Ajit R. Jadhav's blog
http://imechanica.org/blog/1150
enExplicit vs. implicit FDM: Could you please suggest a reference?
http://imechanica.org/node/19490
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/4847">Explicit algorithm</a></div><div class="field-item odd"><a href="/taxonomy/term/10988">explicit vs implicit</a></div><div class="field-item even"><a href="/taxonomy/term/1010">FDM</a></div><div class="field-item odd"><a href="/taxonomy/term/10989">local vs. global support of solution</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The context is the finite difference modeling (FDM) of the transient diffusion equation (the <em>linear</em> one: $\dfrac{\partial T}{\partial t} = \alpha \dfrac{\partial^2 T}{\partial x^2}$).</p>
<p>Two approaches are available for modeling the evolution of $T$ in time: (i) explicit and (ii) implicit (e.g., the Crank-Nicolson method).</p>
<p>It was obvious to me that the explicit approach has a local (or compact) support whereas the implicit approach has a global support.</p>
<p>However, with some simple Google searches (and browsing through some 10+ books I could lay my hands on), I could not find any prior paper/text to cite by way of a reference.</p>
<p>I feel sure that it must have appeared in some or the paper (or perhaps even in a text-book); it's just that I can't locate it.</p>
<p>So, here is a request: please suggest me a reference where this observation (about the local vs. global support of the solution) is noted explicitly. Thanks in advance.</p>
<p>Best,</p>
<p>--Ajit</p>
<p>[E&OE]</p>
</div></div></div>Thu, 18 Feb 2016 07:42:55 +0000Ajit R. Jadhav19490 at http://imechanica.orghttp://imechanica.org/node/19490#commentshttp://imechanica.org/crss/node/19490Expansion of a function into a basis set
http://imechanica.org/node/18322
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/10489">basis sets</a></div><div class="field-item odd"><a href="/taxonomy/term/8709">Fourier series</a></div><div class="field-item even"><a href="/taxonomy/term/10490">polynomial expansion</a></div><div class="field-item odd"><a href="/taxonomy/term/10491">function expansion</a></div><div class="field-item even"><a href="/taxonomy/term/10492">expansion of a function</a></div><div class="field-item odd"><a href="/taxonomy/term/10055">computational mechanics; finite elements</a></div><div class="field-item even"><a href="/taxonomy/term/1612">CFD</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Consider a ``neat'' function such as what an engineer is most likely to use in his typical theory/work. Such a function would typically be: (i) defined over a single finite interval, (ii) continuous throughout, and (iii) smooth enough. In other words, the sort of a function they used when they introduced the very idea of a graph of a function to you, back in high-school. ... Feel free to add any other qualifications you like, but note them explicitly, e.g., (iv) bounded throughout, and (v) periodic. [I don't mind any additional qualifications; basically, as of now, reaching even just the low-hanging fruit would be gratifying.]</p>
<p>Suppose we want to represent this function, may be only in a limiting sense, as a sum of some linearly independent functions. (The basis set itself may be infinite.)</p>
<p>From whatever I know of numerical analysis and computational science and engineering, speaking off-hand, I can think of (only) two methods:</p>
<p>Expansion into <br /><strong>(A)</strong> a polynomial basis set<br /><strong>(B)</strong> the Fourier-theoretical basis set</p>
<p>These respectively cover the cases of $latex \sum a_n x^n$ and $latex \sum a_n e^{inx}$. </p>
<p>I now have two questions:</p>
<p><strong>Q1.</strong> Why don't we use <strong>(C)</strong> $latex \sum a_n e^{nx}$ in our routine numerical analysis/computational modeling work (e.g. FEM)? Can you think of some application/reference where someone has used it?</p>
<p><strong>Q2.</strong> Theoretically, a much more interesting (and demanding) question: </p>
<p>Do these three categories exhaust all the possibilities? Can we logically say so? Is there a mathematical theorem that allows us to say so? <br />If the answer is a no: What is the obvious objection if I were to formulate this conjecture as a lemma?<br />If the answer is a yes: Can such a theorem be proved? </p>
<p><em>What are your thoughts on Q1 and Q2? Care to share? Thanks in advance.</em></p>
<p>As to me, here I swiftly go:</p>
<p>Q1: Never seen one in a real application; don't know the reason; can't even guess why.</p>
<p>Q2: As one who mathematically has always been much more starry- and dewy-eyed than rigorous, when compared to an average engineer, I am inclined towards immediately jumping to the second conclusion, viz. ``yes.'' The ``reason'' which appeals to me here is that theorem proved by Gauss in 1800, viz., that complex numbers form an algebraically closed field. ... But, rigorously speaking, is this theorem even relevant here? To me, it does seem so; but rigorously speaking, I have no clue.</p>
<p>Anyway, over to you; am listening.</p>
<p>Best,</p>
<p>--Ajit<br />[E&OE]</p>
</div></div></div>Tue, 19 May 2015 02:47:33 +0000Ajit R. Jadhav18322 at http://imechanica.orghttp://imechanica.org/node/18322#commentshttp://imechanica.org/crss/node/18322I am [still] confused about gradients, vectors, deformation gradient, etc.
http://imechanica.org/node/17619
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>I am creating this blog entry to have my confusions about gradients, vectors, and deformation gradient, etc., straightened out once (and hopefully for all time!) ... My confusions got exposed (even to me) while commenting on a thread started by Prof. Suo here [<a href="http://imechanica.org/node/17565" target="_blank">^</a>]. In particular, I realized my confusions after writing this comment [<a href="http://imechanica.org/comment/26661#comment-26661" target="_blank">^</a>] there.</p>
<p>So, let me note down my confusions in the form of a few questions; the order may not be exactly from the simplest/basic to the complex:</p>
<ol><li>What is a vector space?</li>
<li>What is a vector field?</li>
<li>(Perhaps a repetition) What is the difference between (a) a function that is defined on a scalar field and outputs a vector-valued "field," and (b) a vector space?</li>
<li>Is the set of traction vectors directed along a given (local) direction, a field? Does it span a vector space? which one?</li>
<li>What exactly is a gradient of a scalar field? is it a vector field?</li>
<li>Is there a gradient of a vector field? What is the nature of its output?</li>
<li>Is the nabla operator always a vector? why or why not? examples?</li>
<li>Is there any technical difference between the terms: displacement gradient and deformation gradient?</li>
<li>Is the deformation gradient a gradient?</li>
<li>Is the deformation gradient a vector?</li>
</ol><p>May be, I will add more questions as they strike me, but this should be enough to get going.</p>
<p>Thanks in advance for clarifying these matters.</p>
<p>Best,</p>
<p>--Ajit</p>
<p>[E&OE]</p>
<p> </p>
</div></div></div>Mon, 08 Dec 2014 13:25:55 +0000Ajit R. Jadhav17619 at http://imechanica.orghttp://imechanica.org/node/17619#commentshttp://imechanica.org/crss/node/17619MWR for the first- and third-order differential equations
http://imechanica.org/node/16093
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/248">finite element analysis</a></div><div class="field-item odd"><a href="/taxonomy/term/3394">Weighted Residual</a></div><div class="field-item even"><a href="/taxonomy/term/7465">Variational Methods</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi all,
</p>
<p>
In engineering sciences, we usually end up using either the second- or the fourth-order differential equations, and the MWR (the method of weighted residuals) works pretty well for them.
</p>
<p>
The question is: how about the first- and the third-order differential equations? Why don't we see any applications of MWR for these odd-ordered differential equations? What gives?
</p>
<p>
If not equations of real application importance, then at least some hypothetical situations or toy applications involving the first- and third-order (and possibly fifth-order as well!) could have been taken up, just to illustrate the method itself. However, that is never found done, not at least in the introductory text-books/course notes. They only mention the issue, if they at all do, but they mostly remain silent about the mathematical reasons behind it.
</p>
<p>
So, here is a request: If you know of any nice introductory passages in a book (or course-notes, or articles) dealing with this issue in sufficient detail, and, preferably, at the final-year undergraduate level, then please leave comments giving references to them. (Treatments at more advanced level too are welcome.)
</p>
<p>
Ditto, if you don't know any references but could work it out and explain the issue to me. (I really don't know the reason. I have tried to think through the issue, but have ended up deriving nothing but some guesses. These guesses could not only be plain naive, they could also easily turn out to be completely besides the point if not outright wrong. And, so, this request. (If you wish, I could easily share my guesses right here; just let me know.))
</p>
<p>
Thanks in advance for your replies.
</p>
<p>
</p>
<p>
--Ajit <br />
PS: Since these days I check blogs etc. at most only once a day or so, I might be a bit late in coming back posting replies, and if so, please bear with me a bit. Thanks!
</p>
<p>
[E&OE]
</p>
</div></div></div>Sat, 15 Feb 2014 14:03:44 +0000Ajit R. Jadhav16093 at http://imechanica.orghttp://imechanica.org/node/16093#commentshttp://imechanica.org/crss/node/16093Those were not waves: A bit historical re. Huygens' principle
http://imechanica.org/node/11825
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/153">waves</a></div><div class="field-item odd"><a href="/taxonomy/term/656">education</a></div><div class="field-item even"><a href="/taxonomy/term/1553">philosophy</a></div><div class="field-item odd"><a href="/taxonomy/term/3117">electromagnetism</a></div><div class="field-item even"><a href="/taxonomy/term/3196">light</a></div><div class="field-item odd"><a href="/taxonomy/term/7053">scientific method</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
A few points that might be of general interest:
</p>
<p>
<strong>1. The dates</strong>: The date of Huygens' first written down material, which was orally presented to the French Academy of Sciences, is 1678---in contrast to the oft-quoted date of 1690. 1690 was the year of the first, French, publication of these notes (plus other material) in the form of a book.
</p>
<p>
Surprisingly, Huygens' book was not translated into English until 1912. Reason: The English mechanicians' dogma favored the Newtonian corpuscular theory of light, warts and all, and, being a dogma---not an exercise in Reason---they either attacked, or at least didn't allow for, the wave theory to be taken any serious notice of. (Such practices are not entirely unnoticeable in our times, either!)
</p>
<p>
Thus, the translation in English of the very beginnings of the first modern wave theory of light occurred some 50 years after the last grand theoretical synthesis involving the wave nature of light by Maxwell (1861 through 1865).
</p>
<p>
Also note, the translation/publication occurred in the USA, not in UK. USA was, back then, an "also ran" country when it came to top few frontline research countries. Germany and UK were the top two superpowers in research back then. Clearly, men like Stokes (a Lucasian Professor at Cambridge) were happy to make do with the original French (or possibly a German translation thereof!)
</p>
<p>
<strong>2. Pulses, not Wave(lets):</strong> When Huygens said waves, he meant pulses. His theoretical arguments never did address a *periodic* or *regularly* repeatative phenomenon. He sure had successive rings in his diagrams, but these were not crests/troughs of waves in his imagination---even if he did allude to waves, actual waves, those on the surface of a pond, right in the same writeups. Instead, per his theory, those successive rings simply were the loci, at successive time instances, of what essentially was a pulse (something like the Dirac delta.)
</p>
<p>
Huygens was concerned with the transmission of momentum from a star to the all-pervading aether surrounding it, in the process of emission of light. Terminology was still evolving back then, and so, the word he used was "movement," not "momentum." He first considered a 1D row of balls, and the propagation of an initially imparted "movement" through that array of balls (rather like in Newton's cradle), in a *finite* amount of time. The Rationalistic philosophic tradition held instantaneous propagation for light. As far as I know, Huygens' argument is the first mechanically based (and hence respectable) departure in asserting finitude to the speed of light.
</p>
<p>
(The premise of instantaneous action at a distance has not completely died yet; indeed it is thriving well: read up any mathematical proof affirming weirdities of quantum entanglement, esp., the proofs involving variational principles/energetics arguments, e.g., von Neumann's proof, or almost any thing after that.)
</p>
<p>
Huygens then extended this logic of movement transfers, via intervening balls, from the 1D situation to 2D and 3D. He continued imagining particles of aethers arranged as systematically arranged spheres (as in an FCC lattice). He thought of an expanding circular locus as if in the process of taking a limit for the decreasing radius for the balls (which, he took to be uniform in size). Throughout this development, however, it was the transmission of a single sharp impulse that he was concerned about, not the propagation of waves. (Indeed, this was the reason why eliminating those backwards propagating "waves" came naturally to him.)
</p>
<p>
<strong>3. Phases:</strong> It was Fresnel who independently rediscovered the Huygens principle on his own, and introduced phases in the theory.
</p>
<p>
Fresnel, in fact, also independently formulated the idea of Young's interference experiment. At first, Fresnel was too isolated to even know about Huygens' or Young's work. During the day, he worked as an on-site engineer building roads (or railroads? I am not sure). He worked on optics and mathematics only in the evenings, as a part-time hobby. (Go through the idea of Fresnel integrals, and try to imagine how a young on-site engineer might have built this theory after coming back tired from the field work in the day, living in tents, with only a handful of mathematics books gathered at radom serving as references, writing with the quilt pen, and of course without electric light.)
</p>
<p>
It was Fresnel's 1815 paper that, for the first time, treated the Huygens waveslets as waves in the modern sense of the term---as something that can interfere/diffract, as something that can have a *phase*.
</p>
<p>
BTW, as far as dates go, note that de Alembert had already formulated and solved the wave PDE by then (around 1757); it was not available in Huygens' time.
