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top99：a 99 line topology optimization code by Ole Sigmund,October 1999 top88：AN 88 LINE TOPOLOGY OPTIMIZATION CODE Nov, 2010master
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%%%% AN 88 LINE TOPOLOGY OPTIMIZATION CODE Nov, 2010 %%%% |
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function top88(nelx,nely,volfrac,penal,rmin,ft) |
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%% MATERIAL PROPERTIES |
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E0 = 1; |
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Emin = 1e-9; |
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nu = 0.3; |
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%% PREPARE FINITE ELEMENT ANALYSIS |
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A11 = [12 3 -6 -3; 3 12 3 0; -6 3 12 -3; -3 0 -3 12]; |
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A12 = [-6 -3 0 3; -3 -6 -3 -6; 0 -3 -6 3; 3 -6 3 -6]; |
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B11 = [-4 3 -2 9; 3 -4 -9 4; -2 -9 -4 -3; 9 4 -3 -4]; |
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B12 = [ 2 -3 4 -9; -3 2 9 -2; 4 9 2 3; -9 -2 3 2]; |
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KE = 1/(1-nu^2)/24*([A11 A12;A12' A11]+nu*[B11 B12;B12' B11]); |
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nodenrs = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx); |
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edofVec = reshape(2*nodenrs(1:end-1,1:end-1)+1,nelx*nely,1); |
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edofMat = repmat(edofVec,1,8)+repmat([0 1 2*nely+[2 3 0 1] -2 -1],nelx*nely,1); |
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iK = reshape(kron(edofMat,ones(8,1))',64*nelx*nely,1); |
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jK = reshape(kron(edofMat,ones(1,8))',64*nelx*nely,1); |
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% DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM) |
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F = sparse(2,1,-1,2*(nely+1)*(nelx+1),1); |
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U = zeros(2*(nely+1)*(nelx+1),1); |
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fixeddofs = union([1:2:2*(nely+1)],[2*(nelx+1)*(nely+1)]); |
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alldofs = [1:2*(nely+1)*(nelx+1)]; |
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freedofs = setdiff(alldofs,fixeddofs); |
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%% PREPARE FILTER |
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iH = ones(nelx*nely*(2*(ceil(rmin)-1)+1)^2,1); |
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jH = ones(size(iH)); |
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sH = zeros(size(iH)); |
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k = 0; |
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for i1 = 1:nelx |
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for j1 = 1:nely |
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e1 = (i1-1)*nely+j1; |
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for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)-1),nelx) |
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for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)-1),nely) |
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e2 = (i2-1)*nely+j2; |
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k = k+1; |
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iH(k) = e1; |
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jH(k) = e2; |
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sH(k) = max(0,rmin-sqrt((i1-i2)^2+(j1-j2)^2)); |
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end |
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end |
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end |
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end |
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H = sparse(iH,jH,sH); |
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Hs = sum(H,2); |
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%% INITIALIZE ITERATION |
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x = repmat(volfrac,nely,nelx); |
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xPhys = x; |
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loop = 0; |
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change = 1; |
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%% START ITERATION |
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while change > 0.01 |
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loop = loop + 1; |
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%% FE-ANALYSIS |
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sK = reshape(KE(:)*(Emin+xPhys(:)'.^penal*(E0-Emin)),64*nelx*nely,1); |
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K = sparse(iK,jK,sK); K = (K+K')/2; |
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U(freedofs) = K(freedofs,freedofs)\F(freedofs); |
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%% OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS |
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ce = reshape(sum((U(edofMat)*KE).*U(edofMat),2),nely,nelx); |
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c = sum(sum((Emin+xPhys.^penal*(E0-Emin)).*ce)); |
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dc = -penal*(E0-Emin)*xPhys.^(penal-1).*ce; |
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dv = ones(nely,nelx); |
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%% FILTERING/MODIFICATION OF SENSITIVITIES |
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if ft == 1 |
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dc(:) = H*(x(:).*dc(:))./Hs./max(1e-3,x(:)); |
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elseif ft == 2 |
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dc(:) = H*(dc(:)./Hs); |
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dv(:) = H*(dv(:)./Hs); |
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end |
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%% OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES |
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l1 = 0; l2 = 1e9; move = 0.2; |
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while (l2-l1)/(l1+l2) > 1e-3 |
