Fluxes, scalars, vectors, tensors…. and, running in circles about them!

0. This post is written for those who know something about Thermal Engineering (i.e., fluid dynamics, heat transfer, and transport phenomena) say up to the UG level at least. [A knowledge of Design Engineering, in particular, the tensors as they appear in solid mechanics, would be helpful to have but not necessary. After all, contrary to what many UGC and AICTE-approved (Full) Professors of Mechanical Engineering teaching ME (Mech – Design Engineering) courses in SPPU and other Indian universities believe, tensors not only appear also in fluid mechanics, but, in fact, the fluids phenomena make it (only so slightly) easier to understand this concept. [But all these cartoons characters, even if they don’t know even this plain and simple a fact, can always be fully relied (by anyone) about raising objections about my Metallurgy background, when it comes to my own approval, at any time! [Indians!!]]]

In this post, I write a bit about the following question:

Why is the flux \vec{J} of a scalar \phi a vector quantity, and not a mere number (which is aka a “scalar,” in certain contexts)? Why is it not a tensor—whatever the hell the term means, physically?

And, what is the best way to define a flux vector anyway?


1.

One easy answer is that if the flux is a vector, then we can establish a flux-gradient relationship. Such relationships happen to appear as statements of physical laws in all the disciplines wherever the idea of a continuum was found useful. So the scope of the applicability of the flux-gradient relationships is very vast.

The reason to define the flux as a vector, then, becomes: because the gradient of a scalar field is a vector field, that’s why.

But this answer only tells us about one of the end-purposes of the concept, viz., how it can be used. And then the answer provided is: for the formulation of a physical law. But this answer tells us nothing by way of the very meaning of the concept of flux itself.


2.

Another easy answer is that if it is a vector quantity, then it simplifies the maths involved. Instead of remembering having to take the right \theta and then multiplying the relevant scalar quantity by the \cos of this \theta, we can more succinctly write:

q = \vec{J} \cdot \vec{S} (Eq. 1)

where q is the quantity of \phi, an intensive scalar property of the fluid flowing across a given finite surface, \vec{S}, and \vec{J} is the flux of \Phi, the extensive quantity corresponding to the intensive quantity \phi.

However, apart from being a mere convenience of notation—a useful shorthand—this answer once again touches only on the end-purpose, viz., the fact that the idea of flux can be used to calculate the amount q of the transported property \Phi.

There also is another problem with this, second, answer.

Notice that in Eq. 1, \vec{J} has not been defined independently of the “dotting” operation.

If you have an equation in which the very quantity to be defined itself has an operator acting on it on one side of an equation, and then, if a suitable anti- or inverse-operator is available, then you can apply the inverse operator on both sides of the equation, and thereby “free-up” the quantity to be defined itself. This way, the quantity to be defined becomes available all by itself, and so, its definition in terms of certain hierarchically preceding other quantities also becomes straight-forward.

OK, the description looks more complex than it is, so let me illustrate it with a concrete example.

Suppose you want to define some vector \vec{T}, but the only basic equation available to you is:

\vec{R} = \int \text{d} x \vec{T}, (Eq. 2)

assuming that \vec{T} is a function of position x.

In Eq. 2, first, the integral operator must operate on \vec{T}(x) so as to produce some other quantity, here, \vec{R}. Thus, Eq. 2 can be taken as a definition for \vec{R}, but not for \vec{T}.

However, fortunately, a suitable inverse operator is available here; the inverse of integration is differentiation. So, what we do is to apply this inverse operator on both sides. On the right hand-side, it acts to let \vec{T} be free of any operator, to give you:

\dfrac{\text{d}\vec{R}}{\text{d}x} = \vec{T} (Eq. 3)

It is the Eq. 3 which can now be used as a definition of \vec{T}.

In principle, you don’t have to go to Eq. 3. In principle, you could perhaps venture to use a bit of notation abuse (the way the good folks in the calculus of variations and integral transforms always did), and say that the Eq. 2 itself is fully acceptable as a definition of \vec{T}. IMO, despite the appeal to “principles”, it still is an abuse of notation. However, I can see that the argument does have at least some point about it.

But the real trouble with using Eq. 1 (reproduced below)

q = \vec{J} \cdot \vec{S} (Eq. 1)

as a definition for \vec{J} is that no suitable inverse operator exists when it comes to the dot operator.


3.

Let’s try another way to attempt defining the flux vector, and see what it leads to. This approach goes via the following equation:

\vec{J} \equiv \dfrac{q}{|\vec{S}|} \hat{n} (Eq. 4)

where \hat{n} is the unit normal to the surface \vec{S}, defined thus:

\hat{n} \equiv \dfrac{\vec{S}}{|\vec{S}|} (Eq. 5)

Then, as the crucial next step, we introduce one more equation for q, one that is independent of \vec{J}. For phenomena involving fluid flows, this extra equation is quite simple to find:

q = \phi \rho \dfrac{\Omega_{\text{traced}}}{\Delta t} (Eq. 6)

where \phi is the mass-density of \Phi (the scalar field whose flux we want to define), \rho is the volume-density of mass itself, and \Omega_{\text{traced}} is the volume that is imaginarily traced by that specific portion of fluid which has imaginarily flowed across the surface \vec{S} in an arbitrary but small interval of time \Delta t. Notice that \Phi is the extensive scalar property being transported via the fluid flow across the given surface, whereas \phi is the corresponding intensive quantity.

Now express \Omega_{\text{traced}} in terms of the imagined maximum normal distance from the plane \vec{S} up to which the forward moving front is found extended after \Delta t. Thus,

\Omega_{\text{traced}} = \xi |\vec{S}| (Eq. 7)

where \xi is the traced distance (measured in a direction normal to \vec{S}). Now, using the geometric property for the area of parallelograms, we have that:

\xi = \delta \cos\theta (Eq. 8)

where \delta is the traced distance in the direction of the flow, and \theta is the angle between the unit normal to the plane \hat{n} and the flow velocity vector \vec{U}. Using vector notation, Eq. 8 can be expressed as:

\xi = \vec{\delta} \cdot \hat{n} (Eq. 9)

Now, by definition of \vec{U}:

\vec{\delta} = \vec{U} \Delta t, (Eq. 10)

Substituting Eq. 10 into Eq. 9, we get:

\xi = \vec{U} \Delta t \cdot \hat{n} (Eq. 11)

Substituting Eq. 11 into Eq. 7, we get:

\Omega_{\text{traced}} = \vec{U} \Delta t \cdot \hat{n} |\vec{S}| (Eq. 12)

Substituting Eq. 12 into Eq. 6, we get:

q = \phi \rho \dfrac{\vec{U} \Delta t \cdot \hat{n} |\vec{S}|}{\Delta t} (Eq. 13)

Cancelling out the \Delta t, Eq. 13 becomes:

q = \phi \rho \vec{U} \cdot \hat{n} |\vec{S}| (Eq. 14)

Having got an expression for q that is independent of \vec{J}, we can now use it in order to define \vec{J}. Thus, substituting Eq. 14 into Eq. 4:

\vec{J} \equiv \dfrac{q}{|\vec{S}|} \hat{n} = \dfrac{\phi \rho \vec{U} \cdot \hat{n} |\vec{S}|}{|\vec{S}|} \hat{n} (Eq. 16)

Cancelling out the two |\vec{S}|s (because it’s a scalar—you can always divide any term by a scalar (or even  by a complex number) but not by a vector), we finally get:

\vec{J} \equiv \phi \rho \vec{U} \cdot \hat{n} \hat{n} (Eq. 17)


4. Comments on Eq. 17

In Eq. 17, there is this curious sequence: \hat{n} \hat{n}.

It’s a sequence of two vectors, but the vectors apparently are not connected by any of the operators that are taught in the Engineering Maths courses on vector algebra and calculus—there is neither the dot (\cdot) operator nor the cross \times operator appearing in between the two \hat{n}s.

But, for the time being, let’s not get too much perturbed by the weird-looking sequence. For the time being, you can mentally insert parentheses like these:

\vec{J} \equiv \left[ \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \right) \right] \hat{n} (Eq. 18)

and see that each of the two terms within the parentheses is a vector, and that these two vectors are connected by a dot operator so that the terms within the square brackets all evaluate to a scalar. According to Eq. 18, the scalar magnitude of the flux vector is:

|\vec{J}| = \left( \phi \rho \vec{U}\right) \cdot \left( \hat{n} \right) (Eq. 19)

and its direction is given by: \hat{n} (the second one, i.e., the one which appears in Eq. 18 but not in Eq. 19).


5.

We explained away our difficulty about Eq. 17 by inserting parentheses at suitable places. But this procedure of inserting mere parentheses looks, by itself, conceptually very attractive, doesn’t it?

If by not changing any of the quantities or the order in which they appear, and if by just inserting parentheses, an equation somehow begins to make perfect sense (i.e., if it seems to acquire a good physical meaning), then we have to wonder:

Since it is possible to insert parentheses in Eq. 17 in some other way, in some other places—to group the quantities in some other way—what physical meaning would such an alternative grouping have?

That’s a delectable possibility, potentially opening new vistas of physico-mathematical reasonings for us. So, let’s pursue it a bit.

What if the parentheses were to be inserted the following way?:

\vec{J} \equiv \left( \hat{n} \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 20)

On the right hand-side, the terms in the second set of parentheses evaluate to a vector, as usual. However, the terms in the first set of parentheses are special.

The fact of the matter is, there is an implicit operator connecting the two vectors, and if it is made explicit, Eq. 20 would rather be written as:

\vec{J} \equiv \left( \hat{n} \otimes \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 21)

The \otimes operator, as it so happens, is a binary operator that operates on two vectors (which in general need not necessarily be one and the same vector as is the case here, and whose order with respect to the operator does matter). It produces a new mathematical object called the tensor.

The general form of Eq. 21 is like the following:

\vec{V} = \vec{\vec{T}} \cdot \vec{U} (Eq. 22)

where we have put two arrows on the top of the tensor, to bring out the idea that it has something to do with two vectors (in a certain order). Eq. 22 may be read as the following: Begin with an input vector \vec{U}. When it is multiplied by the tensor \vec{\vec{T}}, we get another vector, the output vector: \vec{V}. The tensor quantity \vec{\vec{T}} is thus a mapping between an arbitrary input vector and its uniquely corresponding output vector. It also may be thought of as a unary operator which accepts a vector on its right hand-side as an input, and transforms it into the corresponding output vector.


6. “Where am I?…”

Now is the time to take a pause and ponder about a few things. Let me begin doing that, by raising a few questions for you:

Q. 6.1:

What kind of a bargain have we ended up with? We wanted to show how the flux of a scalar field \Phi must be a vector. However, in the process, we seem to have adopted an approach which says that the only way the flux—a vector—can at all be defined is in reference to a tensor—a more advanced concept.

Instead of simplifying things, we seem to have ended up complicating the matters. … Have we? really? …Can we keep the physical essentials of the approach all the same and yet, in our definition of the flux vector, don’t have to make a reference to the tensor concept? exactly how?

(Hint: Look at the above development very carefully once again!)

Q. 6.2:

In Eq. 20, we put the parentheses in this way:

\vec{J} \equiv \left( \hat{n} \hat{n} \right) \cdot \left( \phi \rho \vec{U} \right) (Eq. 20, reproduced)

What would happen if we were to group the same quantities, but alter the order of the operands for the dot operator?  After all, the dot product is commutative, right? So, we could have easily written Eq. 20 rather as:

\vec{J} \equiv \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \hat{n} \right) (Eq. 21)

What could be the reason why in writing Eq. 20, we might have made the choice we did?

