cube.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "d806def4-6f53-495b-ba43-310241d9d572",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function now() {\n",
       "    return new Date();\n",
       "  }\n",
       "\n",
       "  var force = true;\n",
       "\n",
       "  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n",
       "    root._bokeh_onload_callbacks = [];\n",
       "    root._bokeh_is_loading = undefined;\n",
       "  }\n",
       "\n",
       "  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n",
       "    root._bokeh_timeout = Date.now() + 5000;\n",
       "    root._bokeh_failed_load = false;\n",
       "  }\n",
       "\n",
       "  function run_callbacks() {\n",
       "    try {\n",
       "      root._bokeh_onload_callbacks.forEach(function(callback) {\n",
       "        if (callback != null)\n",
       "          callback();\n",
       "      });\n",
       "    } finally {\n",
       "      delete root._bokeh_onload_callbacks\n",
       "    }\n",
       "    console.debug(\"Bokeh: all callbacks have finished\");\n",
       "  }\n",
       "\n",
       "  function load_libs(css_urls, js_urls, js_modules, callback) {\n",
       "    if (css_urls == null) css_urls = [];\n",
       "    if (js_urls == null) js_urls = [];\n",
       "    if (js_modules == null) js_modules = [];\n",
       "\n",
       "    root._bokeh_onload_callbacks.push(callback);\n",
       "    if (root._bokeh_is_loading > 0) {\n",
       "      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n",
       "      return null;\n",
       "    }\n",
       "    if (js_urls.length === 0 && js_modules.length === 0) {\n",
       "      run_callbacks();\n",
       "      return null;\n",
       "    }\n",
       "    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n",
       "\n",
       "    function on_load() {\n",
       "      root._bokeh_is_loading--;\n",
       "      if (root._bokeh_is_loading === 0) {\n",
       "        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n",
       "        run_callbacks()\n",
       "      }\n",
       "    }\n",
       "\n",
       "    function on_error() {\n",
       "      console.error(\"failed to load \" + url);\n",
       "    }\n",
       "\n",
       "    for (var i = 0; i < css_urls.length; i++) {\n",
       "      var url = css_urls[i];\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",
       "    }\n",
       "\n",
       "    var skip = [];\n",
       "    if (window.requirejs) {\n",
       "      window.requirejs.config({'packages': {}, 'paths': {'vtk': 'https://cdn.jsdelivr.net/npm/vtk.js@20.0.1/vtk', 'gridstack': 'https://cdn.jsdelivr.net/npm/gridstack@4.2.5/dist/gridstack-h5', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'vtk': {'exports': 'vtk'}, 'gridstack': {'exports': 'GridStack'}}});\n",
       "      require([\"vtk\"], function() {\n",
       "\ton_load()\n",
       "      })\n",
       "      require([\"gridstack\"], function(GridStack) {\n",
       "\twindow.GridStack = GridStack\n",
       "\ton_load()\n",
       "      })\n",
       "      require([\"notyf\"], function() {\n",
       "\ton_load()\n",
       "      })\n",
       "      root._bokeh_is_loading = css_urls.length + 3;\n",
       "    } else {\n",
       "      root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length;\n",
       "    }    if (((window['vtk'] !== undefined) && (!(window['vtk'] instanceof HTMLElement))) || window.requirejs) {\n",
       "      var urls = ['https://cdn.holoviz.org/panel/0.14.4/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['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n",
       "      var urls = ['https://cdn.holoviz.org/panel/0.14.4/dist/bundled/gridstack/gridstack@4.2.5/dist/gridstack-h5.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/0.14.4/dist/bundled/notificationarea/notyf@3/notyf.min.js'];\n",
       "      for (var i = 0; i < urls.length; i++) {\n",
       "        skip.push(urls[i])\n",
       "      }\n",
       "    }    for (var i = 0; i < js_urls.length; i++) {\n",
       "      var url = js_urls[i];\n",
       "      if (skip.indexOf(url) >= 0) {\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) >= 0) {\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",
       "    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/0.14.4/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-2.4.3.min.js\", \"https://unpkg.com/@holoviz/panel@0.14.4/dist/panel.min.js\"];\n",
       "  var js_modules = [];\n",
       "  var css_urls = [\"https://cdn.holoviz.org/panel/0.14.4/dist/css/alerts.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/loading.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/widgets.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/markdown.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/json.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/dataframe.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/card.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/debugger.css\"];\n",
       "  var inline_js = [    function(Bokeh) {\n",
       "      inject_raw_css(\"\\n    .bk.pn-loading.arc:before {\\n      background-image: url(\\\"data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHN0eWxlPSJtYXJnaW46IGF1dG87IGJhY2tncm91bmQ6IG5vbmU7IGRpc3BsYXk6IGJsb2NrOyBzaGFwZS1yZW5kZXJpbmc6IGF1dG87IiB2aWV3Qm94PSIwIDAgMTAwIDEwMCIgcHJlc2VydmVBc3BlY3RSYXRpbz0ieE1pZFlNaWQiPiAgPGNpcmNsZSBjeD0iNTAiIGN5PSI1MCIgZmlsbD0ibm9uZSIgc3Ryb2tlPSIjYzNjM2MzIiBzdHJva2Utd2lkdGg9IjEwIiByPSIzNSIgc3Ryb2tlLWRhc2hhcnJheT0iMTY0LjkzMzYxNDMxMzQ2NDE1IDU2Ljk3Nzg3MTQzNzgyMTM4Ij4gICAgPGFuaW1hdGVUcmFuc2Zvcm0gYXR0cmlidXRlTmFtZT0idHJhbnNmb3JtIiB0eXBlPSJyb3RhdGUiIHJlcGVhdENvdW50PSJpbmRlZmluaXRlIiBkdXI9IjFzIiB2YWx1ZXM9IjAgNTAgNTA7MzYwIDUwIDUwIiBrZXlUaW1lcz0iMDsxIj48L2FuaW1hdGVUcmFuc2Zvcm0+ICA8L2NpcmNsZT48L3N2Zz4=\\\");\\n      background-size: auto calc(min(50%, 400px));\\n    }\\n    \");\n",
