# Amira Abdel-Rahman
# (c) Massachusetts Institute of Technology 2020
# Topology Optimization Implementation in Julia from various sources
# based on https://paulino.ce.gatech.edu/conferences/papers/11pereira_anefficientandcompact.pdf and http://www.topopt.mek.dtu.dk/apps-and-software and https://github.com/blademwang11/Topopt/blob/master/top.jl
#######################################2d##################################################################
function topologyOptimization(nelx,nely,volfrac,rmin,penal)
anim=Animation()
display("Minimum compliance problem with OC")
display("ndes: $nelx x $nely")
display("volfrac: $volfrac rmin: $rmin penal: $penal")
# Max and min stiffness
Emin=1e-9
Emax=1.0
# dofs:
ndof = 2*(nelx+1)*(nely+1)
# Allocate design variables (as array), initialize and allocate sens.
x=volfrac * ones(Float64,nely,nelx)
xold=copy(x)
xPhys=copy(x)
g=0 # must be initialized to use the NGuyen/Paulino OC approach
dc=zeros(Float64,(nely,nelx))
# FE: Build the index vectors for the for coo matrix format.
KE=lk()
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))
# DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
F = sparse([2],[1],[-1.0],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(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)
###################################################
loop = 0
change = 1
maxIter=1000
# Start iteration
for i =1:maxIter
if (change > 0.01)
# Start iteration
loop += 1
# FE-ANALYSIS
sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),64*nelx*nely,1)
K = sparse(vec(iK),vec(jK),vec(sK)); K = (K+K')/2
@timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
# Objective function and sensitivity analysis
ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx)
c = sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce))
dc = -penal*(Emax-Emin)*xPhys.^(penal-1).*ce
dv = ones(nely,nelx)
dc[:] = H*(dc[:]./Hs)
dv[:] = H*(dv[:]./Hs)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES
l1 = 0; l2 = 1e9; move = 0.2; xnew = 0
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.((0.0.-dc)./dv./lmid)))))
xPhys[:] = (H*xnew[:])./Hs
if sum(xPhys[:]) > volfrac*nelx*nely
l1 = lmid
else
l2 = lmid
end
end
change = maximum(abs.(xnew[:].-x[:]))
x = xnew
m=mean(xPhys[:])
display(" It:$loop Obj:$c Vol:$m ch:$change ")
if loop<20||mod(loop,10)==0
xxx=1 .- clamp.(xPhys,0,1)
display(heatmap(xxx,xaxis=nothing,yaxis=nothing,legend=nothing,fc=:grays,clims=(0.0, 1.0),aspect_ratio=:equal))
heatmap(xxx,xaxis=nothing,yaxis=nothing,legend=nothing,fc=:grays,clims=(0.0, 1.0),aspect_ratio=:equal)
frame(anim)
end
xPhys = copy(x)
end
end
return xPhys,anim
end
#########################################################################################################
function CompliantTopologyOptimization(nelx,nely,volfrac,rmin,penal,maxIter,Load,Support,Spring,DOut)
anim=Animation()
display("Minimum compliance problem with OC")
display("ndes: $nelx x $nely")
display("volfrac: $volfrac rmin: $rmin penal: $penal")
# Max and min stiffness
Emin=1e-9
Emax=1.0
change = 1
# dofs:
ndof = 2*(nelx+1)*(nely+1)
# Allocate design variables (as array), initialize and allocate sens.
x=volfrac * ones(Float64,nely,nelx)
xold=copy(x)
xPhys=copy(x)
g=0 # must be initialized to use the NGuyen/Paulino OC approach
dc=zeros(Float64,(nely,nelx))
# FE: Build the index vectors for the for coo matrix format.
KE=lk()
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))
# DEFINE LOADS AND SUPPORTS
F = sparse(2 .*Load[:,1] .-2 .+ Load[:,2],ones(size(Load,1)),Load[:,3],2*(nely+1)*(nelx+1),2)
DofDOut = 2 * DOut[1] - 2 +DOut[2]; #only one
F=Array(F)
F[DofDOut,2]=-1
fixeddofs = 2 .*Support[:,1] .-2 .+ Support[:,2]
NSpring = size(Spring,1);
s = sparse(2 .*Spring[:,1] .-2 .+ Spring[:,2],ones(size(Spring,1)),Spring[:,3],2*(nely+1)*(nelx+1),2)
m=Array(s)[:,1]
S= sparse(diagm(m))
U = zeros(2*(nely+1)*(nelx+1),2)
alldofs = 1:(2*(nely+1)*(nelx+1))
freedofs = setdiff(alldofs,fixeddofs)
# 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)
change= 1
loop = 0
changeStable=0
# Start iteration
for penal in 1.0:0.5:4
display(" Penalty: $penal")
loop = 0
for i =1:maxIter
if (change < 0.01)
changeStable+=1
display(" Change Stable for $changeStable iterations")
else
changeStable=0
end
if (changeStable<10)
# Start iteration
loop += 1
# FE-ANALYSIS
sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),64*nelx*nely,1)
K = sparse(vec(iK),vec(jK),vec(sK))+S;K = (K+K')/2
@timed U[freedofs,:] = K[freedofs,freedofs] \ Array(F[freedofs,:])
U1=U[:,1]
U2=U[:,2]
# Objective function and sensitivity analysis
#ce = reshape(sum((U1[edofMat]*KE).*U2[edofMat],dims=2),nely,nelx)
ce=reshape(-sum((U1[edofMat]*KE).*U2[edofMat],dims=2),nely,nelx)
#c = sum((Emin.+xPhys[:].^penal*(Emax-Emin)).*ce)
c=U[DofDOut,1]
dc = -penal.*(Emax-Emin).*xPhys.^(penal-1).*ce
dv = ones(nely,nelx)
dc[:] = H*(dc[:]./Hs)
dv[:] = H*(dv[:]./Hs)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES
l1 = 0; l2 = 1e9; move = 0.05; xnew = 0 #move=0.2
while (l2-l1)/(l2+l1) > 1e-4 && l2>1e-40
#while (l2-l1)/(l1+l2) > 1e-3
lmid = 0.5*(l2+l1)
xnew = max.(0,max.(x.-move,min.(1,min.(x.+move,x.*sqrt.(max.(1e-10,-dc./dv./lmid))))))
#xnew = max.(0,max.(x.-move,min.(1,min.(x.+move,x.*sqrt.(-dc./dv./lmid)))))
xPhys[:] = (H*xnew[:])./Hs
if sum(xPhys[:]) > volfrac*nelx*nely
l1 = lmid
else
l2 = lmid
end
end
change = maximum(abs.(xnew[:].-x[:]))
x = xnew
# print result
m=mean(xPhys[:])
display(" It:$loop Obj:$c Vol:$m ch:$change ")
if loop<20||mod(loop,5)==0
xxx=vcat(xPhys[end:-1:1,:],xPhys)
xxx=1 .- xxx
display(heatmap(xxx,xaxis=nothing,yaxis=nothing,legend=nothing,fc=:grays,clims=(0.0, 1.0)))
heatmap(xxx,xaxis=nothing,yaxis=nothing,legend=nothing,fc=:grays,clims=(0.0, 1.0))
frame(anim)
end
xPhys = copy(x)
end
end
end
# heatmap(xPhys, legend=false, axis=nothing, foreground_color_subplot=colorant"white",fc=:grays,clims=(0.0, 1.0))
# heatmap(xPhys, legend=false,fc=:grays,clims=(0.0, 1.0))
return xPhys,anim
end
#########################################3d################################################################
function topologyOptimization3d(nelx,nely,nelz,volfrac,rmin,penal)
anim=Animation()
display("Minimum compliance problem with OC")
display("ndes: $nelx x $nely")
display("volfrac: $volfrac rmin: $rmin penal: $penal")
# Max and min stiffness
Emin=1e-9
Emax=1.0
nu=0.3
# dofs:
ndof = 3*(nelx+1)*(nely+1)*(nelz+1)
