Quasi-RVE Contact Modeling of Rough Flange–Gasket Interfaces for Micro-Leakage Channel Geometry Characterization

Abstract

This paper focuses on the characterization of the micro-leakage channel geometry in the flange-gasket rough contact interface of hazardous chemicals transport vehicles. This work represents the first step in a multi-physics simulation framework for optical-fiber-based micro-leakage monitoring. Directly establishing a full-scale contact model from micron-scale rough peaks and valleys to the decimeter-scale flange structure would lead to extremely high computational costs; a nonlinear contact model based on quasi-representative volume element (quasi-RVE) and quasi-periodic boundary condition (quasi-PBC) is proposed in this paper. Quasi-RVE refers to a local region selected from the overall rough surface. Unlike a traditional RVE that requires strict geometric periodicity, the quasi-RVE is only approximately consistent with the overall surface with respect to key morphological parameters and volume parameters. Quasi-PBC only imposes in-plane displacement compatibility constraint on the relative side boundary without imposing periodic constraints in the peak-valley height direction. In this paper, the average interface gap and its distribution are selected as the geometric descriptors of the micro-leakage channel, and the reliability of the contact model is verified by comparing with the existing experimental and numerical results. On this basis, the influences of surface roughness, gasket material and loading conditions on the geometric characteristics of the micro-leakage channel are further analyzed. The results show that the lower stiffness gasket is easier to fit with the rough flange surface under the same load conditions, so as to obtain a larger contact area and a smaller average gap. The quasi-RVE contact model established in this paper can effectively reduce the computational scale of contact analysis of the rough sealing interface, and provide reliable channel geometric information for subsequent micro-leakage fluid simulation and optical fiber signal response simulation.

1. Introduction

The transportation of hazardous chemicals via road tankers presents significant safety challenges, with flange leakage incidents posing serious risks to public safety, health, and the environment. Effective monitoring of micro-leakage, often below 1 mL/min, is critical for prevention, yet conventional detection methods (e.g., pressure drop [1], acoustic emission [2,3,4,5,6], or gas sensing [7]) typically lack the sensitivity to identify such low flow rates [8]. This study proposes an optical fiber sensing system capable of detecting micro-leakage as low as 1 mL/min by measuring localized changes in refractive index and temperature at the flange interface. To validate this approach, a numerical simulation framework is developed to establish predictive models for temperature, strain and refractive index response in optical fiber under leakage conditions.

Simulating the full physical process involves multiple strongly coupled phenomena: nonlinear contact of rough flange-gasket surfaces, fluid–structure interaction during micro-leakage, coupled heat transfer, and optical fiber multi-physics response. A fully coupled, high-fidelity simulation across all relevant scales, from micrometer surface micro convex bodies to decimeter-scale flange geometry (see Figure 1), would require prohibitively large computational resources. To overcome this challenge, a sequential multi-step simulation strategy is adopted. This strategy decomposes the problem into three tractable, sequentially coupled steps, as illustrated in Figure 2. Step I. Geometric characterization of micro-leakage channels (see Figure 2 Step I): A nonlinear contact analysis of rough flange/gasket surfaces is performed to correlate surface roughness, bolt preload, and material properties with leakage channel characteristics (average gap and gap distribution). Step II. Full-flange heat and mass transfer simulation (see Figure 2 Step II): Using the leakage channel geometry from Stage 1, this step models the fluid flow and heat transfer to determine how channel characteristics, fluid properties, and operating conditions affect leakage rate and temperature distribution. Step III. Optical fiber response simulation (see Figure 2 Step III): This final stage predicts the optical signal variation in the fiber sensor in response to the refractive index and thermal-strain fields resulting from the micro-leakage. This approach ensures computational feasibility while capturing the essential physics needed to develop a reliable predictive model for fiber-optic monitoring of flange micro-leakage in hazardous chemical tankers.

As the first step, this paper focuses on the microscopic contact behavior of flange–gasket interfaces, detailing the characterization and geometric reconstruction of rough surfaces, the development of contact models, and the analysis of the geometric characteristics of micro-leakage channels. To investigate microscopic contact behavior at mating interfaces, dozens of methods have been proposed for reconstructing and simulating surface topography based on observations of micro-scale features, including approaches grounded in fractal geometry, stochastic processes, and statistical descriptors [9,10,11,12,13]. Among these, fractal geometry has proven particularly effective in characterizing distributional properties across multiple scales and is well suited to complex, self-similar surface morphologies [14,15,16,17,18]. Consequently, it has been widely adopted in the reconstruction and analysis of rough surfaces. For example, Cai [19] introduced an efficient fractal-based technique for building micro-surface models from irregular topographical features. Similarly, Zhang [20] applied fractal theory to the simulation of machined surfaces, establishing a quantitative relationship between roughness parameters and fractal dimension through measurements of the roughness and fractal attributes of various finishes. Meanwhile, several studies have employed related mathematical frameworks to perform three-dimensional numerical simulations of surfaces, though most adhere to specific assumed distribution laws. In this context, Tomov [21] constructed rough-surface models from prescribed roughness parameters, directly linking the theoretical representations to the actual machining conditions and analyzed the statistical behavior of key morphological descriptors. Reizer [22] combined time-series models, stochastic-process theory, and digital-filtering methods to model and analyze rough surfaces, generating surfaces that conform to Gaussian-distribution characteristics. Wang [23] fused nonparametric probability-distribution characterization with spectral-reconstruction algorithms to generate rough-surface models that accurately reproduce the statistical features of real surfaces.

The accurate evaluation of contact performance fundamentally relies on constructing contact models that effectively approximate actual assembly characteristics. According to microscopic contact theory for rough-surface sealing, actual contact occurs exclusively between micro convex bodies on opposing rough surfaces. Unlike the simplified case of contact between ideally smooth planes, the stress field generated by sealing contact between rough surfaces exhibits high complexity. Research further indicates that plastic deformation during rough-surface contact is concentrated within micro convex regions, while elastic deformation occurs predominantly in the bulk material beneath the rough surface. In micro-leakage analysis, the plastic deformation of micro convex bodies governs the morphology of the interfacial gap, thereby determining the characteristic dimensions of leakage channels; in contrast, the contribution of bulk elastic deformation can be neglected. Although researchers including Sayles [24] have established two-dimensional micro convex contact models and revealed interfacial deformation and stress distribution patterns through numerical simulations, such simplified two-dimensional representations remain inadequate for fully capturing the stress-field distribution in actual random three-dimensional sealing interfaces. To overcome this limitation, subsequent studies have proposed various improved approaches: Brizmer [25] derived deformation expressions for two rough bodies and systematically investigated the elastoplastic response in three-dimensional spherical micro convex-plane contact; Wang [26] introduced an incremental equivalent circular contact model for gaussian rough surfaces, established the theoretical relationship between contact area and load, and calculated the stiffness characteristics of microscopic contact regions. These morphological simplifications and inherent assumptions lead to observable deviations between computational results and actual contact behavior. Consequently, advances in real-surface reconstruction techniques have not only significantly enhanced the efficiency of microscopic contact mechanics analysis but also profoundly expanded the theoretical framework of surface microcontact. Li [27] employed W-M fractal theory and experimental profile-height measurement methods to compute the characteristic length, fractal dimension, and scale-free intervals of surface profiles, established a microscale contact model for flange sealing surfaces, and analyzed the influence of working conditions and material properties on the sealing performance of the contact interface. Mu [28] utilized non-gaussian theory and modeling engineering to generate mating-surface models and contact models that conform to specific morphological characteristics, providing a new pathway for accurately establishing the relationship between microscopic surface topography and macroscopic mechanical properties. However, existing studies on rough-surface contact are often confined to partial surface regions, and the adoption of symmetry boundary conditions [27,29] or free lateral boundary conditions [28,30] in contact models may introduce discrepancies relative to the actual contact behavior of real rough surfaces. Specifically, symmetry boundary conditions may constrain the transverse deformation of asperities near the boundary and the underlying substrate, resulting in an artificially stiff local contact response. In contrast, free lateral boundary conditions may amplify the lateral deformation near the boundary. For rough sealing interfaces, the differences in contact response caused by boundary treatment can further affect the predicted real contact area, mean contact pressure, and average interface gap, which are directly related to the characteristic size and spatial distribution of micro-leakage channels. Therefore, a more appropriate lateral boundary treatment is required in local rough-surface contact models to reduce artificial boundary effects introduced by conventional symmetry or free boundary conditions.

