CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > Main CFD Forum

Question about multigrid method theory

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Display Modes
Old   August 16, 2012, 10:18
Question Question about multigrid method theory
  #1
lnk
Senior Member
 
lnk
Join Date: Feb 2011
Location: Switzerland
Posts: 118
Rep Power: 6
lnk is on a distinguished road
Hi, everyone

There's one thing i don't understand about the multigrid method. I'm wondering why we smooth our residual from fine to coarse? It's part of updating the equation. Why don't we calculate the residual just at the coarse mesh?

Thanks!

Best,
lnk
lnk is offline   Reply With Quote

Old   August 16, 2012, 10:32
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,582
Rep Power: 20
FMDenaro will become famous soon enough
Quote:
Originally Posted by lnk View Post
Hi, everyone

There's one thing i don't understand about the multigrid method. I'm wondering why we smooth our residual from fine to coarse? It's part of updating the equation. Why don't we calculate the residual just at the coarse mesh?

Thanks!

Best,
lnk

the change of grid and the computational stencil can be seen as a sort of "filtering process" for the Fourier components of the residual...the high wavenumbers are rapidly smoothed in this way
FMDenaro is offline   Reply With Quote

Old   August 16, 2012, 10:39
Question
  #3
lnk
Senior Member
 
lnk
Join Date: Feb 2011
Location: Switzerland
Posts: 118
Rep Power: 6
lnk is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
the change of grid and the computational stencil can be seen as a sort of "filtering process" for the Fourier components of the residual...the high wavenumbers are rapidly smoothed in this way

Does it mean if I calculate the residuals of fine mesh then use the obtained fine mesh residuals to calculate the residuals for coarse mesh, the residual for coarse mesh can be more accurate than calculated directly from the coarse mesh?
lnk is offline   Reply With Quote

Old   August 16, 2012, 10:59
Default
  #4
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,582
Rep Power: 20
FMDenaro will become famous soon enough
Quote:
Originally Posted by lnk View Post
Does it mean if I calculate the residuals of fine mesh then use the obtained fine mesh residuals to calculate the residuals for coarse mesh, the residual for coarse mesh can be more accurate than calculated directly from the coarse mesh?

I'm not sure I understand exactly what you mean ... going from fine (hf) to coarse (hc) grid you have automatically a grid filter (cut-off) at the corresponding Nyquist frequency pi/hc< pi/hf . Therefore less components (at high wavenumbers) of the errors. Additional smoothing of the Fourier components can be in effect depending on the algorithm ...
Maybe someone else can better address the issue ...
FMDenaro is offline   Reply With Quote

Old   August 16, 2012, 23:33
Default
  #5
Member
 
Join Date: Jul 2011
Posts: 59
Rep Power: 5
rmh26 is on a distinguished road
From what I understand the typical smoother will quickly reduce high frequency errors on a grid but will slowly reduce low frequency errors. The idea then is to remove the high frequency errors on a fine grid with a few sweeps of a smoother then make the grid course so that the low frequencies on the fine grid now appear to be high frequencies which can be quickly reduced. The process repeats
rmh26 is offline   Reply With Quote

Old   August 17, 2012, 02:53
Default
  #6
Senior Member
 
Arjun
Join Date: Mar 2009
Location: Nurenberg, Germany
Posts: 366
Rep Power: 10
arjun is on a distinguished road
The idea is to represent the fine level problem on as coarse as possible coarse level. Physically if the curve is smooth you can represent it using little number of points but if it has local features you can not represent it with low number of points. Similar principle applies to multigrid too. You want to create as small coarse problem as possible to save cost.

Further a good smoother is one that would smooth it with as little iteration as possible.

Also note that what looks physically smooth to you may not be algebraically smooth to multigrid.
arjun is offline   Reply With Quote

Old   August 17, 2012, 02:56
Default
  #7
Senior Member
 
Arjun
Join Date: Mar 2009
Location: Nurenberg, Germany
Posts: 366
Rep Power: 10
arjun is on a distinguished road
I would strongly suggest to buy this book if really serious about learning multigrid.

https://computation.llnl.gov/casc/pe...t/welcome.html


There are free notes also available but book is worth much much more than what you might pay for it. I bought this book recently even after reading all the free notes of this available and it taught me a lot.
arjun is offline   Reply With Quote

Old   August 17, 2012, 04:24
Default
  #8
Senior Member
 
Mehdi
Join Date: Jan 2011
Location: Iran
Posts: 130
Rep Power: 6
mb.pejvak is on a distinguished road
I have question about applying Multigrid method in SIMPLER algorithm in collocated grid. In fact, for calculating mass flux in coarse grid we should corrected with mass flux correction and mass flux correction is calculated by using Rhie and Chow interpolation for the difference between the current approximation Uk and the restricted approximation Uk.
this is the way that some papers and books, for example Peric, are introduced. but I can not understand what exactly should be done. in fact, Rhie and Chow method we calculated face velocity with two terms, a pressure term and pseudo velocity. in second term we have source term that contains velocity. my question is if this source term should calculated by difference velocity or current approximation Uk. and also for pressure term, if it shoud be calculated with current approximation pressure or difference between the current approximation pressure and the restricted approximation pressure?
I hope I can convey my purpose.

