??? Nanjing City

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??? Nanjing City
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???? -Hohai University
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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

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Modeling of Anomalous Behaviors of Soft Matter
Wen Chen (??) Institute of Soft Matter
Mechanics Hohai University, Nanjing, China 3
September 2007
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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.

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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

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Lattice in ideal solids
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Polymer macromolecules fractal mesostructures
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Fractured microstructures
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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
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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.

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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.

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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.

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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.

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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.

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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.

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Typical anomalous (complex) behaviors

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

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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.
• 
  

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Outline

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

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What in Part I?

• Field and experimental observations
• Statistical descriptions
• Mathematical physics modelings

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Field and experimental observations
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Anomalous electronic transport

• Non-dissipation

Normal dissipation
Anormalous dissipation
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The absorption of many materials and tissues
obeys a frequency-dependent power law
Courtesy of Prof. Thomas Szabo
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Statistical descriptions
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Anomalous diffusion

• ? 1, Normal (Brownian) diffusion
• ? ?1, Anomalous (? gt1 superdiffusion, ? lt1
  subdiffusion)

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Random walks
Left Brownian motion Right Levy flight with
the same number (7000) of steps.
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Levy self-similar random walks

• A characteristic Levy walk

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

Gaussian diffusion
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Levy stable distribution
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Two cases of Levy distributions
Gaussian (?2)
Cauchy distribution (?1)
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Tsallis distribution (nonequlibrium system)
Tsallis non-extensive entropy
Max s
Boltzmann-Gibbs entropy
Tsallis distribution
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Tsallis distribution cases
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A comparison of diverse distributions
A. Komnik, J. Harting, H.J. Herrmann
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Progresses in statistical descriptions

• Continuous time random walk, fractional Brownian
  motion, Levy walk, Levy flight
• Levy distribution, stretched Gaussian, Tsallis
  distribution.

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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?

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Partial differential equation modeling
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Anomalous diffusion equation in fractional
derivatives
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Physics behind normal diffusion

• Darcys law (granular flow)
• Fourier heat conduction law
• Ficks law
• Ohlm law

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• Continuity

• Fick diffusion

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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
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Anomalous diffusion equation in Fractional
calculus

• Master equation (phenomenological)

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Fractional time derivative in Fourier domain
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Physical significances

• Histroy dependency (memory, non-Markovian)
  corresponding to fractional Brownian motion.
• Singular Volterra integral equation.
• Numerical truncation is risky!

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Operation case
Equation case
Solution
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Weierstrass function (differentiability order
0.5)
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Constitutive relationships

• Hookian law in ideal solids
• Ideal Newtonian fluids
• Newtonian 2nd law for rigid solids
• One model of soft matter

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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

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Numerical fractional space derivative

• Full numerical discretization matrix
• Boundary condition treatments
• Fast algorithm (e.g., fast multipole method).

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Progresses in PDE modeling

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

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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

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Fractional vs. Nonlinear systems

• History dependency
• Global interaction
• Fewer physical parameters (simple beautiful)
• Competition or complementary

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Part II Our works
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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.

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Positive fractional time derivative
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Definitions based on Fourier transform
Fractional derivative

• Positive fractional derivative

Positivity requirments in modeling of dissipation
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• Definitions in time domain

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New definition of fractional Laplacian
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Definitions via Fourier transform
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Traditional definition in space
0lt? lt1
Samko et al. 1993. Fractional Integrals and
Derivives Theory and Applications
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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
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Fractional derivative modelings of
frequency-dependent dissipative medical
ultrasonic wave propagation
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Imaging Comparisons
Courtesy of Prof. Thomas Szabo
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Medical ultrasound

• Imaging (sonography) and ablating the objects
  inside human body for medical diagnosis and
  therapy.

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• 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.

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Time-space wave equations of integer-order
partial derivative only exist for y0, 2

• Thermoviscous wave equation (y2)

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

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

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

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

Phenomenological master equation
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Fourier transform of probability density function
of Lévy ?-stable distribution is the
characteristic function of solution of
anomalous diffusion equation
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• 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.

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Fractal time-space transforms, Hausdorff
derivative, fractional quantum and phonon
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Perplexing issues in anomalous diffusion

• Levy stable process and fractional Brownian
  motion
• The mean square displacement dependence on time

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Fractional (fractal) time-space transforms
Special relativity transforms
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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.

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Fractional quantum in complex fluids

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

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Fractional phonon and vibrational absorption
energy spectrum?
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Hausdorff derivative under fractal
Generalized velocity
Hausdorff derivative diffusion equation
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Statistical and Reynolds equation modelings of
turbulence via fractional derivative
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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)

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Energy spectrum (obtained from Kim,Moins DNS
database)
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Turbulence distributionGauss vs. Levy
Levy distribution
Nature, 409, 10171019, 2001
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Plasma turbulence Oak Ridge National Laboratory
Power law decay of Levy distribution
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Richardson superdiffusion
Richardson diffusion consistent with Kolmogorov
scaling
Fractional Laplace statistical equation
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Fractional derivative Reynolds equation
Navier-Stokes equation
Reynolds decomposition
Reynolds equation
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Fractional Reynolds equation
Three order of magnitude turbulence vs. molecule
viscousity
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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.

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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

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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.

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A Journal Proposal

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

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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.

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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)

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Scattered 3D geologicial data reconstruction
471,031 scattered data made by U.S. Geological
Survey
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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

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2D Helmholtz (k80,N160) problem
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Multiple-edged outdoor noise barrier design
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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
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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.

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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)

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

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• For details
• http//em.hhu.edu.cn/wenchen/english.html
• For contact
• chenwen_at_hhu.edu.cn