topggp.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# This is the Julia transcription of the Matlab code GGP\n",
    "\n",
    "\n",
    "First, here are the necessary imports to run the code\n",
    "\n",
    "Robin Grapin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using SparseArrays\n",
    "using LinearAlgebra\n",
    "using Plots\n",
    "using Statistics\n",
    "using SuiteSparse\n",
    "using VectorizedRoutines # for meshgrid"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Auxiliar functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "lgwt (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function lgwt(N,a,b) #ok\n",
    "    # lgwt.m This script is for computing definite integrals using Legendre-Gauss Quadrature.Computes the Legendre - Gauss nodes and weights on an interval [a, b] with truncation order N\n",
    "    # Suppose you have a continuous function f(x) which is defined on[a, b]  which you can evaluate at any x in [a, b].Simply evaluate itat all of the values contained in the x vector to obtain a vector f.Then compute the definite integral using sum(f. * w);\n",
    "    # Written by Greg von Winckel - 02 / 25 / 2004\n",
    "    N = N - 1\n",
    "    N1 = N + 1\n",
    "    N2 = N + 2\n",
    "    xu = range(-1,stop =  1, length =N1)\n",
    "    xu = xu'\n",
    "\n",
    "    # Initial guess\n",
    "    y = cos.((2 * (0:N)'.+1)*pi/(2*N+2)).+(0.27/N1)*sin.(pi*xu*N/N2)\n",
    "    # Legendre-Gauss Vandermonde Matrix\n",
    "    L=zeros(N1, N2) \n",
    "\n",
    "    # Compute the zeros of the N + 1 Legendre Polynomial using the recursion relation and the Newton - Raphson method\n",
    "    y0 = 2\n",
    "    eps = 2.2204e-16\n",
    "    Lp = 0\n",
    "    # Iterate until new points are uniformly within epsilon of old points\n",
    "    while maximum(abs.(y .- y0)) > eps\n",
    "        L[:, 1].=1\n",
    "        L[:, 2]=y\n",
    "        for k = 3:N2\n",
    "            L[:, k]=((2*(k-1) -1 ) * y[:] .* L[:, k-1] - (k-2) * L[:, k-2] )/ (k-1) ; \n",
    "        end\n",
    "        Lp = N2 * (L[:, N1] - y[:] .* L[:, N2])./ (1 .- y[:].^2);\n",
    "        y0 = y\n",
    "        y = y0[:] .- L[:, N2]./ Lp[:];\n",
    "    end\n",
    "    # Linear map from [-1, 1] to[a, b]\n",
    "    x = (a * (1 .- y) + b * (1 .+ y)) / 2\n",
    "    # Compute the weights\n",
    "    w = (b - a) ./ ((1 .- y[:].^2) .* Lp[:].^2) * (N2 / N1)^2\n",
    "    return x, w\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "acoscplx (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function acoscplx(x)#prolongation of acos on C\n",
    "    y = complex(x)\n",
    "    z = y + im*(1-y*y)^0.5\n",
    "    return -im*log(z)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Wgp (generic function with 6 methods)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Wgp(x,y,Xc,method = \"MMC\",alp=1, epsi =0.866, bet=1e-3,r = 0.5)\n",
    "#  Evaluate characteristic function in each Gauss point\n",
    "    End = length(Xc)\n",
    "    ii=1:length(x);   \n",
    "    X=Xc[1:6:End];\n",
    "    Y=Xc[2:6:End];\n",
    "    L=Xc[3:6:End];\n",
    "    h=Xc[4:6:End];\n",
    "    T=Xc[5:6:End];\n",
    "    jj=1:length(X);\n",
    "    I,J=Matlab.meshgrid(ii,jj)\n",
    "    xi=reshape(x[I],size(I));\n",
    "    yi=reshape(y[I],size(I));\n",
    "    rho=sqrt.((X[J]-xi).^2+(Y[J]-yi).^2);\n",
    "    drho_dX=(X[J]-xi)./(rho.+(rho==0));\n",
    "    drho_dY=(Y[J]-yi)./(rho.+(rho==0));\n",
    "    phi=atan.(-Y[J]+yi,-(X[J]-xi))-T[J]; \n",
    "    dphi_dX=((-Y[J]+yi)./(rho.^2 .+(rho==0)));\n",
    "    dphi_dY=(X[J]-xi)./(rho.^2 .+(rho==0));\n",
    "    dphi_dT=-ones(size(J));\n",
    "    upsi=sqrt.(rho.^2+L[J].^2/4-rho.*L[J].*abs.(cos.(phi))).*(((rho.*cos.(phi)).^2) .>=(L[J].^2/4))+ .!(((rho.*cos.(phi)).^2).>=(L[J].^2/4)).*abs.(rho.*sin.(phi));\n",
    "    dupsi_drho=(2*rho-L[J].*abs.(cos.(phi)))/2 ./(upsi .+(upsi==0)).*((((rho.*cos.(phi)).^2) .>=(L[J].^2/4)))+ .~(((rho.*cos.(phi)).^2) .>=(L[J].^2/4)).*abs.(sin.(phi));\n",
    "    dupsi_dphi=(L[J].*rho.*sign.(cos.(phi)).*sin.(phi))/2 ./(upsi .+(upsi==0)).*((((rho.*cos.(phi)).^2) .>=(L[J].^2/4)))+ .~(((rho.*cos.(phi)).^2) .>=(L[J].^2/4)).*rho.*sign.(sin.(phi)).*cos.(phi);\n",
    "    dupsi_dL=(L[J]/2-rho.*abs.(cos.(phi)))./2 ./(upsi .+(upsi==0)).*((((rho.*cos.(phi)).^2) .>=(L[J].^2/4)) .&reshape([ i != 0 for i in upsi],size(upsi)));\n",
    "    if method ==\"MMC\"\n",
    "        chi0=1 .-(4*upsi.^2 ./h[J].^2).^alp;\n",
    "        dchi0_dh=8*alp*upsi.^2 .*(4*upsi.^2 ./h[J].^2).^(alp-1)./h.^3;\n",
    "        dchi0_dupsi=-8*alp*upsi.*(4*upsi.^2 ./h[J].^2).^(alp-1)./h.^2;\n",
    "        chi,dchi=Aggregation_Pi(chi0); \n",
    "        dchi_dh=(dchi0_dh.*dchi);\n",
    "        dchi_dupsi=(dchi0_dupsi.*dchi);\n",
    "        chi[chi .<=-1e6] .=-1e6;#seuil min\n",
    "        W=(chi .>epsi)+((chi .<=epsi) .& (chi .>=-epsi)).*(3/4*(1-bet)*(chi/epsi .-chi.^3/3/epsi^3).+(1+bet)/2)+(chi .<-epsi)*bet;\n",
    "        dW_dchi=-3/4*(1/epsi .-chi.^2/epsi^3).*(bet-1).*(abs.(chi) .<epsi);\n",
    "        dW_dupsi=repeat(dW_dchi,size(dchi_dh,1)).*dchi_dupsi;\n",
    "        dW_dh=repeat(dW_dchi,size(dchi_dh,1)).*dchi_dh;\n",
    "        dW_dX=dW_dupsi.*(dupsi_dphi.*dphi_dX+dupsi_drho.*drho_dX);\n",
    "        dW_dY=dW_dupsi.*(dupsi_dphi.*dphi_dY+dupsi_drho.*drho_dY);\n",
    "        dW_dL=dW_dupsi.*dupsi_dL;\n",
    "        dW_dT=dW_dupsi.*dupsi_dphi.*dphi_dT;\n",
    "    elseif method == \"GP\" \n",
    "        deltamin=1.0000e-06;\n",
    "        #r=p.r;\n",
    "        zetavar=upsi-h[J]/2;\n",
    "        dzetavar_dupsi=ones(size(upsi));\n",
    "        dzetavar_dh=-0.5*ones(size(J));\n",
    "        deltaiel=(1/pi/r^2*(r^2*acoscplx.(zetavar/r))-zetavar.*sqrt.(complex.(r^2 .-zetavar.^2))).*(abs.(zetavar) .<=r)+((zetavar .<-r));\n",
    "        ddetlaiel_dzetavar=(-2*sqrt.(complex.(r^2 .-zetavar.^2)/pi/r^2)).*(abs.(zetavar) .<=r);\n",
    "        deltaiel = real.(deltaiel)\n",
    "        ddetlaiel_dzetavar = real.(ddetlaiel_dzetavar)\n",
    "        W=deltamin .+(1-deltamin)*deltaiel;\n",
    "        dW_ddeltaiel=(1-deltamin);\n",
    "        dW_dh=dW_ddeltaiel*ddetlaiel_dzetavar.*dzetavar_dh;\n",
    "        dW_dupsi=dW_ddeltaiel*ddetlaiel_dzetavar.*dzetavar_dupsi;\n",
    "        dW_dX=dW_dupsi.*(dupsi_dphi.*dphi_dX+dupsi_drho.*drho_dX);\n",
    "        dW_dY=dW_dupsi.*(dupsi_dphi.*dphi_dY+dupsi_drho.*drho_dY);\n",
    "        dW_dL=dW_dupsi.*dupsi_dL;\n",
    "        dW_dT=dW_dupsi.*dupsi_dphi.*dphi_dT;\n",
    "    elseif method == \"MNA\" #ok\n",
    "        #epsi=p.sigma;\n",
    "        epsi = sigma\n",
    "        ds=upsi;\n",
    "        d=abs.(upsi);\n",
    "        l=h[J]/2 .-epsi/2;\n",
    "        u=h[J]/2 .+epsi/2;\n",
    "        a3= -2 ./((l - u).*(l.^2 - 2*l.*u + u.^2));\n",
    "        a2=   (3*(l + u))./((l - u).*(l.^2 - 2*l.*u + u.^2));\n",
    "        a1=    -(6*l.*u)./((l - u).*(l.^2 - 2*l.*u + u.^2));\n",
    "        a0=(u.*(- u.^2 + 3*l.*u))./((l - u).*(l.^2 - 2*l.*u + u.^2));\n",
    "        W=1*(d .<= l)+(a3.*d.^3+a2 .*d.^2+a1.*d+a0).*((d .<=u) .& (d .>l));\n",
    "        dW_dupsi=sign.(ds).*(3*a3.*d.^2+2*a2 .*d+a1).*((d .<=u) .&(d .>l));\n",
    "        da3_du=- 2 ./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) - (2*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        da2_du=3 ./((l - u).*(l.^2 - 2*l.*u + u.^2)) + (3*(l + u))./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) + (3*(l + u).*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        da1_du=- (6*l)./((l - u).*(l.^2 - 2*l.*u + u.^2)) - (6*l.*u)./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) - (6*l.*u.*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        da0_du=(- u.^2 + 3*l.*u)./((l - u).*(l.^2 - 2*l.*u + u.^2)) + (u.*(- u.^2 + 3*l.*u))./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) + (u.*(3*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2)) + (u.*(- u.^2 + 3*l.*u).*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        dWf_du=(da3_du.*d.^3+da2_du.*d.^2+da1_du.*d+da0_du).*((d.<=u).&(d.>l));\n",