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lmid = 0.5*(l2+l1); |
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xnew = max(0,max(x-move,min(1,min(x+move,x.*sqrt(-dc./dv/lmid))))); |
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if ft == 1 |
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xPhys = xnew; |
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elseif ft == 2 |
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xPhys(:) = (H*xnew(:))./Hs; |
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end |
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if sum(xPhys(:)) > volfrac*nelx*nely, l1 = lmid; else l2 = lmid; end |
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end |
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change = max(abs(xnew(:)-x(:))); |
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x = xnew; |
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%% PRINT RESULTS |
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fprintf(' It.:%5i Obj.:%11.4f Vol.:%7.3f ch.:%7.3f\n',loop,c, ... |
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mean(xPhys(:)),change); |
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%% PLOT DENSITIES |
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colormap(gray); imagesc(1-xPhys); caxis([0 1]); axis equal; axis off; drawnow; |
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end |
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% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% This Matlab code was written by E. Andreassen, A. Clausen, M. Schevenels,% |
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% B. S. Lazarov and O. Sigmund, Department of Solid Mechanics, % |
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% Technical University of Denmark, % |
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% DK-2800 Lyngby, Denmark. % |
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% Please sent your comments to: sigmund@fam.dtu.dk % |
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% % |
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% The code is intended for educational purposes and theoretical details % |
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% are discussed in the paper % |
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% "Efficient topology optimization in MATLAB using 88 lines of code, % |
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% E. Andreassen, A. Clausen, M. Schevenels, % |
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% B. S. Lazarov and O. Sigmund, Struct Multidisc Optim, 2010 % |
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% This version is based on earlier 99-line code % |
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% by Ole Sigmund (2001), Structural and Multidisciplinary Optimization, % |
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% Vol 21, pp. 120--127. % |
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% % |
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% The code as well as a postscript version of the paper can be % |
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% downloaded from the web-site: http://www.topopt.dtu.dk % |
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% % |
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% Disclaimer: % |
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% The authors reserves all rights but do not guaranty that the code is % |
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% free from errors. Furthermore, we shall not be liable in any event % |
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% caused by the use of the program. % |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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@ -0,0 +1,117 @@ |
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% a 99 line topology optimization code by Ole Sigmund,October 1999 |
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clear |
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nelx=60; |
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nely=40; |
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volfrac=0.5; |
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penal=3.; |
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rmin=1.5; |
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% initialize |
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x(1:nely,1:nelx)=volfrac; |
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loop=0; |
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change=1; |
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% start ineration |
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while change>0.01 |
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loop=loop+1; |
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xold=x; |
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% FE analysis |
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[U]=FE(nelx,nely,x,penal); |
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% objective function and sensitivity analysis |
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[KE]=lk;; |
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c=0.; |
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for ely=1:nely |
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for elx=1:nelx |
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n1=(nely+1)*(elx-1)+ely; |
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n2=(nely+1)*elx +ely; |
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Ue=U([2*n1-1;2*n1;2*n2-1;2*n2;2*n2+1;2*n2+2;2*n1+1;2*n1+2],1); |
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c=c+x(ely,elx)^penal*Ue'*KE*Ue; |
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dc(ely,elx)=-penal*x(ely,elx)^(penal-1)*Ue'*KE*Ue; |
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end |
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end |
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% filtering of sensitivities |
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[dc]=check(nelx,nely,rmin,x,dc); |
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% design update by the optimality criteria method |
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[x]=oc(nelx,nely,x,volfrac,dc); |
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% print result |
