{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using LinearAlgebra\n",
"using Plots\n",
"import JSON\n",
"using SparseArrays\n",
"using Images\n",
"using StaticArrays, BenchmarkTools\n",
"using VectorizedRoutines\n",
"using NLopt\n",
"using Statistics\n"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
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},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"showlibrary(:cmocean)\n",
"# :Plots\n",
"# :cmocean\n",
"# :misc\n",
"# :colorcet\n",
"# :colorbrewer"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"getIndex (generic function with 1 method)"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"include(\"./julia/FEM.jl\")\n",
"include(\"./julia/data_FEM.jl\")\n",
"include(\"./julia/MBB.jl\")\n",
"include(\"./julia/topologyOptimization.jl\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## FEM vs Topology"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"nelx=60\n",
"nely=20\n",
"Emin=1e-9\n",
"Emax=1.0"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [],
"source": [
"volfrac=0.5\n",
"problem=MBB(nelx,nely)\n",
"E,f,g,idb,ien,iplane,ndf,nel,nen,nnp,nsd,snu,t,xn,Ke=problem;\n",
"ρ=ones(nel).*volfrac;\n",
"K,F,d,de,Fe=FEM(problem,ρ);\n",
"# ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx)\n",
"# c = sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce))\n",
"\n",
"sK = reshape(Ke[:,:,1][:]*(Emin.+ρ'.^penal*(Emax-Emin)),64*nelx*nely,1);"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [],
"source": [
"# dofs:\n",
"ndof = 2*(nelx+1)*(nely+1)\n",
"# Allocate design variables (as array), initialize and allocate sens.\n",
"x=volfrac * ones(Float64,nely,nelx)\n",
"xold=copy(x)\n",
"xPhys=copy(x)\n",
"g=0 # must be initialized to use the NGuyen/Paulino OC approach\n",
"dc=zeros(Float64,(nely,nelx))\n",
"\n",
"# FE: Build the index vectors for the for coo matrix format.\n",
"KE=lk()\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",
"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",
"# DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)\n",
"F = sparse([2],[1],[-1.0],2*(nely+1)*(nelx+1),1)\n",
"U = zeros(2*(nely+1)*(nelx+1),1)\n",
"fixeddofs = union(1:2:2*(nely+1),2*(nelx+1)*(nely+1))\n",
"alldofs = 1:(2*(nely+1)*(nelx+1))\n",
"freedofs = setdiff(alldofs,fixeddofs)\n",
"sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),64*nelx*nely,1)\n",
"K = sparse(vec(iK),vec(jK),vec(sK)); \n",
"K = (K+K')/2\n",
"@timed U[freedofs] = K[freedofs,freedofs] \\ Array(F[freedofs])\n",
"# Objective function and sensitivity analysis\n",
"ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx)\n",
"c = sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce));"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2562×1 SparseMatrixCSC{Float64,Int64} with 1 stored entry:\n",
" [2, 1] = -1.0"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"F"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8×8 Array{Float64,2}:\n",
" 0.494505 0.178571 -0.302198 … -0.178571 0.0549451 0.0137363\n",
" 0.178571 0.494505 0.0137363 -0.247253 -0.0137363 -0.302198\n",
" -0.302198 0.0137363 0.494505 -0.0137363 -0.247253 0.178571\n",
" -0.0137363 0.0549451 -0.178571 -0.302198 0.178571 -0.247253\n",
" -0.247253 -0.178571 0.0549451 0.178571 -0.302198 -0.0137363\n",
" -0.178571 -0.247253 -0.0137363 … 0.494505 0.0137363 0.0549451\n",
" 0.0549451 -0.0137363 -0.247253 0.0137363 0.494505 -0.178571\n",
" 0.0137363 -0.302198 0.178571 0.0549451 -0.178571 0.494505"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"KE"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2562×2562 SparseMatrixCSC{Float64,Int64} with 32330 stored entries:\n",
" [1 , 1] = 0.0618132\n",
" [2 , 1] = -0.0223214\n",
" [3 , 1] = 0.00686813\n",
" [4 , 1] = -0.00171703\n",
" [43 , 1] = -0.0377747\n",
" [44 , 1] = 0.00171703\n",
" [45 , 1] = -0.0309066\n",
" [46 , 1] = 0.0223214\n",
" [1 , 2] = -0.0223214\n",
