high fidelity in the simulation
results and manageable disk storage and simulation times to accommodate the
extensive computational requirements. The drag coefficient, $C_{d}$, is
determined by the equation:

$C_{d}=\frac{F_{d}}{\frac{1}{2}\rho u_{\infty}^{2}A_{\text{ref}}}$ (1)

The drag force, denoted by $F_{d}$, experienced by a body is a function of its
effective frontal area $A_{ref}$, the freestream velocity $u_{\infty}$, and
the air density $\rho$. This force comprises both pressure and frictional
components.

The evaluation encompassed not only the drag coefficients but also the mesh
size and the computational resources required. The simulations were performed
on a machine equipped with AMD EPYC 7763 64-Core Processors, totaling 256 CPU
cores with 4 Nvidia A100 80GB GPUs.

Our analysis reveals a consistent correlation between our simulations and both
the benchmark experimental data and reference simulations, with the 8 and 16
million cell meshes showing particularly good alignment. The discrepancy
observed in the 40 million cell mesh could stem from differences in mesh
granularity, as the reference simulations utilized a 16 million cell mesh.
Finer meshes capture more intricate flow dynamics, which may not be
represented in coarser meshes, leading to divergences. In computational fluid
dynamics, especially in the preliminary stages of design, an error margin of
up to 5$\%$ is generally considered acceptable for engineering purposes. In
addition, implementing "RANS fine" for all 4000 designs within the DrivAerNet
dataset would require approximately 120TB of storage, posing significant
challenges to data sharing and reproducibility due to the immense storage
demands. Therefore, considering the balance between accuracy and computational
resource allocation, we decided to conduct our simulations using the 8 million
and 16 million cell meshes. These configurations provide a compromise between
computational efficiency and the level of detail necessary for accurate
aerodynamic analysis.

##### Leveraging Multi-Fidelity Data and Transfer Learning for Efficient
Surrogate Model Development:

As shown by [15], leveraging multi-fidelity CFD simulations proves to be a
robust strategy for accurate 3D flow field estimation. This approach involves
using a dataset that combines RANS, which are relatively easier and cheaper to
obtain and can capture the general flow behavior, with Direct Numerical
Simulation (DNS) data, known for its detailed flow information despite being
computationally expensive. Training a deep learning model with this diverse
dataset not only enables the model to generalize effectively to real-world
scenarios, as confirmed by wind tunnel tests, but also facilitates a two-phase
training process. This process starts with medium-fidelity RANS data to grasp
general flow patterns and then transitions to fine-tuning with high-fidelity
DNS data, thereby enhancing the model’s accuracy and real-world applicability.
Similar results leveraging multi-fidelity datasets for training surrogate
models have been demonstrated by [33, 35], underscoring the approach’s
efficacy in aerodynamic analysis. The DrivAerNet dataset can be similarly
utilized, allowing for integration with datasets of either lower or higher
fidelity to augment model training and refine predictive capabilities.

Figure 5: 3D streamlines representing the magnitude of velocity around the
car, alongside the distribution of velocity magnitudes on the car’s wheels.
The gradient observed at the wheels is indicative of angular velocity,
reflecting the rotation of the wheels. Units are in (m/s).

(a) Velocity field distribution around the DrivAer model, with velocity
magnitudes expressed in (m/s).  (b) Pressure coefficient ($C_{p}$)
distribution on the DrivAer model, illustrating the pressure variation across
the car’s surface.

Figure 6: The DrivAerNet dataset includes detailed 3D fields for velocity,
pressure, and wall-shear stresses, alongside aerodynamic coefficients as well
as the detailed 3D meshes of the car bodies and both front and rear wheels in
each of its entries.

####  3.1.3 CFD Simulation Results

##### Inclusion of Diverse Car Dimensions and Complex Flow Dynamics:

In contrast to the approach by [36] where all car models are standardized to a
uniform length of 3.5 meters to fit a predefined computational domain, our
dataset allows diversity in car dimensions, adjusting the mesh, bounding
boxes, and additional layers accordingly for each design. This flexibility is
crucial for capturing the intricate flow dynamics around cars including
phenomena like flow separation, reattachment, and recirculation zones, and
ensuring precise aerodynamic coefficient estimation. This approach addresses
limitations observed in some studies that prioritize dataset size over
simulation fidelity, often overlooking the importance of convergence, accurate
modeling, and appropriate boundary conditions for complex 3D models.

