CN112699508B - Fourier interpolation-based three-dimensional rough surface contact model construction method - Google Patents
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Abstract
The invention discloses a three-dimensional rough surface contact model construction method based on Fourier interpolation, which specifically comprises the following steps: s1: measuring the engineering three-dimensional rough surface by a topography instrument to obtain NxN data points; taking the plurality of data points as original rough surface data; s2: interpolating an NxN matrix formed by the original rough surface data by utilizing a Fourier interpolation method to obtain an Nt xNt matrix; s3: carrying out surface characteristic evaluation on rough surface data before and after interpolation; s4: and importing the original rough surface data into a commercial PE program, and carrying out grid division to construct a Fourier interpolation three-dimensional rough surface contact model. The method solves the problems of unit deformity, stress concentration, instability in the solving process and the like caused by peaks before interpolation, can improve the accuracy of results, and can provide reference for further calculation of the problems of contact, friction, resistance, heat transfer and the like of the three-dimensional surface.
Description
Technical Field
The invention relates to the technical field of smoothing engineering rough surfaces by using an interpolation method, in particular to a three-dimensional rough surface contact model construction method based on Fourier interpolation.
Background
Surface effects are often ignored in performing structural analysis, in part because surface effects are assumed to be minimal. However, in certain engineering applications, the effects caused by surface topography are of major concern, and especially when contact problems are involved, it becomes critical, for example: the contact area between the surfaces plays a very important role in determining the friction and adhesion between the surfaces. Thus, in many engineering fields, including tribology and microelectromechanical systems, attempts have been made to use or alter the behavior of the system at surfaces or interfaces between surfaces. For example: in tribological applications, real surfaces, by virtue of being subjected to a combination of normal and shear loads, will cause asperities on the surfaces to interact in a non-uniform manner, exerting forces on each other and creating friction and wear; in fluid sealing applications, the asperities must be compressed enough to close all potential leak paths across the sealing surface; in mems applications, surface finish or surface damage is on the same order of magnitude as the desired geometry and can also degrade device performance.
In these applications, it is desirable to model the system with a structured (or measured) surface topography to determine the effect of surface roughness on the system itself (friction, wear rate, seal capacity, failure, etc.). It follows that current research is increasingly trending towards numerical analysis using directly measured surface data, and this trend will continue in the future. An effective method also ensues, introducing the topography of the measured real surface into a Finite Element (FE) model. As these technologies become more prevalent and widely adopted by the industry, a high-level and executable tool (e.g., ANSYS, a commercial finite element software program that can perform surface analysis, etc.) will be pursued. The key to predicting the effect of surface topography is the ability of the surface relief structure in the system to deform in response to applied loads and the ability of the program to account for these relief structure deformations with as little solution error as possible.
In general, the surface data measured by the three-dimensional topographer is composed of a series of discrete points located at intervals, but the method of connecting the points of contact directly in the FE model makes the slope of the surface geometry where each data point is located discontinuous, and the degree of discontinuity depends on the roughness of the surface. The rougher the surface, the sharper the peaks of the asperities will be. These spikes will cause stress concentrations in the contact pattern between the rough surfaces. And in some cases may also result in a numerical solution that is singular or not completely solved.
The document shows that the rough surface is continuous under the micro-nano scale. However, in the process of measuring the engineering surface, many inevitable problems occur, for example, due to the space and precision limitations of the instrument, all points of the surface cannot be measured, and the missing data points may possibly cause distortion of the surface topography. Thus, smoothing the surface using interpolation or fitting techniques can avoid the generation of sharp surface roughness peaks, and the surface smoothing will distribute the contact load over a wider area than a single sharp rough surface.
In the documents collected and consulted currently, only the spline interpolation method is applied to smooth rough surfaces and to apply and simulate, but when the spline interpolation method is used, when sampling points are close and values are greatly different, spline curve interpolation does not work, in addition, the surface morphology after smoothing depends on the order of the spline seriously, surfaces obtained by spline interpolation of different orders are different, and the uncertainty of the surface after interpolation is caused.
Therefore, it is necessary to find an interpolation method for smoothing the rough surface and compensating for the topography distortion caused by insufficient accuracy of the instrument to some extent, which is a problem of great attention by researchers.
