20171121_ElementsFinis2D.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ce notebook reproduit une partie du polycopié de travaux numériques de mécanique."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Recherche de la solution approchée "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On va chercher une solution analytique approchée au problème du barrage à l'aide d'un calcul symbolique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "D'abord, définissons quelques symboles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, U, V, l, h = sympy.symbols('x, y, U, V, l, h')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ceci nous permet de définir le champ de déplacement $\\xi$ :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{U y}{h}\\\\\\frac{V y}{h}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡U⋅y⎤\n",
       "⎢───⎥\n",
       "⎢ h ⎥\n",
       "⎢   ⎥\n",
       "⎢V⋅y⎥\n",
       "⎢───⎥\n",
       "⎣ h ⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = sympy.Matrix((U * y/h, V * y / h))\n",
    "\n",
    "xi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut calculer l'énergie potentielle associée à ce champ de déplacement, constituée de deux termes. Le premier est l'énergie des forces extérieures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "gamma_e, gamma = sympy.symbols('gamma_e, gamma')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La première composante vient des forces de gravité :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAXCAYAAABzjqNHAAAABHNCSVQICAgIfAhkiAAAA8VJREFU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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{V \\gamma}{6} h l$$"
      ],
      "text/plain": [
       "-V⋅γ⋅h⋅l \n",
       "─────────\n",
       "    6    "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi1 = sympy.integrate(-gamma * xi[1], (y, 0, h - x/l*h), (x, 0, l))\n",
    "phi1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La deuxième vient des forces hydrostatiques de l'eau :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = sympy.symbols('m')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAaCAYAAABW6GksAAAABHNCSVQICAgIfAhkiAAABZNJREFU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\nepIuCHRKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{U \\gamma_{e}}{6} h^{2} m^{3}$$"
      ],
      "text/plain": [
       "      2  3\n",
       "U⋅γₑ⋅h ⋅m \n",
       "──────────\n",
       "    6     "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi2 = sympy.integrate(gamma_e * (m * h - y) * xi[0], (y, 0, m*h))\n",
    "phi2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il faut maintenant calcul le travail élastique. Construisons le tenseur epsilon :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{U}{h}\\\\0 & \\frac{V}{h}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   U⎤\n",
       "⎢0  ─⎥\n",
       "⎢   h⎥\n",
       "⎢    ⎥\n",
       "⎢   V⎥\n",
       "⎢0  ─⎥\n",
       "⎣   h⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grad_xi = sympy.Matrix([[sympy.diff(xi[0], x), sympy.diff(xi[0], y)],\n",
    "                       [sympy.diff(xi[1], x), sympy.diff(xi[1], y)]])\n",
    "grad_xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\frac{U}{2 h}\\\\\\frac{U}{2 h} & \\frac{V}{h}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡      U ⎤\n",
       "⎢ 0   ───⎥\n",
       "⎢     2⋅h⎥\n",
       "⎢        ⎥\n",
       "⎢ U    V ⎥\n",
       "⎢───   ─ ⎥\n",
       "⎣2⋅h   h ⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon = 1 / sympy.S(2) * (grad_xi + grad_xi.transpose())\n",
    "epsilon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "lambda_, mu = sympy.symbols('lambda, mu')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfAAAAAaCAYAAABIBA62AAAABHNCSVQICAgIfAhkiAAACkRJREFU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\n+D/VYmyLLvjy9QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{l}{4 h} \\left(- U^{2} \\mu - V^{2} \\lambda - 2 V^{2} \\mu\\right) + \\frac{l}{2 h} \\left(U^{2} \\mu + V^{2} \\lambda + 2 V^{2} \\mu\\right)$$"
      ],
      "text/plain": [
       "  ⎛   2      2        2  ⎞     ⎛ 2      2        2  ⎞\n",
       "l⋅⎝- U ⋅μ - V ⋅λ - 2⋅V ⋅μ⎠   l⋅⎝U ⋅μ + V ⋅λ + 2⋅V ⋅μ⎠\n",
       "────────────────────────── + ────────────────────────\n",
