??? Nanjing City - PowerPoint PPT Presentation

1 / 121
About This Presentation
Title:

??? Nanjing City

Description:

Nanjing City College of Civil Engineering in Hohai University Basic facts of the College (largest in our university): Close to 200 staffs 4 departments ... – PowerPoint PPT presentation

Number of Views:72
Avg rating:3.0/5.0
Slides: 122
Provided by: mechatron1
Category:
Tags: city | nanjing

less

Transcript and Presenter's Notes

Title: ??? Nanjing City


1
??? Nanjing City
2
???? -Hohai University
3
(No Transcript)
4
(No Transcript)
5
College of Civil Engineering in Hohai University
  • Basic facts of the College (largest in our
    university)
  • Close to 200 staffs
  • 4 departments (civil engineering, survey
    mapping, earth sciences engineering,
    engineering mechanics)
  • 1 department-scale institute (geotechnic
    institute)
  • Around 3000 undergraduate students 1000
    graduate students

6
Modeling of Anomalous Behaviors of Soft Matter
Wen Chen (??) Institute of Soft Matter
Mechanics Hohai University, Nanjing, China 3
September 2007
7
Soft matter?
  • Soft matters, also known as complex fluids,
    behave unlike ideal solids and fluids.
  • Mesoscopic macromolecule rather than microscopic
    elementary particles play a more important role.

8
Typical soft matters
  • Granular materials
  • Colloids, liquid crystals, emulsions, foams,
  • Polymers, textiles, rubber, glass
  • Rock layers, sediments, oil, soil, DNA
  • Multiphase fluids
  • Biopolymers and biological materials
  • highly deformable, porous, thermal fluctuations
    play major role, highly unstable

9
Lattice in ideal solids
10
(No Transcript)
11
Polymer macromolecules fractal mesostructures
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
Fractured microstructures
20
(No Transcript)
21
(No Transcript)
22
Soft Matter Physics
  • Pierre-Gilles de Gennes proposed the term
    in his Nobel acceptance speech in 1991.
  • widely viewed as the beginning of the soft
    matter science.

_ P. G. De Gennes
23
Why soft matter?
  • Universal in nature, living beings, daily life,
    industries.
  • Research is emerging and growing fast, and some
    journals focusing on soft matter, and reports in
    Nature Science.

24
Engineering applications
  • Acoustic wave propagation in soft matter,
    anti-seismic damper in building,geophysics,vibrati
    on and noise in express train
  • Biomechanics,heat and diffusion in textiles,
    mechanics of colloids, emulsions, foams,
    polymers, glass, etc
  • Energy absorption of soft matter in structural
    safety involving explosion and impact
  • Constitutive relationships of soil, layered
    rocks, etc.

25
Challenges
  • Mostly phenomenological and empirical models,
    inexplicit physical mechanisms, often many
    parameters without clear physical significance
  • Computationally very expensive
  • Few cross-disciplinary research, less emphasis on
    common framework and problems.

26
Characteristic behaviors of soft matter
  • Gradient laws cease to work, e.g., elastic
    Hookean law, Fickian diffusion, Fourier heat
    conduction, Newtonian viscoustiy, Ohlm law
  • Power law phenomena, entropy effect
  • Non-Gaussian non-white noise, non-Markovian
    process
  • In essence, history- and path-dependency,
    long-range correlation.

27
More features (courtesy to N. Pan)
  • Very slow internal dynamics
  • Highly unstable system equilibrium
  • Nonlinearity and friction
  • Entropy significant
  • a jammed colloid system, a pile of sand,
  • a polymer gel, or a folding protein.

28
Major modeling approaches
  • Fractal (multifractal), fractional calculus,
    Hausdorff derivative, (nonlinear model?)
  • Levy statistics, stretched Gaussian, fractional
    Brownian motion, Continuous time random walk
  • Nonextensive Tsallis entropy, Tsallis
    distribution.

29
Typical anomalous (complex) behaviors
  • Anomalous diffusion(heat conduction, seepage,
    electron transport, diffusion, etc.)
  • Frequency-dependent dissipation of vibration,
    acoustics, electromagnetic wave propagation.

30
Mechanics of Soft Matter
  • Basic postulates of mechanics.
  • conservation of mass, momentum and energy
  • Basic concepts of mechanics
  • stress, strain, energy and entropic elasticity
  • Constitutive relations and initialboundary-value
    problems.


