Hi,
i have the problem clearly detailed in a document. i shall provide you
with the link to the document. please save the document and go through
it. The problem is basically a numerical method for solving pairs of
differential equations.
link to the document: http://www.etsu.edu/ospa/rso/lekhalax/document.html
link to additional paper mentioned in the above document :
http://www.etsu.edu/ospa/rso/lekhalax/multigrid.html
I would really appreciate your help! I shall tip you with a decent amount too!! |
Request for Question Clarification by
mathtalk-ga
on
05 Feb 2005 15:32 PST
Hi, perukav-ga:
What is missing, at least from your discussion of the problem data,
are the boundary conditions.
Are we to assume "homogeneous" boundary data, ie. that the sum of
stress is zero on the boundary? Or is there a different condition?
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
05 Feb 2005 19:59 PST
Hi,
sorry for the late reply.
regarding the boundary conditions....on the free boundaries, the
principal stress perpendicular to that boundary is zero. and therefore
the difference of principal stresses(which is the input (or) the
given data) is equal to the other principal stress which is parallel
to the boundary.
also, could you please concentrate on overcoming the problems
asociated with the noise data. (you can find this in equations 10-14
in the paper provided)
please let me know if you need any other information.
i hope you undertsand that i am looking for the program(coding) not
just the algorithm. just wanted to make sure!!!
I really appreciate your help!
Thank you very much!
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
05 Feb 2005 20:29 PST
Hi, perukav-ga:
Thanks for the prompt Clarification. However since the role of the
multigrid or other numerical PDE method is to determine the _sum_ of
the principal stresses, we need a boundary condition for that.
It seems to me that in saying the "free boundaries" have a normal
component of the stress vector zero, you are specifying a varying
linear combination of the principal stress components (the
coefficients depending on the "inclination" of the stress components
at the boundary).
It may be that if the "difference" of the principal stresses is known
at each at each grid point, then a preprocessing step can specify the
sum of the principle stress at the boundary points.
I think it quite ambitious to implement a multigrid method for this
specific application on arbitrary domains, with special attention to
noise reduction. Such a project could easily constitute a masters
thesis or more in terms of effort.
If you are interested in developing the code yourself in iterative
steps from a simple starting point (rectangular domain, Poisson
solver, no concerns about noise reduction), then I'd be glad to advise
you on how to begin.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
05 Feb 2005 20:56 PST
Hi,
actually i need the output from this program to apply for further analysis.
anyways, could you do the same for a rectangular domain with no
concerns on poisson reduction?
please let me know your opinion. I shall get back you depending on your response.
Thank you
Perukav
|
Clarification of Question by
perukav-ga
on
05 Feb 2005 21:07 PST
Hi,
sorry i typed "no concerns on poisson reduction?". I actually meant
"no concerns on noise reduction".
i would appreciate if you can do the program excluding the noise
reduction issue and the arbitrary domain. I need the output for
further analysis and i am stuck at this part.
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
05 Feb 2005 21:57 PST
Hi, Mathtalk-ga,
as i mentioned in my document, the arbitrary domain and noise
reduction are optional. it is perfectly ok if you like to ignore
those.
please let me know your opinion.
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
06 Feb 2005 07:55 PST
Hi, perukav-ga:
Perhaps you can post a small batch of sample input data?
For illustration a 10x10 grid would be nice. It will be helpful to
know precisely what input data you are starting from at this stage.
If you cannot easily reduce a "real world" example to a grid this
size, perhaps you could post a link to a larger example.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
06 Feb 2005 09:21 PST
Hi, Mathtalk,
Thank you for our reply. i shall get the data and get back to you in a while.
regards,
Perukav
|
Clarification of Question by
perukav-ga
on
06 Feb 2005 11:44 PST
Hi, Mathtalk,
here is the data for a simple 3*3 grid. i guess we can change the
dimensions later while working on the actual larger grid.
NODE sigma1-sigma2 alpha(orientation)
1 11414 .758
2 22955 18.33
3 17530 42.08
4 22955 18.33
5 7257 34.26
6 11414 .758
7 17530 42.08
8 9590 0
9 8729 0
please assume a value for K.
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
06 Feb 2005 12:28 PST
Hi,
the third and 7th values for alpha is 3.72. please make a note this.
also, assume K=2.75
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
06 Feb 2005 12:51 PST
Hi, Mathtalk,
i apologise for not giving this information earlier. i dont know if
you need this information or not. if you do , here it is
the body is a square(rectangle) 1-3-2-5-4-7-6-8 ....with 9 as its midpoint
side 6-8-1 is fixed
side 1-3-2 is free boundary
side 2-5-4 ..force applied
side 4-7-6 is free boundary. I hope you can visualise the figure. i
could not upload it, so i am giving you the notations so that you can
visualise it.
