Understanding tensors (of engineering sciences)—part 2: yet another DIY experiment

I continue from my last post.

There is another simple DIY experiment that you can perform at home. The idea of this experiment had occurred quite some time back, but I had completely forgotten it. (I had forgotten it even while delivering lectures for the FEM courses which I taught in 2009 and 2012). Last night, I happened to recall the idea once again, and thought of immediately sharing it with you via this blog post.

A Fun DIY Experiment # 2:

Get that piece of men’s innerware which is known in India as the “banian” or “banyan,” and in American English as the “vest” [^].  (If not sure, check out the “aaraam kaa maamla” ads.) Basically, a banyan (at least these days) is a cotton garment like a T-shirt, but it’s bit smaller in size, and as an inner-ware, it is also meant to be more closely fitting to the body. That also makes it more easily stretchable, and therefore, better suited to our purposes. It’s also very inexpensive.

Start with a new (i.e. unused and never washed (i.e. never stretched/wrinkled)) banyan. The cloth should be easily stretchable. The fabric should be plain and simple, and without any special knitting pattern; e.g. no “self-stripes” etc. Cut it open and lay the cloth flat on a table. Mark a set of regular Cartesian grid-points on it with the help of an ink pen. You can easily make a bigger grid (say of the size 15 cm X 15 cm) at a regular spacing, say of 1 cm.

Lay the cloth flat and unstretched on the glass surface of a computer scanner (or even a Xerox machine), and obtain an image, say PH1. Next, with the help of a friend, stretch the cloth non-uniformly, by pulling unevenly along many directions. Make sure that the stretch is non-uniform but completely planar, and, of course, that there are no wrinkles. Scan it in this stretched state, and thus obtain the second image, PH2.

Advantages of this second experiment are easy to see: (i) As compared to the balloon rubber, is easier to lay the banyan cloth flat and without wrinkles. (ii) It is easier to stretch it in many directions. (iii) It is easier to mark out a regular grid—the regularity of the fabric of the cloth actually helps in ensuring regularity.

Also, even if you manage to get a good piece of a large rubber balloon, it should anyway be easier to obtain the image of a grid on it using an image scanner/Xerox machine rather than using a digital camera—the issues of having to maintain the same zoom and distance don’t arise.

Process the images as mentioned in the previous post, and keep them ready.

In the meanwhile, also consult the references mentioned in the last post, and make sure to go through the following concepts in particular: (i) position vector for a point-particle; (ii) displacement vector for a point-particle; (iii) the position vectors for an infinity of points in a continuum—i.e. the position vector field; (iv) the line segment, i.e., the relative position vector (i.e., the difference between two position vectors); (v) the translation and rotation of a line segment; (vi) the relative displacement vector of a line segment (i.e., the relative displacement vector of a relative position vector!); (vii) the rigid-body translation and rotation vs. the change of size and shape of a continuum body; (viii) the displacement gradient tensor at a point in a continuum; (ix) the rotation tensor at a point in a continuum body vs. its rigid-body rotation as a whole; (viii) the strain tensor at a point in a continuum body; etc. …

We will of course look into all these concepts—in fact, we will calculate the particular values that all these quantities assume in our simple experiment, using the basic data of the two images that our simple experiment generates. That will be our topic in the next post.

But before coming to it, let’s take a pause for a moment to recall what the purpose of this whole exercise is. It is: to know the physical meanings/correspondents of the mathematical concepts; to try and develop a proper hierarchical order the concepts; to develop a physical “feel” for the more abstract concepts involved. And, as far as the last is concerned (developing a physical feel for abstract concepts), there’s no substitute to realizing what the more concrete context of a given more abstract concept is. In understanding the proper context of mathematical concepts, there is no substitute to physical observation. That’s why, no matter how ridiculously simple these experiments might look like, do not skip the step of actually performing one of these two experiments.

And, BTW, in this series, more DIY experiments and fun ideas are going to follow.

More, later.

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A Song I Like:

(Hindi) “ek ghar banaoonga, tere ghar ke saamane…”
Singers: Mohammad Rafi and Lata Mangeshkar
Music: S. D. Burman
Lyrics: Hasrat Jaipuri

[E&OE]

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So, it is a QC (at least this week!)

I wanted to write on tensors etc., but a few very fresh inputs concerning the D-Wave device have appeared, all barely within the past 24 hours or less.

First, it was Prof. David Poulin commenting at Prof. Scott Aaronson’s blog once again [^], alerting some new work from Prof. Troyer. Unlike in his last comment (on the same post, when he thought that it was not a QC), Poulin has now come closer towards (or has started) supporting the position that the D-Wave device is a QC:

“…the problem instances that are easy for the D-wave device can sometimes be hard for the SS model. This is interesting new evidence supporting the quantum nature of the D-wave device.”

Next, a very valuable comment by one Bill Kaminsky appeared on Aaronson’s blog, very neatly explaining the Smolin and Smith model [^], and then contrasting it with the new result by Troyer. [Guess this Bill Kaminsky is the same as one William Kaminsky, who, in turn, is a PhD student in QIS at MIT. (… Just a Google search, that’s all!)] … Incidentally, more explanatory material concerning the adiabatic quantum optimization, quantum annealing, and classical annealing, written by Kaminsky, had already been put up last week at Henning Dekant’s blog; see here [^].

Finally, while idly thinking about all these things, even as idly browsing Prof. Poulin’s home page, I just idly happened to hit the “New on quant-ph” link [^] at its bottom, and thereby landed at the arXiv site; and once there, I noticed a new paper by Troyer (and (eight!) pals): [^].

Essentially, what Troyer et al. now say is that the D-Wave device does something that the classical devices apparently don’t, and so, the D-Wave device must be quantum! … If not all the classical devices, then at least the two devices: one, considered by they themselves, and the other, considered by Smolin and Smith. The D-Wave device behaves unlike both.

