elements/FEM_4Nodes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analytic integration of 4-nodes Finite Element\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.2 (Python 3.6.8-64-bit) (ground types: gmpy)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.2/\n",
      "\n"
     ]
    }
   ],
   "source": [
    "init_session()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "E, rho = symbols('E rho', positive=True)\n",
    "nu, r, s = symbols('nu r s')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Constitutive equations in Voigt's notation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\nu + 1 & \\nu & 0\\\\\\nu & 1 & 0\\\\0 & 0 & - \\nu + \\frac{1}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-ν + 1  ν     0    ⎤\n",
       "⎢                   ⎥\n",
       "⎢  ν     1     0    ⎥\n",
       "⎢                   ⎥\n",
       "⎣  0     0  -ν + 1/2⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "factor = E/((1+nu)*(1-2*nu))\n",
    "factor = 1\n",
    "C = factor*Matrix([\n",
    "     [1 - nu,nu,0], \n",
    "     [nu,1,0], \n",
    "     [0,0,(1-2*nu)/2]])\n",
    "display(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interpolation functions\n",
    "\n",
    "    2-----1\n",
    "    |     |\n",
    "    |     |\n",
    "    3-----4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = S(1)/4*Matrix([(1 + r) * (1 + s),\n",
    "                  (1 - r) * (1 + s),\n",
    "                  (1 - r) * (1 - s),\n",
    "                  (1 + r) * (1 - s)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interpolation matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\left(r + 1\\right) \\left(s + 1\\right)}{4} & 0 & \\frac{\\left(- r + 1\\right) \\left(s + 1\\right)}{4} & 0 & \\frac{\\left(- r + 1\\right) \\left(- s + 1\\right)}{4} & 0 & \\frac{\\left(r + 1\\right) \\left(- s + 1\\right)}{4} & 0\\\\0 & \\frac{\\left(r + 1\\right) \\left(s + 1\\right)}{4} & 0 & \\frac{\\left(- r + 1\\right) \\left(s + 1\\right)}{4} & 0 & \\frac{\\left(- r + 1\\right) \\left(- s + 1\\right)}{4} & 0 & \\frac{\\left(r + 1\\right) \\left(- s + 1\\right)}{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡(r + 1)⋅(s + 1)                   (-r + 1)⋅(s + 1)                    (-r + 1\n",
       "⎢───────────────         0         ────────────────         0          ───────\n",
       "⎢       4                                 4                                   \n",
       "⎢                                                                             \n",
       "⎢                 (r + 1)⋅(s + 1)                    (-r + 1)⋅(s + 1)         \n",
       "⎢       0         ───────────────         0          ────────────────         \n",
       "⎣                        4                                  4                 \n",
       "\n",
       ")⋅(-s + 1)                     (r + 1)⋅(-s + 1)                  ⎤\n",
       "──────────          0          ────────────────         0        ⎥\n",
       " 4                                    4                          ⎥\n",
       "                                                                 ⎥\n",
       "            (-r + 1)⋅(-s + 1)                    (r + 1)⋅(-s + 1)⎥\n",
       " 0          ─────────────────         0          ────────────────⎥\n",
       "                    4                                   4        ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H = zeros(2, 8)\n",
    "for i in range(4):\n",
    "    H[0, 2*i] = N[i]\n",
    "    H[1, 2*i + 1] = N[i]\n",
    "\n",
