notebooks/elasticity.ipynb

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    "# The equations of linear elasticity\n",
    "\n",
    "Linear elasticity is the study of how solid objects deform and become \n",
    "internally stressed due to prescribed loading conditions. It is an important problem\n",
    "in modern engineering. Its corresponding PDE is a generalization of the\n",
    "Poisson equation, and it is among one of the most popular PDEs in \n",
    "engineering. We now study its variational formulation and how to solve\n",
    "this problem using FEniCS in 2D and 3D."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PDE problem\n",
    "\n",
    "The equation governing elastic deformations of a body $\\Omega$ can be written as\n",
    "\n",
    "\\begin{equation}\n",
    "\\label{ftut-elast-varform-equilibrium}\n",
    "-\\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma} = \\boldsymbol{f}\\hbox{ in }\\Omega,\n",
    "\\end{equation}\n",
    "\n",
    "where $\\boldsymbol{\\sigma}$ is the *stress tensor*, and $\\boldsymbol{f}$ is the *body force per unit\n",
    "volume*. For isotropic materials, the stress tensor is further related to the deformation by \n",
    "the following two equations:\n",
    "\\begin{align}\n",
    "\\boldsymbol{\\sigma} &= \\lambda\\,\\hbox{tr}\\,(\\boldsymbol{\\varepsilon}) \\boldsymbol{I} + 2\\mu\\boldsymbol{\\varepsilon},\n",
    "\\label{ftut-elast-varform-stresstrain}\\\\\n",
    "\\boldsymbol{\\varepsilon} &= \\frac{1}{2}\\left(\\boldsymbol{\\nabla} \\boldsymbol{u} + (\\boldsymbol{\\nabla} \\boldsymbol{u})^{\\top}\\right),\n",
    "\\label{ftut-elast-varform-strainu}\n",
    "\\end{align}\n",
    "where $\\boldsymbol{\\varepsilon}$ is the *symmetric strain-rate tensor* (symmetric gradient), \n",
    "and $\\boldsymbol{u}$ is the *displacement vector field*, $\\boldsymbol{I}$ denotes the *identity tensor*, \n",
    "$\\mathrm{tr}$ denotes the *trace operator* on a tensor, and $\\lambda$ and $\\mu$ \n",
    "are material properties known as *Lamé's elasticity parameters*.\n",
    "\n",
    "We can combine (\\ref{ftut-elast-varform-stresstrain}) and\n",
    "(\\ref{ftut-elast-varform-strainu}) to obtain\n",
    "\\begin{equation}\n",
    "\\label{ftut-elast-varform-stressu}\n",
    "\\boldsymbol{\\boldsymbol{\\sigma}} = \\lambda(\\boldsymbol{\\nabla}\\cdot \\boldsymbol{u})\\boldsymbol{I} + \\mu(\\boldsymbol{\\nabla} \\boldsymbol{u} + (\\boldsymbol{\\nabla} \\boldsymbol{u})^{\\top})\n",
    "\\end{equation}\n",
    "\n",
    "Note that\n",
    "(\\ref{ftut-elast-varform-equilibrium})--(\\ref{ftut-elast-varform-strainu})\n",
    "can easily be transformed to a single vector PDE for $\\boldsymbol{u}$, which is the\n",
    "governing PDE for the unknown $\\boldsymbol{u}$ (Navier's equation).  In the\n",
    "derivation of the variational formulation, however, it is convenient\n",
    "to keep the equations split as above."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variational formulation\n",
    "\n",
    "The variational formulation of\n",
    "(\\ref{ftut-elast-varform-equilibrium})--(\\ref{ftut-elast-varform-strainu})\n",
    "consists of forming the inner product of\n",
    "(\\ref{ftut-elast-varform-equilibrium}) and a *vector* test function\n",
    "$\\boldsymbol{v}\\in \\hat{V}$, where $\\hat{V}$ is a vector-valued test function space, and\n",
    "integrating over the domain $\\Omega$:\n",
    "\n",
    "\\begin{equation} -\\int_\\Omega (\\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma}) \\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{x} =\n",
    "\\int_\\Omega \\boldsymbol{f}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{x},\\label{eq:elast:varform:variational}\n",
    "\\end{equation}\n",