# Allocate design variables (as array), initialize and allocate sens.
x=volfrac * ones(Float64,nely,nelx,nelz)
xold=copy(x)
xPhys=copy(x)
g=0 # must be initialized to use the NGuyen/Paulino OC approach
dc=zeros(Float64,(nely,nelx,nelz))
# FE: Build the index vectors for the for coo matrix format.
KE=lk_H8(nu)
nele = nelx*nely*nelz
nodenrs = reshape(1:(1+nelx)*(1+nely)*(1+nelz),1+nely,1+nelx,1+nelz)
edofVec = reshape(3*nodenrs[1:end-1,1:end-1,1:end-1].+1,nelx*nely*nelz,1)
edofMat = repeat(edofVec,1,24).+repeat([0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1 3*(nely+1)*(nelx+1).+[0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1]],nelx*nely*nelz,1)
iK = convert(Array{Int},reshape(kron(edofMat,ones(24,1))',24*24*nele,1))
jK = convert(Array{Int},reshape(kron(edofMat,ones(1,24))',24*24*nele,1))
# USER-DEFINED LOAD DOFs
m= Matlab.meshgrid(nelx:nelx, 0:0, 0:nelz)
il=m[1]
jl=m[2]
kl=m[3]
loadnid = kl.*(nelx+1).*(nely+1).+il.*(nely+1).+(nely+1 .-jl)
loaddof = 3 .*loadnid[:] .- 1;
# USER-DEFINED SUPPORT FIXED DOFs
m= Matlab.meshgrid(0:0,0:nely,0:nelz)# Coordinates
iif=m[1]
jf=m[2]
kf=m[3]
fixednid = kf.*(nelx+1).*(nely+1) .+iif .*(nely .+1) .+(nely .+1 .-jf) # Node IDs
fixeddof = [3 .*fixednid[:]; 3 .*fixednid[:].-1; 3 .*fixednid[:].-2] # DOFs
# DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
F= sparse(loaddof,fill(1,length(loaddof)),fill(-1,length(loaddof)),ndof,1);
U = zeros(ndof,1)
alldofs = 1:ndof
freedofs = setdiff(1:ndof,fixeddof)
# Prepare filter
#iH = ones(convert(Int,nele*(2*(ceil(rmin)-1)+1)^2),1)
#jH = ones(Int,size(iH))
#sH = zeros(size(iH))
iH=[]
jH=[]
sH=[]
k = 0
for k1 = 1:nelz
for i1 = 1:nelx
for j1 = 1:nely
e1 = (k1-1)*nelx*nely + (i1-1)*nely+j1
for k2 = max(k1-(ceil(rmin)-1),1):min(k1+(ceil(rmin)-1),nelz)
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 = (k2-1)*nelx*nely + (i2-1)*nely+j2;
k = k+1;
append!(iH, e1 )
append!(jH, e2 )
append!(sH, max(0.0,rmin-sqrt((i1-i2)^2+(j1-j2)^2+(k1-k2)^2) ))
end
end
end
end
end
end
iH=reshape(iH,length(iH),1)
jH=reshape(jH,length(jH),1)
sH=reshape(convert(Array{Float64}, sH),length(sH),1)
H = sparse(vec(iH),vec(jH),vec(sH))
Hs = sum(H,dims=2)
###################################################
loop = 0
change = 1
maxIter=1000
# Start iteration
for i =1:maxIter
if (change > 0.01)
# Start iteration
loop += 1
# FE-ANALYSIS
sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),24*24*nelx*nely*nelz,1)
K = sparse(vec(iK),vec(jK),vec(sK)); K = (K+K')/2
@timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
# Objective function and sensitivity analysis
ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx,nelz)
c = sum(sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce)))
dc = -penal*(Emax-Emin)*xPhys.^(penal-1).*ce
dv = ones(nely,nelx,nelz)
dc[:] = H*(dc[:]./Hs)
dv[:] = H*(dv[:]./Hs)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES
l1 = 0; l2 = 1e9; move = 0.2; xnew = 0
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.((0.0.-dc)./dv./lmid)))))
xPhys[:] = (H*xnew[:])./Hs
if sum(xPhys[:]) > volfrac*nelx*nely
l1 = lmid
else
l2 = lmid
end
end
change = maximum(abs.(xnew[:].-x[:]))
x = xnew
m=mean(xPhys[:])
display(" It:$loop Obj:$c Vol:$m ch:$change ")
if mod(loop,2)==0
display(topplot3d(xPhys))
topplot3d(xPhys)
frame(anim)
end
xPhys = copy(x)
end
end
return xPhys,anim
end
#########################################################################################################
function compliantOptimization3d(nelx,nely,nelz,volfrac,rmin,penal)
anim=Animation()
display("Compliance sythesyz problem with OC")
display("ndes: $nelx x $nely x $nelz")
display("volfrac: $volfrac rmin: $rmin penal: $penal")
# Max and min stiffness
Emin=1e-9
Emax=1.0
nu=0.3
# dofs:
ndof = 3*(nelx+1)*(nely+1)*(nelz+1)
# Allocate design variables (as array), initialize and allocate sens.
x=volfrac * ones(Float64,nely,nelx,nelz)
xold=copy(x)
xPhys=copy(x)
g=0 # must be initialized to use the NGuyen/Paulino OC approach
dc=zeros(Float64,(nely,nelx,nelz))