    "display(H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mass matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{4 \\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{\\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0\\\\0 & \\frac{4 \\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{\\rho}{9} & 0 & \\frac{2 \\rho}{9}\\\\\\frac{2 \\rho}{9} & 0 & \\frac{4 \\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{\\rho}{9} & 0\\\\0 & \\frac{2 \\rho}{9} & 0 & \\frac{4 \\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{\\rho}{9}\\\\\\frac{\\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{4 \\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0\\\\0 & \\frac{\\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{4 \\rho}{9} & 0 & \\frac{2 \\rho}{9}\\\\\\frac{2 \\rho}{9} & 0 & \\frac{\\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{4 \\rho}{9} & 0\\\\0 & \\frac{2 \\rho}{9} & 0 & \\frac{\\rho}{9} & 0 & \\frac{2 \\rho}{9} & 0 & \\frac{4 \\rho}{9}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡4⋅ρ       2⋅ρ        ρ        2⋅ρ     ⎤\n",
       "⎢───   0   ───   0    ─    0   ───   0 ⎥\n",
       "⎢ 9         9         9         9      ⎥\n",
       "⎢                                      ⎥\n",
       "⎢     4⋅ρ       2⋅ρ        ρ        2⋅ρ⎥\n",
       "⎢ 0   ───   0   ───   0    ─    0   ───⎥\n",
       "⎢      9         9         9         9 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢2⋅ρ       4⋅ρ       2⋅ρ        ρ      ⎥\n",
       "⎢───   0   ───   0   ───   0    ─    0 ⎥\n",
       "⎢ 9         9         9         9      ⎥\n",
       "⎢                                      ⎥\n",
       "⎢     2⋅ρ       4⋅ρ       2⋅ρ        ρ ⎥\n",
       "⎢ 0   ───   0   ───   0   ───   0    ─ ⎥\n",
       "⎢      9         9         9         9 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢ ρ        2⋅ρ       4⋅ρ       2⋅ρ     ⎥\n",
       "⎢ ─    0   ───   0   ───   0   ───   0 ⎥\n",
       "⎢ 9         9         9         9      ⎥\n",
       "⎢                                      ⎥\n",
       "⎢      ρ        2⋅ρ       4⋅ρ       2⋅ρ⎥\n",
       "⎢ 0    ─    0   ───   0   ───   0   ───⎥\n",
       "⎢      9         9         9         9 ⎥\n",
       "⎢                                      ⎥\n",
       "⎢2⋅ρ        ρ        2⋅ρ       4⋅ρ     ⎥\n",
       "⎢───   0    ─    0   ───   0   ───   0 ⎥\n",
       "⎢ 9         9         9         9      ⎥\n",
       "⎢                                      ⎥\n",
       "⎢     2⋅ρ        ρ        2⋅ρ       4⋅ρ⎥\n",
       "⎢ 0   ───   0    ─    0   ───   0   ───⎥\n",
       "⎣      9         9         9         9 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M_inte = H.T * H\n",
    "M = integrate(integrate(rho*M_inte, [r, -1, 1]), [s, -1, 1])\n",
    "display(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Derivatives interpolation matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{s}{4} + \\frac{1}{4} & 0 & - \\frac{s}{4} - \\frac{1}{4} & 0 & \\frac{s}{4} - \\frac{1}{4} & 0 & - \\frac{s}{4} + \\frac{1}{4} & 0\\\\0 & \\frac{r}{4} + \\frac{1}{4} & 0 & - \\frac{r}{4} + \\frac{1}{4} & 0 & \\frac{r}{4} - \\frac{1}{4} & 0 & - \\frac{r}{4} - \\frac{1}{4}\\\\\\frac{r}{4} + \\frac{1}{4} & \\frac{s}{4} + \\frac{1}{4} & - \\frac{r}{4} + \\frac{1}{4} & - \\frac{s}{4} - \\frac{1}{4} & \\frac{r}{4} - \\frac{1}{4} & \\frac{s}{4} - \\frac{1}{4} & - \\frac{r}{4} - \\frac{1}{4} & - \\frac{s}{4} + \\frac{1}{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡s   1           s   1           s   1           s   1         ⎤\n",
       "⎢─ + ─    0    - ─ - ─     0     ─ - ─    0    - ─ + ─     0   ⎥\n",
       "⎢4   4           4   4           4   4           4   4         ⎥\n",
       "⎢                                                              ⎥\n",
       "⎢       r   1             r   1         r   1             r   1⎥\n",
       "⎢  0    ─ + ─     0     - ─ + ─    0    ─ - ─     0     - ─ - ─⎥\n",
       "⎢       4   4             4   4         4   4             4   4⎥\n",
