Multi-Scale Digital Modeling of Precision Assembly Interfaces for Tolerance Analysis Using a Fractal-Wavelet Approach

Abstract

The assembly interface topography of precision machinery exhibits complex multi-scale geometric features, including roughness, waviness, and form error, which critically influence assembly accuracy and tolerance analysis. To address the lack of adaptivity in existing separation criteria, this paper proposes a multi-scale digital modeling approach oriented toward tolerance analysis of precision assembly interfaces, based on a fractal-wavelet framework. Firstly, multiple Weierstrass–Mandelbrot functions with independent fractal dimensions are superposed to construct a multi-fractal topography model with controllable multi-scale characteristics, grounded in the power spectral density energy additivity property. Subsequently, wavelet functions are employed to hierarchically decompose the topography height field information. The effects of the compact support length and vanishing moments of the wavelet functions on the decomposition performance are analyzed to establish a clear basis for their selection. Finally, an adaptive multi-scale separation criterion based on wavelet energy K-means clustering is then proposed, with the optimal number of scale classes determined by maximizing the silhouette coefficient, eliminating reliance on empirical thresholds. Case study results show that the fused waviness-and-form-error model retains 94.8% of the original energy while reducing convex peak count by over 90%, significantly simplifying the interface microstructure for downstream tolerance computation. The proposed method provides a high-fidelity, adaptive digital foundation for assembly accuracy prediction of precision interfaces.

1. Introduction

Precision mechanical systems are typically composed of multiple assembled parts that establish functional connections through the mating of assembly interfaces; consequently, the final assembly accuracy directly dictates the service performance of the mechanical system [1,2,3]. Constrained by the limitations of manufacturing processes, actual assembly interfaces are not ideal planes but rather non-ideal surfaces characterized by complex geometric features. These topographical features alter the actual contact conditions between components, subsequently inducing spatial deviations in the assembly pose [4,5,6]. To evaluate and control these deviations, tolerance analysis must conventionally be performed at the design stage. However, since physical parts have not yet been produced, actual measured surface data are unavailable, making the simulation-based generation of interface topography a necessary standard approach rather than a mere simplification [7,8,9]. Therefore, establishing a high-fidelity digital modeling framework for assembly interfaces holds significant engineering value. Such a framework would not only enable design-stage tolerance analysis using simulated topography but also serve as a crucial bridge, allowing models that represent actual topographical features to be integrated into Monte Carlo-based virtual assemblies, thereby enabling the prediction and control of product accuracy before physical prototyping, as illustrated in Figure 1.

Existing research and engineering practice have demonstrated that assembly interface topography typically exhibits a characteristic multi-scale structure, comprising low-frequency form errors, medium-frequency waviness, and high-frequency microscopic roughness [10,11,12]. From the perspective of tolerance analysis, these scales play distinct roles in determining assembly deviation: form errors are the dominant contributor to macroscopic geometric deviations, waviness governs the contact patterns and positioning stability, while roughness primarily contributes to local elastic-plastic deformation with negligible direct impact on macroscopic pose [4,10,12]. Consequently, directly applying original topography with full-scale details to assembly accuracy analysis may result in substantial computational burdens and inaccurate predictions. In particular, if these densely distributed high-frequency features are treated as perfectly rigid interference points in purely geometric calculations, the solution algorithm may be trapped in local optima, leading to spurious assembly poses that deviate from the actual physical state. Therefore, the multi-scale decomposition of interface topography is not merely a signal-processing exercise but a physically motivated necessity. It constitutes a fundamental prerequisite for high-fidelity analysis by filtering out high-frequency computational disturbances while accurately extracting the waviness and form-error components that primarily govern macroscopic assembly interference.

To digitally model these complex interfaces, various approaches have been developed. While traditional methods like degree-of-freedom variations and discrete decomposition are limited to macroscopic errors or are computationally cumbersome [13,14,15,16,17,18,19,20,21,22], the fractal method has emerged as a superior approach for surface simulation. Leveraging the Weierstrass–Mandelbrot (W-M) function, researchers have established parametric models to characterize surface deviations [23,24,25,26] and further explored their influence on contact mechanics and assembly performance [27,28,29,30,31]. These studies confirm the fractal method’s strong capability in representing intrinsic micro-geometric structures. However, a critical drawback is that fractal-generated surfaces inherently contain excessive fine-scale details, which, without appropriate preprocessing, can trigger numerical instabilities in assembly analysis.

Consequently, multi-scale separation techniques are essential for preprocessing these simulated surfaces. Standard Gaussian filtering (ISO 16610-21 [32]), though simple, lacks adaptivity due to its fixed cutoff wavelengths [33,34,35]. Data-driven methods like empirical mode decomposition (EMD) often suffer from issues such as mode mixing and end effects [36,37]. In contrast, wavelet analysis offers a balanced combination of multi-resolution capability and mathematical rigor. It has been effectively applied to characterize surface profiles, filter unwanted features from worn surfaces, and even construct ‘skin model shapes’ that integrate multi-scale information [38,39,40,41,42,43]. Nevertheless, a significant gap remains: the efficacy of wavelet reconstruction heavily depends on the choice of the wavelet basis function, for which systematic justification in the context of assembly interfaces is insufficient.

In summary, generating interface data via fractal modeling and subsequently performing separation via wavelet analysis is a highly promising technical route for Monte Carlo-based assembly simulations. However, the realization of this strategy still faces critical challenges:

(1) The lack of an objective scale classification criterion. For a high-fidelity assembly analysis, it is essential to accurately isolate the topographical components, as each scale plays a distinct role in determining the final assembly pose. However, current engineering practice often relies on rigid, empirical thresholds (e.g., <1 mm for roughness, >10 mm for form error). Such fixed criteria lack rigorous algorithmic justification and adaptability, making them insufficient for guiding the accurate multi-scale separation of complex or previously unknown virtual topographies.

(2) The lack of an optimization method for wavelet basis selection. The fidelity of multi-scale decomposition using wavelet analysis is critically dependent on the chosen basis function, as its mathematical properties directly determine the accuracy of feature reconstruction. Despite its importance, the selection process in the context of surface topography remains largely heuristic, relying on researcher experience rather than objective criteria. This introduces considerable subjectivity and uncertainty, hindering the repeatability and reliability of the reconstructed topography models.

To address the aforementioned challenges, this paper proposes a multi-scale digital modeling approach for precision assembly interfaces. The remainder of this paper is organized as follows. In Section 2, a multi-fractal simulation method based on superposed W-M functions is established to generate virtual topographies with controllable multi-scale characteristics. In Section 3, an adaptive wavelet-based separation method is developed. This includes a rational criterion for wavelet basis selection and a K-means wavelet energy clustering algorithm to objectively identify the boundaries of roughness, waviness, and form errors without empirical thresholds. Section 4 validates the method’s robustness and superiority through comparative studies.

2. Interface Topography Modeling Based on Multiple Fractal Dimensions

2.1. Multiple Fractal Simulation

To establish a multi-scale model of assembly interface topography and investigate the influence of topographical features on assembly accuracy, it is essential to first simulate the microscopic surface computationally. Constrained by actual machining processes, manufactured surfaces inevitably exhibit non-ideal, complex topographical characteristics. Experimental evidence reveals that the height variations in such surfaces behave as non-stationary stochastic processes, characterized by randomness, irregularity, and multi-scale features. Furthermore, these surfaces are inherently self-affine, being continuous everywhere but differentiable nowhere. Consequently, conventional geometric methods are inadequate for accurately characterizing their roughness. In this context, fractal geometry offers distinct advantages: by employing fractal dimensions, it enables the cross-scale characterization of surface topography while maintaining descriptive robustness against variations in experimental conditions, such as measurement resolution and sampling length.

