## Friday, May 21, 2010

### Bike to work week

Generally, "Hallmark events", which seem to have strangely proliferated in recent years, evoke a mildly derisive response in me, for their crass commercialism. That way they can sell more cards, restaurants can populate slow week-nights, and essentially somebody, somewhere, can get the wheels of businesses rolling.

Last week (May 17-21) was national "bike to work" week, and it was a pleasure to see more than the usual number of folks grinding away the streets of Tallahassee. As for myself, I could not avoid driving on Tuesday, but still 4/5 is not too shabby.

Here are a few nice local bicycling related links:

1. Capital City Cyclists: a Tallahassee based cycling club, mostly focused on biking for recreation, and competitions - but useful local information for commuters as well.

2. Tallahassee Mountain Bike Association: maps of trails and pictures, which are useful because you don't have to commute the same route as you drive.

3. SharetheRoad: a Canadian site with plenty of interesting information (so not really local :) )

## Monday, May 10, 2010

### Vizualizing Orientation Tensors with Matlab/Octave

Recently, I had to vizualize the standard fiber orientation tensor A=<pp> where p is the orientation of a single fiber as shown in the figure below, and the angular brackets denote an average over the distribution function. Although, I explicitly refer to the fiber orientation tensor in this blog, the discussion below is valid for any symmetric tensor, such as a stress tensor, for example.

For some simple cases, the tensor has a direct physical meaning:

For non-simple cases, it can be tricky to mentally visualize the tensor. However, if we calculate the eigenvalues and eigenvectors of the matrix A, visualizing the distribution can be done easily by drawing an ellipsoid. The ellipsoid points in a direction determined by the eigenvectors and the shape and size of the ellipsoid is determined by the eigenvalues.

I modified some code (VizOrient.m) that I found online. It works fine with any relatively recent version of Matlab and Octave 3.0 and higher. Below are five examples:

A=[0.45 0 0; 0 0.45 0; 0 0 0.1];
VizOrient(A);
A=[0.90 0 0; 0 0.05 0; 0 0 0.05];
VizOrient(A);
A=[0.33 0 0; 0 0.33 0; 0 0 0.33];
VizOrient(A);
A=[0.33 0.1 0; 0.1 0.33 0; 0 0 0.33];
VizOrient(A);
A=[0.45 0.2 0; 0.2 0.45 0; 0 0 0.1];
VizOrient(A);

The only problem with Octave is actually a problem with gnuplot. The statement "axis equal" is not strictly implemented, and hence some mental gymnastics could be required. Works great on Matlab, though.

References:
1. Chung and Kwon, Korea-Australia Rheology Journal, 14(4), 2002, 175-188

Code:
function VizOrient(A)
%
% take a 3x3 fiber orientation (symmetric) matrix A, and visualize the
% resulting ellipsoid. using a program from Nima Moshtagh, and erasing the 2D
% lines - also correcting the meaning of a, b, and c axes (1/a)
%
%------------------------------------------------------------------------------
% Copyright (c) 2009, Nima Moshtagh
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions are
% met:
%
%    * Redistributions of source code must retain the above copyright
%      notice, this list of conditions and the following disclaimer.
%    * Redistributions in binary form must reproduce the above copyright
%      notice, this list of conditions and the following disclaimer in
%      the documentation and/or other materials provided with the distribution
%
%THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
%AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
%IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
%ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
%LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
%CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
%SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
%INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
%CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
%ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
%POSSIBILITY OF SUCH DAMAGE.
%------------------------------------------------------------------------------
N       = 20;
[U D V] = svd(A);

%----------------------------------
% generate the ellipsoid at (0,0,0)
%----------------------------------
a = D(1,1);
b = D(2,2);
c = D(3,3);

[X,Y,Z] = ellipsoid(0,0,0,a,b,c,N);

%-------------------------
% rotate and the ellipsoid
%-------------------------

XX = zeros(N+1,N+1);
YY = zeros(N+1,N+1);
ZZ = zeros(N+1,N+1);

for k = 1:length(X),
for j = 1:length(X),
point = [X(k,j) Y(k,j) Z(k,j)]';
P = V * point;
XX(k,j) = P(1);
YY(k,j) = P(2);
ZZ(k,j) = P(3);
end
end

%-----------------
% Plot the ellipse
%-----------------
surf(XX,YY,ZZ);
xlabel('x');
ylabel('y');
zlabel('z');
axis equal
hidden on

end

## Saturday, May 1, 2010

### Why do you follow the news?

A few years ago, a friend I played ultimate frisbee with in Ann Arbor, asked me that question.

"Why do you follow the news?"

I bumbled a five-minute response, which was sprinkled with phrases like "it is a good habit", "we do not live in isolation", "it allows me to respond to emergencies", and other such platitudes.

Truth be revealed: I thought my answer was pathetic, and figured that if I really thought about the question for sometime, I would be able to come up with something less wimpy.

Part of the obsession to stay on top of the news was certainly an addiction. Reading the news/blogs was a daily fix. But addictions are hard to defend. So I gave up on that.

Part of the motivation was very pedestrian: reading the news helped with small-talk, which, while easy to belittle, lubricates conversations with people, just outside your first circle of friends. For example, following college football closely, allowed me to engage people I shared nothing else with, for long hours. Following a little celebrity gossip, likewise, had some twisted merit.

"Hmm, I may be getting somewhere with this",  I thought. I tried to generalize that sentiment into something grand: "following the news helps us participate in a shared culture", or some such BS.

But even that approach had a problem.

If that "shared culture" was being synthesized by the popular media (which I don't have a very high regard for), then perhaps it really wasn't that desirable!

In the end, I came up with something that went along the lines of, "We live in a democracy. Leaders that we elect, influence our futures. The process of choosing our leaders is best served, if we are aware of the world. Ergo."

Wimpy, I know.