       "⎢                                                              ⎥\n",
       "⎢r   1  s   1    r   1    s   1  r   1  s   1    r   1    s   1⎥\n",
       "⎢─ + ─  ─ + ─  - ─ + ─  - ─ - ─  ─ - ─  ─ - ─  - ─ - ─  - ─ + ─⎥\n",
       "⎣4   4  4   4    4   4    4   4  4   4  4   4    4   4    4   4⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B = zeros(3, 8)\n",
    "for i in range(4):\n",
    "    B[0, 2*i] = diff(N[i], r)\n",
    "    B[1, 2*i + 1] = diff(N[i], s)\n",
    "    B[2, 2*i] = diff(N[i], s)\n",
    "    B[2, 2*i + 1] = diff(N[i], r)\n",
    "\n",
    "display(B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stiffness matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "K_inte = B.T * C * B\n",
    "K = simplify(integrate(integrate(K_inte, [r, -1, 1]), [s, -1, 1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{2 \\nu}{3} + \\frac{1}{2} & \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{\\nu}{2} - \\frac{1}{8} & \\frac{\\nu}{3} - \\frac{1}{4} & - \\frac{1}{8} & \\frac{\\nu}{6} & - \\frac{\\nu}{2} + \\frac{1}{8}\\\\\\frac{1}{8} & - \\frac{\\nu}{3} + \\frac{1}{2} & - \\frac{\\nu}{2} + \\frac{1}{8} & \\frac{\\nu}{3} & - \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{\\nu}{2} - \\frac{1}{8} & - \\frac{\\nu}{6} - \\frac{1}{4}\\\\\\frac{\\nu}{6} - \\frac{1}{4} & - \\frac{\\nu}{2} + \\frac{1}{8} & - \\frac{2 \\nu}{3} + \\frac{1}{2} & - \\frac{1}{8} & \\frac{\\nu}{6} & \\frac{\\nu}{2} - \\frac{1}{8} & \\frac{\\nu}{3} - \\frac{1}{4} & \\frac{1}{8}\\\\\\frac{\\nu}{2} - \\frac{1}{8} & \\frac{\\nu}{3} & - \\frac{1}{8} & - \\frac{\\nu}{3} + \\frac{1}{2} & - \\frac{\\nu}{2} + \\frac{1}{8} & - \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4}\\\\\\frac{\\nu}{3} - \\frac{1}{4} & - \\frac{1}{8} & \\frac{\\nu}{6} & - \\frac{\\nu}{2} + \\frac{1}{8} & - \\frac{2 \\nu}{3} + \\frac{1}{2} & \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{\\nu}{2} - \\frac{1}{8}\\\\- \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{\\nu}{2} - \\frac{1}{8} & - \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{1}{8} & - \\frac{\\nu}{3} + \\frac{1}{2} & - \\frac{\\nu}{2} + \\frac{1}{8} & \\frac{\\nu}{3}\\\\\\frac{\\nu}{6} & \\frac{\\nu}{2} - \\frac{1}{8} & \\frac{\\nu}{3} - \\frac{1}{4} & \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4} & - \\frac{\\nu}{2} + \\frac{1}{8} & - \\frac{2 \\nu}{3} + \\frac{1}{2} & - \\frac{1}{8}\\\\- \\frac{\\nu}{2} + \\frac{1}{8} & - \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{1}{8} & \\frac{\\nu}{6} - \\frac{1}{4} & \\frac{\\nu}{2} - \\frac{1}{8} & \\frac{\\nu}{3} & - \\frac{1}{8} & - \\frac{\\nu}{3} + \\frac{1}{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  2⋅ν   1             ν   1     ν   1     ν   1                 ν        ν   \n",
       "⎢- ─── + ─    1/8      ─ - ─     ─ - ─     ─ - ─     -1/8        ─      - ─ + \n",
       "⎢   3    2             6   4     2   8     3   4                 6        2   \n",
       "⎢                                                                             \n",
       "⎢             ν   1     ν   1      ν                 ν   1     ν   1      ν   \n",
       "⎢   1/8     - ─ + ─   - ─ + ─      ─       -1/8      ─ - ─     ─ - ─    - ─ - \n",
       "⎢             3   2     2   8      3                 6   4     2   8      6   \n",
       "⎢                                                                             \n",
       "⎢  ν   1      ν   1    2⋅ν   1               ν       ν   1     ν   1          \n",
       "⎢  ─ - ─    - ─ + ─  - ─── + ─   -1/8        ─       ─ - ─     ─ - ─      1/8 \n",
       "⎢  6   4      2   8     3    2               6       2   8     3   4          \n",
       "⎢                                                                             \n",