Q. 6.3:

We wanted to define the flux vector for all fluid-mechanical flow phenomena. But in Eq. 21, reproduced below, what we ended up having was the following:

\vec{J} \equiv \left( \phi \rho \vec{U} \right) \cdot \left( \hat{n} \otimes \hat{n} \right) (Eq. 21, reproduced)

Now, from our knowledge of fluid dynamics, we know that Eq. 21 seemingly stands only for one kind of a flux, namely, the convective flux. But what about the diffusive flux? (To know the difference between the two, consult any good book/course-notes on CFD using FVM, e.g. Jayathi Murthy’s notes at Purdue, or Versteeg and Malasekara’s text.)

Q. 6.4:

Try to pursue this line of thought a bit:

Start with Eq. 1 again:

q = \vec{J} \cdot \vec{S} (Eq. 1, reproduced)

Express \vec{S} as a product of its magnitude and direction:

q = \vec{J} \cdot |\vec{S}| \hat{n} (Eq. 23)

Divide both sides of Eq. 23 by |\vec{S}|:

\dfrac{q}{|\vec{S}|} = \vec{J} \cdot \hat{n} (Eq. 24)

“Multiply” both sides of Eq. 24 by \hat{n}:

\dfrac{q} {|\vec{S}|} \hat{n} = \vec{J} \cdot \hat{n} \hat{n} (Eq. 25)

We seem to have ended up with a tensor once again! (and more rapidly than in the development in section 4. above).

Now, looking at what kind of a change the left hand-side of Eq. 24 undergoes when we “multiply” it by a vector (which is: \hat{n}), can you guess something about what the “multiplication” on the right hand-side by \hat{n} might mean? Here is a hint:

To multiply a scalar by a vector is meaningless, really speaking. First, you need to have a vector space, and then, you are allowed to take any arbitrary vector from that space, and scale it up (without changing its direction) by multiplying it with a number that acts as a scalar. The result at least looks the same as “multiplying” a scalar by a vector.

What then might be happening on the right hand side?

Q.6.5:

Recall your knowledge (i) that vectors can be expressed as single-column or single-row matrices, and (ii) how matrices can be algebraically manipulated, esp. the rules for their multiplications.

Try to put the above developments using an explicit matrix notation.

In particular, pay particular attention to the matrix-algebraic notation for the dot product between a row- or column-vector and a square matrix, and the effect it has on your answer to question Q.6.2. above. [Hint: Try to use the transpose operator if you reach what looks like a dead-end.]

Q.6.6.

Suppose I introduce the following definitions: All single-column matrices are “primary” vectors (whatever the hell it may mean), and all single-row matrices are “dual” vectors (once again, whatever the hell it may mean).

Given these definitions, you can see that any primary vector can be turned into its corresponding dual vector simply by applying the transpose operator to it. Taking the logic to full generality, the entirety of a given primary vector-space can then be transformed into a certain corresponding vector space, called the dual space.

Now, using these definitions, and in reference to the definition of the flux vector via a tensor (Eq. 21), but with the equation now re-cast into the language of matrices, try to identify the physical meaning the concept of “dual” space. [If you fail to, I will sure provide a hint.]

As a part of this exercise, you will also be able to figure out which of the two \hat{n}s forms the “primary” vector space and which \hat{n} forms the dual space, if the tensor product \hat{n}\otimes\hat{n} itself appears (i) before the dot operator or (ii) after the dot operator, in the definition of the flux vector. Knowing the physical meaning for the concept of the dual space of a given vector space, you can then see what the physical meaning of the tensor product of the unit normal vectors (\hat{n}s) is, here.

Over to you. [And also to the UGC/AICTE-Approved Full Professors of Mechanical Engineering in SPPU and in other similar Indian universities. [Indians!!]]

A Song I Like:

[TBD, after I make sure all LaTeX entries have come out right, which may very well be tomorrow or the day after…]

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See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—3

I was looking for a certain book on heat transfer which I had (as usual) misplaced somewhere, and while searching for that book at home, I accidentally ran into another book I had—the one on Classical Mechanics by Rana and Joag [^].

After dusting this book a bit, I spent some time in one typical way, viz. by going over some fond memories associated with a suddenly re-found book…. The memories of how enthusiastic I once was when I had bought that book; how I had decided to finish that book right within weeks of buying it several years ago; the number of times I might have picked it up, and soon later on, kept it back aside somewhere, etc.  …

Yes, that’s right. I have not yet managed to finish this book. Why, I have not even managed to begin reading this book the way it should be read—with a paper and pencil at hand to work through the equations and the problems. That was the reason why, I now felt a bit guilty. … It just so happened that it was just the other day (or so) when I was happily mentioning the Poisson brackets on Prof. Scott Aaronson’s blog, at this thread [^]. … To remove (at least some part of) my sense of guilt, I then decided to browse at least through this part (viz., Poisson’s brackets) in this book. … Then, reading a little through this chapter, I decided to browse through the preceding chapters from the Lagrangian mechanics on which it depends, and then, in general, also on the calculus of variations.

It was at this point that I suddenly happened to remember the reason why I had never been able to finish (even the portions relevant to engineering from) this book.

The thing was, the explanation of the \delta—the delta of the variational calculus.

The explanation of what the \delta basically means, I had found right back then (many, many years ago), was not satisfactorily given in this book. The book did talk of all those things like the holonomic constraints vs. the nonholonomic constraints, the functionals, integration by parts, etc. etc. etc. But without ever really telling me, in a forth-right and explicit manner, what the hell this \delta was basically supposed to mean! How this \delta y was different from the finite changes (\Delta y) and the infinitesimal changes (\text{d}y) of the usual calculus, for instance. In terms of its physical meaning, that is. (Hell, this book was supposed to be on physics, wasn’t it?)

Here, I of course fully realize that describing Rana and Joag’s book as “unsatisfactory” is making a rather bold statement, a very courageous one, in fact. This book is extraordinarily well-written. And yet, there I was, many, many years ago, trying to understand the delta, and not getting anywhere, not even with this book in my hand. (OK, a confession. The current copy which I have is not all that old. My old copy is gone by now (i.e., permanently misplaced or so), and so, the current copy is the one which I had bought once again, in 2009. As to my old copy, I think, I had bought it sometime in the mid-1990s.)

It was many years later, guess some time while teaching FEM to the undergraduates in Mumbai, that the concept had finally become clear enough to me. Most especially, while I was going through P. Seshu’s and J. N. Reddy’s books. [Reflected Glory Alert! Professor P. Seshu was my class-mate for a few courses at IIT Madras!] However, even then, even at that time, I remember, I still had this odd feeling that the physical meaning was still not clear to me—not as as clear as it should be. The matter eventually became “fully” clear to me only later on, while musing about the differences between the perspective of Thermodynamics on the one hand and that of Heat Transfer on the other. That was some time last year, while teaching Thermodynamics to the PG students here in Pune.

Thermodynamics deals with systems at equilibria, primarily. Yes, its methods can be extended to handle also the non-equilibrium situations. However, even then, the basis of the approach summarily lies only in the equilibrium states. Heat Transfer, on the other hand, necessarily deals with the non-equilibrium situations. Remove the temperature gradient, and there is no more heat left to speak of. There does remain the thermal energy (as a form of the internal energy), but not heat. (Remember, heat is the thermal energy in transit that appears on a system boundary.) Heat transfer necessarily requires an absence of thermal equilibrium. … Anyway, it was while teaching thermodynamics last year, and only incidentally pondering about its differences from heat transfer, that the idea of the variations (of Cov) had finally become (conceptually) clear to me. (No, CoV does not necessarily deal only with the equilibrium states; it’s just that it was while thinking about the equilibrium vs. the transient that the matter about CoV had suddenly “clicked” to me.)

In this post, let me now note down something on the concept of the variation, i.e., towards understanding the physical meaning of the symbol \delta.

Please note, I have made an inline update on 26th December 2016. It makes the presentation of the calculus of variations a bit less dumbed down. The updated portion is clearly marked as such, in the text.


The Problem Description:

The concept of variations is abstract. We would be better off considering a simple, concrete, physical situation first, and only then try to understand the meaning of this abstract concept.

Accordingly, consider a certain idealized system. See its schematic diagram below:

mechanicalengineering_1d_cov

 

 

 

 

There is a long, rigid cylinder made from some transparent material like glass. The left hand-side end of the cylinder is hermetically sealed with a rigid seal. At the other end of the cylinder, there is a friction-less piston which can be driven by some external means.

Further, there also are a couple of thin, circular, piston-like disks (D_1 and D_2) placed inside the cylinder, at some x_1 and x_2 positions along its length. These disks thus divide the cylindrical cavity into three distinct compartments. The disks are assumed to be impermeable, and fitting snugly, they in general permit no movement of gas across their plane. However, they also are assumed to be able to move without any friction.

Initially, all the three compartments are filled with a compressible fluid to the same pressure in each compartment, say 1 atm. Since all the three compartments are at the same pressure, the disks stay stationary.

Then, suppose that the piston on the extreme right end is moved, say from position P_1 to P_2. The final position P_2 may be to the left or to the right of the initial position P_1; it doesn’t matter. For the current description, however, let’s suppose that the position P_2 is to the left of P_1. The effect of the piston movement thus is to increase the pressure inside the system.

The problem is to determine the nature of the resulting displacements that the two disks undergo as measured from their respective initial positions.

There are essentially two entirely different paradigms for conducting an analysis of this problem.


The “Vector Mechanics” Paradigm:

The first paradigm is based on an approach that was put to use so successfully by Newton. Usually, it is called the paradigm of vector analysis.

In this paradigm, we focus on the fact that the forced displacement of the piston with time, x(t), may be described using some function of time that is defined over the interval lying between two instants t_i and t_f.

For example, suppose the function is:
x(t) = x_0 + v t,
where v is a constant. In other words, the motion of the piston is steady, with a constant velocity, between the initial and final instants. Since the velocity is constant, there is no acceleration over the open interval (t_i, t_f).

However, notice that before the instant t_i, the piston velocity was zero. Then, the velocity suddenly became a finite (constant) value. Therefore, if you extend the interval to include the end-instants as well, i.e., if you consider the semi-closed interval [t_i, t_f), then there is an acceleration at the instant t_i. Similarly, since the piston comes to a position of rest at t = t_f, there also is another acceleration, equal in magnitude and opposite in direction, which appears at the instant t_f.

The existence of these two instantaneous accelerations implies that jerks or pressure waves are sent through the system. We may model them as vector quantities, as impulses. [Side Exercise: Work out what happens if we consider only the open interval (t_i, t_f).]

We can now apply Newton’s 3 laws, based on the idea that shock-waves must have begun at the piston at the instant t = t_i. They must have got transmitted through the gas kept under pressure, and they must have affected the disk D_1 lying closest to the piston, thereby setting this disk into motion. This motion must have passed through the gas in the middle compartment of the system as another pulse in the pressure (generated at the disk D_1), thereby setting also the disk D_2 in a state of motion a little while later. Finally, the pulse must have got bounced off the seal on the left hand side, and in turn, come back to affect the motion of the disk D_2, and then of the disk D_1. Continuing their travels to and fro, the pulses, and hence the disks, would thus be put in a back and forth motion.

After a while, these transients would move forth and back, superpose, and some of their constituent frequencies would get cancelled out, leaving only those frequencies operative such that the three compartments are put under some kind of stationary states.

In case the gas is not ideal, there would be damping anyway, and after a sufficiently long while, the disks would move through such small displacements that we could easily ignore the ever-decreasing displacements in a limiting argument.

Thus, assume that, after an elapse of a sufficiently long time, the disks become stationary. Of course, their new positions are not the same as their original positions.

The problem thus can be modeled as basically a transient one. The state of the new equilibrium state is thus primarily seen as an effect or an end-result of a couple of transient processes which occur in the forward and backward directions. The equilibrium is seen as not a primarily existing state, but as a result of two equal and opposite transient causes.

