JuliaFEM.jl/notebooks/2015-06-25-elasticity-solver-example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving elasticity problems using JuliaFEM\n",
    "\n",
    "Author(s): Jukka Aho\n",
    "\n",
    "**Abstract**: Solving elasticity equations using JuliaFEM.\n",
    "\n",
    "### Weak form\n",
    "\n",
    "Given function spaces\n",
    "\\begin{align}\n",
    "\\boldsymbol{\\mathcal{U}} & =\\left\\{ \\boldsymbol{u}\\in H^{1}\\left(\\Omega\\right)|\\boldsymbol{u}\\left(\\boldsymbol{X},t\\right)=\\hat{\\boldsymbol{u}}\\left(\\boldsymbol{X},t\\right)\\text{ on }\\Gamma_{\\mathrm{u}}\\right\\} ,\\\\\n",
    "\\boldsymbol{\\mathcal{V}} & =\\left\\{ \\delta\\boldsymbol{u}\\in H^{1}\\left(\\Omega\\right)|\\delta\\boldsymbol{u}\\left(\\boldsymbol{X}\\right)=0\\text{ on }\\Gamma_{\\mathrm{u}}\\right\\} ,\n",
    "\\end{align}\n",
    "find $\\boldsymbol{u}\\in\\boldsymbol{\\mathcal{U}}$ such that\n",
    "\\begin{equation}\n",
    "\\delta\\mathcal{W}:=\\int_{\\Omega_{0}}\\rho_{0}\\ddot{\\boldsymbol{u}}\\cdot\\delta\\boldsymbol{u}\\,\\mathrm{d}V_{0}+\\int_{\\Omega_{0}}\\boldsymbol{S}:\\delta\\boldsymbol{E}\\,\\mathrm{d}V_{0}-\\int_{\\Omega_{0}}\\hat{\\boldsymbol{b}}_{0}\\cdot\\delta\\boldsymbol{u}\\,\\mathrm{d}V_{0}-\\int_{\\Gamma_{\\sigma}}\\hat{\\boldsymbol{t}}_{0}\\cdot\\delta\\boldsymbol{u}\\,\\mathrm{d}A_{0} =0 \\qquad\\forall\\delta\\boldsymbol{u}\\in\\boldsymbol{\\mathcal{V}}\n",
    "\\end{equation}\n",
    "\n",
    "### Some formulas\n",
    "\\begin{align}\n",
    "J & =\\det\\left(F\\right)\\\\\n",
    "I_{c} & =\\mbox{tr}\\left(C\\right)\\\\\n",
    "\\mathbf{C} & =\\mathbf{F}^{\\mathrm{T}}\\mathbf{F}\\\\\n",
    "\\mathbf{F} & =\\mathbf{I}+\\nabla\\mathbf{u}\\\\\n",
    "\\mathbf{E} & =\\frac{1}{2}\\left(\\mathbf{F}^{\\mathrm{T}}\\mathbf{F}-\\mathbf{I}\\right)\n",
    "\\end{align}\n",
    "\n",
    "### Potential energy\n",
    "\n",
    "\\begin{equation}\n",
    "\\underset{u\\in\\boldsymbol{\\mathcal{U}}}{\\min}\\Pi\\left(\\mathbf{u}\\right)\n",
    "\\end{equation}\n",
    "\\begin{equation}\n",
    "\\Pi\\left(\\mathbf{u}\\right)=\\int_{\\Omega}\\psi\\left(\\mathbf{u}\\right)-\\int_{\\Omega}\\hat{\\mathbf{b}}_{0}\\cdot\\mathbf{u}-\\int_{\\Gamma_{\\sigma}}\\hat{\\mathbf{t}}_{0}\\cdot\\mathbf{u}\\,\\mathrm{d}A_{0}\n",
    "\\end{equation}\n",
    "\n",
    "### Material models\n",
    "\n",
    "https://en.wikipedia.org/wiki/Strain_energy_density_function\n",
    "\n",
    "Saint-Venant-Kirchhoff model https://en.wikipedia.org/wiki/Hyperelastic_material\n",
    "\\begin{equation}\n",
    "\\psi\\left(\\mathbf{E}\\right)=\\frac{\\lambda}{2}\\left[\\mbox{tr}\\left(\\mathbf{E}\\right)\\right]^{2}+\\mu\\mbox{tr}\\left(\\mathbf{E}^2\\right)\n",
    "\\end{equation}\n",
    "\n",
    "neo-Hookean material https://en.wikipedia.org/wiki/Neo-Hookean_solid\n",
    "\\begin{equation}\n",
    "\\psi=\\frac{\\mu}{2}\\left(I_{c}-3\\right)-\\mu\\ln\\left(J\\right)+\\frac{\\lambda}{2}\\ln\\left(J\\right)^{2}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "using JuliaFEM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: Registered handlers: Any[\"ELEMENT\",\"NODE\",\"NSET\",\"ELSET\"]\n",
      "INFO: Parsing elements\n",
      "INFO: 31437 elements found\n",
      "INFO: Creating ELSET PISTON\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Dict{Any,Any} with 3 entries:\n",
       "  \"nodes\"    => Dict{Any,Any}(2843=>[-27.41338,-2.18093,-16.93988],4495=>[-14.0…\n",
       "  \"elements\" => Dict{Any,Any}(43367=>[4492,4602,4494,6450],35510=>[2796,2798,80…\n",
       "  \"elsets\"   => Dict{Any,Any}(\"BC3\"=>[3975,3976,3977,3978,3979,3980,3981,3982,3…"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: Parsing elements\n",
      "INFO: 5894 elements found\n",
      "INFO: Creating element set BC1\n",
      "INFO: Creating element set BC2\n",
