Re: Isotropisation af a matrix

On Mar 29, 4:12 am, ravindrakumar.bactavatcha...@xxxxxxxx wrote:
On 29 mar, 02:35, "Timothy Golden BandTechnology.com"



<tttppp...@xxxxxxxxx> wrote:
On Mar 28, 9:46 am, ravindrakumar.bactavatcha...@xxxxxxxx wrote:

On 26 mar, 15:57, "Timothy Golden BandTechnology.com"

<tttppp...@xxxxxxxxx> wrote:
On Mar 26, 6:07 am, ravindrakumar.bactavatcha...@xxxxxxxx wrote:

Hi,

I have a problem: I would like to isotropize a matrix.
In fact,I have first one matrix. I would like to give the conditions
that this matrix stay invariant even if i applied a rotationnal
process.
For this one, I will get normally conditions in the constants of this
matrix.

But I don´t know how can I do it with mapple or mathematica.

My matrix is a tensor (6*6)

thank you verymuch for the answer. If somebody have one idea.

Ravi

I think your context is false.
The matrix is already inherently isotropic.
So the 'isotropization' of such is not available.
Instead you would need to seek a form that is demonstrably anisotropic
and then 'isotropize' it.
As I seek such forms the meaning of isotropy starts to seem less and
less plausible. The context of this word has little value and the
dependence upon it within the matrix goes against the grain of
relativity theory, where an interesting anisotropic form is evident
with the unidirectional quality of time. My own conclusion is that the
matrix representation and therefor the tensor representation is
invalid. Upon developing an anisotropic form the matrix representation
will no longer be valid. The possibility of 'isotropization' will
inherently produce conflict with the general dimensional system.

-Tim- Masquer le texte des messages précédents -

- Afficher le texte des messages précédents -

I understand your meaning. But if you considered a material which is
composed of a sum of crystals with a very tiny dimension (nano) ,which
are randomly oriented: you get at the end a macroscopical istropic
material. The isotropy conditions of macroscopical mechanical
properties is done in this way: they take the elastic tensor for
example and put the condition the orientation of this tensor invariant
is.
For the cubic crystal, people gets for example the isotropy relation:
c11=c12+2c44

And I would like to get such relations for the other symmetry
materials??

This sounds like an interesting problem but I don't understand the
procedures you use. You said you are up in 6D. Do you have a model of
your material? If so then it may be possible to take random projection
points and connect them by lines and minimize or maximize a measure by
projecting from points along that line. This then can be iterated and
because the randomization takes care of projection you can sit back
and drink tea. This is really just a guess of what you are trying to
do and this method could be time intensive. Data density is
challenging in high D.

Do the crystals take on random orientations at the macro scale? I
suppose no but that the actual pattern is unknown? Otherwise you
wouldn't have a macro crystal would you? It seems that in effect you
are asking to find the orientations that some qualities will hold
constant even when the crystal is rotated. I have worked with high
dimensional data and nonorthogonal data as well and am happy to learn
something but I will understand if you don't want to answer all of my
questions. Here is an instance of the projection style that I am
speaking of:
http://bandtechnology.com/PolySigned/Lattice/P5Signon.gif
This is a 4D projection (4D->4D->2D). Do you expect continuous results
or are you going to get discrete nodes of equivalent behavior?

-Tim- Masquer le texte des messages précédents -

- Afficher le texte des messages précédents -

Very interesting, I will try to give an answer of your questions. I
hope that I can...

Take experimentally nanopouders which consist on crystal cubic
symmetry.
And then compact it: you will get a (macroscopical) material with unit
which are cubic.

Now, look at the elastic properties: the mechanical properties of each
unit are caracterised by the knowledge of 3 constants: c11, c12 and
c44.

But the elastic constants of the material <c11>, <c12> and <c44> can
be related to the elastic constants of each unit (c11, c12 and c44).
It is my hypothesis....

To calculate it, we can isotropise the matrix (6*6), I mean to give
the hypothesis that this one has to be invariant in rotation. And I
have to get relations between <c11>, <c12> and <c44> and c11, c12 and
c44.

My Problem ist more mathematical; If C is my matrix, I would like to
do Rot(C)=C with C a matrix (6*6). Under these conditions, I will get
relation between the elements of the matrix C.

I hope that I had give you some answers of your questions. If you know
how can I do this simple mathematically calculations, it can be very
helpfull for me...
Thanks...

Ravi

I'm not sure that this is a good solution to your problem but perhaps
it will stoke some thoughts for you or others:

If C were your object i.e. the crystal albeit in elastic
representation then we could take projections of it and compare those
projections. Then we would hopefully still see:
Rot( C ) = C .
Could one then take say a 2D projection Proj2():
Proj2( Rot( C )) = Proj2( C ) ?
Would the zero results of these two projections still be zero? Yes, I
believe so.
But
Proj2( Rot())
is just another Proj2 unique from the noncomposite Proj2(C) so we
introduce notation
Proj2A, etc.
These projections are as I outlined in the previous post. One can
sweep the space in lines and seek the minima of
Proj2A(C) - Proj2(C).
While there is no guarantee that the minima will be correct in the
higher dimension they may be candidates and those that were filtered
out are justly rejected. If the solution is continuous then neighbors
will help find the minima.

Should the progression upward one dimension to Proj3() reuse the data
found in Proj2() then a path toward Proj6() is found.

This is all a bit loose and I cannot guarantee its accuracy but the
process I would call inverse dimensional reduction. I am having a hard
time seeing the jump upward in dimension and the validity of the
rejection process yet informationally these seem sensible. As I review
my code for the 4D projection I see that my claim of doing a 4D->4D-
2D composite is false. I actually go straight from 4D->2D so the
orthogonalization is greatly simplified. The code that does this is at
the end of this post, but my point is that this chore is going to be
computationally more intensive and it is right there in
orthogonalizing the vectors upward in dimension that I am struggling
to see at the moment.

I'm sorry if your investment in time here is wasted but food for
thought can come in many forms including my mistakes. Do you work
physically with these materials?

-Tim

RandomRotatingProjection::RandomRotatingProjection( int dim, int
frameCount )
:Projection( dim, 2 ), c1(dim), c2(dim), dc(dim)
{
frames = frameCount;
curFrame = 0;
srand(time(0));
c1.Random();
c2.Random();
dc = c2 - c1;
dc = dc / frames;
*proj[0] = c1;
proj[1]->Random();
proj[1]->Orthogonalize( *proj[0] );

}




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Relevant Pages

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