Fantastic video! You are able to explain a difficult concept in an order that makes sense without glossing over the math. Textbooks usually go over the weak formulation in detail before expressing how what one really wants is an ansatz on a properly defined basis to solve the problem - I like how you start out with that, present how it won’t work naively, and then proceed to motivate weak formulations. I think a lot of people, myself included, would appreciate if this evolved into a series on FEM and its intricacies!
Yes, I thought the way this video divided the ideas up was extremely useful. In most introductions those things are all just "rammed together," and while you still can see that all the math is "technically correct," it's easy to lose sight of those boundaries.
This is an outstanding work, you made a lot of effort for that, unfortunately, this type of content will not have a lot of audience, but the amount of work realise there is just phenomenal, thank you to spend your time worrying about people like us not mastering mathematical and physical principle. I cannot not thank you enough man, this is unbelievable.
However I came here already knowing what an interpolation or form function Ni is, knowing also lagrange approximation and hermite approximation, maybe it would be interesting to add that in the course.
Wow, wonderful video, I came to the same intuition last year when I got to know PDEs, howerver I clearly could not explain it with such beautifuls images and great explanations: you truly are the boss!!!
Bravo! I studied computational mechanics for my PhD and this is one of the best explanations i have seen. And one of the best videos on TH-cam. A second part would be great!
Your video is so intuitive in explaining the weak and strong formulations. 👍 It is especially beneficial to engineering students or those who are not majoring in mathematics. The common difficulty in reading FEM textbooks is that the content is rigorously written from a mathematical point of view, making the concepts sometimes too abstract for beginners to grasp. I hope you will lecture us in the future on further details of FEM, such as matrix formulation, through simple examples on topics like the mechanics of solids or even vibration. 🙏
That’s a 12/10 vid, extremely grateful. Thanks for adding an example and a short look into finite elements!! Looking forward to a full finite elements playlist😊
I would love further information on FEM in higher dimensions, in particular deriving a weak formulation for various PDEs and how to choose a good test function. Thank you for the video!
Awesome video! In my computational science masters program we mainly focus on mathematical proofs but I never quite got the intuition. This video helped me a lot! Glad to see more about FEA from you!
This is so so useful. The explaination is easy to follow and the animations are beautiful! I definitely would love to see a video on FEM and FEM at higher dimension than 1D. Amazing job! Thank you so much!
Would love to see more vids like this on variational methods, functional analysis for PDEs, and more! This video helped me a ton for getting some intuition on how we set up FEM problems and why they turn into linear systems. Awesome job!
Sorry, I was not precise there. What I meant was adding a linear function to the function, which looks a bit like a rotation if the slope of the added linear function is small :)
Finally i have found a helpful video on weak formulations after months of searching. Thanks for the great explanation. Looking forward to more content from you!
Maybe he is from Germany, as in Germany we call integration by parts "Partielle Integration", which translates to "partial integration" if translates verbatim.
This video is amazing, as a Master student who is currently studying Linear and Non Linear Continuum Mechanics, this video is very helpful. Also, I would also prefer to continue to upload videos of more advanced topics please
the only problem with the explaination that such test functions are not allowed since the left hand side can not be integrated by parts. I think it is easier and more mathematically correct to explain the weak formulation using that residual (u''(x)-f(x)) must be a L^2 orthogonal to the test functions and if we have enough test functions, it actually forces the residual to be zero pointwise.
I am using the discontinuous test functions here only for developing some graphical intuition about the meaning of the fundamental lemma of the calculus of variations (before even talking about partial integration). It should also not be taken as a rigorous proof of the lemma. For those interested in more mathematical details, please refer to Theorem 0.1.4 in "The Mathematical Theory of Finite Element Methods" by Brenner and Scott, where continuous test functions with compact support and partial integration are considered.
Very good explanation, thank you. I would like to ask a question that some viewers may have had when you started explanation for the motivation of the weak formulation at around 7:20 time stamp. That question is, why not use a quadratic shape function instead of linear shape function, then the 2nd derivative exists (a constant)? Then, you would not need to use the weak formulation, i.e., the motivation explained at around 7:20 time stamp?
I think this was very good. You completely dodged the messy business of coordinate system transformations, where you bring the "real" coordinates of each element into a common "local coordinates" formulation. I think that's important, of course, when really learning finite elements, but it is unnecessary if your goal is to motivate intuition. So - good call. I think even in a video aimed at teaching finite elements it would be best to treat those two aspects separately - the local coordinate thing is more of a "computational optimization" than it is critical to the core concept. It lets you think in terms of "universal shape functions" that get transformed (via Jacobians) to fit each element in turn. But this is completely separate from grasping the general idea that you can transform the continuous original problem into a parameterized linear algebra problem.
Yes, I am 100 percent with you! To understand the core concept of FEM it is not necessary to learn about the reference element. Of course later it is necessary to understand why the reference element is so useful...
@@DrSimulate Yes - it's useful from the "practical computation" standpoint. I was fairly fortunate in graduate school; in my first "introductory" class the subject was presented very mechanically - the professor sort of "took us by the nose" and dragged us through it. But then I took a "topics in FEM" class that was taught by Eric Becker, who was a fairly prominent FEM "guy" and had written textbooks on the subject. His style was very interesting; he'd just wander into the lecture hall, stand there and ponder for a minute, and then just start talking about some aspect of it all. Kind of whatever happened to be on his mind that day. He chose well, and wound up showing us a lot of interesting things. The "informality" of that approach would have been disastrous in the first class, I think, but in a "follow-up" class it just worked extremely nicely. I always looked forward to those lectures. Wow - that was... so long ago. Back around 1990 or so, maybe the late 1980's. Dr. Becker actually sat on my PhD committee. I felt privileged to learn from and be exposed to such a knowledgeable person. This was at The University of Texas at Austin. It also helped a lot that I'd taken a linear algebra class prior to studying FEM.