      "INFO: Creating element set BC3\n"
     ]
    }
   ],
   "source": [
    "model = open(JuliaFEM.parse_abaqus, \"../geometry/piston/piston_8789_P1.inp\")\n",
    "model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "perm = Dict{Int64, Int64}()\n",
    "for (i, nid) in enumerate(keys(model[\"nodes\"]))\n",
    "    perm[nid] = i\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: 37331 elements created.\n"
     ]
    }
   ],
   "source": [
    "using JuliaFEM: Element, Tri3, Tet4\n",
    "elements = Dict{Int64, Element}()\n",
    "for (elid, node_ids) in model[\"elements\"]\n",
    "    connectivity = Int64[perm[nid] for nid in node_ids]\n",
    "    coords = Vector{Float64}[model[\"nodes\"][nid] for nid in node_ids]\n",
    "    if length(coords) == 3\n",
    "        element = Tri3(connectivity)\n",
    "        element[\"geometry\"] = coords\n",
    "        elements[elid] = element\n",
    "    elseif length(coords) == 4\n",
    "        element = Tet4(connectivity)\n",
    "        element[\"geometry\"] = coords\n",
    "        elements[elid] = element\n",
    "    else\n",
    "        warn(\"unknown element dim = $(length(connectivity))\")\n",
    "    end\n",
    "end\n",
    "info(\"$(length(elements)) elements created.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "37331"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "31437+5894"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using JuliaFEM: ElasticityProblem, DirichletProblem\n",
    "\n",
    "piston = ElasticityProblem()\n",
    "for elid in model[\"elsets\"][\"PISTON\"]\n",
    "    elements[elid][\"youngs modulus\"] = 210.0e3\n",
    "    elements[elid][\"poissons ratio\"] = 0.3\n",
    "    push!(piston, elements[elid])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "load = Vector{Float64}[[0.0, 0.0, -10.0], [0.0, 0.0, -10.0], [0.0, 0.0, -10.0]]\n",
    "for elid in model[\"elsets\"][\"BC1\"]\n",
    "    elements[elid][\"displacement traction force\"] = load\n",
    "    push!(piston, elements[elid])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "encastre = DirichletProblem(\"displacement\", 3)\n",
    "for elid in model[\"elsets\"][\"BC2\"]\n",
    "    elements[elid][\"displacement\"] = 0.0\n",
    "    push!(encastre, elements[elid])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "using JuliaFEM: DirectSolver\n",
    "solver = DirectSolver()\n",
    "push!(solver, piston)\n",
    "push!(solver, encastre);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: # of field problems: 1\n",
      "INFO: # of boundary problems: 1\n",
      "INFO: Starting iteration 1\n",
      "INFO: # of dofs: 26367, # of interface dofs: 1350\n",
      "INFO: solved. length of solution vector = 52734\n",
      "INFO: timing info for non-linear iteration:\n",
      "INFO: boundary assembly      : 1.4367611408233643\n",
      "INFO: field assembly         : 17.380300998687744\n",
      "INFO: create sparse matrices : 0.38140082359313965\n",
      "INFO: solution of system     : 3.159921169281006\n",
      "INFO: update element data    : 3.3958511352539062\n",
      "INFO: non-linear iteration   : 25.75425100326538\n",
      "INFO: Starting iteration 2\n",
      "INFO: # of dofs: 26367, # of interface dofs: 1350\n",
      "INFO: solved. length of solution vector = 52734\n",
      "INFO: timing info for non-linear iteration:\n",
      "INFO: boundary assembly      : 0.1028587818145752\n",
      "INFO: field assembly         : 15.321197032928467\n",
      "INFO: create sparse matrices : 0.11679887771606445\n",
      "INFO: solution of system     : 2.1216089725494385\n",
      "INFO: update element data    : 3.484022855758667\n",
      "INFO: non-linear iteration   : 21.146512031555176\n",
      "INFO: Starting iteration 3\n",
