Motivation

Emissions from traffic pose a health threat to humans and animals [1]. With the implementation of strict regulations, the amount of exhaust emissions from traffic has been drastically reduced [2]. Non-exhaust emissions, which have their origins in brake wear, road wear, and tire wear, now make up the largest part of particulate matter emitted by traffic [3,4]. Wear from tires can be found as fine dust in the air or as larger particles at the sides of roads, from where they can be washed into water bodies [5,6]. Airborne particulate matter has diverse health effects directly where it is created [7–9]. In water bodies, larger wear particles decompose into microplastics. In total, 28% of the microplastics released into the oceans stem from tire wear [10]. In the environment, the wear particles release chemicals that were originally used in the tire production process but now seem to be abundant in all kinds of water reservoirs, where they can further react into more harmful substances [11]. These substances show acute toxicity to marine animals [12] and have already been found in humans [13]. Therefore, care should be taken to reduce emissions of microplastics caused by tire wear. However, the current trend towards electric cars may even increase tire wear because of the vehicles’ increased weight and drive torque [14,15].

To address the issue of tire wear at its source, it is essential to consider tire wear reduction as a key objective in the suspension system design process. By optimizing suspension elasto-kinematics, tire positioning on the road can be improved, potentially reducing tire wear across a wide range of everyday driving conditions, including scenarios with particularly high wear rates. Such optimizations can lead to longer-lasting tires, lower maintenance costs, and a reduction in environmental impact due to less frequent tire replacements. However, the optimization process is highly complex, involving numerous design variables, possible adaptations, and a variety of driving situations to consider. Automated numerical optimization is therefore necessary to explore the design space effectively. This approach requires a large number of tire wear predictions across different driving conditions for various suspension designs. To keep computation time feasible while maintaining the accuracy required to provide valuable design insights, efficient tire wear prediction models are crucial. Surrogate models offer an effective balance between computational efficiency and prediction accuracy, making them ideal for this type of large-scale design optimization in an industrial context.

Prediction of Tire Wear—Initial Situation

In previous publications [16–18], prediction of tire wear was based on the contact results produced by the tire model FTire [19]; see Fig. 1. When running virtual test drives using FTire, output of the results of the contact model can be enabled to produce contact output (.cfo) files. These contain data, such as tangential forces $F_{\text{x}}$ and $F_{\text{y}}$ and slip velocities $v_{\text{sx}}$ and $v_{\text{sy}}$, for every contact element e in contact with the road at discrete times t_i for every output time step i. Using these data, the transmitted frictional power can be calculated per contact element e. Since the contact elements are arranged in so-called tread strips over the tire width, the frictional powers can be summed for every tread strip, yielding a frictional power distribution over the tire width. This can be used as an input for a wear law predicting tire wear.

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For further clarification, Fig. 2 gives an overview of how the contact elements and tread strips are arranged on the tire model. The tire’s belt is divided into circumferential belt segments (Fig. 2, left). The number of belt segments $n_{\text{seg}}$ can be chosen arbitrarily; a higher number yields smoother contact results. Along the tire’s width, the surface is divided into tread strips (Fig. 2, middle). These tread strips do not necessarily correspond to the actual tread profile; they only serve as a logical element. The number of tread strips $n_{\text{str}}$ can be chosen arbitrarily as well, with a higher number providing finer frictional power distributions. The actual contact calculations are based on so called tread blocks, each with a distinct contact element for tire–road interaction on its surface. These tread blocks are attached to the belt segments and arranged in the defined number of tread strips (Fig. 2, right). For this arrangement to be possible, the number of tread blocks per belt segment $n_{\text{blocks}}/n_{\text{seg}}=z·n_{\text{str}}$ must be a multiple of the number of tread strips, where $n_{\text{blocks}}$ is the total number of tread blocks and $z\in \mathbb{Z}$ is an arbitrary integer.

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In the following, the upright subscript $\text{f}$ denotes quantities related to friction; $\text{el}$ signifies calculations at the element level; $\text{str}$ refers to the tread strip level, aggregating data across multiple elements; $\text{tot}$ represents results aggregated over the entire contact patch; and $\text{s}$ is used with slip quantities.

Using the contact discretization concept from above, the first step of the post-processing is to calculate the frictional power (1) $$P_{\text{f},\text{el},e,i}=-{\overrightarrow{F}}_{\text{f},\text{el},e,i}·{\overrightarrow{v}}_{\text{s},\text{el},e,i}$$ (2) $$=-{\left[\begin{array}{c}{F}_{\text{fx},\text{el},e,i}\\ F_{\text{fy},\text{el},e,i}\end{array}\right]}^{\text{T}}\left[\begin{array}{c}{v}_{\text{sx},\text{el},e,i}\\ v_{\text{sy},\text{el},e,i}\end{array}\right]$$ (3) $$=-F_{\text{fx},\text{el},e,i}{v}_{\text{sx},\text{el},e,i}-F_{\text{fy},\text{el},e,i}{v}_{\text{sy},\text{el},e,i}$$per contact element e at time step i that is dissipated between tire and road. It results from scalar product of the shear force ${\overrightarrow{F}}_{\text{f},\text{el},e,t}$ transmitted by the contact element and the relative velocity ${\overrightarrow{v}}_{\text{s},\text{el},e,t}$ between element and road. Since we want to look at the frictional power from the rubber’s perspective, we additionally multiply by −1 to get positive values for $P_{\text{f},\text{el},e,i}$ and hence a flow of energy out of the tire–road interface into the material. For direct handling of the contact results, this can also be calculated component-wise using the longitudinal friction force $F_{\text{fx},\text{el}}$, the lateral friction force $F_{\text{fy},\text{el}}$, the longitudinal slip velocity $v_{\text{sx},\text{el}}$ and the lateral slip velocity $v_{\text{sy},\text{el}}$.

