An Introduction to Multiscale
Modeling
Scientific Computing
and Numerical Analysis Seminar
CAAM 699
Outline
• Multiscale Nature of Matter
– Physical Scales
– Temporal Scales
•
•
•
•
•
Different Laws for Different Scales
Computational Difficulties
Homogeneous Elastic String
Inhomogeneous Elastic String
Overview of Seminar Topics
Physical Scales
• Discrete Nature of
Matter
– Multiple Physical
(Spatial) scales Exist
– Example: River
– Physical Scale:
km = 103 m
http://ak.water.usgs.gov/yukon/index.php
Physical Scales
• Water Drops
• Physical Scale:
http://www.btinternet.com/~martin.ch
aplin/clusters.html
– mm = 10-3 m
http://eyeofthefish.org/leakyleushke/
• Water Cluster
• Physical Scale:
– 5 nm = 5 x 10-9 m
Physical Scales
• Water Molecule
• Physical Scale:
0.278 nm = 2.78 x 10-10 m
http://commons.wikimedia.org/wiki/File:
Water_molecule.png
Temporal Scales
• Multiple Time Scales in Matter
• Time Scale of Interest Depends
on Phenomenon of Interest
• Fluid Time Scales:
– River Flow: hours
– Rain Drop Falling: 30-60 min
– Water Molecule Interactions:
fractions of a second
Different Scales, Different Laws
• Governing Equations different for different
scales
• Example: Modeling a Fluid:
– River Flow: Navier-Stokes Equations
– Interactions between fluid particles: Newton’s
Molecular Dynamics
– Atomic, Subatomic Description of Fluid Molecule:
Schrödinger’s equations
Model Choice
• Could represent river as discrete fluid
particles, and utilize molecular dynamics to
model its flow
• More details included in the model, the more
accurate your model will likely be
• What’s the problem???
Good Luck trying to do this
computationally!!!
Computational Difficulties
• Number of elements
• Smaller Spatial Scale may warrant a smaller
time scale in order to keep numerical methods
stable
– Example: CFL number for hyperbolic PDEs
Model Choice
• Balance detail and computational complexity
• Choice often made to model a material as a
continuum
• Goal is to then find a constitutive law that can
explain how the material behaves
• If the material is homogeneous, the
continuum assumption is typically acceptable
and constitutive laws can be found
• Heterogeneous materials are more difficult
to model, and motivate the need for
multiscale models
Homogeneous Elastic String
• Discrete Scale: Mass-Spring system
•
point masses of mass
• Springs between each mass have spring
constant
• In zero strain state, springs are length
• Derive Equation for Longitudinal Motion
Homogeneous Elastic String
• Let
be the displacement of mass
from its zero strain state at time
• Equation of motion for mass
can be
written using Newton’s Law:
• The
can be written as
Homogeneous Elastic String
• Forces felt by mass
come from mass
and mass
• Net force on mass
difference in forces from
the left and right mass
Homogeneous Elastic String
• Full equation for mass
Homogeneous Elastic String
•
•
•
Elasticity Modulus
Linear Mass Density
• Take Limit as
Homogeneous Elastic String
• 1D Wave Equation
• Continuum-Level model, limit of the
microscopic (discrete) model
• Wave speed determined by
does NOT
depend on location in the string
• Hyperbolic PDE
easy to simulate
Inhomogeneous Elastic String
• Discrete Model, Mass-Spring system
• Number of springs between each point mass
can vary
Inhomogeneous Elastic String
•
point masses of mass
• Springs between each mass have spring
constant
• In zero strain state, springs are length
•
displacement of mass
•
and
= number of springs between mass
at time
Inhomogeneous Elastic String
• Equation of Motion for mass
Inhomogeneous Elastic String
• Equation of Motion for mass
• Take Limit as
Inhomogeneous Elastic String
• Wave equation with locally varying
wave speed:
• To solve this wave equation you need
to know
but this is a
microscopic quantity! (Local density
of springs)
• Micro quantity needed in a
continuum equation
Inhomogeneous Elastic String
Put another way in the form of a constitutive law
(relation between stress and strain)
Dividing by h and taking the limit h  0
Inhomogeneous Elastic String
Equating these two quantities gives:
Utilizing:
Elastic properties of the spring vary spatially
Practical Example
• Rupturing String:
• Assume springs break if the segment length
for some distance
• Microscopic Model:
• Continuum Model:
Practical Example
• Begin with string in zero strain state attached
at one end to wall
• This string is stretched at the other end by a
constant strain rate
• 11 point masses, 100 springs between each
pair of masses
• When distance between masses exceeds
then springs break
1D Rupturing String
Force
Ruptured Strings
1D Rupturing String
Displacement
Time Steps
Challenges
• General problems with trying to couple a
microscopic and continuum model
– Number of elements in the microscopic scale
– Carrying out the microscopic model for full
continuum level time scale
– If you only do a spatial or time sample of the
microscopic evolution, how would you represent
the micro-state at a later point in time?
Overview of Seminar Topics
• Interesting Medical/Biological Problems that
would benefit from multiscale modeling
• Models of the Cytoskeleton
• Continuum Microscopic (CM) Methods
• Probability Theory, PDF Estimation
• CM modeling with statistical sampling
• Solution Methods to Non-Linear Systems of
Equations
• And more…..
References
• E W, Engquist B, “Multiscale Modeling and
Computation”, Notices of the AMS, Vol 50:9,
p. 1062-1070