Here are a few interesting links I browsed recently, listed in no particular order:

“Mathematicians Tame Turbulence in Flattened Fluids” [^].

The operative word here, of course, is: “flattened.” But even then, it’s an interesting read. Another thing: though the essay is pop-sci, the author gives the Navier-Stokes equations, complete with fairly OK explanatory remarks about each term in the equation.

(But I don’t understand why every pop-sci write-up gives the NS equations only in the Lagrangian form, never Eulerian.)

“A Twisted Path to Equation-Free Prediction” [^]. …

“Empirical dynamic modeling.” Hmmm….

“Machine Learning’s Amazing’ Ability to Predict Chaos” [^].

Click-bait: They use data science ideas to predict chaos!

8 Lyapunov times is impressive. But ignore the other, usual kind of hype: “…the computer tunes its own formulas in response to data until the formulas replicate the system’s dynamics. ” [italics added.]

“Your Simple (Yes, Simple) Guide to Quantum Entanglement” [^].

Click-bait: “Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.”

Don’t dismiss the description in the essay as being too simplistic; the author is Frank Wilczek.

“A theoretical physics FAQ” [^].

Click-bait: Check your answers with those given by an expert! … Do spend some time here…

Tensor product versus Cartesian product.

If you are engineer and if you get interested in quantum entanglement, beware of the easily confusing terms: The tensor product and the Cartesian product.

The tensor product, you might think, is like the Cartesian product. But it is not. See mathematicians’ explanations. Essentially, the basis sets (and the operations) are different. [^] [^].

But what the mathematicians don’t do is to take some simple but non-trivial examples, and actually work everything out in detail. Instead, they just jump from this definition to that definition. For example, see: “How to conquer tensorphobia” [^] and “Tensorphobia and the outer product”[^]. Read any of these last two articles. Any one is sufficient to give you tensorphobia even if you never had it!

You will never run into a mathematician who explains the difference between the two concepts by first directly giving you a vague feel: by directly giving you a good worked out example in the context of finite sets (including enumeration of all the set elements) that illustrates the key difference, i.e. the addition vs. the multiplication of the unit vectors (aka members of basis sets).

A third-class epistemology when it comes to explaining, mathematicians typically have.

A Song I Like:

(Marathi) “he gard niLe megha…”
Music: Rushiraj
Lyrics: Muralidhar Gode

[As usual, a little streamlining may occur later on.]

# Some suggested time-pass (including ideas for Python scripts involving vectors and tensors)

Actually, I am busy writing down some notes on scalars, vectors and tensors, which I will share once they are complete. No, nothing great or very systematic; these are just a few notings here and there taken down mainly for myself. More like a formulae cheat-sheet, but the topic is complicated enough that it was necessary that I have them in one place. Once ready, I will share them. (They may get distributed as extra material on my upcoming FDP (faculty development program) on CFD, too.)

While I remain busy in this activity, and thus stay away from blogging, you can do a few things:

1.

Think about it: You can always build a unique tensor field from any given vector field, say by taking its gradient. (Or, you can build yet another unique tensor field, by taking the Kronecker product of the vector field variable with itself. Or, yet another one by taking the Kronecker product with some other vector field, even just the position field!). And, of course, as you know, you can always build a unique vector field from any scalar field, say by taking its gradient.

So, you can write a Python script to load a B&W image file (or load a color .PNG/.BMP/even .JPEG, and convert it into a gray-scale image). You can then interpret the gray-scale intensities of the individual pixels as the local scalar field values existing at the centers of cells of a structured (squares) mesh, and numerically compute the corresponding gradient vector and tensor fields.

Alternatively, you can also interpret the RGB (or HSL/HSV) values of a color image as the x-, y-, and z-components of a vector field, and then proceed to calculate the corresponding gradient tensor field.

Write the output in XML format.

2.

Think about it: You can always build a unique vector field from a given tensor field, say by taking its divergence. Similarly, you can always build a unique scalar field from a vector field, say by taking its divergence.

So, you can write a Python script to load a color image, and interpret the RGB (or HSL/HSV) values now as the $xx$-, $xy$-, and $yy$-components of a symmetrical 2D tensor, and go on to write the code to produce the corresponding vector and scalar fields.

Yes, as my resume shows, I was going to write a paper on a simple, interactive, pedagogical, software tool called “ToyDNS” (from Toy + Displacements, Strains, Stresses). I had written an extended abstract, and it had even got accepted in a renowned international conference. However, at that time, I was in an industrial job, and didn’t get the time to write the software or the paper. Even later on, the matter kept slipping.

