Circular geometry.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.ma as ma\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Coordinate matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# number of nodes\n",
    "nnos = 101\n",
    "# number of elements\n",
    "nel  = nnos - 2\n",
    "\n",
    "# coordinate matrix pre-allocation\n",
    "coord = np.zeros((nnos, 3))  \n",
    "coord[0][0] = 1\n",
    "\n",
    "# theta value\n",
    "theta = (np.pi/2)/(nnos-2)\n",
    "\n",
    "for i in range(0, nnos-1):\n",
    "    coord[i + 1][0] = i + 2             # Node reference  \n",
    "    coord[i + 1][1] = np.cos(theta*i)   # X coordinate\n",
    "    coord[i + 1][2] = np.sin(theta*i)   # Y coordinate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plot "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y')"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "plt.xlim([0,1])\n",
    "plt.ylim([0,1])\n",
    "plt.scatter(coord[:,1], coord[:,2], marker = 'o', c = 'black', alpha = 0.85)\n",
    "plt.xlabel('x', fontsize = 'xx-large')\n",
    "plt.ylabel('y', fontsize = 'xx-large')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.patches.Polygon at 0x7f7eb46cb990>,\n",
       " <matplotlib.patches.Polygon at 0x7f7eb55d7190>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "plt.xlim([0,1])\n",
    "plt.ylim([0,1])\n",
    "ax.fill(coord[:,1], coord[:,2],0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Incidence matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# incidence matrix pre-allocation\n",
    "inci = np.zeros((nel, 3))  \n",
    "\n",
    "for i in range(0,nel):\n",
    "    # central node\n",
    "    inci[i][0] = 1\n",
    "    # first node\n",
    "    inci[i][1] = i + 2\n",
    "    # second node\n",
    "    inci[i][2] = i + 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Area"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error =  0.00419577941351902 %\n"
     ]
    }
   ],
   "source": [
    "# somatory\n",
    "A = 0 \n",
    "\n",
    "# central node coordinates\n",
    "x1 = coord[int(inci[0,0] - 1), 1]\n",
    "y1 = coord[int(inci[0,0] - 1), 2]\n",
    "\n",
    "f = np.zeros(nnos-1)\n",
    "\n",
    "for i in range(0,nel):\n",
    "    #     second node coordinates\n",
    "    x2 = coord[int(inci[0,1] - 1), 1]\n",
    "    y2 = coord[int(inci[0,1] - 1), 2]\n",
    "    \n",
    "    #     third node coordinates\n",
    "    x3 = coord[int(inci[0,2] - 1), 1]\n",
    "    y3 = coord[int(inci[0,2] - 1), 2]\n",
    "    \n",
    "    a = 0.5*((x2 - x1)*(y3-y1) - (x3 - x1)*(y2 - y1))\n",
    "    \n",
    "    f[i] = a\n",
    "    \n",
    "    #     area summation\n",
    "    A = A + a\n",
    "    \n",
    "# relative error\n",
    "error_Area = (np.pi/4 - A)/(np.pi/4)*100\n",
    "# display('Error = ',error_Area)\n",
    "\n",
    "print('Error = ', error_Area,'%')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Centroid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error centroid X =  -57.06974669011344 %\n",
      "Error centroid Y =  98.75389011436114 %\n"
     ]
    }
   ],
   "source": [
    "centx = 0\n",
    "centy = 0\n",
    "\n",
    "for i in range(0,nel):\n",
    "    #     second node coordinates\n",
    "    x2 = coord[int(inci[0,1] - 1), 1]\n",
    "    y2 = coord[int(inci[0,1] - 1), 2]\n",
    "    \n",
    "    #     third node coordinates\n",
    "    x3 = coord[int(inci[0,2] - 1), 1]\n",
    "    y3 = coord[int(inci[0,2] - 1), 2]   \n",
    "    \n",
    "    centx = centx + f[i]*((x1+x2+x3)/3)\n",
    "    centy = centy + f[i]*((y1+y2+y3)/3)\n",
    "    \n",
    "# centroid X\n",
    "X = centx/A\n",
    "\n",
    "# centroid Y\n",
    "Y = centy/A\n",
    "\n",
    "# exact centroid \n",
    "exact_cent = 4/(np.pi*3)\n",
    "\n",
    "# error X\n",
    "error_x = (exact_cent - X)/exact_cent*100\n",
    "# error Y\n",
    "error_y = (exact_cent - Y)/exact_cent*100\n",
    "\n",
    "print('Error centroid X = ', error_x,'%')\n",
    "print('Error centroid Y = ', error_y,'%')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Inertia moment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ix = "
   ]
  }
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