The Finite Element Method (FEM) - A Beginner's Guide

Рет қаралды 119,124

Күн бұрын

📈 APEX Consulting: theapexconsulting.com
🌎 Website: jousefmurad.com
In this first video, I will give you a crisp intro to the Finite Element Method! If you want to jump right to the theoretical part, timestamps are in the description! :)
📋 PDF of the presentation: / introduction-to-33013510
ONLINE PRESENCE
================
🌍 My website - jousefmurad.com/
💌 My weekly science newsletter - jousef.substack.com/
📸 Instagram - / jousefmrd
🐦 Twitter - / jousefm2
SUPPORT MY WORK
=================
👕 Science Merch: engineered-mind.creator-sprin...
👉 Become my Patreon: / theengiineer
🧠 Subscribe for more free videos: bit.ly/2RLmMxq
RECOMMENDATIONS
==================
🎥 1 Month Free of Skillshare Premium: bit.ly/3E3bXwC
📕 Books on AI, Data Science & Programming: bit.ly/2I1lUD1 - use "podengineered20" to get a 40% discount!
🎵 Mahmut Orhan - Game Of Thrones (Original Mix)
------------------------------
As a member of SimScale, some pictures in this video have been used from the SimScale platform. SimScale is the world’s first production-ready SaaS application for engineering simulation, completely cloud-based and easily accessible through a web browser. Try CFD and FEA now by creating a free Community account in minutes: bit.ly/316Aqgt
Contact:
If you need help or have any questions or want to collaborate feel free to reach out to me via email: support@jousefmurad.com
#FEM
#FiniteElementMethod
#Engineering
Time Stamps
----------------------
0:00 - 0:12 : Intro
0:13 - 0:36 : Agenda
0:37 - 4:40 : History of the FEM
4:41 - 5:41 : What is the FEM?
5:42 - 6:12 : Why do we use FEM?
6:13 - 6:43 : How does the FEM help?
6:44 - 9:14 : Divide & Conquer Approach
9:15- 9:26 : 1-D Axially Loaded Bar
9:27 - 13:24 : Derivation of the Stiffness Matrix [K]
13:25 - 14:09 : Global Assembly
14:10 - 14:44 : Dirichlet Boundary Condition
14:45 - 15:00 : Neumann Boundary Condition
15:01 - 16:12 : Element Types
16:13 - 16:33 : Dirichlet Boundary Condition
16:34 - 16:46 : Neumann Boundary Condition
16:47 - 17:06 : Robin Boundary Condition
17:07 - 18:13 : Boundary Conditions - Physics
18:14 - End : Outlook & Outro

Пікірлер: 154

@JousefM 4 жыл бұрын

🎓 My Science Courses - courses.jousefmurad.com/ ✍️ Latest blog posts: jousefmurad.com 📥 My Newsletter - jousef.substack.com/ Time Stamps ---------------------- 0:00 - 0:12 : Intro 0:13 - 0:36 : Agenda 0:37 - 4:40 : History of the FEM 4:41 - 5:41 : What is the FEM? 5:42 - 6:12 : Why do we use FEM? 6:13 - 6:43 : How does the FEM help? 6:44 - 9:14 : Divide & Conquer Approach 9:15- 9:26 : 1-D Axially Loaded Bar 9:27 - 13:24 : Derivation of the Stiffness Matrix [K] 13:25 - 14:09 : Global Assembly 14:10 - 14:44 : Dirichlet Boundary Condition 14:45 - 15:00 : Neumann Boundary Condition 15:01 - 16:12 : Element Types 16:13 - 16:33 : Dirichlet Boundary Condition 16:34 - 16:46 : Neumann Boundary Condition 16:47 - 17:06 : Robin Boundary Condition 17:07 - 18:13 : Boundary Conditions - Physics 18:14 - End : Outlook & Outro

@humane11 3 жыл бұрын

hey bhaijaan, PLEASE PLEASE PLEASE MAKE A VIDEO soon on the future of Finite element method/ Computer aided Engineering , is it gonna die or demand gonna decrease drastically due to Artificial intelligence, in the upcoming 5-10 years ?? Please make a detailed specific video...

@JousefM 3 жыл бұрын

@@humane11 I cannot predict the future but something I can definitely cover soon. :)

@humane11 3 жыл бұрын

@@JousefM eagerly waiting for it !

