Approximate Solutions - The Galerkin Method

[@pauloedmachado9137]

Nice video, but I just don't get how the load f(x) was taken out of the integral as a constant, if it's a function of x

[@Freeball99]

I realize that I wasn't clear enough about this, but I made the assumption that f(x) was constant (at the time when I took it out of the integral) - so this is the result for a constantly distributed load. I should have made a point of mentioning that! Thanks for catching it.

[@pauloedmachado9137]

@@Freeball99 oh ok Thanks!

[@Stone2home]

@@Freeball99 I had the same question; now I can quit scritching my head.

[@jpsmathematicsworld9883]

Good explanation, but I was having the same question that initially he took f as a function of x and finally converted it into constant without explanation. But he answered you so.....

[@veal4]

I have a feeling distributed load in real life also depends on displacement: f = f(x, w)

[@jimmypostwala8741]

You saved so much of my time … I heard my professor for weeks strong form weak form …. Dint understand at all what is he tryna accomplish…. What a breadth of fresh air !

[@Freeball99]

For more on weak forms and strong forms, watch this: kzbin.info/www/bejne/apeUZ2NnrJmmr6s

[@blackguardian89]

The day just become better! A long waited video and I hope that there will be many more! Thank you!

[@Freeball99]

I hope so too!

[@nickgenin21]

the best video on this topic here in KZbin. Great job, professor. You make it so clear and really easy to understand

[@hnrwagner]

best explanation I have ever seen on Galerkin Method

[@delfipotters3161]

Incredible video. I have been searching through textbooks to find an explanation for why we would want our basis functions to be orthogonal to the residual, and I couldn't find anything. Thank you so much. I understand the concept so much better now.

[@Freeball99]

From my experience, almost nobody seems to explain this.

[@sridharsingh4545]

The best explanation I have ever seen on Galerkin's Method. Thank you sir.

[@POPO-kk6nh]

Fantastic explanation!!! I couldn't resist waiting for the end of the video to leave a comment. It couldn't be more explicit and I REALLY needed to learn this method. Thank you very much.

[@omarsaleem1352]

Your Videos are awesome. I spent this weekend watching and applying the tutorials in which you explained coding by python. I can't find the right words to describe how much I'm impressed with your work. Thanks a lot for your efforts and time and I hope you will add more in the direction of python coding.

[@kenankenan6371]

Your explanation methods your approach is brilliant!!! Thank you dear professor!

[@Freeball99]

You are most welcome!

[@lonesnyder]

I have been waiting for more of your videos.😍

[@TarshitSehgal]

crystal clear explanation, u r a really good teacher!!

[@lujqian]

Very nice video and the great lecturer. Explained clearly. Thanks a lot.

[@Freeball99]

Glad you liked it

[@JavArButt]

Now I understand what I was reading for some time - thank you for this lecture

[@iuryt]

I am really impressed on how pedagogical this is

[@behzadtaghipour6394]

Thanks alot! specially the explanation of the concept of orthogonality. that was what i was looking for great job!

[@AmericanDream_VNPride]

Thank you Sir! This is a beautiful explanation that I could not ask for more. It clears my mind and give an aha moment! Salute.

[@twa5010]

Hello thank you for great visually explanation of orthogonality

[@zillenjunge]

The reference to the weighted residuals was really helpful. Interesting generalization of collocation and least squares methods, which were familiar to me.

[@frannieves8495]

Fantastic video; congratulations and many thanks.

[@Freeball99]

Glad you enjoyed it

[@xino951]

I really really should search explanations in KZbin, I needed this 3 weeks ago for my finals!

[@wangAo-gw4xu]

THANKS FOR YOUR SPEECH!

[@paulbytheriver4976]

thanks, excellent presentation with good insights. i actually derived to what amounts to the same approach many years ago, without the mathematical background using least squares curve fitting to a known function integrating over the domain instead of discrete points. rationalizing that to minimize the error in each coefficient meant finding partial derivatives in the approximating function. forms a set of linear equations.

[@nazmdar]

Fantastic. Thank you very much for this nice explanation.

