A Neural Network-Based Feature Recognition Method in Adaptive Refinement for Efficient Reynolds Equation Solving

Abstract

This study proposes an adaptive refinement method based on feature recognition to rapidly obtain solutions of the Reynolds equation. Leveraging an isogeometric analysis (IGA) solution framework supporting local refinement, three natural refinement features tailored to solving the Reynolds equation in fluid lubrication, including two physical features, a pressure value, a pressure gradient, and an element size feature for discretization, are introduced first to identify mesh elements. Then a neural network model is trained on feature data to predict element classifications effectively. Finally, this model is integrated into the adaptive refinement solution framework and validated through simulations. Comparative validation was conducted on two distinct Reynolds equation instances, with the results demonstrating that the proposed algorithm can effectively evaluate refinement regions globally, avoiding issues such as mesh non-conformity often caused by conventional independent element marking algorithms. The distribution of degrees of freedom is more rational, and the parallel prediction model enhances the speed of the refinement solution.

1. Introduction

The Reynolds equation is a fundamental equation in the field of fluid lubrication, derived from the Navier–Stokes equations under specific lubrication conditions [1], and it is extensively applied in the analysis and design of lubrication systems. However, fluid lubrication problems typically exhibit unsteady characteristics, such as time-dependent terms [2], dynamic loads [3], and multi-field parameter coupling [4]. Consequently, lubrication analysis or the design of related components often necessitates multiple computations of the Reynolds equation (e.g., for time-varying unsteady problems) or iterative computations (e.g., for multi-field coupling problems). Although current methods for solving the Reynolds equation are relatively well-developed, including the finite difference method (FDM) [5], finite element method (FEM) [6], finite volume method (FVM) [7], and the increasingly popular IGA [8], the computational efficiency of a single solution process remains a critical concern when addressing repeated solving demands. Improvements in solving efficiency for coupled or time-varying problems can generally be approached from two perspectives: optimizing the solution of the Reynolds equation itself [9] or enhancing the computation of coupled solutions [10]. Clearly, improvements in the former are more universally applicable and can also support the latter.

In a study by Yang, H., various numerical algorithms for solving the Reynolds equation are comprehensively discussed [11], highlighting that the IGA method is particularly well-suited for solving simplified forms of the Reynolds equation. For instance, the normalized computational domain is relatively regular [12], facilitating discretization, while the unrestricted order of IGA basis functions [13] is especially valuable for solving partial differential equations, particularly those with variable coefficients. Consequently, researchers have successfully adapted IGA for the Reynolds equation [14], demonstrating its advantages in this context. In fact, the IGA method closely resembles the finite element method (FEM) in its overall process, and efficiency improvements can similarly be approached from three aspects: discretization quality of the computational domain (meshing), coefficient matrix computation (stiffness matrix assembly), and equation system solving. Among these, the meshing process significantly impacts both solution accuracy and computational time [15]. Generally, mesh size directly correlates with computational time, while mesh layout affects solution accuracy. Therefore, optimizing mesh layout can enhance computational efficiency. In FEM, adaptive refinement techniques [16] control mesh size by allocating denser or higher-quality elements only in regions with larger computational errors, achieving accurate large-scale global refinement with a reduced mesh size. This approach minimizes mesh size while maintaining accuracy, thereby reducing computational time and improving efficiency [17]. IGA can adopt a similar strategy to enhance computational efficiency. Our previous work has explored this approach, yielding promising results.

It is worth noting that most existing adaptive refinement algorithms rely on threshold-based criteria, confining their scope to specific elements or fields [18]. This approach presents two limitations: firstly, thresholds restrict applicability across varying operating conditions, parameter adjustments, or coupling relationships, and secondly, effective mesh layout optimization requires both local adjustments to specific elements and global considerations across the computational domain [19]. Compounding these issues, the pronounced nonlinearity of the Reynolds equation [20] renders conventional error estimation algorithms inadequate for accurately capturing errors in complex flows. This stems from a priori and a posteriori error estimation relying on theoretical models [21], which is undermined by continuously varying equation coefficients (e.g., viscosity and pressure), leading to error estimates that are highly sensitive to convergence point distributions and exhibiting poor stability [22], particularly under significant spatial or temporal variations [23]. To overcome these limitations, this paper proposes classifying mesh elements using natural refinement features, thereby eliminating threshold-related constraints. Furthermore, by employing a neural network model for global-level element recognition and classification within the computational domain, the proposed method enhances mesh layout optimization efficiency while supplanting computationally intensive error calibration techniques.

This paper is structured as follows: Initially, leveraging an IGA framework, three natural refinement features are introduced to classify mesh elements. Subsequently, a neural network model is trained with feature data to predict element categories across the computational domain. This model is then incorporated into an adaptive refinement framework, establishing a feature recognition-based adaptive refinement algorithm. Finally, the algorithm’s effectiveness is validated through simulations and comparative analysis.

2. PHT-Based IGA and Neural Network Feature Recognition

This section first briefly introduces the basic process of discretely solving the Reynolds equation using IGA, then discusses the refinement method of PHT splines, and finally summarizes the advantages of feature recognition in the refinement process.

2.1. Discrete Solution of the Reynolds Equation Using IGA

The most general form of the mathematical model of the Reynolds equation, which describes the pressure distribution of a lubricating oil film, is as follows [24]:

$${\displaystyle \frac{\partial }{\partial \mathrm{x}}}\left({\displaystyle \frac{\mathsf{\rho }{\mathrm{h}}^3}{12\mathsf{\eta }}}{\displaystyle \frac{\partial \mathrm{p}}{\partial \mathrm{x}}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{y}}}\left({\displaystyle \frac{\mathsf{\rho }{\mathrm{h}}^3}{12\mathsf{\eta }}}{\displaystyle \frac{\partial \mathrm{p}}{\partial \mathrm{y}}}\right)={\displaystyle \frac{\partial }{\partial \mathrm{x}}}\left({\displaystyle \frac{\mathsf{\rho }\mathrm{h}\left({\mathrm{u}}_1+{\mathrm{u}}_2\right)}{2}}\right)+{\displaystyle \frac{\partial }{\partial \mathrm{y}}}\left({\displaystyle \frac{\mathsf{\rho }\mathrm{h}\left({\mathrm{v}}_1+{\mathrm{v}}_2\right)}{2}}\right)$$

(1)

where $\mathrm{p}$ is the pressure; $\mathrm{h}$ is the film thickness; $\mathsf{\eta }$ is the viscosity; $\mathsf{\rho }$ is the density; and ${\mathrm{u}}_1$ and ${\mathrm{u}}_2$ and ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$ represent the tangential velocities of the two lubricated surfaces in the x- and y-directions, respectively.

After performing appropriate variable substitutions and normalization, Equation (1) can be expressed as the following second-order partial differential equation:

$$\nabla \left(C\nabla p\right)=f$$

(2)

where $p$ is the pressure field to be solved; $C$ is a second-order diffusion tensor (matrix) describing the flow characteristics of the medium, determined by the oil film thickness $\mathrm{h}$ and dynamic viscosity $\mathsf{\eta }$; $\nabla$ is the gradient operator; and $f$ is the source term induced by the relative motion of lubricated surfaces.

