{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Shape function and integration points\n",
    "\n",
    "Author(s): Jukka Aho\n",
    "\n",
    "**Abstract**: Shape functions and element descriptions used in JuliaFEM."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "#init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "xi = DeferredVector(r\"xi\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1D shape function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear 2-node segment (Lagrange family)\n",
    "\n",
    "|       | $\\xi_1$ |\n",
    "| ----- | -------:|\n",
    "| $N_1$ |      -1 |\n",
    "| $N_2$ |       1 |\n",
    "\n",
    "\\begin{equation}\n",
    "    \\left(\\mathbf{P}\\boldsymbol{\\alpha}\\right)\\left(\\xi_1\\right) = \\alpha_{1}+\\alpha_{2}\\xi_{1}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [-xi[1]/2 + 1/2],\n",
       " [ xi[1]/2 + 1/2]]), Matrix([\n",
       " [-1/2],\n",
       " [ 1/2]]))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, -1], [1, 1]])\n",
    "P = Matrix([1, xi[1]]).T\n",
    "N = (P*A.inv()).T\n",
    "dN = Matrix([N.diff(xi[1]).T]).T\n",
    "N, dN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quadratic 3-node segment (Lagrange family)\n",
    "\n",
    "|       | $\\xi_1$ |\n",
    "| ----- | -------:|\n",
    "| $N_1$ |      -1 |\n",
    "| $N_2$ |       1 |\n",
    "| $N_3$ |       0 |\n",
    "\n",
    "\\begin{equation}\n",
    "    \\left(\\mathbf{P}\\boldsymbol{\\alpha}\\right)\\left(\\xi_1\\right) = \\alpha_1 + \\alpha_2\\xi_1 + \\alpha_3\\xi_1^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [xi[1]**2/2 - xi[1]/2],\n",
       " [xi[1]**2/2 + xi[1]/2],\n",
       " [       -xi[1]**2 + 1]]), Matrix([\n",
       " [xi[1] - 1/2],\n",
       " [xi[1] + 1/2],\n",
       " [   -2*xi[1]]]))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, -1, (-1)**2],\n",
    "            [1,  1,    1**2],\n",
    "            [1,  0,    0**2]])\n",
    "P = Matrix([1, xi[1], xi[1]**2]).T\n",
    "N = (P*A.inv()).T\n",
    "dN = Matrix([N.diff(xi[1]).T]).T\n",
    "N, dN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P-elements"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2D shape functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear triangle\n",
    "\n",
    "|       | $\\xi_1$ | $\\xi_2$ |\n",
    "| ----- | -------:| -------:|\n",
    "| $N_1$ |       0 |       0 |\n",
    "| $N_2$ |       1 |       0 |\n",
    "| $N_3$ |       0 |       1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [-xi[1] - xi[2] + 1],\n",
       " [             xi[1]],\n",
       " [             xi[2]]]), Matrix([\n",
       " [-1, -1],\n",
       " [ 1,  0],\n",
       " [ 0,  1]]))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, 0, 0], [1, 1, 0], [1, 0, 1]])\n",
    "P = Matrix([1, xi[1], xi[2]]).T\n",
    "N = (P*A.inv()).T\n",
    "dN = Matrix([N.diff(xi[1]).T, N.diff(xi[2]).T]).T\n",
    "N, dN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quadratic triangle\n",
    "\n",
    "|       | $\\xi_1$ | $\\xi_2$ |\n",
    "| ----- | -------:| -------:|\n",
    "| $N_1$ |       0 |       0 |\n",
    "| $N_2$ |       1 |       0 |\n",
    "| $N_3$ |       0 |       1 |\n",
    "| $N_4$ |     1/2 |       0 |\n",
    "| $N_5$ |     1/2 |     1/2 |\n",
    "| $N_6$ |       0 |     1/2 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1,   0,   0,   0,   0,   0],\n",
       "[1,   1,   0,   1,   0,   0],\n",
       "[1,   0,   1,   0,   1,   0],\n",
       "[1, 1/2,   0, 1/4,   0,   0],\n",
       "[1, 1/2, 1/2, 1/4, 1/4, 1/4],\n",
       "[1,   0, 1/2,   0, 1/4,   0]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = Matrix([1, xi[1], xi[2], xi[1]**2, xi[2]**2, xi[1]*xi[2]]).T\n",
