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 (at iMechanica 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.