I could so the aim really of
the objectives of coming here today is
to maybe introduce potential collaborators
to this first topic which is modeling.
I've been doing this for carbon nanotubes
we're looking at proteins D.N.A.
could be microtubules.
Give you the freshman physics understand
this is a computing crowd and not going to
really want to hear too much of about
the mechanical engineering aspects.
I'll try to focus on capabilities and I'll
give you an idea about where we're going
to tell you a little bit about
collaborators that are working together.
Well seeking potential collaborators
to look at cellular automata modeling
approach for last dynamics.
This is looking at pressure and
shear ways in surface ways and
plates lamb modes various ways I think
there could be potential for looking
electromagnetics but a program manager
a white paper exactly on that topic.
So again there may be a collaborator
interested electromagnetics or
maybe a collaborator from
the standpoint of computing perils
simulation that may take an interest
in this over time to work OK.
So first protein modeling what
you're looking at is hemoglobin.
And what struck me when I
first looked at him a Global
was the presence of these alpha heal a cs.
So there's a few shapes and
a protein one of the dominant shapes
is the so-called helix will shape or
alpha helix.
You'll see a number of them here blue and
red you'll see some beta sheets and
you'll see some other shapes in there as
well I'm going to describe an approach for
interpellation curvature in a continuum.
And using a numerical scheme on
top of that modeling approach and
I think that's going to be very efficient
for describing for example the portions of
this protein molecule the beta sheets
as well and really anything else.
That just happens to do a very nice
job with Yoko shapes with a look at
flat shapes.
So here that is the equations of
motion that I'll start with your look.
Can't slide that prepared when
this model is first developed for
carbon nanotubes and
that's what this is depicting here.
Now this start of the formulation
I should note that do we.
Hodges an arrow he was the one that
came up to most of the equation them in
a present for you for intrinsic continua
well then talk about how you can use them
in a multi scale approach and I can do
a few other things with them as well.
If you're familiar with tangent
normal by normal coordinates and
you should because I taught it to you.
I think actual long time ago but if
you're looking at an under form state in
a deformed state we would typically
call maybe this little B.
want to tangent vector we
would have a normal vector and
A by normal between these be two and
be three.
And then in that court system we know
things like this error in a formula that
say how those that will change
with respect to a spatial
length or an arc length.
That's what's captured here.
These are all vectors so we've got three
of these I ranging from one to three.
That's a spatial derivative and
we say well if we know
this this thing called curvature we take
the cross product of the basis vectors.
We're going to get the change in
those basis vectors as we move
through the continuum.
OK So
this is sort of a classic formula you'd
see in a differential geometry course.
Now what we're interested in here
is not only what's happening with
the center line.
So over here we're
depicting a center line.
Going through.
Let's say the centroid of that shape.
Well we're also interested in how
maybe the cross-section strains.
It can do a couple strands of constrain
like this it could strain this way so
we can strain that way
we can strain this way.
This is to go be in theory central if
you know the kinematics behind that and
it can also extend actually.
And what that does is makes our
description of the centerline
change spatial change a little more
complex instead of this derivative just
simply being be one the tangent direction
the derivative of position back to respect
the space being the tangent and stead
we have a term to actually extension.
We have a term dude.
One of the cross-sectional and
one of the other.
OK that's just kinematics describing
what's happening with the center line and
what's happening with these basis vectors.
We have to bring in F. because I'm a using
this coordinate system the Sarah from
a coordinate system which is captured by
those basis vectors now is like to
point out a few of these terms.
This is that because I'm
a Newton's second law.
This is oil or second law moments
changing in your momentum.
We have a cross-section here we don't just
have a line element
that's why we see both.
If we took a little element here
on this continue this carbon
in a tube we would see an F.
in a D.F. D.F.T. S.T.S. and
really the only thing that would survive
would be the D.F.T. Yes when we wrote
our equation of motion so this is the
Acting on a small differential element.
We might have a small
force per unit length
acting on its surface some type attraction
and that's going to change our momenta P.
is linear momenta this is them V So
if we take a derivative we get there.
And now that we've got this system.
This system moving through space and
changing its orientation.
We tend to get some terms out
a little harder to into it.
This is the curvature cross
product with the internal force.
This is the anger of the last of
the cross product with the moment but
otherwise that's Africa as I may.
Now we can do the same thing with the
moment here is the little bit of moment
as we move in our differential element.
This is our anger man and you know I know
everyone here took physics so I mean.
