conditions. For hybrid PINNs, both data loss and physics-informed loss are considered. Using automatic differentiation (AD), the output gradient with respect to the specific input is computed and passed to the physics-informed loss. The different losses are scaled, summed, and used to update the neural network’s parameters. This process is referred to as training. PINNs have already been successfully applied to various areas of tribology, such as lubricant prediction and wear and damage prediction / Mar19/ , / Mar23/ . Compared to model order reduction (MOR) techniques for EHL simulations / Mai15/ , PINNs offer several advantages. They provide a flexible structure, allowing for the quick addition of parameters and modification of the network architecture. Furthermore, PINNs can be applied to hyperbolic problems with discontinuities, such as cavitation problems involving pressure values of zero. These scenarios are challenging for MOR techniques like the Reduced Basis method. Another benefit of Science and Research 26 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 Introduction The tribological behavior of components significantly impacts the efficiency and lifetime of technical systems. Elastohydrodynamic lubrication (EHL) simulation models offer a feasible approach to characterizing these interfaces compared to time-consuming and costly experimental investigations. These models compute the friction in tribological contacts, such as seals and journal bearings, by solving hydrodynamics within lubricated contacts, deformation, and contact mechanics. One main drawback of EHL models is their high computational cost. Purely data-driven machine learning algorithms can accelerate computations but often suffer from their black-box behavior. As hybrid solvers, physics-informed neural networks (PINNs) combine datadriven and physics-based methods to solve partial differential equations (PDEs) / Rai19/ . Thus, PINNs can accurately compute the pressure build-up and cavitation in sealing contacts and journal bearings determined by variations of the Reynolds equation. Figure 1 illustrates a schematic PINN with two inputs x and t and one output u. The inputs are passed through the neural network to determine the output. Classical neural networks use databased loss functions, whereas PINNs incorporate physics-informed losses. These losses include the governing equations of the investigated system, specifically, the averaged Reynolds equation and the Fischer-Burmeister equation in this paper, along with boundary and initial Physics-Informed Deep Learning for Lubricated Contacts with Surface Roughness as Parameter Faras Brumand-Poor, Michael Rom, Nils Plückhahn, Katharina Schmitz* Submitted: 30.08.2024 accepted: 19.11.2024 (peer review) Presented at the GfT Conference 2024 Physics-informed neural networks (PINNs) are developed to solve variants of averaged Reynolds equations for accurately and time-efficiently modeling pressure build-up and cavitation in sealing contacts and journal bearings with rough surfaces. We use microscale coefficients provided through Patir and Cheng’s average flow model or homogenization to integrate roughness or texture height into these macroscale equations. Based on these equations we implement parameter-dependent PINNs to solve multi-case scenarios with varying roughness or texture heights, thus investigating the adaptability and generalizability of PINNs for modeling rough lubricated interfaces. The results demonstrate the promising potential of PINNs to accelerate tribological system computations. Keywords Physics-Informed Machine Learning; Hydrodynamics; Reciprocating Seals; Journal Bearings; Surface Roughness; Flow Factors; Homogenization Abstract * Faras Brumand-Poor Orcid-ID: https: / / orcid.org/ 0009-0006-7442-8706 Dr. rer. nat. Michael Rom Orcid-ID: https: / / orcid.org/ 0000-0002-2963-9081 Nils Plückhahn Orcid-ID: https: / / orcid.org/ 0009-0000-0789-6194 Univ.-Prof. Dr.