1. Mech300 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Four Optimization

2. Mech300 Numerical Methods, Hong Kong University of Science and Technology. 2 Motivation Given a function f(x), find an x 0 where f(x) is the maximum or minimum. Noncomputer method: derive the exact formula of the first derivative f’(x) using calculus and then solve the equation f’(x) = 0. Question: How about find (x 1, x 2, x 3, x 4, x 5 ) such that the following function is minimized?

3. Mech300 Numerical Methods, Hong Kong University of Science and Technology. 3 Example: Optimization of Parachute Cost For each chute: A = 2πr 2 l = √2r c = k c Am = M t /n Cost = c 0 + c 1 l + c 2 A 2 Task: Given the total weight M t, the impact speed limit v c, and the initial height z 0, determine the size (r) and the number of chutes (n) such that the total cost n(c 0 + c 1 l + c 2 A 2 ) is the minimum while the impact speed of the chutes when reaching the ground is less than v c. Minimize C = n(c 0 + c 1 l + c 2 A 2 ) Subject to v ≤ v 0 and n ≥ 1 where A = 2πr 2 l = √2r c = k c A m = M t /n t = root [ z 0 – gmt/c + (gm 2 /c 2 )(1-e -(c/m)t )] v = (gm/c)(1-e -(c/m)t )

4. Mech300 Numerical Methods, Hong Kong University of Science and Technology. 4 Mathematical Background Optimization or mathematical programming problem Find x, which minimizes or maximizes f(x) Subject to d i (x) ≤ a i i = 1, 2, …, m c i (x) = b i i = 1, 2, …, p where x: an n-dimensional design vector f(x): the objective function d i (x): inequality constraints c i (x): equality constraints If f(x) and the constraints are linear, it is linear programming. If f(x) is quadratic and the constraints are linear, it is quadratic programming. Otherwise, it is non-linear programming.

5. Mech300 Numerical Methods, Hong Kong University of Science and Technology. 5 Overall Structure