" [2 , 2] = 0.0618132\n",
" [3 , 2] = 0.00171703\n",
" [4 , 2] = -0.0377747\n",
" ⋮\n",
" [2520, 2561] = 0.00171703\n",
" [2559, 2561] = 0.00686813\n",
" [2560, 2561] = -0.00171703\n",
" [2561, 2561] = 0.0618132\n",
" [2562, 2561] = -0.0223214\n",
" [2517, 2562] = 0.0223214\n",
" [2518, 2562] = -0.0309066\n",
" [2519, 2562] = -0.00171703\n",
" [2520, 2562] = 0.00686813\n",
" [2559, 2562] = 0.00171703\n",
" [2560, 2562] = -0.0377747\n",
" [2561, 2562] = -0.0223214\n",
" [2562, 2562] = 0.0618132"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {},
"outputs": [
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" 2232.76,1486.45 2232.76,1355.96 2256.76,1355.96 2232.76,1355.96 2232.76,1207.49 2256.76,1207.49 2232.76,1207.49 2232.76,1059.02 2256.76,1059.02 2232.76,1059.02 \n",
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" \"/>\n",
"</svg>\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"df=6\n",
"display(heatmap(reshape(U[edofMat][:,df],nely,nelx)))\n",
"display(heatmap((reshape(de'[:,df],nelx,nely)')))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"9600"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K = sparse(vec(iK),vec(jK),vec(sK)); K = (K+K')/2\n",
"@timed U[freedofs] = K[freedofs,freedofs] \\ Array(F[freedofs])\n",
"# Objective function and sensitivity analysis\n",
"ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx)\n",
"c = sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Topology with new FEM"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"topologyOptimizationFEMCode (generic function with 2 methods)"
]
},
"execution_count": 113,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function topologyOptimizationFEMCode(MBB,nelx,nely,volfrac,rmin ,penal)\n",
" anim=Animation()\n",
" \n",
" display(\"Minimum compliance problem with OC\")\n",
" display(\"ndes: $nelx x $nely\")\n",
" display(\"volfrac: $volfrac rmin: $rmin penal: $penal\")\n",
" # Max and min stiffness\n",
" Emin=1e-9\n",
" Emax=1.0\n",
" # dofs:\n",
" ndof = 2*(nelx+1)*(nely+1)\n",
" # Allocate design variables (as array), initialize and allocate sens.\n",
" x=volfrac * ones(Float64,nelx,nely)\n",
" xold=copy(x)\n",
" xPhys=copy(x)\n",
" g=0 # must be initialized to use the NGuyen/Paulino OC approach\n",
" dc=zeros(Float64,(nelx,nely))\n",
" \n",
" \n",
" problem=MBB(nelx,nely)\n",
" E,f,g,idb,ien,iplane,ndf,nel,nen,nnp,nsd,snu,t,xn,Ke=problem;\n",
" ρ=ones(nel).*volfrac;\n",
" \n",
" \n",
" # FE: Build the index vectors for the for coo matrix format.\n",
"# KE=lk()\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",
"# 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",
"# # DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)\n",
"# F = sparse([2],[1],[-1.0],2*(nely+1)*(nelx+1),1)\n",
"# U = zeros(2*(nely+1)*(nelx+1),1)\n",
"# fixeddofs = union(1:2:2*(nely+1),2*(nelx+1)*(nely+1))\n",
"# alldofs = 1:(2*(nely+1)*(nelx+1))\n",
"# freedofs = setdiff(alldofs,fixeddofs)\n",
" \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",
" k = 0;\n",
" for i1 = 1:nely\n",
" for j1 = 1:nelx\n",
" e1 = (i1-1)*nelx+j1\n",
" for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)-1),nely)\n",
" for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)-1),nelx)\n",
" e2 = (i2-1)*nelx+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",
" loop = 0\n",
" change = 1\n",
" maxIter=1000\n",
" # Start iteration\n",
" for i =1:maxIter\n",
" if (change > 0.01)\n",
" # Start iteration\n",
" loop += 1\n",
" # FE-ANALYSIS\n",
"# sK = reshape(KE[:]*(Emin.+xPhys[:]'.^penal*(Emax-Emin)),64*nelx*nely,1)\n",
"# K = sparse(vec(iK),vec(jK),vec(sK)); K = (K+K')/2\n",
"# @timed U[freedofs] = K[freedofs,freedofs] \\ Array(F[freedofs])\n",
" # Objective function and sensitivity analysis\n",
"# ce = reshape(sum((U[edofMat]*KE).*U[edofMat],dims=2),nely,nelx)\n",
"# c = sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce))\n",
" \n",
" \n",