    "A = Matrix([\n",
    "        P.subs({xi[1]: 0, xi[2]: 0}),\n",
    "        P.subs({xi[1]: 1, xi[2]: 0}),\n",
    "        P.subs({xi[1]: 0, xi[2]: 1}),\n",
    "        P.subs({xi[1]: Rational(1,2), xi[2]: 0}),\n",
    "        P.subs({xi[1]: Rational(1,2), xi[2]: Rational(1,2)}),\n",
    "        P.subs({xi[1]: 0, xi[2]: Rational(1,2)}),\n",
    "    ])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [2*xi[1]**2 + 4*xi[1]*xi[2] - 3*xi[1] + 2*xi[2]**2 - 3*xi[2] + 1],\n",
       " [                                             2*xi[1]**2 - xi[1]],\n",
       " [                                             2*xi[2]**2 - xi[2]],\n",
       " [                          -4*xi[1]**2 - 4*xi[1]*xi[2] + 4*xi[1]],\n",
       " [                                                  4*xi[1]*xi[2]],\n",
       " [                          -4*xi[1]*xi[2] - 4*xi[2]**2 + 4*xi[2]]]), Matrix([\n",
       " [ 4*xi[1] + 4*xi[2] - 3,  4*xi[1] + 4*xi[2] - 3],\n",
       " [           4*xi[1] - 1,                      0],\n",
       " [                     0,            4*xi[2] - 1],\n",
       " [-8*xi[1] - 4*xi[2] + 4,               -4*xi[1]],\n",
       " [               4*xi[2],                4*xi[1]],\n",
       " [              -4*xi[2], -4*xi[1] - 8*xi[2] + 4]]))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = (P*A.inv()).T\n",
    "dN = Matrix([N.diff(xi[1]).T, N.diff(xi[2]).T]).T\n",
    "N, dN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3D shape functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear tetrahedra, **tet4**\n",
    "\n",
    "|       | $\\xi_1$ | $\\xi_2$ | $\\xi_2$ |\n",
    "| ----- | -------:| -------:| -------:|\n",
    "| $N_1$ |       0 |       0 |       0 |\n",
    "| $N_2$ |       1 |       0 |       0 |\n",
    "| $N_3$ |       0 |       1 |       0 |\n",
    "| $N_4$ |       0 |       0 |       1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0, 0],\n",
       "[1, 1, 0, 0],\n",
       "[1, 0, 1, 0],\n",
       "[1, 0, 0, 1]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = Matrix([1, xi[1], xi[2], xi[3]]).T\n",
    "A = Matrix([\n",
    "        P.subs({xi[1]: 0, xi[2]: 0, xi[3]: 0}),\n",
    "        P.subs({xi[1]: 1, xi[2]: 0, xi[3]: 0}),\n",
    "        P.subs({xi[1]: 0, xi[2]: 1, xi[3]: 0}),\n",
    "        P.subs({xi[1]: 0, xi[2]: 0, xi[3]: 1}),\n",
    "    ])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Matrix([\n",
       " [-xi[1] - xi[2] - xi[3] + 1],\n",
       " [                     xi[1]],\n",
       " [                     xi[2]],\n",
       " [                     xi[3]]]), Matrix([\n",
       " [-1, -1, -1],\n",
       " [ 1,  0,  0],\n",
       " [ 0,  1,  0],\n",
       " [ 0,  0,  1]]))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = (P*A.inv()).T\n",
    "dN = Matrix([N.diff(xi[1]).T, N.diff(xi[2]).T, N.diff(xi[3]).T]).T\n",
    "N, dN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quadratic Lagrange tetrahedral element, 10 nodes, **tet10**\n",
    "\n",
    "|        | $\\xi_1$ | $\\xi_2$ | $\\xi_2$ |\n",
    "| -----  | -------:| -------:| -------:|\n",
    "| $N_1$  |       0 |       0 |       0 |\n",
    "| $N_2$  |       1 |       0 |       0 |\n",
    "| $N_3$  |       0 |       1 |       0 |\n",
    "| $N_4$  |       0 |       0 |       1 |\n",
    "| $N_5$  |     1/2 |       0 |       0 |\n",
    "| $N_6$  |     1/2 |     1/2 |       0 |\n",
    "| $N_7$  |       0 |     1/2 |       0 |\n",
    "| $N_8$  |       0 |       0 |     1/2 |\n",
    "| $N_9$  |     1/2 |       0 |     1/2 |\n",
    "| $N_{10}$ |       0 |     1/2 |     1/2 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1,   0,   0,   0,   0,   0,   0,   0,   0,   0],\n",
       "[1,   1,   0,   0,   1,   0,   0,   0,   0,   0],\n",
