“Math rules”?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And with that “vision,” began my fall.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Two reasons:

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

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

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

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

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

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

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

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

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

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

I Song I Like:

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

[As usual, may be another editing pass…]

[E&OE]

6 thoughts on ““Math rules”?”

1. First, u is spatial quantity u = u(x,t), where (small letter) x is fixed in space, i.e., x could be your nodal coordinate of your FDM grid. What the operator means physically is that the change in u at that fixed point is affected by both the local change at that point (given by partial derivative with respect time t) and the fact that the fluid is moving past (convected through) that point.

If you assume that there exists a deformation mapping mapping the material coordinate (capital letter) X (attached to the fluid) to its spatial coordinate x at some time t:

x = phi(X,t)

then you can write U(X,t) = u(phi(X,t),t). Then, the time derivative of U(X,t) is

dU(X,t)/t = del u / del t (fix x) + grad u dot v

where v is the spatial velocity v=v(x,t). As in u, v is a spatial quantity located at your FDM node in space. You don’t need to think in terms of Lagrangian or Eulerian. It is all the same.

Thanks for your comment and sorry for the delay in reply. I do not check the blog comments queue every day (and don’t know how to set up emails notifications for comments).

What you write about is something like this: https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field.

Mapping is a higher-level abstraction that connects the Eulerian and Lagrangian viewpoints. Implication: epistemologically, it is not possible to conceive of such a notion unless you have already pursued thought previously in terms of the separate Eulerian and Lagrangian descriptions. Historically, that anyway has been the order: mappings came only later.

My personal preference is to continue thinking in those two separate ways. I find that my mind is loaded less this way. I can always go higher up to the mapping later on, if the need be. In engineering, I have not so far found any need to go to the mapping.

Even if there is an objective need to refer to mappings (e.g., in simplifying presentations of such things as ALE) I still am not going to introduce such mappings at the more primary level—certainly not even in physics, and also not, for the most part, in engineering. [That’s my way of keeping out the relativistic nonsense of the spacetime nonentity looping simultaneously around, through, within and out of itself at no time and at all times.]

But, yes, coming back to the point you make, the idea can perhaps be seen in an even simpler manner in the context of the integral equation i.e. the Reynolds Transport Theorem. Refer to Ferziger and Peric, p. 4. Using their notation, for the Eulerian frame, $\vec{v}_b = \vec{0}$, for the Lagrangian frame, $\vec{v}_b = \vec{v}$, and then, there is an infinity of possible values: $\vec{0} < \vec{v}_b < \vec{v}$ where the inequalities are enforced strictly.

Realize that even if you choose $\vec{v}_b$ to be neither Eulerian nor Lagrangian, you still don't gain any new insight, nor do you simplify the calculations. There is no particularly valuable cognitive gain to be made by preferentially referring to the higher-level abstract mapping, and then (as if condenscendingly) saying that the Eulerian and the Lagrangian and all is really all the same. No, mathematically, they are not the same: Mathematically, there are differences (as captured by the value of $\vec{v}_b$). The sameness is only in reference to the broad underlying physical principle—neither mathematics-wise nor physical imagination-wise

Anyway, the rules of thumb (with good grounding in epistemology) which I follow are: never invert the proper hierarchy; never lionize mathematics; never elevate maths above the physical imagination in which it is grounded. That's why, I don't care for the mapping. Neither in physics nor in CFD.

Anyway, thanks again for your comment.

Best,

–Ajit
[E&OE]

2. The use of mappings has practical implications. As an example that may be closer to your field, you may have heard of Total Lagrangian vs Updated Lagrangian approach in calculations of large deformations of solids. In the former, you map everything back to the reference (fixed) configuration and do your calculations there (e.g. calculate your stiffness matrix, etc), and in the latter you work with current spatial quantities. in the TL approach certain quantities such as density are constant (hence, the mass matrix can be computed once and for all from the start), whereas if you try to work with quantities in the current coordinate frame (the mass matrix would be recomputed each time). Also shape functions in the TL approach are computed once and for all from the start, whereas in the UL approach, the shape functions need to be recalculated each time. However, both should yield the same results. One is not better than the other but for practical purposes, one may prefer to use one or the other.

Not sure what you mean that they are mathematically not the same. The use of mapping is essentially a coordinate transformation (implicitly assuming that the mapping between the 2 coordinate systems exists). You have done so in elementary calculus where certain integrals are easier to do in some coordinate system. It is really a generalization of this concept.

