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Amira Abdel-Rahman
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# Amira Abdel-Rahman
# (c) Massachusetts Institute of Technology 2020
#############################################2d Periodic############################################################
# Based on https://link.springer.com/article/10.1007/s00158-015-1294-0
## PERIODIC MATERIAL MICROSTRUCTURE DESIGN
function topX(nelx,nely,volfrac,penal,rmin,ft,optType="bulk")
## 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 = repeat(edofVec,1,8).+repeat([0 1 2*nely.+[2 3 0 1] -2 -1],nelx*nely,1)
iK = convert(Array{Int},reshape(kron(edofMat,ones(8,1))',64*nelx*nely,1))
jK = convert(Array{Int},reshape(kron(edofMat,ones(1,8))',64*nelx*nely,1))
## PREPARE FILTER
iH = ones(convert(Int,nelx*nely*(2*(ceil(rmin)-1)+1)^2),1)
jH = ones(Int,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(vec(iH),vec(jH),vec(sH))
Hs = sum(H,dims=2)
## PERIODIC BOUNDARY CONDITIONS
e0 = Matrix(1.0I, 3, 3);
ufixed = zeros(8,3);
U = zeros(2*(nely+1)*(nelx+1),3);
alldofs = [1:2*(nely+1)*(nelx+1)];
n1 = vcat(nodenrs[end,[1,end]],nodenrs[1,[end,1]]);
d1 = vec(reshape([(2 .* n1 .-1) 2 .*n1]',1,8));
n3 = [vec(nodenrs[2:(end-1),1]');vec(nodenrs[end,2:(end-1)])];
d3 = vec(reshape([(2 .*n3 .-1) 2 .*n3]',1,2*(nelx+nely-2)));
n4 = [vec(nodenrs[2:end-1,end]');vec(nodenrs[1,2:end-1])];
d4 = vec(reshape([(2 .*n4 .-1) 2 .*n4]',1,2*(nelx+nely-2)));
d2 = setdiff(vcat(alldofs...),union(union(d1,d3),d4));
for j = 1:3
ufixed[3:4,j] = [e0[1,j] e0[3,j]/2 ; e0[3,j]/2 e0[2,j]]*[nelx;0];
ufixed[7:8,j] = [e0[1,j] e0[3,j]/2 ; e0[3,j]/2 e0[2,j]]*[0;nely];
ufixed[5:6,j] = ufixed[3:4,j] .+ ufixed[7:8,j];
end
wfixed = [repeat(ufixed[3:4,:],nely-1,1); repeat(ufixed[7:8,:],nelx-1,1)];
## INITIALIZE ITERATION
qe = Array{Any,2}(undef, 3, 3);
Q = zeros(3,3);
dQ = Array{Any,2}(undef, 3, 3);
x = volfrac.*ones(nely,nelx)
for i = 1:nelx
for j = 1:nely
vall=3
if optType=="poisson"
vall=6
end
if sqrt((i-nelx/2-0.5)^2+(j-nely/2-0.5)^2) < min(nelx,nely)/vall
x[j,i] = volfrac/2.0;
end
end
end
xPhys = copy(x);
change = 1;
loop = 0;
xnew=zeros(size(x))
## START ITERATION
while (change > 0.01)
loop = loop +1;
## FE-ANALYSIS
sK = reshape(KE[:]*(Emin .+ xPhys[:]'.^penal*(80 .- Emin)),64*nelx*nely,1);
K = sparse(vec(iK),vec(jK),vec(sK));
K = (K.+K')./2.0;
Kr = vcat(hcat(K[d2,d2] , K[d2,d3]+K[d2,d4]),hcat((K[d3,d2]+K[d4,d2]),(K[d3,d3]+K[d4,d3]+K[d3,d4]+K[d4,d4])));
U[d1,:] .= ufixed;
U[[d2;d3],:] = Kr\(-[K[d2,d1]; K[d3,d1]+K[d4,d1]]*ufixed-[K[d2,d4]; K[d3,d4]+K[d4,d4]]*wfixed);
U[d4,:] = U[d3,:]+wfixed;
## OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS
for i = 1:3
for j = 1:3
U1 = U[:,i]; U2 = U[:,j];
qe[i,j] = reshape(sum((U1[edofMat]*KE).*U2[edofMat],dims=2),nely,nelx)./(nelx*nely);
Q[i,j] = sum(sum((Emin .+ xPhys.^penal*(E0 .-Emin)).*qe[i,j]));
dQ[i,j] = penal*(E0-Emin)*xPhys.^(penal-1).*qe[i,j];
end
end
if optType=="bulk"
#bulk
c = -(Q[1,1]+Q[2,2]+Q[1,2]+Q[2,1]);
dc = -(dQ[1,1]+dQ[2,2]+dQ[1,2]+dQ[2,1]);
elseif optType=="shear"
#shear
c=-Q[3,3];
dc=-dQ[3,3];
elseif optType=="poisson"
c = Q[1,2]-(0.8^loop)*(Q[1,1]+Q[2,2]);
dc = dQ[1,2]-(0.8^loop)*(dQ[1,1]+dQ[2,2]);
end
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);
end
## OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES
if optType=="poisson"
l1=0; l2=1e9; move = 0.1;
else
l1 =0; l2 = 1e9; move = 0.2;
end
while (l2-l1 > 1e-9)
lmid = 0.5*(l2+l1);
if optType=="poisson"
xnew = max.(0.0,max.(x.-move,min.(1.0,min.(x.+move,x.*(-dc./dv./lmid)))));
else
xnew = max.(0.0,max.(x.-move,min.(1.0,min.(x.+move,x.*sqrt.(0.0.-dc./dv./lmid)))));
end
if ft == 1
xPhys = copy(xnew);
elseif ft == 2
xPhys[:] = (H*xnew[:])./Hs;
end
if mean(xPhys[:]) > volfrac
l1 = lmid;
else
l2 = lmid;
end
end
change = maximum(abs.(xnew[:].-x[:]))
x = xnew;
## PRINT RESULTS
display(" It:$loop Obj:$c Vol:$(mean(xPhys[:])) ch:$change ")
## PLOT DENSITIES
heatmap(xPhys, aspect_ratio=:equal, legend=false, axis=nothing, foreground_color_subplot=colorant"white",fc=:grays,clims=(0.0, 1.0))
frame(anim)
end
return xnew
end
######################3d Periodic#########################################
#INCORRECT!! FIX LATER
function topX3D(nelx,nely,nelz,volfrac,penal,rmin,ft,optType="bulk")
nel=nelx*nely*nelz
# MATERIAL PROPERTIES
E0 = 1;
Emin = 1e-9;
nu = 0.3;
lx=0.1;ly=0.1;lz=0.1;
vert_cor = [0 lx lx 0 0 lx lx 0;
0 0 ly ly 0 0 ly ly;
0 0 0 0 lz lz lz lz];
cellVolume = lx*ly*lz;
# PREPARE FINITE ELEMENT ANALYSIS
# KE=lk_H8(nu)
# the initial definitions of the PUC
D0 = E0/(1+nu)/(1-2*nu)*
[ 1-nu nu nu 0 0 0 ;
nu 1-nu nu 0 0 0 ;
nu nu 1-nu 0 0 0 ;
0 0 0 (1-2*nu)/2 0 0 ;
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