flyline_dynamics.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#import packages we need\n",
    "import os\n",
    "import imageio\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from numba import jit, njit\n",
    "from tqdm.notebook import tqdm\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import solve_ivp\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ignore all numbda warnings\n",
    "# The only warnings that are left are about not being able to parallelize\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "@jit(parallel=True, fastmath=True, forceobj=True)\n",
    "def fun(t, Y, npts, mxf, pbar, state):\n",
    "    last_t, dt, final_t = state\n",
    "    phi = Y[0:npts]\n",
    "    phi_dot = Y[npts:npts*2]\n",
    "\n",
    "    n = (t - last_t) // dt\n",
    "    this_t = last_t + dt * n\n",
    "    state[0] = this_t\n",
    "\n",
    "    Y_dot = dYdt(t, phi, phi_dot, npts)\n",
    "\n",
    "    if this_t == final_t:\n",
    "        if mxf:\n",
    "            df_A = pd.DataFrame(data=A[0:,0:])\n",
    "            df_A.to_excel(\".//matrices//big_a.xlsx\")\n",
    "\n",
    "            df_a = pd.DataFrame(data=a[0:,0:])\n",
    "            df_a.to_excel(\".//matrices//little_a.xlsx\")\n",
    "\n",
    "            df_Q = pd.DataFrame(data=Q[0:,0:])\n",
    "            df_Q.to_excel(\".//matrices//Q.xlsx\")\n",
    "\n",
    "            df_ainv = pd.DataFrame(data=ainv[0:,0:])\n",
    "            df_ainv.to_excel(\".//matrices//ainv.xlsx\")\n",
    "\n",
    "            df_M = pd.DataFrame(data=M[0:,0:])\n",
    "            df_M.to_excel(\".//matrices//M.xlsx\")\n",
    "            \n",
    "            df_b = pd.DataFrame(data=b[0:,0:])\n",
    "            df_b.to_excel(\".//matrices//b.xlsx\")\n",
    "\n",
    "            df_phi = pd.DataFrame(data=phi[0:])\n",
    "            df_phi.to_excel(\".//matrices//phi.xlsx\")\n",
    "\n",
    "            df_phi_dot = pd.DataFrame(data=phi_dot[0:])\n",
    "            df_phi_dot.to_excel(\".//matrices//phi_dot.xlsx\")\n",
    "\n",
    "            df_Y_dot = pd.DataFrame(data=Y_dot[0:])\n",
    "            df_Y_dot.to_excel(\".//matrices//Y_dot.xlsx\")\n",
    "\n",
    "    pbar.update(n)\n",
    "\n",
    "    return Y_dot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit(parallel=True, fastmath=True)\n",
    "def dYdt(t, phi, phi_dot, npts):\n",
    "    #get rod tip data (pos vel acc)\n",
    "    xt = get_xt(t)\n",
    "    yt = get_yt(t)\n",
    "    xt_dot = get_xt_dot(t)\n",
    "    yt_dot = get_yt_dot(t)\n",
    "    xt_ddot = get_xt_ddot(t)\n",
    "    yt_ddot = get_yt_ddot(t)\n",
    "    \n",
    "    #build little a, then invert and put into M\n",
    "    #a needs to be n/2 by n/2 np.array/matrix\n",
    "    A = build_bigA(phi, phi_dot, mass)\n",
    "    a = build_a(phi, phi_dot, A, length, mass)\n",
    "    ainv = np.linalg.inv(a)\n",
    "\n",
    "    #build RHS vector Q\n",
    "    Q = build_Q(phi, phi_dot, mass, length, radius, xt, yt, xt_dot, yt_dot, xt_ddot, yt_ddot, A)\n",
    "    \n",
    "    #matrix [I 0;0 ainv]  #see eqn 32 of wang paper\n",
    "    M = np.identity(npts*2)\n",
    "    M[npts:, npts:] = ainv\n",
    "\n",
    "    b = np.zeros((npts*2, 1))\n",
    "    b[:npts] = phi_dot.reshape((npts, 1))\n",
    "    b[npts:] = Q\n",
    "\n",
    "    Y_dot = M@b\n",
    "\n",