Great video! I really enjoyed it. However, I have a question that came to mind after your explanation. You explained that we cannot rely on the formulation involving the second derivative because our ansatz, based on linear shape functions, always has a zero second derivative. But what if we change the shape functions and use ones that are not linear-for example, polynomial functions of a degree higher than 1? This is probably a silly question, but I would really appreciate an explanation in the same spirit as the one you gave in the video. Thank you in advance if you’re able to answer!
Hi, thank you for the great tutorial. I have a couple of questions. 1. At 7:19 you explain that we cannot insert solution function into the strong formulation because the second derivative would be zero everywhere. That is true if you choose linear shape functions, what if we use quadratic or higher order shape functions, what would happen then? Is there some kind of rule that we can use any shape function so that it has to be solvable even for the first order shape function? 2. You explain how weak formulation is made and that it is mathematically correct. BUT you do not specifically say why they transform it in that way? Why do they multiply it with test function and integrate it? Is partial integration their main motivation or is it something else? I mean, why exactly they do it like this?
Hi, these are great questions! I will try my best to answer them: 1. This has been also discussed in other comments, maybe you can find them. Here is a short answer: Even if we take higher order ansatz functions, it is very likely that there exists no realization of parameters u_i such that the strong form is exactly fulfilled at all points x. This means substituting the higher order ansatz in the strong form and solving for the parameters u_i is not possible. A remedy to this problem would be to select a bunch of points x and minimize the sum of squared residuals of the strong form at these points. This is called collocation and there has been research on this, but the finite element method has been more sucessful. 2. I totally understand why multiplying with test functions and integrating seems very random and out of the blue. I'm afraid that I cannot give you a completely satisfying explanation. But here are some thought on this. As explained in answer 1. there usually exists no realization of parameters u_i such that the strong form is exactly fulfilled at all points x. Therefore the strong formulation is a too strong requirement. We can weaken this requirement by integrating both sides of the strong form. In this way we make sure that the integrals of u'' and f are equal. But this requirement is too weak. There are many functions u that fulfill this requirement. By multiplying with the test functions before integrating, we somehow make sure that the integral of u'' and the intergal of f are similar over arbitrary intervals of x. So we again have a stronger requirement. Apologies that this explanation is not very mathematically rigorous, maybe I will make a future video about it. I will also make a video in the future about variational calculus, where I show that the weak form is the necessary condition of a minimization problem. This will hopefully add to a better understanding of the weak form. I think the weak form is a difficult topic because it feels like coming from out of the blue and in many years of studying and reasearch I have not yet found an explanation that is completely satisfactory from a didactical perspective.
@@DrSimulate thank you for the answer. I am a phd student of mechanical engineering and I am working on axial flux motors, both electromagnetic and mechanical design. Currently, I am trying to understand FEM because most of the things I simulate is done by FEM. When I was learning about electric machines I came to conclusion that most of the concepts can only be understood reading older textbooks, because, modern engineers often take the conclusions of the concepts without understanding basic idea. Conclusion is often enough to make something work, but for me personally, I like to understand ideas. So I will try to find the answers in the textbooks by the people who invented FEM. If I find anything meaningful I will share with you since this video is really great.
Cool Video! But I am a bit confused about what you say at 22:35 - 22:55. Does that mean you get different results by using the weak formulation instead of the strong formulation? This must be caused at the last step at 20:10 during the partial integration, but why does this change the solution?
Thanks Peter, the analytical solution (the quadratic function) is a solution to both the strong and the weak form. But when we introduce the piecewise linear finite element ansatz, we cannot use the strong form anymore. So, the numerical solution that we find in the end is not a solution to the strong form. You are right, the part of the video you are referring to is confusing! All I wanted to say is that when we choose a parameterization of u, it is better to consider the weak form because it allows for the piecewise linear parametric ansatz whose second derivative is zero almost everywhere.
This not an area of Mathematics I've been remotely involved in at all, but I do have a good background from PDEs/ODEs in general and damn... what an insightful video. The intuition and explanations were so damn good, I was able to see that Integration-By-Parts was going to be necessary as soon as I saw you multiply and constrain the original strong form of the equation by v(x). Was plainly obvious (and no I did not skip forward!). The video lead the conclusions at each stage super well. Lovely job! Not everyone that uses this software to produce Mathematical animations has such a clear talent for demonstrating this tricky concept as you do. I forgot the name of the software, so if you could drop a name for it, that would be greatly appreciated. Best, Vhaanzeit
Thanks :) Manim files are not public yet. This was my first time using manim, so the code is a big mess. Maybe for future videos, I will share manim codes 🙂
26:35 What would happen if you over-defined the system of equations by an additional test function? As you said, it's always true for ANY test function so I'm wondering if the resulting system of equations would have no solutions or infinite solutions. Great video by the way! Regarding the question to viewers at 29:50, all topics sound very interesting. I came to this video from the one you did on Finite Element and it was very good. Looking forwards to your future videos.
