The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each l...The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.展开更多
This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intens...This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length ξ and the roughness exponent α, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with α= 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.展开更多
The measured profiles of laboratory fractured rocks should be self-affine fractal.The scaling properties of these profiles are described by two parameters-the fractal dimension D and the crossover length tc The D valu...The measured profiles of laboratory fractured rocks should be self-affine fractal.The scaling properties of these profiles are described by two parameters-the fractal dimension D and the crossover length tc The D values of eight profiles are calculated by the ruler method and by the standard deviation method respectively.It is shown that if tc is far greater than the sampling step tc two methods yield the same results,although if it is far smaller than r,the D by the standard method will be about 1.20,while D by the ruler method will very close to 1.0,because two fractal dimensions,local and global,exist on two sides of tc In order to obtain the local fractal dimension which may be close to that of the standard deviation method,the ruler method must be modified.We propose a way to estimate the tc and to modify the ruler method.Finally,a profile having given D is generated in terms of the principle of non-integer order differential,through which the above two methods are verified and lead to the same展开更多
In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-S...In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.展开更多
The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><...The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><em>μ<sub>M,D</sub></em></span><sub> </sub>is to estimate the number of orthogonal exponentials in <em>L</em><sup>2</sup><span style="white-space:normal;">(</span><em>μ<sub>M,D<span style="white-space:normal;">)</span></sub></em>. In the present paper, we establish some relations inside the zero set <img src="Edit_2196df81-d10f-4105-a2a9-779f454a56c3.png" width="55" height="23" alt="" /> by the Fourier transform of the self-affine measure <em>μ<sub>M,D</sub></em>. Based on these facts, we show that <em>μ<sub>M,D</sub></em> is a non-spectral measure<em><sub> </sub></em>and there exist at most 4 mutually orthogonal exponential functions in <em style="white-space:normal;"><em style="white-space:normal;">L</em><sup style="white-space:normal;">2</sup><span style="white-space:normal;">(</span><span style="white-space:normal;"></span><em style="white-space:normal;">μ<sub>M,D)</sub></em></em>, where the number 4 is the best possible. This extends several known conclusions.展开更多
Iterated function system (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wi...Iterated function system (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a "mapping partial derivative", and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in R^n. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.展开更多
Modeling the real contact area plays a key role in every tribological process,such as friction,adhesion,and wear.Contact between two solids does not necessarily occur everywhere within the apparent contact area.Consid...Modeling the real contact area plays a key role in every tribological process,such as friction,adhesion,and wear.Contact between two solids does not necessarily occur everywhere within the apparent contact area.Considering the multiscale nature of roughness,Persson proposed a theory of contact mechanics for a soft and smooth solid in contact with a rigid rough surface.In this theory,he assumed that the vertical displacement on the soft surface could be approximated by the height profile of the substrate surface.Although this assumption gives an accurate pressure distribution at the interface for complete contact,when no gap exists between two surfaces,it results in an overestimation of elastic energy stored in the material for partial contact,which typically occurs in many practical applications.This issue was later addressed by Persson by including a correction factor obtained from the comparison of the theoretical results with molecular dynamics simulation.This paper proposes a different approach to correct the overestimation of vertical displacement in Persson’s contact theory for rough surfaces with self-affine fractal properties.The results are compared with the correction factor proposed by Persson.The main advantage of the proposed method is that it uses physical parameters such as the surface roughness characteristics,material properties,sliding velocity,and normal load to correct the model.This method is also implemented in the theory of rubber friction.The results of the corrected friction model are compared with experiments.The results confirm that the modified model predicts the friction coefficient as a function of sliding velocity more accurately than the original model.展开更多
