cantilever_beam

% Exercise 4.31: Design of a cantilever beam (non-recursive formulation)
% (For a detailed explanation see section 4.5.4, pp. 163-165)
% 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
%                6*F/(E*w_i*h_i^3) == d_i
%                (2*i - 1)*d_i + v_(i+1) <= v_i
%                (i - 1/3)*d_i + v_(i+1) + y_(i+1) <= y_i
%                y_1 <= y_max
%
% with variables w_i, h_i, d_i, (i = 1,...,N) and v_i, y_i (i = 1,...,N+1).
% (Consult the book for other definitions and a recursive formulation of
% this problem.)
%
% Almir Mutapcic 01/25/06

% optimization variables
N = 8;
gpvar w(N) h(N) v(N+1) y(N+1);

% 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;

% non-recursive formulation
d = 6*F*ones(N,1)./(E*ones(N,1).*w.*h.^3);
constr_v = [];
constr_y = [];
for i = 1:N
  constr_v = [constr_v; (2*i-1)*d(i) + v(i+1) <= v(i)];
  constr_y = [constr_y; (i-1/3)*d(i) + v(i+1) + y(i+1) <= y(i)];
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);
  constr_v; constr_y;
  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])

 
Iteration     primal obj.         gap        dual residual     previous step.
 
   1         5.51086e+00       1.41000e+02       4.78e+02               Inf
   2         7.51982e+00       6.05762e+01       7.43e+00       8.68550e-01
   3         8.54854e+00       2.94849e+01       1.20e-01       8.74857e-01
   4         6.68816e+00       2.00658e+01       1.62e-02       6.21831e-01
   5         5.12341e+00       1.50260e+01       4.49e-03       5.00000e-01
   6         3.73928e+00       9.98356e+00       8.17e-05       1.00000e+00
   7         1.26815e+00       4.99359e+00       1.33e-06       1.00000e+00
   8         2.74551e-01       2.49745e+00       5.75e-05       1.00000e+00
   9         1.13439e-01       1.87348e+00       1.48e-05       5.00000e-01
  10        -3.31042e-03       1.24953e+00       4.57e-07       1.00000e+00
 
Iteration     primal obj.         gap        dual residual     previous step.
 
   1         6.01601e+00       1.41000e+02       5.80e-01               Inf
   2         5.99011e+00       7.04860e+01       6.02e-04       1.00000e+00
   3         5.93268e+00       3.52434e+01       7.50e-04       1.00000e+00
   4         5.80743e+00       1.76225e+01       6.64e-04       1.00000e+00
   5         5.51852e+00       8.81364e+00       3.09e-04       1.00000e+00
   6         4.89586e+00       4.42267e+00       2.99e-05       1.00000e+00
   7         4.29210e+00       2.21572e+00       5.85e-06       1.00000e+00
   8         3.99034e+00       1.11031e+00       1.18e-06       1.00000e+00
   9         3.86425e+00       5.56771e-01       5.11e-07       1.00000e+00
  10         3.80503e+00       2.79235e-01       1.69e-07       1.00000e+00
  11         3.77556e+00       1.40061e-01       4.32e-08       1.00000e+00
  12         3.76101e+00       7.02076e-02       1.69e-08       1.00000e+00
  13         3.75402e+00       3.51772e-02       5.82e-09       1.00000e+00
  14         3.75057e+00       1.76216e-02       1.11e-09       1.00000e+00
  15         3.74882e+00       8.82381e-03       1.88e-10       1.00000e+00
  16         3.74794e+00       4.41635e-03       2.72e-11       1.00000e+00
  17         3.74751e+00       2.20954e-03       2.94e-12       1.00000e+00
  18         3.74729e+00       1.10515e-03       2.40e-13       1.00000e+00
  19         3.74718e+00       5.52672e-04       1.64e-14       1.00000e+00
  20         3.74712e+00       2.76361e-04       1.05e-15       1.00000e+00
  21         3.74709e+00       1.38187e-04       6.95e-17       1.00000e+00
  22         3.74708e+00       6.90950e-05       5.41e-18       1.00000e+00
  23         3.74707e+00       3.45479e-05       6.43e-19       1.00000e+00
  24         3.74707e+00       1.72741e-05       1.24e-19       1.00000e+00
  25         3.74707e+00       8.63707e-06       3.06e-20       1.00000e+00
  26         3.74707e+00       4.31855e-06       8.08e-21       1.00000e+00
  27         3.74707e+00       2.15928e-06       2.16e-21       1.00000e+00
  28         3.74707e+00       1.07964e-06       5.78e-22       1.00000e+00
  29         3.74707e+00       5.39821e-07       1.54e-22       1.00000e+00
  30         3.74707e+00       2.69911e-07       4.11e-23       1.00000e+00
  31         3.74707e+00       1.34956e-07       1.09e-23       1.00000e+00
  32         3.74707e+00       6.74779e-08       2.90e-24       1.00000e+00
  33         3.74707e+00       3.37390e-08       7.68e-25       1.00000e+00
  34         3.74707e+00       1.68695e-08       2.03e-25       1.00000e+00
  35         3.74707e+00       8.43477e-09       5.38e-26       1.00000e+00
Solved
Problem succesfully solved.
The optimal widths and heights are: 

w =

    0.6214
    0.7830
    0.9060
    1.0124
    1.1004
    1.1762
    1.2000
    1.3333


h =

    3.1072
    3.9149
    4.5298
    5.0620
    5.5019
    5.8812
    6.0000
    6.0000

The optimal minimum volume of the beam is 42.3965