To obtain a numerical solution, Equation (2) is transformed into an equivalent weak integral form. Based on the standard Galerkin method and integration by parts [25], the bilinear form $a\left(p,v\right)$ and the linear form $l\left(v\right)$ for solving the Reynolds equation can be defined as follows:

$$a\left(p,v\right)=-{{\iint }_D\nabla p}^{T}C\nabla vdxdy$$

(3)

$$l\left(v\right)={\iint }_{D}fvdxdy$$

(4)

To facilitate the computational handling of complex physical geometries, IGA introduces the concept of a parameter domain $D_0$. The parameter domain is typically a simple, regular rectangle (e.g., the unit square $\left[0,1\right]\times [0,1]$), with coordinates denoted by $\left(U,V\right)$. The bridge connecting this simple parameter domain to the actual physical domain $D$ is a geometric mapping $F$. This mapping $F$ precisely describes how any coordinates $\left(x,y\right)$ in the physical domain are determined by the parameter coordinates $\left(U,V\right)$, as follows [26]:

$$F:D_0\to D,F\left(U,V\right)=\left(\begin{array}{c}x\\ y\end{array}\right)$$

In IGA, using PHT-spline functions, $F$ can be defined as

$$F\left(U,V\right)={\sum }_{i=1}^m{\sum }_{j=1}^n{R}_{ij}\left(U,V\right)G_{ij}$$

(5)

where $R_{ij}$ represents the spline basis functions [27], which can be selected from options such as NURBS, HB, THB, or PHT. This study adopts PHT spline basis functions, which support refinement. Their relevant properties are referenced in studies by Bazilevs et al. [28] and Yuan et al. [29], and the next section will also discuss their refinement process in IGA.

Based on the chain rule for $p\left(x,y\right)=p\left(F\left(U,V\right)\right)$, the differential form can be expressed as

$${\nabla }_{\left(x,y\right)}p\left(x,y\right)=DF{\left(U,V\right)}^{-T}\nabla \left(U,V\right)p\left(U,V\right)$$

(6)

where $DF\left(U,V\right)$ denotes a 2 × 2 Jacobian matrix:

$$DF\left(U,V\right)=\left(\begin{array}{c}{\displaystyle \frac{\partial F_1}{\partial U}}{\displaystyle \frac{\partial F_1}{\partial V}}\\ {\displaystyle \frac{\partial F_2}{\partial U}}{\displaystyle \frac{\partial F_2}{\partial V}}\end{array}\right)$$

(7)

The approximate solution $p$ can be expressed using PHT basis functions and points $q_{ij}\in R$ as follows:

$$p\left(U,V\right)={\sum }_{i=1}^m{\sum }_{j=1}^n{R}_{ij}\left(U,V\right)q_{ij}$$

(8)

According to the Galerkin method, the test function $V$ is selected from the same basis function space as the approximate solution; i.e., $V$ sequentially takes all basis functions $R_{ij}$. Substituting the pressure approximation (5) and the coordinate transformation relation into Equation (2) yields a large sparse linear algebraic system:

$$\mathrm{A}\mathrm{q}=\mathrm{b}$$

(9)

where $\mathrm{q}$ is a column vector composed of all unknown control variables $q_{ij}$.

It should be emphasized that the general Reynolds Equation (2) provides the governing equation for a class of problems. For specific physical models, the physical domain $D$, the specific form of the diffusion tensor $C$, and the source term determined by different kinematics will vary. Therefore, for each specific model, the general Equation (2) must be appropriately specialized to establish a mathematical model that accurately describes its lubrication characteristics.

2.2. Boundary Conditions

First, on the fixed geometric boundaries of the solution domain, conventional Dirichlet or Neumann conditions must be applied to ensure the problem is well-posed.

Several common models exist for handling boundary conditions in the Reynolds equation, primarily aimed at addressing cavitation issues [30]:

• Gumbel condition: This model employs a simple post-processing approach, setting the negative pressure regions obtained from the solution to zero.
• Reynolds boundary condition: This theory requires that both the pressure and its gradient be zero at the unknown oil film rupture boundary.
• JFO mass conservation theory: The JFO theory unifies the full-film and cavitation regions under a mass conservation framework by introducing the oil film fractional content θ.

However, different cavitation models have their own advantages and disadvantages in terms of computational complexity and physical accuracy. To simplify computations, this study primarily adopts the Reynolds boundary condition in subsequent numerical solutions, while more complex models, such as the JFO mass conservation theory, can be further explored in future work.

2.3. PHT-Based IGA Refinement

A core advantage of PHT splines lies in their support of efficient local refinement. When combined with appropriate refinement strategies, they can significantly enhance the computational efficiency of the algorithm while maintaining accuracy. For detailed information on PHT splines, refer to the article by Wen, Z. [31]. The following takes a two-dimensional problem as an example to briefly introduce the refinement process of PHT splines.

As shown in Figure 1, an example of local refinement using PHT in IGA features three colored points divided into two categories: interior points and exterior points. Blue points are exterior points, while green and red points are interior points, with interior points further classified as cross points (red points) and T-points (green points). In practice, the PHT method involves inserting a cross at the center of the current element to divide a parent element into four child elements.

The following Figure 2 provides a PHT-based IGA refinement process:

During the refinement process, leveraging the properties of PHT splines, it is only necessary to mark the elements to be refined according to the strategy to reconstruct the parametric domain and complete the refinement process in the physical domain. It should be noted that the IGA method is essentially a parameterized finite element approach, and the PHT-based IGA method, by default, maps the physical domain to ${\left[01\right]}^n$, where $n$ denotes the number of independent parameters or dimensions and $n=2$ in this context.

2.4. Neural Network-Based Feature Recognition

From the preceding refinement process, it is evident that an appropriate element marking algorithm is critical for optimizing mesh layout. Numerous existing methods enable element marking for region recognition or local refinement [32]. However, these methods often exhibit low efficiency and lack a global perspective, making it challenging to achieve effective adaptability in complex geometric and physical problems. Moreover, they typically permit adjustments only within confined regions, failing to respond effectively to large-scale variations, particularly in handling complex variable-coefficient problems [33], where comprehensive optimization is often unattainable. Most critically, the majority of these algorithms rely on threshold-based specific refinement rules, which impose significant limitations on their applicability.

To address these limitations, employing feature recognition to identify elements for refinement provides an effective solution. Feature recognition methods overcome the constraints of local computation by analyzing key attributes or patterns in data to perform pattern classification and prediction. Currently, mainstream feature recognition methods, represented by machine learning [34], involve algorithms that learn patterns from data to make predictions or decisions. Unlike traditional threshold-based algorithms, machine learning does not rely on explicitly defined rules but adapts models by extracting features and patterns from data. Among various machine learning approaches, neural networks [35] are a particularly popular method. This study employs neural networks for feature recognition, utilizing supervised learning, although other machine learning methods are also applicable.

3. Adaptive Refinement Algorithm Based on Neural Network Feature Recognition

To effectively recognize elements for refinement and achieve the refinement process, this study builds on prior research by first proposing three natural refinement features for solving the Reynolds equation in fluid lubrication: pressure values and pressure gradients as physical characteristics and element-scale features for discrete representation. After extracting these element features, traditional sorting and computational classification are performed on each mesh element, assigning refinement labels based on specific refinement criteria. Subsequently, a neural network model is trained using the existing feature data and corresponding labels, enabling accurate classification and prediction of all mesh elements in the current discrete solution domain. Finally, this model is integrated into an adaptive refinement framework.

