Lec 20 MIT 18.085 Computational Science and Engineering I

should we have a go on lecture 20 I hate to interrupt all these great conversations uh I promise to stop after an hour uh um so this is lecture 20 which is where the main theme is the finite element method but I have uh continuation completion to to to do of lecture 19 and the you'll see the link right away uh when finite elements come I just indicated up here section 36 is where this calculus of variations is summarized and then finite elements you'll easily identify that section further along uh but this was where we were so at the end of lecture 19 we're looking for the best U of x to make this quantity small and we identified the key principle is move away a little from from U of X move away by some probably I I'm thinking of v ofx as a small perturbation so V ofx I'm thinking of as a small function in any direction positive negative and our quantity would become this but this is supposed if if U is the winner any V is supposed to give us something bigger and now what do I do just the way in calculus it was always a subtraction so I so I look to see okay what's the difference between this one and this one and you see because practically I mean that term is copied here and this term is copied here so and that term was copied there so what's left what's the V ofx term well there's the 1/2 C ofx X DX dvdx with the two because we square that sum so the 1/2 cancels the two and leaves us with that so that's the that's the linear term involving dvdx and then of course there's the v ofx f ofx term and then finally you could say well what about the 1/2 dvdx and that's the quadratic term okay and that's supposed to be if you is the winner then for any V this is positive you see that this equation I'm writing down is the same as the equation in calculus would be 0 is less than or equal to F ATX + Delta x minus F ofx that's that's what's identifying the the best best guy and it will lead us to the equation it calculus it led us to the equation DF DX equals z at at at at the winner now coming back to here what can I conclude for any V ofx this stuff has to be positive and out of that I trying to produce an equation okay well this has has v ofx fine that has the derivative of V ofx so any suggestion how can I how can I deal with that I I would rather see V ofx there that would be beautiful because then I have a v ofx in this term and I'd have a v of X in that term excellent yes right so I want to yes so what's the opposite of a integrate by Parts exactly exactly that's what I want to do I want to integrate by part so this term I want to replace this by now what happens so I'm going to I'll separate it into these are my two parts right I integrate by parts that always produces a minus sign and it takes the derivative off of V so it leaves V alone and towards the derivative onto this so it's it's the derivative of that right and then you could say wait a minute there's some boundary terms in integrating by Parts there's a boundary term there's a there's a c of C of x uh Dux V ofx between between zero and one that I should watch for okay I I admit that all right but let me just be sure let me let me deal with that later was that okay for integration by parts and then now I've got a v of X now I can combine that with this V ofx and and and put them together so that all that stuff is multiplying V ofx and integrated and then this is the boundary term that I'm fudging on for the moment okay now what do you say I'm damn near done that's right because what this quantity is supposed to be always positive so what's my conclusion it's it's supposed to be this thing is this thing is never negative no matter what of X I choose it has to be zero it has to be zero it has to be zero that's the conclusion I want that that's the key that's the fundamental limma of the calculus of variations when you get this expression that that has that has that can never be negative only positive or zero then since V ofx has sign either sign the only way you know it if this is ever nonzero we can give V of X which we're free to choose we can give V of x the sign the same sign or the opposite sign we can choose the sign of v ofx and and get a contradiction so that's the conclusion this has to be zero and that this quantity this integral this integral oh no this Quant sorry not the integral it's the it has to be zero at every point because we could choose V of X that's the this is the dpd that we were looking for only I'm using it's the first variation rather than the first derivative we just use the word variation as a reminder that uh we're dealing with functions we're in continuous variables so the big moment has just is just there okay there is good what happened to this guy yes no you're absolutely right so there there are two points that I have not dealt with one is that one so let's deal with that one okay now how to deal with that the reasoning is always the same that this thing we the reasoning is that this thing has to be greater or equal Z for all V and out of that we want to conclude that this thing has to be zero that's this fundamental fact now how what's our how's what's our thinking going to be why why can we forget this which well yeah but it it it's always positive that that's