Hi, Mathtalk,

this is Perukav, i posted a question regarding "the integration of
pairs of diferential equations-numerical method". I was looking for
your response,  posted a couple of comments for you, but  did not
receive any response, i thought may be you weren't notified of those
since my question has expired.  Thought, i would contact you through
another post.

I am looking forward to your response for that question, i have much
work pending(that depends on this program) and would really appreciate
your help!

I would appreciate if you could get back to me as early as possible. 

Thank you,
Perukav

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Request for Question Clarification by mathtalk-ga on 18 Mar 2005 04:39 PST

Hi, Perukav-ga:

I apologize for not having gotten back to you more quickly.  The time
I've been able to do programming for you this past week was very
limited because of my "day job".  I hope this weekend will provide the
opportunity to get you a framework of code to test your ideas.

regards, mathtalk-ga

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Clarification of Question by perukav-ga on 20 Mar 2005 09:16 PST

Hi, Mathtalk,

Thank you very much for your response. I can understand your time constraints!
I look forward to your solution and really appreciate your help!

Regds,
Perukav

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Clarification of Question by perukav-ga on 23 Mar 2005 02:54 PST

Hi, Mathtalk,

wondering if you are working on the easiest method i have proposed or 
the multigrid method. In the paper for the easiest method, i guess we
have enough information on the available data along with the equations
that need to be integrated, also the initial boundary conditions.

I would appreciate if you could let me know which method you are
working on! I shall provide you with the links for the easiest method
once again, incase you would like to have a look at those.

http://www.etsu.edu/ospa/rso/lekhalax/image1.jpg
http://www.etsu.edu/ospa/rso/lekhalax/image2.jpg

also, please let me know if you still are looking for a tutorial on
stress tensors, i had difficulty finding a better one(soft copy), but
shall try again.

I look forward to your reply!
Thank you,
Perukav

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Request for Question Clarification by mathtalk-ga on 27 Mar 2005 16:33 PST

Hi, perukav-ga:

Let's check my understanding of the "measurement" data and what
requires calculation.  From the Quiroga/Gonzalez-Cano paper:

The sum ? of principal stresses is both (according to Trace Theorem):

  ? = ?_1 + ?_2 = ?_x + ?_y

where the latter subscripts indicate "Cartesian components" of a
stress tensor, and are _not_ to be understood as derivatives.

In what follows we will be concerned with the determination of the sum
?, because the difference of principal stresses is already known
directly (more or less) from experimental measurements.

Their paper contemplates that you can easily derive from the
photoelastic measurements two functions of (x,y) coordinates at grid
positions:

  ? = K ? cos(2?)

  ? = K ? sin(2?)

which satisfy two first-order differential equations:

  ??/?x = -??/?x - ??/?y  (6a)

  ??/?y =  ??/?y - ??/?x  (6b)

On a simple rectangular domain, say 0 < x < L and 0 < y < M, we could
solve either of these equations (assuming the right hands sides are
"known") uniquely by specifying a respective boundary value, e.g.

   ?(0,y) for y in [0,M]
or
   ?(x,0) for x in [0,L]

In the case that we have no boundary values, but instead both
equations above with "known" right hand sides, then for a simple
rectangular domain the problem is both over- and under-determined.  It
is overdetermined in the sense that the right hand side of either
differential equation allows for "degrees of freedom" in specifying
the values over the entire rectangle, while the "missing" boundary
value for either equation constitutes only a specification of values
over a one-dimensional variety.

On the other hand the problem is under-determined in a fairly benign
sense that any constant c can be added to a solution ? to obtain
another function ?+c that equally satisfies both differential
equations.

My proposal is therefore to approach this problem with a sparse-matrix
linear least squares solver.  Such an approach is pretty flexible, in
that one can proceed entirely without boundary conditions on ?, or one
can easily add them to the problem formulation _if_ they can be stated
in terms of ?.  Perhaps it is a matter of confusing the "normal
derivatives" of ? along the boundary with the (coincidentally)
principal component ?_x or ?_y there, but I had trouble understanding
exactly what a "free boundary" condition tells us about ?, or what is
separately known (to take the other tack) about ?_x and ?_y in the
domain's interior.

