So, welcome back.

Welcome to our tutorial today.

We will discuss today exercise 7 and exercise 8.

These are already exercises dealing with the one-dimensional case.

And let us start with exercise 7.

Here we have to derive the shape functions for a bi-quadratic quadrilateral reference element.

I will just copy that to my OneNote document.

So, we have today the fourth tutorial already.

And maybe for those of you who attended the lecture, it is not that surprising what we have to do here.

How would you derive these shape functions for this bi-quadratic element?

Or did we do that in the bilinear case this morning?

In the bilinear case, we considered the one-dimensional case in both directions.

And we do that in the same manner here in the case of bi-quadratic elements.

We have here the one-dimensional representation as well.

And as the local coordinates, we had only xi, and now we have here xi1 and xi2.

What were the local node numbers in the one-dimensional case for this quadratic truss element?

One? Okay, one is fine.

And I put here eta hat equals one.

The second local node is located where?

It's the rightmost node.

And the third node? Okay, there is only one node remaining. It's in the middle.

And likewise, but here I use eta tilde.

We have it here in the second direction.

Okay, would you please look up and tell me the 1D shape functions for the quadratic truss element?

Pardon? Half?

Xi? Yes.

That's it.

And the third one?

Like this? Okay.

So, and now let us consider specific nodes here.

We do not derive the shape function for all nodes.

We start here with the first node located here.

What shape functions do we have to involve here?

Which one?

The first and?

And the other first?

Okay, one is for Xi1 and one is for Xi2.

Okay, so, and with that we get Ne1 of boldface Xi, which is then 1 by 4.

Xi1, Xi1 minus 1, Xi2, Xi2 minus 1.

And then it's a matter of taste how to write that.

One can maybe write this like this.

Then this is the first shape function.

Then let us have a look at this node, node 2.

What do we have to consider here as the shape functions?

The second and the first?

So here we have Ne2 boldface Xi.

What do we have then?

Which function gets the Xi1 and which one the Xi2?

So we are at this node.

This node gets which contribution from Xi1 and which contribution from Xi2?