Data-Driven Multi-Objective Optimization Design of Micro-Textured Wet Friction Pair

Abstract

Friction pairs in heavy-duty power-shift tractor wet clutches operate under complex conditions, making them vulnerable to damage and reducing reliability. Optimizing their tribological performance requires a trade-off between a high coefficient of friction (COF) for torque transmission and a low temperature rise ($∆T$) to prevent thermal damage. Surface texturing is an effective method for improving the tribological performance of friction pairs. This study simulated the friction of wet clutch pairs via pin-on-disk tests and designed micro-textures on the pin surface to enhance tribological performance. Based on the experimental data, a Gaussian Process Regression (GPR) surrogate model was developed to accurately predict COF and $∆T$ as a function of the clutch’s operating and micro-texture’s geometric parameters. A Multi-Objective Particle Swarm Optimization (MOPSO) algorithm was then employed to obtain the optimal set of solutions. The obtained pareto front clearly revealed the COF–temperature rise trade-off. From the optimal solution set, optimal micro-texture parameters for two typical operating conditions of different clutches were extracted. Compared with the untextured surface, the optimal solutions increased COF by 2.6%/1.2% and reduced $∆T$ by 39.2%/12.1%. Relative to neighboring experimental points, COF further increased by 11.3%/2.7% and $∆T$ decreased by 16.6%/1.7%. This work establishes a method for balancing the frictional and thermal performance of friction pairs.

1. Introduction

Wet clutches are key components in the transmission systems of heavy-duty tractors, responsible for power transmission and smooth shifting [1]. Their core elements, the wet clutch friction pairs, endure high loads, significant relative velocities, and intense thermal shock during frequent engagement and disengagement cycles [2]. The tribological characteristics of these pairs are paramount, as they directly govern the clutch’s torque capacity, operational lifespan, and the reliability of the entire machine [3].

Researchers have carried out numerous studies on clearance lubrication and friction characteristics to cope with the complex mechanical and thermal loads that wet clutch friction pairs encounter during operation. For example, Zhang et al. [4] employed a modified Reynolds equation with an elastic contact model to numerically simulate how material parameters affect the thermal characteristics of the friction pair and ATF during engagement. Regarding surface structures, Jang et al. [5] built a mathematical model of the heat generation and transfer process to investigate the influence of various groove patterns and their structural parameters on the clutch’s thermal behavior. Kong et al. [6] analyzed convective cooling and temperature distribution for waffle and radial-arc groove designs using a CFD–thermal coupled analysis. Furthermore, Xie et al. [7] modeled the oil film flow field considering the friction plate grooves to numerically determine their impact on oil film velocity and pressure. Zhou et al. [8] proposed a semi-Y labyrinth seal (SYLS) to reduce leakage and enhance the stability of centrifugal pump. Compared to the traditional interlocking labyrinth seal (ILS) structure, the leakage of the SYLS was reduced significantly. Yang et al. [9] used a model based on the Reynolds equation, average flow model, and Greenwood model and friction test based on SAE#2 testing machine to study the COF of different groove types. The results showed that waffling groove or double arc groove are suitable for the friction plate of the wet clutch. This body of work highlights the positive effect of optimizing friction parameters to improve clutch performance. Indeed, optimizing the design of the wet clutch friction pair to achieve a stable COF while managing frictional heat is crucial for ensuring stable torque transmission and minimizing thermal damage [1]. While these studies mainly focused on designing macroscopic grooves on the friction pair, the impact of surface micro-texture is equally significant.

Surface micro-texturing technology, which involves fabricating micro-scale dimples or grooves with specific geometries on a frictional surface, has been proven to be an effective way to improve tribological performance [10]. Fesanghary et al. [11] used mathematical optimization methods to obtain the optimum periodic surface grooves (elongated “heart-like” shape for small aspect ratios, and spiral shape for high aspect ratios) with the highest load-carrying capacity (LCC) in parallel flat surface bearings. Agrawal et al. [12] deals with the performance of micro-grooved hole-entry hybrid spherical thrust bearing (HSTB) considering Non-Newtonian behavior of the lubricant. The result showed that the performance of the hole-entry hybrid spherical thrust bearing was enhanced by the micro-grooves and was affected by the geometry of the micro-groove. Through a series of pioneering studies, Etsion proved that Laser Surface Texturing (LST) technology can significantly improve the performance of mechanical seals [13], journal bearings [14], piston rings [15,16], and cylinder liners [17].

However, surface micro-textures come in various forms, and determining the appropriate micro-texture pattern and optimizing its structural parameters is key to enhancing the overall performance of the friction pair. Chen et al. [18] studied the influence of the structural parameters of hexagonal grooves on the wet friction characteristics on glass and compared them with square, rhombic, and triangular patterns, demonstrating the high-friction advantage of the hexagonal structure. Iturri et al. [19] prepared hexagonal specimens from PDMS material with different aspect ratios and found that, compared to untextured specimens, the hexagonal specimens showed significantly improved tribological performance on both wetted and non-wetted surfaces. Liu et al. [20] introduced hexagonal micro-textures to the friction pair formed between the stator and rotor of a screw pump, which enhanced the oil film load-carrying capacity of the friction pair.