    "where $\\mathrm{d}\\boldsymbol{x}$ denotes the volume measure.\n",
    "\n",
    "Since $\\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma}$ contains second-order derivatives of the primary\n",
    "unknown $\\boldsymbol{u}$, we integrate this term by parts:\n",
    "\n",
    "\\begin{equation}\n",
    "-\\int_\\Omega (\\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma}) \\cdot \\boldsymbol{v} \\ \\mathrm{d}\\boldsymbol{x}\n",
    "= \\int_\\Omega \\boldsymbol{\\sigma} : \\boldsymbol{\\nabla} \\boldsymbol{v} \\ \\mathrm{d}\\boldsymbol{x} - \\int_{\\partial\\Omega}\n",
    "(\\boldsymbol{\\sigma}\\cdot \\boldsymbol{n})\\cdot \\boldsymbol{v} \\ \\mathrm{d}\\boldsymbol{s},\n",
    "\\label{eq:elast:varform:integbypart}\n",
    "\\end{equation}\n",
    "\n",
    "where the colon operator is the inner product between tensors (summed\n",
    "pairwise product of all elements), $\\boldsymbol{n}$ is the outward unit normal\n",
    "at the boundary, and $\\mathrm{d}\\boldsymbol{s}$ is a measure in surface area.\n",
    "\n",
    "The quantity $\\boldsymbol{\\sigma}\\cdot \\boldsymbol{n}$ is known as the\n",
    "*traction* or stress vector at the boundary, and is often prescribed\n",
    "as a boundary condition on a part $\\partial_T\\Omega$ of the boundary \n",
    "as $\\boldsymbol{\\sigma}\\cdot \\boldsymbol{n} = \\boldsymbol{T}$, whereas the \n",
    "remaining part of the boundary would have the value of the displacements prescribed.\n",
    "\n",
    "Substituting (\\ref{eq:elast:varform:integbypart}) into (\\ref{eq:elast:varform:variational}), \n",
    "we thus obtain\n",
    "\\begin{equation}\n",
    "\\int_\\Omega \\boldsymbol{\\sigma} : \\boldsymbol{\\nabla} \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{x} =\n",
    "\\int_\\Omega \\boldsymbol{f}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{x}\n",
    "+ \\int_{\\partial\\Omega} \\boldsymbol{T}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{s}.\n",
    "\\label{eq:elast:varform:varform_original}\n",
    "\\end{equation}\n",
    "\n",
    "Inserting the expression (\\ref{ftut-elast-varform-stressu}) for\n",
    "$\\boldsymbol{\\sigma}$ gives the variational form with $\\boldsymbol{u}$ as unknown. "
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symmetrizing $\\boldsymbol{\\nabla} \\boldsymbol{v}$\n",
    "\n",
    "One can show that the inner product of a symmetric tensor $\\boldsymbol{A}$ and an\n",
    "anti-symmetric tensor $\\boldsymbol{B}$ vanishes. Since $\\boldsymbol{\\sigma}$ is a\n",
    "symmetric tensor, if we express $\\boldsymbol{\\nabla} \\boldsymbol{v}$ as a sum\n",
    "of its symmetric and anti-symmetric parts, only the symmetric part will\n",
    "survive in the product $\\boldsymbol{\\sigma} :\\boldsymbol{\\nabla} \\boldsymbol{v}$. Thus replacing $\\boldsymbol{\\nabla} \\boldsymbol{u}$ by the symmetric gradient gives rise to the slightly different variational form\n",
    "\n",
    "\\begin{equation}\n",
    "\\int_\\Omega \\boldsymbol{\\sigma} : \\boldsymbol{\\epsilon}(\\boldsymbol{v})\\ \\mathrm{d}\\boldsymbol{x} =\n",
    "\\int_\\Omega \\boldsymbol{f}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{x}\n",
    "+ \\int_{\\partial\\Omega} \\boldsymbol{T}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{s}\n",
    "\\label{ftut-elast-varform-sigma_inner_gradv}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\boldsymbol{\\epsilon}(\\boldsymbol{v})$ is the symmetric part of $\\boldsymbol{\\nabla} \\boldsymbol{v}$:\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\epsilon}(\\boldsymbol{v}) = \\frac{1}{2}\\left(\\boldsymbol{\\nabla} \\boldsymbol{v} + (\\boldsymbol{\\nabla} \\boldsymbol{v})^{\\top}\\right).\n",
    "\\label{ftut-elast-varform-sigma_inner_eps}\n",
