Novel three-dimensional surface topography fractal dimension calculation method

Abstract

The invention provides a novel method (3D-Sa) for calculating fractal dimensions of a three-dimensional surface, and belongs to the field of fractal geometry of mathematics. Firstly, fractal models of isotropic and anisotropic curved surfaces are established based on a W-M function; the three-dimensional morphology of the fractal curved surface under different fractal dimensions is simulated by utilizing MATLAB (Matrix Laboratory) software. The surface parameter of the fractal curved surface is calculated and analyzed to obtain the mapping relation between the fractal dimension and the traditional surface parameter. The invention provides a surface arithmetic mean height method (3D-Sa) capable of accurately calculating the fractal dimension of the machined surface. In combination with a surface arithmetic mean height formula and the definition of the fractal dimension, a calculation formula of the fractal dimension is deduced by taking the surface arithmetic mean height Sa as a main body. And then, optimizing the formula by using a simulation fitting mode to realize fitting parameter correction. The method has relatively low relative errors for anisotropic and isotropic curved surfaces, and the calculation precision is stable.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of fractal theory and mechanical engineering. In particular, it relates to a novel method for calculating the fractal dimension of three-dimensional surface morphology. Background Technology

[0002] To make the technical problems to be solved, the technical solutions and the beneficial effects of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings.

[0003] The presence of mating surfaces reduces mechanical strength, directly impacting the development of the equipment manufacturing industry. Therefore, accurately analyzing the contact characteristics of mating surfaces has always been a topic of interest for many scholars. However, the statistical parameters commonly used are affected by factors such as the resolution of roughness measuring instruments, and rough surfaces exhibit multi-scale characteristics, making their description of rough surfaces insufficiently comprehensive and accurate.

[0004] Therefore, we consider using fractal geometry to analyze mechanical contact surfaces using scale-independent parameters. Fractal dimension, as an important parameter in fractal geometry, can quantitatively describe the complexity of geometric shapes and is a key parameter for describing fractal surfaces.

[0005] There are two approaches to studying fractal dimension. The first is the relationship between the two-dimensional surface profile dimension and the surface fractal dimension proposed by Mandelbrot in his research. However, because it is difficult to determine which cross-sectional profile and measurement direction can represent the surface, estimating the fractal dimension of anisotropic surfaces based on the two-dimensional profile fractal dimension is not accurate.

[0006] The second approach aims to avoid this error by directly calculating the surface fractal dimension using surface data. Key methods include the Difference Box Count (DBC) method, the Triangular Prism Area Method (TPSA), and the Structure Function Method. While these methods directly calculate the fractal dimension using surface data, avoiding calculations based on contours measured along a specific direction, they also have their own drawbacks. Given these issues, it is necessary to research a more accurate, efficient, and realistic method for calculating the three-dimensional fractal dimension.

[0007] This invention aims to improve the accuracy of fractal dimension calculation. It conducts theoretical, simulation and experimental research on fractal surfaces, and proposes a new method for calculating fractal dimension based on the arithmetic mean height of the surface. Summary of the Invention

[0008] The purpose of this invention is to provide a novel method for calculating the fractal dimension of three-dimensional surface topography, in order to solve the problem of how to directly calculate the high-precision fractal dimension D from surface data.

[0009] To achieve the above objectives, the present invention is implemented through the following solution:

[0010] Based on fractal theory, we conduct simulation analysis on the traditional surface parameters of fractal surfaces.

[0011] To investigate the relationship between fractal dimension D and three-dimensional surface parameters, MATLAB was used to simulate fractal surfaces with different D values ​​and calculate their traditional surface parameters. It can be found that there is a strong mapping relationship between fractal dimension and traditional surface parameters.

[0012] Derivation of the 3D-Sa method: Using the WM fractal function, the mean square increment of the WM fractal surface and its increment follow the following power law:

[0013]

[0014] Furthermore, according to the definition of the mean square increment, the surface increment

[0015]

[0016] Therefore, the root mean square of the fractal surface can be expressed as:

[0017]

[0018] Based on the definition of root mean square (RMS) and its connection to the definition of surface arithmetic mean height, using the expression for RMS containing fractal dimension as an intermediary to link fractal dimension and surface arithmetic mean height, we can obtain the following:

[0019]

[0020] Simplifying the expression for Sa using polar coordinate transformation yields:

[0021]

[0022] To ensure calculation accuracy, the formula was corrected through simulation, based on equation (1).

