## 1 Introduction

Formula 1 engineering requires a very high degree of accuracy since the slightest improvement can have a decisive impact of the aerodynamics and hence, on performance. Racing teams’ Engineers use high-ﬁdelity numerical simulations to assess the eﬀect of design variations, without actually building the real prototype.

## 2 Method Overview

### 2.1 Numerical optimization using Neural Concept Shape

Computational Fluid Dynamics simulations are at the heart of the car development process of all Formula 1 teams, and one of the key performance factors for the race. Even though teams can generate thousands of simulations every year, and explore as many design variations, they are limited both by the computational cost and the reglementation from the FIA. Among this large number of designs, it is critical to be able to converge quickly towards designs that correspond to a reasonable compromise. Hence, running approximate simulation results without re-running expensive simulations every time can be critical in this phase. In the case of Formula 1, low ﬁdelity simulations are not an option since the very complex nature are the phenomena can only be captured by high-end CFD resolutions.

A classical approach to reduce the number of simulations and quickly map the search space is to use surrogate modelling via Gaussian Process (GP) regressors. This interpolator is then used as a proxy for the true objective to speed-up the computation, which is referred to as Kriging in the literature. However, those regressors are only eﬀective for shape deformations that can be param- eterized using relatively few parameters and their performance therefore hinges on a well-designed parameterization. Furthermore, Formula 1 teams generate hundreds of simulations per day and have therefore sometimes accumulated a very large dataset of existing simulation, which do not share a common paramterization and therefore cannot be directly exploited by such parametric approaches. Finally, all previous available approaches fail to accurately reconstruct 3D ﬂow or ﬁelds on a surface or a plane. They all tend to over-smooth the solutions and produce outputs for which the structure looks nowhere near the one of the initial ﬂow. This may be explained by the linear nature of those approximation methods, which goes against the strong non-linearities of the underlying physics.

A recently developed software, Neural Concept Shape (NCS), opens the door to a solution to this problem through Geometric Convolutional Neural Networks, a new way to build surrogates of numerical simulator. It is agnostic to the shape parameters as it processes directly the mesh representation of the design. Hence, design parameters are decoupled from the learning problem and a single predictor can be trained with a large amount of data and used for many optimization tasks. Unlike Kriging methods, the engineer does not have to choose and stick to a speciﬁc parametrization from the beginning to the end of experiments. Furthermore, it can leverage on transfer learning abilities of Deep Learning models to blend simulations from multiple sources and diﬀerent parametrizations. To date, NCS is the only CAE-oriented Deep-Learning code that is able to process raw unstructured 3D data directly withing the scale of what is used in Formula 1 simulations. It uses multi-scale geometric neural networks, through a combination of surface — such as geodesic — and euclidean network architectures. It is able to learn to emulate simulators and reproduce multiple outputs such as integrated scalar values — e.g. drag forces or energy consumption — or surface and volumetric ﬁeld values —e.g. pressure or temperature.

### 2.2 Objectives

In this project, we evaluate the performances of the NCS’s ProjectiveRegressor Neural Network model and provide a ﬁrst element of answer with respect to the feasibility of the use of NCS in F1 racing. We focused our attention on the front side of a Formula 1, from the front wing to the bargeboard, including the nose of the car.

Due to conﬁdentiality and accessibility reasons, note that the data and simulations used within this project are a simpliﬁed version of the more complex geometries and simulations found in operational conditions in Formula 1. The CAD design does not oﬀer all the details and subtleties found on nowaday’s cars, and the CFD was resolved at a lower resolution than what is done in the most competitive teams. However, the problem presented here exhibits most of the practical challenges found in operations, including long range interactions, complex vortex structures, multiple connected components and importance of geometrically ﬁne details.

### 2.3 CAD design

OnShape, a CAD software hosted on the cloud was used to design the Formula 1. This geometry was generated by varying up to 56 diﬀerent variables, detailed below:

This parametric space allows to perform very ﬁne design variations on the car, and ultimately to assess the inﬂuence of each of these parameters on the aerodynamic performances of the car. To generate the training dataset, we automated the geometries generation by using the OnShape API. We deﬁned lower and higher values for each of the parameters, and a random value was chosen at each iteration for the parameters. We were able to build a database of 1644 geometries using this method. It ensured a good distribution of the geometries, and each of them is unique in the dataset. An example of the foils variability within the dataset is shown with ﬁgure 8 where it is possible to see a sub-group of 5 shapes representing the variation of the shapes of foils that were generated.

