{ "metadata": { "name": "", "signature": "sha256:9fe4e792829b1eac848064ee9ceb5fa71421055ed2a4d0ff1ef4faef63704aa4" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 5, "metadata": {}, "source": [ "TP 2 -- M\u00e9thodes Num\u00e9riques pour l\u2019Ing\u00e9nieur CM3 -- Mars 2015 " ] }, { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Equations diff\u00e9rentielles ordinaires" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Le texte de cette session de travaux pratiques est \u00e9galement disponible ici\n", "\n", "http://nbviewer.ipython.org/github/ecalzavarini/numerical-methods-at-polytech-lille/blob/master/MNI-TP2-2015.ipynb " ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Instructions pour ce TP" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pendant ce TP vous aurez \u00e0 \u00e9crire plusieurs scripts (nous vous sugg\u00e9rons de les nommer script1.py , script2.py ,...) \n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Les scripts doivent \u00eatre accompagn\u00e9s par un document descriptif unique ( README.txt ). \n", "Dans ce fichier, vous devrez d\u00e9crire le mode de fonctionnement des scripts et, si besoin, mettre vos commentaires.\n", "Merci d'y \u00e9crires aussi vos nomes et pr\u00e9noms complets.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Tous les fichiers doivent etre mis dans un dossier appel\u00e9 TP1-nom1-nom2 et ensuite \u00eatre compress\u00e9s dans un fichier TP1-nom1-nom2.tgz . \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Enfin vous allez envoyer ce fichier par email \u00e0 l'enseignant : \n", "\n", "soit Enrico (enrico.calzavarini@polytech-lille.fr) ou Stefano (stefano.berti@polytech-lille.fr)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vous avez une semaine de temps pour compl\u00e9ter le TP, c\u2019est-\u00e0-dire que la date limite pour envoyer vos travaux est 7 jours apr\u00e8s la date du TP courant." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ " Objectif" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ecrire un script Python permettant de r\u00e9soudre une \u00e9quation diff\u00e9rentielle ordinaire du premier\n", "ordre $y' = f (x, y(x))$ avec la condition initiale $y(x_0 ) = y_0$ par la m\u00e9thode :\n", "\n", "$a$) d'Euler (de la tangente) : $y_{i+1} = y_i + h\\ f(x_i,y_i)\\ $, $\\ y_0 = y_0(x_0)\\ $, $\\ i = 0 , 1, \\ldots$\n", "\n", "$b$) du point au milieu : $y_{i+1} = y_i + h\\ f(x_i + \\frac{h}{2},y_i+\\frac{h}{2}\\ f(x_i,y_i), )\\ $, $\\ y_0 = y_0(x_0)\\ $, $\\ i = 0 , 1, \\ldots$\n", "\n", "Faire en sorte que le pas d\u2019int\u00e9gration puisse \u00eatre rentr\u00e9 par l\u2019utilisateur.\n", "La fonction $f(x)$ pourra \u00eatre d\u00e9finie \u00e0 l\u2019aide d\u2019une $function$ dans le script.\n" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Programmation et validation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "R\u00e9soudre l\u2019\u00e9quation $y' = \\cos(x)$ avec $y(0) = 0$ pour $x$ variant de $0$ \u00e0 $2\\pi$. Faire les calculs en\n", "faisant varier le pas: $h = \\pi/5, \\pi/10, \\pi/20, \\ldots$ et comparer la valeur de $y$ au point $x = 3\\pi/2$ (valeur th\u00e9orique: -1). \n", "\n", "Les valeurs de $y(3\\pi/2)$ obtenues num\u00e9riquement doivent \u00eatre affich\u00e9es de fa\u00e7on claire sur l'\u00e9cran par le script et dans le compte-rendu dans un tableau.\n", "\n", "Repr\u00e9senter sur un graphique la courbe exacte $y_e = \\sin(x)$ et les solutions obtenues \u00e0 l\u2019aide des deux m\u00e9thodes (pour $x\\in \\left[ 0,2\\pi \\right]$).\n", "\n", "Formuler une conclusion en comparant les r\u00e9sultats obtenus par les deux m\u00e9thodes." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Application" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Un projectile de masse $m$ est lanc\u00e9 \u00e0 la vitesse initiale $\\vec{v}_0$\n", "et \u00e0 l'angle $\\alpha_0$ de l'horizontale (voir figure). Pendant son\n", "mouvement il est soumis \u00e0 la force de pesanteur \n", "\n", "$\\vec{P} = m \\vec{g}$\n", "\n", "et \u00e0 la r\u00e9sistance de l'air \n", "\n", "$\\vec{F}_r = \u2212k \\vec{v}$ \n", "\n", "o\u00f9 $\\vec{v} = d\\vec{x}/dt$ est le vecteur vitesse, $k$ une constante et $\\vec{x}$ le vecteur position." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import Image\n", "Image(filename='rocket.jpg')" ], "language": "python", "metadata": {}, "outputs": [ { "jpeg": "/9j/4AAQSkZJRgABAQEAlgCWAAD/4gJASUNDX1BST0ZJTEUAAQEAAAIwQURCRQIQAABtbnRyUkdC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\n", "metadata": {}, "output_type": "pyout", "prompt_number": 1, "text": [ "" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "a) En appliquant le Principe Fondamental de la Dynamique, trouver analytiquement les \u00e9quations diff\u00e9rentielles v\u00e9rifi\u00e9es par les composantes du vecteur $\\vec{x}$.\n", "\n", "b) Par un changement de variables, transformer les \u00e9quations (et les conditions initiales) obtenues en un syst\u00e8me de quatre \u00e9quations diff\u00e9rentielles d'ordre 1 de forme g\u00e9n\u00e9rale :\n", "\n", "$$y_1'(x) = f_1(x,y_1,y_2,y_3,y_4) $$\n", "$$y_2'(x) = f_2(x,y_1,y_2,y_3,y_4) $$\n", "$$y_3'(x) = f_3(x,y_1,y_2,y_3,y_4) $$\n", "$$y_4'(x) = f_4(x,y_1,y_2,y_3,y_4) $$\n", "avec les conditions initiales\n", "$$y_1(x_0)=y_{1,0} $$\n", "$$ y_2(x_0)=y_{2,0} $$\n", "$$y_3(x_0)=y_{3,0} $$\n", "$$ y_4(x_0)=y_{4,0} $$\n", "\n", "o\u00f9 l\u2019expression des fonctions $f_1$ , $f_2$ , $f_3$, $f_4$ doit \u00eatre d\u00e9termin\u00e9e.\n", "\n", "On adoptera les notations suivantes : $x$ (le temps $t$) , $y_1$ (l'abscisse $x$), $y_2$ (l'ordonn\u00e9e $z$), $y_3$ (la vitesse horizontale $v_x$) et $y_4$ (la vitesse verticale $v_z$)." ] }, { "cell_type": "heading", "level": 4, "metadata": {}, "source": [ "Question 1 :" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Trouver la trajectoire de l'objet (et les vitesses horizontale et verticale). Les valeurs de $t,x,z,v_x,v_z$ obtenues num\u00e9riquement doivent \u00eatre affich\u00e9es de fa\u00e7on claire sur l'\u00e9cran par le script et dans le compte-rendu dans un tableau. \n", "Trouver aussi le point d\u2019impact en utilisant les deux m\u00e9thodes num\u00e9riques d\u00e9finies pr\u00e9c\u00e9demment.\n", "\n", "\n", "Donn\u00e9es : \n", "\n", "la vitesse initiale $\\ v_0 = 10$ $[m/s]$,\n", "\n", "la masse $\\ m = 5$ $[Kg]$,\n", "\n", "l\u2019acc\u00e9l\u00e9ration terrestre $\\ g = 9.8$ $[m/s^2]$,\n", "\n", "l'angle $\\ \\alpha_0 = 45$ $[degr\u00e9s]$,\n", "\n", "le coefficient de r\u00e9sistance $\\ k = 0.5$ $[1/s]$,\n", "\n", "On prendra un pas d\u2019int\u00e9gration $\\ \\Delta t = 10^{-2}$ $[s]$." ] }, { "cell_type": "heading", "level": 4, "metadata": {}, "source": [ "Question 2 :" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Comparer les positions du point d\u2019impact calcul\u00e9es par les diff\u00e9rentes m\u00e9thodes pour\n", "plusieurs valeurs du pas d\u2019int\u00e9gration $\\Delta t$ (on prendra successivement les pas $\\Delta t$ suivants : $10^{-1}, 10^{-2}, 10^{-3}, \\ldots$ secondes). Les r\u00e9sultats pourront \u00eatre pr\u00e9sent\u00e9s sous forme d\u2019un tableau.\n", "\n" ] }, { "cell_type": "heading", "level": 4, "metadata": {}, "source": [ "Bonus: " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Appliquer la m\u00e9thode de Runge-Kutta d'ordre 4 ($RK4$) afin de r\u00e9soudre l\u2019\u00e9quation\n", "diff\u00e9rentielle\n", "$ y' = \u2212ry$ , avec $y(0) =1$ et $r = 8$, pour $x \\in \\left[0,2\\right]$. Repr\u00e9senter sur un graphique la solution num\u00e9rique obtenue ainsi que la solution analytique (exacte)\n", "$y_e = \\exp(\u2212rx)$. Motiver le choix du pas d\u2019int\u00e9gration $h$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.core.display import HTML\n", "def css_styling():\n", " styles = open('custom.css', 'r').read()\n", " return HTML(styles)\n", "css_styling()" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", "\n", "\n", "\n", "" ], "metadata": {}, "output_type": "pyout", "prompt_number": 2, "text": [ "" ] } ], "prompt_number": 2 } ], "metadata": {} } ] }