[blockMesh] Grading calculation equations

ishan_ae · April 6, 2023, 09:52

I am trying to understand the equations behind the total expansion ratio calculation and simpleGrading in general. I know that the grading is based on geometric progression.

The following quantities are of interest to us:

= edge length
$\delta_s$ = length of the start cell

= number of cells
$\delta_e$ = length of the end/last cell

= common ratio or cell-to-cell expansion ratio(i think so)

= total expansion ratio

The common ratio is:
$r = \frac{\delta_e}{\delta_s} ^{\frac{1}{N-1}}$

In the above equation, $e_r = \delta_e/\delta_s$ is the the total expansion ratio which we need to use in simpleGrading input.

The summation of all the geometric terms( cell lengths in this case) is the length of the edge

:
$L = \frac {\delta_s (1 - r^N)} {1 -r}$

If we know $\delta_s$ ,

and

, we can re-formulate the above equation into a polynomial in terms of

:
$\delta_s \cdot r^{N} - r\cdot L + L = 0$

We can numerically find the root of the above polynomial which will give us the value of

. Once we know the know the common ratio, the calculation of $\delta_e$ and finally the much needed

is straightforward.

Now the issue is that when I tried to solve the above polynomial equation numerically(Newton's method), it finds the root but it's values does not match with the grading calculator script here https://openfoamwiki.net/index.php/S...ng_calculation.I tried to dig deeper into the code of the calculator but I understood very little.

The reason why I want to know more about this is because I want to automate a portion of my grading using a Python script since I have a lot of blocks to generate the grading for.

Can someone please point me in the right direction ?

hjasak · April 7, 2023, 10:34

Sure. My old friend Cedo says the most human way to define the grading is the size ration of the first to last cell. This, grading ratio of 5 means that the last cell is 5 times larger than the first. For the opposite effect, you would use 0.2

ishan_ae · April 7, 2023, 17:57

I just checked the user guide and this explanation matches with the definition of the total expansion ratio. Thank you !

How would we approach the problem if we have the first cell height , the edge length and the number of cells? Mathematically, I don't see any other way other than solving for the root which, would be the common ratio.

William_Sewell · January 8, 2024, 00:42

The grading calculation formulae employed by educational institutions can differ depending on the grading system and the specific criteria applied.

https://www.calculatorcube.com/

April 6, 2023, 09:52	Grading calculation equations	#1
ishan_ae Member ishan Join Date: Oct 2017 Posts: 77 Rep Power: 8	I am trying to understand the equations behind the total expansion ratio calculation and simpleGrading in general. I know that the grading is based on geometric progression. The following quantities are of interest to us: = edge length $\delta_s$ = length of the start cell = number of cells $\delta_e$ = length of the end/last cell = common ratio or cell-to-cell expansion ratio(i think so) = total expansion ratio The common ratio is: $r = \frac{\delta_e}{\delta_s} ^{\frac{1}{N-1}}$ In the above equation, $e_r = \delta_e/\delta_s$ is the the total expansion ratio which we need to use in simpleGrading input. The summation of all the geometric terms( cell lengths in this case) is the length of the edge : $L = \frac {\delta_s (1 - r^N)} {1 -r}$ If we know $\delta_s$ , and , we can re-formulate the above equation into a polynomial in terms of : $\delta_s \cdot r^{N} - r\cdot L + L = 0$ We can numerically find the root of the above polynomial which will give us the value of . Once we know the know the common ratio, the calculation of $\delta_e$ and finally the much needed is straightforward. Now the issue is that when I tried to solve the above polynomial equation numerically(Newton's method), it finds the root but it's values does not match with the grading calculator script here https://openfoamwiki.net/index.php/S...ng_calculation.I tried to dig deeper into the code of the calculator but I understood very little. The reason why I want to know more about this is because I want to automate a portion of my grading using a Python script since I have a lot of blocks to generate the grading for. Can someone please point me in the right direction ?

April 7, 2023, 10:34		#2
hjasak Senior Member Hrvoje Jasak Join Date: Mar 2009 Location: London, England Posts: 1,905 Rep Power: 33	Sure. My old friend Cedo says the most human way to define the grading is the size ration of the first to last cell. This, grading ratio of 5 means that the last cell is 5 times larger than the first. For the opposite effect, you would use 0.2 __________________ Hrvoje Jasak Providing commercial FOAM/OpenFOAM and CFD Consulting: http://wikki.co.uk

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April 7, 2023, 17:57		#3
ishan_ae Member ishan Join Date: Oct 2017 Posts: 77 Rep Power: 8	I just checked the user guide and this explanation matches with the definition of the total expansion ratio. Thank you ! How would we approach the problem if we have the first cell height , the edge length and the number of cells? Mathematically, I don't see any other way other than solving for the root which, would be the common ratio.

January 8, 2024, 00:42		#4
William_Sewell New Member Join Date: Jan 2024 Posts: 1 Rep Power: 0	The grading calculation formulae employed by educational institutions can differ depending on the grading system and the specific criteria applied. https://www.calculatorcube.com/