@abrorabyyu6221 3 жыл бұрын

I'm always loved someone who tell history before explaining content

@JousefM 3 жыл бұрын

Appreciate it buddy :)

@karmayogi1439 2 жыл бұрын

Same here

@karmayogi1439 2 жыл бұрын

Same here

@w.k8624 Жыл бұрын

@@JousefM why you are showing direct stiffness method, and not telling it is direct stiffness method.

@mukulbondade9559 Жыл бұрын

Amazing video Jousef, keep it going, these videos are very interesting

@xaverpfk 2 жыл бұрын

Very good overview about FEM. Thanks!

@alihaider5027 Жыл бұрын

Thank You SO Much Jousef, An excellent piece of information in a very organized manner. Thank You!

@JousefM Жыл бұрын

Cheers bud!

@ingGS 3 жыл бұрын

This was a beautiful introduction. Keep uploading videos!

@JousefM 3 жыл бұрын

Appreciate your kind words my friend 👊

@jorgealejandroaguirrelara 2 жыл бұрын

Of all videos I've watched about FEM, for me this was the easiest to fllow. Thank you for this great and clear explanation!!!

@JousefM 2 жыл бұрын

Thanks my friend, appreciate it 🙂

@A5Gold 3 жыл бұрын

Great Video! Can't wait for the next one on FEM.

@JousefM 3 жыл бұрын

Thank you David :) Stay tuned!

@nikkig1146 3 жыл бұрын

Thanks for this clear and helpful video! I like that you focus on the mathematics of FEM because some other videos do not go into that much detail.

@JousefM 3 жыл бұрын

Thanks a lot :) there’s more to come on fem maths soon

@subashchandarn6667 3 жыл бұрын

Video was very helpful in understanding the basic working of FEM as a newbie to FEM .I appreciate your efforts in making this video and I expect more videos in this FEM series

@JousefM 3 жыл бұрын

Thanks a lot :) there’s more to come!

@nouraldeenalmalohe2644 Жыл бұрын

شكرا لك من أبسط وأوضح الشروح التي شاهدتها

@insertnamehere3836 4 жыл бұрын

Thank you for this video! Really excellent tutorial

@JousefM 4 жыл бұрын

Thanks a lot, really appreciate it!

@saleemjaveed6226 4 жыл бұрын

Great effort Jousef. Waiting for more upcoming videos with lil more indepth topics like hourglass, shear locking instability of elements, negative volume, and common errors while running the analysis. Thanks.

@JousefM 4 жыл бұрын

Appreciate it Saleem 🙂 Stay tuned my friend!

@MDougan98 3 жыл бұрын

Clear and concise, thanks very much!

@JousefM 3 жыл бұрын

You’re welcome! 🙂

@markomarkomarko 4 жыл бұрын

Good video. Errors are always very useful to learn about.

@JousefM 4 жыл бұрын

Already on my to-do list! :) Thanks for your comment mate - stay tuned and feel free to invite your friends as well! Enjoy your weekend!

@OlliLukkari 4 жыл бұрын

Great video, thank you! Subscribed!

@JousefM 4 жыл бұрын

Thanks Olli! Appreciate it :)

@smurface549 2 жыл бұрын

I can't understand why this video doesn't have more views. It's a concise introduction to FEM and easy to follow along.

@JousefM 2 жыл бұрын

Thanks mate :)

@sandrotorres1298 3 жыл бұрын

Awesome! Please keep doing what you do. God bless!

@JousefM 3 жыл бұрын

Thanks a lot for your kind words Sandro :)

@LaraxHUN16 4 жыл бұрын

Fantastic video, keep it up! :)

@JousefM 4 жыл бұрын

Thanks a lot bro :) Appreciate your comment!

@MrBLAZE707 4 жыл бұрын

Danke dir für die ganz einfach erklärt :)

@JousefM 4 жыл бұрын

Danke Yousef! :) Es kommt noch mehr, stay tuned!

@shihanjanith1804 3 жыл бұрын

Nice one.this video is my first step on finite element world . Got a good idea about it..thankz again.👍👍

@JousefM 3 жыл бұрын

Sure :) new podcast on FEM dropping in a few hours. Stay tuned!