-Ing. Katharina Schmitz Orcid-ID: https: / / orcid.org/ 0000-0002-1454-8267 RWTH Aachen University Institute for Fluid Power Drives and Systems (ifas) Campus-Boulevard 30, 52074 Aachen, Germany RWTH Aachen University Institut für Geometrie und Praktische Mathematik Templergraben 55, 52056 Aachen, Germany PINNs is their fast computation after the initial training phase. In this work, the PINNs could compute pressure and cavitation for one scenario in less than 0.01 s. Almqvist was the first to implement a PINN to determine the pressure build-up described by a simplified version of the 1D Reynolds equation / Alm21/ . Further research was conducted by Li et al., Yadav et al., and Zhao et al. for the 2D Reynolds equation applied to gas bearings, journal bearings, and linear sliders, respectively, see / Li22/ , / Yad22/ , / Zha23/ . The first successful application of PINNs for solving the Reynolds equation with Jakobsson-Floberg-Olsson (JFO) cavitation modeling was done by Rom by integrating soft constraints and collocation point adaptation to accurately model cavitation and areas with high gradients / Rom23/ . Cheng et al. implemented PINNs to solve the JFO and Swift-Stieber (SS) cavitation models / Che23/ . In recent studies, Xi et al. improved the accuracy of the PINN for solving the Reynolds equation by integrating hard and soft constraints in the training process / Xi24/ . Regarding 1D sealing gaps, Brumand-Poor et al. implemented a hydrodynamic framework to solve the stationary Reynolds equation for scenarios with and without cavitation and considering interpolation and extrapolation problems / Bru24a/ , / Bru24b/ , / Bru24c/ , / Bru24d/ . Research on PINNs for solving a complete EHL model was conducted by Rimon et al., who solved a simplified version of the stationary Reynolds equation without cavitation in combination with the Lamé equation to describe seal deformation / Rim23/ . This contribution applies a PINN framework to solve averaged Reynolds equations for rough or textured surfaces for two scenarios (1D sealing gap and 2D journal bearing). Roughness or texture effects are incorporated into these macroscale equations via coefficients determined on the microscale through homogenization or Patir and Cheng’s average flow model. Multi-case scenarios are solved by parameter-dependent PINNs, simultaneously considering several roughness or texture heights, i.e., several flow factors or homogenization coefficients. The PINNs are validated against finite difference, Dynamic Description of Sealings (DDS), and finite element solutions, demonstrating their capability for computationally efficient modeling of tribological interfaces and their adaptability to new test cases not used during training and, therefore, their generalizability. Averaged Reynolds Equations for Hydrodynamic Lubrication with Cavitation Both Patir and Cheng’s average flow model and homogenization aim to reduce the computational cost of solving the Reynolds equation for rough or textured surfaces. This is done by incorporating the microscale effects of the roughness and textures in small reference problems, which can be solved efficiently to determine shear and pressure flow factors Φ τ and Φ p , determined by the standard deviation of the roughness σ / Pat78/ or, similarly, coefficients a 11 , a 12 , a 21 , a 22 , b 1 and b 2 / Rom21/ . These enter an averaged macroscale equation, for Patir and Cheng’s average flow model in 1D given by (1) or derived by homogenization in 2D resulting in (2) The first equation contains the velocity of the counter surface v, the gap height h, and the dynamic viscosity of the lubricant η. This equation is utilized for the first scenario, the 1D sealing gap. In the second equation, which is used for the second scenario investigating the journal bearing, these values are incorporated into the coefficients a 11 , …, b 2 . Both equations are solved for the averaged pressure p and cavity fraction θ. This requires satisfying the constraints p ≥ 0 and 0 ≤ θ ≤ 1 (θ = 0 if p > 0 and θ > 0 if p = 0). Hence, the Fischer-Burmeister equation Science and Research 27 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 Figure 1: Illustration of a PINN. / Bru24a/ A divergent gap with cavitation is investigated for a range of σ between 0.5 and 1, resulting in a h/ σ range displayed in Figure 2, which spans from 0.5 to 2.0. Noteworthy is that a value less than 3.0 is considered to describe a rough surface. In this case, a purely physics-based PINN is trained; therefore, no simulated or measured data are provided to the PINN to solve the pressure build-up and cavitation for several rough surfaces. In Figure 3 the pressure and shear flow factors for different σ are illustrated over the investigated position of Science and Research 28 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 (3) must be satisfied. Both geometries are investigated for stationary rough or textured surfaces, thus the stationary averaged Reynolds equations are implemented. PINNs for Solving the Averaged Reynolds Equation Table 1 shows the first scenario dealing with the 1D gap and its parameters. Note that all quantities are nondimensional. Table 1: The investigated scenario for the first test case Scenario ! "# $ Boundary Conditions Gap Geometry Stationary Cavitation %# % & '()*#+,-.