# “Math rules”?

My work (and working) specialization today is computational science and engineering. I have taught FEM, and am currently teaching both FEM and CFD.

However, as it so happens, all my learning of FEM and CFD has been through self-studies. I have never sat in a class-room and learnt these topics that way. Naturally, there were, and are, gaps in my knowledge.

The most efficient way of learning any subject matter is through traditional formal learning—I mean to say, our usual university system. The reason is not just that a teacher is available to teach you the material; even books can do that (and often times, books are actually better than teachers). The real advantage of the usual university education is the existence of those class-mates of yours.

Come the week of those in-semester unit tests, at least in the hostels of Indian engineering schools, every one suddenly goes in the studies mode. In the hostel hallways, you casually pass someone by, and he puts a simple question to you. It is, perhaps, his genuine difficulty. You try to explain it to him, find that there are some gaps in your own knowledge, too. After a bit of a discussion, some one else joins the discussion, and then you all have to sheepishly go back to the notes or books or solve a problem together. It helps all of you.

Sometimes, the friend could be even just showing off to you—he wouldn’t ask you a question if he knew you could answer it. You begin answering in your usual magnificently nonchalant manner, and soon reach the end of your wits. (A XI standard example: If the gravitational potential inside the earth is constant, how come a ball dropped in a well falls down? [That is your friend’s question, just to tempt you in the wrong direction all the way through.]… And what would happen if there is a bore all through the earth’s volume, assuming that the earth’s core is solid all the way through?) His showing off helps you.

No, not every one works in this friendly a mode. But enough of them do that one gets [too] used to this way of studying/learning.

And, it is this way of studying which is absent not only in the learning by pure self-studies alone, but also in those online/MOOC courses. That is the reason why NPTEL videos, even if downloaded and available on the local college LAN, never get referred to by individual students working in complete isolation. Students more or less always browse them in groups even if sitting on different terminals (and they watch those videos only during the examination weeks!)

Personally, I had got [perhaps excessively] used to this mode of learning. [Since my Objectivist learning has begun interfering here, let me spell the matter out completely: It’s a mix of two modes: your own studies done in isolation, and also, as an essential second ingredient, your interaction with your class-mates (which, once again, does require the exercise of your individual mind, sure, but the point is: there are others, and the interaction is exposing the holes in your individual understanding).]

It is to this mix that I have got too used to. That’s why, I have acutely felt the absence of the second ingredient, during my studies of FEM and CFD. Of course, blogging fora like iMechanica did help me a lot when it came to FEM, but for CFD, I was more or less purely on my own.

That’s the reason why, even if I am a professor today and am teaching CFD not just to UG but also to PG students, I still don’t expect my knowledge to be as seamlessly integrated as for the other things that I know.

In particular, one such a gap got to the light recently, and I am going to share my fall—and rise—with you. In all its gloriously stupid details. (Feel absolutely free to leave reading this post—and indeed this blog—any time.)

In CFD, the way I happened to learn it, I first went through the initial parts (the derivations part) in John Anderson, Jr.’s text. Then, skipping the application of FDM in Anderson’s text more or less in its entirety, I went straight to Versteeg and Malasekara. Also Jayathi Murthy’s notes at Purdue. As is my habit, I was also flipping through Ferziger+Peric, and also some other books in the process, but it was to a minor extent. The reason for skipping the rest of the portion in Anderson was, I had gathered that FVM is the in-thing these days. OpenFOAM was already available, and its literature was all couched in terms of FVM, and so it was important to know FVM. Further, I could also see the limitations of FDM (like requirement of a structured Cartesian mesh, or special mesh mappings, etc.)

Overall, then, I had never read through the FDM modeling of Navier-Stokes until very recent times. The Pune University syllabus still requires you to do FDM, and I thus began reading through the FDM parts of Anderson’s text only a couple of months ago.

It is when I ran into having to run the FDM Python code for the convection-diffusion equation that a certain lacuna in my understanding became clear to me.

Consider the convection-diffusion equation, as given in Prof. Lorena Barba’s Step No.8, here [^]:

$\dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right) \\ \dfrac{\partial v}{\partial t} + u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} = \nu \; \left(\dfrac{\partial ^2 v}{\partial x^2} + \dfrac{\partial ^2 v}{\partial y^2}\right)$

I had never before actually gone through these equations until last week. Really.

That’s because I had always approached the convection-diffusion system via FVM, where the equation would be put using the Eulerian frame, and it therefore would read something like the following (using the compact vector/tensor notation):

$\dfrac{\partial}{\partial t}(\rho \phi) + \nabla \cdot (\rho \vec{u} \phi) = \nabla \cdot (\Gamma \nabla \phi) + S$
for the generic quantity $\phi$.

For the momentum equations, we substitute $\vec{u}$ in place of $\phi$, $\mu$ in place of $\Gamma$, and $S_u - \nabla P$ in place of $S$, and the equation begins to read:
$\dfrac{\partial}{\partial t}(\rho \vec{u}) + \nabla \cdot (\rho \vec{u} \otimes \vec{u}) = \nabla \cdot (\mu \nabla \vec{u}) - \nabla P + S_u$

For an incompressible flow of a Newtonian fluid, the equation reduces to:

$\dfrac{\partial}{\partial t}(\vec{u}) + \nabla \cdot (\vec{u} \otimes \vec{u}) = \nu \nabla^2 \vec{u} - \dfrac{1}{\rho} \nabla P + \dfrac{1}{\rho} S_u$

This was the framework—the Eulerian framework—which I had worked with.

