Question about the Newton Raphson Method

[fran.cesc]

Hi all,

I have got a question about the way the NR method works. I am reading lots of books but I cannot figure it out.

Let's consider the non linear graph of the picture(

http://nptel.ac.in/courses/105108072/mod07/image/lec1_clip_image026_0000.jpg).

My question is: does the solver already know the equation of that non linear curve, before starting the iterative procedure? If not:

My first guess is the origin of the coordinate system. Then I apply a load equal to the load step. Using linear equations, I obtain the tangent. I intersect the tangent with the horizontal line of equation y=Load Step.

Now??? How can I project the point on the horizontal line down to the non linear curve if we do not know the equation of that curve?

On the other hand, if the solver already knows the equation of the non linear curve, then how has it calculated it? And why do we need an iterative procedure to get the displacement at y= load step when we know the equation of the curve? Cannot the solver replace y=Load step into the non linear equation?

Thanks very much

 

[prex]

Of course you must be able to obtain a value of F for any given value of d (as in your graph), but you are supposed to not be able to do the inverse: calculate d for a given F. That's where NR comes in.

prex
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[fran.cesc]

Of course you must be able to obtain a value of F for any given value of d (as in your graph), but you are supposed to not be able to do the inverse: calculate d for a given F. That's where NR comes in.
prex

So I do not originally have the non linear curve. You are saying that:
I start with a certain value of F and d (on the picture (0;0)). I apply the force equal to the load step (on the picture the force F): now, as you say, I cannot calculate the d from F (where F is the force equal to the load step amplitude), so I have to use linear assumptions to get what the d would be if F was applied linearly. Now I have got the point (d1; F). So I have got the point on the horizontal line. Now I can calculate the real Force which is associated to d1 through the use of non linear equations because, as you say, I can obtain F from d. In this way I obtain the projection of the point (d1,F) on the non linear curve. And then I can restart. Is it correct?

Thanks

 

[prex]

Yes, correct.

prex
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[rb1957]

I guess it took me a long time to figure out the sketch.

Given F, find d ...
first estimate is based on the initial slope (d1)
but then you find the slope at d1 is much lower than the initial slope
and when you evaluate the function for d1 you get F1<F
so you can extrapolate the slope at d1 to F and get d2
evaluate, extrapolate, repeat ...

another day in paradise, or is paradise one day closer ?

 

[fran.cesc]

Of course you must be able to obtain a value of F for any given value of d (as in your graph), but you are supposed to not be able to do the inverse: calculate d for a given F. That's where NR comes in.

I am sorry, but there is something which I don't get.

You are saying that there is a way through which I can go from d to F with non linear equations? I thought FEA software could work only with linear equations. So the solver can deal with non linear equation when it goes from d to F, but it cannot do that when it goes from F to d?

 

[IRstuff]

All numerical methods use some form of quasi-linearization for non-linear cases. That means it's implicitly based on an assumption that the equation looks linear in a local neighborhood. It's the discretization step in differential calculus before you take the limit as Δh goes to zero.


TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

https://www.youtube.com/watch?v=BKorP55Aqvg

faq731-376 forum1529

 

[EngineerMickeyMouse]

Extending fran.cesc description, lets annotate projection of point (d1,f) at non-linear curve will be (d1,F1). As such all the magic behind this process is hidden in F1 calculation procedure, which in fact is as per linear FEA. Non-linearity discussed here for NR Method is covered by next approximations of plenty linear equations, until convergence limit will be met. This is my understanding only, please double check my words.