       "    },    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "function(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",
       "      }} 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",
       "  }\n",
       "\n",
       "  if (root._bokeh_is_loading === 0) {\n",
       "    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(css_urls, js_urls, js_modules, function() {\n",
       "      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
       "    });\n",
       "  }\n",
       "}(window));"
      ],
      "application/vnd.holoviews_load.v0+json": "(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  var force = true;\n\n  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, js_modules, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n    if (js_modules == null) js_modules = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls.length === 0 && js_modules.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error() {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (var i = 0; i < css_urls.length; i++) {\n      var url = css_urls[i];\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    }\n\n    var skip = [];\n    if (window.requirejs) {\n      window.requirejs.config({'packages': {}, 'paths': {'vtk': 'https://cdn.jsdelivr.net/npm/vtk.js@20.0.1/vtk', 'gridstack': 'https://cdn.jsdelivr.net/npm/gridstack@4.2.5/dist/gridstack-h5', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'vtk': {'exports': 'vtk'}, 'gridstack': {'exports': 'GridStack'}}});\n      require([\"vtk\"], function() {\n\ton_load()\n      })\n      require([\"gridstack\"], function(GridStack) {\n\twindow.GridStack = GridStack\n\ton_load()\n      })\n      require([\"notyf\"], function() {\n\ton_load()\n      })\n      root._bokeh_is_loading = css_urls.length + 3;\n    } else {\n      root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length;\n    }    if (((window['vtk'] !== undefined) && (!(window['vtk'] instanceof HTMLElement))) || window.requirejs) {\n      var urls = ['https://cdn.holoviz.org/panel/0.14.4/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['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n      var urls = ['https://cdn.holoviz.org/panel/0.14.4/dist/bundled/gridstack/gridstack@4.2.5/dist/gridstack-h5.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/0.14.4/dist/bundled/notificationarea/notyf@3/notyf.min.js'];\n      for (var i = 0; i < urls.length; i++) {\n        skip.push(urls[i])\n      }\n    }    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      if (skip.indexOf(url) >= 0) {\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) >= 0) {\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    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/0.14.4/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-2.4.3.min.js\", \"https://unpkg.com/@holoviz/panel@0.14.4/dist/panel.min.js\"];\n  var js_modules = [];\n  var css_urls = [\"https://cdn.holoviz.org/panel/0.14.4/dist/css/alerts.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/loading.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/widgets.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/markdown.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/json.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/dataframe.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/card.css\", \"https://cdn.holoviz.org/panel/0.14.4/dist/css/debugger.css\"];\n  var inline_js = [    function(Bokeh) {\n      inject_raw_css(\"\\n    .bk.pn-loading.arc:before {\\n      background-image: url(\\\"data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHN0eWxlPSJtYXJnaW46IGF1dG87IGJhY2tncm91bmQ6IG5vbmU7IGRpc3BsYXk6IGJsb2NrOyBzaGFwZS1yZW5kZXJpbmc6IGF1dG87IiB2aWV3Qm94PSIwIDAgMTAwIDEwMCIgcHJlc2VydmVBc3BlY3RSYXRpbz0ieE1pZFlNaWQiPiAgPGNpcmNsZSBjeD0iNTAiIGN5PSI1MCIgZmlsbD0ibm9uZSIgc3Ryb2tlPSIjYzNjM2MzIiBzdHJva2Utd2lkdGg9IjEwIiByPSIzNSIgc3Ryb2tlLWRhc2hhcnJheT0iMTY0LjkzMzYxNDMxMzQ2NDE1IDU2Ljk3Nzg3MTQzNzgyMTM4Ij4gICAgPGFuaW1hdGVUcmFuc2Zvcm0gYXR0cmlidXRlTmFtZT0idHJhbnNmb3JtIiB0eXBlPSJyb3RhdGUiIHJlcGVhdENvdW50PSJpbmRlZmluaXRlIiBkdXI9IjFzIiB2YWx1ZXM9IjAgNTAgNTA7MzYwIDUwIDUwIiBrZXlUaW1lcz0iMDsxIj48L2FuaW1hdGVUcmFuc2Zvcm0+ICA8L2NpcmNsZT48L3N2Zz4=\\\");\\n      background-size: auto calc(min(50%, 400px));\\n    }\\n    \");\n    },    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      }} 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  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, js_modules, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "\n",
       "if ((window.PyViz === undefined) || (window.PyViz instanceof HTMLElement)) {\n",
       "  window.PyViz = {comms: {}, comm_status:{}, kernels:{}, receivers: {}, plot_index: []}\n",
       "}\n",
       "\n",
       "\n",
       "    function JupyterCommManager() {\n",
       "    }\n",
       "\n",
       "    JupyterCommManager.prototype.register_target = function(plot_id, comm_id, msg_handler) {\n",
       "      if (window.comm_manager || ((window.Jupyter !== undefined) && (Jupyter.notebook.kernel != null))) {\n",
       "        var comm_manager = window.comm_manager || Jupyter.notebook.kernel.comm_manager;\n",