right but we we this business how do we how do we deal with it well you probably say I don't want to hear any more about calculus but uh how did we how did we deal with the difference between these was was Delta X fime of X Plus then there were really more terms weren't there there was Delta x^ 2ar over 2 fpre Prime that's that's the kind of term we're looking at here but how why was it that that this thing had to be zero and that that I didn't have to worry about this stuff let me give let me answer the question take V since V is arbitrary take it very small take V to be whatever sign we want it to have but quite make its magnitude really small then what that's so that's the good idea there think of v as just a little change then if this wasn't zero we would get something right we'd get something in this part and we'd get something here but it would be second order so so it the sign of this would control there's a subtle not deep but but there you ask exactly the right question and the answer is that because this has a v squared in it when I think of v as being just a small peration this guy can't doesn't affect our reasoning it it it's it it's true that it could be positive but positive and so small compared to the other one that it's the other one that's that's that's uh in charge sorry yeah I that's true maybe I want to take V's smooth as well as small the the beauty is I'm allowed any V and and so that's what the limit says when you get that much Freedom the only way out is for the thing that multiplies V to be zero yeah that's that's right that's a you're making a good point that well I could V could be small but but I mean a mathematician might say replace V by Epsilon time V so let replace V by Epsilon V and then this would be an would have an Epsilon squared in it do you want to see any of that no no I sorry oh right oh yeah but is C negative yeah no I I built that in way back at the original problem C was positive just the way our Matrix Capital C was was had to be a symmetric positive definite diagonal thing yeah yeah yeah otherwise you know OHS law would go backwards and the world would blow up okay right um yeah yeah but that's you're you're absolutely right that that that if C in other words let me let me say because you're quite right if C of X is not positive then we don't have a minimum we've got a saddle point we've got weird stuff yeah yeah yeah so C is definitely positive here okay and then this H how about this fudge this I shouldn't have used that word on tape how about this point uh well um what's going to deal with the boundary terms the boundary conditions I I intentionally didn't mention those but of course this problem comes with boundary conditions it comes it might come with the boundary condition U zero at both ends and then what would what would happen happen then the V would have to be zero at both ends the the admissive the the functions that are allowed in the competition have to be zero at both ends and then that boundary term goes away I let me just not because that because the boundary part could take could extend this part of the lecture let me just say that it's the boundary conditions that as we saw in in in in the A and A transpose discussion a month ago it's the boundary terms when we integrate by Parts there's boundary terms and it's boundary conditions that deal with them so they knock this part out this part is small because it's Epsilon squared and this is the conclusion then this has to be zero and this is called the oiler equation okay actually I I can even say one more thing before we before we got to the conclusion that this had to be zero we were at the step where this integral with the V of X in it has to be zero for any V satisfying the boundary conditions can I can I do a distinction here between three statements of the problem one is is the minimum statement minimize a quadratic over a whole bunch of use the second is this this is I'm going to call this the weak form of the equation weak because it's still got the V hanging in there that the integral of this quantity times any V is zero is the weak form and then I'll call this Oiler equation I'll call this the strong form where V is now out of the picture and this quantity and the integral is not g is gone we're just at every point this has to be zero do do you see a a distinction let me I'm going to move pretty quickly now to finite elements and gurk principle and guren's principle is so basic and it's built on this weak form that's why I wanted to to pause uh and mention this okay let me just step back a second I think I've probably said everything I want to about the calculus of variations only I've only done one example but that example does really exemplify the typical situation a typical so here's here's c a calculus of variations minimize some integral of some function of U of X Dux maybe X itself DX now would be there is a problem a more General problem in the calculus of variation so it minimize any function of of a function you see where we have a function of a function the unknown is is an unknown function and it comes in its derivative comes in its second derivative might come in the into the problem it partial derivatives might come in if we if it was a function of several variables so that's what