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{
"cells": [
{
"cell_type": "code",
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"execution_count": 134,
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"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
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"import matplotlib.pyplot as plt"
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]
},
{
"cell_type": "code",
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"execution_count": 135,
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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",
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" n: nombre de points par cote du care => Npts points de discretisation au total\n",
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"\n",
" Returns:\n",
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" 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",
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" \"\"\"\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": 136,
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"metadata": {},
"outputs": [],
"source": [
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"def show(coordinates, u, title) -> None:\n",
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" \"\"\"Fonction d'affichage de la solution u sur le maillage defini par elements3, coordinates.\n",
"\n",
" Args:\n",
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" 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",
" 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",
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" title: le titre de la figure\n",
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"\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",
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" plt.title(title)\n",
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" plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
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"## Partie I : maillage triangulaire et conditions de Dirichlet\n",
"\n",
"$$\n",
"\\left\\{\n",
"\\begin{array}{rll}\n",
"\n",
"\\displaystyle -\\delta u (x, y) &= f(x, y) &\\text{sur } \\Omega \\\\\n",
"\\displaystyle u (x, y) &= u_d(x, y) &\\text{sur } \\partial\\Omega_d \\\\\n",
"\\displaystyle \\frac{\\partial u (x, y)}{\\partial n} &= g(x, y) &\\text{sur } \\partial\\Omega_n\n",
"\n",
"\\end{array}\n",
"\\right.\n",
"$$\n"
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]
},
{
"cell_type": "code",
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"execution_count": 137,
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"metadata": {},
"outputs": [],
"source": [
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"def f(x, y) -> np.ndarray:\n",
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" return 2 * np.pi ** 2 * np.sin(np.pi * x) * np.sin(np.pi * y)\n",
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"\n",
"\n",
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"def u_ex(x, y) -> np.ndarray:\n",
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" return np.sin(np.pi * x) * np.sin(np.pi * y)\n",
"\n",
"\n",
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"def u_d(x, y) -> np.ndarray:\n",
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" return np.zeros(x.shape[0])"
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]
},
{
"cell_type": "code",
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"execution_count": 138,
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"metadata": {},
"outputs": [
{
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"name": "stdout",
"output_type": "stream",
"text": [
"coords = [[0. 0. ]\n",
" [0.5 0. ]\n",
" [1. 0. ]\n",
" [0. 0.5]\n",
" [0.5 0.5]\n",
" [1. 0.5]\n",
" [0. 1. ]\n",
" [0.5 1. ]\n",
" [1. 1. ]]\n",
"\n",
"elems3 = [[0 1 3]\n",
" [4 3 1]\n",
" [1 2 4]\n",
" [5 4 2]\n",
" [3 4 6]\n",
" [7 6 4]\n",
" [4 5 7]\n",
" [8 7 5]]\n",
"\n",
"dirichlet = [[0]\n",
" [1]\n",
" [2]\n",
" [5]\n",
" [8]\n",
" [7]\n",
" [6]\n",
" [3]]\n",
"\n",
"neumman = []\n"
]
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}
],
"source": [
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"# affichage d'un petit maillage\n",
"coords, elems3, dirichlet, neumann = maillage_carre(3)\n",
"print(\n",
" f\"coords = {coords}\",\n",
" f\"elems3 = {elems3}\",\n",
" f\"dirichlet = {dirichlet}\",\n",
" f\"neumman = {neumann}\",\n",
" sep=\"\\n\\n\"\n",
")"
]
},
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{
"cell_type": "code",
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"execution_count": 139,
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"metadata": {},
"outputs": [],
"source": [
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"def calcul_alpha(x, y) -> float:\n",
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" \"\"\"Calcul du coefficient alpha.\n",
"\n",
" Args:\n",
" x (np.array): les coordonnées x du triangle.\n",
" y (np.array): les coordonnées y du triangle.\n",
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"\n",
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" Returns:\n",
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" alpha: le coefficient alpha.\n",
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" \"\"\"\n",
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" 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",
"\n",
