Welcome back. At this point we've completed our derivation of the finite element method for one d linear elliptic p d e's. And we've also spent a little time in understanding the basic mathematical properties of the finite element method. And especially with application to this problem. As well as how that pans out into studies of the consistency and conversions of the method and therefore to error analysis really, right. We looked at how the, the error of the finite element solution in the, the error in the finite element solution converges with element size tending to zero. [COUGH] What I'd like to do in this segment and maybe the next one or two, is look at an alternate way in which the weak form can be obtained. And this approach is a a subset of a type of calculus that may be sometimes called variational calculus or more broadly. Various known methods. Okay? So the topic of this segment is going to be we're, we're working towards variation of methods so uh,so let me start out with talking about. Variational methods, okay? Before we can get there, however, we, of course, have to motivate it, right? And in order to motivate let us consider the following, all right. Consider the following integral. Okay. Consider, let me call this, i func, i depending on u. Okay? Written in this form. And I'm going to write it as, I'm going to define it here. It is the integral, over, the domain of the following quantity. One half E A, u comma x. The whole squared. D x. All right? Now, when you stare at this integral, if you, remind yourself, of, one dimensional linearized elasticity, you will recognize E to be the modulus. Right? And you will recognize u comma x. Right? That quantity to be the strain. Okay? And then, when you stare at this integral, you would recognize it to be the, the strain energy, right? The strain energy in. Linearized elasticity. The strain energy in linearized elasticity of course in one d. Right. So, so this is all well and good. Now what one can do is, is the following. So we, we, we, we make this recognition first and then. The, you know, once, once we recognize that there is this notion of a strain energy, we can ask ourselves, well, what does it do for elasticity problems, right? Of course it gives us a notion of, of the, of the energy stored in the elasticity problem. But it also allows us to construct a different kind of quantity, okay? It allows us. To construct. The following integral. Okay. Now, now I'm going to start putting down the kind of notation that we need for the development ahead of us. All right? It allows us to construct the following in a, integral. Or to construct the following integral which I'm going to denote as pi. It also depends upon you. Right? And in the context of elasticity, this is the, well, this is just a displacement field. Right? U is the displacement field. And pi of u. I'm now going to define by a few terms, by using a few terms, the first is going to be the same, term, that we wrote above, the integral that we wrote above, the, strain energy. Okay? Now, recalling the other data of the strong form of the problem, all right? I am going to add on a few more integrals, I'm going to add on integral over omega f, which you recall is our distributed forcing. Right? Times u, multiplied by A for the area, d x minus I am also going to consider the traction that we apply in the context of a Dirichlet Neumann problem. Okay? So t A would be the force, right? The, that, that makes up the traction. That, multiplied by u at the position l. Okay? So I want us to consider this. And consider this integral where we're seeing as we did for the strong form of the problem that u belongs to the space s, right? Our usual space of functions for the for the exact solution, right? The, the space's function, functions from, from which we draw the exact solution, right? So u belongs to s, which now consists of, all functions, u, such that, we're thinking, we're thinking of this in the context of our Dirichlet problem. Right? So we're thinking of this in the context of the Dirichlet condition. U at zero equals u nought. Okay. All right. And f t and the constitutive relation that we're familiar with. Sigma equals E u comma x. Right, all of these are given. All right. Consider the following, consider the setting. Now what I want you to do is con is focus your attention upon this integral that I've defined that I've written as pi. And ask yourself what it is, right? What does it tell us physically at least for this problem if we're thinking of an elasticity problem. Right? Take a few seconds to look at it. Note that the first term was the strain energy. This term, I'll point out to you, the gives you the total work done by the force f on the displacement. And this recognizing that t A is also force. Is the work done by the traction specified for the Neumann boundary condition under displacement at that end. All right? So, in the context of elasticity, if you've studied that problem, this is something that you probably recognize as a potential energy. All right? More broadly, pi is pi of u. Is something we actually can call the Gibbs free energy. Okay? When we restrict our attention to mechanics only. Right? The Gibbs free ener, energy for purely mechanical systems. Or, purely mechanical problems. All right. Also called, like I said, earlier, the potential energy in the context of mechanics. It turns out, as many of you may know, that the Gibbs free energy is actually applicable to general processes and physics. It's typical to have a contribution from chemistry. Maybe from, from, from temperature and so on. Also in the Gibbs free energy. However if we restrict ourselves to only mechanical problems as we're are trying, as we're doing in this case, then pi of u would be recognized to be the Gibbs free energy restricted to that. Okay. All right, so that, that, that's something to note. Now I want you to also observe that as a matter of notation. I've written u in rectangular brackets there, right? What I'm trying to point out here is that there is the, the, the reason I'm writing u in in rectangular brackets is that I want you to think of this quantity pi. As not just a function. Okay? So I'm going to state that here. Pi of u is not a function. Okay? The reason it's not a function is because a function, properly defined, mathematically defined, takes on a, point value of it's argument. And returns to us another point value. Okay. Or more typically, a function is a mapping from, typically, real numbers into real numbers. Okay. Let me just, just, just state this. So, supposing we do indeed have a function. G of x, okay? This may typically be a mapping from the space of real numbers, right? From which you may take x. On to real numbers, right. And graphically this can be represented as follows. So on this axis I have x, I have g. And in order to get a particular value of g, you would go on to the horizontal axis, choose a particular value of x, and say, all right, what is the value of g? Maybe it is that one. You take another value of x, you pick your value of g, that's that. All right, maybe this is this, and. The process goes on. Right? Now the function that we have in mind is what we get by connecting all of these dots. Right? And maybe it does something else out here. Okay? Nevertheless, the important idea is that when we take a particular value of x. Right? We get back a, well, actually, what I've drawn is yeah, it's, it's okay. We get back a particular value of g. All right? Okay? So, in this sense, a function takes a point value of its argument and returns a point value. Or it takes, in this case, a real number and, and returns a real number. Pi, however, is a little different. Okay? And I want you to think for a few seconds, or maybe a little longer, in what manner pi, as defined in the previous slide, is different. Is it, in what manner is pi different from this function g that I've specified here. Right? So, how is pi of u different? You'd know the answer immediately if you've gone through this sort of exercise and you may still know the answer. You may have figured it out. But the response is that in order to evaluate pi, we need more than just a point value of u. Because pi is an integral, we actually need the entire field u in order to evaluate pi. And then, in fact, when we supply the entire field u defined on our domain omega, we get back a real number for pi. Okay? So, let me state that here. Pi of u is a mapping of a field, right? The field u, which is a, u is properly a function of x, right? Because you, you pick a particular val, value on your domain x, and you get a particular value for the displacement, right? You get back a point value for the displacement, right? You get back a real number for the displacement, right? So that's the nature u, u is a function. However pi is a mapping of that entire field u, okay? To, the real numbers, All right? And, why to real numbers? Because we have recognized pi to be an energy of sorts. It's a free energy or potential energy. Which after all is a scalar. Right? It's a number. Right? It gives us a real number. Mathematically stated, we have the following. Pi of u is a mapping from something that tells us where we draw u from. Where do we draw u from. Right? We draw it from our function space s. Okay? So prize and mapping s, to the real numbers. Okay?