Whenever I went through the literature mentioning FDM for NS equations (e.g. the computer graphics papers on fluids), I more or less used to skip looking at the maths sections, simply because there is such a variety of reporting NS, and those initial sections of the papers all are covering the same background material. (Ferziger and Peric, I recall off-hand, mention of some 72 ways of writing down the NS equations.)  The meat of the paper comes only later.

The trouble occurred when, last week, I began really reading through (as in contrast to rapidly glancing over) Barba’s Step No. 8 as mentioned above. Let me copy-paste the convection-diffusion equations once again here, for ease of reference.

$\dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right) \\ \dfrac{\partial v}{\partial t} + u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} = \nu \; \left(\dfrac{\partial ^2 v}{\partial x^2} + \dfrac{\partial ^2 v}{\partial y^2}\right)$

Look at the left hand-side (LHS for short). What do you see?

What I saw was an application of the following operator—an operator that appears only in the Lagrangian framework:

$\dfrac{\partial}{\partial t} + (\vec{u} \cdot \nabla)$

Clearly, according to what I saw, the left hand-side of the convection-diffusion equation, as written above, is nothing but this operator, as applied to $\vec{u}$.

And with that “vision,” began my fall.

“How can she use the Lagrangian expression if she is going to use a fixed Cartesian—i.e. Eulerian—grid? After all, she is doing FDM here, isn’t she?” I wondered.

If it were to be a computer graphics paper using FDM, I would have skipped over it, presuming that they would sure transform this equation to the Eulerian form some time later on. But, here, I was dealing with a resource for the core engineering branches (like mech/aero/met./chem./etc.), and I also had a lab right this week to cover this topic. I couldn’t skip over it; I had to read it in detail. I knew that Prof. Barba couldn’t possibly make a mistake like that. But, in this lesson, even right up to the Python code (which I read for the first time only last week), there wasn’t even a hint of a transformation to the Eulerian frame. (Initially, I even did a search on the string: “Euler” on that page; no beans.)

There must be some reason to it, I thought. Still stuck with reading a Lagrangian frame for the equation, I then tried to imagine a reasonable interpretation:

Suppose there is one material particle at each of the FDM grid nodes? What would happen with time? Simplify the problem all the way down. Suppose the velocity field is already specified at each node as the initial condition, and we are concerned only with its time-evolution. What would happen with time? The particles would leave their initial nodal positions, and get advected/diffused away. In a single time-step, they would reach their new spatial positions. If the problem data are arbitrary, their positions at the end of the first time-step wouldn’t necessarily coincide with grid points. If so, how can she begin her next time iteration starting from the same grid points?

I thought through it twice, but with the same result. I searched through her other steps. (Idly browsing, I even looked up her CV: PhD from CalTech. “No, she couldn’t possibly be skipping over the transformation,” I distinctly remember telling myself for the $n$th time.)

Faced with a seemingly unyielding obstacle, I had to fall back on to my default mode of learning—viz., the “mix.” In other words, I had to talk about it with someone—any one—any one, who would have enough context. But no one was available. The past couple of days being holidays at our college, I was at home, and thus couldn’t even catch hold of my poor UG students.

But talking, I had to do. Finally, I decided to ask someone about it by email, and so, looked up the email ID of a CFD expert, and asked him if he could help me with something that is [and I quote] “seemingly very, very simple (conceptual) matter” which “stumps me. It is concerned with the application of Lagrangian vs. Eulerian frameworks. It seems that the answer must be very simple, but somehow the issue is not clicking-in together or falling together in place in the right way, for me.” That was yesterday morning.

It being a week-end, his reply came fairly rapidly, by the yesterday afternoon (I re-checked emails at around 1:30 PM); he had graciously agreed to help me. And so, I rapidly wrote up a LaTeX document (for equations) and sent it to him as soon as I could. That was yesterday, around 3:00 PM. Satisfied that finally I am talking to someone, I had a late lunch, and then crashed for a nice ciesta. … Holidays are niiiiiiiceeeee….

Waking up at around 5:00 PM, the first thing I did, while sipping a cup of tea, was to check up on the emails: no reply from him. Not expected this soon anyway.

Still lingering in the daze of that late lunch and the ciesta, idly, I had a second look at the attached document which I had sent. In that problem-document, I had tried to make the comparison as easy for the receiver to see, and so, I had taken care to write down the particular form of the equation that I was looking for:

$\dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial uv}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right) \\ \dfrac{\partial v}{\partial t} + \dfrac{\partial uv}{\partial x} + \dfrac{\partial v^2}{\partial y} = \nu \; \left(\dfrac{\partial ^2 v}{\partial x^2} + \dfrac{\partial ^2 v}{\partial y^2}\right)$

“Uh… But why would I keep the product terms $u^2$ inside the finite difference operator?” I now asked myself, still in the lingering haze of the ciesta. “Wouldn’t it complicate, say, specifying boundary conditions and all?” I was trying to pick up my thinking speed. Still yawning, I idly took a piece of paper, and began jotting down the equations.

And suddenly, way before writing down the very brief working-out by hand, the issue had become clear to me.

Immediately, I made me another cup of tea, and while still sipping it, launched TexMaker, wrote another document explaining the nature of my mistake, and attached it to a new email to the expert. “I got it” was the subject line of the new email I wrote. Hitting the “Send” button, I noticed what time it was: around 7 PM.