    "    return Y_dot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#helper routines to build RHS vectors and matrices\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def build_Q(phi, phi_dot, mass, length, radius, xt, yt, xt_dot, yt_dot, xt_ddot, yt_ddot, A):\n",
    "    n = len(phi)\n",
    "    Q = np.zeros((n, 1))\n",
    "    rhoa = 1.225 * np.pi * np.power(0.0005205,2) * 0.1 # roughly 1.225 kg/m^3\n",
    "    CS = 0.005\n",
    "\n",
    "    for i in range(n):\n",
    "        y_force = get_y_force(phi, i, yt, length)\n",
    "            \n",
    "        s = mass[i] * 0.5 * length[i] * (y_force * np.cos(phi[i]) - np.sin(phi[i]) * xt_ddot+np.cos(phi[i]) * yt_ddot)\n",
    "\n",
    "        for k in range(i, n):\n",
    "            y_force = get_y_force(phi, k, yt, length)\n",
    "\n",
    "            if k != i:\n",
    "                s += mass[k] * length[i] * (y_force * np.cos(phi[i]) - np.sin(phi[i]) * xt_ddot + np.cos(phi[i]) * yt_ddot)\n",
    "        Q[i] = -s\n",
    "\n",
    "        s = 0\n",
    "        for k in range(i, n):\n",
    "            if k != i:\n",
    "                s += A[i, k] * length[i] * length[k] * phi_dot[k] ** 2 * np.sin(phi[k] - phi[i])\n",
    "        Q[i] += s\n",
    "\n",
    "        # cos(phi[k] - phi[i]) = 1 when i = k therefore cos is not needed\n",
    "        s = get_CD(get_Re(phi, phi_dot, i, radius, length, xt_dot, yt_dot)) * rhoa * radius[i] *length[i] * get_ViD(xt_dot, yt_dot, phi, i) * \\\n",
    "            np.abs(get_ViD(xt_dot, yt_dot, phi, i)) * 0.5 * length[i]\n",
    "        \n",
    "        for k in range(i, n):\n",
    "            if k != i:              \n",
    "                s += get_CD(get_Re(phi, phi_dot, k, radius, length, xt_dot, yt_dot)) * rhoa * radius[k] * length[k] * get_ViD(xt_dot, yt_dot, phi, k) * \\\n",
    "                    np.abs(get_ViD(xt_dot, yt_dot, phi, k)) * length[i] * np.cos(phi[k] - phi[i])\n",
    "        Q[i] += -s\n",
    "\n",
    "        # sin(phi[i] - phi[k]) = 0 when i = k therefore s = 0\n",
    "        s = 0\n",
    "        for k in range(i, n):\n",
    "            if k != i:\n",
    "                s += CS * rhoa * np.pi * radius[k] * length[k] * get_ViS(xt_dot, yt_dot, phi, k) * \\\n",
    "                    np.abs(get_ViS(xt_dot, yt_dot, phi, k)) * length[i] * np.sin(phi[i] - phi[k])\n",
    "        Q[i] += -s\n",
    "    return Q\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def build_a(phi, phi_dot, A, length, mass):\n",
    "    #build little a matrix from wang paper\n",
    "    #mass is vector of segment masses\n",
    "    #L is vector of segment lengths\n",
    "    #A is big A matrix \n",
    "    n = len(phi)\n",
    "    a = np.zeros((n, n))\n",
    "    for i in range(n):\n",
    "        for j in range(n):\n",
    "            if i == j:\n",
    "                s = mass[j] * length[j] * length[j] * 0.25\n",
    "                for k in range(i,n):\n",
    "                    s+= mass[k] * length[k] * length[k]\n",
    "                a[i, j] = s\n",
    "            else:\n",
    "                a[i, j] = A[i, j] * length[i] * length[j] * np.cos(phi[i] - phi[j])\n",
    "    return a\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def build_bigA(phi, phi_dot, mass):\n",
    "    #build big A matrix from wang paper\n",
    "    #mass is vector of masses of segments\n",
    "    n = len(phi)\n",
    "    A = np.zeros((n, n))\n",
    "    for i in range(n):\n",
    "        for j in range(n):\n",
    "            if j > i:\n",
    "                s = 0\n",