31
Outline
  • Part I Progresses and problems a personal view
  • Part II Our works in recent five years

32
What in Part I?
  • Field and experimental observations
  • Statistical descriptions
  • Mathematical physics modelings

33
Field and experimental observations
34
Anomalous electronic transport
  • Non-dissipation

Normal dissipation
Anormalous dissipation
35
The absorption of many materials and tissues
obeys a frequency-dependent power law
Courtesy of Prof. Thomas Szabo
36
Statistical descriptions
37
Anomalous diffusion
  • ? 1, Normal (Brownian) diffusion
  • ? ?1, Anomalous (? gt1 superdiffusion, ? lt1
    subdiffusion)

38
Random walks
Left Brownian motion Right Levy flight with
the same number (7000) of steps.
39
Levy self-similar random walks
  • A characteristic Levy walk

40
Stretched Gaussian Distribution
41
Measured probability density of changes of the
wind speed over 4 sec
42
  • Stretched Gaussian diffusion

Gaussian diffusion
43
Levy stable distribution
44
Two cases of Levy distributions
Gaussian (?2)
Cauchy distribution (?1)
45
(No Transcript)
46
Tsallis distribution (nonequlibrium system)
Tsallis non-extensive entropy
Max s
Boltzmann-Gibbs entropy
Tsallis distribution
47
Tsallis distribution cases
48
A comparison of diverse distributions
A. Komnik, J. Harting, H.J. Herrmann
49
Progresses in statistical descriptions
  • Continuous time random walk, fractional Brownian
    motion, Levy walk, Levy flight
  • Levy distribution, stretched Gaussian, Tsallis
    distribution.

50
Problems in statistical descriptions
  • Relationship and difference between Levy
    distribution, stretched Gaussian, and Tsallis
    distribution?
  • Calculus corresponding to stretched Gaussian and
    Tsallis distrbiution?
  • Infinite moment of Levy distribution?

51
Partial differential equation modeling
52
Anomalous diffusion equation in fractional
derivatives
53
Physics behind normal diffusion
  • Darcys law (granular flow)
  • Fourier heat conduction law
  • Ficks law
  • Ohlm law

54
  • Continuity
  • Fick diffusion

55
Nonlinear Modelings
Multirelaxation models, nonlinear models
varied models for different media with quite a
few parameters having no explicit physical
significance. For instance, nonlinear power law
fluids
56
Anomalous diffusion equation in Fractional
calculus
  • Master equation (phenomenological)

57
Fractional time derivative in Fourier domain
58
Physical significances
  • Histroy dependency (memory, non-Markovian)
    corresponding to fractional Brownian motion.
  • Singular Volterra integral equation.
  • Numerical truncation is risky!

59
Operation case
Equation case
Solution
60
Weierstrass function (differentiability order
0.5)
61
Constitutive relationships
  • Hookian law in ideal solids
  • Ideal Newtonian fluids
  • Newtonian 2nd law for rigid solids
  • One model of soft matter

62
Numerical fractional time derivative
  • Volterra integral equation
  • Finite difference formulationGrunwald-Letnikov
    definition
  • Short memory approach (truncation and
    stability)
  • Something new?
  • I. Podlubny, Fractional Differential equation,
    Academic Press, 1999

63
Numerical fractional space derivative
  • Full numerical discretization matrix
  • Boundary condition treatments
  • Fast algorithm (e.g., fast multipole method).

64
Progresses in PDE modeling
  • Fractional time derivative, fractional Laplacian
  • Hausdorff derivative
  • Growing PDE models in various areas.

65
Problems in PDE modelings
  • Relationship and difference between fractional
    calculus and Hausdorff derivative?
  • Fractional time and space modelings?
  • Computing cost
  • Nonlinear vs. fractional modeling
  • Physical foundation of phenomenological modelings

66
Fractional vs. Nonlinear systems
  • History dependency
  • Global interaction
  • Fewer physical parameters (simple beautiful)
  • Competition or complementary

67
Part II Our works
68
Summary
  • New definition of fractional Laplacian
  • Introduction of positive fractional time
    derivative, and modified Szabo dissipative wave
    equations
  • Mathematical physics explanation of 0,2
    frequency power dependency via Levy statistics
  • Fractal time-space transforms underlying
    anomalous physical behaviors, and two
    hypotheses concerning the effect of fractal
    time-space fabric on physical behaviors,
  • Introduction of Hausdorff fractal derivative
  • Fractional derivative modeling of turbulence.