the calculus of variations is is about how do you minimize functionals and and the way what happens is you go to you move U by Little V you look at the linearized term it has to be zero and that gives you the oiler equation so that this is the this is the more general not necessarily quadratic like if you wanted to find the shortest distance between two points you could use the calc variations to show sure enough it's a straight line yes right that's right that's true when you when I move it by a little bit the difference has to be greater or equal zero but then what's that next crucial step that when I look at the linearized part the part that just multiplies V alone that has to be zero yeah and that's the the tricky point the that's the tricky point that that that this higher order stuff can be forgotten because because because I look at very small V's yeah okay that's so uh so I'm just describing the whole calculus of variations here in this tiny corner of the board the main point is that it's the general case of which this is the best example okay are we ready for finite elements yes and I'll work with that same example so what's the finite element method the finite element method makes this problem finite dimensional by not allowing every function U and every Mo movement V but only a finite family of U's and V's so that's the that's the key idea of finite elements and and that's the big decision is what what uh so now I'm ready for finite elements so here come finite elements or the gurin method in general gurin takes we want to get a finite problem and how do I do it I only allow these U's which are combinations of with unknown coefficients I of some basis functions n of them so I only I only minimize you see I'm I'm sort of down to this finite dimensional space instead of minimizing over every U which led me to this differential equation for the absolute winner I'm now going to take a space winner a limited winner by choosing just a few functions or maybe a 100 functions but not all functions and I'll only allow those winners so now these are chosen we hope chosen we hope well that's the big big decision it's it's so much of Applied mass is okay what are the good trial functions these are the trial functions because those are the ones we're going to try and other functions if if the if the absolute winner has they could be like sinx sin 2x sin 3x up to s 100x that would give us a 100 function and then this would be sort of like a fora sign series of the winner but if I choose that if I make that is that a good choice to take those functions sinx sin 2x up to sin 100x what I'm going to do then I'm going to plug this into the problem right I'm going to plug that into the minimum thing and minimize but minimize I've only got these are my unknowns now the weights are the unknowns and unknowns so I've got 100 unknowns I'm not too crazy about that choice of sinx sin 2x up to sin 100x why not because the calculations would take a long time I'd have to be able to do to compute this integral for functions like sin 100x and then minimize over this 100 dimensional space and the idea of finite elements was a better choice instead of using signs and cosiness what's a what's a better choice of trial functions and exponentials you could say okay plug those guys in but still you have these integrals to do uh so when gurkin suggested this idea well the computer of course didn't exist so n n was three at the most probably two right in other words so gurin would have spent a lot of time trying to get because he's only choosing two functions he would try to get two functions that were pretty close to the to the real winner of course the real winner is unknown but he would he would work hard to try to find functions that were close to it if we if we deal with a 4A series and we take a hundred or a thousand terms well we don't have to worry are we have we got a good approximation we can expect yes we probably have if if our if we've taking a thousand for8 terms and we find the best thousand weights we can probably come darn close to the true U the but my complaint about the forier choice was the cost of computation all right so let me let's invent the finite element method here uh give me suggest some other functions sorry pooms okay so that I could take the functions 1 x x^2 x Cub up to X 99th that would be then my space would be all the polinomial of degree 99 that 100 dimensional space I could minimize okay you're going to okay so with with that polinomial what do you think is is that a is that a good choice I think not the best okay so you have another so you're going to take functions yeah now you're going in the right direction okay so you're going to take functions that is this the kind of function you're putting in okay so this is what Walsh thought of so this is the Walsh's functions sorry that somebody we'll we'll add your name also Walsh BL all right okay all right but yeah this you're you're absolutely right can I am I allowed that function the derivative is a problem so you're you you cut me out actually I said any function but I I really overdid it because that because the step functions have the derivative is a Delta and here we're squaring the darn thing and we would get infinite energy so I really should