A complete rough-surface contact model must retain a large number of micron-scale peak–valley morphological features. If such a model is directly used for flange–gasket interface contact analysis, it would lead to a high computational cost. Meanwhile, real rough surfaces do not satisfy strict geometric periodicity, making conventional RVE/PBC methods difficult to apply directly to rough-surface contact problems. However, the height distribution and key topographic statistical parameters of a rough surface can usually exhibit a certain degree of statistical stability and representativeness within a sufficiently large local region. Therefore, in this study, a statistically representative local rough region is extracted from the overall rough surface obtained by measurement or reconstruction, and a quasi-representative volume element (quasi-RVE) model is constructed to characterize the main micro-topographic features of the overall rough surface. Subsequently, considering that the quasi-RVE model is statistically equivalent but not strictly geometrically periodic, quasi-periodic boundary condition (quasi-PBC) is introduced. In-plane displacement compatibility constraints are imposed only on opposite lateral boundaries in the length and width directions, while no periodic constraints are applied in the peak–valley height direction. Based on the proposed model, the nonlinear contact response of the rough flange–gasket interface is analyzed using the finite element method, and the effectiveness of the quasi-RVE contact model is verified by comparison with existing experimental and numerical results. Finally, the method is applied to flange–gasket contact analyses under different surface roughness levels, gasket materials, and loading conditions. The average interface gap and gap distribution are used as geometric descriptors of the micro-leakage channel, and the spatial distribution characteristics and characteristic size parameters of the micro-leakage channel during contact are systematically investigated. The obtained geometric information provides input for subsequent leakage transport simulation and optical fiber signal response simulation. The research procedure is shown in Figure 3.

2. Rough-Surface Generation and Quasi-RVE Selection

To reduce the computational cost of full rough-surface contact simulations, this study proposes a quasi-representative volume element (quasi-RVE) selection framework based on statistically equivalent local surfaces. In this framework, a complete random rough surface is first obtained using numerical reconstruction methods. Local regions with different candidate window sizes are then extracted, and the errors between each candidate region and the full surface are calculated in terms of height-distribution parameters and volume parameters. On this basis, by comparing the parameter errors and their convergence trends under different window sizes, the local region with relatively small parameter errors, stabilized parameter variations, and a relatively small computational domain is selected as the quasi-RVE.

2.1. Numerical Generation of Rough Surfaces

In this study, actual rough surface topography is characterized by constructing numerical randomly rough surfaces. For machined sealing rough contact surfaces, numerical simulations based on given parameters are primarily implemented through two methods: the random rough surface generation method based on height probability distribution (HPD) and power spectral density (PSD) [23], and the rough surface generation method based on the Weierstrass–Mandelbrot fractal function [31].

2.1.1. Rough-Surface Reconstruction Based on Prescribed HPD and PSD

The calculation method for HPD involves using measured rough surface height samples, which are arranged in ascending order to construct an empirical cumulative distribution function (ECDF) that directly represents the probability of each roughness height. The ECDF is defined as follows [23]:

$${\widehat{F}}_n(z)={\displaystyle \frac{N_z}{n}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{1}_{X_i\le z}}$$

(1)

where z is roughness data in ascending order, ${\widehat{F}}_n(z)$ is the ECDF, N_z is the number of points in the measured roughness less than or equal to z, $1_{X_i\le z}$ is the indicator of the event, $X_i\le z$ is the height of each data point, and n is the total number of points in the measured roughness.

To enhance the smoothness of the ECDF, linear interpolation is employed to generate a piecewise continuous function. The smoothed values are given by [23]:

$${\widehat{F}}_n^s(z_i)={\displaystyle \frac{{\widehat{F}}_n(z_i)+{\widehat{F}}_n(z_{i+1})}{2}}$$

(2)

Due to the sparsity of roughness height data points in the tail regions, the generalized pareto distribution (GPD) is used to further improve tail smoothness. The probability density function of the GPD is as follows [23]:

when k ≠ 0:

$$f(z|k,\sigma ,\theta )={\displaystyle \frac{1}{\sigma }}{\left(1+{\displaystyle \frac{k(z-\theta )}{\sigma }}\right)}^{-{\displaystyle \frac{1}{k}}-1}$$

(3)

when k = 0:

$$f(z|0,\sigma ,\theta )={\displaystyle \frac{1}{\sigma }}{e}^{-{\displaystyle \frac{(z-\theta )}{\sigma }}}$$

(4)

where k is the shape parameter, σ is the scale parameter, and θ is the threshold parameter.

The calculation method for PSD involves performing a transform on the measured roughness data (a matrix whose size is M × N) to obtain frequency information, which reflects the energy distribution of the roughness at different frequencies. The formula for calculating PSD is as follows [23]:

$$S_z(k\Delta {\omega }_x,l\Delta {\omega }_y)=FFT[z(i\Delta x,j\Delta y)]\cdot conj(FFT[z(i\Delta x,j\Delta y)])$$

(5)

where i = 0, 1,…, M − 1, j = 0, 1,…, N − 1, k = 0, 1,…, $\frac{M}{2}$ − 1, and l = 0, 1, …, $\frac{N}{2}$ − 1, S_z is the PSD, Δx and Δy are the sampling intervals in the spatial domain, Δω_x and Δω_y are the corresponding sampling intervals in the frequency domain, FFT denotes the fast Fourier transform, and conj represents the complex conjugate operation. Through this process, the resulting PSD is a real-valued matrix related to frequency and exhibits central symmetry.

The reconstruction method begins by generating two initial rough surfaces, denoted as $Z_{\mathrm{h}}^0$ and $Z_{\mathrm{s}}^0$, where $Z_{\mathrm{h}}^0$ satisfies the given HPD, while $Z_{\mathrm{s}}^0$ conforms to the given PSD. Specifically, $Z_{\mathrm{h}}^0$ is generated using the inverse cumulative distribution function, with the formula as follows [23]:

$$Z_h^0=F^{-1}(Y)$$

(6)

where F⁻¹ denotes the inverse cumulative distribution function of the given HPD, and Y is a random number matrix following a uniform distribution. This generation process ensures that the numerical distribution of $Z_{\mathrm{h}}^0$ conforms to the target HPD.

For $Z_{\mathrm{s}}^0$, the spectral representation method (SRM) is used to generate a rough surface that satisfies the PSD. Assuming the target PSD is a matrix S, the generation formula for $Z_{\mathrm{s}}^0$ is as follows [23]:

$$Z_0^s=FF{T}^{-1}\left(\sqrt{S}\odot e^{i\mathsf{\Phi }}\right)$$

(7)

where $\sqrt{S}$ is the amplitude matrix, Φ is a phase matrix with elements uniformly distributed in the interval [0, 2π]. The uniformly distributed phase Φ ensures that the generated $Z_{\mathrm{s}}^0$ has a gaussian HPD.

Subsequently, an iterative process is adopted to gradually update $Z_{\mathrm{h}}^{\mathrm{i}}$ and $Z_{\mathrm{s}}^{\mathrm{i}}$ to achieve dual matching of both PSD and HPD.

Each iteration consists of the following two steps:

• Update the PSD of $Z_{\mathrm{s}}^{\mathrm{i}}$: By retaining the phase information of the current $Z_{\mathrm{h}}^{\mathrm{i}}$ and replacing the amplitude, a rough surface $Z_{\mathrm{s}}^{\mathrm{i}+1}$ with the specified PSD is generated. The calculation formula is [23]:

$$Z_s^{i+1}=FF{T}^{-1}\left(\sqrt{S}\odot e^{i{\mathsf{\Phi }}_i^h}\right)$$

(8)

where

$${\mathsf{\Phi }}_h^i={\tan }^{-1}\left({\displaystyle \frac{\mathrm{Im}({\widehat{Z}}_h^i)}{\mathrm{Re}({\widehat{Z}}_h^i)}}\right)$$

(9)

where ${\widehat{Z}}_h^i$ is the Fourier transform of the $Z_{\mathrm{h}}^{\mathrm{i}}$. By doing so, the $Z_{\mathrm{s}}^{\mathrm{i}+1}$ has the desired PSD

2.

Update the HPD of $Z_{\mathrm{s}}^{\mathrm{i}+1}$: To ensure the new $Z_{\mathrm{i}+1}^{\mathrm{s}}$ conforms to the given HPD, the points of $Z_{\mathrm{i}+1}^{\mathrm{s}}$ are sequentially replaced with the corresponding height values of $Z_{\mathrm{h}}^{\mathrm{i}+1}$ in descending order of height values, thereby generating $Z_{\mathrm{h}}^{\mathrm{i}+1}$ that satisfies the HPD.

The above two steps are conducted iteratively until the $Z_{\mathrm{h}}^{\mathrm{i}+1}$ and $Z_{\mathrm{s}}^{\mathrm{i}+1}$ converge to a certain level, the workflow is illustrated in the flowchart as shown in Figure 4.

