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   "source": [
    "# Implementation\n",
    "Author: Jørgen S. Dokken\n",
    "\n",
    "In this tutorial, you will learn how to:\n",
    "- Use a vector function space\n",
    "- Create a constant boundary condition on a vector space\n",
    "- Visualize cell wise constant functions\n",
    "- Compute Von Mises stresses\n",
    "\n",
    "## Test problem\n",
    "As a test example, we will model a clamped beam deformed under its own weigth in 3D. This can be modeled, by setting the right-hand side body force per unit volume to $f=(0,0,-\\rho g)$ with $\\rho$ the density of the beam and $g$ the acceleration of gravity. The beam is box-shaped with length $L$ and has a square cross section of width $W$. we set $u=u_D=(0,0,0)$ at the clamped end, x=0. The rest of the boundary is traction free, that is, we set $T=0$. We start by defining the physical variables used in the program."
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Scaled variable\n",
    "import pyvista\n",
    "from dolfinx import mesh, fem, plot, io, default_scalar_type\n",
    "from dolfinx.fem.petsc import LinearProblem\n",
    "from mpi4py import MPI\n",
    "import ufl\n",
    "import numpy as np\n",
    "L = 1\n",
    "W = 0.2\n",
    "mu = 1\n",
    "rho = 1\n",
    "delta = W / L\n",
    "gamma = 0.4 * delta**2\n",
    "beta = 1.25\n",
    "lambda_ = beta\n",
    "g = gamma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then create the mesh, which will consist of hexahedral elements, along with the function space. We will use the convenience function `VectorFunctionSpace`. However, we also could have used `ufl`s functionality, creating a vector element `element = ufl.VectorElement(\"Lagrange\", mesh.ufl_cell(), 1)\n",
    "`, and intitializing the function space as `V = dolfinx.fem.FunctionSpace(mesh, element)`."
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "domain = mesh.create_box(MPI.COMM_WORLD, [np.array([0, 0, 0]), np.array([L, W, W])],\n",
    "                         [20, 6, 6], cell_type=mesh.CellType.hexahedron)\n",
    "V = fem.VectorFunctionSpace(domain, (\"Lagrange\", 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Boundary conditions\n",
    "As we would like to clamp the boundary at $x=0$, we do this by using a marker function, which locate the facets where $x$ is close to zero by machine prescision."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def clamped_boundary(x):\n",
    "    return np.isclose(x[0], 0)\n",
    "\n",
    "\n",
    "fdim = domain.topology.dim - 1\n",
    "boundary_facets = mesh.locate_entities_boundary(domain, fdim, clamped_boundary)\n",
    "\n",
    "u_D = np.array([0, 0, 0], dtype=default_scalar_type)\n",
    "bc = fem.dirichletbc(u_D, fem.locate_dofs_topological(V, fdim, boundary_facets), V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we want the traction $T$ over the remaining boundary to be $0$, we create a `dolfinx.Constant`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = fem.Constant(domain, default_scalar_type((0, 0, 0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also want to specify the integration measure $\\mathrm{d}s$, which should be the integral over the boundary of our domain. We do this by using `ufl`, and its built in integration measures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "ds = ufl.Measure(\"ds\", domain=domain)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variational formulation\n",
    "We are now ready to create our variational formulation in close to mathematical syntax, as for the previous problems."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def epsilon(u):\n",
    "    return ufl.sym(ufl.grad(u))  # Equivalent to 0.5*(ufl.nabla_grad(u) + ufl.nabla_grad(u).T)\n",
    "\n",
    "\n",
    "def sigma(u):\n",
    "    return lambda_ * ufl.nabla_div(u) * ufl.Identity(len(u)) + 2 * mu * epsilon(u)\n",
    "\n",
    "\n",
    "u = ufl.TrialFunction(V)\n",
    "v = ufl.TestFunction(V)\n",
    "f = fem.Constant(domain, default_scalar_type((0, 0, -rho * g)))\n",
    "a = ufl.inner(sigma(u), epsilon(v)) * ufl.dx\n",
    "L = ufl.dot(f, v) * ufl.dx + ufl.dot(T, v) * ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{note}\n",
    "Note that we used `nabla_grad` and optionally `nabla_div` for the variational formulation, as oposed to our previous usage of\n",