@@5eurosenelsuelo Good question! If the additional test functions are linearly dependent on the other test functions, then nothing changes. If not, the system may have no solution and the problem needs to be approached by minimizing the sum of the squared residuals of the system. I don't know, how this will affect the solution 🙄
@@DrSimulate Interesting. My intuition would be that the additional equations will be linearly dependent because it wouldn't add any new information to the system of equations but I don't have enough experience in the topic to test the hypothesis myself. If the new equation makes the system unsolvable it'd be so strange... That new solution found by minimizing residuals would be a more accurate solution compared to the real analytical solution? Is accuracy dependent of the test functions chosen? Computing speed definitely is but the result should be the same and if that's the case, adding more equations should add no new information as previously mentioned.
@@5eurosenelsuelo I think, as the discretization of u has an influence on the accuracy, the choice of v will also have an influence on the accuracy. For example, if you consider a two-dimensional domain, I am sure that choosing many test functions that are zero at the interesting parts of the domain (e.g., at edges or holes) is not benificial. Unfortunately, I don't have any experience with this. It would be interesting to dig in the literature and see if there is any research on this.. 🤔
You would create an over-determined system that has no solution. This is the math telling you that your ansatz that the solution is a linear combination of hat functions is false. But we already knew that the ansatz was an approximation and so is rigorously false. So there is really no point to adding more test functions without also increasing the resolution of the approximation ansatz. This is exactly what is done when one wants more resolution in a specific area for a certain problem -- put more grid points there.
sir, can I take v(x) any function that satisfy the condtion there but other than the combination of N_i . what is the advantages of taking v as N_i. and how can you gurantee the matrix is uniquely solvable.
@@soumyadas9896 You can take other functions than N_i, but they should be linearly independent. If you take linearly dependent functions then the system is not uniquely solvable.
Loved the video but I have a bit of a question. If the reason we need the weak form is because our shape functions can only be differentiated once, why do we not use quadratic shape function and stick with the strong form? Thanks!
This was discussed also in another comment. I don't know how to link it here. Maybe you can find it under this video or under the other video on FEM. But your are right, I am not telling the full story in the video and I understand the confusion. Even when using quadratic shape functions, the derivatives of u are not continuous at the nodes. Therefore, the strong form could not be fulfilled at the nodes. To understand why this is not a problem in the weak formulation, one would need to study which function space u belongs to and study a bit of measure theory, which was not the purpose of the video. There are indeed methods that work with the strong form. These are called collocation methods. You assume an ansatz, insert it in the strong form and minimize the residuals of the strong form a set of points x. Such methods have in general not as nice properties as the FEM based on the weak form.
For anyone else interested, if you go to the FEM video th-cam.com/video/1wSE6iQiScg/w-d-xo.html and crtl+F for this: "I have a question though. If you had chosen 2nd or higher order polynomials for the shape functions N(x), u''(x) would not necessarily be 0 everywhere." The comment should come up
How is one sure that if you evaluate the weak form for N test functions the solution we get satisfies the weak form for *any* function , after all solving the linear equation in the end just show that it satisfies for the N test functions one has chosen , the solution we get from this may not solve for some other test function that i might come up with ?
The discretized weak form will be satisfied for the N test functions. But it will also be satisfied for linear combinations of these test functions, e.g., if the weak form is satisfied for v=N1 and v=N2, it will also be satisfied for a*N1+b*N2, where a and b are some scalar values. So after all, we at least know that the discretized weak form is satisfied for quite many functions...
i would be interested in a follow-up video/notes explaining a-priori bounds for | u - u_approx |, where u = the weak solution, and u_approx = the linear combination of shape functions at 24:00. yes I see the reference in the description, but i would like *your* take on this. =)
Don't see a problem with other parameterizations. Some people recently tried to use neural networks as parameterizations. However, the power of the ansatz functions with local support (i.e., functions that are zero at many nodes) is that the matrix K has a lot of zero entries because many of the integrals vanish. This reduces the computational costs for computing the integrals as well as for solving the final linear system of equation, which is one of the reasons why the FEM is so powerful.
@@DrSimulate Good point on the locality; The advantage of other parameterizations would be requiring fewer paramters to start with and thus a smaller matrix K to start with. Wavelets may be interesting because they retain some degree of locality
If I remember correctly, for problems with periodic boundary conditions (for example multiscale homogenization problems) a Fourier-type ansatz is very popular.
In this video, the functions to be integrated are very simple. They can be computed by hand. Of course for higher dimensions and for higher degree polynomials numerical integration (Gauss integration) should be used. This video had not the focus on FEM. This will be covered in the future.
Thanks very much for this wonderful and clear explanation of the weak form. Is it possible to make a vedio on how to solve the Poisson equation using FEM by python programming, thus to help master the concept!
what is "UNS" in subtitles? I cannot get it when he pronounces a word like "unzerts" and it is showing as UNS in the subtitle. Can someone tell me what is that?
Yes, Manim. In other Videos, I also use matplotlib. I do the video editing with OpenShot and the audio recording with Audacity. So everything is open source or free (in case you are also interested in making videos). :)
So if I understand correctly, for every strong formulation there is one and only one weak formulation, but that does not mean that both have the same answer?
