分形理论的若干应用
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摘要
近20多年来,分形几何理论及其应用发展迅速。分形几何理论主要研究在一定条件下,大量具有标度不变性、随机性等特性的复杂现象,现已应用到自然科学和社会科学的几乎所有领域。本文采用分形几何理论和Monte Carlo等方法相结合,从理论的角度来研究和认识分形物体的结构特点和输运性质。
本文根据分形几何理论,首先研究了多孔介质的结构特点。提出了两个描述多孔介质孔隙空间分布的分形结构模型,并根据其构造方法,求出了周长、面积、孔隙度和比表面积,并在此基础上讨论了周长、面积、孔隙度和Menger海绵的比表面积随分形维数的变化规律,分析了模拟孔隙结构时对各参数的要求。此外,还提出了两个描述多孔介质的统一表达式,其结果与已有模型符合得很好。本文提出的这种方法,为模拟具有不同微结构、不同分维数和孔隙度的实际多孔介质提供了新的途径。
本文还根据多孔介质里孔隙大小分布具有分形幂规律和具有随机性这些特点,采用分形几何理论和蒙特卡罗方法,分别推导了孔隙大小和渗透率的几率模型。以双弥散分形多孔介质为例,计算了其渗透率。所得渗透率的预测值与已有分析解和实验结果做了比较,得到了一致的结果。本文提出的这种模拟方法,可以用来研究(饱和或不饱和)多孔介质的其他输运性质(如热导率、弥散率、电导率和介电常数等)。
接下来,本文根据粗糙表面微凸体(或凹坑)大小分布具有分形幂规律和表面粗糙具有自仿射特征这些特点,提出了一种具有分形特征的粗糙表面的蒙特卡罗模拟方法,并推导了用来产生表面形貌的基于随机数的微凸体大小几率模型。所提出的递推迭代方法能够模拟具有上述特征的粗糙表面。结果表明,表面拓扑结构的变化与粗糙表面轮廓的常数G和分形维数D有关。D值越大,或G值越小,意味着表面拓扑越平坦。本方法可用来预测粗糙表面的其它输运性质,如摩擦力、磨损、润滑、渗透率和热导率或电导率等。
最后,本文在分形几何理论的基础上,提出了一个计算粗糙表面接触热导(TCC)的随机数模型。为了研究真空下接触界面的传热机理及其影响因素,本文从几何和力学两个方面入手展开研究。基于固体的弹塑性理论研究了塑性守恒条件下的表面粗糙度在法向载荷作用下的变形问题,推导了基于分形几何理论的接触热阻网络模型理论.通过对参数的研究发现,分形维数D和特征长度参数G对接触热导有着重要的影响。通过调整参数D和G,本文计算得到的接触热导与实验结果十分吻合。结果显示,在D和G取较大数值时,在已有模型中通常被忽略不计的基体电阻对接触热导的影响不可忽略。本文提出的方法和技巧,可进一步用来研究粗糙表面的其它输运性质。
Over the past two decades, fractal geometry theory, which describes the fractal features such as randomicity and scale-independence, etc, has been received much attention in a variety of sciences sand engineering subjects and has extensively been studied both theoretically and experimentally. In order to theoretically identify and recognize the fractal features of structures or patterns existing in nature, some models, methods and techniques have been developed.
Firstly, the structural properties of porous media are investigated theoretically based on fractal geometry theory. Construction methods of two types of fractal structures (the Sierpinski carpet and the Menger sponge) are presented, and the expressions for the surface and volume of the structures are derived. Furthermore, two unified models for describing the fractal characters of porous media are derived. The results from the proposed unified models are found to be in good agreement with the available models. The present analysis allows for simulating real porous media with different microstructures and different fractal dimensions, porosity and specific surface area.
Secondly, the permeability of porous media is simulated by Monte Carlo technique in this work. Based on the fractal character of pore size distribution in porous media, the probability models for pore diameter and for permeability are derived. Taking the bi-dispersed fractal porous media as examples, the permeability calculations are performed by the proposed Monte Carlo technique. The results show that the present simulations present a good agreement compared with the existing fractal analytical solution in the general interested porosity range. The proposed simulation method may have the potential in prediction of other transport properties (such as thermal conductivity, dispersion conductivity and electrical conductivity) in fractal porous media, both saturated and unsaturated.
Thirdly, a new Monte Carlo method is presented for simulating rough surfaces with fractal behavior. The simulation is based on power-law size distribution of asperity diameter and self-affine property of roughness on surfaces. A probability model based on random number for asperity sizes is developed to generate the surfaces. By iteration, this method can be used to simulate surfaces that exhibit the aforementioned properties. The results indicate that the variation of the surface topography is related to the effects of scaling constant G and the fractal dimension D of the profile of rough surface. The larger value of D or smaller value of G signifies the smoother surface topography. This method may have the potential in prediction of the transport properties (such as friction, wear, lubrication, permeability, thermal and electrical conductivity, etc.) on rough surfaces.
Finally, a random number model based on fractal geometry theory is developed to calculate the thermal contact conductance (TCC) of two rough surfaces in contact. This study is carried out by geometrical and mechanical investigations. The present parametric study reveals that the fractal parameters D and G have the important effects on TCC. The TCC by the proposed model is compared with the existing experiment data, and good agreement is observed by fitting parameters D and G. It is also shown that the effect of the bulk resistance on TCC, which is often neglected in existing models, should not be neglected for the relatively larger G and D. The present results show a better agreement with the practical situation than other models’.