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
06 Feb 2005 16:48 PST
Hi, Mathtalk,
i just drew it again according to my model, it basically is the same
as yours with numbers in a different positions. however, here is my
model
free bdry
a
f 6 -- 7 -- 4 f p
i | | | o p
x 8 -- 9 -- 5 r l
e | | | c i
d 1 -- 3 -- 2 e e
d
free bdry
force applied = 3000n at each of three nodes.
E of the material = 3*10e10
poisson's ratio = 0.3 ..... incase you need these
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
06 Feb 2005 16:50 PST
Hi,
force is 5000n(not 3000n)
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
08 Feb 2005 06:04 PST
Hi, perukav-ga:
The sum of principle stresses:
? = ?_1 + ?_2
satisfies Laplace's equation:
?? ? ?²?/?x² + ?²?/?y² = 0
A finite difference approach would then "discretize" this partial
differential equation by replacing the second derivatives at each
interior node of your rectangular region by a second difference
approximation:
?²?/?x² ? [ ?(x+h,y) - 2?(x,y) + ?(x-h,y) ]/h²
?²?/?y² ? [ ?(x,y+h) - 2?(x,y) + ?(x,y-h) ]/h²
adjusting as necessary for possible variation in the grid-spacing h.
This gives a system of homogeneous linear equations to solve, but we
needed to supplement these with boundary conditions. In fact the
boundary conditions are the "source" of the nonhomogenous data that
drives us to a nonzero solution of Laplace's equation.
In the three by three example the discretization is simply that the
one interior node, which you've labelled 9, is the average of the four
adjacent nodes up, down, left, and right of it (which you've labelled
7,3,8,5 respectively).
So the issue becomes how to assign a value to the sum of principle
stresses ? at these boundary points. Clearly a distinction is to be
drawn between conditions at the "free" boundaries and at the
fixed/force applied boundaries.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
08 Feb 2005 12:32 PST
Hi, Mathtalk,
i really couldn't find out the necessary boundary conditions. but,
since i plan to apply this method to residual stresses, all the
boundaries in that case would be free boundaries with no external
force applied or no boundaries fixed.
so wondering if we can consider, sum of the stresses on any boundary
would be the same as difference of stresses(since only one stress
exists on the boundary). the difference stresses as you know is the
known value. anyways, the input data will be the same, difference of
stresses and angles.
also, actually i got a simpler method of separating the stresses based
on equlibrium equations. final goal is to find out component or
principal stresses. so, i guess it would be much easier for you to do
and isn't complicated in terms of the boundary stresses and so.
here are the links for the notes i have prepared on the method. please
go through those. i am sure you would find it easier. i might have
gone wrong in signs as i did that in a hurry.
www.etsu.edu/ospa/rso/lekhalax/image1.jpg
www.etsu.edu/ospa/rso/lekhalax/image2.jpg
please let me know if you got any questions..
also, if you would like to follow this easier method, since you spent
some time on the other method, i strongly feel you deserve more than
what i have offered in the beginning. I shall be very happy to offer
you $100 more once you are done with the program. i guess it should be
in the form of tips since i already offered maximum limit of $200.
Thank you, i look forward to your comments.
Perukav
|
Clarification of Question by
perukav-ga
on
08 Feb 2005 12:36 PST
Hi, Mathtalk,
i guess if you look at the equilibrium equations, understand that the
shear stress(towxy) is known at all the points(equation (2b) in the
paper provided earlier). you can figure out how this method works by
simple integration.
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
08 Feb 2005 13:21 PST
Hi, Mathtalk,
i guess sum of the principal stresses on a boundary is an invariant,
and i dont think we can know those values on the boundaries.
i look forward to your opinion on my earlier post regarding the shear
difference method based on equilibrium equations.
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
12 Feb 2005 06:04 PST
Hi, perukav-ga:
Okay, I see you are saying that you have the shear stress at every
point (see Comment below), so I guess you have the "delta" information
referred to in the multigrid paper.
BTW, was my use of Greek symbols in an earlier Request for
Clarification (RFC) legible to you? (see above, about five posts)
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
12 Feb 2005 10:47 PST
Hi, Mathtalk,
"delta" ...do you refer to the "isochromatic phase"(2nd line below
equation(2b) in the paper)? if so, yes i have the delta.
but the formula for shear stress is given by equation(2b) in the
paper. we need to know "difference of stresses" and the orientation
"alpha" to calculate the shear stress. both these values are known.
since the difference of stresses is known, delta can be calculated
from equation (3). but i guess we don't need that.
yes, the greek symbols were legible to me. please let me know if i
gave you the needed information.
