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; w <= wmax;
    hmin <= h; h <= hmax;
    Smin <= h./w; 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])
Building recursive relations for index: 8
Building recursive relations for index: 7
Building recursive relations for index: 6
Building recursive relations for index: 5
Building recursive relations for index: 4
Building recursive relations for index: 3
Building recursive relations for index: 2
Building recursive relations for index: 1
 
Calling SeDuMi: 595 variables (19 free), 353 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 19 free variables
eqs m = 353, order n = 285, dim = 945, blocks = 23
nnz(A) = 1181 + 0, nnz(ADA) = 4577, nnz(L) = 2548
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            8.22E-001 0.000
  1 :  2.60E+001 3.22E-001 0.000 0.3925 0.9000 0.9000   4.99  1  1  1.1E+000
  2 :  9.65E+000 9.72E-002 0.000 0.3015 0.9000 0.9000   2.09  1  1  2.1E-001
  3 :  6.13E+000 3.45E-002 0.000 0.3550 0.9000 0.9000   1.54  1  1  6.2E-002
  4 :  4.54E+000 1.24E-002 0.000 0.3582 0.9000 0.9000   1.27  1  1  2.0E-002
  5 :  4.07E+000 5.30E-003 0.000 0.4287 0.9000 0.9000   1.14  1  1  8.5E-003
  6 :  3.89E+000 2.28E-003 0.000 0.4295 0.9000 0.9000   1.12  1  1  3.7E-003
  7 :  3.80E+000 8.25E-004 0.000 0.3624 0.9000 0.9000   1.09  1  1  1.4E-003
  8 :  3.76E+000 2.11E-004 0.000 0.2558 0.9000 0.9000   1.06  1  1  3.5E-004
  9 :  3.75E+000 1.39E-005 0.000 0.0661 0.9900 0.9900   1.03  1  1  2.3E-005
 10 :  3.75E+000 5.13E-007 0.000 0.0368 0.9900 0.9900   1.02  1  1  8.5E-007
 11 :  3.75E+000 2.38E-008 0.189 0.0463 0.9900 0.9900   1.00  2  2  4.0E-008
 12 :  3.75E+000 2.24E-009 0.034 0.0943 0.9900 0.9900   1.00  4  5  3.8E-009

iter seconds digits       c*x               b*y
 12      0.3   8.1  3.7475131070e+000  3.7475130804e+000
|Ax-b| =  7.8e-008, [Ay-c]_+ =  8.3E-010, |x|= 4.1e+001, |y|= 3.4e+000

Detailed timing (sec)
   Pre          IPM          Post
3.004E-002    2.604E-001    0.000E+000    
Max-norms: ||b||=1.422363e+001, ||c|| = 2.202759e+000,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 47.8685.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +42.4155
The optimal widths and heights are: 

w =

    0.6214
    0.7830
    0.9063
    1.0130
    1.1003
    1.1766
    1.2000
    1.3333


h =

    3.1072
    3.9149
    4.5313
    5.0648
    5.5014
    5.8828
    6.0000
    6.0000

The optimal minimum volume of the beam is 42.4078.