Rough Surfaces Simulation and Its Contact Characteristic Parameters Based on Ubiquitiform Theory

Abstract

Ubiquitiform is a new theory of finite-order self-similar physical structure and it is more reasonable to describe real engineering surfaces by ubiquitiform rather than fractal. In this paper, by introducing the frequency truncation criterion, a new analytical expression of the two-dimensional W–M function based on the ubiquitiform theory is firstly derived and constructed and the two-dimensional ubiquitiformal curve characterization under different contact characteristic parameters is achieved. On this basis, the anisotropic three-dimensional surface W–M function with ubiquitiformal features is constructed, and the evolution law of the anisotropic three-dimensional surface morphology under the regulation of the ubiquitiformal complexity is investigated. Then, an improved adaptive box counting algorithm is proposed, and the lower limit of the metric scale in the self-similarity region of the asperities on the rough surface is determined and then the computation method of the ubiquitiformal complexity is established. At last, the validity and accuracy of the method are confirmed by the Koch curves. Key findings include: (1) higher ubiquitiformal complexity D corresponds to increased surface irregularity and complexity; (2) the characteristic scale factor G affects surface height only; (3) reducing the lower limit of metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ increases surface undulation frequency, revealing finer details. This research provides a rationale and quantitative guidance for the matching design of critical joint interfaces in modern precision machinery.

1. Introduction

The interface is a boundary layer formed between the materials of parts in contact with each other, which is used to describe the mechanical behavior of two solids after contact. The surface quality of components and the mechanical behavior of interfaces has an important impact on the performance of the whole machine, the sealing of contact materials, and the transfer of product dynamics [1,2], so the research related to the mechanics of interfacial contact has received great attention from scholars [3,4,5,6,7,8]. Figure 1a shows the three-dimensional topography of the gray cast iron surface after grinding processing within the area of 1 μm × 1 μm, which was obtained by scanning with an atomic force microscope. In the microscale of sampling length, the interface contact is actually the contact and interaction between the asperities on it (as shown in Figure 1b), and the study of thermoelastic plastic contact properties of asperities at the microscale is the prerequisite and basis for further research on the interface macroscale contact problems in engineering applications. However, the randomness, multiscale, and irregularity of the interface morphology lead to the determination of the contact characteristic parameters, which is still difficult, resulting in theoretical modeling and high-precision prediction of interfacial contact parameters remaining a bottleneck in the design and performance prediction of the whole machine.

Currently, the dominant solution method for expressing and analyzing the contact characteristics of interfaces is the fractal theory. Fractal is a theory that describes the multiscale, self-affine, and irregular properties of the interface profile structure, and its research results are abundant [9,10,11,12,13]. However, the actual physical object does not have an infinitely fine structure; that is to say, when the same physical object is observed with different scale coefficients, the structure seen is recognizably similar in shape rather than precisely similar, so the corresponding iterative process cannot be infinite. It can be seen that there are inherent defects in the fractal theory. When the dimensions of the fractal physical quantities are integers, there are many irrationalities and inherent difficulties such as singularity and non-conformity to the actual physical object properties in its engineering applications; for example, in the fractal model established by Davey et al. [14], the integer dimensions of the fractal lengths, fractal stresses, and fractal areas, etc., are all theoretically computed and they cannot be actually measured and obtained. Because the fractal physical quantities with integer dimensions are not measurable, it also leads to the related fractal models [14,15] not being able to be experimentally validated. For this reason, some scholars have proposed redefining the density-like physical quantities in the unit fractal measure [16], but such physical quantities are usually difficult to determine and there is a lack of clear physical significance in practical applications. Other scholars have tried to adopt the methods of reorganization [17], correction, or redefinition of some physical quantities [18], etc. However, these methods still have the problems of complicated processing and a lack of physical basis. In addition, the fractal edge value is difficult to establish and the scope of application of fractal theory, i.e., the difference between physical fractals and mathematical fractals in practical applications, and other issues have also attracted the attention of researchers. In this regard, some scholars [18,19] use similar homogenization approximation methods, but the relevant studies only simplify the computational difficulty, do not give a clear physical significance, and have a lack of sufficient theoretical basis. The above problems directly lead to difficulties such as limited applicability, inconvenient use, large calculation error, and difficulty in guaranteeing the prediction accuracy of the interface contact characteristic parameter prediction model based on fractal theory in the practical application of precision machinery products.