have required the function that I need the thing to stay finite so now you're you're take one more step and you've invented finite elements yeah so instead of peie wise constants which have a jump and the derivative is a is a Delta which which blows us up what shall I take now Peace by linear Peace by linear so now switch to linear finite elements they would be Called Peace wise linear linear pieces linear elements that's where that word elements is coming in okay so my my functions are are the functions I'm allowing then are are any PE wise linear like so and now everybody sees that we've made our life much easier as far as calculating these integrals as we got pretty darn simple functions and we can integrate a piece at a time an element at a time and in that element the thing is actually linear so so I can do these integrals that's the beauty of it so peie wise linear is good now let me ask you what's the basis What in in this language what would the what's the base what's you've given me all the functions at once there all piece suppose the boundary conditions are U of 0 equals 0 0 and U of 1 equals 0 okay and suppose I chop the thing into one two three four five elements so that I had really so I really have four degrees of freedom so n is four here but now tell me the four basis functions that you want me to use yeah okay that's what ped it he made them look like Signs and cosiness so so maybe one of his functions would have gone up and then another one would have had an oscillation another one had two I'm looking for another option another basis for another set of four fees whose combinations give those sorry I I'd be happy if they were but I'm more that's good but um the most important is that the calculation should be fast so I'm looking make them as simple as you can and and the and your choice of making them look like Signs and cosiness was pretty simple but let's make it okay that's possible that's possible well but yeah okay I could yeah I'm I'm GNA let's look some more because this is exactly what what the any any more suggest interval by interval by interval piece by piece that's the winning idea that that here's one of our functions and then and here's a second one and they well they they didn't yeah sorry I better yeah so they do overlap and I don't get that orthogonality but I I just I prepared to live without it for the for this local to be local is is is is to be up to date I mean that that's really local stuff that's what you'll see with wavelets too that they're a local basis so there's one guy the second guy Peaks at Point 2 the third guy Peaks at three the fourth guy Peaks at point four and their combinations give anything let me no I hope not let's see what uh it's a it's a shift it's a shift it's true a shift is okay but if I If I multiply that by two it would just make it bigger yeah if you're all this is the right stuff to think about exactly the right stuff to think about if I if I combine this with this then I can get anybody who does this and then stops but then I've got number three number four they'll pick up somehow okay that's one way to see that these four guys are independent that in fact the the C's actually have a meaning the the C the C1 the one that multiplies this guy is actually the value there this that's the C1 and this is the C2 that height and and this is the C3 happen to be negative and this is the C4 do you see that that if I if I take those things so the these weights are actually Point values which is kind of nice so all I do now well yeah what I want to do now then is I I'm limiting myself to this four-dimensional space of of simple functions and these would be peace wise these would be linear finite elements and now where is my equation going to come from that determines these C's right now we've made the big decision of the basis and and those other bases you mentioned I mean like you know there there those other bases are not did it's just finite elements is that choice well finite elements don't have to be linear we could we could imagine Parabola little pieces of parabolas or little pieces of cubics so those will be quadratic elements cubic elements uh those are all it's highly interesting I mean there was a like a golden age of finite elements when everybody was creating new bases and of course everybody hoped that his basis would be you know his name would be like Immortal uh and a few people probably well PE by linear somehow I guess nobody's name got associated with it and that's that's uh a very important choice just because it's so simple okay come back to that question where are my equations for these four C's how do I find the best C's I I you see i' I've decided okay I'm only going to allow functions that are combinations of these four C's that look look like that now what's the best one how do I get the best one sorry I need a cost function okay so what's my cost function I gu it's this guy okay so so one way to do it is plug this in there and minimize it it's a function of four variables I'm back to the beginning of the last lecture n is four you imagine doing that I've decided what these functions are so those are are fixed those are my trial function those are the ones I'm trying with what I don't know is the four coefficients so I plug it in to