[CFD] Multi-Grid for CFD (Part 2): Restriction and Prolongation

An introduction to the multi-grid method that is used in the majority of finite volume based CFD codes to solve sets of linear algebraic equations.
Time stamps
0:00 Introduction
0:45 Recap
2:54 Agglomeration
5:17 Restriction
10:06 Restriction Example
11:48 Restriction Matrix
15:18 Prolongation
19:55 Prolongation Example
20:57 Prolongation Matrix
24:11 Coarse Mesh A Matrix
26:49 Derivation of A'
30:41 A' Example
34:12 Algebraic Multi-Grid
36:13 Summary
37:42 Outro
References
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C. Greenshields, H. Weller 'Notes on Computational Fluid Dynamics: General Principles', Section 5.18. 2022.
doc.cfd.direct/notes/cfd-gene...
Y. Saad. Iterative Methods for Sparse Linear Systems. 2nd edition. Society
for Industrial and Applied Mathematics, 2003. isbn: 978-0898715347
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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.

[@tonecukon91]

Best way to spend a monday morning. Thanks for the insight!

[@sammartens1090]

Awesome explanation, thank you very much for your work!

[@hadis9109]

This is great stuff. Your channel is the reason I got interested in CFD during my undergrad a few years ago, and now as a professional and a grad student, your videos are greatly insightful. I hope you keep up the great work - looking forward to part 3!

[@18yyt]

This video is great for understanding the main principles behind GMG and AMG. Thank you very much for this explanation! I find it very useful and crystal clear, especially with the examples of its physical meaning.

[@mahmoudsaleh9006]

Very great explanation. Much better than any reference I checked

[@sadeghahmadi6551]

Excellent video to learn basics of Multigrid method!

[@emkaydee6048]

Subbed! looking forward to going through your channel's videos on my journey to work! Yes the examples are really nice to have included.

[@k0185123]

This is amazing! Thank you Dr. Aidan! This helps me so much. I'm a PhD student working on semiconductor simulation. Recently I started looking into various iterative solver for a large sparse matrix (2 million x 2million), and multigrid method seems to be recommended to be a preconditioner with block Krylov method. This is why I found your video! And it'd be wonderful if you could talk a little bit about Krylov subspace method as well. Thank you so much anyway!! Looking forward to your future videos on iterative solvers!

[@oskarelmgren]

Good review! It would be interesting to hear your take on multi-grid strategies for different types of problems. How to best utilize multi-grid in cases where large differences between adjacent cells drive global flow patterns for example (like boundary layers in diffusers that are close to flow separation for instance).

[@thunder852za]

Good lecture, for finite volume I would go further. The new diagonal entry is the sum of the central diagonal of the agglomerated cells subtracted by the cell faces which were 'merged/removed' during the process of forming the new course cell. This is true because the diagonal entries have an opposite sign to the diagonals (ensuring FV conservation). Of course this is all true for the example and R and P provided.

[@sebleaf8433]

Great video. Multigrid is simpler than I thought. Idk if you plan on making a part 3, but you could maybe talk about various types of multigrid cycles.

[@fluidmechanics101]

Yes, I was thinking about doing another part on the various cycles, as it is the next logical part. However, I will wait and see if people would find this helpful, as this might be a level of detail too far for most CFD users. Let me know what you think 😃

[@jarnomastomaki1671]

It would be great to see a follow-up video about different cycles.

[@sebleaf8433]

@@fluidmechanics101 I think it may be too many details for CFD users but greatly useful for grad students. Explaining how the restriction/interpolation steps are practically implemented in a cfd code through cycles, how those cycles compare in terms of convergence speed and computational/memory cost. I think the series would be incomplete without part 3! :D

[@thunder852za]

I think the big challenge is doing it in parallel. Just consider: How do you agglomerate over processor boundaries? How do you ensure load balancing? How do you make sure the new courser grids are correctly communicating over processor boundaries? And all this is being done on complex geometries, with complex connectivity graphs, in 3D! I remember a PhD in my post grad office, a few years back, trying to sort out Parallel AMG and he never got it to work. I will say though that this channel does a good job of pulling out the core aspects of methods and presenting them in a very digestable way.

[@fluidmechanics101]

Thank you, much appreciated. Yes multigrid in parallel is difficult!

[@kamaukingora5074]

great!!!

[@anirbandas4201]

Great Video. It would be fantastic if you can make more videos on topics like steepest and conjugate gradient methods.

[@fluidmechanics101]

Coming soon 😃