       "           4⋅h                         2⋅h           "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W = sympy.integrate(lambda_ / sympy.S(2) * epsilon.trace()**2 + mu * (epsilon * epsilon).trace(), (y, 0, h-x/l*h), (x, 0, l))\n",
    "W"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAAaCAYAAACnz/DkAAAABHNCSVQICAgIfAhkiAAACAxJREFU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\nUEb8C8NfooNdAQGiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{l}{4 h} \\left(U^{2} \\mu + V^{2} \\lambda + 2 V^{2} \\mu\\right)$$"
      ],
      "text/plain": [
       "  ⎛ 2      2        2  ⎞\n",
       "l⋅⎝U ⋅μ + V ⋅λ + 2⋅V ⋅μ⎠\n",
       "────────────────────────\n",
       "          4⋅h           "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(W)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut maintenant minimiser en U et V la fonctionnelle suivante :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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5B/N2mUXbSJPXN25E3oxq0mo/SifmRneZz5+A6uuLVtZ6QP5Gd6tpPcC9zJ24aR3K0XvV\nWu8kW/nL1Hpl6K872vZ0nIT9s579mdOQmxv9+EB/J8kfiqI/2ndzZLpTKzIduR8T6s6Ip+nI4xub\nINvjhemP2nehrvrCa91jw2u9ZAYgvYIFMedsiEzhuAF59bAtMqd2Ok3em8jJQrJv99PqpPGHouiv\n9p2G+xqCOnkQ6VwPTTrR03bk8Y2T6PtluP6qfRfqqC+81j02vNZLRs8JmpFw3m7I64VXkUbZ45i3\n6PO0Nmn9wWNnCPIZ3rq3msvCIOSV4T/qzoin6cjjG6OQ19EeO1XXF17rHhte6xXwFaSRFf16kKc9\n8f5QDCNx+/JqXeyA3HeXLyV6+jd5fGMirdX5rIsq6wuvdY8Nr3WPx+PxeDwej8fj8Xg8Ho/H4/F4\nPB6Px+PxeDweT7vxf4UQMwCKfW4pAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{U \\gamma_{e}}{6} h^{2} m^{3} + \\frac{V \\gamma}{6} h l + \\frac{l}{4 h} \\left(- U^{2} \\mu - V^{2} \\lambda - 2 V^{2} \\mu\\right) + \\frac{l}{2 h} \\left(U^{2} \\mu + V^{2} \\lambda + 2 V^{2} \\mu\\right)$$"
      ],
      "text/plain": [
       "        2  3               ⎛   2      2        2  ⎞     ⎛ 2      2        2  ⎞\n",
       "  U⋅γₑ⋅h ⋅m    V⋅γ⋅h⋅l   l⋅⎝- U ⋅μ - V ⋅λ - 2⋅V ⋅μ⎠   l⋅⎝U ⋅μ + V ⋅λ + 2⋅V ⋅μ⎠\n",
       "- ────────── + ─────── + ────────────────────────── + ────────────────────────\n",
       "      6           6                 4⋅h                         2⋅h           "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "func = W - phi1 - phi2\n",
    "func"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On écrit qu'au point minimum, la dérivée s'annule, ce qui donne un système de deux équations à deux inconnues :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALkAAAAaCAYAAADrJf0cAAAABHNCSVQICAgIfAhkiAAAB59JREFU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\ngQIFChRYK/AvlxNHfs+t3LkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{U l \\mu}{2 h} - \\frac{\\gamma_{e} h^{2}}{6} m^{3}$$"
      ],
      "text/plain": [
       "            2  3\n",
       "U⋅l⋅μ   γₑ⋅h ⋅m \n",
       "───── - ────────\n",
       " 2⋅h       6    "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(func, U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAasAAAAXCAYAAAC1bmDXAAAABHNCSVQICAgIfAhkiAAACqFJREFU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\ngg==\n",
      "text/latex": [
       "$$\\frac{\\gamma h}{6} l + \\frac{l}{4 h} \\left(- 2 V \\lambda - 4 V \\mu\\right) + \\frac{l}{2 h} \\left(2 V \\lambda + 4 V \\mu\\right)$$"
      ],
      "text/plain": [
       "γ⋅h⋅l   l⋅(-2⋅V⋅λ - 4⋅V⋅μ)   l⋅(2⋅V⋅λ + 4⋅V⋅μ)\n",
       "───── + ────────────────── + ─────────────────\n",
       "  6            4⋅h                  2⋅h       "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(func, V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Que l'on peut résoudre :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPcAAAAqCAYAAABvLM2iAAAABHNCSVQICAgIfAhkiAAACbBJREFU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\n",
      "text/latex": [
       "$$\\left \\{ U : \\frac{\\gamma_{e} h^{3} m^{3}}{3 l \\mu}, \\quad V : - \\frac{\\gamma h^{2}}{3 \\lambda + 6 \\mu}\\right \\}$$"
      ],
      "text/plain": [
       "⎧       3  3           2  ⎫\n",
       "⎪   γₑ⋅h ⋅m        -γ⋅h   ⎪\n",
       "⎨U: ────────, V: ─────────⎬\n",
       "⎪    3⋅l⋅μ       3⋅λ + 6⋅μ⎪\n",
       "⎩                         ⎭"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = sympy.solve([sympy.diff(func, U),\n",
    "             sympy.diff(func, V)], (U, V))\n",