Thanks. (I knew this was coming, but hated to edit my above comment after its publication.)

Well, mapping isn’t the word I should have used at each place where I used it. It was a good word to use in some places, and wasn’t a very good choice at others. …

Anyway, the point I wanted to make is the following (and please see if you agree or disagree as we go along):

First, there is this nice Newtonian space—a fixed, absolute space, which, admittedly, is a rationalistic construct, but at a common-sense level, it works very nicely for the most part, esp. in engineering. You basically imagine as if infinitely thin lines making up a non-material grid is already laid out in the physical reality out there. You imagine that the grid exists as a separate physical object in its own right, in the physical reality out there. But you also then imagine, quite un-physically, that this grid remains entirely unaffected by anything happening in the physical world. Being immutable (unaffected by anything and any action), absolute, it also would be independent of any observer, and so, would be the same for every observer. God laid out a Cartesian grid for us, so to speak. The grid is remarkably like the Aristotelian God (Who perhaps made it along with the rest of the world). Along with Him, it simply sits there and does nothing. It does not participate in anything that happens in this world. The grid is an expression of the Aristotelian God’s kindliness—it allows us to coordinate our physical measurements.

The Eulerian description follows most naturally on the basis of this kind of an idea of the space.

Indeed, in the Eulerian description, there are no separate objects at all. Everything that exists is just one object (for instance, just one fluid, one aether). But there can be quantitative differences in its attributes at different places and at different times. These quantitative measurements of the attributes (e.g. fluid velocities, pressure, temperature) are measured in reference to the God-given single, absolute, space. Thus, there is a God-given space, and there is a God-given single entity (the fluid) which is embedded in that space, and then, there is a God-given grid with which to measure that space and that entity’s local attributes here and there, now and then.

Thus in the Eulerian approach, the reference space thus at least coincides with, if not is the same as, the ambient space.

The Lagrangian description focuses instead on the individual material objects. It in fact starts out by recognizing that there are many different objects in the universe (say fluid parcels or particles)—not just one all-encompassing entity (“the” fluid). To define dynamical attributes of these multiple objects, the Lagrangian description, too, needs to make reference to the idea of space. It’s just that in the Lagrangian description, the reference space is not the same as the ambient space.

For the reference space, the Lagrangian description chooses to attach a different local frame to each material object. Thus, there are a number of reference frames, and the frames themselves can move (and possibly even undergo internal deformations) along with the objects to which they are attached. The objects of course suffer changes of shapes and sizes, not just positions.

Now, a subtle point here is that a physical theory may have multiple objects and frames, but still needs to be universal. To describe the working of the universe as a whole, you have to into account all of these multiple objects in one go. But, the multiple objects do interact with each other. The only way we could at all quantitatively capture their interactions would be in reference to a frame that would be common to them all. Thus, for the sake of universality of the theory, the Lagrangian particles, too, must still be seen in reference to a single distinguished frame; the Lagrangian particles too must be seen as being embedded in some or the other ambient space which they all share in common. Simply by admitting “the many” in respect of particles, you can’t get away from “the one” (which is common to them all).

So, you see, you can’t ever get away from having to use the notion of the ambient or embedding space—even if your reference space(s) aren’t the same as the ambient space.

Now the issue becomes: Should it therefore be the same as the Newtonian space? Or are there any choices?

Mathematically, as the Galilean and Lorentzian invariance criteria indicate, objective choices are not only possible, they actually help simplify calculations. Coming back to CFD, you can pick up any local Lagrangian frame (including fixing its internal measures i.e. “shape” at a certain instant of time as the standard initial configuration) and regard it as your ambient frame, and proceed with your calculations. Thus, one frame is singled out; it serves as both a local frame as well as a global, ambient frame.

It is in this context that the issue of mapping comes up.

The mapping is really speaking required only because dynamical description of each attribute of each Lagrangian particle occurs in its own frame, and these quantities must be subject to certain changes before they can be carried across to some other similar frame. The mapping is nothing but this “carrying through” process. Taken in this sense, of course, mapping is not only practical but in fact even indispensable. In its absence, physical calculations would go wrong.