       "        comm_manager.register_target(comm_id, function(comm) {\n",
       "          comm.on_msg(msg_handler);\n",
       "        });\n",
       "      } else if ((plot_id in window.PyViz.kernels) && (window.PyViz.kernels[plot_id])) {\n",
       "        window.PyViz.kernels[plot_id].registerCommTarget(comm_id, function(comm) {\n",
       "          comm.onMsg = msg_handler;\n",
       "        });\n",
       "      } else if (typeof google != 'undefined' && google.colab.kernel != null) {\n",
       "        google.colab.kernel.comms.registerTarget(comm_id, (comm) => {\n",
       "          var messages = comm.messages[Symbol.asyncIterator]();\n",
       "          function processIteratorResult(result) {\n",
       "            var message = result.value;\n",
       "            console.log(message)\n",
       "            var content = {data: message.data, comm_id};\n",
       "            var buffers = []\n",
       "            for (var buffer of message.buffers || []) {\n",
       "              buffers.push(new DataView(buffer))\n",
       "            }\n",
       "            var metadata = message.metadata || {};\n",
       "            var msg = {content, buffers, metadata}\n",
       "            msg_handler(msg);\n",
       "            return messages.next().then(processIteratorResult);\n",
       "          }\n",
       "          return messages.next().then(processIteratorResult);\n",
       "        })\n",
       "      }\n",
       "    }\n",
       "\n",
       "    JupyterCommManager.prototype.get_client_comm = function(plot_id, comm_id, msg_handler) {\n",
       "      if (comm_id in window.PyViz.comms) {\n",
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       "      } else if (window.comm_manager || ((window.Jupyter !== undefined) && (Jupyter.notebook.kernel != null))) {\n",
       "        var comm_manager = window.comm_manager || Jupyter.notebook.kernel.comm_manager;\n",
       "        var comm = comm_manager.new_comm(comm_id, {}, {}, {}, comm_id);\n",
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       "        }\n",
       "      } else if ((plot_id in window.PyViz.kernels) && (window.PyViz.kernels[plot_id])) {\n",
       "        var comm = window.PyViz.kernels[plot_id].connectToComm(comm_id);\n",
       "        comm.open();\n",
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       "      } else if (typeof google != 'undefined' && google.colab.kernel != null) {\n",
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       "        var sendClosure = (data, metadata, buffers, disposeOnDone) => {\n",
       "          return comm_promise.then((comm) => {\n",
       "            comm.send(data, metadata, buffers, disposeOnDone);\n",
       "          });\n",
       "        };\n",
       "        var comm = {\n",
       "          send: sendClosure\n",
       "        };\n",
       "      }\n",
       "      window.PyViz.comms[comm_id] = comm;\n",
       "      return comm;\n",
       "    }\n",
       "    window.PyViz.comm_manager = new JupyterCommManager();\n",
       "    \n",
       "\n",
       "\n",
       "var JS_MIME_TYPE = 'application/javascript';\n",
       "var HTML_MIME_TYPE = 'text/html';\n",
       "var EXEC_MIME_TYPE = 'application/vnd.holoviews_exec.v0+json';\n",
       "var CLASS_NAME = 'output';\n",
       "\n",
       "/**\n",
       " * Render data to the DOM node\n",
       " */\n",
       "function render(props, node) {\n",
       "  var div = document.createElement(\"div\");\n",
       "  var script = document.createElement(\"script\");\n",
       "  node.appendChild(div);\n",
       "  node.appendChild(script);\n",
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       "\n",
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       "    html_node.innerHTML = output.data[HTML_MIME_TYPE];\n",
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       "    }\n",
       "    output_area._hv_plot_id = id;\n",
       "    if ((window.Bokeh !== undefined) && (id in Bokeh.index)) {\n",
       "      window.PyViz.plot_index[id] = Bokeh.index[id];\n",
       "    } else {\n",
       "      window.PyViz.plot_index[id] = null;\n",
       "    }\n",
       "  } else if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n",
       "    var bk_div = document.createElement(\"div\");\n",
       "    bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
       "    var script_attrs = bk_div.children[0].attributes;\n",
       "    for (var i = 0; i < script_attrs.length; i++) {\n",
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       "    // store reference to server id on output_area\n",
       "    output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n",
       "  }\n",
       "}\n",
       "\n",
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       " */\n",
       "function handle_clear_output(event, handle) {\n",
       "  var id = handle.cell.output_area._hv_plot_id;\n",
       "  var server_id = handle.cell.output_area._bokeh_server_id;\n",
       "  if (((id === undefined) || !(id in PyViz.plot_index)) && (server_id !== undefined)) { return; }\n",
       "  var comm = window.PyViz.comm_manager.get_client_comm(\"hv-extension-comm\", \"hv-extension-comm\", function () {});\n",
       "  if (server_id !== null) {\n",
       "    comm.send({event_type: 'server_delete', 'id': server_id});\n",
       "    return;\n",
       "  } else if (comm !== null) {\n",
       "    comm.send({event_type: 'delete', 'id': id});\n",
       "  }\n",
       "  delete PyViz.plot_index[id];\n",