That's essentially what's happening here.
We might have a moment per unit
length on the surface attraction and
we have been equals each dot.
OK Now there are some kinda magical
relationships going on that are on
the bottom and they say things like
the change in spatial the spatial
change in the anger of philosophy will
give us time rate of changes of curvature.
So I use this I think it.
Demonstrates things nicely.
If I was to look at this.
I could say there's three curvature so
to speak.
There's bending this way it's
bending that way and there's twist.
All right.
And if I was to hold one end and
I was to change the angular
velocity let's do the spinning or
velocity make it linearly change while
going to store more and more twist.
If I take those anger
velocities make them equal and
I'm not storing any more twist my twist
with respect to time isn't changing.
That's what this first
equation is saying here.
A spatial change is
an anger of a lost city.
Plus a term that's harder to into it will
give us time rate of changes of curvature.
And you can do something similar and
you'll recover this bottom equation
spatial changes in velocity
will give us time rate of change of
strain that's what's happening here.
OK now that's how we're
describing this continuum and
in fact that will describe
any configuration in space.
Not talking at all about displacement so
rotations.
That's the key.
I told you Piers.
Momenta ages Ingo matter.
We can relate that to velocities and
anger velocities and
at that point you're almost looking
at a completed formulation.
If you like we have six unknowns
in these four equations here.
We have internal forces in internal
moments we have curvature K.
We have strain and we have philosophy and
we having a momentum.
We can eliminate two more in other words
we can introduce two more equations that
would relate these internal forces to
strains and I skip ahead a little bit.
And you'll see here those constituent
relationships that closes
the formulation Now up until this time
I've said nothing about proteins or
carbon nanotubes I'm just
giving you kinematics.
But this is going to be the basis for how
we go about modeling carbon into proteins
D.N.A. anything you like
that can be described.
Is in three space but having one spatial
degree of freedom in our cooling.
OK So there's a problem with
these equations if you want
to use them in a multi-skilled sense and
let me describe that problem a little bit.
Ultimately if I want to model a carbon
nanotube or D.N.A. or protein.
I'm going to want to know.
Well if there's two atoms and
this is a fictitious cross-section but
if there's two atoms one being
a which you see up here.
Our star will get to a two and
a three that's Adam a.
And we've got Adam be here.
What is the pond length and
if we had three bodies what would be
the bonding that if we had four bodies
would be for example the dihedral angle.
How would all of that change
with my defamation my curvature
changes in my strange changes.
I don't know anything about this place
since I can't locate anything in space and
so what was needed was a way of
getting after that information only
knowing the expansions that
were on that previous slide.
So here is the idea.
I can locate the center line at X.
one I can locate a change in the center
line at X. one plus the X. one
by the way that relies on that expansion
you saw earlier where I said if we didn't
have shoes trains we would just have
the one we just have the tangent vector.
I need to know where things are off the
center line so that the center line pull
position plus the A two in the B.
two in the community three.
I do this similarly here for B.
but I do that at X. one plus the X. one.
I tailor expand and when I tell expand I
need expansions on the center line and
I need expansions of
the basis vectors themselves.
This is now getting me off
the centerline at some distance dx one.
And so my basis vectors have changed
because I kind of moved in space and
if you carry out all
the Terra expansions and
you carom out to any order
you like really every D.X.
one here but you can think about is
delta X. one it's a finite distance.
We get an expert.
Russian for the relative vector position
vector between two points A and
B. purely in terms of our state
our curvature is in strain that's
parameterize of course by the positions
of the two atoms we're interested in.
OK so in the in the work I described
this was sort of a key thing was to have
this position vector for
any two points separated by distance D. X.
one and
differing locations on cross sections.
Now I can tell you how
we get to atomistic So
what you're looking at now is
the constituent of law required for
these equations something term type
realized this idiot says that if
we want the internal force we take the
ribs with respect to strain of the energy
strain energy moments are derivatives
of energy expected curvature.
And for a carbon nanotube or
for a protein or
any other molecular biological system this
strain energy you is related to the inner
atomic potential energy through a length
scale is to strain energy per unit length.
We take our energy we divide by length and
we evaluated at a representative
volume element for our system.
And that's what was done first
in these carbon in a tube So
here you're looking at what's
termed a graph in she and
in a tube is this graphene sheet it's
these carbon atoms the carbon atoms being
located at the vertices the bonds
are represented by these line segments.
We have to say that gets curved up first
into a sheet forming a carbon inner tube
and then it undergoes some defamation.
