Video Tutorial : How To Implement Finite Element Method 4

In the previous post I discussed the procedure for finding out the global stiffness matrix through local elements. In this post we are going to take this process further, develop the entire linear system, elaborate on finding out the force vector and finally solve the system by applying boundary conditions.

Linear System: Ku=F

We have already seen how to formulate the global stiffness matrix. This was one part of the entire linear system which is also called the left hand side. Traditionally, the unknowns are kept on the left hand side and the known values are put on the right hand side. In the animation which follows, the same right hand side has been discussed. This is nothing but the force vector. In all our animations, we have tried to keep things simple by assuming parameters to be constants. The same has been applied in this animation. The objective here is to build an intuition. Complex cases would be addressed later.

Let us now see this animation:

Solving the System: Boundary Conditions Application

Once the linear system is developed, the next step is to solve it (of course!). But this step also requires the application of boundary conditions. We will see in what follows that without boundary conditions, the linear system is not resolvable. The matrix is rank deficient and physically this means that the system is not constrained and so any external stimulus may cause it to undergo rigid body motion, however, it will certainly not deform.

Let us now look at the animation below to see how the boundary conditions are applied and how the linear system is resolved.

In the animation above, we also showed the stresses in the bar,

$$\sigma=E\frac{du}{dx}\ $$

Since we approximated the deformation by linear functions, therefore, the derivatives of those linear functions would be a constant. The stresses can still be recovered by joining the mid points of the constant slopes over all the elements.

Instructions for Viewing the Animations Full Screen

Here is how to play this animation full screen. Right click and save file as a gif file. If you do not already have storm codec Media Player Classic then download it using this link. Open the saved gif file with this media player and play and pause as you like. If you do not play this gif file with media player classic then you can not pause the frames and some frames might be too fast to read properly. This animation could also be played with Quicktime Player. You will find it here.

See you next post.