As mentioned before, the contact elements are arranged in tread strips. With E_s being the set of elements belonging to the tread strip with number s, the frictional power (4) $$P_{\text{f},\text{str},s,i}={\displaystyle \sum _{e\in E_s}{P}_{\text{f},\text{el},e,i}}$$per tread strip s can be calculated by summing the frictional powers of the respective contact elements $e\in E_s$. To get from frictional power distributions to tire wear distributions, a wear law is needed. A commonly used law for the wear rate is (5) $$\frac{\Delta m}{\Delta t}\hspace{0.17em}\approx \hspace{0.17em}{k}_1{P}_{\text{f}}^{k_2}$$based on the work by Lupker et al. [20]. Here, m is the wear mass, and k₁ and k₂ are coefficients depending on the tire rubber–road surface combination under certain environmental conditions. Assuming linearity, that is, setting coefficient k₂ to 1, friction work can be directly used as a measure for tire wear. The transmitted friction work (6) $$W_{\text{f},\text{str},s}={\displaystyle \sum _i{P}_{\text{f},\text{str},s,i}}\Delta t$$per tread strip during the whole test drive is therefore calculated by summing up all friction powers per tread strip over time and multiplying with the output time step size $\Delta t$ of the simulation. For additional evaluations, the total friction power (7) $$P_{\text{f},\text{tot},i}={\displaystyle \sum _s{P}_{\text{f},\text{str},s,i}}$$transmitted by the whole tire per time step and the total transmitted friction work (8) $$W_{\text{f},\text{tot}}={\displaystyle \sum _s{W}_{\text{f},\text{str},s}}$$can be calculated by summing up over all tread strips s.

Using the evaluation method described above, simulations for the prediction of tire wear run with real-time factors (9) $$r_{\text{RTF}}=\frac{t_{\text{CPU}}}{t_{\text{sim}}}\gg 1,$$with $t_{\text{CPU}}$ being the CPU time needed to run the simulation and $t_{\text{sim}}$ being the time simulated. For instance, the real driving emissions test drive from [18] with $t_{\text{sim}}\approx 105\text{min}$ using a simple vehicle model equipped with four FTire tire models—of which one generates a .cfo file for post-processing—takes $t_{\text{CPU}}\approx 2000\text{min}\approx 33.3\text{h}$ of calculation time, corresponding to a real-time factor of $r_{\text{RTF}}\approx 19$ on an Intel® Core™ i7-8700 CPU.

Large portions of the calculation time needed stem from the 3D-structural tire model that is solved in the background and that allows for a locally resolved evaluation of friction between tire and road. Also, just the generation of the needed contact outputs adds significantly to the calculation time. A simulation without contact output reduces the calculation time by about 17.5% to $t_{\text{CPU}}\approx 1650\text{min}\approx 27.5\text{h}$ and the real-time factor to $r_{\text{RTF}}\approx 16$.

Additionally, the .cfo files become very large because of the potentially high number of contact elements, as well as output steps and the amount of data associated with every contact element. For the given simulation, the .cfo file reached a file size of 26.4 GB. This not only requires high post-processing effort, but also consumes a large amount of disk space. The disk space can be freed up only after post-processing is done, potentially limiting the number of simulations that can be run at the same time.

Such an evaluation approach in combination with long simulations, which are needed for estimating tire wear under everyday driving conditions, is therefore infeasible for an automated optimization. And even when the overall simulation time is drastically reduced by reducing the simulated test drive to separate short maneuvers as suggested in [18], the calculation time would still be too large for a multi-objective optimization where potentially 1000+ objective function evaluations are necessary. Therefore, the model for prediction of frictional power distributions has to be sped up.

Literature Review

Available tire models stretch from simple formula-based or empirical tire models to complex finite element (FE)–based tire models. A very prominent and widely used example for the former is the model developed by Pacejka [21] based on the so-called magic formula. In the commercial multi-body simulation software MSC Adams Car it is implemented as the PAC2002 tire model [22]. Its main advantages are low complexity and accordingly low computational effort, as well as an accurate representation of the tire behavior relevant for handling simulations. With modern model extensions, it can be used with short obstacle wavelengths and is accurate up to excitation frequencies of 70 to 80 Hz [22]. As a downside, it does not provide detailed contact calculations over the tire width. Therefore, it cannot be used for an evaluation of tire wear distributions.

On the other end of the complexity scale are FE–based tire models. They work with a detailed representation of the tire geometry as well as its material, and hence allow a detailed investigation of the tire–road contact, even in high-frequency applications such as on rough roads. Therefore, FE-based tire models are widely used for the assessment of tire wear [23–27]. Because of their complexity, they come with high computational effort and can hardly be used in a full vehicle simulation covering long representative test drives.