##### Modeling of Wheels, Side Mirrors, and the Underbody:

As previously highlighted, the majority of literature and available datasets
tend to neglect the modeling of wheels, side mirrors, and the underbody, as
detailed in Table 1. Our methodology, in contrast, includes detailed modeling
of these components. Figure 5 illustrates the velocity distribution on the
car: here, the car body exhibits zero velocity due to the no-slip boundary
condition, whereas the wheels show nonzero velocity. Additionally, the figure
visualizes streamlines around the car, offering insight into the flow dynamics
influenced by the inclusion of these features. The DrivAerNet dataset features
full 3D flow field information, as shown in Figure 6(a) with the velocity
data, and in addition, it provides the pressure distribution on the car’s
surface. The pressure coefficient, denoted as $C_{p}$, is calculated as the
ratio of the pressure differential $p-p_{\infty}$ to the dynamic pressure,
$\frac{1}{2}\rho u^{2}$, and is given by:

$C_{p}=\frac{p-p_{\infty}}{\frac{1}{2}\rho u^{2}}.$ (2)

The distribution of $C_{p}$ on the car surface is depicted in Figure 6(b).

###  3.2 Geometric Feasibility

##### Ensuring Geometric Integrity in Automated Design Morphing:

In our approach to generating large diverse automotive designs through
automated morphing with ANSA®, ensuring the geometric quality and feasibility
of each variant is crucial. To address potential issues such as non-watertight
geometries, surface intersections, or internal holes resulting from morphing
operations, we employ an automated mesh quality assessment and repair process.
This procedure not only identifies but also rectifies common geometric
anomalies, ensuring that only simulation-ready models are included in our
dataset. Geometries failing to meet these criteria are systematically excluded
from simulations. The DrivAerNet dataset employs a diverse array of parameters
(total of 50 parameters) to morph car geometries, encompassing modifications
to the side mirror placement, muffler position, windscreen, rear window
length/inclination, the size of the engine undercover, offsets for the doors
and fenders, hood positioning, and the scale of headlights as well as
alterations to the overall car length and width. Additionally, adjustments are
made to the car’s upper and underbody scaling, as well as pivotal angles such
as the ramp, diffusor, and trunk lid angles, all crucial for exploring the
effects of different design modifications on car aerodynamics. For a detailed
account of the morphing parameters, including their lower and upper bounds,
please refer to our GitHub repository.

As we morph the entire car’s geometry, wheel positioning is adjusted in the
$x$, $y$, and $z$ axes during the morphing process. For all simulations, we
use front and rear wheels of the same shape. To accurately simulate the wheel
rotation, we export them as separate .stl files post-morphing, which allows us
to apply the rotatingWallVelocity physical boundary condition. Moreover,
morphing affects the car’s vertical positioning, necessitating a calculation
of the $z$-axis displacement to ensure the car body and wheels are properly
aligned with the ground plane. For simulation purposes, we supply three
distinct .stls: one for the car body, one for the front wheels, and one for
the rear wheels, to model their interactions accurately.

###  3.3 DrivAerNet Dataset Characteristics

Figure 7: Scatter plots displaying the relationship between drag coefficient
($C_{d}$) and lift coefficients ($C_{l}$, $C_{l,f}$, $C_{l,r}$) for the
DrivAerNet dataset. The dataset represents unique design variants generated
via the Optimal Latin Hypercube sampling method, utilizing the Enhanced
Stochastic Evolutionary Algorithm (ESE). Data points are categorized into
training, validation, and test sets (70$\%$, 15$\%$, 15$\%$).

For our simulations, we employed OpenFOAM® version 11, executing the
computational tasks across 128 CPU cores and 4 Nvidia A100 80GB GPUs. This
resulted in a total computational cost of approximately 352,000 CPU hours. We
make available the complete suite of data, encompassing both the raw CFD
outputs and the derived post-processed datasets.