Here is the “development” I had noted in that document:

Start with the equation for momentum along the $x$-axis, expressed in the Eulerian (conservation) form:

$\dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial uv}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right)$

Consider only the left hand-side (LHS for short). Instead of treating the product terms $u^2$ and $uv$ as final variables to be discretized immediately, use the product rule of calculus in the same Eulerian frame, rearrange, and apply the zero-divergence property for the incompressible flow:

$\text{LHS} = \dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial uv}{\partial y} \\ = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + u\dfrac{\partial u}{\partial x} + u \dfrac{\partial v}{\partial y} + v \dfrac{\partial u}{\partial y} \\ = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + u \left[\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} \right] + v \dfrac{\partial u}{\partial y} \\ = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + u \left[ 0 \right] + v \dfrac{\partial u}{\partial y}; \qquad\qquad \because \nabla \cdot \vec{u} = 0 \text{~if~} \rho = \text{~constant} \\ = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y}$

We have remained in the Eulerian frame throughout these steps, but the final equation which we got in the end, happens to be identical in its terms to that for the Lagrangian frame—when the flow is incompressible.

For a compressible flow, the equations should continue looking different, because $\rho$ would be a variable, and so would have to be accounted for with a further application of the product rule, in evaluating $\frac{\partial}{\partial t}(\rho u)$, $\frac{\partial}{\partial x}(\rho u^2)$ and $\frac{\partial}{\partial x}(\rho uv)$ etc.

But as it so happens, for the current case, even if the final equations look exactly the same, we should not supply the same physical imagination. We don’t imagine the Lagrangian particles at nodes. Our imagination continues remaining Eulerian throughout the development, with our focus not on the advected particles’ positions but on the flow variables $u$ and $v$ at the definite (fixed) points in space.

Sometimes, just expressing your problem to someone else itself pulls you out of your previous mental frame, and that by itself makes the problem disappear—in other words, the problem gets solved without your “solving” it. But to do that, you need someone else to talk to!

But how could I make such stupid and simple a mistake, you ask? This is something even a UG student at an IIT would be expected to know! [Whether they always do, or not, is a separate issue.]

Two reasons:

First: As I said, there are gaps in my knowledge of computational mechanics. More gaps than you would otherwise expect, simply because I had never had class-mates with whom to discuss my learning of computational  mechanics, esp., CFD.

Second: I was getting deeper into the SPH in the recent weeks, and thus was biased to read only the Lagrangian framework if I saw that expression.

And a third, more minor reason: One tends to be casual with the online resources. “Hey it is available online already. I could reuse it in a jiffy, if I want.” Saying that always, and indefinitely postponing actually reading through it. That’s the third reason.

And if I could make so stupid a mistake, and hold it for such a long time (a day or so), how could I then see through it, even if only eventually?

One reason: Crucial to that development is the observation that the divergence of velocity is zero for an incompressible flow. My mind was trained to look for it because even if the Pune University syllabus explicitly states that derivations will not be asked on the examinations, just for the sake of solidity in students’ understanding, I had worked through all the details of all the derivations in my class. During those routine derivations, you do use this crucial property in simplifying the NS equations, but on the right hand-side, i.e., for the surface forces term, in simplifying for the Newtonian fluid. Anderson does not work it out fully [see his p. 66] nor do Versteeg and Malasekara, but I anyway had, in my class… It was easy enough to spot the same pattern—even before jotting it down on paper—once it began appearing on the left hand-side of the same equation.

Hard-work pays off—if not today, tomorrow.

CFD books always emphasize the idea that the 4 combinations produced by (i) differential-vs-integral forms and (ii) Lagrangian-vs-Eulerian forms all look different, and yet, they still are the same. Books like Anderson’s take special pains to emphasize this point. Yes, in a way, all equations are the same: all the four mathematical forms express the same physical principle.

But seen from another perspective, here is an example of two equations which look exactly the same in every respect, but in fact aren’t to be viewed as such. One way of reading this equation is to imagine inter-connected material particles getting advected according to that equation in their local framework. Another way of reading exactly the same equation is to imagine a fluid flowing past those fixed FDM nodes, with only the nodal flow properties changing according to that equation.

Exactly the same maths (i.e. the same equation), and of course, also the same physical principle, but a different physical imagination.

And you want to tell me “math [sic] rules?”

I Song I Like:

(Hindi) “jaag dil-e-deewaanaa, rut jaagee…”
Music: Chitragupt
Lyrics: Majrooh Sultanpuri

[As usual, may be another editing pass…]

[E&OE]

# “They don’t even touch a good text-book!”

“They don’t even touch a good text-book!”

This line is a very common refrain that one often hears in faculty rooms or professors’ cabins, in engineering colleges in India.

Speaking in factual terms, there is a lot of truth to it. The assertion itself is overwhelmingly true. The fact that the student has never looked into a good (or “reference” or “foreign authors'”) text is immediately plain and clear to anyone who has ever graded their examination papers, or worked as an examiner on the oral/viva voce examinations.

The undergraduate Indian students these days, esp. those in Pune and Mumbai, and esp. those in the private engineering colleges, always refer to only a locally published text for all their studies.

These texts are published by a few local publishers well known to the students (and their professors). I wouldn’t mind dropping a few names: Nirali, Pragati, TechMax, etc. The books are published at almost throw-away prices (e.g. Rs. 200–300). (There also exists a highly organized market for the second-hand books. No name written, no pencil marks? Some 75% of the cost returned. Etc. There is a bold print, too—provided, the syllabus hasn’t changed in the meanwhile. In that case, there is no resale value whatsoever!)

The authors of these texts themselves are professors in these same private engineering colleges. They know the system in and out. No, I am not even hinting at any deliberate fraud or malpractice here. Quite on the contrary.