Structural tire models, such as CDTire [28] and FTire [19], are in the middle between formula-based and FE-based tire models, both in terms of complexity and calculation time. They model the tire geometry using a limited number of degrees of freedom, which allows the consideration of relatively high-frequency effects and locally resolved results for the tire–road contact. However, as shown before, the calculation time is still too high to be feasible for usage in a multi-objective optimization.

Promising results have been achieved using machine learning to conquer the complexity of tire models. Dye and Lankarani [29] fitted a model for the calculation of tire reaction forces using an artificial neural network (ANN) and real measurement data. This way they achieved functionality similar to a magic formula–based tire model and show that general tire behavior can be learned from measurement data using machine learning methods. A similar approach was presented by Guarneri et al. [30], who trained a recurrent neural network for prediction of forces acting on a vehicle’s chassis through the suspension when driving over a cleat. They were able to reproduce the suspension system’s behavior, including tire and elastic bushings, with high accuracy. However, neither approach provides detailed insights into tire–road contact that could be used for the prediction of tire wear.

Burger and Steidel [31] used machine learning for the prediction of rolling resistance and tread wear coefficients. They generated training data using CDTire in basis events with constant normal force $F_{\text{z}}$, camber angle γ, slip angle α, drive/braking torque $M_{\text{y}}$, and longitudinal speed $v_{\text{x}}$. CDTire returns a rolling resistance indicator and a tread wear indicator for these situations. They were used as targets in the fitting of ANNs and support vector machines. Burger and Steidel [31] showed that a rather simple neural network (three hidden layers with 100 neurons each) can predict these indicators with high accuracy and low computation time. A shortcoming of their approach is that the model yields only the two integral outputs mentioned before. An analysis of the wear distribution over the tire width is not possible.

Approach

To solve the shortcomings of the respective tire models and the approach by Burger and Steidel [31], this paper presents a methodology to derive a combination of models that needs a significantly shorter time for the prediction of tire wear than structural tire models, but is still able to predict frictional power distributions and therefore tire wear over the tire width. As a basis and reference, an FTire model will be used. FE-based tire models or other tire models providing locally resolved contact results could equally be a starting point. To reduce calculation time, the complex tire model will be replaced by a PAC2002 tire model fitted to its handling characteristics.

Additionally, a machine learning–based friction model is generated that predicts frictional power distributions based on the tire’s kinematic quantities that are calculated by the PAC2002 tire model. This way, the combination of both models is still able to predict frictional power distributions and consequently tire wear. Together, the PAC2002 tire model and the artificial intelligence (AI) friction model will be a surrogate for the FTire model.

The PAC2002 model and the machine learning–based friction model will be generated in a training step independent of the virtual test drive in which they will be finally used; see Fig. 3. The result will be a PAC2002 model file and a saved friction model pipeline. The PAC2002 model file can be used to initialize a PAC2002 tire model in any multi-body simulation. Here, it will be used in a virtual representative test drive for a realistic estimation of tire wear. During the simulation, the tire model will output tire states. These can be fed into the saved AI friction model to yield frictional power distributions. The training and prediction steps will be explained in more detail in the following.

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Tire Kinematics

In this work, quantities will be used in accordance with the definitions of the PAC2002 model contained in MSC Adams Car [22]. Specifically, the TYDEX W-axis system [32] will be used to describe tire forces and moments; see Fig. 4.

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This coordinate system’s origin is located at the intersection of the tire’s vertical axis with the road surface. From there the coordinate axes are oriented as follows: longitudinal axis x along the tire’s longitudinal axis and tangential to the road surface; lateral axis y perpendicular to x and also tangential to the road surface; and vertical axis z perpendicular to both other axes, pointing up. All axes together form a right-hand system.

Using these axes, the tire center’s velocity $v_{\text{C}}$ can be projected to get a component $v_{\text{x}}$ along the longitudinal axis and a component $v_{\text{y}}$ along the lateral axis. Also, the tire has a rotational velocity ω around the wheel axis. Based on these velocities, slip velocities between tire and road can be calculated. The longitudinal slip velocity (10) $$v_{\text{sx}}=v_{\text{x}}-\text{ω}{r}_{\text{e}}$$is essential for longitudinal forces and results from the difference between the tire’s longitudinal motion and its circumferential velocity in the tire–road contact. The latter results from the rotational velocity ω and the tire’s effective radius $r_{\text{e}}$, which has a value between the undeformed radius r₀ and the deformed radius r₁. The lateral slip velocity (11) $$v_{\text{sy}}=v_{\text{y}}$$is directly equal to the tire’s lateral velocity because rotation about the longitudinal axis is neglected.

The tire’s state can then be described by the kinematic quantities longitudinal slip κ, slip angle α, camber angle γ, rotational velocity ω, and tire deflection δ. The longitudinal slip κ is a nondimensional measure for the relative motion between tire and road in longitudinal direction and is defined as (12) $$\text{κ}=-\frac{v_{\text{sx}}}{v_{\text{x}}},$$

relating the tire’s longitudinal slip velocity $v_{\text{sx}}$ to its longitudinal velocity $v_{\text{x}}$. Similarly, the dimensionless measure for the side slip (13) $$\tan \text{α}=\frac{v_{\text{sy}}}{\left|v_{\text{x}}\right|}$$results from relating the tire’s lateral slip velocity $v_{\text{sy}}$ to its longitudinal velocity $v_{\text{x}}$. Finally, the camber angle γ is the angle between the wheel center plane and the x–z plane of the TYDEX W-axis system. We will use $\mathit{K}=\left[\text{κ,α,γ,ω,δ}\right]$ to refer to the set of tire’s kinematic quantities.