    "                for k in range(j, n):\n",
    "                    s += mass[k] - 0.5 * mass[j]\n",
    "                A[i, j] = s\n",
    "            else:\n",
    "                s = 0\n",
    "                for k in range(i, n):\n",
    "                    s += mass[k] - 0.5 * mass[i]\n",
    "                A[i, j] = s\n",
    "    return A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Reynolds number, drag, and dissipation calculations\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_Re(phi, phi_dot, i, radius, length, xt_dot, yt_dot):\n",
    "    #return the Reynolds number for segment i\n",
    "    #r is the radius of the segment\n",
    "    #L is a vector of segment lengths\n",
    "    Re = 1.364e5 * radius[i] * np.sqrt(get_x_dot(phi, phi_dot, i, xt_dot, length)**2 + get_y_dot(phi, phi_dot, i, yt_dot, length)**2)\n",
    "    return Re\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_CD(Re):\n",
    "    #return drag coefficient in normal direction\n",
    "    #Re is Reynolds number\n",
    "    CD = 0.0\n",
    "    if Re < 1:\n",
    "        return 7.16\n",
    "    if Re < 34:\n",
    "        return 7.16 * Re ** -0.42\n",
    "    if Re < 1580:\n",
    "        return 3.02 * Re ** -0.165\n",
    "    else:\n",
    "        return 0.9\n",
    "    return CD\n",
    "#note CS = 0.005 constant dont need a function here\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_y_force(phi, i, yt, length):\n",
    "    y_force = 9.8 #gravity\n",
    "    ground_spring_constant = 10\n",
    "    ground_level = 0.\n",
    "\n",
    "    y_level = get_y(phi, i, yt, length)\n",
    "\n",
    "    if y_level <= ground_level:\n",
    "        y_force = y_level * ground_spring_constant\n",
    "\n",
    "    # print(round(y_level,3), np.sign(y_force))\n",
    "    return y_force"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "#routines to get positions and velocities\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_x(phi, i, xt, length):\n",
    "    #get x at position i\n",
    "    #rod tip x coordinate at xt\n",
    "    #L is vector of segment Lengths\n",
    "    s = xt\n",
    "    for j in range(i):\n",
    "        s += length[j] * np.cos(phi[j])\n",
    "    s+= length[i] * np.cos(phi[i]) * 0.5 \n",
    "    return s\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_y(phi, i, yt, length):\n",
    "    #get y at position i\n",
    "    #rod tip y coordinate at yt\n",
    "    #L is vector of segment Lengths\n",
    "    s = yt\n",
    "    for j in range(i):\n",
    "        s += length[j] * np.sin(phi[j])\n",
    "    s+= length[i] * np.sin(phi[i]) * 0.5 \n",
    "    return s\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_x_dot(phi, phi_dot, i, xt_dot, length):\n",
    "    #get xdot at position i\n",
    "    #rod tip x coordinate movign with velocity xt_dot\n",
    "    #L is vector of segment Lengths\n",
    "    s = xt_dot\n",
    "    for j in range(i):\n",
    "        s += -length[i] * np.sin(phi[j]) * phi_dot[j]\n",
    "    s+= -length[i] * 0.5 * np.sin(phi[i]) * phi_dot[i] \n",
    "    return s\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_y_dot(phi, phi_dot, i, yt_dot, length):\n",
    "    #get Y_dot at position i\n",
    "    #rod tip y coordinate moving with velocity yt_dot\n",
    "    #L is vector of segment Lengths\n",
    "    s = yt_dot\n",
    "    for j in range(i):\n",
    "        s += length[j] * np.cos(phi[j]) * phi_dot[j]\n",