We have some curvature and some strain
right nonzero because it's a dynamic
simulation maybe we're applying forces
in this is moving all through space or
maybe in the protein there it's
in the presence of blood flow
in the blood flows forcing it in
some way and potentially giving
a different conformation a different shape
by the way different equilibrium shape.
That's of interest.
So we have to take that energy
we have to use that are A B.
that expansion I told you
about to get beyond links.
Now we can.
About how that bombing stretches
as defamation occurs we can build that
back into our forces in our moments and
we can integrate in time or
that's kind of the big picture.
Defra me it forces extra forces lead
to defamation defamation leads to these
internal forces kind of that's our
simulation just stepping through time.
I want spend very much time here at all
just to say OK we take the what's termed
the strong form and we make a weak
form and a killer can approach
our way to residual approach using
some terms called virtual velocities.
Here's what's important.
I'm going to Inter plate curvature and
I'm an interpreted strain and
this is rather this is unique.
Actually most people would go
about Inter plating position.
But what I'm going to say is that given a
continuous Maybe I'll make a little helix
to make the helix but
I have to do is twisted and bend.
It so I made this helix.
I'm going to keep track of curvature and
strain and that means for example
helix and this could have any arbitrary
number of corals it's going to be hard for
me to make that OK but you can
imagine there's like sixty coils or
that that protein where you saw sixty
wound up like a telephone cord this
formulation will allow us to capture that
with only a single element because in our
mind that's constant
curvature at each node.
So I store maybe two nodes storing the
same value of curvature that I capture any
arbitrary number of corals I like and
that's because I'm interpreting curvature.
And so that's really the power here.
I can do an alpha helix
with one element and
that alpha helix can have
however many corals I like and
that's why I saw such kind of
opportunity with the protein modeling.
In that that interpellation is here.
I've got some shape function
this is basically a lawn.
Continue and
this is these are my nodal curvatures.
I now.
How might this work.
I mean if this is a new formulation we
probably should compare it to what's been
done in the past.
This again is going to get away comply.
The from any molecular or biological
system I'm just going to look at a street
beam and I can build this an abacus it's
a very good commercial fine all my code
it of course is going to use nodal
displacements and rotations.
It's not wrapped up in
this this curvature space.
And what I did in the simulation is
rapidly put a load on and I took
a straight beam and I rapidly suddenly
applied a load of bent over as a horseshoe
and bent it back and there's all these
dynamic all this dynamics going on.
And what you're looking at
now are the actual strain.
And the two shear strains as a function of
time during this very dynamic event and
my code of these kind of little
circles in abacus or the solid lines.
I see you see some very
high frequency content.
I should point out that these
strains are derived quantities were
abacus would be literally taking to rivet
is of it's degrees of freedom of spatial
degrees of freedom.
So you know even comparing these two
things at first I thought maybe this
comparison wouldn't look so great.
And I should mention in this international
journal for numerical methods and
engineering paper.
There were zero energy modes that
were uncovered you know uncovered
some zero energy most I had to take
care of them in order to get this
this nice representation.
But once you know about those
modes you know how to handle them.
You know you can move forward from there.
This is looking at the curvature.
These are the two bending curvatures
again you see that very nice comparison
between the two codes.
I know how does it work in this
multi-skilled approach for
carbon nanotubes this is looking
at adding a lot of damping to
get the equilibrium response of what's
termed an armchair nanotube and
based on various loading States
this is a bending moment.
This is an actual extension and
this is a twisting
taking from the fine Adamic which is
sampling in atomic potential energy and
getting a prediction for
certain bending rigidities and
the Youngs modulus Now how does this
compare to what's been done in the past
with maybe some simpler approaches
you can see we have a pretty good.
Comparison and bending.
Rigidity.
One to two Tear Pascals if you know
anything about nanotubes people
are saying while they're roughly one tear
Pascal in Young's modulus you know this
is an amazing number by the way compared
to for example steel and the code
is predicting one point three and I see
a lot of numbers right around that mark so
that was a good mark and then
the torsional stiffness between analytical
models in this model is again
a kind of a close match.
And there's some in the papers published
on the carbon in tube there's some
information on frequencies and how they
compare and they compare in a similar way.
I know the last thing I mention about this
is our ongoing work in protein folding.
Again here is hemoglobin.
