top88_AD.ipynb

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    "using SparseArrays\n",
    "using LinearAlgebra\n",
    "using Plots\n",
    "using Statistics\n",
    "using SuiteSparse\n",
    "E0 = 1;\n",
    "Emin = 1e-9;\n",
    "nu = 0.3;\n",
    "using ForwardDiff"
   ]
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    "## There is the function top88 using AD\n",
    "\n",
    "*This code uses the Matlab code \"top88\" from Ole Sigmund, to which I add the uautomatic differentiation tool*\n",
    "\n",
    "What is needed to know :\n",
    " -  This code has been written by Robin Grapin for he's work at Supaéro, under the supervision of Joseph Morlier\n",
    " -  The modules above must be imported before running the code\n",
    " -  The goal here is to compare the efficiency of the code and to compare it to the version without AD.\n",
    "\n",
    "\n",
    "The given objective function must take two arguments of dimension nelx x nely : \n",
    " - The matrix of the mass repartition x\n",
    " - ce = u K u\n",
    "\n",
    "The modifications in camparation to the topo88 code I wrote are between the lines 64 and 74, where the process is not making a difference in function of the objective function. In these lines, we use the module \"ForwardDiff\" to calculate the derivative of the objective function."
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       "top88AD (generic function with 5 methods)"
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   "source": [
    "function top88AD(nelx,nely,objectif ,volfrac = 0.4 ,penal = 3,rmin = 2.5 ,ft = 1)\n",
    "        \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",
    "    nodenrs = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx); #number of the nodes in columns\n",
    "    edofVec = reshape(2*nodenrs[1:end-1,1:end-1].+1,nelx*nely,1) ;#1st dof of each element (x top left)\n",
    "    edofMat = zeros(nelx*nely, 8); #every line i contains the 8 dof of the ith element\n",
    "    noeudsvoisins = [0 1 2*nely.+[2 3 0 1] -2 -1];\n",
    "    for i = 1:8\n",
    "        for j = 1:nelx*nely\n",
    "            edofMat[j,i]= edofVec[j]+ noeudsvoisins[i] ;\n",
    "        end\n",
    "    end\n",
    "    #maillage\n",
    "    iK = reshape(kron(edofMat,ones(8,1))',64*nelx*nely,1);# line to build  K\n",
    "    jK = reshape(kron(edofMat,ones(1,8))',64*nelx*nely,1);# columns\n",
    "    #F = spzeros(2*(nely+1)*(nelx+1),1); F[2,1] = -1;   impossible to solve with F sparse in Julia\n",
    "    F = zeros(2*(nely+1)*(nelx+1),1); F[2,1] = -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",
    "    #filtering is linear + chain's rule -> matricial opration with operator H\n",
    "    iH = ones(nelx*nely*(2*(convert(Int64,ceil(rmin)-1))+1)^2);#lines H\n",
    "    jH = ones(size(iH));# columns\n",
    "    sH = zeros(size(iH));# values\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(iH,jH,sH);# matrix of the Hei\n",
    "    Hs = [sum(H[i,:]) for i = 1:(size(H)[1])];# sum of the Hei for e fixed -> coeffs used to divide in the filter\n",
    "    #variables\n",
    "    x = volfrac*ones(nely,nelx);\n",
    "    xPhys = x;#new variable for the density filter\n",
    "    loop = 0;\n",
    "    change = 1;\n",
    "    cValues = []\n",
    "    while change > 0.01\n",
    "        loop = loop + 1;\n",
    "        \n",
    "        sK = [j*((i+Emin)^penal) for i in ((E0-Emin)*xPhys[:]') for j in KE[:] ];#new values EF\n",
    "        K = sparse(iK[:],jK[:],sK); K = (K+K')/2;# force to be symmetric\n",
    "        KK = cholesky(K[freedofs,freedofs]);\n",
    "        U[freedofs] = KK\\F[freedofs];\n",
    "        \n",
    "        # analyse, objective function \n",
    "        edM = [convert(Int64,i) for i in edofMat] ;#to index by int\n",
    "        mat = (U[edM]*KE).*U[edM];\n",
    "        ce = reshape([sum(mat[i,:]) for i = 1:(size(mat)[1])],nely,nelx);\n",
    "\n",
    "        #new\n",
    "        c = objectif(xPhys,ce) #sum(sum((Emin*ones(size(xPhys)).+(xPhys.^penal)*(E0-Emin)).*ce));\n",
    "        push!(cValues,c)\n",
    "        \n",
    "        #here we use AD :      \n",
    "          #to make of it an unary function\n",
    "        dc = -ForwardDiff.gradient(x -> objectif(x,ce), xPhys)#-penal*(E0-Emin)*xPhys.^(penal-1).*ce;\n",
    "        dv = ones(nely,nelx);\n",
    "        \n",