This approach considers both local and global perspectives: locally, it calculates feature information, such as the physical properties of sampling points within each element and element-scale representations (e.g., parametric element area), and globally, it defines labels for each element through data processing and classification criteria, marking elements and defining refinement based on label classification. This feature-based refinement approach circumvents threshold constraints, while the introduction of neural networks replaces cumbersome iterative sorting computations, enhancing refinement efficiency and robustness.

3.1. Definition of Element Feature Information

To effectively align with the natural characteristics of lubrication problems, this study adopts physical characteristics (e.g., pressure and pressure gradients) and geometric features (e.g., element size) as evaluation criteria for marking refinement regions. These physical quantities not only possess clear physical significance but also intuitively reflect the impact of different regions on the solution results [36]. Through feature recognition [37], the refinement process extends beyond local optimization, enabling rational judgments based on physical features across the global domain.

The feature information in this study is categorized into physical features and geometric features. Physical feature information, including pressure and pressure gradients, reflects the physical properties of sampling points within each element. Geometric feature information, represented by the absolute size of elements, pertains to regional information.

Considering the correlation between element size and physical information, this study integrates physical and geometric information to propose the concept of relative element scale, aiding in the global assessment of elements.

In summary, the four types of element feature information are shown in

Table 1. Element feature information.

Parameter	Component Structure	Definition
$P$	$p_1,\dots ,p_l$	Pressure
$dP$	${d{P}^U(dp}_1^U,\dots ,{dp}_l^U),d{P}^V({dp}_1^V,\dots ,{dp}_l^V)$	Pressure gradient
$S$	$∆U\times ∆V$	Element absolute scale
$S_{\Pi }$	$P{W}^T\mathrm{S}$	Element relative scale

below.

Here, $l$ represents the number of Gauss quadrature points.

When extracting feature information, Gaussian quadrature points are used to place sampling points within the element. In IGA, for the physical field to be solved (taking the two-dimensional Reynolds equation as an example), the numerical solution $p$ at any parameter point $\left(U,V\right)$ can be obtained using Equation (8). Furthermore, the pressure gradient can be computed as follows:

The pressure gradients in the x-direction (u-direction) and y-direction (v-direction) can be expressed as

$${dp}_U={\displaystyle \frac{\partial p}{\partial U}}=\sum {\displaystyle \frac{\partial R}{\partial U}}DF{\left(U,V\right)}^{-T}\nabla \left(U,V\right)p\left(U,V\right)G_{ij}$$

(10)

$${dp}_V={\displaystyle \frac{\partial p}{\partial V}}=\sum {\displaystyle \frac{\partial R}{\partial V}}DF{\left(U,V\right)}^{-T}\nabla \left(U,V\right)p\left(U,V\right)G_{ij}$$

(11)

Here, $D$ denotes the physical domain, $G_{ij}\in D$, and $\nabla \left(U,V\right)$ represents the derivative vector of the basis function, directly computed from the spline function definition. Similarly, $DF\left(U,V\right)$ is a 2 × 2 Jacobian matrix, as shown in Equation (7).

Specifically, the Jacobian matrix becomes an identity matrix I in the mapping from physical space to parametric space after PHT discretization, primarily because the derivation of the Reynolds equation involves a regularization of the solution domain that produces a normalization-like effect.

For the geometric feature information $S$, this paper uses the parametric element area, which can be calculated from the coordinates of the element’s four vertices.

The relative scale integrates geometric and physical feature information by multiplying the element pressure value by the element’s absolute scale. Here, the element pressure value $N$, representing the pressure integral over the element domain, can be expressed as

$$\begin{gathered} \mathsf{\Pi }={\iint }_{D}p\left(U,V\right)dUdV \\ ={\sum }_{i=1}^l{p}_i{w}_i·\left|D{F}_i\right| \end{gathered}$$

(12)

Here, $p_i$ and $w_i$ represent the pressure value and integration weight calculated at the internal integration points of the element, respectively, while $D{F}_i$ denotes the Jacobian determinant corresponding to the Gaussian integration points; after mapping, it becomes an identity matrix, and the determinant $\left|D{F}_i\right|$ has a value of 1. Assuming all elements adopt the same sampling rule, $w_i$ can be regarded as a constant coefficient, identical for all elements.

$\mathsf{\Pi }$ approximates the pressure integral of the physical domain, representing the total pressure contribution of the element, which is influenced by the element area $S$. Therefore, the concept of relative scale ($S_{\Pi }$) is proposed to represent the pressure contribution of the element area.

As shown in Figure 3, under the same solution distribution, the physical and geometric features of Element 1 remain unchanged across both discretization scenarios. However, in Scenario 2, the refinement features of Element 1 become more pronounced. This paper enhances the refinement features of Element 1 in Scenario 2 by introducing the element pressure value into the absolute scale to construct a relative scale.

The element feature information can be organized into a set of element feature vectors, as shown in Equation (13), providing a basis for subsequent element classification and foundational data for neural network training.

$$Q_z=\left[P_z,{dP}_z,S_z,S_{\Pi z}\right]$$

(13)

In this equation, $z$ denotes the element index.

3.2. Refinement Method Based on Feature Information

This section primarily introduces the process of defining labels for each feature of elements based on element feature data, marking elements for refinement according to a unified refinement classification criterion, and thereby achieving adaptive refinement. Furthermore, a neural network is introduced to perform learning and training on the existing element data and labels, enabling the classification and recognition of element refinement features.

3.2.1. Data Processing of Feature Information

$P$, $dP$, and $S_{\Pi }$ belong to different data types and exhibit significant differences in magnitude; therefore, they should be standardized [38] separately by category before label definition. Within $dP$, $d{P}^U$ and $d{P}^V$ are also standardized individually due to directional differences. The absolute scale $S$ is obtained from the mapped parametric coordinates and does not require further standardization. This paper adopts the Z-score standardization method, although other methods may also be applied. The general expression for the Z-score is [39]

$$Z={\displaystyle \frac{T-\mu }{\sigma }}$$

(14)

In Equation (14), $Z$ represents the standardized value, $T$ is the raw data value, $\mu$ is the mean of the raw data, and $\sigma$ is the standard deviation of the raw data. After standardization, the Z-scores for most data points fall within the range of [−3, 3].

After processing the feature information, the element feature vector $\left[\overline{P},\overline{d{P}^U},\overline{d{P}^V},S,\overline{S_{\Pi }}\right]$ can be obtained.

In particular, it should be noted that the pressure and pressure gradient in the feature information are multidimensional data, whereas the scale information is one-dimensional data. To ensure consistent standards and enhance the generalization performance of subsequent neural network training, the pressure and pressure gradient data are transformed into one-dimensional data using a method similar to Equation (13), and a new gradient magnitude $\overline{dN}$ is introduced as follows:

$$N=P{W}^T$$

$${dN}^U={dP}^U{W}^T$$

$${dN}^V={dP}^V{W}^T$$

$$\overline{dN}=\sqrt{{\left({dN}^U\right)}^2+{\left({dN}^V\right)}^2}$$

However, Z-score normalization assumes that the data distribution is approximately normal and only performs linear scaling of relative differences between features, which cannot effectively amplify the differences in certain key features, particularly when the data distribution is asymmetric or contains extreme values. Therefore, before applying Z-score normalization, a nonlinear piecewise transformation based on the median is employed here to preprocess the one-dimensional pressure and pressure gradient data, enhancing their differences and facilitating feature recognition.

$$H=\left\{\begin{array}{c}M+{(T-M)}^2,T>M\\ M-{(M-T)}^2,T\le M\end{array}\right.$$

(15)

Equation (15) represents the basic formula for the nonlinear piecewise transformation based on the median, where $H$ is the transformed value, $T$ is the original data, and $M$ is the median of the data.