2.1.2. W-M Fractal Function Method

The rough surface generated by the W-M function can be expressed as [31]:

$$\begin{array}{l}z\left(x,y\right)=L{\left({\displaystyle \frac{G}{L}}\right)}^{D\mathrm{s}-2}{\left({\displaystyle \frac{\ln \gamma }{W}}\right)}^{{\displaystyle \frac{1}{2}}}{\displaystyle \sum _{m=1}^W{\displaystyle \sum _{n=0}^{n_{\max }}{\gamma }^{(D_{\mathrm{s}}-3)n}}}\\ \left\{\cos {\theta }_{m,n}-\cos \left\{{\displaystyle \frac{2\pi {\gamma }^n(x^2+y^2)}{L}}\cdot \cos \left[\arctan \left({\displaystyle \frac{y}{x}}\right)-{\displaystyle \frac{\pi m}{W}}\right]+{\theta }_{m,n}\right\}\right\}\end{array}$$

(10)

where z(x, y) represents the random surface height, L denotes the sample length, G is the characteristic length scale coefficient governing the surface roughness, D_s is the fractal dimension characterizing surface complexity, γ is the frequency density scaling ratio, W is the number of superimposed waves at different frequencies, n_max is the upper limit of the frequency index calculated as $n_{\max }=\mathrm{int}[\ln (L/L_s)/\ln \gamma ]$, L_s is the highest resolution, and θ_m,n is the random phase shift.

2.2. Quasi-Representative Volume Element Selection

The morphological characteristics of rough surfaces are critical factors influencing contact performance. Effective characterization of the morphological parameters is essential for accurately simulating the contact behavior of rough sealing interfaces. According to reference [32], the key morphological parameters include: the arithmetic mean roughness (R_a) quantifies the surface unevenness and is suitable for describing height variations; the root mean square roughness (R_q) measures the dispersion of the surface height distribution relative to the arithmetic mean height, reflecting the amplitude of surface fluctuations; kurtosis (K_u) characterizes the sharpness of the height distribution peak; and skewness (S_k) indicates the asymmetry of the height distribution about its mean. In addition, considering that the micro-leakage channel of the rough sealing interface is closely related to the residual void capacity after compressive contact, this paper further introduces the volume parameters based on the material ratio curve [33], including the void volume (V_v), the core void volume (V_vc) and the valley void volume (V_vv).

Direct numerical modeling that explicitly preserves the complete height distribution and morphological features of a rough surface poses significant computational challenge. Although real rough surfaces are not strictly geometrically periodic, their height distribution, key morphological parameters and volume parameters exhibit statistical periodicity in space, as shown in Figure 5. Therefore, this study proposes a statistically equivalent local surface extraction strategy to select local regions that can represent the overall topography from the overall rough surface. First, several candidate local regions are randomly extracted from the whole surface, and the morphological parameters such as R_a, R_q, S_k, K_u and the volume parameters V_v, V_vc and V_vv based on the material ratio curve are calculated respectively. The relative errors of the morphological parameters and volume parameters between each candidate region and the overall surface are compared. The local region with relatively small parameter errors, good consistency in void-volume characteristics with the full surface, and a relatively small computational domain is preferentially selected as the quasi-representative volume element (quasi-RVE). The selected quasi-RVE can significantly reduce the computational scale of the subsequent finite element contact model while preserving the main statistical characteristics of the full surface.

2.2.1. Quasi-RVE Selection from the Surface Data in Reference [23]

Taking the lapping and honing surfaces provided in reference [23] as examples, this study first uses the morphological parameters of the overall surface as the reference, as summarized in

Table 1. Morphological and volume parameters of the overall surface [23].

Scope	Area Size (μm)	Ra (μm)	Rq (μm)	Sk	Ku	Vv (μm3/μm2)	Vvc (μm3/μm2)	Vvv (μm3/μm2)	Processing Mode
Overall	1024 × 1024	0.321	0.424	−0.422	4.051	0.561	0.500	0.061	Lapping
Overall	1024 × 1024	0.251	0.337	−1.109	4.973	0.358	0.297	0.061	Honing

.

To determine an appropriate quasi-representative volume element size, multiple candidate local regions with progressively increasing window sizes (40 × 40, 50 × 50, 60 × 60, 70 × 70, and 80 × 80 μm²) were extracted from the overall data. At each window size, multiple local candidate regions are randomly extracted from the overall rough surface to reduce the randomness associated with selecting a single region. Subsequently, the key morphological parameters R_a, R_q, S_k, and K_u, as well as the volume parameters V_v, V_vc and V_vv are calculated for each candidate region. By comparing the relative errors of these parameters between each candidate region and the overall surface, the optimal local region at each window size is determined. Its morphological parameters and volume parameters are listed in

Table 2. Morphological and volume parameters of the optimal local regions at different window sizes for the lapping and honing surfaces.

Scope	Area Size (μm)	Ra (μm)	Rq (μm)	Sk	Ku	Vv (μm3/μm2)	Vvc (μm3/μm2)	Vvv (μm3/μm2)	Processing Mode
Quasi-RVE	40 × 40	0.350	0.453	−0.460	4.826	0.604	0.546	0.057	Lapping
Quasi-RVE	50 × 50	0.333	0.441	−0.440	4.226	0.587	0.526	0.060	Lapping
Quasi-RVE	60 × 60	0.325	0.431	−0.433	3.962	0.566	0.506	0.061	Lapping
Quasi-RVE	70 × 70	0.325	0.427	−0.434	4.035	0.564	0.502	0.062	Lapping
Quasi-RVE	80 × 80	0.326	0.430	−0.430	3.962	0.568	0.506	0.061	Lapping
Quasi-RVE	40 × 40	0.244	0.323	−1.155	4.759	0.389	0.325	0.064	Honing
Quasi-RVE	50 × 50	0.253	0.350	−1.137	4.827	0.372	0.310	0.057	Honing
Quasi-RVE	60 × 60	0.252	0.346	−1.128	4.880	0.353	0.292	0.062	Honing
Quasi-RVE	70 × 70	0.250	0.350	−1.130	4.899	0.363	0.302	0.061	Honing
Quasi-RVE	80 × 80	0.252	0.347	−1.128	4.879	0.364	0.301	0.062	Honing

. The surface height data were sampled on a uniform grid with a spatial resolution of 1 μm in both directions (Δx = Δy = 1 μm). Therefore, a window of N × N μm² can be represented as an N × N height matrix: each grid location (x_i, y_j) corresponds to a height value z_ij (e.g., a 60 × 60 μm² window corresponds to 60 × 60 height samples).

The results show that the selected local regions of 40 × 40 and 50 × 50 μm² still exhibit relatively obvious differences from the overall surface in some morphological parameters and volume parameters, indicating that smaller window sizes inadequately represent the height distribution characteristics and void volume characteristics of the overall rough surface. As the candidate window size increases to 60 × 60 μm², the relative errors of the main parameters with respect to the overall surface decrease significantly. When the window size further increases to 70 × 70 and 80 × 80 μm², the variations in the parameters become small, suggesting that the statistical representativeness of the local region gradually stabilizes. Considering the convergence trend of parameter errors at different window sizes, the statistical representativeness of the local region with respect to the overall surface, and the computational cost of subsequent finite element contact analysis, the local region of 60 × 60 μm² is finally selected as the quasi-RVE size. For the finally selected quasi-RVE of 60 × 60 μm², the deviations of its main parameters relative to the overall surface remain within a small range, maintaining good consistency with the overall surface. This indicates that this local region can adequately represent the height distribution characteristics and void volume characteristics of the overall surface. Figure 6 summarizes the height-distribution statistics of the selected 60 × 60 μm² quasi-RVE and compares them with those of the overall surface. The two distributions agree well, with only minor discrepancies observed in skewness and kurtosis.

2.2.2. Quasi-RVE Selection from the Surface Data in Reference [34]

The reference [34] measured multiple distinct regions on the specimen surfaces, providing comprehensive data on microscopic morphological details. The overall surface morphological parameters are presented in

Table 3. Morphological and volume parameters of the overall surface [34].

Scope	Area Size (μm)	Ra (μm)	Rq (μm)	Sk	Ku	Vv (μm3/μm2)	Vvc (μm3/μm2)	Vvv (μm3/μm2)	Processing Mode
Overall	20,000 × 20,000	9.688	18.117	−0.019	2.963	24.067	22.049	2.0170	sanding and shot-blasting

.