Thanks in advance
__________________
Best Regards;
Mehdi
E-mail: mb.pejvak[at]Gmail[dot]com
mb.pejvak is offline   Reply With Quote

Old   August 17, 2012, 07:28
Default
  #9
Senior Member
 
Arjun
Join Date: Mar 2009
Location: Nurenberg, Germany
Posts: 366
Rep Power: 10
arjun is on a distinguished road
I think you are little bit confused. To make things clearer :

1. You are talking about Full Or Geometric multigrid. In principle they are same as algebraic multigrid (some way) but they are really different.

2. In full multigrid, you have to descretize equations again at coarser levels.

3. For momentum and pressure equations you will need mass fluxes. This is where you main confusion is.

4. Mass fluxes at the coarser levels could be computed by just summing the fluxes enclosing coarser control volume by using fine level fluxes. (think about it).

5. Once momentum equations are solved you get fine level fluxes and again coarser level fluxes are calculated as mentioned in (4).

6. The real challenge of full multigrid is to desretize on coarser level.

7. Multigrid for u,v,w is rarely needed (because of diagonal dominance).



Quote:
Originally Posted by mb.pejvak View Post
I have question about applying Multigrid method in SIMPLER algorithm in collocated grid. In fact, for calculating mass flux in coarse grid we should corrected with mass flux correction and mass flux correction is calculated by using Rhie and Chow interpolation for the difference between the current approximation Uk and the restricted approximation Uk.
this is the way that some papers and books, for example Peric, are introduced. but I can not understand what exactly should be done. in fact, Rhie and Chow method we calculated face velocity with two terms, a pressure term and pseudo velocity. in second term we have source term that contains velocity. my question is if this source term should calculated by difference velocity or current approximation Uk. and also for pressure term, if it shoud be calculated with current approximation pressure or difference between the current approximation pressure and the restricted approximation pressure?
I hope I can convey my purpose.

Thanks in advance
arjun is offline   Reply With Quote

Old   August 17, 2012, 10:46
Default
  #10
Senior Member
 
Mehdi
Join Date: Jan 2011
Location: Iran
Posts: 130
Rep Power: 6
mb.pejvak is on a distinguished road
Dear Arujan;
Thank you for your answer. in fact, I Used FAS multigrid in NS equation, and I know equation should be descritized in coarser grid. in fact in coarse grid a source term is added to solve equations in fine grid instead of coarse grid, and also in calculating mass flux in coarse grid, restricted mass flux claculated by sum of the faces of the child grids that make the parent's face. and then rectify in each iteration in coarse grid by difference between the current approximation velocity and the restricted approximation velocity.
as I mention in my question, my problem is calculating mass flux correction in Rhie and Chow method by this difference velocity. in rhie and chow method, how should be behaved for velocity and pressure. does pressure should be approximation pressure or difference between the current approximation pressure and the restricted approximation pressure? and also in in source term of pseudo velocity term. my problem is calculating face velocity in coarse iteretion by the velocity that calculated difference between the current approximation velocity and the restricted approximation velocity.

Quote:
Originally Posted by arjun View Post
I think you are little bit confused. To make things clearer :

1. You are talking about Full Or Geometric multigrid. In principle they are same as algebraic multigrid (some way) but they are really different.

2. In full multigrid, you have to descretize equations again at coarser levels.

3. For momentum and pressure equations you will need mass fluxes. This is where you main confusion is.

4. Mass fluxes at the coarser levels could be computed by just summing the fluxes enclosing coarser control volume by using fine level fluxes. (think about it).

5. Once momentum equations are solved you get fine level fluxes and again coarser level fluxes are calculated as mentioned in (4).

6. The real challenge of full multigrid is to desretize on coarser level.

7. Multigrid for u,v,w is rarely needed (because of diagonal dominance).
__________________
Best Regards;
Mehdi
E-mail: mb.pejvak[at]Gmail[dot]com
mb.pejvak is offline   Reply With Quote

Old   August 20, 2012, 03:34
Default
  #11
Senior Member
 
Arjun
Join Date: Mar 2009
Location: Nurenberg, Germany
Posts: 366
Rep Power: 10
arjun is on a distinguished road
I am afraid I can not be much help there. Because I never applied Full multigrid to Navier Stokes and thus never had to solve for velocity at coarser levels.