Notice that throughout this process, Newton’s laws can be applied directly. The nature of the analysis is such that the quantities in question—viz. the displacements of the disks—always are real, i.e., they correspond to what actually is supposed to exist in the reality out there.

The (values of) displacements are real in the sense that the mathematical analysis procedure itself involves only those (values of) displacements which can actually occur in reality. The analysis does not concern itself with some other displacements that might have been possible but don’t actually occur. The analysis begins with the forced displacement condition, translates it into pressure waves, which in turn are used in order to derive the predicted displacements in the gas in the system, at each instant. Thus, at any arbitrary instant of time t > t_i (in fact, the analysis here runs for times t \gg t_f), the analysis remains concerned only with those displacements that are actually taking place at that instant.

The Method of Calculus of Variations:

The second paradigm follows the energetics program. This program was initiated by Newton himself as well as by Leibnitz. However, it was pursued vigorously not by Newton but rather by Leibnitz, and then by a series of gifted mathematicians-physicists: the Bernoulli brothers, Euler, Lagrange, Hamilton, and others. This paradigm is essentially based on the calculus of variations. The idea here is something like the following.

We do not care for a local description at all. Thus, we do not analyze the situation in terms of the local pressure pulses, their momenta/forces, etc. All that we focus on are just two sets of quantities: the initial positions of the disks, and their final positions.

For instance, focus on the disk D_1. It initially is at the position x_{1_i}. It is found, after a long elapse of time (i.e., at the next equilibrium state), to have moved to x_{1_f}. The question is: how to relate this change in x_1 on the one hand, to the displacement that the piston itself undergoes from P_{x_i} to P_{x_f}.

To analyze this question, the energetics program (i.e., the calculus of variations) adopts a seemingly strange methodology.

It begins by saying that there is nothing unique to the specific value of the position x_{1_f} as assumed by the disk D_1. The disk could have come to a halt at any other (nearby) position, e.g., at some other point x_{1_1}, or x_{1_2}, or x_{1_3}, … etc. In fact, since there are an infinity of points lying in a finite segment of line, there could have been an infinity of positions where the disk could have come to a rest, when the new equilibrium was reached.

Of course, in reality, the disk D_1 comes to a halt at none of these other positions; it comes to a halt only at x_{1_f}.

Yet, the theory says, we need to be “all-inclusive,” in a way. We need not, just for the aforementioned reason, deny a place in our analysis to these other positions. The analysis must include all such possible positions—even if they be purely hypothetical, imaginary, or unreal. What we do in the analysis, this paradigm says, is to initially include these merely hypothetical, unrealistic positions too on exactly the same footing as that enjoyed by that one position which is realistic, which is given by x_{1_f}.

Thus, we take a set of all possible positions for each disk. Then, for each such a position, we calculate the “impact” it would make on the energy of the system taken as a whole.

The energy of the system can be additively decomposed into the energies carried by each of its sub-parts. Thus, focusing on disk D_1, for each one of its possible (hypothetical) final position, we should calculate the energies carried by both its adjacent compartments. Since a change in D_1‘s position does not affect the compartment 3, we need not include it. However, for the disk D_1, we do need to include the energies carried by both the compartments 1 and 2. Similarly, for each of the possible positions occupied by the disk D_2, it should include the energies of the compartments 2 and 3, but not of 1.

At this point, to bring simplicity (and thereby better) clarity to this entire procedure, let us further assume that the possible positions of each disk forms a finite set. For instance, each disk can occupy only one of the positions that is some -5, -4, -3, -2, -1, 0, +1, +2, +3, +4 or +5 distance-units away from its initial position. Thus, a disk is not allowed to come to a rest at, say, 2.3 units; it must do so either at 2 or at 3 units. (We will thus perform the initial analysis in terms of only the integer positions, and only later on extend it to any real-valued positions.) (If you are a mechanical engineering student, suggest a suitable mechanism that can ensure only integer relative displacements.)

The change in energy E of a compartment is given by
\Delta E = P A \Delta x,
where P is the pressure, A is the cross-sectional area of the cylinder, and \Delta x is the change in the length of the compartment.

Now, observe that the energy of the middle compartment depends on the relative distance between the two disks lying on its sides. Yet, for the same reason, the energy of the middle compartment does depend on both these positions. Hence, we must take a Cartesian product of the relative displacements undergone by both the disks, and only then calculate the system energy for each such a permutation (i.e. the ordered pair) of their positions. Let us go over the details of the Cartesian product.

The Cartesian product of the two positions may be stated as a row-by-row listing of ordered pairs of the relative positions of D_1 and D_2, e.g., as follows: the ordered pair (-5, +2) means that the disk D_1 is 5 units to the left of its initial position, and the disk D_2 is +2 units to the right of its initial position. Since each of the two positions forming an ordered pair can range over any of the above-mentioned 11 number of different values, there are, in all, 11 \times 11 = 121 number of such possible ordered pairs in the Cartesian product.

For each one of these 121 different pairs, we use the above-given formula to determine what the energy of each compartment is like. Then, we add the three energies (of the three compartments) together to get the value of the energy of the system as a whole.

In short, we get a set of 121 possible values for the energy of the system.

You must have noticed that we have admitted every possible permutation into analysis—all the 121 number of them.

Of course, out of all these 121 number of permutations of positions, it should turn out that 120 number of them have to be discarded because they would be merely hypothetical, i.e. unreal. That, in turn, is because, the relative positions of the disks contained in one and only one ordered pair would actually correspond to the final, equilibrium position. After all, if you conduct this experiment in reality, you would always get a very definite pair of the disk-positions, and it this same pair of relative positions that would be observed every time you conducted the experiment (for the same piston displacement). Real experiments are reproducible, and give rise to the same, unique result. (Even if the system were to be probabilistic, it would have to give rise to an exactly identical probability distribution function.) It can’t be this result today and that result tomorrow, or this result in this lab and that result in some other lab. That simply isn’t science.

Thus, out of all those 121 different ordered-pairs, one and only one ordered-pair would actually correspond to reality; the rest all would be merely hypothetical.

The question now is, which particular pair corresponds to reality, and which ones are unreal. How to tell the real from the unreal. That is the question.

Here, the variational principle says that the pair of relative positions that actually occurs in reality carries a certain definite, distinguishing attribute.

The system-energy calculated for this pair (of relative displacements) happens to carry the lowest magnitude from among all possible 121 number of pairs. In other words, any hypothetical or unreal pair has a higher amount of system energy associated with it. (If two pairs give rise to the same lowest value, both would be equally likely to occur. However, that is not what provably happens in the current example, so let us leave this kind of a “degeneracy” aside for the purposes of this post.)

(The update on 26 December 2016 begins here:)

Actually, the description  given in the immediately preceding paragraph was a bit too dumbed down. The variational principle is more subtle than that. Explaining it makes this post even longer, but let me give it a shot anyway, at least today.

To follow the actual idea of the variational principle (in a not dumbed-down manner), the procedure you have to follow is this.

First, make a table of all possible relative-position pairs, and their associated energies. The table has the following columns: a relative-position pair, the associated energy E as calculated above, and one more column which for the time being would be empty. The table may look something like what the following (partial) listing shows:

(0,0) -> say, 115 Joules
(-1,0) -> say, 101 Joules
(-2,0) -> say, 110 Joules

(2,2) -> say, 102 Joules
(2,3) -> say, 100 Joules
(2,4) -> say, 101 Joules
(2,5) -> say, 120 Joules

(5,0) -> say, 135 Joules

(5,5) -> say 117 Joules.

Having created this table (of 121 rows), you then pick each row one by and one, and for the picked up n-th row, you ask a question: What all other row(s) from this table have their relative distance pairs such that these pairs lie closest to the relative distance pair of this given row. Let me illustrate this question with a concrete example. Consider the row which has the relative-distance pair given as (2,3). Then, the relative distance pairs closest to this one would be obtained by adding or subtracting a distance of 1 to each in the pair. Thus, the relative distance pairs closest to this one would be: (3,3), (1,3), (2,4), and (2,2). So, you have to pick up those rows which have these four entries in the relative-distance pairs column. Each of these four pairs represents a variation \delta on the chosen state, viz. the state (2,3).

In symbolic terms, suppose for the n-th row being considered, the rows closest to it in terms of the differences in their relative distance pairs, are the a-th, b-th, c-th and d-th rows. (Notice that the rows which are closest to a given row in this sense, would not necessarily be found listed just above or below that given row, because the scheme followed while creating the list or the vector that is the table would not necessarily honor the closest-lying criterion (which necessarily involves two numbers)—not at least for all rows in the table.

OK. Then, in the next step, you find the differences in the energies of the n-th row from each of these closest rows, viz., the a-th, b-th, c-th and c-th rows. That is to say, you find the absolute magnitudes of the energy differences. Let us denote these magnitudes as: \delta E_{na} = |E_n - E_a|\delta E_{nb} = |E_n - E_b|\delta E_{nc} = |E_n - E_c| and \delta E_{nd} = |E_n - E_d|.  Suppose the minimum among these values is \delta E_{nc}. So, against the n-th row, in the last column of the table, you write the value \delta E_{nc}.

Having done this exercise separately for each row in the table, you then ask: Which row has the smallest entry in the last column (the one for \delta E), and you pick that up. That is the distinguished (or the physically occurring) state.

In other words, the variational principle asks you to select not the row with the lowest absolute value of energy, but that row which shows the smallest difference of energy from one of its closest neighbours—and these closest neighbours are to be selected according to the differences in each number appearing in the relative-distance pair, and not according to the vertical place of rows in the tabular listing. (It so turns out that in this example, the row thus selected following both criteria—lowest energy as well as lowest variation in energy—are identical, though it would not necessarily always be the case. In short, we can’t always get away with the first, too dumbed down, version.)

Thus, the variational principle is about that change in the relative positions for which the corresponding change in the energy vanishes (or has the minimum possible absolute magnitude, in case the positions form a discretely varying, finite set).

(The update on 26th December 2016 gets over here.)

And, it turns out that this approach, too, is indeed able to perfectly predict the final disk-positions—precisely as they actually are observed in reality.

If you allow a continuum of positions (instead of the discrete set of only the 11 number of different final positions for one disk, or 121 number of ordered pairs), then instead of taking a Cartesian product of positions, what you have to do is take into account a tensor product of the position functions. The maths involved is a little more advanced, but the underlying algebraic structure—and the predictive principle which is fundamentally involved in the procedure—remains essentially the same. This principle—the variational principle—says:

Among all possible variations in the system configurations, that system configuration corresponds to reality which has the least variation in energy associated with it.

(This is a very rough statement, but it will do for this post and for a general audience. In particular, we don’t look into the issues of what constitute the kinematically admissible constraints, why the configurations must satisfy the field boundary conditions, the idea of the stationarity vs. of a minimum or a maximum, i.e., the issue of convexity-vs.-concavity, etc. The purpose of this post—and our example here—are both simple enough that we need not get into the whole she-bang of the variational theory as such.)

Notice that in this second paradigm, (i) we did not restrict the analysis to only those quantities that are actually taking place in reality; we also included a host (possibly an infinity) of purely hypothetical combinations of quantities too; (ii) we worked with energy, a scalar quantity, rather than with momentum, a vector quantity; and finally, (iii) in the variational method, we didn’t bother about the local details. We took into account the displacements of the disks, but not any displacement at any other point, say in the gas. We did not look into presence or absence of a pulse at one point in the gas as contrasted from any other point in it. In short, we did not discuss the details local to the system either in space or in time. We did not follow the system evolution, at all—not at least in a detailed, local way. If we were to do that, we would be concerned about what happens in the system at the instants and at spatial points other than the initial and final disk positions. Instead, we looked only at a global property—viz. the energy—whether at the sub-system level of the individual compartments, or at the level of the overall system.