69
Positive fractional time derivative
70
Definitions based on Fourier transform
Fractional derivative
  • Positive fractional derivative

Positivity requirments in modeling of dissipation
71
  • Definitions in time domain

72
New definition of fractional Laplacian
73
Definitions via Fourier transform
74
Traditional definition in space
0lt? lt1
Samko et al. 1993. Fractional Integrals and
Derivives Theory and Applications
75
Our definition
0lt? lt1
  • Merits
  • Weak vs. strong singularity ,
  • Accurate vs. approximate,
  • Finite domain with boundary conditions vs.
    infinite domain

Journal of Acoustic Society of America, 115(4),
1424-1430, 2004
76
Fractional derivative modelings of
frequency-dependent dissipative medical
ultrasonic wave propagation
77
Imaging Comparisons
Courtesy of Prof. Thomas Szabo
78
Medical ultrasound
  • Imaging (sonography) and ablating the objects
    inside human body for medical diagnosis and
    therapy.

79
(No Transcript)
80
  • Conventional nonlinear and multirelaxation
    models
  • Material-dependent models
  • Quite a few artifical (non-physical) parameters,
    in essence, empirical and semi-empirical models.
  • Our fractional calculus models
  • Few phyiscally explicit parameters,
  • Parameters available from experimental data
    fitting.

81
Time-space wave equations of integer-order
partial derivative only exist for y0, 2
  • Thermoviscous wave equation (y2)

Damped wave equation (y0)
82
  • Linear positive time fractional derivative wave
    equation

where
Note
83
  • Linear fractional Laplacian wave equation for
    arbitrary frequency dependency

84
??
??
??
?
85
  • Dr. Richters New clinical approach
  • stabilize each deformable breast between two
    plates,
  • detect breast cancer via speed change
    attenuation.

86
Courtesy of Prof. Thomas Szabo
87
3D configuration
88
Levy Statistical explanation of frequency
dependent power
89
  • Anomalous diffusion equation for frequency
    dependent dissipation

Phenomenological master equation
90
Fourier transform of probability density function
of Lévy ?-stable distribution is the
characteristic function of solution of
anomalous diffusion equation
91
  • To satisfy the positive probability density
    function, the Lévy stable index ? must obey
  • 1) In terms of Lévy statistics, the media having
    ? gt2 power law attenuation are not statistically
    stable in nature
  • 2) ?0 is simply an ideal approximation.

92
Fractal time-space transforms, Hausdorff
derivative, fractional quantum and phonon
93
Perplexing issues in anomalous diffusion
  • Levy stable process and fractional Brownian
    motion
  • The mean square displacement dependence on time

94
Fractional (fractal) time-space transforms
Special relativity transforms
95
Two hypotheses for anomalous physical processes
  • The hypothesis of fractal invariance the laws of
    physics are invariant regardless of the fractal
    metric spacetime.
  • The hypothesis of fractal equivalence the
    influence of anomalous environmental fluctuations
    on physical behaviors equals that of the fractal
    time-space transforms.

96
Fractional quantum in complex fluids
  • Fractional quantum relationships between energy
    and frequency, momentum and wavenumber(fractional
    Schrodinger equation)

97
Fractional phonon and vibrational absorption
energy spectrum?
98
Hausdorff derivative under fractal
Generalized velocity
Hausdorff derivative diffusion equation
99
Statistical and Reynolds equation modelings of
turbulence via fractional derivative
100
Kolmogorov -5/3 scaling of turbulence
  • Validation in sufficiently high Reynolds number
    tubulence
  • Narrow spectrum of -5/3 scaling in finite
    Reynolds number turbulence, i.e., intermittency
    (non-Gaussian distribution)