Disclosure of Invention
In order to solve the technical problem, the invention provides a three-dimensional rough surface contact model construction method based on Fourier interpolation, which is characterized in that interpolation is realized on measured engineering three-dimensional rough surface data by applying the Fourier interpolation method, and the same spectrum of the surface data can be ensured through multiple times of interpolation. And the interpolated three-dimensional surface data can be imported into commercial finite element software (such as ANSYS), linear interpolation is carried out on the coordinate data in the X direction and the Y direction according to the interpolation times, and the grid is further divided finely to establish a simulation model. Through tests, the simulation model established by the method solves the problems of unit deformity, stress concentration, instability in the solving process and the like caused by the peak before interpolation, and meanwhile, the accuracy of the result can be improved. And the method can provide reference for further calculation of contact, friction, resistance, heat transfer and other problems of the three-dimensional surface.
In order to achieve the purpose, the invention adopts the technical scheme that:
a three-dimensional rough surface contact model construction method based on Fourier interpolation specifically comprises the following steps:
s1: measuring the engineering three-dimensional rough surface by a topographer to obtain NxN data points; taking the data points as original rough surface data;
s2: interpolating an NxN matrix formed by the original rough surface data by utilizing a Fourier interpolation method to obtain an Nt x Nt matrix ； Where Nt is a positive integer multiple of N, i.e. Nt/N =2 m ,m＝1、2、3……；
S3: carrying out surface characteristic evaluation on rough surface data before and after interpolation;
s4: importing the interpolated rough surface data into a commercial PE program, and carrying out grid division to construct a three-dimensional rough surface contact model based on Fourier interpolation;
s5: the contact problem of the rough surface by normal pressure was tested.
Preferably, the optical topographer in S1 is a nanovast 400 white light optical topographer.
Preferably, the raw rough surface data comprises: highest peak, lowest valley and average roughness.
Preferably, S2 is specifically:
step 1: constructing the raw rough surface data points into an N matrix Z 0 ；
And 2, step: for the matrix Z 0 Executing the MATLAB command: z is a linear or branched member 1 ＝ifftshift(Z 0 ) Will matrix Z 0 Performing spectrum inversion operation on the internal elements to obtain a matrix Z 1 ；
And step 3: for the matrix Z 1 Executing the MATLAB command: z is a linear or branched member 2 ＝fft2(Z 1 ) Will matrix Z 1 Performing two-dimensional fast Fourier transform to obtain a matrix Z 2 ；
And 4, step 4: for the matrix Z 2 Executing the MATLAB command: z 3 ＝fftshift(Z 2 ) To obtain a matrix Z 3 ；
And 5: will NXN matrix Z 3 Expansion into Nt × Nt matrix Z 4 In the middle of the matrixPart is Z 3 The other elements are filled with 0; wherein, nt is the length of the last required matrix singles, is a power of 2 positive integers of N, and the specific procedure is as follows:
Z 4 ＝zeros(Nt,Nt)；
Z 4 ((Nt/2)-N/2+1:(Nt/2)+N/2,(Nt/2)-N/2+1:(Nt/2)+N/2)＝Z 3 ；
step 6: for the matrix Z 4 Executing the MATLAB command: z 5 ＝fftshift(Z 4 ) To obtain a matrix Z 5 ；
And 7: for the matrix Z 5 Executing the MATLAB command: z is a linear or branched member 6 ＝ifft(Z 5 ) To obtain a matrix Z 6 ；
And step 8: for the matrix Z 6 Executing the MATLAB command: z 7 ＝fftshift(Z 6 ) To obtain a matrix Z 7 ；
And step 9: for the matrix Z 7 Executing the MATLAB command:
Z 8 ＝2^2*sqrt(real(Z 7 )^2+imag(Z 7 ) ^ 2) to obtain a matrix Z 8 。
Preferably, Z is 8 ＝2^2*sqrt(real(Z 7 )^2+imag(Z 7 ) ^ 2) is to find the matrix Z 7 The amplitude of each element is multiplied by a matrix coefficient 2 2*(Nt/N) 。
Preferably, the commercial CAE simulation software in S4 uses a mechanical module in ANSYS.
Preferably, the grid division is to perform grid refinement on the interpolated rough surface by using a hierarchical point shifting method.