Further, Troyer et al. offer the following conjecture to account for the difference between the D-Wave chip and the [semi-]classical models:

“…The question of why SQA and semi-classical spin models correlate so differently with the D-Wave device is obviously important and interesting. We note that while SQA captures decoherence in the instantaneous energy eigenbasis of the system, so that each energy eigenstate—in particular the ground state—is itself a coherent superposition of computational basis states, semi-classical spin models assume that each qubit decoheres locally, thus removing all coherence from the ground state. We conjecture that the fact that the D-Wave machine succeeds with high probability on certain instances which the semi-classical models finds hard, can be understood in terms of this difference.”

[emphasis mine]

So, looks like, it is a quantum computer, after all. … At least, for this week!

* * *

Clearly, more studies required. So, here are a few questions to the QC research community:

What needs to be done to study the above conjecture more closely? Would some simple and special-purpose simulations that directly allow for a parametric control of the degrees of decoherence, help at least to illustrate (if not to fully support) the above conjecture? Such simulations could be highly simplified (say involving just a linear graph) but, still, sufficiently complete so as to be able to isolate, study, and possibly help settle, this issue.

How do you square off the quantum-ness of the D-Wave chip, and the “absence” of a speed-up, as discussed on Aaronson’s blog?

What measures would you suggest to capture the “percentage quantum-ness” of a QC? of an adiabatic quantum device such as D-Wave’s?

On these measures, how quantum are the current two D-Wave chips (D-Wave One and Two)? What is your estimate?

* * *

May be, more, later. (Who knows, it might once again collapse back to being a simple classical computer, next Monday!)

 

[May be I will come back (right today) and edit this post a bit, so as to make the write-up a bit more streamlined.]

 

[E&OE]

 

Is it a QC?

This post began its life as a supposedly brief update to my earlier post [^] on D-Wave’s paper, but the text soon grew long enough to become a separate post by itself. So, here we go.

Predictably, a controversy concerning the D-Wave paper (and its coverage in the media) came up soon later, at Prof. Scott Aaronson’s blog [^]. At 300+ comments (as of publishing this post), there is a lot of speculation, skepticism, and hilarity of the usenet/slashdot kind going on over there, apart from also some commentary.

However, as far as I am concerned, the most interesting part in (re) examining the paper and the related claims, was the following doubt which the controversy helped highlight: whether this particular D-Wave device had actually succeeded in exploiting, at least in part, the specifically quantum-mechanical effects, or not; whether there was an engineering success in controlling, at least in part (and to a practically significant extent), the quantum decoherence effects, or not.

The controversy was not entirely unexpected; recall this bit from the first New York Times story [^]:

““There is no sense in which this is the definitive statement about quantum computing,” Ms. McGeoch said. “I’m more interested in how well it works, not whether or not it is quantum.””

Though they called it a “quantum computer” (and I repeated the term), the term obviously was being used in a somewhat loose sense.

And, yes, I will admit it: without going through the paper well, I rather relied on the peer-review process, and so certainly thought, at least at the time of writing my earlier post, that D-Wave had a more impeccable and comprehensive result than what now seems to be the case.

But returning to who is interested in what: Well, as far as I am concerned, the issue of whether they got any speed-up or not, is strictly secondary—it’s “just” a consequence!

(In fact, I even don’t care if a QC research group cannot factor any composite beyond some single digit number, as of today. So long as they demonstrate a practically significant control of decoherence, and some clue about how they expect to scale it up, even their success in factoring only a small number would still make sense to me. Any future value of a QC in cracking open secret codes, or in designing better drugs through quantum chemical modeling, would be “just” a consequence, as far as I am concerned.)

To my mind, the real issue is: whether D-Wave succeeded in building a quantum computer (with some promise of some significant levels of a future scalability), or not.

So, from this angle, the most significant comment at Aaronson’s blog has been this one [^] by Prof. David Poulin, alerting the appearance of a paper by John Smolin and Graeme Smith, both of IBM, at arXiv, yesterday [^]. In case you are wondering whether to give this paper a read or not, let me remind you that IBM is a (corporate-sector) competitor to D-Wave. And, if that isn’t going to help, let me quote a bit from the main text of the paper:

“Since classical simulated annealing is intrinsically random and ‘quantum annealing’ is not…”

[emphasis mine]

and a line from their conclusions section:

“The deterministic nature of quantum annealing leads to rather different behaviors than the random processes of simulated annealing.”

[emphasis mine]

Interesting, no? (LOL!)

Of course, my own interests are in the foundations of QM, in providing a proper conceptual explanation for (and even mathematical expression to) the specifically quantum-mechanical effects/paradoxes/oddities, and not in the details of this or that quantum-mechanical process, whether it has some/a lot of/very great merit in building a scalable QC, or not.

So, I am not going to look too closely into this IBM paper either. Or provide a commentary on the position(s) it takes, its merits, or any polemical value it provides in this controversy (or in any other!). Or, add in any other way, to this D-Wave-related  controversy. … That way, I am not totally averse to controversies, but as far as this one goes, I find that it is a greater fun taking a ring-side view, here.

For another thing, these days, I am also thinking of quite different (and between them, somewhat unrelated) things: diffusion, small dams and water resources engineering/management, and tensors. Expect a post or two on these topics, soon enough.

So, all in all, even if I am having fun watching this controversy develop and grow, I guess I am going to sign off blogging about it. I won’t write any further on this topic, unless, of course something even more funny (or definitive, even if a bit serious) emerges from it.

[E&OE]

The QC pulls ahead of the CC

The first peer-reviewed paper to demonstrate that a quantum computer (QC) outperforms a conventional computer (CC), is here (PDF) [^]. [HT to Henning Dekant [^]].

The New York Times’ story is here [^].

Oh, BTW, this is one of those rare occasions when a peer-reviewed PDF of a scientific paper is being made available from a newspaper’s commercial servers—not from a server at some government-run Important National Lab, or a taxpayer-funded Wonderful State University, or for that matter, even arXiv! An interesting bit by itself, don’t you think?

(And, BBTW, I am old enough that as soon as I read this news, I instinctively slipped into wondering as to the time when the Russians might come forward with some “evidence” to show that they had accomplished the same thing some a few years earlier. … I guess I should go and enquire with the folks at the JNU New Delhi, ISI Kolkata, or IIT Bombay—they should know.)

