出版社:上海社会科学院出版社
出版日期:2014-2
ISBN:9787552004748
作者:瞿波
作者简介
本书是国内也是国际第一本分形几何在流体中的应用的参考书。本书介绍的方法不仅可以用于流体, 还可以用于其他任何有关连续随机的运动轨迹的模拟,用于粒子云的随机散布轨迹。
本书是瞿波博士在英国的博士学位论文的核心成果。书中深入浅出地介绍了分形及其在流体中的应用。详细论述了如何用分形中的分数布朗运动(fBm)模拟流水中污染物的轨迹,包括对海湾和海洋中的污染物传播轨迹的模拟。
书中介绍的方法是基于分数布朗运动(fBm), 这是带有记忆的著名的布朗运动(随机散步)的推广。 作者对著名的fBm作了改进,创建了分数布朗运动粒子跟踪模型,并推广到加速分数布朗运动粒子跟踪模型。基于豪斯特指数(H)的灵活性,分数布朗运动粒子追踪模型的应用非常广泛。可以用于经融、股市、脑电图、岩石的裂缝,道路的分布、海洋的浮标轨迹、粒子的散布、医学上人体肺的分布及毛细胞血管、脑电图曲线(图14)等具有分形的特征物体和现象中。
本书的第一部分介绍了分形。不同于其他书介绍分形时所用的复杂的数学工具,使人望而生畏,书中介绍的分形及分形维数的计算都是用最通俗易懂的方法。这是一本实用性强、浅显易懂的应用数学学习和研究参考用书。书中还附有程序供直接使用。相信此书对大学生、研究生、大学青年教师搞科研有一定的实用和参考价值。
书籍目录
序
自序
分形几何介绍(只需要初等数学的知识)
博士论文
The Use of Fractional Brownian Motion in the Modelling of the Dispersion
of Contaminants in Fluids
Chapter 1:
Outline of Project
1
1.1
Introduction
1
1.2
Fractal and Fractional Brownian Motion
1
1.3
Aim and Objectives
2
1.4
Structure of Thesis
3
Chapter 2:
Diffusion and Dispersion in Fluids
-- A Literature Review
4
2.1
Introduction
4
2.2
Molecular Diffusion: Fick’s Law and the Diffusion
Equation
5
2.3
Statistical Theory of Diffusion: Brownian Motion
8
2.4
Turbulent Diffusion
11
2.4.1
Introduction
11
2.4.2
Eddies
12
2.4.3
Taylor’s Theorem
13
2.4.4
The Relationship Between Lagrangian and
Eulerian Measurement
15
2.4.5
Relative Diffusion and Richardson’s Law
17
2.4.6
Okubo’s Oceanic Diffusion Diagrams
19
2.5
Shear Dispersion
22
2.5.1
Introduction
22
2.5.2
Taylor and Elder’s Shear Dispersion Results
22
2.5.3
Dispersion in Rivers
24
2.5.3.1
Dispersion in Uniform Depth Open Channel
25
2.5.3.2
The Three-Dimensional Diffusion
Coefficients in an Open Channel
28
2.5.3.3
Dispersion in a Natural Channel
30
2.5.4
Dispersion in the Sea
31
2.5.4.1
Introduction
31
2.5.4.2
Relative Diffusion on the Ocean Surface
32
2.5.4.3
Coastal Region
36
2.6
Numerical Model of Dispersion
38
2.6.1
Solution of the Advection-Diffusion Equation
38
2.6.2
The Disadvantage of Solving the
Advection-Diffusion Equation
40
2.7
Particle Tracking Methods
42
2.7.1
Traditional Particle Tracking Methods
42
2.8
Summary
46
Chapter 3
Brownian Motion, Fractional Brownian Motion
and Fractal Geometry
47
3.1
Brownian Motion
47
3.1.1
The Definition of Brownian Motion
47
3.1.2
Two Simple Random Walks
48
3.1.3
Brownian Motion Generation
51
3.1.3.1
Central Limit Theorem Method
52
3.1.3.2
The Box-Muller Method
53
3.1.4
The Properties of a One-Dimensional
Brownian Motion Time Trace
54
3.1.5
The Skewness and Kurtosis of Random Walks
57
3.1.6
Random Walks in Two Dimensions
59
3.1.6.1
Delta Random Walks in Two Dimensions
60
3.1.6.2
Constant Random Walks in Two
Dimensions
60
3.1.6.3
Brownian Motion in Two Dimensions
61
3.1.7
The Last Steps of the Random Walks in Two
Dimensions
62
3.2- Fractional Brownian Motion
63
3.2.1
Introduction
63
3.2.1.1
Fractional Brownian Motion:
A Generalisation of Brownian Motion
63
3.2.1.2
Applications of Fractal Brownian Motion
64
3.2.1.3
The Definition of Fractional Brownian
Motion
67
3.2.1.4
Properties of Fractional Brownain Motion
68
3.2.1.5
Methods for the Generation of Fractional
Brownian Motion
70
3.2.2
FBM Model
71
3.2.3
FBMINC Model
76
3.2.4
The Comparison of the FBM and FBMINC Models
80
3.2.5
fBm Plots in One Dimension
85
3.2.5.1
Fractional Random Walk Plots for the
FBM Model
85
3.2.5.2
The Effect of the Different Random
Number Sequences
89
3.2.5.3
The Mean Absolute Separation of an
fBm Trace
90
3.2.6
The Relationship Between M, NSTEP and P
92
3.2.6.1
Relationship Between NSTEP and M
92
3.2.6.2
The Effect of the Number of Particles
in a Diffusing Cloud
94