3.2.2. Label Definition of Feature Information

The feature information of elements is processed through piecewise linear transformation based on median values and standardization (data processing) to obtain the element feature vector $\left[\overline{N},\overline{d{N}^U},\overline{d{N}^V},\overline{dN},S,\overline{S_{\Pi }}\right]$ used for classification. This study classifies four features—pressure, pressure gradient, relative scale, and absolute scale—into three categories, mapped to labels 1 (small), 2 (medium), and 3 (large). For pressure, pressure gradient, and relative scale, the hierarchical agglomerative clustering algorithm is employed for classification [40]. However, the classification approach for the absolute scale differs. Due to the binary refinement property of PHT splines, the absolute scale of a sub-element is a fixed quarter of its parent element. Therefore, after several refinements, the absolute scale of elements tends to follow a fixed numerical sequence, allowing for direct division using predefined thresholds without requiring clustering.

Ultimately, through the above methods, labels corresponding to the four types of feature information of the element can be obtained. These four labels are combined, as shown in Figure 4, to form a four-digit label, which serves as the final feature label for the element.

3.2.3. Element Type Definition and Refinement Criteria

Once the feature labels of the element are determined, the elements are classified based on these labels, and those requiring further refinement are explicitly identified to provide a basis for refinement recognition in the neural network. During the classification process, attention must be paid not only to the final feature labels of the elements but also to a comprehensive evaluation incorporating refinement criteria to ensure the accuracy of classification and the rationality of refinement. The core principle of refinement prioritizes scale, with pressure and pressure gradient as supplementary factors. The refinement criteria are as follows:

First, when the absolute scale label of an element is 1 (or 3), regardless of the other feature labels, the element must not be refined (or must be refined). When the absolute scale label is 2, the refinement decision is no longer determined solely by scale but is assisted by the combined strength of the feature labels. When the sum of the label values is greater than or equal to 7, the element is considered to exhibit high nonlinearity or significant gradient changes across multiple dimensions, demonstrating pronounced non-uniformity and instability, and thus requires refinement ($E_r=1$). Otherwise, it is not refined ($E_r=0$).

The classification table is summarized as follows (

Table 2. Element classification.

$\mathit{S}$	$\overline{\mathit{N}}$	$\overline{\mathit{d}\mathit{N}}$ ($\overline{\mathit{d}{\mathit{N}}^{\mathit{U}}},\overline{\mathit{d}{\mathit{N}}^{\mathit{V}}}$)	$\overline{{\mathit{S}}_{\mathit{\Pi }}}$	Final Label (${\mathit{E}}_{\mathit{f}}$)	Category (${\mathit{E}}_{\mathit{c}}$)	Refinement State (${\mathit{E}}_{\mathit{r}}$)
3	1, 2, 3	1, 2, 3	1, 2, 3	3111 (3×××,…)	1	1
1	1, 2, 3	1, 2, 3	1, 2, 3	1111 (1×××,…)	2	0
2	1	1	1	2111	3	0
2	1	1	2	2112	4	0
2	1	1	3	2113	5	1
2	1	2	1	2121	6	0
2	1	2	2	2122	7	1
2	1	2	3	2123	8	1
2	1	3	1	2131	9	1
2	1	3	2	2132	10	1
2	1	3	3	2133	11	1
2	2	1	1	2211	12	0
2	2	1	2	2212	13	1
2	2	1	3	2213	14	1
2	2	2	1	2221	15	1
2	2	2	2	2222	16	1
2	2	2	3	2223	17	1
2	2	3	1	2231	18	1
2	2	3	2	2232	19	1
2	2	3	3	2233	20	1
2	3	1	1	2311	21	1
2	3	1	2	2312	22	1
2	3	1	3	2313	23	1
2	3	2	1	2321	24	1
2	3	2	2	2322	25	1
2	3	2	3	2323	26	1
2	3	3	1	2331	27	1
2	3	3	2	2332	28	1
2	3	3	3	2333	29	1

):

Ultimately, all elements are classified into 29 categories, with corresponding refinement results for each category (0 indicating no refinement; 1 indicating refinement). The number of categories can be expanded or adjusted as needed. The entire framework for defining labels and classification is structured as shown in Figure 5 below:

The current solution space is computed using the PHT method, followed by the extraction of feature information for element points and overall element feature information using Gaussian quadrature point techniques. These feature data are then integrated into a set of element feature vectors $\left\{Q_z\right\}$, which undergo standardization and other processing to generate an optimized group of element feature vectors. Finally, feature labels are defined, and element categories, feature labels, and refinement decisions are determined based on specific refinement criteria. Thus, a feature-based refinement method is fully established.

3.2.4. Parameter Optimization for Nonlinear Transformation and Clustering Methods

The parameter selection for nonlinear transformations and clustering methods is critical to the effectiveness of element classification. By comparing the element classification results under different parameter configurations with the expected outcomes, their performance can be effectively evaluated and optimized. Taking the piston–cylinder system as an example, the element feature labels (blue—1; green—2; yellow—3) are displayed in the format of Figure 4, with the results from the first to the fourth classifications recorded as shown below (this figure and all subsequent similar figures are plotted in the nondimensional computational domain, with the horizontal and vertical axes representing the previously defined nondimensional coordinates $X$ and $Y$, respectively, both ranging from $[0,1]$. The z-axis of the cloud maps represents the oil film pressure with physical units (Pa).

In Figure 6, by examining the cloud maps in the left two columns, the feature characteristics of a specific element can be clearly assessed, while the color blocks of feature labels in the right four columns are obtained through nonlinear transformation and hierarchical clustering methods. By comparing the left and right sides, the reasonableness of element classification can be evaluated, thereby selecting the optimal parameter combination for the current model. For example, in the right four columns of (1), the element located in the lower right corner has an absolute scale color block of 3 (yellow), a pressure color block of 1 (blue), a pressure gradient color block of 2 (green), and a relative scale color block of 1, resulting in a feature label of 3121, corresponding to an $m$-value of 1, indicating the need for refinement. Observing the cloud maps in the left two columns, it is evident that this element has a large pressure gradient, a low pressure, and a large absolute scale, clearly requiring refinement. Only when both assessments are consistent can the classification be deemed reasonable and accurate.

3.3. Neural Network Model Construction

The element refinement process in traditional IGA typically employs a serial approach, requiring algorithm reconfiguration when the physical field or parameters change, leading to low efficiency. Although parallel methods can accelerate computation, their lack of generality still necessitates algorithm adjustments for new conditions.

To overcome the bottleneck of flexibility and efficiency, this study proposes replacing the traditional classification decision process with a machine learning approach. Its core advantage lies in the model’s ability to automatically adapt to changes in the physical field or parameters based on standardized element feature data, determining whether refinement is needed, thereby enhancing both flexibility and efficiency.

