# The network wave equation  (Page 6/11)

 Page 6 / 11

If we define $\overline{\rho }={\rho }_{1}+{\rho }_{2}+{\rho }_{3}$ and $\overline{P}={P}_{1}+{P}_{2}+{P}_{3}$ , we are then left with the relation

$\overline{\rho }I\sum _{j=1}^{N}\frac{{\partial }^{2}{c}_{j}\left(t\right)}{\partial {t}^{2}}〈{\phi }_{j},,,{\phi }_{{n}_{1}}〉=-\overline{P}\sum _{j=1}^{N}{c}_{j}\left(t\right)a\left({\phi }_{j},,,{\phi }_{{n}_{1}}\right)$

Together, equations ( ) and ( ) provide us with a system of equations $M{c}^{\text{'}\text{'}}=Kc$ from which to determine our coefficients ${c}_{k}\left(t\right)$ , where $M$ and $K$ are

$M=\left[\begin{array}{cccc}{\rho }_{11}I〈{\phi }_{1},,,{\phi }_{1}〉& {\rho }_{12}I〈{\phi }_{1},,,{\phi }_{2}〉& ...& {\rho }_{1N}I〈{\phi }_{1},,,{\phi }_{N}〉\\ {\rho }_{21}I〈{\phi }_{2},,,{\phi }_{1}〉& {\rho }_{22}I〈{\phi }_{2},,,{\phi }_{2}〉& ...& {\rho }_{2N}I〈{\phi }_{2},,,{\phi }_{N}〉\\ ⋮& & \ddots & \\ {\rho }_{N1}I〈{\phi }_{N},,,{\phi }_{1}〉& {\rho }_{N2}I〈{\phi }_{N},,,{\phi }_{2}〉& ...& {\rho }_{NN}I〈{\phi }_{N},,,{\phi }_{N}〉\end{array}\right]$
$K=\left[\begin{array}{cccc}{P}_{11}a\left({\phi }_{1},,,{\phi }_{1}\right)& {P}_{12}a\left({\phi }_{1},,,{\phi }_{2}\right)& ...& {P}_{1N}a\left({\phi }_{1},,,{\phi }_{N}\right)\\ {P}_{21}a\left({\phi }_{2},,,{\phi }_{1}\right)& {P}_{22}a\left({\phi }_{2},,,{\phi }_{2}\right)& ...& {P}_{2N}a\left({\phi }_{2},,,{\phi }_{N}\right)\\ ⋮& & \ddots & \\ {P}_{N1}a\left({\phi }_{N},,,{\phi }_{1}\right)& {P}_{N2}a\left({\phi }_{N},,,{\phi }_{2}\right)& ...& {P}_{NN}a\left({\phi }_{N},,,{\phi }_{N}\right)\end{array}\right],$

where ${\rho }_{jk}$ and ${P}_{jk}$ are linear combinations of ${\rho }_{i}$ and ${P}_{i}$ that are determined by the geometry of the network.

If we assume ${\ell }_{i}=1$ for $i=1,2,3$ , then the inner product of two non-joint hat functions is exactly the same as in the one-dimensional case, where

$〈{\phi }_{i},,,{\phi }_{j}〉=\left\{\begin{array}{cc}2h/3,\hfill & \phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}i=j\text{;}\hfill \\ h/3,\hfill & \phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}|i-j|=1\text{;}\hfill \\ 0,\hfill & \phantom{\rule{1.em}{0ex}}\text{otherwise.}\hfill \end{array}\right)$

and

$a\left({\phi }_{i},,,{\phi }_{j}\right)=\left\{\begin{array}{cc}-2/h,\hfill & \phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}i=j\text{;}\hfill \\ 1/h,\hfill & \phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}|i-j|=1\text{;}\hfill \\ 0,\hfill & \phantom{\rule{1.em}{0ex}}\text{otherwise.}\hfill \end{array}\right)$

Let us take ${n}_{1}=4$ and ${n}_{2}={n}_{3}=3$ . For this network , if we assume all the legs of the tritar lie at equal angles from each other, we can define the orientation ${v}_{1}=\left[1,0,0\right]$ , ${v}_{2}=\left[.5,.5,0\right]$ , ${v}_{3}=\left[.5,-.5,0\right]$ . Suppose ${s}_{i}=2$ , ${k}_{i}=1$ , ${\rho }_{i}=1$ . Then,

