tp07-EF2D-correction.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# TP 7: Éléments finis en 2D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dans ce TP, on s'intéresse à la résolution approchée du problème aux limites suivant, avec conditions aux limites de Neumann homogène :\n",
    "\n",
    "$$\n",
    "(PN)\\left\\{\\begin{aligned}\n",
    "&-\\Delta u+u=f,\\ \\textrm{dans }\\Omega,\\\\\n",
    "&\\frac{\\partial u}{\\partial n}=0, \\textrm{dans }\\partial\\Omega\n",
    "\\end{aligned}\n",
    "\\right.\n",
    "$$\n",
    "où $f:\\overline{\\Omega}\\longrightarrow\\mathbb{R}$ est une fonction de classe $\\mathcal{C}^2$ donnée et où $\\Omega$ est un ouvert de $\\mathbb{R}^2$ donné. \n",
    "\n",
    "**Q1)** Écrire la formulation variationnelle du problème $(P)$ sous la forme \n",
    "\\begin{equation*}\n",
    "(PNv)\\ \\ \\ \n",
    "\\begin{cases}\n",
    "u\\in V  \\,\\textrm{tel que}\\\\\n",
    "a(v,u)=L(v),\\ \\,\\forall\\ v\\in V ,\n",
    "\\end{cases}\n",
    "\\end{equation*}\n",
    "où $V=H^1(\\Omega)$, $a$ est une forme bilinéaire dans $V$  et $L$ une forme linéaire de $V,$ que l'on explicitera. Montrer que le problème $(PNv)$ admet une unique solution.\n",
    "\n",
    "Soit $T_h$ une triangulation du domaine $\\Omega$ et $V_h$ l'approximation de $H^1(\\Omega)$ par des éléments finis $P^1$ associés à la triangulation $T_h.$ On note $\\{T_N\\}_{N=1,\\dots,Ntri}\\ $ les triangles de $T_h,$ $\\{S_I\\}_{I=1,\\dots,NSom}\\ $ les sommets de la triangulation et $\\{\\Phi_I\\}_{I=1,\\dots,NSom}\\ $ les fonctions de la base de $V_h$ définies par $\\Phi_I(S_J)=\\delta_{IJ},\\ I,J=1,\\dots,NSom.$ \n",
    "\n",
    "Le problème discret consiste alors à chercher $u_h\\in V_h$ tel que \n",
    "$$\n",
    "(PNv_h)\\ \\ \\ \\ \\ \\ \\ \\ \\ a(\\Phi_I,u_h)=L(\\Phi_I),\\ \\,\\forall\\ I=1,\\dots,NSom. \n",
    "$$\n",
    "\n",
    "**Q2)** Soit $u_h=\\displaystyle{\\sum_{I=1}^{{NSom}}u_I\\Phi_I}.$ Montrer que $u_h$ est solution de $(PNv_h)$ si et seulement si le vecteur $U=(u_1,\\dots,u_{NSom})^T$ est solution d'un système linéaire \n",
    "$$\n",
    "KU+MU=F,\n",
    "$$\n",
    "où $K$ et $M$ sont les matrices de $\\mathcal{M}^{NSom\\times NSom}(\\mathbb{R})$ définies par $K_{I,J}=\\int_{\\Omega}\\nabla\\Phi_I\\cdot\\nabla\\Phi_J,$ $M_{I,J}=\\int_{\\Omega}\\Phi_I\\Phi_J,$ et où $F$ est le vecteur de $\\mathbb{R}^{NSom}$ défini par $F_I=\\int_{\\Omega}f\\Phi_I.$ \n",
    "\n",
    "** Maillage de $\\Omega.$ **\n",
    "\n",
    "Dans un premier temps on considère $\\Omega$ le carré $[0,2]\\times[0,2].$ \n",
    "\n",
    "La triangulation $T_h$ de $\\Omega$ est représentée par deux matrices. La première, que l'on appellera dans le programme python $TabTri,$ contient la liste des triangles. Il s'agit d'une matrice de taille $NTri\\times3$ que, dans chaque ligne $N$, contient l'indice $I$ de chaque sommet $S_I$ appartenant au triangle $T_N.$ La deuxième, que l'on appellera $TabSom,$ contient les coordonnées de chaque sommet de la triangulation. C'est une matrice de taille $NSom\\times2$ que, dans chaque ligne $I$ contient les coordonnées du sommet $S_I.$ \n",
    "Le code suivant crée le maillage et ces matrices.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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lFxaal/nJJ3JrMXCgNAGLi+Vk9aJuXTG/aZlTpkiZ+vWBww+X/5nK//Wvsm6vUGtmGZw1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\nIiEhwZNEEAkJCZ78PxPEltvx/oKtAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb6c2970c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import matplotlib.tri as tri\n",