</p>
<p>
<strong>4. A quirk in the development of mathematics:</strong> Huygens was brilliant enough to solve the problem of finding the curvature for the guide-walls of a short pendulum of uniform oscillations. It therefore is especially intriguing that he still didn't think of including anything like a phase in his account of this principle discovered by he himself.
</p>
<p>
This curious omission is sort of like Einstein publishing original arguments on both Brownian movement (leading in part to Perin's Nobel) and the photoelectric effect (his own Nobel), both in the same year (1905), but without connecting the two, in putting forth an explanation regarding the nature of light.
</p>
<p>
<strong>5. The Correspondence with the Action/Variational principles:</strong> Yet, it is interesting to note that Huygens had, right back then, explicitly noted the correspondence between his principle and Fermat's principle. A local, transient, propagating process had thus already been shown to produce the same results as those by a "static," global---and, IMO, very dull sort of---principles, viz., variational principles, in particular, the action principle.
</p>
<p>
Do note that this argument was put forth way before even Maupertius' times, let alone those of Euler, Lagrange, or Hamilton.
</p>
<p>
If you read modern (20th century) accounts, you are likely to come out with the impression (i) that there has been this fantastically fundamental development of the action and variational principles, beginning with Hero's or Fermat's and culminating with Hamilton's (or someone else's still later on), (ii) that the fundamental way to justify Huygens' principle to derive it within the PDE formalism, which itself is to be derived within a topological and variational formalism (with fond allusions to be made to either non-viability or outright triviality of Huygens' principle in comparing 1D vs 2D vs 3D vs nD situations, and (iii) that a rigorous proof of Huygens' principle has been established only in the second half of the 20th century---together with its enormous "limitations" or triviality, of course.
</p>
<p>
Nothing is farther from the truth. Variational principles (and their hodge-podge equivalent of the weighted residuals etc.) are among the high favorites of modern mathematicians (esp. the mathematically inclined Indians, Russians, Frenchmen, and Americans) perhaps only because it's so hard (at least for them) to associate a neat physical mechanism with these principles. (It's the same story as with the Fourier/spectral analysis. They love anything for which a physical system (or "picture") is hard or impossible to conceive of.) However, the outright failures of such mathematicians to supply physical systematic explanation does not make variational/action/topological formalism any more fundamental than Huygens'.
</p>
<p>
It might be best to end this post with a sentence shamelessly lifted from Encyclopaedia Britannica's biographic entry on Fresnel:<cite></cite>
</p>
<p>
<cite>"Although his work in optics received scant public recognition during his lifetime, Fresnel maintained that <strong>not even acclaim from distinguished colleagues could compare with the pleasure of discovering a theoretical truth or confirming a calculation experimentally.</strong>"</cite> [<strong>emphasis</strong> mine]
</p>
<p>
BTW, have a happy Randsday!
</p>
<p>
--Ajit
</p>
<p>
[E&OE]
</p>
</div></div></div>Thu, 02 Feb 2012 06:55:01 +0000Ajit R. Jadhav11825 at http://imechanica.orghttp://imechanica.org/node/11825#commentshttp://imechanica.org/crss/node/11825Journals in Physics and Engineering, and Preprint Servers Like arXiv
http://imechanica.org/node/10895
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/162">computational mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/238">journals</a></div><div class="field-item even"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/920">physics</a></div><div class="field-item even"><a href="/taxonomy/term/948">Computational Science</a></div><div class="field-item odd"><a href="/taxonomy/term/973">software</a></div><div class="field-item even"><a href="/taxonomy/term/1044">engineering</a></div><div class="field-item odd"><a href="/taxonomy/term/4617">arXiv</a></div><div class="field-item even"><a href="/taxonomy/term/6513">preprints</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi all,
</p>
<p>
</p>
<p>
1. In the past, we have had quite some discussion regarding both open-access and open-access journals. However the slant in this blog post is different. I am not concerned here much about open-access journals per say.
</p>
<p>
Here, I am concerned about the policies that the prominent commercial journals keep regarding posting preprints on the Internet before these articles are submitted to them. I would like to know about policies kept in this regard by the commercial journals in the fields of physics, mechanics, and engineering (including software engineering, computational science and engineering, etc.).
</p>
<p>
</p>
<p>
2. The place of "Nature" among journals is pre-eminent. People, even those at highest ranked universities, with pride state acceptance of their work at "Nature." It also happens, I guess, to be the oldest continuously published scientific journal (older than its closest competitors, e.g., Science).
</p>
<p>
"Nature Physics," these days, does allow putting your pre-prints on arXiv. Since 2010, Nature Physics has a policy that says that:
</p>
<p>
<font color="#0000FF">"...any submission to <em>Nature Physics</em> or its sister journals may be posted, in that original submitted form, on the preprint server (although we do ask that the final, revised and accepted version is not posted until six months after publication in the journal; the published version, in the <em>Nature Physics</em> layout, may not be posted)."</font> [<a href="http://blogs.nature.com/nautilus/2010/03/nature_physics_calls_for_suppo_1.html" target="_blank">^</a>]
</p>
<p>
Also that:
</p>
<p>
<font color="#0000FF">"You are welcome to post pre-submission versions or the original submitted version of the manuscript on a personal blog, a collaborative wiki or a preprint server at any time (but not subsequent pre-accept versions that evolve due to the editorial process)." </font>[<a href="http://www.nature.com/authors/policies/confidentiality.html" target="_blank">^</a>].
</p>
<p>
Reasonable enough! (BTW, no, I am not a socialist. In fact, I consider myself a capitalist, in Ayn Rand's sense of the term.)
</p>
<p>
</p>
<p>
3. Now I know that many other journals of a similar standing---most notably, "Science"---do not have a clear general policy that allows for doing so:
</p>
<p>
<font color="#0000FF">"Posting of a paper on the Internet may be considered prior publication that could compromise the originality of the <em>Science</em> submission, although we do allow posting on not-for-profit preprint servers in many cases. Please contact the editors for advice about specific cases."</font> [<a href="http://www.sciencemag.org/site/feature/contribinfo/faq/#prioronline_faq" target="_blank">^</a>]
</p>
<p>
</p>
<p>
4. I tried to locate for myself if other journals had any policy statement on this matter. These journals most notably included: "PRL," "Foundations of Physics," "PNAS," etc. I could not succeed doing so. (The information may be there, but it is hard to find. Even on "Science" mag Web site, it's a few links deeper and not at all obvious, whereas at "Nature," it's more or less very easily accessible starting from the home page.)
</p>
<p>
</p>
<p>
<strong>5. So, here is my request:</strong>
</p>
<p>
Are you aware of policies in this regard maintained by the journals which you help edit or to which you often submit your articles---esp. the journals from the mechanics and engineering fields? What are these policies like? Care to share (about those policies)? BTW, here, also the journals on computational mechanics and those dealing with software in engineering, are to be included.
</p>
<p>
Does posting a preprint of a paper at iMechanica disqualify submitting it to the journals that iMechanicians often use? How about posting it at arXiv?
</p>
<p>
What if I discuss the basic germ of an idea itself here at iMechanica, even though it's not written in the format of a paper? How strict or lenient are the views regarding such pre-submission publication that the journals in Mechanics field take? How do you know---in the sense, to what extent could I be reassured?
</p>
<p>And, finally, how does the iMechanica Creative Commons License work out for this situation? I guess that at iMechanica it's the CC BY-NC-SA license [<a href="http://creativecommons.org/licenses/by-nc-sa/3.0/" target="_blank">^</a>] that we follow/require, and not the CC BY-NC-ND one [<a href="http://creativecommons.org/licenses/by-nc-nd/3.0/" target="_blank">^</a>]. What if I wish to publish at iMechanica, but only with the latter (more restrictive) license? (However, please note, it's not just the policy of iMechanica that is important to me; I actually am more concerned with the preceding questions.) </p>
<p>
</p>
<p>
Thanks in advance for any information and clarifications.
</p>
<p>
--Ajit
</p>
<p>
[E&OE]
</p>
<p>
</p>
</div></div></div>Mon, 08 Aug 2011 09:21:52 +0000Ajit R. Jadhav10895 at http://imechanica.orghttp://imechanica.org/node/10895#commentshttp://imechanica.org/crss/node/10895What would you choose as the Top 5 Equations? Top 10?
http://imechanica.org/node/10188
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/920">physics</a></div><div class="field-item even"><a href="/taxonomy/term/1577">equation</a></div><div class="field-item odd"><a href="/taxonomy/term/1867">mathematical modeling</a></div><div class="field-item even"><a href="/taxonomy/term/6221">top 5 equations</a></div><div class="field-item odd"><a href="/taxonomy/term/6222">top 10 equations</a></div><div class="field-item even"><a href="/taxonomy/term/6223">fundamental equations</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Equations are of central importance in all of science and engineering, but especially so in mechanics.</p>
<p>Even leaving aside algebraic equations, handbooks on PDEs alone list hundreds of equations. However, a few of these do stand out, either because they encapsulate some fundamental aspect of physics/science/engg., or because they serve as simpler prototypes for more complex situtations, or simply because they are so complex as to be fascinating by themselves. There might be other considerations too... But the fact is, some equations really do stand out as compared to others.</p>
<p>If so, what equations would you single out as being most important or interesting? To make the matters more interesting, first, please think of making a short list of only 5 equations. Then, if necessary, make it one of 10 equations---but no more, please! :)</p>
<p>As to me, here is my list, put together in a completely off-hand manner:</p>
<p>Top five:<br />
(1-3) The linear wave-, diffusion- and potential-equations.<br />
(4) The Schrodinger equation<br />
(5) The Navier-Stokes equation</p>
<p>Additionally, perhaps, these equations:<br />
(6) The Maxwell Equations<br />
(7) The equation defining the Fourier transform<br />
(8) Newton's second law (dp/dt = F)<br />
(9) The Lame equation (of elasticity)</p>
<p>Am I already nearing the limit or what... Hmm... But, nope, I am not sure whether I want to include E = mc^2. ... I will give this entire matter a second thought some time later on.</p>
<p>But, how about you? What would be your choices for the top 5/10 equations? Why?
</p>
<p>
</p>
<p>
--Ajit<br />
PS: Also posted today in the Computational Scientists & Engineers group at LinkedIn, and also will post soon at my personal blog. </p>
<p>[E&OE]
</p>
<p>
</p>
</div></div></div>Sat, 30 Apr 2011 12:03:13 +0000Ajit R. Jadhav10188 at http://imechanica.orghttp://imechanica.org/node/10188#commentshttp://imechanica.org/crss/node/10188An interesting arXiv paper: "Precession optomechanics"
http://imechanica.org/node/10166
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/283">Mechanics of Photonic Devices</a></div><div class="field-item odd"><a href="/taxonomy/term/838">quantum mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/3403">quantum</a></div><div class="field-item odd"><a href="/taxonomy/term/6207">photon</a></div><div class="field-item even"><a href="/taxonomy/term/6208">photon mechanics</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi all,
</p>
<p>Just thought that the following paper archived at the arXiv yesterday could be of general interest to any mechanician:</p>
<p>
Xingyu Zhang, Matthew Tomes, Tal Carmon (2011) "Precession optomechanics," arXiv:1104.4839 [<a href="http://arxiv.org/abs/1104.4839" target="_blank">^</a>]
</p>
<p>
The fig. 1 in it makes the matter conceptually so simple that the paper can be recommended to any mechanician for his general reading, and not only to a specialist in the field.
</p>
<p>
<br />
--Ajit
</p>
<p>
[E&OE]
</p>
</div></div></div>Wed, 27 Apr 2011 07:26:09 +0000Ajit R. Jadhav10166 at http://imechanica.orghttp://imechanica.org/node/10166#commentshttp://imechanica.org/crss/node/10166Any tips/comments regarding the latest version of the C++ library: Eigen (v. 3.0)?
http://imechanica.org/node/9987
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/2864">solvers</a></div><div class="field-item odd"><a href="/taxonomy/term/3255">linear solvers</a></div><div class="field-item even"><a href="/taxonomy/term/6112">Eigen 3.0</a></div><div class="field-item odd"><a href="/taxonomy/term/6113">eigenvalues computations</a></div><div class="field-item even"><a href="/taxonomy/term/6114">CPP</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi all,
</p>
<p>
<strong>1. A new version of Eigen (v 3.0 now) is out</strong> (on March 23, 2011), and it seems promising. First, a few links:
</p>
<p>
The main page for the project is here: [<a href="http://eigen.tuxfamily.org/index.php?title=Main_Page" target="_blank">^</a>]. The page for v.3.0 is here: [<a href="http://eigen.tuxfamily.org/index.php?title=3.0" target="_blank">^</a>]. It seems to be <em>very</em> fast: [<a href="http://eigen.tuxfamily.org/index.php?title=Benchmark" target="_blank">^</a>].
</p>
<p>
<br /><strong>2. No compilation issues on the Win32 platform:</strong> I just downloaded it today, and found that it compiles OK with VC++ 10.0 on Win32 (the BUILD_ALL project). I haven't had the time to try anything further, though.
</p>
<p>
<br /><strong>3. Please share your experience:</strong> For whatever reasons, if anyone here eventually decides to use it, or not to use it, or has any comments or tips to offer, I would like to hear from him/them, esp. in respect of the following points:
</p>
<p>
<strong>3.1</strong> <strong>For eigenvalue computations</strong>, comparative advantages/disadvantages, including speed comparisons, with the existing libraries like ARPACK, Trilinios, etc.
</p>
<p>
Assume that the system to solve has origins in a typical structural/solid mechanics FEM. If possible, please share your experience for global [A] matrix orders of 3k, 10k, 100k, and bigger, the matrices being the typical FEM ones: sparse, symmetric, etc.
