Monday, January 25, 2010

"Optimal Design of Composite Granular Protectors"

This paper, which was accepted for publication in October 2008, has finally appeared in print. About bloody time.

Title: Optimal Design of Composite Granular Protectors

Authors: Fernando Fraternali, Mason A. Porter, and Chiara Daraio

Abstract: We employ an evolutionary algorithm to investigate the optimal design of composite protectors using one-dimensional granular chains composed of beads of various sizes, masses, and stiffnesses. We define a fitness function using the maximum force transmitted from the protector to a "wall" that represents the body to be protected and accordingly optimize the topology (arrangement), size, and material of the chain. We obtain optimally randomized granular protectors characterized by high-energy equipartition and the transformation of incident waves into interacting solitary pulses. We consistently observe that the pulses traveling to the wall combine to form an extended (long-wavelength), small-amplitude pulse.

Note that this paper is in many senses an engineering paper. Some of the results are definitely things that would be nice to understand in a more "fundamental" fashion.


GFreak said...

Regarding "understanding in a fundamental fashion": I assume you mean "find a mathematical model", or similar.

Are you aware of the tome (1200 pages) by Wolfram (2002) A New Kind of Science? He proposes that certain types of complex problems may not have a mathematical explanation (at least, not under currently understood methods) and yet may be well-defined by simple algorithmic rules.

At one point he asserts that certain problems may have iteration requirements which are irreducible, i.e., it is not possible to predict the outcome at say, k=100, yet k=100 is well-defined by time-marching.

I just discovered this book, and the implications for me in the study of swarm robotics, i.e., the emergence of complex behavior based on simple rules. I'm (of course) attempting to find models that predict swarm behavior, and yet Wolfram (=smarter than me) says I may not be able to.

Mason said...

Wow, NKS coming up for the second day in a row. I think I don't want to go into a full-blown rant here (and some of the stuff that Wolfram says is actually legitimate even though he's full of shit in many other situations), but some of my readers know just how high Wolfram is on my shitlist---much higher than Bill Gates for instance.

Can anybody remember if I ranted about Wolfram in a previous blog entry? Surely I must have done so by now, given that I've had this blog for over 4 years?

GFreak said...

Wow, so "yes".

I didn't have to read very much to discover his level of hubris, for sure.

Before discovering this work (from your link to Charles Lee and Frank Ling's radio show), all I knew of Wolfram was that he is a Caltech alum and (obviously) the father of Mathematica. I have since learned more.

Mason said...

I believe that when NKS came out, Wolfram compared himself favorably to Newton... That's probably my favorite, though that is just ridiculously funny and there isn't anything morally wrong---his taking credit for work that was done by others, on the other hand... (One of the preface type sections of NKS also seems to include implications that he owns the copyright for any research that comes out of things that build on the stuff in NKS.) And then the monickers "pure NKS" and "applied NKS" that he used during a SIAM talk were also kind of amusing---especially given that he was taking himself seriously.

I'll stop now. :)

Zifnab said...

Putting aside the Wolfram reference (I'm definitely on the same vein as Mason w.r.t. Wolfram), I think GFreak is asking an interesting question about modeling. Do you have any thoughts on whether it's possible to have irreducible algorithmic complexity to describe a system? My (CS) instinct is that it must be true (or we'd end up with a decidability machine,etc), but not necessarily for real-world systems.

Mason said...

I have seen that kind of discussion with respect to things such as chaotic dynamics. I think there are actually a large number of papers pertaining to this, though I am unfamiliar with the general conclusions from the broader conclusions of this particular corner of the chaos literature. I've basically run into things from time to time and didn't look deeply at most of it. One paper that I read years ago that I remember liking is a quantum chaos paper by Ford and Ilg (and there have been several follow-ups to this).

I believe it is this paper:

However, I think other papers on looking at chaos via algorithmic complexity are perhaps better examples. I believe that they do discuss in algorithmic terms how long it would take to resolve certain calculations (such as the symbolic dynamics of a particular trajectory out to a certain number of digits to maybe be able to distinguish it from a nearby trajectory).

Also, I never answered this (because my eyes went red when I saw 'Wolfram'), but I meant understand either via modelling or via first principles physics. A more mathematical understanding would be good as well, but I was actually just thinking of a derivation via the laws of mechanics of why the results should be as they are (as opposed to numerical results and some a posteriori justification that doesn't tell the whole story).

Mason said...

The Ford and Ilg paper isn't actually the one I had in mind. I misremembered (though I read that paper as well as part of the same immediate research thread). The one that I think I really meant is this one:

J. Ford and G. Mantica, Am. J. Phys. 60, 1086-1992.

(I can't find the paper online now with a quick search, so I haven't confirmed this.)

The Ford and Ilg paper is still relevant, but the Ford and Mantica one is expository in nature rather than in a research journal.