101
Energy spectrum (obtained from Kim,Moins DNS
database)
102
Turbulence distributionGauss vs. Levy
Levy distribution
Nature, 409, 10171019, 2001
103
Plasma turbulence Oak Ridge National Laboratory
Power law decay of Levy distribution
104
Richardson superdiffusion
Richardson diffusion consistent with Kolmogorov
scaling
Fractional Laplace statistical equation
105
Fractional derivative Reynolds equation
Navier-Stokes equation
Reynolds decomposition
Reynolds equation
106
Fractional Reynolds equation
Three order of magnitude turbulence vs. molecule
viscousity
107
Relevant Publications
  • W. Chen, S. Holm, Modified Szabos wave equation
    models for lossy media obeying frequency power
    law, J. Acoustic Society of America, 2570-2574,
    114(5), 2003.
  • W. Chen, S. Holm, Fractional Laplacian time-space
    models for linear and nonlinear lossy media
    exhibiting arbitrary frequency dependency, J.
    Acoustic Society of America, 115(4), 1424-1430,
    2004.
  • W. Chen, Lévy stable distribution and 0,2 power
    law dependence of acoustic absorption on
    frequency in various lossy media, Chinese Physics
    Letters,22(10),2601-03, 2005.
  • W. Chen, Time-space fabric underlying anomalous
    diffusion, Soliton, Fractal, Chaos, 28(4),
    923-929, 2006. .
  • W. Chen,. A speculative study of 2/3-order
    fractional Laplacian modeling of turbulence Some
    thoughts and conjectures, Chaos, (in press),
    2006.

108
Keywords
  • Geophysics, bioinformatics, soft matter, porous
    media
  • frequency dependency, power law
  • Fractal, microstructures, self-similarity,
  • Fractional calculus (Abel integral equation
    Volterra integral equation)
  • Entropy irreversibility

109
Thinking future?
  • Phenomenological models physics mechanisms of
    soft matter
  • Time-space mesostructures and statistical models
  • Numerical solution of fractional calculus
    equations
  • Verification and validation of models and
    engineering applications.

110
A Journal Proposal
  • Title Journal of Power Laws and Fractional
    Dynamics
  • Publisher Springer
  • Proposers W. Chen, J. A. T. Machado, Y. Chen

111
Research issues covered in this journal
  • Empirical and theoretical models of a variety of
    anomalous behaviors characteristic in power law
    such as history-dependent process,
    frequency-dependent dissipation etc.
  • Novel physical concepts, mathematical modeling
    approaches and their applications such as
    fractional calculus, Levy statistics, fractional
    Brownian motion, 1/f noise, non-extensive Tsallis
    entropy, continuous time random walk, dissipative
    particle dynamics, etc.
  • Numerical algorithms to solve the relevant
    modeling equations, which often involve non-local
    time-space integro-differential operators
  • Real-world applications in all engineering and
    scientific branches such as mechanics,
    electricity, chemistry, biology, economics,
    control, robotic, image and signal processing.

112
Something else
  • Non-stationary data processing
  • Large-scale multivariate scattered data
    processing (radial basis functions)
  • Meshfree computing and software (e.g., high
    wavenumber acoustics and vibration)

113
Scattered 3D geologicial data reconstruction
471,031 scattered data made by U.S. Geological
Survey
114
Difficulties in simulation of high-dimensional,
high wavenumber and frequency
Wavenumber
  • Ultrasonics(1-100MHz),microwaves
    0.1GHz-100GHz,seismics
  • High wavenumber for 2D problems Ngt100,3D problems
    Ngt20
  • FEM requires at least 12 points in each wavenumber

115
2D Helmholtz (k80,N160) problem
116
Multiple-edged outdoor noise barrier design
117
200 millions DOFs matrix for 2D FEM engineering
precision 30000 full matrix 13.5Gb storage for
the standard BEM
2D case
S Langdon, Lecture notes on Finite element
methods for acoustic scattering, July 11,
2005 S. Chandler-Wilde S. Langdon, Lecture
notes on Boundary element methods for
acoustics, July 19, 2005
118
Advantages
  • 1/100 storage,1/1000 computing cost of the BEM
  • High accuracy, simple program, non numerical
    integration, meshfree, suitable for inverse
    problems
  • Irregularly-shaped boundary, high-dimensional
    problems, symmetric matrix.

119
Drawbacks
  • For extremely high-wavenumber(high frequency
    and/or large domain), full and ill-conditioned
    matrix fast algorithm is desirable, e.g., fast
    multipole method.
  • Exterior problems.
  • Nonlinear? Software package for real-world
    problems (killer applications)

120
  • There is nothing so practical as a sound
    scientific theory.

121
  • For details
  • http//em.hhu.edu.cn/wenchen/english.html
  • For contact
  • chenwen_at_hhu.edu.cn
Write a Comment
User Comments (0)
About PowerShow.com