The Two Paradigms Contrasted from Each Other:

If we were to follow Newton’s method, it would be impossible—impossible in principle—to be able to predict the final disk positions unless all their motions over all the intermediate transient dynamics (occurring over each moment of time and at each place of the system) were not be traced. Newton’s (or vectorial) method would require us to follow all the details of the entire evolution of all parts of the system at each point on its evolution path. In the variational approach, the latter is not of any primary concern.

Yet, in following the energetics program, we are able to predict the final disk positions. We are able to do that without worrying about what all happened before the equilibrium gets established. We remain concerned only with certain global quantities (here, system-energy) at each of the hypothetical positions.

The upside of the energetics program, as just noted, is that we don’t have to look into every detail at every stage of the entire transient dynamics.

Its downside is that we are able to talk only of the differences between certain isolated (hypothetical) configurations or states. The formalism is unable to say anything at all about any of the intermediate states—even if these do actually occur in reality. This is a very, very important point to keep in mind.


The Question:

Now, the question with which we began this post. Namely, what does the delta of the variational calculus mean?

Referring to the above discussion, note that the delta of the variational calculus is, here, nothing but a change in the position-pair, and also the corresponding change in the energy.

Thus, in the above example, the difference of the state (2,3) from the other close states such as (3,3), (1,3), (2,4), and (2,2) represents a variation in the system configuration (or state), and for each such a variation in the system configuration (or state), there is a corresponding variation in the energy \delta E_{ni} of the system. That is what the delta refers to, in this example.

Now, with all this discussion and clarification, would it be possible for you to clearly state what the physical meaning of the delta is? To what precisely does the concept refer? How does the variation in energy \delta E differ from both the finite changes (\Delta E) as well as the infinitesimal changes (\text{d}E) of the usual calculus?


Note, the question is conceptual in nature. And, no, not a single one of the very best books on classical mechanics manages to give a very succinct and accurate answer to it. Not even Rana and Joag (or Goldstein, or Feynman, or…)

I will give my answer in my next post, next year. I will also try to apply it to a couple of more interesting (and somewhat more complicated) physical situations—one from engineering sciences, and another from quantum mechanics!

In the meanwhile, think about it—the delta—the concept itself, its (conceptual) meaning. (If you already know the calculus of variations, note that in my above write-up, I have already supplied the answer, in a way. You just have to think a bit about it, that’s all!)


An Important Note: Do bring this post to the notice of the Officially Approved Full Professors of Mechanical Engineering in SPPU, and the SPPU authorities. I would like to know if the former would be able to state the meaning—at least now that I have already given the necessary context in such great detail.

Ditto, to the Officially Approved Full Professors of Mechanical Engineering at COEP, esp. D. W. Pande, and others like them.

After all, this topic—Lagrangian mechanics—is at the core of Mechanical Engineering, even they would agree. In fact, it comes from a subject that is not taught to the metallurgical engineers, viz., the topic of Theory of Machines. But it is taught to the Mechanical Engineers. That’s why, they should be able to crack it, in no time.

(Let me continue to be honest. I do not expect them to be able to crack it. But I do wish to know if they are able at least to give a try that is good enough!)


Even though I am jobless (and also nearly bank balance-less, and also cashless), what the hell! …

…Season’s greetings and best wishes for a happy new year!


A Song I Like:

[With jobless-ness and all, my mood isn’t likely to stay this upbeat, but anyway, while it lasts, listen to this song… And, yes, this song is like, it’s like, slightly more than 60 years old!]

(Hindi) “yeh raat bhigee bhigee”
Music: Shankar-Jaikishan
Singers: Manna De and Lata Mangeshkar
Lyrics: Shailendra


[E&OE]

My planning for the upcoming summer vacation

0. Yes, I have deleted my previous post. As I took a second look at it, I thought it was a bit too on-the-fly, and perhaps not worth keeping. (It was about these Lok Sabha elections!) Though I have deleted it, if the need be, I will write a better post touching on the same topic, including my further thoughts about the matter.

For the time being, let me get back to engineering.

* * *

As this academic term nears its end, I have already begun planning for things to do this summer vacation. A few things are on the top of my mind. Let me jot down these, so that I could look back a couple of months hence and see how I did on those matters (or, how the matters turned out anyway).

1. Journal papers on my past research: I need to convert at least one or two of my conference papers into journal papers. This is really on the top of the list because I haven’t had a journal publication during my Ph.D. The reason for that, in turn, wasn’t that my research wasn’t worth publishing in journals. In fact, not to immediately publish in journals was a deliberate choice, which was decided after discussion with my guide, the late Prof. S. R. Kajale.

The reason was twofold: (i) Journal papers tend to undergo a more thorough peer-review, and even if not, in any case, are longer. Since I am naturally so talkative (in a way almost carefree), I was afraid whether I might not end up giving out too many details if it is a journal paper, and at that time (mid-naughties) as now, IPR was (and is) an important consideration. (ii) I didn’t have very good library (eJournals) access back then. I was jobless, would take trips to IIT Bombay for literature review, and both money and the time to go through eJournals was very severely limited (a few hours on one or two days at the most, at a time).

The situation has changed since. I now do have a job in hand, and in fact, I now work in Mumbai. So, more frequent trips to the IIT Bombay library for a longer period of literature review is an easier possibility.

Anyway, the above two reasons are not independent; they are inter-related. As it turns out, I learnt after publishing my conference papers, that an approach very close to what I had taken, had already been developed to much more extent than I was aware of, back then. The method in question is: LBM (the lattice Boltzmann method.) LBM, as some of you might know, has since my PhD times been commercialized, with at least two commercial software packages and at least one Open Source + consulting model software having come on the scene. (And, thus, it turns out that the prudence in withholding details was right—there was commercial value to those ideas, even if it turns out that I was not the first to think of them. (Of course, since I honestly can say that I developed my approach fully independently, there happen to be a few (relatively minor) ideas which I had, and which still haven’t been published.))

Another thing. I have derived greater confidence about the new observation that I had made regarding the diffusion equation. This could come about only after a better literature search.

All in all, I think I am ready to write my journal paper on the diffusion equation now.

2. Journal papers on some more recent ideas: Since my PhD (2009), I also had a few extended abstracts accepted at international conferences (some 4 papers in 3 different conferences), but for some reason or the other, I had to withdraw. (Lack of time, or lack of money to complete the experimental part.) I could begin directly writing journal papers on these ideas now.

3. Short-term vacation courses: I am also proposing to conduct a couple of short-term courses on FEM and CFD.

3.1 On FEM: By now, I have taught introductory courses on FEM 4 times: twice to UG, once to PG, and once to practising engineers. I have enjoyed teaching my latest offering this semester. Since the syllabus at the University of Mumbai was different, there was an opportunity for me to look at FEM from a different perspective than what I had taken. I think I could now synthesize my understanding in a (really) improved (if not “new”) short-term course.

So, I am planning to offer a short-term course of about 7–10 days duration. The audience could be any graduate engineer: (i) PG students, (ii) working engineers, (iii) junior faculty from engineering colleges.

3.2 A novel course on CFD: Another course which I have never taught but which I am deeply interested in, is, of course, CFD. So, I am planning to offer a special vacation-time and short-term course on that topic, too.

Ideally, I would like to keep this course more for those who are interested in deeper insights, via self-study. If there are enough people interested in such a course, then I would rather like to keep the number of topics few, and the focus more on the fundamentals.

Of course, fewer topics doesn’t mean less material. Indeed, in many ways, my planned CFD short-term course would have much more material than a traditional one.

I would be ready cover all three methods side by side: FDM, FVM, and FEM—provided the audience already knows FEM in the context of the usual linear structural (or self-adjoint) kind of problems.

Similarly, in my course, I would like to include at least conceptual introductions to what are considered to be “advanced” topics like moving boundary problems, multiphase (VOF) problems, etc.

Thus, my planned CFD course wouldn’t be tied to (or, actually, be subservient to the needs of) only the aerodynamics problems of the aerospace department. It could easily apply to issues like free-surface flows and cavity-filling issues (if not also droplet formation/interaction—which could perhaps be covered, though I am not sure. (It would have been easier to cover if LBM were to be a part of the course offering, but I guess for an introductory/first course that is also short-term, introducing all the main continuum-based methods of FDM, FVM and FEM is a challenge by itself. No need to complicate it further by also introducing a particles-based approaches like LBM/SPH.)

4. More about the above short-term vacation courses:

4.1 My current view is that for a one week course, 4 hours of class-room teaching in the morning and 1–2 hours of hands on sessions in the afternoon for 3–4 days, will be enough.

4.2. The fees will be reasonable, by today’s market standards (though not just a few hundred rupees, if that’s what I understand by the word “reasonable.”). Since I do have a professor’s job, I am not looking at these courses as my primary career. The fees mainly have to cover the course organization expenses, most of which are beyond my control. On my part, an honorarium sort of payment also would be OK by me—strictly because, to repeat, I do have a continuing job that does pay me now.  That’s why. And, the course-fees do stand to drop if the audience is bigger, though I plan not to take more than 25–30 students per course.

4.3. So there. Drop me a line if you are from Mumbai and are interested in attending one of these courses this summer vacation.

Yet, some final clarifications still are due:

4.4 The courses will not follow the syllabus of any university. Drop me a line or follow this blog if you wish to know the details of the course contents. But, essentially, these are not your usual vacation-time coaching classes.

(There! Right there I kill my entire potential market of student-customers.)

4.5 No software package will at all be covered. If you wish to learn, say, ANSYS, or Fluent, there are numerous vendors out there. For OpenFOAM, there is a group in IIT Bombay, and a company in Pune. Contact them directly. (And no, I don’t even know who are better, or just more reputable, among them. (As far as I am concerned neither ANSYS nor Fluent nor OpenFOAM nor ESI gave me a job even if I was competent, when I was most desparate. Now, I couldn’t care less for them bastards. (And, in a class-room, I usually am far more cultured and civilized than expressions of that sort.))) In my course, I may use some programs written by me in C++ or Python or so. (No, Java continues to be a “no” as far as I am concerned!) But no training on software packages as such.

(There! Right there I kill my entire potential market of working engineers looking for in-house company trainings!)

Alright. More, later. [Of course, as in the recent past, my blogging will continue to remain rather infrequent. But what I mean to say here is that once the ideas of the short-term courses take a more concrete form, I will sure write another blog post to give you those details.]

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A Song I Like:
(Hindi) “chandaa ki kiranon se liptee hawaayen”
Singer: Kishore Kumar
Music: Chitragupt
Lyrics: Verma Malik

[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:
(Marathi) “yaa bakuLichyaa zaaDaakhaali…”
Singer: Sushma Shreshtha
Lyrics: Vasant Bapat
Music: Bhanukant Luktuke

[E&OE]

 

Part-time Teaching, Again…

Recently, I have accepted the position of a Visiting Professor at the Symbiosis Institute of Technology, Lavale, Pune, and have just begun teaching an introductory course on Finite Element Analysis to the students of MTech (Mechanical — CAD/CAM/CAE specialization) there, on a part-time basis.

The classes are held in the CAD/CAM/CAE lab., wherein they have everything one would have liked to see: an audio projector hooked to a PC + a networked PC for every student right in the class-room + no acoustic reverbaration problem in the class-room at all + timings convenient to me (OK to conduct classes on weekends and all). It’s a small class (< 20 students), and though it might be too early to form a judgment, the students do seem to be mature and responsive. (A couple of them have even worked in industry for a year or two.) I think I am going to enjoy teaching this course, even though my interest in FEM has been rapidly sliding down in the recent past—I am getting into the CFD field now.

