The network wave equation  (Page 5/11)

We will investigate this example further using a discretization of the network.

Finite element discretization of the network wave equation

To model behavior and structure of a continuous network, we discretize and solve our equations using the finite element method. For the most part, applying FEM to our network model is the same as applying it to a simple string - the hat functions overlap and form a basis for the structure of each leg. The exception is at a joint, which has a new type of hat function, with its support spanning a small section of each string connected at that joint.

The tritar example case

Let us write out the discretization for the example net in . If we take a uniform discretization of each string into $n_1,n_2$ , and $n_3$ pieces (with $N=n_1+n_2+n_3$ ), respectively, we can again derive a system of differential equations to describe the evolution of the coefficients $c_k\left(t\right)$ over time. Define the $N$ basis hat functions as being . Consider first the $k$ th hat function on string $i$ , where $k\ne n_1$ . We multiply each side of the network wave equation by the non-joint hat functions ${\phi }_k$ and integrate over the support of that function. After integration by parts, we have the relation

analagous to the one dimensional finite element discretization of a string. If we substitute in our approximation from the basis of hat functions

we arrive at the relation

Let $L$ be the number of connections in our web; $L=3$ for our tritar. Defining our inner products $〈·,,,·〉$ and $a\left(·,,,·\right)$ as

we see these inner products behave much like the simple string inner products on the topology our network. This gives the relation

The joint is a different case. Let us our joint hat function be ${\phi }_{n_1}\left(x\right)$ . Then, since integration by parts moves a derivative from one function to another with the addition of a boundary value, we get

after integrating over each string where the joint hat function is nonzero. If we recall that our force balance equation was

however, we can sum these equations together to achieve the relation

Conveniently, the force balance equation allows us to generalize this condition to joints with multiple legs as well. Next, substituting in $u_N={\sum }_{j=1}^N{c}_j\left(t\right){\phi }_j\left(x\right)$ , we get