3.2.6.3- A Check on Random Number Seeds
95
3.2.7
Fractional Brownian Motion in Two Dimensions
96
3.2.8
Projection of Two-Dimensional Fractional Brownian
Motion
98
3.2.9
The Use of Simpler Probability Distributions to
Reduce CPU Time
100
3.2.10
Long Term Fickian Behaviour
104
3.3
fBm as a Random Fractal Function
106
3.3.1
Fractal Geometry and Fractal Curves
106
3.3.2
Fractal Dimension
109
3.3.3
Fractal Properties of fBm
110
3.3.3.1
The Box Counting Dimension
111
3.3.3.2
The Dimension of an fBm Trace
111
3.3.3.3
The Dimension of fBm Trajectories
113
3.3.4
Method for Determining H from Real Data
116
3.4- Summary
121
Chapter 4
Coastal Bay Modelling
122
4.1
Introduction
122
4.2
New Particle Tracking Method Using in the Bay
122
4.2.1
Advection
123
4.2.2
Diffusion
124
4.2.2.1
Traditional Random Walk Model
124
4.2.2.2
Diffusion Using Fractional Brownian
Motion Model
125
4.2.2.3
The New fBm Particle Tracking Model
127
4.2.3
Choosing a Time Interval
128
4.2.4
Choosing a Diffusion Coefficient
129
4.2.5
Boundary Reflection
131
4.2.5.1
important Note on FBM Reflection
133
4.2.6
The Particle Tracking Model
133
4.2.6.1
The Particle Tracking Algorithm
133
4.2.6.2
Typical Particle Trajectory Plots for the
Bay Model
136
4.2.7
Particles Clouds
137
4.2.7.1
Computational Effort
137
4.2.8
Concentration Calculation and Plots
139
4.2.8.1
Algorithm for Calculation of Pollution
Concentration
140
4.2.8.2
Contour Plots and 3D Surface Plots
141
4.2.9
Further Reported Results
142
4.3
Shear Dispersion
143
4.3.1
Simple Shear Dispersion (Brownian Motion)
144
4.3.2
Shear Dispersion with Fractional Brownian Motion
147
4.3.3
Shear Dispersion in the Coastal Bay Model
Recirculation Zone
150
4.4
Summary
153
Chapter 5
Simulation of Observed Coastal Dispersion
189
5.1
Introduction
189
5.2
Northumbrian Coastal Water Data Sets
190
5.3
Three Methods for Calculating the Standard Deviation of
the Dye Patch Concentrations
191
5.3.1
The SQ-Method
192
5.3.2
The R-Method
193
5.3.3
The SR-Method
194
5.3.4
Estimation of the Direction of the Mean Advective
Velocity Vector for Each Patch
194
5.4
Comparison of the Three Methods
195
5.4.1
The Reason for Introducing the SR-Method
195
5.4.2
Comparison of the Results Using the Three
Methods
196
5.5
Accuracy of the Results
197
5.5.1
The Sensitivity of the Centre
197
5.5.2
The Concentration Function Calculation
198
5.6
Simulation of the Observed Dye Patches Using an fBm
Based Particle Tracking Model
198
5.6.1
The Accelerated Fractional Brownian Motion
(AFBM) Model
199
5.6.2
Simulation Using the FBMINC and AFBM
Models
202
5.6.3
Concentration Calculations
202
5.6.4
Contour Plots
203
5.7
Summary
205
Chapter 6
Conclusions, Discussion and Recommendations
243
6.1
Introduction
243
6.2
Achievement of Objectives
243
6.3
Discussion
247
6.4
Recommendations for Future Work
249
Appendix 1
FORTRAN 77 Programs
253
References
293
分形应用论文选
1.
分数布朗运动的简化和应用
317
2.
从分形维数到海洋表面漂浮物轨迹的模拟
328
3.
流体中污染物扩散的分形模拟
335
4.
用分数型布朗运动模拟海湾的剪切湍流分散
343
5.
Development of FBMINC model for particle diffusion
in fluids
354
7 加速分数型布朗运动粒子追踪模型在水面污染扩散中的应用
387
内容概要
瞿波 博士
江苏南通人。
1983年华东师范大学数学学士。
1986年华东师范大学数学硕士。
1992年赴英国在爱丁堡龙比亚大学(Edingburgh, Napier University)攻读计算机硕士课程和流体力学博士学位。研究方向是分形在流体中的应用。1999年获英国博士学位(PhD degree)。
1999年英国贝尔法斯特女王大学(Queen’s University of Belfast)研究助理。
2000年香港大学(Hong Kong University)土木工程系博士后。
2003年澳大利亚格里菲里斯大学(Griffith University)研究员。
2008年回国,在南通大学任教。硕士生导师。承担国家自然科学基金(2012年度)“北极的生态系统和二甲基硫对当地气候的影响”等多项课题研究。
20年来致力于分形在流体力学中的应用研究,以及环境模型,水利模型等国际国内课题研究。热衷于数学分形的普及推广,有多项成果在《国际流体数值方法》、《极地生物学》等国际权威杂志发表。