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May 18, 2003, 13:59 
Grid generation...

#1 
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Hi,
I am trying to do some grid generation. I want to create a distribution along a line such that the spacing from cell to cell increases smoothly AND where the distance from the first to the last cell is known AND where the size of the first and last cell are known. so lets say my first cell is 1 long and my last cell is 3 long. The spacing between them is 10. These numbers are not necessariliy integers. How do I get a smooth transition from the cell of size 1 to the cell of size 3. I need to know the number of points in between or else the expansion ratio. If I specify the number of points, I need to ensure its within a particular range. Can anyone tell me how to do solve my problem? I tired to figure out a way of doing it using a geometric progression, but failed. Perhaps someone else knows how. Any info would be much appreciated. Thanks. 

May 18, 2003, 23:04 
Re: Grid generation...

#2 
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In general there is no solution to your problem. To see this, let F and L be the lengths of the first and last cells and D the total segment length. Then if L+F=D there is obviously a solution. Then there is a gap, no solutions until L+sqrt(L*F)+F=D, where the ratio is sqrt(L/F). Then there is another gap, and so on.


May 19, 2003, 01:05 
Re: Grid generation...

#3 
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Given length of first cell h<sub>1</sub>, length of last cell h<sub>N</sub> and total length D. Assume that length of the cells varies linearly
h<sub>i</sub> = h<sub>1</sub> + (i1)d where d is not known. Then h<sub>N</sub> = h<sub>1</sub> + (N1)d and Σ h<sub>i</sub> = D Using the last two equations can you determine N and d? 

May 19, 2003, 02:40 
Re: Grid generation...

#4 
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Hi Praveen, it is good idea to write in this Forum normal mathematical formulas. Could you tell how have you written, for example, a symbol Σ? Do you use a converter TeX > HTML or any other ways?
Levi 

May 19, 2003, 08:53 
Re: Grid generation...

#5 
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You can type html code directly into the message box. The formulae are generated that way. You can generate math symbols in html itself.


May 19, 2003, 22:02 
Re: Grid generation...

#6 
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This doesn't address "Question's" conditions. Since he mentions "expansion ratio" he is looking for a geometric progression. Pardon my crude math expressions, but the "solution" would go like this:
Let F and L be the width of the first and last cells, and D the length of the interval. Let R be the expansion ratio and n the number of intervals. Then (1): L=F*R^n, or R^n=L/F, asserting that the first and last are part of a geometric progression. (2): (1R^n)/(1R)=(1L/F*R)/(1R)=D/F, asserting that a progression of n terms fills the interval D. Solving for R gives R=(DF)/(DL). Solving for n gives n=log(L/F)/log((DF)/(DL)). BUT, in general, n is NOT an INTEGER, and thus there is no solution in general. It CAN be an integer for certain values of the parameters. 

May 20, 2003, 10:33 
Re: Grid generation...

#7 
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Thanks for your very informative answers. If I vary "n" (as an integer) and "R", is it possible to satisfy any "L" and "R" and "D" ? Of course the requirement is that all intermediate cells are of a length between "L" and "R". Also, they value needs to increase from "L" to "R" if "L" is smaller than "R".
The problem can be solved where, for example we have "L" = 2 and "" = 32 and the intermediate distance, "D" = 4 + 8 +16 = 28. This is for an integer solution of R = 2. Luckily n is also an integer in this case. I don't think so the method described above can do what I want it to do, but it is a very appealing idea. Does anyone know how to overcome this meshing problem by another method...other than chossing points by hand? Any references? 

May 21, 2003, 20:54 
Re: Grid generation...

#8 
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Something has to give way to find a useful solution to your problem. I haven't really thought about it, but perhaps you should give up the constraint of a fixed geometric progression. Maybe something like cubic interpolation. Even this would be unsatisfactory is some cases. Maybe you would have to use an arithmetic progression. I think there is a useful solution to your problem as long as you use reasonable values for L, F, and D.


May 22, 2003, 02:38 
Re: Grid generation...

#9 
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I'm not fully sure that the following reference answer your question 100% but to my knowledge is one of the best thing I've seen around: you will be able to derive a provedly smooth point distribution that spans the required distance with however only an approximate specification (if I remember correctly) of the size of the first and last cell.
Vinokur, M., "On OneDimensional Stretching Functions for FiniteDifference Calculations", NASA CR 3313, 1980 Hope it helps you. 

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