Hey, regarding the second part of your question: if no finite element discretization is considered, the strong form and the weak form have the same solution. For the example in the video, the analytical solution is a solution to both the strong and the weak form. But things change after introducing a finite element discretization. The solution that we obtain from finite element analysis, i.e., the numerical solution, is a solution of the discretized weak form, but it is not a solution of the strong form. Regarding the first part of your question: Whether there always exists one and only one weak form for any PDE is a tough question. It can be shown that some PDEs like the one in the video can be written as minimization problems (a.k.a. variational problems). The weak form can then be interpreted as the necessary condition for a minimum of the variational problem. Unfortunately, I don't know whether it is always possible to find a proper variational problem for any arbitrary PDE given in its strong form. Maybe a mathematician is following this thread and can help. I would be highly interested! :) A video on variational calculus is planned for the future...
responding to the second part of the answer: probably no, especially if you are thinking of non-linear PDEs, as they are studied on an ad-hoc basis. Sometimes, it is easy to write down the weak form, e.g. Navier-Stokes, where the non-linear part is actually bilinear, so it is not too bad.
@@strikeemblem2886 I just learned that you can see the integral as a scalar product between two functions. Then under some mild conditions, what you have in the weak form is an approximation of the solution of the strong force !! Really beautifull math, if you want I can show you the proof
I've watched this several times and I still don't 100% get it, but I feel like I will eventually. Textbook explanations are close to meaningless to me, I just don't have the mathematical background for it.
But to be more rigorous, we have to make sure u is indeed two times weakly differentiable. But as what you constructed, u cannot be two times weakly differentiable because there is no continuous representative of the weak derivative of u (a continuous function that agrees with the the weak derivative of u except on a null set, which also admits the same integration by part formula) , which guarantees u is not two times weakly differentiable, hence no two times weak derivatives exists for such u unless u is constant.
Are you referring to the piecewise linear ansatz? These are commonly used and I don't see a problem with this ansatz. Can you be more specific or give a reference? Thanks :)
@@DrSimulate Maybe put it this way, it is well known that if strong derivative exists, then the weak derivative, if it exists, agrees with the strong derivative. But we see that the second derivative of u is 0, which does not recover u' at all when integrated, it must not be second weakly differentiable. This made it hard to justify why integration by part can applied to u' at all, hence it is hard to understand why this approximation works.
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as a computational physicist i have to rate this a 10/10 youtube vid.
Thx!!! 🥰
I have searched high and low for videos to explain this over the years; this is the one!
Thanks a lot! Glad to hear that making the video was worth the efforts. :)
amazing!
Please, I would like a second part that focuses on the finite element method, this is incredible
It's on the to-do-list! :)
@@DrSimulate thank!!
@ComputationalModelingExpert Very much looking forward to it!!! You are making history!
It's now been released and it's great!
This is one of the best explanation that i have ever heard up to now. Please come back with the FEM. Over thousand people are waiting for that!
Thank you so much! The next video about FEM is already in preparation :)
Fantastic video! You are able to explain a difficult concept in an order that makes sense without glossing over the math. Textbooks usually go over the weak formulation in detail before expressing how what one really wants is an ansatz on a properly defined basis to solve the problem - I like how you start out with that, present how it won’t work naively, and then proceed to motivate weak formulations.
I think a lot of people, myself included, would appreciate if this evolved into a series on FEM and its intricacies!
Thanks a lot! 🤗🤗🤗 Yes, more content on FEM is planned! :)
Yes, I thought the way this video divided the ideas up was extremely useful. In most introductions those things are all just "rammed together," and while you still can see that all the math is "technically correct," it's easy to lose sight of those boundaries.
This is an outstanding work, you made a lot of effort for that, unfortunately, this type of content will not have a lot of audience, but the amount of work realise there is just phenomenal, thank you to spend your time worrying about people like us not mastering mathematical and physical principle. I cannot not thank you enough man, this is unbelievable.
However I came here already knowing what an interpolation or form function Ni is, knowing also lagrange approximation and hermite approximation, maybe it would be interesting to add that in the course.
Did you build a pyramid or a tower? Where did you get your inspiration from? What did you drink while making this video? Look at this. Perfect!!
🙏🙏🙏
We took a look at the weak formulation in a math class in passing. This is really interesting and nice to hear it again here.
The easiest to understand explaination I've ever come across. Each one of your videos so far is incredible please keep it up!
Thanks, Joe! This means a lot to me. :)
Wow, wonderful video, I came to the same intuition last year when I got to know PDEs, howerver I clearly could not explain it with such beautifuls images and great explanations: you truly are the boss!!!
Thanks 🙏♥
Thank you so much for this amazing video! Really helped me with my studies!
Bravo! I studied computational mechanics for my PhD and this is one of the best explanations i have seen. And one of the best videos on TH-cam. A second part would be great!
Yay, computational mechanics rocks 😁 I did also my PhD in computational mechanics :)
Im speechless how good this video is! The visualisazions are helping a lot! Best video about this topic on TH-cam
@@tobiasl3517 Thank you so much!
Your video is so intuitive in explaining the weak and strong formulations. 👍 It is especially beneficial to engineering students or those who are not majoring in mathematics. The common difficulty in reading FEM textbooks is that the content is rigorously written from a mathematical point of view, making the concepts sometimes too abstract for beginners to grasp.
I hope you will lecture us in the future on further details of FEM, such as matrix formulation, through simple examples on topics like the mechanics of solids or even vibration. 🙏
@@albajasadur2694 Thanks a lot. The next video on FEM will come in a few weeks :)
This video is nothing short but amazing for getting the intuition behind the weak form and FEM!
