Introduction to finite elements/Model finite element approximation

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Finite Element Formulation for Problem 2.[edit]

The domain for this problem is and the boundary consists of two points . Let us use nodes in the domain so that they divide the domain into nonoverlapping, two-noded elements.

Consider the term in equation (22). The integral can be written as a sum of integrals over each element as

In this equation, and are node numbers. Therefore there are possible values of .

Assume that node = 2 and node = 4. Then at node 2 and zero at all the other nodes. Similarly, at node 4 and zero at all other nodes. Also, is non-zero only between nodes 1, 2, and 3 while is nonzero only between nodes 3, 4, and 5. Since the domains of and do not overlap in this case, all the integrals must be zero.

In general, if and are separated by more than one node, at least one of the basis functions has a zero value within each integral. The same holds for the derivatives of the basis functions. Therefore if and are separated by more than one node.

Therefore, there are three non-trivial cases that need to be looked at

  1. .
  2. .
  3. .

For the first case, set in equation (29). That means

After substituting the values of the basis functions (25) and their derivatives (26) into equation (30) and integrating, we get

For the second case, set in equation (29). In this case, the only non-zero integrals in equation (29) are the ones between and . Hence

After substituting the basis functions and their derivatives into equation (31) and integrating, we get

For the third case, set in equation (29). In this case, the only non-zero integrals in equation (29) are the ones between and . Hence

After substituting the basis functions and their derivatives into equation (\ref{eq:Integralij+1}) and integrating, we get

The same process can be followed for the integral

Now that we know the components and , we can solve the system of equations (23) for the unknowns . This gives us our finite element solution for Problem 2.

Assembly process.[edit]

In the above we did not go through the assembly process that you are familiar with from introductory finite elements. We can simplify things if we use just compute the integrals over each element and assemble them to get the final and matrices.

To see how the assembly process works, let us recall equation (21)

We can rewrite this equation as

where

Let us define

Then the first of the equations in (34) can be written as

From equation (29) we can see that the integral over the entire domain can be written as a sum of integrals over the elements . Therefore, we can write equation (36) as

Let be the local basis functions in an element. Then equation (37) can be written as

Similarly, the second equation in (34) can be written as

where indicates that the integral is over the element.

Therefore, the matrix and the vector can be expressed as a sum over the elements in the form

This is the familiar assembly process. From this process it is clear that if we can find the weak form for one element, then the finite element system of equations for any combination of such elements can be computed by assembly.