Compared with the prior art, the invention has the beneficial effects that:
(1) The invention applies Fourier interpolation to realize interpolation of measured engineering three-dimensional rough surface data, and can ensure the same spectrum of the surface data through multiple times of interpolation. And importing the interpolated three-dimensional surface data into commercial finite element software, performing linear interpolation on the coordinate data in the X and Y directions according to the interpolation times, and further finely dividing the grids to establish a simulation model. Through testing, the simulation model established by the method solves the problems of unit deformity, stress concentration, instability in the solving process and the like caused by the peak before interpolation;
(2) The method predicts data points which are not acquired by the instrument, can reduce the distortion of the surface topography to a certain extent, and improves the accuracy of the result. And the method can provide reference for further calculation of contact, friction, resistance, heat transfer and other problems of the three-dimensional surface.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a diagram of a Fourier interpolation implementation method;
FIG. 2 is a graph of initial rough surface topography;
FIG. 3 is a three-dimensional and two-dimensional topographical map after interpolation; wherein, a, c and e are three-dimensional rough topography maps which are subjected to double, quadruple and octave interpolation respectively; b. d and f are respectively the shape diagrams of the lines at the same position selected from the graphs a, c and e;
FIG. 4 is a comparison of the initial surface and the two, four, eight fold interpolated line shapes within an interval;
fig. 5 is an ANSYS model diagram of the surface after applying the initial surface and interpolation.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Example 1
The embodiment 1 provides a method for constructing a three-dimensional rough surface contact model based on fourier interpolation, which specifically comprises the following steps:
s1: measuring the engineering three-dimensional rough surface by a topography instrument to obtain NxN data points; taking the data points as original rough surface data;
s2: interpolating an NxN matrix formed by the original rough surface data by utilizing a Fourier interpolation method to obtain an Nt x Nt matrix ； Where Nt is a positive integer multiple of N, i.e. Nt/N =2 m ,m＝1、2、3……；
Step 1: constructing the raw rough surface data points into an N matrix Z 0 ；
Step 2: for the matrix Z 0 Executing the MATLAB command: z 1 ＝ifftshift(Z 0 ) Will matrix Z 0 Performing spectrum inversion operation on the internal elements to obtain a matrix Z 1 ；
And step 3: for the matrix Z 1 Executing the MATLAB command: z 2 ＝fft2(Z 1 ) Will matrix Z 1 Performing two-dimensional fast Fourier transform to obtain a matrix Z 2 ；
And 4, step 4: for the matrix Z 2 Executing the MATLAB command: z is a linear or branched member 3 ＝fftshift(Z 2 ) To obtain a matrix Z 3 ；
And 5: will NXN matrix Z 3 Expansion into Nt × Nt matrix Z 4 The middle part of the matrix is Z 3 The other elements are filled with 0; wherein, nt is the length of the last required matrix singles, is a power of 2 positive integers of N, and the specific procedure is as follows:
Z 4 ＝zeros(Nt,Nt)；
Z 4 ((Nt/2)-N/2+1:(Nt/2)+N/2,(Nt/2)-N/2+1:(Nt/2)+N/2)＝Z 3 ；
step 6: for is toThe matrix Z 4 Executing the MATLAB command: z is a linear or branched member 5 ＝fftshift(Z 4 ) To obtain a matrix Z 5 ；
And 7: for the matrix Z 5 Executing the MATLAB command: z 6 ＝ifft(Z 5 ) To obtain a matrix Z 6 ；
And 8: for the matrix Z 6 Executing the MATLAB command: z 7 ＝fftshift(Z 6 ) To obtain a matrix Z 7 ；
And step 9: for the matrix Z 7 Executing the MATLAB command:
Z 8 ＝2^2*sqrt(real(Z 7 )^2+imag(Z 7 ) ^ 2) to obtain a matrix Z 8 。
S3: carrying out surface characteristic evaluation on rough surface data before and after interpolation;
s4: importing the interpolated rough surface data into a commercial PE program, and carrying out grid division to construct a three-dimensional rough surface contact model based on Fourier interpolation;
s5: the contact problem of the rough surface by normal pressure was tested.
Referring to fig. 1, the present application takes a 2 × 2 matrix as an example to illustrate an implementation method of fourier interpolation, and the specific steps are as follows:
and in the operation process, MATLAB software is used for programming.
The method comprises the following steps: original matrix Z 0 The elements are respectively a 11 、a 12 、a 21 And a 22 Wherein each element is a real number;
step two: to matrix Z 0 Executing the MATLAB command: z 1 ＝ifftshift(Z 0 ) So that the elements in the matrix perform the inverse operation of the spectrum, realizing a ij And a ji Exchanging elements to obtain a matrix Z 1 ；
Step three: to matrix Z 1 Executing the MATLAB command: z is a linear or branched member 2 ＝fft2(Z 1 ) Obtaining the matrix Z by implementing a two-dimensional fast Fourier transform 2 Wherein b is 22 、b 21 、b 12 And b 11 Are all imaginary numbers;
step four: to matrix Z 3 Executing the MATLAB command: z 3 ＝fftshift(Z 2 ) So that the elements in the matrix are subjected to spectrum calculation to realize b ij And b ji Exchanging elements to obtain a matrix Z 3 ；
Step five: expanding the 2 x 2 matrix to a 4 x 4 matrix Z 4 Wherein a plurality of elements b 11 、b 12 、b 21 And b 22 And the matrix is arranged at the central position of the matrix, and all the rest vacant positions are filled with zeros.
