# The network wave equation

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$〈{\phi }_{i},,,{\phi }_{j}〉=\frac{{h}_{ij}}{6}$

since two different hat function can overlap on at most one leg (otherwise two legs of a hat function could cover the same support).

Next, we can create our $K$ matrix, which requires knowledge of both $a\left({\phi }_{i},,,{\phi }_{j}\right)$ and our ${P}_{i}$ matrix between nodes $i$ and $j$ , ${P}_{ij}$ . Starting with ${P}_{ij}$ , if we have done our bookkeeping correctly, we have all the variables we need to compute

${P}_{ij}={k}_{ij}\left[\left({s}_{ij}-1\right)I-{v}_{ij}{v}_{ij}^{T}\right],$

after which we only need $a\left({\phi }_{i},,,{\phi }_{j}\right)$ for $|i-j|\le 1$ . For $i=j$ on our main diagonal, the energy inner product is just the sum of the integrals $\int {\left({\phi }_{i}^{\text{'}\text{'}}\left({x}_{i}\right)\right)}^{2}\phantom{\rule{0.166667em}{0ex}}d{x}_{i}$ evaluated for each leg of the hat function. If each leg lives on a support of length ${h}_{ij}$ , the energy inner product is

$a\left({\phi }_{i},,,{\phi }_{i}\right)=\sum _{j\in {N}_{i}}\frac{1}{{h}_{ij}}.$

For $|i-j|=1$ , two hat functions can share support on at most one leg, so our energy inner product is

$a\left({\phi }_{i},,,{\phi }_{j}\right)=-\frac{1}{{h}_{ij}}$

which again is analogous to our single-string case.

## Damping

The case of the damped network wave equation is worth examining as well, especially in the mathematical modeling of a spider's web. The material properties of spiderwebs also make it ideal for simulation via the second order wave equation. These include minimal torsion (twisting) in vibrations, low stiffness, no hysteresis under small strains, and a loss of energy primarily through aerodynamic damping. The wave equation assumes negligible torsion and low stiffness, is meant to model string movement specifically under small strains, and is easy to add a constant aerodynamic/viscous damping term to.

Since the structure of our damping matrix $G$ is built from the same inner products as our $M$ matrix; the only difference is that we now have to keep track of one more constant, the damping coefficient on a connection between two nodal points ${a}_{ij}$ . The $ij$ th block of $G$ is then just the $ij$ th block of $M$ scaled by ${a}_{ij}$ . This allows us to again vary damping from connection to connection, which proves useful in the simulation of spider webs, since the radial and axial fibers of a spiderweb are often subject to different levels of damping.

## Matlab gui

With this last bit of information, we know each block entry of our $N$ blocks by $N$ blocks discretization matrices, and can construct a finite element discretization for a web given only a list of nodes, their positions, and their connectivity. To implement this in an accessible way, a Matlab“point-and-click" GUI was developed to allow users to trace and experiment with their own webs through numerical simulations of web motion and analysis of the eigenvalues and fundamental modes.

## Setting up the web

Using a GUI to wrap around our framework which allows the user to point and click to place nodes down, then to click from one node to another to specify the connection pattern. Endpoints (where the nodes are pinned down, enforcing Dirichlet boundary conditions) are assumed to be nodes with only one neighbor (i.e., not a link in a chain). Once the initial pattern is set, the user can change the discretization fineness as desired, as well as rescale the size of the web to a larger or small grid. When the user is done, the positions and connection pattern of the nodes can be used to create a finite element discretization of the network of strings.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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