I did however apply full multigrid to pressure poisson problems on cartesian type grid system.

I do have a book at home that might have answer to your question, i will look in it and let you know.

Quote:
Originally Posted by mb.pejvak View Post
Dear Arujan;
Thank you for your answer. in fact, I Used FAS multigrid in NS equation, and I know equation should be descritized in coarser grid. in fact in coarse grid a source term is added to solve equations in fine grid instead of coarse grid, and also in calculating mass flux in coarse grid, restricted mass flux claculated by sum of the faces of the child grids that make the parent's face. and then rectify in each iteration in coarse grid by difference between the current approximation velocity and the restricted approximation velocity.
as I mention in my question, my problem is calculating mass flux correction in Rhie and Chow method by this difference velocity. in rhie and chow method, how should be behaved for velocity and pressure. does pressure should be approximation pressure or difference between the current approximation pressure and the restricted approximation pressure? and also in in source term of pseudo velocity term. my problem is calculating face velocity in coarse iteretion by the velocity that calculated difference between the current approximation velocity and the restricted approximation velocity.
arjun is offline   Reply With Quote

Old   August 20, 2012, 19:25
Default
  #12
Senior Member
 
Mehdi
Join Date: Jan 2011
Location: Iran
Posts: 130
Rep Power: 6
mb.pejvak is on a distinguished road
Dear arujan;
Thank you so much. I'm looking forward to hearing from you.

Regards;
Mehdi


Quote:
Originally Posted by arjun View Post
I am afraid I can not be much help there. Because I never applied Full multigrid to Navier Stokes and thus never had to solve for velocity at coarser levels.

I did however apply full multigrid to pressure poisson problems on cartesian type grid system.

I do have a book at home that might have answer to your question, i will look in it and let you know.
__________________
Best Regards;
Mehdi
E-mail: mb.pejvak[at]Gmail[dot]com
mb.pejvak is offline   Reply With Quote

Old   August 22, 2012, 07:32
Default
  #13
Senior Member
 
Arjun
Join Date: Mar 2009
Location: Nurenberg, Germany
Posts: 366
Rep Power: 10
arjun is on a distinguished road
sorry I could not find anything useful for you.
i have this book ,
http://books.google.de/books/about/m...IC&redir_esc=y

which has collection of papers where people used multigrid but i could not find anything that explains what you are looking for.

However please have a look at
http://www.tfd.chalmers.se/~lada/

there read

JOHANSSON, P.
"A Three-Dimensional Laminar Multigrid Method Applied to the SIMPLEC Algorithm", MSc thesis, Rept. 92/5, Thermo and Fluid Dynamics, Chalmers University of Technology, Göteborg, 1992.



This is not quite what you asked but it is interesting read.

Quote:
Originally Posted by mb.pejvak View Post
Dear arujan;
Thank you so much. I'm looking forward to hearing from you.

Regards;
Mehdi
arjun is offline   Reply With Quote

Old   August 23, 2012, 01:12
Default
  #14
Senior Member
 
Mehdi
Join Date: Jan 2011
Location: Iran
Posts: 130
Rep Power: 6
mb.pejvak is on a distinguished road
Dear Arujan;
Thank you so much for your help. I already read the last one, it is really helpful and solved some of my problems. but unfortunately, I can not get the formers. I try to obtain it, and I hope it can fix other problems.

best regards;
Mehdi

Quote:
Originally Posted by arjun View Post
sorry I could not find anything useful for you.
i have this book ,
http://books.google.de/books/about/m...IC&redir_esc=y

which has collection of papers where people used multigrid but i could not find anything that explains what you are looking for.

However please have a look at
http://www.tfd.chalmers.se/~lada/

there read

JOHANSSON, P.
"A Three-Dimensional Laminar Multigrid Method Applied to the SIMPLEC Algorithm", MSc thesis, Rept. 92/5, Thermo and Fluid Dynamics, Chalmers University of Technology, Göteborg, 1992.



This is not quite what you asked but it is interesting read.
__________________
Best Regards;
Mehdi
E-mail: mb.pejvak[at]Gmail[dot]com
mb.pejvak is offline   Reply With Quote

Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Question about immersed solid method in CFX spwater CFX 6 May 24, 2012 13:05
Why does the errors oscillate strongly in my multigrid method? hadesmajesty Main CFD Forum 0 October 19, 2009 03:29
problem with multigrid method Venkatesh Main CFD Forum 2 October 8, 2003 21:59
Pressure Possion equ.and multigrid method Tony Main CFD Forum 16 April 28, 2001 12:08
Multigrid applied to k-e models Paulo Zandonade Main CFD Forum 9 May 24, 1999 08:10


All times are GMT -4. The time now is 11:00.