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},
{
"data": {
"text/plain": [
"\" It:91 Obj:43.18057670963216 Vol:0.40000350839179255 ch:0.009711189576916357 \""
]
},
"metadata": {},
"output_type": "display_data"
},
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"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# minimum compliance \n",
"nelx=40\n",
"nely=26\n",
"volfrac=0.4\n",
"rmin=2\n",
"penal=3.0\n",
"\n",
"xPhys,anim=topologyOptimizationFEMCode(data_FEM,nelx,nely,volfrac,rmin,penal)\n",
"display(heatmap(xPhys',c=:YlGnBu,aspect_ratio=:equal))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"function topX(nelx,nely,volfrac,penal,rmin,ft,optType=\"bulk\")\n",
" ## MATERIAL PROPERTIES\n",
" E0 = 1;\n",
" Emin = 1e-9;\n",
" nu = 0.3;\n",
" ## PREPARE FINITE ELEMENT ANALYSIS\n",
" A11 = [12 3 -6 -3; 3 12 3 0; -6 3 12 -3; -3 0 -3 12]'\n",
" A12 = [-6 -3 0 3; -3 -6 -3 -6; 0 -3 -6 3; 3 -6 3 -6]'\n",
" B11 = [-4 3 -2 9; 3 -4 -9 4; -2 -9 -4 -3; 9 4 -3 -4]'\n",
" B12 = [ 2 -3 4 -9; -3 2 9 -2; 4 9 2 3; -9 -2 3 2]'\n",
" KE = 1/(1-nu^2)/24*([A11 A12;A12' A11]+nu*[B11 B12;B12' B11])\n",
"\n",
" \n",
" nodenrs = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx)\n",
" edofVec = reshape(2*nodenrs[1:end-1,1:end-1].+1,nelx*nely,1)\n",
" edofMat = repeat(edofVec,1,8).+repeat([0 1 2*nely.+[2 3 0 1] -2 -1],nelx*nely,1)\n",
"\n",
" iK = convert(Array{Int},reshape(kron(edofMat,ones(8,1))',64*nelx*nely,1))\n",
" jK = convert(Array{Int},reshape(kron(edofMat,ones(1,8))',64*nelx*nely,1))\n",
" \n",
" ## PREPARE FILTER\n",
" iH = ones(convert(Int,nelx*nely*(2*(ceil(rmin)-1)+1)^2),1)\n",
" jH = ones(Int,size(iH))\n",
" sH = zeros(size(iH))\n",
" \n",
" k = 0;\n",
" for i1 = 1:nelx\n",
" for j1 = 1:nely\n",
" e1 = (i1-1)*nely+j1\n",
" for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)-1),nelx)\n",
" for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)-1),nely)\n",
" e2 = (i2-1)*nely+j2\n",
" k = k+1\n",
" iH[k] = e1\n",
" jH[k] = e2\n",
" sH[k] = max(0,rmin-sqrt((i1-i2)^2+(j1-j2)^2))\n",
" end\n",
" end\n",
" end\n",
" end\n",
" H = sparse(vec(iH),vec(jH),vec(sH))\n",
" Hs = sum(H,dims=2)\n",
" \n",
" ## PERIODIC BOUNDARY CONDITIONS\n",
" e0 = Matrix(1.0I, 3, 3);\n",
" ufixed = zeros(8,3);\n",
" \n",
" U = zeros(2*(nely+1)*(nelx+1),3);\n",
" alldofs = [1:2*(nely+1)*(nelx+1)];\n",
" \n",
" n1 = vcat(nodenrs[end,[1,end]],nodenrs[1,[end,1]]);\n",
" d1 = vec(reshape([(2 .* n1 .-1) 2 .*n1]',1,8));\n",
" n3 = [vec(nodenrs[2:(end-1),1]');vec(nodenrs[end,2:(end-1)])];\n",
" d3 = vec(reshape([(2 .*n3 .-1) 2 .*n3]',1,2*(nelx+nely-2)));\n",
" n4 = [vec(nodenrs[2:end-1,end]');vec(nodenrs[1,2:end-1])];\n",
" d4 = vec(reshape([(2 .*n4 .-1) 2 .*n4]',1,2*(nelx+nely-2)));\n",
" d2 = setdiff(vcat(alldofs...),union(union(d1,d3),d4));\n",
" \n",
" \n",
" \n",
" for j = 1:3\n",
" ufixed[3:4,j] = [e0[1,j] e0[3,j]/2 ; e0[3,j]/2 e0[2,j]]*[nelx;0];\n",