FEM would, of course, continue to remain a matter of interest. Yet, the primary interest for the rest of my career, I have now firmed up my mind, now on is going to be: CFD. CFD using FVM and FDM and FEM, sure,  but what I especially have in mind are some of the more recent particles-based approaches like LBM (the lattice-Boltzmann method) and SPH (the smoothed particle hydrodynamics). Especially, LBM.

Both LBM and SPH are the particles-based approaches that are essentially (i) local, (ii) transient, and (iii) spatially cascading and temporally propagating in nature—so close to the very core of how, I realized only recently, I have always had approached understanding nature.  So much so that I have this “it’s me—it’s my way of doing things” feeling about it.

Indeed, my first memories of wanting a transient, local and propagating description goes as far back as at least the X/XI standard, right at the time when I was being taught Newton’s laws and calculus for the first time, and, unlike many other smart guys, had found the theoretical apparatus wanting when it came to exactly and directly capturing the physical reality the way I wished to. I didn’t have the words back then, though I would describe my differences animatedly. Now, I know the words, too. Things like: micro-dynamics, local, transient, cascading, propagating, feedbacks, emergent, etc. Back then, I didn’t have the words but still found the existing formalism wanting when it came to applying it to even simplest cases like the carrom-striker’s rebound, or the hitting of a ball by a bat. More on this all, again, later! (Do remind me!!)

Enough to say here, and for the time being, that the approach I developed during my PhD—an approach that is local and transient and propagating in nature—was a development that was very, very natural to me—my way of thinking. That way, I have always had fascination for fluids. Yet, comparatively, that’s secondary. The primary thing is the nature of this local and transient approach itself—something that can only become feasible when you are doing computational modeling, not otherwise.

And, so, it simply is marvelous for me to now find that there are these methods that are not only fairly well developed, but also capable of addressing questions of practically important engineering situations. Here, do realize that MSC Software has just last year entered into a partnership/marketing arrangement for the Next Limit’s LBM-based product: XFlow. A fortuitous circumstance, all in all. And, not only that, but also something more, further: given my approach to understanding physical reality in general, and my novel approach in QM in particular, LBM should also allow me to more easily span my interests from engineering to QM (or vice versa). That is a major attraction for me.

Anyway, that’s rather for the long-term future. Coming back to this course on FEM I am right now teaching.

I think I am going to make a departure from using C++. Instead, I am going to try using SciLab right from the beginning of this course. Since the course is the very first one on FEA for most students, I have little choice when it comes to the contents. For the most part, it’s going to be the usual linear and static FEA with simple isoparametric elements. However, I might try to introduce an exposure to diverse topics via assignment of student-delivered seminars. Such topics might include (all at a simple level): nonlinear FEA (as in fluids), plate and shell elements, FEA for fracture mechanics, FEA for simple coupled problems, etc. … It might also be a good idea to introduce a free (free of cost) FE software in this course. Though I have not yet finalized my mind, I am thinking of choosing Elmer.

Also, when it comes to evaluation, I might try out different approaches like open-book examinations, etc.  Otherwise, for a topic like FEM, it can be excruciatingly boring to do all those calculations by hand. Not only that, the traditional examination format restricts you to only very simple, almost artificial sort of problems. People’s grades get determined to 50% extent or more by their skills in doing only simple truss and frame analysis using FEM, simply because no better questions can at all be posed. That, in turn, is because any better question would require more than 3 hours of examination to solve by hand. If you are going to conduct examination in 3 hours, then, probing for the things that really matter, goes out the window, and it becomes a testing for learning by the rote. Instead, I would like to see if I can allow them to use SciLab and refer to openly accessible codes, right during their formal written examinations. If things turn out OK, I might even consider assigning them more ambitious projects via open-book examination mode, for the final examination. Let’s see how things progress. … After all, it’s not just the teacher but also the students—they, too, have to grow comfortable with such ideas, including the idea that their grades may get determined, at least in part, via some approach with which they have had no prior experience… And, the university administration and management also has to be satisfied that sufficient quality was maintained during the evaluation procedures, too. That’s why, I will have to see how things progress…

And, yes, I am still very much on the look out for my primary job. Don’t forget that part.

[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]

Wanted: Fast FEA Solvers…

Summary:

I am thinking of informally conducting a specific case-study concerning the FEA solvers. The reference problem is a very simple but typical problem from stress analysis, leading of course to the linear systems: Ax = b and Ax = Lx.

I seek advise as to what software libraries currently available in the public domain would be best to use—the ones that would be fastest in terms of execution time for the reference problem.

I have a personal and longer-term research interest with certain issues related to the solvers technologies.

Suggestions and comments are welcome!

(1.) The Reference Problem:

(1.1) Consider a homogeneous thin rectangular plate made of MS, say of the size 200 mm X 100 mm, with a thickness of, say, 1 mm.

For the initial requirement, the plate carries no hole, though a small 60 mm dia. hole at the center might be introduced later on, during a separate phase of this study.

(1.2) For static analysis, the plate is loaded with a uniform traction acting on the two shorter sides of the plate, whereas the longer sides are kept free. For modal frequency analysis, the plate is considered clamped on all the four sides.

(1.3) Simple, standard finite elements are to be used: (a) CST and LST for the static analysis, and (b) DKT flat-shell element for the modal analysis.

(1.4) The domain is to be meshed using high-quality irregular triangles, the smallest allowed angle being ~34 degrees as in Shewchuk’s Triangle library [^] or Niceno’s EasyMesh [^].

To obtain a medium-fine mesh, the triangle side may be restricted to < 5 mm. This choice leads to about 2,500 triangles, 1,200 corner nodes, and 4,000 edges—i.e. about 1,200 nodes for CST analysis and 5,200 nodes for LST analysis.

However, if the upper bound on the triangle side is halved (< 2.5 mm), then we obtain a very fine mesh of about 10,000 triangles, 5,000 corner nodes, and 15,000 edges—i.e. about 5,000 nodes for CST and 20,000 nodes for LST.

Note that these numbers refer to the geometry nodes. In the FE model, each such a node would carry several DOFs.

(1.5) The linear systems resulting after the FE-discretization are to be solved for both static and modal analyses.

(2.) The Software/Hardware to be used:

(2.1) The linear system is to be solved using C/C++ callable and fairly well-tested open-source libraries (libraries of the kind: LAPACK, ARPACK, Taucs, etc.).

(2.2) The library itself might have been written in FORTRAN; the only requirement is that compiled binaries and C/C++ wrappers should be readily available.

(2.3) Dependencies on open-source libraries/platforms such as GoToBlas, Boost, MTL, etc. are OK.

(2.4) Assume this (lower-end) software-hardware platform: A single 32-bit desktop PC, Intel Core2 Duo @ ~3 Ghz main clock, 1 MB L2 cache, 2 GB of RAM. Assume the OS to be Windows 2K/XP.

(2.5) The compiler of preference is VC++ 6. However, other free compilers like VC++ Express Edition 2008 can be considered. Also, I am open to using GCC or other compilers, with or without their CMake, MinGW requirements etc.

(2.6) The sequential mode execution is assumed. No parallel processing, whether using shared memory, clusters (MPI), or GPUs. For the same reason, it’s OK if the solver library is not parallel processing-enabled, and does not take advantage of an additional core. Thus, for this study, it is OK even if the total CPU usage on a double-core machine doesn’t exceed 50%.

(2.7) All the solver operations are expected to occur in-core (not out-of-core).

(2.8) Assume that all mathematical operations would be peformed in double precision (8 bytes).

(3.) What Is Being Sought:

(3.1) Considering the above requirements, please suggest the libraries and methods that might provide the highest performance (the least execution time) for the following categories of solvers:
— direct solver for static analysis (Ax = b)
— iterative solver for static analysis (Ax = b)
— direct solver for eigenvalues computations (Ax = Lx)
— iterative solver for eigenvalues computations (Ax = Lx)

For iterative solvers, assume the usual kind of convergence requirements (error norms).

(3.2) The total execution time is to be measured (a) from the tick that the reading of all the disk files containing all the input matrices to RAM is complete, (b) to the tick that the solution is first fully ready in RAM, waiting to be written to the output disk files.

(3.3) Please provide any additional information like the assumption of a specific pre-conditioner, the reason why you recommend a particular algorithm for this type of problem, etc.

(3.4) Not very important right now, but any side suggestions you might have for nonsymmetric A matrices would also be welcome.

(3.5) A general point of reference for this query is this URL:
http://www.netlib.org/utk/people/JackDongarra/la-sw.html

(4.) Why This Study:

The purpose is something like this. I have some preliminary ideas concerning solvers.

I would like to test my ideas against the available state of the art/cutting-edge solver implementations, in the context of the above kind of applications—viz. that the K matrix wouldn’t be tridiagonal but would be banded SPD, having a topology implied by the above category of problems.

It’s easily possible that my ideas may not work out. I wish to put them to the testing ground anyway. (I really am just at a very preliminary stage.)

(5.) Your Suggestions/References:

Well thought-out comments/suggestions w.r.t the point (3.1) are sought.

Since I am not affiliated to any institution having e-Journals access, in case you provide links to research papers, I would greatly appreciate if you could also send e-copies to me by email: aj175tp[ at ]yahoo[ dot ]co[ dot ]in.

Thanks in advance!

PS: Posted also at iMechanica [^]

–  –  –  –  –
A Song I Like

(Hindi/Urdu) “woh nahin milataa mujhe…”
Singer: Chitra Singh
Music: Jagjit Singh
Lyrics: [?]

[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…)

– – – – –

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!!]

A Rapid Update and (Equally Rapid) Comments…

There has been a flurry of activities… I barely find the time to list them here…

1. I have conducted a 7-day course on FEM for a group of about 20-25 working engineers. The trainees were a mix of both highly experienced engineers (with 2 to 3 decades of work experience) and 4–5 IIT trained MTechs. All of them came from a couple of government organizations active in civil engineering design and research.  The course, though nominally meant for 7 days, actually ran into almost 10 calender days. It was a big success. … In conducting this course, a very senior faculty-member from IIT Bombay had also very graciously joined me for a couple of days. (I esp. appreciated it because, these days, normally speaking, I find IIT Bombay hateworthy—for a very good set of reasons.) The course happened in end-May—early June. More on it all, later…

Both these organizations were government organizations. I am still looking to receive my paycheck. However, being government organizations, it is guaranteed, in a way, that the check will certainly arrive some day… (How I wish the organizations were not being run by the government!)

2. About my earlier undergraduate course on FEM at COEP. I finished teaching it. And, also grading the students for their performance… There are times when one wishes god existed so that he could be on one’s side in performing tasks like these—I mean, grading… In the end, one makes as best choices as possible, though!

This course, too, was a wonderful experience for me, and, if informal student reactions is anything to go by, it too was a great success.

The students themselves took a lot of interest… There was a query, rather, a couple of them, right at the beginning of the course, which had caught me not just “unprepared” but actually “ignorant.” … Somehow, I had always associated the word “ignorance” with the word “disease” (not to mention “darkness” etc.)… Little did I know that the same word could be associated, in a way, with “joy,” too!

Anyway, despite such brilliant querries, I had enough of a “teacher” in me to sail smoothly through the course… More on those queries and all, later. (I will surely share them, but later on… You know, this is supposed to be a real *rapid* update.)

3.  More serious. All concerning Congress (I) and Times of India… (If you know me, you expect this off me.)

Kapil Sibbal, my favorite debater on TV (and if I let my emotions interfere, more favorite than his enemy Arun Jaitley), has recently become HRD Cabinet minister. Our PM ManMohan Singh, the Cambridge graduate, has a way of learning, albeit late—he should have removed Arjun Singh long time back.

Immediately after assuming his charge, Kapil has done something about 10th standard examinations. … Now, I do have a lot to say against exams and ranks; e.g., see my informal writing on my Web site (and also the earlier entries on this blogs)… Yet, this decision left me, say, wondering.