[@johnsmithsu310]

Explanation was Awestruck!!! I hope in future video you will share us about explanation and problem solving Space Frame 🙏

[@masoudmirzaei8243]

Excellent

[@minnimau1]

That really helps me for my exam! Watching the video makes it so much clear and when Im now reading the chapter in the book everything makes sense! Keep it up!!

[@rezamadoliat2074]

Thank you very much for your comprehensive and complete explanation.

[@uqyge]

brilliant.every stem student should take a look. best on garlerkin method

[@SeverSpanulescu]

Execllent tutorial, I would recommend this to all students. Just a small correction (non-important here): at 7:02, the Dirac function is not 1 at x=xn. You intended to say that Its integral from minus to plus inf is 1.And the integral of a function multiplied by Dirac function gives the value othe function in xn. This gives indeeed the well-known relation 4.

[@Freeball99]

Yes, you're exactly right! I misspoke. I should have said that the integral of the function is 1. The math, however, is correct.

[@ibrahimozturk0671]

THIS IS GOLD

[@هيثمماجدخزيم]

I salute you very much

[@mohamedat5169]

Great video At 17:28, why do we care that the residual which is in terms of forces, be orthogonal to our functions for displacement which describe space? Intuitively it would make more sense if we had the difference between the true solution and our approximate solution be orthogonal to our approxmate solution. Could you clear this up?

[@Freeball99]

We are attempting to solve the equations of motion, which is why we apply the orthogonality condition there. As a result, we get a constraint on the derivatives of the approximate solution (4th derivative in this case). This is actually a more (much much more) stringent constraint than any constraint involving simply w itself. This is because differentiating an approximate solution increases the error and integrating it smooths out the error.

[@nicolasramirez3944]

Awesome! Just took a finite element course last semester, and this explanation was wonderful

[@nicolasramirez3944]

Thank you for working out the example, but this is only the solution for a constant load f correct? The discussion on orthogonality was helpful.

[@Freeball99]

Yes. The final result is for a uniformly distributed load. I make this assumption at 29:35 where I take f outside of the integral because it is constant. Up to that point, it is valid for any distributed load.

[@nicolasramirez3944]

@@Freeball99 Thank you!

[@NguyenDinh23]

That's great video. Thank you so much

[@Freeball99]

You are welcome!

[@meysamjafari]

This course has been incredibly enlightening! 🌟At 24:15, just to clarify, when we introduce w tilde, does it imply that x, x^2, x^3, and x^4 are our shape functions?Instead, it appears that we're making assumptions about its functional form based on the number of boundary conditions, as highlighted in the steps. Then, we proceed to determine its coefficients in terms of one of these conditions. Subsequently, utilizing d(w tilde) / d(ai), we derive the Phi'i. But do we predefine Phi's or derive them? It's a fascinating process! 🤔💡

[@Andy-hy8px]

No, x, x^2, x^3, and x^4 are not our shape functions because our shape functions need to satisfy ALL the boundary conditions. Instead I am showing here how one can easily find a shape function using a polynomial approach and then applying the boundary conditions to determine (some of) the constants. Since we have 4 boundary conditions, we need at least 5 constants; 4 of them in order to satisfy the BC's and then at least one more which will be determined using the Galerkin Method. After substituting the boundary conditions to eliminate 4 of the constants, I arrived at the form in equation 25 (27:35). This now in the form of a constant multiplied by a shape function (which satisfies all BCs).

[@meysamjafari]

Thank you for the explanations and for taking the time to share your insights! Much appreciated! 🙏 @@Andy-hy8px

[@cp3408]

Keep it up!!

[@PATHMINDER]

Thanks in advance.

[@thedorantor]

Great video, thank you! I was wondering what the limitations are for this method? Can it be used for every DE?

[@Freeball99]

While the Galerkin method can theoretically be applied to many DEs, its effectiveness varies. It's particularly well-suited for linear and some nonlinear problems in structural mechanics, heat transfer, and fluid dynamics. For very complex systems, hybrid or specialized methods might be more appropriate.

[@Shivankaes]

Fantastic Video, Can you suggest some of the books where i can find these type of crux and make my knowledge of FEM stronger.

[@Freeball99]

For classical FEM, anything by Klaus-Jurgen Bathe and for more modern approaches, meshless methods, etc. I'd recommend the books by Satya Atluri.

