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{
"cells": [
{
"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"import mpl_toolkits.mplot3d\n",
"import numpy as np\n",
"import scipy.sparse as scps\n",
"import scipy.sparse.linalg as ssl\n",
"import math"
]
},
{
"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
"outputs": [],
"source": [
"def maillage_carre(n: int):\n",
" \"\"\"\n",
" Une discrétisation possible d'une EDP elliptique sur le domaine ]0,1[ x ]0,1[.\n",
" Le carre [0,1]x[0,1] est maille uniquement avec des triangles.\n",
" Les conditions limites sont de type Dirichlet uniquement -> `neumann=[]`.\n",
"\n",
" Args:\n",
" n (int): nombre de points par cote du care => Npts points de discretisation au total\n",
"\n",
" Returns:\n",
" coordinates : matrice a deux colonnes. Chaque ligne contient les coordonnes 2D d'un des points de la discretisation. Ces sommets seront identifies a l'indice de la ligne correspondante dans la matrice coordinates.\n",
" elements3 : matrice a trois colonnes. Chaque ligne contient les indices des sommets d'un element triangle, dans le sens antihoraire.\n",
" dirichlet : vecteur colonne des indices des sommets de la frontiere de Dirichlet.\n",
" neumann : matrice a deux colonnes. Chaque ligne contient les indices des deux sommets d'une arete de la frontiere de Neumann. (neumann est vide sur cet exemple)\n",
" \"\"\"\n",
"\n",
" h = 1 / (n - 1)\n",
" n_pts = n * n\n",
" n_elm = 2 * (n - 1) * (n - 1)\n",
" coordinates = np.zeros((n_pts, 2))\n",
" elements3 = np.zeros((n_elm, 3), dtype=int)\n",
" neumann = []\n",
" dirichlet = np.zeros((4 * n - 4, 1), dtype=int)\n",
"\n",
" # Coordonnees et connectivites :\n",
" e = -1\n",
" p = -1\n",
" x = np.zeros((n + 1, 1))\n",
" x[n, 0] = 1.0\n",
"\n",
" for l in range(n + 1):\n",
" x[l, 0] = l * h\n",
"\n",
" for j in range(n):\n",
" for i in range(n):\n",
" p = p + 1\n",
" coordinates[p, 0] = x[i, 0]\n",
" coordinates[p, 1] = x[j, 0]\n",
" if (i != n - 1) & (j != n - 1):\n",
" p1 = p\n",
" p2 = p1 + 1\n",
" p3 = p1 + n\n",
" p4 = p2 + n\n",
" e = e + 1\n",
" elements3[e, 0] = p1\n",
" elements3[e, 1] = p2\n",
" elements3[e, 2] = p3\n",
" e = e + 1\n",
" elements3[e, 0] = p4\n",
" elements3[e, 1] = p3\n",
" elements3[e, 2] = p2\n",
"\n",
" # Liste des sommets de la frontiere de Dirichlet:\n",
" p = -1\n",
" for j in range(n):\n",
" p = p + 1\n",
" dirichlet[p, 0] = j\n",
"\n",
" for j in range(n * 2 - 1, n * (n - 1), n):\n",
" p = p + 1\n",
" dirichlet[p, 0] = j\n",
"\n",
" for j in range(n * n - 1, n * n - n - 1, -1):\n",
" p = p + 1\n",
" dirichlet[p, 0] = j\n",
"\n",
" for j in range(n * n - 2 * n, n - 1, -n):\n",
" p = p + 1\n",
" dirichlet[p, 0] = j\n",
"\n",
" return coordinates, elements3, dirichlet, neumann\n"
]
},
{
"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
"outputs": [],
"source": [
"def show(coordinates, u) -> None:\n",
" \"\"\"Fonction d'affichage de la solution u sur le maillage defini par elements3, coordinates.\n",
"\n",
" Args:\n",