       "⎢  ν   1       ν                  ν   1     ν   1     ν   1              ν   1\n",
       "⎢  ─ - ─       ─       -1/8     - ─ + ─   - ─ + ─   - ─ - ─     1/8      ─ - ─\n",
       "⎢  2   8       3                  3   2     2   8     6   4              6   4\n",
       "⎢                                                                             \n",
       "⎢  ν   1                 ν        ν   1    2⋅ν   1             ν   1     ν   1\n",
       "⎢  ─ - ─     -1/8        ─      - ─ + ─  - ─── + ─    1/8      ─ - ─     ─ - ─\n",
       "⎢  3   4                 6        2   8     3    2             6   4     2   8\n",
       "⎢                                                                             \n",
       "⎢            ν   1     ν   1      ν   1               ν   1     ν   1      ν  \n",
       "⎢  -1/8      ─ - ─     ─ - ─    - ─ - ─     1/8     - ─ + ─   - ─ + ─      ─  \n",
       "⎢            6   4     2   8      6   4               3   2     2   8      3  \n",
       "⎢                                                                             \n",
       "⎢    ν       ν   1     ν   1               ν   1      ν   1    2⋅ν   1        \n",
       "⎢    ─       ─ - ─     ─ - ─      1/8      ─ - ─    - ─ + ─  - ─── + ─   -1/8 \n",
       "⎢    6       2   8     3   4               6   4      2   8     3    2        \n",
       "⎢                                                                             \n",
       "⎢   ν   1     ν   1              ν   1     ν   1       ν                  ν   \n",
       "⎢ - ─ + ─   - ─ - ─     1/8      ─ - ─     ─ - ─       ─       -1/8     - ─ + \n",
       "⎣   2   8     6   4              6   4     2   8       3                  3   \n",
       "\n",
       "1⎤\n",
       "─⎥\n",
       "8⎥\n",
       " ⎥\n",
       "1⎥\n",
       "─⎥\n",
       "4⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       "1⎥\n",
       "─⎥\n",
       "2⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gauss integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "wts = [2, [1,1], [S(5)/9, S(8)/9, S(5)/9], \n",
    "        [(18+sqrt(30))/36,(18+sqrt(30))/36, (18-sqrt(30))/36, (18-sqrt(30))/36]];\n",
    "pts = [0, [-sqrt(S(1)/3), sqrt(S(1)/3)], [-sqrt(S(3)/5), 0, sqrt(S(3)/5)],\n",
    "[-sqrt(S(3)/7 - S(2)/7*sqrt((6)/5)), sqrt(S(3)/7 - S(2)/7*sqrt(S(6)/5)),\n",
    "-sqrt(S(3)/7 + S(2)/7*sqrt(S(6)/5)), sqrt(S(3)/7 + S(2)/7*sqrt(S(6)/5))]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{\\nu}{4} + \\frac{3}{16} & \\frac{1}{16} & - \\frac{1}{16} & \\frac{\\nu}{4} - \\frac{1}{16} & \\frac{\\nu}{4} - \\frac{3}{16} & - \\frac{1}{16} & \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{1}{16}\\\\\\frac{1}{16} & - \\frac{\\nu}{8} + \\frac{3}{16} & - \\frac{\\nu}{4} + \\frac{1}{16} & \\frac{\\nu}{8} + \\frac{1}{16} & - \\frac{1}{16} & \\frac{\\nu}{8} - \\frac{3}{16} & \\frac{\\nu}{4} - \\frac{1}{16} & - \\frac{\\nu}{8} - \\frac{1}{16}\\\\- \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{3}{16} & - \\frac{1}{16} & \\frac{1}{16} & \\frac{\\nu}{4} - \\frac{1}{16} & \\frac{\\nu}{4} - \\frac{3}{16} & \\frac{1}{16}\\\\\\frac{\\nu}{4} - \\frac{1}{16} & \\frac{\\nu}{8} + \\frac{1}{16} & - \\frac{1}{16} & - \\frac{\\nu}{8} + \\frac{3}{16} & - \\frac{\\nu}{4} + \\frac{1}{16} & - \\frac{\\nu}{8} - \\frac{1}{16} & \\frac{1}{16} & \\frac{\\nu}{8} - \\frac{3}{16}\\\\\\frac{\\nu}{4} - \\frac{3}{16} & - \\frac{1}{16} & \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{3}{16} & \\frac{1}{16} & - \\frac{1}{16} & \\frac{\\nu}{4} - \\frac{1}{16}\\\\- \\frac{1}{16} & \\frac{\\nu}{8} - \\frac{3}{16} & \\frac{\\nu}{4} - \\frac{1}{16} & - \\frac{\\nu}{8} - \\frac{1}{16} & \\frac{1}{16} & - \\frac{\\nu}{8} + \\frac{3}{16} & - \\frac{\\nu}{4} + \\frac{1}{16} & \\frac{\\nu}{8} + \\frac{1}{16}\\\\\\frac{1}{16} & \\frac{\\nu}{4} - \\frac{1}{16} & \\frac{\\nu}{4} - \\frac{3}{16} & \\frac{1}{16} & - \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{1}{16} & - \\frac{\\nu}{4} + \\frac{3}{16} & - \\frac{1}{16}\\\\- \\frac{\\nu}{4} + \\frac{1}{16} & - \\frac{\\nu}{8} - \\frac{1}{16} & \\frac{1}{16} & \\frac{\\nu}{8} - \\frac{3}{16} & \\frac{\\nu}{4} - \\frac{1}{16} & \\frac{\\nu}{8} + \\frac{1}{16} & - \\frac{1}{16} & - \\frac{\\nu}{8} + \\frac{3}{16}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  