    "import math\n",
    "import numpy.random as rd\n",
    "\n",
    "\n",
    "Nx=18\n",
    "Ny=18\n",
    "\n",
    "x_m=0.\n",
    "x_M=2.\n",
    "y_m=0.\n",
    "y_M=2.\n",
    "\n",
    "x=np.linspace(x_m,x_M,Nx+2)\n",
    "y=np.linspace(y_m,y_M,Ny+2)\n",
    "\n",
    "X,Y=np.meshgrid(x,y)\n",
    "\n",
    "X=X.flatten()\n",
    "Y=Y.flatten()\n",
    "\n",
    "triang = tri.Triangulation(X, Y)\n",
    "\n",
    "NTri=np.shape(triang.triangles)[0]\n",
    "NSom=np.shape(triang.x)[0]\n",
    "\n",
    "#Tableau avec coordonnes des noeuds\n",
    "TabSom=np.zeros([NSom,2])\n",
    "TabSom[:,0]=triang.x\n",
    "TabSom[:,1]=triang.y\n",
    "\n",
    "# Tableau avec noeuds des triangles\n",
    "TabTri=triang.triangles\n",
    "\n",
    "plt.figure(1)\n",
    "plt.gca().set_aspect('equal')\n",
    "plt.triplot(X,Y,triang.triangles, 'b-', lw=0.5)\n",
    "plt.title('maillage')\n",
    "\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** Les matrices élémentaires **\n",
    "\n",
    "Pour construire les matrices de masse M et de rigidité K, on commence par définir des matrices élémentaires qui permettent, localement sur un triangle $T_N$ de noeuds $S_{I_1},\\ S_{I_2}$ et $S_{I_3},$ de calculer \n",
    "$$\n",
    "\\int_{T_N}\\nabla\\Phi_{I_i}\\dot\\nabla\\Phi_{I_j}\\ \\ \\ \\textrm{et}\\ \\ \\ \\ \\int_{T_N}\\Phi_{I_i}\\dot\\Phi_{I_j} \n",
    "$$\n",
    "pour $i,\\ j=1,2,3.$ Ces intégrales vont contribuer respectivement à l'élément $K_{I_i,I_j}$ et $M_{I_i,I_j}$ des matrices globales.\n",
    "\n",
    "** Q3)** Construire deux fonctions $M_{elem}(\\mathcal{S1},\\mathcal{S2},\\mathcal{S3})$ et $K_{elem}(\\mathcal{S1},\\mathcal{S2},\\mathcal{S3})$ calculant les matrices de masse et de rigidité élémentaires sur un triangle $T$ de sommets $\\mathcal{S1}=(x_1,y_1),\\ \\mathcal{S2}=(x_2,y_2)$ et $\\mathcal{S3}=(x_3,y_3).$  \n",
    "Pour calculer la matrice de masse élémentaire sur le triangle $T,$ on peut utiliser les formules ci-dessous, que l'on vérifiera. \n",
    "Les fonctions de base associées à chaque sommet de $T$ sont localement données par\n",
    "$$\n",
    "\\lambda_1(x,y)=\\frac{1}{D}(y_{23}(x-x_3)-x_{23}(y-y_3)),\\ \\ \\ \\lambda_2(x,y)=\\frac{1}{D}(y_{31}(x-x_1)-x_{31}(y-y_1)),\\ \\ \\ \\lambda_3(x,y)=\\frac{1}{D}(y_{12}(x-x_2)-x_{12}(y-y_2)),\n",
    "$$\n",
    "où $x_{ij}=x_i-x_j,\\ y_{ij}=y_i-y_j$ et $D=x_{23}y_{31}-x_{31}y_{23}.$ $|D|$ est égal à deux fois la surface du triangle.\n",
    "\n",
    "Pour $k_1,\\ k_2,\\ k_3\\in\\{0,1,2\\},$ on a \n",
    "$$\n",
    "\\int_{T}\\lambda_1^{k_1}\\lambda_2^{k_2}\\lambda_3^{k_3}=2\\frac{k_1!k_2!k_3!}{(k_1+k_2+k_3+2)!}aire(T).\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "\n",
    "def M_elem(S1,S2,S3):\n",
    "    x1 = S1[0]\n",
    "    y1 = S1[1]\n",
    "    x2 = S2[0] \n",
    "    y2 = S2[1]\n",
    "    x3 = S3[0]\n",
    "    y3 = S3[1]\n",
    "    D = ((x2-x1)*(y3-y1) - (y2-y1)*(x3-x1))\n",
    "    M=(1.*np.abs(D)/24)*np.ones([3,3])\n",
    "    M[range(3),range(3)]=1.*np.abs(D)/12\n",
    "    return M\n",
    "\n",
    "def K_elem(S1,S2,S3):\n",
    "    x1 = S1[0]\n",
    "    y1 = S1[1]\n",
    "    x2 = S2[0] \n",
    "    y2 = S2[1]\n",
    "    x3 = S3[0]\n",
    "    y3 = S3[1]\n",
    "    norm = np.zeros([3, 2])\n",
    "    norm[0, :] = np.array([y2-y3, x3-x2])\n",
    "    norm[1, :] = np.array([y3-y1, x1-x3])\n",
    "    norm[2, :] = np.array([y1-y2, x2-x1])\n",
    "    D = ((x2-x1)*(y3-y1) - (y2-y1)*(x3-x1))\n",
