The network wave equation  (Page 6/11)

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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.

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