${P}_{1}=\left[\begin{array}{ccc}2& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\phantom{\rule{1.em}{0ex}}{P}_{2}=\left[\begin{array}{ccc}5/4& 1/4& 0\\ 1/4& 5/4& 0\\ 0& 0& 1\end{array}\right],\phantom{\rule{1.em}{0ex}}{P}_{3}=\left[\begin{array}{ccc}5/4& -1/4& 0\\ -1/4& 5/4& 0\\ 0& 0& 1\end{array}\right]$

and we can assemble $M$ and $K$ as follows

$M=\frac{h}{6}\left[\begin{array}{cccccccccc}4{\rho }_{1}I& {\rho }_{1}I& 0& 0& 0& 0& 0& 0& 0& 0\\ {\rho }_{1}I& 4{\rho }_{1}I& {\rho }_{1}I& 0& 0& 0& 0& 0& 0& 0\\ 0& {\rho }_{1}I& 4{\rho }_{1}I& {\rho }_{2}I& 0& 0& 0& 0& 0& 0\\ 0& 0& {\rho }_{2}I& 2\overline{\rho }I& {\rho }_{2}I& 0& 0& {\rho }_{3}I& 0& 0\\ 0& 0& 0& {\rho }_{2}I& 4{\rho }_{2}I& {\rho }_{2}I& 0& 0& 0& 0\\ 0& 0& 0& 0& {\rho }_{2}I& 4{\rho }_{2}I& {\rho }_{2}I& 0& 0& 0\\ 0& 0& 0& 0& 0& {\rho }_{2}I& 4{\rho }_{2}I& 0& 0& 0\\ 0& 0& 0& {\rho }_{3}I& 0& 0& 0& 4{\rho }_{3}I& {\rho }_{3}I& 0\\ 0& 0& 0& 0& 0& 0& 0& {\rho }_{3}I& 4{\rho }_{3}I& {\rho }_{3}I\\ 0& 0& 0& 0& 0& 0& 0& 0& {\rho }_{3}I& 4{\rho }_{3}I\end{array}\right]$
$K=\frac{1}{h}\left[\begin{array}{cccccccccc}-2{P}_{1}& {P}_{1}& 0& 0& 0& 0& 0& 0& 0& 0\\ {P}_{1}& -2{P}_{1}& {P}_{1}& 0& 0& 0& 0& 0& 0& 0\\ 0& {P}_{1}& -2{P}_{1}& {P}_{2}& 0& 0& 0& 0& 0& 0\\ 0& 0& {P}_{2}& -\overline{P}& {P}_{2}& 0& 0& {P}_{3}& 0& 0\\ 0& 0& 0& {P}_{2}& -2{P}_{2}& -{P}_{2}& 0& 0& 0& 0\\ 0& 0& 0& 0& {P}_{2}& -2{P}_{2}& {P}_{2}& 0& 0& 0\\ 0& 0& 0& 0& 0& {P}_{2}& -2{P}_{2}& 0& 0& 0\\ 0& 0& 0& {P}_{3}& 0& 0& 0& -2{P}_{3}& {P}_{3}& 0\\ 0& 0& 0& 0& 0& 0& 0& {P}_{3}& -2{P}_{3}& {P}_{3}\\ 0& 0& 0& 0& 0& 0& 0& 0& {P}_{3}& -2{P}_{3}\end{array}\right]$

We can reverse engineer some of the geometry of our network from examination of these matrices - notice that each leg has 3 blocks assigned to it, corresponding to the 3 non-joint hat functions on each string. The far off-diagonal terms capture the connection of the first string to the third string, and the presence of $\overline{\rho }$ and $\overline{P}$ on the diagonal stems from the inner product of the joint hat function with the hat functions on each of the strings.

Unfortunately, for larger and more complex webs, writing the system out by hand becomes far too tedious. We seek a more systematic and flexible way of producing our finite element discretizations. We should note two things about finite element discretizations. First, if we stay consistent, a reordering of the nodes does not affect our discretization, though it may change the structure of our matrix. Secondly, our hat functions are not required to be either uniform or symmetric - they can vary in width depending on index, and one side can have a different width than another. This idea is known as $h$ -adaptivity; advanced finite element methods tend to adapt their discretizations by using error estimates from iteration to iteration to pinpoint areas where a coarse discretization should be refined to allow for greater accuracy.

Knowing this, it is possible to produce a generalized finite element discretization of a web given only physical constants, a set of nodal points and each point's neighbors. Given this, we can calculate the step size $h$ and orientation $v$ from node to node, and thus reconstruct our ${P}_{i}$ matrices. Knowing the neighbors of each node would allow us to reconstruct the structure of our $M$ and $K$ matrices as well. If we examine $M$ and $K$ , we can see they are formed out of an $N$ by $N$ block grid, where each block is a 3 by 3 matrix. Previously, the index $i$ differentiated between constants and different legs/connections in our network. In this generalized scheme, we allow $i$ and $j$ to reference different nodes in our discretization instead. Thus, a connection between a node $i$ and $j$ implies a nonzero entry in the $ij$ th block, and ${k}_{ij}$ , and ${s}_{ij}$ refers to the value of the physical constants on the shared support of the hat functions ${\phi }_{i}$ and ${\phi }_{j}$ . Utilizing this generalized scheme allows for much more flexibility in terms of our physical constants as well; for example, if the stiffness $k$ varied as function of ${x}_{i}$ , we could capture this by varying our stiffness $k$ from node to node. We go into more detail on this in the next section.

what is variations in raman spectra for nanomaterials
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
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
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 did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!       By By By Rhodes 