* ! / # ! 0 '()*#+,-.* ! # ! % 1 ! % Figure 3: Pressure and shear flow factors for the investigated divergent gap for different values of sigma Figure 2: Shear and pressure flow factors for the investigated sealing gap over the height divided by the roughness the divergent gap. The pressure flow factors behave similarly for the different values of the roughness. However, the shear flow factors exhibit different trends over the gap geometry depending on the value of σ. In Table 2 the parameters for the training of the PINN solving the averaged Reynolds equation for the 1D gap are illustrated. Prior work / Bru24b/ , / Bru24d/ provides a detailed training process procedure. The hyperparameters (e.g. layer size, layer width, initial learning rate) are determined by a Bayesian tuner for 30 trials. The tuner selects a set of hyperparameters and trains a PINN for 25,000 epochs. Afterward, the PINN with the best results is trained for 80,000 epochs. In each epoch, the PINN is trained on 200 different roughness values with σ varying from 0.5 to 1. The best performing PINN is implemented with 7 layers with 32 neurons per layer and the following activation functions [2x elu, gelu, tanh, swish, softplus, sigmoid] are utilized / Ten15/ . Depending on the activation function, either the Glorot / Glo10/ or the He / He15/ initialization is used and for the network optimization the ADAM optimizer is implemented / Kin15/ . The PINN is trained with the residual loss of the averaged Reynolds equation and the Fischer-Burmeister equation, the boundary conditions for the cavity fraction and the pressure and the soft constraints. The soft constraints are activated if the PINN computes pressure and cavitation at the same position and if these values exceed the threshold, which is a similar approach to prior research using the soft constraints / Rom23/ , / Bru24c/ . For each loss the meansquared error is computed, and the different losses are balanced during the training based on the Relative Loss Balancing with Random Lookback / Bis22/ . After 15,000 epochs extra collocation points are added to areas with high gradients of θ. The second scenario is a journal bearing with diameter d = 25.5 mm and width w = 20 mm. The bearing is textured with 160 × 40 diagonally arranged textures as described in / Rom21/ . The computational domain can be decomposed into 160 × 40 elements of size 500 μm × 500 μm, where each element is designed as depicted in Figure 4. The hatched area illustrates the texture. Further parameters are lubricant density ρ = 820 kg/ m 3 , dynamic viscosity η = 0.014 Pa s, radial clearance c = 17.5 μm, relative eccentricity e rel = 0.8 and surface velocity (shaft) v = 0.2 m/ s. The bearing is unfolded such that computations can be conducted in a Cartesian domain. At the sides of the bearing, i.e., y = 0 and y = w, the pressure is 0.1 MPa. Due to the nonsymmetric dia- Science and Research 29 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 Table 2: Parameters set in the training procedure for the 1D gap Parameter Symbol Value Collocation Points 2 3 4! ! Pressure Boundaries & '()*#+,-.* ! / # ! 0 Cavitation Boundaries '()*#+,-.* ! # ! Pressure Threshold & *.+(5 ! ! % Cavitation Threshold *.+(5 ! ! 0 Added Coll. Pts. 2 6789: : (: %; Dynamic Viscosity < % Standard Deviation of Roughness = ! ! ; # % Figure 4: Texture element for journal bearing case gonal textures, Patir and Cheng’s average flow model is not applicable, and the homogenized Reynolds equation (2) is used instead. The gap height (without textures) is given by (4) The inputs of the parameter-dependent PINN are the coordinates x and y and the coefficients a 11 , a 12 , a 21 , a 22 , b 1 and b 2 . The PINN is trained for three different settings regarding the texture height h t , namely h t = 10 μm, h t = 15 μm and h t = 20 μm, which leads to different sets # > ? @? A BCD E PINN and choose the mean-squared error (MSE) such that the general form of the loss functions is given by (5) Here, γ is a place holder for p, θ, ∂p ⁄ ∂x, ∂p ⁄ ∂y or f, where f is the PDE residual according to (2), and α γ denotes respective parameters for loss balancing. Hence, apart from the error in the PDE residual f, we