Springs and Assemblies of  Basic Ideas - ``123  Engineering Analysis

A Simple Problem - after JN

The essence of the finite element method is to take a complex problem whose solution may be difficult if not impossible to obtain, and decompose it into pieces upon each of which a simple approximation of the solution may be constructed, and then put the local approximate solutions together to obtain a global approximate solution.

We will first use a simple problem used by Reddy [] to illustrate the basic concept. Consider the problem of computing the perimeter of a circle of radius R.

   
Figure 1.1: Circle discretized into 5 ``elements'' and 5 ``nodes''
• Step 1: Finite Element Discretization

Discretize the circle into a ``mesh'' of 5 elements and 5 nodes. We will define elements and nodes formally later on, but from Fig 1.1 you can see that elements are simply pieces of the circle in this case and nodes are the points where they meet.

•  Step 2: Local Approximation of Solution

On each element let us attempt to compute the length. We assume that the length of each arc can be approximated by the length of the chord i.e. we approximate the arc using a straight line.



  
Figure 1.2: Local approximation of arc length on each ``element''

where  is the length of each segment,  is the angle subtended at the center by each element. Thus,  .

•  Step 3: Assemble the Element Equations

Collect the element equations to get a representation of the whole system.



• Step 4: Obtain the Solution

n=5. Thus,



• Step 5: Analyse the Solution

Let n=10 then



Let n=100



In the limiting case as 



The finite element solution converges to the true solution as the number of elements is increased.

Mon Mar 15 10:37:42 EST 1999