Welcome to our lecture, Introduction to the finite element method.

First of all, I have an announcement, because I'm not available on June the 4th to give a lecture and tutorial.

That's the reason why I'm asking you whether one of these dates given here would fit to your schedule.

Then I would give a lecture and tutorial in advance.

Of course, we also have the video recordings.

Maybe you can think about that. I will ask once again at the end of today's lecture and I will also ask in today's tutorial.

So possible options are Wednesday next week in our seminar room at the Institute of Applied Mechanics from 10.15 on.

Then another option would be Thursday next week from 8.15 in this lecture hall.

Or on Friday we have several options. Actually, it's the whole morning and there we could have a lecture and tutorial together.

And another option would be Wednesday, May 29th in our seminar room.

So we will need in any case two time slots here.

And I would very much appreciate if this shifting would be possible.

Of course, in turn, then on June the 4th there won't be any lecture and tutorial.

So please think about that. I will ask once again later about this.

Okay, then let me please briefly recapitulate what we did last time.

Last time we first considered quadrature rules and focused very much on the Gauss-Legendre quadrature,

which is the typical quadrature rule one uses within the finite element framework.

And after that we discussed how to consider boundary conditions and how to solve the system of equations

that results from the finite element discretization here.

We distinguished between Dirichlet and Neumann boundary conditions.

And then we counted the number of unknowns and we figured out that the number of unknowns is the same like the number of equations,

which is good news, but we have to apply a two-stage procedure because the stiffness matrix,

as it appears without incorporating the boundary conditions, is not invertible,

meaning that the system of equations cannot be solved.

And due to this we identified the first set of equations.

These are all the equations associated with the unknown displacements.

And here we can solve for the unknown displacements, here given in red.

And then a second step is to compute the still unknown reaction forces.

Here we consider now the equations associated with the prescript or given displacements,

so where we have Dirichlet boundary conditions, and this leads to this description here for the reaction forces.

Then after this we considered the case that we have both, we have Neumann and Dirichlet boundary conditions,

but this is in principle the same procedure.

So we always have to subdivide the system of equations into equations associated with the unknown displacements and associated with the known displacements.

Then we have to solve for the unknown displacements and in the second step we can solve for the reaction forces.

Today I would like to conclude the finite element treatment in one dimension

by means of discussing issues of post-processing.

And then we will consider a numerical example before we continue with the two-dimensional case.

Is there anything from your side that is still to be discussed?

If it's not the case, then let me get back to my whiteboard here.

And we discussed some issues concerning post-processing.

What do we actually do within the post-processing?

Exactly, this is the main purpose.

We compute stresses and strains, and then based on this we can assess whether we have any critical regions in our system

where we have to modify maybe the geometry due to stress concentrations and things like this.

The formula which we need for that we already know, but still I would like to summarize that here a little bit.

This is section 3.9.

This is post-processing.

So what we know from the processing step, we know the unknown displacements.

So it's UHU and the reaction forces, F reaction P.

Now we derive stresses and strains, or strains and stresses.