# The network wave equation  (Page 7/11)

 Page 7 / 11

Many of the concepts from the single-string case carry over to networks.

## Assembling theAndMatrices

We begin by describing the notation of the information represented by our data structures. We denote the $i$ th node to have $xyz$ position vector ${p}_{i}$ . For each node at ${p}_{i}$ , there is an associated set of the indices of connected neighbor nodes ${N}_{i}$ and a set ${C}_{i}$ containing the physical constants ${k}_{ij},{s}_{ij}$ pertaining to the connection between nodes $i$ and $j\in {N}_{i}$ . Since we can divide through by ${\rho }_{i}$ on both sides of the network wave equation, we can assume without loss of generality that the constant ${\rho }_{i}=1$ , and that any data carried by the density ${\rho }_{i}$ is now contained in ${k}_{ij}$ .

Assuming we are given a set of $N$ nodes, along with the $xy$ positions of each node (the $z$ positions are assumed to be 0, such that the web is planar in the $xy$ plane at rest), our first goal is to compute our step sizes ${h}_{ij}$ and orientation vectors ${v}_{ij}$ for connections between two nodes $i$ and $j\in {N}_{i}$ . To account for Dirichlet boundary conditions, we also create an anchored node for each endpoint. In this implementation, if a node has only one neighbor, we assume it is connected to a pinned endpoint whose position is in the opposite direction but the same distance away as the only neighbor (this is required to calculate an inner product). For a node connected to an endpoint, we append the index 0 to ${N}_{i}$ .

Given ${p}_{i}$ , ${N}_{i}$ , ${C}_{i}$ , we proceed as follows

• for each $i$ from $1,2,...,N$
• for all $j\in {N}_{i}$ , calculate ${h}_{ij}={\parallel {p}_{i}-{p}_{j}\parallel }_{2}$
• if the number of elements in any set ${N}_{i}=1$
• create an“endpoint" node at position ${p}_{0}=2{p}_{i}-{p}_{j}$ , with step size ${h}_{i0}={\parallel {p}_{i}-{p}_{0}\parallel }_{2}$
• set ${N}_{i}=\left\{{N}_{i},0\right\}$ to indicate the $i$ th node is connected to an anchor

In practice, we normalize the positions of our nodes such that the web lies within a box of a desired arbitrary size $s$ . We do this by calculating the maximum distance ${d}_{max}$ between the anchored endpoints of a web, then scaling the positions ${p}_{i}$ of every node by the factor $s/{d}_{max}$ . Since the absolute positions of the nodes don't affect our discretization, we don't need to worry about subtracting off the centroid of all the node positions.

With all our variables now in place, we can now proceed to the actual construction of our discretization matrices. This requires knowing $〈{\phi }_{i},,,{\phi }_{j}〉$ , $a\left({\phi }_{i},,,{\phi }_{j}\right)$ , and ${P}_{ij}$ . We deal first with constructing the $M$ matrix, which requires only knowledge of $〈{\phi }_{i},,,{\phi }_{j}〉$ . Just as in the case of finite element on the single string, most of the basis functions ${\phi }_{i}$ and ${\phi }_{j}$ don't share support and their inner products are zero. However, in addition to calculating inner products of regular hat functions, we need to compute the inner products with joint, generalized and nonsymmetric hat functions as well.

Starting with our $M$ matrix, we only need to calculate the inner product $〈{\phi }_{i},,,{\phi }_{j}〉$ for $j=i$ and $j\in {N}_{i}$ . For the diagonal case $j=i$ , we note that our inner product $〈u,,,v〉={\sum }_{i=1}^{L}{\int }_{0}^{{\ell }_{i}}u\left({x}_{i}\right)v\left({x}_{i}\right)\phantom{\rule{0.166667em}{0ex}}d{x}_{i}$ needs only be calculated on the support of ${\phi }_{i}$ . For given $i$ ,

$〈{\phi }_{i},,,{\phi }_{i}〉=\sum _{j\in {N}_{i}}{\int }_{{p}_{i}}^{{p}_{i}+{h}_{j}}{\phi }_{i}^{2}\left(x\right)\phantom{\rule{0.166667em}{0ex}}d{x}_{i}=\sum _{j\in {N}_{i}}\frac{{h}_{ij}}{3}$

The last part is a generalization of our inner product for a uniform grid on a single string. For the off-diagonal case $j\in {N}_{i}$ , $j\ne i$ , the inner product is analogous to the single-string case.

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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