04b_Generalized_Midpoint_Newton_Nonlinear.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "<div style=\"display: flex; align-items: center;\">\n",
    "    <img src=\"https://github.com/nagelt/Teaching_Scripts/raw/9d9e29ecca4b04eaf7397938eacbf116d37ddc93/Images/TUBAF_Logo_blau.png\" width=\"500\" height=\"auto\" height=\"auto\" style=\"margin-right: 100px;\" />\n",
    "    <div>\n",
    "        <p><strong>Prof. Dr. Thomas Nagel</strong></p>\n",
    "        <p>Chair of Soil Mechanics and Foundation Engineering<br>Geotechnical Institute<br>Technische Universität Bergakademie Freiberg.</p>\n",
    "        <p><a href=\"https://tu-freiberg.de/en/soilmechanics\">https://tu-freiberg.de/en/soilmechanics</a></p>\n",
    "    </div>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 4 - Generalized midpoint rule, Newton iterations\n",
    "\n",
    "The ODE $\\dot{y} = k (y_\\text{eq} - y)^2$ is to be integrated numerically.\n",
    "\n",
    "First, we import some libraries for numerical basics and for plotting."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "#Some plot settings\n",
    "#plt.style.use('seaborn-deep')\n",
    "%run plot_functions/plot_settings.py"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_eq = 1.\n",
    "k = 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def y_analytical(t):#assuming y0=0\n",
    "    return k*t*y_eq**2./(1.+k*t*y_eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to the ODE definition $\\dot{y}=f[y(t),t]$ we define the rate function itself:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dy_dt(y):\n",
    "    return k*(y_eq - y)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the function and its rate for some exemplary values ($y_0 = 0$) in a time interval $t \\in [0,5]$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "time = np.linspace(0,5,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(time,y_analytical(time),label=r'$y$')\n",
    "plt.plot(time,dy_dt(y_analytical(time)),label=r'$\\dot{y}$')\n",
    "plt.legend()\n",
    "plt.xlabel(r'$t$ / s')\n",
    "plt.ylabel(r'$y$, $\\dot{y}$ / s$^{-1}$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we want to get the same result with a generalized midpoint rule. It can be considered as a weighted average between an explicit forward Euler and an implicit backward Euler scheme. For the generalized $\\alpha$-scheme we find the approximation of the time-discrete rate as\n",
    "\n",
    "$$\\displaystyle \\frac{y_{n+1}- y_n}{t_{n+1} - t_n} = [(1-\\alpha) f(y_n) + \\alpha f(y_{n+1})]$$\n",
    "\n",
    "This can be re-cast in a residual form, i.e based on an approximation error:\n",
    "\n",
    "$$r(y_{n+1}) = \\displaystyle \\frac{y_{n+1}- y_n}{t_{n+1} - t_n} - [(1-\\alpha) f(y_n) + \\alpha f(y_{n+1})]$$\n",
    "\n",
    "where we would like to find the solution $y_{n+1}$ such that $r(y_{n+1}) = 0$.\n",
    "\n",
    "Due to the non-linear dependence of $r$ on the sought solution $y_{n+1}$, the equation cannot be solved directly. We thus use the concepts of a Taylor series expansion truncated after the first member to arrive at a recursive expression:\n",
    "\n",
    "$$r(y_{n+1}) \\approx r(y_n) + \\left. \\frac{\\partial r}{\\partial y} \\right|_n \\underbrace{(y_{n+1} - y_n)}_{\\displaystyle \\Delta y_{n+1}} \\overset{!}{=} 0$$\n",
    "\n",
    "Thus, with the help of the linearization we proceed from the known solution to the new one in a linear step\n",
    "\n",
    "$$\n",
    "    y_{n+1} = y_n - \\left( \\left. \\frac{\\partial r}{\\partial y} \\right|_n \\right)^{-1} r(y_n)\n",
    "$$\n",
    "\n",
    "Because this is only a linear step, we make an error (truncation error of the Taylor series expansion). Thus, we take the new guess for $y_{n+1}$ to repeat the above step recursively until we arrive at a satisfactory solution (linearization always around the newly found state):\n",
    "\n",
    "$$\n",
    "    y_{n+1}^{i+1} = y_{n+1}^{i} - \\left( \\left. \\frac{\\partial r}{\\partial y} \\right|_{i} \\right)^{-1} r(y_{n+1}^i)\n",
    "$$\n",
    "\n",
    "To perform this update, we need two incredients: a residual evaluation and a Jacobian evaluation (aka \"linear slope\")."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def residual(y_i,y_n,alpha,dt):\n",
    "    return (y_i - y_n)/dt - ((1.-alpha) * dy_dt(y_n) + alpha * dy_dt(y_i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def df_dy(y):\n",
    "    return -2.*k*(y_eq - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def jacobian(y,alpha,dt):\n",
    "    return 1./dt - alpha * df_dy(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note, that the Jacobian is constant for a given time step size (usually it is not!!). Thus, in the present case a distinction between Newton-Raphson, modified Newton, or initial stiffness methods is immaterial. In other words, an initial stiffness approach can be chosen here without any loss of convergence quality. We will demonstrate the full Newton-Raphson scheme below.\n",
