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117 lines
3.5 KiB
117 lines
3.5 KiB
% 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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% 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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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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