the thing that I'm trying to minimize and I minimize but I've only got four variables so i' I'm there okay that's that's the ra Ritz that's associated with the names of RA and Ritz gurin had another idea gurin worked with this weak form gurin said okay now I'm looking at this this is this whole thing in yellow let me put the zero in yellow too so that we're seeing the whole weak form in yellow okay so what and the integral is there um so now what's goin's idea going to be I I've got I've got this U is now a combination of from 1 to four of c i fi of X and I've made a smart choice of these fees and smart means that I can do the calculations quickly oh hold on a minute hold on a minute if I put the peace wise linear functions into this form what is going to happen problems are going to H arise right because the derivative of peie wise linear is is PE wise constant and here I've got another derivative yeah that's right so what should I have done I I I just screwed up but but in one line by changing that Arrow I can make myself correct let me get this term that was the so I'm going to put equals z here okay you see what I want this is a weak form that's the weak form with before the integration by parts and now I could take U to be my piecewise linear and the derivative would be PE wise constant that's fine and what about the V what shall I take for V's well I can I I could take the same things for V's or I could take some other choice for V's but but this is so I'm sorry that that's in yellow and this is in white because this is the one that I should have been pointing to that's the form before I integrated by parts and and uh impose this additional derivative on you this was much better so well I think probably the simplest idea is let this has to be true so U is a combination of these now like of 1 to four of these CI F right and what shall I take for v i I'm looking for four equations so so I'm just going to take four v's four test functions yeah and let them be each of these right it's true for every V so I pick any four v's and one Natural Choice in this sort of symmetric problem is let the test functions be the trial function but I don't have to make that choice so can I summarize this guran idea he's got n he chooses n trial functions fi and he chooses n test functions say s well if I don't know how to say that word but J X and those could be the same very often they are the same but they needn't be the same and then his equations are this weak he takes the weak form where he takes V to be in turn one of these size so this is for J = 1 2 up to n we got to n equations and N unknowns that was our job we've got to n equations in N unknowns what are the unknowns they're these weights what are the equations they're the weak form for each uh test function so we're testing it end times and this this is going to that's four equations and four unknowns that's K C equals F somehow I let me you see this C is my four unknowns my Vector of four unknown my K is somehow a matrix that's going to come out of these Integrations my f is somehow the M the the vector that's come going to come out of integrating the linear terms and everything depends on the choice of the fees and the test the trial functions and the test function but do you see that is a pretty reasonable principle we we can't gurin says you've got to make a finite problem and the way to make it finite is choose some functions in advance and look for the best combination then then you have only n unknowns and then how does he get n equations he tests the the the equal zero condition against n test functions and those test functions could be identical to the trial function that that's guren's idea and the result is um a set of linear equations okay let me let me now say okay let's step back all right so suppose we're going to write a finite element code can we let's not actually write it but imagine we're writing one what what would be the subprograms what would be the subot the the the pieces of a of a finite element code maybe we could list those pieces if if we were writing a code what would we uh and you know that the finite element that there are giant finite element codes that have had hundreds of of people years in creating them nastran is one name you may know a Dena is created by a guy at MIT and a and a generation of graduate students um and there there are 10 maybe five to 10 really big big time codes so what do they have well one step is going to be choose the choose the fees and and maybe and and seeds there'll be a library of choices and the and the first guy in every library is this one it would be fun to tell you about some of the other possible fees but but if but all right so we choose these fees now what what what do we have to do what's the next job we've made this choice we know what where our equations are but we've got to turn them into a linear system so the second job is assemble F assemble the Matrix K and the vector and the vector F maybe I'll yeah and the vector F in other words plug these things into the integrals and do the integration that's the expensive part of the finite element code and that's that's why we chose simple functions to keep the cost down we have to assemble these matrices by doing the integrals then thirdly we