@ginersergioadrian1489 2 жыл бұрын

Excellent, clearly explained. My goal is to apply FEM to Chemical Engineering problems, but so far I have not found a course or a book that I can understand, even when I am a PhD in simulation!. I hope with your classes I would be able to improve. Thanks very much

@JousefM 2 жыл бұрын

Thanks Giner my friend! As for courses I’m working on general ones in the beginning and go also for things such as OpenFOAM or SimScale. Make sure to follow me on my socials. My learning Academy will go live in a few days :)

@salsabilahmedshrestho960 9 ай бұрын

Absolutely fantastic video joseph. I always want to know some historical background so that I can appreciate that subject more. Thank you so much for arranging all of the data so neatly in one video. Please don't delete this video ever.

@JousefM 9 ай бұрын

Thanks my friend!

@gregdtlite7106 3 жыл бұрын

very nice job, continue your work!!!

@JousefM 3 жыл бұрын

Thanks a lot :)

@olevel_maths1470 2 жыл бұрын

FEM video 01 - Introduction and bar element kzfaq.info/get/bejne/i66Cot1j08_Tg4k.html FEM video 02 - Bar element example kzfaq.info/get/bejne/bNp7e6tp39bYco0.html FEM video 03 - beam element kzfaq.info/get/bejne/sM-feqV-2deWgHU.html FEM video 04 - beam element example 01 kzfaq.info/get/bejne/sM-feqV-2deWgHU.html FEM video 05 - beam element example 02 with UDL kzfaq.info/get/bejne/kLmDhruDyZaqk4U.html

@Akkachchi 4 ай бұрын

I searched alot about finite element on you tube , this only provides me required information.If possible please upload some more about finite element analysis.Thank you

@Arman-go6mo 3 жыл бұрын

Thanks a lot Jousef. may I know when will you upload the videos covering the topics which you mentioned at the end of this video ?

@JousefM 3 жыл бұрын

Hey Arman! Actually my viewers decide that. If you follow me on my socials, you might have seen that I am currently working on a small Cousera "Guided Project" which I will release soon, that will be FEA for absolute beginners. In the upcoming guided projects, I will include some of the topics listed at the end of the video. If you want to stay up-to-date with my progress there, either follow me on Instagram and/or Twitter :)

@ihebchouikh9816 4 жыл бұрын

Nice video ,thank you :)

@JousefM 4 жыл бұрын

Thanks a lot my friend! :)

@shreyanshajmera6297 4 жыл бұрын

Really good introduction to FEA for someone who already has some background about it. I hope you will get into mathematics with this same ease. Wishing you all the best for all upcoming videos.

@JousefM 4 жыл бұрын

Will do my best Shreyansh, thanks for your kind words! A dedicated series on Linear Algebra is planned :)

@shreyanshajmera6297 4 жыл бұрын

@@JousefM I was talking about FEA mathematics. Well good that you are thinking of Linear Algebra as well. Also you mentioned Gilbert Strang, so till now his videos are the best place for me for Linear Algebra.

@JousefM 4 жыл бұрын

We need a bit of Linear Algebra but FEM maths will follow I know what you mean, like weak form etc. - Gilbert is the best, will be very Hard to beat the quality!

@CMEAcademyforPPSC_FPSC 8 ай бұрын

Really good review

@JousefM 8 ай бұрын

Thanks mate!

@ayyappajammula730 4 жыл бұрын

Good information 👌👌 Thanks

@JousefM 4 жыл бұрын

Thanks my friend :)

@btyan1 4 жыл бұрын

Nice Video on FEM

@JousefM 4 жыл бұрын

Hi Yannick, thanks a lot my friend! Stay tuned for more and make sure to activate the notification bell :)

@artyquantum7283 3 жыл бұрын

Your way of teaching is really great. Can you please provide some example using calculations of actual function like e.g: f(x, t) = x^t etc. or a simple differential equation like a wave motion. And for next lesson if you can extend it to a 3D lattice approximation that would be awesome.

@JousefM 3 жыл бұрын

Appreciate it! And for these specific topics it might take a bit but I’ll definitely write that down on my to-do list 😌

@artyquantum7283 3 жыл бұрын

@@JousefM Thanks a lot :)

@vinven7 3 жыл бұрын

Loved the video Jousef. Clearly explained. What is the model of the skull you showed in the video - the one with a circle in the middle of the forehead?