    "    K = np.zeros([3,3])\n",
    "    for i in range(3):\n",
    "        for j in range(3):\n",
    "            K[i,j] = np.dot(norm[i,:],norm[j,:])\n",
    "    return (1./(2*abs(D)))*K\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** Assemblage des matrices $M$ et $K.$ **\n",
    "\n",
    "Dans cette partie on met en oeuvre un algorithme pour construire les matrices $M$ et $K.$ Pour ce faire on remarque que, par exemple, \n",
    "$$\n",
    "M_{I,J}=\\sum_{N=1}^{NTri}\\int_{T_N}\\Phi_I\\Phi_J=\\sum_{N : S_I,S_J\\in T_N}\\int_{T_N}\\Phi_I\\Phi_J.\n",
    "$$\n",
    "\n",
    "L'algoritme de construction des matrices consiste alors à faire la boucle suivante :\n",
    "\n",
    "    Pour N= 1...NTri\n",
    "         Détérmination des sommets S_I1, S_I2 et S_I3 du triangle T_N\n",
    "         Calcul des matrices élémentaires associées au triangle T_N\n",
    "             Pour i=1..3\n",
    "                Pour j=1...3 \n",
    "                    M(Ii,Ij)=M(Ii,Ij)+M_elem(i,j)\n",
    "\n",
    "**Q4)** Compléter dans le programme l'assemblage des matrices M et K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = np.zeros([NSom,NSom])\n",
    "M = np.zeros([NSom,NSom])\n",
    "\n",
    "for N in range(0,NTri):\n",
    "    S1=TabSom[TabTri[N,0],:]\n",
    "    S2=TabSom[TabTri[N,1],:]\n",
    "    S3=TabSom[TabTri[N,2],:]\n",
    "    Kel=K_elem(S1, S2, S3)\n",
    "    Mel=M_elem(S1, S2, S3)\n",
    "    #print Kel\n",
    "    #print Mel\n",
    "    for i in range(0,3): \n",
    "        I = TabTri[N,i]\n",
    "        for j in range(0,3): \n",
    "            J = TabTri[N,j]\n",
    "            M[I,J] = M[I,J] + Mel[i,j]\n",
    "            K[I,J] = K[I,J] + Kel[i,j]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Calcul du second membre $F.$**\n",
    "\n",
    "On peut calculer exactement les composantes $F_I$ du vecteur $F$ ou approcher ses valeurs en utilisant par exemple des formules de quadrature.\n",
    "\n",
    "Ici on fait le choix de remplacer $f$ par son interpolé $P^1$ aux points du maillage, *i.e.* on approche $f$ par la fonction $\\displaystyle{\\sum_{I=1}^{NSom}f(S_I)\\Phi_I}.$ \n",
    "\n",
    "**Q5)** En approchant $f$ par son interpolé, donner une approximation du second membre $F$ faisant intervenir la matrice de masse.  \n",
    "\n",
    "On admet que cette approximation ne change pas l'approximation par éléments finis du problème. \n",
    "\n",
    "**Validation : calcul d'une solution connue**\n",
    "\n",
    "On considère $f$ tel que la fonction\n",
    "$$\n",
    "u(x,y)=\\cos(\\pi x)\\cos(2\\pi y)\n",
    "$$\n",
    "est solution du problème (PN) dans $\\Omega=[0,2]\\times[0,2].$ \n",
    "\n",
    "**Q6)** Construire dans le programme une fonction $f(x,y)$ définissant le second membre $f$ et calculer l'approximation du vecteur $F$ obtenue comme expliqué dessus. Calculer le vecteur $U$ des coefficients de la solution approchée donnée par la méthode des élements finis et utiliser le code suivant pour la visualiser."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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93t/peMcWxvH53vr/9xy3e+zSCvkJMTpC55K2FSDgToJB+Qm2Epyb6jn5wlaCGchPsJVg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\nQcG6pRBgQUHBuqUQYEFBwbrl/wfU/4BBPWm0pQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb4c594ec>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F = np.zeros([NSom,1])\n",
    "\n",
    "def f(x,y):\n",
    "    return (1.+5*np.pi**2)*np.cos(np.pi*x)*np.cos(2*np.pi*y)\n",
