Hello, and welcome to the series of lectures on finite element methods. What we will be doing here is developing the basic introductory finite element methods applied to a certain number of problems in physics. For those of you who are going to be taking the series of lectures as a MOOC from somewhere in cyberspace hopefully this experiment will be an interesting and more importantly useful one. For the others of you who also have access to real lectures in class, this should all serve as a as an enhancement to the in class experience. Okay, so let's get started and what I'm going to do is give you some introduction to what we will be doing during these lectures. The point here is to introduce you to the development of finite element methods, or the mathematical background to them. And to, how to code them up, right? And eventually, and, and then use those, use that code to solve problems. This series of lectures is not about a specific software, whether that it's a commercial software or something that's open source. We will be using something like that, but this is not the, the point of these lectures is not so much learn the software, as to learn the mathematics and the, the computation algorithms that are required to develop finite element methods. As far as background is concerned, I would expect that this should be accessible. These classes should be accessible to the advanced undergraduate student with an understanding of differential equations, but perhaps more importantly, a grasp of linear algebra. So, it's expected that you know what matrices and vectors are and how to multiply matrices and vectors, maybe compute inverses of them, and so on. With regard to differential equations we will use the terminology of differential equations right through this series of lectures, but again, you're not expected to know classical solution methods for differential equations. You're not expected to recall things like methods of characteristics or separation of variables, asymptotic methods or anything like that, okay? We will refer to differential equations. We will refer to some of the, some of the machinery that goes with them, but we will mostly be developing everything that we need. Let's see, and yes, this is going to be a series of lectures with units, and the units are, are, are, are, are all ready laid out, I will be filling them in with segments. And we'll see how it goes. We'll start right away now and one thing I want to point out is that we are not going to develop these lectures as addressed to specific problems only. And in order to do that, we have to go back and recall what the underlying differential equations are for each particular phenomenon that we want to address, okay? I will focus more upon the nature of the differential equation for which we're developing a particular finite element method. And along the way, wherever appropriate and perhaps also often for the development, I will say that while this particular set of methods that we are developing has a particular phenomenon as a canonical example. So we will often refer to elasticity, or linear elasticity in one dimension or multiple dimensions. We will refer to transport problems like the heat transport problem, and, and so on. And right, the other important thing to note is that because this is meant to be an introductory level class and to finite element methods, it will focus on linear problems only, 'kay? I will try to state that as often as possible, but occasionally I may forget to do so. But you, it will, it will be pretty clear that we're not looking at non-linear problems, okay? I think that's about it and we will just get started now. So, to begin, we are going to [SOUND] consider a particular differential equation, and this is the set of we, we, we're, we're going to start with things in 1D. And we're going to look at a type of differential equation that we will, that I will refer to as linear elliptical differential equations in one dimension. There are at least a couple of examples of phenomena that are governed by this particular differential equation. And, let me straightaway put those down, so that it gives us something to something more concrete to think about as we're developing these methods. Probably the most common one is 1D heat conduction. At steady state. So when we talk of the 1D heat conduction equation at steady state it is actually the same mathematical equation as one-dimensional diffusion at steady state. Okay, so this would be mass diffusion. So we also have, one-dimensional mass diffusion. At steady state. So, whe, when we talk of this, what we have in mind is the following. So, this is my little prop for a one-dimensional domain, all right? So, we're talking of how heat is conducted along this in this case, set of Lego blocks. Or, alternately, if, if this were a one-dimension domain and we were talking of mass transport or mass diffusion along this deme, domain, we may consider maybe introducing a drop of dye at one end and watch as it makes its way by diffusion through the bar or, or through the, through this one-dimensional domain, okay? So those are the two, the two types of problems that I've written down here. In addition, there is also the problem of one-dimensional elasticity, all right? And depending upon where you come from, you may think that is more canonical than either one-dimensional heat conduction or one-dimensional mass diffusion. So, in the case of one-dimensional elasticity, we make, we may look at this as a, as, as a, as representing a bar. And talk of maybe holding it at one end keeping the displacement fixed equal to zero at one end or at where I'm holding it with my right hand. And either apply a load at the other end or specify the displacement of the other end. Okay, and then we would have the problem of solving for the displacement field over the bar, okay? In both cases so whether, whether it's heat conduction or mass diffusion or, or, or elasticity there are other fields that we also need to talk about and which we will here as, as we start developing the, the material. So let me write down also one-dimensional, one-dimensional elasticity, also at steady state. Okay, and as I said just a few minutes ago, all these problems are, that we will consider will be linear problems. In fact, in the series of lectures, except perhaps at the very end, we are not truly going to consider non-linear problems at all. Okay let's actually dive into it and lay down our differential equation, and begin thinking about what it takes to solve it, okay? So. All right, so as, as I develop it just in order to have something to, to, to, to hold on to that's a little more concrete, let's just think of ma, maybe you want to think of one of these three problems in your in your mind, as, as we develop it. I, and, and I will try to refer to these problems as we develop the equations and see what each of these you know, as, at, see what the quantities would work out to in each of these cases. Okay, so let's let's start with this, and, and do the following. Okay, I'm going to develop this first by thinking of a problem in one dimension elasticity for my own purposes, 'kay? So let's suppose we have a bar, okay? And let's say that this is our x-direction, okay? At this end we have 0 and the bar extends up to the point L, okay? Let us suppose that because the way I've drawn it here you know immediately that at the left end I am seeing that the bar is fixed. All right, it's built into some sort of wall here. So, getting back to this, at, at, at well the right end for me, the left end for you, the, the, the bar is fixed, okay? So, so this, this, this what we mean by the support. At the right end, we may either specify the displacement, and I will denote that as U sub g, okay, for given displacement. Or we may do something else with it. We may specify the force at this end. And for various reasons that have to do with how we're going to write things out as we go deeper into this into the, into this series of lectures, I'm going to denote it as t. Okay, so what we have here is U sub g is the specified. Displacement. At x equals L, sorry, x equals capital L or t is the specified. We, we could call it the specified force. But, again, for reasons of generality, with what we're, we will develop as we go deeper into the series of lectures, I'm going to call it the specified traction, 'kay, at x equals L, okay? In addition, what we may also have is a distributed force over this, over this bar. And I will denote that force by just a series of arrows here. And I'm going to label that as f. So what we want to think about here is the foll, is, is, the following idea. As far as this distributed force is concerned we have our bar in one dimension. And what the force is doing is, effect, is, is, it's a force acting on every little volume or mass element of the, of the bar, okay? So, let me write that down as well. F is a distributed Body force as we often call it. Okay, all right, so, so this is the setting and I, I, I'm now going to state our problem in more mathematical terms.