In this paper we calculate the local fractal dimension values D of the self-affine feature of earthquake time series by RMS (root-mean-square) error method, and express the fractal dimensionality by the normalized cor...In this paper we calculate the local fractal dimension values D of the self-affine feature of earthquake time series by RMS (root-mean-square) error method, and express the fractal dimensionality by the normalized correlation coefficient R. The fractal dimension values are given for earthquakes occurred in Tangshan, Haicheng, Songpan, Longling, Changshu, I.iyang in China and its vicinity by the moving scanning method with different magnitude thresholds and the fixed-window length (100 events). The results show the D values are characterized by decreasing, continued low level in values or by decreasing first and then increasing before moderate earthquakes.展开更多
Based on the extreme value theory, self-affinity, and scale invariance, we studied the temporal and spatial relationship and the variation of water level and established a model of Gumbel-Pareto distribution for desig...Based on the extreme value theory, self-affinity, and scale invariance, we studied the temporal and spatial relationship and the variation of water level and established a model of Gumbel-Pareto distribution for designed flood calculation. The model includes the previous extreme value models, the over-threshold data, and the fractal features shared by previous extreme value models. The model was simplified into a logarithmic normal distribution and a Pareto distribution for specific parameter values, and was used to calculate the designed flood values for the Shanghai Wusong Station in 100- and 1 000-year return periods. The calculated results show that the value of the designed flood height calculated in the Gumbel-Pareto distribution is between those in the Gumbel and Pearson-Ⅲ distributions. The designed flood values in the 100- and 1 000-year return periods of the model were 0.03% and 0.11% lower, respectively, than the Gumbel distribution and 0.06% and 1.54% higher, respectively, than the Pearson-Ⅲ distribution. Compared to the traditional model based solely on extreme probability, the Gumbel-Pareto distribution model could better describe the probabilistic characteristics of extreme marine elements and better use the data.展开更多
In this study, the lattice Boltzmann method (LBM) was used to simulate the solute transport in a single rough fracture. The self-affine rough fracture wall was generated with the successive random addition method. T...In this study, the lattice Boltzmann method (LBM) was used to simulate the solute transport in a single rough fracture. The self-affine rough fracture wall was generated with the successive random addition method. The ability of the developed LBM to simulate the solute transport was validated by Taylor dispersion. The effect of fluid velocity on the solute transport in a single rough fracture was investigated using the LBM. The breakthrough curves (BTCs) for continuous injection sources in rough fractures were analyzed and discussed with different Reynolds numbers (Re). The results show that the rough frac~'e wall leads to a large fluid velocity gradient across the aperture. Consequently, there is a broad distribution of the immobile region along the rough fracture wall. This distribution of the immobile region is very sensitive to the Re and fracture geometry, and the immobile region is enlarged with the increase of Re and roughness. The concentration of the solute front in the mobile region increases with the Re. Furthermore, the Re and roughness have significant effects on BTCs, and the slow solute molecule exchange between the mobile and immobile regions results in a long breakthrough tail for the rough fracture. This study also demonstrates that the developed LBM can be effective in studying the solute transport in a rough fracture.展开更多
As mass transport mechanisms,the spreading and mixing(dilution) processes of miscible contaminated compounds are fundamental to understanding reactive transport behaviors and transverse dispersion.In this study,the sp...As mass transport mechanisms,the spreading and mixing(dilution) processes of miscible contaminated compounds are fundamental to understanding reactive transport behaviors and transverse dispersion.In this study,the spreading and dilution processes of a miscible contaminated compound in a three-dimensional self-affine rough fracture were simulated with the coupled lattice Boltzmann method(LBM).Moment analysis and the Shannon entropy(dilution index) were employed to analyze the spreading and mixing processes,respectively.The corresponding results showed that the spreading process was anisotropic due to the heterogeneous aperture distribution.A compound was transported faster in a large aperture region than in a small aperture region due to the occurrence of preferential flow.Both the spreading and mixing processes were highly dependent on the fluid flow velocity and molecular diffusion.The calculated results of the dilution index showed that increasing the fluid flow velocity and molecular diffusion coefficient led to a higher increasing rate of the dilution index.展开更多