</p>
<p>
[This note is not really necessary, but just in the interest of clarity: If a matrix has 10 rows and 10 columns, then its order is taken to be 10, not 100.]
</p>
<p>
<strong>3.2 For solving a linear FEM-generated system</strong> using a direct solver, speed comparisons with the existing FORTRAN/C++ code as implemented in GOTO (latest version), Taucs, etc. Again, matrix orders go from 3k to 100k and bigger, and the matrices, of course, are sparse, symmetric, etc.
</p>
<p>
<strong>4.</strong> Any <strong>other tips</strong> you care to share would also be appreciated.
</p>
<p>
</p>
<p>
Thanks in advance.
</p>
<p>
--Ajit
</p>
<p>
[E&OE]
</p>
</div></div></div>Thu, 24 Mar 2011 13:51:35 +0000Ajit R. Jadhav9987 at http://imechanica.orghttp://imechanica.org/node/9987#commentshttp://imechanica.org/crss/node/9987Open House: Can you define FEM in one line?
http://imechanica.org/node/9919
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/248">finite element analysis</a></div><div class="field-item odd"><a href="/taxonomy/term/370">finite element methods</a></div><div class="field-item even"><a href="/taxonomy/term/447">Finite Element Method</a></div><div class="field-item odd"><a href="/taxonomy/term/846">FEM</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Can you define FEM in one line?
</p>
<p>
If yes, what would it be? And, in that case, permit me a second question: How?
</p>
<p>
</p>
<p>
...Really interested in knowing what the members of this community think (of this matter), if they do...
</p>
<p>
</p>
<p>
--Ajit
</p>
<p>
[E&OE]
</p>
<p>
</p>
</div></div></div>Thu, 10 Mar 2011 06:40:59 +0000Ajit R. Jadhav9919 at http://imechanica.orghttp://imechanica.org/node/9919#commentshttp://imechanica.org/crss/node/9919What would you like for an undergraduate book on QM to explain to you?
http://imechanica.org/node/9791
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/838">quantum mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/976">undergraduate education</a></div><div class="field-item even"><a href="/taxonomy/term/1036">books</a></div><div class="field-item odd"><a href="/taxonomy/term/6006">classical physics</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
</p>
<p><strong>1. Background:</strong></p>
<p>A couple of things concerning books happened recently, in the last week or two.</p>
<p>(i) Dr. Biswajit Banerjee announced last week that a new book on metamaterials and waves in composites authored by him is coming out in print within a few months.</p>
<p>(ii) In one of my regular visits to bookshops, I noticed a hardcover copy of Prof. Allan Bower's book on mechanics of solids. Someone (or more likely, some institution) in Pune had ordered it, and the copy happened to be lying near the top of the stack. The book had been online for some time and I had browsed through it already. So, now, when I opened the printed version out of curiosity, I directly went to the Preface part. This is how Prof. Bower opens his Preface:</p>
<p>
</p>
<p> <font color="#0000ff">Ronald Rivlin, a pioneer in the field of nonlinear elasticity, was asked once whether he intended to write a treatise on his field. "Why should I write a book?" he replied. "<strong>People write books to learn a subject.</strong> I already know it."<br /></font></p>
<p>[Emphasis in <strong>bold</strong> is mine.]</p>
<p>
</p>
<p>
<strong>2. "People write books to learn a subject:"</strong></p>
<p>That line really hit me---in two different ways.</p>
<p>Firstly, it has always been a part of the folklore among engineering college teachers that the best way to learn a subject is to teach it. I first heard it in 1984 in the teachers' room in Bharati Vidyapeeth's College of Engineering. I had then realized, first-hand, how true the saying was. And, I had wondered right back then: if it is not possible to teach a course, how about writing a book on it---mainly for learning the subject.</p>
<p>Secondly, it occurred to me now that if I were to pick out just one subject that I wish to understand better, it would undoubtedly have to be Quantum Mechanics. And, so the thought became: why not write a book on QM---in order to learn it. The idea is not so objectionable after all. Especially if you consider that most physicists (and all mainstream physicists) assure all the rest of us (and very probably also to themselves) that nobody understands quantum mechanics.</p>
<p>Actually, I was planning to write 2--3 journal papers that would extend and better explain some of the fundamental features of quantum physics as using my new approach. I had been gathering my thoughts and background material together. Yet, the more I thought about it, the better it began looking to me that perhaps it makes sense to first write a book on QM before writing those articles.</p>
<p>
</p>
<p>
<strong>3. A question for you:</strong></p>
<p>So, with those thoughts in mind, I would like to raise the title question to you:</p>
<p><font color="#ff0000"><strong>What would you like an undergraduate book on QM to explain to you?</strong></font> Or, better still (because it makes it more personal): <font color="#ff0000"><strong>What do you want me to explain to you, concerning the introductory topics of QM?</strong></font></p>
<p>By introductory topics, I mean the topics covered in (roughly in the order of increasing depth or complexity): Eisberg and Resnick, Hameka, Phillips, Scherrer, Liboff, Gasiorowicz, etc. Also, many other books falling within this same range, e.g.: Griffiths, Zettilli, etc.</p>
<p>If you have any points to raise in this regard, feel free to do so. I will keep this particular post open for comments until I finish writing the book (whenever I come to do that).</p>
<p>Any one may write me. However, a word about the intended audience and the nature of treatment is in order.</p>
<p>
</p>
<p>
<strong>4. The intended audience and the intended treatment of the book:</strong></p>
<p>The primary intended audience is: 3rd/4th year undergraduate students in engineering and applied sciences. Especially, those who did not have a course on electromagnetic fields beforehand (such as those majoring in mechanical, materials, chemical, aeronautical, etc. areas).</p>
<p>The treatment will follow the historical order of developments. I will begin with a summary of the pre-requisites, including very rapid (and perhaps rather conceptual) surveys (done from my own viewpoint, sort of "an engineer's viewpoint") of such topics as: Newton's laws; partial differential equations; The beginning of the energy- and fields-based reformulation of Newton's mechanics by Leibniz; complex numbers and Euler's identity; Lagrangian reformulation of Newtonian mechanics; relevant matrix theory; Fourier theory; relevant probability theory; Hamilotonian reformulation of Newtonian mechanics; electromagnetic field theory from say Coulomb to Maxwell and Hertz; the eigenvalue problem in classical physics; cavity radiation; special relativity (taken as highlighting certain features of the classical electrodynamics).</p>
<p>The QM proper will begin with Planck, of course, and will closely follow the historical sequence, though the notation might be modern---however, I wish to emphasize that I will not introduce Dirac's notation until he introduces it, so to say. Similarly, I will introduce Heisenberg's matrix mechanics before Schrodinger's wave mechanics. I intend to leave the reader at about 1935, though an appendix or two on entanglement is possible.</p>
<p>I will try to keep the length at about 300 pages at the most. I would love to see if it can be done within 250 pages, but doing so seems not easily possible. I will leave out many conceptual explanations, esp. of the prior theories, primarily because that burden has already been relieved for me in the form of Manjit Kumar's book. In a way, I do see my intended book as being complementary to Kumar's book.</p>
<p>I will cover neither Feynman's reformulation nor Bohm's ideas.</p>
<p>The book will also not be a vehicle to introduce my own approach. However, it is impossible for any author to keep aside his viewpoint, while thinking or writing. In this case, I will try to restrict myself to highlighting the wonderful series of ridiculous conclusions to which the earlier theories lead (often isolated and put forth by the formulators of those theories themselves), and providing some explicit hints for getting out of them. However, in the planned book, I will not go beyond providing hints alone. ... Yes, I will be willing to give out some of the material or thoughts that, properly, should have come in the research articles first. However, as far as journal articles go, frankly, I do not care!</p>
<p>The book <strong>will </strong>carry mathematics. (It won't be addressed to the layman.) It will carry derivations too, but only in simple and essential terms. (By simple, I do not mean: devoting inordinate time to one-dimensional and time-independent cases. Adopting this policy may mean that the book ends up being suitable only to the beginning graduate students of engineering. If so, that would be OK by me.)</p>
<p>Further, I would often provide the derivations in an order other than what is found in the usual treatments. For instance, the prerequisites part itself will cover spherical harmonics---right in the context of classical physics. Also, the angular momentum of the EM field. (Yes, the prerequisites part will be a major part in this book, perhaps 40% of the total material.) The prerequisites part will also point out the issue of the instantaneous action-at-a-distance, right in the PDE section.</p>
<p>In short, it will almost be a university text-book. Except that I don't expect any university to adopt it for their classroom usage! Therefore, there won't be any routine kind of chapter-end exercises, nor a section at the beginning of the chapter motivating the student. However, some pointers for further thought might be provided via notes at the end of the book.</p>
<p>Most readers here at iMechanica (and many of the readers of my personal blog) come from engineering and applied sciences background. They are likely to have run into QM as a part of their courses on modern physics, solid physics, nanomaterials, etc. They might have had run into issues concerning the real QM. I would love it if I can provide answers to their questions. And, it goes without saying that students of "pure" sciences---physics, chemistry, etc.---would be as welcome as those of engineering sciences. This book would be directed at them, not at the layman---or at the philosophers.</p>
<p>Of course, as far as raising the questions go, any one may feel free to submit his query via a comment. (I may not reply every comment at iMechanica; I do not moderate anything here.)</p>
<p>All in all, it would be a book written by an engineer, and primarily for engineering/applied science undergraduates/beginning postgraduate students. There already is an excellent book in this space: Prof. Leon van Dommelen's online book. I really like it, but thought that I would have approached many things differently, and so, thought of writing my book. Most important difference, to my mind, is that I would stick to the historical approach throughout. But, yes, as far as undertaking this huge an effort goes, Prof. van Dommelen's book would remain a kind of an inspiration for me.</p>
<p>
</p>
<p>
<strong>5. One final point: Sample questions:</strong></p>
<p>Some time ago, I had written a list of questions that I thought UG students should ask their professors. However, there were also other topics in that post. For ease of direct referencing, here I am copy-pasting those questions below (with a bit of editing). Go through them and see if you have any other questions you wish to raise:</p>
<p>
</p>
<ol><li>
Why are quantum-mechanical forces conservative?</li>
<li>Does the usual time-dependent Schrodinger’s equation (TDSE) apply to propagation of photons? If yes, why does no textbook ever illustrate TDSE involving photons? If not, what principle goes against applicability of TDSE to photons?</li>
<li>What kind of physics would result if the QM wavefunction were not to be complex-valued but real (scalar)-valued? What if the wavefunction were to be deterministic rather than probabilistic? What contradictions would result in each case?</li>
<li>Does QM at all need an interpretation? If yes, why? Why is it that no other theory of physics seems to need special efforts at interpreting it but only QM does, esp. so if all physics theories ultimately describe the same reality? If QM does not need an interpretation, why do people talk about the phrase: “interpretation of QM”? What do they mean by that phrase?</li>
<li>What, precisely, is the physical meaning of an operator? Please don’t simply repeat for us its definition. Instead, please give us the physical meaning of the concept. Or is it the case that no physical meaning is possible for this concept and that it is doomed to remain an exclusively mathematical concept? If yes, why use it in the postulates of a physical theory—without ever taking the care to define its physical correspondents?</li>
<li>Are all quantum-theoretical operators Hermitian? If yes, why? What physical fact does this property indicate/highlight/underscore? What if they are not Hermitian?</li>
<li>Give one example of an important eigenvalue problem from classical mechanics in which the differential equation formalism is very clearly shown to be equivalent to the matrix formalism.</li>
<li>Does the QM theory necessarily require the concept of a wavepacket when it comes to detailing what a QM particle is? If yes, why? What would happen if it were not a packet of waves but instead just a monochromatic wave? If the theory does not necessarily require packets of waves, then can you suggest us any alternative treatment—if there is one?</li>
<li>In every differential equation we have studied thus far, the primary unknown always carried some or the other physical units/dimensions. For example, for mechanical waves, the primary variable would be the displacement from the equilibrium position. But the QM wavefunction seems to be a dimensionless quantity; at least, textbooks don’t seem to note down any units for it. Is it a dimensionless quantity? Why? What important things does this tell us about the nature of theorization followed in QM?</li>
<li>Is QM an action-at-a-distance theory?</li>
<li>How, precisely, does QM relate to the classical EM? Is the term V(x,y,z,t) in Schrodinger’s equation to be understood in the classical sense? If yes, why do people say that between the two, QM is more basic to EM?</li>
<li>Explain precisely how the Newtonian mechanics is implied by QM.</li>
<li>And, one question I raised yesterday, via a tweet: In the mainstream interpretation---taught to all undergraduates world-wide---it is not meaningful to speak of emission of particles. Particles are never emitted, only absorbed---because only absorption can be "observed." The whole world is a series of absorptions, so to speak. True or false? (Hint: In Keynesian economics, there are only consumers, no producers!)
</li>
</ol><p>
<br />
Those were the questions I thought of, some time ago. ... I am sure you can do better.
</p>
<p>
I now look forward to hearing from you.
</p>
<p>
</p>
<p>
--Ajit
</p>
<p>
Also posted at my personal blog [<a href="http://ajitjadhav.wordpress.com/2011/02/12/what-would-you-like-an-undergraduate-book-on-qm-to-explain-to-you/" target="_blank">^</a>].
