Finite Element Method Explained in 3 Levels of Difficulty

[@eduardoimaz1591]

In the first 20 minutes you already explained the FE way better than my proffesors could in an entire semester, thank you so much :)

[@aqibrasheed4874]

Best explanation of FEM one could find on internet. Looking forward for more such videos on FEM.

[@DrSimulate]

@@aqibrasheed4874 Thank you!! :)

[@yondaimenamikaze8793]

By far the best Video on FEM that i could find !! Well structured and illustrated, thank you for the effort :)

[@kialakungalgal6860]

Brilliant just brilliant. You are truly right, trying to understand every concept at once is difficult, but as you explain concept by concept it makes it so obvious. Thank you!! one of the best videos on this topic.

[@DrSimulate]

Thank you so much!! 😃

[@5eurosenelsuelo]

This is a great video. I won't be surprised when it eventually explodes within the Math & Eng communities. I'm definitely checking the rest of the videos from this channel.

[@DrSimulate]

@@5eurosenelsuelo Thanks a lot! Sharing the video with your friends would help a lot 😁🤗

[@edspace3699]

The quality of your videos is insane, thanks a lot!!!

[@Prashanth-yn9zd]

This is the best. 4 months of the course is beautifully summarized in 40 mins!! Please make a video on the Boundary element method if possible!

[@JavierMeseguerdePaz]

What a gem of a video! Best explanation I've seen of FEM in... Forever. Thank you very very much for this masterpiece.

[@hannibal4567]

Omg best video on finite element ever. Ive only learn up to level 2 and got brief introduction to level 3 for structural engineering. This video give me better insight on the latter

[@carlosgiovanardi8197]

Great explanation! I found today this channel. Explained in an easy and understandable way. Congrats!!

[@5eurosenelsuelo]

I am now taking a master's where one of the courses is FEM. I remembered this video and it was extremely helpful. I already said it months ago when I discovered the video but I'll say it again. THANK YOU for uploading this. I will share it with my class mates.

[@DrSimulate]

Thanks a lot for spreading the word! :)

[@mshriharikrishnan8362]

Amazing video with accurate animations! Thanks a lot, hope to see more such amazing videos from you.

[@DrSimulate]

THANKS!

[@dm7893-p2t]

Talking about weak formulation needs to include theory of distributions and the space of test functions (continuously differentiable functions of order n and compactly supported )and Schwartz space. Nevertheless it is an excellent video. I’m an engineer myself but since I decided to get also a degree in applied math and having taken courses on real analysis and functional analysis while studying as well PDEs more rigorously, things make sense .

[@DrSimulate]

You are right. The video should not be seen as a profound mathematical analysis of the problem. It rather intends to create some intuition for engineers. I might do a more mathematical video in the future :)

[@vegetablebake]

Thanks!

[@vegetablebake]

Please don't stop making videos. Have a drink on me :)

[@DrSimulate]

@@vegetablebake Thanks for the support! Cheers 🍻

[@dgg728]

Excellent video, thank you for your work and your time. Your pedagogy is really very good. It will surely be very useful to many people. Thank you very much.

[@DrSimulate]

Thank you so much! I'm glad you liked it

[@cooler291991]

Thank you for your effort in making such an insightful video. More cheers to you. Waiting for more such videos from your channel

[@DrSimulate]

Thanks :D

[@arischai1433]

Thanks so much for your great effort in explaning the stuff beautifully.

[@abcdefghijklm9697]

The explanation is so much more intuitive in this way. Beginning by explaining reference shape function and gaussian quadrature without having grasped the concept of what we're trying to do is so dumb. Thanks a lot , the video is so clear, the quality is crazy 🙏 Maybe one point that had me bugging because Linear algebra was quite some time ago is the fact N'(x)N't(x) define a matrix K s.t. Kij=n'i(x)n'j(x), I haven't tried but I think it can be demonstrated quite easily tho 👍

[@DrSimulate]

THANKS! Yes, you are right. This is the outer product of two vectors :)

[@rezaafshar3832]

Very impressive!.. thanks a lot for sharing 🙏🙏

[@aziz0x00]

i have an exam on it tomorrow, and this boosted my understanding! THANKS!