This study selects a neural network (NN) as the core model due to its ability to automatically learn complex refinement patterns from high-dimensional data through nonlinear activation functions, such as Sigmoid [41], and multilayer structures, without requiring manual feature design. During training, key information such as element pressure, pressure gradient, and size is used to construct the feature vector $\overline{Q_z}$ as input, with all elements collectively forming the dataset $\left\{\overline{Q_z}\right\}$. To ensure the model’s generalization capability, the training data encompass various physical field information generated through parameter variations. This approach significantly simplifies the computational process and substantially improves processing efficiency through CPU parallel computing.

The data generated by the method of refining elements based on feature labels are used as input for neural network training. Meanwhile, to enhance the generalization of the neural network model, model parameters need to be adjusted to incorporate element information from various physical fields for training.

In Figure 7, $E_l$, $J$, and $K$ represent the element hierarchy, different lubrication model parameters, and different lubrication models, respectively. Transformations under these three conditions yield numerous element feature vectors, which are used as input for training the neural network model to enhance its applicability and accuracy.

Once the neural network training achieves the desired performance, the model can be applied to the algorithm, as shown in the process diagram below.

In Figure 8, the flow of feature data and the feature recognition process based on neural networks are clearly illustrated in a left-to-right sequence. The process begins with an initial mesh, where a feature vector ${\left\{\overline{Q}\right\}}_{28\times 9}$ is first extracted for each element. These feature vectors are then input into a pre-trained BP neural network for processing. The neural network outputs a predicted category for each element and generates a binary refinement state based on predefined criteria. Ultimately, only elements marked with a value of 1 are refined, resulting in a new adaptively refined mesh. Compared to Figure 5, this approach effectively replaces the classification and labeling process with a neural network.

Neural Network Model Parameter Selection

The performance of neural network models heavily depends on the selection of model parameters, including network architecture, optimizer, learning rate, regularization strategy, and batch size, among other key factors. The network architecture, typically determined by the number of hidden layers and their neuron counts, directly affects the model’s ability to capture data features. The optimizer (e.g., Adam, SGDM, or RMSProp) dictates the weight update strategy, influencing convergence speed and stability. The learning rate controls the step size of parameter updates, requiring a balance between convergence speed and training stability. Additionally, regularization methods (e.g., L2 regularization) and batch size play critical roles in the model’s generalization ability and training efficiency. Properly configuring these parameters is essential for enhancing the predictive accuracy and generalization capability of neural networks.

First, a baseline model (Model 1) is established with the following parameter configuration: two hidden layers with 16 and 8 neurons, respectively, a learning rate of 0.001, a batch size of 40, the Adam optimizer, an L2 regularization coefficient of 0, and a dataset split of 70%/10%/20% (7:1:2). The input data consists of 28-dimensional row vectors. The two-layer hidden structure, with 16 neurons in the first layer, effectively extracts key features from the 28-dimensional input, while the second layer with 8 neurons further compresses information, reducing the risk of overfitting while maintaining moderate model complexity. A learning rate of 0.001 is a commonly used initial value, and a batch size of 40 balances training efficiency and gradient stability. The Adam optimizer, leveraging the advantages of momentum and RMSProp, ensures fast and stable convergence. The dataset split of 70% for training, 10% for validation, and 20% for testing follows standard practice, with 70% ensuring sufficient training, 10% for parameter tuning, and 20% for evaluating generalization, providing a reliable baseline. Based on this, five groups of neural network models with different parameter configurations are compared to systematically evaluate the impact of each parameter on model performance, thereby selecting more optimal parameter settings (

Table 3. Settings of neural network models with different parameter configurations.

Model ID	Number of Hidden Layers	Number of Neurons per Layer	Learning Rate	Batch Size	Optimizer	L2 Regularization Coefficient	Dataset Split
1	2	16, 8	0.001	40	Adam	0	7:1:2
2	3	32, 24, 16	0.01	80	SGD with Momentum	0.001	6:2:2
3	1	48	0.0001	20	RMSProp	0	8:1:1
4	4	64, 32, 16, 8	0.001	160	Adam	0.01	7:1:2
5	2	24, 12	0.1	40	SGD with Momentum	0	6:2:2

).

As shown in Figure 9, Model 1 achieves an output accuracy of 96.12% on the test set, significantly outperforming other models. To further enhance the neural network model’s performance, a Bayesian optimization algorithm was introduced to fine-tune key parameters, such as the number of neurons and hidden layers. However, the optimized model’s accuracy was comparable to Model 1, with no significant improvement. Therefore, considering both performance and stability, Model 1’s parameter configuration was adopted as the current optimal choice. Following the process depicted in Figure 8, the element feature label result diagram was generated to intuitively validate the label prediction accuracy of Model 1.

The layout of Figure 10 is consistent with that of Figure 6, and it is based on the same physical field lubrication model as Figure 6, representing four adaptive refinement iterations performed according to neural network predictions. Compared to Figure 6, the labels predicted by the neural network show significant differences from those defined in Figure 6. However, as the refinement level increases, the number of elements requiring prediction also rises, as shown in (4), and the prediction accuracy of the neural network gradually improves. From the final element refinement states, the performance of the neural network method converges with that of the feature label method. Therefore, in cases with a large number of elements, Model 1 demonstrates superior performance.

In addition to the above parameters, batch normalization is a critical technique in neural network architecture design, effectively enhancing training stability and prediction accuracy. Therefore, a batch normalization layer is introduced during the neural network construction process to further optimize model performance. By comparing the differences in refinement results between models with and without batch normalization, the specific impact of batch normalization on model accuracy can be systematically evaluated.

In Figure 11, the left column displays the element refinement results predicted by the neural network model without batch normalization, while the right column shows the results predicted by the neural network model with batch normalization. A comparison of the refinement result figures indicates that the two are highly similar, suggesting that batch normalization has a limited impact on the predictive performance and element refinement results of the current neural network. However, the refinement results in the left column are structurally more standardized and rational. Therefore, based on the performance of the current model, batch normalization is not used.

4. Validation and Comparative Studies

Through the aforementioned steps, a neural network model can be successfully constructed and trained to accurately perform feature recognition from element information and predict element categories. To validate the accuracy and robustness of the proposed algorithm, two independent neural network models (BP-1 and BP-2) were trained using the lubrication model of the piston–cylinder system and the Hydrodynamic sliding bearing model, respectively, for performance evaluation. To further investigate the generalization ability and cross-scenario adaptability of neural networks in refinement tasks for different lubrication problems, BP-1 was used to predict element categories for the hydrodynamic sliding bearing model, and BP-2 was similarly used to predict element categories for the piston–cylinder system lubrication model, with a comparison of model prediction accuracy and refinement results. Additionally, the data from both lubrication models were combined into a single dataset to train a new neural network model (BP-3). The generalization validation results demonstrate that BP-3, as a single model, can directly adapt to refinement tasks for various lubrication problems without requiring scenario-specific data preparation or dedicated model training, thereby significantly enhancing the model’s versatility and application efficiency.

4.1. Case Study I: Piston–Cylinder Lubrication

Figure 12a shows the piston–cylinder liner configuration, and Figure 12b illustrates the oil film lubrication. $R$ denotes the crank radius; $e_t$ and $e_b$ denote the distances between the center of the piston skirt and the cylinder liner axis at the top and bottom of the skirt, respectively. $L_{Sk}$ denotes the piston skirt length.

The morphology diagram in Figure 12a is obtained by solving a specific simplified Reynolds equation [42] based on the Navier–Stokes equations and the continuity equation.

