[CFD] Conjugate Gradient for CFD (Part 2): Optimum Distance and Directions

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Күн бұрын

An introduction to the conjugate gradient method and other gradient descent based methods (steepest descent method) for CFD. This is Part 2 of a 2 Part lecture series.
Time stamps
0:00 Introduction
0:50 Basic algorithm summary
3:54 Iteration error (ek)
7:06 The key idea
9:20 Orthogonal directions
10:21 Conjugate directions
12:34 Optimum distance (alphak)
15:54 Alphak derivation
20:37 Next direction (pk+1)
26:39 Slider (betak+1)
28:59 First direction (p0)
30:14 Simplifications (alphak, betak+1)
31:08 Algorithm summary
33:32 Outro
References
J. Shewchuk 'An Introduction to the Conjugate Gradient Method Without the Agonizing Pain', Edition 1 1/4, 1994.
www.cs.cmu.edu/~quake-papers/...
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Disclaimer
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The methods, algorithms, equations, formulae, diagrams and explanations in this talk are for educational and demonstrative purposes only. They should never be used to analyse, design, accredit or validate real scientific / engineering / mathematical structures and flow systems. For such applications, appropriate trained, qualified and accredited (SQEP) engineers / scientists should be consulted along with the appropriate documentation, procedures and engineering standards. Furthermore, the information contained within this talk has not been verified, peer reviewed or checked in any way and is likely to contain several errors. It is therefore not appropriate to use this talk itself (or any of the algorithms, equations, formulae, diagrams and explanations contained within this talk) as an academic or technical reference. The reader should consult the original references and follow the verification and validation processes adopted by your company / institution when carrying out engineering calculations and analyses. Fluid Mechanics 101 and Dr. Aidan Wimshurst are not accountable or liable in any form for the use or misuse of the information contained in this talk beyond the specific educational and demonstrative purposes for which it was intended.

Пікірлер: 29

@youngchulsung975 9 ай бұрын

Nice work! One comment to be clearer is that X_{k+1} (= X_k + alpha_k p_k) is the point in the direction p_k from X_k that miniizes the A-norm of the error vector, i.e., ||e_{k+1}||_A^2. Thanks.

@ablalabla 3 ай бұрын

This is the best explanation I could find. Thank you very much

@theophilusdsouza5993 5 ай бұрын

Kudos to the efforts you put into making this content. Your teaching methodology is really brilliant 🎉. Thanks a lot

@XebionVoker 10 ай бұрын

Nicely explained, it conveys the idea in a way that is easy to remember. It seems in the algorithm summary there might be a mistake for the next search direction. I think it should be r_k+1 + beta_k+1 p_k.

@fluidmechanics101 10 ай бұрын

Well spotted. Thank you!

@sammartens1090 3 ай бұрын

This is great explantion. Thank you very much Aidan!

@berkaysimsek3011 10 ай бұрын

brilliant work! keep it going. You are helping me so much

@jaydubz9053 3 ай бұрын

Great job on this video, however, I have new confusions after watching the video and hope you would answer them. 1. If the definition of iteration error is that it points directly to the solution, why would you say the A matrix multiply by ek+1points to the solution? If ek+1 is transformed by A, wouldn't it actually points away from the solution? 2. In the simplification for alphak and betak+1, their expression have identical denominator, how come you substituted rk to simplify the denominator for betak+1 and not for alphak (second to last slide)? Many thanks!

@changwei2604 10 ай бұрын

Great work!!😊

@netaligirl 6 ай бұрын

Amazingly explained! thank you

@graphicsRat 8 ай бұрын

Utterly brilliant!!!

@hodbadihi631 10 ай бұрын

Great work!

@alibahrami5245 9 ай бұрын

Thanks a lot for this amazing video.

@YuhangGu-b3i Ай бұрын

Excellent talk!

@kleibertenoriodesousa902 2 ай бұрын

Nice video. Thanks!! 😃

@mohammadjadidi233 10 ай бұрын

Thank you !

@sinarezaei218 8 ай бұрын

thank you

@marouaderahmoune1897 8 ай бұрын

thank u sm

@matthewjames7513 9 ай бұрын

outstanding content, thanks so much. Just one small typo at 31:36 your formula for alpha is wrong. You have p_k^TA p_k^T when it should be p_k^T A p_k

@sebleaf8433 9 ай бұрын

Part 3: BiCGStab ? Hehehe

@aravindpalaparthi4629 9 ай бұрын

Hi, i have one doubt. In natural convection or buyoancy flow, which set of equations are used? Does all mass, momentum and energy equations are used to solve natural convection flow?

@fluidmechanics101 9 ай бұрын

All of them (mass, momentum and energy). Your flow solution needs to satisfy conservation of mass, momentum and energy 😄

@aravindpalaparthi4629 9 ай бұрын

@@fluidmechanics101 thankyou. Will keep in touch.i have so many doubts😄

@westonohl1421 10 ай бұрын

Excellent work! Do you still have a Discord server for your channel?

@fluidmechanics101 10 ай бұрын

discord.gg/27d74Kxy

@anirbanbhattacharjee8093 3 ай бұрын

Shouldn't the direction of the vector e (iteration error) be opposite of how it is shown in 6:14? If it is the case and we keep the direction same as shown, then e_{k} = e_{k+1} + alpha_{k} p_{k} instead and alpha becomes positive, the residual becomes Ae_{k} and so on. Edit: alpha_k comes out to be the same value/sign when expressed in terms of the residual and the search direction p.

@jamesm.shihua9288 2 ай бұрын

I think, there is a contradiction point. If e is pointing to the minima, then e=x_f-x. So there should not be a negative sign at 18:40.

@jamesm.shihua9288 2 ай бұрын

You are right I guess. If we follow 6:14, then 16:11 is wrong and we don't need the negative sign. And the rest will be all consistent.