    "        da3_dl=2 ./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) + (2*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        da2_dl= 3 ./((l - u).*(l.^2 - 2*l.*u + u.^2)) - (3*(l + u))./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) - (3*(l + u).*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        da1_dl=    (6*l.*u)./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) - (6*u)./((l - u).*(l.^2 - 2*l.*u + u.^2)) + (6*l.*u.*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        da0_dl= (3*u.^2)./((l - u).*(l.^2 - 2*l.*u + u.^2)) - (u.*(- u.^2 + 3*l.*u))./((l - u).^2 .*(l.^2 - 2*l.*u + u.^2)) - (u.*(- u.^2 + 3*l.*u).*(2*l - 2*u))./((l - u).*(l.^2 - 2*l.*u + u.^2).^2);\n",
    "        dWf_dl=(da3_dl.*d.^3+da2_dl.*d.^2+da1_dl.*d+da0_dl).*((d.<=u).&(d.>l));\n",
    "        dW_dh=0.5*sign.(ds).*(dWf_du+dWf_dl);\n",
    "        dW_dX=dW_dupsi.*(dupsi_dphi.*dphi_dX+dupsi_drho.*drho_dX);\n",
    "        dW_dY=dW_dupsi.*(dupsi_dphi.*dphi_dY+dupsi_drho.*drho_dY);\n",
    "        dW_dL=dW_dupsi.*dupsi_dL;\n",
    "        dW_dT=dW_dupsi.*dupsi_dphi.*dphi_dT;\n",
    "    end\n",
    "    return W,dW_dX,dW_dY,dW_dT,dW_dL,dW_dh\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "doublons (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# gives m with one occur of each line, and give the indexes of the first occur\n",
    "#return r = (a without double line) and li such as r[li,:]=a\n",
    "function doublons(m)\n",
    "    index = []\n",
    "    m2 = []\n",
    "    l = length(m[1])\n",
    "    indexmax = 1\n",
    "    for i in 1:(size(m,1))\n",
    "        ligne = m[i,:]\n",
    "        isin = false\n",
    "        j = 1\n",
    "        while j <= size(m2,1)\n",
    "            if m2[j,:][1]==ligne\n",
    "                isin = true\n",
    "                break\n",
    "            end\n",
    "            j+=1\n",
    "        end    \n",
    "        append!(index,j)\n",
    "        if !isin\n",
    "            push!(m2,ligne)  \n",
    "        end\n",
    "    end\n",
    "    r = zeros(size(m2,1),size(m,2) )\n",
    "\n",
    "    for i = 1:size(m2,1)\n",
    "\n",
    "        r[i,:] =m2[i]\n",
    "    end\n",
    "    return r,index\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Aggregation_Pi (generic function with 4 methods)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Aggregation_Pi(z,aggregation =\"KSl\",ka = 10,zp =1)\n",
    "# function that make the aggregation of the values in z and also compute sensitivities\n",
    "    sz1,sz2 = size(z);\n",
    "    zem = zeros(size(z,2))\n",
    "    for i = 1:size(z,2)\n",
    "        zem[i]=maximum(z[:,i])\n",
    "    end\n",
    "    zeem =zem'\n",
    "    zm = repeat(zeem,sz1) #tableau des m^colonnes du max de la colonne\n",
    "    col = ones(1,sz2)\n",
    "    \n",
    "    if aggregation == \"asymptotic\" #ok\n",
    "        Wa=sum!(ones(1,sz1),z);\n",
    "        dWa=ones(size(z));\n",
    "    elseif aggregation == \"boolean\" #ok\n",
    "        Wa=1 .-prod(1 .-z,dims= 1);\n",
    "        dWa=repeat(prod(1 .-z.* (z .!=1), dims = 1),sz1)./(1 .-z.*(z .!=1));\n",
    "    \n",
    "    elseif aggregation == \"p-norm\"#(sum((z./exp(zm)).^ka,1)).^(1/ka) value different than matlab\n",
    "        zm=zm .+zp;\n",
    "        z=z .+zp;          \n",
    "        Wa=exp.(zm[1,:])'.*(sum!(col,z./exp.(zm).^ka)).^(1/ka) .-zp;\n",
    "        dWa=(z./exp.(zm)).^(ka-1).*repeat((sum!(col,z./exp.(zm).^ka)).^(1/ka-1),sz1);\n",
    "    elseif aggregation == \"p-mean\" \n",
    "        zm=zm .+zp;\n",
    "        z=z.+zp;\n",
    "        Wa=exp.(zm[1,:])'.*(mean!(col,(z./exp.(zm)).^ka)).^(1/ka) .-zp;\n",
    "        dWa=1/size(z,1)^(1/ka)*(z./exp.(zm)).^(ka-1).*repeat(sum!(col,z./exp.(zm).^ka).^(1/ka-1),sz1);\n",
    "    elseif aggregation == \"KS\" #ok\n",
    "        Wa=zm[1,:]'+1/ka*log.(sum!(col,exp.(ka*(z-zm))));\n",
    "        dWa=exp.(ka*(z-zm))./repeat(sum!(col,exp.(ka*(z-zm))),sz1);\n",
    "    elseif aggregation == \"KSl\" #ok\n",
    "        e = exp.(ka*(z-zm))\n",
    "        m = mean!(col,e)\n",
    "        Wa=zm[1,:]' .+1/ka*log.(m);\n",
    "        dWa=exp.(ka*(z-zm))./repeat(sum!(col,e),sz1);\n",
    "        else #ok\n",
    "        e = exp.(ka*(z-zm))\n",
    "        Wa=sum!(col,z.*e)./sum!(col,e);\n",
    "        dWa=((e+ka*z.*e).*repeat(sum!(col,e),sz1)-repeat(sum!(col,z.*e),sz1)*ka.*e)./repeat(sum!(col,e).^2,sz1);\n",
    "        \n",
    "    end\n",
    "    return Wa,dWa,z,zm,sz1,ka\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fdelta (generic function with 2 methods)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fdelta(w,gauss_w,idgp,Ngp,q=1)#to make the optim loop more readable\n",
    "    #delta=sum(reshape(W(:,idgp).*repmat(gauss_weight(:)',size(W,1),1),size(W,1),[],Ngp^2),3)...\n",
    "     #   ./sum(reshape(repmat(gauss_weight(:)',size(W,1),1),size(W,1),[],Ngp^2),3);\n",
    "    sz1 = size(w,1)\n",
    "    agglo_gauss = repeat(gauss_w[:]', sz1)\n",
    "    n = convert(Int64,length(agglo_gauss)/sz1/(Ngp^2))\n",
    "    num = reshape(w[:,idgp].^q .*agglo_gauss,sz1,n,Ngp^2) #size a obtenir 18x300x4\n",
    "    numer = sum!(ones(sz1,n,1),num)\n",
    "    den = reshape(agglo_gauss, sz1,n,Ngp^2)\n",
    "    denom = sum!(ones(sz1,n,1),den)\n",
    "    return reshape(numer ./ denom,sz1,n)\n",
    "    end "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fdelta_c (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fdelta_c(w,dw,gauss_w,idgp,Ngp,q,q2)\n",
    "    sz1 = size(w,1)\n",
    "    agglo_gauss = repeat(gauss_w[:]', sz1)\n",
    "    n = convert(Int64,length(agglo_gauss)/sz1/(Ngp^2))\n",
    "    num = reshape(q*w[:,idgp].^q2 .*dw[:,idgp].*agglo_gauss,sz1,n,Ngp^2) #size a obtenir 18x300x4\n",
    "    numer = sum!(ones(sz1,n,1),num)\n",
    "    den = reshape(agglo_gauss, sz1,n,Ngp^2)\n",
    "    denom = sum!(ones(sz1,n,1),den)\n",
    "    return reshape(numer ./ denom,sz1,n)\n",
    "    end "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "model_updateM (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function model_updateM(delta,X,method,saturation,aggreg)\n",
    "#update the Young Modulus on the base of delta\n",
    "    m=X[6:6:length(X)];\n",
    "    nc=length(m);\n",
    "    m=repeat(m[:],1,size(delta,2));\n",
    "    gammac = 3\n",
    "    E0 = 1\n",
    "    if method == \"MMC\"\n",
    "        E=E0*delta;\n",
    "        dE_ddelta=E0*ones(size(delta));\n",
    "        dE_ddelta=repeat(dE_ddelta,size(m,1),1);\n",
    "        dE_dm=0*m;\n",
    "        \n",
    "    elseif method == \"GP\"\n",
    "        hatdelta=delta.*m.^gammac;\n",
    "        E,dE_dhatdelta=Aggregation_Pi(hatdelta);\n",
    "        if saturation\n",
    "            E,ds=smooth_sat(E,nc,aggreg);\n",
    "            dE_dhatdelta=ds.*dE_dhatdelta;\n",
    "        end\n",
    "        E=E.*E0;\n",
    "        dhatdelta_ddelta=m.^gammac;\n",
    "        dhatdelta_dm=gammac*delta.*m.^(gammac-1);\n",
    "        dE_ddelta=E0*dhatdelta_ddelta.*dE_dhatdelta;\n",
    "        dE_dm=E0*dE_dhatdelta.*dhatdelta_dm;\n",
    "        \n",
    "    else #MNA\n",
    "        hatdelta=delta.*m.^gammac;\n",
    "        rho,drho_dhatdelta=Aggregation_Pi(hatdelta);\n",
    "        if saturation\n",
    "            rho,ds=smooth_sat(rho,nc,aggreg);\n",
    "            drho_dhatdelta=ds.*drho_dhatdelta;\n",
    "        end\n",
    "        E=rho.^penalty.*(E0-Emin)+Emin;\n",
    "        dhatdelta_ddelta=m.^gammac;\n",
    "        dhatdelta_dm=gammac*delta.*m.^(gammac-1);\n",
    "        dE_ddelta=penalty*(E0-Emin)*dhatdelta_ddelta.*drho_dhatdelta.*rho.^(penalty-1);\n",
    "        dE_dm=penalty*(E0-Emin)*dhatdelta_dm.*drho_dhatdelta.*rho.^(penalty-1);\n",
    "    end\n",