The Reynolds equation for this model can similarly be derived through specialization from the general Equation (2). The key difference lies in the fact that the oil film thickness function $h$ varies across different lubrication models. For the piston–cylinder model, the solution for the oil film thickness $h$ is expressed as follows:

$$h\left(\phi ,y\right)=R-{\displaystyle \frac{x_0{r}^2\mathit{cos}\left(\phi \right)+ar\sqrt{\left(a^2-x_0^2{\mathit{sin}}^2(\phi \right)+r^2{\mathit{cos}}^2\left(\phi \right)}}{\left[a^2{\mathit{sin}}^2\left(\phi \right)+r^2{\mathit{cos}}^2\left(\phi \right)\right]}}$$

(16)

where

$$a={\displaystyle \frac{r{L}_{SK}}{\sqrt{{L_{SK}}^2-{\left(e_t-e_b\right)}^2}}},x_0=e_b+{\displaystyle \frac{y}{L_K}}\left(e_t-e_b\right)$$

Furthermore, to enhance computational efficiency, this study leverages the symmetry of the model, solving only one-quarter of the lubrication region. In summary, due to the unique oil film thickness function $h$ and the coordinate transformation tailored to specific regions, the resulting diffusion tensor $C$ and source term $f$ also take on specific forms:

$$C=\left(\begin{array}{cc}{\displaystyle \frac{h^3}{12\eta R^2{\phi }_R^2}}& \\ & {\displaystyle \frac{h^3}{12\eta L_{SK}^2}}\end{array}\right) ,f={\displaystyle \frac{v}{2}}{\displaystyle \frac{\partial h}{\partial y}}-{\displaystyle \frac{\partial h}{\partial t}}$$

(17)

As a specific numerical example, we first consider the piston–cylinder system, using the Reynolds boundary conditions [43]. Other boundary conditions are treated similarly, and the dynamic calculations are omitted for brevity. The input parameters, based on a specific single-cylinder gasoline engine, are listed in

Table 4. Basic input parameters in the calculation example.

Parameter	Value	Unit
$r$	$21.725\times 10^{-3}$	m
$R$	$21.75\times 10^{-3}$	m
$L_{sk}$	$22.5\times 10^{-3}$	m
$\eta$	$0.01295$	Pa · s
$e_t$	$0.78565\times 10^{-5}$	m
$e_b$	$-0.24924\times 10^{-4}$	m
$d_{et}$	$-0.24335\times 10^{-2}$	m · s−1
$d_{eb}$	$-0.84461\times 10^{-2}$	m · s−1
$v_{pr}$	$-11.6543$	m · s−1
$c_b$	$6\times 10^{-3}$	m
$c_c$	$0$	m

[44].

As observed in the figures, the refinement outcomes are similar in the early stages for both methods. However, as the number of refinement iterations increases, as shown in Figure 13c, some elements that should have been refined were not, indicating inaccurate classification. Nevertheless, through the data processing method, each refinement prioritizes identifying critical elements for refinement and progressively expands outward, ensuring no elements requiring refinement are missed. This validates the necessity of data processing (referring to standardization and differential amplification here), as discussed in Section 3.2.

As shown in

Table 5. Comparison of degrees of freedom, pressure values, and refinement time for the two cases (the total force value here represents the total integrated pressure over the entire lubrication domain, which is the bearing load capacity).

Number of Refinements	Without Data Processing			With Data Processing
	DOFs	Total Force (N)	Time (s)	DOFs	Total Force (N)	Time (s)
3	217	183.3025	3.1273	217	183.3025	2.3790
4	769	183.5318	4.3321	727	183.5213	3.4891
5	2825	183.5509	6.1079	919	183.5328	4.7209
6	3829	183.5515	24.7455	1281	183.5443	6.1425
7	11,703	183.5529	31.1459	1841	183.5499	7.8066
8	12,869	183.5532	40.8508	2181	183.5510	9.5794

, the pressure value solutions obtained by the two refinement methods are similar. However, the method with data processing achieves comparable pressure value solutions with fewer degrees of freedom, indicating higher efficiency. Moreover, as the number of refinements increases, the time consumed by the method without data processing grows significantly, reaching approximately four times that of the method with data processing by the eighth refinement, which substantially reduces refinement efficiency.

In comparing these two methods, the input parameter $v_{pr}=-10.6543$, while other parameters remained consistent with Table 4. In Figure 14, the left column displays the 2D refinement results, while the right column shows the corresponding 3D results. A comparison of the left column reveals that in regions with large pressure gradients, the local method fails to adequately refine these areas. Similarly, for regions with high pressure, refinement by the local method is insufficient.

From

Table 6. Comparison of data results.

Number of Refinements	Local Consideration PHT-IAG Method			Feature Classification
	DOFs	Total Force (N)	Time (s)	DOFs	Total Force (N)	Time (s)
1	25	178.5365	0.0414	25	162.5501	0.0394
2	81	181.1382	1.9617	81	180.7430	1.6458
3	189	181.2513	3.2149	217	183.3025	2.3790
4	287	181.2525	4.3925	727	183.5213	3.4891

, it can be observed that, under the same number of refinements, the feature classification-based method achieves a solution closer to the optimal pressure value and requires relatively less time per refinement. In comparison, the feature classification-based method demonstrates superior refinement performance.

Next, when comparing the feature classification method and the neural network-based feature recognition method, new feature data are directly used as input to the neural network to predict refinement results and conduct a comparison.

In Figure 15, the upper half presents the results of the feature classification method, while the lower half shows the results of the neural network-based feature recognition method, with all results obtained after six refinement iterations. As observed from the figure, the stable element pressure values obtained by both methods are very similar, indicating that the neural network model can accurately predict the refinement of unknown elements.

Once the neural network model has been completed, it can also be applied to mesh refinement in other physical fields. The parameter values are modified as shown in

Table 7. New parameters for the piston–cylinder system model.

Parameter	Value	Unit
$r$	$31.725\times 10^{-3}$	m
$R$	$31.75\times 10^{-3}$	m
$L_{sk}$	$30.5\times 10^{-3}$	m
$\eta$	$0.01395$	Pa · s

.

Figure 16 illustrates the refinement results for different lubrication model parameters under the same refinement iteration and neural network model. The upper part presents the results obtained by the neural network-based feature recognition method under the old lubrication model parameters, while the lower part corresponds to the new lubrication model parameters. This comparison reveals that both the element pressure and pressure gradient changed across the entirety of the new physical field. However, the neural network-based feature recognition method effectively identified and refined regions in the new physical field with high absolute pressure values and significant pressure gradients. This demonstrates the strong generalization capability of the neural network model.

The previous section mentioned that this example represents an initial attempt to apply a neural network method, but other machine learning methods are also applicable. In this example, Support Vector Machine (SVM) [45] is introduced as a new machine learning approach. SVM is a machine learning method based on margin maximization, which maps data to a high-dimensional space through the selection of appropriate kernel functions (e.g., linear kernel, Gaussian kernel, or polynomial kernel), making it suitable for both regression and classification tasks. Its key parameters include the kernel function type, kernel scale, regularization parameter, ε-insensitive loss parameter (Epsilon), and whether to standardize data (standardize), among others. The choice of these parameters directly determines the model’s fitting capability and generalization performance. Similarly to the parameter tuning of neural networks, we designed five sets of controlled simulation experiments to systematically compare the impact of different parameter combinations on SVM model performance, thereby determining the optimal parameter settings. The specific parameter configurations are shown in

Table 8. SVM settings for different parameter models.