In order to effectively solve the shortcomings in the application of fractal theory, Ou et al. [20] proposed a new concept, “ubiquitiform”. Ubiquitiform is the shape that exists widely in nature, which is a new concept generated and developed on the basis of fractal theory, and it is defined as a self-similar or self-affine physical structure with a finite order, which can be generated by a finite number of iterations. It is believed that it is more reasonable to describe the physical objects in nature by ubiquitiform instead of fractals, and the irrationality of approximating ubiquitiform by fractal is proved theoretically. The ubiquitiform theory not only preserves the self-similarity property of fractals but also overcomes its assumption of infinite fine structure. By introducing ubiquitiform, interface contact parameters can be described more realistically and with clearer physical significance, especially in the microscale contact problem, which can provide a more accurate modeling method.

The W–M function (Weierstrass–Mandelbrot function), as a core mathematical tool of fractal geometry, derives its academic value from the accurate characterization of the multiscale properties of surfaces. In the 1980s, Mandelbrot, Peitgen, and others proposed the W–M function to describe the statistical self-affinity of surface profiles, while Berry systematically demonstrated its strict mathematical properties, such as continuity and self-affinity [21], laying the theoretical foundation. This function has become the preferred tool for fractal modeling of rough surfaces due to its clear physical meaning of the spectrum and flexible and controllable time-domain-scaling parameters. Majumdar et al. constructed a fractal contact model based on the W–M function, and for the first time quantitatively correlated the multiscale rough structures with macroscopic mechanical behaviors [22], and the subsequent studies were further extended to the static friction prediction [23], and the subsequent studies were further extended to the static friction prediction [24]. The subsequent studies were further extended to the fields of static friction prediction [23], wear rate analysis [24], and adhesion effects in microelectromechanical systems [25]. Compared with the traditional statistical methods, the W–M function provides a key mathematical framework for the breakthrough of interface mechanics theory by characterizing complex features such as nonstationarity and irregularity through fractal dimension parameterization, which is especially good at revealing the cross-scale correlation mechanism between the microscopic roughness peaks and the macroscopic contact behavior. However, the traditional fractal W–M functional model has the deficiency of describing the infinite fine structure, which is not in line with the finite-scale characteristics of the ubiquitiform. On the other hand, the lower limit of the ubiquitiformal metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ can characterize the microscale contact mechanics of ubiquitiformal rough surfaces, which is a key parameter for describing the self-similarity characteristics of interfaces, and its value depends on the microscale structure of the interfaces, but so far there is limited research and there is no conclusive conclusion on how to determine the value of this parameter.

In terms of surface properties, rough surfaces can be categorized into isotropic and anisotropic surfaces. Currently, many contact mechanics studies are based on the isotropic assumption [4,5,6,7,8], i.e., the surface profile is considered to have the same characteristics in different directions. In fact, most machined surfaces have anisotropic features, i.e., the surface profile has different geomorphic characteristics or mechanical properties in different directions. The complex surfaces generated by existing algorithms are mostly isotropic, which limits the true description of the surface properties of actual engineering materials.

In order to better fit the actual engineering physics, this paper introduces the traditional W–M function into the rough surface contact problem based on the ubiquitiform theory and corrects its iterative parameters by designing the upper and lower limits of the metric scale.

Combined with the study of the anisotropic characteristics of the rough surface, the 3D ubiquitiformal surfaces are constructed, which improve the realism and accuracy of the 3D morphology simulation of the ubiquitiformal surface. Then, the effect of the ubiquitiformal complexity on the simulation results of anisotropic 3D surfaces is explored. A method for calculating the ubiquitiformal complexity is constructed by determining the lower limit of metric scale or the self-similar region of the asperities on the rough surface and the validity of the method is confirmed.

2. W–M Two-Dimensional Ubiquitiformal Curve Simulation

The W–M function simulation method is simple and straightforward, and has been widely used in the modeling and characterization of the surface of construction machinery, which is based on the fractal theory of the expression as follows:

$$\mathrm{z}\left(\mathrm{x}\right)={\mathrm{G}}^{\left(\mathrm{D}-1\right)}{\sum }_{\mathrm{n}={\mathrm{n}}_1}^{\mathrm{\infty }}{\mathsf{\gamma }}^{\left(\mathrm{D}-2\right)\mathrm{n}}\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi {\mathsf{\gamma }}^{\mathrm{n}}\mathrm{x}\right)$$

(1)

The W–M function’s mathematical essence, as derived from the equation, lies in the superposition of cosine waves with varying periods to form geometric sequences. Once all parameters are defined, its contour shape becomes fixed. Further surface information can then be extracted by adjusting the interval length of x. This also shows that the W–M function satisfies the self-similarity property of the ubiquitiform theory. Where $\mathrm{z}\left(\mathrm{x}\right)$ is the rough surface contour height function; x is the coordinate along the horizontal direction of the contour; and γ (a constant greater than 1) determines the spatial frequency density of the contour. Since the phase of the actual rough surface contour is random, in order to reflect this feature, it is usually taken as $\mathsf{\gamma }=1.5$ [26], which can be applied to the high spectral density and the randomness of the phase; n is the frequency exponent, which controls the number of iterations of generating the surface; ${\mathsf{\gamma }}^{\mathrm{n}}$ is the spatial frequency of the random profile, corresponding to the reciprocal of the wavelength of the rough surface, which determines the frequency spectrum of the rough surface; and ${\mathrm{n}}_1$ is the minimum frequency index of the surface topography, which depends on the sampling length of the simulated surface. This sampling length is precisely the half-wavelength of the cosine waves, which can be determined by ${\mathsf{\gamma }}^{{\mathrm{n}}_1}=1/\mathrm{L}$.

