岩石裂隙网络的分形特征

分形特征与分形维数广泛应用于岩石裂隙网络的量化,及与工程参数的关系模型建立.然而,严格的分形维数的极限定义形式难以直接应用,工程应用中多用近似分形维数值代替,近似的结果在建立量化关系模型时会产生蝴蝶效应,在量化及预测过程中产生巨大偏差.本文回顾了分形研究一系列的发展过程,并基于最新的分形定义提出了一种新的分形维数计算方法.通过对于十个岩石裂隙网络分形维数的计算,证明该方法能够准确有效的计算出图形的复杂度,避免了以往计算分形维数所产生的问题.     

分形统计测度构建及其在投资组合中的应用研究

准确测量证券的风险和收益无论是对投资管理,还是对金融理论研究,甚至对理论成果向实践应用转化都至关重要。本文在证券价格具有分形特征的现实背景下,基于分形理论构建了分形期望和分形方差两个分形统计测度,以克服非分形统计测度在风险收益方面测不准或不可测的缺陷。在此基础上,应用分形统计测度构建了投资组合模型,给出了分形组合模型的解析解;随后,利用实证分析验证了分形统计测度在投资组合应用中的有效性。本文创新之处在于针对证券价格具有分形特征的现实背景构建了分形期望和分形方差两个分形统计测度;并基于分形统计测度构建了投资组合模型,将证券价格普遍存在的分形特征纳入投资组合的研究框架。     

求多重分形谱的滑动格子法在深浅地表元素分布中的应用研究

在多重分形理论和特征判定法的基础上,构造了求多重分形谱的滑动格子计算法,计算出了研究区域4种元素深、浅层的多重分形谱f(α)的图像.结果显示浅层元素的分布不具备多重分形特征;深层元素分布符合多重分形特征.就三种分形维数——格子维数、信息维数、关联维数对深层元素的分布做出了大小排序解释;后就多重分形谱f(α)的跨度、对称性和两端差值Δf做出了对应于深层元素分布概率分布集中差异、高低浓度分布差异、稳定性的解释.最后根据上述分析的结果指出应用求多重分形谱的滑动格子法研究深浅地层元素分布是一快速、实用、有效的方法,具有良好的应用前景.     

矩形域上分形插值研究

该文给出了矩形域上分形插值数学模型, 分形插值曲面的计算公式, 证明了分形插值曲面迭代函数系唯一性定理, 导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究. 为工程中长期寻求的粗糙表面模拟提供了理论基础和实用方法.     

量化分形图形的新指数——简便分形指数

分形的广泛存在性已被普遍接受,然而分形维数的现有定义计算得到的结果是:不同的分形维数定义得到不同的分形维数值,甚至会出现不同的变化趋势,且在应用时使用的最小二乘回归结果不稳定,导致数值应用也会受影响,出现这些现象主要归咎于现有分形维数定义的严格性、抽象性以及分形图形的码尺效应.为避免这些问题,本文结合分形图形的长尾分布特征及自相似性提出一个新的分形量化形式——简便分形指数,并阐述了该定义背后的分形原理及计算方法,简便分形指数越大,形状复杂程度越高.最后本文利用岩石裂隙图像说明简便分形指数对不同裂隙网络复杂性描述的准确性,验证其作为分形图形量化方法的合理性及便利性.     

由几何反演导出的分形几何

本文介绍了由几何反演而导出的自反分形概念.构造了一个对称的,分叉自反分形曲线和一个自反分形尘集.推广了分形密切的概念而且提出了一个新的概念——分形包络.最后给出了“肥皂”和“鸡蛋”的两个自反分形的实例.     

分形插值曲面理论及其应用*

本文叙述了分形曲面的生成原理,给出了分形插值曲面的计算公式,证明了分形插值曲面迭代函数系唯一性定理,导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究。     

粗糙面分形计算理论研究进展

为提出一种工程上适用可靠的粗糙面分形维数计算方法,在分形曲线的维数计算方法(码尺法,盒维法)基础上,先后提出了星积分形曲面的维数计算方法、三角形棱柱表面积法、投影覆盖法、立方体覆盖法、改进的立方体覆盖法、分形的增变量描述法等曲面分形维数理论.鉴于上述方法的共有缺陷——获取三维坐标的激光表面仪器的扫描尺度限制,研究者提出了粗糙面图像维数计算理论,包括二值化图像维数、灰度图像维数、RGB图像维数计算理论.最后,本文展望了分形维数计算理论领域内亟待解决的三大问题.     

具有有限记忆的随机分形的重分形分解

Dryakhlov和Tempelman对具有有限记忆的随机分形集的Hausdorff维数进行了研究,本文对具有有限记忆的随机分形集K(ω)的重分形分解集Kα(ω)进行研究,得到了在一定条件下,这种随机分形集重分形分解集Kα(ω)的Hausdorff维数表达式.     