I'm going to show you a simulation
that just looked at the numerics and
how stable it was and did it have
the ability to go from a very polled
straight configuration of a protein and
recover the conform shape
which will predisposed to be alpha helix
and I'll show that simulation in a moment.
I'm going to feed it in a potential that's
very simplistic and just going to say that
it's parameterized by some equilibrium
strain which will make you zero and
some equilibria curvature
which will make non-zero.
And if we make these constants which they
are in this potential we should recover
an alpha helix.
All right so let's see how that does.
OK here and here is dynamics
being predicted from that code.
There's a little bit of damping.
That stamping out.
The frequency content slowly over time but
you can see this protein tends
to wind itself up from its And.
At some point I'm going to come in and
blow this up going to zoom in so that.
So the configuration gets bigger.
I actually there's nothing Rand Yeah
there's nothing random in her.
We can add that but we're going to there's
no electrostatics in here at the at
the moment it's just right
now it's in it's kind of like protein
in vacuum we don't even have a fluid.
Although I had damping in here
to somewhat simulate a fluid.
And so
by time this gets to about one twenty.
It's projecting pretty slowly.
Now the video card I guess is
kind of slow for this machine.
But you can see it starting to
reach in out the heel shape.
It's doing this.
Only with five elements
simulating with five elements and
by the time we got to about one twenty
one I stopped the simulator and
said OK I'm satisfied that it
recovered this is Alpha He'll shape.
And as I mentioned we haven't
built in true potentials yet
that's what a student is doing now.
This was just the prototype to show that
we would have the ability to do that.
OK So essentially you can see
that it reached the South.
He'll call shape dynamically.
OK.
Any questions about that before I move on.
I make a few conclusions.
By the way the time step because we're
in a continuous setting we're not it.
We're not limited by what energy has
to do market dynamics are on and
they were offensive seconds right.
And there are times that we're going
to have a much larger time step.
We're going to discretized have
many many many fewer degrees of
freedom some the sense
we're highly efficient.
We know that the technique itself in more
traditional settings does very well we
compared it to abacus.
You know and really I stress this we have
a significantly larger time step in time
durations So for example we can
put a protein in blood in and
study its dynamics with blood flow right
Andy has a very hard time solving that
problem because the shape change of
the protein occurs over such a long period
of time that it really can't crunch
the numbers and get to that time period.
OK.
Well.
Well in part I showed you all that
because what I'm about to show you is so
simplistic.
I thought you'd laugh if I didn't
show you what I can really do.
OK now.
This ends up being so
simplistic but it works and
that's really kind of the beauty of
what I'm about to talk talk about next.
I just briefly.
We're all familiar with a number of
numerical solution techniques there's more
than just what I'm putting up here.
There's even more than I'm
putting down on the bottom here.
All right.
But primarily as an engineer we
see a lot of finite difference.
And find a difference.
Works great for looking at modeling waves.
You know the grid based it's very uniform.
It's easily paralyzed and you know I'm not
a parallel person so I should be careful
and I say things like that but then on
the other hand as engineers we really like
this method called the find an element
method because it allows us to do
arbitrary geometries at a bridge is not
just a rectangular plate rectangular
plates work very nice to find a difference
but they don't work very fine a difference
does not work very nicely when the plate
becomes curved or shaped or has a hole
in it and the find an element of math
that allows us to do that by mashing.
Now that's very well known in
traditional funded Amen approaches that
we've mechanics can be difficult in C.
spurious oscillations you see oscillations
that are in front of the wave you know
that's very non-causal nothing really
should be in front of the wave.
There's oscillations behind it.
If you let the waves go back and forth
these isolation to build and build and
build and you know there's some more
sophisticated fine element methods
that alleviate some of that.
OK so it would be nice to have something
with the utility of the funded element
method but still did a very good
job with waves like to find a different
technique and find it different.
So I think it's good with waves in my
mind because it's an it is continuous
function space.
It's a very simplistic view point but
it seems to be appropriate.
I taunt to talk about cellular automata
I was exposed to this when I was at
Miter and
I was doing some disease spread my.
Modeling.
Is credited to John von Neumann I'm
sure everyone in here knows.
New stability analysis
all sorts of stuff right.
I mean he's very prolific person he
was at Princeton to study study for
Institute for Advanced Studies.
He developed the first what's agreed upon
to be the first time a technique in
the late one nine hundred forty S..
Let me just see if I show a hand.
So when I say this is everyone
know what I'm talking about.
OK so I can skip through this.