Ubiquitiform theory describes that the actual physical objects are not arbitrarily small details; that is, the actual simulation of rough surfaces cannot make the above fractal W–M function of n tend to infinity, so it is necessary to limit the high-frequency part of Formula (1), so that there is an upper limit ${\mathrm{n}}_2$ of the frequency exponent n. Under the ubiquitiform theory, the two-dimensional W–M function can be rewritten as follows:

$$\mathrm{z}\left(\mathrm{x}\right)={\mathrm{G}}^{\left(\mathrm{D}-1\right)}{\sum }_{\mathrm{n}={\mathrm{n}}_1}^{{\mathrm{n}}_2}{\mathsf{\gamma }}^{\left(\mathrm{D}-2\right)\mathrm{n}}\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi {\mathsf{\gamma }}^{\mathrm{n}}\mathrm{x}\right)$$

(2)

For a given ubiquitiform, there exists an accompanying fractal corresponding to its iterative process and the original fractal dimension can be replaced by the ubiquitiformal complexity to characterize the complexity of the rough surface. D in Equation (2) is the ubiquitiformal complexity, which is an important parameter describing the complex nonlinear characteristics of the ubiquitiformal contour. G denotes the characteristic scale factor of the size of z(x), which influences the height change in the detailed part of z(x), and can be adjusted by the characteristic scale factor G to adjust the height of microbumps of the rough surface during the surface simulation process. The height of the asperity on the rough surface can be adjusted by the characteristic scale factor G.

$\mathrm{n}=\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}\left[{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(1/\mathrm{L}\right)}{\ln \mathsf{\gamma }}}\right]$ can be obtained from ${\mathsf{\gamma }}^{\mathrm{n}}=1/\mathrm{L}$. The half-wavelength L is the scale factor δ of the surface measurement, i.e., the metric scale. According to the ubiquitiform theory, the half-wavelength should have an upper definite bound ${\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and lower definite bound ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$. ${\mathrm{n}}_1$ and ${\mathrm{n}}_2$ are the lower and upper limits of the frequency index, which establish the frequency truncation criterion. $\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}$ denotes the floor function (rounding down).

$${\mathrm{n}}_1=\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}\left[{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(1/\mathsf{\delta }\mathrm{m}\mathrm{a}\mathrm{x}\right)}{\ln \mathsf{\gamma }}}\right]$$

(3)

$${\mathrm{n}}_2=\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{r}\left[{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(1/\mathsf{\delta }\mathrm{m}\mathrm{i}\mathrm{n}\right)}{\ln \mathsf{\gamma }}}\right]$$

(4)

The lower limit of the ubiquitiformal metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ characterizes the microscale contact mechanical properties of the ubiquitiformal surface, which is a key parameter in the accurate calculation of the ubiquitiformal complexity and the theoretical modeling of the ubiquitiformal interfacial contact problem, and its value depends on the micro- and nanosize structure of the ubiquitiformal surface. For a given ubiquitiformal rough surface, the values of the above parameters, such as the lower limit of metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$, have also been determined, and then the values of the iterative parameters ${\mathrm{n}}_1$, ${\mathrm{n}}_2$ can be adjusted to obtain results similar to the actual rough surface. In summary, it can be seen that the rough surface morphology of the main parameters of the influence of the ubiquitiformal complexity D, the characteristic scale factor G, and the lower limit of metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$, will not change with the observation of the rough surface of the instrument resolution changes, collectively referred to as the rough surface of the ubiquitiformal characteristics of the parameter.

Figure 2a shows the fractal contours generated by MATLAB R2024a programming according to Equation (1) when $\mathrm{G}={10}^{-10 }\mathrm{m}$ and D is taken as 1.2, 1.5, and 1.8, respectively. Figure 2b shows the ubiquitiformal contours generated by MATLAB programming according to Equations (2)–(4) when $\mathrm{G}={10}^{-10 }\mathrm{m}$, ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}={10}^{-8 }\mathrm{m}$, and D is taken as 1.2, 1.5, and 1.8, respectively. As illustrated in Figure 2, the contour simulation results of both the fractal theory and the ubiquitiformal theory exhibit self-affine similarity. Notably, the higher the complexity D is, the higher the degree of irregularity and complexity of the surface contour is; as the complexity D increases, the amplitude of change in the rough surface contour, i.e., the height, decreases, which means that the surface roughness decreases consequently. The difference between the simulation result of ubiquitiformal contours and that of fractal theory is that the physical objects described by the ubiquitiformal theory do not have the arbitrarily small details as those described by the fractal theory. When the lower limit of the metric scale approaches zero, the rough surface will exhibit an infinitely fine profile. That is, the ubiquitiformal contours will transition to fractal rough contours.