solve K Cal F that's if we've taken 10,000 elements or 100 or a million elements that might take some time but but uh it's under control and then finally would be uh display uh this the our our best our our uh our approximate U our our approximate answer uh this combination of of trial functions is yes that's that's right it's nothing to do with that c ofx yeah would I be better I would be better to call these unknowns UI yeah that would actually be a lot better should I could I it would it would be a lot better to to be to be uh for the UI to be the the weights of the fees why is that better because finite elements actually has this wonderful property that these guys yeah you would have been a heck of a lot better there because because those because 's are actually the values of my approximate so you could say this is the approximate guy and we hope it's true and close to the to the real one you see what we did we minimized or we worked in over this finite dimensional space so we arrived at a finite problem KU equal F instead of a differential equation so instead of a differential equation we got to a finite Matrix equation but the job was figure out the Matrix so that's the stiffness Matrix and this is the load vector and and and to see what K is can I just say what what's a typical member of K and then you've really it's been a long uh day I'm not going to go over long on this but what's a what's a typical what's the IJ entry of this Matrix K it will be it comes from this it's the integral of C ofx D fi D SI J see I'm I'm plugging into the weak form and the K part comes from the quadratic part what's the what's the load uh K it would be the integral that will come from the linear part it'll be the integral of s k is it s k yeah I think times F ofx DX all I've done is plug in the the use and uh and the v's and and uh did the integrals so it's these integrals that that are involved in assembling the in producing The Matrix and hopefully those integrals are not hard I mean how would you actually do these integrals what's what is DF DX it's just piece byse constant it's just two little constants over a little piece of the interval and what's oh what's what's the derivative let's take the fees as and the size as the same so this will be another peie wise constant okay so what's up what kind of a matrix K am I going to have here yeah oh boy you got it you got it first time absolutely I thought when I asked that question I thought I haven't made it clear enough but it thanks it's why is it try diagonal Dare I ask the because they partially overlap because they only write this guy only overlaps with when if I is different from J by more than one there's no overlap so that that local bases paid off they weren't quite orthogonal because this guy and this guy do overlap but it's Tri diagonal and that's that's fine I mean if everything was orthogonal then we would have a diagonal matrix because all the other inner products would give zero but that's Tri diagonal is just as good as diagonal practically yeah thanks okay so that's so it's these it's these calculations that uh all right now what what are we going to do with this C of X are we going to actually do the integral exactly probably not if C of if we do have some variable medium so that c of X really depends on X the the the conductivity or or whatever that physically represents if it's if it's changing continuously hey we're not going to we would approximate that by peie wise constant or PE wise linear or something we all we want is a good approximation to those integrals these and these then we would solve KU equal F and we would have a darn good approximation to the answer so that's the the whole idea of finite elements was making this kind of a choice of trial functions that's that's really and real and then creating the the multi-million dollar systems that would do it um do all the calculations for us yeah can I just ask you to move for these last minutes move into two Dimensions suppose we had a partial differential equation these are double integrals DX Dy and there would be a there would be a y derivatives in there our functions are functions now of X and Y can you invent finite elements in 2D what what's a good what's a what's an invention of a 2dimensional finite element these were great one-dimensional finite elements this one and the next one and the next one and the next one now let me draw a 2d region I'll make it a okay some some two well some two dimensional region might be an air aircraft uh uh surface or something pyramids brilliant right pyramids you you would cook this you would now here's the beauty of it though I don't have to use this equally SPAC mesh I'll just create any kind of mesh it fits the edge of the that fits the surface of the of the plane in a nice way or maybe where I expect shock waves I'll put in a whole bunch of little elements that the the great thing about this idea is it's so local that I can that the geometry can be very general okay and now where's where's tell me again what what's the pyramid just before I because it's exactly the right word yeah yeah we've got it's it's got triangular faces and I don't care if it's for of them it's it's at every at every node we raise a pyramid so we have peie wise linear in in every in