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" return np.linalg.det(mat_alpha)"
]
},
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$[M^A_T]_{ij} = \\displaystyle \\int_T \\nabla \\eta_i (x, y)^\\top \\eta_j (x, y) \\ dx \\ dy$$"
]
},
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{
"cell_type": "code",
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"execution_count": 140,
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"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 1. , -0.5, -0.5],\n",
" [-0.5, 0.5, 0. ],\n",
" [-0.5, 0. , 0.5]])"
]
},
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"execution_count": 140,
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"metadata": {},
"output_type": "execute_result"
}
],
"source": [
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"def raideur(triangle) -> np.ndarray:\n",
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" \"\"\"Construction de la matrice de raideur ́elementaire relative à un ́élément triangle.\n",
"\n",
" Args:\n",
" triangle: les coordonnées x et y des trois points formant le triangle.\n",
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"\n",
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" Returns:\n",
" M: La matrice de raideur ́elementaire.\n",
" \"\"\"\n",
" M = np.zeros((3, 3))\n",
" x = triangle[:, 0]\n",
" y = triangle[:, 1]\n",
"\n",
" alpha = calcul_alpha(x, y)\n",
"\n",
" # calcul de la matrice M\n",
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" 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",
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" return M / alpha / 2\n",
"\n",
"# on affiche la première matrice de raideur pour vérifier\n",
"raideur(coords[elems3[0]])"
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]
},
{
"cell_type": "code",
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"execution_count": 141,
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"metadata": {},
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"outputs": [],
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"source": [
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"def assemblage(coordinates, elements3) -> np.ndarray:\n",
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" \"\"\"Assemblage de la matrice A dans le cas d'un maillage constitué uniquement d'éléments triangles.\n",
"\n",
" Args:\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",
"\n",
" Returns:\n",
" A: matrice nécéssaire à la résolution de la formulation variationnelle du problème.\n",
" \"\"\"\n",
" Ns = len(coordinates)\n",
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" A = np.zeros((Ns, Ns))\n",
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"\n",
" for triangle in elements3:\n",
" M = raideur(coordinates[triangle])\n",
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" for i, a in enumerate(triangle):\n",
" for j, b in enumerate(triangle):\n",
" A[a, b] += M[i, j]\n",
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" \n",
" return A"
]
},
{
"cell_type": "code",
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"execution_count": 142,
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"metadata": {},
"outputs": [],
"source": [
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"def second_membre(coordinates, elements3) -> np.ndarray:\n",
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" \"\"\"Calcul le second membre.\n",
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"\n",
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" Args:\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",
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"\n",
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" Returns:\n",
" b: vecteur b nécéssaire à la résolution de la formulation variationnelle du problème, sans les conditions de Dirichlet.\n",
" \"\"\"\n",
" Ns = len(coordinates)\n",
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" b = np.zeros(Ns)\n",
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" for triangle in elements3:\n",
" coords_triangle = coordinates[triangle]\n",
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" centre = np.mean(coords_triangle, 0)\n",
" x = coords_triangle[:, 0]\n",
" y = coords_triangle[:, 1]\n",
"\n",
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" alpha = calcul_alpha(x, y)\n",
"\n",
" # approximation pour la quadrature du second membre\n",
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" b[triangle] += alpha / 6 * f(centre[0], centre[1])\n",
"\n",
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" return b"
]
},
{
"cell_type": "code",
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"execution_count": 143,
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"metadata": {},
"outputs": [],
"source": [
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"def calcul_Ud(coords, dirichlet) -> np.ndarray:\n",
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" \"\"\"Calcul le vecteur Ud nécéssaire à l'application des conditions de Dirichlet.\n",
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"\n",
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" Args:\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",
" dirichlet: vecteur colonne des indices des sommets de la frontiere de Dirichlet.\n",
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"\n",
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" Returns:\n",