Step six: to matrix Z 4 Executing the MATLAB command: z is a linear or branched member 5 ＝fftshift(Z 4 ) So that the elements in the matrix are spectrally inverted to convert complex elements b 11 、b 12 、b 21 And b 22 Changing to the positions of four corners of the matrix to obtain a matrix Z 5 。
Step seven: to matrix Z 5 Executing the MATLAB command: z is a linear or branched member 6 ＝ifft(Z 5 ) Obtaining the matrix Z by implementing a two-dimensional inverse fast Fourier transform 6 Wherein b is 22 、b 21 、b 12 And b 11 Are all imaginary numbers.
Step eight: to matrix Z 6 Executing the MATLAB command: z 7 ＝fftshift(Z 6 ) The elements in the matrix are subjected to spectrum solving, the exchange among the four two-dimensional matrices is realized, the positions of the elements among the two-dimensional matrices are unchanged, and the matrix Z is obtained 7 。
Step nine: to matrix Z 7 The elements in (1) execute the MATLAB command:
Z 8 ＝2^2*sqrt(real(Z 7 )^2+imag(Z 7 ) 2). To matrix Z 7 The amplitude of each element is calculated, for example: the amplitude of a + bi is
Then multiplying each element by a matrix coefficient, 2 2m And m is the interpolation degree. In this example, the number of interpolation times is 1, and therefore the coefficient of the matrix is 4. Through the above calculation, a 4 × 4 matrix Z is obtained 8 。
The invention applies fft2 and ifft in MATLAB software to carry out fast Fourier transform and inverse operation thereof to replace discrete Fourier transform and inverse operation thereof, and specifically comprises the following steps:
the original signal is converted into a time-domain signal by a discrete fourier transform:
after interpolation, inverse discrete Fourier transform is carried out
In the process of fourier interpolation, the elements of the matrix need to be changed in position. The specific MATLAB command is as follows:
the MATLAB commands are sequentially as follows: z 1 ＝ifftshift(Z 0 )→Z 2 ＝fft2(Z 1 )→Z 3 ＝fftshift(Z 2 )→Z 4 ＝zeros(Z 1 ,N 1 )；Z 4 ((N 1 /2)-N 0 /2+1:(N 1 /2)+N 0 /2,(N 1 /2)-N 0 /2+1:(N 1 /2)+N 0 /2)＝Z 3 ；→Z 5 ＝fftshift(Z 4 )→Z 6 ＝ifft(Z 5 )→Z 7 ＝fftshift(Z 6 )→Z 8 ＝2^2*abs(Z 7 )。
As the interpolation factor increases, the rough surface becomes progressively smoother while maintaining the spectra consistent. The method predicts data points which are not acquired by the instrument, and can reduce the distortion of the surface topography to a certain degree.
To verify the technical effect, the invention selects the original rough surface from a standard microfabricated comparison plate (63M sample), and measures the surface using a NANOVEA ST400 white light optical topographer with a lateral resolution (according to the range between points) of 0.1 μ M. The surface data was measured with a reduced lateral resolution of 2 μm. For the convenience of interpolation and subsequent application operations, only 32 × 32 data points were collected here as the original surface topography, with an average roughness of 0.613 μm, and a peak and valley of 1.82 μm and-1.67 μm, respectively. A three-dimensional view thereof is shown with reference to fig. 2.
The original rough representation is subjected to double, quadruple and eight times interpolation in sequence by using a Fourier interpolation method, a three-dimensional topography is shown in a figure 3a, a figure 3b and a figure 3c, lines at the same positions are selected in the figure 3a, the figure 3b and the figure 3c to form shape diagrams like the figure 3d, the figure 3e and the figure 3f, and then the shape diagrams are put into an interval for comparison, as shown in the figure 4. The surface characteristics of the rough surface subjected to interpolation were evaluated, and the results of the measurements are shown in table 1: (original surface and interpolated surface parameters)
TABLE 1
Raw surface data was imported into a commercial FE program (ANSYS) and a model was created in an ANSYS machine module using APDL (ANSYS Parametric Design Language). A model using the original rough surface is shown in fig. 5 a. In models that apply interpolated surfaces, the vectors in the x and y directions need to be linearly interpolated, and more rescaling in the z direction is necessary to prevent the solution problem caused by cell distortion. The surface-built models applying double, quadruple and octave interpolation are in turn depicted in fig. 5b, 5c and 5 d.