本文根据分形几何理论,首先研究了多孔介质的结构特点。提出了两个描述多孔介质孔隙空间分布的分形结构模型,并根据其构造方法,求出了周长、面积、孔隙度和比表面积,并在此基础上讨论了周长、面积、孔隙度和Menger海绵的比表面积随分形维数的变化规律,分析了模拟孔隙结构时对各参数的要求。此外,还提出了两个描述多孔介质的统一表达式,其结果与已有模型符合得很好。本文提出的这种方法,为模拟具有不同微结构、不同分维数和孔隙度的实际多孔介质提供了新的途径。
本文还根据多孔介质里孔隙大小分布具有分形幂规律和具有随机性这些特点,采用分形几何理论和蒙特卡罗方法,分别推导了孔隙大小和渗透率的几率模型。以双弥散分形多孔介质为例,计算了其渗透率。所得渗透率的预测值与已有分析解和实验结果做了比较,得到了一致的结果。本文提出的这种模拟方法,可以用来研究(饱和或不饱和)多孔介质的其他输运性质(如热导率、弥散率、电导率和介电常数等)。
接下来,本文根据粗糙表面微凸体(或凹坑)大小分布具有分形幂规律和表面粗糙具有自仿射特征这些特点,提出了一种具有分形特征的粗糙表面的蒙特卡罗模拟方法,并推导了用来产生表面形貌的基于随机数的微凸体大小几率模型。所提出的递推迭代方法能够模拟具有上述特征的粗糙表面。结果表明,表面拓扑结构的变化与粗糙表面轮廓的常数G和分形维数D有关。D值越大,或G值越小,意味着表面拓扑越平坦。本方法可用来预测粗糙表面的其它输运性质,如摩擦力、磨损、润滑、渗透率和热导率或电导率等。
最后,本文在分形几何理论的基础上,提出了一个计算粗糙表面接触热导(TCC)的随机数模型。为了研究真空下接触界面的传热机理及其影响因素,本文从几何和力学两个方面入手展开研究。基于固体的弹塑性理论研究了塑性守恒条件下的表面粗糙度在法向载荷作用下的变形问题,推导了基于分形几何理论的接触热阻网络模型理论.通过对参数的研究发现,分形维数D和特征长度参数G对接触热导有着重要的影响。通过调整参数D和G,本文计算得到的接触热导与实验结果十分吻合。结果显示,在D和G取较大数值时,在已有模型中通常被忽略不计的基体电阻对接触热导的影响不可忽略。本文提出的方法和技巧,可进一步用来研究粗糙表面的其它输运性质。
Over the past two decades, fractal geometry theory, which describes the fractal features such as randomicity and scale-independence, etc, has been received much attention in a variety of sciences sand engineering subjects and has extensively been studied both theoretically and experimentally. In order to theoretically identify and recognize the fractal features of structures or patterns existing in nature, some models, methods and techniques have been developed.
Firstly, the structural properties of porous media are investigated theoretically based on fractal geometry theory. Construction methods of two types of fractal structures (the Sierpinski carpet and the Menger sponge) are presented, and the expressions for the surface and volume of the structures are derived. Furthermore, two unified models for describing the fractal characters of porous media are derived. The results from the proposed unified models are found to be in good agreement with the available models. The present analysis allows for simulating real porous media with different microstructures and different fractal dimensions, porosity and specific surface area.
Secondly, the permeability of porous media is simulated by Monte Carlo technique in this work. Based on the fractal character of pore size distribution in porous media, the probability models for pore diameter and for permeability are derived. Taking the bi-dispersed fractal porous media as examples, the permeability calculations are performed by the proposed Monte Carlo technique. The results show that the present simulations present a good agreement compared with the existing fractal analytical solution in the general interested porosity range. The proposed simulation method may have the potential in prediction of other transport properties (such as thermal conductivity, dispersion conductivity and electrical conductivity) in fractal porous media, both saturated and unsaturated.
Thirdly, a new Monte Carlo method is presented for simulating rough surfaces with fractal behavior. The simulation is based on power-law size distribution of asperity diameter and self-affine property of roughness on surfaces. A probability model based on random number for asperity sizes is developed to generate the surfaces. By iteration, this method can be used to simulate surfaces that exhibit the aforementioned properties. The results indicate that the variation of the surface topography is related to the effects of scaling constant G and the fractal dimension D of the profile of rough surface. The larger value of D or smaller value of G signifies the smoother surface topography. This method may have the potential in prediction of the transport properties (such as friction, wear, lubrication, permeability, thermal and electrical conductivity, etc.) on rough surfaces.
Finally, a random number model based on fractal geometry theory is developed to calculate the thermal contact conductance (TCC) of two rough surfaces in contact. This study is carried out by geometrical and mechanical investigations. The present parametric study reveals that the fractal parameters D and G have the important effects on TCC. The TCC by the proposed model is compared with the existing experiment data, and good agreement is observed by fitting parameters D and G. It is also shown that the effect of the bulk resistance on TCC, which is often neglected in existing models, should not be neglected for the relatively larger G and D. The present results show a better agreement with the practical situation than other models’.
引文
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