    "    return E,dE_ddelta,dE_dm\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "smooth_sat (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function smooth_sat(y,nc,aggreg)\n",
    "    #Smooth saturation function\n",
    "    #face issues related to aggregation\n",
    "    if aggreg == \"asymptotic\"\n",
    "        xt=1;\n",
    "    elseif aggreg == \"boolean\"\n",
    "        xt=1;\n",
    "    elseif aggreg == \"p-norm\"\n",
    "       xt=1;\n",
    "    elseif aggreg == \"p-mean\"\n",
    "        zp =1\n",
    "        ka =10\n",
    "       xt=(((nc-1)*zp^ka+(1+zp)^ka)/nc)^(1/ka)-zp;\n",
    "    elseif aggreg == \"KS\"\n",
    "         xt=1;\n",
    "    elseif aggreg == \"KSl\"\n",
    "        ka =10\n",
    "         xt=1+1/ka*log.((1+(nc-1)*exp.(-ka))/nc);\n",
    "    elseif aggreg == \"IE\"\n",
    "        ka =10\n",
    "        xt=1+1/ka*log.((1+(nc-1)*exp.(-ka))/nc);\n",
    "    end\n",
    "    pp=100;\n",
    "    s0=-log(exp(-pp)+1.0./(exp((pp.*0)./xt)+1.0))./pp;\n",
    "    s=(-log.(exp.(-pp) .+1.0./(exp.((pp.*y)./xt) .+1.0))./pp .-s0)/(1-s0)\n",
    "    ds=(exp.((pp.*y)./xt).*1.0./(exp.((pp.*y)./xt) .+1.0).^2)./(xt.*(exp.(-pp) .+1.0./(exp.((pp.*y)./xt) .+1.0)))/(1-s0)\n",
    "    return s,ds\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "model_updateV (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function model_updateV(delta,X,method,saturation,aggreg)\n",
    "# update local densities\n",
    "    m=X[6:6:length(X)];\n",
    "    nc=length(m);\n",
    "    m=repeat(m[:],1,size(delta,2));\n",
    "    gammav = 1\n",
    "    if method == \"MMC\"\n",
    "        rho=delta;\n",
    "        drho_ddelta=ones(size(delta));\n",
    "        drho_ddelta=repmat(drho_ddelta,size(m,1),1);\n",
    "        drho_dm=0*m;\n",
    "        \n",
    "        elseif method == \"GP\"\n",
    "        hatdelta=delta.*m.^gammav;\n",
    "        rho,drho_dhatdelta=Aggregation_Pi(hatdelta);\n",
    "        if saturation\n",
    "            rho,ds=smooth_sat(rho,nc,aggreg);\n",
    "            drho_dhatdelta=ds.*drho_dhatdelta;\n",
    "        end\n",
    "        dhatdelta_ddelta=m.^gammav;\n",
    "        dhatdelta_dm=gammav*delta.*m.^(gammav-1);\n",
    "        drho_ddelta=dhatdelta_ddelta.*drho_dhatdelta;\n",
    "        drho_dm=drho_dhatdelta.*dhatdelta_dm;\n",
    "        \n",
    "        elseif method == \"MNA\"\n",
    "        hatdelta=delta.*m.^gammav;\n",
    "        rho,drho_dhatdelta=Aggregation_Pi(hatdelta);\n",
    "        if saturation\n",
    "            rho,ds=smooth_sat(rho,nc,aggreg);\n",
    "            drho_dhatdelta=ds.*drho_dhatdelta;\n",
    "        end\n",
    "        dhatdelta_ddelta=m.^gammav;\n",
    "        dhatdelta_dm=gammav*delta.*m.^(gammav-1);\n",
    "        drho_ddelta=dhatdelta_ddelta.*drho_dhatdelta;\n",
    "        drho_dm=drho_dhatdelta.*dhatdelta_dm;\n",
    "    end\n",
    "    return rho,drho_ddelta,drho_dm\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "norato_bar (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function norato_bar(xi,eta,L,h)\n",
    "    a=h./2 .*sqrt.(complex(1 .+xi.^2 ./(eta.^2 .+(eta==0)))).*((xi.^2 ./(xi.^2+eta.^2) .<(L.^2 ./(h.^2+L.^2))))\n",
    "    a.*=(xi!=0||eta!=0) .+sqrt(2)/2*h.*(xi==0&&eta==0)\n",
    "    return real.(a)\n",
    "   \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "norato_bar2 (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function norato_bar2(xi,eta,L,h)\n",
    "    a = L/2 .*sqrt.(xi.^2 ./(xi.^2+eta.^2))\n",
    "    a+= sqrt.(complex(h.^2/4-L.^2/4 .*eta.^2 ./(xi.^2+eta.^2)))\n",
    "    a.*=(xi.^2 ./(xi.^2+eta.^2) .>=(L.^2 ./(h.^2+L.^2)))\n",
    "\n",
    "    return real.(a)\n",
    "   \n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transcription of MMA : \"method of mooving asymptotes\"\n",
    "\n",
    "The 3 following functions come from :\n",
    "\n",
    "Copyright (C) 2006 Krister Svanberg\n",
    "\n",
    "    This file, kktcheck.m / mmasub.m / subsolv.m, is part of GCMMA-MMA-code.\n",
    "    \n",
    "    GCMMA-MMA-code is free software; you can redistribute it and/or\n",
    "    modify it under the terms of the GNU General Public License as \n",
    "    published by the Free Software Foundation; either version 3 of \n",
    "    the License, or (at your option) any later version.\n",
    "    \n",
    "    This code is distributed in the hope that it will be useful,\n",
    "    but WITHOUT ANY WARRANTY; without even the implied warranty of\n",
    "    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the\n",
    "    GNU General Public License for more details.\n",
    "    \n",
    "    You should have received a copy of the GNU General Public License\n",
    "    (file COPYING) along with this file.  If not, see \n",
    "    <http://www.gnu.org/licenses/>.\n",
    "    \n",
    "    You should have received a file README along with this file,\n",
    "    containing contact information.  If not, see\n",
    "    <http://www.smoptit.se/> or e-mail mmainfo@smoptit.se or krille@math.kth.se.\n",
    "\n",
    "    Version Dec 2006."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mmasub (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mmasub(m,n,iter,xval,xmin,xmax,xold1,xold2,f0val,df0dx,fval,dfdx,low,upp,a0,a,c,d)# ok\n",
    "    \n",
    "    epsimin = 10^(-7);\n",
    "    raa0 = 0.00001;\n",
    "    move = 0.5;\n",
    "    albefa = 0.1;\n",
    "    asyinit = 0.5;\n",
    "    asyincr = 1.2;\n",
    "    asydecr = 0.7;\n",
    "    eeen = ones(n,1);\n",
    "    eeem = ones(m,1);\n",
    "    zeron = zeros(n,1);\n",
    "\n",
    "    # Calculation of the asymptotes low and upp :\n",
    "    if iter < 2.5\n",
    "      low = xval - asyinit*(xmax-xmin);\n",
    "      upp = xval + asyinit*(xmax-xmin);\n",
    "    else\n",
    "      zzz = (xval-xold1).*(xold1-xold2);\n",
    "      factor = eeen;\n",
    "        factor = reshape([ (asyincr*(x > 0) + asydecr*(x<0)) for x in zzz[:] ],size( zzz))\n",
    "      low = xval - factor.*(xold1 - low);\n",
    "      upp = xval + factor.*(upp - xold1);\n",
    "      lowmin = xval - 10*(xmax-xmin);\n",
    "      lowmax = xval - 0.01*(xmax-xmin);\n",
    "      uppmin = xval + 0.01*(xmax-xmin);\n",
    "      uppmax = xval + 10*(xmax-xmin);\n",
    "      low = max.(low,lowmin);\n",
    "      low = min.(low,lowmax);\n",
    "      upp = min.(upp,uppmax);\n",
    "      upp = max.(upp,uppmin);\n",
    "    end\n",
    "\n",
    "    # Calculation of the bounds alfa and beta :\n",
    "\n",
    "    zzz1 = low + albefa*(xval-low);\n",
    "    zzz2 = xval - move*(xmax-xmin);\n",
    "    zzz  = max.(zzz1,zzz2);\n",
    "    alfa = max.(zzz,xmin);\n",
    "    zzz1 = upp - albefa*(upp-xval);\n",
    "    zzz2 = xval + move*(xmax-xmin);\n",
    "    zzz  = min.(zzz1,zzz2);\n",
    "    beta = min.(zzz,xmax);\n",
    "\n",
    "    # Calculations of p0, q0, P, Q and b.\n",
    "\n",
    "    xmami = xmax-xmin;\n",
    "    xmamieps = 0.00001*eeen;\n",
    "    xmami = max.(xmami,xmamieps);\n",
    "    xmamiinv = eeen./xmami;\n",
    "    ux1 = upp-xval;\n",
    "    ux2 = ux1.*ux1;\n",
    "    xl1 = xval-low;\n",
    "    xl2 = xl1.*xl1;\n",
    "    uxinv = eeen./ux1;\n",
    "    xlinv = eeen./xl1;\n",
    "    \n",
    "    p0 = zeron;\n",
    "    q0 = zeron;\n",
    "    p0 = max.(df0dx,0);\n",
    "    q0 = max.(-df0dx,0);\n",
    "    pq0 = 0.001*(p0 + q0) + raa0*xmamiinv;\n",
    "    p0 = p0 + pq0;\n",
    "    q0 = q0 + pq0;\n",
    "    p0 = p0.*ux2;\n",
    "    q0 = q0.*xl2;\n",
    "\n",
    "\n",
    "    P = max.(dfdx,0);\n",
    "    Q = max.(-dfdx,0);\n",
    "    P = sparse(P);\n",
    "    Q = sparse(Q);\n",
    "    PQ = 0.001*(P + Q) + raa0*eeem*xmamiinv';\n",
    "    P = P + PQ;\n",
    "    Q = Q + PQ;\n",
    "    P = P * sparse(Diagonal(ux2[:,1]))\n",
    "    Q = Q * sparse(Diagonal(xl2[:,1]))#spdiags(xl2,0,n,n);\n",
    "    b = P*uxinv + Q*xlinv .- fval ;\n",