I now plan to surely take this up on priority, as soon as I am done with (i) the notes currently in progress, and immediately thereafter, (ii) my upcoming stress-definition paper (see my last couple of posts here and the related discussion at iMechanica).

Anyway, the ideas in the points 1. and 2. above were, originally, a part of my planned “ToyDNS” paper.

3.

You can induce a “zen-like” state in you, or if not that, then at least a “TV-watching” state (actually, something better than that), simply by pursuing this URL [^], and pouring in all your valuable hours into it. … Or who knows, you might also turn into a closet meteorologist, just like me. [And don’t tell anyone, but what they show here is actually a vector field.]

4.

You can listen to this song in the next section…. It’s one of those flowy things which have come to us from that great old Grand-Master, viz., SD Burman himself! … Other songs falling in this same sub-sub-genre include, “yeh kisine geet chheDaa,” and “ThanDi hawaaein,” both of which I have run before. So, now, you go enjoy yet another one of the same kind—and quality. …

A Song I Like:

[It’s impossible to figure out whose contribution is greater here: SD’s, Sahir’s, or Lata’s. So, this is one of those happy circumstances in which the order of the listing of the credits is purely incidental … Also recommended is the video of this song. Mona Singh (aka Kalpana Kartik (i.e. Dev Anand’s wife, for the new generation)) is sooooo magical here, simply because she is so… natural here…]

(Hindi) “phailee huyi hai sapanon ki baahen”
Music: S. D. Burman
Lyrics: Sahir
Singer: Lata Mangeshkar

But don’t forget to write those Python scripts….

Take care, and bye for now…

/

# Exactly what does this script show?

Update on 02 March 2018, 15:34 IST: I have now added another, hopefully better, version of the script (but also kept the old one intact); see in the post below. The new script too comes without comments.

Here is a small little Python script which helps you visualize something about a state of stress in 2D.

If interested in understanding the concept of stress, then do run it, read it, try to understand what it does, and then, if still interested in the concept of stress, try to answer this “simple” little question:

Exactly what does this script show? Exactly what it is that you are visualizing, here?

I had written a few more notes and inline comments in the script, but have deliberately deleted most of them—or at least the ones which might have given you a clue towards answering the above question. I didn’t want to spoil your fun, that’s why.

Once you all finish giving it a try, I will then post another blog-entry here, giving my answer to that question (and in the process, bringing back all the deleted notes and comments).

Anyway, here is the script:


'''
A simple script to help visualize *something* about
a 2D stress tensor.

--Ajit R. Jadhav. Version: 01 March 2018, 21:39 HRS IST.
'''

import math
import numpy as np
import matplotlib.pyplot as plt

# Specifying the input stress
# Note:
# While plotting, we set the x- and y-limits to -150 to +150,
# and enforce the aspect ratio of 1. That is to say, we do not
# allow MatPlotLib to automatically scale the axes, because we
# want to appreciate the changes in the shapes as well sizes in
# the plot.
#
# Therefore, all the input stress-components should be kept
# to within the -100 to +100 (both inclusive) range.
#
# Specify the stress state in this order: xx, xy; yx, yy
# The commas and the semicolon are necessary.

sStress = "-100, 45; 90, 25"

axes = plt.axes()
axes.set_xlim((-150, 150))
axes.set_ylim((-150, 150))
plt.axes().set_aspect('equal', 'datalim')
plt.title(
"A visualization of *something* about\n" \
"the 2D stress-state [xx, xy; yx, yy] = [%s]" \
% sStress)

mStress = np.matrix(sStress)
mStressT = np.transpose(mStress)

mUnitNormal = np.zeros((2, 1))
mTraction = np.zeros((2, 1))

nOrientations = 18
dIncrement = 360.0 / float(nOrientations)
for i in range(0, nOrientations):
mTraction = mStressT.dot(mUnitNormal)
if i == 0:
plt.plot((0, mTraction[0, 0]), (0, mTraction[0, 1]), 'black', linewidth=1.0)
else:
plt.plot((0, mTraction[0, 0]), (0, mTraction[0, 1]), 'gray', linewidth=0.5)
plt.plot(mTraction[0, 0], mTraction[0, 1], marker='.',
markeredgecolor='gray', markerfacecolor='gray', markersize=5)
plt.text(mTraction[0, 0], mTraction[0, 1], '%d' % dThetaDegrees)
plt.pause(0.05)

plt.show()



Update on 02 March 2018, 15:34 IST:

Here is a second version of a script that does something similar (but continues to lack explanatory comments). One advantage with this version is that you can copy-paste the script to some file, say, MyScript.py, and invoke it from command line, giving the stress components and the number of orientations as command-line inputs, e.g.,

python MyScript.py "100, 0; 0, 50" 12


which makes it easier to try out different states of stress.