Anyway, coming back to this exciting bit of news itself: at least at the time of going to wordpress, far too many American blogs on quantum computing still were completely silent. Especially those being maintained by the American academics. Several days over, and still not even a cursory acknowledgment!

Yet, this bit of news is not a hype; the advancement is for real.  Check out the following links (many of which were mentioned in Henning’s post, anyway): New Scientist [^], MIT’s Technology Review [^], IEEE [^], and even Nature [^].

So, an exciting news item, this one surely is. But what is comprehensively missing is one thing: that American (Hindi word) “taDkaa.”

The MIT Technology Review story, for instance, has this as the subtitle of its online story:

“Tests suggest that a CIA-backed quantum computing technology can be very powerful for some kinds of problems.”

Very careful.

“A” quantum computing technology—not the first to get a definite practical success.  “CIA-backed”—which means, this hint: the CIA has the money to pour into some potentially wasteful projects, and also have the means to choke out any adverse news reports if they fail, unlike the real innovative, open, democratic institutions like certain US universities. And, only “some” kind of problems would become solvable—it’s certain that with more research at MIT and Berkeley, the hardware is bound to get intelligent, but don’t expect it to be omniscient, that’s all. (Parenthetically: the company is Canadian.)

Sooooooo careful.

So, all in all, what I am missing out on is that American “taDkaa.” Even if Lockheed Martin, an American firm, already has gone ahead with the plans to use it [^], and an American by name Bo Ewald has become involved with the DWave [^]. [Full disclosure: I worked with e-Stamp roughly around the same time that Bo Ewald did. [Hi Bo!]]

The major reason I want to see some real American “masaalaa” and “taDkaa” on and around this topic, and if not that, then at least some ordinary hype on it, is: so that people get mysterious about this whole thing. Remember, the field of quantum computing carries two highfalutin words: “quantum” and “computing.” Even if the second word has lost a bit of a shine (Steve Jobs is no longer around, Chairman Bill is no longer the Chairman, and even the DC threatens Google only once in a while—there is no real DoJ action), it still carries a lot of aura. And, till date, they have managed to keep the first word, neatly wrapped up in a thick, impenetrable kind of an aura of a mystery.

When you combine the two together, there should be a multiplicative/exponential kind of a synergy. “Quantum Computing,” you know, should sound big. BIG. VERY BIG.

It, then, would be such a fun to step in on to the scene, and begin explaining how quantum computing is such a simple thing, after all! … How it is not all that big a mystery; how it really works. Explaining quantum computing on the basis of [clears the throat] my novel approach, would be fun, provided there is a preceding American “taDkaa” to it. In sufficient quantities. Together with “masaalaa.” To make it all mysterious in the first place.

There is no fun carrying just a pin around, no matter how sharp it may be. It’s no fun if you do have the pin, but there is no balloon in the first place—or, as in this case, there is that balloon, but still, no one is willing to inflate it.

* * *

Congratulations to the engineers and physicists at the D-Wave, anyway!

 

[E&OE]

Where the Mind Stops—Not!

The way people use language, changes.

In the mid- and late-1990s, when the Internet was new, when blogs had yet to become widespread, when people would often use their own Web sites (or the feedback forms and “guestbooks” at others’ Web sites) to express their own personal thoughts, opinions and feelings—in short, when it still was Web 1.0—one would often run into expressions of the title sort. For example: XYZ is a very great course—NOT! XYZ university has a very great student housing—NOT! XYZ is a very cute product—NOT! … You get the idea—you really do! (NO not!!)… That’s the sense in which the title of this post is to be taken.

For quite some time, I had been thinking of a problem, a deceptively simple problem, from engineering sciences and mechanics. Actually, it’s not a problem, but a way of modeling problems.

Consider a body or a physical object, say a piece of chalk. Break it into two pieces. Easy to do so physically? … Fine. Now, consider how you would represent this scenario mathematically. That is the problem under consideration. … Let me explain further.

The problem would be a mere idle curiosity but for the fact that it has huge economic consequences. I shall illustrate it with just two examples.

Example 1: Consider hot molten metal being poured in sand molds, during casting. Though “thick,” the liquid metal does not necessarily flow very smoothly as it runs everywhere inside the mold cavity. It brushes against mold-walls, splashes, and forms droplets. These flying droplets are more effective than the main body of the molten metal in abrading (“scrubbing”) the mold-walls, and thereby dislodging sand particles off the mold walls. Further, the droplets themselves both oxidize fast, and cool down fast. Both the oxidized and solidified droplets, and the sand particles abraded or dislodged by the droplets, fall into the cooling liquid metal. Due to oxidized layer the solidified droplets (or due to the high melting point of silicates, the sand particles) do not easily remelt once they fall into the main molten metal. The particles remain separate, and thus get embedded into the casting, leading to defective castings. (Second-phase particles like oxidized droplets and sand particles adversely affect the mechanical load-carrying capacity of the casting, and also lead to easier corrosion.) We need the flow here to be smooth, not so much because laminar flow by itself is a wonderful to have (and mathematically easier to handle). We need it to remain smooth mainly in order to prevent splashing and to reduce wall-abrasion. The splashing part involves separation of a contiguous volume of liquid into several bodies (the main body of liquid, and all the splashed droplets). If we can accurately, i.e. mathematically, model how droplets separate out from a liquid, we would be better equipped to handle the task of designing the flow inside a mold cavity.

Example 2: Way back in mid-1980s, when I was doing my MTech at IIT Madras, I had already run into some report which had said that the economic losses due to unintended catastrophic fractures occurring in the US alone were estimated to be some $5 billion annually. … I quote the figure purely from my not-so-reliable memory. However, even today, I do think that the quoted figure seems reasonable. Just consider just one category of fractures: the loss of buildings and human life due to fractures occurring during earthquakes. Fracture mechanics has been an important field of research for more than half a century by now. The process of fracture, if allowed to continue unchecked, results in a component or an object fragmenting into many pieces.

