# Regarding notation for tensor components

51 views

Nov 18, 2021, 10:13:21 AM11/18/21
Dear All

I have a few questions about the notation used for various tensor components used in Basilisk.

1) From the definition of a tensor in the header file  "common.h", I came to know that a tensor is represented with the help of three vectors  x, y, and z.
My question is whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

-------------------------------------

Whether \nabla U = [du/dx  du/dy]         OR     \nabla U = [du/dx  dv/dx]
[dv/dx   dv/dy]                                    [du/dy  dv/dy]

3) The viscous stress tensor
----------------------------------

Let     \tau_{ij} be the viscous stress tensor
then whether the subscript  j denotes the plane normal    OR   the force component?
Accordingly, the subscript i will denote either the plane normal or the force component?

4) The eigenvector matrix  R  of conformation tensor for the viscoelastic flow solver
-----------------------------------

Whether the eigenvectors are stored column-wise or row-wise?

5) Lastly, how is the product between two matrices, say A and B, defined?
This will depend on how the components of tensors appear in the matrix form.

Thank you all.
Regards

--
Assistant Professor
Mechanical Engineering
BITS Pilani
09902346342

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### jmlopez...@gmail.com

Nov 19, 2021, 7:53:13 AM11/19/21
to basilisk-fr
Hi Shyam,

El jueves, 18 de noviembre de 2021 a la(s) 16:13:21 UTC+1, ss.y...@pilani.bits-pilani.ac.in escribió:
Dear All

I have a few questions about the notation used for various tensor components used in Basilisk.

1) From the definition of a tensor in the header file  "common.h", I came to know that a tensor is represented with the help of three vectors  x, y, and z.
My question is whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

Not sure...

-------------------------------------

Whether \nabla U = [du/dx  du/dy]         OR     \nabla U = [du/dx  dv/dx]
[dv/dx   dv/dy]                                    [du/dy  dv/dy]

I struggle a little bit with it. Finally, I realized that in most of the viscoelatic literature  it is used
\nabla U = [du/dx  dv/dx]
[du/dy  dv/dy]

3) The viscous stress tensor
----------------------------------

Let     \tau_{ij} be the viscous stress tensor
then whether the subscript  j denotes the plane normal    OR   the force component?
Accordingly, the subscript i will denote either the plane normal or the force component?

Does not matter because it is symmetric.

4) The eigenvector matrix  R  of conformation tensor for the viscoelastic flow solver
-----------------------------------

Whether the eigenvectors are stored column-wise or row-wise?

by column
[v1 | v2]
each column is an eigenvector.

5) Lastly, how is the product between two matrices, say A and B, defined?
This will depend on how the components of tensors appear in the matrix form.

c= a*b being
a =[axx axy]
[ayx ayy] [
b =[bxx bxy]
[byx byy]

would be the usual "row by column"
cxx = axx*bxx + axy *byx
....
Message has been deleted

Nov 19, 2021, 11:30:02 AM11/19/21
to basilisk-fr
Thanks Prof. Jose!

Just like your previous posts/replies, the current one is also really helpful...

I have following observations, I will go by the order of my original questions...

1) Whether the individual vectors are assumed column-wise or row-wise in the matrix for the tensor?

Consider the following product in the log-conform.h:    Ψ=R⋅log(Λ)⋅R^T

If we represent the eigenvectors column-wise then  R = [R.x.x  R.y.x]               and          log(Λ) = [ log(Λ1)        0]
[R.x.y  R.y.y]                                               [0         log(Λ1)]

Therefore  Ψ=R⋅log(Λ)⋅R^T =    [ log(Λ1)*R.x.x^2 + log(Λ2)*R.y.x^2                    R.x.x*R.x.y* log(Λ1) + R.y.x*R.y.y* log(Λ2)]
[ R.x.x*R.x.y* log(Λ1) + R.y.x*R.y.y* log(Λ2)                    log(Λ1)*R.x.y^2 + log(Λ2)*R.y.y^2]

But we have the following in log-conform.h

Ψ=R⋅log(Λ)⋅R^T =    [ log(Λ1)*R.x.x^2 + log(Λ2)*R.x.y^2                    R.x.x*R.y.x* log(Λ1) + R.x.y*R.y.y* log(Λ2)]
[ R.x.x*R.y.x* log(Λ1) + R.x.y*R.y.y* log(Λ2)                    log(Λ1)*R.y.x^2 + log(Λ2)*R.y.y^2]

This raises the 5th point: how is the product between two matrices, say A and B, defined?

If we do the    Ψ=R⋅log(Λ)⋅R^T    product in the following way:    ( R^T * log(Λ) )*R   then the answer is as mentioned in the log-conform.h file.

\nabla U = [du/dx  dv/dx]              Thanks for the clarification Prof.
[du/dy  dv/dy]

3)  Notation for the viscous stress tensor?

In log-conform.h, we have the x direction force due to divergence of elastic stresses defined as:        d \taup.x.x/dx  +  d \taup.x.y/dy ​​

From the above it appears that in \taup_{ij},    j  indicates the plane normal   and  i   indicates the force direction...

=>    d \taup.x.y/dy  =  the   x  direction stress component on a plane normal to   y   axis  should vary with respect to    y    so as to create a force along   x.

This is the case with divergence of Maxwell stress tensor as well in the EHD solver...

4) Whether the eigenvectors are stored column-wise or row-wise?

The notation is important for point 1 above.

5) How is the product between two matrices, say A and B, defined?

Again, important for point 1 above.

Again, Thank you Prof. Jose for the reply.

Regards