Among the various fractal models available, the Weierstrass–Mandelbrot (W-M) function is adopted in this study based on four primary considerations: First, the W-M function generates profiles that are continuous everywhere but differentiable nowhere, which aligns perfectly with the mathematical properties of actual machined surfaces [23]. Second, its fractal parameters possess distinct physical interpretations: the fractal dimension D characterizes the structural complexity and space-filling capacity of the surface, whereas the roughness parameter G governs the amplitude of height fluctuations. These parameters have been shown to correlate with machining process conditions [44], which not only enables the generation of surfaces with controllable and physically meaningful topographic characteristics, but also provides a solid foundation for simulating and predicting surface topography through the adjustment of machining parameters during the product design phase. Third, the W-M function ensures scale-invariant characterization. Because the fractal dimension remains constant regardless of changes in measurement resolution or observation scale, it is highly suitable for robust surface modeling across different measurement conditions. Finally, the W-M model has been extensively validated in the fields of tribology [23,27,28], contact mechanics [31], and assembly interface modeling [29,30]. Compared to alternative approaches like the random midpoint displacement or spectral synthesis methods, the W-M function offers superior parametric controllability and a more established linkage to physical machining mechanisms.

Building upon these advantages, the classical two-dimensional W-M function can be extended to characterize the three-dimensional contact interfaces of mechanical components. By incorporating multiple variables to represent a three-dimensional stochastic process, the Ausloos-Berman (A-B) function can be derived in the polar coordinate system. This function effectively describes the spatial height distribution of rough surfaces, with its expression formulated as follows:

$$z(\rho ,\theta )={\left({\displaystyle \frac{\ln \gamma }{M}}\right)}^2{\displaystyle \sum _{m=1}^M{A}_m}{\displaystyle \sum _{n=-\infty }^{\infty }{(k{\gamma }^n)}^{(D-3)n}}\cdot \left\{\cos {\varphi }_{m,n}-\cos \left[k{\gamma }^n\rho \cos (\theta -{\alpha }_m)+{\varphi }_{m,n}\right]\right\}$$

(1)

where $\rho$ and $\theta$ denote the polar coordinates of an arbitrary point on the part surface; ${\alpha }_m=\raisebox{1ex}{$\pi m$}\!\left/ \!\raisebox{-1ex}{$M$}\right.$ is the random orientation angle, M represents the number of superposed profile ridges used in surface construction; $k={\displaystyle \raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$L$}\right.}$ is the wavenumber; L is the maximum length of the part surface in a single direction, $L=\max \left(L_1,L_2\right)$; $L_1$ and $L_2$ denote the surface dimensions in the two generating directions, respectively; for an isotropic surface, $A_m=2\pi {\left(2\pi /G\right)}^{2-D}$, $D$ is the fractal dimension, with $2<D<3$; $G$ denotes the scale-independent fractal roughness parameter; $n$ is the summation index; $\gamma$ is the scale parameter, with $\gamma >1$, which governs the spectral density of the rough surface; and ${\varphi }_{m,n}$ is the random phase, distributed within the interval $\left[0,2\pi \right]$.

By substituting $\rho =\sqrt{x^2+y^2}$ and $\theta =\arctan \left({\displaystyle \raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)$ into Equation (1), the spatial surface height distribution function in the Cartesian coordinate system can be obtained, namely, the A-B function in the Cartesian coordinate system, which is expressed as follows:

$$z(x,y)=L{\left[{\displaystyle \frac{G}{L}}\right]}^{(D-2)}{\left({\displaystyle \frac{\ln \gamma }{M}}\right)}^2{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=0}^{n_{max}}{\gamma }^{(D-3)n}}}\cdot \left\{\cos {\varphi }_{m,n}-\cos \left[{\displaystyle \frac{2\pi {\gamma }^n{(x^2+y^2)}^{\frac{1}{2}}}{L}}\cos \left[{\tan }^{-1}\left({\displaystyle \frac{y}{x}}\right)-{\displaystyle \frac{\pi m}{M}}\right]+{\varphi }_{m,n}\right]\right\}$$

(2)

where $x$ and $y$ denote the planar Cartesian coordinates of an arbitrary point on the part surface, $z$ denotes the calculated height value, and $n$ is the summation index. The maximum summation number is defined as $n_{\max }=\mathrm{int}\left[\log \left({\displaystyle \raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$L_s$}\right.}/\log \gamma \right)\right]$, in which $L_s$ is the cutoff length determined by the resolution of the measuring instrument and also represents the sampling interval for generating the part surface.

An important property of the W-M function is its self-affine scaling behavior. For a surface generated by Equation (2), the height field $z(x,y)$ satisfies the following scaling relation: $z(\lambda x,\lambda y)~{\lambda }^{3-D}Z(x,y)$, where $\lambda$ is an arbitrary scaling factor. This relation indicates that W-M surfaces exhibit statistical self-affinity across scales: a magnification of spatial coordinates by a factor of $\lambda$ results in height fluctuations scaling by ${\lambda }^{3-D}$. Consequently, the fractal dimension $D$ remains invariant under changes in measurement resolution or observation scale, a fundamental advantage of fractal-based surface characterization over resolution-dependent parameters such as arithmetic mean roughness. This scale-invariance property also provides a natural connection to wavelet analysis, as wavelets are inherently multi-scale decomposition tools particularly well-suited for analyzing fractal surfaces whose characteristics span a continuum of scales.

Equation (2) can represent a wide variety of geometrical features of interface topography; however, for high-precision machining processes, such as milling, grinding, and turning, its representation capability still requires further improvement. Existing studies have indicated that surfaces and interfaces produced by precision machining exhibit characteristics associated with multiple fractal dimensions. To this end, Equation (2) is extended so that it can simulate the micro-topographical structure with multiple fractal-dimensional features. Specifically, multiple A-B functions with independent fractal dimensions and fractal roughness parameters are superposed to achieve the representation of such features. The corresponding expression is given as follows:

$$h_n\left(x,y\right)=L{\left[{\displaystyle \frac{\ln \gamma }{M}}\right]}^{1/2}{{\displaystyle \sum _{i=1}^{i_{\max }}\left[\frac{G_i}{L}\right]}}^{\left(D_i-2\right)}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=n_i}^{n_{i+1}}{\gamma }^{\left(D_i-3\right)n}}}\left\{\cos {\varphi }_{m,n}-\cos \left[{\displaystyle \frac{2{\pi }^n{\left(x^2+y^2\right)}^{1/2}}{L}}\cos \left[{\tan }^{-1}\left({\displaystyle \frac{y}{x}}\right)-{\displaystyle \frac{\pi m}{M}}\right]+{\varphi }_{m,n}\right]\right\}$$

(3)

where $D_i$ and $G_i$ denote the fractal dimension and fractal roughness corresponding to each spectral interval, respectively; $i_{\max }$ denotes the number of fractal components; and $h_n(x,y)$ is the computed height field of the interface topography. In comparison with Equation (2), Equation (3) can reproduce more complex topographical structures and offers greater flexibility in parameter adjustment.

The validity of the additive superposition in Equation (3) can be justified from both a spectral-theoretic and a physical perspective. From the spectral viewpoint, each W-M component with fractal dimension $D_i$ generates surface features predominantly within a distinct frequency band. Specifically, a component with a lower fractal dimension (closer to 2) produces surface features dominated by low-frequency, large-amplitude undulations, while a component with a higher fractal dimension (closer to 3) produces features dominated by high-frequency, small-amplitude fluctuations. Because the spectral energy of each component is concentrated in a different frequency range, the cross-spectral interaction between components is negligible. This can be expressed formally through the power spectral density (PSD) of the W-M function:

$$S_i(f)\propto f^{-(8-2D_i)}$$

(4)

where $S_i(f)$ is the PSD of the i-th fractal component and f is the spatial frequency [45]. When $D_i$ values are sufficiently distinct, the PSD functions exhibit minimal overlap in the frequency domain. Consequently, the total energy of the superposed surface is approximately equal to the sum of the energies of the individual components:

$$E_{total}={\displaystyle {\int }_0^{\infty }{S}_{total}}(f)df\approx {\displaystyle \sum _{i=1}^P{\displaystyle {\int }_0^{\infty }{S}_i}}(f)df={\displaystyle \sum _{i=1}^P{E}_i}$$

(5)

This energy additivity property ensures that the multi-fractal superposition preserves the spectral characteristics of each component without mutual interference, providing a physically sound basis for the subsequent wavelet-based separation. From the physical perspective, this superposition is motivated by the observation that real machining processes generate surface features at multiple characteristic scales: tool feed marks create periodic waviness at a specific spatial frequency, while cutting-edge micro-geometry and material grain boundaries introduce finer-scale roughness features with different fractal characteristics. The multi-fractal superposition of Equation (3), thus, provides a more physically faithful representation of such multi-mechanism surface generation processes.

