I agree with Jean. Well done

BTW, the posed problem gives the difference equations not differential (as this given solution imposed). Actually, I think it does not matter and it might be more complicated that way (errors, time steps, singularity etc.), not sure.

Regarda,
Radovan

Wrote

I agree with Jean. Well done

BTW, the posed problem gives the difference equations not differential (as this given solution imposed). Actually, I think it does not matter and it might be more complicated that way (errors, time steps, singularity etc.), not sure.

Regarda,
Radovan

Thank you

Does that mean that you think the last approach i did is the best way to solve the problem? The trick i think is to establish boundry conditions for ODE2,3,4 which is done by iteration till t.xy in the input box matches t.xy in the calculation after all the constants is established.

Thanks again for your help :d

Wrote

Does that mean that you think the last approach i did is the best way to solve the problem? The trick i think is to establish boundry conditions for ODE2,3,4 which is done by iteration till t.xy in the input box matches t.xy in the calculation after all the constants is established

Radovan is right: the functional relationship is in term of finite differences,
i.e: for pre-computer age. In fact you just need to integrate the RHS member.
You could think of the integrator operator, that would be a great mistake
on two counts: the operator is horribly slow and may be unusable wrt accuracy.
The "RK" integrator is hyperfast [probably from borrowed code] and accurate.

The other question is if RK supports conditional ... answer is YES

The attached confirms, unfortunately a single conditional.
I don't understand the conditional of your project.
@ 735 °C, steel is pink, visible in the dark, soft, not black smith works.
@ 900 °C, red hot for forging. Why is the conditional infinity @ 735 ?

Just observation(S) ...

Sorry, can't upload the work sheet at the moment

Wrote

Sorry, can't upload the work sheet at the moment

ODE Aminohippuric MCD.sm (44 KiB) downloaded 57 time(s).

... finally, your project is completely correct as it looks.
Observe few modifications to get equally spaced data result,
1. re-arranged the number of points
2. submatrix to get rid of duplicates
3. spline "global"
4. discretise 1,2,3 ... or at will as you wish.

Saved in my personal file, thanks ... Jean

ODE Multiconditional [Steel fire].sm (37 KiB) downloaded 59 time(s).

... two pages of Engineer's stuff has been added [page: 3, 4]
The two solvers are fully automated, i.e: given Y on the red curve
they both solve for their respective values on the X axis.
The function 'θ(t)' does not have an explicit "inverse". No problem
for solving with the Smath solve bloc. We can solve both iteratively,
not offered ... just ask for ... will be pleased to provide.

Conserve preciously the "root" technique ... sometimes more involved.
This project could not have been done so neatly Mathcad 11.
Title, X, Y axes need some identifier(s).

ODE Multiconditional [Steel fire].sm (53 KiB) downloaded 55 time(s).

Wrote

... two pages of Engineer's stuff has been added [page: 3, 4]
The two solvers are fully automated, i.e: given Y on the red curve
they both solve for their respective values on the X axis.
The function 'θ(t)' does not have an explicit "inverse". No problem
for solving with the Smath solve bloc. We can solve both iteratively,
not offered ... just ask for ... will be pleased to provide.

Conserve preciously the "root" technique ... sometimes more involved.
This project could not have been done so neatly Mathcad 11.
Title, X, Y axes need some identifier(s).

ODE Multiconditional [Steel fire].sm (53 KiB) downloaded 55 time(s).

Hi Jean!

Thanks for still working on this project. Interessering things you have done with it. I will have to study it to better understand it.

Just wanted to let you know (you probably already know) your approach still aquires the user to iterate the set of DE's to establish the correct initial conditions for DE2/3/4. If you change the input (ratio.Am_V and ratio.Am.b_V is the ones you will likely change), the DE's gets disconnected.

Regards,

Hans Kristian

A more compact calculation.



Testing4odeCompacto.sm (15 KiB) downloaded 48 time(s).

Best Regards

Carlos

Wrote

A more compact calculation.



Testing4odeCompacto.sm (15 KiB) downloaded 48 time(s).

Best Regards

Carlos

Interessting How do you "calibrate" the T-matrix when the input changes?

It's great to see all the attention this project has gotten, seeing as it is a highly relevant problem for engineers.

Wrote

If you change the input (ratio.Am_V and ratio.Am.b_V is the ones you will likely change), the DE's gets disconnected.

You are right there ! The system is not homogeneous, it depends upon
those parameters you mention ... like a monkey business. You must recast
the system like you did wrt critical temperature ... Oh ! that's your
competence. That stuff is Chinese to me.
BTW: attached the discrete solvers. All in all a huge/interesting project.

Cheers, Jean

ODE Multiconditional [Steel fire].sm (63 KiB) downloaded 45 time(s).

Wrote

Interessting How do you "calibrate" the T-matrix when the input changes?

Yes, but you don't get the same DE shape, not looking like the Excel snippet.
You can join the segments "bing/bang" technique. You still have to submatrix
to get rid of duplicates at junctions.
Good question: can we plug a vector of the start/end/number of points per segment
just to imitate my previous attachment ?

ODE Multiconditional [Carlos Compactum].sm (27 KiB) downloaded 40 time(s).

