So welcome everyone to our afternoon session so to say of the finite element method and

first of all let me a little bit recapitulate what we did this morning. We based on the

consideration of the Gyorgin approach which we did last week, we continued with the finite element

derivation and we remember that the shape functions in case of the Gyorgin framework

we considered last week, they had to vanish at the theoretical boundary and had to be non-zero in

the entire domain and could be somehow arbitrary otherwise. And now our idea was to choose simpler

shape functions and to this end we first subdivided the domain into these finite elements. We

introduced a global point of view and a local point of view with a global node numbering,

you're given in pink, and the local node numbering given here in blue and we approximated the entire

domain by the domain omega H and yeah this is the union of the individual finite elements. Then we

introduced shape functions which are very simple because they are just piecewise linear functions

and first we had still this global point of view and we introduced the shape function associated

for instance with a node i which is phi i and likewise for all the other nodes and we identified

that the coefficients we used last time in the Gyorgin framework now become the nodal values both

of the displacement and of the test function or in a more general way here not the displacement

but the unknown function u. So then we did some term manipulations, we substituted the integration

over the entire interval by the integration over the finite elements and furthermore we discussed

this issue that the shape functions do not necessarily vanish anymore at the boundary

and due to this we have to distinguish between nodes which are part of the Dirichlet boundary

here gamma d or not. So for nodes which are not part of the Dirichlet boundary here this equation

is or this term on the left hand side is still zero and for all the other nodes we have to introduce

this term f reaction which is or which will later turn out to be the reaction force due to a given

Dirichlet boundary condition in a mechanical case. Yeah here is the substitution by the integration

over the elements which I just mentioned and then we considered an example to see what this actually

means when the shape functions vanish in large parts of the domain and are only nonzero in a

very very small part namely here in the one-dimensional case into two adjacent elements.

Yeah and by means of this example we derived that yeah rather than taking the sum over all elements

we just have to take the sum over the elements where the shape functions do not vanish and finally

we introduced this local point of view for the shape functions meaning that yeah we substitute

the global shape functions which have these hat like appearance by local shape functions which

of course render the same values in each element but are here defined element wise. Yeah and here

is the specification of these local shape functions and as far as the labeling is concerned here

we have the local shape function with a superscript E and indicating the element number and the

subscript eta indicating the local node and for the coordinate of a node we use this labeling

scheme the element number is the first superscript and the local node number is the second superscript

this is maybe more like bookkeeping and you will find various ways when you have a look at the

textbooks but for me it has turned out to be a suitable way to label this. Okay are there any

comments from your side or doubts before we continue? No? Should we wait 20 seconds or

should we just continue? Okay maybe some questions will arrive later will arise later on so then let

us continue here so we now have this local point of view for the shape functions and yeah maybe

later we will need this here please take these and we continue with our little example

so

Oh

Okay.

Oh, thank you.

Okay, now let us come back to our equations here.

Or maybe I take them better from here.

So we have this here.

As we discussed in the morning.

And now we have this new point of view for the shape functions.