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 = 10
3
m
http://ak.water.usgs.gov/yukon/index.php
Physical Scales
•
Water Cluster
•
Physical Scale:
–
5 nm = 5 x 10

9
m
http://eyeofthefish.org/leaky

leushke/
http://www.btinternet.com/~martin.ch
aplin/clusters.html
•
Water Drops
•
Physical Scale:
–
mm = 10

3
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
•
= number of springs between mass
and 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
1D Rupturing String
Displacement
Force
Ruptured Strings
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
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