Our dataset serves as a benchmark for evaluating deep learning models,
designed to facilitate effective model testing. To manage the large volumes of
data from CFD simulations, we employ a data reduction strategy that focuses on
key areas of the flow field. This involves retaining data from regions
immediately in front of and behind the car, defined within a specific bounding
box, which helps to reduce the overall data size significantly. Additionally,
we provide a script that converts CFD simulation data into a format suitable
for training deep learning models. Given the widespread use of data
visualization tools such as ParaView and VisIt, which rely on the
Visualization Toolkit (vtk), our dataset is made available in vtk format. This
ensures that the data is easily accessible and usable within these common
visualization environments, supporting a broad range of research and
application needs.

The DrivAerNet dataset offers a comprehensive suite of aerodynamic data
pertinent to car geometries, including key metrics such as the total moment
coefficient $C_{m}$, the total drag coefficient $C_{d}$, total lift
coefficient $C_{l}$, front lift coefficient $C_{l,f}$, and rear lift
coefficient $C_{l,r}$. Included within the dataset are crucial parameters like
wall-shear stress, and the $y^{+}$ metric, integral for mesh quality
evaluations. Further, the dataset provides insights into flow trajectories and
detailed cross-sectional analyses of pressure and velocity fields along the
$x$ and $y$-axes, enriching the understanding of aerodynamic interactions.

The dataset includes:

  * •

Comprehensive CFD simulation data $\sim$ 16TB

  * •

A curated version of CFD simulations $\sim$ 1TB

  * •

3D meshes of 4000 car designs and the corresponding aerodynamic performance
coefficients ($C_{d}$, $C_{l}$, $C_{l,r}$, $C_{l,f}$, and $C_{m}$) $\sim$ 84GB

  * •

2D slices include the car’s wake in the $x$-direction and the symmetry plane
in the $y$-direction $\sim$ 12GB.

Figure 8: Comparative Kernel Density Estimation (KDE) and violin plots of drag
coefficients from two aerodynamic datasets. The blue curve represents the
dataset from Song et. al 2023 [36], and the orange curve corresponds to the
DrivAerNet dataset. DrivAerNet focuses on conventional car designs,
emphasizing the influence of minor geometric modifications on aerodynamic
efficiency.

##### Aerodynamic Performance Variability Amongst Car Designs in DrivAerNet:

Figure 7 displays three scatter plots that map out the relationships between
drag coefficient ($C_{d}$) and various lift coefficients ($C_{l}$, $C_{l,f}$,
and $C_{l,r}$) in the DrivAerNet dataset. The data has been split into
training, validation, and test sets, with a division of 70$\%$ for training
and 15$\%$ each for validation and testing. Such a division is critical for
the integrity of the model training process and subsequent performance
evaluation.

The Kernel Density Estimation (KDE) plot, illustrated in Figure 8, compares
the distribution of drag coefficients across two aerodynamic datasets. Here,
we compare the dataset from [36], which spans a broad range of drag values,
reflecting a wide variety of car designs in ShapeNet. In contrast, our
DrivAerNet dataset targets conventional car designs, accounting for more
detailed geometric modifications. This focus is particularly relevant in the
context of the engineering design process, where an initial car design is
typically refined through incremental changes to optimize aerodynamic
performance. The DrivAerNet dataset, therefore, provides a more specific
examination of subtle design adjustments and their impact on aerodynamic
performance.

In Figure 9, we present the aerodynamic performance across various designs.
The top left illustrates the design with lowest drag coefficient $C_{d}$. In
contrast, the top right shows the design with the highest $C_{d}$, identifying
opportunities for aerodynamic refinement. The bottom left reveals the design
with the lowest lift coefficient $C_{l}$ (indicating the largest downforce),
which is advantageous for stability at high speeds, while the bottom right
exposes design the highest $C_{l}$, potentially complicating aerodynamic
stability.

Figure 9: Aerodynamic performance of car designs from DrivAerNet showcasing a
range of coefficients. Top left: Design with minimum drag coefficient $C_{d}$,
indicating optimal aerodynamic efficiency. Top right: Design with maximum
$C_{d}$. Bottom left: Design with minimum lift coefficient $C_{l}$ (highest
downforce). Bottom right: Design with maximum $C_{l}$.