From Onshape, we can export the shape as an STL ﬁle. The .stl is then remeshed using Openfoam remesher blockmesh. Similarly, the volumetric mesh is created and following its discretization the mesh of the vertical plane just behind the tyre. Note that due to its irregular discretization we are allowed to have a more ﬁne-grained reﬁnement close to the car shape and a coarser one far away from the car.

### 2.4 Numerical Simulation with Openfoam

OpenFoam was used to perform the numerical simulations. It is a free, open source software for Computational Fluid Dynamics simulations. The car is run at atmospheric conditions, at 250 kph (70m/s) in straight ahead condition (no crosswind). We therefore use half of the car to spare compute power. The wheel is deﬁned as rotating according to the car speed. The turbulence model used is SST k-omega transport. More details on this model can be found here.

Simulations were performed on the surface of the car, as shown with ﬁgure 9, but also on a plane slice taken at the last iteration of ﬂow ﬁeld at 1.7m, i.e just behind the tyre. Indeed, usually, taking the whole volumetric prediction is too expensive for high-ﬁdelity simulations. Hence using plane slices is a good trade-oﬀ and allows to assess the inﬂuence of slight geometry variations more precisely. This vertical plane slice also allows to observe some very important ﬂow phenomenon that are happening alongside the Formula 1, and that aerodynamicists want to control as good as possible. A visual example is given with ﬁgure 10.

## 3 Setup in Neural Concept Shape

### 3.1 Geometric Neural Network

We use the recent ”ProjectiveRegressor” deﬁned by the NCS framework as our Geometric Neural Net- work regressor. This model allows for a query mesh diﬀerent than the original shape to be speciﬁed. In this case it is the vertical plane just behind the tyre, in which vortexes are present. Note that the network is unaware of the underlying parametrization and shapes created with other parametrizations can easily be predicted or added to the dataset. Only the similarity to the shape present on the train- ing set will have an eﬀect on the accuracy.

The part using convolution on the shape is illustrated in ﬁgure 13. The ﬁrst part of the model pre-processes the input and constructs a set of features by means of the previously introduced geodesic convolution operations. These features are then used to predict the global scalars via average pooling and two dense layers. The second branch of the network generates pressure ﬁelds relying on an additional set of geodesic convolutions and point-wise operations. The new network takes advantage of a GPU eﬃcient implementation of geodesic convolutions.

## 4 Results

### 4.1 Predictions

As a ﬁrst step, we predict on the surface of the car and evaluate the performances. The solution on the surface of the car is quite stable with respect to the deformations of the shape. The all-point-mixed R2 coeﬃcient on the pressure of the surface car is: 0.916. The distribution of the R2-score over the samples is given with ﬁgure 14 and an visual comparison is given with ﬁgure 15

As a second step, we can predict the pressure on the vertical plane just behind the tyre. The all-points-mixed R2 coeﬃcient on the plane is: 0.9674995.The distribution of the R2-score over the samples is given with ﬁgure 16 in Appendix A. Some visual comparisons between the ground truth (simulation done with OpenFoam) and instantaneous predictions of the neural network are provided with the ﬁgures 17,18,19 and 20 located in Appendix A. 1:

## 5 Conclusion

In this report, we analyse the capabilities of the neural network to perform instantaneous predictions on the surface and on a vertical plane located in the volume behind the wheel of the car. This end- to-end workﬂow demonstrates the abilities of the NCS’s ProjectiveRegressor module to predict ﬂow ﬁelds (pressure and velocity in the three-directions) with a good accuracy and by using as only input the geometry of the car. A sensitivity study was brieﬂy performed, to unveil the potential of such a method in terms of design optimization.

Although the simulations considered here are a simpliﬁed version of the higher ﬁdelity and more detailed models used by competitive teams, it opens the door to the perspective of re-using all the legacy simulations produced and stored by Formula 1 teams over the years. It will make it possible to explore more geometries and converge faster towards really promising designs, without re-running an undue number of simulations.

## 6 Appendix A