Bug in the programmer's guide: divergence of a tensor field

sebastian · July 17, 2012, 11:01

Hi everybody!

I wonder if there is a bug in the programmer's guide of OpenFOAM.

In chapter 2.1.2 the divergence of a tensor field is given as

$\nabla \bullet T = \partial_i T_{ij}=\left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{13}}{\partial x_1}\\ \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{23}}{\partial x_2}\\ \frac{\partial T_{31}}{\partial x_3} + \frac{\partial T_{32}}{\partial x_3} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right)$

In my opinion this is wrong. After the convention in chapter 1.3.1 (inner product) it should be instead

$\nabla \bullet T = \left( \begin{matrix} \frac{\partial }{\partial x_1}, & \frac{\partial }{\partial x_2}, & \frac{\partial}{\partial x_3} \end{matrix} \right) \bullet \left( \begin{matrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{33} \\ T_{31} & T_{32} & T_{33} \\ \end{matrix} \right) =\partial_j T_{ji}=\left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{31}}{\partial x_3}\\ \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{33}}{\partial x_3}\\ \frac{\partial T_{13}}{\partial x_1} + \frac{\partial T_{32}}{\partial x_2} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right)$

Can anybody confirm that or do I make something wrong here?
How is the implementation, does anybody know?

Sebastian

l_r_mcglashan · July 17, 2012, 14:54

It depends on if you multiply a row vector by a tensor or a tensor by a column vector. We want the tensor to act on the vector. What you've written should be:

$\nabla \bullet T = \left( \begin{matrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{33} \\ T_{31} & T_{32} & T_{33} \\ \end{matrix} \right) \left( \begin{matrix} \frac{\partial }{\partial x_1} \\ \frac{\partial }{\partial x_2} \\ \frac{\partial }{\partial x_3} \end{matrix} \right)$

sebastian · July 17, 2012, 15:35

Dear Laurence, thanks for your post.

When I look up for the definition of the divergence of a tensor field, on wikipedia for instance this is
$\nabla \bullet T = \frac{\partial T_{ki}}{\partial x_i} e_k$
This does not agree with the programmer's guide. Why should it be different in OpenFOAM?

Sebastian

l_r_mcglashan · July 17, 2012, 17:15

The programmer's guide is not maintained anymore.

Anyway, how about:

$\partial_j T_{ji} = \partial_i T_{ij}$

sebastian · July 18, 2012, 02:38

I am not talking about if it's a i, j or k.

$\text{div}T= \left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{31}}{\partial x_3}\\ \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{32}}{\partial x_3}\\ \frac{\partial T_{13}}{\partial x_1} + \frac{\partial T_{23}}{\partial x_2} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right) \neq \left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{13}}{\partial x_1}\\ \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{23}}{\partial x_2}\\ \frac{\partial T_{31}}{\partial x_3} + \frac{\partial T_{32}}{\partial x_3} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right)$

The programmer's guide is obviously wrong.

l_r_mcglashan · July 18, 2012, 05:53

Ah, sorry! Yes, the programmer's guide is incorrect.

hhouston · January 18, 2013, 22:03

Hey, the following eq from wikipedia posted higher up in this thread has a typo:

"When I look up for the definition of the divergence of a tensor field, on wikipedia for instance this is

"
and further it is not consistent with the other parts of this thread - it should be Tik not Tki .

Also there are some typos in the matrices you put up, but they are not too hard to figure out. And the overall thrust of the thread is correct.

hhouston · January 18, 2013, 22:05

that is,

July 17, 2012, 11:01	Bug in the programmer's guide: divergence of a tensor field	#1
sebastian Member Sebastian Saegeler Join Date: Nov 2009 Location: Munich Posts: 70 Rep Power: 5	Hi everybody! I wonder if there is a bug in the programmer's guide of OpenFOAM. In chapter 2.1.2 the divergence of a tensor field is given as $\nabla \bullet T = \partial_i T_{ij}=\left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{13}}{\partial x_1}\\ \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{23}}{\partial x_2}\\ \frac{\partial T_{31}}{\partial x_3} + \frac{\partial T_{32}}{\partial x_3} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right)$ In my opinion this is wrong. After the convention in chapter 1.3.1 (inner product) it should be instead $\nabla \bullet T = \left( \begin{matrix} \frac{\partial }{\partial x_1}, & \frac{\partial }{\partial x_2}, & \frac{\partial}{\partial x_3} \end{matrix} \right) \bullet \left( \begin{matrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{33} \\ T_{31} & T_{32} & T_{33} \\ \end{matrix} \right) =\partial_j T_{ji}=\left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{31}}{\partial x_3}\\ \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{33}}{\partial x_3}\\ \frac{\partial T_{13}}{\partial x_1} + \frac{\partial T_{32}}{\partial x_2} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right)$ Can anybody confirm that or do I make something wrong here? How is the implementation, does anybody know? Sebastian