The revised code is here:


'''
A simple script to help visualize *something* about
a 2D stress tensor.

History:
06 March 2018, 10:43 IST:
In computeTraction(), changed the mUnitNormal code to make it np.matrix() rather than python array
02 March 2018, 15:39 IST; Published the code
'''

import sys
import math
import numpy as np
import matplotlib.pyplot as plt

# Specifying the input stress
# Note:
# While plotting, we set the x- and y-limits to -150 to +150,
# and enforce the aspect ratio of 1. That is to say, we do not
# allow MatPlotLib to automatically scale the axes, because we
# want to appreciate the changes in the shapes as well sizes in
# the plot.
#
# Therefore, all the input stress-components should be kept
# to within the -100 to +100 (both inclusive) range.
#
# Specify the stress state in this order: xx, xy; yx, yy
# The commas and the semicolon are necessary.
# If you run the program from a command-line, you can also
# specify the input stress string in quotes as the first
# command-line argument, and no. of orientations, as the
# second. e.g.:
# python MyScript.py "100, 50; 50, 0" 12
##################################################

gsStress = "-100, 45; 90, 25"
gnOrientations = 18

##################################################

dx = round(vTraction[0], 6)
dy = round(vTraction[1], 6)
if not (math.fabs(dx) < 10e-6 and math.fabs(dy) < 10e-6):
axes.annotate(xy=(dx, dy), s='%d' % dThetaDegs, color=clr)

##################################################

mUnitNormal = np.reshape(vUnitNormal, (2,1))
mTraction = mStressT.dot(mUnitNormal)
vTraction = np.squeeze(np.asarray(mTraction))
return vTraction

##################################################

def main():
axes = plt.axes()
axes.set_label("label")
axes.set_xlim((-150, 150))
axes.set_ylim((-150, 150))
axes.set_aspect('equal', 'datalim')
plt.title(
"A visualization of *something* about\n" \
"the 2D stress-state [xx, xy; yx, yy] = [%s]" \
% gsStress)

mStress = np.matrix(gsStress)
mStressT = np.transpose(mStress)
vTraction = computeTraction(mStressT, 0)
plotArrow(vTraction, 0, 'red', axes)
dIncrement = 360.0 / float(gnOrientations)
for i in range(1, gnOrientations):
plt.pause(0.05)
plt.show()

##################################################

if __name__ == "__main__":
nArgs = len(sys.argv)
if nArgs > 1:
gsStress = sys.argv[1]
if nArgs > 2:
gnOrientations = int(sys.argv[2])
main()

`

OK, have fun, and if you care to, let me know your answers, guess-works, etc…..

Oh, BTW, I have already taken a version of my last post also to iMechanica, which led to a bit of an interaction there too… However, I had to abruptly cut short all the discussions on the topic because I unexpectedly got way too busy in the affiliation- and accreditation-related work. It was only today that I’ve got a bit of a breather, and so could write this script and this post. Anyway, if you are interested in the concept of stress—issues like what it actually means and all that—then do check out my post at iMechanica, too, here [^].

… Happy Holi, take care to use only safe colors—and also take care not to bother those people who do not want to be bothered by you—by your “play”, esp. the complete strangers…

OK, take care and bye for now. ….

A Song I Like:

(Marathi [Am I right?]) “rang he nave nave…”
Singer: Shasha Tirupati
Lyrics: Yogesh Damle

# Stress is defined as the quantity equal to … what?

Update on 01 March 2018, 21:27, IST: I had posted a version of this post also at iMechanica, which led to a bit of a very interesting interaction there [^] too. Check it out, if you want… Also see my today’s post concerning the idea of stress, here [^].

In this post, I am going to note a bit from my personal learning history. I am going to note what had happened when a clueless young engineering student that was me, was trying hard to understand the idea of tensors, during my UG years, and then for quite some time even after my UG days. May be for a decade or even more….

There certainly were, and are likely to be even today, many students like [the past] me. So, in the further description, I will use the term “we.” Obviously, the “we” here is the collegial “we,” perhaps even the pedagogical “we,” but certainly neither the pedestrian nor the royal “we.”

What we would like to understand is the idea of tensors; the question of what these beasts are really, really like.

As with developing an understanding of any new concept, we first go over some usage examples involving that idea, some instances of that concept.