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change=max(max(x-xold)) |
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disp(['It.:' sprintf( '%4i',loop) ' Obj.:' sprintf(' %10.4f',c) ... |
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' Vol.:' sprintf('%6.3f',sum(sum(x))/(nelx*nely)) ... |
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' ch.:' sprintf('%6.3f',change)]) |
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% plot densities |
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colormap(gray);imagesc(-x);axis equal;axis tight; axis off;pause(1e-6); |
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end |
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|
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% FE analysis |
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function [U]=FE(nelx,nely,x,penal) |
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[KE]=lk; |
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K=sparse(2*(nelx+1)*(nely+1),2*(nelx+1)*(nely+1)); |
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F=sparse(2*(nely+1)*(nelx+1),1); |
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U=sparse(2*(nely+1)*(nelx+1),1); |
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for elx=1:nelx |
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for ely=1:nely |
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n1=(nely+1)*(elx-1)+ely; |
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n2=(nely+1)*elx +ely; |
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edof=[2*n1-1;2*n1;2*n2-1;2*n2;2*n2+1;2*n2+2;2*n1+1;2*n1+2]; |
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K(edof,edof)=K(edof,edof)+x(ely,elx)^penal*KE; |
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end |
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end |
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% define loads and supports |
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ip=(nelx+1)*(nely+1); |
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F(2*ip,1)=-1; |
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|
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fixeddofs =[1:2*(nely+1)]; |
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alldofs =[1:2*(nely+1)*(nelx+1)]; |
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freedofs =setdiff(alldofs,fixeddofs); |
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% solving |
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U(freedofs,:)=K(freedofs,freedofs)\F(freedofs,:); |
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U(fixeddofs,:)=0; |
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end |
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% mesh-independency filter |
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function [dcn]=check(nelx,nely,rmin,x,dc) |
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dcn=zeros(nely,nelx); |
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for i=1:nelx |
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for j=1:nely |
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sum=0.0; |
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for k=max(i-floor(rmin),1):min(i+floor(rmin),nelx) |
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for l=max(j-floor(rmin),1):min(j+floor(rmin),nely) |
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fac=rmin-sqrt((i-k)^2+(j-l)^2); |
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sum=sum+max(0,fac); |
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dcn(j,i)=dcn(j,i)+max(0,fac)*x(l,k)*dc(l,k); |
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end |
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end |
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dcn(j,i)=dcn(j,i)/(x(j,i)*sum); |
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end |
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end |
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end |
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% Element stiffness matrix |
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function [KE]=lk |
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E=1.; |
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nu=0.3; |
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k=[1/2-nu/6 1/8+nu/8 -1/4-nu/12 -1/8+3*nu/8 ... |
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-1/4+nu/12 -1/8-nu/8 nu/6 1/8-3*nu/8]; |
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KE=E/(1-nu^2)*[ k(1) k(2) k(3) k(4) k(5) k(6) k(7) k(8) |
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k(2) k(1) k(8) k(7) k(6) k(5) k(4) k(3) |
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k(3) k(8) k(1) k(6) k(7) k(4) k(5) k(2) |
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k(4) k(7) k(6) k(1) k(8) k(3) k(2) k(5) |
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k(5) k(6) k(7) k(8) k(1) k(2) k(3) k(4) |
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k(6) k(5) k(4) k(3) k(2) k(1) k(8) k(7) |
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k(7) k(4) k(5) k(2) k(3) k(8) k(1) k(6) |
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k(8) k(3) k(2) k(5) k(4) k(7) k(6) k(1)]; |
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end |
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% optimality criteria update |
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function [xnew]=oc(nelx,nely,x,volfrac,dc) |
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l1=0; |
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l2=100000; |
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move=0.2; |
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while (l2-l1>1e-4) |
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lmid=0.5*(l2+l1); |
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xnew =max(0.001,max(x-move,min(1.,min(x+move,x.*sqrt(-dc./lmid))))); |
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if sum(sum(xnew))-volfrac*nelx*nely>0; |
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l1=lmid; |
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else |
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l2=lmid; |
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end |
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end |
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end |
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