To determine an appropriate quasi-representative volume element size, multiple candidate local regions with dimensions of 2500 × 2500, 3125 × 3125, 3750 × 3750, 4375 × 4375, and 5000 × 5000 μm² were extracted from the overall data. With Δx = Δy = 62.5 μm, the window sizes correspond to 40 × 40, 50 × 50, 60 × 60, 70 × 70, and 80 × 80 height-sampling points, respectively (e.g., 3750 × 3750 μm² → 60 × 60 samples). Using the same screening procedure as described in Section 2.2.1, a statistical analysis was performed on the morphological parameters and volume parameters of local regions under different window sizes, and representative local regions with smaller parameter errors were preferentially selected for each size. The morphological parameters and volume parameters of the selected representative local regions under different window sizes are presented in

Table 4. Morphological and volume parameters of the optimal local regions at different window sizes for the sanding and shot-blasting surface.

Scope	Area Size (μm)	Ra (μm)	Rq (μm)	Sk	Ku	Vv (μm3/μm2)	Vvc (μm3/μm2)	Vvv (μm3/μm2)	Processing Mode
Quasi-RVE	2500 × 2500	9.855	19.218	−0.031	3.826	25.146	22.950	2.195	sanding and shot-blasting
Quasi-RVE	3125 × 3125	9.786	18.651	−0.027	3.356	22.939	20.989	1.950	sanding and shot-blasting
Quasi-RVE	3750 × 3750	9.757	18.223	−0.019	2.986	24.478	22.483	1.995	sanding and shot-blasting
Quasi-RVE	4375 × 4375	9.761	18.428	−0.019	3.005	24.411	22.354	2.057	sanding and shot-blasting
Quasi-RVE	5000 × 5000	9.695	18.347	−0.019	2.990	23.741	21.763	1.977	sanding and shot-blasting

.

The results show that the measured surface data exhibit a similar window-size convergence behavior as the previously described surface data. As the candidate region size increases from 2500 × 2500 μm² (i.e., 40 × 40 height-sampling points) to 3750 × 3750 μm² (i.e., 60 × 60 height-sampling points), the differences between the key parameters of the local region and those of the entire surface decrease significantly. When the size further increases to 4375 × 4375 μm² and 5000 × 5000 μm², the parameter variations become gradual. Based on this convergence behavior, the 3750 × 3750 μm² local region is ultimately selected as the quasi-RVE for this measured surface. As shown in Figure 7, the histogram of the height distribution of the selected quasi-RVE agrees well with the fitted height distribution curve of the overall surface, indicating that the local region is statistically representative in terms of height distribution.

To further evaluate the influence of quasi-RVE location selection on the subsequent contact-analysis results, after determining 3750 × 3750 μm² as the representative window size, three local regions with the same size and with the morphological and volume parameters close to those of the full surface were further selected from the full surface as candidate quasi-RVEs. Their main parameters are listed in

Table 5. Morphological and volume parameters of three candidate quasi-RVEs.

Scope	Area Size (μm)	Ra (μm)	Rq (μm)	Sk	Ku	Vv (μm3/μm2)	Vvc (μm3/μm2)	Vvv (μm3/μm2)	Processing Mode
Quasi-RVE_1	3750 × 3750	9.833	18.222	−0.020	2.986	23.760	21.729	2.031	sanding and shot-blasting
Quasi-RVE_2	3750 × 3750	9.872	18.034	−0.019	2.941	24.411	22.354	2.057	sanding and shot-blasting
Quasi-RVE_3	3750 × 3750	9.514	18.489	−0.020	3.028	24.135	22.159	1.975	sanding and shot-blasting

.

3. Implementation of Quasi-Periodic Boundary Conditions and Development of Contact Models

This study proposes and establishes quasi-periodic boundary conditions for quasi-RVE models of rough surfaces. In Section 2, a statistically equivalent local region is selected as a quasi-RVE to represent the overall rough surface; conventional boundary treatments (e.g., symmetric boundaries) may introduce boundary effects and lead to deviations from the contact response of an infinitely extended interface, whereas conventional periodic boundary conditions require strict geometric periodicity that is not satisfied by random roughness. Therefore, unlike conventional periodic boundary conditions, the proposed quasi-periodic boundary conditions impose periodic displacement constraints only on the in-plane directions (X and Y), while no periodic restriction is applied in the thickness direction (Z). Meanwhile, to ensure robust implementation in finite element analysis, this section develops a boundary-expansion and mesh-pairing strategy to guarantee one-to-one correspondence of nodes on opposite boundaries, and the quasi-periodic constraints are enforced through equation constraint in the INP file.

3.1. Quasi-Periodic Boundary Conditions and Implementation for Quasi-RVE Models

3.1.1. Conventional Periodic Boundary Conditions

In displacement-based finite element analysis of RVE models, periodic boundary conditions (PBC) are commonly used to represent periodic microstructures. PBC comprises displacement continuity and stress continuity; Stress continuity is naturally satisfied through the minimum potential energy principle, and thus PBC implementation reduces to enforcing periodic displacement constraints for solution uniqueness. A practical prerequisite is the one-to-one correspondence of mesh nodes on opposing parallel faces, which enables linear constraint equations to be imposed between paired nodes. Following the unified PBC formulation proposed by Xia et al. [35], the periodic displacement field on a pair of opposing faces can be expressed as:

$$u_x^{j+}-u_x^{j-}={\overline{\epsilon }}_x(x_i^{j+}-x_i^{j-})={\overline{\epsilon }}_x\Delta x_i^j$$

(11)

The superscripts j⁺ and j⁻ respectively denote the positive direction and negative direction along the X-axis, $\overline{{\epsilon }_{\mathrm{x}}}$ is the average strain in the X-direction. For a given $\overline{{\epsilon }_{\mathrm{x}}}$, the displacement difference on the right-hand side of (the above formula) becomes a constant value. Therefore, (the above formula) can be rewritten as:

$$u_x^{j+}-u_x^{j-}=c^j$$

(12)

In the following, this classical PBC framework is extended to develop quasi-periodic boundary conditions for statistically equivalent quasi-RVE of rough surfaces.

3.1.2. Proposed Quasi-Periodic Boundary Conditions

Although conventional periodic boundary conditions provide an effective way to represent an infinitely extended microstructure using a representative volume element (RVE), they inherently rely on strict geometric periodicity and thus cannot be directly applied to the quasi-representative volume element (quasi-RVE) of rough surface. In the present work, the rough surface is represented by a statistically equivalent local region (quasi-RVE) selected from the overall surface. Building on this quasi-RVE concept, we propose quasi-periodic boundary conditions, in which an infinitely extended rough interface is approximated by tiling statistically equivalent units rather than perfectly identical periodic replicas. The proposed quasi-periodic boundary conditions impose periodic displacement constraints only in the in-plane directions (X and Y) to ensure displacement compatibility between adjacent quasi-RVEs. No periodic restriction is applied in the thickness direction (Z), which preserves the rough-surface morphological characteristics and the out-of-plane deformation induced by contact.

As shown in Figure 8, the three-dimensional quasi-representative volume element model of the rough surface has a length of L_x, a width of L_y, and a height of h, with the coordinate origin located at point O. Since the model does not exhibit periodic distribution along the Z-axis direction, quasi-periodic boundary conditions are applied only in the X and Y directions.

Implementations of periodic displacement constraints for stochastic RVE in 2D and 3D finite element analyses have been reported in the literature [36,37]. Building on these implementation principles, we develop the proposed quasi-PBC constraints for the rough-surface quasi-RVE by enforcing in-plane displacement compatibility on opposite X/Y faces only (with no periodic restriction along Z), leading to the following system of equation constraints:

On opposite faces perpendicular to the X-axis

$$\left\{\begin{array}{ll}u{|}_{x=L_x}-u{|}_{x=0}=L_x{\epsilon }_x\hfill & \hfill \\ v{|}_{x=L_x}-v{|}_{x=0}=0\hfill & \hfill \end{array}\right.$$

(13)

On opposite faces perpendicular to the Y-axis

$$\left\{\begin{array}{ll}u{|}_{y=L_y}-u{|}_{y=0}=L_y{\gamma }_{xy}\hfill & \hfill \\ v{|}_{y=L_y}-v{|}_{y=0}=L_y{\epsilon }_y\hfill & \hfill \end{array}\right.$$

(14)

In the equations above, the planes located at x = L_x and y = L_y are designated as master planes, while the planes parallel and opposite to them are regarded as slave planes. For corresponding node pairs on the master–slave planes (e.g., X₁–X₂ and Y₁–Y₂ in Figure 8), the proposed quasi-PBC are enforced by applying the above equation constraints. Here, ε_x, ε_y and γ_xy represent the macroscopic in-plane strain components applied in the general form of quasi-periodic boundary conditions, which are used to describe the overall tensile or shear deformation of quasi-RVE in X and Y directions. In the normal contact problem of rough surface studied in this paper, the external load is mainly applied in the form of displacement or nominal pressure along the Z direction, and no additional macroscopic in-plane tensile or shear deformation is applied to the quasi-RVE. Therefore, ε_x = ε_y = γ_xy = 0 is taken in each contact example.