The professors who write these local text-books often are enthusiastic teachers themselves. You would have to be very enthusiastic, because the royalties they “command” could be as low as a one-time payment of Rs. 50,000/- or so. The payment is always only a one-time payment (meaning, there are no recurring royalties even if a text book becomes a “hit”), and it never exceeds Rs. 1.5 lakhs lump-sum or so. (My figures are about 5 year old.) Even if each line is copied verbatim from other books, the sheer act of having to write down (and then proof-read) some 200 to 350 pages requires for the author to invest, I have been told, between 2 to 4 months, working overtime, neglecting family and all. The monthly salary of these professors these days can easily approach or exceed Rs. 1 lakh. So, clearly, money is not the prime motivation here. It has to be something else: Enthusiasm, love of teaching, or even just the respect or reputation that an author hopes to derive in the sub-community of these local engineering colleges!

These professors—the authors—also often are well experienced (15–40 years of teaching experience is common), and they know enough to know what kind of examination questions are likely to come up on the university examinations. (They themselves have gone through the same universities.) They write these books targeting only task: writing the marks-scoring answers on those university examinations. Thus, these “text” books are more or less nothing but a student aid (or what earlier used to be called the “guide” books).

It in fact has evolved into a separate genre by itself. Contrary to an impression wide-spread among professors of private engineering colleges in India, there in fact are somewhat similar books also used heavily by the students in the USA. Thus, these local Indian books are nothing but an improvised version of the Schaums’ series in science and engineering (or the Sparks Notes in the humanities, in the US schools).

But there is a further feature here. There is a total customization thrown in here. These local books are now-a-days written (or at least adapted) to exactly match the detailed syllabus of each university separately. So, there are different books, by the same author and for the same subject, but one for Mumbai University, and the other for Pune University, etc. Students never mix up the universities.

The syllabus for each university is followed literally, down to dividing the text into chapters as per the headings of the modules mentioned in the syllabus (usually six per course), and dividing each chapter into sections, with the headings and order of these sections strictly following the order and the letter of the syllabus. The text in each section is followed by a compilation of the past university examination questions (of that same university) pertaining to that particular section alone. Most of these past examination questions are solved in the text—that’s the bulk of the book. When the opening page of a chapter lists the sections in it, the list also carries, in the parentheses, whether this section is “theory” or “numericals”.

Overall, the idea is, even just looking at the “text” book, a student can easily anticipate whether a question is likely to be asked on a given section or not, and if yes, of what kind. The students also work out many logics: “Every semester, they have asked a question on this section. So we have to mug it up well.” Or, playing the “contra”: “Last three semesters, not a single question here? It’s going to come this time round.” Etc. (Yes, I followed this practice in my lectures, too—I did want my students to score well on the final university examinations, after all!)

The customization, for each revision of the syllabus of each university, is done down to that level of detail. So, for the first year course on electrical engineering, you have one text-book of title, say, Electrical Engg. (FE), Pune University, 2012 course, and another text book, now of the title, say, Basic Electrical Technology (FE), Mumbai University, 2011 course. Etc.

That’s what I mean, when I use the phrase the “local” text-books.

I certainly don’t mean the SI Units editions of American texts, or the Indian Standards-adapted editions of reputed texts (such as, say, Shigley’s on design or Thomson and Dahleh’s on vibrations). I don’t mean the inexpensive Indian editions of foreign texts (such as those by Pearson, Wiley, ELBS, etc.) I also don’t mean the text-books written by the well-known Indian authors working right in India (such as those by IIT professors, and published by, say, Universities Press, Narosa, or PHI). I don’t even mean the more general text-books written for Indian universities and/or the AMIE examinations (such as those by S. Chand, Khanna, CBS, etc.). When I say “local” text-books, I specifically mean the books of the kind mentioned above.

Undergraduate students in Pune and Mumbai these days refer only to these local books.

They (really) don’t even bother to touch a good reference text, even if it’s available on the college library shelf.

In contrast, in our times, the problem was, we simply didn’t have the “foreign authors'” texts available to us—not always even in the COEP library. In those days, sometimes, such books happened to be too expensive, even for COEP’s library. And, even back then, Shahani’s text-books anyway were available. But at least, they didn’t cater to only the Pune university (they would list problems from universities as far flung as Madras, Gorakhpur, Agra, Allahabad, etc.) And, in fact, these books were generally looked down upon. Even by the students themselves.

The contrast to today’s situation is too glaring. Naturally, professors sometimes do end up saying the title line with a tone of exasperation.

Yes, I used to sometimes say that line myself, of course with sarcasm, when I taught in the late ’80s in the Pune of those days. (The situation back then was not so acute.) Almost as if by habit, I also repeated the line when I more recently taught a course at COEP (2009, FEM). However, observing students, somehow, my line had somehow begun to lose that cutting edge it once had. First, at COEP, I had the freedom to design this course (on FEM), and they did buy at least Logan and/or Cook. (Even if I was distributing my PDF notes.) And, there was something else to it, too. I somehow got a vague feel that it somehow wouldn’t be fully right to blame students (I mean COEP students in general). However, my COEP stint was only for one semester, only for one course, and only as a visiting faculty. So, the vague feel simply remained what it was—just a vague feel.

Then, recently in 2014, when I began teaching at a private engineering college in Mumbai, I once again heard this line from the other professors. And, I used it myself too. With the usual sarcasm. I did that perhaps for the most part of my first semester there.