    "        #filtering : maybe other filters are needed if the objective function is not the sensitivity\n",
    "        if ft == 1\n",
    "            dc[:] = H*(x[:].*dc[:])./Hs./max(1e-3,maximum(x[:]));\n",
    "        elseif ft == 2\n",
    "            dc[:] = H*(dc[:]./Hs);\n",
    "            dv[:] = H*(dv[:]./Hs);\n",
    "        end\n",
    "        #Lagrangian multiplicators research, upload of the density x\n",
    "        l1 = 0; l2 = 1e9; move = 0.2;\n",
    "        xnew = zeros(nely, nelx);\n",
    "        while (l2-l1)/(l1+l2) > 1e-3\n",
    "            lmid = 0.5*(l2+l1);\n",
    "            RacBe = sqrt.(-dc./dv/lmid) ;\n",
    "            XB = x.*RacBe ;\n",
    "            for i = 1:nelx\n",
    "                for j = 1:nely\n",
    "                    xji = x[j,i];\n",
    "                    xnew[j,i]= max(0.001,max(xji-move,min(1,min(xji+move,XB[j,i]))));\n",
    "                end\n",
    "            end  \n",
    "            if ft == 1\n",
    "                xPhys = xnew;\n",
    "            elseif ft == 2 # if we chose the density filter\n",
    "                xPhys[:] = (H*xnew[:])./Hs;\n",
    "            end\n",
    "            if sum(xPhys[:]) > volfrac*nelx*nely\n",
    "                l1 = lmid;\n",
    "            else \n",
    "                l2 = lmid; \n",
    "            end\n",
    "        end\n",
    "        change = maximum(abs.(x-xnew));\n",
    "        x = xnew;\n",
    "        println(\"loop = \", loop, \"Change = \", change ,\" c = \", c , \" densité struct = \", mean(x) )\n",
    "        if loop == 25\n",
    "            return heatmap(x), loop, cValues\n",
    "        end\n",
    "    end\n",
    "    return heatmap(x ), loop , cValues\n",
    "end"
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      "loop = 1Change = 0.2 c = 240.55295821333357 densité struct = 0.4998520540677216\n",
      "loop = 2Change = 0.20000000000000007 c = 142.29012812657385 densité struct = 0.4999542563374929\n",
      "loop = 3Change = 0.20000000000000007 c = 101.29042745368754 densité struct = 0.5002055990575159\n",
      "loop = 4Change = 0.20000000000000007 c = 80.66559467700088 densité struct = 0.5000701665639053\n",
      "loop = 5Change = 0.20000000000000007 c = 70.50380077730274 densité struct = 0.4999593512378573\n",
      "loop = 6Change = 0.20000000000000007 c = 62.76373407505786 densité struct = 0.5001263852460752\n",
      "loop = 7Change = 0.20000000000000007 c = 57.5694569236679 densité struct = 0.5001217271799541\n",
      "loop = 8Change = 0.20000000000000007 c = 53.325317688747006 densité struct = 0.4999988517633573\n",
      "loop = 9Change = 0.20000000000000007 c = 50.258491516120166 densité struct = 0.5000234485734584\n",
      "loop = 10Change = 0.20000000000000007 c = 48.81073539248547 densité struct = 0.499969988891163\n",
      "loop = 11Change = 0.20000000000000007 c = 48.11940887677639 densité struct = 0.4999908170307767\n",
      "loop = 12Change = 0.20000000000000007 c = 47.706743719101354 densité struct = 0.49999294864108174\n",
      "loop = 13Change = 0.20000000000000007 c = 47.471181076797436 densité struct = 0.5000210417377724\n",
      "loop = 14Change = 0.20000000000000007 c = 47.343485675724 densité struct = 0.49999779464938876\n",
      "loop = 15Change = 0.20000000000000007 c = 47.23921449545898 densité struct = 0.5000143110161004\n",
      "loop = 16Change = 0.20000000000000007 c = 47.12797376728905 densité struct = 0.4999847219638031\n",
      "loop = 17Change = 0.20000000000000007 c = 46.99585044363865 densité struct = 0.5000125225560839\n",
      "loop = 18Change = 0.20000000000000007 c = 46.796700693473355 densité struct = 0.5000083286539801\n",
      "loop = 19Change = 0.18506658738056747 c = 46.53871598799836 densité struct = 0.499999557245842\n",
      "loop = 20Change = 0.19999999999999996 c = 46.30448641143059 densité struct = 0.5000122818824392\n",
      "loop = 21Change = 0.14493080658563318 c = 46.2349692855021 densité struct = 0.49999427568539834\n",
      "loop = 22Change = 0.16818799250766336 c = 46.184144301023274 densité struct = 0.5000030432254582\n",
      "loop = 23Change = 0.1592282877208433 c = 46.137793927841486 densité struct = 0.4999955966007355\n",
      "loop = 24Change = 0.1702934199179139 c = 46.09535343500255 densité struct = 0.4999986841813314\n",
      "loop = 25Change = 0.20000000000000007 c = 46.05217114719637 densité struct = 0.5000025072101734\n"