[@DrSimulate]

@@aziz0x00 Good luck!

[@aziz0x00]

@DrSimulate it was great :D ! Thank you.

[@andresyesidmorenovilla7888]

Brilliant explanation, thank you so much!

[@praveenraj8704]

This is pure GOLD

[@DrSimulate]

Thanks :D

[@bobbyshoemaker3817]

excellent video, will be showing my classmates

[@DrSimulate]

Thanks for the advertisement :D

[@camlainguyen3545]

Thank you for your great explanation.

[@hamzazahoil48]

Truly Amazing Thank you so much ❤

[@courageouslyencouraged]

This is a master class!

[@courageouslyencouraged]

I'm prepping for my Candidacy exam and this cleared up a few things I was a bit iffy on!

[@angupraveen]

Learning from the best ;)

[@DrSimulate]

Thanks Angu 😁

[@vybhav345]

Thanks for the video ☺️

[@peasant12345]

Enlightening lecture. Thank you. Would you make a follow-up video about FEM in 3 dimensions?

[@DrSimulate]

Yes, this is definitely planned in the future. But it will take some time, thanks for your patience :)

[@amatoallahouchen5894]

not all heroes wear a cap! thanks a million

[@rajeevnaik3877]

Oh man o man. This is so soothing to watch. Your videos are exact proof why pictures are worth thousand words and why textbooks Sometimes are not that great and simple graphics can replace tons of pages. Where did you learn these? I'm pretty sure there isn't a book that lays down math and intuition so clearly. FEM authors are the most obtuse writers where they assume readers know everything. Which book do you refer or recommend to understand these things so clearly?

[@DrSimulate]

Thanks a lot for your nice comment! I can recommend the lecture notes by Dennis Kochmann: ethz.ch/content/dam/ethz/special-interest/mavt/mechanical-systems/mm-dam/documents/Notes/IntroToFEA_red.pdf

[@hamzazaheer3783]

Thank you for the fantastic videos! I appreciate the effort put into explaining FEM at different levels. Could you recommend resources on Shear Locking, Reduced Integration, and Hourglassing? I understand the theory but struggle with visualizing these concepts.❤❤

[@DrSimulate]

I don't know about a video explaining this, but you may check out these lecture notes, which are very didactic and have one section about this: ethz.ch/content/dam/ethz/special-interest/mavt/mechanical-systems/mm-dam/documents/Notes/IntroToFEA_red.pdf

[@hamzazaheer3783]

@@DrSimulate Thanks

[@schobihh2703]

nettes Video. "Ansatz" kann man m.E. mit approach übersetzen.

[@yugalsharma2773]

ohhh 😂, I was confused by the same, and I dont know german, I thought he is saying "unddasz" something

[@lionelmartinez6810]

Thank you very much for the video. Could you provide a practical example, such as the electric field of a parallel plate capacitor, to observe edge effects? It would be cool to see the solution of electric field in Manim.

[@DrSimulate]

That's an interesting suggestion. I definitely want to show more applications in future videos. The nice thing about the Poisson problem is that it appears in many different disciplines.

[@dodo-js5gw]

Niceeee manim

[@cleisonarmandomanriqueagui9176]

What about shape functions in 3 dimentions ? and how we input any geometry ? . then how the equation gets modified to solve problems with damping and nonlinear terms ? . excellent video , thanks for sharing

[@DrSimulate]

@@cleisonarmandomanriqueagui9176 I plan to cover all these questions in future videos :)

[@amirh1078]

Great!

[@jakeaustria5445]

Thank you

[@KolibriGaming]

Great video! 🎉 But I have a question. If I now had a second order derivative, my mapping function would become zero. How could I get around this?