The tire forces $F_{\text{x}},F_{\text{y}}$, and $F_{\text{z}}$, as well as the moments $M_{\text{x}},M_{\text{y}}$, and $M_{\text{z}}$, are oriented along the respective axes of the W-axis system. Similarly to the kinematic quantities, $\mathit{F}=\left[F_{\text{x}},F_{\text{y}},F_{\text{z}},M_{\text{x}},M_{\text{y}},M_{\text{z}}\right]$ is the set of the tire’s forces and moments.

Training—Overview

The goal of the surrogate model is to generate frictional power distributions over the tire’s tread strips for every time step. The power distributions can be summed up to yield friction work distributions as a measure for tire wear, as described in the introduction. To accelerate the generation of frictional power distributions, the complex FTire model will be replaced by a simple but fast magic formula-based PAC2002 tire model and a machine learning–based friction model. Both parts of the surrogate model need to be fitted/trained based on training data generated with the original FTire model; see Fig. 5.

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To make the training completely independent of the virtual test drive in which the surrogate model will potentially be used, the training data are generated using generic situations on a virtual tire test rig. Using the training data, the PAC2002 model file can be fitted to represent the handling characteristics of the input model, that is, the relationship between tire states and the corresponding tire forces and moments. Additionally, machine learning is used to learn the relationship between tire states and the corresponding frictional power distributions, yielding a machine learning–based friction model.

Generation of Training Data

To be independent of any specific virtual test drive and vehicle model, MSC Adams Car’s virtual tire test rig is used to generate training data; see Fig. 6. Starting at the top, its first component is a mount (gray) where an overall translational motion $v_{\text{C}}$ and vertical forces $F_{\text{z}}$ can be applied. Rigidly attached is one of two concentric cylinders (blue). The cylinders can slide into each other through a prismatic joint. A spring-damper element (red) is placed between them. Spring stiffness and damping coefficient can be arbitrarily specified to dampen out potential vibrations. At the lower end of the second cylinder, the tire model is mounted using three hinge joints. These joints allow three rotations, which can also be prescribed: a rotating velocity ω or a braking/driving torque about the wheel axis resulting in a longitudinal slip κ, a slip angle α with respect to the prescribed translational motion, and a camber angle γ with respect to the vertical plane. Vertical force, rotating velocity/driving torque, slip angle, and camber angle can be functions of time.

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On the tire test rig, the FTire model is used to generate training data for the surrogate model. The training data consist of the tire’s kinematic quantities $\mathit{K}=\left[\text{κ,α,γ,ω,δ}\right]$, tire forces and moments $\mathit{F}=\left[F_{\text{x}},F_{\text{y}},F_{\text{z}},M_{\text{x}},M_{\text{y}},M_{\text{z}}\right]$, and the corresponding frictional power distributions ${\mathit{P}}_{\text{f},\text{str}}=\left[P_{\text{f},\text{str},1},\mathrm{...},P_{\text{f},\text{str},n}\right]$, with n being the total number of tread strips. The tire state variables K can be read directly from the simulation results. The same holds for the tire forces and moments F. The frictional power distributions are calculated based on FTire’s contact outputs (post-processing of .cfo files) as described above.

Fitting of Tire Model

For fitting of the tire model, 75 simulations are run with the virtual tire test rig. These include situations with pure slip in either the longitudinal or lateral direction at different loads and camber angles. In these, either a κ or α sweep is conducted. Further simulations are done for combined slip situations in which α is fixed at certain values and a κ sweep is conducted. Again, different loads and camber angles are applied. Finally, the free-rolling tire ( $\text{κ}=$ 0%) without side slip ( $\text{α}=0^{°}$) is simulated at different speeds and camber angles with increasing load. The resulting data show high-frequency fluctuations because of the discretization of the used FTire model. Therefore, it is smoothed by a moving average, which acts as a low-pass filter.

Using the generated training data containing the tire’s kinematic quantities K and corresponding target tire forces and moments $\widehat{\mathit{F}}$, a PAC2002 model can be fitted. Generally, the PAC2002 tire model receives tire states K as an input and generates tire forces and moments F from these through a functional dependency (14) $$\mathit{F}=\mathit{f}\left(\mathit{K},\mathit{p}\right),$$where p is the set of model parameters. The parameters p are adapted using the Levenberg–Marquardt algorithm implemented in SciPy’s curve-fitting module [33] according to the optimization problem (15) $$\underset{\mathit{p}}{\min }{\displaystyle \sum _i{\left(\mathit{f}\left({\mathit{K}}_i,\mathit{p}\right)-{\widehat{\mathit{F}}}_i\right)}^2},$$which minimizes the sum of the squared errors between model predictions f and target forces $\widehat{\mathit{F}}$ over all time steps i. To conduct the fitting in Python, the PAC2002 tire model is implemented according to MSC Adams Car’s implementation [22], which is based on the original publication of the tire model by Pacejka [21]. The implementation used for this work is limited to the steady-state model parts because of expected relatively slow changes of the tire states in everyday driving.

