# An Introduction to Multiscale

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Nov 16, 2013 (4 years and 6 months ago)

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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