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update exercise 2: enhance heat equation simulation in Jupyter notebook, improve matrix construction and add temperature profile visualization
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Félix MARQUET
@@ -1,18 +1,83 @@
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# Eq en régime permanent
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# d2T / dx2 = 0
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# T(x =0) = T0
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# T(x = 1) = T1
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# T(i)i E [0, N+1]
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# x(i+1) - x(i) = deltaX = 1/N
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# d2T/dx2 = (T(i-1) - 2*T(i) + T(i+1))/deltaX**2
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import numpy as np
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import matplotlib.pyplot as plt
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import matplotlib.cm as cm
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# Question 1: Résoudre cette équation pour les conditions aux bord
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# d2T / dx2 => dT / dx = A => T(x) = Ax + b
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# Ici b = T0 d'où T(x) = Ax + T0
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# T1 - T0 / 1 - 0 = A = T1 - T0 donc T(x) = (T1 - T0)x + T0
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# Question 2: Ecrire l'équation de la chaleur discrétisée pour i E [1, N-2], puis pour i = 0 et i = N
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# d2T/dx2 = 0
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# T(i+1) - 2*T(i) + T(i-1) = 0
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# Pour i = 0
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# -2T0 + T(1) = -T0-
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# Pour i = N
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# Constantes et paramètres
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N = 1000
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delta_x = 1 / N
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delta_x2 = delta_x**2
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D = 1
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delta_t = 0.9 * (delta_x2 / 2 * D)
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T0 = 0
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T1 = 0
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if delta_t < (delta_x2 / 2 * D):
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M = np.zeros((N + 1, N + 1))
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# Remplissage de la matrice M
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for i in range(N + 1):
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if i == 0:
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M[i, i] = 1 - 2 * delta_t * D / delta_x2
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M[i, i + 1] = delta_t * D / delta_x2
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elif i == N:
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M[i, i - 1] = delta_t * D / delta_x2
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M[i, i] = 1 - 2 * delta_t * D / delta_x2
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else:
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M[i, i - 1] = delta_t * D / delta_x2
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M[i, i] = 1 - 2 * delta_t * D / delta_x2
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M[i, i + 1] = delta_t * D / delta_x2
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# Vecteur spatial pour tracer
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x = np.linspace(0, 1, N + 1)
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# Condition initiale T₀(x) = sin(πx)
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T = np.sin(np.pi * x)
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T[0] = T0
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T[N] = T1
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# Temps de simulation
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temps_final = 0.1
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nb_iterations = int(temps_final / delta_t)
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# Nombre de courbes souhaité
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nb_courbes = 100000
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# Sauvegarder plus de profils à différents temps
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profils = [T.copy()]
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iterations_sauvegarde = np.linspace(0, nb_iterations, nb_courbes, dtype=int)
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temps_sauvegarde = [0] + [i * delta_t for i in iterations_sauvegarde[1:]]
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# Simulation de l'évolution temporelle
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for n in range(nb_iterations):
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T = M @ T
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if n in iterations_sauvegarde:
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profils.append(T.copy())
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# Tracer les profils de température avec un dégradé de couleurs
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plt.figure(figsize=(12, 8))
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colormap = cm.viridis
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# Affichage des courbes avec un dégradé de couleurs
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for i, t in enumerate(temps_sauvegarde):
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if i < len(profils):
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couleur = colormap(i / len(temps_sauvegarde))
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plt.plot(x, profils[i], color=couleur, alpha=0.7)
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# Afficher seulement quelques légendes pour éviter l'encombrement
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indices_legende = np.linspace(0, len(temps_sauvegarde) - 1, 10, dtype=int)
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for i in indices_legende:
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if i < len(profils):
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couleur = colormap(i / len(temps_sauvegarde))
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plt.plot([], [], color=couleur, label=f"t = {temps_sauvegarde[i]:.4f}s")
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plt.xlabel("Position x")
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plt.ylabel("Température T")
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plt.title("Évolution de la température au cours du temps (50 courbes)")
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plt.legend(loc="upper right")
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plt.grid(True)
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plt.colorbar(plt.cm.ScalarMappable(cmap=colormap), label="Temps (s)", ax=plt.gca())
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plt.show()
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else:
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print("Constante incorrect")
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print(f"delta_t = {delta_t}")
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print(f"(delta_x2 / 2 * D) = {(delta_x2 / 2 * D)}")
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@@ -250,22 +250,30 @@
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]
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},
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{
