beamUnderDistributedLoad-transient.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from fenics import *\n",
    "from ufl import nabla_div\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from fenicsadapter import *\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Dimensionless Geometry and material properties\n",
    "d=2 #number of dimensions\n",
    "H = 1\n",
    "W = 0.3\n",
    "rho = 0.2\n",
    "E=1000.0\n",
    "nu= 0.3\n",
    "\n",
    "mu    = Constant(E / (2.0*(1.0 + nu)))\n",
    "\n",
    "lambda_ = Constant(E*nu / ((1.0 + nu)*(1.0 - 2.0*nu)))\n",
    "\n",
    "# create Mesh\n",
    "mesh = RectangleMesh(Point(0,0), Point(W,H), 3,10)\n",
    "\n",
    "#create Function Space\n",
    "V = VectorFunctionSpace(mesh, 'P', 2)\n",
    "\n",
    "#BCs\n",
    "tol=1E-14\n",
    "\n",
    "def clamped_boundary(x, on_boundary):\n",
    "    return on_boundary and abs(x[1])<tol\n",
    "\n",
    "def Neumann_Boundary(x,on_boundary):\n",
    "    return on_boundary and not(abs(x[1]<tol))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#alpha method parameters\n",
    "\n",
    "alpha_m = Constant(0.2)\n",
    "alpha_f = Constant(0.4)\n",
    "gamma   = Constant(0.5+alpha_f-alpha_m)\n",
    "beta    = Constant((gamma+0.5)**2/4.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "#parameters for Time-Stepping\n",
    "T = 1.0\n",
    "Nsteps = 80\n",
    "dt = Constant(T/Nsteps)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Loading by an Expression: sinus loading dependend on x_0\n",
    "#p = Expression(('1*sin(2*pi*t) * x[0]','0'),degree=1, t=0)\n",
    "p = Expression( ('t<1 ? 0.01 : 0.01','0'),degree=1, t=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Trial and Test Functions\n",
    "du = TrialFunction(V)\n",
    "v = TestFunction(V)\n",
    "\n",
    "u = Function(V)\n",
    "\n",
    "# function known from previous timestep\n",
    "u_old = Function(V)\n",
    "v_old = Function(V)\n",
    "a_old = Function(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Maybe there should be some BCs here that i don't understand yet\n",
    "boundary_subdomains = MeshFunction(\"size_t\", \n",
    "                                   mesh, mesh.topology().dim()-1)\n",
    "boundary_subdomains.set_all(0)\n",
    "force_boundary = AutoSubDomain(Neumann_Boundary)\n",
    "force_boundary.mark(boundary_subdomains, 3)\n",
    "\n",
    "#Define measure for boundary conditions integral\n",
    "dss=ds(subdomain_data=boundary_subdomains)\n",
    "\n",
    "\n",
    "# clamp the beam at the bottom\n",
    "bc = DirichletBC(V, Constant((0,0)), clamped_boundary)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Define strain \n",
    "def epsilon(u):\n",
    "    return 0.5*(nabla_grad(u) + nabla_grad(u).T)\n",
    "\n",
    "# Define Stress tensor\n",
    "def sigma(u):\n",
    "    return lambda_*nabla_div(u)*Identity(d) + 2*mu*epsilon(u)\n",
    "\n",
    "# Define Mass form\n",
    "def m(u,v):\n",
    "    return rho*inner(u,v)*dx\n",
    "\n",
    "# Elastic stiffness form\n",
    "def k(u,v):\n",
    "    return inner(sigma(u), sym(grad(v))) * dx\n",
    "\n",
    "# External Work\n",
    "def Wext(u_):\n",
    "    return dot(u_,p)*dss(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Update functions\n",
    "\n",
    "# Update accelleration\n",
    "def update_a(u, u_old,v_old, a_old, ufl=True):\n",
    "    if ufl:\n",
    "        dt_=dt\n",
    "        beta_=beta\n",
    "    else:\n",
    "        dt_=float(dt)\n",
    "        beta_=float(beta)\n",
    "    \n",
    "    return ((u - u_old - dt_*v_old)/beta/dt_**2 \n",
    "            - (1-2*beta_)/2/beta_*a_old)\n",
    "\n",
    "# Update velocity\n",
    "def update_v(a,u_old,v_old,a_old, ufl = True):\n",
    "    if ufl:\n",
    "        dt_=dt\n",
    "        gamma_=gamma\n",
    "    else:\n",
    "        dt_=float(dt)\n",
    "        gamma_=float(gamma)\n",
    "    \n",
    "    return v_old + dt_*((1-gamma_)*a_old + gamma_*a)\n",
    "\n",
    "def update_fields(u, u_old, v_old, a_old):\n",
    "    '''Update all fields at the end of a timestep.'''\n",
    "    \n",
    "    u_vec, u0_vec = u.vector(), u_old.vector()\n",