    "\\end{equation}\n",
    "\n",
    "The formulation (\\ref{ftut-elast-varform-sigma_inner_gradv}) is what naturally\n",
    "arises from the minimization of elastic potential energy, and is a more\n",
    "popular formulation than (\\ref{eq:elast:varform:varform_original}).\n",
    "\n",
    "Here, $\\boldsymbol{\\epsilon}$ is a useful operator. The symmetric strain-rate tensor $\\boldsymbol{\\varepsilon}$ in (\\ref{ftut-elast-varform-strainu}) is equal to $\\boldsymbol{\\epsilon}(\\boldsymbol{u})$. "
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Enforcing boundary conditions\n",
    "\n",
    "Now let us consider how to enforce boundary conditions. \n",
    "For Dirichlet boundaries, we will enforce boundary-conditions strongly.\n",
    "For these points, no test functions are associated with the Dirichlet nodes.\n",
    "\n",
    "For traction boundary conditions, we will enforce the boundary condition\n",
    "weakly using the variational form (\\ref{ftut-elast-varform-sigma_inner_gradv}).\n",
    "Similar to the Poisson equation, we require their corresponding test\n",
    "functions $\\boldsymbol{v}$ vanish along $\\partial \\Omega$ for interior points.\n",
    "Then, the boundary integral above has no effects for points on\n",
    "$\\partial\\Omega\\setminus\\partial\\Omega_T$."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Summary of variational form\n",
    "In summary, the variational problem is to find $\\boldsymbol{u}$ in a vector function space $\\hat{V}$ such that\n",
    "\\begin{equation}\n",
    "a(\\boldsymbol{u},\\boldsymbol{v}) = L(\\boldsymbol{v})\\quad\\forall \\boldsymbol{v}\\in\\hat{V},\n",
    "\\end{equation}\n",
    "where \n",
    "\\begin{align}\n",
    "a(\\boldsymbol{u},\\boldsymbol{v}) &= \\int_\\Omega\\sigma(\\boldsymbol{u}) :\\varepsilon(\\boldsymbol{v})\\ \\mathrm{d}\\boldsymbol{x},\\\\\n",
    "L(\\boldsymbol{v}) &= \\int_\\Omega \\boldsymbol{f}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{x} + \\int_{\\partial\\Omega_T} \\boldsymbol{T}\\cdot \\boldsymbol{v}\\ \\mathrm{d}\\boldsymbol{s},\n",
    "\\end{align}\n",
    "and \n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\sigma}(\\boldsymbol{u}) = \\lambda(\\boldsymbol{\\nabla}\\cdot \\boldsymbol{u})\\boldsymbol{I} + \\mu(\\boldsymbol{\\nabla} \\boldsymbol{u} + (\\boldsymbol{\\nabla} \\boldsymbol{u})^{\\top}).\\\\\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## FEniCS implementation\n",
    "\n",
    "To demonstrate the implementation, we will model a clamped beam deformed under its\n",
    "own weight in 3D. This can be modeled by setting the right-hand side\n",
    "body force per unit volume to $\\boldsymbol{f}=(0,0,-\\rho g)$ with $\\rho$ the\n",
    "density of the beam and $g$ the acceleration of gravity. The beam is\n",
    "box-shaped with length $L$ and has a square cross section of width $W$. We\n",
    "set $\\boldsymbol{u}=(0,0,0)$ at the clamped end, $x=0$. The rest of the boundary is\n",
    "traction free; that is, we set $\\boldsymbol{T} = 0$. Therefore,\n",
    "$$L(\\boldsymbol{v}) = \\int_\\Omega \\boldsymbol{f}\\cdot \\boldsymbol{v} \\mathrm{d}\\boldsymbol{x}$$\n",
    "for this problem.\n",
    "\n",
    "<img src=\"https://fenicsproject.org/pub/tutorial/html/fig/elasticity.png\" width=\"400\">\n",
    "\n",
    "### Import packages\n",
    "\n",
    "We start by importing fenics and enforcing matplotlib. In addition, we import `mshr` for mesh generation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from fenics import *\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generate the mesh and function spaces\n",
    "\n",
    "Our action startes by generating meshes and defining function spaces."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create mesh and define function space\n",