However, some way down the line, I once again got that vague feel that, may be, something was “wrong” somewhere, even here, in Mumbai: these kids really were trying to be sincere, and yet, for some reason unknown to me, they still wouldn’t at all refer to good texts.

This is an aside, but I can tell you that it’s very easy to read the faces of the insincere people, esp. when they are young. There are some insincere students too. But, at least going by my own experience, they are in a minority. (It is a headache-some minority. Yet, by numerical magnitude alone, it certainly is in a small minority.) I am not saying this to be politically correct, or to win points from students. What I said is the factual case. In fact, my experience is that when it comes to in-sincerity, parents easily outperform their children. May be because, the specific parents that we mostly end up seeing in college are those whose kids have some problem—low attendance, fee payments, other issues, etc. The parents with whom we get to interact really well, thus, happens to be a self-selected sub-group. They aren’t necessarily representative of all parents… Yet, I am also sure that that’s not the real reason why I think parents can easily be more insincere. I think the real reason is that, at their age, the kids are actually unable to fake too much. It’s far easier for them to be sincere than to be a fake and still get away with it. They just can’t manage it, regardless of their desire. And, looking at it in a better light, I here remember what Ayn Rand had once said in a somewhat similar context, “one doesn’t start out in life by spitting on one’s own face—it’s not in the essential nature of life” or something like that. (Off-hand, I think, it was in the preface to the 25th anniversity edition of The Fountainhead.) So, the kids, by and large, are sincere. … By the time they themselves become parents—well, let’s leave that story right here. (We need them to make all those fee payments, anyway…)

So, coming back to the main thread, I would anyway generally chat with the students, and so, I started asking, esp. some of the more talkative students, the reason why they might not be referring to good texts. After all, in my lectures, I would try to provide very specific references: specific section numbers or even page numbers, in a specific edition of a specific reference text. (And these texts were available in the college library.) Why, I once had even distributed an original research paper. (It was Griffth’s seminal 1920 paper starting the field of fracture mechanics. Griffith’s argument here is rather conceptual, and the paper has surprisingly very little maths. Whatever the maths there is, it is very easily accessible to the SE students, too.)

The result of my initial attempts to understand the reason (why students don’t read good texts) was not so encouraging. The talkative students began dropping by my cabin once in a while, asking which section to use while answering a certain assignment question or so. However, they still only rarely used those better texts, when it came to actually completing their assignments. And, in the unit tests (and in the final end-sem examination), they invariably ended up quoting only the local text books (whether verbatim or not).

The exercise was, thus, futile. And yet, the students’ sincerity—at least the sincerity of their desire, as in contrast to their actions—could not be put in doubt.

So, I took it as a challenge. I set this as a problem for myself: To discover the main reason(s) why my students don’t refer to good text-books. The real underlying reason(s), regardless of whatever they otherwise did to impress me.

It took a while for me to crack the problem. I would anyway generally chat with them, enquiring where they lived, what their parents did, about their friends and brothers and sisters, etc. In addition, I would also observe, now with this new challenge somewhere at the back of my mind, how they behaved (or rushed around) in college: in hallways, labs, canteen, college ground, even at the bus-stop just outside the college, etc.

…Finally, I got it! At least one reason, a main reason, a systemic reason that applied even to those who otherwise were good, talented, curious, or just plain sincere.

As soon as I discovered the reason, I shared it with every one. In fact, I first shared it with my students, before I did with my colleagues or superiors. The answer lies in an Excel spreadsheet, here [^]. (It actually was created in OpenOffice Calc, on Windows 7.)  Go ahead, download it, and play with it a bit. The embedded formulae should be self-explanatory.

The numbers used in the spreadsheet may differ. The specific numbers I have used in the spreadsheet refer to my estimates while working at a college in Mumbai, in particular, in Navi Mumbai. In Mumbai, the time lost commuting is really an issue. If a student lives in Thane or Andheri and attends a college in Navi Mumbai, he easily spends about 3–4 hours in the daily commute (home->bus->railway station/second bus/metro–>another bus or six-seater, all of it taking about 1.5 hours one way, or more). In Pune, the situation is much more heterogeneous. One student could be spending 3 hours commuting both ways (think: from Nigdi to VIT) whereas some other student could be just happily walking to the college campus (think: Paud Phata residents, and MIT). It all depends. In Pune, many students would be using two-wheelers. In any case, for a professor, the only practical guideline for the entire class that he can at all use, would have to be statistical in nature. So, it’s the class average for the daily commute time that matters. For Mumbai in general, it could be 2–3 hours, for Pune students, it could be, say, between 1 to 2 hours (both ways put together).

So Pune is a bit easier on students. In contrast, for many of my Mumbai students, the situation was bad (or even very bad), and they were trying hard (or very hard) to make the best of it. It must have been at least a bit frustrating to them when professors like me, on the top of everything, were demanding making references to good foreign texts, and openly using a sarcastic tone—even if generously laced with humor—if they didn’t. It must have been frustrating to at least 40–60% of them. (The number is my estimate of those who were genuinely interested in referring to good books, even if only for the better-drawn and colorful diagrams, photographs, and also mathematical proofs that came without errors or without arbitrary replacement of $\partial$ by $d$.)

And why do I say that it must have been frustrating? Why didn’t I say it might have been frustrating?

Because, I cannot ever forget that look of that incredibly honest appreciation which slowly appeared on all their faces (including the faces of the “back-benchers”), as I shared my discovery in detail with them.

