Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational methods and structural optimization Jan.-May, 2008
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in
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Homework #1
Assigned: Jan. 8th, 2008
Due: Jan. 17th, 2008
Points: 20

  1. 10 points
    Two points A and B are separated by a horizontal distance of L and vertical distance of V. The two points are in a continuously changing medium. In particular, the speed of light changes as a function of the vertical distance from A to be linearly from a value c1 and c2. Assuming that light takes the path of least time, find the path that a light ray would take in going from A to B.

    Divide the horizontal interval into N points and make the vertical height at each point a variable in the finite variable optimization. Pose the problem mathematically and write down the necessary and sufficient conditions analytically.

  2. 10 points
    Solve the above problem numerically using Matlab's Optimization Toolbox. "fminunc" or "fmincon" would be useful. Sumit a printout of your Matlab script and a figure of the optimal path. Make a table indicating the times taken by each path for different values of N. Do you see a smooth path?