cantilever_beam.m

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% Design of a cantilever beam (recursive formulation)
% Section 4.5.4, pp. 163-165 in Boyd & Vandenberghe "Convex Optimization"
% (a figure is generated)
%
% 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.)
%
% Almir Mutapcic 01/25/06

% optimization variables
N = 8;
gpvar w(N) h(N);

% constants
wmin = .1; wmax = 100;
hmin = .1; hmax = 6;
Smin = 1/5; Smax = 5;
sigma_max = 1;
ymax = 10;
E = 1; F = 1;

% 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)
obj = w'*h; 

% recursive formulation
v = posynomial; y = posynomial; % create empty posynomials
v(N+1,1) = 0; y(N+1,1) = 0;
for i = N:-1:1
  disp(['Processing recursion number: ' num2str(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

% constraint set
constr = [ ...
  wmin*ones(N,1) <= w; w <= wmax*ones(N,1);
  hmin*ones(N,1) <= h; h <= hmax*ones(N,1);
  Smin*ones(N,1) <= h./w; h./w <= Smax*ones(N,1);
  6*F*[1:N]'./(w.*(h.^2)) <= sigma_max*ones(N,1);
  y(1) <= ymax;
];

% solve GP and compute the optimal volume
[obj_value, solution, status] = gpsolve(obj, constr);
assign(solution);

% 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
close all;
plot_cbeam([h; w])