    "length = 1; width = 0.2\n",
    "\n",
    "mesh = BoxMesh(Point(0, 0, 0), Point(length, width, width), 10, 3, 3)\n",
    "V = VectorFunctionSpace(mesh, 'P', 2)\n",
    "\n",
    "vtkfile = File('elasticity/mesh.pvd')\n",
    "vtkfile << mesh"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Define the variational problem\n",
    "\n",
    "The primary unknown is now a vector field $\\boldsymbol{u}$ and not a scalar field,\n",
    "so we need to work with a vector function space. We will use \n",
    "piecewise-linear basis functions for all the components.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "u = TrialFunction(V)\n",
    "v = TestFunction(V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With `u = TrialFunction(V)` we get `u` as a vector-valued finite element\n",
    "function with three components for this 3D problem.\n",
    "\n",
    "Next, we define the stress tensor and $a$. The gradient and divergence operators \n",
    "now have a prefix `nabla_`.\n",
    "This is not strictly necessary in the present problem, but is\n",
    "recommended in general for vector PDEs arising from continuum mechanics,\n",
    "if you interpret $\\boldsymbol{\\nabla}$ as a vector in the PDE notation. See\n",
    "the notes on `grad(u)` vs. `nabla_grad(u)` below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define strain and stress\n",
    "def epsilon(u):\n",
    "    return 0.5*(nabla_grad(u) + nabla_grad(u).T)\n",
    "    #return sym(nabla_grad(u))\n",
    "\n",
    "beta = 1.25\n",
    "lambda_ = beta\n",
    "mu = 1\n",
    "d = u.geometric_dimension()  # space dimension\n",
    "\n",
    "def sigma(u):\n",
    "    return lambda_*nabla_div(u)*Identity(d) + 2*mu*epsilon(u)\n",
    "\n",
    "# Define a\n",
    "a = inner(sigma(u), epsilon(v))*dx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To define $L$, $\\boldsymbol{f}=(0, 0, \\rho*g)$ is a constant vector, instead of a scalar.\n",
    "Such a vector constant is specified as `Constant(0, 0, -rho*g)` in FEniCS."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define L\n",
    "rho = 1.\n",
    "delta = width/length\n",
    "gamma = 0.4*delta**2\n",
    "g = gamma\n",
    "\n",
    "#T = Constant((0,0,0))\n",
    "f = Constant((0, 0, -rho*g))\n",
    "L = dot(f, v)*dx # + dot(T, v)*ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Define boundary conditions\n",
    "\n",
    "We only specify the Dirichlet boundary condition. For the boundary condition \n",
    "$u=(0, 0, 0)$, we must set a vector value to zero, not just a scalar. \n",
    "We specify the vector constant as `Constant((0, 0, 0))`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define boundary condition\n",
    "tol = 1E-14\n",
    "\n",
    "def clamped_boundary(x, on_boundary):\n",
    "    return on_boundary and x[0] < tol\n",
    "\n",
    "bc = DirichletBC(V, Constant((0, 0, 0)), clamped_boundary)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solve the variational problem\n",
    "\n",
    "Finally, we can solve the problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute solution\n",
    "u = Function(V)\n",
    "solve(a == L, u, bc)\n",
    "\n",
    "vtkfile = File('elasticity/solutions.pvd')\n",
    "vtkfile << u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Post-processing\n",
    "\n",
    "### Plot the solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the displacements `u`, we can compute various stress measures. \n",
    "We will compute the von Mises stress defined as\n",
    "$$\\boldsymbol{\\sigma}_M = \\sqrt{\\frac{3}{2}\\boldsymbol{s}:\\boldsymbol{s}},$$\n",
    "where $\\boldsymbol{s}$ is the deviatoric stress tensor\n",