In the next section, we propose a new deep learning method to utilize the
dataset for surrogate modeling.

##  4 Dynamic Graph Convolutional Neural Network for Regression

Figure 10: Architecture of the RegDGCNN for aerodynamic drag prediction. The
model processes a 3D mesh by converting it into a point cloud representation.
It takes $n$ input points, calculates an edge feature set of size $k$ for each
point at an EdgeConv layer, and aggregates features within each set to compute
EdgeConv responses for the corresponding points. The output features of the
last EdgeConv layer are aggregated globally to form a 1D global descriptor,
which is then used to predict the aerodynamic drag coefficient $C_{d}$,
enabling direct learning from the 3D geometry of the object. The EdgeConv
block ingests an input tensor of dimensions $n\times f$, where it determines
edge features for each point utilizing a multi-layer perceptron (MLP). Post-
MLP application, the block outputs a tensor of dimensions $n\times a_{n}$ by
conducting a pooling operation over the neighboring edge features.

Geometrical deep learning has demonstrated significant promise in addressing
fluid dynamics challenges involving irregular geometries, as demonstrated in
studies by [34, 26, 1, 20, 32, 30, 29].

In this work, we extend the Dynamic Graph Convolutional Neural Network (DGCNN)
framework [42], traditionally associated with PointNet [27] and graph CNN
methodologies, to address regression tasks, marking a significant departure
from its conventional applications in classification. Our contribution lies in
adapting the DGCNN architecture to predict continuous values, specifically
focusing on aerodynamic coefficients critical in fluid dynamics and
engineering design. Leveraging the spatial encoding capabilities of PointNet
and the relational inferences provided by graph CNNs, our proposed RegDGCNN
model (as shown in Figure 10) aims to capture the complex interactions of
fluid flow around objects, offering a novel method for accurate estimation of
crucial aerodynamic parameters. This approach harnesses local geometric
structures by constructing a local neighborhood graph and applying
convolution-like operations on the edges linking pairs of neighboring points,
aligning with graph neural network principles. The technique, termed edge
convolution (EdgeConv), is shown to exhibit properties that bridge translation
invariance and non-locality. Uniquely, unlike in standard graph CNNs, the
graph of RegDGCNN is not static but is dynamically updated after each layer of
the network, allowing the graph structure to adapt to evolving feature spaces.

We start by initializing the graph $G$ with node features $X$, along with the
parameters for the EdgeConv layers $\theta$ and the Fully Connected (FC)
layers $\phi$. A distinctive feature of the RegDGCNN is its dynamic graph
construction within each EdgeConv layer, where the $k$-nearest neighbors of
each node are identified based on the Euclidean distance in the feature space,
thereby adaptively updating the graph’s connectivity to reflect the most
significant local structures. The EdgeConv operation is defined as:

$h_{ij}=\Theta\left(x_{i},x_{j}-x_{i}\right)$ (3)

enhances node features by aggregating information from these neighbors using a
shared Multi-Layer Perceptron (MLP), which processes both the individual node
features and their differences with adjacent nodes, capturing the local
geometric context effectively.

Following the EdgeConv transformations, global feature aggregation is
performed, pooling features from across all nodes into a singular global
feature vector:

$x^{\prime}_{i}=\max_{j\in\mathcal{N}(i)}h_{ij}$ (4)

Here, max pooling is employed to encapsulate the graph’s holistic information.
This global feature vector is subsequently processed through several FC
layers, with the inclusion of non-linear activation functions like ReLU and
dropout to introduce non-linearity and prevent overfitting, respectively. The
architecture culminates in an output layer, designed to suit the specific task
at hand, for instance, employing a linear activation for regression tasks.

$h_{ij}=\text{MLP}\left(\left[x_{i},x_{j}-x_{i}\right]\right)$ (5)

$X^{\prime}=\max_{i\in\mathcal{G}}x^{\prime}_{i}$ (6)

The model’s performance is quantified by calculating the loss between its
predicted outputs and the ground truth drag values using the mean squared
error (MSE), with the backpropagation algorithm adjusting the model parameters
$\theta$ and $\phi$ through optimization algorithms such as Adam [21] to
minimize this loss. This iterative refinement process highlights the
RegDGCNN’s capability to dynamically leverage and integrate hierarchical
features from graph-structured data.