The strength of rock structures strongly depends inter alia on surface irregularities of rock joints. These irregularities are characterized by a coefficient of joint roughness. For its estimation, visual comparison i...The strength of rock structures strongly depends inter alia on surface irregularities of rock joints. These irregularities are characterized by a coefficient of joint roughness. For its estimation, visual comparison is often used. This is rather a subjective method, therefore, fully computerized image recognition procedures were proposed. However, many of them contain imperfections, some of them even mathematical nonsenses and their application can be very dangerous in technical practice. In this paper, we recommend mathematically correct method of fully automatic estimation of the joint roughness coefficient. This method requires only the Barton profiles as a standard.展开更多
The Levy-stability of self-affine random cascading model is investigated in some detail. It is found that the Levy-stability indices of cascading process depend upon not only the dynamical fluctuation parameter a but ...The Levy-stability of self-affine random cascading model is investigated in some detail. It is found that the Levy-stability indices of cascading process depend upon not only the dynamical fluctuation parameter a but also the way of shrinking phase space when calculating scaled factorial moments, which propose a quantitative criterion to determine whether the dynamical fluctuation is anisotropic. Besides, the puzzle of Levy-stability violation of NA22 data is comprehensively solved under the hypothesis of anisotropic dynamical fluctuation.展开更多
For each positive integer k,the radix representation of the complex numbers in the base -k+i gives rise to a lattice self-affine tile T_k in the plane,which consists of all the complex numbers that can be expressed in...For each positive integer k,the radix representation of the complex numbers in the base -k+i gives rise to a lattice self-affine tile T_k in the plane,which consists of all the complex numbers that can be expressed in the form∑_(j≥1)d_j(-k+i)^(-j),where d_j∈{0,1,2,...,k^2}.We prove that T_k is homeomorphic to the closed unit disk{z∈C:|z|≤1}if and only if k≠2.展开更多
基金This research is partly supported by NNSF of China (60204001) the Youth Chengguang Project of Science and Technology of Wuhan City (20025001002)
文摘The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.
基金Project supported by the National Natural Science Foundation of China (Grant No 69978012), and by the National Key Basic Research Special Foundation (NKBRSF) of China (Grant No G1999075200).
文摘This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length ξ and the roughness exponent α, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with α= 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.
文摘The measured profiles of laboratory fractured rocks should be self-affine fractal.The scaling properties of these profiles are described by two parameters-the fractal dimension D and the crossover length tc The D values of eight profiles are calculated by the ruler method and by the standard deviation method respectively.It is shown that if tc is far greater than the sampling step tc two methods yield the same results,although if it is far smaller than r,the D by the standard method will be about 1.20,while D by the ruler method will very close to 1.0,because two fractal dimensions,local and global,exist on two sides of tc In order to obtain the local fractal dimension which may be close to that of the standard deviation method,the ruler method must be modified.We propose a way to estimate the tc and to modify the ruler method.Finally,a profile having given D is generated in terms of the principle of non-integer order differential,through which the above two methods are verified and lead to the same
文摘In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.
文摘The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on <span style="white-space:nowrap;"><em>μ<sub>M,D</sub></em></span><sub> </sub>is to estimate the number of orthogonal exponentials in <em>L</em><sup>2</sup><span style="white-space:normal;">(</span><em>μ<sub>M,D<span style="white-space:normal;">)</span></sub></em>. In the present paper, we establish some relations inside the zero set <img src="Edit_2196df81-d10f-4105-a2a9-779f454a56c3.png" width="55" height="23" alt="" /> by the Fourier transform of the self-affine measure <em>μ<sub>M,D</sub></em>. Based on these facts, we show that <em>μ<sub>M,D</sub></em> is a non-spectral measure<em><sub> </sub></em>and there exist at most 4 mutually orthogonal exponential functions in <em style="white-space:normal;"><em style="white-space:normal;">L</em><sup style="white-space:normal;">2</sup><span style="white-space:normal;">(</span><span style="white-space:normal;"></span><em style="white-space:normal;">μ<sub>M,D)</sub></em></em>, where the number 4 is the best possible. This extends several known conclusions.