    "    s+= length[i]* 0.5 * np.cos(phi[i]) * phi_dot[i] \n",
    "    return s\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_ViD(xt_dot, yt_dot, phi, i):\n",
    "    # get VD at i\n",
    "    return -xt_dot * np.sin(phi[i]) + yt_dot * np.cos(phi[i])\n",
    "\n",
    "@njit(parallel=True, fastmath=True)\n",
    "def get_ViS(xt_dot, yt_dot, phi, i):\n",
    "    # get VS at i\n",
    "    return xt_dot * np.cos(phi[i]) + yt_dot * np.sin(phi[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#rod tip and related functions \n",
    "\n",
    "### static rod for now - can replace with analytic functions or read from data file later \n",
    "\n",
    "@njit(fastmath=True)\n",
    "def get_xt(t):\n",
    "    #get rod tip x position at time t\n",
    "    A = 1\n",
    "    xt = A * t\n",
    "    return xt\n",
    "\n",
    "@njit(fastmath=True)\n",
    "def get_yt(t):\n",
    "    #get rod tip y position at time t\n",
    "    yt = 3  #3 meter high rod for now\n",
    "    return yt\n",
    "\n",
    "@njit(fastmath=True)\n",
    "def get_xt_dot(t):\n",
    "    #get rod tip x velocity at time t\n",
    "    A = 1\n",
    "    xt_dot = A\n",
    "    return xt_dot\n",
    "\n",
    "@njit(fastmath=True)\n",
    "def get_yt_dot(t):\n",
    "    #get rod tip y velocity at time t\n",
    "    yt_dot = 0\n",
    "    return yt_dot\n",
    "\n",
    "@njit(fastmath=True)\n",
    "def get_xt_ddot(t):\n",
    "    #get rod tip x acceleration at time t\n",
    "    xt_ddot = 0\n",
    "    return xt_ddot\n",
    "\n",
    "@njit(fastmath=True)\n",
    "def get_yt_ddot(t):\n",
    "    #get rod tip y acceleration at time t\n",
    "    yt_ddot = 0\n",
    "    return yt_ddot\n",
    "\n",
    "def get_time_array():\n",
    "    return False\n",
    "\n",
    "def get_max_time():\n",
    "    max_time = 0.05\n",
    "    return max_time\n",
    "\n",
    "def get_total_time_intervals():\n",
    "    total_time_intervals = 101\n",
    "    return total_time_intervals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "#line functions\n",
    "\n",
    "def get_number_of_points(df):\n",
    "    npts = len(df.index)\n",
    "    return npts\n",
    "\n",
    "def get_length_array(df):\n",
    "    #length = 0.1 * np.ones(npts)\n",
    "    length = df[\"Length\"].to_numpy()\n",
    "    return length\n",
    "\n",
    "def get_radius_array(df):\n",
    "    #radius = 0.001 * np.ones(npts)\n",
    "    radius = df[\"Radius\"].to_numpy()\n",
    "    return radius\n",
    "\n",
    "def get_mass_array(df):\n",
    "    #mass = 0.01 * np.ones(npts)\n",
    "    mass = df[\"Mass\"].to_numpy()\n",
    "    return mass\n",
    "\n",
    "def get_phi_init_array(df, npts):\n",
    "    #phi_init = np.zeros((npts, 1))\n",
    "    phi_init = df[\"Phi Init\"].to_numpy()\n",
    "    phi_init = np.reshape(phi_init, (npts, 1))\n",
    "    return phi_init\n",
    "\n",
    "def get_phi_dot_init_array(df, npts):\n",
    "    #phi_dot_init = np.zeros((npts, 1))\n",
    "    phi_dot_init = df[\"Phi Dot Init\"].to_numpy()\n",
    "    phi_dot_init = np.reshape(phi_dot_init, (npts, 1))\n",
    "    return phi_dot_init"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_xlsx_files():\n",
    "    xlsx_files = False\n",
    "    return xlsx_files\n",
    "\n",
    "def get_dataframe(location):\n",
    "    df = pd.read_csv(location)\n",
    "    return df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "#plot data\n",
    "@jit(parallel=True, forceobj=True)\n",