... the ODE compactum has just been doctored.
The suite will be same as my previous attachment,
submatrix to remove duplicates, discretize for
regular spacing tabulation ... solvers.

Jean

ODE Multiconditional [Carlos Compactum].sm (19 KiB) downloaded 40 time(s).

... all done up to the two solve/inverse.

ODE Multiconditional [Carlos Compactum].sm (25 KiB) downloaded 49 time(s).

Wrote

Wrote

Interessting How do you "calibrate" the T-matrix when the input changes?

Yes, but you don't get the same DE shape, not looking like the Excel snippet.
You can join the segments "bing/bang" technique. You still have to submatrix
to get rid of duplicates at junctions.
Good question: can we plug a vector of the start/end/number of points per segment
just to imitate my previous attachment ?

ODE Multiconditional [Carlos Compactum].sm (27 KiB) downloaded 40 time(s).

I haven't had much time to progress on this project during the weekend. I will work more on it this week (for instance stack all the solutions and submatrix the junctions like you showed us )

Is this what your were looking for regarding automating equal steps?

Best regards,
Hans Kristian




Edit: Upon closer inspection, computational times using your method of stacking all the DE's increases dramcatically. To calculate temperature at a given time takes ~1,3sec using your method.

Using my method (granted much more primitive, and it does give some unatural results in the crossover between the DE's, the computational times are ~0.15sec.

Wrote

Upon closer inspection, computational times using your method of stacking all the DE's increases dramcatically. To calculate temperature at a given time takes ~1,3sec using your method.

Using my method (granted much more primitive, and it does give some unatural results in the crossover between the DE's, the computational times are ~0.15sec.

You have several points in one remark:

1. The compactum method is not shorter in work sheet space. It is slower
because of the counter cumulative in "for loop" ... just interesting.

2. Your "steps*(,-,) computes more points 603 vs mine 491
thus slower up to the result. No interest as it produces the duplicates.

3. unatural results in the crossover between the DE's
You have put the finger in the wound !!!
There are very many more "splines" than Smath "ainterp [Akima], cinterp"
Of all the splines [lot] I'm familiar, NONE will accept duplicate
points ... worse: in this work, the duplicates are zombies , Ah !
So, you must strip for duplicates [submatrix does it well]. By same token,
once striped and stacked, the piecewise "ainterp" is completely useless.
As a remark, Smath "ainterp, cinterp" are hyperfast ... they spit 1000 data points
before it seems to have digested .
"cinterp" looks 1/1 universal vs others [Mathcad ...]
"Akima ainterp", I have not seen under the hood of it. From using, it reveals
very excellent in some works, but also pure scrap in other type applications.

4. After stripping, the f(x):=ainterp(X,Y,x) is passed in the reconstruct
discretiser to re-tabulate for equal points spacing as an export. For
interpolation, just spline over this data table.

Thanks Collab for this most interesting DE application.

Jean

ODE Multiconditional [Carlos Compactum] Copy.sm (26 KiB) downloaded 42 time(s).

ODE Multiconditional [Steel fire] Copy.sm (64 KiB) downloaded 41 time(s).

Wrote

... last version for my projects file.
Number of points is ~ immaterial wrt computation timing.
The offered is + than enough for spline

ODE Multiconditional [Steel fire] Copy.sm (53 KiB) downloaded 43 time(s).

Cool stuff. The "discrete solver/inverser" takes approx 12 seconds to solve on my computer. None of the graphs start out at 20degC though, why is that?

Edit: Stupid question; what is the purpose of the "root" expression in your workbook? Couldn't the original theta.g expression be used in the solver instead?

Best regards,

Wrote

Cool stuff. The "discrete solver/inverser" takes approx 12 seconds to solve on my computer. None of the graphs start out at 20degC though, why is that?

Edit: Stupid question; what is the purpose of the "root" expression in your workbook? Couldn't the original theta.g expression be used in the solver instead?

1. Copy/paste the last collapsed area in new sheet.
Both methods start plotting @ 20 °C as specified from "if/otherwise".

2. Often, "maple/solve" returns the inverse function.
This one though it looks simple does not, alternately "root" does.
and by same token result is the simplified expansion.

This refreshed version looks like nearly the "last brick in the wall".

Please: don't hesitate for more questions ... Jean

ODE Multiconditional [Steel fire].sm (64 KiB) downloaded 54 time(s).

This project has been re-organised a bit for publishing.
Few things are missing: an abstract, identify the X,Y plot arguments.
The discrete solver/collector can be set incremental for finer tabulation.

Jean



ODE Multiconditional [Steel fire] Copy.sm (67 KiB) downloaded 50 time(s).

I have organized my worksheet somewhat now, and collected only the relevant parts for my specific problem at the top. The other stuff that although it displays the power of SMath - it's not relevant for this workbook, is gathered at the bottom.


I really appreciate all the help you have given med Jean!

Best regards,

Staltemperaturer ved brann.sm (286 KiB) downloaded 87 time(s).

I have organized my worksheet somewhat now, and collected only the relevant parts for my specific problem at the top. The other stuff that although it displays the power of SMath - it's not relevant for this workbook, is gathered at the bottom.

Something mysterious, it does not download *.sm, only some sort of "Chrome" document.
Nothing to see, that's OK unless I can help more.

Cheers, Jean