    "sols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut maintenant dessiner dans un cas simple la déformée :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAnCAYAAABdRFVFAAAABHNCSVQICAgIfAhkiAAAA9NJREFU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\nGjWOWv4DFnTdKfJhcUoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{\\gamma h y}{3 \\lambda + 6 \\mu}$$"
      ],
      "text/plain": [
       " -γ⋅h⋅y  \n",
       "─────────\n",
       "3⋅λ + 6⋅μ"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi[1].subs(sols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_dam_solution():\n",
    "    \"\"\"Plots the dam problem solution.\"\"\"\n",
    "    l_, h_ = 0.5, 0.3\n",
    "    gamma_e_ = 0.5\n",
    "    gamma_ = 0.1\n",
    "    mu_ = 0.2\n",
    "    m_ = 1.\n",
    "    lambda__ = 0.1\n",
    "    A = (0,  0)\n",
    "    B = (l_, 0)\n",
    "    C = (0, h_)\n",
    "    points = np.c_[A, B, C, A].T\n",
    "    plt.plot(points[:, 0], points[:, 1], '-o', label='dam (original)')\n",
    "    Cd = np.array(C)\n",
    "    Cd[0] += xi[0].subs(sols).subs({gamma_e:gamma_e_, h:h_, l:l_, mu:mu_, m: m_, y:h_})\n",
    "    Cd[1] += xi[1].subs(sols).subs({gamma_e:gamma_e_, h:h_, l:l_, mu:mu_, m: m_, y:h_, gamma: gamma_, lambda_: lambda__})\n",
    "    points2 = np.c_[A, B, Cd, A].T\n",
    "    plt.plot(points2[:, 0], points2[:, 1], '-o', label='dam (displaced)')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SpF8IiNlEJdg8LT6vA2uSuW76IEk/2fK6OTOjpyfTetTgetgj/L7fwfh1f/Ew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\nvn3sR7tHH3Mz0hF+9oe172Wq0s7MnfS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      "text/plain": [
       "<matplotlib.figure.Figure at 0xa63c940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_dam_solution()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut également manipuler la solution de manière interactive :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "from ipywidgets import interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "ba3a29a121f741d2886c953d587a7da1",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>interactive</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in Jupyter Notebook or JupyterLab, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another notebook frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "interactive(children=(FloatSlider(value=0.5, description='l_', max=1.5, min=-0.5), FloatSlider(value=0.3, description='h_', max=0.8999999999999999, min=-0.3), FloatSlider(value=0.5, description='gamma_e_', max=1.5, min=-0.5), FloatSlider(value=0.1, description='gamma_', max=0.30000000000000004, min=-0.1), FloatSlider(value=0.2, description='mu_', max=0.6000000000000001, min=-0.2), FloatSlider(value=1.0, description='m_', max=3.0, min=-1.0), FloatSlider(value=0.1, description='lambda__', max=0.30000000000000004, min=-0.1), Output()), _dom_classes=('widget-interact',))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact\n",
    "def plot_dam_solution(l_=0.5, \n",
    "                      h_=0.3,\n",
    "                      gamma_e_ = 0.5,\n",
    "                      gamma_ = 0.1,\n",
    "                      mu_ = 0.2,\n",
    "                      m_ = 1.,\n",
    "                      lambda__ = 0.1):\n",
    "    \"\"\"Plots the dam problem solution.\"\"\"\n",
    "    \n",
    "    A = (0,  0)\n",
    "    B = (l_, 0)\n",
    "    C = (0, h_)\n",
    "    points = np.c_[A, B, C, A].T\n",
    "    plt.plot(points[:, 0], points[:, 1], '-o', label='dam (original)')\n",
    "    Cd = np.array(C)\n",
    "    Cd[0] += xi[0].subs(sols).subs({gamma_e:gamma_e_, h:h_, l:l_, mu:mu_, m: m_, y:h_})\n",
    "    Cd[1] += xi[1].subs(sols).subs({gamma_e:gamma_e_, h:h_, l:l_, mu:mu_, m: m_, y:h_, gamma: gamma_, lambda_: lambda__})\n",
    "    points2 = np.c_[A, B, Cd, A].T\n",
    "    plt.plot(points2[:, 0], points2[:, 1], '-o', label='dam (displaced)')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comme noté dans le guide, on peut remarquer que la solution trouvée n'est pas compatible avec les conditions cinématiques et statiques..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Recherche de la solution approchée par éléments finis "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Todo : continuer la description pour l'élément de type 1 du maillage, puis l'élément 2 dans lequel on traite : l'énergie élastique, les efforts imposés et enfin le lagrangien pour imposer des déplacements dans la résolution du système."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}