@JousefM 3 жыл бұрын

Thanks Vineeth! That’s a fem simulation of a skull with and without a helmet which you can find in SimScale’s public projects. :)

@andrenedaei965 3 жыл бұрын

Very nice video and good explanation. I could not find your next video about other topics such as singularity,Linear and Non-linear?, Would you be able to send a link here?

@JousefM 3 жыл бұрын

These were all suggestions from my side but I will definitely will work on them soon :) I am planning to release a course on FEM soon so stay tuned for that as well :)

@andrenedaei965 3 жыл бұрын

Great Jousef and I am looking forward to participating in the course.

@sriti_hikari 4 жыл бұрын

Find's super!

@JousefM 4 жыл бұрын

Thanks! ♥️

@timeajozsa3806 2 жыл бұрын

Did you cover the other topics mentioned at end of the video. I cannot find more videos related to FEA. I would be interested in some of the topics, hourglass, shearlocking, manage projects etc.

@JousefM 2 жыл бұрын

Coming in the future. Getting back to more theoretical videos from FEA,CFD and AI :)

@richardbrewin4936 3 жыл бұрын

Can you create an FEA report from an engineered blueprint? I have a couple of pieces of equipment I’d like to get a report on

@JousefM 3 жыл бұрын

Could you contact me via email please? Send a request to engineeredmindbusiness@gmail.com

@atefrod680 2 жыл бұрын

Where can the follow up videos be found? Very informative video btw!

@JousefM 2 жыл бұрын

In planning! Currently working on more Coursera courses and my own ones, but I have already my video production list for FEA ready. What would you be personally interested in?

@atefrod680 2 жыл бұрын

@@JousefM Im currently studying mechanical engineering and would like to obtain a more fundemental understanding of FEM. But also get familiair with ots limitations.

@patelpavan5479 10 ай бұрын

Which method is best for boundry value problem fem or fdm

@afaqahmed7591 3 жыл бұрын

HI Murad. Very well described video. When is the next video coming if I may ask? cheers

@JousefM 3 жыл бұрын

Plenty in the pipeline - as I am currently finishing up my first coursera course I’m a bit behind for videos. What would you personally like to see? :)

@afaqahmed7591 3 жыл бұрын

@@JousefM thanks for your reply.. I would love to see more videos related to FEM

@JousefM 3 жыл бұрын

Sure! Planned 😌

@seifsalem2484 3 жыл бұрын

amazing video put i want to understand some abbreviations what is U stands for and the L ? epsilon

@JousefM 3 жыл бұрын

Good point Seif! u = displacement L = Length of the beam epsilon = strain For more details, have a look at this page: www.engineeringarchives.com/les_mom_truestresstruestrainengstressengstrain.html#:~:text=Engineering%20strain%20is%20the%20amount,length%20as%20given%20by%20Eq4.

@realisticmonk804 4 жыл бұрын

Very informative. Is there any MS program for FEM specifically or perhaps a course.

@JousefM 4 жыл бұрын

Are you a beginner? The upcoming videos from me will definitely help you out, I planned a course for the future with applications as well as programming exercises which does not exist in that sense so far. Let me know what you think!

@realisticmonk804 4 жыл бұрын

@@JousefM thank you for a quick reply. Actually I am working in preprocessing stage as meshing, connection and assembly of full vehicle for crash domain for 2 years. I wanted to know if any specific MS program is there.

@JousefM 4 жыл бұрын

@@realisticmonk804 For crash simulatin in general I would recommend LS-Dyna, they are quite famous in that field and companies like Porsche use their software to simulate crash tests.

@realisticmonk804 4 жыл бұрын

@@JousefM I am waiting for your next video

@JousefM 4 жыл бұрын

@@realisticmonk804 Thanks for being such an active subscriber :) I already uploaded another one about 2nd order differential equation and how to solve it in MATLAB. By the way, I have started a poll which you can find on my channel in the "Community" tab. Feel free to vote what you want to see next! :)

@zss8290 2 жыл бұрын

Moin, Jousef, also habe ich mich schon bei Cousera an Ihre Projekt teilgenommen, aber welche Kontaktmedien ist besser, wenn ich Frage zu dem detalisierte Dinge des Projekts Ihnen stellen könnte? Also Hilfe brauche. Danke schön! :)

@JousefM 2 жыл бұрын

Moin und danke :) Am Besten über Instagram oder Twitter. Versuche dann so schnell wie möglich zu antworten! :) 🐦 Twitter: @jousefm2 // twitter.com/Jousefm2 📸 IG: instagram.com/jousefmrd/