[@ghufranullahkhan7479]

Thanks a lot for this. It will be great if we get videos on the finite element implementation in Matlab or python.

[@Freeball99]

We're going to get to finite element theory soon in the context of variational principles. However, in the meantime, I do have a playlist with some introductory FEM videos and Python code to accompany it. kzbin.info/aero/PL2ym2L69yzkaue8Ly2Oz51LALRzUV8LZ0

[@Chaabast]

Hi ! Thanks for this very clear explanation. Just a question, how do you assume the shape functions such as phi = [1 X X^2 X^3 X^4], and how am I supposed to know what phi must be in more complex situations ?

[@Freeball99]

Note that [1 x x^2 x^3 x^4] is not our shape function because our shape functions need to satisfy ALL the boundary conditions. Instead I am showing here how one can easily find a shape function using a polynomial approach and then applying the boundary conditions to eliminate some of the constants (there are various are other approaches too - textbooks have been written on the subject of picking shape functions for different systems). Since we have 4 boundary conditions, we need a minimum of 5 constants for the polynomial approach - 4 of them in order to satisfy the BC's and then at least one more which will be determined using the Galerkin Method. This is why we choose a 4th order polynomial. After substituting the 4 boundary conditions to eliminate 4 of the constants, we arrived at the approximate displacement field in equation 25 (27:35). From this approximate Φ_0 = 0 too).

[@jv2781]

Great video as always. Are you using the "Solid Mechanics a Variational Approach" by Clive and Shames for this like your previous videos?

[@Freeball99]

Nope. I found Dym [sic] and Shames to be somewhat lacking in their explanation of the Galerkin Method. For this video I used a combination of my class notes and some articles that I found online. I didn't follow a single source, but rather just gathered a bunch of information from various places and tried to include what I thought were the best parts.

[@maxip.4380]

Great video. I got lost in the beginning. What is phi_i? And what does it stand for?

[@Freeball99]

The φ_i's are the shape functions. The process begins by assuming an approximate displacement field consisting of constants and shape functions - i.e. w_approx = a_1 φ_1 + a_2 φ_2 + a_3 φ_3 +... with the φ_i's satisfing the boundary conditions.

[@insainsin]

The dirac delta function is equal to infinity, not 1, at x=x_i.

[@Freeball99]

I misspoke. I should have said that the integral of the function is 1.

[@andyvald3s]

What books are good for introducing both the Galerkin method, the variational method and weighted residue?

[@Freeball99]

For Variational Calculus, you can try Dym & Shames, "Solid Mechanics: A Variational Approach" I don't really have any good ones for Galerkin and weighted residuals on the top of my head. Many textbooks explain this stuff rather poorly, which is why I wanted to make a video - although it's going to require several more to really cover the topic well. Perhaps your could try KJ Bathe's "Finite Element Procedures" if you can find a copy of it somewhere. Anything by Bathe on the subject should be good.

[@mihkelkorgesaar4368]

How was this video made? It seems that you have prewritten the slide and here u somehow show the specific equations while recording your voice? Or did you use some video making software (premiere pro?). Whst soft are u making notes? I am asking cause i need to do the same with min effort. I did premiere pro once but this was quire an effort… anyway great job, love the videos

[@Freeball99]

The app is called "Paper" by WeTransfer. It is running on an iPad Pro 13 inch and I am using an Apple Pencil. Parts of the slide can be deleted and then recovered by using the "undo" function. By selectively deleting pieces of the slide (in reverse order) and then undoing this, the pieces of the slide can be displayed as I present it. I am using Quicktime to capture the screen recordings and audio and then I edit the videos using iMovie to allow them to flow. Pro Tip: Make sure to duplicate your slides before attempting this. The app can be a little buggy at times and quit midway - in which case the deleted parts of the slide cannot be undone and remain deleted and you can end up losing your slide.

[@rezamadoliat2074]

many thanks for your excellent explanations. hopefully, the extension of your Galerkin's method for a real problem will appear later. I mean those problems which do not have exact solutions. for the solved problem, how we can use a trial solution of lower degrees such as a third-order polynomial. In finite elements, we mainly look for an approximate solution. getting a higher-order degree polynomial would be fine, but what about considering the lower-order polynomial?