文摘Iterated function system (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a "mapping partial derivative", and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in R^n. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.
文摘Modeling the real contact area plays a key role in every tribological process,such as friction,adhesion,and wear.Contact between two solids does not necessarily occur everywhere within the apparent contact area.Considering the multiscale nature of roughness,Persson proposed a theory of contact mechanics for a soft and smooth solid in contact with a rigid rough surface.In this theory,he assumed that the vertical displacement on the soft surface could be approximated by the height profile of the substrate surface.Although this assumption gives an accurate pressure distribution at the interface for complete contact,when no gap exists between two surfaces,it results in an overestimation of elastic energy stored in the material for partial contact,which typically occurs in many practical applications.This issue was later addressed by Persson by including a correction factor obtained from the comparison of the theoretical results with molecular dynamics simulation.This paper proposes a different approach to correct the overestimation of vertical displacement in Persson’s contact theory for rough surfaces with self-affine fractal properties.The results are compared with the correction factor proposed by Persson.The main advantage of the proposed method is that it uses physical parameters such as the surface roughness characteristics,material properties,sliding velocity,and normal load to correct the model.This method is also implemented in the theory of rubber friction.The results of the corrected friction model are compared with experiments.The results confirm that the modified model predicts the friction coefficient as a function of sliding velocity more accurately than the original model.
文摘In this paper we calculate the local fractal dimension values D of the self-affine feature of earthquake time series by RMS (root-mean-square) error method, and express the fractal dimensionality by the normalized correlation coefficient R. The fractal dimension values are given for earthquakes occurred in Tangshan, Haicheng, Songpan, Longling, Changshu, I.iyang in China and its vicinity by the moving scanning method with different magnitude thresholds and the fixed-window length (100 events). The results show the D values are characterized by decreasing, continued low level in values or by decreasing first and then increasing before moderate earthquakes.
基金Supported by the NSFC-Shandong Joint Fund "Study on the DisasterCausing Mechanism and Disaster Prevention Countermeasures of MultiSource Storm Surges"(No.U1706226)the Program of Promotion Plan for Postgraduates’Educational Quality "Paying More Attention to the Study on the Cultivation Mode of Mathematical Modeling for Engineering Postgraduates"(No.HDJG18007)
文摘Based on the extreme value theory, self-affinity, and scale invariance, we studied the temporal and spatial relationship and the variation of water level and established a model of Gumbel-Pareto distribution for designed flood calculation. The model includes the previous extreme value models, the over-threshold data, and the fractal features shared by previous extreme value models. The model was simplified into a logarithmic normal distribution and a Pareto distribution for specific parameter values, and was used to calculate the designed flood values for the Shanghai Wusong Station in 100- and 1 000-year return periods. The calculated results show that the value of the designed flood height calculated in the Gumbel-Pareto distribution is between those in the Gumbel and Pearson-Ⅲ distributions. The designed flood values in the 100- and 1 000-year return periods of the model were 0.03% and 0.11% lower, respectively, than the Gumbel distribution and 0.06% and 1.54% higher, respectively, than the Pearson-Ⅲ distribution. Compared to the traditional model based solely on extreme probability, the Gumbel-Pareto distribution model could better describe the probabilistic characteristics of extreme marine elements and better use the data.