July 17, 2012, 14:54		#2
l_r_mcglashan Senior Member Laurence R. McGlashan Join Date: Mar 2009 Posts: 345 Rep Power: 12	It depends on if you multiply a row vector by a tensor or a tensor by a column vector. We want the tensor to act on the vector. What you've written should be: $\nabla \bullet T = \left( \begin{matrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{33} \\ T_{31} & T_{32} & T_{33} \\ \end{matrix} \right) \left( \begin{matrix} \frac{\partial }{\partial x_1} \\ \frac{\partial }{\partial x_2} \\ \frac{\partial }{\partial x_3} \end{matrix} \right)$ __________________ Laurence R. McGlashan :: Website

July 17, 2012, 17:15		#4
l_r_mcglashan Senior Member Laurence R. McGlashan Join Date: Mar 2009 Posts: 345 Rep Power: 12	The programmer's guide is not maintained anymore. Anyway, how about: $\partial_j T_{ji} = \partial_i T_{ij}$ __________________ Laurence R. McGlashan :: Website

July 18, 2012, 02:38		#5
sebastian Member Sebastian Saegeler Join Date: Nov 2009 Location: Munich Posts: 70 Rep Power: 5	I am not talking about if it's a i, j or k. $\text{div}T= \left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{31}}{\partial x_3}\\ \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{32}}{\partial x_3}\\ \frac{\partial T_{13}}{\partial x_1} + \frac{\partial T_{23}}{\partial x_2} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right) \neq \left( \begin{matrix} \frac{\partial T_{11}}{\partial x_1} + \frac{\partial T_{12}}{\partial x_1} + \frac{\partial T_{13}}{\partial x_1}\\ \frac{\partial T_{21}}{\partial x_2} + \frac{\partial T_{22}}{\partial x_2} + \frac{\partial T_{23}}{\partial x_2}\\ \frac{\partial T_{31}}{\partial x_3} + \frac{\partial T_{32}}{\partial x_3} + \frac{\partial T_{33}}{\partial x_3} \end{matrix} \right)$ The programmer's guide is obviously wrong. Last edited by sebastian; July 19, 2012 at 09:24.

July 18, 2012, 05:53		#6
l_r_mcglashan Senior Member Laurence R. McGlashan Join Date: Mar 2009 Posts: 345 Rep Power: 12	Ah, sorry! Yes, the programmer's guide is incorrect. __________________ Laurence R. McGlashan :: Website

July 17, 2012, 15:35		#3
sebastian Member Sebastian Saegeler Join Date: Nov 2009 Location: Munich Posts: 70 Rep Power: 5	Dear Laurence, thanks for your post. When I look up for the definition of the divergence of a tensor field, on wikipedia for instance this is $\nabla \bullet T = \frac{\partial T_{ki}}{\partial x_i} e_k$ This does not agree with the programmer's guide. Why should it be different in OpenFOAM? Sebastian

January 18, 2013, 22:03	error in wikipedia	#7
hhouston New Member heidi Join Date: Jan 2013 Posts: 2 Rep Power: 0	Hey, the following eq from wikipedia posted higher up in this thread has a typo: "When I look up for the definition of the divergence of a tensor field, on wikipedia for instance this is " and further it is not consistent with the other parts of this thread - it should be Tik not Tki . Also there are some typos in the matrices you put up, but they are not too hard to figure out. And the overall thrust of the thread is correct.

January 18, 2013, 22:05		#8
hhouston New Member heidi Join Date: Jan 2013 Posts: 2 Rep Power: 0	that is,

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