ν   3                        ν   1     ν   3                          ν   1\n",
       "⎢- ─ + ──    1/16     -1/16     ─ - ──    ─ - ──    -1/16      1/16    - ─ + ─\n",
       "⎢  4   16                       4   16    4   16                         4   1\n",
       "⎢                                                                             \n",
       "⎢            ν   3     ν   1    ν   1               ν   3     ν   1      ν   1\n",
       "⎢  1/16    - ─ + ──  - ─ + ──   ─ + ──    -1/16     ─ - ──    ─ - ──   - ─ - ─\n",
       "⎢            8   16    4   16   8   16              8   16    4   16     8   1\n",
       "⎢                                                                             \n",
       "⎢            ν   1     ν   3                        ν   1     ν   3           \n",
       "⎢ -1/16    - ─ + ──  - ─ + ──   -1/16      1/16     ─ - ──    ─ - ──     1/16 \n",
       "⎢            4   16    4   16                       4   16    4   16          \n",
       "⎢                                                                             \n",
       "⎢ ν   1     ν   1                ν   3     ν   1     ν   1              ν   3 \n",
       "⎢ ─ - ──    ─ + ──    -1/16    - ─ + ──  - ─ + ──  - ─ - ──    1/16     ─ - ──\n",
       "⎢ 4   16    8   16               8   16    4   16    8   16             8   16\n",
       "⎢                                                                             \n",
       "⎢ ν   3                          ν   1     ν   3                        ν   1 \n",
       "⎢ ─ - ──    -1/16      1/16    - ─ + ──  - ─ + ──    1/16     -1/16     ─ - ──\n",
       "⎢ 4   16                         4   16    4   16                       4   16\n",
       "⎢                                                                             \n",
       "⎢           ν   3     ν   1      ν   1               ν   3     ν   1    ν   1 \n",
       "⎢ -1/16     ─ - ──    ─ - ──   - ─ - ──    1/16    - ─ + ──  - ─ + ──   ─ + ──\n",
       "⎢           8   16    4   16     8   16              8   16    4   16   8   16\n",
       "⎢                                                                             \n",
       "⎢           ν   1     ν   3                          ν   1     ν   3          \n",
       "⎢  1/16     ─ - ──    ─ - ──     1/16     -1/16    - ─ + ──  - ─ + ──   -1/16 \n",
       "⎢           4   16    4   16                         4   16    4   16         \n",
       "⎢                                                                             \n",
       "⎢  ν   1     ν   1              ν   3     ν   1     ν   1                ν   3\n",
       "⎢- ─ + ──  - ─ - ──    1/16     ─ - ──    ─ - ──    ─ + ──    -1/16    - ─ + ─\n",
       "⎣  4   16    8   16             8   16    4   16    8   16               8   1\n",
       "\n",
       " ⎤\n",
       "─⎥\n",
       "6⎥\n",
       " ⎥\n",
       " ⎥\n",
       "─⎥\n",
       "6⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       " ⎥\n",
       "─⎥\n",
       "6⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "K_1pts = simplify(K_inte.subs({r:0, s:0})*wts[0])\n",
    "display(K_1pts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "K_2pts = zeros(8, 8)\n",
    "for row in range(8):\n",
    "    for col in range(8):\n",