" elements3 : matrice a trois colonnes contenant les elements triangles de la discretisation, identifies par les indices de leurs trois sommets.\n",
" coordinates : matrice a deux colonnes contenant les coordonnes 2D des points de la discretisation.\n",
" u : vecteur colonne de longueur egale au nombre de lignes de coordinates contenant les valeurs de la solution a afficher aux points de la discretisation.\n",
"\n",
" Returns:\n",
" None, plots a figure\n",
" \"\"\"\n",
"\n",
" ax = plt.figure().add_subplot(projection=\"3d\")\n",
" ax.plot_trisurf(\n",
" coordinates[:, 0], coordinates[:, 1], u, linewidth=0.2, antialiased=True\n",
" )\n",
" plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Partie I : maillage triangulaire et conditions de Dirichlet**\n"
]
},
{
"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
"outputs": [],
"source": [
"def f(x, y):\n",
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" return 2 * np.pi ** 2 * np.sin(np.pi * x) * np.sin(np.pi * y)\n",
" # return np.ones(x.shape[0])\n",
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"\n",
"\n",
"def u_ex(x, y):\n",
" return np.sin(np.pi * x) * np.sin(np.pi * y)\n",
"\n",
"\n",
"def u_d(x, y):\n",
" return np.zeros(x.shape[0])\n",
"\n",
"\n",
"def g(x):\n",
" return np.cos(x)"
]
},
{
"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"n = 10\n",
"coords, elems3, dirichlet, neumann = maillage_carre(n)\n",
"show(coords, f(coords[:, 0], coords[:, 1]))\n",
"# show(coords, u_ex(coords[:, 0], coords[:, 1]))\n",
"# print(maillage_carre(3))"
]
},
{
"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
"outputs": [],
"source": [
"def raideur(triangle):\n",
" M = np.zeros((3, 3))\n",
" x = triangle[:, 0]\n",
" y = triangle[:, 1]\n",
"\n",
" # calcul de alpha\n",
" mat_alpha = np.array(\n",
" [\n",
" [x[1] - x[0], x[2] - x[0]],\n",
" [y[1] - y[0], y[2] - y[0]]\n",
" ]\n",
" )\n",
" alpha = np.linalg.det(mat_alpha)\n",
"\n",
"\n",
" for i in range(3):\n",
" grad_eta_i = np.array(\n",
" [\n",
" y[(i+1)%3] - y[(i+2)%3],\n",
" x[(i+2)%3] - x[(i+1)%3]\n",
" ]\n",
" )\n",
" for j in range(3):\n",
" grad_eta_j = np.array(\n",
" [\n",
" y[(j+1)%3] - y[(j+2)%3],\n",
" x[(j+2)%3] - x[(j+1)%3]\n",
" ]\n",
" )\n",
"\n",
" M[i, j] = np.dot(grad_eta_i, grad_eta_j)\n",
"\n",
" return M / alpha / 2"
]
},
{
"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
"outputs": [
{
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"data": {
"image/png": "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
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
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}
],
"source": [
"def assemblage(coords, elems3):\n",
" Ns = len(coords)\n",
" A = np.zeros((Ns, Ns))\n",
" for triangle in elems3:\n",
" M = raideur(coords[triangle])\n",
" for i, a in enumerate(triangle):\n",
" for j, b in enumerate(triangle):\n",
" A[a, b] += M[i, j]\n",
" return A\n",
"\n",
"\n",
"def second_membre(coords, elem3, f):\n",
" Ns = len(coords)\n",
" b = np.zeros(Ns)\n",
" for triangle in elem3:\n",