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    "penal = 3\n",
    "sensi( x, ce)= sum(sum((Emin*ones(size(x)).+(x.^penal)*(E0-Emin)).*ce));# example of the sensitivity to see if the results do match\n",
    "\n",
    "\n",
    "x,l,values= top88AD(50,33,sensi ,0.5,3,2)\n",
    "\n",
    "x"
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    "plot(1:l,values,title = \"sensity evolution\", label = \"sensity c\") #evolution of c during the process\n",
    "xlabel!(\"iteration\")"
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     "text": [
      "loop = 1Change = 0.2 c = 181.02944558470028 densité struct = 0.49995144886389187\n",
      "loop = 2Change = 0.20000000000000007 c = 87.70747378137004 densité struct = 0.5001614594571799\n",
      "loop = 3Change = 0.20000000000000007 c = 51.26685653381925 densité struct = 0.49987273213909994\n",
      "loop = 4Change = 0.20000000000000007 c = 37.107211620138884 densité struct = 0.5000877840246694\n",
      "loop = 5Change = 0.20000000000000007 c = 33.4722698714435 densité struct = 0.500032469187293\n",
      "loop = 6Change = 0.20000000000000007 c = 31.7012718049617 densité struct = 0.500038081949538\n",
      "loop = 7Change = 0.20000000000000007 c = 30.680779048340334 densité struct = 0.4999725989287503\n",
      "loop = 8Change = 0.20000000000000007 c = 30.056612408082696 densité struct = 0.4999948254122206\n",
      "loop = 9Change = 0.20000000000000007 c = 29.697007899596613 densité struct = 0.49999010865046273\n",
      "loop = 10Change = 0.20000000000000007 c = 29.509405391612148 densité struct = 0.4999967663976441\n",
      "loop = 11Change = 0.20000000000000007 c = 29.383561398743222 densité struct = 0.5000018543166495\n",
      "loop = 12Change = 0.20000000000000007 c = 29.292020865470853 densité struct = 0.5000056006524927\n",
      "loop = 13Change = 0.19999999999999996 c = 29.24240510439929 densité struct = 0.49999839612737335\n",
      "loop = 14Change = 0.20000000000000007 c = 29.207364309609275 densité struct = 0.5000011934646181\n",
      "loop = 15Change = 0.20000000000000007 c = 29.187038379615093 densité struct = 0.5000010560455185\n",
      "loop = 16Change = 0.19999999999999996 c = 29.17646552587641 densité struct = 0.49999900360725485\n",
      "loop = 17Change = 0.19999999999999996 c = 29.16804630459132 densité struct = 0.5000009111491156\n",
      "loop = 18Change = 0.19999999999999996 c = 29.159652759678835 densité struct = 0.500003645722454\n",
      "loop = 19Change = 0.20000000000000007 c = 29.15149759678352 densité struct = 0.49999977118178457\n",
      "loop = 20Change = 0.20000000000000007 c = 29.14484798084566 densité struct = 0.5000036966265997\n",
      "loop = 21Change = 0.20000000000000007 c = 29.138851524867256 densité struct = 0.5000031425180504\n",
      "loop = 22Change = 0.19999999999999996 c = 29.133919112304977 densité struct = 0.5000032434818593\n",
      "loop = 23Change = 0.19999999999999996 c = 29.129077261109185 densité struct = 0.5000028234793086\n",
      "loop = 24Change = 0.20000000000000007 c = 29.123976206454618 densité struct = 0.49999588458437483\n",
      "loop = 25Change = 0.20000000000000007 c = 29.11866208700512 densité struct = 0.5000009009004683\n",
      " 68.512863 seconds (180.65 M allocations: 70.440 GiB, 14.24% gc time)\n"
     ]
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   "source": [
    "@time top88AD(100,100,sensi,0.5,3,2);"
   ]
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  {
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   "source": [
    "## Comparation of the algorithms\n",
    "\n",
    "About the speed, the \"classic\" algorithm top88 took, for the \"relatively big\" example 100x100 :\n",
    "\n",
    "12.845634 seconds (156.44 M allocations: 6.358 GiB, 10.76% gc time)\n",
    "\n",
    "against 58.238602 seconds (182.39 M allocations: 70.522 GiB, 13.31% gc time) for top88AD (stopping both of them after 25 iteration if the stopping condition on the change isn't reached.).\n",
    "\n",
    "top88 has already the explicit formulae of the derivative of the sensity, that is why it is quicker. But we can appreciate the precicion of top88_AD, which gave the same results with 11 exact decimals. This algorithm may also be used with other functions than the sensity."
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