    "        K_2pts[row, col] = sum(wts[1][kx]*wts[1][ky]*\n",
    "                               K_inte[row, col].subs({r: pts[1][kx], s:pts[1][ky]})\n",
    "                               for kx in range(2) for ky in range(2))\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  0  0  0  0  0  0⎤\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎣0  0  0  0  0  0  0  0⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(K_2pts - K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "K_3pts = zeros(8, 8)\n",
    "for row in range(8):\n",
    "    for col in range(8):\n",
    "        K_3pts[row, col] = sum(wts[2][kx]*wts[2][ky]*\n",
    "                        K_inte[row, col].subs({r: pts[2][kx], s:pts[2][ky]})\n",
    "                        for kx in range(3) for ky in range(3))\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  0  0  0  0  0  0⎤\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎢0  0  0  0  0  0  0  0⎥\n",
       "⎢                      ⎥\n",
       "⎣0  0  0  0  0  0  0  0⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(K_3pts - K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# References\n",
    "\n",
    "1. Bathe, Klaus-Jürgen. Finite element procedures. Klaus-Jurgen Bathe, 2006."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
       "<style>\n",
       "/* Based on Lorena Barba template available at: https://github.com/barbagroup/AeroPython/blob/master/styles/custom.css*/\n",
       "@font-face {\n",
       "font-family: \"Computer Modern\";\n",
       "src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "}\n",
       "div.cell{\n",
       "width:800px;\n",
       "margin-left:16% !important;\n",
       "margin-right:auto;\n",
       "}\n",
       "h1 {\n",
       "font-family: 'Alegreya Sans', sans-serif;\n",
       "}\n",
       "h2 {\n",
       "font-family: 'Fenix', serif;\n",
       "}\n",
       "h3{\n",
       "font-family: 'Fenix', serif;\n",
       "margin-top:12px;\n",
       "margin-bottom: 3px;\n",
       "}\n",
       "h4{\n",
       "font-family: 'Fenix', serif;\n",
       "}\n",
       "h5 {\n",
       "font-family: 'Alegreya Sans', sans-serif;\n",
       "}\t\n",
       "div.text_cell_render{\n",
       "font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "line-height: 135%;\n",
       "font-size: 120%;\n",
       "width:600px;\n",
       "margin-left:auto;\n",
       "margin-right:auto;\n",
       "}\n",
       ".CodeMirror{\n",
       "font-family: \"Source Code Pro\";\n",
       "font-size: 90%;\n",
       "}\n",
       "/* .prompt{\n",
       "display: None;\n",
       "}*/\n",
       ".text_cell_render h1 {\n",
       "font-weight: 200;\n",
       "font-size: 50pt;\n",
       "line-height: 100%;\n",
       "color:#CD2305;\n",
       "margin-bottom: 0.5em;\n",
       "margin-top: 0.5em;\n",
       "display: block;\n",
       "}\t\n",
       ".text_cell_render h5 {\n",
       "font-weight: 300;\n",
       "font-size: 16pt;\n",
       "color: #CD2305;\n",
       "font-style: italic;\n",
       "margin-bottom: .5em;\n",
       "margin-top: 0.5em;\n",
       "display: block;\n",
       "}\n",
       ".warning{\n",
       "color: rgb( 240, 20, 20 )\n",
       "}\n",
       "</style>\n",
       "<script>\n",
       "MathJax.Hub.Config({\n",
       "TeX: {\n",
       "extensions: [\"AMSmath.js\"]\n",
       "},\n",
       "tex2jax: {\n",
       "inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "},\n",
       "displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "\"HTML-CSS\": {\n",
       "styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "}\n",
       "});\n",
       "</script>\n",
       "\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = open('./styles/custom_barba.css', 'r').read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}