Model ID	Kernel Function	Kernel Scale	Standardization	BoxConstraint (C)	Epsilon	Dataset Partition
1	Rbf	Auto	True	1	0.037	7:1:2
2	Linear	-	True	0.1	0.05	6:2:2
3	Polynomial	3	False	10	0.2	8:1:1
4	Rbf	1	True	5	0.01	7:1:2
5	Polynomial	2	True	0.5	0.5	6:2:2

below.

The SVM model is evaluated based on two metrics: predicted residual distribution and test set output accuracy. The results are shown in the figure below.

Figure 17 and Figure 18 illustrate the performance evaluation of five SVM models. Figure 17, which depicts the residual distribution, shows that Model 1 and Model 4 exhibit the smallest prediction bias, with their residuals highly concentrated between −0.1 and 0.1 and a peak frequency approaching 80. In contrast, Model 5 has the largest bias, with the widest residual distribution, spread primarily across the range of −0.3 to 0.3. Figure 18, a bar chart of the m-value accuracy, indicates that Model 1 achieves the highest accuracy (97.53%), followed by Model 4 (96.91%), Model 2 (94.44%), and Model 3 (92.68%). Model 5 has the lowest accuracy at 72.22%, which is consistent with the trend observed in the residual distribution. This analysis suggests that the parameter settings for Model 1 are optimal. Therefore, the SVM model parameters for this study are set according to those of Model 1.

Under the same input conditions, the final cell refinement performance of the neural network and SVM methods is compared, with the results shown in Figure 19.

In the figure, the upper part represents the neural network method, while the lower part represents the SVM method. It can be observed from the figure that the grid sparsity differs between the two methods. The SVM method fails to identify the lower left region during refinement, but the overall cell refinement trend is similar to that of the neural network method, indicating that both methods can effectively capture data features. Based on the current simulation results, the neural network method demonstrates superior performance.

4.2. Case Study II: Hydrodynamic Journal Bearing

Figure 20a depicts the bearing in a non-operating state where the shaft center $O_2$ coincides with the bearing center $O_1$. Figure 20b illustrates the bearing in an operating state, where an external load $W_F$ is applied and the shaft rotates with an angular velocity ω. This results in the shaft center being displaced by an eccentricity $e_o$ at an attitude angle $\mathsf{\phi }$. Figure 20a depicts the bearing in a non-operating state, Figure 20b illustrates the bearing in an operating state, and Figure 20c shows a schematic diagram of the oil film pressure distribution.

Similarly, this model can be derived through specialization from Equation (2), with the solution for $h$ expressed as follows:

$$h=c_a+e_o·\mathit{cos}(\phi )$$

(18)

This model solves for half of the lubrication region, resulting in the following diffusion tensor $C$ and source term $f$:

$$C=\left(\begin{array}{cc}{\displaystyle \frac{h^3}{r^2{\phi }_R^2}}& \\ & {\left({\displaystyle \frac{L_{bw}}{2}}\right)}^2\end{array}\right),f=600\pi \eta ·e_o\mathit{sin}({\phi }_R·X)$$

(19)

The parameters of the radial sliding bearing model for this example are shown in

Table 9. Parameters of the hydrodynamic sliding bearing model.

Parameter	Value	Unit
$c_a$	$5\times 10^{-5}$	m
$n$	$3\times 10^3$	rps
$\lambda$	$5.6\times 10^{-2}$	m
$L_{bw}$	$4.8\times 10^{-2}$	m
$\eta$	$5\times 10^{-2}$	Pa · s
$e_o$	$0.6$
$w$	$100·\mathsf{\pi }$	rad/s
${\mathsf{\phi }}_{\mathrm{R}}$	$\mathsf{\pi }$	rad

below.

The figure results and tables presented next (Figure 21, Figure 22, Figure 23 and Figure 24,

Table 10. Comparison of degrees of freedom and pressure for unprocessed and processed data methods.

Number of Refinements	Without Data Processing			With Data Processing
	DOFs	Total Force (N)	Time (s)	DOFs	Total Force (N)	Time (s)
1	25	1053.4	0.0414	25	1053.4	0.0519
2	81	1523	1.4326	81	1523	1.3354
3	203	1529.1	3.8336	153	1530.5	3.0211
4	557	1529.6	4.4012	487	1529.9	4.2153
5	1967	1529.4	6.3218	881	1529.6	5.1384
6	7205	1529.4	28.2372	1161	1529.6	6.3547

) are consistent with Section 4.1 in terms of methods and content, with the only difference being the lubrication model used.

The model parameters were adjusted as shown in

Table 11. New parameters.

Parameter	Value	Unit
$c_a$	4.8 × 10−5	m
$n$	4 × 103	rps
$e_o$	0.2
$\eta$	4 × 10−2	Pa · s

below, and the modified model was used to generate a new physical field for comparison.

The figures and tables presented above illustrate comparative refinement results under various scenarios. These results demonstrate that the neural network-based feature recognition method exhibits superior adaptability and generalization capability across diverse model configurations. It is noteworthy that, although the SVM and neural network methods exhibit significant differences in terms of the degrees of freedom of the solved elements after the same number of refinement iterations, the computed pressure values are remarkably similar.

4.3. Generalization Validation

This section aims to validate the generalization ability and cross-scenario adaptability of neural networks in refinement tasks for different lubrication problems. Element feature data from two lubrication models (approximately 800 samples each) were collected, denoted as Dataset 1 and Dataset 2, respectively. First, Dataset 1 was used as input to invoke BP-2 for predicting element categories; similarly, Dataset 2 was used as input to invoke BP-1 for predicting element categories. The prediction accuracy was compared with that of the neural network model trained on the corresponding lubrication model. The results are shown in

Table 12. Comparison of prediction accuracy for BP models across different lubrication models.

Neural Network Model	Lubrication Model	Prediction Accuracy
BP-1	Piston–cylinder system	94.55%
BP-2	Hydrodynamic sliding bearing system	70.85%
BP-1	Piston–cylinder system	75.25%
BP-2	Hydrodynamic sliding bearing system	96.95%

below.

Based on the results shown in this table, the current BP models demonstrate a certain degree of cross-scenario adaptability. BP-1 achieves an accuracy of 94.55% for the piston–cylinder system (Dataset 1), but this drops to 70.85% when predicting the hydrodynamic sliding bearing system (Dataset 2). Similarly, BP-2 achieves an accuracy of 96.95% for the hydrodynamic sliding bearing system (Dataset 2) but only 75.25% when predicting the piston–cylinder system (Dataset 1). The decline in prediction accuracy in new scenarios indicates that the generalization ability of the BP models is still limited. By optimizing the model structure or training with a mixed dataset, the adaptability and prediction accuracy of neural networks across different lubrication problems can be further enhanced.

Following this approach, a neural network can be trained using mixed data. Datasets 1 and 2 were combined as training data, ensuring consistent proportions of data from both lubrication models across training, testing, and validation sets, to train BP-3. Subsequently, the parameters of the two lubrication models were adjusted to generate Datasets 4 and 5 (approximately 750 samples each). BP-3 was applied to predict element categories in these datasets, and the model’s accuracy was evaluated. The results are shown in

Table 13. Comparison of prediction accuracy for BP-3 across different lubrication models.