" Ud: vecteur pour appliquer les conditions de Dirichlet.\n",
" \"\"\"\n",
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" Ns = len(coords)\n",
" U = np.zeros(Ns)\n",
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"\n",
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" U[dirichlet.T] = u_d(coords[dirichlet, 0], coords[dirichlet, 1])\n",
"\n",
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" return U"
]
},
{
"cell_type": "code",
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"execution_count": 144,
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"metadata": {},
"outputs": [],
"source": [
"def tildage(A, b, coordinates, dirichlet):\n",
" \"\"\"Permet de retirer les parties de A et b soumis au conditions de Dirichlet, nécéssaire avant la résolution numérique.\n",
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"\n",
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" Args:\n",
" A: La matrice A de la résolution numérique.\n",
" b: Le vecteur b de la résolution numérique.\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",
" dirichlet: vecteur colonne des indices des sommets de la frontiere de Dirichlet.\n",
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"\n",
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" Returns:\n",
" A: La matrice A de la résolution numérique tildée.\n",
" b: Le vecteur b de la résolution numérique tildé.\n",
" coordinates: matrice a deux colonnes. Chaque ligne contient les coordonnes 2D d'un des points de la discretisation non soumis ausx conditions de Dirichlet. Ces sommets seront identifies a l'indice de la ligne correspondante dans la matrice coordinates.\n",
" \"\"\"\n",
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" A_tild = np.delete(A, dirichlet, 0)\n",
" A_tild = np.delete(A_tild, dirichlet, 1)\n",
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" \n",
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" b_tild = np.delete(b, dirichlet, 0)\n",
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" \n",
" coords_tild = np.delete(coordinates, dirichlet, 0)\n",
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"\n",
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" return A_tild, b_tild, coords_tild"
]
},
{
"cell_type": "code",
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"execution_count": 145,
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"metadata": {},
"outputs": [],
"source": [
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"def untildage(x, dirichlet, U_d) -> np.ndarray:\n",
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" \"\"\"Opération inverse de la fonction tildage, place dans le vecteur x aux coordonnées de dirichlet les valeurs des conditions\n",
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"\n",
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" Args:\n",
" x: le vecteur solution trouvé après résolution.\n",
" dirichlet: vecteur colonne des indices des sommets de la frontiere de Dirichlet.\n",
" Ud: vecteur pour appliquer les conditions de Dirichlet.\n",
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"\n",
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" Returns:\n",
" x: le vecteur solution complet, avec les conditions aux bords.\n",
" \"\"\"\n",
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" x_untild = np.zeros(U_d.shape[0])\n",
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" not_dirichlet = np.setdiff1d(range(U_d.shape[0]), dirichlet)\n",
"\n",
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" x_untild[dirichlet] = U_d[dirichlet]\n",
" x_untild[not_dirichlet] = x\n",
"\n",
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" return x_untild"
]
},
{
"cell_type": "code",
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"execution_count": 146,
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"metadata": {},
"outputs": [
{
"data": {
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"image/png": "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"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
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"image/png": "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"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
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"n = 50\n",
"coords, elems3, dirichlet, neumann = maillage_carre(n)\n",
"\n",
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"# calcul du premier membre de l'équation\n",
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"A = assemblage(coords, elems3)\n",
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"\n",
"# calcul du second membre de l'équation\n",
"b = second_membre(coords, elems3)\n",
"\n",
"# calcul du vecteur des conditions de dirichlet\n",
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"U_d = calcul_Ud(coords, dirichlet)\n",
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"\n",
"# on modifie b pour vérifier les conditions \n",
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"b -= np.dot(A, U_d)\n",
"\n",
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"# on enlève les conditions aux bords avant résolution\n",
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"A_tild, b_tild, coords_tild = tildage(A, b, coords, dirichlet)\n",
"\n",
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"# on résoud le système\n",
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"x = np.linalg.solve(A_tild, b_tild)\n",