    "    v0_vec, a0_vec = v_old.vector(), a_old.vector()\n",
    "    \n",
    "    #call update functions\n",
    "    a_vec = update_a(u_vec, u0_vec, v0_vec, a0_vec, ufl=False)\n",
    "    v_vec = update_v(a_vec, u0_vec, v0_vec, a0_vec, ufl=False)\n",
    "    \n",
    "    #assign u->u_old\n",
    "    v_old.vector()[:], a_old.vector()[:] = v_vec, a_vec\n",
    "    u_old.vector()[:] = u.vector()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def avg(x_old, x_new, alpha):\n",
    "    return alpha*x_old + (1-alpha)*x_new\n",
    "\n",
    "# residual\n",
    "a_new = update_a(du, u_old, v_old, a_old, ufl=True)\n",
    "v_new = update_a(a_new, u_old, v_old, a_old, ufl=True)\n",
    "\n",
    "res = m(avg(a_old,a_new,alpha_m), v) \\\n",
    "+ k(avg(u_old,du, alpha_f), v) - Wext(v)\n",
    "\n",
    "a_form= lhs(res)\n",
    "L_form= rhs(res)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "#optional precomputation of LU decomposition of K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Prepare for time-stepping\n",
    "time = np.linspace(0,T,Nsteps+1)\n",
    "u_tip = np.zeros((Nsteps+1,1))\n",
    "E_ext = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time: 0.0125\n",
      "Time: 0.025\n",
      "Time: 0.037500000000000006\n",
      "Time: 0.05\n",
      "Time: 0.0625\n",
      "Time: 0.07500000000000001\n",
      "Time: 0.08750000000000001\n",
      "Time: 0.1\n",
      "Time: 0.1125\n",
      "Time: 0.125\n",
      "Time: 0.1375\n",
      "Time: 0.15000000000000002\n",
      "Time: 0.1625\n",
      "Time: 0.17500000000000002\n",
      "Time: 0.1875\n",
      "Time: 0.2\n",
      "Time: 0.21250000000000002\n",
      "Time: 0.225\n",
      "Time: 0.23750000000000002\n",
      "Time: 0.25\n",
      "Time: 0.2625\n",
      "Time: 0.275\n",
      "Time: 0.28750000000000003\n",
      "Time: 0.30000000000000004\n",
      "Time: 0.3125\n",
      "Time: 0.325\n",
      "Time: 0.3375\n",
      "Time: 0.35000000000000003\n",
      "Time: 0.36250000000000004\n",
      "Time: 0.375\n",
      "Time: 0.3875\n",
      "Time: 0.4\n",
      "Time: 0.41250000000000003\n",
      "Time: 0.42500000000000004\n",
      "Time: 0.4375\n",
      "Time: 0.45\n",
      "Time: 0.4625\n",
      "Time: 0.47500000000000003\n",
      "Time: 0.48750000000000004\n",
      "Time: 0.5\n",
      "Time: 0.5125000000000001\n",
      "Time: 0.525\n",
      "Time: 0.5375\n",
      "Time: 0.55\n",
      "Time: 0.5625\n",
      "Time: 0.5750000000000001\n",
      "Time: 0.5875\n",
      "Time: 0.6000000000000001\n",
      "Time: 0.6125\n",
      "Time: 0.625\n",
      "Time: 0.6375000000000001\n",
      "Time: 0.65\n",
      "Time: 0.6625000000000001\n",
      "Time: 0.675\n",
      "Time: 0.6875\n",
      "Time: 0.7000000000000001\n",
      "Time: 0.7125\n",
      "Time: 0.7250000000000001\n",
      "Time: 0.7375\n",
      "Time: 0.75\n",
      "Time: 0.7625000000000001\n",
      "Time: 0.775\n",
      "Time: 0.7875000000000001\n",
      "Time: 0.8\n",
      "Time: 0.8125\n",
      "Time: 0.8250000000000001\n",
      "Time: 0.8375\n",
      "Time: 0.8500000000000001\n",
      "Time: 0.8625\n",
      "Time: 0.875\n",
      "Time: 0.8875000000000001\n",
      "Time: 0.9\n",
      "Time: 0.9125000000000001\n",
      "Time: 0.925\n",
      "Time: 0.9375\n",
      "Time: 0.9500000000000001\n",
      "Time: 0.9625\n",
      "Time: 0.9750000000000001\n",
      "Time: 0.9875\n",
      "Time: 1.0\n"
     ]
    }
   ],
   "source": [
    "#time loop\n",
    "for n in range(Nsteps):\n",
    "    \n",
    "    t=time[n+1]\n",
    "    print(\"Time:\",t)\n",
    "    \n",
    "    #evaluate Force field at alpha_f-average\n",
    "    p.t=t-float(alpha_f*dt)\n",
    "    \n",
    "    #solve for new displacements\n",
    "    solve(a_form==L_form, u, bc)\n",
    "    \n",
    "    update_fields(u, u_old, v_old, a_old)\n",
    "    \n",
    "    p.t=t\n",
    "    \n",
    "    u_tip[n+1] = u(0.1,1.)[0]\n",
    "    \n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "# Plot tip displacement evolution\n",
    "plt.figure()\n",
    "plt.plot(time, u_tip)\n",
    "plt.xlabel(\"Time\")\n",
    "plt.ylabel(\"Tip displacement\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}