    "`div` and `grad`. This is because for scalar functions $\\nabla u$ has a clear meaning\n",
    "$\\nabla u = \\left(\\frac{\\partial u}{\\partial x}, \\frac{\\partial u}{\\partial y}, \\frac{\\partial u}{\\partial z} \\right)$.\n",
    "\n",
    "However, if $u$ is vector valued, the meaning is less clear. Some sources define $\\nabla u$ as a matrix with the elements $\\frac{\\partial u_j}{\\partial x_i}$, while other  sources prefer\n",
    "$\\frac{\\partial u_i}{\\partial x_j}$. In DOLFINx `grad(u)` is defined as the amtrix with element $\\frac{\\partial u_i}{\\partial x_j}$. However, as it is common in continuum mechanics to use the other definition, `ufl` supplies us with `nabla_grad` for this purpose.\n",
    "```\n",
    "\n",
    "## Solve the linear variational problem\n",
    "As in the previous demos, we assemble the matrix and right hand side vector and use PETSc to solve our variational problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "problem = LinearProblem(a, L, bcs=[bc], petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"})\n",
    "uh = problem.solve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As in the previous demos, we can either use Pyvista or Paraview for visualization. We start by using Pyvista. Instead of adding scalar values to the grid, we add vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "      if (!reloading && (!bokeh_loaded || is_dev)) {\n",
       "\troot.Bokeh = undefined;\n",
       "      }\n",
       "      load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n",
       "\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "\trun_inline_js();\n",
       "      });\n",
       "    }\n",
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       "  // Give older versions of the autoload script a head-start to ensure\n",
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i < links.length; i++) {\n      var link = links[i]\n      if (link.href != null) {\n\texisting_stylesheets.push(link.href)\n      }\n    }\n    for (var i = 0; i < css_urls.length; i++) {\n      var url = css_urls[i];\n      if (existing_stylesheets.indexOf(url) !== -1) {\n\ton_load()\n\tcontinue;\n      }\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }    if (((window['vtk'] !== undefined) && (!(window['vtk'] instanceof HTMLElement))) || window.requirejs) {\n      var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js'];\n      for (var i = 0; i < urls.length; i++) {\n        skip.push(urls[i])\n      }\n    }    if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n      var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n      for (var i = 0; i < urls.length; i++) {\n        skip.push(urls[i])\n      }\n    }    if (((window['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n      var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/gridstack/gridstack@7.2.3/dist/gridstack-all.js'];\n      for (var i = 0; i < urls.length; i++) {\n        skip.push(urls[i])\n      }\n    }    if (((window['Notyf'] !== undefined) && (!(window['Notyf'] instanceof HTMLElement))) || window.requirejs) {\n      var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/notificationarea/notyf@3/notyf.min.js'];\n      for (var i = 0; i < urls.length; i++) {\n        skip.push(urls[i])\n      }\n    }    var existing_scripts = []\n    var scripts = document.getElementsByTagName('script')\n    for (var i = 0; i < scripts.length; i++) {\n      var script = scripts[i]\n      if (script.src != null) {\n\texisting_scripts.push(script.src)\n      }\n    }\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n\tif (!window.requirejs) {\n\t  on_load();\n\t}\n\tcontinue;\n      }\n      var element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n    for (var i = 0; i < js_modules.length; i++) {\n      var url = js_modules[i];\n      if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n\tif (!window.requirejs) {\n\t  on_load();\n\t}\n\tcontinue;\n      }\n      var element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.async = false;\n      element.src = url;\n      element.type = \"module\";\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n    for (const name in js_exports) {\n      var url = js_exports[name];\n      if (skip.indexOf(url) >= 0 || root[name] != null) {\n\tif (!window.requirejs) {\n\t  on_load();\n\t}\n\tcontinue;\n      }\n      var element = document.createElement('script');\n      element.onerror = on_error;\n      element.async = false;\n      element.type = \"module\";\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      element.textContent = `\n      import ${name} from \"${url}\"\n      window.