3.2. Implementation of Quasi-Periodic Boundary Conditions for Quasi-Rve

During the reconstruction of the three-dimensional finite-element model of the quasi-representative volume element, the height-data matrix of the selected local rough surface was extended at the boundaries by padding the periphery with values equal to the mean height. This boundary expansion enables one-to-one correspondence of mesh nodes on opposing boundaries, thereby facilitating the imposition of quasi-periodic displacement constraints via equation constraints [38]. As shown in Figure 9, contact within the flange rough-sealing interface occurs primarily between the roughness peaks of the surfaces, with resulting deformation and displacement primarily attributed to deformation of these peaks [39]. The padded band is introduced only to facilitate node pairing for the quasi-PBC implementation. It is excluded from statistical post-processing, and contact is found to remain within the original quasi-RVE region for the investigated cases.

The extended local rough surface data were subjected to interpolated surface fitting using MATLAB R2022b routines. To ensure the accuracy of the finite element computations, the element size within the contact region and its adjacent area must be constrained to the actual contact half-width, with an optimal size being less than half of the contact half-width [40]. The rough surface was reconstructed by generating an INP file. The fitted surface data matrix contains the height values of all nodes along the Z-axis, which, combined with the corresponding X and Y coordinates, generated the spatial positions of the top-layer nodes. Intermediate and bottom-layer nodes were created through layered extrusion along the Z-axis. Finally, by defining element types and establishing nodal connectivity, a three-dimensional solid finite element model incorporating the rough surface was constructed, as shown in Figure 10.

4. Quasi-RVE Contact Model

4.1. Comparison with Reference [23]

In reference [23], the authors experimentally measured the metal rough surface data and used numerical simulations to study the contact behavior between the rough surface and a rigid plane. In this study, a quasi-representative volume element (quasi-RVE) model was constructed based on a local region selected from the rough surface data in reference [23] in Section 2.2.1, and the same simulation method as used in reference [23] was applied. Subsequently, the computational results of the quasi-RVE model were compared with the results of the full surface model in reference [23].

4.1.1. Contact Model for the Lapping and Honing Surfaces

The generated quasi-representative volume element (quasi-RVE) INP file was imported into ABAQUS. The rough-surface solid was discretized using C3D8R continuum elements with a mesh size of 0.25 μm, resulting in a total of 1,428,864 elements. In comparison, the overall rough surface has a size of 1024 × 1024 μm², whereas the selected quasi-RVE has a size of 60 × 60 μm². Under the same in-plane mesh density and through-thickness discretization strategy, the full-surface model would contain approximately (1024/60)² ≈ 291.3 times more elements than the quasi-RVE model, corresponding to an estimated element number of about 4.16 × 10⁸. Therefore, the proposed quasi-RVE model reduces the finite-element model scale by approximately 99.66% compared with a full-surface model constructed at the same mesh density. The rough surface solid adopts a linear elastic material model with equivalent elastic properties, and the Poisson’s ratio is 0.30. According to the setting of the rough surface and rigid plane contact example in reference [23], E* = 25 MPa is used as the reference equivalent modulus for load normalization and dimensionless contact response comparison. The corresponding dimensionless load is expressed as p/E* for load normalization and dimensionless contact response comparison. In the contact definition, the rigid plane was assigned as the master surface and the rough surface as the slave surface. A finite-sliding surface-to-surface hard contact formulation was employed, and the friction coefficient between the contact surfaces was set to 0.15 [23].

To evaluate the effect of boundary treatments on the contact response, two models were established that are identical in all aspects except for the boundary conditions: (1) Quasi-periodic boundary condition model: quasi-periodic displacement constraints were imposed on the two pairs of opposite boundary faces in the 1 and 2 directions (implemented via equation constraints in the INP file); (2) Symmetry boundary condition model: symmetry constraints were applied on the corresponding side boundary faces in the 1 and 2 directions. The contact model is illustrated in Figure 11. The applied load was normalized using an equivalent modulus of E* = 25 MPa. The dimensionless load ranged from 0.0001 E* to 0.9 E*. An implicit solver was employed, and the contact problem was solved using the full Newton method.

4.1.2. Results and Discussion for the Lapping and Honing Surfaces

Three key parameters were calculated to characterize the contact behavior between a rough surface and a rigid plane: the relative contact area (a_r), the mean contact pressure (p_c), and the mean gap (u_g). The relative contact area a_r is defined as the ratio of the actual contact area to the nominal contact area. The mean contact pressure p_c represents the arithmetic average of the contact pressure values at all contacting nodes. The mean gap u_g represents the mean value of the gap between the deformed rough surface and the rigid plane.

Figure 12 and Figure 13 compare the results obtained from the quasi-representative volume element (quasi-RVE) models with quasi-periodic boundary conditions and symmetry boundary conditions, respectively, with the numerical contact results obtained using the full measured surface topography reported in reference [23]. The results show that the relative contact area and mean contact pressure are more sensitive to the boundary treatment. The symmetry boundary model predicts a smaller relative contact area and a higher mean contact pressure. This is mainly because the symmetry boundary condition restricts the lateral deformation of the substrate and asperities near the side boundaries, leading to stronger lateral constraint and a higher apparent local stiffness. Under the same external normal loading, such enhanced boundary constraint suppresses the local deformation of some asperities and the expansion of contact areas, resulting in a smaller real contact area. Consequently, the applied normal load is carried by a smaller actual contact region, which increases the mean contact pressure at the contacting nodes. In comparison, the quasi-periodic boundary condition imposes in-plane displacement compatibility on opposite lateral boundaries and can reduce the artificial boundary effect caused by excessive lateral constraint, leading to predictions of the relative contact area and mean contact pressure that are closer to the full-surface reference model.

Compared with the full model, although the quasi-RVE model captures most of the morphological features and height-distribution characteristics of the rough surface, it cannot fully represent extremely high micro convex bodies and deep valleys with very low occurrence probabilities. This limitation is particularly evident in the low-load range (0.0001 E*–0.01 E*) of the mean contact pressure and relative contact area curves, because contact in this regime is dominated by a small number of the highest micro convex bodies and the response is highly sensitive to extreme heights. The loss of extreme peak information in the simplified model reduces the number of effective load-bearing contact spots at the onset of contact, leading to a smaller predicted contact area and, consequently, a higher mean contact pressure and a lower relative contact area. Meanwhile, in this load range, micro convex bodies near the model boundaries and the underlying material tend to expand laterally. Since the symmetry boundary condition model imposes strict displacement constraints on the side boundaries and suppresses lateral deformation, it produces a slightly higher contact pressure and thus exhibits a larger equivalent surface stiffness.

As the load further increases (>0.01 E*), more micro convex bodies become engaged and are progressively flattened. The contact plane approaches the mean height of the rough surface, and the contact state tends toward full contact, so boundary-localized effects are significantly weakened. Consequently, the simplified quasi-RVE models with quasi-periodic and symmetry boundary conditions both show good agreement with the reference data in terms of mean contact pressure and relative contact area, thereby validating the effectiveness of the proposed quasi-RVE model based on quasi-periodic boundary conditions.

4.2. Comparison with Reference [34]

In reference [34], the authors experimentally measured the surface topography of a rough copper specimen and conducted flat-punch compression tests to obtain the relationship between the normal force and the compression displacement. The compression displacement was controlled by the prescribed rigid displacement of the flat punch, and loading was terminated when the normal force reached 200 N. In the compression tests, the surface roughness of the flat punch was approximately 1 μm and was relatively smoother than that of the specimen; therefore, it was approximated as a smooth rigid plane in the numerical simulations.

In this study, a quasi-representative volume element (quasi-RVE) model was constructed using a local region selected from the rough surface data reported in reference [34] in Section 2.2.2, and contact simulations were performed under the same loading protocol as in reference [34]. In the post-processing, the normal force was calculated as the resultant normal force over the contact region of the rough surface in the compression direction. Specifically, the normal contact-force components of the contacting nodes or contact elements on the rough surface were extracted and summed to obtain the total normal force at each prescribed compression displacement. Subsequently, the simulation results of the quasi-RVE model were compared with the normal force–compression displacement curve reported in reference [34] to validate the effectiveness of the proposed quasi-RVE model.

