# Machine Learning for Film Thickness Prediction in Elastohydrodynamic Lubricated Elliptical Contacts

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{x}and R

_{y}, in the x- and y- space directions, respectively. Under unloaded and dry contact conditions, these solids share one point of contact, therefore with the name “point contacts”. When loaded, the contact patch is elliptical (or circular, when R

_{x}= R

_{y}), hence with the name “elliptical contacts” (or “circular contacts”, respectively). Two types of elliptical contacts can be recognized: wide and narrow (or slender). For wide contacts, the lubricant entrainment direction is perpendicular to the major semi-axis of the contact ellipse. Conversely, for narrow contacts, the lubricant entrainment direction is perpendicular to the minor semi-axis.

_{x}and R

_{y}). A more recent set of analytical formulas with greater range was presented by Nijenbanning et al. [17]. Both sets of equations are restricted to circular and wide elliptical contacts only. Chittenden et al. [18] were the first to extend their equations to both wide and narrow elliptical contacts, but the range of application of these equations is rather limited. An extensive review of analytical formulas developed for isothermal point contacts and their respective range of interest is provided by Wheeler et al. [19].

## 2. Finite Element Model

#### 2.1. Governing Equations

#### 2.2. Overall Numerical Procedure

#### 2.3. Experimental Validation

## 3. Machine Learning

#### 3.1. Data Generation

#### 3.2. Feature Selection

#### 3.3. Gaussian Process Regression (GPR)

## 4. Results and Discussion

**Remark 1.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\alpha}^{*}$ | Reciprocal asymptotic isoviscous pressure coefficient (Pa^{−1}) |

${\beta}_{K}$ | Murnaghan EoS isothermal bulk modulus temperature coefficient (K^{−1}) |

$\mu $ | Lubricant low-shear/Newtonian viscosity (Pa·s) |

$\overline{\mu}$ | Dimensionless lubricant low-shear/Newtonian viscosity |

${\mu}_{g}$ | Lubricant viscosity at glass transition temperature (Pa·s) |

${\mu}_{0}$ | Lubricant low-shear/Newtonian viscosity at ambient pressure (Pa·s) |

${\mu}_{tr}$, ${\sigma}_{tr}$ | Mean and standard deviation of features within the training dataset |

$\upsilon $ | Equivalent solid Poisson coefficient |

${\upsilon}_{1}$, ${\upsilon}_{2}$ | Poisson coefficient of solids 1 and 2 |

$\mathsf{\Omega}$ | Equivalent solid computational domain |

${\mathsf{\Omega}}_{c}$ | Contact computational domain |

$\partial {\mathsf{\Omega}}_{c}$ | Boundaries of ${\mathsf{\Omega}}_{c}$ |

$\partial {\mathsf{\Omega}}_{b}$ | Fixed boundary of $\mathsf{\Omega}$ |

$\partial {\mathsf{\Omega}}_{s}$ | Symmetry boundary of $\mathsf{\Omega}$ |

${\mathsf{\Psi}}_{1}$ | Complete elliptic integral of the first kind |

$\rho $ | Lubricant density (kg/m^{3}) |

$\overline{\rho}$ | Lubricant dimensionless density |

${\rho}_{0}$ | Lubricant density at ambient pressure (kg/m^{3}) |

${\sigma}_{n}$ | Normal component of 3D stress tensor (Pa) |

${\sigma}_{f}$, $l$, $\alpha $, $\nu $ | GPR model hyperparameters |

$\left\{{\sigma}_{t}\right\}$ | Vector of tangential components of 3D stress tensor (Pa) |

$\theta $ | Contact ellipticity ratio |

${\tau}_{ij}$ | Shear stress in the j-direction within a plane having i as normal (Pa) |

${a}_{x}$, ${a}_{y}$ | Hertzian elliptical contact semi-axes in the x, y-directions (m) |

${A}_{1}$, ${C}_{2}$ | Modified Yasutomi-WLF viscosity model parameters (°C) |

${A}_{2}$, ${b}_{1}$ | Modified Yasutomi-WLF viscosity model parameters (Pa^{−1}) |