2.2. Fractal Parameter Calibration and Sensitivity Analysis

The practical application of the W-M model requires the determination of appropriate fractal parameters. Among these, the fractal dimension D and the roughness parameter G are the two most critical parameters governing the topographic characteristics of the generated surface. This subsection addresses two key questions: (1) how these parameters can be calibrated from machining process conditions or history data, and (2) how sensitive the resulting surface characteristics are to variations in these parameters.

Fractal parameter calibration from machining parameters. Following the methodology established by Qiu et al. [44], the fractal dimension D can be expressed as a quadratic function of the machining process parameters:

$$D=\left\{\begin{array}{l}{a}_1{f}_z^2+b_1{f}_z+c_1\hfill \\ a_2{a}_p^2+b_2{a}_p+c_2\hfill \\ a_3{n}_r^2+b_3{n}_r+c_3\hfill \end{array}\right.$$

(6)

where $f_z$ is the feed rate, $a_p$ is the cutting depth, and $n_r$ is the spindle speed. The nine fitting coefficients were determined through systematic experiments: $a_1=5.467\times {10}^{-4}$, $b_1=-1.883\times {10}^{-2}$, $c_1=2.617$, $a_2=-3.106\times {10}^{-3}$, $b_2=2.342\times {10}^{-2}$, $c_2=2.467$, $a_3=7.653\times {10}^{-3}$, $b_3=-4.185\times {10}^{-4}$, $c_3=2.541$.

These expressions establish a mapping from measurable machining parameters to the fractal dimension, enabling the generation of surfaces with fractal characteristics that are consistent with specific manufacturing processes. For the case study in Section 4, the machining parameters are set as $f_z=8.2 \mathrm{\mu }\mathrm{m}/\mathrm{rev}$, $a_p=5.7 \mathrm{\mu }\mathrm{m}$, $n_r=420 \mathrm{rev}/\min$, yielding a calibrated fractal dimension of $D=2.50$. Figure 2 displays the calibration curves, illustrating how D varies with each individual machining parameter around this reference point.

Furthermore, the roughness parameter G, which serves as an independent scaling factor for the overall surface amplitude in the W-M function, needs to be determined from experimental data (e.g., by fitting the generated surface’s $S_a$ to actual measurements). The remaining W-M structural parameters are set as: scale parameter $\gamma =1.5$, number of superposed ridges $M=10$, sampling step $L_s=5\times {10}^{-5} \mathrm{m}$, and surface length $L=0.025 \mathrm{m}$.

Sensitivity analysis of fractal dimension. To investigate the influence of the fractal dimension D on the topographic characteristics of the generated surface, a parametric study is conducted by varying D from 2.1 to 2.8 in steps of 0.1. To isolate the effect of D, all other parameters are kept fixed, with the roughness parameter G maintained at a constant value of $1\times {10}^{-10}$. For each value of D, a W-M surface is generated, and the following topographic parameters are computed: arithmetical mean height $S_a$, root mean square height $S_q$, maximum peak height $S_p$, maximum pit height $S_v$, maximum height $S_z$, and number of summit peaks $N_p$ (defined as local maxima higher than all 8 neighbors). The results of the sensitivity analysis, normalized by their initial values at $D=2.1$ for clear comparison, are presented in Figure 3. The following observations can be made:

(1) All height-related parameters ($S_a,S_q,S_p,S_v$, and $S_z$) exhibit a monotonic and non-linear increase as D increases from 2.1 to 2.8. In the W-M function, the amplitude of each frequency component is proportional to ${\gamma }^{n(D-3)}$. As D increases towards 3, the exponent $(D-3)$ becomes less negative, which means the amplitude of high-frequency components decays more slowly. Consequently, the surface retains more energy in its high-frequency microscopic details without losing its low-frequency base, leading to an overall magnification of height fluctuations and extreme peak/pit values.

(2) The number of summit peaks increases significantly with increasing D (from less than 1500 at $D=2.1$ to over 5000 at $D=2.8$). A higher fractal dimension introduces denser fine-grained features and sharper micro-peaks, as illustrated in Figure 4. This trend aligns perfectly with the physical interpretation of the fractal dimension as a quantitative measure of space-filling capacity and geometric complexity.

These results demonstrate that the fractal dimension D has a pronounced and systematic influence on all major topographic parameters. Furthermore, the monotonic relationships between D and the topographic parameters ensure that the surface generation process is well-conditioned: controlled adjustments in D produce predictable changes in the resulting surface characteristics, confirming the physical consistency of the W-M model for parametric surface generation.

3. Wavelet-Based Multi-Scale Decomposition of Interface Topography

Assembly interface topography refers to the geometric surface morphology formed under the combined effects of multiple machining parameters and influencing factors during the manufacturing process. It contains various components, including roughness, waviness, and form error, and exhibits pronounced multi-scale characteristics. Direct use of the interface topography simulated by Equation (3) in assembly accuracy studies may render the problem excessively complex and may even lead the analysis to converge to a local solution; for instance, in assembly positioning analysis, contact points may occur only in local regions of the interface topography. Therefore, it is necessary to extract and reconstruct the multi-scale information of the assembly interface topography to better satisfy the requirements of assembly accuracy analysis. In this section, wavelet analysis is adopted to extract topographical information from the interface, thus providing a foundation for the multi-scale decomposition and reconstruction of assembly interface topography.

3.1. Wavelet Analysis Procedure for the Interface Topography Height Field

Wavelet analysis is essentially a filtering process. By decomposing a signal into wavelet basis functions at different scales, it enables multi-resolution analysis and the extraction of characteristic information across multiple scales. When applied to the extraction and analysis of interface topography, wavelet analysis hierarchically decomposes the height information of a random spatial surface into a series of sub-signals with different frequency and spatial localization characteristics. These sub-signals correspond to the detailed features of the interface topography at different scales, thereby achieving multi-level and multi-scale feature extraction. This approach not only reveals the macroscopic characteristics of the interface topography but also captures microscopic details that have a significant influence on assembly accuracy. For the multi-scale decomposition of assembly interface topography, the choice of a computationally efficient and mathematically rigorous decomposition algorithm is essential. In this study, the Mallat pyramid algorithm [46] is adopted for the following reasons:

(1) Computational efficiency. The Mallat algorithm implements the discrete wavelet transform through a cascade of conjugate mirror filter (CMF) banks followed by downsampling, achieving a computational complexity of $O(N)$ for one-dimensional signals and $O(N^2)$ for two-dimensional data of size $N\times N$. This is significantly more efficient than direct computation of wavelet coefficients, which would require $O(N^2\log N)$ operations. For the assembly interface topography height fields considered in this study, this efficiency advantage is critical for enabling repeated decomposition within Monte Carlo tolerance analysis loops.

(2) Compatibility with regular grid data. The Mallat algorithm operates on uniformly sampled discrete data, which is naturally compatible with the regular grid structure of the height field generated by the W-M fractal function in Equation (3). The decimated Mallat decomposition produces a critically sampled representation, meaning that the total number of wavelet coefficients equals the number of original data points. This property ensures: (a) no redundancy in the representation, which facilitates unambiguous scale classification; and (b) exact reconstruction capability, which is essential for building multi-scale surface models from the decomposition results.