</p>
<p>
[E&OE]
</p>
<p>
</p>
</div></div></div>Sat, 12 Feb 2011 09:54:34 +0000Ajit R. Jadhav9791 at http://imechanica.orghttp://imechanica.org/node/9791#commentshttp://imechanica.org/crss/node/9791How about having a special Mechanics Gallery here?
http://imechanica.org/node/9752
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/643">iMechanica</a></div><div class="field-item odd"><a href="/taxonomy/term/5986">Mechanics Gallery</a></div><div class="field-item even"><a href="/taxonomy/term/5987">What is mechanics</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Here is an idea I submit for consideration by all iMechanicians, but esp. so by the admins and moderators. Discussion is welcome.
</p>
<p>
</p>
<p>
<strong>Idea: Why not have a Mechanics Gallery section here? </strong>
</p>
<p>
Origin: Recently, I was browsing for some OpenGL-encapsulating C++ class libraries, e.g. OpenSceneGraph, VTK, the game development libraries, etc. The Web sites of all such libraries always carry a "Gallery" page which is designed to attract the potential users. The Gallery page shows the capabilities and advantages of that library/framework.
</p>
<p>
A similar idea is possible for our field too.
</p>
<p>Many iMechanicians do excellent work in mechanics. Their own Web sites often carry attractive illustrations and explanations of their research and education programs. Thus, many of us could very easily contribute exhibits for the iMechanica Gallery.</p>
<p>
We can perhaps have a <strong>main Gallery Section page</strong> which will carry a list of the resources being exhibited. The main page may also carry brief (say 100 words) description for each exhibit, preferably with an<br />
attractive graphic, with hyperlinks provided for further information (Web URLs, PDF files, graphics, software, etc.).
</p>
<p>
<strong>Each </strong><strong>exhibit item</strong> should identify the broad area of mechanics via our usual tags. Further, the item should also explicitly identify <strong>the</strong> <strong>assumed level of the target audience, </strong>e.g.: layman, undergraduate, graduate students, advanced researchers, etc.
</p>
<p>
The resource items should be rather <strong>general-purpose</strong> in nature; they should avoid the tunnel-vision syndrome. The brief description should avoid equations as far as possible, though I do agree that in certain cases using equations would be unavoidable. The idea is to keep the focus more on the main concepts being illustrated.
</p>
<p>
The exhibits may come from professionals working in <strong>industry</strong> as well as from <strong>lab. researchers</strong> and <strong>academics (professors as well as students).</strong>
</p>
<p>
I sugget that at least in the beginning, there could be a limit on the number of exhibit items that an individual might submit, say, <strong>at most 2 exhibits per person</strong>. The limit is expected to help the member think hard as to what item of <strong>general </strong>interest he might submit.
</p>
<p>
The visual format of the gallery may be finalized after further discussion.
</p>
<p>
<strong>Over a period of time, the Gallery could easily become a good initial place of contact between the layman and professionals from other fields on the one hand, and we iMechanicians on the other.<br /></strong>
</p>
<p><strong>Feel free to add further ideas, suggestions and discussions in this regard.<br /></strong></p>
<p>
</p>
<p>
- - - - -
</p>
<p>
History: This idea was initially posted as a comment at Bisjwajit's blog a few days ago. I wanted to edit the text before posting it to a separate thread, but find no time, and so I have just copy-pasted it here.
</p>
<p>
Biswajit noted that Drupal may not be the ideal platform for a gallery. At the Drupal site, they say: "Convert <strong>any</strong> website layout or template into a Drupal theme - easily!" [<a href="http://drupal.org/node/313510" target="_blank">^</a>] (<strong>bold</strong> emphasis added.) I have no idea how easy it would be to create a special Gallery page, but at Drupal they say that it is possible to customize only a section of the overall Web site. Admins/IT Support people may be able to tell us better.
</p>
<p>
<strong>Over to you all.<br /></strong>
</p>
<p>
--Ajit
</p>
</div></div></div>Mon, 07 Feb 2011 09:40:50 +0000Ajit R. Jadhav9752 at http://imechanica.orghttp://imechanica.org/node/9752#commentshttp://imechanica.org/crss/node/9752How to supply a visualization for the displacement gradient tensor
http://imechanica.org/node/9159
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/973">software</a></div><div class="field-item odd"><a href="/taxonomy/term/5262">Tensor Visualization</a></div><div class="field-item even"><a href="/taxonomy/term/5263">Visualization</a></div><div class="field-item odd"><a href="/taxonomy/term/5550">displacement</a></div><div class="field-item even"><a href="/taxonomy/term/5689">toy</a></div><div class="field-item odd"><a href="/taxonomy/term/5690">displacement gradient</a></div><div class="field-item even"><a href="/taxonomy/term/5691">rotation</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi all,
</p>
<p>
[Warning: The writing is long, as is usually the case with my posts :)]
</p>
<p>
It all began with a paper that I proposed for an upcoming conference in India. The extended abstract got accepted, of course, but my work is still in progress, and today I am not sure if I can meet the deadline. So, I may perhaps withdraw it, and then submit a longer version of it to a journal, later.
</p>
<p>
Anyway, here is a gist of the idea behind the paper. I am building a very small pedagogical software called "toyDNS." DNS stands for <strong>D</strong>isplacement, strai<strong>N</strong>, and stre<strong>S</strong>s, and the order of the letters in the acronymn emphasizes what I (now) believe is the correct hierarchical order for the three concepts. Anyway, let's keep the hierarchical order aside and look into what the software does---which I guess could be more interesting.
</p>
<p>
The sofware is very very small and simple. It begins by showing the user a regular 2D grid (i.e. squares). The user distorts the grid using the mouse (somewhat similar to the action of an image-warping software). The software then, immediately (in real time, without using menus etc.) computes and shows the following fields in the adjacent windows: (i) the displacement vector field, (ii) the displacement gradient tensor field, (iii) the rotation field, (iv) the strain field, (v) and the stress field. The software assumes plane-stress, linear elasticity, and uses static configuration data for material properties like nu and E. The software also shows the boundary tractions (forces) that would be required to maintain the displacement field that the user has specified.
</p>
<p>
Basically, the idea is that the beginning undergraduate student encountering the mechanics of materials for the first time, gets to see the importance of the rotation field (which is usually not emphasized in textbooks or courses), and thereby is able to directly appreciate the reason why an arbitrary displacement field does uniquely determines the corresponding stress fields but why the converse is not true---why an arbitrary stress/strain field cannot uniquely determine a corresponding displacement field. To illustrate this point (call it the compatibility issue if you wish) is the whole rationale behind this toy software.
</p>
<p>
Now, when it comes to visualizing the fields, I can always use arrows for showing the vector fields of displacements and forces. For strains and stresses, I can use Lame's ellipse (in 2D). In fact, since the strain and stress fields are symmetric, in <em>2D</em>, they each have only 3 components, which means that the symmetric tensor object as a whole can directly map onto an RGB (or HLS) color-space, and so, I can also show a single, full-color field plot for the strain (or stress) field.
</p>
<p>
Ok. So far, so good.
</p>
<p>
The problem is with the displacement gradient tensor (DG for short here). Since the displacement field is arbitrary, there is no symmetry to the DG tensor. Hence, even in 2D, there are 4 independent components to it---i.e. one component too many than what can be accomodated in the three-component color-space. So, a direct depiction of the tensor object taken as a whole is not possible, and something else has to be done. So, I thought of the following idea.
</p>
<p>
First, the notation. Assume that the DG tensor is being described thus:
</p>
<p>
DG11 DG12<br />
DG21 DG22
</p>
<p>
=
</p>
<p>
du/dx du/dy<br />
dv/dx dv/dy
</p>
<p>
where DGij are the components of the DG tensor, u and v are the x- and y-components of the displacement field, and the d's represent the <em>partial</em> differentation. (Also imagine as if the square brackets of the matrix notation are placed around the components listing above.)
</p>
<p>
Consider that DGij can be taken to represent a component of a vector that refers to the i-th face and j-th direction. Understanding this scheme is easier to do for the stress tensor. For the stress tensor, Sij is the component of the traction vector acting across the i-the face and pointing in the j-th direction. For instance, in fig. 2.3 here: <a href="http://en.wikipedia.org/wiki/Stress_(mechanics">http://en.wikipedia.org/wiki/Stress_(mechanics</a>), T^{e_1} is the vector acting across the face normal to the 1-axis.
</p>
<p>
Even if the DG tensor is not symmetric, the basic idea would still apply, wouldn't it?
</p>
<p>
Thus, each row in the DG tensor represents a vector: the first row is a vector acting on the face normal to the x-axis, and the second is another vector (which, for DG, is completely indpendent of the first) acting on the face normal to the y-axis. For 2D, subsitute "line" in place of "face."
</p>
<p>
If I now show these two vectors, they would completely describe the DG tensor. This representation would be somewhat similar to the "cross-bars" visualization commonly used in engineering software for the stress tensor, wherein the tensor field is shown using periodically cross-bars---very convenient if the grid is regular and uniform and has square elements.
</p>
<p>
Notice a salient difference, however. Since the DG tensor is <em>asymmetric</em>, the two vectors will not in general lie at right-angles to each other. The latter is the case only with the symmetric tensors such as the strain and stress tensors.
</p>
<p>
My question is this: Do you see any issues with this kind of visualization for the DG tensor? Is there any loss of generality by following this scheme of visualization? I mean, I read some literature on visualization of asymmetric tensors, and noticed that they sometimes worry about the eigenvalues being complex, not real. I think that complex eigenvalues would not be a consideration for the above kind of depiction of the DG tensor---the rotation part will be separately shown in a separate window anyway. But, still, I wanted to have the generality aspect cross-checked. Hence this post. Am I missing something? assuming too much? What are the other things, if any, that I need to consider? Also: Would you be "intuitively" comfortable with this scheme? Can you think of or suggest any alternatives?
</p>
<p>
Comments are welcome.
</p>
<p>
--Ajit
</p>
<p>
[E&OE]
</p>
</div></div></div>Mon, 25 Oct 2010 08:06:29 +0000Ajit R. Jadhav9159 at http://imechanica.orghttp://imechanica.org/node/9159#commentshttp://imechanica.org/crss/node/9159Mohr's Circle---When Was the Last Time You Used It in Your Professional Engineering Work?
http://imechanica.org/node/8341
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item odd"><a href="/taxonomy/term/5261">Mohr's Circle</a></div><div class="field-item even"><a href="/taxonomy/term/5262">Tensor Visualization</a></div><div class="field-item odd"><a href="/taxonomy/term/5263">Visualization</a></div><div class="field-item even"><a href="/taxonomy/term/5264">Post-Processor</a></div><div class="field-item odd"><a href="/taxonomy/term/5265">Professional Engineering</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
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.)</p>
<p>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:</p>
<p>
<strong>Question 1:<br /></strong>
</p>
<p>
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?</p>
<p>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.</p>
<p>In case you are curious, my own answer to this question is: No, never. I would like to know yours.
</p>
<p>
<br /><strong>Question 2:<br /></strong>
</p>
<p>
The second question just pursues one of the lines indicated in the first.</p>
<p>In a modern FEM postprocessor, visualizations of stress/strain patterns are provided, usually via field plots and contour lines.</p>
<p>For instance, they show field plots of individual stress tensor components, one at a time.</p>
<p>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.</p>
<p>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.</p>
<p>The questions here are:</p>
<p><strong>(2.a)</strong> 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?
</p>
<p>
<strong>(2.b)</strong> 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.
</p>
<p>
<strong>(2.c)</strong> 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.</p>
<p>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.</p>
<p>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.</p>
<p>Thanks in advance for your comments.</p>
<p>PS: Also posted in my other, personal blog here [<a href="http://ajitjadhav.wordpress.com/2010/06/03/mohrs-circle-when-was-the-last-time-you-used-it-in-your-professional-engineering-work/" target="_blank">^</a>]
</p>
<p>
[E&OE]
</p>
</div></div></div>Thu, 03 Jun 2010 17:01:13 +0000Ajit R. Jadhav8341 at http://imechanica.orghttp://imechanica.org/node/8341#commentshttp://imechanica.org/crss/node/8341Use Only the Angular Quantities in Analysis? Three Sample Problems to Consider...
http://imechanica.org/node/8313
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/277">energy</a></div><div class="field-item odd"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/989">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/5216">linear momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5217">momentum</a></div><div class="field-item odd"><a href="/taxonomy/term/5218">angular momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5219">conservation</a></div><div class="field-item odd"><a href="/taxonomy/term/5220">independence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
A recent discussion at iMechanica following my last post here [<a href="node/8288" target="_blank">^</a>] leads to this post. The context of that discussion is assumed here.
</p>
<p>
I present here three sample problems, thought of almost at random, just to see how the suggestions made by Jaydeep in the above post work out.
</p>
<p>
<strong>Problem 1.</strong> For simplicity of analysis, assume a gravity-free, frictionless, airless, field-free, etc. kind of a physical universe, and set up a suitable Cartesian reference frame in it such that the xy-plane forms the ground and the z-axis is vertical. (Assume a right-handed coordinate system.)</p>
<p>Consider a straight rigid rod of zero thickness and length 2a, initially positioned parallel to the x-axis and at a height of h, translating with a uniform linear velocity, v, in the positive y-direction.</p>
<p>Suppose that the translating rod runs into a rigid vertical pole of infinite strength, zero thickness and a height > h so that a collision between the two is certain. </p>
<p>If the collision were to occur at the midpoint of the rod (i.e. at its CM), it would simply begin translating back with a -v velocity. </p>
<p>However, assume that the collision occurs at a distance d (0 < d < a) from the CM. This will impart an angular motion to the rod after the collision.</p>
<p>Derive the equations of motion for the rod before and after the impact, using (a) the usual method of analysis (having both the linear and angular quantities in it), and (b) the method / lines of analysis suggested by Jaydeep.</p>
<p>Comment on the linear and angular quantities before and after the impact in both the cases.</p>
<p>Make any additional assumptions as necessary.