    "    #Solving the subproblem by a primal-dual Newton method\n",
    "    xmma,ymma,zmma,lam,xsi,eta,mu,zet,s = subsolv(m,n,epsimin,low,upp,alfa,beta,p0,q0,P,Q,a0,a,b,c,d);\n",
    "\n",
    "    return xmma,ymma,zmma,lam,xsi,eta,mu,zet,s,low,upp\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "subsolv (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function subsolv(m,n,epsimin,low,upp,alfa,beta,p0,q0,P,Q,a0,a,b,c,d)\n",
    "\"\"\"\n",
    " This function subsolv solves the MMA subproblem:\n",
    "         \n",
    " minimize   SUM[ p0j/(uppj-xj) + q0j/(xj-lowj) ] + a0*z +\n",
    "          + SUM[ ci*yi + 0.5*di*(yi)^2 ],\n",
    "\n",
    " subject to SUM[ pij/(uppj-xj) + qij/(xj-lowj) ] - ai*z - yi <= bi,\n",
    "            alfaj <=  xj <=  betaj,  yi >= 0,  z >= 0.\n",
    "        \n",
    " Input:  m, n, low, upp, alfa, beta, p0, q0, P, Q, a0, a, b, c, d.\n",
    " Output: xmma,ymma,zmma, slack variables and Lagrange multiplers.\n",
    "\"\"\"\n",
    "\n",
    "    een = ones(n,1);\n",
    "    eem = ones(m,1);\n",
    "    epsi = 1;\n",
    "    epsvecn = epsi*een;\n",
    "    epsvecm = epsi*eem;\n",
    "    x = 0.5*(alfa+beta);\n",
    "    y = eem;\n",
    "    z = 1;\n",
    "    lam = eem;\n",
    "    xsi = een./(x-alfa);\n",
    "    xsi = max.(xsi,een);\n",
    "    eta = een./(beta-x);\n",
    "    eta = max.(eta,een);\n",
    "    mu  = max.(eem,0.5*c);\n",
    "    zet = 1;\n",
    "    s = eem;\n",
    "    itera = 0;\n",
    "    while epsi > epsimin\n",
    "      epsvecn = epsi*een;\n",
    "      epsvecm = epsi*eem;\n",
    "      ux1 = upp-x;\n",
    "      xl1 = x-low;\n",
    "      ux2 = ux1.*ux1;\n",
    "      xl2 = xl1.*xl1;\n",
    "      uxinv1 = een./ux1;\n",
    "      xlinv1 = een./xl1;\n",
    "      plam = p0 + P'*lam ;\n",
    "      qlam = q0 + Q'*lam ;\n",
    "      gvec = P*uxinv1 + Q*xlinv1;\n",
    "      dpsidx = plam./ux2 - qlam./xl2 ;\n",
    "      rex = dpsidx - xsi + eta;\n",
    "      rey = c .+ d.*y - mu - lam;\n",
    "      rez = a0 - zet .- a'*lam;\n",
    "      relam = gvec .- a*z - y + s - b;\n",
    "      rexsi = xsi.*(x-alfa) - epsvecn;\n",
    "      reeta = eta.*(beta-x) - epsvecn;\n",
    "      remu = mu.*y - epsvecm;\n",
    "      rezet = zet*z - epsi;\n",
    "      res = lam.*s - epsvecm;\n",
    "      residu1 = [rex' rey' rez]';\n",
    "      residu2 = [relam' rexsi' reeta' remu' rezet res']';\n",
    "      residu = [residu1' residu2']';\n",
    "      residunorm = sqrt.(residu'*residu)[1,1];\n",
    "      residumax = maximum(abs.(residu));\n",
    "      ittt = 0;\n",
    "      while (residumax > 0.9*epsi) && (ittt < 200)\n",
    "        ittt=ittt + 1;\n",
    "        itera=itera + 1;\n",
    "        ux1 = upp-x;\n",
    "        xl1 = x-low;\n",
    "        ux2 = ux1.*ux1;\n",
    "        xl2 = xl1.*xl1;\n",
    "        ux3 = ux1.*ux2;\n",
    "        xl3 = xl1.*xl2;\n",
    "        uxinv1 = een./ux1;\n",
    "        xlinv1 = een./xl1;\n",
    "        uxinv2 = een./ux2;\n",
    "        xlinv2 = een./xl2;\n",
    "        plam = p0 + P'*lam ;\n",
    "        qlam = q0 + Q'*lam ;\n",
    "        gvec = P*uxinv1 + Q*xlinv1;\n",
    "        GG = P*sparse(Diagonal(uxinv2[:,1])) -Q*sparse(Diagonal(xlinv2[:,1]))#P*spdiags(uxinv2,0,n,n) - Q*spdiags(xlinv2,0,n,n);\n",
    "        dpsidx = plam./ux2 - qlam./xl2 ;\n",
    "        delx = dpsidx - epsvecn./(x-alfa) + epsvecn./(beta-x);\n",
    "        dely = c .+ d.*y - lam - epsvecm./y;\n",
    "        delz = a0 .- a'*lam .- epsi/z;\n",
    "        dellam = gvec .- a*z - y - b .+ epsvecm./lam;\n",
    "        diagx = plam./ux3 + qlam./xl3;\n",
    "        diagx = 2*diagx + xsi./(x-alfa) + eta./(beta-x);\n",
    "        diagxinv = een./diagx;\n",
    "        diagy = d .+ mu./y;\n",
    "        diagyinv = eem./diagy;\n",
    "        diaglam = s./lam;\n",
    "        diaglamyi = diaglam+diagyinv;\n",
    "        if m < n\n",
    "          blam = dellam .+ dely./diagy - GG*(delx./diagx);\n",
    "          bb = [blam' delz]';\n",
    "          Alam = sparse(Diagonal(diaglamyi[:,1]))+GG*sparse(Diagonal(diagxinv[:,1]))*GG'#spdiags(diaglamyi,0,m,m) + GG*spdiags(diagxinv,0,n,n)*GG';\n",
    "          AA = [Alam     a\n",
    "                a'    -zet/z ];\n",
    "          solut = AA\\bb;\n",
    "          dlam = solut[1:m];\n",
    "          dz = solut[m+1];\n",
    "          dx = -delx./diagx - (GG'*dlam)./diagx;\n",
    "        else\n",
    "          diaglamyiinv = eem./diaglamyi;\n",
    "          dellamyi = dellam + dely./diagy;\n",
    "          Axx = saprse(Diagonal(diagx[:,1])) + GG'*sparse(Diagonal(diaglamyiinv[:,1]))*GG#spdiags(diagx,0,n,n) + GG'*spdiags(diaglamyiinv,0,m,m)*GG;\n",
    "          azz = zet/z + a'*(a./diaglamyi);\n",
    "          axz = -GG'*(a./diaglamyi);\n",
    "          bx = delx .+ GG'*(dellamyi./diaglamyi);\n",
    "          bz  = delz .- a'*(dellamyi./diaglamyi);\n",
    "          AA = [Axx   axz\n",
    "                axz'  azz ];\n",
    "          bb = [-bx' -bz]';\n",
    "          solut = AA\\bb;\n",
    "          dx  = solut[1:n];\n",
    "          dz = solut[n+1];\n",
    "          dlam = (GG*dx)./diaglamyi - dz*(a./diaglamyi) + dellamyi./diaglamyi;\n",
    "        end\n",
    "    \n",
    "        dy = -dely./diagy + dlam./diagy;\n",
    "        dxsi = -xsi .+ epsvecn./(x-alfa) - (xsi.*dx)./(x-alfa);\n",
    "        deta = -eta .+ epsvecn./(beta-x) + (eta.*dx)./(beta-x);\n",
    "        dmu  = -mu .+ epsvecm./y - (mu.*dy)./y;\n",
    "        dzet = -zet .+ epsi/z - zet*dz/z;\n",
    "        ds   = -s .+ epsvecm./lam - (s.*dlam)./lam;\n",
    "        xx  = [ y'  z  lam'  xsi'  eta'  mu'  zet  s']';\n",
    "        dxx = [dy' dz dlam' dxsi' deta' dmu' dzet ds']';\n",
    "        \n",
    "        stepxx = -1.01*dxx./xx;\n",
    "        stmxx  = maximum(stepxx);\n",
    "        stepalfa = -1.01*dx./(x-alfa);\n",
    "        stmalfa = maximum(stepalfa);\n",
    "        stepbeta = 1.01*dx./(beta-x);\n",
    "        stmbeta = maximum(stepbeta);\n",
    "        stmalbe  = max.(stmalfa,stmbeta);\n",
    "        stmalbexx = max.(stmalbe,stmxx);\n",
    "        stminv = max.(stmalbexx,1);\n",
    "        steg = 1/stminv;\n",
    "    \n",
    "        xold   =   x;\n",
    "        yold   =   y;\n",
    "        zold   =   z;\n",
    "        lamold =  lam;\n",
    "        xsiold =  xsi;\n",
    "        etaold =  eta;\n",
    "        muold  =  mu;\n",
    "        zetold =  zet;\n",
    "        sold   =   s;\n",
    "    \n",
    "        itto = 0;\n",
    "        resinew = 2*residunorm;\n",
    "        while resinew > residunorm && itto < 50\n",
    "            itto = itto+1;\n",
    "            x   =   xold + steg*dx;\n",
    "            y   =   yold + steg*dy;\n",
    "            z   =   zold + steg*dz;\n",
    "            lam = lamold + steg*dlam;\n",
    "            xsi = xsiold + steg*dxsi;\n",
    "            eta = etaold + steg*deta;\n",
    "            mu  = muold  + steg*dmu;\n",
    "            zet = zetold + steg*dzet;\n",
    "            s   =   sold + steg*ds;\n",
    "            ux1 = upp-x;\n",
    "            xl1 = x-low;\n",
    "            ux2 = ux1.*ux1;\n",
    "            xl2 = xl1.*xl1;\n",
    "            uxinv1 = een./ux1;\n",
    "            xlinv1 = een./xl1;\n",
    "            plam = p0 + P'*lam ;\n",
    "            qlam = q0 + Q'*lam ;\n",
    "            gvec = P*uxinv1 + Q*xlinv1;\n",
    "            dpsidx = plam./ux2 - qlam./xl2 ;\n",
    "            rex = dpsidx - xsi + eta;\n",
    "            rey = c .+ d.*y - mu - lam;\n",
    "            rez = a0 - zet .- a'*lam;\n",
    "            relam = gvec .- a*z - y + s - b;\n",
    "            rexsi = xsi.*(x-alfa) - epsvecn;\n",
    "            reeta = eta.*(beta-x) - epsvecn;\n",
    "            remu = mu.*y - epsvecm;\n",
    "            rezet = zet*z - epsi;\n",
    "            res = lam.*s - epsvecm;\n",