[0023]

[0024] It was found that under multiple values ​​of L, the calculated value of the initial formula (named LSa in the formula) and the theoretical value obtained by the definition of Sa have a power-law relationship trend, that is, they have a good linear relationship after taking the logarithm. Therefore, L is chosen as the variable to determine the slope k and intercept b of the modified formula after taking the logarithm.

[0025] After a series of optimizations, the final simplified formula is as follows:

[0026]

[0027] Where A, B, m, and n are correction coefficients related to L. , , , .

[0028] Compared with the DBC method and the structural method, the 3D-Sa method has better stability and higher accuracy than the other two methods.

[0029] A surface morphology acquisition test rig was constructed using 45 steel as the material. Microscopic surface data were collected from specimens processed by turning, milling, and grinding. The fractal dimension values ​​of the actual processed specimens were obtained for each method. Comparison with fractal dimension values ​​obtained from other calculation methods showed that the simulation results were consistent with the actual processing results. Attached Figure Description

[0030] Figure 1 This is a flowchart illustrating a novel method for calculating the fractal dimension of three-dimensional surface topography, a preferred embodiment of the present invention.

[0031] Figure 2 This refers to the relationship between the isotropic surface D and the traditional surface parameters in the relationship between fractal dimension and fractal surface parameters in this invention;

[0032] Figure 3 This refers to the relationship between the anisotropic surface D and the traditional surface parameters in the relationship between fractal dimension and fractal surface parameters in this invention;

[0033] Figure 4 This refers to the calculated value of Sa under different L values ​​in the formula fitting optimization of this invention. Detailed Implementation Plan

[0034] The purpose of this invention is to provide a novel method for calculating the fractal dimension of three-dimensional surface topography, thereby solving the problem of how to directly calculate a high-precision fractal dimension D from surface data. It includes:

[0035] 1. Relationship between different three-dimensional surface parameters and fractal dimension, where Sa is the arithmetic mean height of the surface, defined as follows:

[0036]

[0037] The Z-coordinate is the arithmetic or geometric mean of the distances between points on the profile surface and the center plane. In the sampling area D, it is the arithmetic mean of the Z-coordinate distances between the measured profile surface and the established reference plane, i.e., the arithmetic mean of the absolute values ​​of the Z-coordinates of the surface roughness surface equation. Sq is the root mean square height, defined as the root mean square value of the distances between each point on the surface within a specified area S and the mean plane. Sp is the maximum value, i.e., the difference between the highest peak and the mean plane. Sv is the deepest value, i.e., the difference between the lowest valley and the mean plane. Sz is the maximum height, i.e., Sp - Sv.

[0038] 2. Relationship between fractal dimension and three-dimensional surface parameters

[0039] To investigate the relationship between fractal dimension D and three-dimensional surface parameters, MATLAB was used to simulate fractal surfaces with different D values, and their conventional surface parameters were calculated. For isotropic surfaces, such as... Figure 2 As shown, with the increase of the fractal dimension D, the surface arithmetic mean height Sa and the root mean mean height Sq gradually decrease. The maximum value Sp and the deepest value Sv gradually approach 0. The maximum height Sz gradually decreases. For anisotropic surfaces, such as Figure 3 As shown, with the increase of the fractal dimension D, the surface arithmetic mean height Sa and the root mean height Sq gradually decrease. The maximum value Sp and the deepest value Sv gradually approach 0. The maximum height Sz gradually decreases. For the same value of D, the values ​​of Sa, Sq, and Sp are larger for isotropic surfaces than for anisotropic surfaces.

[0040] The study found a strong mapping relationship between fractal dimension and traditional surface parameters, and the same pattern applies to both isotropic and anisotropic surfaces. Theoretically, the characteristics of a microscopic surface can be measured by the magnitude of the fractal dimension.

[0041] 3.3 Derivation and simplification of the D-Sa method formula;

[0042] WM fractal functions are primarily used to simulate the contour curves of rough surfaces with fractal characteristics. They are continuous everywhere but not differentiable anywhere. Therefore, they must exist. Make Equal to the height of the average plane, at this time Establish a coordinate system with the origin to analyze the data points. The mean square increment of the WM fractal surface can then be expressed as:

[0043]

[0044] In the formula, Let be the process increments in the x and y directions, respectively; and represent the mathematical expectation of the data. The mean square increment of the WM fractal curve and the process increment follow a power law relationship as follows:

[0045]

[0046] According to the definition of the mean square increment, the surface increment It can be written as:

[0047]

[0048] Therefore, the root mean square of a fractal surface can be expressed as:

[0049]