The research above indicates that introducing hexagonal textures on a wet friction surface can improve its tribological performance. However, exploring the coupled effects of micro-texture geometric parameters and operating conditions on friction performance is a complex, multi-variable, and non-linear problem. Traditional numerical and experimental methods are often limited to specific conditions and cannot capture the full complexity of these interactions [21,22]. Here, the rise of data-driven methods, particularly machine learning, offers a powerful alternative [23]. By creating data-driven surrogate models, researchers have successfully predicted complex frictional behaviors in cam-followers [24], calibrated dynamic models of piezoelectric drives [25], and improved torque simulation accuracy in wet clutches [26]. Such approaches enable the efficient prediction of system performance across a wide range of parameter combinations [27].

Simultaneously, to ensure the wet clutch can transmit stable torque with minimal thermal damage, the parameter optimization of the friction pair requires a trade-off between the conflicting objectives of coefficient of friction (COF) and temperature rise ($∆T$). This is a classic Multi-Objective Optimization (MOP) problem [28]. While various evolutionary algorithms, such as the NSGA-II, are applied to such engineering trade-offs [29], the swarm intelligence-based Multi-Objective Particle Swarm Optimization (MOPSO) algorithm was selected for this study due to its distinct advantages [30,31,32]. Compared to other methods, MOPSO generally exhibits faster convergence [33]—a critical feature when integrated with the GPR surrogate model used in this work. Furthermore, by effectively balancing global search (Exploration) and local refinement (Exploitation), MOPSO efficiently navigates complex, high-dimensional search spaces and avoids being trapped in local optima [34]. These characteristics make it an ideal fit for this complex, data-driven tribological optimization that involves exploring the coupled effects of multiple variables.

However, previous research has typically optimized structural parameters for a single, fixed operating condition. In recent years, several studies have begun to recognize the necessity of incorporating operating conditions into the decision space. For example, Kong et al. [35] explicitly examined the influence of rotational speed, clearance, and eccentricity on the optimal texture design in the topology optimization of textured journal bearings. Similarly, Zhang et al. [36] compared optimal groove distributions under different speeds and eccentricities. These results collectively highlight the strong coupling between geometry and operating conditions; nevertheless, most of these efforts still adopt a “scenario-by-scenario” analysis rather than a unified optimization. This approach is insufficient for applications like heavy-duty power-shift tractors, where different wet clutches encounter variable sliding conditions. An optimization design that neglects operating parameters fails to achieve optimal tribological performance for the wet clutch friction pair under specific working conditions. To overcome this limitation, it is necessary to include both structural and operating parameters as decision variables in a synchronous optimization. Hingawe et al. [37] proposed a multi-objective framework for a square-textured parallel slider, in which sliding velocity was introduced together with texture density and aspect ratio, yielding a Pareto front between load-carrying capacity enhancement and friction reduction based on response surface methodology and gray relational analysis. Abbas and Metwalli [38] extended this idea to rolling element bearings under elastohydrodynamic lubrication (EHL) by formulating a joint decision space of geometry and operating conditions, and used a hybrid NSGA-II with supervised machine learning to obtain a Pareto front validated by industrial case studies. Collectively, these studies confirm that integrating both structural and operating parameters can generate more comprehensive Pareto fronts and improve the reliability of optimization outcomes. However, in the field of wet clutch friction pairs, such unified optimization has rarely been reported. GPR provides a high-precision surrogate model capable of capturing complex non-linear relations from limited experimental data [39], while MOPSO is known for its faster convergence and efficiency in high-dimensional spaces. By integrating these two methods, the present study addresses the above limitation and establishes a framework for synchronously optimizing micro-texture geometry and operating parameters of wet clutch friction pairs.

This paper proposes to simulate the friction experiments of wet clutch friction pairs using pin-on-disk tests and to incorporate operating parameters themselves as decision variables into a multi-objective optimization model. The objective is no longer to find a single optimal solution, but rather to provide optimal design schemes for the surface micro-texture geometric parameters of wet clutch friction pairs under different sliding friction states. This approach proposes an integrated framework that combines experimental data, machine learning, and multi-objective optimization. Firstly, a high-precision surrogate model is constructed from the experimental data using Gaussian Process Regression (GPR). Subsequently, the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is employed to obtain the optimal solution set based on the GPR model. Finally, this yields representative optimal solutions for the micro-textures of clutch friction pairs under two different sliding friction states. This study provides a new method for enhancing the tribological performance and reducing the damage of the wet clutch friction pairs in heavy-duty power-shift tractors.

