Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May., 2016
Instructor: G. K. Ananthasuresh , Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Lectures: Tu, Th: 08:30 AM - 10:00 AM; Venue: ME Lecture Hall

Homework #3
Assigned: Jan. 28th, 2016
Due: Feb. 2nd, 2016
Points: 50
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

1. 20 points
   The aim of this problem is to understand the notion of a local minimum in calculus of variations. A beam under loading takes a stable deformed shape that assume minimum potential energy. You are given Matlab finite element codes: ( beamFEAfiles ).
   solveBeam.m finds the stable equilibrium solution w*(x) for given loading and boundary conditions.
   PEBeam.m computes the potential energy for any w(x) for given loading and boundary conditions.
   Your task is to verify that w*(x) indeed has the local minimum for the potential energy. You do this by perturbing w*(x) slightly in whatever way you wish. Then, plot ||w(x) - w*(x)|| (take the least squares norm) on the x-axis and PE(w(x)) on the y-axis. Then, see if PW(w*(x)) has the leadt value locally. Explore also globally for extra credit!
2. 30 points
   The aim of this problem is to understand how to verify if a given function is a minimizer or not.
   We take a classic problem of minimizing the surface of revolution of a curve y(x) about the x-axis and passing through two givenpoints (x1,y1) and (x2,y2). The information you need is shown below.
   
   The minimizing curve y(x) is a catenary, which is shown above (just below the optimization problem statement). Now, take a = -0.4 and b = 2 and 3. So, you are given two catenaries. Which one of them is the local (global) minimizer? Provide numerical evidence to your answer by doing what you did in the first problem.
   Explore sufficiency condition for a minimum, now or later in the semester, for extra credit.