Thanks :)
That’s a 12/10 vid, extremely grateful. Thanks for adding an example and a short look into finite elements!! Looking forward to a full finite elements playlist😊
Thanks :)
I would love further information on FEM in higher dimensions, in particular deriving a weak formulation for various PDEs and how to choose a good test function. Thank you for the video!
Thank you! Many seem to be interested in more details on FEM. I will definitely do a video about it in the future! :)
Awesome video! In my computational science masters program we mainly focus on mathematical proofs but I never quite got the intuition. This video helped me a lot! Glad to see more about FEA from you!
Wow, this explains it so well, I’m amazed
Great Video !!! This is a really helpful visualization. Keep going like this :D
Thanks a lot! Glad you enjoyed! :)
I have taken multiple course in FEA but your explanation incredible and amazing!!! Please continue this series of videos.
Thanks for the kind words! :)
Gold.
This is gold!
Thank you so much!! :D
Really Really well done. Having an intuition makes learning rigous material more easier.
You are a trully amazing teacher!
Beautiful explanation! This video deserves going viral!
Thanks :)
This is so so useful. The explaination is easy to follow and the animations are beautiful! I definitely would love to see a video on FEM and FEM at higher dimension than 1D. Amazing job! Thank you so much!
Thank you so much!! 🙏🙏🙏
Great!! I'm studying FEM for engineering. I think this video is very easy to understand for beginners.
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Would love to see more vids like this on variational methods, functional analysis for PDEs, and more! This video helped me a ton for getting some intuition on how we set up FEM problems and why they turn into linear systems. Awesome job!
Thanks! :)
Incredible explanation!! Now I can visualise FEA concepts better. Thank You!!
Thanks for sharing, it's now very easy to understand.
wow, crystal clear, you did amazing job, you deserve a medal for this video bro
Thank you! :DD
Small correction: at 3:56, rotating the function does change its second derivative! Translating it side to side rather, doesn't
Sorry, I was not precise there. What I meant was adding a linear function to the function, which looks a bit like a rotation if the slope of the added linear function is small :)
So well explained! Thank you!
Finally i have found a helpful video on weak formulations after months of searching. Thanks for the great explanation. Looking forward to more content from you!
Thanks :D
Excellent job!
Fantastic explanations accompanied by excellent visual aids!
Thank you so much!
Wonderful video. I wish I had visualizations like these when I was at uni.
Beautiful explanation! Thanks 😀🙌👏
Great video. Looking forward for the next ones. Thanks.
where was this video 4 years ago when I was taking the FEA class? Haha I got thru it but this video would have been tremendously helpful. Well done!
20:11 "partial integration" == "integration by parts" (just in case anyone was confused and thinking of undoing partial differentiation like I was).
Maybe he is from Germany, as in Germany we call integration by parts "Partielle Integration", which translates to "partial integration" if translates verbatim.
This video is amazing, as a Master student who is currently studying Linear and Non Linear Continuum Mechanics, this video is very helpful. Also, I would also prefer to continue to upload videos of more advanced topics please
Thanks for the suggestion! There will be definitely more advanced content down the line... :)
Thank you very much.
Would definitely be interested in further videos into the finite element method
It's definitely on the list ✅ Will take some time unfortunately
What a watch! - greatly explained can't wait for more videos :)
Thank you so much Nathan!!
Oh my god, what a great video!
the only problem with the explaination that such test functions are not allowed since the left hand side can not be integrated by parts. I think it is easier and more mathematically correct to explain the weak formulation using that residual (u''(x)-f(x)) must be a L^2 orthogonal to the test functions and if we have enough test functions, it actually forces the residual to be zero pointwise.
I am using the discontinuous test functions here only for developing some graphical intuition about the meaning of the fundamental lemma of the calculus of variations (before even talking about partial integration). It should also not be taken as a rigorous proof of the lemma. For those interested in more mathematical details, please refer to Theorem 0.1.4 in "The Mathematical Theory of Finite Element Methods" by Brenner and Scott, where continuous test functions with compact support and partial integration are considered.
Very good explanation, thank you. I would like to ask a question that some viewers may have had when you started explanation for the motivation of the weak formulation at around 7:20 time stamp. That question is, why not use a quadratic shape function instead of linear shape function, then the 2nd derivative exists (a constant)? Then, you would not need to use the weak formulation, i.e., the motivation explained at around 7:20 time stamp?
This is beautifully done. Would love a follow-up video on the finite element method:)
Coming ;)
I think this was very good. You completely dodged the messy business of coordinate system transformations, where you bring the "real" coordinates of each element into a common "local coordinates" formulation. I think that's important, of course, when really learning finite elements, but it is unnecessary if your goal is to motivate intuition. So - good call. I think even in a video aimed at teaching finite elements it would be best to treat those two aspects separately - the local coordinate thing is more of a "computational optimization" than it is critical to the core concept. It lets you think in terms of "universal shape functions" that get transformed (via Jacobians) to fit each element in turn. But this is completely separate from grasping the general idea that you can transform the continuous original problem into a parameterized linear algebra problem.
Yes, I am 100 percent with you! To understand the core concept of FEM it is not necessary to learn about the reference element. Of course later it is necessary to understand why the reference element is so useful...