Figure 3 shows the ubiquitiformal profile curves generated by MATLAB programming according to Equations (2)–(4) at $\mathrm{D}=1.5$, ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}={10}^{-8 }\mathrm{m}$, and $\mathrm{G}={10}^{-8},{10}^{-10},{10}^{-12 }\mathrm{m}$, respectively. From the figure, it can be seen that the height of the rough surface becomes smaller as the ubiquitiformal feature scale factor G decreases. However, the undulation of the rough surface profile is basically unchanged and the density and complexity of its surface profile are basically the same. This indicates that the ubiquitiformal feature scale factor G only affects the height of the rough surface and does not affect the complexity of the rough surface.

Figure 4 shows the ubiquitiformal contour curves generated by MATLAB programming according to Equations (2)–(4) at $\mathrm{D}=1.5$, $\mathrm{G}={10}^{-10 }\mathrm{m}$, and ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}={10}^{-4},{10}^{-5},{10}^{-6 }\mathrm{m},\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$. From the figure, it can be seen that as the lower limit of the ubiquitiformal metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ decreases, the frequency of surface contour undulations increases, and the two-dimensional rough surface contour will show a smaller portion of details. It can be inferred that when ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ tends to 0, the two-dimensional rough surface then shows infinitely fine contours when the ubiquitiformal rough surface transitions to the fractal rough surface, which corresponds to the fractal theory results.

3. Simulation of Anisotropic Three-Dimensional Ubiquitiformal Surface Morphology

In this section, anisotropic three-dimensional ubiquitiformal surfaces will be simulated based on the anisotropic ubiquitiform theory.

According to the literature [27], anisotropic fractal surfaces can be generated by simulating a bivariate W–M function with random phase, which can be expressed in the polar coordinate system as

$$W\left(\rho ,\theta \right)=\sqrt{{\displaystyle \frac{\ln \gamma }{M}}}\times {\displaystyle \sum _{m=1}^M{A}_m}{\displaystyle \sum _{n=-\infty }^{+\infty }\frac{1-\exp \left[j{k}_0{\gamma }^n\rho \cos \left(\theta -{\alpha }_m\right)\right]}{{\left(k_0{\gamma }^n\right)}^{2-D}}}\exp \left(j{\varphi }_{m,n}\right)$$

(5)

where M is the number of overlapping rumble sections on the constructed surface; $A_m$ is the quantity controlling the geometric anisotropy of the surface; j is the imaginary unit, $j=\sqrt{-1}$; $k_0$ is the number of spatial waves on the fractal rough surface; ρ is the polar diameter; θ is the polar angle; ${\alpha }_m$ is the arbitrary angle in the azimuthal direction biasing the rumble section; and ${\varphi }_{m,n}$ is the random phase uniformly distributed in the range $\left(0,2\pi \right)$.

If a fractal surface has randomness in each plane direction, its surface height function can be described by the real part of Equation (5)

$$z\left(\rho ,\theta \right)=\mathrm{Re}W\left(\rho ,\theta \right)=\sqrt{{\displaystyle \frac{\ln \gamma }{M}}}\times {\displaystyle \sum _{m=1}^M{A}_m}{\displaystyle \sum _{n=-\infty }^{+\infty }\frac{\cos {\varphi }_{m,n}-\cos \left[k_0{\gamma }^n\rho \cos \left(\theta -{\alpha }_m\right)+{\varphi }_{m,n}\right]}{{\left(k_0{\gamma }^n\right)}^{2-D}}}$$

(6)

In order to transform to the expression in the Cartesian coordinate system, some parameters in Equation (6) are transformed as follows

$$\rho =\sqrt{x^2+y^2}$$

(7)

$$\theta =\arctan {\displaystyle \frac{y}{x}}$$

(8)

$$A_m={\left(2\pi \right)}^{2-D}{G}^{D-1}$$

(9)

$$k_0={\displaystyle \frac{2\pi }{L}}$$

(10)

$${\alpha }_m={\displaystyle \frac{\pi m}{M}}$$

(11)