${b}_{2}$, ${C}_{1}$ | Modified Yasutomi-WLF viscosity model parameters |

$D$ | Ratio of contact equivalent radii of curvature ${R}_{x}$ and ${R}_{y}$ |

$E$ | Equivalent solid Young’s modulus of elasticity (Pa) |

${E}_{1}$, ${E}_{2}$ | Young’s moduli of elasticity of solids 1 and 2 (Pa) |

$F$ | Contact external applied load (N) |

$G$, $U$, $W$ | Hamrock and Dowson material, speed, and load dimensionless groups |

$h$ | Lubricant film thickness (m) |

${h}_{c}$ | Central film thickness (m) |

${h}_{m}$ | Minimum film thickness (m) |

${H}_{c}$, ${H}_{c}^{*}$ | Dimensionless central film thickness |

${H}_{m}$, ${H}_{m}^{*}$ | Dimensionless minimum film thickness |

${H}_{0}$ | Dimensionless rigid-body separation |

$H$, ${H}^{*}$ | Dimensionless lubricant film thickness |

${K}_{00}$ | Isothermal bulk modulus at zero absolute temperature (Pa) |

${K}_{0}^{\prime}$ | Pressure rate of change of isothermal bulk modulus at zero pressure |

$L$, $M$ | Moes dimensionless material properties and load parameters |

$m$, $k$ | Mean and kernel functions |

$\tilde{n}$, $\widehat{n}$ | Sizes of sample datasets $\tilde{X}$, $\widehat{X}$ |

${N}_{f}$ | Number of input features |

${N}_{tr}$, ${N}_{te}$ | Number of samples in the training and testing datasets |

$p$ | Pressure (Pa) |

${p}_{h}$ | Hertzian contact pressure (Pa) |

$P$ | Dimensionless pressure |

${R}_{x1}$, ${R}_{x2}$ | Principal radii of curvature of solids 1 and 2 in the xz-plane (m) |

${R}_{y1}$, ${R}_{y2}$ | Principal radii of curvature of solids 1 and 2 in the yz-plane (m) |

${R}_{x}$ | Radius of curvature of equivalent elastic solid in the xz-plane (m) |

${R}_{y}$ | Radius of curvature of equivalent elastic solid in the yz-plane (m) |

$\overline{R}$ | Equivalent radius of curvature of reduced contact geometry (m) |

${T}_{g}$ | Glass transition temperature (K) |

${T}_{g0}$ | Glass transition temperature at zero pressure (K) |

${T}_{0}$ | Ambient temperature (K) |

$u$, $v$, $w$ | Equivalent solid deformation components in the x, y, z-directions (m) |

${u}_{1}$, ${u}_{2}$ | Surface velocities of solids 1 and 2 in the x-direction (m/s) |

${u}_{m}$ | Contact mean entrainment speed in the x-direction (m/s) |

$U$, $V$, $W$ | Solid dimensionless deformation components in x, y, z-directions |

$x$, $y$, $z$ | Space coordinates (m) |

$y$, ${y}_{tr}$, ${y}_{te}$ | Gaussian distribution for standard, training and testing subsets |

${\widehat{y}}_{te}$ | Prediction function of GPR model for testing samples |

$\tilde{x}$, $\widehat{x}$ | Sample input features |

${\tilde{x}}_{i}$, ${\widehat{x}}_{i}$ | Input i of $\tilde{x}$, $\widehat{x}$ |

${x}_{i}$ | Input feature number i |

${\stackrel{\u2323}{x}}_{i}$ | Normalized value of input feature number i |

${y}_{i}$, ${\widehat{y}}_{i}$, $\overline{y}$ | Output variable i, its predicted and mean values in the testing dataset |

$X$, $Y$, $Z$ | Dimensionless space coordinates |

${X}_{tr}$, ${X}_{te}$ | Training and testing sample datasets |

$\tilde{X}$, $\widehat{X}$ | Sample datasets |

## Appendix A. Kernel Function Definitions

## Appendix B. Data Standardization and Performance Metrics

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**Figure 1.**The contact geometry of (

**a**) an actual elliptical contact and (

**b**) a reduced elliptical contact.

**Figure 3.**Comparison of numerical and experimental central and minimum film thickness variations as a function of the entrainment speed for (

**a**) wide contacts and (

**b**) narrow contacts.

**Figure 4.**Distribution of data points across the input space for unconstrained (

**a**) wide and (

**b**) narrow contacts, as well as constrained (

**c**) wide and (

**d**) narrow contacts.

**Figure 5.**Diagram illustrating the dataset generation, dimensionality reduction, and feature selection for this study.