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"\n",
"# on remet les conditions aux bords\n",
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"x_untild = untildage(x, dirichlet, U_d)\n",
"\n",
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"# on affiche le résultat\n",
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"show(coords, x_untild, \"solution calculée\")\n",
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"\n",
"# on compare avec le résultat théorique exacte\n",
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"show(coords, u_ex(coords[:, 0], coords[:, 1]), \"résultat théorique\")"
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]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
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"## Partie II : maillage mixte et ajoût des conditions de Neumann\n"
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]
},
{
"cell_type": "code",
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"execution_count": 147,
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"metadata": {},
"outputs": [],
"source": [
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"def f(x, y) -> int:\n",
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" return 1\n",
"\n",
"\n",
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"def u_d(x, y) -> int:\n",
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" return 1\n",
"\n",
"\n",
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"def g(x) -> int:\n",
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" return 1"
]
},
{
"cell_type": "code",
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"execution_count": 148,
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"metadata": {},
"outputs": [],
"source": [
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"# Création d'un maillage mixte\n",
"\n",
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"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",
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" [0, 0],\n",
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" [1 / 3, 0],\n",
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" [16 / 30, 0],\n",
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" [2 / 3, 1 / 3],\n",
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" [1, 14 / 30],\n",
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" [1, 2 / 3],\n",
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" [1, 1],\n",
" [2 / 3, 1],\n",
" [1 / 3, 1],\n",
" [0, 1],\n",
" [0, 2 / 3],\n",
" [0, 1 / 3],\n",
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" [1 / 3, 1 / 3],\n",
" [1 / 3, 2 / 3],\n",
" [2 / 3, 2 / 3],\n",
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" [1, 0],\n",
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" ]\n",
")"
]
},
{
"cell_type": "code",
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"execution_count": 149,
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"metadata": {},
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"outputs": [
{
"data": {
"text/plain": [
"array([[ 0.66666667, -0.16666667, -0.33333333, -0.16666667],\n",
" [-0.16666667, 0.66666667, -0.16666667, -0.33333333],\n",
" [-0.33333333, -0.16666667, 0.66666667, -0.16666667],\n",
" [-0.16666667, -0.33333333, -0.16666667, 0.66666667]])"
]
},
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"execution_count": 149,
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"metadata": {},
"output_type": "execute_result"
}
],
"source": [
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"def raideur_quadrangle(quadrangle) -> np.ndarray:\n",
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" \"\"\"Construction de la matrice de raideur ́elementaire relative à un ́élément quadrangle.\n",
"\n",
" Args:\n",
" quadrangle: les coordonnées x et y des quatres points formant le quadrangle.\n",
"\n",
" Returns:\n",
" M: La matrice de raideur ́elementaire.\n",
" \"\"\"\n",
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" x = quadrangle[:, 0]\n",
" y = quadrangle[:, 1]\n",
"\n",
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" # calcul de la jacobienne et de son déterminant\n",
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" J_kk = np.array([[x[1] - x[0], x[3] - x[0]], [y[1] - y[0], y[3] - y[0]]])\n",
" det_J_kk = np.linalg.det(J_kk)\n",
"\n",
" # on récupère les coefficients\n",
" coeffs = np.linalg.inv(np.matmul(J_kk.T, J_kk))\n",
" a = coeffs[0, 0]\n",
" b = coeffs[0, 1]\n",
" c = coeffs[1, 1]\n",
"\n",
" # on calcul M (on a calculé toutes les intégrales au préalable)\n",
" M = np.array(\n",
" [\n",
" [2 * a + 3 * b + 2 * c, -2 * a + c, -a - 3 * b - c, a - 2 * c],\n",
" [-2 * a + c, 2 * a - 3 * b + 2 * c, a - 2 * c, -a + 3 * b - c],\n",
" [-a - 3 * b - c, a - 2 * c, 2 * a + 3 * b + 2 * c, -2 * a + c],\n",
" [a - 2 * c, -a + 3 * b - c, -2 * a + c, 2 * a - 3 * b + 2 * c],\n",
" ]\n",
" )\n",
"\n",
" return det_J_kk / 6 * M\n",
"\n",
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"# on affiche la première matrice de raideur pour vérifier\n",
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"raideur_quadrangle(ccs[e4[0]])"
]
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},
{
"cell_type": "code",
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"execution_count": 150,
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"metadata": {},
2022-04-03 20:18:44 +00:00
"outputs": [],
2022-03-16 14:44:07 +00:00
"source": [
2022-04-03 20:42:26 +00:00
