Welcome to our, it's already the fifth lecture on the finite element method.

First of all I would like to recapitulate a little bit what we did last time.

Last time we started to derive the relations for the finite element framework

and we started with the finite element shape functions.

So this is the recapitulation number two here.

And first of all we provided the global formulation of the shape functions,

which is given here and here is also given a specific element

from node i to node i plus one.

So we have two shape functions here.

The first shape function is associated with node i

and the second shape function is here associated with node i plus one.

And they are piecewise linear functions

and they are characterized here by the Kronecker delta

and they are one at the node they are assigned to and vanish at all the other nodes.

And yeah, with these shape functions we can set up the approximation,

which we already considered in the Gyorgyan framework,

and we identified the coefficients of the linear combinations of the shape functions

as the nodal values of the displacement and of the test function.

So with that we proceeded with the weak form,

inserted this new approximation of test function and function u,

and we ended up here with this equation.

And the difference to the Gyorgyan framework is now that

since we do not require the shape functions to vanish at the Dirichlet boundary anymore,

we have to distinguish two cases namely,

this term here is zero for all i,

well the associated nodes are not part of the Dirichlet boundary,

and we have a term, a non-zero term, F reaction i,

if the node is at the Dirichlet boundary.

And accidentally one names this as the reaction force term

and indeed it will turn out to be the reaction force due to a Dirichlet boundary condition.

Yeah, this was the global point of view for the shape functions

and we changed this point of view by considering the shape functions now from a local viewpoint,

meaning that this is the global point of view and here on the right hand side this is the local point of view.

We introduce local node numbers here eta1 and eta2 of a specific element

and we introduce now only those shape functions or parts of the shape functions

which are inside the element domain and the right part of phi2 is here labeled as n21.

The superscript indicates the element and the subscript indicates the local node number

and the same happens for the shape function phi3 where we consider here the left part and this is labeled as n22.

Based on that we identified the entries of the stiffness matrix and the load vector still in global coordinates.

We discussed the assembly procedure which is straightforward to implement for a computer code

and then this was the last step we did.

We changed from global coordinates to local coordinates,

meaning that here on the left hand side we still have the global coordinates with the coordinate xe1 indicating the first local node of the element e

and xe2 indicating the second local node of the same element.

And here we change to a local coordinate system that is the same for all elements

and we have a variable xi running from minus one to one and we formulate the shape functions here based on this new variable xi

and this leads to this formulation of the two shape functions here.

Beyond the linear shape functions we can consider for instance quadratic shape functions.

They are given here and we will consider this in more detail in the tutorial.

Okay and the very last step is the isoparametric concept and the idea here is that we approximate the geometry in the same way as the solution and the test function