    "def plot_data(x, y, x_max, x_min, y_max, title):\n",
    "    #change the plot labels and size\n",
    "    fontsize = 20\n",
    "    plt.title(title, fontsize=fontsize)\n",
    "    plt.xlabel(\"Length\", fontsize=fontsize)\n",
    "    plt.ylabel(\"Height\", fontsize=fontsize)\n",
    "    plt.rcParams[\"figure.figsize\"] = (20, 15)\n",
    "    plt.xlim([x_min - 0.5, x_max + 0.5])\n",
    "    plt.ylim([-0.5, y_max + 0.5])\n",
    "    plt.xticks(fontsize=fontsize)\n",
    "    plt.yticks(fontsize=fontsize)\n",
    "    plt.plot(x, y, 'bo-')\n",
    "    plt.plot(x[0], y[0], 'ro-')\n",
    "    plt.plot([-sum(length) - 0.5, sum(length) + 0.5], [0,0], 'k-')\n",
    "    return plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_x_non_jit(phi, i, xt, length):\n",
    "    #get x at position i\n",
    "    #rod tip x coordinate at xt\n",
    "    #L is vector of segment Lengths\n",
    "    s = xt\n",
    "    for j in range(i):\n",
    "        s += length[j] * np.cos(phi[j])\n",
    "    s+= length[i] * np.cos(phi[i]) * 0.5 \n",
    "    return s\n",
    "\n",
    "def get_y_non_jit(phi, i, yt, length):\n",
    "    #get y at position i\n",
    "    #rod tip y coordinate at yt\n",
    "    #L is vector of segment Lengths\n",
    "    s = yt\n",
    "    for j in range(i):\n",
    "        s += length[j] * np.sin(phi[j])\n",
    "    s+= length[i] * np.sin(phi[i]) * 0.5 \n",
    "    return s\n",
    "\n",
    "def plot_init_data(phi, length, t=0):\n",
    "    n = len(phi) #L also has length n\\n\",\n",
    "    x = np.zeros(n)\n",
    "    y = np.zeros(n)\n",
    "    xt = get_xt(t)\n",
    "    yt = get_yt(t)\n",
    "    for i in range(n):\n",
    "        x[i] = get_x_non_jit(phi, i, xt, length)\n",
    "        y[i] = get_y_non_jit(phi, i, yt, length)\n",
    "\n",
    "    x_max = np.max(x)\n",
    "    x_min = np.min(x)\n",
    "    y_max = np.max(y)\n",
    "\n",
    "    #change the plot labels and size\n",
    "    fontsize = 20\n",
    "    plt.title(\"Initial Condition\", fontsize=fontsize)\n",
    "    plt.xlabel(\"Length\", fontsize=fontsize)\n",
    "    plt.ylabel(\"Height\", fontsize=fontsize)\n",
    "    plt.rcParams[\"figure.figsize\"] = (20, 15)\n",
    "    plt.xlim([x_min - 0.5, x_max + 0.5])\n",
    "    plt.ylim([-0.5, y_max + 0.5])\n",
    "    plt.xticks(fontsize=fontsize)\n",
    "    plt.yticks(fontsize=fontsize)\n",
    "    plt.plot(x, y, 'bo-')\n",
    "    plt.plot(x[0], y[0], 'ro-')\n",
    "    plt.plot([-sum(length) - 0.5, sum(length) + 0.5], [0,0], 'k-')\n",
    "    return plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_gif(sol, length): #Make rod tip red and the line nodes blue\n",
    "    images = []\n",
    "    x_vals = np.zeros((sol.y.shape[1], length.shape[0]))\n",
    "    y_vals = np.zeros((sol.y.shape[1], length.shape[0]))\n",
    "\n",
    "    # Create the images directory if not already created\n",
    "    try:\n",
    "        cwd = os.getcwd()\n",
    "        os.mkdir(f\"{cwd}//images\") \n",
    "    except Exception:\n",
    "        pass\n",
    "\n",
    "    for i, time, in enumerate(tqdm(sol.t, desc=\"Progress on converting phi to x and y\")):\n",
    "        phi = sol.y[0:length.shape[0], i]\n",
    "        xt = get_xt(time)\n",
    "        yt = get_yt(time)\n",
    "        for j in range(sol.y.shape[0]//2):\n",
    "            x_vals[i][j] = get_x(phi, j, xt, length)\n",
    "            y_vals[i][j] = get_y(phi, j, yt, length)\n",
    "\n",