</p>
<p>
<br /><strong>Problem 2.</strong> Repeat the Problem 1., using both the usual analysis and Jaydeep's method, now assuming that the rod is linear elastic with infinite strength. All other assumptions and data remain the same; in particular, the pole continues to remain rigid.
</p>
<p>
<br /><strong>Problem 3.</strong> Provide a detailed derivation for an FE model of the beam element, replacing the three rotations and three translations by six rotations so as to conform to Jaydeep's method.
</p>
<p>
</p>
<p>
<strong>Asides:</strong> BTW, I plan to solve none. But I will make sure to have a look at any solution(s) that are offered, but without promising in advance that I will also comment on them.
</p>
<p>
With that said, solutions and comments are most welcome!
</p>
<p>
--Ajit
</p>
<p>
PS: Also posted at my blog here [<a href="http://ajitjadhav.wordpress.com/2010/05/27/use-only-the-angular-quantities-in-analysis-three-sample-problems-to-consider/" target="_blank">^</a>]
</p>
<p>
<em>Update on May 29:</em> Corrected a typo. Now the Problem 1 reads: "...a uniform linear velocity, v, in the positive y-direction. ..." This is the way it was intended. Earlier, it incorrectly read: "...a uniform linear velocity, u, in the positive x-direction. ..."
</p>
<p>
[E&OE]
</p>
<p>
</p>
</div></div></div>Thu, 27 May 2010 16:06:49 +0000Ajit R. Jadhav8313 at http://imechanica.orghttp://imechanica.org/node/8313#commentshttp://imechanica.org/crss/node/8313Are Linear and Angular Momenta Interconvertible?
http://imechanica.org/node/8288
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/277">energy</a></div><div class="field-item odd"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/989">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/5216">linear momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5217">momentum</a></div><div class="field-item odd"><a href="/taxonomy/term/5218">angular momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5219">conservation</a></div><div class="field-item odd"><a href="/taxonomy/term/5220">independence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>To the best of my knowledge, the two momentum conservation principles, namely, the conservation of linear- and angular-momentum, operate completely independent of each other. For an isolated object, there is no possibility of conversion of one form of momentum to the other.</p>
<p>Today, when I Googled on this topic, I found that most pages agree with my position above. Yet, to my great surprise, I did run into a page written by an engineer here [<a href="http://cnx.org/content/m14342/latest/" target="_blank">^</a>] claiming that the linear momentum is only a special case of the angular momentum... Here is the relevant excerpt (bold emphasis mine):</p>
<p> <font color="#0000ff">Conservation of angular momentum is generally believed to be the counterpart of conservation of linear momentum as studied in the case of translation. This perception is essentially flawed. As a matter of fact, this is a generalized law of conservation applicable to all types of motions. We must realize that conservation law of linear momentum is a <strong>subset </strong>of more general conservation law of angular momentum. <strong>Angular quantities are all inclusive of linear and rotational quantities.</strong> As such, conservation of angular momentum is also all inclusive. However, this law is regarded to suit situations, which involve rotation. This is the reason that we tend to identify this conservation law with rotational motion.</font></p>
<p>
<br />
Of course, I think the above argument is flawed, but that's a different story... (I do plan to alert the author right away.)</p>
<p>Coming back to the main issue I want to discuss here: As far as I know, we have these two forms of momenta, and their conservation is independent of each other.</p>
<p>Now, as far as conservation principles go, we also have the principle of conservation of (to simplify the matters, say, the mechanical) energy.</p>
<p>Most, if not all, college-level physics text-books begin illustrating the principle of conservation of energy in reference to only the translational motion. Then, typically, these books go on to directly generalize the energy principle to all types of motion. A few text-books might spend some (though inadequate amount of) time to point out the relation between the two conservation principles: energy and momentum. But most authors (and teachers) simply skip over this entire issue, and directly or indirectly deploy the Argument from the Authority in impressing upon the student that the energy principle is at least desirable if not superior to the momenta principles. (Even Resnick and Halliday, e.g., think of energy conservation as one of the unifying themes of physics, and base the organization of their text around this theme, and so.)</p>
<p>Consider the motion of an object of non-symmetrical shape and non-uniform matter density, e.g., an inter-galactic probe, and suppose that its batteries (or other power-sources) have run out (or have become defunct). Let us also neglect any external forces such as gravity and friction for the time being. For the motion of an isolated object like this, in the general case, there would be both rotations and translations.</p>
<p>Now, such an object can be assumed to have a constant mechanical energy. But that is only one part of the story. The fact is, since each linear and angular momentum of this object is conserved, the respective components of its linear and angular energy also would be conserved independent of each other.</p>
<p>The question I had in mind is this: Not just in the above example, but considering the totality of your knowledge, do you know of any single physical situation or context in which the linear form of momentum transforms into the angular form, or vice versa?</p>
<p>One notable example I can think of is that frustrating problem of pushing an egg with the sharp tip of a pencil. The egg refuses to get pushed; it simply rotates in place. ... Even a point-sharp pencil could have imparted a translational motion if there were no friction between the egg and the supporting surface. It's this friction which helps convert all your applied point force into a torque. ... But even otherwise, whatever be the effectively applied torques and forces, once they set an object in motion in a frictionless world, it would be impossible to convert rotations/spins into translations, and vice versa. ...</p>
<p>...If you wish to have a present-day media-friendly sound-byte about it: the spin will keep spinning without getting anywhere. (Apart from being media-friendly, it also describes the media rather well, doesn't it?)</p>
<p>Or, if you prefer to think about the whole situation in terms of energy, let me ask: for an isolated (unconstrained and unforced) physical object, what is it that prevents a transformation of the translational form of the energy into the rotational form or vice-versa?</p>
<p>So, it's one matter to say that the total energy is conserved; it's another matter to say that the specific fractions of the linear and angular parts of the total energy also stay unaltered.</p>
<p>I think that physical scientists and engineers should have already acknowledged the second part as a principle in its own right: the principle of impossibility of interconversion of the linear and angular parts of motion of an object i.e. its momenta/energies.</p>
<p>Here is a request: If you find any explicit recognition of this principle in the prior literature (easily possible right since Isaac Newton, Jr.,'s Cambridge days), then please do point it out to me. Thanks in advance.</p>
<p>Incidentally, let me add one more observation. Why the above cited author might have thought that the angular momentum is more general.</p>
<p>Consider a solid object having its center of mass (CM) at a point P. If you apply a "force" F at a finite perpendicular distance s from CM, then what the "force" actually acts as is a torque, of magnitude T = Fs (and appropriate vector direction). Consider a smaller distance, and the torque gets reduced even if the applied "force" is the same. In differential terms, we may say: dT = Fds. When ds tends to zero, the torque dT tends to zero, but the F remains finite. At ds = 0, the object is imparted a force, not a torque. Considering this mathematical part, there is likely to be a kinetic argument that the concept of torque encompasses that of force, or, in kinematical terms, angular motion includes linear motion.</p>
<p>The argument is false. There is a certain reason why I have been putting quote marks around "force." When you consider application of a "force" with a finite s, in theory, you are considering the effect that the "force" applying agent, if acting singly on another hypothetical object, would have on that hypothetical object---the effect of linear acceleration. But this entire situation still is hypothetical---not actual. In the given actual situation, we assume that there is a constraint or at least a friction-providing surface on the other side of CM, and this couple of forces then goes on to produce a torque, which in turn produces angular acceleration. To treat just one of the forces out of context, is an error.</p>
<p>Further, as I keep pointing out in my writings, there also is a subtle point about the essential difference between the infinitesimal, the finite, and the zero. Most people tend to confuse between the case that the ds tends to zero and the case that the ds is zero. Not just beginning students and common engineers but also mechanicians with PhDs. By way of example, consider my explicit position in the discussion on point vs particle at iMechanica here [<a href="node/5214" target="_blank">^</a>], and also the lack of any explicit support to this position in that discussion. If you consider a material particle to have zero volume (or mass), you are essentially committing the same error as that involved in directly setting ds equal to (and not tending to) zero, in the above example.</p>
<p>Both errors arise from misunderstanding the difference of an infinitesimal from a zero. Further, I also suspect, an over-emphasis in looking merely at the mathematics also is a reason.</p>
<p>In general: you cannot substitute mathematics in place of physics. I am "inherently" very cautious of any characterization of physics as being inherently mathematical---as famously done by Dick (Richard) Feynman, and also as stated in the Table of Contents of a forthcoming book by David Harriman [<a href="http://www.amazon.com/Logical-Leap-Induction-Physics/dp/0451230051" target="_blank">^</a>]. I do look forward to reading it.
</p>
<p>
</p>
<p>
In the meanwhile, comments and corrections regarding my position(s) in this blog post are certainly welcome.
</p>
<p>
</p>
<p>
Note added on May 24, 2010: Also posted at my other blog [<a href="http://ajitjadhav.wordpress.com/2010/05/23/are-linear-and-angular-momenta-interconvertible/" target="_blank">^</a>]
</p>
<p>
- - - - - <br />
[E&OE]
</p>
</div></div></div>Sun, 23 May 2010 09:48:43 +0000Ajit R. Jadhav8288 at http://imechanica.orghttp://imechanica.org/node/8288#commentshttp://imechanica.org/crss/node/8288Wanted: Fast FEA Solvers...
http://imechanica.org/node/8240
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/719">sparse solver</a></div><div class="field-item odd"><a href="/taxonomy/term/751">Shell</a></div><div class="field-item even"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item odd"><a href="/taxonomy/term/880">open source</a></div><div class="field-item even"><a href="/taxonomy/term/938">Fast</a></div><div class="field-item odd"><a href="/taxonomy/term/1032">eigenvalue</a></div><div class="field-item even"><a href="/taxonomy/term/2864">solvers</a></div><div class="field-item odd"><a href="/taxonomy/term/4872">cst</a></div><div class="field-item even"><a href="/taxonomy/term/5181">direct</a></div><div class="field-item odd"><a href="/taxonomy/term/5182">iterative</a></div><div class="field-item even"><a href="/taxonomy/term/5183">LST</a></div><div class="field-item odd"><a href="/taxonomy/term/5184">DKT</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<strong>Summary:<br /></strong><br />
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.</p>
<p>I seek advice 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. </p>
<p>I have a personal and longer-term research interest with certain issues related to the solvers technologies. </p>
<p>Suggestions and comments are welcome!</p>
<p><strong>(1.) The Reference Problem:</strong></p>
<p>(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. </p>
<p>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.</p>
<p>(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.</p>
<p>(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.</p>
<p>(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 [<a href="http://www.cs.cmu.edu/~quake/triangle.html" target="_blank">^</a>] or Niceno's EasyMesh [<a href="http://www-dinma.univ.trieste.it/nirftc/research/easymesh/" target="_blank">^</a>]. </p>
<p>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.</p>
<p>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.</p>
<p>Note that these numbers refer to the geometry nodes. In the FE model, each such a node would carry several DOFs.</p>
<p>(1.5) The linear systems resulting after the FE-discretization are to be solved for both static and modal analyses.</p>
<p><strong>(2.) The Software/Hardware to be Used:</strong></p>
<p>(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.). </p>
<p>(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.</p>
<p>(2.3) Dependencies on open-source libraries/platforms such as GoToBlas, Boost, MTL, etc. are OK.</p>
<p>(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. </p>
<p>(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.</p>
<p>(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%.</p>
<p>(2.7) All the solver operations are expected to occur in-core (not out-of-core).</p>
<p>(2.8) Assume that all mathematical operations would be peformed in double precision (8 bytes).</p>
<p>
<strong>(3.) What Is Being Sought:<br /></strong><br />
(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:<br />
-- direct solver for static analysis (Ax = b)<br />
-- iterative solver for static analysis (Ax = b)<br />
-- direct solver for eigenvalues computations (Ax = Lx)<br />
-- iterative solver for eigenvalues computations (Ax = Lx)</p>
<p>For iterative solvers, assume the usual kind of convergence requirements (error norms).</p>
<p>(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.</p>
<p>(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.</p>
<p>(3.4) Not very important right now, but any side suggestions you might have for nonsymmetric A matrices would also be welcome.</p>
<p>(3.5) A general point of reference for this query is this URL: <br /><a href="http://www.netlib.org/utk/people/JackDongarra/la-sw.html">http://www.netlib.org/utk/people/JackDongarra/la-sw.html</a></p>
<p>
<strong>(4.) Why This Study:</strong></p>
<p>The purpose is something like this. I have some preliminary ideas concerning solvers. </p>
<p>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. </p>
<p>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.)</p>
<p>
<strong>(5.) Your Suggestions/References:</strong></p>
<p>Well thought-out comments/suggestions w.r.t the point (3.1) are sought.</p>
<p>
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.</p>
<p>
Thanks in advance!</p>
<p>
--Ajit
</p>
<p>
<em>PS:</em> Posted also at my blog here [<a href="http://ajitjadhav.wordpress.com/2010/05/14/wanted-fast-fea-solvers/" target="_blank">^</a>].