"def assemblage_quadrangle(coordinates, elements4) -> np.ndarray:\n",
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" \"\"\"Assemblage de la matrice A dans le cas d'un maillage constitué uniquement d'éléments quadrangles.\n",
"\n",
" Args:\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",
" elements4: matrice a quatre colonnes. Chaque ligne contient les indices des sommets d'un element quadrangle, dans le sens antihoraire.\n",
"\n",
" Returns:\n",
" A: matrice nécéssaire à la résolution de la formulation variationnelle du problème.\n",
" \"\"\"\n",
" Ns = len(coordinates)\n",
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" A = np.zeros((Ns, Ns))\n",
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"\n",
" for quadrangle in elements4:\n",
" M = raideur_quadrangle(coordinates[quadrangle])\n",
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" for i, a in enumerate(quadrangle):\n",
" for j, b in enumerate(quadrangle):\n",
" A[a, b] += M[i, j]\n",
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" \n",
" return A"
]
},
{
"cell_type": "code",
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"execution_count": 151,
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"metadata": {},
"outputs": [],
"source": [
2022-04-03 20:42:26 +00:00
"def second_membre_quadrangle(coordinates, elements4) -> np.ndarray:\n",
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" \"\"\"Calcul le second membre.\n",
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"\n",
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" Args:\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",
" elements4: matrice a quatre colonnes. Chaque ligne contient les indices des sommets d'un element quadrangle, dans le sens antihoraire.\n",
"\n",
" Returns:\n",
" b: vecteur b nécéssaire à la résolution de la formulation variationnelle du problème, sans les conditions de Dirichlet.\n",
" \"\"\"\n",
" Ns = len(coordinates)\n",
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" b = np.zeros(Ns)\n",
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" for quadrangle in elements4:\n",
" coords_quadrangle = coordinates[quadrangle]\n",
" centre = np.mean(coords_quadrangle, 0)\n",
" x = coords_quadrangle[:, 0]\n",
" y = coords_quadrangle[:, 1]\n",
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"\n",
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" alpha = calcul_alpha(x, y)\n",
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"\n",
" b[quadrangle] += alpha / 4 * f(centre[0], centre[1])\n",
"\n",
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" return b"
]
},
{
"cell_type": "code",
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"execution_count": 152,
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"metadata": {},
"outputs": [],
"source": [
2022-04-03 20:42:26 +00:00
"def condition_neumann(coordinates, neumann) -> np.ndarray:\n",
2022-04-03 20:18:44 +00:00
" \"\"\"Calcul le vecteur nécéssaire à l'application des conditions de Neumann.\n",
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"\n",
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" Args:\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",
" neumann: vecteur colonne des indices des sommets de la frontiere de Neumann.\n",
"\n",
" Returns:\n",
" Ud: vecteur pour appliquer les conditions de Neumann.\n",
" \"\"\"\n",
" Ns = len(coordinates)\n",
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" coeffs = np.zeros(Ns)\n",
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" for i, j in neumann:\n",
" point1 = coordinates[i]\n",
" point2 = coordinates[j]\n",
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" \n",
" valeur = np.linalg.norm(point1 - point2) / 2 * g((point1 + point2) / 2)\n",
" coeffs[i] += valeur\n",
" coeffs[j] += valeur\n",
"\n",
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" return coeffs"
]
},
{
"cell_type": "code",
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"execution_count": 153,
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"metadata": {},
"outputs": [
{
"data": {
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"image/png": "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"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# calcul de premier membre de l'équation\n",
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"A3 = assemblage(ccs, e3)\n",
"A4 = assemblage_quadrangle(ccs, e4)\n",
"A = A3 + A4\n",
"\n",
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"# calcul du second membre de l'équation\n",
"b3 = second_membre(ccs, e3)\n",
"b4 = second_membre_quadrangle(ccs, e4)\n",
"b = b3 + b4\n",
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"\n",
2022-04-03 20:18:44 +00:00
"# calcul de Ud pour les conditions de Dirichlet\n",
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"U_d = calcul_Ud(ccs, dds)\n",
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"\n",
"# modifiction de b pour vérifier Dirichlet\n",
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"b -= np.dot(A, U_d)\n",
"\n",
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"# modification de b pour vérifier Neumann\n",
"b += condition_neumann(ccs, nns)\n",
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"\n",
2022-04-03 20:18:44 +00:00
"# on enlève les conditions aux bords (Dirichlet) avant résolution\n",
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"A_tild, b_tild, ccs_tild = tildage(A, b, ccs, dds)\n",
"\n",