# FE: Build the index vectors for the for coo matrix format.
KE=lk_H8(nu)
nele = nelx*nely*nelz
nodenrs = reshape(1:(1+nelx)*(1+nely)*(1+nelz),1+nely,1+nelx,1+nelz)
edofVec = reshape(3*nodenrs[1:end-1,1:end-1,1:end-1].+1,nelx*nely*nelz,1)
edofMat = repeat(edofVec,1,24).+repeat([0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1 3*(nely+1)*(nelx+1).+[0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1]],nelx*nely*nelz,1)
iK = convert(Array{Int},reshape(kron(edofMat,ones(24,1))',24*24*nele,1))
jK = convert(Array{Int},reshape(kron(edofMat,ones(1,24))',24*24*nele,1))
# # USER-DEFINED LOAD DOFs
# m= Matlab.meshgrid(nelx:nelx, 0:0, 0:nelz)
# il=m[1]
# jl=m[2]
# kl=m[3]
# loadnid = kl.*(nelx+1).*(nely+1).+il.*(nely+1).+(nely+1 .-jl)
# loaddof = 3 .*loadnid[:] .- 1;
# # USER-DEFINED SUPPORT FIXED DOFs
# m= Matlab.meshgrid(0:0,0:nely,0:nelz)# Coordinates
# iif=m[1]
# jf=m[2]
# kf=m[3]
# fixednid = kf.*(nelx+1).*(nely+1) .+iif .*(nely .+1) .+(nely .+1 .-jf) # Node IDs
# fixeddof = [3 .*fixednid[:]; 3 .*fixednid[:].-1; 3 .*fixednid[:].-2] # DOFs
# # DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
# F= sparse(loaddof,fill(1,length(loaddof)),fill(-1,length(loaddof)),ndof,1);
# U = zeros(ndof,1)
# alldofs = 1:ndof
# freedofs = setdiff(1:ndof,fixeddof)
# USER-DEFINED LOAD DOFs
il = [0 nelx]; jl = [nely nely]; kl = [0 0];
loadnid = kl .*(nelx+1) .*(nely+1) .+il .*(nely .+1) .+(nely+1 .-jl);
loaddof = 3 .*loadnid[:] .- 2;
din = loaddof[1]; dout = loaddof[2];
F = sparse(loaddof,[1;2],[1;-1],ndof,2);
# USER-DEFINED SUPPORT FIXED DOFs
# Top face
m = Matlab.meshgrid(0:nelx,0:nelz)
iif=m[1]
kf=m[2]
fixednid = kf .*(nelx+1) .*(nely+1) .+iif .*(nely+1) .+1;
fixeddof_t = 3 .*fixednid[:] .-1;
# Side face
m = Matlab.meshgrid(0:nelx,0:nely)
iif=m[1]
jf=m[2]
fixednid = iif .*(nely+1) .+(nely+1 .-jf);
fixeddof_s = 3*fixednid[:];
# Two pins
iif = [0;0]; jf = [0;1]; kf = [nelz;nelz];
fixednid = kf .*(nelx+1) .*(nely .+1) .+iif .*(nely .+1) .+(nely+1 .-jf)
fixeddof_p = [3 .*fixednid[:]; 3 .*fixednid[:].-1; 3 .*fixednid[:].-2] # DOFs
# Fixed DOFs
fixeddof = union(fixeddof_t,fixeddof_s);
fixeddof = union(fixeddof, fixeddof_p);
fixeddof = sort(fixeddof);
# Displacement vector
U = zeros(ndof,2);
freedofs = setdiff(1:ndof,fixeddof)
# Prepare filter
#iH = ones(convert(Int,nele*(2*(ceil(rmin)-1)+1)^2),1)
#jH = ones(Int,size(iH))
#sH = zeros(size(iH))
iH=[]
jH=[]
sH=[]
k = 0
for k1 = 1:nelz
for i1 = 1:nelx
for j1 = 1:nely
e1 = (k1-1)*nelx*nely + (i1-1)*nely+j1
for k2 = max(k1-(ceil(rmin)-1),1):min(k1+(ceil(rmin)-1),nelz)
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 = (k2-1)*nelx*nely + (i2-1)*nely+j2;
k = k+1;
append!(iH, e1 )
append!(jH, e2 )
append!(sH, max(0.0,rmin-sqrt((i1-i2)^2+(j1-j2)^2+(k1-k2)^2) ))
end
end
end
end
end
end
iH=reshape(iH,length(iH),1)
jH=reshape(jH,length(jH),1)
sH=reshape(convert(Array{Float64}, sH),length(sH),1)
H = sparse(vec(iH),vec(jH),vec(sH))
Hs = sum(H,dims=2)
###################################################
loop = 0
change = 1
maxIter=1000
# Start iteration
for i =1:maxIter
if (change > 0.01)
# Start iteration
loop += 1
# FE-ANALYSIS
sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),24*24*nelx*nely*nelz,1)
K = sparse(vec(iK),vec(jK),vec(sK))
K[din,din] = K[din,din] + 0.1
K[dout,dout] = K[dout,dout] + 0.1
K = (K+K')/2
# @timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
@timed U[freedofs,:] = K[freedofs,freedofs] \ Array(F[freedofs,:])
U1 = U[:,1]
U2 = U[:,2]
# Objective function and sensitivity analysis
#ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx,nelz)
ce = reshape(sum((U1[edofMat]*KE).*U2[edofMat],dims=2),nely,nelx,nelz);
#c = sum(sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce)))
c = U[dout,1]
dc = penal*(Emax-Emin)*xPhys.^(penal-1).*ce
dv = ones(nely,nelx,nelz)
dc[:] = H*(dc[:]./Hs)
dv[:] = H*(dv[:]./Hs)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES
#l1 = 0; l2 = 1e9; move = 0.2; xnew = 0
l1 = 0; l2 = 1e9; move = 0.1;xnew = 0
#while (l2-l1)/(l1+l2) > 1e-3
while (l2-l1)/(l1+l2) > 1e-4 && l2 > 1e-40
lmid = 0.5*(l2+l1)
#xnew = max.(0,max.(x.-move,min.(1,min.(x.+move,x.*sqrt.((0.0.-dc)./dv./lmid)))))
xnew = max.(0,max.(x.-move,min.(1,min.(x.+move,x.*(max.(1e-10,-dc./dv./lmid)).^0.3))));
xPhys[:] = (H*xnew[:])./Hs
if sum(xPhys[:]) > volfrac*nelx*nely
l1 = lmid
else
l2 = lmid
end
end
change = maximum(abs.(xnew[:].-x[:]))
x = xnew
m=mean(xPhys[:])
display(" It:$loop Obj:$c Vol:$m ch:$change ")
if mod(loop,5)==0
display(topplot3d(xPhys))
topplot3d(xPhys)
frame(anim)
end
xPhys = copy(x)
end
end
return xPhys,anim
end
#############################################multimaterial############################################################