    "\n",
    "def uex(x,y):\n",
    "    return np.cos(np.pi*x)*np.cos(2*np.pi*y)\n",
    " \n",
    "ff=f(TabSom[:,0],TabSom[:,1])\n",
    "   \n",
    "F=np.dot(M,ff)\n",
    "\n",
    "A=M+K\n",
    "\n",
    "u=np.linalg.solve(A,F)\n",
    "ue=uex(TabSom[:,0],TabSom[:,1])\n",
    "\n",
    "plt.figure(2)\n",
    "plt.gca().set_aspect('equal')\n",
    "#plt.tripcolor(triang, U, shading='flat')\n",
    "plt.tripcolor(triang.x,triang.y,triang.triangles, u, shading='flat')\n",
    "plt.colorbar()\n",
    "plt.title('solution approchee par EF P1')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q7)** Soit $\\Pi_h u=\\displaystyle{\\sum_I=1^{NSom} u(S_I)\\Psi_i}$ l'interpolé $P1$ de la solution exacte $u$ de $(PN)$ aux points $S_I.$ Calculer, pour différentes valeurs de $h,$ la norme $L^2$ de l'erreur $u_h-\\Pi_h u.$ Remarquer que cette intégrale peut se calculer en utilisant la matrice de masse $M.$ Evaluer l'ordre de précision de la méthode en norme $L^2.$ \n",
    "\n",
    "**Q8) Une autre structure de maillage** \n",
    "Remplacez le debut de votre programme par le code ci-dessus, pour générer un maillage non structuré du carré."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Nx=18\n",
    "Ny=18\n",
    "\n",
    "x_m=0.\n",
    "x_M=2.\n",
    "y_m=0.\n",
    "y_M=2.\n",
    "\n",
    "Rx=rd.random([Nx+2,Nx+2])\n",
    "Rx[:,0]=0\n",
    "Rx[:,-1]=0\n",
    "Ry=rd.random([Ny+2,Ny+2])\n",
    "Ry[0,:]=0\n",
    "Ry[-1,:]=0\n",
    "x=np.linspace(x_m,x_M,Nx+2)#+4*((x_M-x_m)/(5*(Nx+2)))*Rx\n",
    "y=np.linspace(y_m,y_M,Ny+2)#+4*((y_M-y_m)/(5*(Ny+2)))*Ry\n",
    "\n",
    "X,Y=np.meshgrid(x,y)\n",
    "X=X+0.9*((x_M-x_m)/((Nx+2)))*Rx\n",
    "Y=Y+0.9*((y_M-y_m)/((Ny+2)))*Ry\n",
    "print X\n",
    "print Y\n",
    "\n",
    "X=X.flatten()\n",
    "Y=Y.flatten()\n",
    "\n",
    "triang = tri.Triangulation(X, Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q9) Domaine coronne** On considère le problème (PN) dans une coronne. Pour cela remplacer le debut de votre code par le code ci-dessous, qui crée le domaine de calcul et le maillage. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def triangulation_couronne(nrad=8,nang=36):\n",
    "    radii = np.linspace(0.25, 1, nrad)\n",
    "    angles = np.linspace(0, 2*np.pi, nang, endpoint=False)\n",
    "    angles = np.repeat(angles[..., np.newaxis], nrad, axis=1)\n",
    "    angles[:, 1::2] += np.pi/nang\n",
    "    x = (radii*np.cos(angles)).flatten()\n",
    "    y = (radii*np.sin(angles)).flatten()\n",
    "\n",
    "    # On construit une triangulation de Delaunay et on filtre les triangles proches de l'origine\n",
    "    triang = tri.Triangulation(x, y)\n",
    "    xmid = x[triang.triangles].mean(axis=1)\n",
    "    ymid = y[triang.triangles].mean(axis=1)\n",
    "    mask = xmid*xmid + ymid*ymid >= .25*.25\n",
    "    T = triang.triangles[mask,:]\n",
    "    return x,y,T\n",
    "\n",
    "[x,y,T] = triangulation_couronne(8,36)\n",
    "\n",
    "\n",
    "NTri=np.shape(T)[0]\n",
    "NSom=np.shape(x)[0]\n",
    "\n",
    "TabSom=np.zeros([NSom,2])\n",
    "TabSom[:,0]=x\n",
    "TabSom[:,1]=y\n",
    "\n",
    "# noeuds des triangles\n",
    "TabTri=T\n",
    "\n",
    "# .... Le programme\n",
    "\n"
   ]
  }
 ],
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   "language": "python",
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    "name": "ipython",
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   "file_extension": ".py",
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