${name} = ${name}\n      window._bokeh_on_load()\n      `\n      document.head.appendChild(element);\n    }\n    if (!js_urls.length && !js_modules.length) {\n      on_load()\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  var js_urls = [\"https://cdn.holoviz.org/panel/1.2.1/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.holoviz.org/panel/1.2.1/dist/panel.min.js\"];\n  var js_modules = [];\n  var js_exports = {};\n  var css_urls = [];\n  var inline_js = [    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\nfunction(Bokeh) {} // ensure no trailing comma for IE\n  ];\n\n  function run_inline_js() {\n    if ((root.Bokeh !== undefined) || (force === true)) {\n      for (var i = 0; i < inline_js.length; i++) {\n        inline_js[i].call(root, root.Bokeh);\n      }\n      // Cache old bokeh versions\n      if (Bokeh != undefined && !reloading) {\n\tvar NewBokeh = root.Bokeh;\n\tif (Bokeh.versions === undefined) {\n\t  Bokeh.versions = new Map();\n\t}\n\tif (NewBokeh.version !== Bokeh.version) {\n\t  Bokeh.versions.set(NewBokeh.version, NewBokeh)\n\t}\n\troot.Bokeh = Bokeh;\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    }\n    root._bokeh_is_initializing = false\n  }\n\n  function load_or_wait() {\n    // Implement a backoff loop that tries to ensure we do not load multiple\n    // versions of Bokeh and its dependencies at the same time.\n    // In recent versions we use the root._bokeh_is_initializing flag\n    // to determine whether there is an ongoing attempt to initialize\n    // bokeh, however for backward compatibility we also try to ensure\n    // that we do not start loading a newer (Panel>=1.0 and Bokeh>3) version\n    // before older versions are fully initialized.\n    if (root._bokeh_is_initializing && Date.now() > root._bokeh_timeout) {\n      root._bokeh_is_initializing = false;\n      root._bokeh_onload_callbacks = undefined;\n      console.log(\"Bokeh: BokehJS was loaded multiple times but one version failed to initialize.\");\n      load_or_wait();\n    } else if (root._bokeh_is_initializing || (typeof root._bokeh_is_initializing === \"undefined\" && root._bokeh_onload_callbacks !== undefined)) {\n      setTimeout(load_or_wait, 100);\n    } else {\n      Bokeh = root.Bokeh;\n      bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n      root._bokeh_is_initializing = true\n      root._bokeh_onload_callbacks = []\n      if (!reloading && (!bokeh_loaded || is_dev)) {\n\troot.Bokeh = undefined;\n      }\n      load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n\trun_inline_js();\n      });\n    }\n  }\n  // Give older versions of the autoload script a head-start to ensure\n  // they initialize before we start loading newer version.\n  setTimeout(load_or_wait, 100)\n}(window));"
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       "  var render_items = [{\"docid\":\"a5f35c6f-2022-4692-a778-082cf977fe6b\",\"roots\":{\"6e0c2e5b-d6e3-4173-8b96-39b01082a910\":\"e15c686d-e4a8-4bf2-8f62-ed7e8494abd1\"},\"root_ids\":[\"6e0c2e5b-d6e3-4173-8b96-39b01082a910\"]}];\n",
       "  var docs = Object.values(docs_json)\n",
       "  if (!docs) {\n",
       "    return\n",
       "  }\n",
       "  const py_version = docs[0].version.replace('rc', '-rc.').replace('.dev', '-dev.')\n",
       "  const is_dev = py_version.indexOf(\"+\") !== -1 || py_version.indexOf(\"-\") !== -1\n",
       "  function embed_document(root) {\n",
       "    var Bokeh = get_bokeh(root)\n",
       "    Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "    for (const render_item of render_items) {\n",
       "      for (const root_id of render_item.root_ids) {\n",
       "\tconst id_el = document.getElementById(root_id)\n",
       "\tif (id_el.children.length && (id_el.children[0].className === 'bk-root')) {\n",
       "\t  const root_el = id_el.children[0]\n",
       "\t  root_el.id = root_el.id + '-rendered'\n",
       "\t}\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "  function get_bokeh(root) {\n",