Substituting Equations (7)–(11) into Equation (6), the height of the fractal rough surface profile is obtained as

$$\begin{array}{l}\mathrm{z}\left(x,y\right)=L{\left({\displaystyle \frac{G}{L}}\right)}^{\left(D-1\right)}{\left({\displaystyle \frac{\ln \gamma }{M}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=0}^{+\infty }{\gamma }^{n\left(D-2\right)}}}\times \\ \left\{\cos {\varphi }_{m,n}-\cos \left[{\displaystyle \frac{2\pi {\gamma }^n{\left(x^2+y^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{L}}\cos \left(\arctan \left({\displaystyle \frac{y}{x}}\right)-{\displaystyle \frac{\pi m}{M}}\right)+{\varphi }_{m,n}\right]\right\}\end{array}$$

(12)

Under the ubiquitiform theory, the evaluation length of the actual surface contour n cannot be infinite. Setting the upper limit of the frequency index as $n_2$ and substituting Equation (4) into Equation (12) yields

$$\begin{array}{l}\mathrm{z}\left(x,y\right)=L{\left({\displaystyle \frac{G}{L}}\right)}^{\left(D-1\right)}{\left({\displaystyle \frac{\ln \gamma }{M}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=0}^{floor\left[\frac{\ln \left(1/{\delta }_{\min }\right)}{\ln \gamma }\right]}{\gamma }^{n\left(D-2\right)}}}\times \\ \left\{\cos {\varphi }_{m,n}-\cos \left[{\displaystyle \frac{2\pi {\gamma }^n{\left(x^2+y^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{L}}\cos \left(\arctan \left({\displaystyle \frac{y}{x}}\right)-{\displaystyle \frac{\pi m}{M}}\right)+{\varphi }_{m,n}\right]\right\}\end{array}$$

(13)

Equation (13) is the W–M function for anisotropic three-dimensional surface with ubiquitiformal features. Given the parameters $L=6.1\times {10}^{-7}$ m, $G=1.36\times {10}^{-11}$ m, $\gamma =1.5$, $M=10,$ $1\le m\le 10,$ m is integer, ${\varphi }_{m,n}=0$, ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}={10}^{-4 }\mathrm{m},$ when D is taken 1.2, 1.5, 1.8, respectively, and the sampling length is taken as 0.01 μm, the anisotropic three-dimensional ubiquitiformal surfaces generated by simulation are shown in Figure 5.

As the ubiquitiformal complexity D increases, it can be seen that: the complex amplitude of the change in the height of the surface contour decreases, i.e., the roughness decreases; the degree of irregularity and complexity of the surface three-dimensional contour increases; and the three-dimensional contour curves have self-affine similarity.

4. Values of the Lower Limit of the Ubiquitiformal Metric Scale and Ubiquitiformal Complexity

A fractal curve is a fractal interface profile curve intercepted from a normalized cross-section. The length of a fractal curve with nominal length l and dimension D (1 < D < 2) is when different observation scales, i.e., the scale factor δ (a key parameter used to characterize the self-similarity of interfaces), converge to 0

$$\mathrm{L}=\underset{\mathsf{\delta }\to 0}{\lim }\mathrm{L}\left(\mathsf{\delta }\right)=\underset{\mathsf{\delta }\to 0}{\lim }{\mathrm{l}}^{\mathrm{D}}{\mathsf{\delta }}^{1-\mathrm{D}}=\mathrm{\infty }$$

(14)

The actual physical object does not have an infinitely fine structure, so the corresponding iterative process cannot be infinite, i.e., the process of δ converging to 0 in Equation (14) cannot be realized, and the length of the ubiquitiformal curve is

$${\mathrm{L}}_{\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{f}}=\underset{\mathsf{\delta }\to {\mathsf{\delta }}_{\min }}{\lim }\mathrm{L}\left(\mathsf{\delta }\right)={\mathrm{l}}^{\mathrm{D}}{\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{1-\mathrm{D}}<\mathrm{\infty }$$

(15)

Equation (15) shows that the fractal scale factor δ tends to the lower limit of the ubiquitiformal metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$; the ubiquitiformal curve has a finite metric, which is fundamentally different from the corresponding fractal curve with an infinite metric. The lower limit of the ubiquitiformal metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ can characterize the microscale contact mechanical properties of ubiquitiformal rough surfaces, which is a key parameter for describing the self-similarity of interfaces and the accurate calculation of the ubiquitiformal complexity, but so far there is limited research, and there are no conclusions on how to determine the value of this parameter.