2. Materials and Methods

2.1. Experimental Apparatus and Procedure

The tribological tests in this study were conducted on a UMT Tribolab tester (Bruker, San Jose, CA, USA) using a Pin-on-Disk configuration to simulate the operating conditions of a wet clutch friction pair. The experimental setup is illustrated in Figure 1. The pin specimens were fabricated from 45 steel (surface roughness Ra 1.6, hardness 170–200 HB without heat treatment; Shufengtai Hardware Manufacturing Co., Ltd., Shenzhen City, Guangdong Province, China) to represent the steel plate, while the disk specimens were made of H62 brass (surface roughness Ra 3.2, hardness 60–80 HB without heat treatment; Shufengtai Hardware Manufacturing Co., Ltd., Shenzhen City, Guangdong Province, China) to represent the friction plate.

The pin specimen was designed with a three-section stepped cylindrical geometry, featuring a 20 mm diameter contact face. For real-time temperature monitoring, a K-type thermocouple (measurement range 0–100 °C, accuracy ± 0.1 °C; Yizhiding Precision Measurement & Control Co., Ltd., Yancheng City, Jiangsu Province, China) was embedded in the pin just beneath the contact surface. The disk, with an outer diameter of 90 mm and an inner diameter of 30 mm, was mounted on a turntable, with the pin sliding against it at a constant rotational radius of 35 mm, and the thermocouple was positioned at a rotational radius of 40 mm. The entire friction assembly was submerged in Bosch DOT4 lubrication oil to maintain a wet friction environment throughout the tests.

Before each test, all specimens were ultrasonically cleaned in acetone and ethanol for 10 min each, rinsed with deionized water, and subsequently dried in a vacuum oven at 60 °C before testing. The procedure began by allowing the lubrication oil to reach a stable temperature, after which the predetermined normal load was applied using a force sensor (measurement range 0–200 N, accuracy ± 0.01 N), and the rotation was initiated. Each experiment continued until both the COF and temperature stabilized, at which point the steady-state data were logged. Prior to testing, each pin was run-in for 15 min under the prescribed load and speed to stabilize the contact. To ensure the reliability of the results, experiments for each condition were performed in triplicate, and the averaged values were used for the final analysis.

2.2. Micro-Texture Design and Parametric Study

In this study, micro-textures with different diagrams (untextured, hexagonal, square, and circular, as shown in Figure 2) were first fabricated on the surface of 45 steel pins using laser etching to comparatively investigate their tribological performance. After laser processing, the average surface roughness of the micro-textured grooves was approximately Ra 2.2. The laser parameters used for this process are detailed in

Table 1. Parameters of laser processing.

Parameters	Values
Wave length (nm)	1064
Laser power (W)	10
Scanning speed (mm/s)	500
Diameter of defocused laser (mm)	0.05
Pulse duration (ns)	100
The times of scanning	2

. Based on these preliminary results, the hexagonal micro-texture was selected for further optimization. A detailed parameter study was then conducted to optimize three key geometric parameters: groove depth, circumcircle diameter, and groove width. Concurrently, the effects of two critical operating parameters, normal load and rotational speed, on the tribological performance were also examined. The specific ranges for these parameters are listed in

Table 2. Geometric parameters of the hexagonal texture and the operating parameters.

Parameters		Values
Geometric parameters	Groove depth ($h/\mathsf{\mu }\mathrm{m}$)	30	60	90	120	150
	Circumcircle diameter ($d/\mathrm{m}\mathrm{m}$)	3	4.5	6	7.5	9
	Groove width ($w/\mathsf{\mu }\mathrm{m}$)	300	400	500	600	700
Operating parameters	Normal load ($F/\mathrm{N}$)	30	35	40	45	50
	Rotational speed ($\omega /\mathrm{r}\mathrm{p}\mathrm{m}$)	130	150	170	190	210

.

A dataset for machine learning modeling was systematically constructed from a total of 468 experiments, performed across various combinations of geometric and operating parameters.

2.3. Data-Driven Modeling and Optimization Framework

2.3.1. Predictive Modeling via Gaussian Process Regression (GPR)

• Theoretical Background of GPR

Gaussian Process Regression (GPR) is a non-parametric regression method based on Bayesian theory. It is particularly well-suited for addressing complex problems characterized by small sample sizes, high dimensionality, and non-linearity, which aligns perfectly with the inherent characteristics of tribological experimental data [28,40]. The core idea of GPR is to define a prior distribution over the target function itself, which is known as a Gaussian Process (GP). A function $f\left(x\right)$ is said to be drawn from a Gaussian Process if, for any finite set of input points $\mathrm{X}=\left\{x_1,\cdots ,x_n\right\}$, the corresponding vector of function values $\mathrm{f}=\left\{f\left(x_1\right),\cdots ,f\left(x_n\right)\right\}$ follows a joint multivariate Gaussian distribution [41]. This process can be expressed as:

$$f\left(x\right)~GP\left(m\left(x\right),k\left(x,x^{\prime }\right)\right)$$

(1)

where $m\left(x\right)$ is the mean function, typically assumed to be zero, and $k\left(x,x^{\prime }\right)$ is the covariance function, also known as the kernel. The kernel defines the correlation between the function values at any two points, $x$ and $x^{\prime }$. The choice of kernel function encodes our prior assumptions about the properties of the target function (e.g., smoothness, periodicity) [42]. This study employs the Squared Exponential (SE) kernel function, which is widely used and effective in engineering applications:

$$k\left(x_i,x_j\right)={\sigma }_f^2\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{{‖x_i-x_j‖}^2}{2l^2}}\right)+{\sigma }_n^2{\delta }_{ij}$$