@@DrSimulate Yes - it's useful from the "practical computation" standpoint. I was fairly fortunate in graduate school; in my first "introductory" class the subject was presented very mechanically - the professor sort of "took us by the nose" and dragged us through it. But then I took a "topics in FEM" class that was taught by Eric Becker, who was a fairly prominent FEM "guy" and had written textbooks on the subject. His style was very interesting; he'd just wander into the lecture hall, stand there and ponder for a minute, and then just start talking about some aspect of it all. Kind of whatever happened to be on his mind that day. He chose well, and wound up showing us a lot of interesting things. The "informality" of that approach would have been disastrous in the first class, I think, but in a "follow-up" class it just worked extremely nicely. I always looked forward to those lectures.
Wow - that was... so long ago. Back around 1990 or so, maybe the late 1980's. Dr. Becker actually sat on my PhD committee. I felt privileged to learn from and be exposed to such a knowledgeable person. This was at The University of Texas at Austin.
It also helped a lot that I'd taken a linear algebra class prior to studying FEM.
Wow. Thank you for all the hard work. Keep posting.
Great video! I really enjoyed it. However, I have a question that came to mind after your explanation. You explained that we cannot rely on the formulation involving the second derivative because our ansatz, based on linear shape functions, always has a zero second derivative. But what if we change the shape functions and use ones that are not linear-for example, polynomial functions of a degree higher than 1?
This is probably a silly question, but I would really appreciate an explanation in the same spirit as the one you gave in the video. Thank you in advance if you’re able to answer!
Outstanding work. Thank you for producing this content.
Thank you so much, Allan! :)
Your channel looks great
Thx!!
Hi, thank you for the great tutorial. I have a couple of questions. 1. At 7:19 you explain that we cannot insert solution function into the strong formulation because the second derivative would be zero everywhere. That is true if you choose linear shape functions, what if we use quadratic or higher order shape functions, what would happen then? Is there some kind of rule that we can use any shape function so that it has to be solvable even for the first order shape function? 2. You explain how weak formulation is made and that it is mathematically correct. BUT you do not specifically say why they transform it in that way? Why do they multiply it with test function and integrate it? Is partial integration their main motivation or is it something else? I mean, why exactly they do it like this?
Hi, these are great questions! I will try my best to answer them:
1. This has been also discussed in other comments, maybe you can find them. Here is a short answer: Even if we take higher order ansatz functions, it is very likely that there exists no realization of parameters u_i such that the strong form is exactly fulfilled at all points x. This means substituting the higher order ansatz in the strong form and solving for the parameters u_i is not possible. A remedy to this problem would be to select a bunch of points x and minimize the sum of squared residuals of the strong form at these points. This is called collocation and there has been research on this, but the finite element method has been more sucessful.
2. I totally understand why multiplying with test functions and integrating seems very random and out of the blue. I'm afraid that I cannot give you a completely satisfying explanation. But here are some thought on this. As explained in answer 1. there usually exists no realization of parameters u_i such that the strong form is exactly fulfilled at all points x. Therefore the strong formulation is a too strong requirement. We can weaken this requirement by integrating both sides of the strong form. In this way we make sure that the integrals of u'' and f are equal. But this requirement is too weak. There are many functions u that fulfill this requirement. By multiplying with the test functions before integrating, we somehow make sure that the integral of u'' and the intergal of f are similar over arbitrary intervals of x. So we again have a stronger requirement. Apologies that this explanation is not very mathematically rigorous, maybe I will make a future video about it. I will also make a video in the future about variational calculus, where I show that the weak form is the necessary condition of a minimization problem. This will hopefully add to a better understanding of the weak form.
I think the weak form is a difficult topic because it feels like coming from out of the blue and in many years of studying and reasearch I have not yet found an explanation that is completely satisfactory from a didactical perspective.
@@DrSimulate thank you for the answer. I am a phd student of mechanical engineering and I am working on axial flux motors, both electromagnetic and mechanical design. Currently, I am trying to understand FEM because most of the things I simulate is done by FEM. When I was learning about electric machines I came to conclusion that most of the concepts can only be understood reading older textbooks, because, modern engineers often take the conclusions of the concepts without understanding basic idea. Conclusion is often enough to make something work, but for me personally, I like to understand ideas. So I will try to find the answers in the textbooks by the people who invented FEM. If I find anything meaningful I will share with you since this video is really great.
Cool Video!
But I am a bit confused about what you say at 22:35 - 22:55. Does that mean you get different results by using the weak formulation instead of the strong formulation?
This must be caused at the last step at 20:10 during the partial integration, but why does this change the solution?
Thanks Peter,
the analytical solution (the quadratic function) is a solution to both the strong and the weak form. But when we introduce the piecewise linear finite element ansatz, we cannot use the strong form anymore. So, the numerical solution that we find in the end is not a solution to the strong form.
You are right, the part of the video you are referring to is confusing! All I wanted to say is that when we choose a parameterization of u, it is better to consider the weak form because it allows for the piecewise linear parametric ansatz whose second derivative is zero almost everywhere.
Amazing explanations. Congratulations for the channel, and well done.
Thank you so much for your kind comment! New video coming today. :)
Extremely good videos, keep it up
Thanks Guiseppe!
Great explanation, thanks!
Welcome :)
Incredible video, thanks
Such a nice explanation! May i ask how did do the dynamic simulation 😢
Thank you sir
Brilliant!
This not an area of Mathematics I've been remotely involved in at all, but I do have a good background from PDEs/ODEs in general and damn... what an insightful video. The intuition and explanations were so damn good, I was able to see that Integration-By-Parts was going to be necessary as soon as I saw you multiply and constrain the original strong form of the equation by v(x). Was plainly obvious (and no I did not skip forward!). The video lead the conclusions at each stage super well.
Lovely job! Not everyone that uses this software to produce Mathematical animations has such a clear talent for demonstrating this tricky concept as you do. I forgot the name of the software, so if you could drop a name for it, that would be greatly appreciated.
Best,
Vhaanzeit
Thank you Vhaanzeit :) The name of the software is Manim (standing for mathematical animation)
Great video, many thanks!
Great Video!
How to learn any numerical analysis
Great video, loved the explanation and animations!
Thank you so much, Daniel!
Great video, just curious how you got this first video into the algorithm? 17k views on a first video is vert impressive, did you advertise anywhere?
Thanks! No, it is very unpredictable. The first weeks, I had almost no views. Then it went up. Now it's stagnating a bit...
Great explanation. Thank you.
Thanks, you're welcome! :)
great video
Very good job!! did you save the Manim files into a repository?
Thanks :) Manim files are not public yet. This was my first time using manim, so the code is a big mess. Maybe for future videos, I will share manim codes 🙂
26:35
What would happen if you over-defined the system of equations by an additional test function? As you said, it's always true for ANY test function so I'm wondering if the resulting system of equations would have no solutions or infinite solutions.
Great video by the way!
Regarding the question to viewers at 29:50, all topics sound very interesting. I came to this video from the one you did on Finite Element and it was very good. Looking forwards to your future videos.
@@5eurosenelsuelo Good question! If the additional test functions are linearly dependent on the other test functions, then nothing changes. If not, the system may have no solution and the problem needs to be approached by minimizing the sum of the squared residuals of the system. I don't know, how this will affect the solution 🙄
@@DrSimulate Interesting. My intuition would be that the additional equations will be linearly dependent because it wouldn't add any new information to the system of equations but I don't have enough experience in the topic to test the hypothesis myself. If the new equation makes the system unsolvable it'd be so strange... That new solution found by minimizing residuals would be a more accurate solution compared to the real analytical solution?
Is accuracy dependent of the test functions chosen? Computing speed definitely is but the result should be the same and if that's the case, adding more equations should add no new information as previously mentioned.
@@5eurosenelsuelo I think, as the discretization of u has an influence on the accuracy, the choice of v will also have an influence on the accuracy. For example, if you consider a two-dimensional domain, I am sure that choosing many test functions that are zero at the interesting parts of the domain (e.g., at edges or holes) is not benificial. Unfortunately, I don't have any experience with this. It would be interesting to dig in the literature and see if there is any research on this.. 🤔
You would create an over-determined system that has no solution. This is the math telling you that your ansatz that the solution is a linear combination of hat functions is false. But we already knew that the ansatz was an approximation and so is rigorously false. So there is really no point to adding more test functions without also increasing the resolution of the approximation ansatz. This is exactly what is done when one wants more resolution in a specific area for a certain problem -- put more grid points there.
sir, can I take v(x) any function that satisfy the condtion there but other than the combination of N_i . what is the advantages of taking v as N_i. and how can you gurantee the matrix is uniquely solvable.
@@soumyadas9896 You can take other functions than N_i, but they should be linearly independent. If you take linearly dependent functions then the system is not uniquely solvable.
Awesome.Thanks.
Plz keep going.
Thanks Farzin! :)
Loved the video but I have a bit of a question. If the reason we need the weak form is because our shape functions can only be differentiated once, why do we not use quadratic shape function and stick with the strong form? Thanks!
This was discussed also in another comment. I don't know how to link it here. Maybe you can find it under this video or under the other video on FEM.
But your are right, I am not telling the full story in the video and I understand the confusion. Even when using quadratic shape functions, the derivatives of u are not continuous at the nodes. Therefore, the strong form could not be fulfilled at the nodes. To understand why this is not a problem in the weak formulation, one would need to study which function space u belongs to and study a bit of measure theory, which was not the purpose of the video.
There are indeed methods that work with the strong form. These are called collocation methods. You assume an ansatz, insert it in the strong form and minimize the residuals of the strong form a set of points x. Such methods have in general not as nice properties as the FEM based on the weak form.
@@DrSimulate Ah I think I understand, it would be great to see a video on that at some point!
For anyone else interested, if you go to the FEM video th-cam.com/video/1wSE6iQiScg/w-d-xo.html and crtl+F for this: "I have a question though. If you had chosen 2nd or higher order polynomials for the shape functions N(x), u''(x) would not necessarily be 0 everywhere." The comment should come up
@@robm624 Yes, I hope I can do a video that is more mathematically rigorous and covers e.g. the function spaces in the future :)
Great explanation!
Thanks!! :)
Great video, would like further videos.
Thanks a lot! Appreciate it! More videos planned.
As clear as 3Blue1Brown ! Thanks a lot !
Wow, this is a big compliment. Thanks! :D
Can you explain the steps programming for the Galerkin method in MATLAB?
I am planning to do programming videos in the future, but for the next weeks/months, I am occupied with another project. Thanks for your patience :)
@DrSimulate thanks🌹🌹
How is one sure that if you evaluate the weak form for N test functions the solution we get satisfies the weak form for *any* function , after all solving the linear equation in the end just show that it satisfies for the N test functions one has chosen , the solution we get from this may not solve for some other test function that i might come up with ?
The discretized weak form will be satisfied for the N test functions. But it will also be satisfied for linear combinations of these test functions, e.g., if the weak form is satisfied for v=N1 and v=N2, it will also be satisfied for a*N1+b*N2, where a and b are some scalar values. So after all, we at least know that the discretized weak form is satisfied for quite many functions...