#Based on: https://link.springer.com/article/10.1007/s00158-013-0999-1
function multitop(nx,ny,tol_out,tol_f,iter_max_in,iter_max_out,p,q,e,v,rf)
anim=Animation()
alpha = zeros(Float64,nx*ny,p)
for i = 1:p
alpha[:,i] .= v[i]
end
# MAKE FILTER
H,Hs = make_filter(nx,ny,rf)
obj=0
change_out = 2*tol_out; iter_out = 0;
while (iter_out < iter_max_out) && (change_out > tol_out)
alpha_old = copy(alpha)
for a = 1:p
for b = a+1:p
obj,alpha = bi_top(a,b,nx,ny,p,v,e,q,alpha,H,Hs,iter_max_in);
end
end
iter_out = iter_out + 1;
change_out = norm(alpha[:].-alpha_old[:],Inf);
display("Iter:$iter_out Obj.:$obj change:$change_out");
# UPDATE FILTER
if (change_out < tol_f) && (rf>3)
tol_f = 0.99*tol_f; rf = 0.99*rf;
H,Hs = make_filter(nx,ny,rf);
end
# SCREEN OUT TEMPORAL TOPOLOGY EVERY 5 ITERATIONS
if mod(iter_out,5)==0
I = make_bitmap(p,nx,ny,alpha);
display(RGB.(I[:,:,1],I[:,:,2],I[:,:,3]))
I = permutedims(I, [3, 1, 2])
img = colorview(RGB, I)
heatmap(img,xaxis=nothing,yaxis=nothing,aspect_ratio=:equal)
frame(anim)
end
end
return anim,alpha
end
function make_filter(nelx,nely,rmin)
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)
return H,Hs
end
function bi_top(a,b,nx,ny,p,v,e,q,alpha_old,H,Hs,maxIter)
alpha = copy(alpha_old)
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]);
nelx=nx
nely=ny
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))
# DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
F = sparse([2],[1],[-1.0],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)
o=0;alpha_a=0
# inner iteration
for i =0: (maxIter-1)
# FE-ANALYSIS
E = e[1]*alpha[:,1].^q
for phase = 2:p
E = E + e[phase]*alpha[:,phase].^q;
end
sK = reshape(KE[:]*E[:]',64*nelx*nely,1)
K = sparse(vec(iK),vec(jK),vec(sK)); K = (K+K')/2
@timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
# Objective function and sensitivity analysis
ce = sum((U[edofMat]*KE).*U[edofMat],dims=2)
o = sum(sum(E.*ce))
# FILTERING OF SENSITIVITIES
dc = 0.0 .-(q .*(e[a].-e[b]).*alpha[:,a].^(q-1)).*ce;
dc = H*(alpha[:,a].*dc)./Hs./max.(1e-3,alpha[:,a])
dc = min.(dc,0.0)
# UPDATE LOWER AND UPPER BOUNDS OF DESIGN VARIABLES
move = 0.2;
r = ones(nx*ny,1)
for k = 1:p
#if (k ~= a) && (k ~= b)
if (k != a) && (k != b)
# display("k $k a $a b $b")
r = r .- alpha[:,k]
end
end
l = max.(0,alpha[:,a] .-move)
u = min.(r,alpha[:,a] .+move)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES
l1 = 0; l2 = 1e9
while (l2-l1)/(l1+l2) > 1e-3
lmid = 0.5*(l2+l1)
alpha_a = max.(l,min.(u,alpha[:,a].*sqrt.(-dc./lmid)));
if sum(alpha_a) > nx*ny*v[a]
l1 = lmid
else
l2 = lmid
end
end
alpha[:,a] = alpha_a
alpha[:,b] = r .-alpha_a
end
return o,alpha
end
function multitop_compliant(nx,ny,tol_out,tol_f,iter_max_in,iter_max_out,p,q,e,v,rf,Load,Support,Spring,DOut)
anim=Animation()
alpha = zeros(Float64,nx*ny,p)
for i = 1:p
alpha[:,i] .= v[i]
end
# MAKE FILTER
H,Hs = make_filter(nx,ny,rf)
obj=0
change_out = 2*tol_out; iter_out = 0;
while (iter_out < iter_max_out) && (change_out > tol_out)
alpha_old = copy(alpha)
for a = 1:p
for b = a+1:p
obj,alpha = bi_top_compliant(a,b,nx,ny,p,v,e,q,alpha,H,Hs,iter_max_in);
end
end
iter_out = iter_out + 1;
change_out = norm(alpha[:].-alpha_old[:],Inf);
display("Iter:$iter_out Obj.:$obj change:$change_out");
# UPDATE FILTER
if (change_out < tol_f) && (rf>3)
tol_f = 0.99*tol_f; rf = 0.99*rf;
H,Hs = make_filter(nx,ny,rf);
end
# SCREEN OUT TEMPORAL TOPOLOGY EVERY 5 ITERATIONS
if mod(iter_out,5)==0
I = make_bitmap_compliant(p,nx,ny,alpha);
display(RGB.(I[:,:,1],I[:,:,2],I[:,:,3]))
I = permutedims(I, [3, 1, 2])
img = colorview(RGB, I)
heatmap(img,xaxis=nothing,yaxis=nothing,aspect_ratio=:equal)
frame(anim)
end
end
return anim,alpha
end
function bi_top_compliant(a,b,nx,ny,p,v,e,q,alpha_old,H,Hs,maxIter,Load,Support,Spring,DOut)
alpha = copy(alpha_old)
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]);
nelx=nx
nely=ny
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))
# # DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
# F = sparse([2],[1],[-1.0],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)
# DEFINE LOADS AND SUPPORTS
F = sparse(2 .*Load[:,1] .-2 .+ Load[:,2],ones(size(Load,1)),Load[:,3],2*(nely+1)*(nelx+1),2)
DofDOut = 2 * DOut[1] - 2 +DOut[2]; #only one
F=Array(F)
F[DofDOut,2]=-1
fixeddofs = 2 .*Support[:,1] .-2 .+ Support[:,2]
NSpring = size(Spring,1);
s = sparse(2 .*Spring[:,1] .-2 .+ Spring[:,2],ones(size(Spring,1)),Spring[:,3],2*(nely+1)*(nelx+1),2)
m=Array(s)[:,1]
S= sparse(diagm(m))
U = zeros(2*(nely+1)*(nelx+1),2)
alldofs = 1:(2*(nely+1)*(nelx+1))
freedofs = setdiff(alldofs,fixeddofs)
o=0;alpha_a=0
# inner iteration
for i =0: (maxIter-1)
# FE-ANALYSIS
E = e[1]*alpha[:,1].^q
for phase = 2:p
E = E + e[phase]*alpha[:,phase].^q;
end