compute errors of the PINN w.r.t. the corresponding finite element solutions for the pressure, the cavity fraction and the first partial derivatives of the pressure. The PDE residual is defined by (6) Note that due to the gap height function being only dependent on x and not on y, see Equation (4), the coeffi- F G H I G J K G LMJJ N # N G OPQ N # N J NR Science and Research 30 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 of homogenization coefficients. In contrast to the sealing gap scenario, simulation data from an in-house finite element solver are used for the training of the PINN in addition to the PDE. After training, the PINN is applied to a new case with h t = 12 μm to test its generalizability. For the neural network within the PINN, we choose a standard network with eight hidden layers and 20 neurons per layer. As extensive tests have shown, this provides a network being deep and wide enough in our setting. The finite element solutions for the texture heights h t = {10,15,20} μm consist of 161 × 41 data points each such that the training data set for the PINN in total has 19,803 entries. Periodicity at the boundaries x = 0 and x = πd is enforced by applying Fourier feature embedding / Don21/ . To avoid vanishing or exploding gradients while training, the inputs of the PINN are scaled to the interval [-1,1] for each input quantity individually. The weights and biases are initialized using Glorot initialization / Glo10/ . We use the hyperbolic tangent as activation function and the L-BFGS algorithm / Liu89/ as optimizer. Overall, we set up five loss functions to train the Figure 6: Pressure and cavitation (left) and the film height (right) for the gap with σ = 0.65 and 0.85 Figure 5: Pressure and cavitation (left) and the film height (right) for the gap with σ = 0.5 and 1 S # H # # # # # # # # cients a 11 , …, b 2 are also only dependent on x / Rom21/ . Hence, ∂a 21 ⁄ ∂y = 0, ∂a 22 ⁄ ∂y = 0 and ∂b 2 ⁄ ∂y = 0. The partial derivatives of the pressure and the cavity fraction are computed by automatic differentiation in case of the PINN and by finite differences in case of the FEM solution. The values for the parameters α γ are automatically set in the first training epoch so that all five losses have the same value at the beginning of the training. The PINN is trained for 30,000 epochs. Results In Figure 5, the results of the PINN for the sealing gap are shown for σ = 0.5 and 1. The results show good agreement regarding the location of pressure and cavitation areas, trends and values. The cavitation can be further investigated through the film height h lubricant of the lubricant, which can be directly computed through the determined cavity fraction θ and the sealing gap h by h lubricant = h · (1 - θ). The film height of PINN and the numerical solver also show good agreement, underlining the capability of the PINN to solve the problem for different roughnesses. In Figure 6 two more results are displayed for σ = 0.65 and 0.85, which are inside the range of the trained roughness. The results yield good agreement for pressure and the cavitation, i.e. film height for the different σ. Small deviations can be observed around high gradients for θ, which can be further improved by tuning the soft constraints. The results of the PINN for the journal bearing test case with h t = 12 μm are displayed in Figure 7 (top left). They are compared with the corresponding reference solution computed using an in-house finite element solver (bottom left). The plots combine pressure and cavitation, with separation at x = 0.043 m (vertical white line in Figure 7). A difference between the solutions is barely noticeable. The right-hand side of Figure 7 shows the absolute deviations between the PINN and the finite Science and Research 31 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 Figure 7: Combined pressure and cavitation solutions of PINN (top left) and finite element solver (bottom left) and deviation between solutions (pressure top right, cavity fraction bottom right) Figure 8: PINN and finite element solutions on the centerline y = 0.01 m element solution for the pressure (top) and the cavity fraction (bottom). With values up to 0.002 MPa, the absolute error in the PINN prediction is small everywhere. The same holds for the cavity fraction apart from the area of the transition from the cavitated region to the full-film region. There, the cavity fraction jumps from a value greater than zero to zero. Resolving this jump is