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"# on résoud le système\n",
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"x = np.linalg.solve(A_tild, b_tild)\n",
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"\n",
"# on remet les conditions aux bords (Dirichlet)\n",
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"x_untild = untildage(x, dds, U_d)\n",
"\n",
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"# on affiche le résultat\n",
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"show(ccs, x_untild, \"résultat numérique\")"
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]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Compléments d'analyse du système\n"
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]
2022-03-24 07:10:59 +00:00
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
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"### Analyse de l’
]
},
{
"cell_type": "code",
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"execution_count": 154,
2022-04-03 20:18:44 +00:00
"metadata": {},
"outputs": [],
"source": [
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"def f(x, y) -> np.ndarray:\n",
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" return 2 * np.pi ** 2 * np.sin(np.pi * x) * np.sin(np.pi * y)\n",
"\n",
"\n",
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"def u_ex(x, y) -> np.ndarray:\n",
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" return np.sin(np.pi * x) * np.sin(np.pi * y)\n",
"\n",
"\n",
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"def u_d(x, y) -> np.ndarray:\n",
2022-04-03 20:18:44 +00:00
" return np.zeros(x.shape[0])"
]
},
{
"cell_type": "code",
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"execution_count": 155,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"98/98\r"
]
}
],
"source": [
"erreurs = []\n",
"hs = []\n",
"range_n = range(3, 100, 5)\n",
"\n",
"for n in range_n:\n",
" print(f\"{n}/{max(range_n)}\", end=\"\\r\")\n",
" coords, elems3, dirichlet, neumann = maillage_carre(n)\n",
"\n",
" A = assemblage(coords, elems3)\n",
" b = second_membre(coords, elems3)\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",
" x_ex = u_ex(coords[:, 0], coords[:, 1])\n",
"\n",
" v = x_untild - x_ex\n",
" h = np.sqrt(1/len(v))\n",
" hs.append(h)\n",
" erreur = h * np.linalg.norm(v)\n",
" erreurs.append(erreur)"
]
},
{
"cell_type": "code",
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"execution_count": 156,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"regression linéaire: \n",
"2.078 x - 0.0363\n"
]
},
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"log_hs = np.log(hs)\n",
"log_erreurs = np.log(erreurs)\n",
"\n",
"# affichage des erreurs\n",
"plt.plot(log_hs, log_erreurs, label=\"numérique\")\n",
"\n",
"# regression linéaire\n",
"coeffs = np.polyfit(log_hs, log_erreurs, 1)\n",
"reg = np.poly1d(coeffs)\n",
"\n",
"# affichage de la regression linéaire\n",
"plt.plot(log_hs, reg(log_hs), alpha=0.5, label=\"regression\")\n",
"plt.xlabel(\"log(h)\")\n",
"plt.ylabel(\"log(erreur)\")\n",
"plt.grid()\n",
"plt.legend()\n",
"\n",
"print(f\"regression linéaire: {reg}\")"
]
},
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"On observe alors que l'on obtient un ordre de discrétisation d'environ 2."
]
},
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Résolution du système linéaire par une méthode directe\n"
]
},
{
"cell_type": "code",
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"execution_count": 157,
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"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"98/98\r"
]
}
],
"source": [
"zeros_A = []\n",
"zeros_L = []\n",
"range_n = range(3, 100, 5)\n",
"\n",
"for n in range_n:\n",
" print(f\"{n}/{max(range_n)}\", end=\"\\r\")\n",
" coords, elems3, dirichlet, neumann = maillage_carre(n)\n",
"\n",
" A = assemblage(coords, elems3)\n",
"\n",
" A_tild = np.delete(A, dirichlet, 0)\n",
" A_tild = np.delete(A_tild, dirichlet, 1)\n",
"\n",
" L = np.linalg.cholesky(A_tild)\n",
"\n",
" zeros_A.append(len(np.where(A == 0)[0]))\n",
" zeros_L.append(len(np.where(L == 0)[0]))"
]
},
{
"cell_type": "code",
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"execution_count": 158,
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"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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"<matplotlib.legend.Legend at 0x7faf9ef04850>"
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]
},
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"execution_count": 158,
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"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.plot(range_n, zeros_A, label=\"A\")\n",
"plt.plot(range_n, zeros_L, label=\"L\")\n",
"plt.xlabel(\"n\")\n",
"plt.ylabel(\"#zéros\")\n",
"plt.grid()\n",
"plt.legend()"
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]
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},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On observe que la matrice L possède moins de zeros que la matrice A, il n'est donc pas bénéfique de l'utiliser pour gagner en espace mémoire lors d'un stockage creux (surtout pour n très grand)."
]
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}
],
"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",
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"version": "3.10.4"
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}
},
"nbformat": 4,
"nbformat_minor": 4
}