Neural Network Model	Lubrication Model	Prediction Accuracy
BP-3	Piston–cylinder system	95.72%
BP-3	Hydrodynamic sliding bearing system	96.69%

below.

According to the results in this table, BP-3, after training on a mixed dataset, demonstrates strong generalization ability. It achieves an accuracy of 95.72% for the piston–cylinder system (Dataset 4) and 96.69% for the hydrodynamic sliding bearing system (Dataset 5), indicating that the single model BP-3 can effectively adapt to different lubrication scenarios with high prediction accuracy.

Next, different neural network models can be applied to a unified adaptive refinement algorithm for more specific comparative validation. First, BP-1 and BP-2 are separately integrated into the adaptive refinement algorithm, using four datasets from the hydrodynamic sliding bearing model (including new parameters). Additionally, a feature classification method is employed, ultimately generating six refinement results for comparison, as shown in Figure 25 below.

In

Table 14. Data comparison table for different parameter models and feature recognition methods after multiple refinements.

Refinement Method	Number of Refinements	Number of Elements After Refinement	Average Prediction Accuracy
BP-2	3	46	82.97%
BP-1	3	25	45.33%
Feature classification method	3	44	-
BP-2 (NPM)	3	49	80.74%
BP-1 (NPM)	3	25	41.33%
Feature classification method (NPM)	3	44	-

, NPM denotes the new parameter model. As shown in the comparison figures and tables, BP-2, trained on data from the hydrodynamic sliding bearing model, can more accurately predict element categories, thereby achieving superior element refinement performance with a higher average prediction accuracy. Although BP-1, which is unrelated to the training data, is slightly inferior to BP-2 in refinement performance and has lower accuracy, it can still effectively perform feature recognition of critical elements and complete refinement. When the parameters of the hydrodynamic sliding bearing model change, both neural networks maintain relatively stable element category prediction capabilities, indicating strong generalization ability.

Subsequently, the lubrication model of the piston–cylinder system was used, with the results shown in Figure 26 below.

By comparing the element refinement results in the figures with

Table 15. Data comparison table for lubrication models based on different parameter models and feature recognition methods after multiple refinements.

Refinement Method	Number of Refinements	Number of Elements After Refinement	Average Prediction Accuracy
BP-1	3	61	81.28%
BP-2	3	31	71.98%
Feature classification method	3	64	-
BP-1 (NPM)	3	61	75.03%
BP-2 (NPM)	3	31	74.55%
Feature classification method (NPM)	3	52	-

, it is evident that BP-1, trained on its own model data, can better predict the element categories of its own model, while BP-2 performs slightly less effectively, but both exhibit stable prediction capabilities. For BP-2, its model demonstrates relatively high prediction accuracy across different lubrication models, indicating superior generalization performance.

Through generalization validation in specific applications, it is confirmed that the BP-1 and BP-2 models can handle the refinement process in various lubrication scenarios, although they may underperform compared to specialized models in certain cases. It should be noted that this generalization validation is based solely on the average prediction accuracy of three refinement results, which does not fully reflect the true performance of the BP models. In the future, further model tuning and parameter optimization are expected to significantly enhance the generalization capabilities of BP-1 and BP-2, thereby achieving superior performance and higher robustness in a broader range of application scenarios.

Based on the above concept, BP-3 is similarly applied to both lubrication models, yielding the refinement result figures and performance comparison table shown below (

Table 16. Comparison of BP-3 refinement performance across different lubrication models.

Lubrication Model	Number of Refinements	Number of Elements After Refinement	Average Prediction Accuracy
Piston–cylinder system	3	64	85.42%
Piston–cylinder system (NPM)	3	61	76.67%
Hydrodynamic sliding bearing model	3	49	80.06%
Hydrodynamic sliding bearing model (NPM)	3	52	78.53%

, Figure 27).

Analysis of the refinement comparison figures and tables reveals that the BP-3 model, trained on mixed data, effectively predicts element categories for both the piston–cylinder lubrication model and the hydrodynamic sliding bearing model, achieving a high average prediction accuracy. Additionally, this model demonstrates robust and stable generalization performance, outperforming the BP-1 and BP-2 models with notably superior overall performance and prediction accuracy.

Although the feature classification method (Method 2) overcomes many limitations of the residual judgment method based on bubble functions (Method 1) and enhances refinement efficiency through parallel operations in nonlinear transformation and hierarchical clustering, these two steps remain time-consuming when the number of elements is large. In contrast, the BP neural network method (Method 3) eliminates these steps by directly generating outputs from inputs through data parallelism, further improving refinement efficiency. To validate its refinement efficiency, full refinement of elements was performed using the piston–cylinder lubrication model, as shown in Figure 28 below:

Using the element data (16,384 elements) obtained from the sixth full refinement as the data to be predicted, the residual judgment algorithm based on bubble functions, the feature classification method, and the BP neural network method were applied to classify these data. The time taken is shown in

Table 17. Comparison of time taken by three methods for element category classification.

Number of Samples Classified	Required Time for Residual-Based Bubble Function Method(s)	Required Time for Feature Classification Method(s)	Required Time for BP Neural Network Method(s)
16,348	51.234466	5.1689	0.2423

below.

As shown in the table, all three methods classified approximately 16,000 elements. Among them, the BP neural network method required the least time, approximately 1/20 of that needed by the feature classification method, and was even faster than the residual judgment algorithm based on bubble functions.

Overall, within the current research framework, BP-3 enables data parallel operations, efficiently predicting element categories under large element datasets. As a highly efficient neural network model, it relies solely on a single model to effectively handle refinement processes across different lubrication scenarios.

5. Conclusions

This study employed a neural network model to achieve recognition of element refinement features, enabling adaptive refinement for solving the Reynolds equation within the PHT-based IGA framework. The proposed method introduced the concept of refinement features, which incorporated key physical quantities in lubrication analysis, such as the pressure and its gradient, alongside the element scale from traditional finite element methods, to facilitate global feature recognition and element refinement.

The method proposed in this study has demonstrated the following key advantages: First, it addressed the essential requirements of lubrication analysis by using physical information for the recognition and marking of critical regions. Second, it overcame the limitations of traditional refinement algorithms, which were confined to individual elements, by holistically evaluating refinement regions from the perspective of the solution domain and adaptively adjusting refinement strategies to meet the specific needs of different problems. Third, unlike conventional threshold-based methods, the incorporation of refinement features, combined with data processing techniques commonly used in machine learning, eliminated threshold constraints, achieving strong generalization performance and enabling effective adaptation to the refinement processes of various lubrication problems without requiring specialized data preparation or model training. Finally, the classification and prediction capabilities of machine learning are highly parallelizable, supporting data parallelism, model parallelism, and operation-level parallelism with numerous implementations, thus providing significant efficiency advantages in large-scale applications.

Admittedly, this study has certain limitations. First, the cavitation issue in lubrication problems was not investigated, which will be a key focus of our future research. Cavitation could serve as a refinement feature, and its corresponding regions should be clearly refined to obtain a reliable oil film pressure distribution. Second, the impact of different classification prediction models and data processing methods on prediction performance was not thoroughly analyzed. Additionally, while the proposed method achieved good accuracy in the refinement process, it lacks a complete mathematical proof, requiring further validation and refinement.