In the following, an improved adaptive box counting algorithm, referred to as the box-counting method, is designed to use squares with different side lengths δ to cover the ubiquitiformal curves to be measured, and the number of meshes occupied by the ubiquitiformal curves to be measured $\mathrm{N}\left(\mathsf{\delta }\right)$ will be changed accordingly when the side lengths of the square meshes are changed. The basic steps are designed as follows: square boxes with different side lengths δ are used to cover the to-be-measured pantomorphic curves in a complete and immediate manner and the number of square boxes N required is obtained through MATLAB R2024a programming; the collected data about the surface profile are converted into double logarithmic coordinate points of the metric scale and the measure, i.e., (lnδ, lnN), which is denoted by $\left({{\mathrm{t}}_{\mathrm{i}},\mathrm{f}}_{\mathrm{i}}\right)$, I = 1,2, …, N. Solve for m and n, $1\le \mathrm{m}<\mathrm{n}\le \mathrm{N}$, such that the sum of the deviations $\left[{\mathrm{t}}_1,{\mathrm{t}}_{\mathrm{m}-1}\right]$, $\left[{\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{n}}\right]$, $\left[{\mathrm{t}}_{\mathrm{n}+1},{\mathrm{t}}_{\mathrm{N}}\right]$ of the fitted data of the three straight lines obtained by the least squares linear regression method $\mathrm{F}\left(\mathrm{m},\mathrm{n}\right)$ is minimized in each of the three regions ${\mathrm{Q}}_1$, ${\mathrm{Q}}_2$, ${\mathrm{Q}}_3$

$${\mathrm{Q}}_1={\sum }_{\mathrm{t}=1}^{\mathrm{m}-1}{\left({\mathrm{f}}_{\mathrm{i}}-{\mathrm{a}}_1-{\mathrm{b}}_1{\mathrm{t}}_{\mathrm{i}}\right)}^2$$

(16)

$${\mathrm{Q}}_2={\sum }_{\mathrm{t}=\mathrm{m}}^{\mathrm{n}}{\left({\mathrm{f}}_{\mathrm{i}}-{\mathrm{a}}_2-{\mathrm{b}}_2{\mathrm{t}}_{\mathrm{i}}\right)}^2$$

(17)

$${\mathrm{Q}}_3={\sum }_{\mathrm{t}=\mathrm{n}+1}^{\mathrm{N}}{\left({\mathrm{f}}_{\mathrm{i}}-{\mathrm{a}}_3-{\mathrm{b}}_3{\mathrm{t}}_{\mathrm{i}}\right)}^2$$

(18)

$$\mathrm{F}\left(\mathrm{m},\mathrm{n}\right)=\mathrm{m}\mathrm{i}\mathrm{n}{\sum }_{\mathrm{k}=1}^3{\mathrm{Q}}_{\mathrm{k}}$$

(19)

The interval $\left({\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{n}}\right)$ obtained by satisfying the above equation to obtain m, n is the possible scale-free zone, and the data points determined in this way and the straight line obtained by fitting have the best linearity. In the above Equations (16)–(18), the values of the fitting parameters ${\mathrm{a}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}}$ are taken as follows

$${\mathrm{a}}_1={\displaystyle \frac{1}{\mathrm{m}-1}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{f}}_{\mathrm{i}}-{\displaystyle \frac{{\mathrm{b}}_1}{\mathrm{m}-1}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{t}}_{\mathrm{i}}$$

(20)

$${\mathrm{b}}_1={\displaystyle \frac{\left(\mathrm{m}-1\right){\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}\left({\mathrm{f}}_{\mathrm{i}}{\mathrm{t}}_{\mathrm{i}}\right)-\left({\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{f}}_{\mathrm{i}}\right)\left({\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{t}}_{\mathrm{i}}\right)}{\left(\mathrm{m}-1\right){\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{t}}_{\mathrm{i}}^2-{\left({\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{t}}_{\mathrm{i}}\right)}^2}}$$

(21)

$${\mathrm{a}}_2={\displaystyle \frac{1}{\mathrm{n}-\mathrm{m}+1}}{\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}{\mathrm{f}}_{\mathrm{i}}-{\displaystyle \frac{{\mathrm{b}}_2}{\mathrm{n}-\mathrm{m}+1}}{\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}$$

(22)

$${\mathrm{b}}_2={\displaystyle \frac{\left(\mathrm{n}-\mathrm{m}+1\right){\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}\left({\mathrm{f}}_{\mathrm{i}}{\mathrm{t}}_{\mathrm{i}}\right)-\left({\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}{\mathrm{f}}_{\mathrm{i}}\right)\left({\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}\right)}{\left(\mathrm{n}-\mathrm{m}+1\right){\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}^2-{\left({\sum }_{\mathrm{i}=\mathrm{m}}^{\mathrm{n}}{\mathrm{t}}_{\mathrm{i}}\right)}^2}}$$

(23)

$${\mathrm{a}}_3={\displaystyle \frac{1}{\mathrm{N}-\mathrm{n}}}{\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}{\mathrm{f}}_{\mathrm{i}}-{\displaystyle \frac{{\mathrm{b}}_3}{\mathrm{N}-\mathrm{n}}}{\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i}}$$