It might surprise many of you (in fact, almost anyone who has not studied fluid mechanics or fracture mechanics) that there simply does not exist any good way to mathematically represent this crucial aspect of droplets formation or fracture: namely, the fact of one body becoming several bodies. More accurately, no one so far (at least to my knowledge) has ever proposed a neat mathematical way to represent such a simple physical fact. Not in any way that could even potentially prove useful in building a better mechanics of fluids or fracture.

Not very surprising. After all, right since Newton’s time, the ruling paradigm of building mathematical models has been: differential equations. Differential equations necessarily assume the existence of a continuum. The region over which a given differential equation is to be integrated, may itself contain holes. Now, sometimes, the existence of holes in a region of space by itself leads to some troubles in some areas of mechanics; e.g., consider how the compatibility criteria of elasticity lose simplicity once you let a body carry holes. Yet, these difficulties are nothing once you theoretically allow the original single body to split apart into two or more fragments. The main difficulty is the following:

A differential equation is nothing but an equation defined in terms of differentials. (That is some insight!) In the sense of its usage in physics/engineering, a differential equation is an equation defined over a differential element. A differential element (or an infinitesimal) is a mathematical abstraction. It begins with a mathematically demarcated finite piece of a continuum, and systematically takes its size towards zero. A “demarcated finite piece” here essentially means that it has boundaries. For example, for a 1D continuum, there would be two separate points serving as the end-points of the finite piece. Such a piece is, then, subjected to the mathematical limiting process, so as to yield a differential element. To be useful, the differential equation has to be integrated over the entire region, taking into consideration the boundary and initial values. (The region must be primarily finite, and it usually is so. However, sometimes, through certain secondary mathematical considerations and tricks involving certain specific kinds of boundary conditions, we can let the region to be indefinitely large in extent as well.)

Since the basic definition of the differential element itself refers to a continuum, i.e. to a continuous region of space, this entire paradigm requires that cuts or holes not existing initially in the region cannot at all be later introduced. A hole is, as I said above, mostly acceptable in mathematical physics. However, the hole cannot grow so as to actually severe a single contiguous region of space into two (or more) separate regions. A cut cannot be allowed to run all the way through. The reason is: (i) either the differential element spanning the two sides of the cut must be taken out of the model—which cannot be done under the differential equations paradigm, (ii) or the entire model must be rejected as being invalid.

Thus, no cut—no boundary—can be introduced within a differential element. A differential element may be taken to end on a boundary, in a sense. However, it can never be cut apart. (This, incidentally, is the reason why people fall silent when you ask them the question of one of my previous posts: can an infinitesimal carry parts?)

You can look at it as a simple logical consistency requirement. If you model anything with differential elements (i.e. using the differential equations paradigm), then, by the logic of the way this kind of mathematics has been built and works, you are not allowed to introduce a cut into a continuum and make fragments out of it, later on.  In case you are wondering about a logically symmetrical scenario: no, you can also not join two continua into one—the differential equation paradigm does not allow you to do that either. And, no, topology does not lead to any actual progress with this problem either. Topology only helps define some aspects of the problem in mathematically precise terms. But it does not even address the problem I am mentioning here.

Such a nature of continuum modeling is indeed was what I had once hinted at, in one of my comments at iMechanica [^]. I had said (and none contradicted me at that forum for it) that:

As an aside, I think in classical mathematics there is no solution to this issue, and there cannot be—you simply cannot model a situation like “one thing becomes two things” or “two infinitesimally close points become separated by a finite distance” within any continuum theory at all…

In other words, this is a situation where, if one wishes to think about it in mathematical terms, one’s mind stops.

Or does it?

Today, I happened to idly go over these thoughts once again. And then, a dim possibility of appending a NOT appeared.

The reason I say it’s a dim possibility is because: (i) I haven’t yet carefully thought it through; (ii) and so, I am not sure if it really does not carry philosophic inconsistencies (philosophy, here, is to be rather taken in the sense of philosophy of science, of physics and mathematics); (iii) I already know enough to know that this possibility would not in any way help at least that basic fracture mechanical problem which I have mentioned above; and (iv) I think an application simpler than the basic problem of fracture mechanics, should be possible—with some careful provisos in place. May be, just may be. (The reason I am being so tentative is that the idea struck me only this afternoon.)

I still need to go over the matter, and so, I will not provide any more details about that dim possibility, right here, right today. However, I think I have already provided a sufficiently detailed description of the problem (and the supposed difficulty about it) that, probably, anyone else (trained in basic engineering/physics and mathematics) could easily get it.

So, in the meanwhile, if you can think of any solution—or even a solution approach—that could take care of this problem, drop me a line or add a comment.  … If you are looking for a succinct statement of the problem out of this (as usual) verbose blog-post, then take the above-mentioned quote from my iMechanica comment, as the problem statement. … For years (two+ decades) I thought no solution/approach to that problem was possible, and even at iMechanica, it didn’t elicit any response indicating otherwise. … But, now, I think there could perhaps be a way out—if I am consistent by basic philosophic considerations, that is. It’s a simple thing, really speaking, a very obvious one too, and not at all a big deal… However, the point is, now the (or my) mind no longer comes to a complete halt when it comes to that problem…

Enough for the time being. I will consider posting about this issue at iMechanica after a little while. … And, BTW, if you are in a mode to think very deeply about it, also see something somewhat related to this problem, viz., the 2011 FQXi Essay Contest (and what its winners had to say about that problem): [^]. Though related, the two questions are a bit different. For the purpose of this post, the main problem is the one I mentioned above. Think about it, and have fun! And if you have something to say about it, do drop me a line! Bye for now!!

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A Song I Like
(Hindi) “nahin nahin koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[PS: Perhaps, a revision to fix simple errors, and possibly to add a bit of content here and there, is still due.]
[E&OE]

What would you choose as the Top 5 Equations? Top 10?

I was going to write my thoughts on the issue of “Universe: Finite or Infinite?” this weekend. Though the weekend has technically already begun, I have been busy at my day-job. Further, though I have been hurriedly jotting down my points in a small pocket notebook (a habit that has come to serve me well), I still have not found the time to put them in a computer file (usually, just a .txt file) and order them. I hope to be doing the upcoming week. However, in the meanwhile, the title question struck me, and so, I decided to “ship it out” this blog post. Read, and do let me know your thoughts…

* * * * *   * * * * *   * * * * *

Equations are of central importance in all of science and engineering, but especially so in physics and mechanics.