We all know that we have a cell and
it stores a state and
it relies on local interactions and
we have an update to
the state got a rule that tells us how to
update that state at the next time step.
That's very very nice.
It's a paradigm it doesn't tell us how
to model proteins doesn't tell us how to
model elastic wave propagation
just a paradigm really and
it can have deterministic and
probabilistic rules that state
variables can be energy or
floats you know discrete or continuous.
And we know it's been used in
a diverse number of applications.
And before I even discuss this is
everyone seen John Conway's Game of Life.
OK OK So for this crowd everyone
knows what I'm talking about.
Now it's massively parallel and
in easily incorporate stochastic city and
heterogeneity you got to remember
I'm thinking as an engineer.
How do we use this
inelastic wave propagation.
One of the main things I
like about it is that.
I'm not writing partial differential
equations you know you saw those in
the protein modeling they have all sorts
of issues when I discretized them and
I don't always know what's
going to happen and
my going to be unconditionally stable and
whatnot.
OK I don't even know if I'm
at spurious oscillations.
I don't know how to take many P.D.
Eason couple.
Right.
I mean six simulations that can be that
can be a bottleneck for
simulation I can have different times
because there's all sorts of things
that I don't really like about P.D.'s.
So your comment on the other
hand is object oriented fits
naturally with modern computing languages
and software practice is more thing.
That I like about it.
And here's just some of
the applications that I know about
I was involved in disease spread.
I know.
Dr Fujimoto is doing traffic I
don't know that you know doing so
Utama lattice Boltzmann by
the way is a very robust and
mature version of cellular Tom and
for food dynamics.
I'm going to talk about a lastic wave
propagation and that's where it's not so
much or.
Skip this you can say
how it's up to twenty.
Now in solid mechanics we're interested
in lastic waves like pressure waves waves
moving like this shear ways moving like
you know surface waves like a water ripple
Lambe waves in plates you know there's
all sorts of ways you're interested in
there's been very sparse treatment
some people have attempted to do
cellular automata I list
a few of their names.
A lot of it's done with these elastic and
plastic springs are like masses and
elastic and plastic Springs.
Sometimes that the masses are like sag and
shape.
But it turns out that we can't
really get at arbitrary points on.
Facts that way.
OK And so what I'm going to do is
describe an approach where we can get
at materials with arbitrary Poisson's
ratio and I'll stay in the two D. regime.
And one of things I'll do is show you that
if the grid is rectangular fifty uniform
grid that will be equivalent to
central difference finite difference.
So that's one of the only kind of
analysis I'll do with the method
everything else will just
be numerical experiments.
So it's fairly immature in that sense.
I'll generalize the approach of trying
your cells and show you how we do
arbitrary geometry and I'll talk about you
know how it compares to other methods.
So I've already mentioned manatees
it vantages we're avoiding P.
simplifying interactions and
I have to show you that we're accurate.
Right.
Well there's a straightforward boundary
condition that comes out of this result.
And it produces a very
accurate seismic simulator.
And then we appear to be avoiding
spurious oscillations I said.
If we do the final bit method we watch
these oscillations drone front of and
behind the wave front.
It's not very satisfying.
And we tend to avoid that and
I can't really tell you yet
why that is OK now here's what
I tell you it's very simple.
I'm going to start with the rectangular
case if we want to study elastic wave
propagation one thing we could do is break
the domain into a bunch of squares or
rectangles and
we can use that because I'm a to keep
track of the forces on the edges.
In fact the language is even simpler
to talk about the right to left
the top on the bottom.
OK.
And before I do that.
Introduce some things
about more neighbors and.
Newman neighbors I think you already
know there are those neighbors.
That's the yellow being a subset the Banu
neighbors of the more neighbors and
for some of these interactions.
I can use my neighboring face to
figure out my neighboring force but
sometimes I have to use
more than one neighbor.
And I'm going to call these type one and
Type two and
I make this more clear in a moment.
Now the rules that's going to come from
a momentum balance and a stepping in time.
All right so the tensile forces.
This is our invoice in what's
termed Voigt notation.
This is how our stress is related
to our strain through this
material matrix tensor.
Right.
C.
And we know that there's
two types of strains.
There's one strain that's called ten Sile.
So you have a good picture of this here.
Are if I'm going to look at the force
in the normal direction so if
I'm concerned with this I'm on the right
face I want to know about this force.
There's two things that
generate that force.
One two cells kind of moving
relative to each other like this.
This is a ten Sile force or stress right.
That's basically D U X dx I
want to go in the X. and Y..