In the process of establishing the models, the invention further finely divides the grids in a layered point shifting mode and tests the contact problem of the rough surface with normal pressure. Analysis and test show that the solving process is more stable through interpolation and grid re-division, undesirable phenomena such as stress concentration can not be generated, and the result can be more accurate.
The above-described embodiments are only intended to illustrate the preferred embodiments of the present invention, and not to limit the scope of the present invention, and various modifications and improvements made to the technical solution of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.
Claims (5)
1. A three-dimensional rough surface contact model construction method based on Fourier interpolation is characterized by comprising the following steps:
s1: measuring the engineering three-dimensional rough surface by a topography instrument to obtain NxN data points; and using the data points as raw rough surface data;
s2: interpolating an NxN matrix formed by the original rough surface data by utilizing a Fourier interpolation method to obtain an Nt xNt matrix; where Nt is a positive integer multiple of N, i.e. Nt/N =2 m M is the interpolation degree, m =1, 2, 3 … …;
s3: carrying out surface characteristic evaluation on rough surface data before and after interpolation;
s4: importing the interpolated rough surface data into commercial finite element software, and carrying out grid division to construct a three-dimensional rough surface contact model based on Fourier interpolation;
s5: testing the contact problem of the rough surface with normal pressure;
the S2 specifically comprises the following steps:
step 1: constructing the raw rough surface data into an NxN matrix Z 0 ；
And 2, step: for the matrix Z 0 Executing the MATLAB command: z 1 ＝ifftshift(Z 0 ) Will matrix Z 0 Performing spectrum inversion operation on the internal elements to obtain a matrix Z 1 ；
And step 3: for the matrix Z 1 Executing the MATLAB command: z 2 ＝fft2(Z 1 ) Will matrix Z 1 Performing two-dimensional fast Fourier transform to obtain a matrix Z 2 ；
And 4, step 4: for the matrix Z 2 Executing the MATLAB command: z 3 ＝fftshift(Z 2 ) To obtain a matrix Z 3 ；
And 5: will N matrix Z 3 Expansion into Nt × Nt matrix Z 4 The middle part of the matrix is Z 3 Each element of (1), whichThe rest elements are filled with 0; wherein, nt is the length of the last required matrix singles, is a power of 2 positive integers of N, and the specific procedure is as follows:
Z 4 ＝zeros(Nt，Nt)；
Z 4 ((Nt/2)-N/2+1：(Nt/2)+N/2，(Nt/2)-N/2+1：(Nt/2)+N/2)＝Z 3 ；
step 6: for the matrix Z 4 Executing the MATLAB command: z 5 ＝fftshift(Z 4 ) To obtain a matrix Z 5 ；
And 7: for the matrix Z 5 Executing the MATLAB command: z is a linear or branched member 6 ＝ifft(Z 5 ) To obtain a matrix Z 6 ；
And step 8: for the matrix Z 6 Executing the MATLAB command: z 7 ＝fftshift(Z 6 ) To obtain a matrix Z 7 ；
And step 9: for the matrix Z 7 Executing the MATLAB command:
Z 8 ＝2^2*sqrt(real(Z 7 )^2+imag(Z 7 ) ^ 2) to obtain a matrix Z 8 。
2. The fourier-interpolation-based three-dimensional rough surface contact model construction method of claim 1, wherein:
the profile instrument in the S1 is a NANOVAST 400 white light optical profile instrument.
3. The fourier-interpolation-based three-dimensional rough surface contact model construction method of claim 1, wherein: the raw rough surface data includes: highest peak, lowest valley and average roughness.
4. The method for constructing the three-dimensional rough surface contact model based on the Fourier interpolation according to claim 1, wherein: the commercial finite element software in S4 employs a mechanical module in ANSYS.
5. The fourier-interpolation-based three-dimensional rough surface contact model construction method of claim 1, wherein:
and the grid division is to adopt a layered point shifting method to carry out grid refinement on the interpolated rough surface.
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