    "            residu1 = [rex' rey' rez]';\n",
    "            residu2 = [relam' rexsi' reeta' remu' rezet res']';\n",
    "            residu = [residu1' residu2']';\n",
    "            resinew = sqrt.(residu'*residu)[1,1];\n",
    "            steg = steg/2;\n",
    "        end\n",
    "      residunorm=resinew;\n",
    "      residumax = maximum(abs.(residu));\n",
    "      steg = 2*steg;\n",
    "      end\n",
    "      if ittt > 198\n",
    "        epsi\n",
    "        ittt\n",
    "      end\n",
    "    epsi = 0.1*epsi;\n",
    "    end\n",
    "    xmma   =   x;\n",
    "    ymma   =   y;\n",
    "    zmma   =   z;\n",
    "    lamma =  lam;\n",
    "    xsimma =  xsi;\n",
    "    etamma =  eta;\n",
    "    mumma  =  mu;\n",
    "    zetmma =  zet;\n",
    "    smma   =   s;\n",
    "    return xmma,ymma,zmma,lamma,xsimma,etamma,mumma,zetmma,smma\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "kktcheck (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function kktcheck(m,n,x,y,z,lam,xsi,eta,mu,zet,s,xmin,xmax,df0dx,fval,dfdx,a0,a,c,d)\n",
    "\"\"\"\n",
    "%  The left hand sides of the KKT conditions for the following\n",
    "%  nonlinear programming problem are calculated.\n",
    "%         \n",
    "%      Minimize  f_0(x) + a_0*z + sum( c_i*y_i + 0.5*d_i*(y_i)^2 )\n",
    "%    subject to  f_i(x) - a_i*z - y_i <= 0,  i = 1,...,m\n",
    "%                xmax_j <= x_j <= xmin_j,    j = 1,...,n\n",
    "%                z >= 0,   y_i >= 0,         i = 1,...,m\n",
    "%*** INPUT:\n",
    "%\n",
    "%   m    = The number of general constraints.\n",
    "%   n    = The number of variables x_j.\n",
    "%   x    = Current values of the n variables x_j.\n",
    "%   y    = Current values of the m variables y_i.\n",
    "%   z    = Current value of the single variable z.\n",
    "%  lam   = Lagrange multipliers for the m general constraints.\n",
    "%  xsi   = Lagrange multipliers for the n constraints xmin_j - x_j <= 0.\n",
    "%  eta   = Lagrange multipliers for the n constraints x_j - xmax_j <= 0.\n",
    "%   mu   = Lagrange multipliers for the m constraints -y_i <= 0.\n",
    "%  zet   = Lagrange multiplier for the single constraint -z <= 0.\n",
    "%   s    = Slack variables for the m general constraints.\n",
    "%  xmin  = Lower bounds for the variables x_j.\n",
    "%  xmax  = Upper bounds for the variables x_j.\n",
    "%  df0dx = Vector with the derivatives of the objective function f_0\n",
    "%          with respect to the variables x_j, calculated at x.\n",
    "%  fval  = Vector with the values of the constraint functions f_i,\n",
    "%          calculated at x.\n",
    "%  dfdx  = (m x n)-matrix with the derivatives of the constraint functions\n",
    "%          f_i with respect to the variables x_j, calculated at x.\n",
    "%          dfdx(i,j) = the derivative of f_i with respect to x_j.\n",
    "%   a0   = The constants a_0 in the term a_0*z.\n",
    "%   a    = Vector with the constants a_i in the terms a_i*z.\n",
    "%   c    = Vector with the constants c_i in the terms c_i*y_i.\n",
    "%   d    = Vector with the constants d_i in the terms 0.5*d_i*(y_i)^2.\n",
    "%     \n",
    "%*** OUTPUT:\n",
    "%\n",
    "% residu     = the residual vector for the KKT conditions.\n",
    "% residunorm = sqrt(residu'*residu).\n",
    "% residumax  = max(abs(residu)).\n",
    "\"\"\"\n",
    "    rex   = df0dx .+ dfdx .* lam - xsi + eta;\n",
    "    rey   = c .+ d.*y - mu - lam;\n",
    "    rez   = a0 - zet .- a'*lam;\n",
    "    relam = fval .- a*z - y + s;\n",
    "    rexsi = xsi.*(x-xmin);\n",
    "    reeta = eta.*(xmax-x);\n",
    "    remu  = mu.*y;\n",
    "    rezet = zet*z;\n",
    "    res   = lam.*s;\n",
    "\n",
    "    residu1 = [rex' rey' rez]';\n",
    "    residu2 = [relam' rexsi' reeta' remu' rezet res']';\n",
    "    residu = [residu1' residu2']';\n",
    "    residunorm = sqrt(residu'*residu);\n",
    "    residumax = maximum(abs.(residu));\n",
    "return residu,residunorm,residumax\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Main function\n",
    "\n",
    "After running this code, the optimal shape SIMP and the result GGP obtained are plotted, as the evolution of the sensitivity and the volume fraction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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     "output_type": "stream",
     "text": [
      " It.: 1 Obj.: 7141.827810763383 Vol.: 0.10948364767099499 kktnorm.: 10.001 change : 1\n",
      " It.: 2 Obj.: 2.411814213982618e7 Vol.: 0.5231096333778478 kktnorm.: [81.05333488829469] change : 2.3291826695712206\n",
      " It.: 3 Obj.: 5.725811082285876e6 Vol.: 0.22203929058025373 kktnorm.: [16.290521793890516] change : 2.691851699550515\n",
      " It.: 4 Obj.: 157.08505706473895 Vol.: 0.5724966397628576 kktnorm.: [50.12685453487609] change : 1.698904921249032\n",
      " It.: 5 Obj.: 5.457677329376988e6 Vol.: 0.5220833111022571 kktnorm.: [41.372342144420315] change : 2.070630106579628\n",
      " It.: 6 Obj.: 21.6354721633516 Vol.: 0.8043320627335929 kktnorm.: [20.689313906378977] change : 1.479219246844641\n",
      " It.: 7 Obj.: 379.06567194472615 Vol.: 0.4818066723998964 kktnorm.: [78.88386958520668] change : 1.3015076850113243\n",
      " It.: 8 Obj.: 6.911329417077098e6 Vol.: 0.43158000066456587 kktnorm.: [148.51700894168502] change : 1.6792574586576072\n",
      " It.: 9 Obj.: 38.87474275127783 Vol.: 0.7711795581200039 kktnorm.: [8.35738325347702] change : 1.2065696423966072\n",
      " It.: 10 Obj.: 681711.334289422 Vol.: 0.4318054511280811 kktnorm.: [155.84908249885328] change : 1.0165616110951887\n",
      " It.: 11 Obj.: 383078.5553941283 Vol.: 0.4676617991546891 kktnorm.: [3.026082160338268] change : 0.8098134334380085\n",
      " It.: 12 Obj.: 1.0902004667275413e6 Vol.: 0.32898578538004924 kktnorm.: [42.403732983716516] change : 1.2205256611542796\n",
      " It.: 13 Obj.: 378807.70764818037 Vol.: 0.5503376242164573 kktnorm.: [26.23651414238863] change : 0.8300553167720995\n",
      " It.: 14 Obj.: 359762.7805972266 Vol.: 0.3613943044819631 kktnorm.: [22.39534348018037] change : 0.9271580405632267\n",
      " It.: 15 Obj.: 4.379375009684838e6 Vol.: 0.34388358777386513 kktnorm.: [19.751825698103826] change : 0.8429417004264003\n",
      " It.: 16 Obj.: 889737.4303468297 Vol.: 0.41347562476758115 kktnorm.: [22.661772016810417] change : 0.32424327443031786\n",
      " It.: 17 Obj.: 2.1504540679710233e6 Vol.: 0.32500020854349104 kktnorm.: [8.16949340755658] change : 0.41588642793630604\n",
      " It.: 18 Obj.: 797343.139061473 Vol.: 0.39587444552819634 kktnorm.: [26.713714460866903] change : 0.44897921715514694\n",
      " It.: 19 Obj.: 9556.629237157782 Vol.: 0.421140311861596 kktnorm.: [11.598929531822304] change : 0.42103821298868815\n",
      " It.: 20 Obj.: 674133.9077614495 Vol.: 0.3564658343332799 kktnorm.: [74.16865900315236] change : 0.772700779229322\n",
      " It.: 21 Obj.: 2.514277787744665e6 Vol.: 0.2226712468838257 kktnorm.: [21.614310654338798] change : 0.5223665624638544\n",
      " It.: 22 Obj.: 220538.89349568996 Vol.: 0.41870329690967933 kktnorm.: [49.27664147024022] change : 0.40857162670520414\n",
      " It.: 23 Obj.: 466.5759475732116 Vol.: 0.2954482366247822 kktnorm.: [6.9513914824991] change : 0.4440260137271392\n",
      " It.: 24 Obj.: 189.90496988965688 Vol.: 0.3854912590915615 kktnorm.: [44.55240970503536] change : 0.6071099140628377\n",
      " It.: 25 Obj.: 57.37498639189128 Vol.: 0.4079449687204674 kktnorm.: [27.22127118698154] change : 0.40195138392913626\n",
      " It.: 26 Obj.: 382826.3977377646 Vol.: 0.3471160371356201 kktnorm.: [15.51118436543824] change : 0.2752847756802959\n",
      " It.: 27 Obj.: 528.3330754532222 Vol.: 0.30368353439632917 kktnorm.: [22.618041413088857] change : 0.37347835144665537\n",
      " It.: 28 Obj.: 84.61141340343465 Vol.: 0.37312632779445254 kktnorm.: [53.897691747900986] change : 0.46028730240179644\n",
