Nonlinear finite elements/Model finite element approximation

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.