All right, welcome back. We'll continue with the delicate task of assembling our global matrix vector equations. From the specific version that we have of the weak form. And that version the weak form comes from about the code to last slide of the previous segment. You probably all have it before you in your notes and I have it here too, so we're going to start working on that. Okay, so, so we're going to write out the matrix vector weak form. In terms of. Global matrices and vectors. Okay. Okay, for completeness and, because there's just, just one more thing I wanted to say about it, let me just, rewrite the weak form that we have, okay? So, we have the following. We have C21 EA over h1 minus 1, 1. D11. D2,1 plus summation, E going from two to number of elements c1e, c2e, 'kay, here we have EA over he. 1, minus 1, minus 1, 1. d1e d2e equals c21fah1 over 2 Plus the sum, e going from 2 to number of elements. C1e, c2e. In the vector, so you're multiplying, the vector, forming a product, actually, with the vector. Eh, with, with components fahe over 2. Fahe over 2. And finally, we have this contribution from the boundary. C2. Nel times t times A. Okay. This is what we have. Now for again notational purposes, I want to state that in, from the conventional finite element literature, from the traditional finite element literature that matrix, including the factor in front of the actual matrix, is traditionally called the stiffness, the element stiffness matrix. Okay? We will denote it as K sub-e, or with K being a general Symbol use for stiffness. Okay, likewise we can call this sort of a force, this sort of vector the the force for vector for element e, okay, all right, so let me just state what these are called. This thing is called the element stiffness matrix. All right? And, that. It's called the element force factor. Now, of course, when we use terms like stiffness and force, this is clearly a a residual from the times when the finite element method was taught off chiefly as a technique for structural mechanics. Okay? So we will, but, but it's convention to continue to sometimes say element stiffness matrix even if one is doing a problem that has nothing to do with elasticity. They probably, it's, it's in the context of heat conduction it would be more appropriate in to call case sub E, the element conductivity matrix and FE as being the element of heating vector or something like that, okay? But, but this is just terminology that you may see around you in, in various spaces. Okay, so, so this is what we have to do. Now we're going to assemble then. And the process of assembly is the following. Okay? All right, this is called finite element assembly, right? So it's often referred to not just as assembly, but as finite element assembly. All right in order to do this, we recognize that because of the sort of mapping, we. Observed on the last slide of the previous segment. We can write these, degrees of freedom that are used to interpret the weight in function, okay? Into a single global vector. We're going write that single global vector as follows, right? And we're going to write it now using global numbering. Because this is what the mapping that we last looked at in the previous segment, this is what that mapping does for us, okay? So the relevant, global degrees of freedom for the waiting function are C2, C3, right, no commas here. C2, C3 so on up to C and E L, and C and E L plus 1. Okay here, I am going to write, way over here, the corresponding vector of global Solution values at the nodes. Right? Global nodal solution values. This would be the vector consisting of d1, d2, and these are global, okay? D3, all the way down to d n e L, and d n e l plus 1. What I've done here, with this vector on the left-hand side consisting of the Cs is collect the terms coming, like C 2 e, all the terms coming from C 1, and C 2 for each element. What I'm doing with this global vector is, collect the contributions coming from here, and there. Recognizing, however, that when we are writing global vectors, we use the global nodal numbering. Okay? Now, the contributions that then come from stiffness terms such as this, or the stiffness matrix K e can be obtained, or can be written, in a big matrix. Okay? Which I will sketch out, and then fill up. The way we do this now, is to observe that there are contributions coming, for instance. C 2 multiplies d 1, and d 2, because of the contributions coming from element 1, okay? All right? So, those would be the contributions E A over h 1, with a minus sign here, and E A over h 2. Sorry, over h 1, again. Okay? So that's, what we get from element 1. Let's go to element two, right? So, element 2 would have would have degrees of freedom, global degrees of freedom, C 2, and C 3, which would come from, C 1 with e equals 1, and C 2, sorry, C 1 with e equals 2, and C 2 with e equals 2, 'kay? Then did the local to global numbering, the global, local to global mapping of the greater freedom numbers would confirm for us. That the contributions from element 2, as far as the degrees of freedom of the waiting function are concerned, would be C 2, and C 3, okay? Now, those contributions come from the entries to this matrix, okay? Right? So, when we put those together, what we observe is that, we get here a, a, an, an additional contribution. And I realize, that I'm already running out of room here. So, let me just move this over a little. Just move this over to E A over h1 here. Okay? All right. The contributions then coming from element number 2, would be the following. We get minus E A by h 1 from element 1, plus E A over h 2, okay? Here we would get minus, E A over h 2. All right? Down