" ufixed[7:8,j] = [e0[1,j] e0[3,j]/2 ; e0[3,j]/2 e0[2,j]]*[0;nely];\n",
" ufixed[5:6,j] = ufixed[3:4,j] .+ ufixed[7:8,j];\n",
" end\n",
" wfixed = [repeat(ufixed[3:4,:],nely-1,1); repeat(ufixed[7:8,:],nelx-1,1)];\n",
" \n",
"\n",
" \n",
" ## INITIALIZE ITERATION\n",
" qe = Array{Any,2}(undef, 3, 3);\n",
" Q = zeros(3,3);\n",
" dQ = Array{Any,2}(undef, 3, 3);\n",
" x = volfrac.*ones(nely,nelx)\n",
" \n",
" for i = 1:nelx\n",
" for j = 1:nely\n",
" vall=3\n",
" if optType==\"poisson\"\n",
" vall=6\n",
" end\n",
" if sqrt((i-nelx/2-0.5)^2+(j-nely/2-0.5)^2) < min(nelx,nely)/vall\n",
" x[j,i] = volfrac/2.0;\n",
" end\n",
" end\n",
" end\n",
" xPhys = copy(x);\n",
" change = 1;\n",
" loop = 0;\n",
" xnew=zeros(size(x))\n",
" ## START ITERATION\n",
" while (change > 0.01)\n",
" loop = loop +1;\n",
" ## FE-ANALYSIS\n",
" sK = reshape(KE[:]*(Emin .+ xPhys[:]'.^penal*(80 .- Emin)),64*nelx*nely,1);\n",
" \n",
" K = sparse(vec(iK),vec(jK),vec(sK)); \n",
" K = (K.+K')./2.0;\n",
" 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])));\n",
" U[d1,:] .= ufixed;\n",
" U[[d2;d3],:] = Kr\\(-[K[d2,d1]; K[d3,d1]+K[d4,d1]]*ufixed-[K[d2,d4]; K[d3,d4]+K[d4,d4]]*wfixed);\n",
" U[d4,:] = U[d3,:]+wfixed;\n",
" \n",
" \n",
" ## OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS\n",
" for i = 1:3\n",
" for j = 1:3\n",
" U1 = U[:,i]; U2 = U[:,j];\n",
" qe[i,j] = reshape(sum((U1[edofMat]*KE).*U2[edofMat],dims=2),nely,nelx)./(nelx*nely);\n",
" Q[i,j] = sum(sum((Emin .+ xPhys.^penal*(E0 .-Emin)).*qe[i,j]));\n",
" dQ[i,j] = penal*(E0-Emin)*xPhys.^(penal-1).*qe[i,j];\n",
" end\n",
" end\n",
" if optType==\"bulk\"\n",
" #bulk\n",
" c = -(Q[1,1]+Q[2,2]+Q[1,2]+Q[2,1]);\n",
" dc = -(dQ[1,1]+dQ[2,2]+dQ[1,2]+dQ[2,1]);\n",
" elseif optType==\"shear\"\n",
" #shear\n",
" c=-Q[3,3];\n",
" dc=-dQ[3,3];\n",
" elseif optType==\"poisson\"\n",
" c = Q[1,2]-(0.8^loop)*(Q[1,1]+Q[2,2]);\n",
" dc = dQ[1,2]-(0.8^loop)*(dQ[1,1]+dQ[2,2]);\n",
" end\n",
" \n",
" dv = ones(nely,nelx);\n",
" ## FILTERING/MODIFICATION OF SENSITIVITIES\n",
" if ft == 1\n",
" dc[:] = H*(x[:].*dc[:])./Hs./max.(1e-3,x[:]);\n",
" elseif ft == 2\n",
" dc[:] = H*(dc[:]./Hs);\n",
" dv[:] = H*(dv[:]./Hs);\n",
" end\n",
" ## OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES\n",
" \n",
" if optType==\"poisson\"\n",
" l1=0; l2=1e9; move = 0.1;\n",
" else\n",
" l1 =0; l2 = 1e9; move = 0.2;\n",
" end\n",
" while (l2-l1 > 1e-9)\n",
" lmid = 0.5*(l2+l1);\n",
" if optType==\"poisson\"\n",
" xnew = max.(0,max.(x.-move,min.(1.0,min.(x.+move,x.*(-dc./dv./lmid)))));\n",
" else\n",
" xnew =max.(0.0,max.(x.-move,min.(1.0,min.(x.+move,x.*sqrt.(0.0.-dc./dv./lmid)))));\n",
" end\n",
" if ft == 1\n",
" xPhys = copy(xnew);\n",
" elseif ft == 2\n",
" xPhys[:] = (H*xnew[:])./Hs;\n",
" end\n",
" if mean(xPhys[:]) > volfrac\n",
" l1 = lmid;\n",
" else\n",
" l2 = lmid;\n",
" end\n",
" end\n",
" change = maximum(abs.(xnew[:].-x[:]))\n",
