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

3d numerical integration over a control volume

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

Reply
 
LinkBack Thread Tools Display Modes
Old   July 29, 2012, 00:29
Default 3d numerical integration over a control volume
  #1
New Member
 
mre
Join Date: Jul 2012
Posts: 1
Rep Power: 0
hitmre is on a distinguished road
I found an equation of numerical integration over of control volume
\Omega_{ij}=((x,y)|x_{x-1/2}<x<x_{x+1/2} and y_{j-1/2}<y<y_{i+1/2})
and it says "use nine point stencil" and this equation
\int _{\Omega_{ij}} g\approx \frac{h^2}{24}(16g_{ij}+g_{i-1,j-1}+g_{i-1,j}
+g_{i-1,j+1}+g_{i,j-1}+g_{i,j+1}
+g_{i+1,j-1}+g_{i+1,j}+g_{i+1,j+1})

I am wondering why it is 16/24 and 1/24.
I would like to get the formula for 3D numerical integration, which should based on 3x3x3 points. What the weight should be?

The equation is from this paper (SIAM J. SCI. COMPUT. Vol. 20, No. 4, pp. 1165-1191) on page 1172 http://diyhpl.us/~bryan/papers2/frey...0algorithm.pdf



Thank you.
Jo
hitmre is offline   Reply With Quote

Old   July 30, 2012, 00:55
Default
  #2
New Member
 
Join Date: Mar 2012
Posts: 6
Rep Power: 6
vanchanh123 is on a distinguished road
Quote:
Originally Posted by hitmre View Post
I found an equation of numerical integration over of control volume
\Omega_{ij}=((x,y)|x_{x-1/2}<x<x_{x+1/2} and y_{j-1/2}<y<y_{i+1/2})
and it says "use nine point stencil" and this equation
\int _{\Omega_{ij}} g\approx \frac{h^2}{24}(16g_{ij}+g_{i-1,j-1}+g_{i-1,j}
+g_{i-1,j+1}+g_{i,j-1}+g_{i,j+1}
+g_{i+1,j-1}+g_{i+1,j}+g_{i+1,j+1})

I am wondering why it is 16/24 and 1/24.
I would like to get the formula for 3D numerical integration, which should based on 3x3x3 points. What the weight should be?

The equation is from this paper (SIAM J. SCI. COMPUT. Vol. 20, No. 4, pp. 1165-1191) on page 1172 http://diyhpl.us/~bryan/papers2/frey/distance/Sussman%20M.,%20An%20efficient%20interface%20prese rving%20level%20set%20redistancing%20algorithm.pdf



Thank you.
Jo
In 3D model, I think you can use any the iterpolation way.

The frist question, I think auther use the interpolation method with some weight.

That mean use can choose some coefficentes a_{i',j',k'}: a_{'i,j',k'}\ge 0, \sum_{(i',j')\in control volume  } a_{i,'j',k'}=1, when we have interpolation's formular
\int _{\Omega_{i,j,k}} g =\sum_{(i',j',k')\in control volume} a_{i',j',k'}g_{i',j',k'}.
vanchanh123 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
microscale modeling using control volume manaspaliwal Main CFD Forum 0 December 29, 2010 19:11
Creating a control volume rodrigoemp NUMECA 1 December 20, 2010 02:46
air bubble is disappear increasing time using vof xujjun CFX 9 June 9, 2009 07:59
CuBit t42 OpenFOAM Meshing & Mesh Conversion 6 July 10, 2008 07:51
Any numerical triple integration program is available in Fortran? Radhakrishnan Main CFD Forum 3 March 4, 1999 02:03


All times are GMT -4. The time now is 23:07.