4.2.1. Contact Model for the Sanding and Shot-Blasting Surfaces

The generated quasi-representative volume element (quasi-RVE) model INP file was imported into ABAQUS. The rough-surface solid was discretized using C3D8R continuum elements with a mesh size of 15.625 μm, resulting in a total of 1,428,864 elements. Compared with directly constructing a full rough-surface model of 20,000 × 20,000 μm², the selected 3750 × 3750 μm² quasi-RVE reduces the in-plane computational domain by a factor of approximately (20,000/3750)² ≈ 28.4. Under the same mesh density and through-thickness discretization strategy, this corresponds to an estimated reduction of approximately 96.5% in the finite-element model scale. Copper was considered as the material of the rough-surface solid, with a Young’s modulus of E = 80 GPa and a Poisson’s ratio of 0.30. An elastic–perfectly plastic constitutive model was adopted, with the yield stress set to σ_y = 266 MPa. In the contact definition, the rigid plane was assigned as the master surface and the rough surface as the slave surface. A finite-sliding surface-to-surface hard contact formulation was employed, and the friction coefficient between the contact surfaces was set to 0.15 [34].

Quasi-periodic boundary conditions (quasi-PBC) were imposed on the two pairs of opposite side faces in the 1 and 2 directions via equation constraints, following the same implementation as in Section 4.1.1 (see Figure 11 for the contact model schematic). The prescribed displacement load ranged from 0 to 17.5 μm and was applied in increments of 2.5 μm. The contact problem was solved using an implicit solver with the full Newton method.

4.2.2. Results and Discussion for the Sanding and Shot-Blasting Surfaces

Figure 14 compares the compression displacement–normal force curves obtained from two sets of compression tests under the same experimental conditions reported in reference [34] with the simulation results of three candidate quasi-RVE models at different regions under the same window size. It can be observed that the simulation results of the three quasi-RVE models are generally in good agreement with the experimental measurements, and all reproduce the overall nonlinear growth trend of the experimental curves. Meanwhile, the simulation results of the three quasi-RVEs at different regions are close to each other without significant dispersion, indicating that under the condition that the representativeness of morphological parameters and volume parameters is satisfied, the influence of the quasi-RVE region selection on the compressive contact response is small, and the model results exhibit good repeatability.

5. Flange-Gasket Rough Sealing Interface Contact

In practical flange-gasket sealing structures, the surface roughness of the flange ranges from R_a = 0.2 to 0.8 μm, while that of the metal sealing ring ranges from R_a = 0.1 to 0.2 μm, indicating that the flange roughness is 2 to 8 times greater than that of the sealing ring [35]. The side with higher roughness (i.e., the flange surface) dominates the majority of the flow-channel gap. Given that the flange and the metal sealing ring materials possess comparable elastic moduli, under the same load, micro convex bodies with similar geometric morphology will undergo comparable levels of deformation. Calculations based on Hertz contact theory further indicate that taller micro-convex bodies exhibit relatively smaller deformation amplitudes. Therefore, preserving the rough morphology of the high-roughness side contributes to maintaining the integrity of the flow-channel structure. In practical engineering applications, flange materials generally exhibit higher hardness than gasket materials. Based on this difference, the present study treats the flange surface as a rough surface with micro convex bodies, while the metal gasket is modeled as an ideally smooth surface.

5.1. Contact Model for the Flange–Gasket Rough Sealing Interface

This section is based on two sets of rough flange sealing-surface data reported in reference [41] (R_a = 0.2 μm and Ra = 0.4 μm). A local region from each data was selected to construct a quasi-representative volume element (quasi-RVE). The selected surface patch was defined as a 60 μm × 60 μm window discretized by a 60 × 60 height-node matrix (Δx = Δy = 1 μm). The quasi-RVE has dimensions of 60 μm × 60 μm × 6 μm and was meshed with a uniform element size of 0.25 μm using C3D8R elements, resulting in a total of 1,428,864 elements. The two flange surfaces with different roughness levels were each combined with two different gasket materials [41,42]; the gasket geometry and meshing strategy were identical to those of the flange, and the material properties of the flange and gasket are listed in

Table 6. The material properties.

Structure	Material	Elastic Modulus [MPa]	Poisson’s Ratio	Yield Strength [MPa]	Constitutive Model
Flange	Superalloy (GH3044)	210,000	0.292	685	Elastic–perfectly plastic
Gasket	Superalloy (GH738)	221,500	0.315	969	Elastic–perfectly plastic
Gasket	Metal-Graphite	207,000	0.28	200	Elastic–perfectly plastic

. In the contact model, the gasket is the upper body and the flange is the lower body. Consistent quasi-periodic displacement constraints (quasi-PBC) were applied to both the flange and gasket on the two pairs of opposite side faces in the 1 and 2 directions. The contact model of the rough flange–gasket sealing interface is shown in Figure 15.

5.2. Normal Displacement Analysis

5.2.1. R_a = 0.2 μm Flange Under Normal Displacement Loading

In the simulation, the initial normal displacement load was set to 0 μm and incrementally increased with a step size of 0.05 μm until a final normal displacement load of 0.4 μm was reached. The resulting nominal contact pressure (p_nom), relative contact area (a_r), maximum gap (u_max), mean gap (u_g), and interfacial gap distribution information of the rough surface are presented in Figure 16 and Figure 17. Here, p_nom is defined as the ratio of the total normal force on the rough surface to the nominal projected contact area. In Section 4, the mean contact pressure p_c was adopted for comparison with reference [23]. This parameter represents the arithmetic average of the contact pressure values at the contacting nodes and was used mainly to remain consistent with the definition and comparison metrics in reference [23]. In contrast, in the present flange–gasket sealing interface analysis, the focus is on the overall sealing contact response under different materials and loading conditions at the engineering scale. Therefore, the nominal contact pressure p_nom is used to characterize the overall pressure level over the nominal sealing region. The relative contact area a_r is defined as the ratio of the actual contact area to the nominal projected contact area and is used to characterize the degree of real contact at the interface. The mean gap u_g represents the spatial mean of the local interfacial gaps between the deformed rough surface and the opposing surface, whereas u_max represents the maximum value of the local interfacial gap field.

As shown in Figure 16a, the nominal contact pressure increases markedly in a nonlinear manner with increasing displacement, and the nominal contact pressure of the superalloy is consistently higher than that of the metal–graphite over the entire displacement range. At the larger displacement end (at 0.4 μm), the difference becomes more pronounced (superalloy reaches approximately 200 MPa, whereas metal–graphite is about 110 MPa). As shown in Figure 16b, the relative contact area increases monotonically with displacement and also exhibits a nonlinear growth trend. Compared with the superalloy, metal–graphite shows a larger relative contact area in the medium-to-high displacement range (at 0.4 μm, metal–graphite is about 0.21, while superalloy is about 0.18). Figure 16c,d present the evolution of the mean gap and the maximum gap with displacement, respectively. Both gap measures decrease approximately linearly as displacement increases, and the curves for the two materials almost overlap. This is because, under the same normal displacement-controlled loading condition, the closure of the rough interface is mainly governed by the initial surface topography and the prescribed displacement. Lorenz and Persson gave the relationship between the imposed displacement and the average gap as u = h_max – s + dσ/E [43], where u is the mean gap, h_max is the maximum asperity height of the initial rough surface measured from the mean plane, s is the imposed normal displacement, and dσ/E represents the overall compression term of the solid. This relationship indicates that, when the imposed displacement s is the same and the rough-surface height characteristics are similar, the average gap is primarily constrained by the imposed closure and the initial surface morphology. In addition, the average gap is a spatially averaged quantity over the entire rough interface, rather than a quantity determined only by local contact spots. Under the present low-roughness condition, the material-dependent deformation difference mainly occurs near local contacting micro-convex bodies, whereas non-contact or slightly deformed valley regions still retain relatively large gaps and make an important contribution to the average gap. Since the two materials have relatively close elastic moduli and the surface roughness is low, the difference in their interface closure behavior under the same displacement-controlled loading condition is limited, leading to similar average gap curves. For the maximum gap, its value usually occurs in deep valley regions that remain out of contact or undergo only slight deformation. Therefore, under the same normal displacement-controlled loading condition, the maximum gap curves of the two materials are similar.