(2)

where ${\sigma }_f^2$ is the signal variance, which controls the vertical variation in the function; $l$ is the characteristic length-scale, which determines the function’s smoothness; ${\sigma }_n^2$ is the noise variance, which describes the level of noise in the observed data; and ${\delta }_{ij}$ is the Kronecker delta function [43]. These hyperparameters are automatically optimized by maximizing the marginal likelihood of the training data. Given a training dataset $\mathrm{D}=\left\{\mathrm{X},\mathrm{y}\right\}$, GPR utilizes Bayesian inference to update the prior distribution into a posterior distribution. For a new test point, $x_{\ast }$, the resulting predictive distribution is also Gaussian, and its mean and variance can be computed analytically. The predictive mean serves as the final prediction, while the predictive variance provides a quantitative measure of the uncertainty in that prediction. This uncertainty quantification is a significant advantage of GPR compared to other ‘black-box’ models.

2.

Model Application;

This study aims to construct two independent GPR surrogate models to facilitate efficient parameter optimization. The model input is a five-dimensional parameter vector comprising three geometric parameters of the micro-texture (groove depth $h$, circumcircle diameter $d$, and groove width $(w$)) and two operating parameters (normal load $F$ and rotational speed $(\omega$)). The model outputs are two key tribological performance metrics: the Coefficient of Friction (COF) and the temperature rise ($∆T$).

A total of 351 valid COF data points and 117 valid $∆T$ data points were obtained from the experiments. To train and validate the models, the dataset was split into a training set and a test set at an 80:20 ratio. The models were trained on the training set, and the kernel’s hyperparameters were optimized by maximizing the marginal likelihood function. Subsequently, the models’ predictive performance was evaluated on the unseen test set, with the primary metrics being the Coefficient of Determination ($R^2$) and the Root Mean Square Error (RMSE).

The model is formulated as follows:

Model for COF: $COF=f_{GPR1}\left(F,\omega ,h,d,w\right)$.

Model for $∆T$: $∆T=f_{GPR2}\left(F,\omega ,h,d,w\right)$.

The parameter values of the GPR models are summarized in

Table 3. Parameters of values of the GPR models.

Methods or Parameters	Functions or Values
Basic function	Linear
Number of iterations	100
Kernel function	Squared Exponential
Kernel scale	0.11544
Sigma	0.0641

.

2.3.2. Multi-Objective Optimization via MOPSO

For a minimization problem, a solution $x$ is Pareto optimal (or non-dominated) if no other solution $x^{\prime }$, exists that is better than or equal to $x$ in all objectives and strictly better in at least one objective. The mapping of all Pareto optimal solutions in the objective function space forms the Pareto Front [44].

• Theoretical Background of MOPSO

Multi-Objective Particle Swarm Optimization (MOPSO) is a population-based swarm intelligence algorithm inspired by the social behavior of bird flocking, which has been proven effective for solving complex MOPs [45]. The algorithm maintains a population of particles, each representing a potential solution in the search space. The movement of each particle (i.e., the updating of the solution) is guided by its own best-known historical position ($pBest$) and the global best-known position of the swarm ($gbest$) [46]. The velocity and position update equations are as follows:

$$v_i\left(t+1\right)=w·v_i\left(t\right)+c_1·r_1·\left(pBest-x_i\left(t\right)\right)+c_2·r_2·\left(gBest-x_i\left(t\right)\right)$$

(3)

$$x_i\left(t+1\right)=x_i\left(t\right){+v}_i\left(t+1\right)$$

(4)

where $v_i\left(t\right)$ and $x_i\left(t\right)$ are the velocity and position of particle $i$ at iteration $t$, $w$ is the inertia weight, $c_1$ and $c_2$ are the learning factors, and $r_1$ and $r_2$ are random numbers in. The key to MOPSO is its handling of multiple objectives. It incorporates an external archive to store and maintain all non-dominated solutions found so far. In each iteration, the global best leader, $gbest$, is selected from this external archive. To promote solution diversity and prevent premature convergence, a strategy based on crowding distance or grid-based density is often used to select $gbest$, giving preference to solutions located in sparser regions of the Pareto front.

2.