As illustrated in Figure 5, the input signal for wavelet analysis is the height field of the interface topography, namely the initial height field $C_{0,k,m}=h_n\left(x,y\right)$. A multilevel wavelet transform is then performed in sequence. For the i-th decomposition level, the procedure is described as follows. First, row-wise filtering is conducted using the low-pass filter ${\overline{h}}_{-k}$ and high-pass filter ${\overline{g}}_{-k}$, respectively. The filtered outputs are then sampled by retaining the even-indexed components, so that the height field is decomposed into two parts: the low-frequency sub-band on the left and the high-frequency sub-band on the right. Subsequently, each sub-band is filtered column-wise by two groups of filters. The outputs are again sampled by retaining the even-indexed samples, resulting in the low-frequency sub-band $C_{j,k,m}$, the horizontal high-frequency sub-band $D_{j,k,m}^1$, the vertical high-frequency sub-band $D_{j,k,m}^2$, and the diagonal high-frequency sub-band $D_{j,k,m}^3$. This completes the first-level wavelet decomposition. The i-th level decomposition process can be expressed as follows:

$$\left\{\begin{array}{c}{C}_{j,k,m}={\displaystyle \sum _{l,n}{\overline{h}}_{l-2k}{\overline{h}}_{n-2m}{C}_{j+1,l,n}}\\ D_{j,k,m}^1={\displaystyle \sum _{l,n}{\overline{h}}_{l-2k}{\overline{g}}_{n-2m}{C}_{j+1,l,n}}\\ D_{j,k,m}^2={\displaystyle \sum _{l,n}{\overline{g}}_{l-2k}{\overline{h}}_{n-2m}{C}_{j+1,l,n}}\\ D_{j,k,m}^3={\displaystyle \sum _{l,n}{\overline{g}}_{l-2k}{\overline{g}}_{n-2m}{C}_{j+1,l,n}}\end{array}\right.$$

(7)

where $j$ represents the current decomposition level, $N$ represents the final decomposition level, $k$ and $m$ denote the indices of the wavelet basis functions associated with the variables $x$ and $y$, respectively. The resulting low-frequency sub-band $C_{j,k,m}$ can be further decomposed at the next level into the low-frequency sub-band $C_{j-1,k,m}$, the horizontal high-frequency sub-band $D_{j-1,k,m}^1$, the vertical high-frequency sub-band $D_{j-1,k,m}^2$ and the diagonal high-frequency sub-band $D_{j-1,k,m}^3$, until further decomposition is no longer necessary.

An important practical consideration in applying the Mallat algorithm to finite-size topography patches is the treatment of boundary effects. Because the convolution operation underlying wavelet filtering requires data values beyond the boundaries of the height field, an appropriate signal extension strategy must be employed. Three commonly used extension modes are:

(1) Zero-padding extension: the signal is extended with zeros beyond the boundaries ($Z(i,j)=0$ for $(i,j)$ outside the domain). This method is simple but introduces artificial discontinuities at the boundaries, which may generate spurious high-frequency coefficients in the wavelet decomposition.

(2) Periodic extension: the signal is treated as periodic, i.e., $Z(i+N_x,j)=Z(i,j)$ and $Z(i,j+N_y)=Z(i,j)$. This mode is appropriate when the surface topography exhibits approximate periodicity, as is the case for surfaces generated by periodic machining processes (e.g., milling with regular feed marks). However, if the topography values at opposite boundaries do not match, this extension may also introduce boundary artifacts.

(3) Symmetric (half-point) extension: the signal is reflected symmetrically about its boundary, i.e., $Z(-i,j)=Z(i,j)$ and $Z(N_x+i,j)=Z(N_x-i,j)$. This extension ensures continuity at the boundaries and is the most widely adopted strategy in surface metrology applications, as it preserves the local trend of the surface height field near the edges.

In this study, the symmetric extension mode is adopted for the following reasons. First, the assembly interface topography generated by the W-M fractal function in Equation (3) does not inherently possess periodicity at its boundaries, making periodic extension inappropriate. Second, symmetric extension preserves the continuity of the first derivative at the boundary, which minimizes the generation of spurious high-frequency wavelet coefficients near the edges. This is particularly important for the wavelet energy analysis, where artificial boundary energy could distort the clustering results.

To quantify the influence of boundary effects, let $N_b$ denote the number of boundary-affected coefficients at each decomposition level. For a wavelet filter of length $L_f$ (e.g., the db12 wavelet has $L_f=24$), the number of boundary-affected samples at level $l$ is approximately: $N_b(l)\approx L_f\cdot 2^{-l}$. As the decomposition level $l$ increases, the sub-band size decreases by a factor of 2 at each level, while $N_b(l)$ also decreases. Consequently, the relative proportion of boundary-affected coefficients is given by: $r_b(l)={\displaystyle \frac{2\cdot N_b(l)}{N\cdot 2^{-l}}}={\displaystyle \frac{2L_f}{N}}$, which is independent of the decomposition level $l$. For the grid size $N=256$ and the db12 wavelet ($L_f=24$) used in this study, $r_b=2\times 24/256\approx 18.8\%$. This indicates that the vast majority (over 80%) of the wavelet coefficients at each level are unaffected by boundary treatment. Moreover, since the symmetric extension is employed, the boundary-affected coefficients exhibit smooth continuity rather than abrupt discontinuities, further mitigating their influence on the overall energy distribution and clustering results.

By repeatedly performing the procedure illustrated in Figure 5 up to the maximum decomposition level, the interface topography information at different levels can be obtained. The information at each level represents different geometric structural characteristics and error components embedded in the interface topography. Combined with the scale decomposition criterion, reconstruction of the information at different levels makes it possible to establish an interface topography model suitable for precision assembly accuracy analysis. The scale decomposition criterion and the topography reconstruction method will be presented in the following sections.

3.2. Wavelet Function Selection for Interface Topography Decomposition

In the extraction of interface topography information using wavelet analysis, the choice of an appropriate wavelet function is a crucial step. As the fundamental mathematical tool in wavelet analysis, the properties of the selected wavelet function directly affect the quality of topography decomposition, the accuracy of feature extraction, and the reliability of subsequent analyses. In practice, however, no wavelet function can be regarded as universally optimal for all engineering problems, owing to methodological constraints and the discrepancy between theoretical assumptions and practical engineering requirements. Although different wavelet functions may exhibit certain advantageous properties, they inevitably possess limitations in other respects. Therefore, in practical applications, the selection of a wavelet function should be made through a comprehensive trade-off among its various properties, in accordance with the specific problem under consideration and the intended objective.

Existing wavelet function selection methods are often characterized by strong subjectivity and uncertainty. In many cases, researchers select wavelet functions on the basis of personal experience, intuitive knowledge of specific application scenarios, or their understanding of the properties of certain wavelet families. Such a selection strategy, lacking systematic and comprehensive evaluation, may easily lead to a suboptimal choice, thereby compromising the quality and effectiveness of the overall research or application process. To overcome this problem, this study develops a wavelet function selection method by considering two factors: the arithmetic mean deviation of the reconstructed topography and the degree of asperity simplification. The aim is to identify wavelet functions suitable for the extraction of interface topography information. Specifically, the arithmetic mean deviation is defined in Equation (8) and is mainly used to evaluate the accuracy of topography separation and reconstruction, whereas the degree of asperity simplification is quantified by the number of reconstructed asperities, as expressed in Equation (9).

$$S_a={\displaystyle \frac{1}{M\times N}}{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N|}}{z}_{i,j}-\mu |$$

(8)

$$\left\{\begin{array}{c}h\left(i-1\right)<h\left(i\right),h\left(i+1\right)<h\left(i\right)\\ h\left(i-1\right)>h\left(i\right),h\left(i+1\right)>h\left(i\right)\end{array}\right.$$

(9)

where $\mu$ is the mean value of the surface height, i.e., $\mu ={\displaystyle \frac{1}{M\times N}}{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^N{z}_{i,j}}}$; $M$ and $N$ denote the number of sampling points in the X- and Y-directions, respectively; $z_{i,j}$ is the topography height matrix; $h\left(i\right)$ is the height value of the i-th point.

This study concentrates on three key factors influencing wavelet function performance, namely compact support length, regularity, and the number of vanishing moments. A comparative analysis is performed on commonly used wavelet basis functions to identify those most suitable for the application requirements. As listed in

Table 1. Common Wavelet Basis Functions and Their Characteristics.

Wavelet Function	Expression Form	Compact Support Length	Regularity	Number of Vanishing Moments
Daubcchics	dbN	2N − 1	Yes	N
Biorthgonal	BiorNr.Nd	Decomposition: 2Nr + 1 Reconstruction: 2Nd + 1	No	Nr
Coiflets	coifN	6N − 1	Yes	2N

, the Daubechies (dbN), Coiflets (coifN), and biorthogonal (BiorNr.Nd) wavelet families are typically representative with respect to these characteristics. Therefore, the present study focuses primarily on these three wavelet families.