</p>
<p>
[E&OE]
</p>
<p>
</p>
</div></div></div>Fri, 14 May 2010 14:18:50 +0000Ajit R. Jadhav8240 at http://imechanica.orghttp://imechanica.org/node/8240#commentshttp://imechanica.org/crss/node/8240An Urgent Appeal for Your Support of My Job Application at COEP's Mechanical Engineering Department
http://imechanica.org/node/7245
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/948">Computational Science</a></div><div class="field-item even"><a href="/taxonomy/term/1302">Mechanical</a></div><div class="field-item odd"><a href="/taxonomy/term/2443">Computational Engineering</a></div><div class="field-item even"><a href="/taxonomy/term/4620">job application</a></div><div class="field-item odd"><a href="/taxonomy/term/4621">multidisciplinary</a></div><div class="field-item even"><a href="/taxonomy/term/4622">interdisciplinary</a></div><div class="field-item odd"><a href="/taxonomy/term/4623">crossdisciplinary</a></div><div class="field-item even"><a href="/taxonomy/term/4624">teaching</a></div><div class="field-item odd"><a href="/taxonomy/term/4627">metallurgy</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear iMechanicians,</p>
<p>I have applied for the job of "Associate Professor" in the Department of Mechanical Engineering at COEP, Pune, India [<a href="http://www.coep.org.in" target="_blank">^</a>]---the same place from where I did my PhD (Mech.) research.</p>
<p>I most earnestly make an appeal to you to provide me with an informal support for my job application by way of a brief email recommendation. My resume may be found here [<a href="http://www.jadhavresearch.info/docs/Detailed%20Resume%20-%20Ajit%20R%20Jadhav%20-%20October%202009.pdf" target="_blank">^</a>].</p>
<p><font color="#ff6600">Time is of essence because the process of short-listing and interviewing has already begun. Please send your emails immediately, preferably within the next 48 hours.</font></p>
<p>
<strong>Details:</strong></p>
<p>Today, via informal discussions with both the HoD Mechanical (Dr. Pande) and the Director (Dr. Sahasrabuddhe) of COEP, I came to know that my application at COEP is not being short-listed, because I have my basic degrees (BE and MTech) in Metallurgy even though my PhD is in Mechanical. </p>
<p>The Director, Dr. Sahasrabuddhe, said that personally he was willing to have me, but that his hands are tied because the Mechanical department today feels that with my Metallurgy background, I am not suitable. (The same Mechanical department had shortlisted me last year; I was ranked 2/20+ even if had no PhD in mechanical last year by their ranking system. I was declined a job subsequently out of the anxiety that my level would be too high for the average COEP students---something which I proved wrong by successfully taking a course on FEM last year.) Today, Dr. Sahasrabuddhe then indicated that if I can arrange for emails/letters of support from leading universities abroad, notably those in the USA and UK, then his "arms will grow stronger."</p>
<p>Please note, COEP is run by the State Government of Maharashtra, India, and as such, broadly speaking, the MPSC (Maharashtra Public Service Commission)'s rules are applicable. The MPSC rules themselves allow even if the bachelor's and master's degrees are not from the same discipline but come from an allied discipline. However, the rules are understandably silent as to what might be considered to be an allied discipline. This is left up to the discretion of the hiring committee(s).</p>
<p>By way of precedents, I can cite several examples. Dr. Y. V. Deshmukh, now retired, had his BE and ME in Metallurgy and PhD in Mechanical (just like in my case), and was allowed to teach both Metallurgy and Mechanical (and also Production Engg.). Dr. P. P. Chikate had his PhD in Mechanical but he retired as HoD of the Production Engg department. One of his PhD students, Dr. B. B. Ahuja, has BE, ME and PhD all in Mechanical, but today he not only heads the Production Engg department at COEP but also currently serves as its Deputy Director. Mr. Sawant, the currently serving HoD of the Computer Science Dept at COEP has both his BE and ME in Electronics and Telecommunications, not in CS, but has taught in the CS department for years as a Full Professor. Similar examples exist at some other notable institutes in India. For instance, Dr. Prashant Date had his BE in Mech, but both his MTech and PhD in Metallurgy, but he has been serving in the Mechanical department of IIT Bombay (today, as a Full Professor). Other similar examples can also be cited, but I restrict myself here only to those persons who I know directly in person. (For example, Dr. Date was my class-mate during our MTech in Metallurgy at IIT Madras, and Dr. Chikate was my first informal guide for the recent PhD, and Dr. Ahuja was on my PhD supervision comittee.)</p>
<p>My field of specialization is Computational Science and Engineering. I have taught a course in FEM to undergraduates in Mechanical last year at COEP, and it was well received. If hired, the cluster of courses that I would teach would be something like: Strength of Materials/Mechanics of Solids (II year UG), Advanced Strength of Materials, FEM (III year UG and also for ME), courses aligned with Numerical Analysis, courses aligned with and CAD, CAM CAE and introductory mechanical design, and possibly, also basic courses such as Heat Transfer. (CFD is an option that I could teach but am not currently very keen on unless I get a few months to develop my notes.)
</p>
<p>
I now make this appeal to you to show your support for my job application at COEP.</p>
<p>While support from any quarters is perfectly welcome, given the fact that this is more a matter of perceptions than of anything substantial, I especially urge the mechanicians associated with the very top-ranked American schools to provide me with the necessary support.
</p>
<p>
Thus, your support would be especially relevant and valued if you have had your own education or a post-doc, or currently have a faculty position with, schools such as: MIT, Harvard, Stanford, Berkeley, CalTech, Brown, Cornell, Santa Barbara, UIUC, Purdue, GeorgiaTech, UT Austin, Northwestern, Michigan (Ann Arbor), Duke, VirginiaTech ..., or Cambridge, Oxford ..., or Max Planck Institutes ..., or similarly for others (many of them not mentioned).
</p>
<p>
(I am separately writing personal emails to my friends and also my PhD thesis evaluators from the faculties of the IITs.)</p>
<p>So, coming back to iMechanica, esp. senior professors who have received high honors and awards, are being most earnestly urged to provide me with a timely support at this juncture of my life.</p>
<p>You may directly write an email in complete confindence to the COEP Director and the HoD at the following addresses: Prof. Dr. Anil Sahasrabuddhe (<a href="mailto:director@coep.org.in">director@coep.org.in</a>) and Prof. Dr. D. W. Pande (<a href="mailto:hod@mech.coep.org.in">hod@mech.coep.org.in</a>).</p>
<p>You may write your email of support any which way you like. All that I wish to point out here is that even the following paragraph would be quite adequate:</p>
<p><em>"I write this email following a public post by Dr. Jadhav at iMechanica. I am aware that Dr. Jadhav has had his bachelor's and master's in Metallurgy, and has recently earned his PhD in Mechanical Engg with a special focus on Computational Science and Engineering. As a fellow iMechanician, I support Dr. Jadhav's current job application for the post of Associate Professor in the Mechanical Engg department at COEP for your current cycle of recruitment."<br /></em><br /><br />
Needless to add, your support for this particular post at COEP, at this point of time, would not be taken by me as a recommendation for a post-doc position abroad some time in future. [As it is, currently, my passport is in the process of renewal; its last renewal had occurred at the Indian Consulate at San Francisco, not in India; therefore, the current renewal would take longer; it would take several months anyway.]
</p>
<p>
It is true that a friend in need is a friend indeed. Yet I myself wish that the basis of your support, if any, ought to refer solely to such things as: my talent, research, professional work, overall achievements, together with my sense of ethics. My resume may be found here [<a href="http://www.jadhavresearch.info/docs/Detailed%20Resume%20-%20Ajit%20R%20Jadhav%20-%20October%202009.pdf" target="_blank">^</a>].
</p>
<p>
Time is of essence because the process of short-listing and interviewing has already begun. Please send your emails immediately, preferably within the next 48 hours.</p>
<p>Thanks in advance and regards,</p>
<p>
--Ajit<br />
PS: A special request (made by me for the first time here) to iMechanica admins: <font color="#ff6600">Please promote this post to the front page at your earliest convenience; thanks in advance. </font>[Update on Dec. 15, 2009: The decision to post this request here has been taken with the foreknowledge of the Director, Dr. Anil D. Sahasrabuddhe, and also of Dr. S. R. Kajale, my guide, with neither of them seeing anything wrong with it because, as one of them put it, this <em>is</em> a public recruitment process. As such, this post may indeed be promoted to the front page for wider circulation on an urgent basis. Thanks in advance.]
</p>
</div></div></div>Mon, 14 Dec 2009 14:27:46 +0000Ajit R. Jadhav7245 at http://imechanica.orghttp://imechanica.org/node/7245#commentshttp://imechanica.org/crss/node/7245Food for Thought: A Few Recent arXiv Papers
http://imechanica.org/node/7240
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/180">thermodynamics</a></div><div class="field-item odd"><a href="/taxonomy/term/920">physics</a></div><div class="field-item even"><a href="/taxonomy/term/2319">Cellular Automata</a></div><div class="field-item odd"><a href="/taxonomy/term/4616">paradox</a></div><div class="field-item even"><a href="/taxonomy/term/4617">arXiv</a></div><div class="field-item odd"><a href="/taxonomy/term/4618">statistical physics</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Since my research touches on the basics of QM, I have developed this habit of visiting arXiv.org every now and then. Last week or so, at arXiv.org, I found a couple of interesting articles on physics in general. I would like to share these with you.
</p>
<p>
<br />
One of these is: Dragoljub A. Cucic, “Types of paradox in physics,” arXiv:0912.1864v1 [<a href="http://arxiv.org/abs/0912.1864" target="_blank">^</a>]. It’s a very comprehensive kind of article. Impressed, I did an author search on Cucic, and found a few more papers on this and related topics by him [<a href="http://arxiv.org/find/physics/1/au:+Cucic_D/0/1/0/all/0/1" target="_blank">^</a>].</p>
<p>The other article I have in mind is: Franco Bagnoli, “From Newton to cellular automata,” arXiv:0912.2056v1 [<a href="http://arxiv.org/abs/0912.2056" target="_blank">^</a>]. Once again, the scope of this article is just wonderfully wide, even though the writing tends to be a bit too terse at places. But Bagnoli compensates for this by including a neat “concept map.”</p>
<p>Both the papers are easily accessible even to undergraduates. Both provide enormous food for thought.</p>
<p>Indeed, I already find myself wondering if I should write an article or two addressing one or two of the many paradoxes that Cucic lists. [I am still getting my thoughts together.]</p>
<p>And, I cannot thank Bagnoli enough for providing a kind of “white paper” material that was so badly needed in explaining to other researchers (not just to laymen) just what kind of research ideas and methods I seem to be pursuing and how these differ from those in the typical PhD researches, esp. those from the engineering sciences. It helps explain why there is this general (and pretty vague) impression to the effect that there is not enough “maths” or “rigour” in my research or in my papers… Bagnoli helps point out the why of it…</p>
<p>I might even write an informal article showing what kind of maths it will look like if an artificial attempt is made to mathematicize these ideas at any cost, using only the classical or traditional way of putting maths… [I would write such an article anyway but especially so if some renowned researcher/mathematician otherwise has problems accepting my research and so asks me to do so (as I had indicated in my post at my personal blog [<a href="http://ajitjadhav.wordpress.com/2009/12/06/somewhat-scientific-somewhat-latest/" target="_blank">^</a>].)]</p>
<p>Anyway, do go over the abovementioned articles, and if you wish to discuss, feel free to leave a comment or two.
</p>
<p>
--Ajit
</p>
</div></div></div>Sat, 12 Dec 2009 15:43:59 +0000Ajit R. Jadhav7240 at http://imechanica.orghttp://imechanica.org/node/7240#commentshttp://imechanica.org/crss/node/7240My Ph.D. Defence
http://imechanica.org/node/6784
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/4382">Ph.D. Defence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
I am pleased to inform you that I will be defending my Ph.D. thesis, formally in mechanical engineering, at COEP, University of Pune, India, on the next Sunday (i.e. 20th September, 2009).
</p>
<p>
The title of my thesis is: "A New Approach to Computer Modeling and Analysis of Certain Fundamental Field Problems from Engineering Sciences."
</p>
<p>
I am attaching the 10 (actually 13) pages long abstract of my thesis for your information. The thesis is based on my published articles which may be downloaded from my Web site [<a href="http://www.JadhavResearch.info/publications.htm" target="_blank">here</a>].
</p>
<p>
If you would like to formally raise some questions on any part of my thesis, to be included during the official defence proceedings, then please send me a message via my profile at iMechanica and I will then let you know the email address of the Defence Committee Chairman. You could then submit your questions in complete confidence directly to the Chairman. This being the Internet, I would request you to kindly include your verification information such as your name and affiliation in your message. If this information is not completely available, I may not be able to respond to your messages. I will check messages until Saturday morning (India time).