    "$$\\boldsymbol{s} = \\boldsymbol{\\sigma} - \\frac{1}{3}\\mathrm{tr}\\,(\\boldsymbol{\\sigma})\\,\\boldsymbol{I}.$$\n",
    "There is a one-to-one mapping between these formulas and the FEniCS code,\n",
    "where we compute the `von_Mises` variable by projecting it to the finite element space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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+8Qmam5tL/n/llVf43Oc+x969e/n0pz/Njh07ePnll3nppZe44447kFLy0EMP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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa233599a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot stress\n",
    "s = sigma(u) - (1./3)*tr(sigma(u))*Identity(d)  # deviatoric stress\n",
    "von_Mises = sqrt(3./2*inner(s, s))\n",
    "V = FunctionSpace(mesh, 'P', 1)\n",
    "von_Mises = project(von_Mises, V)\n",
    "plot(von_Mises, title='Stress intensity')\n",
    "\n",
    "vtkfile = File('elasticity/stress_intensity.pvd')\n",
    "vtkfile << von_Mises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also evaluate the magnitude of displacements."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "min/max u: -0.00126709539513 0.240872921329\n"
     ]
    },
    {
     "data": {
      "image/png": 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AqoEE3zt8CL/mQQEikci8hL5kMYzrWb55ahuDhSR+WeNjPVexoe6V1c5Mn3K7bbOsrZ14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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa234650f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute magnitude of displacement\n",
    "u_magnitude = sqrt(dot(u, u))\n",
    "u_magnitude = project(u_magnitude, V)\n",
    "plot(u_magnitude, 'Displacement magnitude')\n",
    "print('min/max u:',\n",
    "      u_magnitude.vector().array().min(),\n",
    "      u_magnitude.vector().array().max())\n",
    "\n",
    "vtkfile = File('elasticity/disp_mag.pvd')\n",
    "vtkfile << u_magnitude"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Note on `grad(u)` vs. `nabla_grad(u)`\n",
    "\n",
    "For scalar functions, $\\boldsymbol{\\nabla} u$ has a clear meaning as the vector\n",
    "$$\\boldsymbol{\\nabla} u =\\left(\\frac{\\partial u}{\\partial x}, \\frac{\\partial u}{\\partial y},\n",
    "\\frac{\\partial u}{\\partial z}\\right).$$\n",
    "\n",
    "However, for a vector-valued function $\\boldsymbol{u}$, the meaning is less clear.\n",
    "Some sources define $\\boldsymbol{\\nabla}\\boldsymbol{u}$ as the matrix with elements\n",
    "$\\partial u_j / \\partial x_i$, especially in continuum mechanics.\n",
    "Some other sources prefer $\\partial u_i / \\partial x_j$, such as in the \n",
    "*Jacobian matrix*.\n",
    "\n",
    "In our notes, $\\boldsymbol{\\nabla}$ always denotes the matrix \n",
    "with elements $\\partial u_j / \\partial x_i$, to be consistent with most PDEs.\n",
    "The corresponding operator in FEniCS is `nabla_grad(u)`.\n",
    "For the Jacobian matrix, its corresponding operator in FEniCS is `grad(u)`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional notes on linear elasticity\n",
    "\n",
    "### Scaling\n",
    "\n",
    "It is often advantageous to scale a problem as it reduces the need for setting\n",
    "physical parameters, and one obtains dimensionsless numbers that\n",
    "reflect the competition of parameters and physical effects. We develop\n",
    "the code for the original model with dimensions, and run the scaled\n",
    "problem by tweaking parameters appropriately. Scaling reduces the\n",
    "number of active parameters from 6 to 2 for the present application.\n",
    "\n",
    "In Navier's equation for $\\boldsymbol{u}$, arising from inserting\n",
    "(\\ref{ftut-elast-varform-stresstrain}) and\n",