Problem Formulation

The multi-objective optimization problem in this study is to find an optimal set of micro-texture geometric parameters and operating parameters that simultaneously maximizes frictional performance and minimizes thermal effects. The establishment of this objective is driven by the dual requirements of heavy-duty clutches in practical applications: high torque transmission capacity and high thermal stability. The problem is formally defined as follows:

(1)

The decision variables consist of a five-dimensional vector $x=\left\{s_1\right.,\left.s_2,s_3,s_4,s_5\right\}$, where $s_1$ is the pressure, $s_2$ is the relative rotational speed of the friction pair, $s_3$ is the groove depth of the hexagonal micro-texture, $s_4$ is the circumcircle diameter of the hexagonal micro-texture, and $s_5$ is the groove width of the hexagonal micro-texture. The value ranges of these variables are strictly constrained within the experimental design space defined in Section 2.2 to ensure the physical feasibility of the optimization results.

(2)

Objective Functions are as follows:

Maximize COF: $\mathrm{m}\mathrm{a}\mathrm{x}{f}_1\left(x\right)={COF}_{GPR}\left(x\right)$.

Minimize $∆T$: $\mathrm{m}\mathrm{i}\mathrm{n}{f}_2\left(x\right)={∆T}_{GPR}\left(x\right)$.

(3)

The constraints are the lower and upper bounds for each decision variable, i.e., $x_{min}\le x\le x_{max}$.

The ultimate objective is to find the set of Pareto optimal solutions that constitutes the Pareto front. Each point on this front represents a non-dominated design solution, where an improvement in any single objective can only be achieved at the expense of degrading at least one other objective. This solution set provides the decision-maker with a spectrum of trade-off options between maximizing torque transmission capacity (via a high COF) and minimizing the risk of thermal damage (via a low $∆T$).

The MOPSO parameters are listed in

Table 4. MOPSO parameter settings.

Parameters	Values
Inertia weight	1.2
Maximum number of iterations	200
Population size	100
Cognitive coefficient	1.8
Social coefficient	2.0
Archive size	350

.

The Data-Driven Modeling and Optimization Framework were performed by MATLAB R2023a (The Mathworks Inc., Natick, MA, USA).

3. Results and Discussion

3.1. Comparison of Tribological Performance for Different Micro-Texture Diagrams

The design of friction pairs for heavy-duty tractor wet clutches requires considering the trade-off between a high COF and the resulting frictional heat. A friction pair design that only considers increasing friction to provide sufficient torque without addressing heat dissipation is not an optimal choice and will inevitably lead to premature failure. In the structural parameter optimization design, a preliminary comparison of the tribological performance of different micro-texture diagrams was conducted, with the results shown in Figure 3. Compared to the COF of the untextured surface (approx. 0.158), the Square and Hexagon diagrams increased it to approximately 0.174 and 0.168, respectively, whereas the Circle diagram decreased it to about 0.152. In terms of $∆T$, all textured surfaces exhibited superior heat dissipation performance compared to the untextured surface (approx. 2.05 °C). Among them, the Circle diagram showed the lowest $∆T$ (approx. 1.6 °C), and $∆T$ for the Hexagon diagram was also significantly reduced to about 1.85 °C.

Comprehensively, the hexagonal micro-texture friction pair exhibits both a relatively high COF and a low $∆T$, demonstrating a trade-off optimization effect. The reason is that the micro-flow field channels formed by the hexagonal micro-texture have a higher average flow velocity. On one hand, this efficient flow discharge characteristic promotes full contact at the friction pair interface, thereby increasing friction. On the other hand, it enhances convective heat transfer, allowing the lubricating oil to carry away more heat and thus reducing the $∆T$ [47]. Compared to other texture diagrams, the hexagonal micro-texture provides a more stable COF through a controllable mixed lubrication mechanism, ensuring sufficient torque transmission. Concurrently, its unique “percolation network” structure endows it with excellent forced convection heat dissipation capabilities, enabling it to maintain a lower operating temperature even while generating more heat. This dual advantage of simultaneously “increasing friction and reducing temperature” makes it an ideal surface design solution for heavy-duty wet clutch applications.

3.2. Influence of Hexagonal Micro-Texture Geometric Parameters on Tribological Performance

To investigate the underlying mechanisms of how the geometric parameters of the hexagonal micro-texture influence frictional performance, this section selects experimental data under representative operating conditions (e.g., 40 N, 210 rpm). It analyzes the influence of the hexagonal micro-texture’s key parameters—including groove depth, circumcircle diameter, and groove width—on the tribological performance of the wet clutch friction pair.