Here, there is not much of a problem; our mind easily picks up the stress as a “simple” and familiar example of a tensor. So, we try to understand the idea of tensors via the example of the stress tensor. [Turns out that it becomes far more difficult this way… But read on, anyway!]

Not a bad decision, we think.

After all, even if the tensor algebra (and tensor calculus) was an achievement wrought only in the closing decade(s) of the 19th century, Cauchy was already been up and running with the essential idea of the stress tensor right by 1822—i.e., more than half a century earlier. We come to know of this fact, say via James Rice’s article on the history of solid mechanics. Given this bit of history, we become confident that we are on the right track. After all, if the stress tensor could not only be conceived of, but even a divergence theorem for it could be spelt out, and the theorem even used in applications of engineering importance, all some half a century before any other tensors were even conceived of, then developing a good understanding of the stress tensor ought to provide a sound pathway to understanding tensors in general.

So, we begin with the stress tensor, and try [very hard] to understand it.

We recall what we have already been taught: stress is defined as force per unit area. In symbolic terms, read for the very first time in our XI standard physics texts, the equation reads:

$\sigma \equiv \dfrac{F}{A}$               … Eq. (1)

But given this way of putting things as the starting point, the only direction which we could at all possibly be pursuing, would be nothing but the following:

The 3D representation ought to be just a simple generalization of Eq. (1), i.e., it must look something like this:

$\overline{\overline{\sigma}} = \dfrac{\vec{F}}{\vec{A}}$                … Eq. (2)

where the two overlines over $\sigma$ represents the idea that it is to be taken as a tensor quantity.

But obviously, there is some trouble with the Eq. (2). This way of putting things can only be wrong, we suspect.

The reason behind our suspicion, well-founded in our knowledge, is this: The operation of a division by a vector is not well-defined, at least, it is not at all noted in the UG vector-algebra texts. [And, our UG maths teachers would happily fail us in examinations if we tried an expression of that sort in our answer-books.]

For that matter, from what we already know, even the idea of “multiplication” of two vectors is not uniquely defined: We have at least two “product”s: the dot product [or the inner product], and the cross product [a case of the outer or the tensor product]. The absence of divisions and unique multiplications is what distinguishes vectors from complex numbers (including phasors, which are often noted as “vectors” in the EE texts).

Now, even if you attempt to “generalize” the idea of divisions, just the way you have “generalized” the idea of multiplications, it still doesn’t help a lot.

[To speak of a tensor object as representing the result of a division is nothing but to make an indirect reference to the very operation [viz. that of taking a tensor product], and the very mathematical structure [viz. the tensor structure] which itself is the object we are trying to understand. … “Circles in the sand, round and round… .” In any case, the student is just as clueless about divisions by vectors, as he is about tensor products.]

But, still being under the spell of what had been taught to us during our XI-XII physics courses, and later on, also in the UG engineering courses— their line and method of developing these concepts—we then make the following valiant attempt. We courageously rearrange the same equation, obtain the following, and try to base our “thinking” in reference to the rearrangement it represents:

$\overline{\overline{\sigma}} \vec{A} = \vec{F}$                  … Eq (3)

It takes a bit of time and energy, but then, very soon, we come to suspect that this too could be a wrong way of understanding the stress tensor. How can a mere rearrangement lead from an invalid equation to a valid equation? That’s for the starters.

But a more important consideration is this one: Any quantity must be definable via an equation that follows the following format:

the quantiy being defined, and nothing else but that quantity, as appearing on the left hand-side
=
some expression involving some other quantities, as appearing on the right hand-side.

Let’s call this format Eq. (4).

Clearly, Eq. (3) does not follow the format of Eq. (4).

So, despite the rearrangement from Eq. (2) to Eq. (3), the question remains:

How can we define the stress tensor (or for that matter, any tensors of similar kind, say the second-order tensors of strain, conductivity, etc.) such that its defining expression follows the format given in Eq. (4)?

Can you answer the above question?

If yes, I would love to hear from you… If not, I will post the answer by way of an update/reply/another blog post, after some time. …

Happy thinking…

A Song I Like:
(Hindi) “ye bholaa bhaalaa man meraa kahin re…”
Singers: Kishore Kumar, Asha Bhosale
Music: Kishore Kumar
Lyrics: Majrooh Sultanpuri

[I should also be posting this question at iMechanica, though I don’t expect that they would be interested too much in it… Who knows, someone, say some student somewhere, may be interested in knowing more about it, just may be…

Anyway, take care, and bye for now…]