% Section 4.5.4: Design of a cantilever beam: recursive formulation (GP)
% Boyd & Vandenberghe "Convex Optimization"
% (a figure is generated)
% Almir Mutapcic 02/08/06
%
% We have a segmented cantilever beam with N segments. Each segment
% has a unit length and variable width and height (rectangular profile).
% The goal is minimize the total volume of the beam, over all segment
% widths w_i and heights h_i, subject to constraints on aspect ratios,
% maximum allowable stress in the material, vertical deflection y, etc.
%
% The problem can be posed as a geometric program (posynomial form)
% minimize sum( w_i* h_i)
% s.t. w_min <= w_i <= w_max, for all i = 1,...,N
% h_min <= h_i <= h_max
% S_min <= h_i/w_i <= S_max
% 6*i*F/(w_i*h_i^2) <= sigma_max
% y_1 <= y_max
%
% with variables w_i and h_i (i = 1,...,N).
% For other definitions consult the book.
% (See exercise 4.31 for a non-recursive formulation.)
% optimization variables
N = 8;
% constants
wmin = .1; wmax = 100;
hmin = .1; hmax = 6;
Smin = 1/5; Smax = 5;
sigma_max = 1;
ymax = 10;
E = 1; F = 1;
cvx_begin gp
% optimization variables
variables w(N) h(N)
% setting up variables relations
% (recursive formulation)
v = cvx( zeros(N+1,1) );
y = cvx( zeros(N+1,1) );
for i = N:-1:1
fprintf(1,'Building recursive relations for index: %d\n',i);
v(i) = 12*(i-1/2)*F/(E*w(i)*h(i)^3) + v(i+1);
y(i) = 6*(i-1/3)*F/(E*w(i)*h(i)^3) + v(i+1) + y(i+1);
end
% objective is the total volume of the beam
% obj = sum of (widths*heights*lengths) over each section
% (recall that the length of each segment is set to be 1)
minimize( w'*h )
subject to
% constraint set
wmin <= w <= wmax;
hmin <= h <= hmax;
Smin <= h./w <= Smax;
6*F*[1:N]'./(w.*(h.^2)) <= sigma_max;
y(1) <= ymax;
cvx_end
% display results
disp('The optimal widths and heights are: ');
w, h
fprintf(1,'The optimal minimum volume of the beam is %3.4f.\n', sum(w.*h))
% plot the 3D model of the optimal cantilever beam
figure, clf
cantilever_beam_plot([h; w])