As shown in Figure 6, to ensure a consistent comparison among different wavelet basis functions, the following parameters are selected according to Equation (3): $L=100 \mathrm{mm}$, $L_s=1 \mathrm{mm}$, $D=2.9$, $G=0.05$, $\gamma =1.5$, $M=10$. Based on these parameters, a complex interface topography sample is generated, whose arithmetic mean deviation is 0.04 mm.

3.2.1. Influence of Compact Support Length on Topography Decomposition

According to Table 1, the db6 and coif3 wavelet functions have comparable numbers of vanishing moments and similar regularity, while the compact support length of the coif3 wavelet is 6 units longer than that of db6. Therefore, these two wavelet functions are selected to investigate the effect of compact support length on the reconstructed signal. The db6 and coif3 wavelet functions are, respectively, applied to perform four-level decomposition and single-level reconstruction of the surface topography. The three-dimensional topographies of the low-frequency information at the four reconstructed scales are shown in Figure 7 and Figure 8, respectively. In addition, the three-dimensional arithmetic mean deviation and the number of asperity peaks before and after wavelet transform are calculated, and the comparison results are presented in Figure 9a,b. By comparing the low-frequency information images obtained by the two wavelet functions, it can be seen that, for two wavelet functions with similar regularity and numbers of vanishing moments, both can produce clear topographies and reflect the shape characteristics of the rough surface.

In terms of the accuracy of three-dimensional roughness evaluation parameters, the arithmetic mean deviation $S_a$ and the number of asperity peaks $N_p$ at the four decomposition levels obtained using the two wavelet functions show only minor differences (relative deviations less than 2%). Similarly, the reconstructed surfaces exhibit only small discrepancies in both arithmetic mean deviation and asperity peak count. In addition, the resulting three-dimensional topographies are clear, with no block-like artifacts or other distortions observed.

These results indicate that, for the machined surface topography examined in this study, the compact support length has a limited influence on the reconstruction accuracy of the wavelet transform, provided that the regularity and vanishing moment properties are comparable. However, it should be noted that this conclusion is drawn from a specific type of interface topography with moderate fractal characteristics. For surfaces with significantly different frequency content distributions—such as highly anisotropic textures, surfaces dominated by periodic tool marks, or topographies with abrupt discontinuities—the influence of compact support length may become more pronounced. In such cases, a longer compact support may provide better localization of sharp features, while a shorter compact support may introduce boundary artifacts at discontinuities. Therefore, while the results of this study suggest that compact support length is not the primary factor governing wavelet selection for typical machined surfaces, its effect should be re-evaluated when the method is applied to surface topographies with substantially different spectral characteristics.

To further assess the robustness of this conclusion, additional numerical tests were conducted using W-M fractal surfaces with different fractal dimensions (D = 2.2, 2.4, 2.6, 2.8). For each surface, the db6 and coif3 wavelets were applied under identical decomposition conditions. The relative differences in $S_a$ remained below 3% across all tested fractal dimensions, and the peak count differences were consistently within 5%. These supplementary results confirm that the limited influence of compact support length is not an artifact of a single test case but holds across a range of surface roughness conditions representative of precision machining.

3.2.2. Influence of Vanishing Moments on Topography Decomposition

The regularity of the dbN series wavelet basis functions increases with the order N of vanishing moments. In contrast, the biorNr.Nd series wavelet basis functions do not possess regularity. Given that the regularity of the dbN series is positively correlated with the order of vanishing moments, the dbN series is adopted in this study to analyze the effects of regularity and vanishing moments on topography decomposition, with the biorNr.Nd series used as a reference for comparison. For the interface topography shown in Figure 6, bior1.5 and the db series wavelet basis functions are specifically selected for analysis. To enable a rapid comparison of the performance of different wavelet functions, a two-level decomposition and single-level reconstruction are first carried out using the different wavelet functions.

As shown in Figure 10, the corresponding topography decomposition results are obtained after two-level decomposition and single-level reconstruction using bior1.5, db1, db2, and db4. The physical interpretation of the observed differences in wavelet performance is rooted in the mathematical relationship between vanishing moments and polynomial representation capability. A wavelet with $N$ vanishing moments satisfies the condition: ${\displaystyle \int t^k}\psi (t)dt=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=0,1,\dots ,N-1$. This mathematical property means that the wavelet is orthogonal to all polynomials of degree less than N. In practical terms, a wavelet with N vanishing moments can perfectly separate and represent low-order polynomial trends. For machined surface topography, the low-frequency components (waviness and form error) are locally approximated by low-order polynomials. When a wavelet function has insufficient vanishing moments to represent these polynomial trends, spectral leakage occurs: the polynomial-like content that the wavelet cannot suppress leaks from the approximation coefficients into the detail coefficients, causing the reconstructed approximation to exhibit discontinuous step-like behavior. Specifically for this study:

bior1.5: This wavelet has only 1 vanishing moment for decomposition, meaning it can only suppress constant (degree-0) polynomials. Any linear, quadratic, or higher-order trends in the topography are not orthogonal to the wavelet, resulting in spectral leakage between approximation and detail coefficients. This manifests as the characteristic block-like artifact distribution observed in the reconstructed surface.

db1: The Haar wavelet is the simplest Daubechies wavelet with only 1 vanishing moment and is a piecewise constant function. Its extremely compact support of length 2 means it captures only point-to-point differences, completely ignoring any polynomial structure. This severe limitation produces the characteristic block-like discontinuities in the reconstruction, as the wavelet has no capacity to represent even linear trends.

db2: With 2 vanishing moments, db2 provides orthogonality to linear polynomials, which eliminates the block artifacts observed with bior1.5 and db1. However, db2’s limited regularity and smaller compact support cause the reconstruction to exhibit oscillatory ringing near sharp features, a manifestation of the Gibbs phenomenon. The wavelet smoothness is insufficient to eliminate these artifacts completely.

db4: With 4 vanishing moments, db4 can suppress polynomials up to degree 3 (cubic). This capability, combined with higher regularity (measured by the Holder exponent, which scales approximately as $\alpha \approx 0.2N$ for dbN), yields smooth reconstructions with less ringing and more accurate height distribution. The block artifacts are eliminated, and the ringing is significantly reduced.

It can be seen from the Figure 10 that the surfaces obtained using the bior1.5 and db1 wavelet functions exhibit a block-like distribution, with blurred topographic features, making it difficult to effectively extract three-dimensional characteristics. The surface obtained using the db2 wavelet function shows sharp peaks and obvious topographic abrupt changes, but distortion is also observed. For the topography decomposed using the db4 wavelet function, although the peaks and valleys remain relatively sharp, the height distribution is more uniform, enabling more effective data simplification and separation of high-frequency detail components, with a relatively lower degree of distortion. Therefore, based on the above analysis, the bior1.5, db1, and db2 wavelet functions are excluded from the subsequent study, and the focus is placed on db4 and higher-order dbN wavelet functions (N > 4).

In this study, multiple wavelet functions from db4 to db20 were subjected to four-level decomposition and single-level reconstruction. The arithmetic mean deviation $S_a$ and the number of asperity peaks of the resulting topographies at each decomposition level were calculated, and the results are presented in Figure 11a,b. The figures show that, with the increase in the vanishing moment order N, $S_a$ exhibits a gradually increasing trend, approaching the preset value, while the number of asperity peaks gradually decreases. However, when N increases to 12, both $S_a$ and the number of asperity peaks become essentially stable. These results indicate that increasing the vanishing moment order N within a certain range can improve the reconstruction accuracy of the topography while reducing the number of features, thereby enabling effective data simplification while preserving the key topographic characteristics.

3.2.3. Unified Optimization Criterion for Wavelet Selection

The separate analyses presented in Section 3.2.1 and Section 3.2.2 clarified how the compact support length, the regularity, and the number of vanishing moments each influence the decomposition performance. In engineering practice, however, these three factors must be balanced simultaneously rather than considered in isolation. To eliminate the residual subjectivity associated with visually inspecting reconstructed surfaces, and to provide a single reproducible decision rule for the choice of wavelet basis, a unified optimization criterion that integrates the three factors into one composite objective is formulated in this section.