@nishantmarkad4519 Жыл бұрын

linear vs non-linear please

@gregreilly326 Жыл бұрын

Thanks for this excellent video. In your example of a 1-d axially loaded bar, I expect what ultimately you want to find is the displacements {u}, right? In on your equation [K]{u}={F}, you know [K] but you don't know {F}, and you're trying to find {u}, so don't you have too many unknowns? If possible I'd like a simple example where you get numeric answers, not just a relationship between {u} and {F}. If I know F4, can I write F2 and F3 in terms of F4, something like F3=-F4 and F2=-F3? Then I know all the Fs, and I can solve for {u}. Is that how you could use this equation?

@JousefM Жыл бұрын

Can do! A draft is currently on my website - hope to push it live in a couple of days/weeks

@gregreilly326 Жыл бұрын

@@JousefM Looking forward to it. Thanks!

@JousefM Жыл бұрын

Here's one blog post: www.jousefmurad.com/fem/the-finite-element-method-derivation-discrete-system-1/ - an arithmetic example will follow.

@tarangmehta4335 4 жыл бұрын

Hi, currently i am studying FEM but i am also suggested by professor to look into thermoelasticity problem... Can you suggest me some sources?

@JousefM 4 жыл бұрын

Nice topic Tarang! I think that this here is a good start: ethz.ch/content/dam/ethz/special-interest/baug/ibk/structural-mechanics-dam/education/femII/presentation_09_thermomechanics_v2.pdf - I can see if I find some good books that I could recommend.

@tarangmehta4335 4 жыл бұрын

Thanks Jousef... Its so nice of to reply this soon... I will definitely look to this presentation... And also eagerly waiting for your next videos so as to revise my FEM...

@JousefM 4 жыл бұрын

@@tarangmehta4335 Thank you very much mate, appreciate your support :)

@md.sarowarhossainrana4787 Жыл бұрын

Don't we need weak form for discrete system?

@marytheraspberry3145 7 ай бұрын

Hello Mr. Jousef. I'm taking the coursera course that you made about FEA, and in the tuto, you clicked a switch that activated 2nd order meshing. I can't find that option though. Maybe there was a simscale update and they removed that. What do I do instead to get second order meshing?

@JousefM 7 ай бұрын

Check out their documentation

@marytheraspberry3145 7 ай бұрын

@@JousefM I haven´t found any article in the documentation about it. Recently, I clicked the event log of what you labeled Mesh Linear and it indicated it was 2nd order. So now I will need to set a first order mesh, but I don´t know how, since the order is automatic.

@JousefM 7 ай бұрын

@marytheraspberry3145 the forum helps! 🙂

@MachineLearningwithPhil 4 жыл бұрын

Just seeing this now. Comment and like for that ALGO.

@JousefM 4 жыл бұрын

Thanks a ton Phil, appreciate it mate! :)

@Danaili6i 4 жыл бұрын

Nice video! Did you study/work in Germany? It looks/sounds like it at least! Freundliche Grüße :)

@JousefM 4 жыл бұрын

Thanks a ton for your feedback, really appreciate it! Yes I work and study in Germany :) Not sure if that’s a good thing if you can hear that 😄😋

@Danaili6i 4 жыл бұрын

@@JousefM Mach dir kein Stress, hat nichts mit Deutsch Akzent zu tun ;) Ich hab selber Maschinenbau in Deutschland studiert...Deine Erklärungen hören sich bekannt an :)

@JousefM 4 жыл бұрын

@@Danaili6i Ah verstehe ;-) Wo hast du studiert? Nicht zufällig am KIT, oder? :D

@longnguyenbuitien3521 2 жыл бұрын

Hi, thank you for your video. I am a newbie in CAE and I want to learn more about FEM/CFD. So, can you recommend for me some courses teaching FEM/CFD basics? Thank you so much.

@JousefLITE 2 жыл бұрын

Sure thing mate! :) Thanks for watching. I can recommend my small Coursera courses for hands-on projects: www.coursera.org/instructor/jousefmrd - these do not go into much theoretical details. I can recommend the courses from Lukasz Skotny or Dominique Madier, experts in their field and very good teachers! I am working on a CAE Masterclass as well as separate CFD & FEM courses here: academy.jousefmurad.com/ - sign up for the course newsletter, no spam :)

@longnguyenbuitien3521 2 жыл бұрын

@@JousefLITE thank you. I'll explore it right now.