       "  if ((window.Bokeh !== undefined) & (id in window.Bokeh.index)) {\n",
       "    var doc = window.Bokeh.index[id].model.document\n",
       "    doc.clear();\n",
       "    const i = window.Bokeh.documents.indexOf(doc);\n",
       "    if (i > -1) {\n",
       "      window.Bokeh.documents.splice(i, 1);\n",
       "    }\n",
       "  }\n",
       "}\n",
       "\n",
       "/**\n",
       " * Handle kernel restart event\n",
       " */\n",
       "function handle_kernel_cleanup(event, handle) {\n",
       "  delete PyViz.comms[\"hv-extension-comm\"];\n",
       "  window.PyViz.plot_index = {}\n",
       "}\n",
       "\n",
       "/**\n",
       " * Handle update_display_data messages\n",
       " */\n",
       "function handle_update_output(event, handle) {\n",
       "  handle_clear_output(event, {cell: {output_area: handle.output_area}})\n",
       "  handle_add_output(event, handle)\n",
       "}\n",
       "\n",
       "function register_renderer(events, OutputArea) {\n",
       "  function append_mime(data, metadata, element) {\n",
       "    // create a DOM node to render to\n",
       "    var toinsert = this.create_output_subarea(\n",
       "    metadata,\n",
       "    CLASS_NAME,\n",
       "    EXEC_MIME_TYPE\n",
       "    );\n",
       "    this.keyboard_manager.register_events(toinsert);\n",
       "    // Render to node\n",
       "    var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
       "    render(props, toinsert[0]);\n",
       "    element.append(toinsert);\n",
       "    return toinsert\n",
       "  }\n",
       "\n",
       "  events.on('output_added.OutputArea', handle_add_output);\n",
       "  events.on('output_updated.OutputArea', handle_update_output);\n",
       "  events.on('clear_output.CodeCell', handle_clear_output);\n",
       "  events.on('delete.Cell', handle_clear_output);\n",
       "  events.on('kernel_ready.Kernel', handle_kernel_cleanup);\n",
       "\n",
       "  OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
       "    safe: true,\n",
       "    index: 0\n",
       "  });\n",
       "}\n",
       "\n",
       "if (window.Jupyter !== undefined) {\n",
       "  try {\n",
       "    var events = require('base/js/events');\n",
       "    var OutputArea = require('notebook/js/outputarea').OutputArea;\n",
       "    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
       "      register_renderer(events, OutputArea);\n",
       "    }\n",
       "  } catch(err) {\n",
       "  }\n",
       "}\n"
      ],
      "application/vnd.holoviews_load.v0+json": "\nif ((window.PyViz === undefined) || (window.PyViz instanceof HTMLElement)) {\n  window.PyViz = {comms: {}, comm_status:{}, kernels:{}, receivers: {}, plot_index: []}\n}\n\n\n    function JupyterCommManager() {\n    }\n\n    JupyterCommManager.prototype.register_target = function(plot_id, comm_id, msg_handler) {\n      if (window.comm_manager || ((window.Jupyter !== undefined) && (Jupyter.notebook.kernel != null))) {\n        var comm_manager = window.comm_manager || Jupyter.notebook.kernel.comm_manager;\n        comm_manager.register_target(comm_id, function(comm) {\n          comm.on_msg(msg_handler);\n        });\n      } else if ((plot_id in window.PyViz.kernels) && (window.PyViz.kernels[plot_id])) {\n        window.PyViz.kernels[plot_id].registerCommTarget(comm_id, function(comm) {\n          comm.onMsg = msg_handler;\n        });\n      } else if (typeof google != 'undefined' && google.colab.kernel != null) {\n        google.colab.kernel.comms.registerTarget(comm_id, (comm) => {\n          var messages = comm.messages[Symbol.asyncIterator]();\n          function processIteratorResult(result) {\n            var message = result.value;\n            console.log(message)\n            var content = {data: message.data, comm_id};\n            var buffers = []\n            for (var buffer of message.buffers || []) {\n              buffers.push(new DataView(buffer))\n            }\n            var metadata = message.metadata || {};\n            var msg = {content, buffers, metadata}\n            msg_handler(msg);\n            return messages.next().then(processIteratorResult);\n          }\n          return messages.next().then(processIteratorResult);\n        })\n      }\n    }\n\n    JupyterCommManager.prototype.get_client_comm = function(plot_id, comm_id, msg_handler) {\n      if (comm_id in window.PyViz.comms) {\n        return window.PyViz.comms[comm_id];\n      } else if (window.comm_manager || ((window.Jupyter !== undefined) && (Jupyter.notebook.kernel != null))) {\n        var comm_manager = window.comm_manager || Jupyter.notebook.kernel.comm_manager;\n        var comm = comm_manager.new_comm(comm_id, {}, {}, {}, comm_id);\n        if (msg_handler) {\n          comm.on_msg(msg_handler);\n        }\n      } else if ((plot_id in window.PyViz.kernels) && (window.PyViz.kernels[plot_id])) {\n        var comm = window.PyViz.kernels[plot_id].connectToComm(comm_id);\n        comm.open();\n        if (msg_handler) {\n          comm.onMsg = msg_handler;\n        }\n      } else if (typeof google != 'undefined' && google.colab.kernel != null) {\n        var comm_promise = google.colab.kernel.comms.open(comm_id)\n        comm_promise.then((comm) => {\n          window.PyViz.comms[comm_id] = comm;\n          if (msg_handler) {\n            var messages = comm.messages[Symbol.asyncIterator]();\n            function processIteratorResult(result) {\n              var message = 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document.createElement(\"div\");\n    bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n    var script_attrs = bk_div.children[0].attributes;\n    for (var i = 0; i < script_attrs.length; i++) {\n      toinsert[toinsert.length - 1].childNodes[1].setAttribute(script_attrs[i].name, script_attrs[i].value);\n    }\n    // store reference to server id on output_area\n    output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n  }\n}\n\n/**\n * Handle when an output is cleared or removed\n */\nfunction handle_clear_output(event, handle) {\n  var id = handle.cell.output_area._hv_plot_id;\n  var server_id = handle.cell.output_area._bokeh_server_id;\n  if (((id === undefined) || !