[@DrSimulate]

Thanks! One typically uses the same functions for the mapping as for discretizing u. So, when you have higher order derivatives in your weak form, you would also use higher order shape functions N and thus higher order functions for the mapping.

[@whmi8498]

May Allah bless you 💖

[@teunschuur7988]

Excelent video, I tried to code along in Julia. But I there is a small mistake. You keep displaying the initial condition of u'(0)=0, where as the solution you show (and also the plot of u'(x)) don't support that. I think u'(0) would be 1? please correct me if I'm wrong. edit: never mind, I now see that is says u'(1)=0, which does indeed work.

[@emirbfitness]

Thank u

[@padaddadada5417]

First thank you so much for this video, I have a question does FEM work only if the starting equation is u(x)´´ = f(x) does the problem or the differential equation has to have this form ?

[@DrSimulate]

Hi. No, I am just using this equation as an example (because it is the most often used example). The FEM can be applied to a variety of different problems.

[@LucasVieira-ob6fx]

I'm impressed how much valuable information you've managed to pack in that video! Congratulations! 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. That implies you could solve the discretized Strong form without the need to recast it into the discretized Weak form. Is that right or am I missing something?

[@hamzazaheer3783]

The purpose of the weak form is to reduce computational effort. Linear functions require less computational power compared to quadratic or higher-order functions. Additionally, the strong form requires both essential and natural boundary conditions, while the weak form only requires essential boundary conditions, thereby reducing the continuity requirements. In conclusion, using the weak form reduces computational power and lowers continuity requirements. If I am mistaken, please correct me.

[@LucasVieira-ob6fx]

@@hamzazaheer3783 thanks. I do agree that linear shape functions and the weak form are computationally less intensive. But my question was more about the way it was framed, i.e. "let's use linear shape functions", and then "oh, these have 2nd derivative = 0", therefore "we must solve the weak form". It seems that this motivation would've been avoided if the shape functions were 2nd order or higher. So I wonder if there would be more hurdles if one would proceed with this approach (use higher order polynomials to avoid the weak form).

[@hamzazaheer3783]

@@LucasVieira-ob6fx Yes, you can solve the strong form without using the weak form, but it requires higher-order shape functions compared to the weak form. Additionally, it must satisfy both essential and natural boundary conditions. You might wonder why we use the weak form if higher-order shape functions are needed. The reason is that, in the weak form, when using higher-order shape functions, you only need to satisfy the essential boundary conditions. The primary advantage of using the weak form is that it lowers the continuity requirements.

[@DrSimulate]

I would like to add another point. Even if the second derivative of the ansatz would not be zero, there may be a problem: It is very likely that it is not possible to tweak the parameters in the ansatz in a way that the strong form is exactly fulfilled at all points x. In other words: the "exact" or "true" solution of the differential equation cannot be expressed by the ansatz. In this case, what people do is to minimize the norm of the residuals of the strong form at some chosen points x. This is called collocation and there is still active research on that. But collocation methods are by far not as successful as methods based on the weak form.

[@hamzazaheer3783]

@@DrSimulate you are talking about galerkin and rayleigh ritz method ?

[@kdo74]

What an incredible video! Your explanation was absolutely fantastic - so clear and easy to follow. The way you broke down the concepts step by step made it so much easier to understand. Your presentation skills are top-notch, and it's obvious how much effort you put into making everything so accessible. If I could make a suggestion, it would be amazing to see another video diving into the Galerkin method. I think your excellent teaching style would make it much easier to grasp such an important topic. Keep up the fantastic work, and thanks again for sharing your knowledge with such passion and dedication!