Three normalized sub-objectives are introduced, each evaluated across all decomposition levels so that the criterion captures the multi-scale nature of the decomposition rather than relying on a single reconstruction scale. Let ${\psi }_N$ denote a candidate wavelet from the dbN family with vanishing-moment order N and compact support length $L_N=2N-1$, and let $l_{\max }$ be the total number of decomposition levels.

(1) Multi-scale reconstruction fidelity. Because no closed-form ground truth exists for the multi-scale decomposition of a measured surface, the ensemble average over all $M$ candidate wavelets at each level is adopted as the best available reference, in a manner analogous to ensemble averaging in uncertainty quantification. The multi-scale fidelity error is defined as:

$$E_f({\psi }_N)={\displaystyle \sum _{j=1}^{l_{\max }}{\alpha }_j}{\displaystyle \frac{|S{a}_j({\psi }_N)-{\overline{Sa}}_j|}{{\overline{Sa}}_j}}$$

(10)

where $S{a}_j({\psi }_N)$ is the arithmetic mean deviation of the reconstructed topography at level $j$, ${\overline{Sa}}_j=M^{-1}{\displaystyle \sum _{k}S}{a}_j({\psi }_{N_k})$ is the ensemble mean at level $j$, and ${\alpha }_j$ are level-dependent weights satisfying ${\displaystyle \sum _j{\alpha }_j}=1$. Because the inter-wavelet differences are negligible at the finest reconstruction scale (Level-1; as shown in Figure 11a) but become progressively more pronounced at deeper levels, which correspond to the waviness and form-error components that dominate the contact-pressure prediction, the level weights are set to $({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4)=(0.1,\hspace{0.33em}0.2,\hspace{0.33em}0.3,\hspace{0.33em}0.4)$ so that deeper levels receive higher priority. A wavelet whose $Sa$ trajectory deviates systematically from the consensus—either by retaining excessive fine-scale detail (under-decomposition) or by removing too much energy (over-decomposition)—incurs a large $E_f$.

(2) Multi-scale feature-simplification index. Similarly, the feature-simplification index is generalized to a weighted sum across decomposition levels:

$$E_s({\psi }_N)={\displaystyle \sum _{j=1}^{l_{\max }}{\beta }_j}{\displaystyle \frac{N{p}_j({\psi }_N)}{N{p}_{j,\max }}}$$

(11)

where $N{p}_j({\psi }_N)$ is the number of asperity peaks at level $j$, $N{p}_{j,\max }={\max }_{k}N{p}_j({\psi }_{N_k})$, and $({\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4)=(0.1,\hspace{0.33em}0.2,\hspace{0.33em}0.3,\hspace{0.33em}0.4)$.

(3) Computational-complexity index. The complexity index remains $E_c({\psi }_N)=L_N/L_{\max }$, $L_{\max }$ is the support length of the largest candidate. All three sub-objectives are dimensionless and lie in the interval $[0,1]$.

The three sub-objectives are combined through a convex weighted aggregation to form the unified criterion:

$$J({\psi }_N)=w_1{E}_f({\psi }_N)+w_2{E}_s({\psi }_N)+w_3{E}_c({\psi }_N),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{w}_1+w_2+w_3=1,\hspace{0.33em}{w}_i\ge 0$$

(12)

Reconstruction fidelity is prioritized because biased $Sa$ values propagate directly into contact-pressure and interference predictions; feature simplification is the second priority because an over-dense asperity field complicates the downstream contact model; and support length is retained as the least-weighted factor so that, among wavelets with comparable fidelity and simplification, the shorter (more computationally efficient) one is preferred. In this study the weights are set to $(w_1,w_2,w_3)=(0.5,\hspace{0.33em}0.3,\hspace{0.33em}0.2)$.

To exclude the severely distorted wavelets identified in Section 3.2.2 (bior1.5, db1, and db2) from the feasible set, the optimization is formally posed as the constrained problem:

$${\psi }^{\ast }=\arg {\min }_{{\psi }_N}J({\psi }_N),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{subject} \mathrm{to}\hspace{0.33em}N\ge 4.$$

(13)

This constraint is the mathematical expression of the minimum regularity requirement and guarantees that only wavelets with at least four vanishing moments—sufficient to suppress polynomials up to degree three, which covers the low-order polynomial trends typical of machined surfaces—enter the candidate set. Within the feasible set the objective $J({\psi }_N)$ is a scalar function of a single discrete parameter $N$, so the optimum can be obtained by direct enumeration.

Applying the unified criterion to the dbN family ($N=4,8,12,16,20$) using the $Sa$ and $Np$ data of Figure 11, the trajectory of $J({\psi }_N)$ exhibits a U-shaped behaviour. For small $N$ (e.g., db4), $E_f$ is elevated because the wavelet under-decomposes at deeper levels, producing $Sa$ values that lie above the ensemble consensus; $E_s$ is also relatively high because fewer asperity peaks are removed. For large $N$ (e.g., db20), $E_f$ rises again because the wavelet over-decomposes, and $E_c$ reaches its maximum. The minimum of $J({\psi }_N)$ is attained at $N=12$, confirming that ${\psi }^{\ast }=\mathrm{db}12$ offers the best compromise among fidelity, simplification, and computational cost, and is therefore selected as the wavelet basis used in the case study of Section 4.

3.2.4. Development of a Multi-Scale Classification Criterion for Interface Topography

Existing studies have shown that interface topography exhibits multi-scale characteristics, including roughness, waviness, and form error. This study focuses on assembly accuracy analysis, and, in view of the characteristics of this problem, the consideration of interface topography is mainly concentrated on waviness and form error. In general, roughness is regarded as a microscopic geometric error, with peak-to-peak spacing less than 1 mm; waviness is considered a microscopic geometric error (between the microscopic and macroscopic scales), with peak-to-peak spacing ranging from 1 to 10 mm; form error, by contrast, is a macroscopic geometric error, with peak-to-peak spacing greater than 10 mm. It should be noted that the above classification criteria are mainly derived from practical experience and lack rigorous theoretical support. Moreover, under different machining conditions or assembly accuracy requirements, the applicability of these ranges may vary. Therefore, in order to more accurately investigate the mechanism by which interface topography affects assembly accuracy, it is necessary to explore a multi-scale classification criterion for interface topography and to establish a multi-scale interface topography model suitable for assembly accuracy analysis.

To overcome the subjective uncertainty associated with empirical judgment, this study investigates a multi-scale separation criterion for assembly interface topography errors based on wavelet energy. The wavelet energy criterion is selected as the basis for multi-scale classification because it satisfies the fundamental principle of energy conservation in signal processing. By Parseval’s theorem, the total energy of a signal is preserved under the wavelet transform: the sum of energies in the approximation (low-frequency) and detail (high-frequency) bands equals the total energy of the original signal. This energy conservation property ensures that the wavelet energy distribution accurately reflects the contribution of each frequency scale to the overall topographic structure without introducing spurious artifacts or losing information. Furthermore, the energy at each decomposition level represents the spectral power concentrated at that particular scale, establishing a direct and physically meaningful correspondence between wavelet energy and scale-specific signal content. Alternative metrics such as entropy or variance do not carry the same energy conservation guarantee: entropy is defined differently for each scale relative to the local signal magnitude and does not satisfy Parseval’s theorem, while variance measures dispersion rather than total power. By contrast, wavelet energy provides a scale-invariant, additive measure of contribution, making it the optimal choice for multi-scale classification.

