# The Fundamental Physical Bases of the WR Approach (and, Consequently, of FEM) in General

It has been quite some time (more than 1.5 years) that I had touched upon the topic of the physical bases of FEM in general, and of the general weighted residual (WR) approach in particular, at iMechanica (see here).

The position I then took was that there is no known physical basis at all for the WR approach---despite its loving portrayals in mathematical terms, or its popularity.

Further, I had also expressed (here and elsewhere) that a basis in physical principles existed for FEM only in a rather limited sense: wherever the energy interpretation was available for the model. (Note, this too is already at variance with what some of the authors have written in books.)

I have not yet changed my opinions.

But, still, it is easy to miss things.

And, it is easy to teach wrong things to one's students too---especially if doing so suits one's own professional goals, research programs, reputation, funding applications in the pipeline, greencard/citizenship applications, or a precisely similar set of things for one's PhD advisor/professional mentor/current superior/group/past alumni associates (sentimental in nature or otherwise).

Nevertheless, I would consider myself immoral if I did not check things out before proceeding to profess or teach them (or create some impression in my students' minds about them---positive or negative).Accordingly, here we go.

Please let me know if anyone has been able to unearth or discover any physical basis whatsoever---i.e. any explanation in terms of any known (or newly discovered) physical principles---over the past 1.5 years (or anytime earlier) whereby the weighted residual approach can be said to possess a fundamental physical interpretation.

And, thereby, also FEM, in its more general "avatar."

Thanks in advance.

- Ajit R. Jadhav's blog
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## FEM vs. Fourier

Dear Ajit,

if you believe that there is at best an insufficient physical basis for WR and FEM: I agree wholeheartedly. Deformation theory is in deep conflict with potential theory.

FEM is a method invented to approximate solutions the conventional deformation theory does not provide by itself. The natural method to solve spatial gradient roblems would be the Fourier method. But that method needs a baseline term which the current deformation theory does not offer. So the grid is built up, and the distance of any two points may serve as a makeshift baseline.

The error is in my view in the Cauchy theory of stress: in his continuity approach Cauchy let his tetrahedron reach zero in order to describe stress at a point Q. Following the laws of potential theory (which in the 1820s was at best in its embryonic state) that should not have happened, because (a) elasticity is a part of thermodynamics, (b) the thermodynamic work function is logarithmic, (c) hence elasticity is logarithmic; besides (d), since logarithmic potentials do not have a natural zero point, some zero state must be defined by convention. Best example is the standard state in thermodynamics, relative to which all other states are then defined. Note, please, that the entity which all other terms refer to is a finite unit mass, or unit volume, the mol; in PV = nRT both n and V are finite.

This appears to contrast sharply with Cauchy's attempt to describe stress at a point. But that is not entirely true. In potential theory it would be done this way: assume some finite volume V with mass n; its external and internal properties, boundary conditions and whatnot can then be understood as a function of the center of mass of V, call that point Q. Thus external gradients can easily be described as a function of a coordinate system in which Q is a point. I mean to say: we do not need Cauchy's continuity approach. Take thermodymanics as a guide. Leave the system finite.

If V is finite, it has a radius r. In the standard state r can be set to have unit magnitude; and suddenly you have the baseline for Fourier methods. This unit distance term (which Cauchy let vanish identically) is in potential theory the zero potential distance. It is of most profound importance because it is required to definie work. Best example is the length of Hookes's spring. Cauchy's theory is wrong because he abolished that distance term.

In the application of my theory (see my blog ) I do not need FEM. I do it all with Fourier.

Regards,

Falk

## Reply to Falk

Dear Falk,

Thanks for writing.

However, I would rather keep this thread for discussion on MWR iteself---its physical bases---and not any topic that is too far away from MWR.

Having said that, nevertheless, let me also answer you in brief---rather, make my position clear with respect to your propositions.

I have gone through your site and papers once, and here is what I had said about them. Since I had carefully thought about the issue at that time itself, and since your recent posts do not address the observations I had made, I suppose there is no reason for me to change my judgment in any way.

As I said, I would like to conduct the rest of this debate/discussion in another thread. And yet, frankly, I don't see any need for having such a discussion for myself either.

But if you must insist, then, I don't mind if you address the following concerns (in some other thread). Please note, I certainly don't promise you (or anyone else) anything like a prolonged debate or discussion. But I will surely read your next reply (in another thread),

providedthat you make sure that it touches upon the following points:(A) The points I have raised earlierhere:Especially, my point nos. (2.1) and (3.0) at that page.

(B) Additional points:(i) Why do you say that a characterization of mechanical response of solids from a thermodynamic perspective is absent in the traditional theory (i.e. all solid mechanics other than what you say of it)? I can clearly see that such a description does exist. What makes you say that it doesn't?

(ii) What part about Cauchy's definition lacks thermodynamic implications?

(iii) In your "theory", do you believe that stress remains a 2nd rank tensor quantity?

If yes, can you give component-wise derivation as to how such a quantity is defined after starting with the thermodynamic laws?

If not, what are the physical units or dimensions of stress according to you? And, why should anyone begin using such a "stress"?

OK. I suppose this would be enough for the time being.

Please post your answers elsewhere (you can always create a new thread)

and let me know so that this thread remains concerned with MWR itself.

- - - - - - - - - -

And, finally, please remember two things:

(i) I will not give up my terms. And, BTW, I do know for myself what they are. Therefore, if you wish me to consider your theory, it would be your responsibility to indicate the physical bases and conceptual connections of your theory to my---i.e. the generally accepted solid mechanical---terms.

(ii) I am nobody. Although a productive researcher, I don't have either a job or a PhD in hand. Further, despite blogging here at iMechania for quite some time, I still am neither a moderator, nor an administrator here. For that matter, I have not even been invited to run just a single edition of the journal club here or whatever---ever... Indeed, I do have a lot of (valid) issues with Americans in general too---their academia (who once failed me without reason---and never made up for it despite my immediate supervisors' apparent willingness to help me), their media (who think my private life belongs either to their own father(s) or to their government---whoever they "respect" more), and their government (read good criticism of this last at the Ayn Rand Institute---who themselves also, to the best of my direct knowledge, have never supported my specific case in any direct way.) And yes, I do think that so far you've written as if you had taken into account some of these things. I appreciate that. I am just spelling it out more fully now (though such things can never be absolutely "complete"). ... Just so you know (I mean to say, I don't care to discuss these things, but then, sometimes, people don't know about these things and so....)

Now, it's your call (addressing the things that are

abovethe dashed line, and in another thread, please).Truly,

Ajit R. Jadhav

## Cauchy vs. potential theory

Ajit,

very well, then come over to my blog .

Falk