An Introduction to Multiscale

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Nov 16, 2013 (3 years and 9 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