" K,F,d,de,Fe=FEM(problem,(Emin.+xPhys[:]'.^penal*(Emax-Emin)));\n",
" \n",
" # Objective function and sensitivity analysis\n",
" ce = reshape(sum((de'*Ke[:,:,1]).*de',dims=2),nelx,nely)\n",
" c = sum(sum((Emin.+xPhys.^penal*(Emax-Emin)).*ce))\n",
"\n",
" \n",
" \n",
" dc = -penal*(Emax-Emin)*xPhys.^(penal-1).*ce\n",
" dv = ones(nelx,nely)\n",
" dc[:] = H*(dc[:]./Hs)\n",
" dv[:] = H*(dv[:]./Hs)\n",
" # OPTIMALITY CRITERIA UPDATE OF DESIGN VARIABLES AND PHYSICAL DENSITIES\n",
" l1 = 0; l2 = 1e9; xnew = 0\n",
" move = 0.2\n",
" while (l2-l1)/(l1+l2) > 1e-3\n",
" lmid = 0.5*(l2+l1)\n",
" xnew = max.(0,max.(x.-move,min.(1,min.(x.+move,x.*sqrt.((0.0.-dc)./dv./lmid)))))\n",
" xPhys[:] = (H*xnew[:])./Hs\n",
" xPhys[:] = (xnew[:])\n",
" if sum(xPhys[:]) > volfrac*nelx*nely\n",
" l1 = lmid\n",
" else\n",
" l2 = lmid\n",
" end\n",
" end\n",
" change = maximum(abs.(xnew[:].-x[:]))\n",
" x = xnew\n",
" m=mean(xPhys[:])\n",
" display(\" It:$loop Obj:$c Vol:$m ch:$change \")\n",
" if loop<20||mod(loop,10)==0\n",
" xxx=1 .- clamp.(xPhys,0,1)\n",
" display(heatmap(xxx',xaxis=nothing,yaxis=nothing,legend=nothing,fc=:grays,clims=(0.0, 1.0),aspect_ratio=:equal))\n",
" heatmap(xxx',xaxis=nothing,yaxis=nothing,legend=nothing,fc=:grays,clims=(0.0, 1.0),aspect_ratio=:equal)\n",
" frame(anim)\n",
" end\n",
"\n",
" xPhys = copy(x)\n",
" end\n",
" end\n",
" return xPhys,anim\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\"Minimum compliance problem with OC\""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"\"ndes: 60 x 20\""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"\"volfrac: 0.4 rmin: 5.4 penal: 3.0\""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"\" It:1 Obj:19.668400255357035 Vol:0.39984401223926097 ch:0.20000000000000007 \""
]
},
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"output_type": "display_data"
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],
"source": [
"# minimum compliance \n",
"nelx=60\n",
"nely=20\n",
"volfrac=0.4\n",
"rmin=5.4\n",
"penal=3.0\n",
"\n",
"xPhys,anim=topologyOptimizationFEMCode(MBB,nelx,nely,volfrac,rmin,penal)\n",
"display(heatmap(xPhys',c=:YlGnBu,aspect_ratio=:equal))"
]
},
{
"cell_type": "code",
"execution_count": 117,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\"Minimum compliance problem with OC\""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"\"ndes: 40 x 26\""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"\"volfrac: 0.4 rmin: 2 penal: 3.0\""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"\" It:1 Obj:323.7045130363365 Vol:0.40007995187112544 ch:0.20000000000000007 \""
]
},
"metadata": {},
"output_type": "display_data"
},
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"metadata": {},
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],
"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": [
{
"data": {
"text/plain": [
"4-element Array{Int64,1}:\n",
" 101\n",
" 10201\n",
" 10101\n",
" 1"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"8-element reshape(::Adjoint{Int64,Array{Int64,2}}, 8) with eltype Int64:\n",
" 201\n",
" 202\n",
" 20401\n",
" 20402\n",
" 20201\n",
" 20202\n",
" 1\n",
" 2"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"198-element Array{Int64,1}:\n",
" 2\n",
" 3\n",
" 4\n",
" 5\n",
" 6\n",
" 7\n",
" 8\n",
" 9\n",
" 10\n",
" 11\n",
" 12\n",
" 13\n",
" 14\n",
" ⋮\n",
" 8989\n",
" 9090\n",
" 9191\n",
" 9292\n",
" 9393\n",
" 9494\n",
" 9595\n",
" 9696\n",
" 9797\n",
" 9898\n",
" 9999\n",
" 10100"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"396-element reshape(::Adjoint{Int64,Array{Int64,2}}, 396) with eltype Int64:\n",
" 3\n",
" 4\n",
" 5\n",
" 6\n",
" 7\n",
" 8\n",
" 9\n",
" 10\n",
" 11\n",
" 12\n",
" 13\n",
" 14\n",
" 15\n",
" ⋮\n",
" 19189\n",
" 19190\n",
" 19391\n",
" 19392\n",
" 19593\n",
" 19594\n",
" 19795\n",
" 19796\n",
" 19997\n",
" 19998\n",
" 20199\n",