    "(\\ref{ftut-elast-varform-strainu}) into\n",
    "(\\ref{ftut-elast-varform-equilibrium}),\n",
    "\n",
    "$$ -(\\lambda + \\mu)\\boldsymbol{\\nabla}(\\boldsymbol{\\nabla}\\cdot \\boldsymbol{u}) - \\mu\\boldsymbol{\\nabla}^2 \\boldsymbol{u} = \\boldsymbol{f},$$\n",
    "\n",
    "we insert coordinates made dimensionless by $L$, and $\\bar{\\boldsymbol{u}}=\\boldsymbol{u}/U$,\n",
    "which results in the dimensionless governing equation\n",
    "\n",
    "$$\n",
    "-\\beta\\bar{\\boldsymbol{\\nabla}}(\\bar{\\boldsymbol{\\nabla}}\\cdot\\bar{\\boldsymbol{u}})-\\bar{\\boldsymbol{\\nabla}}^{2}\\bar{\\boldsymbol{u}}=\\bar{\\boldsymbol{f}},\\quad\\bar{\\boldsymbol{f}}=(0,0,\\gamma)\n",
    ",$$\n",
    "where $\\beta = 1 + \\lambda/\\mu$ is a dimensionless elasticity parameter and\n",
    "\n",
    "$$ \\gamma = \\frac{\\rho gL^2}{\\mu U}$$\n",
    "is a dimensionless variable reflecting the ratio of the load\n",
    "$\\rho g$ and the shear stress\n",
    "term $\\mu\\boldsymbol{\\nabla}^2 \\boldsymbol{u}\\sim \\mu U/L^2$ in the PDE.\n",
    "\n",
    "One option for the scaling is to choose $U$ such that $\\gamma$ is of\n",
    "unit size ($U = \\rho gL^2/\\mu$). However, in elasticity, this leads\n",
    "to displacements of the size of the geometry, which makes plots\n",
    "look very strange. Therefore, we want the characteristic displacement\n",
    "to be a small fraction of the characteristic length of the geometry.\n",
    "This can be achieved by choosing $U$ equal to the maximum deflection\n",
    "of a clamped beam, for which there actually exists a formula: $U =\n",
    "\\frac{3}{2}\\rho gL^2\\delta^2/E$, where $\\delta = L/W$ is a\n",
    "parameter reflecting how slender the beam is, and $E$ is the modulus\n",
    "of elasticity. Thus, the dimensionless parameter $\\delta$ is very\n",
    "important in the problem (as expected, since $\\delta\\gg 1$ is what\n",
    "gives beam theory!). Taking $E$ to be of the same order as $\\mu$,\n",
    "which is the case for many materials, we realize that $\\gamma \\sim\n",
    "\\delta^{-2}$ is an appropriate choice.  Experimenting with the code to\n",
    "find a displacement that ``looks right'' in plots of the deformed\n",
    "geometry, points to $\\gamma = 0.4\\delta^{-2}$ as our final choice of\n",
    "$\\gamma$.\n",
    "\n",
    "The simulation code implements the problem with dimensions and\n",
    "physical parameters $\\lambda$, $\\mu$, $\\rho$, $g$, $L$, and $W$.\n",
    "However, we can easily reuse this code for a scaled problem: just set\n",
    "$\\mu = \\rho = L = 1$, $W$ as $W/L$ ($\\delta^{-1}$), $g=\\gamma$, and\n",
    "$\\lambda=\\beta$."
   ]
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    "### Anisotropic elasticity\n",
    "We can easily allow for anisotropic materials since\n",
    "(\\ref{ftut-elast-varform-sigma_inner_eps}) is just the inner product of\n",
    "the stress tensor and the strain tensor. Anisotropic elasticity is\n",
    "easiest to deal with by having a 6-vector with the unique components of\n",
    "stress and strain and relate the 6-vectors through a $6\\times 6$\n",
    "elasticity matrix $\\boldsymbol{D}$: $\\boldsymbol{\\sigma} = \\boldsymbol{D}\\boldsymbol{\\varepsilon}$. We then form the\n",
    "inner product of $\\boldsymbol{\\sigma}$ and $\\boldsymbol{\\varepsilon}$ to be $\\boldsymbol{\\varepsilon}^T \\boldsymbol{D}\\boldsymbol{\\varepsilon}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Acknowledgment**: This notebook was adapted from [The FEniCS Tutorial Volume I](https://fenicsproject.org/pub/tutorial/sphinx1/) by Hans Petter Langtangen and Anders Logg, released under CC Attribution 4.0 license."
   ]
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