[@Freeball99]

I will certainly be making additional video showing examples of the Galerkin Method. Was impossible to cram all of it into one video. So I began with something simple for which we knew an exact result and will extend it from there. This will include (likely next) showing how to solve this same problem using a weak formulation (including 1st order shape functions) and also the effect of increasing the number of terms in the approximation of the displacement. Will also include a video showing example(s) for which exact results are not possible. Just trying to break this all down into bite-sized chunks.

[@nebiyoukassahun618]

Dear sir this method checks not only orthogonality of two function but co linearity of two functions as long as they dont make up an area in between them, they will satisfy this equation.

[@Freeball99]

Not sure I follow this. How does this check for co-linearity?

[@nicolasramirez3944]

Any suggested further reading on orthogonality of functions? I am TOTALLY with you in your "analogy" of minimizing error when it's thought of as a vector resulting in an orthogonal projection... But to make the jump into functions I just follow by faith. Is this a lienar algebra topic?

[@Freeball99]

Yes, it is a linear algebra topic. Try to find something on function spaces.

[@Freeball99]

This might be of interest... math.stackexchange.com/questions/1209408/why-is-a-function-space-considered-to-be-a-vector-space-when-its-elements-are

[@steveshaver4000]

Hi! What are the units of the third derivative? Can you explain how you obtain shear from approximate displacement? Also, what are the units of the other derivatives?

[@Freeball99]

With each derivative, you are effectively dividing by distance. The unit of displacement is the unit of distance (like m or ft). So w, x is the slope and has units of radians (which are dimensionless). The 2nd derivative w, xx has units of "per unit length", but really it's radians/length and is a measure of curvature. When multiplying by EI (which has units like Force x distance^2), then this becomes a moment (force x distance). Taking the 3rd derivative of the displacement, w,xxx gives us units of radians/(length^2) and multiplying this by EI give us the shear force. So we get from the approximate displacement to the shear force by taking the 3rd derivative and multiplying by EI.

[@bettercallsha0]

Great content! In orthogonal projection part, the explanation seems to correspond better to least square method and more assumptions are necessary to connect the dots to galerkin. In the projection examples, the residual is linearly dependant on the shape function (basis vectors), thus least square and galerkin are identical. In the strong form of the beam equation, the residual is linearly dependent on the 4th x derivative of the shape function, thus least square demands we use d(w_xxxx)/d(a_i) as the basis vectors as those are the linearly contributing components to the residual, which makes perfect sense to me. But galerkin instead uses d(w)/d(a_i) as the basis, the reason for this substitution is unclear in the explanation (also unclear to me). I assume some additional assumptions are made about the nature of the beam equation? I hope I am making sense😅 Thank you for the great content!

[@Freeball99]

Yes, your comment makes sense. I am unsure too how Galerkin arrived at his choice of shape function. Since his paper was written in Russian (which I don't understand), and since I have not found any suitable translations or explanations for why he chose this weight function, I too don't have the answer to this. Interestingly, I haven't been able to find any articles or papers that explain it. All of them seem to just accept it rather than offer an explanation as to why it is. I tried working through it mathematically where I assumed that the weight could be treated as the variation of the displacement (even though all the texts tell us that the Method of Weighted Residuals is not based on a variational principle). I seemed to get close to proving that using Galerkin's weighting method was the equivalent to converting the Method of Weighted Residuals into a variational method (like the Ritz Method), but I couldn't quite get the math to work out. This is why, the best answer I could arrive at is that the shape functions provide a basis to the space and that the orthogonal error projection means there is no error in the directions of each basis vector and thus no error within that space spanned by the basis vectors. BTW - I don't believe this has to do with assumptions regarding the nature of the beam since this method can be used for solving differential equations in general for all sorts of problems - not just beams. Thanks for your feedback. If you manage to figure it out, I would love to hear about it.

[@mayureshsalunkhe3732]

How to solve coupled differential equations in 2-D using the Galerkin method, which contains two dependent variables and two independent variables?

[@Freeball99]

I will need to make a video on this. It's going to be impossible to explain it in the comments I'm afraid.