here we would get E A over h 2, the minus sign, and here we would get, E A over h 2. Okay? All right? And then, we go onto element 3. Element 3 would have C degrees of freedom, C 3, and C 4. Okay? It would be this one, and another, and the other entry which comes just next to it. Okay? Those would map onto the matrix, to rows two, and three. Okay, as far as degrees of freedom, from the displacement are concerned. We would have contributions here coming from, hang on. Am I doing this right? Not quite. Okay. We're going to have to back up, a little. Okay? So, I'm just going to Okay. I'm going to erase this stuff. I'm going to go back to some point, and when Okay. All right. I'm, I'm, I'm going to go back even more. Sorry. Sorry guys. Okay. So we would, we, we would have to continue again from the point, where I wrote out assembly, okay? And then, went ahead. Okay. In order to do assembly, we need to assemble the local nodal degrees of freedom, into global degrees of freedom. And the way we do this is with recognizing, that we get here C 2, C 3, so on, up to C N e L, C N e L plus 1. We have for the degrees of freedom corresponding to the solution, we have contributions from the global numbering which are, d 1, d 2, d 3, and so on, up to d N e L, d N e L plus 1. So and here, we will write out a matrix in which we will con, collect the contributions coming from the stiffness matrix, right, from each element. Okay, let's look at what happens with element 1. Okay? Because element 1 has a single contribution as far as the C 2 degrees of freedom are concerned. Okay? It will fill out only the first row, of this matrix, right? As far as the solution degrees of freedom are concerned, it has contributions from d 1, and d 2. Okay? So, if we now, look at the line above us, what we see is that the contributions that come in, are going to be are going to be the following. And, and just for I'm going to pull E A out of this matrix. Okay? All right. So, E A is common for, for all the elements, right? And we, we are assuming that. Okay. So now, what that let's us do, is write out the contributions from element 1, as being minus 1 over h 1. And, one over h 1. Okay? So, minus 1 over h1 is is will, will multiply d1, 1 over h1 will multiply d2, okay? All right. Now, then we go into the contributions that come from that sum, okay? And remember, the very first element that makes a contribution from that sum is element 2, okay? Now, recalling our global, our local to global map, we recognized that contributions that come to that come from the c vector, right, in element 2 are from c2 and c3. This means, that we are going to fill out as far as c2 and c3 are concerned, we are going to be filling out the we're going to be, again, using rows 1 and 2 from this matrix, okay? But as far as, as the contributions from the d vector are concerned, the contributions of element 2 are in d2 and d3. So, this means that we are going to be getting contributions from columns 2 and 3. Okay, or co, contributions to columns 2 and 3 in this matrix, okay? So what we get here then, are we have 1 over h1 here, we get a contribution of the form 1 over h2. Over in the next column, we get minus 1 over h2, okay? In row 2 column 2, we get 1 over h2, with the minus sign, and row 2, column 3, we get 1 over h2. Okay? We go on then to element 3, element 3 has contributions coming from c3 and c4, which is just next to it. Okay? As far as the d degrees of freedom of concern, it has contributions coming from d3 and d4, which are next, okay? So, it is going to occupy it's going to provide contributions to our matrix in rows 2 and 3, and columns 3 and 4. Okay? In rows 2 and 3 the contributions are going to be of the form this, plus 1 over h3, right? So that's row 2, column 3. Row 2, column 4 is going to be minus 1 over h3, okay? We will have row 3, column 3, right? Which will be minus 1 over h3, and row 3 column 4 is going to be 1 over h3, okay? This process continues, all right? Continues all the way down, okay? Until we come to the very last element, the very last element will have a contribution coming from, sorry. The, the very last element, and, and actually, the last but one element are the ones who will contribute to the last two rows, and the last two columns of this matrix, okay? The last but one element is going to contribute a term on the diagonal, here, which will be 1 over h n e l minus 1, okay? All right, the very last element will contribute a term here, which will be this plus 1 over h n e l. I need a little more room, and I think I can get just a little more room by moving this over a little. Okay, so, here we will get a contribution minus 1 over h n e l. Here, we get a contribution, also from the element will be, which will be minus 1 over h n e l, and here, we get a contribution, which is 1 over h n e l, okay? So, these are all the contributions that come from the stiffness matrices, from the yeah from the individual element's stiffness matrices, okay? What I'm going to do is so write then, what I'm going to do is to write this out in sort of, operator notation. Now, what is often done here is, in, in the finite element literature is to say, that this is the result of assembling over all elements, e equals 1 to, n e l, okay? It's the process of assembling over all elements contributions of the form c1 e, c2 e times the stiffness matrix times d1 e, d2 e, okay? This is just an abstract way of writing it, and what is involved in this operator eh, in, in, this sort of an operation, is the detailed sort of book keeping that we followed in, by using the nodal and global degree of freedom numbering, right? In order to fill out this matrix.