</p>
<p>
Thanks in advance,
</p>
<p>
--Ajit
</p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
<thead><tr><th>Attachment</th><th>Size</th> </tr></thead>
<tbody>
<tr class="odd"><td><span class="file"><img class="file-icon" alt="" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="http://imechanica.org/files/Ajit%20R%20Jadhav%20-%20Ten%20Pages%20Abstract%20of%20the%20PhD%20Thesis.pdf" type="application/pdf; length=124109" title="Ajit R Jadhav - Ten Pages Abstract of the PhD Thesis.pdf">Ajit R Jadhav - Ten Pages Abstract of the PhD Thesis.pdf</a></span></td><td>121.2 KB</td> </tr>
</tbody>
</table>
</div></div></div>Wed, 16 Sep 2009 12:35:16 +0000Ajit R. Jadhav6784 at http://imechanica.orghttp://imechanica.org/node/6784#commentshttp://imechanica.org/crss/node/6784A Different Kind of a Book Involving Electromagnetism and Potential Theory
http://imechanica.org/node/6751
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/277">energy</a></div><div class="field-item odd"><a href="/taxonomy/term/4299">Potential</a></div><div class="field-item even"><a href="/taxonomy/term/4366">Carol White</a></div><div class="field-item odd"><a href="/taxonomy/term/4367">Riemann</a></div><div class="field-item even"><a href="/taxonomy/term/4368">Electromagnetic Theory</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Unlike other blog-posts of mine, I am not going "own" this particular thread. By that, I mean to say: I am going to only begin this thread and immediately turn it over to you completely. I am not going to watch over whether the discussion here continues to stick to its main theme or not, whether it slides into some minor side issues, whether it deserts the main theme altogether, etc., the way I usually do.</p>
<p>- - - - -</p>
<p>This thread is meant to be about the following book:<br />
<br />
Carol White , "Energy Potential: Toward a New Electromagnetic Field Theory," (with essays by Bernhard Riemann trans. from German by J. J. Cleary, Jr.), Campaigner Publications, New York, 1977. </p>
<p>I ran into it in mid-August 2009. It can be downloaded for free from here: <a href="http://www.archive.org" target="_blank">http://www.archive.org</a>. (Search for it at this site.)</p>
<p>Unfortunately, I could not find the time to go through all of it. (One reason is that in order to do so, I will have to take a paper printout of it (my reading habits are old-fashioned), and somehow <em>that</em> has not happened so far.) Yet, I find that the book has been written very interestingly.</p>
<p>In our times, there is this widespread tendency to write in a self-censoring way so as not to offend anybody, to try to be as sensitive to as many desires (even whims) of as many other people as possible, to try to be "politically correct" even while writing for/on issues of hard science. Against this background, Carol White's book comes across as a breath of fresh air. (And to think that it was published barely 32 years---one generation---ago!)</p>
<p>Of course, I don't think I am going to concurr with every idea or opinion which she expresses in this book. But that hardly matters.</p>
<p>I still strongly recommend this book to you because of its directness, its freshness, its willingness to pick up philosophical issues for examination right while working through the things scientific.... All of this is so unlike our present times. Also, the engaging style in which the text has been written... You might pick up virtually any page at random and see what I mean. For that one reason alone---call it the "style" of the book if you wish---it makes for a very interesting reading. </p>
<p>It's precisely because the author writes with such a passion, candor and directness, in such an opinionated manner, that her content becomes so very interesting to read. </p>
<p>And that was the biggest point I wanted to make here. </p>
<p>[And yes, Riemann's essays might form an additional/major attraction for some of you.]</p>
<p>From this point on, in this thread (alone), I am going to ignore it even if you begin any comment by addressing it to me---remember, I no longer "own" this thread.
</p>
<p>
All in all, very highly recommended... And, I am eager to know your reactions to it/opinions about it. </p>
<p>Over to you all! ... </p>
<p>
</p>
</div></div></div>Thu, 10 Sep 2009 14:17:44 +0000Ajit R. Jadhav6751 at http://imechanica.orghttp://imechanica.org/node/6751#commentshttp://imechanica.org/crss/node/6751The Meaning of the Concept of Potential in Mechanics (and in Physics)
http://imechanica.org/node/6634
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/277">energy</a></div><div class="field-item even"><a href="/taxonomy/term/4299">Potential</a></div><div class="field-item odd"><a href="/taxonomy/term/4300">Potential Function</a></div><div class="field-item even"><a href="/taxonomy/term/4301">Field Theory</a></div><div class="field-item odd"><a href="/taxonomy/term/4302">Force</a></div><div class="field-item even"><a href="/taxonomy/term/4303">Calculus of Variations</a></div><div class="field-item odd"><a href="/taxonomy/term/4304">Energy Methods</a></div><div class="field-item even"><a href="/taxonomy/term/4305">Kinetic</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
If someone knows of books/articles dealing with the meaning of the concept of potential in physics (or concerning the physical bases underlying the energy methods of mechanics) then I would very much appreciate getting to know about these.
</p>
<p>
Please note, when I say physical bases, I mean <em>physical</em> bases---not "simpler/prior mathematical notions/procedures, very easy to work out." Thus, my query is for material that is primarily conceptual, not mathematical. (As an aside: Mathematical material on this topic is so easy to get that, speaking metaphorically, a stone's throw would yield a dozen references if not 1200. ... But I was talking about treatment that is not exclusively mathematical. Essentially, a counterbalance to Lagrange is what I was looking for.)
</p>
<p>
Also note, by potential, I do not mean the limited context of electromagnetism (EM) alone. Indeed, if you ask me, energy methods are far more valuable in mechanics than in EM primarily because the (statically) indeterminate case is so easy to run into, in mechanics. The momentum approach isn't, therefore, most convenient.
</p>
<p>
I have already browsed through Lanczos (The Variational Principles of Mechanics) and find it helpful. Just the right sort of book, even though if I were to have the material to write this book, I wouldn't present it in the order that he does. ... Anyway, apart from this book, is there any other source? That's the question I have here.
</p>
<p>
I might as well mention here that for my purpose here, Goldstein (Classical Mechanics) has been a big let down (both in terms of the contents as well as their ordering) and so has been Weinstok (Calculus of Variations). I remember having browsed very rapidly through Morse and Feschback a few years back, but without finding anything directly useful in this context.
</p>
<p>So, there. Any indicators/links other than Lanczos would be very much appreciated. If there aren't any, I guess I might myself write up a research article on this topic.</p>
<p>
Thanks in advance for any links/references.
</p>
</div></div></div>Sat, 15 Aug 2009 10:08:59 +0000Ajit R. Jadhav6634 at http://imechanica.orghttp://imechanica.org/node/6634#commentshttp://imechanica.org/crss/node/6634Wondering about the Mechanics of Bacterial Death
http://imechanica.org/node/6633
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/19">biomechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/131">stress</a></div><div class="field-item even"><a href="/taxonomy/term/1407">bursting</a></div><div class="field-item odd"><a href="/taxonomy/term/4293">bacteria</a></div><div class="field-item even"><a href="/taxonomy/term/4294">viruses</a></div><div class="field-item odd"><a href="/taxonomy/term/4295">osmosis</a></div><div class="field-item even"><a href="/taxonomy/term/4296">cytolysis</a></div><div class="field-item odd"><a href="/taxonomy/term/4297">van der Waals forces</a></div><div class="field-item even"><a href="/taxonomy/term/4298">salt-water</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
0. I was idly thinking about the current H1N1 flue pandemic, and the following things occurred to me. Please note, I know very little about this subject matter. So, please consider descriptions in the following as, at best, tentative.</p>
<p>1. There is a basic difference between how alcohol kills viruses and how salt-water kills bacteria. [Alcohol is used in the hand-cleaners they use in hospitals. Girgling with salt-water is the first line of defense (and an unexpectedly highly effective one) which is well known for millenia.] </p>
<p>But the mechanisms involved are different. </p>
<p>Alcohol, I guess, oxidizes organic material that comes in physical contact with it. When the "organic material" is a virus, the extent of the oxidation is sufficient that the virus gets completely burnt. (BTW, is oxidation the reason why you see white patch on your hand after handling alcohol or kerosene?) </p>
<p>On the other hand, in the case of bacteria in salt-water, osmotic pressure results in diffusion of water molecules across the permeable bacterium skin. As water moves inside, it bloats the bacterium. Eventually, the bacterium becomes so bloated that, like an inflated baloon, its skin cannot take the mechanical stress, and so it bursts open. I don't know, but this is what I had heard from a biophysicist friend once. (And, the entry for "cytolysis" on Wikipedia tells something similar.) If the explanation is correct, then what ultimately kills bacteria in salt-water is the mechanical stress. </p>
<p>A few questions: </p>
<p>(i) By any chance, is there any way that not just bacteria but also viruses could get killed due to mechanical stress?
</p>
<p>
One way I imagine it might come to happen is if the van der Waals forces cause a few surrounding particles to get a grip on the skin of the virus, and then if these particles, for any reason, move across sufficiently differently that high stresses get induced within the virus skin/body. ... Yet, one has never heard of such a mechanism of killing viruses. The action almost always is chemical in nature. (Or radiational. Radiation gets absorbed and heats up the virus body, or it can cause breakages in the RNA strands). Is the virus skin too tough, and the relevant van der Waals forces too weak, or the gradients in the shear forces in the surrounding fluid too small, that the virus may break open mechanically? Any idea?</p>
<p>(ii) Has anyone done any modeling, preferably computational modeling, of the phenomenon of bloating of bactria? Also: Can the model explain why bacteria do survive in sea-water? How about fruit juices?
</p>
</div></div></div>Sat, 15 Aug 2009 09:48:24 +0000Ajit R. Jadhav6633 at http://imechanica.orghttp://imechanica.org/node/6633#commentshttp://imechanica.org/crss/node/6633A couple of upcoming international conferences in India
http://imechanica.org/node/6552
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/74">conference</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/151">Conference</a></div><div class="field-item odd"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/1570">theoretical</a></div><div class="field-item odd"><a href="/taxonomy/term/1571">applied</a></div><div class="field-item even"><a href="/taxonomy/term/1747">computational</a></div><div class="field-item odd"><a href="/taxonomy/term/2787">ISTAM</a></div><div class="field-item even"><a href="/taxonomy/term/4253">IndACM</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
(i) 3rd International Congress on Computational Mechanics and Simulation (ICCMS09) to be held this year at IIT Bombay, on December 1--5, 2009. Abstracts due by July 31, 2009:
</p>
<p>
<a href="http://www.civil.iitb.ac.in/help/114_iccms09/iccms09.html">http://www.civil.iitb.ac.in/help/114_iccms09/iccms09.html</a>
</p>
<p>
(ii) 54th Congress of the Indian Society of Theoretical and Applied Mechanics (An International Meet), to be held this year at the Netaji Subhas Institute of Technology, New Delhi, during December 18--21, 2009. Abstracts due by September 30, 2009:
</p>
<p>
<a href="http://www.webmath.iitkgp.ernet.in/~istam/">http://www.webmath.iitkgp.ernet.in/~istam/</a>
</p>
<p>
<a href="http://www.nsit.ac.in">http://www.nsit.ac.in</a>
</p>
<p>
</p>
</div></div></div>Tue, 28 Jul 2009 14:23:01 +0000Ajit R. Jadhav6552 at http://imechanica.orghttp://imechanica.org/node/6552#commentshttp://imechanica.org/crss/node/6552What Platform Would You Prefer for a Software That Helps in Learning FEM---Windows or Java?
http://imechanica.org/node/5461
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item odd"><a href="/taxonomy/term/973">software</a></div><div class="field-item even"><a href="/taxonomy/term/1355">C++</a></div><div class="field-item odd"><a href="/taxonomy/term/1356">Java</a></div><div class="field-item even"><a href="/taxonomy/term/3898">Windows</a></div><div class="field-item odd"><a href="/taxonomy/term/3899">Training</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Ideally, this post of mine should carry a poll, but I guess as an ordinary user, I cannot insert one.
</p>
<p>
Currently, I am writing a small software program that is especially designed to help learn FEM. For instance, I will be providing detailed listings for every intermediate step, e.g. all those [D], [B], [k], etc. matrices for each element, as well as the final assembled global system {F} = [K]{d} and its solution separately at each Gauss point. Only linear static problems for the time being; will add transients/eigenvalue problems in near future.
</p>
<p>
Needless to add, to make it all easy to use, I am providing a GUI as well.
</p>
<p>
Currently, I am implenting it all in VC++ 6 on Windows. (That's the latest version of MFC I have with me.) However, I am open to change the platform and/or language should the need arise.
</p>
<p>
This post is to ascertain whether you would rather prefer a C++/Windows version or should it be Java-based. (A third possibility is a QT based version, but I haven't seriously considered it---I would rather do it in Java than compile on many platforms.)
</p>
<p>
Please, this is not about platform wars. I am honestly interested in knowing what you would honestly find more convenient to use---as an end-user. And, before you answer, please note that this will not be an open source thing. There will be a small charge, say through a PayPal sort of arrangement. (Currently, I plan to keep the cost at about US $ 25/- per copy or less.)
</p>
<p>
So, please indicate what version you would prefer: Java or Windows. Thanks in advance.
</p>
<p>
--Ajit
</p>
<p>
- - - - -<br />
I remain jobless---and I remain being targeted by the Americans on a daily basis, including psychically, but to a noticeably lesser extent since I began talking about it here at iMechanica.
</p>
</div></div></div>Fri, 15 May 2009 15:47:12 +0000Ajit R. Jadhav5461 at http://imechanica.orghttp://imechanica.org/node/5461#commentshttp://imechanica.org/crss/node/5461Bending and 2D Elasticity: Going Back in Time
http://imechanica.org/node/5040
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/34">textbooks</a></div><div class="field-item odd"><a href="/taxonomy/term/179">solid mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/464">bending</a></div><div class="field-item even"><a href="/taxonomy/term/3642">couple stresses</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The following is a (relatively minor) question which had occurred to me more than two decades ago. By now I have forgotten precisely when it was... It could have been when I was in my TE (third year engineering) at COEP. ... Or, perhaps, it was later on, when I as at IIT Madras (studying stress analysis on my own). ... I don't remember precisely when it occurred to me, only *how* it did---it was when I was poring over the first part of Dieter's book.</p>
<p>IMHO, a matter like this should have been explicitly dealt with by the undergraduate texts on solid mechanics / elasticity. But, none does. Without straining your curiosity any further, let me tell you what that (minor) problem is:</p>
<p>Consider a horizontal cantilever beam as shown in the accompanying figure (A).
