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-   -   Matrix storage and diagonal coeffients calculation (https://www.cfd-online.com/Forums/openfoam-programming-development/72456-matrix-storage-diagonal-coeffients-calculation.html)

santiagomarquezd February 5, 2010 13:40

Matrix storage and diagonal coeffients calculation
 
Hi all, here we're dealing with lduMatrix coefficients storage and calculation. In classical books for FVM like the one written by Patankar the diagonal coefficient arises as the summation of all the neighbour's coefficients, i.e. the coefficients in the same row, talking about matrices. But looking inside in the implementation of laplacian and full upwind schemes, both uses the negSumDiag method, that sums by column not by row to obtain the diagonal coefficient. Might be the matrices transposed for storage?

Regards.

hjasak February 6, 2010 07:31

Nope: the negSumDiag() will sum up on rows. What may be confusing you is the names of the addressing arrays, but the operation is correct:
void Foam::lduMatrix::negSumDiag()
{
const scalarField& Lower = const_cast<const lduMatrix&>(*this).lower();
const scalarField& Upper = const_cast<const lduMatrix&>(*this).upper();
scalarField& Diag = diag();

const unallocLabelList& l = lduAddr().lowerAddr();
const unallocLabelList& u = lduAddr().upperAddr();

for (register label face=0; face<l.size(); face++)
{
Diag[l[face]] -= Lower[face];
Diag[u[face]] -= Upper[face];
}
}

Lower stand for "lower addressing index"; since the addressing is done through the upper triangle, the lower index is the index for a row.

Enjoy,

Hrv

santiagomarquezd February 8, 2010 10:15

I know I'm wrong, but I can't figure where is my error, I can't completely understand the phrase "Lower stand for "lower addressing index"; since the addressing is done through the upper triangle, the lower index is the index for a row." Lower is Upper Triangular? Maybe an example can help my point of view and my error. Suppose we have a mesh with two layers of four hexahedron, representing a 2D mesh.

*-------*-------*-------*-------*
| 0 | 1 | 2 | 3 |
*-------*-------*-------*-------*
| 4 | 5 | 6 | 7 |
*-------*-------*-------*-------*

the sides of this hexahedron have unitary areas. When the mesh is obtained we have the following arrays:

Owner={0, 0, 1, 1, 2, 2, 3, 4, 5, 6, ...}
Neigbour={1, 4, 2, 5, 3, 6, 7, 5, 6, 7}

Using an advective velocity of 1 in horizontal direction (to the right) and applying the FVM method we can assemble for cell 1 (equation #1, starting in 0) the following equation:

-phi_0 + phi_1=0

due the direction of the velocity, the negative coefficient (-1) will be ever in the lower triangular.
Running the example with scalarTransportFoam and debugging (in my case with gdb) we can can stop the program when negSumDiag is called within the advective term calculation.
At this point we have the following advective matrix (M):

0 0 0 0 0 0 0 0
-1 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 0
0 0 -1 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 -1 0 0 0
0 0 0 0 0 -1 0 0
0 0 0 0 0 0 -1 0

l array has: {0, 0, 1, 1, 2, 2, 3, 4, 5, 6}, so l (from lowerAddr method) has the same data of Owner (at least the first 10 terms),
u array has: {1, 4, 2, 5, 3, 6, 7, 5, 6, 7}, so u (from upperAddr method) has the same data of Neighbour.
On the other hand, we have for Lower: {-1, -0, -1, -0, -1, -0, -0, -1, -1, -1} (from the lower attribute of the class), it says that Lower is the Lower Triangular, and for Upper: {0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (from the upper attribute of the class), the Upper Triangular. Finally for Diag: {0, 0, 0, 0, 0, 0, 0, 0}.
Looking at the Lower Triangular l array give us the columns of elements, and u the rows. So for element number 0, i.e. -1, we have row: 1, column: 0, like in the matrix.
Running the loop in negSumDiag, for example for the equation presented before (equation #1), calculation of diagonal coefficient is given in loops 0, 2 and 3 (i.e. face index in loop takes values 0, 2 and 3), in loop 0 we have:

Diag[l[0]] -= Lower[0];
Diag[u[0]] -= Upper[0];

Second line operates on Diag[1] M[1][1] subtracting the first element of Upper, i.e M[0][1] (row is given by l and column by u). In loop 2:

Diag[l[2]] -= Lower[2];
Diag[u[2]] -= Upper[2];

First line operates on M[1][1] subtracting M[2][1] (the third element of Lower whose row is given by u and column by l). Finally in loop 3 we have M[1][1]=M[1][1]-M[5][1]. Finally the diagonal coefficient 1 was calculated from elements in the same column but in different rows when I had expected that this calculation would be made from elements in the same row. i.e. M[1][1]=M[1][1]-M[1][0].