Groove depth directly determines the volume of the micro-textures, thereby affecting their oil storage and debris trapping capacities [48]. Figure 4a shows the trends of COF and $∆T$ as a function of groove depth under fixed operating conditions. As can be seen from the figure, when the depth increases from 30 $\mathsf{\mu }\mathrm{m}$ to 120 $\mathsf{\mu }\mathrm{m}$, the COF shows a trend of first increasing and then decreasing, while the $∆T$ exhibits the opposite pattern, reaching its minimum at a depth of around 90–120 $\mathsf{\mu }\mathrm{m}$. As the depth increases, the micro-textures can store more lubricant and generate significant hydrodynamic pressure support during the sliding process, partially separating the frictional surfaces and thus reducing both COF and $∆T$ [49]. In the range of 30–60 $\mathsf{\mu }\mathrm{m}$, COF and $∆T$ show opposite trends, which can be explained by the transition in load sharing, and stronger hydrodynamic effects promote heat dissipation (lower $∆T$) while the increased contribution of fluid shear may temporarily raise COF [50].

The circumcircle diameter of the texture determines its area density, which in turn affects the ratio between the load-bearing area of the friction pair and the hydrodynamic pressure generation area [51]. As shown in Figure 4b, when the diameter increases from 3.0 mm to 9.0 mm, both COF and $∆T$ fluctuate, generally decreasing at first and then increasing, reaching their minimum values at a diameter of 7.5 mm. As the circumcircle diameter increases moderately, the ratio of the load-bearing area to the hydrodynamic pressure area is optimized, which can produce the strongest hydrodynamic effect and effectively separate the frictional surfaces, thereby lowering $∆T$. In the smaller range of 3.0–4.5 mm, however, COF decreases while $∆T$ increases. This opposite trend arises because the broader continuous lands reduce asperity contact and lower COF, but the reduced texture density weakens lubricant renewal and increases viscous shear heating, leading to a rise in $∆T$ [6].

From Figure 4c, we can see that the COF and $∆T$ exhibit a complex, non-monotonic trend with changes in the texture’s groove width. When the width increases from 0.3 mm to 0.4 mm, both COF and $∆T$ decrease. A significant rebound occurs at 0.5 mm. Subsequently, as the width increases from 0.5 mm to 0.7 mm, both COF and $∆T$ show a continuous decreasing trend. As the groove width increases moderately, the hydrodynamic effect is enhanced, leading to improved lubrication [52]. At larger groove widths (600–700 $\mathsf{\mu }\mathrm{m}$), COF and $∆T$ move in opposite directions because stronger hydrodynamic entrainment improves heat dissipation (lower $∆T$), while fluid shear in a thicker film partly offsets the reduction in asperity friction, limiting COF reduction [50].

3.3. Influence of Operating Parameters on Tribological Performance

For friction pairs with hexagonal micro-textures of different groove depths, the trends of COF and $∆T$ with varying rotational speed and normal load are shown in Figure 5 and Figure 6. It can be observed that the $∆T$ generally increases monotonically with both speed and load, whereas the COF exhibits a complex non-linear relationship with the operating parameters. This phenomenon can be explained by the direct influence of operating conditions on the oil film thickness and contact state. In the mixed lubrication regime, higher speeds and loads imply greater shear rates and energy input, causing increases in both viscous shear heating within the oil film and asperity contact heating. Macroscopically, this manifests as a continuous increase in $∆T$. The complex fluctuation of the COF is because the total friction force is the sum of the fluid viscous shear force and the asperity contact friction force. As the operating conditions change, the proportion of these two forces also dynamically changes [53]. For instance, an increase in speed may enhance the hydrodynamic effect but will also increase fluid shear friction. Because the friction pair operates primarily in the mixed or boundary lubrication states, which are highly sensitive to operating parameters, the change in the total COF appears non-monotonic.

Table 5. Optimal values of a single geometric parameter under 40 N, 210 rpm.

Parameters	$\mathbf{Optimal}\mathbf{Value}\mathbf{for}∆\mathit{T}$ Minimization	Optimal Value for COF Maximization
Groove depth	120$\mathsf{\mu }\mathrm{m}$	60$\mathsf{\mu }\mathrm{m}$
Circumcircle diameter	7.5$\mathrm{m}\mathrm{m}$	3$\mathrm{m}\mathrm{m}$
Groove width	600$\mathsf{\mu }\mathrm{m}$	700$\mathsf{\mu }\mathrm{m}$

presents the micro-texture geometric parameters, manually selected from the experimental data, that correspond to the minimum $∆T$ and maximum COF under fixed operating conditions. However, this optimization approach is inefficient and cannot obtain a comprehensively optimal solution within the full range of the experimental parameters. Therefore, it is necessary to further establish a mapping relationship between the parameters (both geometric and operating) and the objectives (COF and $∆T$). Based on this relationship, the optimal solution set can be obtained using a multi-objective optimization method.