My X. displacement see how it changes
now on the other hand there's
this idea of a poison effect if
I'm stretched in the transverse
direction I found in the Y. direction.
I tend to compress in this is
called the police on effect.
And if my neighbors are doing that too.
Then they're kind of compressing in and so
if I'm on this face if
I'm on the right face.
I need to know.
Well what are these three blocks
doing in the water in motion and
one of my neighboring three
blocks doing the wire motion and
that should give me a very good idea
about this this poison effect right.
That's the type to.
Type one uses just the von Neumann
neighbor type who's going to use
some more neighbors.
And that's what's shown here.
I type one that's moving this way
type two is the police on effect.
I'm just looking at a finite
difference of the strain essentially
on the right hand side.
You can do the same thing was sheer.
Forces.
You can add them all up for a cell and
you can write the balance of momentum and
I'm going to do this on first order form.
So I may potentially have some
extra a force in the X. and the Y.
going to some all these
faces right top left bottom.
There's going be some type of state
evolution causes Thomas we talk about
states basically there's to be
a time step and that's the rules.
Right now the boundary conditions come
out very simply because I'm talking about
faces storing a displacement state you
actually knew was in the center but
I have these faces that I'm free
to use to describe boundaries.
So I can talk about
a boundary attraction and
a stress applied to a Face
it could be a free stress.
You know it could be zero stress or
free surface.
It could be a fixed surface where
we have zero cell displacement.
And you can assemble
these equations now what I've done is
to show that this is in fact the same
essential difference this very simple
scheme where the cells are rectangular.
This is just a depiction that in fact
we're going to recover the central
difference approximations to what we know
to be the partial differential equation
governing this system into
the Lamaze equations in two D..
Don't go over it you just take all
the assembled equations you recognize what
looks like central difference quantities
essential difference quantities
around the bottom and so
we know for this simple case we're
replicating an already well known and.
Well studied method.
I saw how does it do this is
looking at last a cast basically
of a free surface the sides
are all going to be fixed.
They can have any other brand you can
actually not even let the simulation
the waves get out to the edges.
And they'll be a differentiated
galaxy impulse applied to the surface
that will produce three waves that
will produce the pressure wave
the differential gas impulse is almost
like a sinusoid it's got two lobes.
So will first see the pressure wave
will see a slower shear wave and
we'll see a rally surface wave up here.
And here's the cellular Tamada prediction.
And here is what I say
is the defacto standard
this approach called staggered grid
that staggered good was developed
in part to have a better
boundary condition and
as I said the sort of Tom really suggests
the most obvious boundary condition.
And they compare within two percent as
long as you take into account that in
the staggered grid actually just as I
said so you can do the free surface.
What the stress of interest
lies right on the surface and
my point is starting to die.
All right so it does have a nice job
compares well with the staggered grid.
Let me show you some further results.
Now staggered grid leads to some asymmetry
and leftward and right we're moving way.
I showed you results for the band we
see stress kind of a cumulative stress
this is velocity vs time.
And if you notice the cell your
comment on the left shows.
Pretty much ideal symmetry the staggered
grid starts to lose symmetry.
It's a consequences.
Fact that the grid is staggered and
it also shows quite a lot of
oscillations in the response if you
look at the surface velocity you see
that very choppy response in our solar
time and appearing to be rather smooth.
And again I'm showing you just
kind of numerical experiments.
I'm not offering a lot of
insight as to why that is.
Now I said in an aside to
Thomas it's very easy to do.
Heterogeneous media you know and
you can do it in a finite differences Well
it's very easy here to talk about one of
the cell properties being the let me
parameters of the elastic coefficients.
Here grading them in the X.
direction and making one the May
parameters increase as I go towards the X.
and what I'm trying to do
is see if the resulting
discrete ization causes
reflections right so grating this
I know in the analytical solution I see
any reflections as a smooth grating.
And when I run the cellular
atomic code I see no
reflections I see a very quick
response as the wave passes through.
That's simply what this is showing just
showing some maybe nice features of of
the simulation.
And now here is actually
material random material.
Yes I agree.
Agree that you can do
this with any simulation.
So this is random material
again very easy because I
specify the I can do this very easily but
it can be really done at any code.
OK so once that was done.
It's very simplistic once we
looked at the rectangular case we
asked ourselves could we extend it in some
way to look at measures of triangles and
if we could do that.