    "    for i, time, in enumerate(tqdm(sol.t, desc=\"Progress on creating gif\")):\n",
    "        plot_data(x_vals[i], y_vals[i], np.max(x_vals), np.min(x_vals), np.max(y_vals), \"Time: {} second(s)\".format(round(time, 5)))\n",
    "        filepath = r\".//images//Flyline at {} seconds.png\".format(round(time, 5))\n",
    "        plt.savefig(filepath)\n",
    "        images.append(imageio.imread(filepath))\n",
    "        plt.clf()\n",
    "\n",
    "    imageio.mimsave('Flyline.gif', images, format=\"GIF\", duration=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "#main driver routine -- \n",
    "##--- set up parameter values/initial conditions and call odeint\n",
    "\n",
    "#parameter values\n",
    "max_time = get_max_time()\n",
    "tti = get_total_time_intervals()\n",
    "mxf = make_xlsx_files()\n",
    "df_flyline = get_dataframe(r\"Flyline.csv\")\n",
    "npts = get_number_of_points(df_flyline) #number of points along fly line\n",
    "t_span = [0, max_time]\n",
    "t = np.linspace(*t_span, tti) #simulate one second in time in steps of 0.01\n",
    "\n",
    "length = get_length_array(df_flyline) \n",
    "radius = get_radius_array(df_flyline)\n",
    "mass = get_mass_array(df_flyline)\n",
    "\n",
    "#inital conditions\n",
    "##rod tip\n",
    "xt = get_xt(0) \n",
    "yt = get_yt(0)\n",
    "##initial angles/velocities\n",
    "phi_init = get_phi_init_array(df_flyline, npts)\n",
    "phi_dot_init = get_phi_dot_init_array(df_flyline, npts)\n",
    "##initial solution vector\n",
    "Yinit = np.append(phi_init, phi_dot_init)\n",
    "\n",
    "#plot initial condition\n",
    "plot_init_data(phi_init, length)\n",
    "\n",
    "if mxf:\n",
    "    try:\n",
    "        cwd = os.getcwd()\n",
    "        os.mkdir(f\"{cwd}//matrices\") \n",
    "    except Exception:\n",
    "        pass\n",
    "\n",
    "#do stuff with solution - plot, analyze, etc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "Progress on ODE:   0%|          | 0/1000 [00:00<?, ?‰/s]",
      "application/vnd.jupyter.widget-view+json": {
       "version_major": 2,
       "version_minor": 0,
       "model_id": "ac48518754dc4b72a5a82a2904b5aa6c"
      }
     },
     "metadata": {}
    }
   ],
   "source": [
    "percent = 1000\n",
    "\n",
    "state = [0, max_time/percent, max_time]\n",
    "\n",
    "#call solve_ivp\n",
    "with tqdm(total=percent, unit=\"‰\", desc=\"Progress on ODE\") as pbar:\n",
    "    sol = solve_ivp(\n",
    "        fun, \n",
    "        t_span, \n",
    "        Yinit, \n",
    "        t_eval=t, \n",
    "        method='LSODA', \n",
    "        args=[npts, mxf, pbar, state]\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "Progress on converting phi to x and y.:   0%|          | 0/101 [00:00<?, ?it/s]",
      "application/vnd.jupyter.widget-view+json": {
       "version_major": 2,
       "version_minor": 0,
       "model_id": "dfa1c98ba1ce4a49a8f3cc27750a49c7"
      }
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "Progress on creating gif:   0%|          | 0/101 [00:00<?, ?it/s]",
      "application/vnd.jupyter.widget-view+json": {
       "version_major": 2,
       "version_minor": 0,
       "model_id": "cbf3406a18884b3d8d28321e1010bfa9"
      }
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 1440x1080 with 0 Axes>"
     },
     "metadata": {}
    }
   ],
   "source": [
    "make_gif(sol, length)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
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