function MMA(nelx,nely,volfrac,penal,rmin);
nelx=60;%x轴向单元数目
nely=20;%y轴向单元数据
volfrac=0.5;%体积比
penal=3;%材料插值的惩罚因子
rmin=2.5;
% INITIALIZE
x = repmat(volfrac,[nely nelx]);
m=1;
n=nely*nelx;
xmin(1:nely,1:nelx)=0.01;
xmin=reshape(xmin,n,1);
xmax(1:nely,1:nelx)=1;
xmax=reshape(xmax,n,1);
low=zeros(n,1);
upp=ones(n,1);
a0=1;
cmma=1000*ones(m,1);
xold1=zeros(n,1);
xold2=zeros(n,1);
loop = 0;
change = 1.;
% START ITERATION
while change > 0.01
loop = loop + 1;
xold = x;
% FE-ANALYSIS
[U]=FE(nelx,nely,x,penal);
% OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS
[KE] = lk;
c = 0.;
for ely = 1:nely
for elx = 1:nelx
n1 = (nely+1)*(elx-1)+ely;
n2 = (nely+1)* elx +ely;
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);
c = c + x(ely,elx)^penal*Ue'*KE*Ue;
dc(ely,elx) = -penal*x(ely,elx)^(penal-1)*Ue'*KE*Ue;
end
end
% FILTERING OF SENSITIVITIES
[dc] = check(nelx,nely,rmin,x,dc);
dfdx=ones(1,n)/n;
dfdx=reshape(dfdx,nely,nelx);
% DESIGN UPDATE BY THE OPTIMALITY CRITERIA METHOD
% [x] = OC(nelx,nely,x,volfrac,dc);
% DESIGN UPDATE BY THE MMA METHOD
xval=x(:);
f0val=c;
df0dx=dc(:);
fval=sum(xval)/(n*volfrac)-1;
dfdx=dfdx(:);
a0=1;a=0;c1=1000;d=0;
% ymma,zmma,lam,xsi,eta,mu,zet,s,
[xmma,low,upp] = mmasub(m,n,loop,xval,xmin,xmax,xold1,xold2, ...
f0val,df0dx,fval,dfdx,low,upp,a0,a,c1,d);
xnew=reshape(xmma,nely,nelx);
xold2=xold1;
xold1=x(:);
x=xnew;
% PRINT RESULTS
change = max(max(abs(x-xold)));
Md=0;
for i = 1:nelx
for j = 1:nely
Md=Md+(4*x(j,i)*(1-x(j,i)));
end
end
MD=100*Md/(nelx*nely);
disp([' It.: ' sprintf('%4i',loop) ' Obj.: ' sprintf('%10.4f',c) ...
' Vol.: ' sprintf('%6.3f',sum(sum(x))/(nelx*nely)) ...
' ch.: ' sprintf('%10.6f',change ) ' MD.: ' sprintf('%10.4f',MD)])
% PLOT DENSITIES
colormap(gray); imagesc(-x); axis equal; axis tight; axis off;pause(1e-6);
end
%%%%%%%%%% OPTIMALITY CRITERIA UPDATE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function [xnew]=OC(nelx,nely,x,volfrac,dc)
% l1 = 0; l2 = 100000; move = 0.2;
% while (l2-l1 > 1e-4)
% lmid = 0.5*(l2+l1);
% xnew = max(0.001,max(x-move,min(1.,min(x+move,x.*sqrt(-dc./lmid)))));
% if sum(sum(xnew)) - volfrac*nelx*nely > 0;
% l1 = lmid;
% else
% l2 = lmid;
% end
% end
%%%%%%%%%% MESH-INDEPENDENCY FILTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [dcn]=check(nelx,nely,rmin,x,dc)
dcn=zeros(nely,nelx);
for i = 1:nelx
for j = 1:nely
sum=0.0;
for k = max(i-floor(rmin),1):min(i+floor(rmin),nelx)
for l = max(j-floor(rmin),1):min(j+floor(rmin),nely)
fac = rmin-sqrt((i-k)^2+(j-l)^2);
sum = sum+max(0,fac);
dcn(j,i) = dcn(j,i) + max(0,fac)*x(l,k)*dc(l,k);
end
end
dcn(j,i) = dcn(j,i)/(x(j,i)*sum);
end
end
%%%%%%%%%% FE-ANALYSIS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [U]=FE(nelx,nely,x,penal)
[KE] = lk;
K = sparse(2*(nelx+1)*(nely+1), 2*(nelx+1)*(nely+1));
F = sparse(2*(nely+1)*(nelx+1),1); U = zeros(2*(nely+1)*(nelx+1),1);
for elx = 1:nelx
for ely = 1:nely
n1 = (nely+1)*(elx-1)+ely;
n2 = (nely+1)* elx +ely;
edof = [2*n1-1; 2*n1; 2*n2-1; 2*n2; 2*n2+1; 2*n2+2; 2*n1+1; 2*n1+2];
K(edof,edof) = K(edof,edof) + x(ely,elx)^penal*KE;
end
end
% DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
F(2,1) = -1;
fixeddofs = union([1:2:2*(nely+1)],[2*(nelx+1)*(nely+1)]);
alldofs = [1:2*(nely+1)*(nelx+1)];
freedofs = setdiff(alldofs,fixeddofs);
% SOLVING
U(freedofs,:) = K(freedofs,freedofs) \ F(freedofs,:);
U(fixeddofs,:)= 0;
%%%%%%%%%% ELEMENT STIFFNESS MATRIX %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [KE]=lk
E = 1.;
nu = 0.3;
k=[ 1/2-nu/6 1/8+nu/8 -1/4-nu/12 -1/8+3*nu/8 ...
-1/4+nu/12 -1/8-nu/8 nu/6 1/8-3*nu/8];
KE = E/(1-nu^2)*[ k(1) k(2) k(3) k(4) k(5) k(6) k(7) k(8)
k(2) k(1) k(8) k(7) k(6) k(5) k(4) k(3)
k(3) k(8) k(1) k(6) k(7) k(4) k(5) k(2)
k(4) k(7) k(6) k(1) k(8) k(3) k(2) k(5)
k(5) k(6) k(7) k(8) k(1) k(2) k(3) k(4)
k(6) k(5) k(4) k(3) k(2) k(1) k(8) k(7)
k(7) k(4) k(5) k(2) k(3) k(8) k(1) k(6)
k(8) k(3) k(2) k(5) k(4) k(7) k(6) k(1)];
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