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
12 Feb 2005 10:57 PST
hi Mathtalk,
by shear stress, i mean "sigmaxy"(the second term in equations (1a)
and (1b) ........the LHS in equation (2b)).
i apologise i referred to shear stress as "towxy"in my previous comments.
Thank you,
Perukav
|
Clarification of Question by
perukav-ga
on
12 Feb 2005 15:31 PST
Hi, mathtalk,
sorry for the confusion on my part. yes i do have the values of
"delta" at every point.
basically i know diffrence of stresses at every point and from those
values "delta" can be calculated(equation 3).
"alpha" - orientation of stresses is known at every point.
from the above values shear stress "sigmaxy" can be
calculated(equation (2b)). but it really doesn't matter u know the
differene of stresses or the "delta", since both are related to each
other. while calculating, we could either use equation (2b) directly
or substitute 1/2*K*delta for the difference of stresses.
i hope you could access the notes i prepared on this method.
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
13 Feb 2005 07:09 PST
Hi, perukav-ga:
To implement these calculations, would it be suitable to provide a C program?
Whatever language is used, one of the early design decisions is about
the representation of the geometry. I gather from the discussion that
while a rectangular geometry is of interest, more general figures are
important to your work.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
13 Feb 2005 08:16 PST
Hi, Mathtalk,
C program is perfectly ok! yes, though a rectangle is of primary
interest, arbitray figures are important to my work as you said.
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
14 Feb 2005 06:57 PST
Okay, my sense is that for rectangular domains it would be fine to write C code.
However once you get into managing the geometry of more general
domains, even simply connected ones in the plane, the design
complexity is sufficient to make C++ or another object-oriented
language attractive.
For the present I'll implement the solution in C for a rectangular
domain, however, because I have no clear picture of what your more
general domains are going to entail.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
14 Feb 2005 22:18 PST
Hi, Mathtalk,
ok...sounds good!!
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
15 Feb 2005 06:22 PST
Here's my impression of the method you propose.
You turn the partial differential equations into some ordinary
differential equations (ODEs), separately for ?_x and ?_y.
Even so you must still tie these differential equations to some data
on the boundary by way of "initial conditions" for the ODEs.
Apparently you have concluded that ?_x is known on the "force applied"
boundary (x=L) and that ?_y is zero on the "free" boundary (y=0).
Naturally this approach is very dependent on the rectangular geometry.
I also think the equation for ?_y at the top of the second page should
have been for the evaluations of ?_y at y, rather than at x, since
your integral there is from y = 0 to y. Please check this.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
15 Feb 2005 10:57 PST
Hi, Mathtalk,
you are exactly correct. also, yes the second page should have been
sigmay at y. i am sorry, i wrote that down in a hurry.
if we like to consider a different geometry other than a rectangle, i
guess we should start from any free boundary, since we know both the
stresses sigmax, sigmay there(one stress is zero and other is the
difference of stresses, which is a known value). now we can use the
same two equations for sigmax and sigmay in which shear stress is
known and the initial values of sigmax and sigmay are known(the
boundary conditions). in that case(arbitrary shape) it might
cumbersome to find the stresses at every point of the body, but doing
that across the centreline or etc. should be easy, i guess.
Thank you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
22 Feb 2005 08:02 PST
Hi, perukav-ga:
I would expect to have some working/tested code for the rectangle in
about a week, assuming the development goes "smoothly". My biggest
concern is that we seem to be dealing with an over-determined problem,
but that this seeming embarrassment of riches is clouded by a
dependence on experimental data that may prove "inconsistent" due to
measurement errors. I'm mindful of an earlier request by you for help
in smooting errors, which I did not grasp clearly at the time.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
05 Mar 2005 09:00 PST
Hi Mathtalk,
my question(post) expires on march 7th i.e., a couple of days from
today. wondering if you could have the program done by then! I would
really appreciate your help!
Thak you,
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
05 Mar 2005 11:34 PST
Hi, Perukav-ga
I hope to have at least a satisfactory program for you to solve
simplified boundary conditions. You can then request a series of
clarifications as appropriate to any more realistic data or as needed
to correct a misunderstanding on my part.
regards, mathtalk-ga
|
Clarification of Question by
perukav-ga
on
07 Mar 2005 08:25 PST
Hi Mathtalk,
just a reminder that my question expires today. wondering if you want
me to repost the question? That would help me request for
clarifications, if any. i don't know if i can ask for clarifications
on an expired post.
please let me know your opinion and i look forward to the solution to my problem.
Thank you!
Perukav
|
Request for Question Clarification by
mathtalk-ga
on
07 Mar 2005 09:42 PST
Actually we _will_ be able to post Comments on the expired Question
here, so I suggest that we take that approach, at least for the next
couple of days. I would rather not post an altogether incomplete
answer at this time.
Thanks, mathtalk-ga
|
