So welcome to our lecture today on a Friday. Thanks for coming. And first of all I would

like to recapitulate a little bit what we did last time. We started to discuss the two-dimensional

case for the finite element method and we did that by means of a stationary heat conduction example.

And here I put here the recommendation number three with the basic considerations. We have a

temperature distribution phi as a function of X. X is a position in the three-dimensional space and

the heat flux vector which is Q of X. Then we introduced the basis and position vectors,

the gradient of a scalar function, the divergence of a vector function and eventually the divergence

of a gradient which is the Laplace operator. And then we set up the model problem. So this

is based on balance of power and we ended up with this so-called strong form where we have

K as the heat conductivity, we have Laplace of the temperature plus the distributed heat source R.

In some this has to vanish and we introduce boundary conditions namely on the Dirichlet boundary the

temperature is given and on the Neumann boundary the heat flux is given. And in the next step we

derived the weak form of this problem by introducing the residuum which is in essence the left-hand

side of the strong form and by integration by parts and applying Gauss theory we ended up with

this formulation. And as in the one-dimensional case we introduced a test function. Here it's a

scalar test function because our variable theta, the temperature, which is our variable of interest,

is also a scalar function. And again we have to require that this formulation, this integral

formulation has to hold for all V of the test function space and we specified also the solution

space and the test function space. So this was more or less what we did last time and today I

would like to continue here but first of all are there any comments from your side, doubts which

we should address before we go on? Yes please. Yes. What do you mean by energy consumed or taken by the system?

Okay this would be an extension of our problem formulation. We did not consider this. We had

only the heat source, the distributed heat source, which could be also a heat sink if you put a minus

sign and the heat flux at the boundaries. So if you have any, as far as I understand your dissipative

effects in the material then you would have to use another material description. So this would be

even more complicated. Yes this is an ideal case.

So in essence you could also set up such a residuum.

Yes, yes, yes, yes. But then most likely you will have maybe a rate of a quantity. So in a

mechanical case typically when you have dissipation in, for instance viscoelasticity, then you would

have a term dependent on the strain which would be the elastic part and the term based on the

strain rate, so the velocity of the deformation which would then be the viscous part. Okay, any

further doubts? Okay, so then let's continue and as in the one-dimensional case now we have to specify

our domain and discretize the domain. This is the main issue now. So and I would like to do that with

the help of some figures from the lecture notes. So please apologize when I just copy things here.

So the next subsection we want to consider here is 4,3 which is the finite element approximation

and the first subsection would be the subdivision into finite elements.

And let us take from the lecture notes the sketch.

You might still remember this sketch because we addressed that in the one-dimensional case

and I'll come back to that in a second. So we have in gray the continuum domain omega which

could be 2D or 3D and then we introduce nodes and finite elements and here if this would be a

three-dimensional case we would just see the outermost faces of the elements which would be in

that case tetrahedrons or in a two-dimensional case this would be triangles. And as you see when

we connect linearly the nodes which is the typical thing when one introduces these finite elements

then the boundary of the domain does not fit anymore to the initial boundary of the

undiscretized problem. So we get a domain omega h which deviates from the initial domain omega.

In the one-dimensional case and this was the reason why we discussed this figure already in

the one-dimensional case. In the one-dimensional case we do not have this issue because there the

discrete domain does not deviate from the initial continuum domain. Here we have this difference and

here is highlighted a single finite element which we also label by omega e as before. And the entire

discretized domain is as before just a union of all the elements and again we assume that the