My Request for Clarification would be to have a data set for a
rectangular domain that provides the functions ? and ? referred to
above in (6a-b).  From this we can in principle "solve" for ? up to
some unknown constant of integration, but this is just a single
unknown value that we should be able to "fit" a posteriori to your
boundary conditions, assuming that my account of the problem above is
sound!

regards, mathtalk-ga

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Clarification of Question by perukav-ga on 01 Apr 2005 15:13 PST

Hi Mathtalk,

sorry for the delay in responding. I was away on some office
assignment. I shall provide you with the data you have asked for! I
got a question though!

if we derive the functions ? and ? from the data set for a
rectangular domain, woudn't that solution be specific only to that 
particular rectangular problem?

may be i got it wrong!! because if the forces applied are the
same(equivalent - i mean force per unit area), any rectangular body
would have the set of ? and ?.

I would appreciate of you could post your response! i shall provide
the data in a couple of days from now.

Thank you,
Perukav

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Request for Question Clarification by mathtalk-ga on 03 Apr 2005 08:23 PDT

Hi, Perukav:

I believe the calculations based on functions ? and ? are roughly
equivalent to basing them on ? and ?, and I'd be happy to have
experimental measurements in the latter form.  Since these depend on
experimental measurements, corresponding to the forces imposed on an
object, they vary with the experimental forces as well as with the
object's geometry (whether rectangular or not).

I've been able to reproduce the formulas in the Quiroga/Gonzalez-Cano
paper with one notable exception, which is their claim that the sum of
principal stresses ? satisfies the Laplace equation, ?? = 0.  I
suspect that this cannot be derived simply from the equilibrium
conditions, and that it depends on assuming the absence of "body
forces", ie. that all the external forces are imposed on the boundary.
 In any case I do not know how to derive it.

regards, mathtalk-ga

I found it interesting that the original problem was a $200 question,
and now the price is $2.00....

But, that's none of my business ;)

Hi, Perukav-ga:

My reading of the paper by J A Quiroga and A Gonz´alez-Cano:

"Stress separation from photoelastic data by a multigrid method"
Meas. Sci. Technol. 9 (1998) 1204?1210.

suggests that they consider the prior art (for separating the
principal stresses) as including a "line integration" approach by
Haake et al:

"2D and 3D separation of stresses using automated photoelasticity"
Exp. Mech. 36 269?76

similar to the last method you proposed, and also identify as a
drawback to this approach the influence of (experimental) errors
propogated along such lines of integration.

One insight that I have concerning their approach is that they mean to
distinguish solving the _pair_ of differential equations (6a-6b) in
their paper from the solution of the Laplace equation (7) by
multi-grid methods.  This distinction was not clear to me at the
outset, but it seems to be an important point to making optimal use of
the experimental data.

To refresh my rather faded memory of components analysis in stress
tensors, and especially to clarify what the notations mean in 2D, I
located the fairly engineering oriented tutorial here:

[Stress, strength, and safety]
http://www.mech.uwa.edu.au/DANotes/SSS/home.html

and especially the link to "Stress resolution and principal stresses;
strain resolution".  Perhaps you can suggest a better tutorial
connecting the real world measurements and the mathematical notation.

regards, mathtalk-ga

Hi, Mathtalk,

I will try to find some good tutorial on that. but as far as i am
concerned, could you please write the program not having much concern
on eliminating the errors. I would be happy if i can have a simple
program based on the method i proposed.
It would be really helpful if errors were eliminated and the program
is made sophisticated, but unfortunately i dont have much time left
now. so, I sincerely request you to provide me with a program for the
method proposed. I appreciate your help!

Thank you so much!
Perukav

A write-up of the relationship between principal stresses and
components of the stress tensor in plane geometries is found here:

[Principal Stress for the Case of Plane Stress]
http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/plane_stress_principal.cfm

The Website also presents a calculator for finding principal stresses
in the plane case on a different page, which illustrates the
connection with Mohr's circle.

regards, mathtalk-ga

Hi Mathtalk,

firstly, i sincerely apologise for not being in contact with you. i
was hospitalized and wasn't in a position to do anything.

wondering if you could solve the question for me. i am willing to pay
a decent amount($400) for the time lost.

looking forward to your reply.

Thanks you,
Perukav