      " It.: 29 Obj.: 59.36632091459543 Vol.: 0.4092390842311099 kktnorm.: [23.95194067632251] change : 0.2734834069372881\n",
      " It.: 30 Obj.: 33.1156091489539 Vol.: 0.4033826503852456 kktnorm.: [23.099535799961608] change : 0.14338456211775522\n",
      " It.: 31 Obj.: 32.071217929229434 Vol.: 0.41478012827156635 kktnorm.: [24.900709776712233] change : 0.21203774547159715\n",
      " It.: 32 Obj.: 29.324315938295967 Vol.: 0.40524368434863633 kktnorm.: [24.91082156369769] change : 0.16385885194791408\n",
      " It.: 33 Obj.: 84.235293469462 Vol.: 0.35292061010243553 kktnorm.: [50.56190936769248] change : 0.31614717517817453\n",
      " It.: 34 Obj.: 31.229046386830557 Vol.: 0.40543303782741125 kktnorm.: [27.87277225036598] change : 0.19089738221507305\n",
      " It.: 35 Obj.: 23.38118331943116 Vol.: 0.4460943195172879 kktnorm.: [10.996981753166713] change : 0.16607977644975386\n",
      " It.: 36 Obj.: 21.05588728780379 Vol.: 0.4371263475330831 kktnorm.: [3.551366663714998] change : 0.17463176607827346\n",
      " It.: 37 Obj.: 20.516063710432732 Vol.: 0.4336046832103384 kktnorm.: [3.7047532839533104] change : 0.1363667552639283\n",
      " It.: 38 Obj.: 20.302934299660663 Vol.: 0.43152981418493686 kktnorm.: [6.626041824134994] change : 0.1536741036332831\n",
      " It.: 39 Obj.: 44.716397154637505 Vol.: 0.4149560513964418 kktnorm.: [8.135935327395446] change : 0.30936223248094513\n",
      " It.: 40 Obj.: 26.161465094552312 Vol.: 0.42700384940713193 kktnorm.: [12.76967131382453] change : 0.16763978768444987\n",
      " It.: 41 Obj.: 19.825543422535095 Vol.: 0.4318549327243845 kktnorm.: [14.06322991409437] change : 0.11627717001437853\n",
      " It.: 42 Obj.: 19.704092181200714 Vol.: 0.45796265701526645 kktnorm.: [6.977614734802684] change : 0.19158250886251055\n",
      " It.: 43 Obj.: 20.521102033563363 Vol.: 0.44272388246929273 kktnorm.: [11.171655446818697] change : 0.23500394950097606\n",
      " It.: 44 Obj.: 17.590229374732218 Vol.: 0.4487317408079813 kktnorm.: [7.635394903183628] change : 0.15324247862080198\n",
      " It.: 45 Obj.: 17.31319645398989 Vol.: 0.4571798799078101 kktnorm.: [1.4066220548756396] change : 0.07748444099886044\n",
      " It.: 46 Obj.: 17.299986720331592 Vol.: 0.4533452269246719 kktnorm.: [1.9826233343605666] change : 0.08650688697744609\n",
      " It.: 47 Obj.: 17.29905222804607 Vol.: 0.45361009293105575 kktnorm.: [1.552217708410156] change : 0.11643871612375717\n",
      " It.: 48 Obj.: 17.907285160873233 Vol.: 0.4531835770731081 kktnorm.: [1.6253239821725796] change : 0.14286009250755716\n",
      " It.: 49 Obj.: 17.24428931261299 Vol.: 0.4506041498525679 kktnorm.: [5.2489927122766815] change : 0.14448069174450154\n",
      " It.: 50 Obj.: 17.114153327848065 Vol.: 0.4529671337877547 kktnorm.: [2.379497734294971] change : 0.12243611622215618\n"
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      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function topggp(nelx,nely,problem = \"MBB\",settings = \"GP\" , volfrac = 0.4,maxoutit = 25  )\n",
    "    starting_guess = \"Crosses\"\n",
    "    stopping_criteria=\"change\"  #Stopping Criteria:  'change'  'kktnorm'\n",
    "    if settings==\"GGP\"\n",
    "        method=\"MMC\"   #Method:   'MMC'  'MNA'  'GP'\n",
    "        q=1                 #q=1\n",
    "        zp=1           #parameter for p-norm/mean regularization\n",
    "        alp=1          #parameter for MMC\n",
    "        epsi=0.866     #parameter for MMC\n",
    "        bet=1e-3       #parameter for MMC\n",
    "        minh=1              # minimum height of a component\n",
    "        aggregation=\"KSl\"   #parameter for the aggregation function to be used\n",
    "        #'IE'-Induced Exponential  'KS'-KS function  'KSl'-Lowerbound KS function 'p-norm'  'p-mean'\n",
    "        ka=10          #parameter for the aggregation constant\n",
    "        saturation=true     #switch for saturation\n",
    "        ncx=1           #number of components in the x direction\n",
    "        ncy=1           #number of components in the y direction\n",
    "        Ngp=2           #number of Gauss point per sampling window\n",
    "        R=0.5          #radius of the sampling window (infty norm)\n",
    "        initial_d=0.5    #component initial mass\n",
    "    elseif settings==\"MMC\"\n",
    "        method=\"MMC\"  #'MMC'  'MNA'  'GP'\n",
    "        q=1                 #q=1\n",
    "        zp=1           #parameter for p-norm/mean regularization\n",
    "        alp=1          #parameter for MMC\n",
    "        epsi=0.866     #parameter for MMC\n",
    "        bet=1e-3       #parameter for MMC\n",
    "        minh=1              # minimum height of a component\n",
    "        aggregation=\"KSl\"  #parameter for the aggregation function to be used\n",
    "        #'IE'-Induced Exponential  'KS'-KS function  'KSl'-Lowerbound KS function 'p-norm'  'p-mean'\n",
    "        ka=10          #parameter for the aggregation constant\n",
    "        saturation=true    #switch for saturation\n",
    "        ncx=1               #number of components in the x direction\n",
    "        ncy=1               #number of components in the y direction\n",
    "        Ngp=2               #number of Gauss point per sampling window\n",
    "        R=0.5    #radius of the sampling window (infty norm)\n",
    "        initial_d=0.5         #component initial mass\n",
    "    elseif settings==\"MNA\"\n",
    "        method=\"MNA\"    #'MMC'  'MNA'  'GP'\n",
    "        q=1             #q=1\n",
    "        zp=1           #parameter for p-norm/mean regularization\n",
    "        minh=1              # minimum height of a component\n",
    "        sigma=1        #parameter for MNA\n",
    "        penalty=3      #parameter for MNA\n",
    "        gammav=1       #parameter for GP\n",
    "        gammac=3       #parameter for GP\n",
    "        aggregation=\"KSl\"    #parameter for the aggregation function to be used\n",
    "        #'IE'-Induced Exponential  'KS'-KS function  'KSl'-Lowerbound KS function 'p-norm'  'p-mean'\n",
    "        ka=10           #parameter for the aggregation constant\n",
    "        saturation=true     #switch for saturation\n",
    "        ncx=1               #number of components in the x direction\n",
    "        ncy=1               #number of components in the y direction\n",
    "        Ngp=2               #number of Gauss point per sampling window\n",
    "        R=0.5               #radius of the sampling window (infty norm)\n",
    "        initial_d=0.5       #component initial mass\n",
    "    elseif settings==\"GP\"\n",
    "        method=\"GP\"    #'MMC'  'MNA'  'GP'\n",
    "        q=1                 #q=1\n",
    "        zp=1           #parameter for p-norm/mean regularization\n",
    "        deltamin=1e-6  #parameter for GP\n",
    "        r=0.5          #parameter for GP\n",
    "        minh=1              # minimum height of a component\n",
    "        gammav=1       #parameter for GP\n",
    "        gammac=3       #parameter for GP\n",
    "        aggregation=\"KSl\"  #parameter for the aggregation function to be used\n",
    "        #'IE'-Induced Exponential  'KS'-KS function  'KSl'-Lowerbound KS function 'p-norm'  'p-mean'\n",
    "        ka=10          #parameter for the aggregation constant\n",
    "        saturation=true   #switch for saturation\n",
    "        ncx=1               #number of components in the x direction\n",
    "        ncy=1               #number of components in the y direction\n",
    "        Ngp=2               #number of Gauss point per sampling window\n",
    "        R=0.5    #radius of the sampling window (infty norm)\n",
    "        initial_d=0.5       #component initial mass\n",
    "    else\n",
    "        print(\"settings string should be a valid entry: GGP, MMC, MNA or GP\") \n",
    "        return        \n",
    "    end\n",
    "    E=1\n",
    "    nu=0.3\n",
    "    E0 = 1;\n",