Regards.

santiagomarquezd February 11, 2010 13:00

Any clues...? Regards.

santiagomarquezd March 26, 2010 07:28

We've studying the matter for an extra while and found that row summation is not done in the same way that in Patankar's book, but the results are correct. Column summation that I explained in past posts is only an algorithmic implementation for the sake of simplification in machine use. Explanation is large, if somebody needs more details, contact me or contact nisi as well. Regards.

ganeshv November 3, 2014 14:07

Could you please post the explanation
 
Hi!

I'm so thankful for finding this post. I've been wondering about this for a while. I see that this implementation is ok for symmetric matrices. But I still don't see it for asymmetric matrices. Could you please post the explanation for all of us.

I also have a follow up question. Even if this implementation works for symmetric matrices... why would it help so much to implement it this way? What do we save by introducing this (for lack of a better word) "confusion" ?

ganesh

sharonyue March 23, 2020 19:17

This topic was discussed at length in this link: http://www.cfd-china.com/topic/547
in Chinese though...

Rvadrabade May 23, 2020 06:09

Few comments.

As everyone is aware of equation after discretization A_{P} \psi_{P}+\sum_{N} A_{N} \psi_{N}=\mathbf{S}_{p}.

For div(phi,U)

A_{P}=-\sum_{N} A_{N}+totalFlux

for 2d: A_{P}=A_{W}+A_{E}+A_{S}+A_{N}+\{phi_e-phi_w +phi_n-phi_s\}

For linear (w=0.5) and upwind (w=1) {w=weight}

A_{W} = -w\bullet phi_{w}
A_{E} = (1-w)\bullet phi_{e}
A_{N} = (1-w)\bullet phi_{n}
A_{S} = -w\bullet phi_{s}

So, yes it seems that OF does sum by column but it is not.


Reference:- Section 5.7.2 Hybrid differencing scheme for multi-dimensional convection-diffusion;
An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Book by H K Versteeg and W Malalasekera

Note that:- Coefficient in the book are calculated assuming equation A_{p} \psi_{p}=\sum_{N} A_{N} \psi_{N}+\mathbf{S}_{p} so they have opposite sign but taken care at matrix construction.

hjasak May 23, 2020 12:22

First, you’re wrong because FOAM does this in face addressing so you don’t have w and (1-w)

Second, this has worked for 27 years without a change, producing perfect results. Did you actually produce the test case that reproduces the error?

Rvadrabade May 24, 2020 01:30

Hi Prof. Hrvoje Jasak

I agree with you and OF implementation is absolutely correct, i don't have any objections.

I would like to convey is that, the diagonal coefficient is not just addition of off diagonal coeffs but totalFlux term should be considered.

for 2D:
A_{P}=A_{W}+A_{E}+A_{S}+A_{N}+\{phi_e-phi_w +phi_n-phi_s\}

With the face addressing it is taken care however with cell addressing (as per books ) the extra term totalFlux needs attention.

Also, regarding use of 'w and (1-w)' probably one can refer here
https://www.openfoam.com/documentati...n-details.html

Thank you

Tobermory December 23, 2020 06:25

Quote:

Originally Posted by santiagomarquezd (Post 251810)
We've studying the matter for an extra while and found that row summation is not done in the same way that in Patankar's book, but the results are correct. Column summation that I explained in past posts is only an algorithmic implementation for the sake of simplification in machine use. Explanation is large, if somebody needs more details, contact me or contact nisi as well. Regards.

Dear Santiago - I have been trying to puzzle this out for a few days ... could you share your insight on how this works? Thanks & regards.

Tobermory January 5, 2021 04:49

Okay, I have finally puzzled this out - the coding in negSumDiag is in effect a piece of "sleight of hand" that makes for efficient coding.

Here is the key: when you write out the matrix coefficients for the divergence term, employing interpolation of the variable to the face and the face flux with the relevant sign, you get a set of off-diagonal and diagonal coefficients for the master and slave cell (owner and neighbour in OF terminology, although the term neighbour is a bit ambiguous) of each face. The trick is to realise that the diagonal coeffs are equal to the negative of the off-diagonal coeffs of the neighbouring cell, and not their own cell. This is why negSumDiag appears to "sum the columns in the matrix" ... it actually is!

I was confused for a while since I had written down the divergence in non-conservative form for some crazy reason, in which case the diagonal coeffs are then the negatives of their own cell's off-diagonal coefficients, and negSumDiag would then have had to sum the rows.


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