# The network wave equation

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$〈{\phi }_{i},,,{\phi }_{j}〉=\frac{{h}_{ij}}{6}$

since two different hat function can overlap on at most one leg (otherwise two legs of a hat function could cover the same support).

Next, we can create our $K$ matrix, which requires knowledge of both $a\left({\phi }_{i},,,{\phi }_{j}\right)$ and our ${P}_{i}$ matrix between nodes $i$ and $j$ , ${P}_{ij}$ . Starting with ${P}_{ij}$ , if we have done our bookkeeping correctly, we have all the variables we need to compute

${P}_{ij}={k}_{ij}\left[\left({s}_{ij}-1\right)I-{v}_{ij}{v}_{ij}^{T}\right],$

after which we only need $a\left({\phi }_{i},,,{\phi }_{j}\right)$ for $|i-j|\le 1$ . For $i=j$ on our main diagonal, the energy inner product is just the sum of the integrals $\int {\left({\phi }_{i}^{\text{'}\text{'}}\left({x}_{i}\right)\right)}^{2}\phantom{\rule{0.166667em}{0ex}}d{x}_{i}$ evaluated for each leg of the hat function. If each leg lives on a support of length ${h}_{ij}$ , the energy inner product is

$a\left({\phi }_{i},,,{\phi }_{i}\right)=\sum _{j\in {N}_{i}}\frac{1}{{h}_{ij}}.$

For $|i-j|=1$ , two hat functions can share support on at most one leg, so our energy inner product is

$a\left({\phi }_{i},,,{\phi }_{j}\right)=-\frac{1}{{h}_{ij}}$

which again is analogous to our single-string case.

## Damping

The case of the damped network wave equation is worth examining as well, especially in the mathematical modeling of a spider's web. The material properties of spiderwebs also make it ideal for simulation via the second order wave equation. These include minimal torsion (twisting) in vibrations, low stiffness, no hysteresis under small strains, and a loss of energy primarily through aerodynamic damping. The wave equation assumes negligible torsion and low stiffness, is meant to model string movement specifically under small strains, and is easy to add a constant aerodynamic/viscous damping term to.

Since the structure of our damping matrix $G$ is built from the same inner products as our $M$ matrix; the only difference is that we now have to keep track of one more constant, the damping coefficient on a connection between two nodal points ${a}_{ij}$ . The $ij$ th block of $G$ is then just the $ij$ th block of $M$ scaled by ${a}_{ij}$ . This allows us to again vary damping from connection to connection, which proves useful in the simulation of spider webs, since the radial and axial fibers of a spiderweb are often subject to different levels of damping.

## Matlab gui

With this last bit of information, we know each block entry of our $N$ blocks by $N$ blocks discretization matrices, and can construct a finite element discretization for a web given only a list of nodes, their positions, and their connectivity. To implement this in an accessible way, a Matlab“point-and-click" GUI was developed to allow users to trace and experiment with their own webs through numerical simulations of web motion and analysis of the eigenvalues and fundamental modes.

## Setting up the web

Using a GUI to wrap around our framework which allows the user to point and click to place nodes down, then to click from one node to another to specify the connection pattern. Endpoints (where the nodes are pinned down, enforcing Dirichlet boundary conditions) are assumed to be nodes with only one neighbor (i.e., not a link in a chain). Once the initial pattern is set, the user can change the discretization fineness as desired, as well as rescale the size of the web to a larger or small grid. When the user is done, the positions and connection pattern of the nodes can be used to create a finite element discretization of the network of strings.

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