[0050] According to the definition of root mean square (RMS), in statistical data analysis, it refers to the sum of the squares of all values, the average of the sums, and then the square root. Connecting this to the definition of the surface arithmetic mean height (Sa), Sa is the surface arithmetic mean height, referring to the arithmetic or geometric mean of the distances between points on the profile surface and the center plane. In the sampling area D, it is the arithmetic mean of the Z-coordinate distances between the measured profile surface and the established reference plane, i.e., the arithmetic mean of the absolute values ​​of the Z-coordinates of the surface roughness surface equation. The mathematical expression is:

[0051]

[0052] In the defined formula, the root mean square (RMS) and the surface arithmetic mean height share the same mean calculation component. The expression for the RMS containing the fractal dimension is used as an intermediary to link the fractal dimension and the surface arithmetic mean height.

[0053]

[0054] For surface data points, we can adjust the sampling length by... This is equivalent to a circle with radius L, which means performing a polar coordinate transformation in the expression for Sa:

[0055]

[0056] After simplification, we get:

[0057]

[0058] 4. Formula fitting

[0059] Since the above formula is based on the power law relationship between the mean square increment S and its increment, we find that most parameter values ​​in the WM function are ignored in the approximation. This leads to numerical deviations when calculating the D value through Sa or in reverse calculation. To ensure calculation accuracy, the formula is simulated and corrected, a large number of calculations are performed using the formula, and the calculation results are analyzed to obtain the following results: Figure 4 As shown.

[0060]

[0061] As shown in equation (1), it can be observed that under multiple values, the calculated value of the initial formula (named LSa in the formula) and the theoretical value obtained by the definition of Sa have a power-law relationship trend, that is, they have a good linear relationship after taking the logarithm. Therefore, L is chosen as the variable to determine the slope k and intercept b of the modified formula after taking the logarithm. A large amount of data simulation is used to determine the values ​​of the slope k and intercept b.

[0062] This invention uses L=0.0001 as the initial value and a step size of 0.0001 to take 100 sets of data with different sampling lengths, and fits them with k and b with L as the measure.

[0063] The value of b is optimized. The result of power-law fitting of b and L is as follows: Compared to other methods, power-law fitting is the most stable and has a sufficiently good fit when solving for the intercept.

[0064] The value of k is optimized. Since the slope k has a greater impact on the formula than the intercept b, and different solution methods each have their advantages in application, three schemes are designed for fitting and simulating the slope. The calculation results of the three fitting correction schemes are compared, and the advantages and disadvantages of each scheme are analyzed.

[0065] The scheme combines logarithmic domain linear polynomial fitting and power-law fitting. The formula is simple, the multiple fitting rate is good, the correction coefficient introduced in the fitting process is small (m1=-1.067, n1=5.968), and the average error is 0.0642%.

[0066] Scheme 2 uses power-law fitting for both slope k and intercept b. The formula is relatively simple, the fitting rate is good and stable, the correction coefficient introduced during the fitting process is small (m2=1.102, n2=-0.06399), and the average error is 0.0361%.

[0067] Scheme 3 combines first-order Fourier series fitting and power-law fitting in the logarithmic field. The formula is relatively complex, but the fitting rate is good. However, the correction coefficients introduced during the fitting process are relatively large (m3=-15.41, n3=45.83), and the average error is 0.1598%.

[0068] In summary, Scheme 2 has better computational accuracy, and its fitting rate and fitting parameters are stable under different data, with a relatively simple formula. Therefore, Scheme 2 is chosen for secondary correction. The specific steps are as follows: according to the formula of Combination 2, the m2 and n2 introduced by the first correction are ignored, i.e.

[0069]

[0070] Where A and B are correction coefficients related to L. , Using the above formula, the initial value of L is 0.0001, with a step size of 0.0001 for 10 sets; the initial value of D is 2.05, with a step size of 0.05 for 19 sets. Then, the calculated values ​​of D under different L values ​​are... L Fitting with the theoretical value D, and

[0071]

[0072] We obtain the fitting parameters ki and bi, analyze the relationship between ki, bi and L, and perform a second-order power-law fitting on k and L, i.e.

[0073]

[0074] In the formula , , In this fitting method, the fitting rate can reach 1. Similarly, for the second-order power-law fitting of b and L, i.e.

[0075]

[0076] In the formula , , The fitting rate can reach 0.9999. After the above steps, the formula after secondary correction can be obtained, and the simplified formula is as follows:

[0077]

[0078] Where A, B, m, and n are correction coefficients related to L. , , , .

[0079] The fractal dimension D was calculated using the method proposed in this invention, and the results were compared with those obtained by the DBC method and the structure function method.

