Homework 11 - Optimal Design of Fiber Nets
First Draft Due 1pm Friday November 9. Final Draft Due 5pm Monday, November 12

We turn to fmincon to design the strongest possible cantilever. You will write a function 

    function bestcant(cx,cy,F,V)

that builds last week's adjacency matrix, A, vector of fiber lengths, L, and force vector, f. This f should be all zeros except for the value F in the component corresponding to vertical force at the center of the right face. (There is no center if cy is not even). Your bestcant will also generate an initial guess for the fiber cross-sectional areas, a0, that jibes with the volume constraint, L'*a0 = V. Your bestcant will then set lower and upper bounds on the permissible a and then call fmincon to find the best a. fmincon will in turn call your compcant function, where the compliance of the fiber net is computed. Each of these steps is a straighforward generalization of our best tent code.

As we are now moving into larger nets we should attempt to give fmincon a hand. fmincon searches for a minimum of the constrained compliance by searching for a critical point, that is an a for which the gradient, i.e., the vector of derivatives of the compliance with respect to each a(j), is zero. Hence, to accelerate fmincon we need only pass along information on the gradient of the compliance. It is hard, but not too hard, to see that 

   the gradient of   x'f   is   -e2/L  where
   e=Ax is the associated elongation.

You will inform fmincon that you are supplying the gradient of your objective function by switching on one of its options and by coding compcant to return both the compliance (1-by-1) and its gradient (m-by-1). In particular, you must follow this protocol 

    function [comp,grad] = compcant(a,A,L,f)

Once fmincon returns the best a it should be plotted, as above. For visual purposes you may wish to use the fat-factor-5 of btent.

Your work will be graded as follows: 

        First draft
                3 pts for thoughts on bestcant
                3 pts for thoughts on compcant

        Final draft 
                6 pts for header CONTAINING detailed USAGE
                8 pts for further comments in code 
                4 pts for indentation
                6 pts for correct bestcant
                4 pts for correct basic compcant
                6 pts for correct compcant with gradient

                set V = 20  and  F = -0.5  in the following

	        4 pts for labeled jpeg plot of best net, cx=4, cy=6
                4 pts for labeled jpeg plot of best net, cx=4, cy=2
                4 pts for labeled jpeg plot of best net, cx=2, cy=4
                4 pts for labeled jpeg plot of best net, cx=2, cy=2

                4 pts for a one paragraph discussion that contrasts,
                and not merely describes, your 4 jpegs

	EXTRA CREDIT - PLEDGED
                15 points for a 500 word essay. The essay must be typed and
                have a title, introduction and body. The backbone of the
                essay should be an overview of the survey by 
                J. Kepler
                Into that foundation you should weave detailed reference to 
                your 3 weeks of fiber net experience.  You may neither
                give nor receive aid from other students. You may consult
                and cite both paper and electronic media. Your citation list
                does not count toward the 500 word minimum.