BTW, why don’t I see a single article from Ramchandra Guha (of Bangalore) or Prof. Dipankar Gupta (of JNU) on this topic—whether the 10th board examinations are to be outright canclled or not? Or, from Gurcharan Das (yet another Harvard fellow to Kapil, apart from PC). Or, others…

My thoughts, once these two (or others) share theirs…

4. I have joined, part-time, with a Pune-based firm, a software producer in the Civil Engg. Design field, as a Consultant in software development. The domain is CAE. … Ashutosh Parasnis of PTC, Most all at Geometric Software (and sister companies like 3D PLM), MSC Sofy, and all others like them ought to find this particular development offensive. (Or, very offensive.)

I am happy about it. … And, about my work. (It does take a lot of my energy though…)

The time of transition is a time of feeling whether one is missing something… Others (many of them actually idiots) may call it a time of opportunity, a time of excitement, a time to be prudish, and so on… But if you are like me, you not only get excited and try to make best of your opportunities but you also tend to grow apprehensive—about the direction in which all the development goes… With my first corporate training program in the CAE field already delivered, and now with this opportunity, these sure are times of transition for me… I have waited long for things like these to materialize.

These opportunities have come after going without a job for 6–7 years, after running my own Web site for 3-4 years, and after running my blog at the Harvard-based iMechanica.org for roughly two years. Clearly, lack of information (including that found on the Internet) couldn’t have been the cause why I didn’t get such opportunities before… Clearly, the reasons had to do with politics—including international politics.

And that’s why I am worried as to what game of international politics I am being subjected to… Why should I get encouraged, by the world (the bold-letters is not an accident), to do in-depth research in Civil, but not in Mechanical…

Hey, Ashutosh (Parasnis, of PTC, working under a lot of BA types in USA), do you have answers? I want to ask the same question to yet another Harvard graduate running Geometric Software–do you have any idea why I was going without job for all these years even when you kept paying Brahmins and Reserved Category alike for all these years? Was the word “competence” ever a part of your processes?

5. That brings me to one more item of news that I consider as nothing but positive and encouraging for me… Mr. Narendra Jadhav has finally given up the much coveted position of being the V/C of University of Pune. Yes, he is gone! Finally!! But not before awarding himself (if the printed rumour is evidence to go by), 10/10 points for his grandiosly poor performance on this particular job.

Hmmm….

(Sometimes, Naryaa, one doesn’t even have the energy left to LOL! But one wishes to!! Honestly!!!)

Let me get back to the business of living my (difficult to live (on several counts)) life though!

Pandit (i.e. Mr. Pandit Vidyasagar), since you were mentioned this morning in ToI, for selection in some committee etc., let me make this public. (And I have already let you know of my feelings—no matter what consequences.)

I think you will make for a very poor V/C. Of, University of Pune—as poor as Mr. Narendra Jadhav was. (One has read about how the Mahamahopadhyay ran the University…)  (Cost? I could give up my PhD degree, though, I already know, this isn’t going to be the case even if I do oppose incompetent Pandit’s nomination/application.) He may add “feathers” to his (possibly existing) cap by being committee member here and there… But he should not be made the V/C… As a student (still) of University of Pune, this is what I wanted to say—and let the (rest of) the world take notice.

6. I also attended a funny interview on being a teacher of Mechanical Engineering in University of Pune… The interviewers did not bother introducing them (I was the first person to be interviewed), and when I enquired about them, they over-emphasized the title “Dr.” in front of their names… I mean, “Dr.” Jain of Padmashree DY Patil College of Engineering in Pune, and also, one “Dr.” Ghanegaokar… That particular stress on having got a PhD was the funniest thing in that interview… If the title was supposed to generate respect in me, exactly the opposite happened.. LOL! (Mr. Jain, Mr. Ghanegaokar, I do hope that you do read this.) … As happens in such interviews, neither of them never ever came to considering (or questioning)  my ability to teach technical mechanical engineering subjects. Yet, they both were insistent on mentioning that there were some “technical” difficulties in hiring me as a teacher of Mechanical Engineering in University of Pune.

Mr. Jain, and Mr. Ghanegaokar, I pity your interviewing skills. And, in response to your emphasizing your doctorate degrees, I must say, I also pity your creativity in the engineering field…  Here, I am not exceeding my limits… I am willing to exchange our respective PhD theses, just for personal reading. I am sure you, too, will come to form the same judgment, no matter in how implicit and unacknowledged terms. And, even without inviting comparisons of that sort, one could always raise the point: Why be so bureaucratic in education, Mr. Jain? Mr. Ghanegaokar? Don’t you think you pull the standards of education down when you engage in that kind of a mindless conformance to the mindless bureaucracy which informs today’s University of Pune? You evidently conform to its mindless norms—with more than a shade of authoritativeness coming forth off you. That is, even while interviewing someone like me. Isn’t it hight time someone pulled you up for that? And nothing in this is personal… The same applies to anyone else like you—anyone else who serves only to extend the mindlessness of the University of Pune.

The fact of the matter is, there are intellectual pygmies staffing the various private engineering colleges (not to mention also the government colleges, but to a somewhat lesser extent), and they all form a closed system, and feel threatened by open talent and merit. That’s what has become of today’s engineering education under University of Pune.

Yeah! Go ahead!! Delay my PhD even further you [expletives not written not even the very first version so that the issue of their deletion does not arise, but a note must be made that they would apply most fittingly here].

7. With that said, if I still get a teacher’s (i.e. an engineering professor’s) post this season, consider it an “Allah Ki Marzi,” “Will of God,” “Devaachi Ichhaa” etc. Wouldn’t that be right, Barak—the first one?

And, why, come to think of it, since it’s July the 4th today, … why is it that Americans don’t call Barak Obama by his first name, or affectionately call him, say, BHO (like “JFK” or “Abbey” or whatever) but instead prefer to mention him by his last name: “Obama”? What gives?

Anyway, I was not thrilled when he was running for Presidency, and I don’t find him very interesting today either. It’s between him and Americans—what to call him. … One just wonders the moral distance between the American presidents of the late-20th/early 21st centuries and the Founding Fathers, that’s all…

8. All this flurry of activity of mine is OK—it gets me money, in the field I have fought years to get in—namely, CAE… But, for the time being the researcher within me is yearning to get out and get going… I don’t find any time at all for my FAQ-related research… There are so many ideas I have in there…

Oh, well… Some other time.

– – – – –

PS: I have already replaced my initial “brain-storm” version for this post with a better written one, and may be I will streamline this present version too, once again in a couple of days’ time… Also, I need to upload some thoughts that I had written each time I finished teaching FEM—both at COEP and as a corporate trainer…. More on this all, later. Hopefully, soon enough…

Something on Betterment of Syllabii at COEP + My Joblessness

The FEM course that I am teaching at COEP is now drawing to a close. The time is, therefore, conducive to take a review; a time to reflect upon what went wrong, how things could have been done better, etc. Naturally, this thinking mode also spills over into other things, and that is what I am going to note down here today.

I believe that COEP should take a major initiative into multidisciplinary areas of computational physics, computational mechanics, computational and engineering. In short, computational science and engineering (CSE).

I mean it in a much deeper sense than just saying that COEP should hire me with all due respect to my background and work, and allow me to pursue my research in CSE without any hassles. The issue is not about my career, primarily (though this aspect, too, is relevant). In fact, the issue is not even about introducing researches in the multidisciplinary area of CSE at the PG level…

What I am actually advocating here is a fairly major revamp in COEP syllabi, and that too, right at the UG level—a basic change right at the very course-structure level. Let me indicate in brief what I mean by that.

Every academic institution (of some note) has a certain unique set of historical, cultural, intellectual, environmental (including geographical) factors which together decide what course of action would be best for it to pursue. Here, given certain special factors specific to Pune in the recent times, I believe that COEP is extremely well-positioned to take a quantum leap into computational science and engineering. Consider some of these factors.

In the recent two to three decades, Pune has seen major developments occur in the field of computer science and software engineering. Two decades ago, Pune had become home to C-DAC, the designers and builders of India’s first parallel supercomputer. Today, Pune remains home to a top-10 supercomputer in the world (Tata Sons’ Eka).

Pune has had one of the two foremost laboratories of CSIR, namely, the National Chemical Laboratory. (The other lab comparable to NCL being the National Physical Laboratory at Ahmedabad.) Today, Pune has compounded the presence of NCL with the IISER—the institutes which are designed to go at par with IITs but with an emphasis on basic sciences. Pune also is home to some other finer talent in the field of physical sciences and engineering: IUCAA, NCRA, DoD labs, etc.

Finally, Pune also is home to many educational institutions. (I do have a great deal of reservations about their quality—but they do keep education going in some sense).

Speaking in overall terms, in India, Pune is exceptionally strong on the two components that CSE requires and depends on, namely: science (including engineering), and computation. For example, Pune has the greatest penetration of PCs for the past decade or so (which means, on a per-capita basis, Pune buys more PCs than any other city in India, large or small, including Bangalore, Bombay, and Hyderabad, and Mysore, Indore or Nasik.) Another point: MCCIA has identified Pune as the next major hub for animation and gaming industry. In any case, if I remember it right, Pune has already overtaken Bangalore in terms of software exports in USD terms. (If not, Pune would be second only to Bangalore, and a very close second at that… Pune does have companies like PTC and Geometric Software who don’t give me any jobs.)

What does it all translate into, for a UG/PG student, you ask? Let me give you a simple example.

Step inside some of the bookstores in Pune (most notably, the Technical Book Services), and you will always find a lot of movement in there—a movement of books, rather than of customers. … You see, what happens is all these labs, universities, institutes, and software companies together order a lot of books and journals. Many of these orders are handled by these small bookshops. Anytime one steps in a bookshop like TBS, you will always get a sense of how ideas possibly are moving in the city.

For example, I have seen the very latest American conference proceedings touching on topics like, say, simulation of fracture in random nanocomposites, or the use of the LB method in modeling fluid dynamical problems from tribology… Titles like these appear in these small shops literally within a few weeks or months of actual holding of these conferences in the USA… (And, I was naming just two topics among an array of them)… Now I am not sure if all these volumes actually get bought by the potential customers who order them for a review or not. But the shop owners do oblige their regular customers and get these proceedings and all from those bigger shops/agents from Mumbai and Delhi, for a quick review by the customer on a returnable basis, no questions asked and no strings attached. … Precisely one of the reasons why I hang around these bookshops a lot. (Some, like the TBS and I have grown older together—15 years is a long time in the human life-span!)

OK, so, people here in Pune are aware about the technological developments… My only concern is that it doesn’t translate into anything tangible for the UG student at COEP.

That, plus the fact that I have always had a lot to say about the way they teach mathematics and other related topics at COEP… I also used to have a lot to say about the stupid if not vengeful way in which they used to examine the hapless UG engineering student in earlier times—some three decades ago. A lot of that has now changed for the better. But still, the teaching of mathematics, even the syllabus, haven’t changed… Also, the UG program composition as such…

Now, whenever someone mention such things, people give some very typical reactions. Some of these reactions are listed below, in no particular order:
(i) Yes, I agree that we should introduce some more career-oriented courses at COEP… We should make the syllabus more practically relevant
(ii) We should cover the very latest instruction sets of the latest chips
(iii) We should introduce biotechnology
(iv) Etc.

Let me tell you my take on these:

(i) “We should change our coursework at COEP to make it more career-oriented.”

An outright stupid idea.

Yes, you read it right. An outright stupid idea.

Engineering education is meant to be theoretical. If, as a student, you cannot digest it, give it up and join a road-side garage to turn yourself into an auto mechanic.