* * * * *   * * * * *   * * * * *

Do you have the time to read good, lengthy, or conceptually clarifying “reference” texts? Say, Timoshenko (app. mech. and strength of materials); Shames, or Popov (strength of materials); White, or Fox & McDonald, or Som & Biswas (Fluid Mech.); Holman, or Nag, or Sukhatme (Heat Transfer)?

And, if you do, do you spend time reading these texts? If yes, did you complete them (I mean only the portion relevant to the syllabus) in the same semester that you were learning or teaching the subject for the first time? Could you have?

And yes, in my last sentence, I have included “teaching” too. My questions are directed to the professors too. In fact, my questions are directed, first and foremost, only at them.

After all, it is the professors—or at least some of us—who are in the driver’s seat here; the students never are. It is the professors who (i) design the syllabii as well as the examination schemes (including the number of tests to have and their nature), (ii) decide on the number of assignments (and leave no opportunity to level criticism in our capacity as External Examiners, if the length or difficulty of an assignment falls short), (iii) decide on the course text-books (and take due care to list more than 5 prescribed text-books, and more than 10 reference books per course) (iv) decide on the student attendance criteria in detail, up to the individual course level, and report on the defaulting students (and follow through with the meetings with their parents) every two weeks or at least once a month, (v) set the examination papers according to the established pattern—after all, it’s only us who is going to check the papers!, (vi) sometimes, write those local text-books!, and (vii) also keep the expectation that students should somehow show in their final university examination answer books, some evidence of having gone through some good, thick, reference texts, too. Whether we ourselves had managed to do that during our own UG years or not!

And, yes, I also want the IIX professors to ponder over these matters. All their students enjoy a fully residential program; these kids from these private engineering colleges mostly don’t. They at IIXs always get to design all their course syllabii and decide on the examination patterns, and they even get to enjoy the sole responsibility to grade their students. The possibility of adopting a marks normalization scheme, after the examination, always lies at hand, with them, just in case a topic took too long with a certain class or so… Are they then being reasonable in their request demand that the students of these “other” engineering colleges in India be well-read enough, at least by the time the students join them at IIXs for ME/MTech studies?

As to me, no, as I indicated in my earlier posts, while being a professor, I could not always find the time to do that—referring to good text-books. I tried, but basically my situation wasn’t much different from that of my students—we both were short on the available time. So, I didn’t always succeed.

[As to my own UG years, it was mixed: I did hunt for months, and got my hands on, the books like Reed-Hill, White, Holman, etc. However, I would be dishonest if I claimed that it was right during my UG years that I had got whatever I did, from books like these. In my case, the learning continued for years. Yes, I even bought and religiously studied once again even Thomas & Finney’s calculus, when I was in my PhD program at UAB. Despite my attempts during the UG years, I really cannot ascribe a large part, or even a significant part of my current understanding to my UG years. Your case may be different; I was just narrating my own experience.]

… And, as far making references to good books goes, now that I do have time at my hand these days, there is another problem: I don’t know what course in particular I will be teaching the next semester, and where—or for that matter, whether some college will even hire me in the first place, or not.

So, I end up “wasting” my time writing blog posts like this one. Thus, I, too, end up not touching a good reference text!

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A Song I Like:

(Hindi) “aane waalaa pal, jaane waalaa hai…”
Lyrics: Gulzaar
Singer: Kishor Kumar
Music: R. D. Burman

[I will go over this post once again, editing it, and may be adding a bit here and there. Done. This post is already too long. So, I will write another post—a brief one—to jot down some tips to make the best possible use of the student’s time—including my suggestions to the engineering colleges as to what they can do to help the students. Also, my take on whether the system as noted above has diluted the quality of education or not—esp. as in contrast to what we had as UG students at COEP more than three decades ago.]

[E&OE]

# University Courses in Mechanical Engineering that I Could Teach

0. Hey fellowbloggers (or at least my blogreaders),

These are the times of recruitment for faculty positions in India. As you know, I, too, am trying (once again!).

In academia, my main interest is neither administration nor for that matter even teaching in the sense the word is often understood in India, but, mainly research, and also, course development (as is possible at autonomous institutes  (but Somaiya, headed by Prof. Shubha Pandit, who as a student was senior to me at COEP by just one year, didn’t even call me for an interview, or bother to reply my query emails.)).

1. A List of the Mechanical Engineering Courses that I Could Teach:

Here is a list of courses in Mechanical Engineering that I am confident I could teach, despite having had my bachelor’s and master’s degrees only in Metallurgy. The list is indicative, and not complete. I have tried to order the list in the decreasing order of my preference. Thus, the courses appearing near the top is what I am currently most interested in teaching, esp. at the master’s level. The level at which these courses get offered is also indicated in parentheses. The courses in italics are the courses I taught most recently.

1. Computational Fluid Dynamics (BE/ME)
2. Finite Element Method (TE/BE/ME)
3. Fluid Mechanics (SE/TE)
4. Heat Transfer (TE)
5. Mechanical Vibrations (TE)
7. Fracture Mechanics (ME)
8. Strength of Materials (SE)
9. Thermodynamics (SE)

As you can see, they span over two sub-divisions currently routine in Indian universities: thermal, and design. But more troublesome to the Indian academia is the fact that I jumped from Metallurgy to Mechanical, and therefore, they insist, I must teach Materials Technology.

2. A Special Note on Why I Should Not Teach Materials Technology:

Despite my academic degrees, I am not at all interested in teaching Materials Science/Engineering/Technology at any level. There are a few reasons for that.