(id in PyViz.plot_index)) && (server_id !== undefined)) { return; }\n  var comm = window.PyViz.comm_manager.get_client_comm(\"hv-extension-comm\", \"hv-extension-comm\", function () {});\n  if (server_id !== null) {\n    comm.send({event_type: 'server_delete', 'id': server_id});\n    return;\n  } else if (comm !== null) {\n    comm.send({event_type: 'delete', 'id': id});\n  }\n  delete PyViz.plot_index[id];\n  if ((window.Bokeh !== undefined) & (id in window.Bokeh.index)) {\n    var doc = window.Bokeh.index[id].model.document\n    doc.clear();\n    const i = window.Bokeh.documents.indexOf(doc);\n    if (i > -1) {\n      window.Bokeh.documents.splice(i, 1);\n    }\n  }\n}\n\n/**\n * Handle kernel restart event\n */\nfunction handle_kernel_cleanup(event, handle) {\n  delete PyViz.comms[\"hv-extension-comm\"];\n  window.PyViz.plot_index = {}\n}\n\n/**\n * Handle update_display_data messages\n */\nfunction handle_update_output(event, handle) {\n  handle_clear_output(event, {cell: {output_area: handle.output_area}})\n  handle_add_output(event, handle)\n}\n\nfunction register_renderer(events, OutputArea) {\n  function append_mime(data, metadata, element) {\n    // create a DOM node to render to\n    var toinsert = this.create_output_subarea(\n    metadata,\n    CLASS_NAME,\n    EXEC_MIME_TYPE\n    );\n    this.keyboard_manager.register_events(toinsert);\n    // Render to node\n    var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n    render(props, toinsert[0]);\n    element.append(toinsert);\n    return toinsert\n  }\n\n  events.on('output_added.OutputArea', handle_add_output);\n  events.on('output_updated.OutputArea', handle_update_output);\n  events.on('clear_output.CodeCell', handle_clear_output);\n  events.on('delete.Cell', handle_clear_output);\n  events.on('kernel_ready.Kernel', handle_kernel_cleanup);\n\n  OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n    safe: true,\n    index: 0\n  });\n}\n\nif (window.Jupyter !== undefined) {\n  try {\n    var events = require('base/js/events');\n    var OutputArea = require('notebook/js/outputarea').OutputArea;\n    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n      register_renderer(events, OutputArea);\n    }\n  } catch(err) {\n  }\n}\n"
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<style>.bk-root, .bk-root .bk:before, .bk-root .bk:after {\n",
       "  font-family: var(--jp-ui-font-size1);\n",
       "  font-size: var(--jp-ui-font-size1);\n",
       "  color: var(--jp-ui-font-color1);\n",
       "}\n",
       "</style>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import ufl\n",
    "\n",
    "from mpi4py import MPI\n",
    "from petsc4py.PETSc import ScalarType\n",
    "\n",
    "from dolfinx import mesh, fem, plot, io\n",
    "from dolfinx.io import XDMFFile, gmshio\n",
    "import dolfinx.geometry as geo\n",
    "import gmsh\n",
    "\n",
    "import pyvista\n",
    "pyvista.set_jupyter_backend('panel');\n",
    "# pyvista.set_jupyter_backend('trame');\n",
    "# pyvista.set_jupyter_backend('pythreejs');\n",
    "# pyvista.set_jupyter_backend('ipygany')\n",
    "# pyvista.set_jupyter_backend('static');\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7793d7b2-dde7-4833-a0fc-a72efdc7d5c6",
   "metadata": {},
   "source": [
    "Defining the phisical properties of fibres"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "66cd7795-695e-407c-9188-8da498b3e48c",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "ORDER = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "9a8eb1a5-9239-4fea-80eb-a9d1b116d8eb",
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div id='1007'>\n",
       "  <div class=\"bk-root\" id=\"43db7970-d1bc-4c9d-b319-08e58413d64c\" data-root-id=\"1007\"></div>\n",
       "</div>\n",
       "<script type=\"application/javascript\">(function(root) {\n",
       "  function embed_document(root) {\n",
       "    var docs_json = 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Application\",\"version\":\"2.4.3\"}};\n",
       "    var render_items = [{\"docid\":\"ddbc8b5b-eec7-4a86-91f1-594241e66948\",\"root_ids\":[\"1007\"],\"roots\":{\"1007\":\"43db7970-d1bc-4c9d-b319-08e58413d64c\"}}];\n",
       "    root.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",
       "  if (root.Bokeh !== undefined && root.Bokeh.Panel !== undefined && ( root['vtk'] !== undefined) && ( root['vtk'] !== undefined) && ( root['vtk'] !== undefined)) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined && root.Bokeh.Panel !== undefined && (root['vtk'] !== undefined) && (root['vtk'] !== undefined) && (root['vtk'] !== undefined)) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else if (document.readyState == \"complete\") {\n",
       "        attempts++;\n",
       "        if (attempts > 200) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 25, root)\n",
       "  }\n",