3.4. GPR Model Validation

The entire experimental dataset was randomly split into a training set and a test set at an 80:20 ratio. Simultaneously, the five input parameters were normalized to ensure the accuracy and regularity of the training process. The GPR models were trained and their hyperparameters optimized on the training set, after which their generalization ability was evaluated on the test set. The predictive performance metrics of the models are as follows: for the COF model, the $R^2$ was 0.88 and the Root Mean Square Error (RMSE) was 0.00547; for the $∆T$ model, the $R^2$ was as high as 0.96 with an RMSE of 0.11338. Figure 7 shows a comparison of predicted versus actual values for the COF and $∆T$ models on the test set. It can be seen from the figure that all data points are tightly clustered around the $\mathrm{y}=\mathrm{x}$ diagonal line, indicating that the GPR models can capture the complex non-linear relationship between the input parameters and the output responses with extremely high precision. In particular, the high fidelity of the $∆T$ model provides a solid foundation for the subsequent multi-objective optimization based on this surrogate model.

3.5. Multi-Objective Optimization and Pareto Analysis

Using the trained GPR models as the objective functions, the MOPSO algorithm was run to perform multi-objective optimization. Following an adequate number of evolutionary generations, the algorithm successfully converged, yielding a set of non-dominated solutions that form the Pareto front, which is illustrated in Figure 8.

The Pareto front clearly illustrates the intrinsic trade-off relationship between maximizing COF (Objective F1) and minimizing $∆T$ (Objective F2). Any point on the curve represents an optimal design solution combination (including both operating and geometric parameters). As can be seen from the figure, Objective F1 (COF) and Objective F2 ($∆T$) are positively correlated, and the Pareto front broadly covers the objective space. The solutions are uniformly distributed across the possible value range of the objective functions, with no significant gaps or overly dense regions, indicating good coverage and diversity. The value range for COF is approximately 0.13 to 0.18, while the range for $∆T$ is between 0.6 and 1.7. The overall trend of the curve indicates that to obtain a higher coefficient of friction, a higher $∆T$ must be accepted, and vice versa. For example, solutions located in the upper-right region of the curve have COF values close to 0.18, but their $∆T$ also exceeds 1.6; these are suitable for applications pursuing maximum torque transmission where thermal management requirements are not stringent. Conversely, solutions in the lower-left region of the curve have a $∆T$ below 0.8, but their COF is correspondingly reduced to below 0.14, making them suitable for operating conditions that demand high durability and low thermal damage. Furthermore, to verify the robustness of the optimization procedure, convergence curves of the MOPSO algorithm are shown in Figure 9. These curves display the evolution of the population’s fitness values with iterations and demonstrate that the algorithm converges smoothly without oscillations, confirming its stability and reliability. Such convergence analysis is consistent with practices recommended in recent optimization studies [31,33].

The parallel coordinate plot in Figure 10 displays the distribution of the normalized values of the decision variables ($s_1$–$s_5$) corresponding to each optimal solution on the Pareto front. Several trends can be observed from this plot:

Operating Parameters ($s_1$, $s_2$): The values for pressure ($s_1$) and rotational speed ($s_2$) are spread across the entire feasible region, indicating that optimal texture designs exist for various combinations of operating conditions. Notably, solutions with high COF and high $∆T$ (corresponding to the upper-right portion of the Pareto front) tend to be associated with higher pressures and rotational speeds.

Geometric Parameters ($s_3$, $s_4$, $s_5$): The optimal values for groove depth ($s_3$) tend to be distributed in the mid-to-high range. The optimal solutions for the circumcircle diameter ($s_4$) are more concentrated in the lower range of values. The values for groove width ($s_5$) exhibit a concentrated distribution at both ends.

3.6. Discussion of Optimal Designs Under Typical Situations

Heavy-duty power-shift tractors have numerous gears and operate under complex conditions. During gear shifts, some wet clutches primarily experience high sliding friction torque at a low speed differential, whereas others operate with low sliding friction torque at a high speed differential. This study aims to provide customized optimal design solutions for wet clutches in different shifting states. In a laboratory environment, a pin-on-disk friction test method was used to obtain the influence laws of different operating parameters and micro-texture geometric parameters on the COF and $∆T$ of the wet friction pair, and to establish a mapping relationship. This enables the selection of corresponding optimal micro-texture designs from the Pareto optimal set according to the requirements of different types of clutches.

Based on this approach, we selected representative optimal solutions for the friction pair micro-texture under typical clutch operating conditions from the Pareto optimal set. This was conducted to simulate the parameter design optimization process for a heavy-duty tractor wet clutch, and the results are presented in

Table 6. Optimal solutions for the friction pair micro-texture under typical situations.

Parameters or Objectives	Situation A	Situation B
Normal load ($s_1$, N)	41.88	33.67
Rotational speed ($s_2$, rpm)	145.55	191.69
Groove depth ($s_3$, $\mathsf{\mu }\mathrm{m}$)	112.56	69.23
Circumcircle diameter ($s_4$, mm)	4.18	5.63
Groove width ($s_5$, $\mathsf{\mu }\mathrm{m}$)	629.58	508.58
Predicted COF (F1)	0.157	0.158
Predicted $∆T$ (F2, $°\mathrm{C}$)	1.22	1.24

.