###  4.1 Implementation Details

##### Network Architecture:

We constructed the graph for the RegDGCNN using the $k$-nearest neighbors
algorithm, with $k$ set to 40. This parameter was critical in defining the
local neighborhood over which convolutional operations were performed. The
RegDGCNN model was instantiated with specific parameters to accommodate the
nature of the regression task. The EdgeConv layers were configured with
channels of sizes {256, 512, 512, 1024}, and the following MLP layers were
{128, 64, 32, 16}. Lastly, the embedding dimension of the network was set at
512, providing a high-dimensional space to capture the complex features
necessary for the regression tasks at hand. Our RegDGCNN model is entirely
differentiable and can seamlessly integrate with 3D generative AI applications
for enhancing design optimization.

##### Model Hyperparameters:

The experiments were conducted using the PyTorch framework. The models were
trained using a batch size of 32, with the number of points per input set to
5000. Training was distributed across 4 NVIDIA A100 80GB GPUs, utilizing data
parallelism to enhance computational efficiency. The network’s learning rate
was initially set to 0.001, and a learning rate scheduler was employed to
reduce the rate upon plateauing of the validation loss, specifically using the
ReduceLROnPlateau scheduler with a patience of 10 epochs and a reduction
factor of 0.1. This approach helped in fine-tuning the models by adjusting the
learning rate in response to the performance on the validation set. The models
were trained for a total of 100 epochs, ensuring sufficient learning while
preventing overfitting. For optimization, we used the Adam optimizer [21] due
to its adaptive learning rate capabilities.

##### Inference Time:

The RegDGCNN model, with its compact size of roughly 3 million parameters and
a storage requirement of about 10MB, performs drag estimation for a car design
with 540k mesh faces in 1.2 seconds on 4 A100 80GB GPUs, a significant
efficiency gain compared to the 2.3 hours taken by a standard CFD simulation
on 128 CPU cores with 4 A100 80GB GPUs.

##  5 Surrogate Modeling of Aerodynamic Drag

In this section, we evaluate our RegDGCNN model on two aerodynamic datasets,
DrivAerNet and ShapeNet, highlight the impact of larger training volumes on
model performance, and investigate the model’s learned features.

Figure 11: Correlation plot of the predicted drag coefficient $C_{d}$ by our
RegDGCNN model against the ground truth for the DrivAerNet unseen test set,
achieving an $R^{2}$ score of 0.9. The dotted line denotes the line of perfect
correlation, representing the ideal prediction scenario.

###  5.1 DrivAerNet: Aerodynamic Drag Prediction of High-Resolution Meshes

The examination of RegDGCNN’s performance on the DrivAerNet dataset, depicted
in Figure 11, reveals a good correlation between predicted values and ground-
truth data from CFD, underscoring the model’s effectiveness. The complexity of
the DrivAerNet dataset is attributed to its inclusion of industry-standard
shapes, varied through 50 geometric parameters, presenting a comprehensive
challenge in aerodynamic prediction. Our model effectively navigated the
complexities of the dataset and directly processed the 3D mesh data, marking a
significant shift from traditional methods that often rely on generating
Signed Distance Fields (SDF) or rendering 2D images. This straightforward
approach enabled us to achieve an $R^{2}$ score of 0.9 on the unseen test set,
emphasizing the model’s ability to accurately discern subtle aerodynamic
differences.

###  5.2 ShapeNet: Aerodynamic Drag Prediction of Arbitrary Vehicle Shapes

To test the generalizability of the proposed RegDGCNN model, we also evaluated
its ability to adapt to complex geometries on an existing benchmark dataset,
utilizing 2,479 diverse car designs from the ShapeNet dataset [36] (see Figure
12), which exhibits a broader range of car shapes than our DrivAerNet dataset.

Figure 12: A selection of car samples from the ShapeNet dataset, demonstrating
the diversity in car shapes and mesh resolutions, utilized to evaluate the
generalization capabilities of RegDGCNN. These samples provide a comparative
baseline against the high-resolution meshes found in our DrivAerNet dataset.