基金supported by the National Natural Science Foundation of China(Grants No.51079043,41172204,and 51109139)the Natural Science Foundation of Jiangsu Province(Grant No.BK2011110)
文摘In this study, the lattice Boltzmann method (LBM) was used to simulate the solute transport in a single rough fracture. The self-affine rough fracture wall was generated with the successive random addition method. The ability of the developed LBM to simulate the solute transport was validated by Taylor dispersion. The effect of fluid velocity on the solute transport in a single rough fracture was investigated using the LBM. The breakthrough curves (BTCs) for continuous injection sources in rough fractures were analyzed and discussed with different Reynolds numbers (Re). The results show that the rough frac~'e wall leads to a large fluid velocity gradient across the aperture. Consequently, there is a broad distribution of the immobile region along the rough fracture wall. This distribution of the immobile region is very sensitive to the Re and fracture geometry, and the immobile region is enlarged with the increase of Re and roughness. The concentration of the solute front in the mobile region increases with the Re. Furthermore, the Re and roughness have significant effects on BTCs, and the slow solute molecule exchange between the mobile and immobile regions results in a long breakthrough tail for the rough fracture. This study also demonstrates that the developed LBM can be effective in studying the solute transport in a rough fracture.
基金supported by the National Natural Science Foundation of China(Grant No.41602239)the Natural Science Foundation of Jiangsu Province(Grant No.BK20160861)+2 种基金the Fundamental Research Funds for the Central Universities(Grant No.2016B05514)the International Postdoctoral Exchange Fellowship Program from the Office of China Postdoctoral Council(Grant No.20150048)the"333 Project"of Jiangsu Province(Grant No.BRA2015305)
文摘As mass transport mechanisms,the spreading and mixing(dilution) processes of miscible contaminated compounds are fundamental to understanding reactive transport behaviors and transverse dispersion.In this study,the spreading and dilution processes of a miscible contaminated compound in a three-dimensional self-affine rough fracture were simulated with the coupled lattice Boltzmann method(LBM).Moment analysis and the Shannon entropy(dilution index) were employed to analyze the spreading and mixing processes,respectively.The corresponding results showed that the spreading process was anisotropic due to the heterogeneous aperture distribution.A compound was transported faster in a large aperture region than in a small aperture region due to the occurrence of preferential flow.Both the spreading and mixing processes were highly dependent on the fluid flow velocity and molecular diffusion.The calculated results of the dilution index showed that increasing the fluid flow velocity and molecular diffusion coefficient led to a higher increasing rate of the dilution index.
基金The Project LO1202 by financial means from the Ministry of Education, Youth ; Sports under the National Sustainability Programme I
文摘The strength of rock structures strongly depends inter alia on surface irregularities of rock joints. These irregularities are characterized by a coefficient of joint roughness. For its estimation, visual comparison is often used. This is rather a subjective method, therefore, fully computerized image recognition procedures were proposed. However, many of them contain imperfections, some of them even mathematical nonsenses and their application can be very dangerous in technical practice. In this paper, we recommend mathematically correct method of fully automatic estimation of the joint roughness coefficient. This method requires only the Barton profiles as a standard.
基金supported in part by the National Natural Science Foundation of China the DYTF of the state Education Commssion of China and the CGP of Wuhan City.
文摘The Levy-stability of self-affine random cascading model is investigated in some detail. It is found that the Levy-stability indices of cascading process depend upon not only the dynamical fluctuation parameter a but also the way of shrinking phase space when calculating scaled factorial moments, which propose a quantitative criterion to determine whether the dynamical fluctuation is anisotropic. Besides, the puzzle of Levy-stability violation of NA22 data is comprehensively solved under the hypothesis of anisotropic dynamical fluctuation.
基金The first author is supported by Youth Project of Tianyuan Foundation(10226031)zhongshan University Promotion Foundation for Young Teachers (34100-1131206)+1 种基金the second author is supported by National science Foundation(10041005)Guangdong Provic
文摘For each positive integer k,the radix representation of the complex numbers in the base -k+i gives rise to a lattice self-affine tile T_k in the plane,which consists of all the complex numbers that can be expressed in the form∑_(j≥1)d_j(-k+i)^(-j),where d_j∈{0,1,2,...,k^2}.We prove that T_k is homeomorphic to the closed unit disk{z∈C:|z|≤1}if and only if k≠2.