       "})(window);</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Scaled variable\n",
    "# cube sizes\n",
    "length = 1.0e-2\n",
    "width = 1.0e-2\n",
    "height = 1.0e-2\n",
    "unit_disp = 1.0e-2;\n",
    "bbox = [-length/2, -width/2, -height/2 , length/2, width/2, height/2]\n",
    "# phisical properties\n",
    "ν = 0.3\n",
    "E = 2.1e10\n",
    "\n",
    "λ = ν*E/(1+ν)/(1-2*ν)\n",
    "μ = E/2/(1+ν)\n",
    "\n",
    "ρ = 7800.\n",
    "\n",
    "# define oputput \n",
    "## 0 - with XDMF file\n",
    "## 1 - with VTK file\n",
    "## 2 - with pyvista\n",
    "OUTPUT = 2\n",
    "\n",
    "# elements order\n",
    "ORDER = 2\n",
    "\n",
    "gmsh.initialize()\n",
    "\n",
    "model = gmsh.model()\n",
    "model.add(\"main_domain\")\n",
    "model.setCurrent(\"main_domain\")\n",
    "\n",
    "# Choose if Gmsh output is verbose\n",
    "gmsh.option.setNumber(\"General.Terminal\", 0)\n",
    "\n",
    "# Set elements order to the specified one\n",
    "gmsh.option.setNumber(\"Mesh.ElementOrder\", ORDER)\n",
    "# Set elements size\n",
    "# gmsh.option.setNumber(\"Mesh.MeshSizeFromCurvature\", 5) # uncomment to use for mesh refinement dependending from its surface curvature\n",
    "gmsh.option.setNumber(\"Mesh.MeshSizeMax\", 10e-3)\n",
    "gmsh.option.setNumber(\"Mesh.MeshSizeMin\", 5e-3)\n",
    "\n",
    "# Set threads number for distrebuted meshing\n",
    "# gmsh.option.setNumber(\"Mesh.MaxNumThreads3D\", 4)\n",
    "\n",
    "# Set mesh algorithm (default is Delaunay triangulation)\n",
    "# see https://gmsh.info/doc/texinfo/gmsh.html#Choosing-the-right-unstructured-algorithm\n",
    "gmsh.option.setNumber(\"Mesh.Algorithm3D\", 3)\n",
    "\n",
    "# gmsh.option.setNumber(\"Mesh.RecombinationAlgorithm\",3)\n",
    "# gmsh.option.setNumber(\"Mesh.Recombine3DAll\",1)\n",
    "\n",
    "# Set the usage of hexahedron elements \n",
    "gmsh.option.setNumber(\"Mesh.SubdivisionAlgorithm\", 2)\n",
    "\n",
    "\n",
    "box_tag = model.occ.add_box(-length/2, -width/2, -height/2 , length, width, height)\n",
    "\n",
    "# Synchronize OpenCascade representation with gmsh model\n",
    "model.occ.synchronize()\n",
    "\n",
    "# Add physical marker for cells. It is important to call this function\n",
    "# after OpenCascade synchronization\n",
    "model.add_physical_group(dim=3, tags=[box_tag])\n",
    "\n",
    "# Generate the mesh\n",
    "# model.mesh.generate(2)\n",
    "# model.mesh.recombine()\n",
    "model.mesh.generate(dim=3)\n",
    "\n",
    "\n",
    "\n",
    "# Create a DOLFINx mesh (same mesh on each rank)\n",
    "msh, cell_markers, facet_markers = gmshio.model_to_mesh(model, MPI.COMM_SELF,0)\n",
    "msh.name = \"Box\"\n",
    "cell_markers.name = f\"{msh.name}_cells\"\n",
    "facet_markers.name = f\"{msh.name}_facets\"\n",
    "\n",
    "# Finalize gmsh to be able to use it again\n",
    "gmsh.finalize()\n",
    "\n",
    "if OUTPUT == 0:\n",
    "    # write mesh to XDMF file\n",
    "    with io.XDMFFile(msh.comm, \"out/mesh.xdmf\", \"w\") as file:\n",
    "        file.write_mesh(msh)\n",
    "        file.write_meshtags(cell_markers)\n",
    "        msh.topology.create_connectivity(msh.topology.dim - 1, msh.topology.dim)\n",
    "        file.write_meshtags(facet_markers)\n",
    "elif OUTPUT == 1:\n",
    "    # write mesh to VTK file\n",
    "    with io.VTKFile(msh.comm, \"out/mesh.pvd\", \"w\") as vtk:\n",
    "        vtk.write_mesh(msh)\n",
    "elif OUTPUT == 2: \n",
    "    # visualize using pyvista\n",
    "    # to use it, ipycanvas and ipyvtklink py needs to be installed\n",
    "    # Just run in terminal:\n",
    "    # ~# pip install ipyvtklink ipycanvas\n",
    "    pyvista.start_xvfb()\n",
    "\n",
    "    # Create plotter and pyvista grid\n",
    "    p = pyvista.Plotter()\n",
    "    topology, cell_types, geometry = plot.create_vtk_mesh(msh)\n",
    "    grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)\n",
    "\n",
    "    actor_0 = p.add_mesh(grid, show_edges=True)#style=\"wireframe\", color=\"k\")\n",
    "\n",
    "    p.show_axes()\n",
    "    if not pyvista.OFF_SCREEN:\n",
    "       p.show()\n",
    "    else:\n",
    "       figure_as_array = p.screenshot(\"mesh.png\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "aaa59723-eb30-44e0-b685-098abe98edba",
   "metadata": {},
   "outputs": [],
   "source": [
    "# T = fem.Constant(msh, ScalarType((0., 0., 0.)))\n",
    "# ds = ufl.Measure(\"ds\", domain=msh)\n",
    "def epsilon(u):\n",
    "    return ufl.sym(ufl.grad(u)) # Equivalent to 0.5*(ufl.nabla_grad(u) + ufl.nabla_grad(u).T)\n",
    "def sigma(u):\n",
    "    return λ * ufl.nabla_div(u) * ufl.Identity(len(u)) + 2*μ*epsilon(u)\n",
    "\n",
    "\n",
    "V = fem.VectorFunctionSpace(msh, (\"CG\", ORDER))\n",
    "u = ufl.TrialFunction(V)\n",
    "v = ufl.TestFunction(V)\n",
    "f = fem.Constant(msh, ScalarType((0., 0., 0.)))\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",
   "id": "ea11b63c-5999-41fc-8b1b-51a66f29872b",
   "metadata": {},
   "source": [
    "Marking nodes on each side of the RVE by following:\n",
    "- left (x = -1) - 1;\n",
    "- right (x = 1) - 2;\n",
    "- bottom (z = -1) - 3;\n",
    "- top (z = 1) - 4;\n",
    "- front (y = -1) - 5;\n",
    "- back (y = 1) - 6;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "dad264d1-708e-4224-8ac6-2f2594195d78",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function numpy.isclose(a, b, rtol=1e-05, atol=1e-08, equal_nan=False)>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eps = np.linalg.norm(np.array(bbox[0:3]) + np.array(bbox[3:]));\n",
