# 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.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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