(i) My Own Personal Reason:

Teaching a course does build a sort of vortex of ideas or an “ideas-sink” in which your mind gets drawn, at least for the duration of one entire semester. But, at my age of 52 (soon 53), I don’t have enough time at hand in my life to still be led away from my core research interests: computational mechanics/engineering.

(ii) Empirically, and statistically, it’s also not very good for all the students :

I also honestly think that the existing professors/others do a better job teaching it, at least at the SE level. This, in fact has actually been the case.

When I taught Materials Technology last (to SE students), the “top” 10% of the class was happy to very happy, with some students on their own coming in and gushing an almost embarassing kind of praise on me. [Drop a line to me and I will give you some quotes, though for reasons of breach of trust and confidentiality, I will not divulge the names of the students themselves]. (The “top” students need not have been class toppers, but they did tend to cluster somewhere towards the top; certainly they were above class average.)

But the in the final University examination, more students failed my course than what has been the historical average at the college where I taught. Reason?

I tend to explain well (even “average” students have told me that, not just the “top”), but in a “theoretical” subject like Materials Technology, what the below-average (and even average) students need is a sequence of those point-by-point model answers, whether explanation accompanies it or not. I try, but tend not to actually deliver very well, on that count.

Further, the average or below-average students also need a lot of “drilling in,” and I am not as good at it as other professors are, because as I focus mainly on supplying explanations: on fundamentals and how they connect together, and how they lead to something of importance in practice. In the process, I either tend to forget the drilling-in part, or the lecture-time simply gets over. All the three parts of (a) finishing the syllabus and, (b) also supplying explanations, and (c) also drilling in for the below-average students, is practically impossible for me. I can do (a) and (b) but not (c). Other professors probably do (a) and (c) and tend to ignore (b). But they are more successful as far as University exam results are concerned.

So, it’s an empirically established fact that I actually do poorly (or at least not as well) on the Materials Technology course at the SE level.

As to the student praise, they have also rushed in and gushed an almost embarassing sort of praise, also for the other courses I taught. But in spite of praising me, their performance on the final University exams was not affected much adversely. In fact, in all these other courses, they performed either slightly better or even noticeably better, than what had been the past historical average at the college. Why? Here is my reasoning.

The University examinations for these courses involve “sums” i.e. quantitative problems. If I still focus more on fundamentals and conceptual explanations in the class, and discuss only an outline of the problem attack strategy (and the reasons why) in the class, and then assign the full solution for home-work, the students do “get it.” They somehow go home, try some “sums,” and thereby manage to get both: a better conceptual knowledge, as well as the development of the examination-taking skills. At least, statistically speaking. And, at least as per my actual, empirical, observation.

In constrast, when it comes to Materials Technology, home-work assignments doesn’t work, because students simply copy from each other, or write some shortened versions of paragraphs from a locally published book, without ever pausing to think what they were writing. They, in essence, they take down a dictation from the local book. (Some had even had their brothers and sisters take down the dictation from the book, complete with a noticeably different hand-writing.) So, when it comes MT, my teaching + home-work is not an effective strategy. (Copying goes on also in other courses, but the fact that the problem on examination would be an unknown “sum” (at least one with different numerical values!) induces them to at least work through some of the assignment problems.)

But for Materials Technology, since the University examination emphasizes descriptions and not quantitative problems, or not even some “theoretical” but objective questions really probing deeper aspects, it requires a different kind of a drilling-in technique on the professor’s part. I am, as I said, not good on that count—and in fact, never was, ever in my life, even when I was a student myself. My lack of the “skill” shows in the results of my students.

This is an indication of the kind of reasons why I should make for a better professor for the listed courses rather than for the one course that seems to be a favorite one for the interview-committees: Materials Technology.

The interview committee members, if they read this blog, would now know how dumb a question it is to ask me why I don’t want to teach MT, and how much even dumber it would be for them to do resource planning or time-table scheduling assuming that I would handle MT. The empirical facts concerning the University examination indicate otherwise, despite my sincere and honest try at it (even if the matter was against my explicitly stated preference). And, if you now doubt my sincerity (as Indians are likely to do), go ask my students of MT—including those who failed in the final University examinations (or on my class tests). They themselves will tell you the real story. Then, if you wish, come back and share it with me. I remain open to that possibility—if you take some effort over and above that requiring to be a Doubting Tom.

3. Guiding Student Projects:

Apart from these, feel free to peruse some 7–8 ideas for student projects at the ME (Mech.) level that I have indicated in this blog recently. … Some of these (and other) ideas, suitably expanded, are good enough to guide at least one or two PhDs in Mechanical Engineering.

I also have quite a few other ideas that I have not even mentioned. For instance, I once wrote extended abstracts for a couple of papers, anticipating that an ME student would join me to work on these, and both these extended abstracts were accepted at a high quality international conference. The papers were based on an idea for an ME project that I haven’t mentioned on this blog. I had to withdraw the papers after acceptance, because I didn’t have an ME student to work with me. (You see, the papers were about CFD, and my friends in Mechanical engineering were busy avoiding polluting their branch with Metallurgy graduates, throwing as many obstacles in my path as humanly possible to them.)

Apart from it all, I could easily co-guide a few projects from the CS and Civil fields.