sK = reshape(KE[:]*E[:]',64*nelx*nely,1)
K = sparse(vec(iK),vec(jK),vec(sK));
K[din,din] = K[din,din] + 0.1;
K[dout,dout] = K[dout,dout] + 0.1;
K = (K+K')/2
# @timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
@timed U[freedofs,:] = K[freedofs,freedofs] \ Array(F[freedofs,:])
U1=U[:,1]
U2=U[:,2]
# Objective function and sensitivity analysis
# ce = sum((U[edofMat]*KE).*U[edofMat],dims=2)
ce =-sum((U1[edofMat]*KE).*U2[edofMat],dims=2)
# o = sum(sum(E.*ce))
o=U[DofDOut,1]
# FILTERING OF SENSITIVITIES
dc = 0.0 .-(q .*(e[a].-e[b]).*alpha[:,a].^(q-1)).*ce;
dc = H*(alpha[:,a].*dc)./Hs./max.(1e-3,alpha[:,a])
dc = min.(dc,0.0)
# UPDATE LOWER AND UPPER BOUNDS OF DESIGN VARIABLES
move=0.1;
r = ones(nx*ny,1)
for k = 1:p
#if (k ~= a) && (k ~= b)
if (k != a) && (k != b)
# display("k $k a $a b $b")
r = r .- alpha[:,k]
end
end
l = max.(0,alpha[:,a] .-move)
u = min.(r,alpha[:,a] .+move)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES
l1 = 0; l2 = 1e9
while (l2-l1)/(l2+l1) > 1e-4 && l2>1e-40
# while (l2-l1)/(l1+l2) > 1e-3
lmid = 0.5*(l2+l1)
alpha_a = max.(l,min.(u,alpha[:,a].*((max.(1e-10,-dc./lmid)).^0.3 )));
if sum(alpha_a) > nx*ny*v[a]
l1 = lmid
else
l2 = lmid
end
end
alpha[:,a] = alpha_a
alpha[:,b] = r .-alpha_a
end
return o,alpha
end
function multitop3D(nx,ny,nz,tol_out,tol_f,iter_max_in,iter_max_out,p,q,e,v,rf)
anim=Animation()
alpha = zeros(Float64,nx*ny*nz,p)
for i = 1:p
alpha[:,i] .= v[i]
end
# MAKE FILTER
H,Hs = make_filter3D(nx,ny,nz,rf)
obj=0
change_out = 2*tol_out; iter_out = 0;
while (iter_out < iter_max_out) && (change_out > tol_out)
alpha_old = copy(alpha)
for a = 1:p
for b = a+1:p
obj,alpha = bi_top3D(a,b,nx,ny,nz,p,v,e,q,alpha,H,Hs,iter_max_in);
end
end
iter_out = iter_out + 1;
change_out = norm(alpha[:].-alpha_old[:],Inf);
display("Iter:$iter_out Obj.:$obj change:$change_out");
# UPDATE FILTER
if (change_out < tol_f) && (rf>3)
tol_f = 0.99*tol_f; rf = 0.99*rf;
H,Hs = make_filter3D(nx,ny,nz,rf)
end
# SCREEN OUT TEMPORAL TOPOLOGY EVERY 5 ITERATIONS
if mod(iter_out,5)==0
I = make_bitmap_3d(p,nx,ny,nz,alpha)
threshold=0.05
display(topplotmulti3d(nx,ny,nz,I,threshold))
frame(anim)
end
end
return anim,alpha
end
function make_filter3D(nelx,nely,nelz,rmin)
iH=[]
jH=[]
sH=[]
k = 0
for k1 = 1:nelz
for i1 = 1:nelx
for j1 = 1:nely
e1 = (k1-1)*nelx*nely + (i1-1)*nely+j1
for k2 = max(k1-(ceil(rmin)-1),1):min(k1+(ceil(rmin)-1),nelz)
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 = (k2-1)*nelx*nely + (i2-1)*nely+j2;
k = k+1;
append!(iH, e1 )
append!(jH, e2 )
append!(sH, max(0.0,rmin-sqrt((i1-i2)^2+(j1-j2)^2+(k1-k2)^2) ))
end
end
end
end
end
end
iH=reshape(iH,length(iH),1)
jH=reshape(jH,length(jH),1)
sH=reshape(convert(Array{Float64}, sH),length(sH),1)
H = sparse(vec(iH),vec(jH),vec(sH))
Hs = sum(H,dims=2)
return H,Hs
end
function bi_top3D(a,b,nx,ny,nz,p,v,e,q,alpha_old,H,Hs,maxIter)
alpha = copy(alpha_old)
nu = 0.3;
# PREPARE FINITE ELEMENT ANALYSIS
KE=lk_H8(nu)
nelx=nx
nely=ny
nelz=nz
nele = nelx*nely*nelz
ndof = 3*(nelx+1)*(nely+1)*(nelz+1)
nodenrs = reshape(1:(1+nelx)*(1+nely)*(1+nelz),1+nely,1+nelx,1+nelz)
edofVec = reshape(3*nodenrs[1:end-1,1:end-1,1:end-1].+1,nelx*nely*nelz,1)
edofMat = repeat(edofVec,1,24).+repeat([0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1 3*(nely+1)*(nelx+1).+[0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1]],nelx*nely*nelz,1)
iK = convert(Array{Int},reshape(kron(edofMat,ones(24,1))',24*24*nele,1))
jK = convert(Array{Int},reshape(kron(edofMat,ones(1,24))',24*24*nele,1))
# USER-DEFINED LOAD DOFs
m= Matlab.meshgrid(nelx:nelx, 0:0, 0:nelz)
il=m[1]
jl=m[2]
kl=m[3]
loadnid = kl.*(nelx+1).*(nely+1).+il.*(nely+1).+(nely+1 .-jl)
loaddof = 3 .*loadnid[:] .- 1;
# USER-DEFINED SUPPORT FIXED DOFs
m= Matlab.meshgrid(0:0,0:nely,0:nelz)# Coordinates
iif=m[1]
jf=m[2]
kf=m[3]
fixednid = kf.*(nelx+1).*(nely+1) .+iif .*(nely .+1) .+(nely .+1 .-jf) # Node IDs
fixeddof = [3 .*fixednid[:]; 3 .*fixednid[:].-1; 3 .*fixednid[:].-2] # DOFs
# DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
F= sparse(loaddof,fill(1,length(loaddof)),fill(-1,length(loaddof)),ndof,1);
U = zeros(ndof,1)
alldofs = 1:ndof
freedofs = setdiff(1:ndof,fixeddof)
o=0;alpha_a=0
# inner iteration
for i =0: (maxIter-1)
# FE-ANALYSIS
E = e[1]*alpha[:,1].^q
for phase = 2:p
E = E + e[phase]*alpha[:,phase].^q;
end
sK = reshape(KE[:]*E[:]',24*24*nelx*nely*nelz,1)
K = sparse(vec(iK),vec(jK),vec(sK)); K = (K+K')/2
@timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
# Objective function and sensitivity analysis
ce = sum((U[edofMat]*KE).*U[edofMat],dims=2)
o = sum(sum(E.*ce))
# FILTERING OF SENSITIVITIES
dc = 0.0 .-(q .*(e[a].-e[b]).*alpha[:,a].^(q-1)).*ce;
dc = H*(alpha[:,a].*dc)./Hs./max.(1e-3,alpha[:,a])
dc = min.(dc,0.0)