分形曲面的构造算法研究

以分形理论为依据 ,根据分形几何描绘自然界景物的基本思想 ,结合解析几何中旋转曲面的构造 ,把已生成的二维平面分形曲线绕着同一平面上的直线旋转 ,获得一类三维旋转曲面的构造算法 ,给出了相应的三维迭代函数系统和三维仿射变换矩阵 ,并进行了深入的理论分析 .本文的研究为分形曲面的构造探求了一种简易算法 ,并为分形曲面的生成和实践应用提供了理论依据 .     

星积分形曲面及其维数

通过分形曲线定义了一类分形曲面(被称为星积分形曲面),讨论了这类分形曲面的分形维数,得出了分形曲线的维数与它们所构造出的分形曲面维数之间的关系。     

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

Node insertion in Coalescence Fractal Interpolation Function

The Iterated Function System (IFS) used in the construction of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) depends on the interpolation data. The insertion of a new point in a given set of interpolation data is called the problem of node insertion. In this paper, the effect of insertion of new point on the related IFS and the Coalescence Fractal Interpolation Function is studied. Smoothness and Fractal Dimension of a CHFIF obtained with a node are also discussed.     

An SDE model for deterioration of rock surfaces

Abstract

A stochastic differential equation (SDE) is derived and examined for approximately modeling the breaking down of rock surfaces through random processes. The rock surfaces include, for example, surfaces of historical monuments, gravestones, or natural rock formations. Rock surfaces break down through wear, weathering, and erosion. During weathering, rocks are worn away and fractured into smaller pieces while in erosion, the rock pieces are transported through actions, for example, of air, water, and gravity. In the mathematical model developed in the present investigation, it is assumed that environmental actions cause particles or pieces of a rock to gradually break off with erosion occurring simultaneously, that is, the rock pieces are transported away immediately after separation.     

Fixed Point Theorems of the Iterated Function Systems

In this paper, we present some fixed point theorems of iterated function systems consisting of α-ψ-contractive type mappings in Fractal space constituted by the compact subset of metric space and iterated function systems consisting of Banach contractive mappings in Fractal space constituted by the compact subset of generalized metric space, which is also extensively applied in topological dynamic system.     

The effect of seepage on the stress–strain state of rock near a borehole

The effect of the strain-strength and seepage properties of rock and the compressibility of the percolating fluid on the dimensions of the rock fracture zones, which occur in oil and gas boreholes when the bottom hole pressure is reduced, is investigated. The seepage is considered basing on the stationary formulation of the problem, which enables the general case to be investigated. It is shown that in the case of unsteady flow, the stresses on the boundary of the rock fracture zone and, as a consequence, on its dimensions, are independent of the nature of the pressure distribution in the stratum, and are determined solely by the pressure of the percolating fluid on the boundary of this zone. It is established that an increase in the compressibility of the percolating fluid leads to an increase in the dimensions of the rock fracture zone.     

锚固岩体参数的等效方法研究

利用岩体中结构体与结构面的刚度系数比较严密地推导了裂隙岩体的等效法向及切向刚度系数;基于岩体小变形、变形可叠加性及强度参数C、φ不相关性假定,利用Coulomb强度理论推导了裂隙岩体等效连续体的等效强度参数C、φ值的表达式.其成果在岩体边坡的整体稳定分析方面具有重要的应用价值,对于推动岩体本构关系的研究具有重要意义.     

The Long Time Behaviors of Non-autonomous Navier-Stokes Equations with Linear Dampness on the Whole R2 Space

In this paper, the long time behaviors of non-autonomous Navier-Stokes equations with linear dampness on the whole R² space are considered. The existence of uniform attractor is proved when the external force terms satisfy suitable conditions. Moreover, the upper bounds of the uniform attractor's Hausdorff and Fractal dimensions are obtained.     

Coarse-scale particle tracking approaches for contaminant transport in fractured rock

Random-walk particle tracking methods are frequently used for modeling contaminant transport, as relevant to radionuclide transport in fractured rock. Standard particle-tracking methods need to be modified for handling discontinuities in velocity and diffusion coefficients as at fracture–matrix interfaces, and handling these discontinuities accurately requires time steps much smaller than the diffusion time scale across narrow fracture apertures. In this work we present coarse-scale particle tracking methods that exploit the contrast in diffusivities between fracture and rock matrix to allow the use of time steps much larger than the diffusion time scale across fracture apertures. Thus, they reduce computational effort by several orders of magnitude. We develop two coarse-scale versions of the standard particle tracking method, one applicable to particles starting in the fracture, and another to particles starting in the rock matrix. The two methods can be used in combination to track particles through individual fractures, including the influence of matrix diffusion. The main advantage of our methods result from the computationally efficient treatment of (two-way) fracture–matrix particle transfer. These methods can also be combined with existing particle tracking approaches for complex advection–diffusion–dispersion in fractures to handle fracture–matrix interactions efficiently.