4. Co-Curricular Development of the Post-graduate Students and Junior Faculty Members:

I could also conduct special short-term courses for final year BE/ME students and/or junior faculty in random areas such as:

• LaTeX and Beamer (including scalable graphics for manuscript submissions)
• Python and Its Ecosystem
• Open Source Packages in CAE
• CFD with FVM. (No commercial packages, even if  available at the college, but with some custom-written simple programs written in Python, or using FiPy (but not OpenFOAM))
• FEM. (No commercial packages, even if  available at the college, but with some custom-written simple programs written in Python, or SfePy, etc.)
• OpenFOAM. (Only introductory, but certainly going a bit beyond the tutorials included in the official documentation, or IIT Bombay’s Spoken Tutorials.)
• GIS using QGIS (Mostly based on Ujaval Gandhi’s tutorials, going just a bit here and there beyond it.)

5. Never Lose the Focus:

But just in case we lose the focus: Please go through the list of the routine courses in Mechanical Engineering proper that I can teach, as mentioned at the beginning of the post. That’s what really counts for the interview process.

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A Song I Like:

Not to be noted for this post [unless I change my mind later]

(Western Classical)
Composer: Franz Liszt
Work: Liebestraum No. 3

[I could not find on the ‘net the rendering that I first heard and still like the most (and as usual, have lost the cassette for it by now). It was quite modern-sounding (it even had drums!), but without ever getting loud, gaudy, or ever overshadowing the original subtlety. Sorry, but I didn’t as much appreciate most of the renderings now thrown up on priority in a Google search on Liszt. (The rendering I heard must have been from ’70s or ’80s, because I had bought the cassette in India, in the late ’80s.)

… The post is technically over, complete with this section on the songs I like too, but I still can’t resist the temptation to share this bit. It’s a quote concerning Liszt which I found being quoted as a part of a doctoral study in music at the Indian University [^]:

“…When Liszt was teaching his famous master classes in Weimar at the end of his life, one day a student brought to him his famous Liebestraüm No. 3. After he played, Liszt was very mad because while performing the cadenza written in small notes (found on the second last page of the piece), the student had played exactly what was written on the printed page. `But you are a pianist now, you have to make your own cadenzas!,’ Liszt spontaneously exclaimed after he played.”

]

[Minor editing after publication, as usual, is still possible.]

[E&OE]

/

# MWR for the first- and third-order differential equations

I am teaching an introductory course on FEA this semester. Teaching always involves learning—at least on the teacher’s side.

No, there was no typo in there. I did mean what I just said. It’s based on my own personal observation. Teaching actually involves (real) learning on the part of the teacher—and hopefully, if he is effective enough in his teaching (and if the student, too, is attentive and hard-working enough), then, also on the part of the student.

When you teach a course, in thinking about how to simplify the ideas involved, how to present them better, you have to mentally go over the topics again and again; you have to think and re-think about the material; you have to see if rearranging the ideas and the concepts involved or seeing them in a different light might make it any easier to “get” it or even just to retain it, and so on. … The end result is that you often actually end up deriving at least new mnemonics if not establishing new connections about the topics. In any case, you derive better conceptual integrations or strengthen them better. You end up mastering the material better than at the beginning of the course. … Or at least that’s what happens to me. I always end up learning at least a bit more about what I am teaching.

And, sometimes, the teacher even ends up deriving completely new ideas this way. At least, it seems, I just did—about the nature of FEA and computational mechanics in general. The idea is new, at least to me. But anyway, talking about this new idea is for some other day. … I have to first rigorously think about it. The idea, as of today, is just at that nascent phase (it struck me right this evening). I plan to put it to the paper soon, work out its details, refine the idea, and put it in a more rigorous form, etc. That will take time. And then, second, I have to also check whether someone has already published something of that kind or not. … As someone—was it Mark Twain?—said, the best of my ideas were stolen by the ancients… So, that part—checking the literature—too, will take quite some time. My own anticipation is that someone must have written something about it. In any case, it’s not all that big an idea. Just a simple something.

But, anyway, in the meanwhile, for this blog post, let me note down something different. An item, not of my knowledge, and not one of even potentially new knowledge, but of my ignorance, which got highlighted recently, during my lecture preparations.

I realized that if one of my students poses a question about it, I don’t know the reason why MWR (the method of weighted residuals) isn’t effective, or at least isn’t often used, and may be even cannot be relied on, for the first- and the third-order differential equations.  (See, see, see, I don’t even know whether it’s a “cannot” or an “isn’t”!) I don’t know the answer to that question.

Of course, as it so happens, most differential equations of engineering importance are only of the second and the fourth order. Whether linear or non-linear, they simply aren’t of the third-order. I haven’t myself seen a single third-order differential equation in any of the course-work I have ever done so far. Sure, I have seen such equations, but only in a mathematical handbook on the differential equations—never in a text-book or a monograph on engineering sciences as such. And, even if of the first-order, in physics and engineering, they often come as coupled equations, and thus, (almost nonchalantly, right in front of your eyes) jump into the usual class of the second-order differential equations—e.g. the partial differential wave equation.

Anyway, coming back to this MWR-related issue, I checked up the text-books by Reddy and Finlayson, but didn’t find the reason mentioned. I hope that someone knows the answer—someone would. So, I am going to raise this issue at iMechanica, right today.

That’s about all for this blog post, folks. Once I post my question at iMechanica, may be I will come back and add a link to it from here, but that’s about it. More, some other time.

[And, yes, I promise to blog about the new idea once I am done working it out and checking about it a bit. It just struck me just today, and it still is purely in the conceptual terms. The idea itself is such that it can (very) easily be translated into proper mathematical terms, but the point is: that’s something I haven’t done yet. Let me do that over, say the next few weeks/months, and then, sure, I will come back and blog about it a bit. I mean, I will sure blog about it way, way before sending any paper to any journal or so. That’s a promise. So, bye for now…]

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I Song I Like: