Variational methods and structural optimization

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 Back to homepage of the course

Homework #1
Assigned: Jan. 6th, 2009
Due: Jan. 15th, 2009
Points: 20

10 points
Galileo Galili (1564-1642) thought of a problem that is now known as the brachistochrone (minimum-time) problem. The credit is often given to John Bernoulli because he formally posed it many years later. Galileo's version is different and has a narrow scope and that is what we will consider here.
"Among all the circular arcs passing through two points in a vertical plane separated by a vertical distance of v and horizontal distance of h, find the circular arc that makes the bead sliding along it reach the lower point starting from rest at the higher point under the action of gravity."
Since we have not yet discussed the calculus of variations, solve this problem using finite-variable optimization
10 points
Formulate and solve two optimization problems that minimize the time of travel for light rays such the laws of reflection and refraction could be derived. In both the statements below, the two points are separated by a horizontal distance h and vertical distance v.
- Reflection: Take two points and a line in a plane so that the two points lie on the same side of the line. Treat the line as a mirror so that the two points lie on the reflecting side of the line. Find the minimum-time path from the first point to the second point and derive the law of reflection.
- Refraction: Take two points and a line in a plane so that the two points lie on either side of the line. Assume that the speed of light on one side of the line is c₁ and that on the other side of the line is c₂. Find, the minimum-time path from one point to the other and derive the Snell's law of refraction.