    "np.isclose"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "ee9a4195-3d0e-4120-a897-50d5cca4b3c4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.005, -0.005, -0.005, 0.005, 0.005, 0.005]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bbox"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "89f4d5e7-3ec0-4fb1-8f14-f0218b552fc9",
   "metadata": {},
   "outputs": [],
   "source": [
    "def left(x):\n",
    "    return np.isclose(x[0], bbox[0], atol = eps);\n",
    "\n",
    "def right(x):\n",
    "    return np.isclose(x[0], bbox[3], atol = eps);\n",
    "\n",
    "def bottom(x):\n",
    "    return np.isclose(x[2], bbox[2], atol = eps);\n",
    "\n",
    "def top(x):\n",
    "    return np.isclose(x[2], bbox[5], atol = eps);\n",
    "\n",
    "def front(x):\n",
    "    return np.isclose(x[1], bbox[1], atol = eps);\n",
    "\n",
    "def back(x):\n",
    "    return np.isclose(x[1], bbox[4], atol = eps);\n",
    "\n",
    "fdim = msh.topology.dim - 1\n",
    "\n",
    "# find all facets on top, bottom and left boundary\n",
    "left_facets = mesh.locate_entities_boundary(msh, fdim, left);\n",
    "right_facets = mesh.locate_entities_boundary(msh, fdim, right);\n",
    "bottom_facets = mesh.locate_entities_boundary(msh, fdim, bottom);\n",
    "top_facets = mesh.locate_entities_boundary(msh, fdim, top);\n",
    "front_facets = mesh.locate_entities_boundary(msh, fdim, front);\n",
    "back_facets = mesh.locate_entities_boundary(msh, fdim, back);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "eda0cc35-7871-4327-b7c4-c384ae34778a",
   "metadata": {},
   "outputs": [],
   "source": [
    "marked_facets = np.hstack([left_facets, \n",
    "                           right_facets, \n",
    "                           bottom_facets,\n",
    "                           top_facets,\n",
    "                           front_facets,\n",
    "                           back_facets,\n",
    "                          ]);\n",
    "\n",
    "markers = np.hstack([np.full_like(left_facets, 1),\n",
    "                     np.full_like(right_facets, 2),\n",
    "                     np.full_like(bottom_facets, 3),\n",
    "                     np.full_like(top_facets, 4),\n",
    "                     np.full_like(front_facets, 5),\n",
    "                     np.full_like(back_facets, 6),\n",
    "                    ]);\n",
    "\n",
    "facets_order = np.argsort(marked_facets);\n",
    "\n",
    "facets_tags = mesh.meshtags(msh, \n",
    "                            fdim, \n",
    "                            marked_facets[facets_order],\n",
    "                            markers[facets_order]);\n",
    "\n",
    "ds = ufl.Measure('ds', domain=msh, subdomain_data=facets_tags);\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6734ca6f-c3c0-4d7f-a6e0-cfdd20b87a22",
   "metadata": {},
   "source": [
    "Firstly, we want to compress across Z direction, so\n",
    "1. facets on bottom should be fixed in all directions $(u_3 = (0.0, 0.0, 0.0)$);\n",
    "2. facets on top should be shifted across Z diraction on unit distance $(u_4 = (0.0, 0.0, 0.1))$;\n",
    "3. facets on other sides should be constrained in XY plane $(u_{i,j} = 0.0$, where $i \\in \\{1, 2, 3, 4\\}$, $j \\in \\{1, 2\\})$;\n",
    "4. *facets on left, right, front and back sides should be shifted according to linear low* $u_{i,3}(z) = \\frac{z-z_{\\min}}{z_{\\max} - z_{\\min}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "cd21f8af-e028-4151-b699-5a1eb6850586",
   "metadata": {},
   "outputs": [],
   "source": [
    "# # apply 1st constraint to the bottom RVE's facet\n",
    "# bottom_dofs = fem.locate_dofs_topological(V, \n",
    "#                                         facets_tags.dim, \n",
    "#                                         facets_tags.find(3)); # since bottom facets has mark 3\n",
    "# ub_3 = np.array([0., 0., 0.], dtype = ScalarType);\n",
    "# bc_bottom = fem.dirichletbc(ub_3, \n",
    "#                        bottom_dofs,\n",
    "#                        V);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "4f8610e3-532a-434a-a717-e96435c3a4fd",
   "metadata": {},
   "outputs": [],
   "source": [
    "# # apply 2nd constraint to the top RVE's facet\n",
    "# top_dofs = fem.locate_dofs_topological(V, \n",
    "#                                            facets_tags.dim, \n",
    "#                                            facets_tags.find(4)); # since top facets has mark 4\n",
    "\n",
    "# ub_4 = np.array([0., 0., unit_disp], dtype = ScalarType);\n",
    "# bc_top = fem.dirichletbc(ub_4,\n",
    "#                        top_dofs,\n",
    "#                        V);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "cd99f3af-928e-4815-b208-391a871a6821",
   "metadata": {},
   "outputs": [],
   "source": [
    "# # apply 3rd constraint\n",
    "# facets = np.hstack([facets_tags.find(1),\n",
    "#                     facets_tags.find(2), \n",
    "#                     facets_tags.find(5),\n",
    "#                     facets_tags.find(6),\n",
    "#                    ]);\n",
    "# side_xdofs = fem.locate_dofs_topological(V.sub(0),\n",
    "#                                          facets_tags.dim,\n",
    "#                                          facets);\n",
    "# side_ydofs = fem.locate_dofs_topological(V.sub(1),\n",
    "#                                          facets_tags.dim,\n",
    "#                                          facets);\n",
    "\n",