In Table 3, we compare the performance of two models: the attn-ResNeXt model
from the study by [36], which implements a self-attention mechanism to boost
the understanding of interactions among various regions of an image. It uses
2D depth/normal renderings as input and has approximately 2 billion
parameters, achieving an $R^{2}$ score of 0.84; our proposed RegDGCNN model,
which directly processes 3D mesh data, significantly reduces the number of
parameters to 3 million, and achieves a superior $R^{2}$ score of 0.87. This
comparison underscores the efficiency and effectiveness of our model in
aerodynamic drag prediction tasks.

Table 3: Comparison of model performance for drag prediction on arbitrary car shapes from the ShapeNet dataset. Model | Input | $R^{2}$ |  $\\#$ Parameters
---|---|---|---
attn-ResNeXt [36] | 2D Depth/Normal Renderings | 0.84 | 2B
RegDGCNN (ours) | 3D Mesh | 0.87 | 3M

###  5.3 Effect of Training Dataset Size

For both ShapeNet444The work expanded their dataset to 9,896 variations
through resizing and flipping augmentations, but considered only 2,474 unique
car designs as independent samples for their surrogate model training to
prevent data leakage across different sets. and DrivAerNet datasets, we first
allocated 70$\%$ for training, and 15$\%$ each for validation and testing.
Subsequently, we experimented with training subsets at 20$\%$, 40$\%$, 60$\%$,
80$\%$, and 100$\%$ of the training portion. The ShapeNet subsets ranged from
1270 to 6352 samples. Meanwhile, for the DrivAerNet dataset, the corresponding
sample sizes were 560, 1120, 1680, 2240, and 2800 samples.

Figure 13 reveals a clear trend where the average relative error in drag
coefficient predictions decreases as the percentage of the dataset used for
training increases. This trend is consistent for both datasets, underscoring
the common machine learning principle that more training data generally leads
to better model performance. The improved performance of the DrivAerNet
Dataset across all sizes of training data highlights the critical role of
bigger datasets in machine learning models for aerodynamics and further
establishes the value of DrivAerNet dataset, which is significantly larger
than previous open-source datasets.

The figure also indicates that RegDGCNN yields better performance on the
DrivAerNet dataset compared to the dataset from ShapeNet. This can be
attributed to several factors:

  * •

The large variation in shapes within ShapeNet does not correspond with an
adequate number of samples to encompass the entire range of aerodynamic drag
values.

  * •

ShapeNet models cars were modeled as single-bodied entities, omitting crucial
details like wheels and underbodies that are vital for accurate aerodynamic
modeling.

  * •

All drag values in ShapeNet were computed using a single reference area, which
does not account for the significant variation in frontal projected areas of
different car designs.

  * •

There is a considerable variation in mesh resolutions across the ShapeNet
dataset, potentially leading to inconsistencies in aerodynamic predictions.

This analysis serves to demonstrate the generalization capabilities of our
model, emphasizing the objective of developing models that effectively
generalize to out-of-domain distributions.

Figure 13: Average relative error in drag coefficient predictions on the
unseen test set for our model RegDGCNN, based on training set size. The
results from the ShapeNet drag dataset [36] are denoted in blue, while results
from the DrivAerNet dataset are shown in orange. The training set sizes vary
from 20$\%$ to 100$\%$. We observe that the increase in dataset size leads to
significant error reduction, showing the necessity of larger datasets in
aerodynamic surrogate modeling.

###  5.4 Features Learned

Figure 14: Visualization of feature importance across an upsampled point cloud
of a car model from DrivAerNet, derived from our RegDGCNN, specifically
focusing on features from the EdgeConv layers. Feature intensities are color-
coded from low (light yellow) to high (dark red), indicating areas of
significant learning focus by the convolutional layers. This mapping
highlights the learned features that contribute to the model’s predictions.