As a refinement of the fitting process and to reduce the number of parameters that are adapted at once, the fitting is done gradually for different terms of the PAC2002 model. For instance, the predicted longitudinal force under pure longitudinal slip (16) $$F_{\text{x0},\text{PAC2002}}=D\hspace{0.17em}\sin \left(\left[\mathrm{C\hspace{0.17em}}\arctan \left\{B\text{κ}-E\left(B\text{κ}-\arctan \left(B\text{κ}\right)\right)\right\}\right]\right)+S_{\text{V}},$$with the coefficients (17) $$B=K/\left(CD\right),$$ (18) $$C=p_{\text{C1}},$$ (19) $$D=\text{μ}{F}_{\text{z}},$$ (20) $$E=\left(p_{\text{E1}}+p_{\text{E2}}\text{d}{F}_{\text{z}}+p_{\text{E3}}\text{d}{F}_{\text{z}}^2\right)\left\{1-p_{\text{E4}}\text{sgn}\left(\text{κ}\right)\right\},$$the longitudinal coefficient of friction (21) $$\text{μ}=\left(p_{\text{D1}}+p_{\text{D2}}\text{d}{F}_{\text{z}}\right)\left(1+p_{\text{p3}}\text{d}{p}_{\text{i}}+p_{\text{p4}}\text{d}{p}_{\text{i}}^2\right)\left(1-p_{\text{D3}}{\text{γ}}^2\right),$$and the longitudinal slip stiffness (22) $$K=F_{\text{z}}\left(p_{\text{K1}}+p_{\text{K2}}\text{d}{F}_{\text{z}}\right){\text{e}}^{p_{\text{K3}}\text{d}{F}_{\text{z}}}\left(1+p_{\text{p1}}\text{d}{p}_{\text{i}}+p_{\text{p1}}\text{d}{p}_{\text{i}}^2\right),$$results from base terms that are active all the time and terms that describe the behavior under non-nominal conditions [22]. Many terms only become active when there is a deviation $\text{d}{F}_{\text{z}}$ of the current load from the nominal load, a deviation $\text{d}{p}_{\text{i}}$ of the current pressure from the nominal pressure, or when the tire’s camber angle γ is nonzero. Therefore, the basic parameters $p_{\text{C1}},p_{\text{D1}},p_{\text{E1}}$, and $p_{\text{K1}}$ for the tire’s behavior at nominal load, nominal pressure, and zero camber can be fitted first and independently of the other parameters, using respective test rig data generated in the nominal state. Then, data gathered at different loads are used to fit the load dependency of the force, and experiments with different cambers are used to fit the camber dependency. Parameters for the pressure dependency are set to zero in our case since the tire model is used only at nominal pressure. Additionally, parameters for asymmetric tire behavior, such as $p_{\text{Ex4}}$, are set to zero since the considered tire has a symmetric geometry and behavior. The parameters for the remaining forces and moments are fitted in an equally stepwise fashion.

Fitting of Neural Network

The fitted PAC2002 tire model works with a point contact and therefore does not generate any distributed contact results that could be used to directly calculate frictional power distributions. Hence, a separate friction model is needed to yield the friction results needed for prediction of tire wear.

To determine which variables can be used as input to the friction model, a look at the final virtual test drive is needed. To yield the exact same driving situations independent of the current vehicle setup, the virtual test drive prescribes a target vehicle speed, and therefore longitudinal acceleration, as well as a lateral acceleration over time. Adams Car’s Driving Machine then controls the vehicle’s steering, braking, and accelerator inputs to yield the tire forces needed to follow the given acceleration trajectories. Therefore, the sum of the tire forces, that is, the resulting force acting on the vehicle, is indirectly controlled and given by the virtual test drive, and will stay the same regardless of the current vehicle setup. Through the suspension setup, however, how these forces are generated by the tires can be affected. Hence, to make the friction model completely independent of the potential virtual test drive, but make it react to changes in the suspension setup, the model is trained to reproduce the relationship between the tire’s kinematic quantities K and the frictional power distributions ${\mathit{P}}_{\text{f},\text{str}}$. Only the tire deflection δ is replaced by the normal force $F_{\text{z}}$ in the set of kinematic quantities, so that the input for the friction model becomes $\mathit{X}=\left[\text{κ,α,γ,ω},F_{\text{z}}\right]$. For coherence with the nomenclature in popular machine learning frameworks, $\mathit{Y}={\mathit{P}}_{\text{f},\text{str}}$ is used for the model’s output in the following.

