%%%% A 110 LINE TOPOLOGY OPTIMIZATION CODE WITH HEAVISIDE FILTERING Nov, 2010%%%% function top110(nelx,nely,volfrac,penal,rmin,ft) %% MATERIAL PROPERTIES E0 = 1; Emin = 1e-9; nu = 0.3; %% PREPARE FINITE ELEMENT ANALYSIS A11 = [12 3 -6 -3; 3 12 3 0; -6 3 12 -3; -3 0 -3 12]; A12 = [-6 -3 0 3; -3 -6 -3 -6; 0 -3 -6 3; 3 -6 3 -6]; B11 = [-4 3 -2 9; 3 -4 -9 4; -2 -9 -4 -3; 9 4 -3 -4]; B12 = [ 2 -3 4 -9; -3 2 9 -2; 4 9 2 3; -9 -2 3 2]; KE = 1/(1-nu^2)/24*([A11 A12;A12' A11]+nu*[B11 B12;B12' B11]); nodenrs = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx); edofVec = reshape(2*nodenrs(1:end-1,1:end-1)+1,nelx*nely,1); edofMat = repmat(edofVec,1,8)+repmat([0 1 2*nely+[2 3 0 1] -2 -1],nelx*nely,1); iK = reshape(kron(edofMat,ones(8,1))',64*nelx*nely,1); jK = reshape(kron(edofMat,ones(1,8))',64*nelx*nely,1); % DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM) F = sparse(2,1,-1,2*(nely+1)*(nelx+1),1); U = zeros(2*(nely+1)*(nelx+1),1); fixeddofs = union([1:2:2*(nely+1)],[2*(nelx+1)*(nely+1)]); alldofs = [1:2*(nely+1)*(nelx+1)]; freedofs = setdiff(alldofs,fixeddofs); %% PREPARE FILTER iH = ones(nelx*nely*(2*(ceil(rmin)-1)+1)^2,1); jH = ones(size(iH)); sH = zeros(size(iH)); k = 0; for i1 = 1:nelx for j1 = 1:nely e1 = (i1-1)*nely+j1; for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)-1),nelx) for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)-1),nely) e2 = (i2-1)*nely+j2; k = k+1; iH(k) = e1; jH(k) = e2; sH(k) = max(0,rmin-sqrt((i1-i2)^2+(j1-j2)^2)); end end end end H = sparse(iH,jH,sH); Hs = sum(H,2); %% INITIALIZE ITERATION x = repmat(volfrac,nely,nelx); beta = 1; if ft == 1 || ft == 2 xPhys = x; elseif ft == 3 xTilde = x; xPhys = 1-exp(-beta*xTilde)+xTilde*exp(-beta); end loopbeta = 0; loop = 0; change = 1; %% START ITERATION while change > 0.01 loopbeta = loopbeta+1; loop = loop+1; %% FE-ANALYSIS sK = reshape(KE(:)*(Emin+xPhys(:)'.^penal*(E0-Emin)),64*nelx*nely,1); K = sparse(iK,jK,sK); K = (K+K')/2; U(freedofs) = K(freedofs,freedofs)\F(freedofs); %% OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS ce = reshape(sum((U(edofMat)*KE).*U(edofMat),2),nely,nelx); c = sum(sum((Emin+xPhys.^penal*(E0-Emin)).*ce)); dc = -penal*(E0-Emin)*xPhys.^(penal-1).*ce; dv = ones(nely,nelx); %% FILTERING/MODIFICATION OF SENSITIVITIES if ft == 1 dc(:) = H*(x(:).*dc(:))./Hs./max(1e-3,x(:)); elseif ft == 2 dc(:) = H*(dc(:)./Hs); dv(:) = H*(dv(:)./Hs); elseif ft == 3 dx = beta*exp(-beta*xTilde)+exp(-beta); dc(:) = H*(dc(:).*dx(:)./Hs); dv(:) = H*(dv(:).*dx(:)./Hs); end %% OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES l1 = 0; l2 = 1e9; move = 0.2; while (l2-l1)/(l1+l2) > 1e-3 lmid = 0.5*(l2+l1); xnew = max(0,max(x-move,min(1,min(x+move,x.*sqrt(-dc./dv/lmid))))); if ft == 1 xPhys = xnew; elseif ft == 2 xPhys(:) = (H*xnew(:))./Hs; elseif ft == 3 xTilde(:) = (H*xnew(:))./Hs; xPhys = 1-exp(-beta*xTilde)+xTilde*exp(-beta); end if sum(xPhys(:)) > volfrac*nelx*nely, l1 = lmid; else l2 = lmid; end end change = max(abs(xnew(:)-x(:))); x = xnew; %% PRINT RESULTS fprintf(' It.:%5i Obj.:%11.4f Vol.:%7.3f ch.:%7.3f\n',loop,c, ... mean(xPhys(:)),change); %% PLOT DENSITIES colormap(gray); imagesc(1-xPhys); caxis([0 1]); axis equal; axis off; drawnow; %% UPDATE HEAVISIDE REGULARIZATION PARAMETER if ft == 3 && beta < 512 && (loopbeta >= 50 || change <= 0.01) beta = 2*beta; loopbeta = 0; change = 1; fprintf('Parameter beta increased to %g.\n',beta); end end % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This Matlab code was written by E. Andreassen, A. Clausen, M. Schevenels,% % B. S. Lazarov and O. Sigmund, Department of Solid Mechanics, % % Technical University of Denmark, % % DK-2800 Lyngby, Denmark. % % Please sent your comments to: sigmund@fam.dtu.dk % % % % The code is intended for educational purposes and theoretical details % % are discussed in the paper % % "Efficient topology optimization in MATLAB using 88 lines of code, % % E. Andreassen, A. Clausen, M. Schevenels, % % B. S. Lazarov and O. Sigmund, Struct Multidisc Optim, 2010 % % This version is based on earlier 99-line code % % by Ole Sigmund (2001), Structural and Multidisciplinary Optimization, % % Vol 21, pp. 120--127. % % % % The code as well as a postscript version of the paper can be % % downloaded from the web-site: http://www.topopt.dtu.dk % % % % Disclaimer: % % The authors reserves all rights but do not guaranty that the code is % % free from errors. Furthermore, we shall not be liable in any event % % caused by the use of the program. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%