Building on this theoretical foundation, wavelet energy can characterize the energy variation in interface topography across scales, from microscopic roughness to macroscopic form error. Specifically, high-frequency wavelet energy peaks correspond to fine surface micro-structures, the concentration region of medium-frequency energy maps waviness features, and low-frequency energy dominates the overall contour trend. By quantifying the gradient variation and cumulative effects of energy at each scale, the critical scales of roughness, waviness, and form error can be objectively identified without preset thresholds, thereby enabling adaptive characterization of interface topography under different process conditions. The specific analysis procedure is as follows:

Step 1: According to Equation (14), calculate the maximum decomposable scale of the original interface topography, and then perform wavelet decomposition level by level until the maximum-scale decomposition is completed.

$$l_{\max }=[{\log }_{2}n]-1$$

(14)

where $l_{\max }$ denotes the maximum decomposition level; $n$ is the number of sampling points; and $[·]$ represents taking the floor of the real number in the brackets.

Step 2: According to Equation (15), calculate the wavelet energy at each level, including both the low-frequency energy and the high-frequency energy at the corresponding level. The energy at the j-th decomposition level is partitioned into low-frequency (approximation) and high-frequency (detail) components. According to reference [46], for a 2D discrete wavelet decomposition of the surface topography matrix, the energy equations are defined as follows:

$$\left\{\begin{array}{l}{E}_j^{low}={\displaystyle \sum _{m=1}^{M_j}{\displaystyle \sum _{n=1}^{N_j}|}}c{A}_j(m,n){|}^2\\ E_j^{high}={\displaystyle \sum _{m=1}^{M_j}{\displaystyle \sum _{n=1}^{N_j}\left[|c{H}_j(m,n){|}^2+|c{V}_j(m,n){|}^2+|c{D}_j(m,n){|}^2\right]}}\end{array}\right.$$

(15)

where $c{A}_j(m,n)$, $c{H}_j(m,n)$, $c{V}_j(m,n)$, and $c{D}_j(m,n)$ are the approximation, horizontal detail, vertical detail, and diagonal detail coefficients, respectively, at the j-th decomposition level; and $M_j\times N_j$ is the coefficient matrix dimension at level j. By Parseval’s theorem, the total energy is conserved:

$$E_{total}=E_j^{low}+{\displaystyle \sum _{i=1}^j{E}_i^{high}}$$

(16)

Step 3: Perform adaptive K-means clustering on the decomposition levels based on the wavelet energy, classifying levels belonging to the same cluster into one scale. The maximum number of candidate cluster centers is set to $K_{max}=3$, grounded in the physical basis of surface metrology. In accordance with ISO 4287 [47] and ISO 25178 [48], interface topography is fundamentally classified into three geometric error scales:

(1) roughness: high-frequency micro-scale features created by material removal or deformation processes;

(2) waviness: mid-frequency features arising from machine tool dynamics, spindle runout, or feed variations;

(3) form error: low-frequency macro-scale deviations from the nominal geometry, determined by machine capability and part clamping.

These three scales are not arbitrary; they reflect the intrinsic multi-scale structure of manufacturing processes and are universally recognized in surface metrology standards. However, not all surfaces exhibit all three scales with equal prominence. Therefore, the optimal number of clusters $K^{*}$ is determined adaptively using the silhouette coefficient, which measures both the cohesion and separation quality of the clustering result:

$$s(i)={\displaystyle \frac{b(i)-a(i)}{\max \{a(i),b(i)\}}}$$

(17)

where $a(i)$ is the mean intra-cluster distance (cohesion: how tightly clustered a point is with other points in its cluster) and $b(i)$ is the mean nearest-cluster distance (separation: how well-separated a point is from the nearest neighboring cluster) for the i-th data point. The silhouette coefficient ranges from −1 to +1: values close to +1 indicate strong clustering structure, values near 0 indicate overlapping or ambiguous clusters, and negative values indicate points assigned to the wrong cluster. The average silhouette coefficient $\overline{s}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}s}(i)$ is computed for $K=1,2,3$. The value of $K$ that maximizes $\overline{s}$ is selected as the optimal number of scale classes $K^{*}$: $K^{*}=\arg {\max }_{K\in \{1,2,3\}}\overline{s}(K)$.

Step 4: Fuse the high-frequency components of all levels belonging to the same scale, and reconstruct the component models of the assembly interface topography at each scale, including the single-scale roughness model, the single-scale waviness model, the single-scale form-error model, and the two-scale fused model of waviness and form error.

4. Case Study

In this study, the experimental data reported in Ref. [44] are adopted. Under the processing parameters of feed rate $f_z=8.2 \mathrm{\mu }\mathrm{m}/\mathrm{rev}$, cutting depth $a_p=5.7 \mathrm{\mu }\mathrm{m}$, and spindle speed $n_r=420 \mathrm{rev}/\min$, the fractal dimension is calculated as $D=2.49$. As shown in Figure 12, this fractal dimension is used to simulate an interface topography with a size of $100 \mathrm{mm}\times 100 \mathrm{mm}$, using a sampling step of $L_s=1 \mathrm{mm}$. The topography height field is normalized to $\pm 0.005 \mathrm{mm}$, and this is then taken as the numerical example for validation of the method proposed in this paper.

4.1. Wavelet Analysis of the Interface Topography Height Field

In this study, the db12 wavelet is employed to perform multilevel decomposition of the interface topography height field. Based on the sampling step and Equation (14), the maximum decomposition level is determined to be 6. After six-level decomposition and single-level reconstruction of the original interface topography shown in Figure 12, the topography models at different levels are obtained, as presented in Figure 13.

The overall trend indicates that, from the 1st to the 6th decomposition level, high-frequency information is progressively filtered out, resulting in a gradual reduction in the complexity of the topographical geometric structure, which is intuitively reflected by a significant decrease in the number of asperity peaks. At the final level, the asperity peaks have nearly disappeared, and the topography tends to become consistent with the nominal design plane. These characteristics demonstrate that wavelet analysis can effectively decompose complex topographical information into components at different scales, level by level, thereby enabling the extraction of geometric structural features at each level and the progressive simplification of these features.

4.2. Multi-Scale Decomposition of Interface Topography

Based on the decomposition results at each level shown in Figure 13, the low-frequency energy and high-frequency energy at each decomposition level are calculated according to Equation (15). The calculated energies are then normalized as ratios to the total energy of the original topography. Subsequently, K-means clustering is performed on the normalized energy proportions, with the number of cluster centers set to 3. The clustering results for the low-frequency energy are presented in Figure 14a, while those for the high-frequency energy are shown in Figure 14b. The analysis is as follows:

(1) Overall analysis: the hierarchical patterns revealed by the clustering results of low-frequency energy and high-frequency energy are consistent. Specifically, Levels 1 and 2 belong to the roughness scale, Levels 3 and 4 belong to the waviness scale, and Levels 5 and 6 belong to the form error scale.

(2) Low-frequency energy analysis (as shown in Figure 14a): The proportions of low-frequency energy at Levels 1 and 2 are both relatively high, and the decreasing gradient between these two levels is gentle, indicating that most of the original energy is retained at this scale, with only a small amount of high-frequency components being filtered out. At Levels 3 and 4, a significantly accelerated decrease is observed; in particular, the decreasing gradient at Level 3 is approximately twice that at Level 2, which leads to its classification into a different category from the first two levels (Levels 1 and 2). The energy decreasing gradients at Levels 5 and 6 are similar; however, the decreasing gradient at Level 5 increases significantly again to approximately twice that at Level 4. Therefore, Levels 5 and 6 are also classified into a different category from Levels 3 and 4.

(3) High-frequency energy analysis (as shown in Figure 14b): The proportion of high-frequency energy shows a gradual increasing trend with the increase in decomposition level. This is mainly because wavelet analysis preferentially extracts the high-frequency roughness components, which are characterized by high feature frequency but low amplitude, and therefore make only a small initial contribution to the total energy. As indicated by the variation in the increasing gradient shown in the figure, the gradient changes gently at Levels 1 and 2, becomes noticeably steeper at Levels 3 and 4, and then tends to level off again at Levels 5 and 6. Regarding the cross-scale transition points, the increasing gradient of the high-frequency energy proportion at Level 3 relative to Level 2, and at Level 5 relative to Level 4, both exhibit an approximately twofold jump. This feature is consistent with the cross-scale gradient-jump characteristic revealed by the variation in low-frequency energy.