" x = xnew;\n",
" ## PRINT RESULTS\n",
" display(\" It:$loop Obj:$c Vol:$(mean(xPhys[:])) ch:$change \")\n",
"\n",
" ## PLOT DENSITIES\n",
" heatmap(xPhys, aspect_ratio=:equal, legend=false, axis=nothing, foreground_color_subplot=colorant\"white\",fc=:grays,clims=(0.0, 1.0))\n",
" frame(anim)\n",
" end\n",
" return xnew\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 160,
"metadata": {},
"outputs": [],
"source": [
"# function topX(nelx,nely,volfrac,penal,rmin,ft,optType=\"bulk\")\n",
"nelx=100\n",
"nely=100\n",
"volfrac=0.5\n",
"penal=3\n",
"rmin=5\n",
"ft=1\n",
"optType=\"bulk\"\n",
"\n",
"\n",
"E0 = 1;\n",
"Emin = 1e-9;\n",
"nu = 0.3;\n",
"## PREPARE FINITE ELEMENT ANALYSIS\n",
"A11 = [12 3 -6 -3; 3 12 3 0; -6 3 12 -3; -3 0 -3 12]'\n",
"A12 = [-6 -3 0 3; -3 -6 -3 -6; 0 -3 -6 3; 3 -6 3 -6]'\n",
"B11 = [-4 3 -2 9; 3 -4 -9 4; -2 -9 -4 -3; 9 4 -3 -4]'\n",
"B12 = [ 2 -3 4 -9; -3 2 9 -2; 4 9 2 3; -9 -2 3 2]'\n",
"KE = 1/(1-nu^2)/24*([A11 A12;A12' A11]+nu*[B11 B12;B12' B11])\n",
"\n",
"\n",
"nodenrs = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx)\n",
"edofVec = reshape(2*nodenrs[1:end-1,1:end-1].+1,nelx*nely,1)\n",
"edofMat = repeat(edofVec,1,8).+repeat([0 1 2*nely.+[2 3 0 1] -2 -1],nelx*nely,1)\n",
"\n",
"iK = convert(Array{Int},reshape(kron(edofMat,ones(8,1))',64*nelx*nely,1))\n",
"jK = convert(Array{Int},reshape(kron(edofMat,ones(1,8))',64*nelx*nely,1))\n",
"\n",
"## PREPARE FILTER\n",
"iH = ones(convert(Int,nelx*nely*(2*(ceil(rmin)-1)+1)^2),1)\n",
"jH = ones(Int,size(iH))\n",
"sH = zeros(size(iH))\n",
"\n",
"k = 0;\n",
"for i1 = 1:nelx\n",
" for j1 = 1:nely\n",
" e1 = (i1-1)*nely+j1\n",
" for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)-1),nelx)\n",
" for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)-1),nely)\n",
" e2 = (i2-1)*nely+j2\n",
" k = k+1\n",
" iH[k] = e1\n",
" jH[k] = e2\n",
" sH[k] = max(0,rmin-sqrt((i1-i2)^2+(j1-j2)^2))\n",
" end\n",
" end\n",
" end\n",
"end\n",
"H = sparse(vec(iH),vec(jH),vec(sH))\n",
"Hs = sum(H,dims=2);\n",
"\n",
" "
]
},
{
"cell_type": "code",
"execution_count": 161,
"metadata": {},
"outputs": [],
"source": [
"## PERIODIC BOUNDARY CONDITIONS\n",
"e0 = Matrix(1.0I, 3, 3);\n",
"ufixed = zeros(8,3);\n",
"\n",
"U = zeros(2*(nely+1)*(nelx+1),3);\n",
"alldofs = [1:2*(nely+1)*(nelx+1)];\n",
"\n",
"n1 = vcat(nodenrs[end,[1,end]],nodenrs[1,[end,1]]);\n",
"d1 = vec(reshape([(2 .* n1 .-1) 2 .*n1]',1,8));\n",
"n3 = [vec(nodenrs[2:(end-1),1]');vec(nodenrs[end,2:(end-1)])];\n",
"d3 = vec(reshape([(2 .*n3 .-1) 2 .*n3]',1,2*(nelx+nely-2)));\n",
"n4 = [vec(nodenrs[2:end-1,end]');vec(nodenrs[1,2:end-1])];\n",
"d4 = vec(reshape([(2 .*n4 .-1) 2 .*n4]',1,2*(nelx+nely-2)));\n",
"d2 = setdiff(vcat(alldofs...),union(union(d1,d3),d4));\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 162,
"metadata": {},
"outputs": [
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