# UPDATE LOWER AND UPPER BOUNDS OF DESIGN VARIABLES
move = 0.2;
r = ones(nx*ny*nz,1)
for k = 1:p
#if (k ~= a) && (k ~= b)
if (k != a) && (k != b)
# display("k $k a $a b $b")
r = r .- alpha[:,k]
end
end
l = max.(0,alpha[:,a] .-move)
u = min.(r,alpha[:,a] .+move)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES
l1 = 0; l2 = 1e9
while (l2-l1)/(l1+l2) > 1e-3
lmid = 0.5*(l2+l1)
alpha_a = max.(l,min.(u,alpha[:,a].*sqrt.(-dc./lmid)));
if sum(alpha_a) > nx*ny*v[a]
l1 = lmid
else
l2 = lmid
end
end
alpha[:,a] = alpha_a
alpha[:,b] = r .-alpha_a
end
return o,alpha
end
function multitop3D_compliant(nx,ny,nz,tol_out,tol_f,iter_max_in,iter_max_out,p,q,e,v,rf)
anim=Animation()
alpha = zeros(Float64,nx*ny*nz,p)
for i = 1:p
alpha[:,i] .= v[i]
end
# MAKE FILTER
H,Hs = make_filter3D(nx,ny,nz,rf)
obj=0
change_out = 2*tol_out; iter_out = 0;
while (iter_out < iter_max_out) && (change_out > tol_out)
alpha_old = copy(alpha)
for a = 1:p
for b = a+1:p
obj,alpha = bi_top3D_compliant(a,b,nx,ny,nz,p,v,e,q,alpha,H,Hs,iter_max_in);
end
end
iter_out = iter_out + 1;
change_out = norm(alpha[:].-alpha_old[:],Inf);
display("Iter:$iter_out Obj.:$obj change:$change_out");
# UPDATE FILTER
if (change_out < tol_f) && (rf>3)
tol_f = 0.99*tol_f; rf = 0.99*rf;
H,Hs = make_filter3D(nx,ny,nz,rf)
end
# SCREEN OUT TEMPORAL TOPOLOGY EVERY 5 ITERATIONS
if mod(iter_out,5)==0
I = make_bitmap_3d(p,nx,ny,nz,alpha)
threshold=0.05
display(topplotmulti3d(nx,ny,nz,I,threshold))
frame(anim)
end
end
return anim,alpha
end
function bi_top3D_compliant(a,b,nx,ny,nz,p,v,e,q,alpha_old,H,Hs,maxIter)
alpha = copy(alpha_old)
nu = 0.3;
# PREPARE FINITE ELEMENT ANALYSIS
KE=lk_H8(nu)
nelx=nx
nely=ny
nelz=nz
nele = nelx*nely*nelz
ndof = 3*(nelx+1)*(nely+1)*(nelz+1)
nodenrs = reshape(1:(1+nelx)*(1+nely)*(1+nelz),1+nely,1+nelx,1+nelz)
edofVec = reshape(3*nodenrs[1:end-1,1:end-1,1:end-1].+1,nelx*nely*nelz,1)
edofMat = repeat(edofVec,1,24).+repeat([0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1 3*(nely+1)*(nelx+1).+[0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1]],nelx*nely*nelz,1)
iK = convert(Array{Int},reshape(kron(edofMat,ones(24,1))',24*24*nele,1))
jK = convert(Array{Int},reshape(kron(edofMat,ones(1,24))',24*24*nele,1))
# USER-DEFINED LOAD DOFs
il = [0 nelx]; jl = [nely nely]; kl = [0 0];
loadnid = kl .*(nelx+1) .*(nely+1) .+il .*(nely .+1) .+(nely+1 .-jl);
loaddof = 3 .*loadnid[:] .- 2;
din = loaddof[1]; dout = loaddof[2];
F = sparse(loaddof,[1;2],[1;-1],ndof,2);
# USER-DEFINED SUPPORT FIXED DOFs
# Top face
m = Matlab.meshgrid(0:nelx,0:nelz)
iif=m[1]
kf=m[2]
fixednid = kf .*(nelx+1) .*(nely+1) .+iif .*(nely+1) .+1;
fixeddof_t = 3 .*fixednid[:] .-1;
# Side face
m = Matlab.meshgrid(0:nelx,0:nely)
iif=m[1]
jf=m[2]
fixednid = iif .*(nely+1) .+(nely+1 .-jf);
fixeddof_s = 3*fixednid[:];
# Two pins
iif = [0;0]; jf = [0;1]; kf = [nelz;nelz];
fixednid = kf .*(nelx+1) .*(nely .+1) .+iif .*(nely .+1) .+(nely+1 .-jf)
fixeddof_p = [3 .*fixednid[:]; 3 .*fixednid[:].-1; 3 .*fixednid[:].-2] # DOFs
# Fixed DOFs
fixeddof = union(fixeddof_t,fixeddof_s);
fixeddof = union(fixeddof, fixeddof_p);
fixeddof = sort(fixeddof);
# Displacement vector
U = zeros(ndof,2);
freedofs = setdiff(1:ndof,fixeddof)
o=0;alpha_a=0
# inner iteration
for i =0: (maxIter-1)
# FE-ANALYSIS
E = e[1]*alpha[:,1].^q
for phase = 2:p
E = E + e[phase]*alpha[:,phase].^q;
end
sK = reshape(KE[:]*E[:]',24*24*nelx*nely*nelz,1)
K = sparse(vec(iK),vec(jK),vec(sK));
K[din,din] = K[din,din] + 0.1;
K[dout,dout] = K[dout,dout] + 0.1;
K = (K+K')/2;
# @timed U[freedofs] = K[freedofs,freedofs] \ Array(F[freedofs])
@timed U[freedofs,:] = K[freedofs,freedofs] \ Array(F[freedofs,:])
U1 = U[:,1]
U2 = U[:,2]
# Objective function and sensitivity analysis
ce = -sum((U1[edofMat]*KE).*U2[edofMat],dims=2)
# o = sum(sum(E.*ce))
o = U[dout,1]
# FILTERING OF SENSITIVITIES
dc = 0.0 .-(q .*(e[a].-e[b]).*alpha[:,a].^(q-1)).*ce;
dc = H*(alpha[:,a].*dc)./Hs./max.(1e-3,alpha[:,a])
dc = min.(dc,0.0)
# UPDATE LOWER AND UPPER BOUNDS OF DESIGN VARIABLES
move = 0.1;
r = ones(nx*ny*nz,1)
for k = 1:p
#if (k ~= a) && (k ~= b)
if (k != a) && (k != b)
# display("k $k a $a b $b")
r = r .- alpha[:,k]
end
end
l = max.(0,alpha[:,a] .-move)
u = min.(r,alpha[:,a] .+move)
# OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES
l1 = 0; l2 = 1e9
# while (l2-l1)/(l1+l2) > 1e-3
while (l2-l1)/(l1+l2) > 1e-4 && l2 > 1e-40
lmid = 0.5*(l2+l1)
# alpha_a = max.(l,min.(u,alpha[:,a].*sqrt.(-dc./lmid)));
alpha_a = max.(l,min.(u,alpha[:,a].*((max.(1e-10,-dc./lmid)).^0.3 )));
if sum(alpha_a) > nx*ny*v[a]
l1 = lmid
else
l2 = lmid
end
end
alpha[:,a] = alpha_a
alpha[:,b] = r .-alpha_a
end
return o,alpha
end
