So welcome to our today's tutorial. Today we will consider exercise number nine, which

is a two-dimensional system with a unit thickness T and it is subjected to a heat flux Q as depicted

and the system is discretized by two linear triangular finite elements with a given connectivity

matrix. So you see here the red arrows, they indicate the heat flux and here the vectorial

description of the heat flux is given. As you see this heat flux extends over two elements

and the link between local and global coordinates is given by the connectivity matrix here on the

right hand side. Q0 is the only quantity that is given and we should compute the heat flow

vector per unit thickness due to the heat flux Q. This is the only task we have here, but yeah,

it's a little bit longer this exercise because we have to consider several issues that we

discussed during the last lectures. Okay, so the heat flow vector there we have to consider in the

recapitulations, the associated formulation we addressed this this morning. So let me just copy that.

So this is the heat flow vector and yeah, when we have a look at this there is only

one part that will play a role here namely the first part with a heat flux. We do not have any

distributed heat sources or see heat sources are so the second part does not appear here.

We have to do several steps to solve this question here or to answer this question.

And I would like to summarize them briefly as already discussed in the morning. We have to

start with the shape functions.

So somehow maybe we have to reopen it. So we start with the shape functions

and closely related to the shape functions we have the element representation. I put

that here in red because this is yet to be discussed. So

then we have to compute the Jacobian.

It's determinant and the inverse of the Jacobian.

Then we have to provide the link between and I put that here in red because this is also a

very important quantity. GEA which links the global sorry the reference and the physical

area elements. At least we will see that later we have to consider some of them.

Then

so we have to consider also Q bar in terms of the local coordinates. It is just given here in the

global coordinate system so we have to translate that into the local coordinates.

So we need QE of X and from that we have to derive QE of Xi and then we have to compute the

integral and eventually to get the vector which is fx for element one and for element two because

we have of course two elements and then we have to assemble the heat flow vector.

So these are the basic steps and we have to do them one after the other.

Let me first start with the element representation in the reference configuration.

So the triangular element we have the coordinates Xi 1 and Xi 2 and then we have a triangular

element and I would like to know from you what are the element nodes where are they located and

what are the associated shape functions. So first what are the local nodes where are they

placed. Let us start here in the corner which node is that?

Say it once again. Someone said that at least I meant to hear that. The third one yes. This one

is first. Okay and this is the second. Okay so and the shape functions which are they?

First shape function.

Which Xi? Xi 1. Second one. Two and third.

Let me go back to the question. Which parts of the question indicate that you indeed have to

use these shape functions? Exactly so this is important linear triangular elements.

This governs the number of element nodes and of course also the shape functions.

Okay let me furthermore copy the connectivity matrix.

This is the link between local and global coordinates and for the consideration of

this heat flux we need the surfaces of the element which are in the two-dimensional case

not the not areas but here edges and in addition we need also the surface normal vectors. Let me

just draw them. I do not specifically label them yet. I just sketch them here and of course they

would require specific labeling which I omit here because they are three different ones.