Figure 17a,b illustrate the evolution of the distribution probability of the interfacial gap during displacement-controlled loading, where the gaps are classified into four ranges (<0.1 μm, 0.1–0.5 μm, 0.5–1.0 μm, and >1.0 μm). Figure 17a corresponds to the superalloy, and Figure 17b corresponds to the metal–graphite. Overall, the two materials exhibit similar trends: as the displacement increases, the gap distribution shifts progressively from larger-gap ranges to smaller-gap ranges, indicating that the sealing interface gradually closes and approaches a more conformal contact state. This is mainly because the variation in the local gap field is approximately dominated by the geometry-driven closure process induced by the initial rough surface morphology and the imposed displacement, while the influence of material differences is relatively limited. Joe et al. developed a theoretical model for the load–displacement relation and the probability density function of local gaps between contacting rough surfaces [44], indicating that the gap probability distribution is related to the mean gap state, the spectral characteristics of the rough surface, and the interface closure state. For a low-roughness surface, the initial height variation is small, and the same imposed displacement leads to a similar overall closure state of the interface for the two materials. The material-dependent deformation differences mainly affect local contacting micro-convex regions, whereas the overall area fractions of different gap ranges are largely constrained by the prescribed displacement, the initial rough-surface morphology, and the resulting interface closure state. This does not imply that the material properties have no influence, but indicates that their influence is not dominant in determining the gap-interval area fractions under the present low-roughness and displacement-controlled condition. Consequently, the gap probability distributions of the two materials exhibit highly similar evolution trends.

Specifically, the distribution probability in the 0.5–1.0 μm range is relatively high at the initial stage and then continuously decreases with increasing displacement, implying that the medium-gap regions shrink during compression and migrate toward smaller-gap ranges. In contrast, the probability in the 0.1–0.5 μm range increases markedly with displacement. Meanwhile, the probability in the <0.1 μm range increases monotonically, with a more pronounced rise at larger displacements, reflecting the progressive expansion of contact. The probability in the >1.0 μm range remains close to zero and shows little variation with displacement, suggesting that large-gap regions constitute only a minor fraction of the interface or diminish rapidly at the early stage of loading.

5.2.2. R_a = 0.4 μm Flange Under Normal Displacement Loading

In the simulation, the initial normal displacement load was set to 0 μm and incrementally increased with a step size of 0.1 μm until a final normal displacement load of 0.8 μm was reached. The resulting nominal contact pressure (p_nom), relative contact area (a_r), maximum gap (u_max), mean gap (u_g), and interfacial gap distribution information of the rough surface are presented in Figure 18 and Figure 19.

When the roughness increases to R_a = 0.4 μm, the overall evolution trends of the contact behavior remain consistent with those observed for R_a = 0.2 μm. With increasing displacement loading, the nominal contact pressure continuously rises (Figure 18a), and the growth rate becomes noticeably higher at larger displacements; the superalloy consistently exhibits higher nominal contact pressure than the metal–graphite material. Correspondingly, the relative contact area increases monotonically with displacement loading (Figure 18b), but the ranking between the two materials is reversed: the metal–graphite material generally shows a higher relative contact area than superalloy. By contrast, both the mean gap and the maximum gap decrease approximately linearly with displacement loading, and the two curves almost overlap (Figure 18c,d), indicating that, under displacement-controlled loading, gap closure is governed primarily by geometric closure. Compared with R_a = 0.2 μm, the mean and maximum gaps at R_a = 0.4 μm are overall higher, suggesting that the larger peak-to-valley micro convex body amplitudes associated with the rougher surface increase the initial void space and reduce the efficiency of interfacial closure.

Both R_a = 0.2 μm and R_a = 0.4 μm rough interfaces exhibit similar contact evolution mechanisms under displacement-controlled loading. At the initial stage, only a small number of higher micro convex bodies first come into contact with the smooth plane, resulting in a discrete real contact pattern; the nominal contact pressure is relatively low and mainly concentrated at the micro convex body summits. As the displacement increases, more micro convex bodies progressively participate in load bearing, and both the real contact area and the nominal contact pressure increase accordingly.

Under the same displacement-controlled condition, the load-carrying behaviors of the metal–graphite and the superalloy on the same rough surface show pronounced differences. The superalloy exhibits higher nominal contact pressure, which is primarily associated with its higher elastic stiffness and yield resistance: the material is less able to blunt summit stresses through local plastic accommodation, and thus is more prone to intensified local stress concentration. In contrast, the metal–graphite material, with lower hardness and stronger plastic compliance, can alleviate stress concentration at micro convex body summits through more extensive local deformation, leading to a relatively more uniform pressure distribution and lower peak pressures.

Correspondingly, in terms of the relative contact area, the superalloy is generally smaller than the metal–graphite material. Owing to its limited conformability to rough topography, the superalloy tends to confine real contact to a small number of higher micro convex body summits. By contrast, the metal–graphite material more readily undergoes elastoplastic deformation and partially fills micro-valleys, thereby increasing the number and extent of load-bearing micro-contacts, forming a larger real contact area, and yielding a higher relative contact area.

In terms of gap characteristics, the two materials exhibit similar evolution trends in both the mean gap and the maximum gap. This similarity arises because gap variations are governed primarily by surface topography and the external loading condition. Under the same displacement-controlled loading and identical surface morphology, the two materials are subjected to essentially the same geometric constraints, leading to consistent gap evolution patterns. Therefore, the influence of material properties on the mean and maximum gaps is relatively limited.

Figure 19a,b illustrate the evolution of the gap distribution for the R_a = 0.4 μm interface under displacement-controlled loading, where Figure 19a corresponds to the superalloy and Figure 19b corresponds to the metal–graphite material. Compared with the R_a = 0.2 μm surface, the difference between the two materials also becomes more noticeable for the R_a = 0.4 μm surface. This can be attributed to the larger height variation and deeper valley structures of the rougher surface, which make the local closure process less uniform and more sensitive to the local elastic response of the material. Although the distribution still shifts overall toward smaller gap ranges, the transition is more gradual, and at the same displacement level the distribution remains more biased toward larger-gap intervals. At the initial stage of loading, the probability density is mainly concentrated in the medium-gap range, dominated by the 0.5–1.0 μm interval, which reflects the larger initial peak-to-valley amplitude and deeper micro-valleys of the R_a = 0.4 μm rough surface. With increasing displacement, the fraction of gaps in the 0.1–0.5 μm interval rises continuously and gradually becomes the dominant component, indicating that interfacial closure proceeds primarily through the shrinkage of medium gaps (0.5–1.0 μm), rather than through an immediate formation of near-intimate contact regions. Meanwhile, the fraction of gaps < 0.1 μm also increases with displacement but remains relatively limited, suggesting that even under high displacement loading the interface has not developed a large area of intimate contact. In addition, the decay of the large-gap tail (>1.0 μm) is slower than that observed for the R_a = 0.2 μm surface, implying that deep-valley regions are more difficult to close under the same displacement constraint.

5.3. Nominal Pressure Analysis

5.3.1. R_a = 0.2 μm Flange Under Nominal Pressure Loading

In the simulation, the initial load was set to 10 MPa and increased with an increment of 10 MPa until reaching a final load of 100 MPa. The nominal contact pressure (p_nom), relative contact area (a_r), maximum gap (u_max), mean gap (u_g), and interfacial gap distribution information of the rough surface are shown in Figure 20 and Figure 21.

As shown in Figure 20a, for the two different materials, the calculated nominal contact pressure is almost identical to the prescribed nominal pressure. This is because a uniformly distributed nominal pressure was applied in this set of simulations, meaning that the prescribed nominal pressure was directly applied to the loading surface of the model. After static equilibrium was reached in the finite-element calculation, the total normal reaction force on the rough surface balanced the total normal force generated by the applied uniform pressure. In the post-processing, the nominal contact pressure was obtained by dividing this total normal reaction force by the nominal projected contact area, which is the same as the loading area. Therefore, the calculated nominal contact pressure is essentially consistent with the prescribed nominal pressure. As shown in Figure 20b, the relative contact area increases monotonically with nominal load and exhibits a nonlinear growth trend. Compared with the superalloy, the metal–graphite material shows a larger relative contact area over the entire load range, and the difference further enlarges in the medium-to-high load regime. Figure 20c,d present the evolution of the mean gap and the maximum gap with load, respectively. Both gap measures decrease in a distinctly nonlinear manner (a rapid reduction at low loads followed by a more gradual decline at higher loads), and the metal–graphite material generally yields smaller gap values (at 100 MPa, the mean gap is approximately 0.25 μm for metal–graphite and about 0.32 μm for the superalloy; the corresponding maximum gaps are about 0.94 μm and 1.02 μm, respectively).

Figure 21a,b illustrate the evolution of the interfacial gap distribution probability under pressure-controlled loading, where the gaps are classified into four intervals (<0.1 μm, 0.1–0.5 μm, 0.5–1.0 μm, and >1.0 μm). Figure 21a corresponds to the superalloy, and Figure 21b corresponds to the metal–graphite material. Overall, both materials exhibit a consistent closure trend: as the nominal load increases, the gap distribution gradually shifts from larger-gap intervals toward smaller-gap intervals, indicating continuous reduction of interfacial voids and a progressive transition toward a more conformal contact state.

