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P 1 u 1 x 1 ( 1 , t ) = P 2 u 2 x 2 ( 0 , t ) + P 3 u 3 x 3 ( 0 , t )

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.

Finite element discretization of a tritar, with a pyramidal hat function φ n 1 at the joint. r 1 , r 2 and r 3 denote the first, second, and third strings, respectively, which are discretized into n 1 , n 2 , and n 3 parts, respectively.

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 ( t ) over time. Define the N basis hat functions as being . Consider first the k th hat function on string i , where k n 1 . We multiply each side of the network wave equation by the non-joint hat functions φ k and integrate over the support of that function. After integration by parts, we have the relation

ρ i I 0 i 2 u i t 2 ( x i , t ) φ k ( x i ) d x i = - P i 0 i u i x i φ k x i d x i

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

u N = j = 1 N c j ( t ) φ j ( x )

we arrive at the relation

ρ i I j = 1 N 2 c j ( t ) t 2 0 i φ j ( x i ) φ k ( x i ) d x i = - P i j = 1 N c j ( t ) 0 i φ j x i φ k x i d x i

Let L be the number of connections in our web; L = 3 for our tritar. Defining our inner products · , · and a · , · as

u , v = i = 1 L 0 i u ( x i ) v ( x i ) d x i , a u , v = i = 1 L 0 i u ( x i ) x i v ( x i ) x i d x i

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

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

The joint is a different case. Let us our joint hat function be φ n 1 ( x ) . Then, since integration by parts moves a derivative from one function to another with the addition of a boundary value, we get

ρ 1 I 0 1 2 u 1 t 2 ( x 1 , t ) φ n 1 ( x 1 ) d x 1 = P 1 u 1 x 1 ( 1 , t ) - P 1 0 1 u 1 x 1 φ n 1 x 1 d x 1 ρ 2 I 0 2 2 u 2 t 2 ( x 2 , t ) φ n 1 ( x 2 ) d x 2 = - P 2 u 2 x 2 ( 0 , t ) - P 2 0 2 u 2 x 2 φ n 1 x 2 d x 2 ρ 3 I 0 3 2 u 3 t 2 ( x 3 , t ) φ n 1 ( x 3 ) d x 3 = - P 3 u 3 x 3 ( 0 , t ) - P 3 0 3 u 3 x 3 φ n 1 x 3 d x 3

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

P 1 u 1 x 1 ( 1 , t ) - P 2 u 2 x 2 ( 0 , t ) - P 3 u 3 x 3 ( 0 , t ) = 0 ,

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

i = 1 3 ρ i I 0 i 2 u i t 2 ( x i , t ) φ n 1 ( x i ) d x i = - i = 1 3 P i 0 i u i x i φ n 1 x i d x i

Conveniently, the force balance equation allows us to generalize this condition to joints with multiple legs as well. Next, substituting in u N = j = 1 N c j ( t ) φ j ( x ) , we get

i = 1 3 ρ i I j = 1 N 2 c j ( t ) t 2 0 i φ j ( x i ) φ n 1 ( x i ) d x i = - i = 1 3 P i j = 1 N c j ( t ) 0 i φ j x i φ n 1 x i d x i

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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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