Even leaving aside algebraic equations, handbooks on PDEs alone list hundreds of equations. However, a few of these do stand out, either because they encapsulate some fundamental aspect of physics/science/engg., or because they serve as simpler prototypes for more complex situtations, or simply because they are so complex as to be fascinating by themselves. There might be other considerations too… But the fact is, some equations really do stand out as compared to others.

If so, what equations would you single out as being most important or interesting? To make the matters more interesting, first, please think of making a short list of only 5 equations. Then, if necessary, make it one of 10 equations—but no more, please! 🙂

As to me, here is my list, put together in a completely off-hand manner:

Top five:
(1-3) The linear wave-, diffusion- and potential-equations.
(4) The Schrodinger equation
(5) The Navier-Stokes equation

Additionally, perhaps, these equations:
(6) The Maxwell Equations
(7) The equation defining the Fourier transform
(8) Newton’s second law (dp/dt = F)
(9) The Lame equation (of elasticity)

Am I already nearing the limit or what… Hmm… But, nope, I am not sure whether I want to include E = mc^2. … I will give this entire matter a second thought some time later on.

But, how about you? What would be your choices for the top 5/10 equations? Why?

(Also posted today in the Computational Scientists & Engineers group at LinkedIn, and at iMechanica [^]).

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A Song I Like:
(Hindi) “wahaan kaun he teraa…”
Singer and Music: S. D. Burman
Lyrics: Shailendra

[PS: Song was “TBD” while publishing this post yesterday. Added the song on May 1, 2011. … Still, I am not sure if I haven’t mentioned this song before. I will check later on, and if so, will replace it with some other song.]

[E&OE]

Mohr’s Circle—When Was the Last Time You Used It in Your Professional Engineering Work?

As a consultant in computational mechanics, I currently help write some FEM-related code, and while doing this job, an episode from a recent past came to my mind. Let me go right on to the technical issue, keeping aside the (not so good) particulars of that episode. (In case you are curious: it happened outside of my current job, during a job interview.)

If you are a design engineer, FE analyst, researcher, or any professional dealing with stress analysis in your work, I seek answers to a couple of questions from you:

Question 1:
When was the last time you used Mohr’s circle of strain/stress in your professional work? Was it a week ago? a month? a year? five years? ten years? longer? In what kind of an application or research context?

Please note, I do not mean to ask whether you directly or indirectly used the coordinate transformation equations—the basis for constructing Mohr’s circle—to find the principal quantities. The question is: whether you spoke of Mohr’s circle itself—and not of the transformation equations—in a direct manner, in a professional activity of yours (apart from teaching Mohr’s circles). In other words, whether, in the late 20th and early 21st century, there was any occasion to plot the circle (by hand or using a software) in the practice of engineering, did it directly illuminate something/anything in your work.

In case you are curious, my own answer to this question is: No, never. I would like to know yours.

Question 2:
The second question just pursues one of the lines indicated in the first.

In a modern FEM postprocessor, visualizations of stress/strain patterns are provided, usually via field plots and contour lines.

For instance, they show field plots of individual stress tensor components, one at a time.

Recently, there also have been some attempts to try to directly show tensor quantities in full directly, via systematically arranged ellipsoids of appropriate sizes and orientations. The view you get is in a way analogous to the arrow plots for visualizing vector fields in those CFD and EM software packages. Other techniques for tensor visualization are not, IMHO, as successful as the ellipsoids. Mostly, all such techniques still are at the research stage and have not yet made to the commercial offerings.

Some convenience can be had by showing some scalar measures of the tensors such as the von Mises measure, in the usual field/contour plots.

The questions here are:

(2.a) Would you like to see an ellipsoids kind of visualization in your engineering FEM software? If yes, would this feature be a “killer” one? Would you consider it to be a decisive kind of advantage?
(2.b) Would a simpler, colored cross-bars kind of visualization do? That is, two arrows aligned with the principal directions. The colors and the lengths of the arrows help ascertain the strength of the principal quantities.
(2.c) Would you like to see Mohr’s circles being drawn for visualization or any other purposes in such a context? If yes, please indicate the specific way in which it would help you.

My own answers to question 2 are: (a) Ellipsoids would be “nice to have” but not “killer.” I wouldn’t be very insistent on them. Having them is not a decisive adavantage. (b) For 2D, this feature should be provided. (c) Not at all.

Please note, the questions are directed rather at experienced professionals, even engineering managers, but not so much at students as such. The reason is that the ability to buy is an important consideration here, apart from the willingness. Of course, experienced or advanced PhD students and post-docs may also feel free to share their experiences, thoughts and expectations.

Thanks in advance for your comments.

Also posted at iMechanica, here [^].

–  –  –  –  –
A Song I Like:
(Marathi) “aabhaas haa…”
Singers: Vaishali Samant and Rahul Vaidya
Music: Nilesh Mohrir
Lyrics: Ashwini Shende

[E&OE]

The Recent Workshop on Advanced Nonlinear FEM at COEP

For the couple of days that just passed by, i.e. on April 9 and 10, I attended a two-day Workshop on Advanced Nonlinear FEM at COEP [^]. It was organized jointly by Pro-Sim, Bangalore [^] and COEP’s Mechanical Engineering Department. However, quite a few people from some other organizations also came in to deliver their talks. These included managers or senior engineers in charge of the CAE departments in Eaton, Mahindras, Tata Motors, CDAC, and others. The new Vice-Chancellor of the University of Pune, Dr. Shevgaonkar, also dropped by for the inaugural function.

BTW, this being COEP, there never was any question of their inviting me to give a lecture/talk as a part of any workshop such as this. I suppose that they would consider it as compromising their [unstated] standards of quality. However, I did pay their registration fees, and attend the event as a regular attendee, just to see what all things were being discussed during the event.

