% 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. sedumi will be called several times to refine the solution. Original size: 127 variables, 88 equality constraints 23 exponentials add 184 variables, 115 equality constraints ----------------------------------------------------------------- Cones | Errors | Mov/Act | Centering Exp cone Poly cone | Status --------+---------------------------------+--------- 8/ 8 | 2.949e+00 6.560e-01 7.202e-08 | Solved 8/ 8 | 2.417e-01 4.179e-03 1.569e-07 | Solved 8/ 8 | 1.641e-02 1.922e-05 1.253e-08 | Solved 8/ 8 | 1.033e-03 8.954e-08 1.041e-08 | Solved 0/ 8 | 6.465e-05 1.084e-08 1.031e-08 | Solved ----------------------------------------------------------------- Status: Solved Optimal value (cvx_optval): +40.9668 The optimal widths and heights are: w = 0.6214 0.7830 0.8963 0.9865 1.0627 1.1292 1.1888 1.3333 h = 3.1072 3.9149 4.4814 4.9324 5.3133 5.6462 5.9439 6.0000 The optimal minimum volume of the beam is 40.9668.