" coords_triangle = coords[triangle]\n",
" centre = np.mean(coords_triangle, 0)\n",
"\n",
" # calcul de alpha\n",
" x = coords_triangle[:, 0]\n",
" y = coords_triangle[:, 1]\n",
" mat_alpha = np.array([[x[1] - x[0], x[2] - x[0]], [y[1] - y[0], y[2] - y[0]]])\n",
" alpha = np.linalg.det(mat_alpha)\n",
"\n",
" b[triangle] += alpha / 6 * f(centre[0], centre[1])\n",
"\n",
" return b\n",
"\n",
"\n",
"def calcul_Ud(coords, dirichlet):\n",
" Ns = len(coords)\n",
" U = np.zeros(Ns)\n",
" # for d in dirichlet:\n",
" # x, y = coords[d].flatten()\n",
" # U[d] = u_d(x, y)\n",
" U[dirichlet.T] = u_d(coords[dirichlet, 0], coords[dirichlet, 1])\n",
"\n",
" return U\n",
"\n",
"\n",
"def tildage(A, b, coords, dirichlet):\n",
" A_tild = np.delete(A, dirichlet, 0)\n",
" A_tild = np.delete(A_tild, dirichlet, 1)\n",
" b_tild = np.delete(b, dirichlet, 0)\n",
" coords_tild = np.delete(coords, dirichlet, 0)\n",
"\n",
" return A_tild, b_tild, coords_tild\n",
"\n",
"\n",
"def untildage(x, dirichlet, U_d):\n",
" x_untild = np.zeros(U_d.shape[0])\n",
" not_dirichlet = np.setdiff1d(range(n*n), dirichlet)\n",
"\n",
" x_untild[dirichlet] = U_d[dirichlet]\n",
" x_untild[not_dirichlet] = x\n",
"\n",
" return x_untild\n",
"\n",
"n = 50\n",
"coords, elems3, dirichlet, neumann = maillage_carre(n)\n",
"\n",
"A = assemblage(coords, elems3)\n",
"b = second_membre(coords, elems3, f)\n",
"U_d = calcul_Ud(coords, dirichlet)\n",
"b -= np.dot(A, U_d)\n",
"\n",
"A_tild, b_tild, coords_tild = tildage(A, b, coords, dirichlet)\n",
"\n",
"x = np.linalg.solve(A_tild, b_tild)\n",
"x_untild = untildage(x, dirichlet, U_d)\n",
"\n",
"# show(coords_tild, x)\n",
"# print(coords.shape, x_untild.shape)\n",
"# print(coords, x_untild)\n",
"show(coords, x_untild)\n",
"show(coords, u_ex(coords[:, 0], coords[:, 1]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Partie II : maillage mixte et ajoût des conditions de Neumann**\n"
]
},
{
"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
"outputs": [],
"source": [
"e3 = np.array(\n",
" [[1, 2, 12], [2, 3, 12], [3, 4, 14], [4, 5, 14], [2, 15, 3], [3, 15, 4]]\n",
").astype(int)\n",
"\n",
"e4 = np.array(\n",
" [\n",
" [0, 1, 12, 11],\n",
" [11, 12, 13, 10],\n",
" [12, 3, 14, 13],\n",
" [10, 13, 8, 9],\n",
" [13, 14, 7, 8],\n",
" [14, 5, 6, 7],\n",
" ]\n",
").astype(int)\n",
"\n",
"dds = np.array([2, 15, 4, 6, 7, 8, 9, 10, 11, 0]).astype(int)\n",
"\n",
"nns = np.array([[4, 5], [5, 6], [0, 1], [1, 2]]).astype(int)\n",
"\n",
"ccs = np.array(\n",
" [\n",
" [0.0, 0.0],\n",
" [1 / 3, 0],\n",
" [0.53333333333333, 0.0],\n",
" [2 / 3, 1 / 3],\n",
" [1.0, 0.47],\n",
" [1, 2 / 3],\n",
" [1.0, 1.0],\n",
" [2 / 3, 1.0],\n",
" [1 / 3, 1.0],\n",
" [0.0, 1.0],\n",
" [0.0, 2 / 3],\n",
" [0.0, 1 / 3],\n",
" [1 / 3, 1 / 3],\n",
" [1 / 3, 2 / 3],\n",
" [2 / 3, 2 / 3],\n",
" [1.0, 0.0],\n",
" ]\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Compléments d'analyse du système**\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
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