One part of my interest in attending this workshop concerned learning. I have never been taught FEM in a class-room, or for that matter by anyone in person as such—I’ve picked up all my FEM on my own, by going through books and writing my own code, and then also by interacting via blogs/emails. (For example, see my grappling of the issue of banding and discontinuity of the derivatives, on iMechanica, here [^], something which I took complete care of soon later on, way before beginning teaching my FEM courses at COEP and CDO/MERI….) Anyway, given that I had never sat in an FEM classroom, I thought that it might be fun to do so, for a change. Another part of my interest in the workshop touched on my professional interests. I have myself begun conducting courses on fundamentals of FEM, and I wanted to compare the cost-to-benefit ratio for my course offering vis-a-vis others’.

Overall, I would say that it was only a barely acceptable deal at Rs. 4,000/- for the two days.  Of course, it certainly was worth more than a thousand bucks a day. I think it would have been a fairly good deal at about Rs. 2,500/- or so.

One doesn’t keep quite the same expectations from a workshop as one would from a training course. Yet, considering the fact that the settings for this workshop would be academic, it would have been better if the topics in this Workshop were to be sequenced better and treated differently. What happened in this workshop was that the individual faculty members were, by and large, actually good and knowledgeable engineers. Yet, the actual amount of knowledge to get transferred was, I am afraid, only minimal.

Many of the speakers could neither pace themselves well nor select their main topics (or subtopics) well. Further, the sequence of these lectures was not very well organized. There was this absence of an integrating theme continuously running through the lectures.

Now, I realize that it is always difficult to ensure a theme even for a small group of speakers. Sticking to a theme would be even more difficult to ensure in a workshop that is delivered by 5+ people. Yet, if you look at say, SIGGRAPH workshops in the USA, or, closer to India, the workshops covered in the NDT-related events, one can clearly see that maintaining an integrating theme, in which people progress from simple topics and fundamentals on to more complex topics and applications, is not as difficult as it might otherwise sound.

Since there was no theme, it had the appearance of a collage, not of a coherent picture. I mean, if you were to catch hold of a typical young attendee (say a BTech/MTech student) and if you were to ask him to identify in one line what distinguishes non-linearity from linearity in the context of FEM, he won’t be able to tell you that it’s all about going from: \begin{bmatrix}A\end{bmatrix} \begin{Bmatrix}x\end{Bmatrix} = \begin{Bmatrix}b\end{Bmatrix} to: \begin{bmatrix}A(x)\end{bmatrix} \begin{Bmatrix}x\end{Bmatrix} = \begin{Bmatrix}b\end{Bmatrix}. … In this workshop, there was an impressive array of topics, many insights, even more colorful pictures… But little reference was made to fundamentals.

So, if such a workshop is to be conducted in future, I think there should be three/four  (at least two/three) short tutorial or review sessions (of 1.5 to 2 hours each, complete with fill-in-the-blank type of worksheets), before the biggies begin to deliver their talks. It would always be helpful to review basics first. And, the matter should not end there. The entire workshop should be a well-ordered progression.

Another matter. The lectures should be interspersed with 30 minute sessions of actually working out simple problems, using an actual software. It would be OK even if such demos did not include hands-on experience.

Yet another matter. A workshop like this should include applications to fracture processes and mechanics. Also, handling the differential kind of non-linearity via FEM, for instance, modeling of the Navier-Stokes equation using FEM. A discussion of this aspect was surprisingly absent.

Also another matter. For an advanced topic like Nonlinear FEM, the discussions must touch upon how to abstract boundary and initial conditions from the given actual situation. This should be done via giving specific references to a few examples, rather than breezing through numerous case studies with the assumption that the audience knows how to specify the constraints. It should be assumed that they don’t. This must be done even if you don’t include topics like well-posedness, dynamic instability-related points, and so on.

One last point. This is not specific to this particular workshop, but to almost any lecture/delivery by almost any Indian researchers/engineers. Namely, that they are either poor on presentation skills. Or, they are *very* poor.

… Among all the lectures, those by Mr. Ashok Joshi (Manager, CAE, Tata Motors), Mr. Anil Gupta (Manager, CAE, Eaton), and Dr. Sundarrajan (Group Coordinator, CDAC) stood out, on this particular point. Especially the one by Mr. Joshi. …

… But many other speakers had just plain unacceptable habits of speaking: not realizing that too much time is being spent on trivia while keeping a single slide open for too long and then rushing through many other more relevant ones; lecture delivery that comes far too haltingly with far too many pauses and breaks; just too much of jumping around the sub-phrases of a single sentence with absolutely indiscriminate levels of “it”s thrown in… In general, far too much mangling of the grammar…  That way, I have no issues with accent—even an outright regional sort of accent—so long as the speaker is clear and audible. I do have a lot of issues with the contents, the grammar, and the general way of delivering statements—regardless of the accent.

I think that if they tape their lecture delivery and listen to it later (or better still: try to transcribe it on paper), they themselves will realize what they need to do. Here is a made-up example:

“… I mean, it is not like, … let me tell you, what I am trying to do it here… As the forces will be applied to it… and… it will not be the same everywhere… I am telling you, it will be different and why it will be happening is… it will not be the same… It will vary… this point, this point… Ok… You can see, it will be different, the displacement.”

The speaker takes so many pauses, so many breaks, before you realize that what he is trying to point out is the spatial non-uniformity of the displacement field—not of the applied traction (a quantity that too is visible, in a colorful manner, in the same diagram, but something which neither the uttered words nor the waved hands make any reference to, even if necessary in this context).

And, BTW, in this made-up example, I have used fewer “it”s and “will”s. I just can’t get why they can’t workout the structure of a sentence just a fraction of a second in advance before proceeding to utter it. Why do they just have to jump in somewhere in the middle of a thought, literally wherever they want, blurt out those pieces, and then haphazardly attempt to connect them with only one constant expression on the face: why are you not getting me?  … What would be so wrong if the speaker were just to take a complete pause (not even those “umms” and “hmmms”), and then just say: “A force is applied over this part of the boundary. We are interested in the displacement field in this region. We are first interested in displacement because it’s the primary unknown. As expected, the displacement field is not uniform. The interesting feature of its non-uniformity is … [so and so]. … Let’s try to understand the causal relation of this pattern with the distribution of the applied traction.”