      "INFO: # of dofs: 26367, # of interface dofs: 1350\n",
      "INFO: solved. length of solution vector = 52734\n",
      "INFO: timing info for non-linear iteration:\n",
      "INFO: boundary assembly      : 0.09190106391906738\n",
      "INFO: field assembly         : 14.772294044494629\n",
      "INFO: create sparse matrices : 0.12086606025695801\n",
      "INFO: solution of system     : 2.260561943054199\n",
      "INFO: update element data    : 3.4714930057525635\n",
      "INFO: non-linear iteration   : 20.717132806777954\n",
      "INFO: Starting iteration 4\n",
      "INFO: # of dofs: 26367, # of interface dofs: 1350\n",
      "INFO: solved. length of solution vector = 52734\n",
      "INFO: timing info for non-linear iteration:\n",
      "INFO: boundary assembly      : 0.09454894065856934\n",
      "INFO: field assembly         : 15.800831079483032\n",
      "INFO: create sparse matrices : 0.10474300384521484\n",
      "INFO: solution of system     : 2.3751449584960938\n",
      "INFO: update element data    : 3.471247911453247\n",
      "INFO: non-linear iteration   : 21.846541166305542\n",
      "INFO: solver finished in 90.86189484596252 seconds.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(4,true)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solver(0.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Saving results to file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Grid Name=\"Grid\">\n",
       "  <Time Value=\"0\"/>\n",
       "</Grid>\n"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xdoc, xmodel = JuliaFEM.xdmf_new_model()\n",
    "temporal_collection = JuliaFEM.xdmf_new_temporal_collection(xmodel)\n",
    "grid = JuliaFEM.xdmf_new_grid(temporal_collection; time=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Save geometry to xdmf file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: number of nodes in model: 8789\n",
      "INFO: Number of elements in model: 31437\n"
     ]
    }
   ],
   "source": [
    "nnodes = length(model[\"nodes\"])\n",
    "info(\"number of nodes in model: $nnodes\")\n",
    "\n",
    "X = zeros(3, nnodes)\n",
    "for nid in keys(model[\"nodes\"])\n",
    "    X[:, perm[nid]] = model[\"nodes\"][nid]\n",
    "end\n",
    "\n",
    "nelements = length(model[\"elsets\"][\"PISTON\"])\n",
    "info(\"Number of elements in model: $nelements\")\n",
    "elmap = zeros(Int64, 5, nelements)\n",
    "#elmap[1,:] = 0x0026\n",
    "elmap[1,:] = 0x6\n",
    "for (i, elid) in enumerate(model[\"elsets\"][\"PISTON\"])\n",
    "    elmap[2:end,i] = Int64[perm[nid] for nid in model[\"elements\"][elid]]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "JuliaFEM.xdmf_new_mesh(grid, X, elmap)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "988949"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "JuliaFEM.xdmf_save_model(xdoc, \"/tmp/piston.xmf\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Save nodal data to model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3,8789)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using JuliaFEM: get_connectivity\n",
    "\n",
    "u = zeros(3, nnodes)\n",
    "\n",
    "for element in values(elements)\n",
    "    isa(element, Element{Tet4}) || continue\n",
    "    connectivity = get_connectivity(element)\n",
    "    field = element[\"displacement\"](0.0)\n",
    "    for (i, nid) in enumerate(connectivity)\n",
    "        u[:, nid] = field[i]\n",
    "    end\n",
    "end\n",
    "size(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "JuliaFEM.xdmf_new_field(grid, \"Displacement\", \"nodes\", u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1573751"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "JuliaFEM.xdmf_save_model(xdoc, \"/tmp/piston.xmf\")"
   ]
  }
 ],
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