4.3. Multi-Scale Reconstruction of Interface Topography

Based on the results shown in Figure 14, the levels belonging to the same category are merged to form the roughness component, waviness component, form error component, and the integrated waviness-form error model, as shown in Figure 15. The analysis is as follows:

Analysis of the rationality of the scale models: The roughness component model accounts for 5.4% of the total energy, with an arithmetical mean deviation of $S_a=0.0003$, a root mean square deviation of $S_q=0.0004$, and 835 peak summits, reflecting pronounced micro-fluctuation characteristics with small amplitude and high frequency. The waviness component model accounts for 21.1% of the total energy, with $S_a=0.0006$, $S_q=0.0008$, and 90 peak summits. Its morphological complexity is greatly reduced, and the fluctuation frequency decreases while regularity is enhanced. The form error component model accounts for 73.7% of the total energy, with $S_a=0.0012$, $S_q=0.0014$, and 13 peak summits. Its morphological features tend to be smoother, with the lowest fluctuation frequency and predominantly large amplitudes. These characteristics are consistent with the basic physical mechanism underlying the formation of topographical errors on sealing surfaces.

Accuracy analysis of the scale-fusion model: The integrated waviness-form error model accounts for 94.8% of the total energy, with an arithmetical mean deviation of $S_a=0.0013$ (actual value $S_a=0.0014$), a root mean square deviation of $S_q=0.0016$ (actual value $S_a=0.0016$), and 80 peak summits. These results indicate that the integrated model captures the major error components of the interface topography. At the same time, while ensuring reconstruction accuracy, it greatly simplifies the micro-structural details of the model, which is highly beneficial for subsequent assembly accuracy analysis.

4.4. Spectral and Statistical Validation of the Decomposition Results

To provide spectral-domain evidence for the effectiveness of the multi-scale separation, a power spectral density (PSD) analysis of the separated components is conducted. Figure 16 presents the PSD of each separated component with its corresponding dominant frequency band highlighted. The adaptive frequency band boundaries were determined using the energy-weighted spectral centroid of each component, yielding a Form error/Waviness boundary at 0.032 mm⁻¹ and a Waviness/Roughness boundary at 0.113 mm⁻¹. As shown in Figure 16a, the roughness component exhibits its dominant energy concentration in the high-frequency region (>0.113 mm⁻¹) with a peak at 0.18 mm⁻¹. As shown in Figure 16b, the waviness component is centered in the mid-frequency band (0.032–0.113 mm⁻¹) with a peak at 0.04 mm⁻¹. As shown in Figure 16c, the form error component dominates the low-frequency region (<0.032 mm⁻¹) with a peak at 0.02 mm⁻¹. The spectral contribution ratio plot in Figure 16d further confirms the clear spectral separation: the form error (blue) contributes nearly unity below 0.1 mm⁻¹, waviness (green) dominates the mid-band around 0.04–0.1 mm⁻¹, and roughness (red) becomes the dominant contributor beyond 0.1 mm⁻¹. These results confirm that the proposed wavelet-clustering separation method effectively isolates components into their physically expected frequency ranges without requiring predefined cutoff frequencies.

Figure 17 provides a comprehensive PSD analysis of the multi-scale separation results. Figure 17a shows the original surface PSD, with the two adaptive frequency band boundaries marked as dashed vertical lines at 0.032 mm⁻¹ and 0.113 mm⁻¹, respectively. Figure 17b overlays the PSD curves of all three separated components against the original surface signal, demonstrating that the spectral content of the original surface is fully decomposed into three distinct frequency-domain components with minimal spectral leakage across band boundaries. Figure 17c displays each component’s PSD on a logarithmic y-axis across the linear spatial frequency range of 0–0.6 mm⁻¹, clearly revealing the frequency separation between components spanning over 15 orders of magnitude in spectral power. Figure 17d quantifies the frequency band energy distribution using grouped bar charts on a logarithmic scale, where the three bands correspond to form error (<0.032 mm⁻¹), waviness (0.032–0.113 mm⁻¹), and roughness (>0.113 mm⁻¹). The diagonal-dominant pattern—where each component’s energy is predominantly concentrated within its corresponding frequency band while remaining orders of magnitude lower in the other bands—provides direct spectral evidence that the proposed separation method achieves effective and clean multi-scale decomposition.

To verify the consistency of the decomposed components with the original surface, ISO 25178 [48] topographic parameters are computed for all surfaces and summarized in Figure 18. Figure 18a compares Sa, Sq, and Sz across all surfaces. The form error component dominates with the largest height parameters (Sa ≈ 0.0012 mm, Sq ≈ 0.0014 mm, Sz ≈ 0.0067 mm), followed by waviness (Sa ≈ 0.0007 mm, Sz ≈ 0.0046 mm) and roughness (Sa ≈ 0.0004 mm, Sz ≈ 0.003 mm), consistent with the physical expectation that large-scale form deviations dominate. The reconstructed surface parameters match the original exactly, confirming perfect reconstruction fidelity (error: 8.67 × 10⁻¹⁹ mm).

Figure 18b compares the proposed method against conventional Gaussian filtering (ISO 16610-21 [32]). A notable discrepancy appears in the roughness component, where the Gaussian filter yields near-zero Sa and Sq values, indicating that fixed cutoff wavelengths fail to capture the actual spectral content. The proposed method, by adaptively determining separation boundaries from the data itself, avoids this subjective bias.

Figure 18c shows that the Sa/Sq ratios of all surfaces cluster tightly around 0.80–0.82, closely matching the theoretical Gaussian reference of 0.798, confirming that the decomposition preserves the statistical distribution of the original surface without introducing distortion or artifacts.

Figure 18d presents normalized parameters relative to the original surface, yielding a clear contribution hierarchy of form error (Sa ≈ 0.85) > waviness (Sa ≈ 0.45) > roughness (Sa ≈ 0.22), consistent with the PSD energy distribution analysis.

Figure 19 presents a statistical validation of the proposed adaptive wavelet energy clustering method across 100 independent runs. The pairwise co-occurrence matrix (Figure 19a) demonstrates that decomposition levels L1–L2, L3–L4, and L5–L6 consistently co-cluster with high frequency, confirming the structural stability of the three-class separation into roughness, waviness, and form error. The cluster membership probability matrix (Figure 19b) further quantifies this stability: L1 and L2 are assigned to the roughness cluster with probability 1.00, L4 to waviness with probability 1.00, and L5–L6 to the form error cluster with probabilities of 0.96 and 1.00, respectively. The sole ambiguity occurs at L3 (waviness probability: 0.71; roughness probability: 0.29), which is physically consistent with the transitional nature of this scale between roughness and waviness regimes. The Adjusted Rand Index (ARI) distribution (Figure 19c) yields a mean of 0.824, indicating strong agreement between repeated clustering outcomes. The mean silhouette score distribution (Figure 19d) is concentrated in the range of 0.60–0.80 (reference value: 0.662), confirming satisfactory intra-cluster compactness and inter-cluster separation. Collectively, these results demonstrate that the proposed clustering criterion is robust to initialization variability and produces reproducible, physically meaningful scale decomposition.

5. Conclusions

This paper proposes a multi-scale digital modeling framework for precision assembly interfaces based on a fractal-wavelet approach, addressing two fundamental challenges: the lack of an objective scale classification criterion and the absence of a systematic wavelet basis selection method. The main contributions and findings are summarized as follows:

(1) A microstructural simulation method for interface topography based on multifractal dimensions was established. Through the superposition and fusion of multiple W–M functions, high-fidelity simulation of complex interface topography errors and geometric structures was achieved.

(2) A wavelet analysis procedure for the height field of interface topography was developed. The influence mechanisms of the compact support length and vanishing moments of wavelet functions on the decomposition of interface topography were systematically analyzed, and the basis for selecting wavelet basis functions was clarified.

(3) A multi-scale separation criterion based on wavelet energy clustering was proposed. To overcome the limitation of traditional scale classification relying on experience, clustering analysis of wavelet energy was employed to achieve the adaptive separation of roughness, waviness, and form error, and an integrated model of waviness and form error was constructed.

This study generated virtual interfaces based on fractal simulation. In the future, extensive comparative validation using measured data is still needed, so as to further optimize the fractal parameters and the threshold values for wavelet energy clustering, thereby improving the engineering applicability of the model.