The following is a discussion of the two optimal design schemes for the clutch friction pair from Table 6, aimed at revealing the optimization process for the micro-texture parameters of wet clutch friction pairs under different sliding states, as well as the performance advantages of the optimal solution set.

3.6.1. Discussion of Solution for Situation A

Solution for Situation A is selected from the high-normal-load ($s_1\approx 41.9\mathrm{N}$), low-rotational-speed ($s_2\approx 146\mathrm{r}\mathrm{p}\mathrm{m}$) region of the design space. It is intended to simulate the clutch shifting state characterized by high sliding torque and low speed difference. Its geometric features include a deeper groove depth ($s_3\approx 112.56\mathsf{\mu }\mathrm{m}$), a relatively smaller circumcircle diameter ($s_4\approx 4.18\mathrm{m}\mathrm{m}$), and a larger groove width ($s_5\approx 629.58\mathsf{\mu }\mathrm{m}$). When compared with the data from neighboring experimental points with similar operating and structural parameters, the optimal solution exhibited an 11.3% increase in the COF and a 16.6% reduction in $∆T$. Compared to the untextured surface, the hexagonal micro-texture corresponding to Solution for Situation A increased COF by 2.6% and reduced $∆T$ by 39.2% under the same operating conditions.

In Solution for Situation A, the smaller texture diameter and larger groove width increase the texture’s area density and reduce the actual load-bearing land area, thereby increasing the local contact stress [54]. This design deliberately enhances the direct interaction of asperities, which is the primary contributor to the higher friction force. Meanwhile, the deeper grooves act as micro-reservoirs that provide secondary lubrication to the contact zone during intense friction, enabling convective heat transfer and carrying away part of the generated heat [55].

3.6.2. Discussion of Solution for Situation B

Solution for Situation B is selected from the low-pressure ($s_1\approx 33.7\mathrm{N}$), high-rotational-speed ($s_2\approx 192\mathrm{r}\mathrm{p}\mathrm{m}$) region of the design space. It is designed to simulate clutch operating conditions with a high speed difference and low sliding torque. Its geometric features are a larger circumcircle diameter ($s_4\approx 5.6\mathrm{m}\mathrm{m}$) combined with a medium-depth groove ($s_3\approx 69.23\mathsf{\mu }\mathrm{m}$). When compared with the data from neighboring experimental points with similar operating and structural parameters, the optimal solution exhibited an 2.7% increase in the COF and a 1.7% reduction in $∆T$. Compared to the untextured surface, the hexagonal micro-texture corresponding to Solution for Situation B increased COF by 1.2% and reduced $∆T$ by 12.1%.

The larger circumcircle diameter in this solution creates a low-texture area density, thus forming a broad, continuous land area on the surface, which greatly promotes the formation of a hydrodynamic oil film. This oil film effectively separates the two frictional surfaces, fundamentally minimizing solid–solid contact that is the primary source of heat generation, and in turn, suppresses $∆T$. Although the friction generated by asperity contact is weakened, the viscous resistance produced by shearing the lubricating oil over the textured surface still exists, but the total friction force, as the sum of asperity contact and fluid viscous shear, remains superior to that of the untextured surface, thereby achieving effective thermal management while maintaining the required friction performance [56].

4. Conclusions

In this study, a data-driven framework integrating experimentation, modeling, and optimization was applied to the design of micro-textures on wet friction pairs. The main conclusions are as follows:

(1)

By including operating parameters (normal load and rotational speed) in addition to geometric parameters, the optimization process provided a set of solutions suitable for different working conditions, rather than a single fixed design.

(2)

A high-precision GPR surrogate model was successfully constructed: The model accurately predicts the complex relationship between micro-texture parameters, operating parameters, and frictional performance. Notably, the $∆T$ model’s predictive $R^2$ exceeded 0.96, proving its reliability as an objective function for optimization.

(3)

The Pareto optimal set was obtained using the MOPSO algorithm: This clearly revealed the trade-off relationship between the conflicting objectives of maximizing COF and minimizing $∆T$. The solution set provides designers with a visual decision-making tool to make trade-off choices based on specific performance requirements.

(4)

Optimal design solutions for specific clutch operating conditions were identified: Through analysis of the Pareto optimal set, this study matched corresponding optimal micro-texture geometric parameters to different typical operating conditions of heavy-duty tractor clutches. Compared to and untextured surface, the hexagonal micro-textures of Solution for Situation A and B increased the COF by 2.6% and 1.2%, respectively, and reduced the $∆T$ by 39.2% and 12.1% under the same operating conditions. Furthermore, when compared with the values corresponding to neighboring experimental data points with similar operating and structural parameters, the hexagonal micro-textures in Situation A and B increased the COF by 11.3% and 2.7%, and reduced the $∆T$ by 16.6% and 1.7%, respectively.