For training of the friction model, each of the input variables is independently sampled from a respective uniform distribution U, that is, $\text{κ}\sim U(-5\%,5\%),\text{α}\sim U(-5°,5°),\text{γ}\sim U(-2°,2°),\text{ω}\sim U(5\text{rad}{\text{s}}^{-\text{1}},\text{120}\text{rad}{\text{s}}^{-1})$, and $F_{\text{z}}\sim U(3500\text{n},5500\text{N})$, corresponding to the estimated operating range of the tire. This is done for 2500 simulations with FTire on the tire test rig in which the sampled tire states are held constant. The simulations run for $t_{\text{sim}}=2\text{s}$ each. Because of an initial settling of the tire at the beginning of the simulations, only the results of the last second are used as training data. The tire states and the frictional power distributions are averaged over this second, yielding the time-averaged quantities $\overline{\mathit{X}}=\left[\overline{\text{κ}},\overline{\text{α}},\overline{\text{γ}},\overline{\text{ω}},\overline{F_{\text{z}}}\right]$ and $\overline{\mathit{Y}}=\left[{\overline{P}}_{\text{f},\text{str},1},\mathrm{...},{\overline{P}}_{\text{f},\text{str},n}\right]$. This results in 2500 pairs of $\overline{\mathit{X}}$ and $\overline{\mathit{Y}}$ that can be used for training; see Fig. 7. Because of the large number of samples, they cover the relatively big range of simulated tire states evenly and densely despite the random sampling method. Since separate simulations are run on the virtual tire test rig, generation of the training data can be parallelized and takes approximately 20 min using all six cores of an Intel® Core™ i7-8700 CPU.

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It should be noted, however, that the frictional powers span multiple orders of magnitude. The frictional powers occurring at tread strip 11 in the middle of the tire, which are shown in Fig. 7, reach from ${\overline{P}}_{\text{f},\text{str},11}<1W$ around $\text{κ}=0\%$ and $\text{α}=0°$ to more than 1000 W for situations with high longitudinal slip and side slip. This can pose a problem for machine learning methods that tend to be better at predicting targets in similar orders of magnitude.

Therefore, the frictional powers in $\overline{\mathit{Y}}$ are transformed using the transformation rule (23) $$y*=\text{θ}(y)=\sqrt{y},$$

which condenses them into a smaller range. The inverse transformation (24) $$y={\text{θ}}^{-1}(\mathrm{y*})={\mathrm{y*}}^2$$is defined to transform the predictions of the friction model back into the original scale later. The quantities in the input vector $\overline{\mathit{X}}$ are transformed as well. They are passed through scikit-learn’s standard scaler [34] (25) $$x^{*}=s(x)=\frac{x-\overline{x}}{{\text{σ}}_x^2},$$where $\overline{x}$ is the respective variable’s mean over all samples and ${\text{σ}}_{\text{x}}^2$ is the respective variable’s variance over all samples. Hence, all input variables are independently scaled to zero mean and unit variance.

Because of the vector shape of Y, the required model is a multiple-input–multiple-output regression model. Since the frictional powers at neighboring tread strips have a strong correlation to each other, a model that naturally reproduces such a relationship is found to be a reasonable approach. Therefore, a fully connected ANN in feed-forward configuration is selected. It can produce multiple outputs based on multiple inputs, and its outputs are related through the common hidden layer before the output layer.

For the problem at hand, neural networks with one to eight hidden layers and 100 to 800 neurons per layer in steps of 100 neurons are set up and trained using Keras [35] with TensorFlow [36], resulting in 64 training runs. Additionally, there is an input layer with five neurons matching the size of the input vector X and an output layer with 22 neurons to output a prediction for the frictional power distribution Y, matching the number of tread strips n of the tire model. Since only positive friction powers can occur, the output neurons’ activation functions are chosen to be rectifying linear units. All other activation functions are set to tangens hyperbolicus. Before the input layer, the input vector is passed through the standard scaler s(x), and after the output layer, the predictions are passed through the back transformation ${\text{θ}}^{-1}(y)$; see Fig. 8.

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As a loss function for training of the neural network, the mean squared error (MSE) (26) $${\mathrm{ϵ}}_{\text{MSE}}=\frac{1}{mn}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{s=1}^n{\left({\widehat{P}}_{\text{f},\text{str},s,i}-{\tilde{P}}_{\text{f},\text{str},s,i}\right)}^2}}$$between the targets ${\widehat{\mathit{Y}}}_i=\left[{\widehat{P}}_{\text{f},\text{str},1,i},\mathrm{...},{\widehat{P}}_{\text{f},\text{str},n,i}\right]$ given by the training data and the predictions ${\tilde{\mathit{Y}}}_i=\left[{\tilde{P}}_{\text{f},\text{str},1,i},\mathrm{...},{\tilde{P}}_{\text{f},\text{str},n,i}\right]$ will be minimized. For evaluation of the model during training, 15% of the data are used as a validation data set. The error ${\mathrm{ϵ}}_{\text{MSE}}$ is calculated using the validation data after each training epoch. Under these conditions, the model containing six hidden layers with 100 neurons per layer and 53 322 trainable parameters in total performs best, producing a validation loss of ${\mathrm{ϵ}}_{\text{MSE}}=0.7{\text{W}}^2$.

Inference

For application of the surrogate model to predict frictional power distributions and tire wear, the original FTire tire model is replaced by the generated PAC2002 tire model. Then, arbitrary experiments can be conducted with the adapted model; see Fig. 9.

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With the PAC2002 model, the virtual experiments still yield the input vectors ${\mathit{X}}_i$ with the tire states at every output time step t_i. They are fed into the friction model to predict the frictional power distributions ${\tilde{\mathit{Y}}}_i=\left[{\tilde{P}}_{\text{f},\text{str},1,i},\mathrm{...},{\tilde{P}}_{\text{f},\text{str},n,i}\right]$. These can then be summed and multiplied by the time step size to get the distribution of the friction work transmitted between tire and road, as presented in Eq. (6).