(24)

$${\mathrm{b}}_3={\displaystyle \frac{\left(\mathrm{N}-\mathrm{n}\right){\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}\left({\mathrm{f}}_{\mathrm{i}}{\mathrm{t}}_{\mathrm{i}}\right)-\left({\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}{\mathrm{f}}_{\mathrm{i}}\right)\left({\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i}}\right)}{\left(\mathrm{N}-\mathrm{n}\right){\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i}}^2-{\left({\sum }_{\mathrm{i}=\mathrm{n}+1}^{\mathrm{N}}{\mathrm{t}}_{\mathrm{i}}\right)}^2}}$$

(25)

where $\left({\mathrm{t}}_{\mathrm{i}},{\mathrm{f}}_{\mathrm{i}}\right)$ are the discrete data points and ${\mathrm{a}}_{\mathrm{i}}$ and ${\mathrm{b}}_{\mathrm{i}}$ are the intercept and slope of the least squares fitted line, respectively.

The linear segment $\left[{\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{n}}\right]$ corresponding to finding the smallest deviation, i.e., when the linear relationship is best, is the region with the best degree of self-similarity, and the minimum box scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and maximum box scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ in that region are the corresponding sums.

For a ubiquitiformal set F, $\mathrm{N}\left(\mathsf{\delta }\right)$ is the number of sets covering F with diameter at most δ. $\mathrm{N}\left(\mathsf{\delta }\right)$ and δ that satisfy a power law relationship

$$\mathrm{N}\left(\mathsf{\delta }\right)\propto {\mathsf{\delta }}^{\mathrm{k}}$$

(26)

Fitting the data between the above linear best parts $\left[{\mathrm{t}}_{\mathrm{m}},{\mathrm{t}}_{\mathrm{n}}\right]$ in double logarithmic coordinates, the resulting slope K is related to the ubiquitiformal complexity D as

$$\mathrm{D}=-\mathrm{K}$$

(27)

5. Method Validation

The Koch curve is one of the three classical self-similar sets. It is a kind of geometric curve like a snowflake, so it is also called snowflake curve, which is a special case of de Rham curve, which is recursively split into 4 identical small segments by a line segment digging out 1/3 of the middle segment, and each segment is 1/3 of the original length, and so on and so forth, so its Hausdorff dimensions $\mathrm{s}={\log }_{3}4$. When the scale is ${\left({\displaystyle \frac{1}{3}}\right)}^{\mathrm{n}}$ (n is the recursive depth), the curve generated by MATLAB programming has a $4^{\mathrm{n}}$ unit.

Based on the self-similar dimension, the dimension D of the Koch curve can be calculated as

$$\mathrm{D}={\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{N}\right)}{\mathrm{l}\mathrm{o}\mathrm{g}(1/\mathrm{r})}}={\displaystyle \frac{\mathrm{l}\mathrm{o}\mathrm{g}4}{\mathrm{l}\mathrm{o}\mathrm{g}3}}\approx 1.262$$

(28)

where N is the number of new segments generated recursively, which is 4 and r is the scaling ratio of each segment with respect to the original segment (for Koch curves, r = 1/3).

In this section, based on the above constructed method of taking the values of the contact characteristic parameters of the ubiquitiformal interface, the correctness of the method is verified by the example of the Koch curve. First, according to the construction process of the Koch curve, MATLAB software is used to write the M function that generates the generating element of the Koch curve, and the recursive depth n is taken as 3, 4, 5, 6, 7, and 8, respectively, and each Koch curve is generated through the recursive call of this function. Then, the plotting function is utilized to generate Koch curve images (e.g., Figure 6), which are stored in digital image format in order to provide inputs for the subsequent box-counting method processing. Finally, to obtain sufficient accuracy, the images are adjusted and it is ensured that they have a high resolution of 4025 × 3146. In this case, the recursion depth n is the number of layers of recursive subdivision in the Koch curve generation process, i.e., the number of times the current line segment is split and replaced with a more complex graph. The recursion depth n determines the level of detail of the Koch curve and the complexity of the final generated graph. It can also be seen from Figure 6 that the larger the value of n, the higher the complexity of the Koch curve and the more line segments.

The computation of the ubiquitiformal complexity D depends on the accurate representation of the objects in the graph, so the generated Koch curve image needs to be preprocessed as necessary to ensure the accuracy of the computation. The color or grayscale image is transformed into a black and white binary image as shown in Figure 7. The original image pixel luminance values are normalized to the range [0, 1], and a global threshold of 0.56 is calculated by using the Otsu method, which then distinguishes the curve (foreground) from the background region. By median filtering, possible noise points in the image are eliminated and the main structure of the Koch curve itself is preserved.