[@DrSimulate]

Thank you so much! I'm very glad that it helped you to understand! :)

[@yugalsharma2773]

At 10:08, what is the word that you are using "Finite Element Undas" ? Please explain this

[@DrSimulate]

Finite element ansatz 😁

[@Edge_Rider]

I have a question but first of all thank you for great work. In many practical FEA training guides I read that the most accurate displacement solutions is at the Gauss points. However as you stated, Gaussian quadrature evaluates the integral exactly on the reference element, it is not an approximation. The integration by substation methods ensures that the integral on the reference element and the global element are the same thing. Also the whole point is to find K and F values so that I can get to my u, which is displacement at the given node. So shouldn't we conclude that the most accurate value of displacement is calculated at the node, rather then the Gaussian point?

[@DrSimulate]

Thanks. The statement that the solution is the most accurate at the GPs is new to me. Can you give me a reference? I think one has to distinguish between error due to integration and error due to discretization. Even if the integrals are computed correctly, the FEM solution is not equal to the exact solution of the differential equation. So, maybe the FEM solution is the closest to the exact solution at the Gauss points? Is this what your reference is saying?

[@Edge_Rider]

@@DrSimulate Hello. Thank you for taking your time to reply to me. Exact documents that I was referring are in-company training documents, design practices or best practices so I'm afraid I can not provide them to you. However I re-read the document and realized that I could have explained it much better, so my apologies. The exact phrasing is that "Strain and stresses are calculated at Gaussian points and than extrapolated to nodes to find strains and stresses at nodes. Therefore the most accurate strain and stress values are the ones at Gaussian points." I guess by "accurate" it means closest to actual strain and stresses. If I go back to the point where I'm puzzled with it is this: I understand that what you are showing here is the general method for any system that can be modeled with a differential equation. So speaking for the special case of static structural analysis -> we calculate the integrals using mappings and integration by substitutions which only give me my K values. K values, for the special case of static structural, have some material property constants multiplied with it but lets assume it is 1 to not complicate it any further. So K is literally a stiffness matrix where the u is displacements of nodes and f is forces on the nodes. Therefore calculation goes as calculate K -> calculate u -> using u and original length calculate strain -> use hooks law to calculate stress. Looking at these steps I don't understand how stress and strain are "calculated at Gaussian points". What is there to calculate at a Gaussian point? We only use it to get to the K matrix and we already know what value the shape function takes at the Gaussian point, because we made up the shape function ourselves in the first place. :) Although I can not give you the original document, I found similar discussions on the internet about evaluating strain and stresses at Gaussian points: www.quora.com/In-FEA-why-is-it-more-accurate-to-compute-element-stress-using-an-average-in-the-Gauss-points-than-an-average-in-the-nodes Second comment here made by the Ansys employee: innovationspace.ansys.com/forum/forums/topic/nodal-or-gauss-point-displacements/ First reply here again state that "The stresses at the integration points are the most accurate." www.eng-tips.com/threads/stress-at-integration-points-or-at-nodes.232206/ Sorry for the lengthy response, best regards.

[@DrSimulate]

@@Edge_Rider I am not sure if I understand correctly, but here are some thoughts. If you have a linear problem, e.g., elasticity, and you use linear ansatz functions, then after you solve the problem with FEM, you got u as a piecewise linear function. As you said, you can compute the strain by differentiating u. Because u is piecewise linear, the strain is piecewise constant. Same for the stress. This means the strain and the stress are constant over each element and they jump at the nodes. This means that the strain/stress is not defined at the nodes. To anyways get an estimate of the strain/stress at the nodes, one can average the different strain/stress values of the adjacent elements at the nodes. Maybe this is the reason why they say that the strains/stresses are less accurate at the nodes, but I don't know. One other thing that came to my mind is that maybe the reference you are talking about is referring to nonlinear problems, e.g., plasticity. For such problems, the material model and the update of the internal variables are computed only at the Gauß points. So in this case, one could maybe expect more accurate stress computations at the GPs.