#############################################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,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
#####################################FEA####################################################################
function 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[0+1] k[1+1] k[2+1] k[3+1] k[4+1] k[5+1] k[6+1] k[7+1];
k[1+1] k[0+1] k[7+1] k[6+1] k[5+1] k[4+1] k[3+1] k[2+1];
k[2+1] k[7+1] k[0+1] k[5+1] k[6+1] k[3+1] k[4+1] k[1+1];
k[3+1] k[6+1] k[5+1] k[0+1] k[7+1] k[2+1] k[1+1] k[4+1];
k[4+1] k[5+1] k[6+1] k[7+1] k[0+1] k[1+1] k[2+1] k[3+1];
k[5+1] k[4+1] k[3+1] k[2+1] k[1+1] k[0+1] k[7+1] k[6+1];
k[6+1] k[3+1] k[4+1] k[1+1] k[2+1] k[7+1] k[0+1] k[5+1];
k[7+1] k[2+1] k[1+1] k[4+1] k[3+1] k[6+1] k[5+1] k[0+1] ];
return (KE)
end
function lk_H8(nu)
A = [32 6 -8 6 -6 4 3 -6 -10 3 -3 -3 -4 -8;
-48 0 0 -24 24 0 0 0 12 -12 0 12 12 12];
k = 1/144*A'*[1; nu];
K1 = [k[1] k[2] k[2] k[3] k[5] k[5];
k[2] k[1] k[2] k[4] k[6] k[7];
k[2] k[2] k[1] k[4] k[7] k[6];
k[3] k[4] k[4] k[1] k[8] k[8];
k[5] k[6] k[7] k[8] k[1] k[2];
k[5] k[7] k[6] k[8] k[2] k[1]];
K2 = [k[9] k[8] k[12] k[6] k[4] k[7];
k[8] k[9] k[12] k[5] k[3] k[5];
k[10] k[10] k[13] k[7] k[4] k[6];
k[6] k[5] k[11] k[9] k[2] k[10];
k[4] k[3] k[5] k[2] k[9] k[12]
k[11] k[4] k[6] k[12] k[10] k[13]];
K3 = [k[6] k[7] k[4] k[9] k[12] k[8];
k[7] k[6] k[4] k[10] k[13] k[10];
k[5] k[5] k[3] k[8] k[12] k[9];
k[9] k[10] k[2] k[6] k[11] k[5];
k[12] k[13] k[10] k[11] k[6] k[4];
k[2] k[12] k[9] k[4] k[5] k[3]];
K4 = [k[14] k[11] k[11] k[13] k[10] k[10];
k[11] k[14] k[11] k[12] k[9] k[8];
k[11] k[11] k[14] k[12] k[8] k[9];
k[13] k[12] k[12] k[14] k[7] k[7];
k[10] k[9] k[8] k[7] k[14] k[11];
k[10] k[8] k[9] k[7] k[11] k[14]];
K5 = [k[1] k[2] k[8] k[3] k[5] k[4];
k[2] k[1] k[8] k[4] k[6] k[11];
k[8] k[8] k[1] k[5] k[11] k[6];
k[3] k[4] k[5] k[1] k[8] k[2];
k[5] k[6] k[11] k[8] k[1] k[8];
k[4] k[11] k[6] k[2] k[8] k[1]];
K6 = [k[14] k[11] k[7] k[13] k[10] k[12];
k[11] k[14] k[7] k[12] k[9] k[2];
k[7] k[7] k[14] k[10] k[2] k[9];
k[13] k[12] k[10] k[14] k[7] k[11];
k[10] k[9] k[2] k[7] k[14] k[7];
k[12] k[2] k[9] k[11] k[7] k[14]];
KE = 1/((nu+1)*(1-2*nu))*[ K1 K2 K3 K4;K2' K5 K6 K3';K3' K6 K5' K2';K4 K3 K2 K1'];
return KE
end
function getDisplacement(xPhys)
display("Minimum compliance problem with OC")
display("ndes: $nelx x $nely")
display("volfrac: $volfrac rmin: $rmin penal: $penal")
# Max and min stiffness
Emin=1e-9
Emax=1.0
nu=0.3
# dofs:
ndof = 3*(nelx+1)*(nely+1)*(nelz+1)
# Allocate design variables (as array), initialize and allocate sens.
x=volfrac * ones(Float64,nely,nelx,nelz)
xold=copy(x)
g=0 # must be initialized to use the NGuyen/Paulino OC approach
dc=zeros(Float64,(nely,nelx,nelz))
# FE: Build the index vectors for the for coo matrix format.
KE=lk_H8(nu)
nele = nelx*nely*nelz
nodenrs = reshape(1:(1+nelx)*(1+nely)*(1+nelz),1+nely,1+nelx,1+nelz)
edofVec = reshape(3*nodenrs[1:end-1,1:end-1,1:end-1].+1,nelx*nely*nelz,1)
edofMat = repeat(edofVec,1,24).+repeat([0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1 3*(nely+1)*(nelx+1).+[0 1 2 3*nely.+[3 4 5 0 1 2] -3 -2 -1]],nelx*nely*nelz,1)
iK = convert(Array{Int},reshape(kron(edofMat,ones(24,1))',24*24*nele,1))
jK = convert(Array{Int},reshape(kron(edofMat,ones(1,24))',24*24*nele,1))
# USER-DEFINED LOAD DOFs
il = [0 nelx]; jl = [nely nely]; kl = [0 0];
loadnid = kl .*(nelx+1) .*(nely+1) .+il .*(nely .+1) .+(nely+1 .-jl);
loaddof = 3 .*loadnid[:] .- 2;
din = loaddof[1]; dout = loaddof[2];
F = sparse(loaddof,[1;2],[1;-1],ndof,2);
# USER-DEFINED SUPPORT FIXED DOFs
# Top face
m = Matlab.meshgrid(0:nelx,0:nelz)
iif=m[1]
kf=m[2]
fixednid = kf .*(nelx+1) .*(nely+1) .+iif .*(nely+1) .+1;
fixeddof_t = 3 .*fixednid[:] .-1;
# Side face
m = Matlab.meshgrid(0:nelx,0:nely)
iif=m[1]
jf=m[2]
fixednid = iif .*(nely+1) .+(nely+1 .-jf);
fixeddof_s = 3*fixednid[:];
# Two pins
iif = [0;0]; jf = [0;1]; kf = [nelz;nelz];
fixednid = kf .*(nelx+1) .*(nely .+1) .+iif .*(nely .+1) .+(nely+1 .-jf)
fixeddof_p = [3 .*fixednid[:]; 3 .*fixednid[:].-1; 3 .*fixednid[:].-2] # DOFs
# Fixed DOFs
fixeddof = union(fixeddof_t,fixeddof_s);
fixeddof = union(fixeddof, fixeddof_p);
fixeddof = sort(fixeddof);
# Displacement vector
U = zeros(ndof,2);
freedofs = setdiff(1:ndof,fixeddof)
iH=[]
jH=[]
sH=[]
k = 0
for k1 = 1:nelz
for i1 = 1:nelx
for j1 = 1:nely
e1 = (k1-1)*nelx*nely + (i1-1)*nely+j1
for k2 = max(k1-(ceil(rmin)-1),1):min(k1+(ceil(rmin)-1),nelz)
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 = (k2-1)*nelx*nely + (i2-1)*nely+j2;
k = k+1;
append!(iH, e1 )
append!(jH, e2 )
append!(sH, max(0.0,rmin-sqrt((i1-i2)^2+(j1-j2)^2+(k1-k2)^2) ))
end
end
end
end
end
end
iH=reshape(iH,length(iH),1)
jH=reshape(jH,length(jH),1)
sH=reshape(convert(Array{Float64}, sH),length(sH),1)