Specifically, at the initial loading stage, the probability is dominated by the 0.5–1.0 μm interval, whereas the 0.1–0.5 μm interval accounts for a smaller fraction. With increasing load, the probability in the 0.5–1.0 μm interval decreases continuously, suggesting that medium gaps are progressively compressed and migrate toward smaller-gap ranges. In contrast, the fraction in the 0.1–0.5 μm interval increases rapidly and becomes the dominant component over a broad load range, indicating that the primary closure pathway is the compression of medium gaps into the 0.1–0.5 μm interval. Meanwhile, the fraction of gaps < 0.1 μm increases monotonically with load, reflecting the gradual expansion of near-intimate contact regions during loading. The fraction in the >1.0 μm interval remains close to zero throughout, implying that large-gap regions constitute only a minor portion of the interface or are rapidly eliminated at the early stage of loading.

In addition, the evolution of the smallest-gap interval reveals a material-dependent difference. Compared with the superalloy, the metal–graphite interface shows a more pronounced increase in the fraction of gaps < 0.1 μm at higher loads, which is consistent with its stronger elastoplastic compliance: the material more readily undergoes local deformation and fills micro-valleys, thereby promoting faster development of near-contact regions. Correspondingly, in Figure 21b, the 0.1–0.5 μm fraction tends to stabilize at high loads, which can be interpreted as a redistribution of gaps from the 0.1–0.5 μm interval to the <0.1 μm interval as the interface continues to close. This behavior is more pronounced under pressure-controlled loading than under displacement-controlled loading. Under displacement-controlled loading, the imposed normal displacement directly constrains the overall closure of the interface, so the gap evolution is mainly governed by the prescribed displacement and the initial surface morphology. In contrast, under pressure-controlled loading, the closure depends on the deformation response generated by the applied nominal pressure, making the gap distribution more sensitive to material properties.

5.3.2. R_a = 0.4 μm Flange Under Nominal Pressure Loading

In the simulation, the initial load was set to 10 MPa and increased with an increment of 10 MPa until reaching a final load of 100 MPa. The nominal contact pressure (p_nom), relative contact area (a_r), maximum gap (u_max), mean gap (u_g), and interfacial gap distribution information of the rough surface are shown in Figure 22 and Figure 23.

As shown in Figure 22a, the nominal contact pressure increases approximately linearly with the applied nominal load, and the nominal-contact-pressure curve almost overlaps with the nominal-load curve; the two materials also exhibit nearly identical macroscopic pressure levels over the entire load range. As shown in Figure 22b, the relative contact area increases monotonically with nominal load and shows a nonlinear growth trend. The metal–graphite consistently exhibits a higher relative contact area than the superalloy, and the difference becomes more evident in the medium-to-high load regime (at 100 MPa, metal–graphite reaches about 0.16–0.17, whereas the superalloy is about 0.08–0.09). Figure 22c,d present the evolution of the mean gap and the maximum gap with load, respectively. Both gap measures decrease in a distinctly nonlinear manner (a rapid reduction at low loads followed by a more gradual decline at higher loads), and metal–graphite generally yields smaller gaps than the superalloy (at 100 MPa, the mean gap is approximately 0.40 μm for metal–graphite and about 0.46 μm for the superalloy; the corresponding maximum gaps are about 1.92 μm and 1.99 μm, respectively). Compared with the R_a = 0.2 μm case, the R_a = 0.4 μm interface exhibits overall smaller relative contact area and larger mean/maximum gaps at the same nominal load, indicating that the larger peak-to-valley micro convex body amplitude of the rougher surface increases the initial void space and reduces the efficiency of interfacial closure under pressure-controlled loading.

Under pressure-controlled loading, the evolution of the gap distribution for the R_a = 0.4 μm rough-surface contact is shown in Figure 23a,b. Overall, as the nominal load increases, the centroid of the distribution gradually shifts from larger-gap intervals toward smaller-gap intervals, indicating continuous compression of interfacial voids and a progressive transition toward a more conformal contact state. At the initial loading stage, the gaps are mainly concentrated in the 0.5–1.0 μm interval, while the 1.0–1.5 μm interval still accounts for a certain fraction. With increasing load, the fraction of gaps larger than 1.0 μm decreases rapidly, suggesting that relatively large gaps are preferentially closed at the early stage of loading. In the medium-to-high load regime, the fraction in the 0.1–0.5 μm interval increases continuously and gradually becomes the dominant component, whereas the 0.5–1.0 μm fraction decreases accordingly. This indicates that subsequent closure is mainly characterized by progressive shrinkage of medium gaps and their redistribution into the 0.1–0.5 μm interval. Meanwhile, the fraction of gaps < 0.1 μm increases monotonically with load but remains limited in magnitude, implying that under the roughness constraint of R_a = 0.4 μm, a non-negligible portion of the interface still does not reach intimate contact even at high loads. For larger-gap intervals (e.g., 1.5–2.0 μm and >2.0 μm), the probabilities remain close to zero throughout the entire loading range, indicating that deep-valley large gaps constitute only a minor fraction of the interface or are rapidly eliminated at the early loading stage. In addition, a material-dependent difference is observed in the evolution of the smallest-gap interval: the metal–graphite material shows a more pronounced increase in the fraction of gaps < 0.1 μm at high loads, suggesting that its stronger elastoplastic compliance facilitates valley closure and promotes earlier development of near-contact regions.

6. Conclusions

This study focuses on the first step of a three-step, cross-scale, multi-physics simulation strategy for optical-fiber-based monitoring of micro-leakage at hazardous chemical tank-truck flange connections, namely, obtaining reliable geometric information of micro-leakage channels. By reconstructing the rough surfaces and introducing a quasi-representative volume element in conjunction with quasi-periodic boundary conditions, a nonlinear finite-element contact analysis framework for rough sealing interfaces is established. Based on this framework, the interfacial gap and its distribution are adopted as key geometric descriptors of micro-leakage channels, and the effects of surface roughness, material properties, and external loading mode on the evolution of the interfacial gap are investigated. The main conclusions are as follows:

(1) Through statistical analysis of the full surface and candidate local regions with different window sizes, the morphological parameters (R_a, R_q, S_k, K_u) and volume parameters (V_v, V_vc, V_vv) were quantified. The results show that the parameter deviations decrease markedly and then gradually stabilize as the window size increases. Based on the convergence trend, void-volume consistency, and computational cost, a local region with relatively small parameter errors was selected as the quasi-RVE. This strategy reduces the computational domain while preserving the main statistical and void-capacity characteristics of the full surface.

(2) For the rough-surface quasi-representative volume element (quasi-RVE), a quasi-periodic boundary condition (quasi-PBC) scheme was developed and implemented, in which only the in-plane displacement compatibility in the X/Y directions is enforced (with no periodic constraint imposed in the Z direction). To enable a robust finite-element implementation, a boundary-expansion and node-pairing strategy was introduced, and the quasi-PBCs were applied via equation constraints. Comparisons with published full-surface simulations and compression experiments show good agreement in the overall evolution trends, particularly in the medium-to-high load regime, thereby supporting the validity of the proposed quasi-RVE model with quasi-PBCs.

(3) A systematic parametric study was performed to examine the sealing contact behavior of flanges with two roughness levels (R_a = 0.2 μm and 0.4 μm) and two gasket materials (superalloy and metal–graphite). The results show that the gap distribution progressively shifts toward smaller-gap intervals with increasing displacement or nominal pressure. Under displacement-controlled loading, the gap evolution is mainly governed by the prescribed displacement and initial surface morphology; therefore, the two materials exhibit similar gap distributions for the low-roughness surface (R_a = 0.2 μm), while the rougher surface (R_a = 0.4 μm) retains more medium and large gaps and shows a slower closure process. Under pressure-controlled loading, the gap distribution becomes more sensitive to material properties, and the more compliant metal–graphite gasket develops a higher fraction of near-contact gaps at high pressure levels. Overall, lower roughness promotes more efficient interfacial closure, whereas higher roughness increases the persistence of residual gaps and requires higher sealing pressure for effective closure. Within the scope of this work, the developed contact model provides a validated mechanical basis and quantitative gap descriptors for subsequent leakage-transport and optical-fiber signal-response simulations.

Limitations of the present study include the use of a limited set of measured rough surfaces and material models. The gasket surface is simplified as smooth, and the influence of gasket roughness on local contact and residual gap morphology is not explicitly considered. In addition, the constitutive model mainly describes the elastoplastic contact response under compression, while viscoelasticity, stress relaxation, and creep are not included. As the first stage of micro-leakage channel geometry characterization, the present model does not yet couple multi-physics effects such as fluid transport, temperature variation, and chemical interactions. Variations in the friction coefficient may also affect local tangential deformation and contact pressure distribution. These factors may further influence the average gap, real contact area, and leakage-channel distribution, and will be considered in future work.