If, as a potential employer you happen to carry the same ideas, then check out what I have to say about the sort of employer you are, below…

Indeed, here, Rahul Bajaj himself (the father of Rajiv and Sanjiv Bajaj) or, if not himself, at least responsible people (managers and all) from his company (and also managers from Bajaj Tempo/Force Motors, and TELCO/Tata Motors) used to say, when I was an undergraduate student at COEP, that they wanted to see engineering disciplines like, say, “Maintenance Engineering” at the UG level … Yes, this is serious—not a joke. Grown up and highly paid managers used to say such things to us at COEP in those times. Indeed, it should not be any surprise; Bania companies actually are known to say things like that…

I mean to say, disciplines like Mechanical / Electrical were not enough, they wanted the Sandwich training program. Then, it was not enough, they wanted super-specialization like Production Engg, right at the undergraduate degree level. Then, it, too, was not enough and so they began wanting to have Maintenance Engineering too, to be made into an engineering discipline… The only way to counter such suggestions was to ask: How’s the idea of introducing a BE in “Boiler Design for Thermax,” and another BE in “CNC Machining with Fanuc” for the smaller components-suppliers to TELCO, and yet another BE in “Maintenance in Plant No. 6 of Bajaj Auto Factory at Waluj…”? All these degrees, of course, to be awarded by the University of Pune?… (BTW, my use of the term “Bania” is more generic than being just carrying a caste-ist kind of interpretation. I don’t care for castes. All that I want to counter is some false tears being shed in certain quarters such as those by Swami in a recent ToI column. So, Bania is to be taken in a generic sense, in exactly the same way that Brahmin/Pandit is.)

Don’t get me wrong. I am not saying that the practical businessmen and engineers running those automobile or engineering or components-manufacturing industries don’t face some really acute problems when it comes to staffing engineers or making use of whatever engineers they do find… The employers do have a lot of problems in finding good engineers. But what I want to point out is that all these problems, essentially, are of their own making.

These Bania idiots have no idea (not even a vague sense of an idea) as to how to employ a talented engineer with them, or how to make his theoretical skills productive in the environment that they themselves have created within the factories/offices which they own. Why, they have no idea that such a thing is even practically possible. (Despite their Harvard MBAs, and regardless of whether they are Parsis or Hindu Marwaris by Religion or caste, they all remain Indian Bania idiots at heart. The inability to perform integration is the chief attribute of whatever substitute they use for a working epistemology.)

The fact is, if you are going to make herd management the primary duty of a well-educated BE engineer, making it appear as if you are doing favor (Hindi: “Upkaar”) to him by giving him a job, it’s very obvious that he is soon enough going to cram English words for a few months so as to get a good GRE score, and then leave you as soon as he gets the I-20 form in hand. If not, similarly try to beat the system by joining a coaching class for the CAT examination… What you are going to be left with, then, are going to be mostly second-rate folks. The engineers themselves might be great personally, at least in the initial years. But the interface that you impose on them permits only so much of their productivity to come out. Over a period of time, they then loose that too, and become thoroughly second-rate themselves… By and large…. In such an environment, both the employer/manager and the employee come to develop a faulty working epistemology for approaching anything in life—their professional work included. Understanding how theory is, or can be, integrated with practice, is a theme that would be far too much for such a mentality to even think of handling. But since you have none better to employ anyway, you continue employing them, the height of creativity in cost-cutting being, what else, to employ a thinner sheet metal for automobiles… (LOL!). As a decade or so elapses, you then make that guy your Manager. Once this idiot (he has become one by now) gets the management power, the only folks he is going to feel comfortable managing are obviously going to be only the second-rate sort of folks… The thing continues. Then, something happens… Our man—the employed “engineer”—crosses three decades of his working in the same company. He has been sent on short trips abroad, has acquired public-speaking skills (including the skill to tie a tie and perhaps a tie with the tie.org organization too), and on strength of these “qualities,” he gets invited to share his “thoughts” on how to shape engineering education in the next century (the first decade of which, BTW, has already passed by.) Now, problems—real problems—do not disappear simply because someone unworthy of managing them has been promoted in his job. Naturally, our guy does carry a vague sense or an awareness that something needs to be done w.r.t. engineering education… This, he thinks, is because the syllabus is bad—it is not practical enough. And so, he advises, in a sufficiently grave and sufficiently civic tones: “Make education practically relevant…” Et cetera… (I am sure you have heard out these idiots often enough, though I am not sure you had the clarity of thought to judge them as idiots—or the inclination to judge anything any time in life at all.)

(ii) As to the next two suggestions (“engineering education is absolutely bogus and worthless if the very latest instruction sets of Intel (or AMD or RISC) processors are not included right in the next semester” or “we should include biotechnology”), you can see that these represent nothing but somewhat more informed kind of mistakes using the same, faulty, working epistemology. (As to the biotechnology-related suggestion, it often results not only from a faulty working epistemology, but also an outright lack of understanding of either biology or technology. It just happens that the speaker has heard that biotech is now in vogue in the USA (after computer science the metaphorical bus for which he missed), and so he wants to talk something about it here, that’s all… What is displayed in such cases, oftentimes, is nothing more than the favorite working mode of Indian Pundits; I call it the Parrot Epistemology. (It’s very favorite with Indians—all it involves is memorization of sounds and their hi-fidelity reproduction, emotional undertones faithfully included.)

May be I will write about them some other time… For the time being, I have to finish this post…

So, coming back to the main theme, what I want to emphasize is that for higher-quality education, we actually need not more of but less of an emphasis on practicality—especially at COEP.

In other words, I am arguing for making the COEP UG education more theoretical—but also more interesting and more solidly grounded in reality…

One way to do this is by including an emphasis on computational physics in the engineering curricula.

I will expand upon this theme (and certain other related matters) some time later on, but for the time being, let me note a few things about them quickly. (I am sure many in the USA and in India will be quick to both understand what I am saying and following it up, but without giving me any due credit—e.g. linking to my blog or dropping an email to me explicitly.)

(1) Concerning Mathematics

What I say is: Thrash away all those “Engineering Mathematics” courses. Yes. Throw them away. Completely. Two reasons:

Firstly, the contents of these courses haven’t changed in any essential way from the times three decades ago when Wartikar brothers’ very poorly written text used to be the gold standard in Pune. (And, nowhere else!) That book is indeed fairly good, but only on that count for which an author can hardly take any credit—namely, the set of (unsolved) problems contained in it. But the main text itself is pathetic or worse on all the other important counts: (a) explanations providing appropriate context and highlighting conceptual understanding, (b) maintaining a good hierarchy in the ordering of topics, (c) the tie to physics and engineering, and (d) the production values of a book. It’s a real pity that this third-class book continues to inform the design of the syllabus in mathematics courses at University of Pune and at COEP.

Secondly, calling them “Mathematics” courses itself gives very wrong ideas to professors in this country—a country which is already so heavy on mysticism, paternalistic attitudes, deductions, etc. The effect of intrinsicism and mysticism could be easily found in any subject, but they leave an especially inescapable imprint on the teaching of mathematics. The reason is, mathematics by its nature is so abstract and “mental.” (The referents of mathematical concepts themselves reside only in the mind—not in the physical reality. When two mangoes exist in the world, what actually exists in the concrete reality is only those separate mangoes. The concept of “two” itself doesn’t exist in reality independent of the consciousness of man who has reached that stage of learning/thinking.) Since mathematics is so mental, it is so easy for the teacher to get carried away into deductive complexity upon complexity without caring anything for either the subject, or its physical correspondents, or its application, or the student learning it all. And that tendency only grows in a mystic country like India. A mathematics professor most directly insulting a student’s mind would be easy to find anywhere in the world; but they are a regular feature in India. (“What, you can’t even derive this? It’s so simple! Start with nonlinear equation and go down to linearity. Yeah, right. Start with NS and derive the Euler equation—not in a revision, but the very first time you run into it! You should be able to do it if you are smart!! Look at Narlikar. He is so smart… He became a Wrangler at Cambridge! And now, look at you… You don’t only study hard enough…” Etc. Etc. Etc. … See, how easy it is for people to get wrong ideas as soon as you mention they are going to teach “mathematics”!)

It is for this reason that I advocate that those Engineering Mathematics courses should be completely abolished. In their place, what I suggest, is to (a) begin calling them mathematical physics (which would work as a temporary band-aid) (b) completely alter the order and sequences of all the topics, (c) reduce the complexity of examination question but go ahead and introduce some more advanced topics, esp. their conceptual treatment, (d) extend the lengths of all these “mathematics” courses.

Don’t get shocked at the last suggestion! Don’t say that we have no place left for an additional course or two in maths. There is. Because, I am also advocating to also do away with all the numerical analysis courses. And also, many others (e.g. the “Applied Science” courses that do no good to anyone.)

Instead, we need to have a sequence of four to five courses on “Mathematical and Computational Physics,” all to be completed by all engineers (including the IT and CS engineers) within the first two years.

Today, at University of Pune and COEP, the situation is so bad that the IT and CS majors have absolutely no idea about, say, 3D boundary value problems, what the term stress means, what Fourier’s law of heat conduction is, and why, even about EM fields, really speaking. (They cannot even properly do visualization of a 3D wave-field… all that they can sketch is a static wave in 1D. And, theywill invariably fail to tell in which direction it would move—to the left or to the right.)

It’s for this reason that all engineers, regardless of their branch, must be taught a common curriculum which is strong on physics and basic engineering sciences, in the first two years.

These courses should emphasize the teaching of mathematics from a conceptual and physics-based viewpoint; they should keep the engineering or technological applications in sight but only as distant ends; and they should make use of computational physics as an indispensably important tool for both pedagogy as well as professional preparation.

For example, currently, COEP has no separate course on differential equations (DEs). Instead, some of the topics on DEs are distributed piecemeal, and are covered without depth. (For instance, COEP/UoP students are not taught the diffusion equation in 3D, only in 1D; most of them cannot tell when they will use Fourier’s method vs. Laplace’s even though thousands of them could easily solve the examination type of questions on either method.) Instead, there should be one complete dedicated course on ODEs and another on PDEs (possibly with vector Analysis), and more: both these courses should have not mathematics as their central focus but mathematical physics. Further, these courses should integrate the computational physics part within them. For instance, not only should the UG student be taught about the well- and ill-posed problems, but he should also be shown, with the help of some simple C++ code snippets based on the simple finite difference method, what kind of unphysicality creeps in if an ODE/PDE problem is made ill-posed. The students should be made to appreciate that they need to learn differential equations to be able to tackle the IV-BV class of problems—not in order to deductively manipulate Euler’s identity so as to satisfy an orthodox MSc in pure mathematics who would smirk disdainfully at the student’s lack of technical proficiency in rapidly performing meaningless manipulations involving it.

Needless to add, the situation at IITs is not very different. However, they are slightly better in that they do refer to Kreyszig or other books while designing their syllabii—not Wartikar—and so, the syllabus at least tends to be somewhat better—even if the students themselves or the professors themselves are not very different anyway. (Indeed, at IITs, the tendency to be rationalistic is even more pronounced.)

(2) Concerning Physics

Contrary to what the Indian Bania industrialists (or the small-scale “industrialists”) come and tell you, reduce the share of technology-specific courses, and instead, increase the share of physics (or basic engineering scienes-related) education in engineering and technology UG curricula.

A room for greater physics can be made by downsizing (or altogether dropping) technological courses. After all, going by my own practical experience (and that of hundreds if not thousands of working engineers), picking up technology is so damn easy if you are sound on fundamentals (and impossible if you are not). And, further, one way or the other, you are going to take some time to pick up technology anyway because you would be working one or two decades later… How is it possible for an e-School to prepare you for the next generation technology when none has any idea about it? So, much time is, really speaking, only wasted in “teaching” technology at e-Schools. Instead, such topics could easily be relegated for self-reading plus technical reports or seminar (say for 1 credit hour).

