run_ideal_gas.ipynb

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    "`Cortix` 2019 **07Aug2019**\n",
    "\n",
    "# Ideal Gas Example\n",
    "* This is part of the [Cortix](https://cortix.org)-on-[Jupyter-Notebook](https://github.com/dpploy/cortix-nb) examples.\n",
    "* You must be in a Jupyter Notebook server to run this notebook.\n",
    "* Select each of the cells below and run them sequentially (use the run button, `>|` on the tool bar or use the `Cell` option on the menu bar).\n",
    "* Alternatively, on the menu bar run all cells: `Cell -> Run All`.\n",
    "\n",
    "$  \n",
    "  \\newcommand{\\Amtrx}{\\boldsymbol{\\mathsf{A}}}\n",
    "  \\newcommand{\\Bmtrx}{\\boldsymbol{\\mathsf{B}}}\n",
    "  \\newcommand{\\Smtrx}{\\boldsymbol{\\mathsf{S}}}\n",
    "  \\newcommand{\\xvec}{\\boldsymbol{\\mathsf{x}}}\n",
    "  \\newcommand{\\vvar}{\\boldsymbol{v}}\n",
    "  \\newcommand{\\fvar}{\\boldsymbol{f}}\n",
    "  \\newcommand{\\Power}{\\mathcal{P}}\n",
    "  \\newcommand{\\bm}[1]{{\\boldsymbol{#1}}}\n",
    "$"
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    "---\n",
    "## Table of Contents\n",
    "* [Introduction](#intro)\n",
    "  - [Elatstic collision model](#collisionmodel)\n",
    "  - [Collision detection model](#collisiondetection)\n",
    "* [Call the Run File](#collisiondetection)\n",
    "* [Display the Animated Results](#net)\n",
    "* [View Kinetic Energy Vs Time](#run)\n",
    "* [Simulation with New Parameters and Geometry](#new)\n",
    "\n",
    "\n",
    "---"
   ]
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   "source": [
    "## Introduction<a id=\"intro\"></a>"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "This Cortix use-case simulates the random motion of particles which simulates the behavior of ideal gas."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Elastic Collision model<a id=\"collisionmodel\"></a>"
   ]
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    "The equations for resultant velocity after a 2D elastic collision between two circular objects can be represented as:\n",
    "    \\begin{equation*}\n",
    "        \\bm{v'}_{1x} = \\frac{\\text{v}_{1} cos(\\theta_1-\\phi){(m_1-m_2)}+2 \\text{m}_2 v_2 cos(\\theta_2-\\phi)}{m_1+m_2} cos(\\phi) + v_1 sin(\\theta_1 - \\phi) cos(\\phi + \\frac{\\pi}{2})\n",
    "    \\end{equation*}\n",
    "    \\begin{equation*}\n",
    "        \\bm{v'}_{1y} = \\frac{\\text{v}_{1} cos(\\theta_1-\\phi){(m_1-m_2)}+2 \\text{m}_2 v_2 cos(\\theta_2-\\phi)}{m_1+m_2} sin(\\phi) + v_1 sin(\\theta_1 - \\phi) sin(\\phi + \\frac{\\pi}{2})\n",
    "    \\end{equation*}\n",
    "Where: $\\theta_1$ and $\\theta_2$ are the movement angles of the object, $v_1$ and $v_2$ are the velocities of the objects, $m_1$ and $m_2$ are the object masses, and $\\phi$ is the contact angle.\n",
    "\n",
    "Upon contact with another particle, the velocity is shifted to a new coordinate system with components in the normal and transverse directions. The collision can then be seperated into 2 1-Dimensional collision problems, then the resultant velocity vector is shifted back to x-y coordinates. "
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   "source": [
    "### Collision Detection<a id=\"collisiondetection\"></a>"
   ]
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   "metadata": {},
   "source": [
    "Each Particle object stores its own identifying information as class attributes. For each timestep, a Particle's trajectory is calculated independently. If a collision event is triggered, then the Particle responds with the appropriate function.\n"
   ]
  },
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   "source": [
    "## Call the Run File and Modify Simulation Parameters <a id=\"runfile\"></a>"
   ]
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  {
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   "source": [
    "# Import various packages; must have the Cortix repository installed\n",
    "\n",
    "try:\n",
    "    import cortix\n",
    "except ImportError:\n",
    "    print('Installing the \"cortix\" package...\\n')\n",
    "    !pip install cortix\n",
    "    import cortix\n",
    "\n",
    "# Import the example modules\n",
    "from cortix.examples import Run_Particle"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "# Create a Cortix object with Python multiprocessing\n",
    "\n",
    "simulation = Run_Particle()\n",
    "simulation.fps = 100\n",
    "simulation.runtime = 6\n",
    "simulation.procs = 2\n",
    "simulation.n_list = 5\n",
    "simulation.run()"
   ]
  },
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   "metadata": {},
   "source": [
    "## Display the Animated Results <a id=\"animation\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
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   "source": [
    "from IPython.display import HTML\n",
    "\n",
    "HTML(\"\"\"\n",
    "<video width=\"600\" height=\"600\" controls>\n",
    "  <source src=\"particle_animation.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## View Kinetic Energy Vs Time<a id=\"run\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.image as mpimg\n",
    "import matplotlib as mpl\n",
    "mpl.rcParams['figure.dpi']= 1500\n",
    "%matplotlib inline\n",
    "\n",
    "fig = plt.figure(figsize=(12,12))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.axis('off')\n",
    "img = mpimg.imread('energy_vs_time.png')\n",
    "ax.imshow(img)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulation with New Parameters and Geometry <a id=\"new\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "simulation = Run_Particle()\n",
    "simulation.fps = 100\n",
    "simulation.runtime = 10\n",
    "simulation.procs = 7\n",
    "simulation.n_list = 15\n",
    "simulation.a=(0,-32.2)\n",
    "simulation.cor = 1\n",
    "simulation.r = 0.5\n",
    "simulation.shape = [(0, 0), (0, 30), (30, 30),(30,25),\n",
    "             (65,25),(65,-20),(85,-20),(85,-40),\n",
    "             (45,-40),(45,-20),(49,-20),(49,15),\n",
    "             (30,15),(30,0),(0,0)]\n",
    "simulation.run()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import HTML\n",
    "\n",
    "HTML(\"\"\"\n",
    "<video width=\"600\" height=\"600\" controls>\n",
    "  <source src=\"particle_animation.mp4\" type=\"video/mp4\">\n",
    "</video>\n",
    "\"\"\")\n",
    "\n"
   ]
  }
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