… More than a mere presentation skills issue, I think there also is something about mental discipline, and more: something about keeping some concern with inductive integration rather than with the deductive jumping around.

I think they should hire professionals from those management/BPO/similar training institutes and undergo a special training course on public speaking. Further, I think they should also introduce some basics of applied epistemology (say, as what even today gets covered in the better among those BEd/MEd courses) in the engineering/science curricula to highlight the importance of ordering, hierarchy, perceptual referents, inductive arguments, integration, and general pacing out the things to be taught. And I think they should make these courses compulsory, the grades being included in the final GPA. Then, the students will take these matters seriously, and then, the future speakers will turn out to be better.

Of course, the above criticism doesn’t mean that there was no value in the workshop. As I said, it certainly was worth about half the price. Also, the above criticism was based not just on this workshop but on virtually all the conferences that I have attended in the past decade in India (including the ISTAM ones). Indian engineers and scientists, in general (exceptions granted), are very poor on presentation skills.

Coming back to this workshop in particular, there indeed was some definite value to it. But still, … how do I put it?… I think the biggest “carry home” point(*) about it was not the contents of the proceedings themselves—it was: those shake-hands and the exchange of the visiting cards before and after the talks. … Sorry, I still can’t call them as my “contacts” yet, but yes, that socializing was, the way I see it, the biggest import of the event for most of the attendees. And that, whether for the good or for the bad, would summarize the nature of this event right.

It was so for me too…. But, apart from it, to me, personally, the event happened to provide one unexpected benefit: it boosted my confidence. (You might want to read it a little differently, too.)

And, there were certain other pleasant moments on the side, too. Dr. Shevgaonkar highlighted the importance of building CAE software in India—as against merely using the packages made abroad. Dr. Arul Selvan tried to drive home the point that materials modeling was right at the core of advanced FEM for mechanical engineers too (though I can’t be sure that the point reached the aforementioned “home”). Dr. Shamasundar indicated how automated optimization was no longer a “hi fi” thing of research but a tool already deployed right here, in Indian industry. Dr. Sreehari Kumar and Dr. Sundarrajan even touched on the issues related to solver technologies, and their discussions of the topic was a welcome addition given the kind of issue that typical Indian mechanical engineers have with any discipline other than their own, e.g. disciplines like computer science, metallurgy, instrumentation, or physics.

(*I can’t recall the informal word they use in such contexts—esp. for conferences—something like “carry home” or “upshot” “take out” or something like that…)

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A Couple of Songs I Like:

1. (Marathi) “daari paaoos paDato, raani paaravaa bhijato…”
Singer: Suman Kalyanpur
Music: Ashok Patki
Lyrics: Ashok G. Paranjape

2. (Marathi) “bolaavaa vithhal, pahaavaa vithhal…”
Lyrics: “sant tukaaraam”
Singer: “prabhaakar kaarekar” [Not sure yet, but it appears to be him. In my guesswork, many clues I gave here earlier turned out to have been incorrect. But I could locate my CD, though not its cover. I still need to check if it’s Karekar, which I could do starting with the publication number they print on the CD itself. And, yes, in any case, IMHO, this rendition is better than any one any other singer, notably: Kishori Amonkar, Jitendra Abhisheki, Aarati Anklikar-Tikekar, Shaunak Abhisheki, others…. If it indeed is Karekar, then the “shishya” obviously rendered it better than the “guru.” I say this even if in the Indian classical music tradition it is a taboo to claim the superiority of the “shishya” if the claimant is not the “guru” in question himself. … Weird! (And let me know if you want the original clues to appear here, possibly scratched out—I hardly care for the “rules” of blogging either!!]

Some Aspects of Modeling with Continua in Physics and Engineering Sciences

Some Ideas Implicit in Modeling Something as a Continuum

This post concerns continua. It touches on certain basic conceptual ideas concerning continua but not with details of their applications. Of course, the ideas here remain relevant for applications too. Many engineers/physicists do not seem to be very clear on these ideas. … Most of these are the observations I have derived myself through self-study and thought. As such, their expression is likely to be a bit immature. However, a blog is an informal medium, and so, there is no harm sharing these…

What is a continuum? For the purposes of our discussion, the real number line, studied right in high-school, provides a suitable example. Take any line segment; no matter how small it may be, there always are an infinity of points lying on that segment. There are no gaps in any part of a line segment. As such, it is a continuously existing object—a continuum. Extend this same idea to a 2D or 3D embedding space, and you can say that there are an infinity of points on any surface or in any 3D volume. … We all learnt it in high-school. However, some of its implications for our more advanced courses in physics/engineering sciences are not always fully grasped.

Consider a simple physical situation concerning fluids.

Light up a mosquito coil, an incense stick (or, if you are a smoker, a cigarette) in a closed lighted room, and watch the smoke spread. (No fans—let the process be as dominated by natural convection as possible.) This is a routine example to illustrate the transition from the laminar to the turbulent flow. We will not go into those details. But what concerns us here is a certain basic conceptual point.

On their upward path, initially, the smoke lines are rather sharply defined. As the smoke goes further up and the turbulence sets in, the streak-lines mix up and become less sharply defined. As the smoke travels still further, the lines also become increasingly thin, and therefore, visually, they are less sharply defined. If you let the smoke fill the room, eventually, you will find that there no longer are smoke lines visible at a remote corner; instead, it’s all almost uniformly foggy out there.

You know that such mechanics of fluids is ruled by the Navier-Stokes equation—an equation that is very much continuum-mechanical. … Now, here is a question.

Consider two fluid (air) particles initially in close vicinity with each other. (You may consider smoke particles too, so long as remember that our basic concern is with the air motion; we consider smoke only because it makes the fluid movement visible.) The question is: as the fluid moves, do the neighboring particles remain neighbors always? Or do they drift apart?