    "    Emin = 1e-6\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",
    "    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",
    "    \n",
    "    # Define the nodal coordinates\n",
    "    x=1\n",
    "    Yy=zeros((nelx+1)*(nely+1))\n",
    "    Xx=zeros((nelx+1)*(nely+1))\n",
    "\n",
    "    for i = 1:(nelx+1)\n",
    "        for j = 1:(nely+1)\n",
    "            Yy[x]= j\n",
    "            Xx[x]= i\n",
    "            x+=1\n",
    "        end\n",
    "    end\n",
    "    Yy=(nely+1).-Yy\n",
    "    Xx=Xx.-1\n",
    "\n",
    "    \n",
    "    #only MBB to begin\n",
    "    fixeddofs = union(1:2:2*(nely+1),[2*(nelx+1)*(nely+1)]);fixednodes=[1:(nely+1);(nelx+1)*(nely+1)];fixed_dir=[ones(nely+1,1);2];\n",
    "    alldofs = 1:2*(nely+1)*(nelx+1);\n",
    "    freedofs = setdiff(alldofs,fixeddofs);\n",
    "    emptyelts=[]; fullelts = [];\n",
    "    F = zeros(2*(nely+1)*(nelx+1),1); F[2,1] = -1; \n",
    "    U = zeros(2*(nely+1)*(nelx+1),1);\n",
    "\"\"\"\n",
    "    if problem == \"MBB\"\n",
    "        excitation_node=1;excitation_direction=2;\n",
    "        amplitude=-1;\n",
    "        F = sparse(2*(excitation_node-1)+excitation_direction,1,amplitude,2*(nely+1)*(nelx+1),1);\n",
    "        fixednodes=[find(Xx==min(Xx));(nelx+1)*(nely+1)];fixed_dir=[ones(nely+1,1);2];\n",
    "        fixeddofs=2*(fixednodes .-1)+fixed_dir;\n",
    "        emptyelts=[]; fullelts = [];\n",
    "    elseif problem == \"Short_Cantilever\"\n",
    "        excitation_node=find((Xx==max(Xx))&(Yy==fix(0.5*min(Yy)+0.5*max(Yy))));excitation_direction=2;\n",
    "        amplitude=-1;\n",
    "        F = sparse(2*(excitation_node-1)+excitation_direction,1,amplitude,2*(nely+1)*(nelx+1),1);\n",
    "        fixednodes=repmat(find(Xx==min(Xx)),2,1);fixed_dir=[ones(nely+1,1);2*ones(nely+1,1)];\n",
    "        fixeddofs=2*(fixednodes-1)+fixed_dir(:);\n",
    "        emptyelts=[]; fullelts = [];\n",
    "    elseif problem == \"L-shape\"\n",
    "        excitation_node=find((Xx==max(Xx))&(Yy==fix(0.5*min(Yy)+0.5*max(Yy))));excitation_direction=2;\n",
    "        amplitude=-1;\n",
    "        F = sparse(2*(excitation_node-1)+excitation_direction,1,amplitude,2*(nely+1)*(nelx+1),1);\n",
    "        fixednodes=repmat(find(Yy==max(Yy)),2,1);fixed_dir=[ones(nelx+1,1),2*ones(nelx+1,1)];\n",
    "        fixeddofs=2*(fixednodes-1)+fixed_dir(:);\n",
    "        emptyelts=find(xc>=(((max(Xx)+min(Xx))/2))&(yc>=((max(Yy)+min(Yy))/2)));\n",
    "        fullelts = [];\n",
    "    else\n",
    "        println(\"BC string should be a valid entry: ''MBB'',''L-Shape'',''Short_Cantilever'\")\n",
    "        return\n",
    "    end\n",
    "\"\"\"\n",
    "\n",
    "        # Compute the centroid coordinates\n",
    "    xc = zeros(nelx * nely)\n",
    "    yc = zeros(nelx * nely)\n",
    "    for xi = 0:(nelx-1)\n",
    "        xc[(xi*nely+1):(xi+1)*nely].=xi+0.5\n",
    "    end\n",
    "    for yi = 0:(nely-1)\n",
    "        yc[(yi+1):nely:nelx*nely].=yi+0.5\n",
    "    end\n",
    "    yc = nely .- yc\n",
    "              \n",
    "    centroid_coordinate=zeros(nelx*nely,2)\n",
    "    for i = 1:nelx*nely\n",
    "        centroid_coordinate[i,1]= xc[i]\n",
    "        centroid_coordinate[i,2]= yc[i]\n",
    "    end\n",
    "    a=-R\n",
    "    b=R\n",
    "    gpc,wc=lgwt(Ngp,a,b)\n",
    "\n",
    "    gpcx,gpcy=Matlab.meshgrid(gpc,gpc) \n",
    "    gauss_w = wc* wc'\n",
    "    gpcx = repeat( gpcx,nelx*nely)[:] #different results than matlab here?\n",
    "    gpcy = repeat( gpcy,nelx*nely)[:]\n",
    "    gauss_weight=repeat(gauss_w,nelx*nely)[:]\n",
    "    cc=repeat(centroid_coordinate,Ngp^2)\n",
    "    gauss_point = cc + vcat(gpcx', gpcy')'\n",
    "    ugp , idgp= doublons(gauss_point)#doublons is my function, encoded higher\n",
    "\n",
    "    #design variables\n",
    "    xp=range(minimum(Xx),stop=maximum(Xx),length=ncx+2)\n",
    "    yp=range(minimum(Yy),stop=maximum(Yy),length=ncy+2); \n",
    "    xx,yy=Matlab.meshgrid(xp,yp);\n",
    "    Xc=repeat(xx[:],2,1);\n",
    "    Yc=repeat(yy[:],2,1); \n",
    "    Lc=2*sqrt.((nelx/(ncx+2))^2+(nely/(ncy+2))^2)*ones(size(Xc)); \n",
    "    OnE = ones(convert(Int64,ceil(length(Xc)/2)))\n",
    "    Tc=atan.(nely/ncy,nelx/ncx)*[OnE;-OnE];# component orientation angle tetha\n",
    "    hc=2*ones(length(Xc),1); # component h\n",
    "    Mc=initial_d*ones(size(Xc)); # component mass (For MNA and GP)\n",
    "    Xg=vcat(Xc',Yc',Lc',hc',Tc',Mc')[:]\n",
    "    xPhys = zeros(nely, nelx)\n",
    "    \n",
    "    #bounds\n",
    "    Xl=minimum(Xx.-1)*ones(size(Xc));Xu=maximum(Xx.+1)*ones(size(Xc));\n",
    "    Yl=minimum(Yy.-1)*ones(size(Xc));Yu=maximum(Yy.+1)*ones(size(Xc));\n",
    "    Ll=0*ones(size(Xc));Lu=sqrt(nelx^2+nely^2)*ones(size(Xc));\n",
    "    hl=minh*ones(size(Xc));hu=sqrt(nelx^2+nely^2)*ones(size(Xc));\n",
    "    Tl=-2*pi*ones(size(Xc));Tu=2*pi*ones(size(Xc));\n",
    "    Ml=0*ones(size(Xc));Mu=ones(size(Xc));\n",
    "    lower_bound=vcat(Xl',Yl',Ll',hl',Tl',Ml')[:] #\n",
    "    upper_bound=vcat(Xu',Yu',Lu',hu',Tu',Mu')[:] \n",
    "    X=(Xg-lower_bound)./(upper_bound-lower_bound);\n",
    "    \n",
    "    #init MMA : not sure that I use all these parameters-> clean later\n",
    "    loop = 0;\n",
    "    m = 1;\n",
    "    n = length(X);\n",
    "    epsimin = 0.0000001;\n",
    "    xval    = X[:];\n",
    "    xold1   = xval;\n",
    "    xold2   = xval;\n",
    "    xmin    = zeros(n);\n",
    "    xmax    = ones(n);\n",
    "    low     = xmin;\n",
    "    upp     = xmax;\n",
    "    C       = 1000*ones(m);\n",
    "    d       = zeros(m);\n",
    "    a0      = 1;\n",
    "    a       = zeros(m);\n",
    "    outeriter = 0;\n",
    "    kkttol  =0.001;\n",
    "    changetol=0.001;\n",
    "    kktnorm = kkttol+10;\n",
    "    outit = 0;\n",
    "    change=1;\n",
    "    \n",
    "    cvec=zeros(maxoutit);\n",
    "    vvec=zeros(maxoutit);\n",
    "    #ovvec=cvec;gvec=cvec;pvec=cvec;#for storage and plots\n",
    "    plot_rate=10;\n",
    "    \n",
    "    #for plots\n",
    "    tt=0:0.1:(2*pi);tt=repeat(tt,1,length(Xc))';\n",
    "    cc=cos.(tt);ss=sin.(tt);\n",
    "    \n",
    "    if stopping_criteria == \"kktnorm\"\n",
    "        stop_cond=outit < maxoutit && kktnorm>kkttol;\n",
    "    else\n",
    "        stop_cond=outit < maxoutit &&change>changetol;\n",
    "    end\n",
    "    \n",
    "    #design loop\n",
    "    while  stop_cond \n",
    "        outit   = outit+1;\n",
    "        outeriter = outeriter+1;\n",
    "        #Compute the smooth characteristic functions and gradients for each component \n",
    "        # on each sampling window Gauss point (Can support GPU)\n",
    "        W,dW_dX,dW_dY,dW_dT,dW_dL,dW_dh=Wgp(ugp[:,1],ugp[:,2],Xg,settings); #l183 matlab, here we get W different than in matlab code\n",
    "        #Compute local volume fractions and gradients using generalized projection\n",
    "        # delta is for densities, deltac for Young modulus\n",
    "        #for next arrays : not same size than matlab because my \"doublons\" is working differently than matlab's one\n",
    "        delta = fdelta(W,gauss_weight,idgp,Ngp)\n",
    "        ddelta_dX = fdelta(dW_dX,gauss_weight,idgp,Ngp)\n",
    "        ddelta_dY = fdelta(dW_dY,gauss_weight,idgp,Ngp)\n",
    "        ddelta_dT = fdelta(dW_dT,gauss_weight,idgp,Ngp)\n",
    "        ddelta_dL = fdelta(dW_dL,gauss_weight,idgp,Ngp)\n",
    "        ddelta_dh = fdelta(dW_dh,gauss_weight,idgp,Ngp)\n",
    "        delta_c = fdelta(W,gauss_weight,idgp,Ngp,q) # q = 1 \n",
    "        ddelta_c_dX = fdelta_c(W,dW_dX,gauss_weight,idgp,Ngp,q,q-1)\n",
    "        ddelta_c_dY = fdelta_c(W,dW_dY,gauss_weight,idgp,Ngp,q,q-1)\n",
    "        ddelta_c_dT = fdelta_c(W,dW_dT,gauss_weight,idgp,Ngp,q,q-1)\n",
    "        ddelta_c_dL = fdelta_c(W,dW_dL,gauss_weight,idgp,Ngp,q,q-1)\n",
    "        ddelta_c_dh = fdelta_c(W,dW_dh,gauss_weight,idgp,Ngp,q,q-1)\n",