H = sparse(vec(iH),vec(jH),vec(sH))
Hs = sum(H,dims=2)
###################################################
# FE-ANALYSIS
sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),24*24*nelx*nely*nelz,1)
K = sparse(vec(iK),vec(jK),vec(sK))
K[din,din] = K[din,din] + 0.1
K[dout,dout] = K[dout,dout] + 0.1
K = (K+K')/2
@timed U[freedofs,:] = K[freedofs,freedofs] \ Array(F[freedofs,:])
return U,ndof,freedofs
end
#####################################Ployt####################################################################
function topplot3d(xPhys)
ix=[]
iy=[]
iz=[]
for j in 1:nely
for i in 1:nelx
for k in 1:nelz
if(xPhys[j,i,k]>0.0)
append!(ix,i)
append!(iy,j)
append!(iz,k)
end
end
end
end
# r = 4.0
# lim = FRect3D((-4,-4,-4*r),(8,8,8*r))
return scatter(ix,iz,iy,color="black",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
end
function plotDisplacement(nelx,nely,nelz,xPhysC)
U,ndof,freedofs=getDisplacement(xPhysC)
displacement=reshape(U[:,2],3,(nely+1),(nelx+1),(nelz+1));
anim1=Animation()
for step in 1:10
ix=[]
iy=[]
iz=[]
exg=0.04*step
for j in 1:nely
for i in 1:nelx
for k in 1:nelz
if(xPhysC[j,i,k]>0.0)
append!(ix,i+exg*displacement[1,j,i,k])
append!(iy,j+exg*displacement[2,j,i,k])
append!(iz,k+exg*displacement[3,j,i,k])
end
end
end
end
# r = 4.0
# lim = FRect3D((-4,-4,-4*r),(8,8,8*r))
scatter(ix,iz,iy,color="black",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60),xlim=(0,nelx),zlim=(0,nely),ylim=(-10,10))#,markershape = :square
frame(anim1)
end
return anim1
end
function make_bitmap(p,nx,ny,alpha)
color = [1 0 0; 0 0 .45; 0 1 0; 0 0 0; 1 1 1];
I = zeros(nx*ny,3);
for j = 1:p
I[:,1:3] = I[:,1:3] + alpha[:,j] .*color[j,1:3]';
end
# I = imresize(reshape(I,ny,nx,3),10,'bilinear');
I=reshape(I,ny,nx,3)
II=hcat(I[:,end:-1:1,:],I)
return II
end
function make_bitmap_compliant(p,nx,ny,alpha)
color = [1 0 0; 0 0 .45; 0 1 0; 0 0 0; 1 1 1];
I = zeros(nx*ny,3);
for j = 1:p
I[:,1:3] = I[:,1:3] + alpha[:,j] .*color[j,1:3]';
end
# I = imresize(reshape(I,ny,nx,3),10,'bilinear');
I=reshape(I,ny,nx,3)
# II=hcat(I[:,end:-1:1,:],I)
II=vcat(I[end:-1:1,:,:],I)
return II
end
function make_bitmap_3d(p,nx,ny,nz,alpha)
color = [1 0 0; 0 0 1; 0 1 0; 0 0 0; 1 1 1];
I = zeros(nx*ny*nz,3);
for j = 1:p
I[:,1:3] = I[:,1:3] + alpha[:,j] .*color[j,1:3]';
end
# I = imresize(reshape(I,ny,nx,3),10,'bilinear');
I=reshape(I,ny,nx,nz,3)
II=hcat(I[:,end:-1:1,:,:],I)
return II
end
function topplotmulti3d(nelx,nely,nelz,I,threshold)
ixr=[]
iyr=[]
izr=[]
ixg=[]
iyg=[]
izg=[]
ixb=[]
iyb=[]
izb=[]
for j in 1:nely
for i in 1:nelx
for k in 1:nelz
if(I[j,i,k,1]>threshold)
append!(ixr,i)
append!(iyr,j)
append!(izr,k)
end
if(I[j,i,k,2]>threshold)
append!(ixg,i)
append!(iyg,j)
append!(izg,k)
end
if(I[j,i,k,3]>threshold)
append!(ixb,i)
append!(iyb,j)
append!(izb,k)
end
end
end
end
p=scatter(ixr,izr,iyr,color="red",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
p=scatter!(ixg,izg,iyg,color="green",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
p=scatter!(ixb,izb,iyb,color="blue",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
return p
end
function topplotmulti3d_mirror(nelx,nely,nelz,I,threshold,vertical)
ixr=[]
iyr=[]
izr=[]
ixg=[]
iyg=[]
izg=[]
ixb=[]
iyb=[]
izb=[]
if(vertical)
for j in 1:nely
for i in 1:nelx
for k in 1:nelz
if(I[j,i,k,1]>threshold)
append!(ixr,i)
append!(ixr,i)
append!(iyr,nely-j)
append!(iyr,j+nely-1)
append!(izr,k)
append!(izr,k)
end
if(I[j,i,k,2]>threshold)
append!(ixg,i)
append!(ixg,i)
append!(iyg,nely-j)
append!(iyg,j+nely-1)
append!(izg,k)
append!(izg,k)
end
if(I[j,i,k,3]>threshold)
append!(ixb,i)
append!(ixb,i)
append!(iyb,nely-j)
append!(iyb,j+nely-1)
append!(izb,k)
append!(izb,k)
end
end
end
end
else
for j in 1:nely
for i in 1:nelx
for k in 1:nelz
if(I[j,i,k,1]>threshold)
append!(ixr,nelx-i+nelx-1)
append!(ixr,i)
append!(iyr,j)
append!(izr,k)
append!(iyr,j)
append!(izr,k)
end
if(I[j,i,k,2]>threshold)
append!(ixg,nelx-i+nelx-1)
append!(ixg,i)
append!(iyg,j)
append!(izg,k)
append!(iyg,j)
append!(izg,k)
end
if(I[j,i,k,3]>threshold)
append!(ixb,nelx-i+nelx-1)
append!(ixb,i)
append!(iyb,j)
append!(izb,k)
append!(iyb,j)
append!(izb,k)
end
end
end
end
end
p=scatter(ixr,izr,iyr,color="red",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
p=scatter!(ixg,izg,iyg,color="green",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
p=scatter!(ixb,izb,iyb,color="blue",label="",markersize =4, aspect_ratio=:equal,markerstrokealpha = 0.2,markeralpha = 0.6,markershape = :square,camera = (30, 60))#,markershape = :square
return p
end
#####################################Utils####################################################################
function getIndex(i,j,nelx,nely)
return (i-1)*(nely+1)+(j-1)+1
end