Maybe we're going to recover these
benefits of finite difference in wave
propagation but have the generality
of a fine and element technique and
that's what's being done here is to say OK
let's move away from a very uniform grid.
And last.
Look at some collection of triangles
we've got measures that will
tessellated domain into
triangles that's no problem.
But there's a few complications.
We want to still use this type one and
Type two insight that we had
about poisons affects right.
We want to borrow all that.
But we now have too many more neighbors in
fact if you look at a vertex of a chart
and you can have as many more neighbors
as you like you can just keep splitting
these triangles bisecting the triangles
and you have any number of more neighbors.
Whereas we are limited to eight more
neighbors with the rectangular approach.
So how do we do type one and Type two.
That was a question.
And furthermore if you look
the centroids of the triangles
don't line up nicely so to speak.
If this is my cell of interest and
these red lines are those type
two calculations that we did.
Now if we were to try to repeat that.
Notice the centroids
are all kind of whacked up.
You know they're not they're not purely
in the transverse direction like we'd
like them to be.
So we had so many questions about how
this would work the first decision
we made was to say OK we have too many
more neighbors so let's only use von
Neumann neighbors of my von Neumann
neighbors in character lesions.
So here is the cell of interest
here is of a Neumann neighbor.
When I do these type two calculations
let's use von Neumann neighbors of
my five Neumann neighbor
that kind of eliminated.
Many of the of the neighbors
made arc computation
at least algorithmically
you know tractable.
And here I'm just kind of showing how you
can have these offsets in the centuries.
All right so this is kind of restating it.
Again we're going to use interactions or
forces at the faces to stay with
the cellular automata paradigm.
We're going to store displacements
velocities external forces
at the central points and
will reference an X.Y. Global X. Y.
axis system much like you would
do in a fun and technique.
The Trying to.
Cells have unique face lengths and angles
you know there's a few complications here
and we've got these
multiple more neighbors.
So the way it's done and
I may pass through this pretty quickly.
Each face has a rotation
associated with it.
So we can always get into a normal and
tangent coordinate system using a rotation
matrix we go through this
same step of type one and
Type two normal strains put stuff that's
kind of associate with a poison effect.
You just have to work in normal
intention tangential coordinate systems.
Here these are calculations
Illustrated graphically.
Again we're using basically a hoax law.
It's a linear elastic material.
And we can get to assemble normal
forces intentional forces on each face.
Which we then use in
a balance of momentum.
Just like I showed previously same
first order form same updated questions
by the way that at the time
integrated extremely simple here.
We haven't done anything like generalized
Alpha like we're using in the protein
simulator.
We're just doing basically the simplest
this could as ation in time that you
can do.
And we have a way in fact of doing all the
boundary treatments Norman Odair chalet.
OK.
So let me just talk briefly about this
a simulation now is very different than
a fine and simulator fundament simulator
most of them that have been written have
sort of this this top hierarchy right
kind of assemble all the forces and
then solve the nodal degrees of freedom.
OK.
This works by if you ignore the fact
that we have to read in a mesh and
put you know that's too germane
to what I'm talking about but.
The simulation itself just says OK you're
a cell you have pointers to your neighbors
update your state and that's the next time
Step Update your state update your state.
It just proceeds in this manner.
So it's very autonomous it has no
higher order authority working with a.
It's very nice.
In fact I think you could
do discrete event this way.
Right because you could just say well
only update if some event triggers you
that you need to update.
There's no waves nearby right this very
question to round you don't bother
updating.
Because it's not necessary.
And now show you are numerical
experiments again I want to stress.
We haven't done analysis of the method.
We've played with the method.
And the work is relatively
a mature in that sense.
But what we did was to say OK the first
first thing we'd better look good in
is some type of interior load.
Let's forget about what's
happening at the boundary.
This shows a very fine mesh we don't need
anything nearly this fine but we use
the commercial code comm saw to generate
a response interior loaded domain.
It's a two dimensional domain.
You're looking at either you accept you.
Y. displacement projected upwards.
And we use the same mash out of
the final Mikko because OK so
we said well there's ways of.
Optimizing our mesh for our method we're
not even to be concerned with that
at this moment we're just going to borrow
a mass from a finite simulator and
these little green dots superimposed on
top of this surface green dots are our
code console is that colored surface.
Right.
So it does very very good job.
Even with these things we recognize
as being almost efficiencies
are approximations centuries
not lining up and whatnot.
And we tend to have about half
the degrees of freedom as com So for
that particular batch.