       "    if (root.Bokeh === undefined) {\n",
       "      return null\n",
       "    } else if (root.Bokeh.version !== py_version && !is_dev) {\n",
       "      if (root.Bokeh.versions === undefined || !root.Bokeh.versions.has(py_version)) {\n",
       "\treturn null\n",
       "      }\n",
       "      return root.Bokeh.versions.get(py_version);\n",
       "    } else if (root.Bokeh.version === py_version) {\n",
       "      return root.Bokeh\n",
       "    }\n",
       "    return null\n",
       "  }\n",
       "  function is_loaded(root) {\n",
       "    var Bokeh = get_bokeh(root)\n",
       "    return (Bokeh != null && Bokeh.Panel !== undefined && ( root['vtk'] !== undefined) && ( root['vtk'] !== undefined))\n",
       "  }\n",
       "  if (is_loaded(root)) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (is_loaded(root)) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else if (document.readyState == \"complete\") {\n",
       "        attempts++;\n",
       "        if (attempts > 200) {\n",
       "          clearInterval(timer);\n",
       "\t  var Bokeh = get_bokeh(root)\n",
       "\t  if (Bokeh == null || Bokeh.Panel == null) {\n",
       "            console.warn(\"Panel: ERROR: Unable to run Panel code because Bokeh or Panel library is missing\");\n",
       "\t  } else {\n",
       "\t    console.warn(\"Panel: WARNING: Attempting to render but not all required libraries could be resolved.\")\n",
       "\t    embed_document(root)\n",
       "\t  }\n",
       "        }\n",
       "      }\n",
       "    }, 25, root)\n",
       "  }\n",
       "})(window);</script>"
      ],
      "text/plain": [
       "VTKRenderWindowSynchronized(vtkXOpenGLRenderWindow, color_mappers=[LinearColorMapper(id='3a9...], orientation_widget=True, sizing_mode='stretch_width')"
      ]
     },
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       "id": "6e0c2e5b-d6e3-4173-8b96-39b01082a910"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyvista.start_xvfb()\n",
    "\n",
    "# Create plotter and pyvista grid\n",
    "p = pyvista.Plotter()\n",
    "topology, cell_types, geometry = plot.vtk_mesh(V)\n",
    "grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\n",
    "\n",
    "# Attach vector values to grid and warp grid by vector\n",
    "grid[\"u\"] = uh.x.array.reshape((geometry.shape[0], 3))\n",
    "actor_0 = p.add_mesh(grid, style=\"wireframe\", color=\"k\")\n",
    "warped = grid.warp_by_vector(\"u\", factor=1.5)\n",
    "actor_1 = p.add_mesh(warped, show_edges=True)\n",
    "p.show_axes()\n",
    "if not pyvista.OFF_SCREEN:\n",
    "    p.show()\n",
    "else:\n",
    "    figure_as_array = p.screenshot(\"deflection.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could also use Paraview for visualizing this.\n",
    "As explained in previous sections, we save the solution with `XDMFFile`.\n",
    "After opening the file `deformation.xdmf` in Paraview and pressing `Apply`, one can press the `Warp by vector button` ![Warp by vector](warp_by_vector.png) or go through the top menu (`Filters->Alphabetical->Warp by Vector`) and press `Apply`. We can also change the color of the deformed beam by changing the value in the color menu ![color](color.png) from `Solid Color` to `Deformation`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "with io.XDMFFile(domain.comm, \"deformation.xdmf\", \"w\") as xdmf:\n",
    "    xdmf.write_mesh(domain)\n",
    "    uh.name = \"Deformation\"\n",
    "    xdmf.write_function(uh)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stress computation\n",
    "As soon as the displacement is computed, we can compute various stress measures. We will compute the von Mises stress defined as $\\sigma_m=\\sqrt{\\frac{3}{2}s:s}$ where $s$ is the deviatoric stress tensor $s(u)=\\sigma(u)-\\frac{1}{3}\\mathrm{tr}(\\sigma(u))I$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = sigma(uh) - 1. / 3 * ufl.tr(sigma(uh)) * ufl.Identity(len(uh))\n",
    "von_Mises = ufl.sqrt(3. / 2 * ufl.inner(s, s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `von_Mises` variable is now an expression that must be projected into an appropriate function space so that we can visualize it. As `uh` is a linear combination of first order piecewise continuous functions, the von Mises stresses will be a cell-wise constant function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "V_von_mises = fem.FunctionSpace(domain, (\"DG\", 0))\n",
