This week, we're entering a totally new realm, and that's the so-called finite element method. The finite element method is one of the oldest numerical approaches that has been used in engineering and science. It was primarily developed in the engineering community, structural engineering. Engineers that are interested in the movement and the forces related to buildings. Here's a very simple example, a drawing of a building made out of individual beams that are connected. Think of steel beams making up a very tall building. Now, how to understand the potential possible motions of such a building. The finite element method takes a very specific approach. Let's look at an individual beam of such a building, and single it out, and ask the question what are the forces inside such a beam. Then you need to connect this beam to the others. So, what are the neighboring beams in this situation, and we highlight them here in the sketch. That's actually called then the assembly. You basically build up everything with a description of the forces inside a single beam, connect them with others, called assembly, and mathematically that will lead to a large system, the linear system of equations that you need to solve. So, that's kind of the finite element method, and it's the first method that actually allows to solve problems with arbitrary geometric complexity, like a building might be very complex, or think of the vibrations of a wing, or think of wave propagation in media with very strong topography, like a volcano. So, the finite element method is very capable for such a media with complex geometry. Now, let's see how this can be developed from a mathematical point of view, and that's going to be the next steps. So, let's get into the math of the finite element method. Actually, the mathematical derivation of finite element method is quite more involved than some of the other methods that we encountered so far. Actually, there are whole volumes of books written. One of the major textbooks has 700 pages, just volume one, the introductory part. Of course, in this course we can only scratch at the surface of this. But let's do it step-by-step, starting with really the simplest case. Let's look again at one-dimensional elastic wave equation. Now, very familiar to you. So, we have the density multiplying the second time derivative of u, the displacement, equaling the, again that this first space derivative multiplying mu, multiplying the first space derivative of u, plus a fourth term. Now, we actually start with the static approximation, the static case in which case the system has no time dependence. So, what does that mean? The left-hand side, the time derivative of u is actually zero. So, we can replace it with a zero on the left-hand side, and we are actually left then with a so-called Poisson equation. So, for the moment we assume that the shear modulus is actually homogeneous constant in the domain, the physical domain we consider. So, we can take it out, and we're left actually with on the left-hand side, minus mu, multiplying the second space derivative of u, equaling a force. That's a Poisson equation as mentioned before, and actually that's the equation that we are going to solve first. But before we do that, let's have a look at a physics example of such a situation. Again we can look at a string instrument. Here, we're in the music department, and there's this beautiful harp. Now, let's again think about the problem of static elasticity first. So, we have a string here that has probably hopefully fairly homogeneous conditions. So, what I'm doing here, and hope the harpists aren't watching. I'm pulling this string quite heavily. So, that's actually the situation. This is like the force that I'm acting, that I'm applying here is like the f, and our Poisson equation on the right-hand side, and the displacement would be u. For example, if we know the shear modulus here, the shear properties basically, the shear modulus is the elasticity of that string, the problem then would be for a given force, we would calculate the displacement. There is also the inverse problem. If we know the displacement, we could derive the shear properties from this. But what happens if I let go, then we have the dynamic problem. So, back to the equation, vector maths. What's the solutions strategy for such an equation, and such a physical problem? So, we will take this step-by-step, and I would like to note again, and this is the practical aspect of this course. We're not going to argue a lot with very mathematical terms here. We're going to show you the steps how to develop a finite element algorithm, and there's lots of additional material that we'll find online to actually explain you a little bit more the mathematical background of the developments that we're going to do in the next couple of minutes.