</p>
<p>
<img src="http://www.jadhavresearch.info/blog_docs/Canti_to_2DElast.jpg" alt="Image of Cantilever and a Plane Stress Problem" width="400" height="512" /></p>
<p>
The beam has the length of L. Suppose that it has a uniform rectangular cross section, say of height h, and thickness t. Suppose the beam is loaded by nothing but a point load P at its free end. </p>
<p>Analysis of stresses/deflections in a cantilever beam like this involves considering the bending moments existing along the length of the beam. Bending moment is nothing but another name for torque. The simple Euler-Bernoulli theory for such a beam is given in any introductory book on solid mechanics.</p>
<p>Now, suppose you increase h such that its magnitude becomes comparable to that of L, say, h = L. This circumstance is shown in the figure (B).
</p>
<p>
Suddenly, the beam problem now looks like one from the plane elasticity.</p>
<p><strong>Three closely related questions follow:</strong>
</p>
<p>
<strong>(A)</strong> Now, checking the formulae or detailed derivations from 2D elasticity theory, we find no mention of the term "bending moment" anywhere in them. Why is it so?
</p>
<p>
<strong>(B)</strong> Why do torques seem to be present in the beam, but not in the plate? Don't the forces in the plate (say those associated with stresses) also form couples? After all, these forces also do act across finite moment-arms, right? If so, precisely where, in the act of "stretching" the beam into the plate (or of "compressing" the plate into the beam), do they torques get vanished (or introduced)? </p>
<p><strong>(C)</strong> To make the matter even more confusing: Does the beam theory include couple-stresses as in contrast to the Cauchy definition (which, obviously, doesn't)? </p>
<p>What would be your own answers to the above questions (A), (B) and (C)?
</p>
<p>
Note that despite the length of the description preceding these questions, one-line answers are possible (though by no means mandatory!)</p>
<p><strong>-----</strong></p>
<p><strong>A little more on it all</strong>
</p>
<p>
Surprising, but I haven't ever found a single person thinking along the above lines---neither a professor, nor a postdoc, nor a student. My personal interactions with mechanicians have been limited, and so, in a way, this is not a big deal.
</p>
<p>
But, still, I found it surprising that no <em>textbooks </em>write about such matters either. Neither Beer (of Lehigh, and guru to more than one Timoshenko winner), nor Popov (of Berkeley, a student of Timoshenko's, I suppose), nor Shames (of SUNY Buffalo, a winner of several outstanding teacher awards) nor Crandall (MIT(?)), nor Timoshenko himself (later, of Stanford), nor AEH Love (of the 19th century, the author of what is probably the longest in-print title in the solid mechanics field) mention any such relation or contrast between these two theories directly and explicitly.
</p>
<p>
I could be wrong, but at least I don't remember having run into a comparison like this during my browsing of any of these books... </p>
<p>So, the question also becomes: Why don't textbooks mention the above matter even if they do cover the two topics separately in great detail and depth? </p>
<p>Is it the case that the matter behind my questions is so trivial and obvious that any competent engineer could be assumed to have known and mastered it if he has mastered the these textbooks? </p>
<p>Or is it that what we bank on, in engineering education, is an indirect implication, namely, that if the student knows how to work out solutions to numerical (i.e. mathematical) problems from each of the two areas taken separately, then all must be well with the state of his <em>overall</em> theoretical integrations, too? ... </p>
<p>Comments on this more general issue, as well as answers to the specific questions (A) through (C) above, are both welcome! </p>
<p>Also, if you remember having seen something like a comparison of the two theories in one of the books mentioned above, or any other book, then do feel absolutely free to correct me---I will appreciate your help.</p>
<p>And also, no, I won't mind being told (even very bluntly) that I was making a mountain out of a mole-hill, if that's what you honestly feel about this issue... </p>
<p>Thanks in advance for your answers/comments!
</p>
<p>
(Update on March 12, 2009 only: Made better my use of the English language, and streamlined the writing.)
</p>
</div></div></div>Thu, 12 Mar 2009 07:19:53 +0000Ajit R. Jadhav5040 at http://imechanica.orghttp://imechanica.org/node/5040#commentshttp://imechanica.org/crss/node/5040Java for High-Performance Computational Engg. / Sciences?
http://imechanica.org/node/4822
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1356">Java</a></div><div class="field-item odd"><a href="/taxonomy/term/2443">Computational Engineering</a></div><div class="field-item even"><a href="/taxonomy/term/3482">High Performance Computing</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
In the past few days at iMechanica, there have been quite a few messages dealing with different aspects of programming, libraries and so on...
</p>
<p>
It would perhaps be timely, therefore, to ask:
</p>
<p>
<span>Do you have any opinion about using Java in numerical analysis (NA) / FEM / CFD etc.---i.e., in computational engineering and sciences (CES)?<br /></span>
</p>
<p>
Do you have any experience or hard data concerning performance of Java vis-a-vis C++ or FORTRAN, esp. for large systems, or for high-performance applications? Any pointers?
</p>
<p>
<span><br /></span>Please leave a note... Thanks in advance...
</p>
<p>
-----
</p>
<p>
A few clarifications:
</p>
<p>
-- This is not about starting platform wars/flames, just an honest query. (However improbable it may sound, it's true).
</p>
<p>
-- I personally have no opinion about Java in CES. My first choice is C++ and that's what I use all the time anyway. I sometimes run into Web pages that claim that C++ templates are now achieving as much performance as FORTRAN (or better). I, therefore, indirectly conclude that FORTRAN must be in general somewhat faster to C++. Yet, I like C++ (esp. the VC++ 6 IDE and all), and since I am so comfortable with it (15 consecutive years by now), I use it---neither FORTRAN nor Java nor C# (and certainly not VB!!).
</p>
<p>
-- I don't use Java, but suspect that it (or C#) would be slow for high-performance applications. This could be either because they are interpreted, or, even if someone produces native code out of them, I think, if C++ itself tends to be slower than FORTRAN, then Java (and C#) couldn't fare much better
</p>
<p>
-- Yet, I also run into things like: JAMA, Java Numerics, MTJ, Colt, etc. These seem serious toolkits... I mean, they also have MPI on Java and so on...
</p>
<p>
Therefore, the above question.
</p>
<p>
-- Also feel free to add any other toolkit that is relevant to CES. (Commercial ones are OK.)
</p>
</div></div></div>Wed, 18 Feb 2009 16:03:17 +0000Ajit R. Jadhav4822 at http://imechanica.orghttp://imechanica.org/node/4822#commentshttp://imechanica.org/crss/node/4822FEM Is Not a Local Method (and It Isn't Global Either)
http://imechanica.org/node/4752
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item odd"><a href="/taxonomy/term/1010">FDM</a></div><div class="field-item even"><a href="/taxonomy/term/2608">BEM</a></div><div class="field-item odd"><a href="/taxonomy/term/3398">local theory</a></div><div class="field-item even"><a href="/taxonomy/term/3399">global theory</a></div><div class="field-item odd"><a href="/taxonomy/term/3400">sub-global theory</a></div><div class="field-item even"><a href="/taxonomy/term/3401">Fourier</a></div><div class="field-item odd"><a href="/taxonomy/term/3402">integral equations</a></div><div class="field-item even"><a href="/taxonomy/term/3403">quantum</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In the literature, FEM has sometimes been characterized as a local approach, but IMO this needs to be corrected.</p>
<p>The piecewise continuous trial-functions of FEM can be looked at from two different viewpoints:</p>
<p>(i) If FEM is seen as an expansion method making use of basis functions, then naturally the comparison is with the Fourier-theoretic approaches (and all the derived, consequent or similar ones). The basis functions for the latter are global in the sense they have supports all over the domain. This, indeed, is unlike the limited (piecewise) support of the FEM trial-functions.</p>
<p>(ii) Yet, on the other hand, if FEM is seen an integral approach, i.e. an approach which is based either on functionals or on domain integrals of the residuals, then notice that both such expressions enjoy support over a finite region—not infinitesimal. Consequently, FEM cannot really be said to be a local approach. Compare it, for instance, with FDM. Or, with the ideas of the differential calculus per say.</p>
<p>I therefore object to characterizations of FEM as a local approach. Also for its characterization as a global approach. I believe that FEM is “midway,” really speaking.</p>
<p>As a relevant aside, also consider here that, in structural FEM for instance, the potential energy (and its stationarity) *is* globally defined, and yet, the displacements are defined only in a piecewise manner—i.e. over only sub-intervals of the overall domain.</p>
<p>I, therefore, suggest that a term like “sub-global” (or "supra-local") might be used to characterize FEM. And also, other methods like FEM.</p>
<p>In between the two candidates considered here, the first (”sub-global”) appears a more complete description as compared to the second (”supra-local”.) The first also sounds more honest and less pseudo-intellectual. (A third candidate, viz. “ultra-local,” seems to imply exactly the opposite of the intended meaning: it seems to doubly emphasize the local nature. Hence, it is unsuitable.)</p>
<p>I, thus, vote for the “sub-global” term.</p>
<p>Other methods where this term becomes relevant and applicable include, for example, BEM. Also MD, wherein, despite the use of point-particles, the potential itself is not a point-phenomenon (just the way it also does not have infinite support due to the cut-off). As such, MD, too, too should be characterized as a “sub-global” method.</p>
<p>BTW, this issue is neither obscurely academic nor pedantic. … Here, ask yourself why quantum entanglement is at all considered to be so “counter-intuitive” or “dramatic” (or gets so hotly debated/discussed).</p>
<p>-----</p>
<p>Well thought-out corrections to my position would be welcome, as also any really relevant complementary observation(s).</p>
<p>Thanks in advance.</p>
</div></div></div>Thu, 05 Feb 2009 07:46:22 +0000Ajit R. Jadhav4752 at http://imechanica.orghttp://imechanica.org/node/4752#commentshttp://imechanica.org/crss/node/4752The Fundamental Physical Bases of the WR Approach (and, Consequently, of FEM) in General
http://imechanica.org/node/4736
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item odd"><a href="/taxonomy/term/920">physics</a></div><div class="field-item even"><a href="/taxonomy/term/3394">Weighted Residual</a></div><div class="field-item odd"><a href="/taxonomy/term/3395">WR</a></div><div class="field-item even"><a href="/taxonomy/term/3396">physical bases</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
It has been quite some time (more than 1.5 years) that I had touched upon the topic of the physical bases of FEM in general, and of the general weighted residual (WR) approach in particular, at iMechanica (see <a href="node/1479" target="_blank">here</a>).
</p>
<p>
The position I then took was that there is no known physical basis at all for the WR approach---despite its loving portrayals in mathematical terms, or its popularity.
</p>
<p>
Further, I had also expressed (here and elsewhere) that a basis in physical principles existed for FEM only in a rather limited sense: wherever the energy interpretation was available for the model. (Note, this too is already at variance with what some of the authors have written in books.)
</p>
<p>
I have not yet changed my opinions.
</p>
<p>
But, still, it is easy to miss things.
</p>
<p>
And, it is easy to teach wrong things to one's students too---especially if doing so suits one's own professional goals, research programs, reputation, funding applications in the pipeline, greencard/citizenship applications, or a precisely similar set of things for one's PhD advisor/professional mentor/current superior/group/past alumni associates (sentimental in nature or otherwise).
</p>
<p>
Nevertheless, I would consider myself immoral if I did not check things out before proceeding to profess or teach them (or create some impression in my students' minds about them---positive or negative).Accordingly, here we go.
</p>
<p>
Please let me know if anyone has been able to unearth or discover any physical basis whatsoever---i.e. any explanation in terms of any known (or newly discovered) physical principles---over the past 1.5 years (or anytime earlier) whereby the weighted residual approach can be said to possess a fundamental physical interpretation.
</p>
<p>
And, thereby, also FEM, in its more general "avatar."
</p>
<p>
Thanks in advance.
</p>
</div></div></div>Mon, 02 Feb 2009 09:54:07 +0000Ajit R. Jadhav4736 at http://imechanica.orghttp://imechanica.org/node/4736#commentshttp://imechanica.org/crss/node/4736A (really) small problem
http://imechanica.org/node/4660
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/3325">quiz problem</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Here's a quick one.</p>
<p>Refer to the accompanying figure.
</p>
<p>
<img src="http://www.JadhavResearch.info/Images/AReallySimpleProblem.jpg" alt="Figure to accompany a really small problem" width="370" height="370" /></p>
<p>
</p>
<p>
It shows a frictionless glass tube lying on a horizontal desktop. The tube is smoothly bent at a couple of places as shown in the figure. </p>
<p>Suppose that a small steel ball (say, one taken from a ball-bearing) enters the tube at the point A, with an initial velocity of v.</p>
<p>The problem is to predict the local speeds of the ball as existing at the following points/sections. </p>
<p>(i) at the point C (the tangential speed)<br />
(ii) at the point F (the tangential speed)<br />
(iii) in the straightline section GH </p>
<p>For each point or section, it would be enough to predict whether the local speed there would be greater/smaller/equal to the initial value v.
</p>
<p>
If the problem cannot be solved as stated, feel free to say so.
</p>
<p>
In any case, it is important that you are able to provide brief reasons for your answers.
</p>
</div></div></div>Tue, 20 Jan 2009 11:52:02 +0000Ajit R. Jadhav4660 at http://imechanica.orghttp://imechanica.org/node/4660#commentshttp://imechanica.org/crss/node/4660