    "\n",
    "The Newton update function is then as follows (returns $\\Delta y_{n+1}^{i+1} = y_{n+1}^{i+1} - y_{n+1}^i$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Newton_Update(res,Jac):\n",
    "    return -res/Jac"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, for every time step we advance the system by applying the generalized midpoint Newton-Raphson update rule. Each new time step value then serves as the initial condition for the following linearization step.\n",
    "\n",
    "We will *recursively* apply the Newton update until the residual is below a specified tolerance:\n",
    "\n",
    "$\\left| r_{n+1}^{i+1}\\right| < \\epsilon_\\text{abs}$\n",
    "\n",
    "with a suitably chosen absolute tolerance set here to $10^{-6}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def integrate(dt,alpha):\n",
    "    #Startwerte\n",
    "    t_end = 5.\n",
    "    absolute_tolerance = 1.e-6\n",
    "    max_iter = 1000\n",
    "    iteration_counter = np.array([0])\n",
    "    y = np.array([0.])\n",
    "    times = np.array([0.])\n",
    "    #\n",
    "    #Jac = jacobian(y[-1],alpha,dt)\n",
    "    while times[-1]+dt < t_end: #repeat the loop as long as the final time step is below the end point\n",
    "        times = np.append(times,times[-1]+dt) #here define the next time point as the previous time point plus the time increment dt\n",
    "        y_old = y[-1] #Starting value for recursive update\n",
    "        i = 0\n",
    "        #\n",
    "        #Jac = jacobian(y[-1],alpha,dt)\n",
    "        while True:\n",
    "            #evaluate residual\n",
    "            res = residual(y_old,y[-1],alpha,dt)\n",
    "            #if residual is below tolerance, above maximum iterations, stop iterations\n",
    "            if (np.abs(res) < absolute_tolerance or i > max_iter): \n",
    "                break\n",
    "            #evaluate Jacobian\n",
    "            Jac = jacobian(y_old,alpha,dt)\n",
    "            #perform linear step\n",
    "            y_new = y_old + Newton_Update(res,Jac)\n",
    "            #update counter\n",
    "            i += 1\n",
    "            y_old = y_new #preparation of next recursion\n",
    "\n",
    "        y = np.append(y,y_new) #append the new found solution to the solution vector\n",
    "        iteration_counter = np.append(iteration_counter,i) #store how much iterations this time step took to converge\n",
    "    return times, y, iteration_counter\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "17a58ac68c4e497a9504e274c212b286",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(BoundedFloatText(value=0.2, description='$\\\\Delta t$ / s', max=1.0, min=0.001, step=0.1)…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from ipywidgets import widgets\n",
    "from ipywidgets import interact\n",
    "\n",
    "@interact(dt=widgets.BoundedFloatText(value=0.2,min=1e-3,max=1,step=0.1,description=r'$\\Delta t$ / s'),\n",
    "          alpha=widgets.RadioButtons(options=[0,0.5,1], value=1, description=r'$\\alpha$'))\n",
    "\n",
    "def plot_CN(dt=0.1,alpha=1):\n",
    "    time_n, y_n, iter = integrate(dt,alpha)\n",
    "    fig, ax = plt.subplots(ncols=2,nrows=2,figsize=(18,12))\n",
    "    #solution\n",
    "    ax[0][0].plot(time,y_analytical(time),label=r'analytical')\n",
    "    ax[0][0].plot(time_n,y_n,label=r'numerical',marker='d',ls=':')\n",
    "    ax[0][0].legend()\n",
    "    ax[0][0].set_xlabel(r'$t$ / s')\n",
    "    ax[0][0].set_ylabel(r'$y$')\n",
    "    #error\n",
    "    ax[0][1].plot(time_n,y_n-y_analytical(time_n),ls=':',marker='d',label=r'absolute')\n",
    "    ax[0][1].plot(time_n[1:],(y_n[1:]-y_analytical(time_n[1:]))/y_analytical(time_n[1:]),\n",
    "               ls=':',marker='d',label=r'relative')\n",
    "    ax[0][1].legend()\n",
    "    ax[0][1].set_xlabel(r'$t$ / s')\n",
    "    ax[0][1].set_ylabel(r'error')\n",
    "    #iterations\n",
    "    ax[1][0].plot(iter)\n",
    "    ax[1][0].set_xlabel(r'time step')\n",
    "    ax[1][0].set_ylabel(r'number of iterations')\n",
    "    \n",
    "    plt.show()"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tasks:\n",
    "\n",
    "- Compare the number of iterations with the Picard implementation (Exercise 3)\n",
    "- Investigate the influence of the time-step size\n",
    "- Repeat the analysis for $\\alpha = 1$ (Backward Euler)\n",
    "- How does the solution change for the modified NR method and the initial stiffness method? Discuss the result in terms of accuracy and iterations required to reach the result.\n",
    "- Repeat the analysis for a function with a non-constant Jacobian (one example is in lecture notes, but you may choose any other)\n"
   ]
  }
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