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If we define ρ ¯ = ρ 1 + ρ 2 + ρ 3 and P ¯ = P 1 + P 2 + P 3 , we are then left with the relation

ρ ¯ I j = 1 N 2 c j ( t ) t 2 φ j , φ n 1 = - P ¯ j = 1 N c j ( t ) a φ j , φ n 1

Together, equations ( ) and ( ) provide us with a system of equations M c ' ' = K c from which to determine our coefficients c k ( t ) , where M and K are

M = ρ 11 I φ 1 , φ 1 ρ 12 I φ 1 , φ 2 ... ρ 1 N I φ 1 , φ N ρ 21 I φ 2 , φ 1 ρ 22 I φ 2 , φ 2 ... ρ 2 N I φ 2 , φ N ρ N 1 I φ N , φ 1 ρ N 2 I φ N , φ 2 ... ρ N N I φ N , φ N
K = P 11 a φ 1 , φ 1 P 12 a φ 1 , φ 2 ... P 1 N a φ 1 , φ N P 21 a φ 2 , φ 1 P 22 a φ 2 , φ 2 ... P 2 N a φ 2 , φ N P N 1 a φ N , φ 1 P N 2 a φ N , φ 2 ... P N N a φ N , φ N ,

where ρ j k and P j k are linear combinations of ρ i and P i that are determined by the geometry of the network.

If we assume 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

φ i , φ j = 2 h / 3 , if i = j ; h / 3 , if | i - j | = 1 ; 0 , otherwise.

and

a φ i , φ j = - 2 / h , if i = j ; 1 / h , if | i - j | = 1 ; 0 , otherwise.

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 = [ 1 , 0 , 0 ] , v 2 = [ . 5 , . 5 , 0 ] , v 3 = [ . 5 , - . 5 , 0 ] . Suppose s i = 2 , k i = 1 , ρ i = 1 . Then,

P 1 = 2 0 0 0 1 0 0 0 1 , P 2 = 5 / 4 1 / 4 0 1 / 4 5 / 4 0 0 0 1 , P 3 = 5 / 4 - 1 / 4 0 - 1 / 4 5 / 4 0 0 0 1

and we can assemble M and K as follows

M = h 6 4 ρ 1 I ρ 1 I 0 0 0 0 0 0 0 0 ρ 1 I 4 ρ 1 I ρ 1 I 0 0 0 0 0 0 0 0 ρ 1 I 4 ρ 1 I ρ 2 I 0 0 0 0 0 0 0 0 ρ 2 I 2 ρ ¯ I ρ 2 I 0 0 ρ 3 I 0 0 0 0 0 ρ 2 I 4 ρ 2 I ρ 2 I 0 0 0 0 0 0 0 0 ρ 2 I 4 ρ 2 I ρ 2 I 0 0 0 0 0 0 0 0 ρ 2 I 4 ρ 2 I 0 0 0 0 0 0 ρ 3 I 0 0 0 4 ρ 3 I ρ 3 I 0 0 0 0 0 0 0 0 ρ 3 I 4 ρ 3 I ρ 3 I 0 0 0 0 0 0 0 0 ρ 3 I 4 ρ 3 I
K = 1 h - 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 - 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

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 ρ ¯ and P ¯ on the diagonal stems from the inner product of the joint hat function with the hat functions on each of the strings.

Generalized numbering scheme and-adaptivity

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 i j th block, and k i j , and s i j refers to the value of the physical constants on the shared support of the hat functions φ i and φ 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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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