@JousefM 2 жыл бұрын

Cool! And let me know if you’d want to see something special in the course such as a topic or concept explained Support@jousefmurad.com

@stauffap 3 жыл бұрын

I think i need simple examples that we can solve by hand to really understand this (for a simple grid). Is that even possible for very simple systems. Maybe solving at least for one timestep?

@JousefM 3 жыл бұрын

Good point! You don’t even need it in terms of time steps but can solve easy FEM systems by hand. Will prep something for the future - stay tuned! 🙂

@stauffap 3 жыл бұрын

@@JousefM Thanks for the reply! Well, i was talking about CFD mainly. I don't understand yet how i would solve some form of the navier stokes equation to go from some initial condition to the condition after one timestep (in a small 2D grid).

@JousefM 3 жыл бұрын

Ah got you! Thought because you were commenting on the FEM video you also were referring to it :) But definitely possible to do that. The easiest and most intuitive way would probably to explain this with Finite-Differences

@deepjyoti9499 3 жыл бұрын

Simscale use CodeAster for FEA.

@JousefM 3 жыл бұрын

That’s running in the back, correct :)

@sarowarhossain9029 Жыл бұрын

Where is the next video?

@BahaaAbdElBaset14 4 жыл бұрын

1.30

@JousefM 4 жыл бұрын

Is that the time stamp you mean Bahaa? If so, I assume you want to learn more about how to construct shape functions? :)

@bravindra4138 2 жыл бұрын

Can you pin the link for the other videos which follows this. Unable to find the playlist

@JousefM 2 жыл бұрын

That’s the only FEM video I have so far. More to come soon 🙂 I’m finishing up coursera courses at the moment on using SimScale for FEA and CFD. Stay tuned mate!

@bravindra4138 2 жыл бұрын

Ok!! A complete course on FEM and CFD may help.

@JousefM 2 жыл бұрын

Coming over at academy.jousefmurad.com soonish 🙂😎

@abrorabyyu6221 3 жыл бұрын

In 9:11 why you put 4 nodes? Why don't 2 or 6 nodes?

@JousefM 3 жыл бұрын

You could use as many nodes as you want. The more nodes you use, the better the accuracy but you need more time to compute the solution. If you use less, you have a less accurate solution but you're faster. For showing how the FEM system works a low number of nodes make sense then you can actually see how easily the matrices can be assembled (assembling process := putting the individual systems together).

@abrorabyyu6221 3 жыл бұрын

@@JousefM thank you, actually i am waiting your content. Cause in here just clicking software fem mindlessly without understanding concept

@JousefM 3 жыл бұрын

@@abrorabyyu6221 Coursera mini course (called "Guided Project" for absolute beginners) is in production, hopefully finished soon :) Stay tuned. Make sure to follow me on Twitter & Instagram where I keep my followers posted about latest updates, course creations etc.

@sudmudmud357 3 ай бұрын

How fantastic is this video. I wished they taught this at university 15 years ago ….

@humane11 3 жыл бұрын

@JousefM 3 жыл бұрын

Good question! :) Might come in the future!

@humane11 3 жыл бұрын

@@JousefM please make it soon brother... I know you're busy in podcasts, but please make a room for this topic soon in your list of podcasts/videos ....maybe arrange a podcast including all these questions with Mr. Lukasz Skotny

@MecaByte 2 жыл бұрын

History of FEM without mentioning Boris Galerkin !?

@JousefM 2 жыл бұрын

Next time 🙂 so many people involved I think the history of FEM needs its own video!

@MecaByte 2 жыл бұрын

@@JousefM Can't agree more, nice video though by the way

@JousefM 2 жыл бұрын

Thanks buddy!! 🙂

@MjahedTV 4 жыл бұрын

Kmfdag

@JousefM 4 жыл бұрын

Danke dir Bruder!

@Oscar-jg9gg 3 жыл бұрын

why do we use it... why do we have it?

@JousefM 3 жыл бұрын

Approximation method to solve partial differential equations - imagine you do a crash test, way too inefficient to do hundreds of crash tests and there is simply no analytical solution for a deformation of a crash per se. Radical but very easy to understand example :)