**Figure 6.**Predicted vs. true values and percentage residual plots of central ((

**a**) and (

**c**), respectively) and minimum film thicknesses ((

**b**) and (

**d**), respectively) for the best-performing ML model.

**Figure 7.**Predicted vs. true values and percentage residual plots of central ((

**a**) and (

**c**), respectively) and minimum film thicknesses ((

**b**) and (

**d**), respectively) for the best-performing ML model, without logarithmic scaling of $M$, $L$, ${H}_{c}^{*}$, and ${H}_{m}^{*}$.

**Figure 8.**Predicted vs. true values and percentage residual plots of central ((

**a**) and (

**c**), respectively) and minimum film thicknesses ((

**b**) and (

**d**), respectively) for the Hamrock and Dowson analytical formulas for wide and circular cases of the testing dataset only.

Parameter | Lower Bound | Upper Bound | Unit | |

Ranges of interest | ${u}_{m}$ | 0.01 | 50 | m/s |

${p}_{h}$ | 0.4 | 4 | GPa | |

$D$ | 1/12 | 12 | - | |

Constraints | $M$ | 10 | 3000 | - |

$L$ | 1 | 20 | - |

**Table 2.**Performance metrics of GPR models based on the testing dataset for different kernel functions.

${h}_{c}$ | ${h}_{m}$ | ||||||

Kernel Function | Adj. R^{2}(-) | MAPE (%) | MAXAPE (%) | Adj. R^{2}(-) | MAPE (%) | MAXAPE (%) | |

ARD-RBF | 0.9871 | 2.28% | 22.02% | 0.9841 | 6.89% | 68.20% | |

RQ | 0.9988 | 0.71% | 5.15% | 0.9979 | 1.88% | 11.74% | |

$\nu =3/2$ | 0.9995 | 0.39% | 5.33% | 0.9987 | 1.39% | 7.66% | |

ARD-Matern | $\nu =5/2$ | 0.9990 | 0.53% | 9.12% | 0.9975 | 1.58% | 12.86% |

$\nu =3/2$$\oplus $$\nu =5/2$ | 0.9999 | 0.31% | 3.05% | 0.9992 | 1.00% | 6.97% |

**Table 3.**Performance metrics of GPR models using the initial scale (instead of logarithmic) of $M$, $L$, ${H}_{c}^{*}$, and ${H}_{m}^{*}$ for different kernel functions.

${h}_{c}$ | ${h}_{m}$ | ||||||

Kernel Function | Adj. R^{2}(-) | MAPE (%) | MAXAPE (%) | Adj. R^{2}(-) | MAPE (%) | MAXAPE (%) | |

ARD-RBF | 0.9420 | 4.65% | 49.15% | 0.8857 | 36.28% | 566.52% | |

RQ | 0.9969 | 1.06% | 8.36% | 0.9927 | 5.38% | 30.63% | |

$\nu =3/2$ | 0.9993 | 0.47% | 5.98% | 0.9974 | 2.11% | 11.89% | |

ARD-Matern | $\nu =5/2$ | 0.9968 | 0.87% | 20.90% | 0.9882 | 6.84% | 239.95% |

$\nu =3/2$$\oplus $$\nu =5/2$ | 0.9998 | 0.52% | 7.48% | 0.9992 | 1.50% | 7.32% |

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## Share and Cite

**MDPI and ACS Style**

Issa, J.; El Hajj, A.; Vergne, P.; Habchi, W.
Machine Learning for Film Thickness Prediction in Elastohydrodynamic Lubricated Elliptical Contacts. *Lubricants* **2023**, *11*, 497.
https://doi.org/10.3390/lubricants11120497

**AMA Style**

Issa J, El Hajj A, Vergne P, Habchi W.
Machine Learning for Film Thickness Prediction in Elastohydrodynamic Lubricated Elliptical Contacts. *Lubricants*. 2023; 11(12):497.
https://doi.org/10.3390/lubricants11120497

**Chicago/Turabian Style**

Issa, Joe, Alain El Hajj, Philippe Vergne, and Wassim Habchi.
2023. "Machine Learning for Film Thickness Prediction in Elastohydrodynamic Lubricated Elliptical Contacts" *Lubricants* 11, no. 12: 497.
https://doi.org/10.3390/lubricants11120497