So welcome to our today's lecture on the finite element method. First of all let me

a little bit summarize what we did last time and this I will do with help of

the notes of last time. If you remember we provided the element wise approximation

of the derivatives of the solution and the test function. This is done in the

similar way like the approximation of the geometry and furthermore we figured

out that we have a derivative of the shape function which is a function of

the local coordinate with respect to the global coordinates and here comes the

derivative due to the chain rule of the local coordinates with respect to the

global coordinates in which is here just the inverse of the Jacobian and

eventually we ended up with this approximation of the derivative of the

solution and same for the derivative of the test function. The similarity with

the approximation of the geometry is obvious here. We started here with this

approximation of the solution and of the test function and these are the same

like the approximation of the geometry. Then we inserted that into the

formulation for an entry for the stiffness matrix and we figured out

that we can have the same format for all elements except the inverse of the

Jacobian brings in the relation between the local and global coordinates

which is in general different for each element. Then we did some examples to

demonstrate that and after that we had a look to the entries of the force vector

which works similarly and when it comes to the force vector we have here the

distributed load either the G or the N in and to provide here a formulation local

coordinates used again the same approximation like for the geometry for

the solution and for the test function. Then we also did some computations we

will come back to that in today's tutorial there we will derive the

stiffness matrix entries for a quadratic element. So after that we considered

numerical integration and we focused first on a very simple approach the

rectangle method then the Newton-Cotes quadrature where we have certain

combinations of quadrature points positions and weights and we also

derived or actually we discussed what polynomials of which degree can be

integrated exactly by using the Newton-Cotes integration and here is once

again given the rule for that. Today I would like to continue here and come

for a short time back to the Newton-Cotes quadrature but first the

question to you is there anything we should discuss of last time? Did you

figure out any issues? So in an examination you could be asked for let's

say please integrate this polynomial using Newton-Cotes or Gauss quadrature so

it is relevant. So when it comes to Newton-Cotes we have actually we have

the same setup for all these quadrature rules and here it is specifically given

for Newton-Cotes because here we have the number of Newton-Cotes points and we

always have a combination. First we have to have this interval from minus

one to one. The finite element framework this is given because we always have

this interval chosen for the local coordinates. So and then we have the

summation over the Newton-Cotes quadrature points and we have to evaluate

our function which we want to integrate so the argument of our integration. This

we have to evaluate at the specific quadrature point and multiply that with

the associated weight. So for instance when we choose one quadrature point then

we have to evaluate f at zero with a weight of two. When we have two points

then we have to evaluate the function at minus one and one with a weight of

one each. So and then you can have more and more quadrature points. Yes, yes.

So that should answer your question. Okay so then let us come back today briefly to this choice

Newton-Cotes with two quadrature points.