Validation

The two parts of the surrogate model can be validated separately and in combination. This gives three validation steps: validation of the generated PAC2002 tire model, validation of the friction model, and validation of the full surrogate consisting of both model parts. The validation is done for an FTire model that was fitted to a Michelin Pilot Sport 3 with dimensions 245/40 R18, load index 97Y, and rim size 8J at a tire pressure of $p_{\text{i}}=2.70\text{bar}$. The corresponding PAC2002 model and the AI friction model are generated using the described methods.

Tire Model Isolated

To validate the generated PAC2002 model, it is used on the virtual tire test rig in the same situations as were run to generate its training data. This allows a validation of its capabilities in very distinct driving situations. Since kinematic quantities such as longitudinal slip κ or slip angle α are prescribed during these experiments, the focus of the evaluation is to compare the forces generated by the PAC2002 model with the forces generated by the FTire model for given tire kinematics. To illustrate, the results for three experiments will be shown here: First, an experiment with pure longitudinal slip is evaluated. Then, we will have a look at the tire reactions under pure side slip. Finally, a combined slip experiment will be evaluated. For usage in a representative test drive, matching force and moment characteristics are important for small amounts of slip in particular. Therefore, the focus will be put on longitudinal slip $\left|\text{κ}\right|<2.5\%$ and slip angles $\left|\text{α}\right|<{2.5}^{°}$, although virtual tests were run through a wider slip range. The camber angle is set to $\text{γ}=0^{°}$ to evaluate the fitting results in a basic situation.

The relevant tire responses under pure longitudinal slip are the longitudinal force $F_{\text{x}}$ and rolling resistance $M_{\text{y}}$. The fitted PAC2002 model matches the results of the FTire model over a wide slip range; see Fig. 10. Only at longitudinal slips close to $\left|\text{κ}\right|=2.5\%$ do the longitudinal forces produced by the PAC2002 model slightly deviate from those generated by FTire. Rolling resistance $M_{\text{y}}$ follows the training data closely but its absolute values are slightly underestimated over the whole slip range. This could lead to lower longitudinal slip throughout the simulation of a virtual test drive with the generated PAC2002 as compared with a simulation with the original FTire model, since less rolling resistance has to be compensated.

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Under pure side slip, lateral force $F_{\text{y}}$, overturning moment $M_{\text{x}}$, and aligning moment $M_{\text{y}}$ are relevant responses. Here again, the fitted PAC2002 model matches the results of the FTire model over a wide slip range; see Fig. 11. The fit of $F_{\text{y}}$ is almost perfect around $\text{α}=0^{°}$, but a small discrepancy becomes apparent for increasing absolute slip angles. The overturning moment $M_{\text{x}}$ is perfectly matched, but the aligning moment $M_{\text{z}}$ shows significant deviations at slip angles close to $\left|\text{α}\right|={2.5}^{°}$. In summary, slightly higher slip angles are expected in the virtual test drive with the generated PAC2002 model than in the test drive with the FTire-based model.

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At combined slip, the force and moment characteristics again match the curves produced by FTire; see Fig. 12. The reduction of lateral force at high longitudinal slips is well reproduced by the PAC2002 model. However, it becomes clearer that there is a discrepancy between the lateral forces generated by FTire and the fitted PAC2002 model at $\text{κ}=0\%$, that is, pure side slip. The PAC2002 model underestimates the absolute value of the lateral force, which should lead to higher slip angles in driving situations with high lateral accelerations. This in turn could lead to an overestimation of the generated tire wear. The moment characteristics match the original tire model well, but as before, the absolute values of $M_{\text{y}}$ are slightly underestimated. Conversely, the aligning moment $M_{\text{z}}$ is slightly overestimated but closely follows the shape as the target curve.

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Neural Network—Isolated

The accuracy of the fitted friction model can be tested independently of the fitted tire model when the tire states produced by the original FTire model are used as an input. The neural network’s output can then be compared with the frictional power distributions and friction work distributions predicted by post-processing of FTire’s contact results. We use the full virtual test drive from [18] for validation here; see Fig. 13. It begins on city roads with speeds up to $v_{\text{lon}}\approx 15\text{m}{\text{s}}^{-\text{1}}$. In the middle, it continues on rural roads with speeds up to $v_{\text{lon}}\approx 25\text{m}{\text{s}}^{-\text{1}}$. The final section is a motorway with speeds up to $v_{\text{lon}}\approx 35\text{\hspace{0.17em}m}{\text{s}}^{-\text{1}}$. Lateral accelerations stay mostly under $\left|a_{\text{lat}}\right|<2.5\text{m}{\text{s}}^{-\text{2}}$. In total, the virtual test drive covers a distance of 103 km in a simulated time of 110 min. Since the neural network of the friction model is trained with data generated on the virtual tire test rig, the training is independent of the test data and the neural network can be used for other test drives as well. As a vehicle model, Adams Car’s MDI Demo Vehicle is used. Its tires are replaced by the FTire model mentioned above. During the simulation, the longitudinal and lateral accelerations prescribed by the test drive are actively controlled to ensure that the vehicle always follows the same route, regardless of the tire model used. Output is generated and validation is done for the rear left tire.