Next, the improved adaptive box counting algorithm designed in the previous section is used to measure and calculate the ubiquitiformal complexity: square boxes with different side lengths δ are selected to cover the Koch curves and the corresponding number of boxes N(δ) is recorded; the bilogarithmic relationship is plotted with lnδ as the horizontal coordinate and lnN(δ) as the vertical coordinate; the data points are fitted by the least squares linear regression given in the previous section; and the whole process and the linear section are calculated separately. The slope of the best part is ${\mathrm{Q}}_2$, the negative of which is the ubiquitiformal complexity D. Taking n = 5 as an example, the values of the box side lengths δ are [1, 2, 4, 8, 16, 32, 64, 128, 256, and 512] pixels, and the corresponding number of boxes N(δ) are [27724, 11088, 4703, 2093, 938, 387, 154, 69, 27, and 10], respectively. The double logarithmic relation is plotted in Figure 8. The serial number on the left side of the ${\mathrm{Q}}_2$ line segment is 4, which corresponds to a box with a side length of 8 pixels, and the serial number on the right side is 7, which corresponds to a box with a side length of 64 pixels. At this time, the negative of the slope of the ${\mathrm{Q}}_2$ line segment is 1.2571, and the error is −0.37%.

Koch curve images with different recursion depths are used for experimental tests to compare the variation rule of their ubiquitiformal complexity. In order to ensure that each line segment corresponds to at least one pixel, the relationship between recursion depth n and resolution (R × R) is $\mathrm{R}=3^{\mathrm{n}}$. Due to limitations in image resolution and computational resources, all images in this study were generated at a uniform maximum resolution of 4025 × 3146 pixels. respectively, the ubiquitiformal complexity of all the points and ${\mathrm{Q}}_2$ line segments is compared with the standard dimensions of the Koch curve of 1.262 to compute the error, and the results are shown in

Table 1. Results of ubiquitiformal complexity calculation with different recursion depths.

Recursion Depth n	Minimum Resolution	All Points Dimension	All Points Error	Q2 Dimension	Q2 Line Segment Number	Q2 Error
3	33 = 27	1.1554	−8.43%	1.1260	6–9	−7.76%
4	34 = 81	1.2032	−4.64%	1.1544	2–7	−4.51%
5	35 = 243	1.2529	−0.71%	1.2571	4–7	−0.37%
6	36 = 729	1.2904	2.27%	1.2758	2–8	1.11%
7	37 = 2187	1.2970	2.79%	1.2748	6–9	1.03%
8	38 = 6561	1.2963	2.73%	1.2748	6–9	1.03%

. The obtained results are shown in Table 1.

From the analysis of Table 1, it can be seen that when the recursive depth n ≥ 5, the dimension of the Koch curve is close to the standard dimension 1.262, and when the recursive depth n = 7 and n = 8, the ubiquitiformal complexity of all the points and ${\mathrm{Q}}_2$ line segments is basically the same as the error obtained from the comparison of the Koch curve’s standard dimension 1.262. The error value of the dimension of the ${\mathrm{Q}}_2$ line segments is smaller than the whole process, which indicates that the ubiquitiformal interface contact characteristics constructed in this paper can obtain a more accurate ubiquitiformal complexity. It shows that the parameter value method constructed in this paper can obtain a more accurate ubiquitiformal complexity, which confirms the validity and rationality of the method. It should be noted that when the value of recursive depth is small, such as n < 5, the Koch curve has fewer iterations and a simpler structure, and it does not have obvious ubiquitiformal characteristics of actual engineering surfaces.

6. Conclusions

In this paper, the following conclusions and outcomes are obtained: (1) the ubiquitiformal contour curve has a certain degree of self-affine similarity; (2) the larger the ubiquitiformal complexity D is, the higher the degree of irregularity and complexity of the surface contour is; (3) the ubiquitiformal feature scale coefficient G only affects the height of the rough surface and does not affect the degree of complexity of the rough surface; (4) as the lower limit of the ubiquitiformal metric scale ${\mathsf{\delta }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ decreases, the frequency of surface contour undulation increases and the rough surface contour will show a smaller part of the details; (5) the new W–M function expressions for 2D and anisotropic 3D surfaces based on the ubiquitiform theory constructed effectively solve the infinite recurrence paradox of the traditional fractal model; and (6) the lower limit of the metric scale for the self-similar region of the asperities on rough surfaces is determined, and a method for calculating the ubiquitiformal complexity is established, which is easy to implement.

The research results in this paper not only theoretically enrich the toolbox of rough surface contact characterization but also provide a more accurate method of predicting contact characteristic parameters for the matching design of precision mechanical interfaces.