Let me give you one specific example in reference to the mechanical engineering curriculum even though the same essential argument can be easily extended to any other branch of engineering as well.

There is no need to have three separate courses, one on Fluid Mechanics, another on Fluid Machinery, and one more on Energy Conversion, and one more on Power Plant Engineering, all with partial overlap of contents on each. (And, we are leaving aside Thermodynamics and Heat Transfer etc. courses too.)

Instead, make it just a two-course sequence. This will become possible if students are already familiar with concepts like differential nonlinearity and differential coupling, via their earlier courses in computational and mathematical physics. (That is, the revised maths course and sequence I spelt out above.) Today, the situation is: mention these two terms and the students would look at you blankly. (They continue to do so until after their BE/BTech graduation.) Forget nonlinearity, they don’t even know that beautiful theorem by Helmholtz which says that you can always split up any arbitrary vector field into a sum of one irrotational field and another one, a solenoidal field. Now, if you go and ask any engineering mathematics professor, he will laugh and say, “no, that topic is far too advanced for an undergraduate; it requires far too much advanced mathematics.” He would say so, in the process completely ignoring (or even evading) the fact that Helmholtz himself was trained only as a medical doctor (i.e. not even as an engineer let alone as a mathematical physicist or a mathematician proper). Now, I fail to understand why is it that a 19th century medical doctor can, with self study, originally invent that theorem, but a 21st century graduate engineer cannot handle it in his curriculum? Why does this impression persist—namely, that Helmholtz’ theorem is too advanced to be presented to UG in engineering at UoP/COEP/many IITs?

The impression persists not because the topic itself is advanced, but because the teacher himself sees nothing but a further bout of meaningless symbolic manipulation which must precede before the theorem can be taught. He foresees that bout of deductive complexity whenever he happens to think of that topic. The teacher himself doesn’t consider the inductive reasoning behind Helmholtz’ theorem, he doesn’t bother visualizing fields or tracing the geometrical and physical lines of thinking about it, he doesn’t consider the simplicity of the essential argument behind that theorem. All that he foresees are those threateningly complex mathematics, because that is what his teachers had made out of that topic by overemphasizing deduction, by indulging in a dance of ideas progressing from one idea to the next each of which was necessarily kept divorced from reality. Naturally, being a well-meaning teacher, he cannot imagine unleashing that kind of atrocity upon his students while they are still in their UG years. (They need to mature, become thick enough, and then, the atrocity could certainly be unleashed against them. That’s what he means. … Everyone likes them young!)

Anyway, to return to this sub-point concerning reduction in the time spent on fluid mechanics, if basic physics (like conservation of angular momentum) and basic engineering sciences (like kinematics of deformation, including a discussion of vorticity right while introducing or discussing strains) has been taught well, then, the subsequent “mathematics” courses can also be sufficiently physical as to include a discussion of the differential nonlinearities. In which case, the strain involved in the teaching of fluid mechanics would get reduced, and so, only a two-course sequence would be enough to cover the overlapping topics from turbomachinery, hydromachinery and whatnot. Further, FM itself could also be made more interesting using CFD software for visualization (i.e. even if the students don’t take a separate course on CFD proper), and by making use of some small, simple codes illustrating and highlighting various features/aspects of nonlinearity.

So, you can see the deeper sense in which I mean to say that computers and computational physics should be made use of, in engineering education…

—–

It’s high time that COEP took advantage of the resonating kind of institutions nearby, and certainly, the availability of well-trained people in the field of computational sciences (or at least the ready availability of such people who could, with some extra effort, be turned into well-trained people spanning computer science to science to engineering) and use both these to enhance the quality of its engineering programs.

Throwing money on buying computers accomplishes nothing. Not if things of the above kind aren’t taken up for implementation.

And, introducing biology courses in engineering programs can be, if you ask me, a very poor idea, indeed. I mean, it’s OK as an elective. But not at the FE level. And not as a means for enhancing the core engineering program itself, in general. (Think: What good would it do to introduce a course on the kinematics of machinery in a dentist’s undergraduate program, simply because he happens to use a rather complicated contraption for a drill? Or, what good would it do to introduce the mechanics of materials in an undergraduate surgeon’s program—on the grounds that he is going to cut tissues and so must know the mechanics involved in the cutting action? Doesn’t it look like an outright laughable idea right on the face of it? If yes, why does the reverse feel so appealing? Are our engineering students so dumb that they can’t look up a bones model and figure out the precise way in which the hip joint does or doesn’t develop contact? Is that the case? Or is it the case that our educators carry a remnant or a vestige of a sense of intellectual inferiority which a lot of Indian engineers feel whenever they run into doctors? (You see, in India, medical admissions would be a shade more competitive in the statistical sense—there would be fewer seats. But if you ask me, the medical admissions would also be a shade less competitive because cracking biology would of course involve more of parrot-like cramming than the sort of on-the-fly application of a few fundamental principles which physics and mathematics involves. So, doctors could be expected, by and large, to be dumber but harder-crammers…. But then, again, people (engineers) aren’t always so sure about even valid observations so long as they aren’t widely accepted in society… It’s no accident that with the IT industry and American money, engineers started earning more, and so, (dumb and beautiful) girls’ parents started running after engineers more than the doctors, and so, engineers started having confidence. But that’s a recent story, and applicable only to IT and CS folks—the same ones who can’t tell the units of stress or one honest application of the concept of gradient.)

If your objective is to enhance the engineering education, you have to increase the emphasis on basic physics, basic engineering sciences, and computational science and engineering—not on biology…

Do the first, and you will see the efforts bearing world-class fruits in a matter of time as short as a decade or less…

(Actually, there are many reasons why I don’t give a damn to the adjective “world-class.” A sidey incremental development translating into nothing of major or lasting value (or of any practical use) can also be world-class… After all, world-class is necessarily a comparative description, not absolute: it tells you about the relation of one man against others—not of the relation of a man with reality. That’s the difference.)

More on all these topics, I mean, expanding on use of computational physics in engineering, the third paradigm and its relevance, and all later on…

For the time being, I guess I have already used this keyboard a lot in one go… Time to take a break (for a few days or so)…

—–

I also consider it my moral obligation to keep reminding you that I am currently jobless, and that all my posts and former emails (and job applications) in this context are relevant…

…Do consider it a shamelessness on the part of all the powers that be (here and in the USA) that I am jobless when many relatively worthless folks (including those graduating BTechs from Kanwal Rekhi and Vinod Khosla and Suhas Patil’s IITs, and the BCS or BE in ITs) have been rich or super-rich.

A dean like Anand Bhalerao also is one of them. (For those who don’t know: He was willing to offer me a job as a full professor of mechanical engineering, but only on the condition that I would not talk about my (what he felt and said were “superior”) achievements to my students or my younger colleagues from the faculty so as not to disturb the “atmosphere” that he had painstakingly set up at his university… I refused, saying that I could not hang my achievements at the gates of his university so as to be able to enter as a suitably meek man once inside that campus. Achievements—real achievements—aren’t hats, I had pointed out.) A Vishwajit Kadam interviewing me and welcoming me in his institution verbally but not actually issuing the appointment letter (so that the issue of paying me salary simply does not arise later on) also falls in exactly the same category… And so do my friends who asked me to get meek at least once because I anyway had no other job in hand—-this, at a time that Anand Bhalerao himself had not noticed this weakness of my situation… With friends like these, who needs enemies? (And, in case you read this, Vishwajit: In retrospect, it is me who is sorry. I am indeed sorry that I could as low as saying, just for getting a job for myself, that I was looking forward to “encouragement” of my work from you… For that one moment, I had lost my bearing only because I wanted to conform as well as I could—I was asked, by my “friends,” to behave as “neatly” as possible. I couldn’t have managed it, and so, I ended up saying what I did—that I appreciated that “encouragement” from you, an encouragement worth 40,000 Rs per month… Sorry, indeed, I am. I should have kept my (actually) better sense compared to my “friends” (as well as my pride) and should have told you on the face that you were doing a good thing in hiring me.)

Anyway, to return to the theme of this point, this neat-looking young nothing called Vishwajit Kadam also is one of them (the aforementioned BTech IITians through to Sakal editors) if he can promise me a job but not release the appointment letter.

And, while on this line, let me also note that it was repugnant to read, just two weeks later, that some Pune educationist’s son had been caught throwing lakhs of rupees in cash at a dance-bar girl near Mumbai over the course of a single night. (I also want to note here that I doubt that it could have been Vishwajit in that dance bar there… I mean, there are many big educationists in Pune, and they have sons whose ideas of life aren’t exactly in line with education in any sense of that word… But then, how do you know that it just couldn’t have been this guy or that guy if you also know that you had been promised an appointment letter but it never landed in your hands? How do you know that such characters can be worthy of your trust?)

Anyway, far more important than the issue of dance bars, here, are the issues of education and of my unemployment… After all, anyone could easily vouch for the fact that Ms. Vidya Yerwadekar simply won’t ever go out to New Mumbai and throw lakhs of her own money on those dance-bar girls, over the course of a single night. Anybody could vouch that Vidya herself, certainly, won’t do that. Yet, how does knowing this fact about Vidya help one if the application one made to her institution, namely, Symbiosis (which spends crores in advertising itself—often also buying editorial influence in ToI), cannot bother to even acknowledge by email the job application?

And, I want to go further: Why can’t Vidya invite me for an interview for her new engineering college, I ask pointedly… After all, I also blog at Harvard-based iMechanica; have a master’s from an IIT (and at 40+ age after competitive examinations, was offered admissions to both IIT and IISc); I do research at COEP; and, Vidya herself is not all that distant in space or age to me—she probably was just a batch or two junior to me but in BJ when I was an undergrad in COEP…

I could go on, but anyway, Swami (of Swaminomics of ToI fame), it is to you that I want to remind: Hindi: “hamaam me sab nange”—whether they are (dear to your heart) Bania “businessmen”, or those “professional” “educationists” from Pune, or those Delhi bureaucrats taking pleasure in running educational institutions, or those educationists vying to become bureaucrats, or, of course, the Americans. “hamaam me sab hai nange.”

—–

An aside: The last few columns in Sunday ToI by M.J.Akbar have been excellent. Better than Swami’s or anyone else’s (in that newspaper)… I don’t know if Akbar has actually fallen out with the Gandhi-Nehru parivar or what… (I don’t follow up too closely on journos but vaguely remember having read in my younger days in 1980s—say in magazines like Outlook, Frontline, India Today or the like—that charges were being leveled against Akbar in those times that he would defend Rajiv Gandhi regardless of season or reason… (I guess the journo in question was Akbar only, but I am not certain.) Has there been a real change in Akbar or what? … I think that even if he has remained as acute as before, packing as many dramatic punches and twists in his writing as before, still, today, I have this vague sense that Akbar might actually have mellowed a little bit now… I mean, his writing has acquired a bit of gravitas of a certain kind that is not very easy for a journo to pick up once he has begun writing… A welcome change it is… All in all, these days, it is his columns which make for the best read (I mean within ToI)… (I can only hope that the quantum measurement effect doesn’t affect his writing style in the immediate future. (Incidentally, it is this fact which heightens my irritation at my being jobless—obviously, the powers that be do read me, take notice of what I write, and yet, continue to make sure that I go jobless. BXXXXXXs…))

[Updated and expanded on 6th and 7th April, 2009. Yes, I have expanded my entry and added even more asides in it… Do you have any problem with that, journos? IB folks? Americans? Others?]

[BTW, WordPress’ AutoSave is a great feature… In Pune the electricity goes off without warning and some 1000+ of my words were accessible thanks only to this AutoSave feature!]

[Correction on April 10, 2009: The dean’s name is Anand Bhalerao, not Rajesh Bhalerao as posted earlier]