What software do you use to generate those animations with the graphs morphing from one shape to another?
I use Manim www.manim.community/
i would be interested in a follow-up video/notes explaining a-priori bounds for | u - u_approx |, where u = the weak solution, and u_approx = the linear combination of shape functions at 24:00.
yes I see the reference in the description, but i would like *your* take on this. =)
Thanks for the suggestion! I hope I can cover this in one of the next videos on FEM. :)
It's a great content
please keep forward
Would this also allow for the computation of solutions parameterized on other basis sets (e.g. Fourier series, wavelets, Chebyshev polynomials, etc.)?
Don't see a problem with other parameterizations. Some people recently tried to use neural networks as parameterizations. However, the power of the ansatz functions with local support (i.e., functions that are zero at many nodes) is that the matrix K has a lot of zero entries because many of the integrals vanish. This reduces the computational costs for computing the integrals as well as for solving the final linear system of equation, which is one of the reasons why the FEM is so powerful.
@@DrSimulate Good point on the locality; The advantage of other parameterizations would be requiring fewer paramters to start with and thus a smaller matrix K to start with. Wavelets may be interesting because they retain some degree of locality
If I remember correctly, for problems with periodic boundary conditions (for example multiscale homogenization problems) a Fourier-type ansatz is very popular.
Noooo, how do you compute those integrals? What programs do you use? :'d
In this video, the functions to be integrated are very simple. They can be computed by hand. Of course for higher dimensions and for higher degree polynomials numerical integration (Gauss integration) should be used. This video had not the focus on FEM. This will be covered in the future.
What happens with other Boundary conditions?
Thanks very much for this wonderful and clear explanation of the weak form. Is it possible to make a vedio on how to solve the Poisson equation using FEM by python programming, thus to help master the concept!
Thanks a lot! For the next few videos, I am planning to mostly focus on theory. At some point in the future, I will also share codes! :)
what is "UNS" in subtitles? I cannot get it when he pronounces a word like "unzerts" and it is showing as UNS in the subtitle. Can someone tell me what is that?
It's "ansatz", which is apparently not as commonly used as I thought :)
Thanks
Thank you! :)
banger vid
I love this kind of presentation, what kind of applications or software did you use to create this type of video? Excellent job
looks like manim
Yes, Manim. In other Videos, I also use matplotlib. I do the video editing with OpenShot and the audio recording with Audacity. So everything is open source or free (in case you are also interested in making videos). :)
amazing ❤ !!
please can you recommend me a reference for the method that is easy to follow just like your videos ? 😅
Good!
this is great
So if I understand correctly, for every strong formulation there is one and only one weak formulation, but that does not mean that both have the same answer?
Hey,
regarding the second part of your question: if no finite element discretization is considered, the strong form and the weak form have the same solution. For the example in the video, the analytical solution is a solution to both the strong and the weak form. But things change after introducing a finite element discretization. The solution that we obtain from finite element analysis, i.e., the numerical solution, is a solution of the discretized weak form, but it is not a solution of the strong form.
Regarding the first part of your question: Whether there always exists one and only one weak form for any PDE is a tough question. It can be shown that some PDEs like the one in the video can be written as minimization problems (a.k.a. variational problems). The weak form can then be interpreted as the necessary condition for a minimum of the variational problem. Unfortunately, I don't know whether it is always possible to find a proper variational problem for any arbitrary PDE given in its strong form. Maybe a mathematician is following this thread and can help. I would be highly interested! :)
A video on variational calculus is planned for the future...
responding to the second part of the answer: probably no, especially if you are thinking of non-linear PDEs, as they are studied on an ad-hoc basis. Sometimes, it is easy to write down the weak form, e.g. Navier-Stokes, where the non-linear part is actually bilinear, so it is not too bad.
@@strikeemblem2886 I just learned that you can see the integral as a scalar product between two functions.
Then under some mild conditions, what you have in the weak form is an approximation of the solution of the strong force !!
Really beautifull math, if you want I can show you the proof
@@chainetravail2439 thanks, but i am familiar with the proof (i work in PDEs). =)
u know that the creator of the video is german when he uses the expression "partial integration" :D
Danke für das Video!
This and the very German accent :D
I've watched this several times and I still don't 100% get it, but I feel like I will eventually.
Textbook explanations are close to meaningless to me, I just don't have the mathematical background for it.
Keep grinding 💪
Dutch accent adds a lot of credibility.
German accent, but not too far off ;)
But to be more rigorous, we have to make sure u is indeed two times weakly differentiable. But as what you constructed, u cannot be two times weakly differentiable because there is no continuous representative of the weak derivative of u (a continuous function that agrees with the the weak derivative of u except on a null set, which also admits the same integration by part formula) , which guarantees u is not two times weakly differentiable, hence no two times weak derivatives exists for such u unless u is constant.
Are you referring to the piecewise linear ansatz? These are commonly used and I don't see a problem with this ansatz. Can you be more specific or give a reference? Thanks :)
@@DrSimulate Maybe put it this way, it is well known that if strong derivative exists, then the weak derivative, if it exists, agrees with the strong derivative. But we see that the second derivative of u is 0, which does not recover u' at all when integrated, it must not be second weakly differentiable. This made it hard to justify why integration by part can applied to u' at all, hence it is hard to understand why this approximation works.