[@Edge_Rider]

@@DrSimulate Ohhh! I think you are on to something. Yes, strain is actually not defined at nodes and constant through the elements! That is quite possibly what they are trying to say but using very confusing jargon. Why not just say strain is calculated for the element rather then saying strain is calculated at the Gaussian point... anyhow thanks for the idea. I didn't want to immediately go to chief engineers to ask before really thinking about it myself first. Although I did ask my closest co-workers, no one has any idea :). Your video and comments were very helpful, thank you again. About the plastic analysis; most of our parts are life critical so we never let them get anywhere near the plastic zone and therefore our analysis are linear. Although there are some exceptions while checking for limit maneuver loads but I don't think that is what they mean.

[@maxhullmann5660]

Great explanation so far, I've learned a lot! What would the calculation look like if the forcing term f is chosen as a complex number?

[@DrSimulate]

Can you be more specific about the problem you are interested in? Is your solution function u also complex?

[@maxhullmann5660]

@@DrSimulate I am sorry, I did it a little bit wrong. It turns out to be a little bit more complicated. Edit: The system looks like: 1 / (i * k) * u'' - u = f; where only u is complex. The physical system contains a time-alternating flux density (field source = forcing term, homogeneously distributed along x) that penetrates an electrically conductive material and therefore induces a voltage in the material that causes eddy currents and damping reaction fields (Lenz's rule). f is the flux density (e. g. 0.1 T) and k can be e. g. 2 * pi * frequency * 4 * pi * 10^-7 * 625000 u = 0 at the left and right boundary.

[@DrSimulate]

@@maxhullmann5660 Mhh. I have never worked with such a system. Did you already derive a weak form? Maybe you can discretize both the real and imaginary part of u and substitute this into the weak form (just a guess).

[@DrSimulate]

Is f periodic in time (e.g., sin or cos)? If yes, you may assume a periodic ansatz for u. If not, you may have to discretize in time (e.g., Euler discetization in time). Is the problem in 1D?

[@maxhullmann5660]

@@DrSimulate 1 / (i * k) * u''(x) - u(x) = f is 1-Dimensional and f is a constant (= flux density amplitude, e.g.). I could figure out a solution: weak form: 1 / (i * k) * Integral u''(x) * v(x) dx - Integral u(x) * v(x) dx = Integral f(x) * v(x) dx. The only difference to your example is the complex factor of u''(x) * v(x) and the additional term - u(x) * v(x). The final solution turns out to be ( i / k * K - K' ) * U = F where K = Integral N' N'T dx and K' = Integral N NT dx = IdentityMatrix * ElementLength with u0 = 0 and uend = 0. The analytical solution can be computed by u_an(x) = f * cosh( sqrt( i * k ) * x) / cosh( sqrt( i * k) * Interval_Length / 2) - f (symetrical interval, e.g. -0.015

[@omdave1008]

Literal God !

[@fadhilfatih1588]

this a very good explanation, just one tip, u really need to change ur thumbnail its not really reflecting the greatness in this video

[@gearstil]

Hello, what software do you use for making the graph ?

[@DrSimulate]

It's manim :)

[@gearstil]

@@DrSimulate Thank you!

[@Bahrzy2002]

Is this galerkin method?

[@DrSimulate]

Yes

[@Bombolbie]

Que software utiliza para sus animaciones por fa

[@DrSimulate]

It's manim :)

[@Bombolbie]

@@DrSimulate gracias

[@RaginCadjinn]

Me watching this having just relearned algebra 👁️👄👁️

[@krokodilvomnil5327]

Gibts das auch auf deutsch ?

[@DrSimulate]

Ist erstmal nicht geplant ... sorry

[@idiosinkrazijske.rutine]

Thank you very much Sir! Can you sent me FEM tutorial video for 3d Nuclear Reactor boiling case for Ansys Fluent 2019 r3 with .m download file? Thank you sir!

[@DrSimulate]

That is a very specific request 😅

[@yugalsharma2773]

@@DrSimulate man is very clear with his interests😂