# Section 4.5.4: Design of a cantilever beam: recursive formulation (GP)

Jump to: Source code Text output Plots Library index

    % 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])

    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

    Successive approximation method to be employed.
       SDPT3 will be called several times to refine the solution.
       Original size: 231 variables, 88 equality constraints
       23 exponentials add 184 variables, 115 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
     15/ 15 | 8.000e+00  6.560e-01  0.000e+00 | Solved
     23/ 23 | 2.725e-01  5.153e-03  0.000e+00 | Solved
     16/ 16 | 1.372e-02  1.293e-05  0.000e+00 | Solved
      6/ 11 | 1.057e-03  6.085e-08  0.000e+00 | Solved
      0/  0 | 8.196e-05  0.000e+00  0.000e+00 | Solved
    -----------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +42.3965

    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
        4.5299
        5.0620
        5.5019
        5.8811
        6.0000
        6.0000

    The optimal minimum volume of the beam is 42.3965.