    "\n",
    "        E,dE,dE_dm=model_updateM(delta_c,X,method,saturation,aggregation);#X l167\n",
    "        dE_dX=dE.*ddelta_c_dX;\n",
    "        dE_dY=dE.*ddelta_c_dY;\n",
    "        dE_dT=dE.*ddelta_c_dT;\n",
    "        dE_dL=dE.*ddelta_c_dL;\n",
    "        dE_dh=dE.*ddelta_c_dh;\n",
    "        E=reshape(E[:],nely,nelx);#useless ?\n",
    "\n",
    "        #compute densities\n",
    "        rho,drho_ddelta,drho_dm=model_updateV(delta,X,method,saturation,aggregation);\n",
    "        drho_dX=drho_ddelta.*ddelta_dX;\n",
    "        drho_dY=drho_ddelta.*ddelta_dY;\n",
    "        drho_dT=drho_ddelta.*ddelta_dT;\n",
    "        drho_dL=drho_ddelta.*ddelta_dL;\n",
    "        drho_dh=drho_ddelta.*ddelta_dh;\n",
    "        xPhys=reshape(rho[:],nely,nelx);#useless ?\n",
    "        \n",
    "        #Take in account passive elements\n",
    "        xPhys[emptyelts] .= 0;\n",
    "        xPhys[fullelts] .= 1;\n",
    "        E[emptyelts] .= Emin;\n",
    "        E[fullelts] .= E0;\n",
    "\n",
    "        # FE-ANALYSIS\n",
    "        sK = reshape(KE[:]*(E[:]'),64*nelx*nely,1);\n",
    "        K = sparse(iK[:],jK[:],sK[:]); K = (K+K')/2;\n",
    "        U[freedofs] = K[freedofs,freedofs]\\F[freedofs];#not same than matlab\n",
    "\n",
    "        # OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS\n",
    "        edM = [convert(Int64,i) for i in edofMat] ;#to index by int\n",
    "        v = (U[edM]*KE).*U[edM]\n",
    "        ce = reshape(sum!(ones(size(v,1)),v),nely,nelx);\n",
    "        c = sum(E.*ce);\n",
    "        v=mean(xPhys);\n",
    "\n",
    "        dc_dE = -ce;\n",
    "        dc_dE[emptyelts] .= 0;\n",
    "        dc_dE[fullelts] .= 0;\n",
    "        dc_dX=dE_dX*dc_dE[:]\n",
    "        dc_dY=dE_dY*dc_dE[:]\n",
    "        dc_dL=dE_dL*dc_dE[:]\n",
    "        dc_dh=dE_dh*dc_dE[:]\n",
    "        dc_dT=dE_dT*dc_dE[:]\n",
    "        dc_dm=dE_dm*dc_dE[:]\n",
    "        lll = length(X)\n",
    "        dc=zeros(size(X));\n",
    "        dc[1:6:lll]=dc_dX;\n",
    "        dc[2:6:lll]=dc_dY;\n",
    "        dc[3:6:lll]=dc_dL;\n",
    "        dc[4:6:lll]=dc_dh;\n",
    "        dc[5:6:lll]=dc_dT;\n",
    "        dc[6:6:lll]=dc_dm;\n",
    "        dv_dxPhys = ones(nely,nelx)/nelx/nely;\n",
    "        dv_dxPhys[emptyelts] .= 0;\n",
    "        dv_dxPhys[fullelts] .= 0;\n",
    "        dv_dX=drho_dX*dv_dxPhys[:]\n",
    "        dv_dY=drho_dY*dv_dxPhys[:]\n",
    "        dv_dL=drho_dL*dv_dxPhys[:]\n",
    "        dv_dh=drho_dh*dv_dxPhys[:]\n",
    "        dv_dT=drho_dT*dv_dxPhys[:]\n",
    "        dv_dm=drho_dm*dv_dxPhys[:]\n",
    "        dv=zeros(size(X));\n",
    "        dv[1:6:lll]=dv_dX;\n",
    "        dv[2:6:lll]=dv_dY;\n",
    "        dv[3:6:lll]=dv_dL;\n",
    "        dv[4:6:lll]=dv_dh;\n",
    "        dv[5:6:lll]=dv_dT;\n",
    "        dv[6:6:lll]=dv_dm;\n",
    "\n",
    "        # store the output for plot\n",
    "        cvec[outit]=c;vvec[outit]=v;\n",
    "        println(\" It.: \",outit , \" Obj.: \",c ,\" Vol.: \",mean(xPhys[:]), \" kktnorm.: \", kktnorm,\" change : \",change);\n",
    "\n",
    "        # pass scaled objective and constraint function and sensitivities to MMA\n",
    "        f0val=log(c+1); #valeur ok\n",
    "        fval=[(v-volfrac)/volfrac]*100; \n",
    "        df0dx=(dc[:]/(c+1).*(upper_bound[:]-lower_bound[:])); \n",
    "        dfdx=(dv[:]'/volfrac)*100 .*(upper_bound[:]-lower_bound[:])';\n",
    "\n",
    "\n",
    "        ##partie mmasub (ligne 366 à 385 du main)\n",
    "        X,ymma,zmma,lam,xsi,eta,mu,zet,S,low,upp=mmasub(m,n,outeriter,xval,xmin,xmax,xold1,xold2, f0val,df0dx,fval,dfdx,low,upp,a0,a,C,d);\n",
    "        xold2 = xold1;\n",
    "        xold1 = xval;\n",
    "        xval  = X;\n",
    "        Xg=lower_bound+(upper_bound-lower_bound).*X;\n",
    "        change=norm(xval .-xold1);\n",
    "        #The residual vector of the KKT conditions is calculated:\n",
    "        residu,kktnorm,residumax = kktcheck(m,n,X,ymma,zmma,lam,xsi,eta,mu,zet,S, xmin,xmax,df0dx,fval,dfdx[:],a0,a,C,d);\n",
    "\n",
    "    \n",
    "        # update the stopping criterion \n",
    "        if stopping_criteria == \"kktnorm\"\n",
    "            stop_cond=outit < maxoutit && kktnorm>kkttol;\n",
    "        else\n",
    "            stop_cond=outit < maxoutit &&change>changetol;\n",
    "        end\n",
    "  \n",
    "        end #of the optim loop\n",
    "        #if(rem(outit, plot_rate)==0) #plot one pic every plot_rate, not activated so far to have only the interesting result at the end\n",
    "    #component plot\n",
    "    \n",
    "    longXg = length(Xg)\n",
    "    Xc=Xg[1:6:longXg];\n",
    "    Yc=Xg[2:6:longXg];\n",
    "    Lc=Xg[3:6:longXg];\n",
    "    hc=Xg[4:6:longXg];\n",
    "    Tc=Xg[5:6:longXg];\n",
    "    Mc=Xg[6:6:longXg];\n",
    "\n",
    "    C0=repeat(cos.(Tc),1,size(cc,2));\n",
    "    S0=repeat(sin.(Tc),1,size(cc,2));\n",
    "    xxx=repeat(Xc[:],1,size(cc,2))+cc;\n",
    "    yyy=repeat(Yc[:],1,size(cc,2))+ss;\n",
    "\n",
    "    xi=C0.*(xxx .-Xc)+S0.*(yyy .-Yc);\n",
    "    Eta=-S0.*(xxx .-Xc)+C0.*(yyy .-Yc);\n",
    "\n",
    "    Lrep = repeat(Lc[:],1,size(cc,2))\n",
    "    hrep = repeat(hc[:],1,size(cc,2))\n",
    "    dd=norato_bar(xi,Eta,Lrep,hrep) + norato_bar2(xi,Eta,Lrep,hrep);\n",
    "    xn=repeat(Xc,1,size(cc,2))+dd.*cc;\n",
    "    yn=repeat(Yc,1,size(cc,2))+dd.*ss;\n",
    "    tolshow=0.1;\n",
    "\n",
    "    Shown_compo=[i for i in 1:length(Mc) if Mc[i]>tolshow ]#find(Mc>tolshow);\n",
    "    \n",
    "    \"\"\"\n",
    "        if strcmp(BC,'L-shape')\n",
    "            fill([fix((min(Xx)+max(Xx))/2),max(Xx),max(Xx),fix((min(Xx)+max(Xx))/2)],[fix((min(Yy)+max(Yy))/2),...\n",
    "                fix((min(Yy)+max(Yy))/2),max(Yy),max(Yy)],'w')\n",
    "        end\n",
    "    \"\"\"\n",
    "    return outit, vvec,cvec, xPhys, xn , yn, Shown_compo,Xx[fixednodes],Yy[fixednodes]\n",
    "end\n",
    "outit, vvec,cvec, xPhys, xn,yn,Shown_compo,Xf,Yf= topggp(20,40,\"MBB\",\"GP\" ,  0.45, 50)\n",
    "\n",
    "heatmap(xPhys)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
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    "plot(1:outit,vvec,label = \"volume\", xlabel = \"iteration\")"
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  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyError",
     "evalue": "KeyError: key :max not found",
     "output_type": "error",
     "traceback": [
      "KeyError: key :max not found",
      "",
      "Stacktrace:",
      " [1] getindex(::Dict{Symbol,Any}, ::Symbol) at .\\dict.jl:477",
      " [2] default(::Symbol) at C:\\Users\\robin\\.julia\\packages\\Plots\\cc8wh\\src\\args.jl:639",
      " [3] warnOnUnsupported_args(::Plots.GRBackend, ::Plots.Attr) at C:\\Users\\robin\\.julia\\packages\\Plots\\cc8wh\\src\\args.jl:1174",
      " [4] _add_the_series(::Plots.Plot{Plots.GRBackend}, ::Plots.Subplot{Plots.GRBackend}, ::Plots.Attr) at C:\\Users\\robin\\.julia\\packages\\Plots\\cc8wh\\src\\pipeline.jl:378",
      " [5] _process_seriesrecipe(::Plots.Plot{Plots.GRBackend}, ::Plots.Attr) at C:\\Users\\robin\\.julia\\packages\\Plots\\cc8wh\\src\\pipeline.jl:407",
      " [6] _plot!(::Plots.Plot{Plots.GRBackend}, ::Dict{Symbol,Any}, ::Tuple{UnitRange{Int64},Array{Float64,1}}) at C:\\Users\\robin\\.julia\\packages\\Plots\\cc8wh\\src\\plot.jl:233",
      " [7] plot(::UnitRange{Int64}, ::Vararg{Any,N} where N; kw::Base.Iterators.Pairs{Symbol,Any,Tuple{Symbol,Symbol,Symbol},NamedTuple{(:label, :xlabel, :ymax),Tuple{String,String,Int64}}}) at C:\\Users\\robin\\.julia\\packages\\Plots\\cc8wh\\src\\plot.jl:57",
      " [8] top-level scope at In[20]:1"
     ]
    }
   ],
   "source": [
    "plot(1:outit,cvec,label = \" sensitivity\",xlabel = \"iteration\", ymax = 10)"
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   "execution_count": null,
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