[0080] Comparison reveals that the DBC method and structure function method have lower accuracy in calculating the fractal dimension D of fractal surfaces because they ignore many fine structures and surface points. The method presented in this paper calculates Sa from fractal surface data points and then calculates the fractal dimension D, resulting in smaller errors. The 3D-Sa method is more stable than the former two, ensuring the accuracy of data calculation. It significantly improves the average relative error of calculations for both isotropic and anisotropic surfaces, providing a theoretical basis for subsequent experimental applications.

[0081] Considering the most commonly used methods in the actual processing of sensor components, using 45 steel as the material, specimens were selected for three processing methods: turning, milling, and grinding, and microscopic surface data were collected from the specimens. The arithmetic mean height (Sa) of the surfaces of turned specimen 1 and turned specimen 2 were 8.596 and 12.84, respectively. The Sa of milled specimen 1 and milled specimen 2 were 9.779 and 23.73, respectively. The Sa of ground specimen 1 and ground specimen 2 were 25.22 and 27.21, respectively. The fractal dimension of the microscopic surfaces was calculated and compared using the DBC method, the structure function method, and the 3D-Sa method. The results are shown in Table 1.

[0082]

[0083] The calculation results show that the differences between the 3D-Sa method, DBC method, and structure function method for the same specimen are within a small range, and the three calculation methods show the same trend for the calculation results of multiple specimens.

Claims

1. A novel method for calculating the fractal dimension of three-dimensional surface topography, characterized in that, Includes the following steps ① Link different three-dimensional surface parameters with fractal dimension; Sa is the arithmetic mean height, Sq is the root mean flatness, Sp is the maximum value, i.e., the difference between the highest peak and the mean plane, Sv is the deepest value, i.e., the difference between the lowest valley and the mean plane, and Sz is the maximum value height, i.e., Sp-Sv. Analyze the relationship between fractal dimension and three-dimensional surface parameters. ② The derivation and simplification of the 3D-Sa method formula; the WM function is selected as the basic function for proposing the new calculation method, which represents the average square increment of the WM fractal curve. It is known that the average square increment and its increment follow a power law relationship. ③ Based on the definition of the mean square increment, the expression formulas for the surface increment and the root mean square are derived. Then, in conjunction with the definitions of the root mean square and the arithmetic mean height of the Sa surface, the mathematical expression for the arithmetic mean height of the Sa surface is derived based on the arithmetic mean of the absolute values ​​of the Z coordinates of the surface roughness surface equation. ④ Using the expression containing the fractal dimension at the root mean square as an intermediary, the fractal dimension is linked to the surface arithmetic mean height. Polar coordinate transformation is performed in the expression for Sa, and finally, after simplification, the final mathematical expression for the surface arithmetic mean height Sa is obtained:

2. The novel method for calculating the fractal dimension of three-dimensional surface topography as described in claim 1, characterized in that... The obtained formula was experimentally fitted and optimized; the mean square increment and its increment follow a power law relationship, which causes most parameter values ​​in WM to be ignored, resulting in some deviation in the calculation process; therefore, in order to ensure accuracy, the formula was simulated and corrected. The initial calculated value LSa and the theoretical value obtained by defining the surface arithmetic mean height Sa show a power-law trend, meaning they have a good linear relationship after taking the logarithm.

3. The novel method for calculating the fractal dimension of three-dimensional surface topography as described in claim 1, characterized in that... The obtained formula was experimentally fitted and optimized. Includes the following steps ①With L as the variable, we choose L as the variable to determine the slope k and intercept b of the modified formula after taking the logarithm. The values ​​of slope k and intercept b are determined through a large number of simulations. ②Optimize b. Compared with other forms, power fitting is the most stable and has a sufficiently good fit. Therefore, it can be directly calculated. ③ The value of k was optimized. Since the slope has a greater impact on the formula than the intercept, and different solution methods have their own advantages in application, three schemes were used for fitting simulation of the slope. The calculation results of the three schemes were compared, the advantages and disadvantages of the schemes were analyzed, and the better scheme was selected for optimization.

4. The novel method for calculating the fractal dimension of three-dimensional surface topography as described in claim 1, characterized in that... In the formula, Sa is the surface arithmetic mean height, which refers to the arithmetic or geometric mean of the distances between points on the contour surface and the center plane. This is the average squared increment of the WM fractal surface, used to relate the fractal dimension to the arithmetic mean height of the surface. This represents the surface increment of the fractal surface. is the root mean square of the fractal surface, used to calculate the arithmetic mean of the absolute values ​​of the Z-coordinates of the surface roughness equation.

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