" 20200"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"198-element Array{Int64,1}:\n",
" 10102\n",
" 10103\n",
" 10104\n",
" 10105\n",
" 10106\n",
" 10107\n",
" 10108\n",
" 10109\n",
" 10110\n",
" 10111\n",
" 10112\n",
" 10113\n",
" 10114\n",
" ⋮\n",
" 8889\n",
" 8990\n",
" 9091\n",
" 9192\n",
" 9293\n",
" 9394\n",
" 9495\n",
" 9596\n",
" 9697\n",
" 9798\n",
" 9899\n",
" 10000"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"396-element reshape(::Adjoint{Int64,Array{Int64,2}}, 396) with eltype Int64:\n",
" 20203\n",
" 20204\n",
" 20205\n",
" 20206\n",
" 20207\n",
" 20208\n",
" 20209\n",
" 20210\n",
" 20211\n",
" 20212\n",
" 20213\n",
" 20214\n",
" 20215\n",
" ⋮\n",
" 18989\n",
" 18990\n",
" 19191\n",
" 19192\n",
" 19393\n",
" 19394\n",
" 19595\n",
" 19596\n",
" 19797\n",
" 19798\n",
" 19999\n",
" 20000"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"19602-element Array{Int64,1}:\n",
" 205\n",
" 206\n",
" 207\n",
" 208\n",
" 209\n",
" 210\n",
" 211\n",
" 212\n",
" 213\n",
" 214\n",
" 215\n",
" 216\n",
" 217\n",
" ⋮\n",
" 20187\n",
" 20188\n",
" 20189\n",
" 20190\n",
" 20191\n",
" 20192\n",
" 20193\n",
" 20194\n",
" 20195\n",
" 20196\n",
" 20197\n",
" 20198"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"display(n1)\n",
"display(d1)\n",
"display(n3)\n",
"display(d3)\n",
"display(n4)\n",
"display(d4)\n",
"display(d2)"
]
},
{
"cell_type": "code",
"execution_count": 163,
"metadata": {},
"outputs": [],
"source": [
"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"
]
},
{
"cell_type": "code",
"execution_count": 164,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8×3 Array{Float64,2}:\n",
" 0.0 0.0 0.0\n",
" 0.0 0.0 0.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 50.0\n",
" 0.0 100.0 50.0\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0"
]
},
"execution_count": 164,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ufixed"
]
},
{
"cell_type": "code",
"execution_count": 165,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"396×3 Array{Float64,2}:\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0\n",
" 100.0 0.0 0.0\n",
" ⋮ \n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0"
]
},
"execution_count": 165,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"wfixed = [repeat(ufixed[3:4,:],nely-1,1); repeat(ufixed[7:8,:],nelx-1,1)];\n",
"wfixed"
]
},
{
"cell_type": "code",
"execution_count": 174,
"metadata": {},
"outputs": [],
"source": [
"## 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"
]
},
{
"cell_type": "code",
"execution_count": 175,
"metadata": {},
"outputs": [],
"source": [
"## START ITERATION\n",
"\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",
"\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"
]
},
{
"cell_type": "code",
"execution_count": 177,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"20402×3 Array{Float64,2}:\n",
" 0.0 0.0 50.0\n",
" 0.0 100.0 0.0\n",
" -3.67405e-11 7.00067e-14 49.9953\n",
" 0.236372 99.0911 -1.00882e-11\n",
" -4.54123e-11 6.15561e-14 49.9892\n",
" 0.471982 98.1818 -1.44058e-11\n",
" -5.17758e-11 6.23073e-14 49.9802\n",
" 0.706067 97.2718 -1.6833e-11\n",
" -5.61746e-11 7.17849e-14 49.967\n",
" 0.937861 96.3608 -1.85503e-11\n",
" -5.96617e-11 7.22162e-14 49.948\n",
" 1.1666 95.4483 -1.98748e-11\n",
" -6.25283e-11 7.66748e-14 49.922\n",
" ⋮ \n",
" 100.0 1.34213e-13 0.0519715\n",
" -1.1666 4.5517 50.0\n",
" 100.0 1.2473e-13 0.0330427\n",
" -0.937861 3.63922 50.0\n",
" 100.0 1.13706e-13 0.0198063\n",
" -0.706067 2.72816 50.0\n",
" 100.0 9.64519e-14 0.0108357\n",
" -0.471982 1.81817 50.0\n",
" 100.0 7.3829e-14 0.00470755\n",
" -0.236372 0.908907 50.0\n",
" 100.0 0.0 0.0\n",
" 0.0 0.0 50.0"
]
},
"execution_count": 177,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.5.2",
"language": "julia",
"name": "julia-1.5"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.5.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
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