every plain surface it's an ax + b y + C it's linear it's piecewise linear piecewise linear and continuous across the pieces across the boundaries between elements yeah and the basis functions are pyramids which is a one at that point and zero at all the other no yeah darn Right add that here mesh big big part of the system is setting up the mesh they didn't have to be absolutely not absolutely not that's you have great freedom in the geometry but you got to keep track of it yeah and then you've got to and then for each each one is local and then of course you got to keep track of where where it is you got a little local coordinates to tell you the the little local integral and then you got to assemble all those pieces so this is actually what you do you you assemble this Matrix K by a bunch of little local integrals it's it's oh it's a it's a big world of uh computational mechanics whole journals Matrix we'll see a matrix oh yeah yeah we'll see a matrix our in this 2D problem our our approximation is a combination of some functions which will be the pyramid Heights at the point times the pyramid functions uh at the nodes of my grid so I have this grid and I should have made it proper triangular grid we could deal with rectangles and create some other functions but piecewise linear it goes with triangles so I create so I take this region and divide it in some way into triangles and then for every node I have a basis function which is a pyramid which is one at that function and zero at all the other nodes one at that node and zero at all the others my that's true we got n nodes right right one to n so we've got the we've got that problem that of you know really it's it's uh two Dimensions being squeezed into one because I have to number yeah well loser out Tri diagonal absolutely right yep we'll lose Tri diagonal because I can't do the numbering in such a way that neighbors stay neighbors right exactly right right exactly it'll be sparse because the only neighbors that affect it the only in the row of the matric that corresponds to this guy the only nonzeros will come from the pyramids the only overlapping pyramids will be the neighboring pyramids right so it'll be sparse it'll be block Tri diagonal if I do it in a beautiful regular order and what I'll get will look like it'll look like the fivepoint scheme actually if I do a re a square mesh and everything very care very sort of uh in the obvious way my K will look like a finite difference Matrix so that's nice you know and there was a lot of discussion back in the 40s and 50s when the when these ideas came up you know that people said is this just the finite differences in Disguise but no it's got so much more freedom it it gives you an automatic way to deal with irregular geometry you you create a mesh you lay a mesh down so you got mesh sub routine number zero here creating the mesh and then you've chosen the elements and and then then here's the cost the assembly and then the solution right so it's not triagonal that's a very you you really have seen so much here well I mean you're yes to n uh let's see um that's true that's true so n will be the number of nodes and the number of triangles is yeah uh there would be some Oiler would have figured out some relation between the number of triangles and the number of nodes um and the number of edges yeah so that's there's some neat facts but n would be the number of nodes here yeah yeah any other thought so we've really you've created finite elements and uh do I want to say anything about the history of the finite element method uh this sort of idea emerged in several different places uh people give uh sometimes give kurant credit for uh uh for it in China I met a wonderful old guy fun Kong who who invented it too um but neither of these and and and other early people had any way to they never thought about the actual steps of the code I mean really this was all like prehistory and then came the structural engineers who actually saw that it was a great idea that it wasn't just a math paper but it was a it was a it was a brilliant thing that's continued to grow so these this these guys in the 50s and 60s did the translation of this idea into uh into something that really worked yeah you I would say usually but wouldn't have to yeah usually would use the same functions for fi and often without especially thinking about it yeah yeah they they when they chose the element they probably uh chose chose that so I'll mention some of the engineers uh um the the classic textbook in engineering uh in this area is uh uh a uh guy who was in Polish you'll see from the name but he was in Wales o zich and now there are other the whole series of books and then math Ians like myself learned about the method said wait a minute what's going on is that just the finite element method in Disguise how can you understand the the accuracy how close is this to the real answer so I wrote an early book called an analysis of the finite element method um along with uh my co-author George fix so that was 1973 so you see mathematicians as usual just ended up kind of coming in too late to to uh and then there are other other math books and many other engineering books um okay that's 4:30 thanks for uh a good afternoon and uh see you next Thursday