But then, as it turns out, e.g as the RTT equation illustrates, for picking out an ambient frame, you are not even required to choose some frame that is fixed to an actually existing particle. You can even construct a local frame such that it is not attached to, and does not move or deform with, any actually existing particle. How come? It’s just that the control volume does not have to coincide even with a moving control mass’s instantaneously occupied volumes, mathematically speaking. The choice of which frame to pick out as defining the ambient space is thus made completely “arbitrary.” In the RTT description, as I said, it is the \vec{v}_b which can range over (\vec{0}, \vec{v}), both endpoints excluded.

This kind of a generalized maths is a good thing, but only so long as it allows us to carry out calculations (and qua an abstraction, helps simplify its presentation). This much part is valid, and I have no issue with it.

The trouble occurs when people go overboard, and begin to say: So, the distinction between Lagrangian and Eulerian is all meaningless.

Meaningless? Really? And if so, what is meaningful then? And how did you get to that meaningful notion in the first place?

Turns out that it would be impossible to construct the generalized abstract notion of mappings, without first having had concepts such as: the embedding space (which is to be distinguished from the unique and universal Newtonian space), the Newtonian frame, the reuse of the Newtonian frame as the Eulerian frame (and hence the coincidence of the reference and the ambient frames in the Eulerian case), the fact of having to embed the multiple Lagrangian frames into a single (universal) embedding space—for the sake of universality of the theory.

You can’t run away from the last. Just because an infinity of ambient spaces can be constructed, you don’t thereby free yourself from having to embed it into a distinguished, universal, embedding space. (The moving control mass too occupies a volume—but how do you define that volume? To avoid circularity, here, you have to first take for granted the existence of a common universal embedding space. Only then can you set up the volume integrals over the non-fixed instantaneous volumes occupied by the control mass.)

At this point, I now say: if you can’t escape from that universal embedding space, why not just use your Eulerian frame for describing it? After all, it is so convenient, what with $\vec{v}_b$ being equal to a zero vector in it. It simplifies maths. Sounds reasonable?

If you say yes, then realize, you basically come to concur to my basic point.

In agreeing to fix the Eulerian frame to the embedding space, you in fact also agree to (i) keeping a Eulerian frame included in your physical description, and (ii) keeping it separate from the Lagrangian frames. You thereby agree to first having both the Eulerian and the Lagrangian as two separate descriptions, and you also agree to constructing a mapping only in reference to these two different notions. You thereby agree to the idea that the existence of the mapping does not make the Eulerian and Lagrangian descriptions the same.

The issue thus boils down to the following choice: In what light do you see the more abstract mathematical formalism of the arbitrarily deformable and arbitrarily choosable frames?

Do you view it as a mathematical object that exists completely all by itself, without any reference to the distinct Eulerian and Lagrangian descriptions? Do you treat it as severed from any consideration of any universal (physical) embedding space? If yes, then you are treating the general mathematical formalism of the mapping as a mathematically primary object. Treating it as the primary object is tantamount to the fallacy of the stolen concept. And, of course, with the result that your maths is cut away from physical reality.

When I objected to mapping, I meant to refer to this kind of an abuse.

But, in fact, even if I was not very clear in my writing, I did use the word “primary” in my above comment; check out the lines wherever the word “primary” appears above. You can’t treat a derived concept as the primary, that was my point.

The mapping is just a higher-level abstract connection between two separate viewpoints/descriptions, and they both lie at its basis. In particular, the Eulerian description is necessary to deriving that generalized notion too, even if the ambient frame of the Lagrangian description is in fact completely arbitrary, simply because the Eulerian frame is the simplest ambient frame that can be attached to the common embedding space, and the common (universal) embedding space is indispensable to physical theory. The idea of a universal embedding space cannot be done away with just because mathematically you can have a choice of an infinity of Lagrangian frames to choose from.

Your writing about the Total Lagrangian and Updated Lagrangian basically refers to something similar. In TL, you are better off having the initial reference frame coincide with the one that actually is Eulerian in nature!

Yes, if the hierarchy is not inverted, if the generalized formalism isn’t treated as a primary, if all its distinct bases are acknowledged to be present even if only implicitly in the context, then, of course, I do find the mappings valid. And, yes, I would use it, if I had to. (It’s just that the range of problems I deal with doesn’t require me to get into that part.)

Too long a reply, probably a bit repetitive, but the hope is that at least this time round I got in some good clarity to my writing.

Feel free to offer further comments.:)

Best,

–Ajit
[E&OE]

3. Your explanation invokes the abstract that I do not know. I only learned enough to implement a nonlinear fem code in a course project. But if it works and clearer for you, ok.