To further assess the model’s performance, we analyze the features learned in
the intermediate layers following the edge convolution operations. Figure 14
illustrates the distribution of feature importance across an upsampled point
cloud of a car sample from DrivAerNet, with color coding from light yellow
(low importance) to dark red (high importance). Initially, RegDGCNN zeroes in
on the car’s frontal and rear area, which are critical in shaping aerodynamic
performance. This focus is notably pertinent for aerodynamic design, as the
frontal area has a substantial effect on pressure drag and the rear area is
particularly relevant due to its role in airflow separation and wake region
formation. As the model delves deeper through its layers, it starts to
recognize more complex geometric details. Conversely, areas such as the roof
and windows are less crucial for drag, underscoring the model’s adeptness at
identifying areas with a more significant aerodynamic influence.

##  6 Limitations and Future Work

This section discusses the limitations of our study. Despite careful selection
to ensure a balance between detail and computational efficiency, the model’s
parameterization faces an inherent limitation. This stems from the trade-off
between the compactness of the representation and the flexibility needed to
capture a broad range of aerodynamic phenomena. Consequently, while our
approach offers valuable insights for many applications, it may not fully
encompass all aerodynamic variabilities pertinent to automotive engineering.
The dataset comprises 4,000 instances, which, while significant, might not
fully capture the broad spectrum of real-world automotive designs.
Additionally, our focus was primarily on drag prediction; however, we plan to
extend the application of RegDGCNN to include surface field predictions in
future work. While our dataset is large and of high-fidelity, it is important
to acknowledge that we are still in the early stages of approaching the scale
and foundational impact seen in AI fields like image processing and natural
language processing, where large datasets are a norm.

One of the key challenges in applying graph-based methods like RegDGCNN is the
significant GPU memory requirements. This is due to the need to compute all
pairwise distances between points, which can be highly memory-intensive.
Moreover, the non-uniform density of point clouds introduces additional
complexities; a fixed $k$-nearest neighbors approach may not be suitable for
areas with varying densities of points.

Another limitation is that, in its current form, RegDGCNN does not reduce the
number of points during the forward pass for large-scale point clouds, leading
to high computational demands and potentially limiting the model’s scalability
to even larger datasets. Addressing these challenges will be crucial for
advancing the capabilities and applications of graph-based neural networks in
processing complex aerodynamic data.

##  7 Conclusion

In our conclusion, we highlight the distinct advantages of DrivAerNet, which,
by focusing on detailed geometric modifications, outperforms broader datasets
such as those referenced in [36, 22, 31], especially in the context of real-
world aerodynamic design applications. Additionally, the compact RegDGCNN
model, with 3 million parameters and 10MB size, efficiently estimates drag in
just 1.2 seconds for industry-standard designs with 540k mesh faces,
significantly outpacing traditional CFD simulations. Moreover, our RegDGCNN
model, in particular, showcases superior performance by directly processing 3D
meshes, thereby eliminating the need for 2D image rendering or the generation
of Signed Distance Functions (SDF), which simplifies the preprocessing stages
and increases the model’s accessibility. Importantly, the RegDGCNN model’s
ability to deliver precise drag predictions without requiring water-tight
meshes highlights its adaptability and effectiveness in leveraging real-world
data. Through the expansion of the DrivAerNet dataset from 560 to 2800
samples, we achieved a remarkable reduction in error by approximately 75$\%$.
Similarly, on the dataset from [36], increasing the training samples from 1270
to 6352 led to a 56$\%$ decrease in error, underscoring the substantial impact
of dataset scale on enhancing the performance of deep learning models in
aerodynamic studies. The inclusion of specific parametric alterations (50
geometric parameters) within our DrivAerNet dataset has significantly improved
model learning, resulting in a notable increase in predictive accuracy,
essential for the refinement of aerodynamic designs. This emphasizes the
critical role of large detailed, high-fidelity datasets in crafting models
capable of adeptly handling the complexities inherent in aerodynamic surrogate
modeling.

##  8 Acknowledgements

The research work of Mohamed Elrefaie was supported by the German Academic
Exchange Service (DAAD) under the IFI - "Internationale Forschungsaufenthalte
für Informatikerinnen und Informatiker" Grant and the Department of Mechanical
Engineering at MIT. The authors thank Florin Morar from BETA CAE Systems USA
Inc for his support with morphing the DrivAer model in ANSA®. We extend our
gratitude to Christian Breitsamter for his role in facilitating the
organizational aspects of this research.

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