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The friction work distribution at the end of the test drive predicted by the AI model closely matches the distribution generated by FTire; see Fig. 14. This can be quantified using the coefficient of determination R² for friction work across tread strips (27) $$R_{\text{Wf},\text{str}}^2=1-\frac{{\displaystyle \sum _{s=1}^{n_{\text{str}}}{\left({\widehat{W}}_{\text{f},\text{str},s}-{\tilde{W}}_{\text{f},\text{str},s}\right)}^2}}{{\displaystyle \sum _{s=1}^{n_{\text{str}}}{\left({\widehat{W}}_{\text{f},\text{str},s}-{\overline{W}}_{\text{f},\text{str}}\right)}^2}},$$where ${\widehat{W}}_{\text{f},\text{str},s}$ represents the friction work distribution predicted by FTire’s contact output, ${\tilde{W}}_{\text{f},\text{str},s}$ is the AI model’s prediction, and ${\overline{W}}_{\text{f},\text{str}}$ is the mean friction work across all tread strips in FTire’s prediction. The R² metric assesses how well the AI model captures the overall variance and distribution generated by FTire. By looking at the distribution across tread strips rather than an absolute error alone, R² offers a possibility to evaluate the AI model’s capability to reproduce FTire’s friction work pattern, which is critical for a suspension optimization towards evenly distributed wear.

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The resulting value of $R^2=0.9279$ indicates a high degree of similarity in shape between the AI model’s predicted distribution and FTire’s output. In particular, the steep incline of the friction work at the outer tread strips (numbers 1 to 5 and 16 to 22) is captured well. Just the region of lower friction work in the middle of the tire is slightly overestimated and shows unnatural kinks. A possible reason for this is that the training data generated with the tire test rig still contain frictional powers over multiple orders of magnitude, despite the transformations applied. The model therefore gets too insensitive for the influence of low variations in slip and camber on the frictional power distribution. During the virtual test drive, both slip angle and camber angle just slightly vary around their static setups of $\text{α}\approx 1°$ and $\text{γ}\approx -1°$.

Additionally, the relative deviation of the total friction work predicted by the trained friction model in comparison with FTire’s output (28) $${\mathrm{ϵ}}_{\text{Wf},\text{tot}}=\frac{{\tilde{W}}_{\text{f},\text{tot}}-{\widehat{W}}_{\text{f},\text{tot}}}{{\widehat{W}}_{\text{f},\text{tot}}}$$is evaluated. This metric quantifies the deviation in the area under the friction work curves, reflecting the resulting total tire wear when using a linear wear law. With a deviation of only ${\mathrm{ϵ}}_{\text{Wf},\text{tot}}=4.7\%$, the model demonstrates high accuracy in predicting overall tire wear. The overestimation in the middle of the tire is compensated by slightly underestimated friction work on the left side of the tire.

To make sure the neural network does not just output an average frictional power distribution during the whole virtual test drive, the accumulated friction work (29) $$W_{\text{f},\text{acc},\text{k}}={\displaystyle \sum _{i=0}^k{P}_{\text{f},\text{tot}}},i\Delta t$$is also evaluated over time; see Fig. 15. During the city section, the AI friction model continuously underestimates the total frictional power, leading to a deviation from the accumulated friction work generated by FTire of ${\mathrm{ϵ}}_{\text{Wf},\text{acc}}=-13.7\%$ at the transition from city to rural road. On the other hand, the friction model overestimates the friction power on rural roads and motorways, compensating the city section and leading to a deviation of ${\mathrm{ϵ}}_{\text{Wf},\text{acc}}=3.0\%$ at the transition from rural road to motorway and a total deviation of ${\mathrm{ϵ}}_{\text{Wf},\text{tot}}=4.7\%$ at the end of the test drive, as mentioned above. This emphasizes again that the friction model is not perfectly able to reproduce the fine nuances between driving situations with low frictional power and high frictional power, but tends to swing to either side.

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Surrogate Model—Combination

Finally, the combination of the fitted PAC2002 model with the AI friction model is compared with the results of the original structural tire model. Therefore, the vehicle model’s tires are replaced by the fitted PAC2002 model and the simulation results are post-processed with the AI friction model. In this combination, the errors of both parts of the surrogate model add up. The surrogate model still reproduces the general shape of the friction work distribution generated by the original FTire model; see Fig. 16. However, it overestimates the friction work to the left and in the center of the tire (tread strips 4 to 13), and slightly underestimates the friction work to the right of this region. This flips the inaccuracies resulting from the friction model alone (as shown in Fig. 14) from left to right and could be a result from too-low lateral forces generated by the PAC2002 model and hence higher required slip angles to produce the same lateral acceleration, as observed in Fig. 12. The resulting coefficient of determination is $R^2=0.8347$ and the relative deviation of the predicted total friction work from the results generated by FTire is ${\mathrm{ϵ}}_{\text{Wf},\text{tot}}=6.1\%$. This is not as accurate as the AI predictions based on the inputs from FTire, but shows that the combined model is able to predict the general wear pattern and overall tire wear.

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The calculation time for the virtual test drive is reduced from $t_{\text{CPU}}\approx 2000\text{min}$ with the original model to $t_{\text{CPU}}\approx 468\text{min}$ when using the surrogate model, resulting in a real-time factor of $r_{\text{RTF}}\approx 5$ and a reduction of calculation time by more than 75%, saving more than 24 hours. This more than compensates the initial effort of approximately 1 hour for generating the surrogate model after a single virtual test drive.