[@riadelhamoud8224]

Thanks a lot !!! Your videos are short but very important. Could you at the end some references, so i can dig more 😁

[@Freeball99]

Try Dym & Shames, "Solid Mechanics. A Variational Approach"

[@legendary_egg]

I believe that once you move the residual outside the integral in Eq. 29 (with the assumption of a constant f) the entire integral could have been cancelled right away. I assume that you performed the subsequent calculations for pedagogical reasons? Also, thank you for the excellent videos!

[@Freeball99]

Yes. I was trying to demonstrate that the integral did, in fact, go to zero as it's supposed to.

[@ΜΙΧΑΗΛΚΑΤΤΗΣ]

Can we see the method on a problem that requires a numeric solution?

[@Freeball99]

Yes. I will be showing several additional examples using the Galerkin Method.

[@christosgeorgiadis]

I'm a little bit confused. What's the difference between mode shape and shape function? Are these two concepts related in a manner?

[@Freeball99]

Using a shape function is a technique to distribute the nodal displacements of the beam along its length. In general, these shape functions need to satisfy exactly the boundary conditions, but beyond that they simply approximate the displacement at all other points along the beam - ie it provides a spatial representation of the displacement and other properties at every point along the beam. The mode shape can be thought of as a special case of the shape function which represents the EXACT distribution of the displacement over the length of the beam. While using the mode shape will always give more accurate (exact) results, it is often hard to find. For this reason, we use shape functions that simply satisfy the boundary conditions in order to provide approximate solutions.

[@gloryforthehord6575]

Hi ! Thank you for this video, it helped a lot ☺ There's something I still don't understand, though. It is quite clear to me that we want the residual R(x) to be orthogonal to the function basis in which we express the force F\tilde, but I don't understand why this can be extended to the function basis phi_i(x) in which we express the approximated displacement W\tilde. I would rather have R(x) orthogonal to the psi_i(x) basis which is the basis used to approximate F\tilde. But then, how can we link the psi_i(x) basis to the phi_i(x) basis ?

[@Freeball99]

It's hard to know exactly why Galerkin came up with this method because I haven't yet found a translation of his paper nor an explanation in English. My understanding, however, is that since we are finding an approximate solution, which exists in the function space defined by the shape functions used in the response, it seems that by ensuring that R(x) is orthogonal to the shape functions, it means that R(x) does not appear in the solution/response space. By adding more shape functions to the response, we thereby drive the residual error to zero.

[@barrysilver2075]

are you from SA ? greetings from an ex-toti character ...great video and nice explanation

[@Freeball99]

Ja. From Durbs originally.

[@barrysilver2075]

@@Freeball99 Lekker by die see ! ... on a more serious note ...do you know of any nice and easy reference books etc that could help me with recasting PDE's into variational form

[@Freeball99]

@@barrysilver2075 My goto text on Calculus of Variations is "Solid Mechanics: A Variational Approach" by Dym & Shames. You can probably find a PDF online. Not sure that it specifically shows how to cast PDEs into variational form, but there's likely enough background on variational principles in there to allow you to figure it out.

[@SuperDeadparrot]

What if you just subtract lines 1/ from 2/? That gives E I ( W,xxxx - W~,xxxx ) = -R( x ) and work from there?

[@Freeball99]

Sorry for the delayed response, but I somehow missed this until here... The problem with this is that we do not know what w is. This is why we are trying to find an approximate solution.

[@sogolahadi-x8l]

Orthognal in vectors is diffrent with functions i don't under stand

[@Freeball99]

Try to find a book or articles on function spaces (which is a topic of linear algebra). Here is an example math.stackexchange.com/questions/1209408/why-is-a-function-space-considered-to-be-a-vector-space-when-its-elements-are

[@comment8767]

Thank you for not mentioning variational calculus.

[@Ed-of8rf]

Great video, but unnecessary parameter changes have made a simple equation confusing. When using ai, you may not change it to ci, or wi, etc.

[@Freeball99]

The a's and c's are just dummy variables which is why they can be readily interchanged.

[@fawgawtten9515]

Please be my professor

[@wangAo-gw4xu]

THANKS FOR YOUR SPEECH!

[@nellyclg8038]

Great video, thank you so much !