difficult for any solver and explains the large deviation of the PINN solution from the FEM solution. Figure 8 shows the solutions on the centerline y = 0.01 m. Only by examining close-ups of the pressure and cavity fraction maxima slight deviations become visible. The relative errors of the PINN solution w.r.t. the reference solution are 0.4 % and 1.0 % for the pressure maximum and the cavity fraction maximum, respectively. multi-case scenarios with varying roughness or texture height. Two variants of the averaged Reynolds equation are solved for a sealing gap and a journal bearing and compared with finite difference and finite element solutions: the pressure distribution and cavitation match for both scenarios with the values computed with classical solvers. The PINN results are provided in a fraction of a second, highlighting the accelerated computation ability of PINNs for tribological interfaces. Furthermore, the different geometries, roughness, and texture heights illustrate the adaptability and generalizability of PINNs for modeling lubricated interfaces. Based on these results, integrating deformation to obtain a solution for the entire EHL simulation will be the subject of future investigations. Research conducted by Nguyen-Thanh et al. / Ngu20/ , / Ngu24/ and Abueidda / Abu21/ successfully applied PINNs to solve hyperelastic and plastic deformation, respectively. Based on these promising results, a complete physics-informed EHL simulation can be implemented, with one PINN computing the hydrodynamic pressure and the second PINN calculating the deformation accordingly. These two PINNs could be integrated into a classical fluid-structure-interaction framework, replacing the classical solvers / Dak20/ . The fluid-structure interaction would greatly benefit from the fast computation of the PINNs since the deformation and pressure must be computed several times to find a suitable solution. Parameter-dependent PINNs can also be developed to solve the reference problems on the microscale to compute the flow factors or homogenization coefficients. Only the gap height on the macroscale enters these problems as parameter. Overall, this would lead to an approach combining two consecutive PINNs, where the first one on the microscale provides the coefficients and, therefore, the inputs of the second one on the macroscale. Science and Research 32 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021 Important performance parameters of a journal bearing are the maximum pressure p max , the load-carrying capacity W and the friction force F. The latter two are given by (7) Table 3 compares the values computed from the PINN solution with those computed from the FEM solution. The maximum relative error of 0.36 % (p max ) again demonstrates the excellent accuracy of the PINN solution. In Figure 9, we compare the PINN solution for the pressure (left) for the whole domain with the solution of a standard artificial neural network (ANN) (right), i.e., a network which is trained with data only and not with any additional physics (PDE) information. The setup of the ANN regarding hyperparameters, initialization, optimizer, etc. is the same as the setup of the PINN. Analogously to the PINN, the ANN is trained with the texture heights h t = {10,15,20} μm and then applied to the test case h t = 12 μm. The solutions are similar, but the ANN solution exhibits oscillations, e.g., around x = 0.015 m. This demonstrates that adding physics information is clearly beneficial to obtain an accurate solution, in particular when it comes to generalizability of a trained network. Conclusion The results presented in this contribution demonstrate the capability of parameter-dependent PINNs to solve T U # E E # Table 3: Performance parameters computed from PINN and FEM solution Parameter PINN FEM Rel. error & W 9 3 [MPa] ! X Y Y % ! X Y X X ! Z X [ \ [N] 0 0 ] ! Y 0 0 / 0 Y ! Z ; [ ^ [N] ! 4 / X ; ] ! 4 / X ] / ! ! X [ Figure 9: Pressure solution of PINN (left) and standard artificial neural network (right) O U # # V # E E Acknowledgement The authors thank the Research Association for Fluid Power of the German Engineering Federation VDMA for its financial support. (grant: FKM No. 7058400). Funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy - EXC-2023 Internet of Production - 390621612. Literature / Rai19/ Raissi, M., et al. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, Vol. 378, Nr. 1, S. 686-707, 2019. / Mar19/ Marian, M., et al. Current Trends and Applications of Machine Learning in Tribology - A Review, Lubricants, Vol. 9, 2019. / Mar23/ Marian, M., et al. Physics-Informed Machine Learning - An Emerging Trend in Tribology, Lubricants, Vol. 11, 2023. / Mai15/ Maier, D., On the Use of Model Order Reduction Techniques for the Elastohydrodynamic Contact Problem, KIT Scientific Publishing, 2015. / Bru24a/ Brumand-Poor, F., et al. Physics-Informed Neural Networks for the Reynolds Equation with Transient Cavitation Modeling, MDPI Lubricants, Vol. 12, 2024. / Alm21/ Almqvist, A. Fundamentals of physics-informed neural networks applied to solve the reynolds boundary value problem, Lubricants, Vol. 9, Nr. 82, 2021. / Li22/ Li, L., et al. Ref-nets: Physics-informed neural network for Reynolds equation of gas bearing, Computer Methods in Applied Mechanics and Engineering, Vol. 391, 2022. / Yad22/ Yadav, S.K., et al. Solution of Lubrication Problems with Deep Neural Network, Advances in Manufacturing, S. 471-477, 2022. / Zha23/ Zhao, Y., et al. Application of physics-informed neural network in the analysis of hydrodynamic lubrication, Friction, Vol. 11, S. 1253-1264, 2023. / Rom23/ Rom, M. Physics-informed neural networks for the Reynolds equation with cavitation modeling, Tribology International, Vol. 179, 2023. / Che23/ Cheng, Y., et al. HL-nets: Physics-informed neural networks for hydrodynamic lubrication with cavitation, Tribology International, Vol. 188, 2023. / Xi24/ Xi, Y., et al. A new method to solve the Reynolds equation including mass-conserving cavitation by physics informed neural networks (PINNs) with both soft and hard constraints, Tribology International, Vol. 12, S. 1165-1175, 2024. / Bru24b/ Brumand-Poor, F., et al. Fast Computation of Lubricated Contacts: A Physics-informed Deep Learning Approach, River Publishers, Vol. 19, 2024. / Bru24c/ Brumand-Poor, F., et al. Extrapolation of Hydrodynamic Pressure in Lubricated Contacts: A Novel Multi-Case Physics-Informed Neural Network Framework, Lubricants, Vol. 12, Nr.4, 2024. / Bru24d/ Brumand-Poor, F., et al. Advancing Lubrication Calculation: A Physics-Informed Neural Network Framework for Transient Effects and Cavitation Phenomena in Reciprocating Seals, 22nd International Sealing Conference, Stuttgart, 2024. / Rim23/ Rimon, M., T., I., et al. A Design Study of an Elasto-Hydrodynamic Seal for sCO 2 Power Cycle by Using Physics Informed Neural Network, ASME, 2023. / Pat78/ Patir, N., Cheng, H. S. An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication, Journal of Lubrication Technology, Vol. 100, Nr. 1, S. 12- 17, 1978. / Rom21/ Rom, M., et al. Why homogenization should be the averaging method of choice in hydrodynamic lubrication, Applications in Engineering Science, Vol. 7,2021. / Ten15/ Abadi M., et al. Tensorflow: Large-Scale Machine Learning on Heterogeneous Systems, https: / / www.tensorflow.org/ api_docs/ python/ tf/ keras/ activations,2015. / Glo10/ Glorot, X., et al. Understanding the difficulty of training deep feedforward neural networks, Proceedings of the 13th International Conference on Artificial Intelligence and Statistics, S. 249-256, 2010. / He15/ He, K., et al. Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification, IEEE International Conference on Computer Vision, 2015. / Kin15/ Kingma, D., et al. ADAM: A Method for Stochastic Optimization, ICLR, 2015. / Bis22/ Bischof, R., et al. Multi-Objective Loss Balancing for Physics-Informed Deep Learning, arXiv: 2110.09813v2, 2022. / Don21/ Dong, S., et al. A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks, Journal of Computational Physics, Vol. 435, 2021. / Liu89/ Liu, D., C., et al. On the limited memory BFGS method for large scale optimization, Mathematical Programming, Vol. 45, S. 503-528, 1989. / Ngu20/ Ngyuen-Thanh. V.M. et al. A deep energy method for finite deformation hyperelasticity, European Journal of Mechanics, Vol. 80, 2020. / Ngu24/ Ngyuen-Thanh. V.M. et al. Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials, ar- Xiv, 2024. / Abu21/ Abueidda, D.W., et al. Meshless physics-informed deep learning method for three-dimensional solid mechanics, International Journal for Numerical Methods in Engineering, Vol. 122, 2021. / Dak20/ Dakov, N. Elastohydrodynamische Simulation von Wellendichtungen am Beispiel der PFTE-Manschettendichtung mit Rückförderstrukturen, Stuttgart: Institut für Maschinenelemente, 2020. Science and Research 33 Tribologie + Schmierungstechnik · volume 71 · issue 4/ 2024 DOI 10.24053/ TuS-2024-0021