So

good morning everyone. Welcome to our today's lecture on the finite element method. And first

of all I would like to recapitulate what we did last time. Last time we focused on the approximation

of the finite element method in two dimensions and we discussed triangular elements, bilinear

quadrilateral elements, we approximated the geometry. This was already two weeks ago and last time we

discussed a little bit more in detail the approximation of the solution, the approximation of the test

function and the approximation of the gradients. What we need for this is first a relation between

line elements which is done by the Jacobian and the relation between volume elements which we need

for the finite element treatment of the integrals. Here the volume element in the physical

representation is linked to the volume element in the reference representation by the determinant

of the Jacobian and today we will also focus on the relation between area elements. The approximation

of the solution test function and of the gradients is done as follows. First the approximation of the

temperature and of the test function. They are done the same way. We can introduce this vector of shape

functions lowercase n and multiply that by the vector of nodal temperatures or the vector of nodal

values of the test function and we get this formulation or this formulation. The next step we

approximated the gradient of the temperature which in essence means we have to take the derivatives

of the shape functions with respect to the global coordinates because the gradient is a physical

quantity defined in the global coordinates and we can introduce a matrix called B where we subsume

all the derivatives of the shape functions with respect to the global coordinates. Of course we

have to consider here that the shape functions are given in local coordinates but we take the

derivative with respect to the global coordinates and this link is done via the entries of the

inverse of the Jacobian. Okay this was more or less what we did last time before we continue. My

question is if you have any doubts or remarks something to be discussed.

If this is not the case then let us continue here and to this end I take a screenshot here.

So this is the definition of the gradient

and in the lecture notes we have an example where we can see how this actually is done.

So let me go there.

So here we go.

So we have given a triangular element. We know that example already. Now we continue here with

the temperature gradient which is to be computed and we have given the nodal coordinates x1,

xe1, xe2, xe3 and the nodal temperature and we have to compute the temperature gradient.

And before we start I would like to put together the individual steps because if you are asked in

an examination perhaps I don't know whether this could happen to compute the gradient then you

should know what are the individual steps without being asked explicitly for the individual steps.

So this means we have to compute the gradient of the temperature in this specific element and my

question is what steps are needed. So and we should put that together now together.

What is the first step? What would you do right at the beginning?

If you are not sure whether this is indeed the first step you can say any step and we will put that into the correct order later on.

We need the shape functions and in addition to the shape functions what do we need?

So with n you mean the vector of shape functions.

Okay so the shape functions are needed.

Yes, yes.

So I put that here so let's say two we need the shape functions.

What else?

Yes, so I put here the shape functions here the shape function vector this is a function of Xi.

Yes, so the conversion from global to local coordinates or the other way around.

How do we or which quantity governs this link between global and local coordinates?

There is a specific quantity the Jacobian yes so we have to compute the Jacobian.

When we compute the Jacobian then of course we also need to compute.

Yes, so shape functions and their derivatives.