    "stress_expr = fem.Expression(von_Mises, V_von_mises.element.interpolation_points())\n",
    "stresses = fem.Function(V_von_mises)\n",
    "stresses.interpolate(stress_expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the previous sections, we have only visualized first order Lagrangian functions. However, the Von Mises stresses are piecewise constant on each cell. Therefore, we modify our plotting routine slightly. The first thing we notice is that we  now set values for each cell, which has a one to one correspondence with the degrees of freedom in the function space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
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       "  var render_items = [{\"docid\":\"6857ce86-eb99-4a4c-8a7e-182cf298cf39\",\"roots\":{\"dcf4dfe8-89df-4d31-90f9-ee6cde29f647\":\"decc304d-af9d-4a86-b3b8-20087914081a\"},\"root_ids\":[\"dcf4dfe8-89df-4d31-90f9-ee6cde29f647\"]}];\n",
       "  var docs = Object.values(docs_json)\n",
       "  if (!docs) {\n",
       "    return\n",
       "  }\n",
       "  const py_version = docs[0].version.replace('rc', '-rc.').replace('.dev', '-dev.')\n",
       "  const is_dev = py_version.indexOf(\"+\") !== -1 || py_version.indexOf(\"-\") !== -1\n",
       "  function embed_document(root) {\n",
       "    var Bokeh = get_bokeh(root)\n",
       "    Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "    for (const render_item of render_items) {\n",
       "      for (const root_id of render_item.root_ids) {\n",
       "\tconst id_el = document.getElementById(root_id)\n",
       "\tif (id_el.children.length && (id_el.children[0].className === 'bk-root')) {\n",
       "\t  const root_el = id_el.children[0]\n",
       "\t  root_el.id = root_el.id + '-rendered'\n",
       "\t}\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "  function get_bokeh(root) {\n",
       "    if (root.Bokeh === undefined) {\n",
       "      return null\n",
       "    } else if (root.Bokeh.version !== py_version && !is_dev) {\n",
       "      if (root.Bokeh.versions === undefined || !root.Bokeh.versions.has(py_version)) {\n",
       "\treturn null\n",
       "      }\n",
       "      return root.Bokeh.versions.get(py_version);\n",
       "    } else if (root.Bokeh.version === py_version) {\n",
       "      return root.Bokeh\n",
       "    }\n",
       "    return null\n",
       "  }\n",
       "  function is_loaded(root) {\n",
       "    var Bokeh = get_bokeh(root)\n",
       "    return (Bokeh != null && Bokeh.Panel !== undefined && ( root['vtk'] !== undefined) && ( root['vtk'] !== undefined))\n",
       "  }\n",
       "  if (is_loaded(root)) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (is_loaded(root)) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else if (document.readyState == \"complete\") {\n",
       "        attempts++;\n",
       "        if (attempts > 200) {\n",
       "          clearInterval(timer);\n",
       "\t  var Bokeh = get_bokeh(root)\n",
       "\t  if (Bokeh == null || Bokeh.Panel == null) {\n",
       "            console.warn(\"Panel: ERROR: Unable to run Panel code because Bokeh or Panel library is missing\");\n",
       "\t  } else {\n",
       "\t    console.warn(\"Panel: WARNING: Attempting to render but not all required libraries could be resolved.\")\n",
       "\t    embed_document(root)\n",
       "\t  }\n",
       "        }\n",
       "      }\n",
       "    }, 25, root)\n",
       "  }\n",
       "})(window);</script>"
      ],
      "text/plain": [
       "VTKRenderWindowSynchronized(vtkXOpenGLRenderWindow, color_mappers=[LinearColorMapper(id='44b...], orientation_widget=True, sizing_mode='stretch_width')"
      ]
     },
     "metadata": {
      "application/vnd.holoviews_exec.v0+json": {
       "id": "dcf4dfe8-89df-4d31-90f9-ee6cde29f647"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "warped.cell_data[\"VonMises\"] = stresses.vector.array\n",
    "warped.set_active_scalars(\"VonMises\")\n",
    "p = pyvista.Plotter()\n",
    "p.add_mesh(warped)\n",
    "p.show_axes()\n",
    "if not pyvista.OFF_SCREEN:\n",
    "    p.show()\n",
    "else:\n",
    "    stress_figure = p.screenshot(f\"stresses.png\")"
   ]
  }
 ],
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