       "[1,   0,   1,   0,   0,   0,   1,   0,   0,   0],\n",
       "[1,   0,   0,   1,   0,   0,   0,   0,   1,   0],\n",
       "[1, 1/2,   0,   0, 1/4,   0,   0,   0,   0,   0],\n",
       "[1, 1/2, 1/2,   0, 1/4, 1/4, 1/4,   0,   0,   0],\n",
       "[1,   0, 1/2,   0,   0,   0, 1/4,   0,   0,   0],\n",
       "[1,   0,   0, 1/2,   0,   0,   0,   0, 1/4,   0],\n",
       "[1, 1/2,   0, 1/2, 1/4,   0,   0,   0, 1/4, 1/4],\n",
       "[1,   0, 1/2, 1/2,   0,   0, 1/4, 1/4, 1/4,   0]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = Matrix([1, xi[1], xi[2], xi[3], xi[1]**2, xi[1]*xi[2], xi[2]**2, xi[2]*xi[3], xi[3]**2, xi[1]*xi[3]]).T\n",
    "A = Matrix([\n",
    "        P.subs({xi[1]: 0, xi[2]: 0, xi[3]: 0}),\n",
    "        P.subs({xi[1]: 1, xi[2]: 0, xi[3]: 0}),\n",
    "        P.subs({xi[1]: 0, xi[2]: 1, xi[3]: 0}),\n",
    "        P.subs({xi[1]: 0, xi[2]: 0, xi[3]: 1}),\n",
    "\n",
    "        P.subs({xi[1]: Rational(1,2), xi[2]: 0,             xi[3]: 0}),\n",
    "        P.subs({xi[1]: Rational(1,2), xi[2]: Rational(1,2), xi[3]: 0}),\n",
    "        P.subs({xi[1]:             0, xi[2]: Rational(1,2), xi[3]: 0}),\n",
    "\n",
    "        P.subs({xi[1]:             0, xi[2]: 0,             xi[3]: Rational(1,2)}),\n",
    "        P.subs({xi[1]: Rational(1,2), xi[2]: 0,             xi[3]: Rational(1,2)}),\n",
    "        P.subs({xi[1]:             0, xi[2]: Rational(1,2), xi[3]: Rational(1,2)}),\n",
    "    ])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[(xi[1] + xi[2] + xi[3] - 1)*(2*xi[1] + 2*xi[2] + 2*xi[3] - 1)],\n",
       "[                                          xi[1]*(2*xi[1] - 1)],\n",
       "[                                          xi[2]*(2*xi[2] - 1)],\n",
       "[                                          xi[3]*(2*xi[3] - 1)],\n",
       "[                         -4*xi[1]*(xi[1] + xi[2] + xi[3] - 1)],\n",
       "[                                                4*xi[1]*xi[2]],\n",
       "[                         -4*xi[2]*(xi[1] + xi[2] + xi[3] - 1)],\n",
       "[                         -4*xi[3]*(xi[1] + xi[2] + xi[3] - 1)],\n",
       "[                                                4*xi[1]*xi[3]],\n",
       "[                                                4*xi[2]*xi[3]]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = (P*A.inv()).T\n",
    "dN = Matrix([N.diff(xi[1]).T, N.diff(xi[2]).T, N.diff(xi[3]).T]).T\n",
    "factor(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[ 4*xi[1] + 4*xi[2] + 4*xi[3] - 3,  4*xi[1] + 4*xi[2] + 4*xi[3] - 3,  4*xi[1] + 4*xi[2] + 4*xi[3] - 3],\n",
       "[                     4*xi[1] - 1,                                0,                                0],\n",
       "[                               0,                      4*xi[2] - 1,                                0],\n",
       "[                               0,                                0,                      4*xi[3] - 1],\n",
       "[-4*(2*xi[1] + xi[2] + xi[3] - 1),                         -4*xi[1],                         -4*xi[1]],\n",
       "[                         4*xi[2],                          4*xi[1],                                0],\n",
       "[                        -4*xi[2], -4*(xi[1] + 2*xi[2] + xi[3] - 1),                         -4*xi[2]],\n",
       "[                        -4*xi[3],                         -4*xi[3], -4*(xi[1] + xi[2] + 2*xi[3] - 1)],\n",
       "[                         4*xi[3],                                0,                          4*xi[1]],\n",
       "[                               0,                          4*xi[3],                          4*xi[2]]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(dN)"
   ]
  }
 ],
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