… Think about it for a while before proceeding further…

If you have caught the drift of this post (and not just the drift of the smoke), you would notice that the question does not so much concern only the NS equations as it does the fundamental assumptions we bring to bear in our continuum—i.e. conceptual and therefore mathematical—modeling.

Let’s make the question more precise: consider two particles that are only an infinitesimal distance apart initially. How about these?

The answer, as far as I know it, based purely on my own reasoning (and I would be happy to correct my reasoning if you point out errors in it) is: in the NS (or any continuum theory), the two particles would always remain neighbors no matter how far away they travel.

Now, here, you are likely to disagree with me, basing your reasoning on the chaos theory—the extraordinary sensitivity to the initial conditions brought about by the differential non-linearity and all…

(Incidentally, from a general philosophic viewpoint, the naming of the non-linear phenomena is a curiosity—the terminology wrongly suggests that things are/can be defined purely by negation of something else—not in reference to some actually existing facts of reality, but by negation alone.)

But coming back to the chaos theory, though I have only surface knowledge (if at all that) concerning it, I, however, also believe (wrongly or rightly—but I think rightly) that the said sensitivity to the initial condition applies only to those particles which are a finite distance apart initially—not infinitesimal. … There is a huge mathematical difference between the two: the finite and the infinitesimal.

If you find it strange that despite chaos (or turbulence) the infinitesimally close particles don’t move a finite distance apart, consider that college physics experiment done with some peculiar jelly-like thick fluid. (I forgot where I read up on this experiment but it could easily be the Feynman Lectures.) Roughly speaking, the experiment goes like this.

You take that jelly-like fluid in a beaker and put a drop of ink or so in it. Then, with a stirrer, you gently stir the fluid, say in the CW sense. What now happens is that the portions near the stirrer keep sticking to it, and so, due to the stirring action, they get stretched and form roughly circular lines. As these fluid layers stretch, the ink particles move along too. (After all, the stickiness applies as much to the ink particles as to the fluid-stirrer interface). After a few rotations, you see a highly mixed up jelly. All the colored ribbons seem entangled with each other and it seems impossible to disentangle them.

However—and here’s the dramatic part of the experiment—if you now slowly rotate the stirrer backwards (in the CCW sense), then the jelly-ribbons actually begin to “shrink” backwards. Once you complete the same number of rotations backwards, you once again get just a localized spot of the ink. The entanglement of the jelly-streaks disappears completely!!

The reason the experiment works even under actual physical (laboratory) conditions is because the fluid in question is thick. However, the principle does get established with this experiment. So long as you have viscosity, in principle, the same behavior can be expected.

During fluid mixing, local sub-regions of a viscous fluid never really tear away from each other (neither do they begin sticking with some other sub-regions). Continuity of the adjacent fluid particles is maintained…. Is “continuity” the right word here? … Actually, what gets preserved is not so much continuity as it is: connectivity.

Two fluid particles connected to each other always remain so. Regardless of the degree of internal mixing of the fluid. Nay, not just that. Regardless of any turbulence within the fluid. That’s the conclusion we seem to be reaching here, don’t we?

If you have seen those CFD simulations of vortex-shedding, you must be wondering: “how come?”After all, a vortex is defined by a finite quantity of fluid rotating within its local region. As vortices get shed, they drift away from each other. If one considers a fluid particle in between two adjacent vortices but closer to one of them, it is an easy guess that it would get sucked into the nearest vortex. Thus, there should be a separation between this particle and its neighbor that went to the other neighbouring vortex.

Right? Wrong?

But rather than answer this question, I would let you figure it how—just in case, of course, you don’t know it already. …

Actually, connectivity or otherwise of adjacent points in vortex-shedding is a fairly well known result. … Typically, students do know the right answer about it. [Something similar used to be a routine orals question at COEP for a second-year course on fluid mechanics + heat transfer for us—the students of metallurgy.] What students don’t realize is that the right answer also applies as a generalization to all continuum phenomena—wherever the assumption of the continuum applies.

[Of course, I am only asserting a generalization here, without really having proved it, and so, I could be wrong. But I can’t think of any good argument why or how this could be a hasty generalization. Please let me know if you know of any.]

Something similar is what I had indicated during one of my past comments at iMechanica. The context, there, was mechanics of solids rather than of fluids, in particular, fracture mechanics. I had pointed out how I had had a discussion about these observations of mine with a graduate student of mathematics (among others) and how I was always told that (in my words) that “one body separates internally and becomes two bodies” is something that simply cannot be dealt with using the mathematics of continua. The issue is one of connectivity, really speaking. A continuum (of the type we conventionally use in physics and engineering to describe real things) locally does conserve connectivity.

I have something to say about implications of this all too, but some other time, may be, my next post here… Also, about singularities in continua—some basic conceptual comments, concerning the way we do our mathematics.

– – – – –

Update on February 1, 2010

No one seems to have caught an obvious error in the above description. (Actually, the statements above are not so much wrong as they are in need of suitable qualifications throughout.). … I had thought that someone would catch me right within a day or so—at least that part of the writing above where I get to making that statement about two infinitesimally distant particles going into two different vortex regions. … In any case, right from the very first draft of this post, I had been dropping hints, too, concerning singularity and all… Now, more than two days later (and many hits later), still, none seems to have caught the error. What’s going on?

Apparently, going by the comments received at this blog (but moderated out) many people (notably including certain students (probably from an engineering department) at IIT Bombay) seem to have been more concerned with swearing at me rather than catching the technical/conceptual/mathematical/reasoning errors in the write-ups…. Hmmm…

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A Few Songs I Like (More or less at random)

1. (Marathi) “tuj saguN mhaNu ki nirguN re…”
Lyrics: Sant Dnyaneshwar
Singer: Pt. Bhimsen Joshi

2. (Marathi) “tyaa phulanchyaa ganDhakoshi…”
Lyrics: Sooryakant Khandekar
Singer: Hridaynath Mangeshkar

3. (Hindi) “chhupaa lo yun dil me pyaar meraa…”
Music: Roshan
Singers: Hemant Kumar, Lata Mangeshkar
Lyrics: Majrooh Sultaanpuri