Now this is quite a bit more.
Difficult problem to solve this is
looking at today domain with a whole.
You're looking again at console versus or
cell or Tom a simulation.
If you flip this like that.
Right.
And look at it in project up to us or
you why your choice.
You'll see this top curve.
And again green dots are calm so
if you are so
tamad a code for colored surfaces calm so.
Now that's an applied boundary
traction and it's a dynamic
problem we looked at a snapshot in
time we compared the two codes.
We're going to do and they're doing
a very good job in comparison.
Now this is the one that got
a little more interesting and
it's leading to some
further study on our part.
Now we're going to apply apply and
impose boundary displacement.
This is an even more
difficult numerical problem
in the sense of you know stability.
So we apply a unit
displacement at the boundary.
And what we noted this
is the response over
over a few times steps a wave
going to propagate to the right.
You're looking at very early on and
some time later and
sometime much much later on the bottom.
You're looking at the two codes and
what we saw was as we know the funded
element code the spirit oscillations
that com So minimize is quite a bit.
It does it with the expense
of some numerical damping.
But it minimizes that I can make
a final make code pull up with the wave
propagation so that the oscillations
are on the order of the wave I put in it.
And notice we don't have
these oscillations.
All the way through the response as we get
close to the wave front you know we're
we're kind of much smoother I suppose.
And we're not quite sure why that is
exactly where we're starting that now to
see why we seem to be avoiding this
problem and I think that's important.
If you're going to study structural health
monitoring which is a big area looking at
waves to detect cracks and if you're going
to simulate and you're going to see all
this high frequency stuff and we tend to
identify cracks with high frequencies.
Then you going to say well maybe there's
a crack there you know there are certain
reasons why you don't want this
high frequency oscillation present.
OK so we seem to be avoiding these
periods oscillations and we do it.
It's just so easy.
All right.
Told you I've presented this protein stuff
because I'd be afraid that you would
just laugh at how easy this is so we're.
Town a method for computing lasted
a much a sponsor doing this a complex to
the media were observing very good
accuracy for interior and boundary loads.
We believe we see better away from
resolution than a conventional phenomena
not a discontinued school or can you know
like I said you can you can do better than
just opening up a commercial code.
And we think there's
might be attractive for
simulating wave propagation
in arbitrary domains.
I'm talking to the Army
about maybe electromagnetics
where they want to do physics simulations
of wireless communication and
they've got scattering they tend
to do a finite difference and
they have these very very fine mesh
measures around the scattering
I think we may be able to do that much
cheaper than they're currently doing it.
And with that I'll open it up for
questions.
Thanks.
So that that's kind of why I'm here
in a way I don't have the expertise
to paralyze my own code.
And that's you know I'm
just not a parallel person.
So if if there's an interest in you know
if there's funding to extend this and
I believe there is and that would be one
type of collaboration I'd be looking for
quite easy.
OK.
Right.
Yeah I think so
yeah in fact if you're looking you know
we do these diffuse wave studies for
structural health monitoring where some
portion of the domain may no longer be.
All important.
You know I may not be seeing any
way France Letterman that it for
radios are turning on and off again why
simulate most of the domain if there's
nothing in that domain so I think you
would want to basically not step through
lots of cells when you don't have to and I
think that could be discrete Aventura than
well to be fair we haven't tried
it on any real proteins yet
we've done carbon into validated that
work and it's appeared in been published.
We're now working on the proteins and
hemoglobin is there something particular
about hemoglobin all these proteins.
Right.
They show the same alpha
helix conformations So
now it could really be any protein.
OK
So actually a lot of protein studies
are based on a lattice which would make
them sell or Tom and I asked this work
that we're doing has nothing to do it on
the protein folding I mean I kind of
put these two together and maybe at
times didn't make a clear separation but
I'm not too insular talent of proteins.
Maybe you
know what I described with this
continuous technique my advisor back in
Michigan who I worked with on study
had nothing to do with D.N.A.
or proteins or R.N.A. or anything
else he happened to go towards D.N.A.
with similar tools and I've been going
towards proteins with my tools and
I really never want to come
back this way with my advisor.
So I stay away from D.N.A. and R.N.A. is.
I think he's done a very nice job and
I want to advise or.
We're all in the same boat.
Yeah they are they are in
fact his work and D.N.A.
is relatively mature at this
point he's got a lot of funding
to look at even therapeutics
associated with D.N.A. So
really OK OK that's interesting.
OK
Thank you thanks.