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"metadata": {},
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"metadata": {
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"jupyter": {
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"is_executing": true
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},
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"ExecuteTime": {
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"start_time": "2025-04-01T09:53:17.236001Z"
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}
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},
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"cell_type": "code",
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import matplotlib.cm as cm\n",
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"\n",
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"# Constantes et paramètres\n",
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"N = 10\n",
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"delta_t = (delta_x2 / 2 * D)\n",
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"delta_x2 = 1/N\n",
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"N = 1000\n",
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"delta_x = 1/N\n",
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"delta_x2 = delta_x**2\n",
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"D = 1\n",
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"delta_t = 0.9 * (delta_x2 / 2 * D)\n",
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"T0 = 0\n",
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"T1 = 0\n",
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"\n",
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"if delta_t < (delta_x2 / 2 * D):\n",
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"\n",
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" M = np.zeros((N + 1, N + 1))\n",
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"\n",
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" # Remplissage de la matrice M\n",
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@@ -281,9 +289,6 @@
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" M[i, i] = (1 - 2 * delta_t * D / delta_x2)\n",
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" M[i, i + 1] = delta_t * D / delta_x2\n",
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"\n",
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" # Affichage de la matrice M\n",
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" print(f\"Matrice M: {M}\")\n",
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"\n",
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" # Vecteur spatial pour tracer\n",
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" x = np.linspace(0, 1, N+1)\n",
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"\n",
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@@ -296,30 +301,44 @@
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" temps_final = 0.1\n",
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" nb_iterations = int(temps_final / delta_t)\n",
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"\n",
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" # Sauvegarder quelques profils à différents temps\n",
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" # Nombre de courbes souhaité\n",
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" nb_courbes = 100000\n",
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"\n",
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" # Sauvegarder plus de profils à différents temps\n",
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" profils = [T.copy()]\n",
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" temps_sauvegarde = [0, 0.01, 0.02, 0.05, 0.1]\n",
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" indices_sauvegarde = [int(t/delta_t) for t in temps_sauvegarde[1:]]\n",
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" iterations_sauvegarde = np.linspace(0, nb_iterations, nb_courbes, dtype=int)\n",
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" temps_sauvegarde = [0] + [i * delta_t for i in iterations_sauvegarde[1:]]\n",
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"\n",
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" # Simulation de l'évolution temporelle\n",
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" for n in range(nb_iterations):\n",
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" T = M @ T\n",
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" if n+1 in indices_sauvegarde:\n",
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" if n in iterations_sauvegarde:\n",
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" profils.append(T.copy())\n",
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"\n",
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" print(f\"Solution profils: {profils}\")\n",
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" # Tracer les profils de température avec un dégradé de couleurs\n",
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" plt.figure(figsize=(12, 8))\n",
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" colormap = cm.viridis\n",
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"\n",
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" # Tracer les profils de température\n",
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" plt.figure(figsize=(10, 6))\n",
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" # Affichage des courbes avec un dégradé de couleurs\n",
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" for i, t in enumerate(temps_sauvegarde):\n",
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" if i < len(profils):\n",
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" plt.plot(x, profils[i], label=f't = {t}s')\n",
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" couleur = colormap(i / len(temps_sauvegarde))\n",
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" plt.plot(x, profils[i], color=couleur, alpha=0.7)\n",
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"\n",
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" # Afficher seulement quelques légendes pour éviter l'encombrement\n",
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" indices_legende = np.linspace(0, len(temps_sauvegarde)-1, 10, dtype=int)\n",
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" for i in indices_legende:\n",
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" if i < len(profils):\n",
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" couleur = colormap(i / len(temps_sauvegarde))\n",
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" plt.plot([], [], color=couleur, label=f't = {temps_sauvegarde[i]:.4f}s')\n",
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"\n",
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" plt.xlabel('Position x')\n",
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" plt.ylabel('Température T')\n",
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" plt.title('Évolution de la température au cours du temps')\n",
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" plt.legend()\n",
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" plt.title('Évolution de la température au cours du temps (50 courbes)')\n",
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" plt.legend(loc='upper right')\n",
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" plt.grid(True)\n",
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" plt.colorbar(plt.cm.ScalarMappable(cmap=colormap),\n",
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" label='Temps (s)', ax=plt.gca())\n",
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" plt.show()\n",
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"else:\n",
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" print(\"Constante incorrect\")\n",
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