Regarding notation for tensor components

[Shyam Sunder Yadav]

Dear All

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

Please help me with respect to the assumptions / notations.

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?

2) The velocity gradient 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. 

Please help me in this regard.

Thank you all.

Regards

--

Dr. Shyam Sunder Yadav
Assistant Professor
Mechanical Engineering
BITS Pilani
09902346342
http://www.bits-pilani.ac.in/pilani/ssyadav/Profile

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[jmlopez...@gmail.com]

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.

Please help me with respect to the assumptions / notations.

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...

2) The velocity gradient tensor

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

    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

....

[Shyam Sunder Yadav]

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.

2)  About velocity gradient in Basilisk?

       \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