    "# bc_sidex = fem.dirichletbc(ScalarType(0),\n",
    "#                            side_xdofs,\n",
    "#                            V.sub(0));\n",
    "# bc_sidey = fem.dirichletbc(ScalarType(0),\n",
    "#                            side_ydofs,\n",
    "#                            V.sub(1));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "973b95d2-a956-464d-88d2-94332ac12cf1",
   "metadata": {},
   "outputs": [],
   "source": [
    "# # apply 4th constraint\n",
    "# ub_sidez = fem.Function(V.sub(2).collapse()[0]);\n",
    "\n",
    "# def side_z(x):\n",
    "#     # values = np.zeros(x.shape[1]);\n",
    "#     values = unit_disp*(x[2] - bbox[2])/(bbox[5] - bbox[2]);\n",
    "#     return values;\n",
    "\n",
    "# ub_sidez.interpolate(side_z);\n",
    "\n",
    "# side_zdofs = fem.locate_dofs_topological(V.sub(2),\n",
    "#                                          facets_tags.dim,\n",
    "#                                          facets);\n",
    "# bc_sidez = fem.dirichletbc(ub_sidez,\n",
    "#                            side_zdofs);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "e369f1c2-33a0-4808-815e-a963f1b4d08b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def KUBC(x, i, j, ud):\n",
    "    values = np.zeros(x.shape);\n",
    "    \n",
    "    values[i,:] += 0.5*ud*(x[j])/(bbox[j+3] - bbox[j]);\n",
    "    values[j,:] += 0.5*ud*(x[i])/(bbox[i+3] - bbox[i]);\n",
    "        \n",
    "        \n",
    "    return values;\n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "be94c336-fdc2-4627-a479-1fbd0669e076",
   "metadata": {},
   "outputs": [],
   "source": [
    "# apply 2nd, 3rd and 4th constraints\n",
    "facets = np.hstack([facets_tags.find(1),\n",
    "                    facets_tags.find(2),\n",
    "                    facets_tags.find(4),\n",
    "                    facets_tags.find(5),\n",
    "                    facets_tags.find(6),\n",
    "                   ]);\n",
    "\n",
    "ub_ = fem.Function(V);\n",
    "\n",
    "full_bc = lambda x: KUBC(x, 1, 2, unit_disp);\n",
    "\n",
    "ub_.interpolate(full_bc);\n",
    "\n",
    "nonbottom_dofs = fem.locate_dofs_topological(V,\n",
    "                                         facets_tags.dim,\n",
    "                                         marked_facets);\n",
    "bc_ = fem.dirichletbc(ub_, nonbottom_dofs);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "a95e31a9-ae80-41aa-853c-38d07bb005f6",
   "metadata": {},
   "outputs": [],
   "source": [
    "# problem = fem.petsc.LinearProblem(a, L, bcs=[bc_bottom, bc_top, bc_sidex, bc_sidey, bc_sidez], petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"})\n",
    "problem = fem.petsc.LinearProblem(a, L, bcs=[ bc_], petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"})\n",
    "\n",
    "uh = problem.solve()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53572ac4-8a02-4d22-bde0-a612ec230ec7",
   "metadata": {},
   "source": [
    "## Visualzation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "c39b3520-013f-4655-99e2-9f98b338b020",
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "<div id='1031'>\n",
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       "</div>\n",
       "<script type=\"application/javascript\">(function(root) {\n",
       "  function embed_document(root) {\n",
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\",\"99d03e2dee3edf0e0bd008a8931fd7cd\":\"UEsDBBQAAAAIAKBZlFYqY0/9PRsAAFgvAAAlAAAAZGF0YS85OWQwM2UyZGVlM2VkZjBlMGJkMDA4YTg5MzFmZDdjZE1ae3yP5ftfOWXkS81PcizSqpUx3/G5bxM6OFQ0UdSUxJxS5FRzyGlOox3MbGOzGcZojOHzeS7nOTSNnC1CEyG+lB9KP/r+rvf1fO65/3heu/Z87uc+XOfrfd2BJ3MoftdteQKZzu9URgdKSoTWMTsouPtaoau9tZ9OvLhfaPM7nqG5myhmRLLQ361ZQtMOJwo982QmNUjLFPqRqVn0W9VYGhkxhwounKRjlR16Liqe7vcvpqEVMsjbMJaio3+liMCTvP7HtHHKQTrVPpf3NIZC7u6h4uYpvOab8vu1SC+PfVeXxUTxfF1V/K4G+kDJlxQVv8vbKPYVfazydNre+zEnv9MQvb33Z1RFDfeMjNivouLnUOeyGAfjO5clUEDAOhnTITeO55/h3AmeoLvXm4x1fZg/LGwhz3neWTairwoISKGi26XO3qR4lXU0iz4prsP7P6cCT6ZRUP6fTnpaXc+yEdm8nydp7jP/9bQ+lEtbKIzU8HWqQ+5quljnFSqLuaW8DVfxPG2o2fbeqrh5Pu/tLT57Ax0Vv5h6eCvzmPp6ZMRKWjbi37x2V53faQmlp9Xi39rp56JyKfZqC+rYoCdFTiT6fUkp/djzSVr/fiZl19hP/ULuOvX3FdCJ3UepelEozQjbQ9XGn6GzT+x0gtpMkW9yjyQ4ZnyNtVsdM+bk4EeJrhykZ949z+M7OGb+4VlHfVfmZ1LbF/ZTYv5UX9+TG+lOyXF6pstjzpi3Z9LU2dtoZ9XFnsYHMqm0035qteorz5AW6+je3SN0qR15Tg6eRxsW7KTvfN2d/3Z7l6qMWce8/h/fxCOD6NLxQt7Hb94TLz5PWz9dTpP7f+OpWbEDvfn0arqQXVHd6/UV7Qv3Uc9lL6iup96jUyfXsS41UXigI6wbTOf4AgIm07xFsR5+fJ3rT6JxbYM8P8bf9yRdGQjdwxgHusZ88GxtMdX5+tk4qtTtZ1+jHd18CW0mUObIDT5+PGPjhlL4gsoOzn/i5gzomzZrPRf1L23W6l7vMub0NIodRqvzntfYR0Tge/TGpAJl6NmzHsF4BZ2tMfp97/k+h53p1RfQ0umhztD0Fc6CwnhqHF5bfVD3njNqQDqd3/inr2KHmjT7j0yMdwasvOb0HJsqtvP8kLXk6bej3O5gn2JflZJo95dLhO4cm0yLkzaV26SxzxlhsfRp7yxqFDtT3wluz3ZzQmWE99TLRmie54panddSRwS+w3YxB2fCecFjnKmcvt+/ru7hDac7wffUjes1dHpaFzqZsw5yKB9zoGSubhR71elcpvTQCn3Zvv52IgJb87thrLfxTuzVSP3qq6/q8UH5DtusVsMba2/DShQWFsbjYb8lTlR8Rw2+Q+/at7hEs4vi6Epvoi3zSqlmv0FU6eIq2n/0B2qbnEBltTKpzW/FdGZpDHnfTKVlVfbxnhL0pIIBTr+QPrx+jL5c+JBzM26gds+V40CeZTHJmvnbZty4cbx+LNOn2vQL+UrGBAS0X+qeO0cH5XfS1yLb+HkyTcdePazc94N4jmNC46x8Pv8Y4Z38ZR/HPKpbPg/7LBkzPugl8EJo1int1ymdET6CfZaS9/y79uss77+DfmNSCw9o9hd67jMv+0BjjwEBjZeC3kKa32/3gc+xVxN5z14FWXQuS+cxARo8mdotTh8aMk2BJ6faz9Qhd19T4AN4AF6AD+ABeIE5b41rqZKadpG9pV7+3Hf4XFXt7meK535/lz68LJp1t63QjcMXOPzIng+fK/a17fOt0DXWRqjgX8UONPsB1bRGc6Fhh70n7JExT35x3eFH6AbHdjr8GF6xztSS8YPrJDieP11+Pn6/gefetY5Cn7i51zdwo7uH6rMeJX6E9vyZ5lToH+Yf87rvb91HeM22i3eeAyUVmf/x7N9zVNHtEP6bzP53v1o2IkIX3Z4F2Sj46qj4TNb7Gho+edmIJSzXx1hnevGTxrz+Q8EnX4vMYnnUhnzUG5PSdL3LtxR8/rHKi3VoSkXm91q2mXjW95Wq9aHnVGhKAsebbxXbgYJe8b7YZ9z2BOXPZBt4EzZVHosR+wx9sOM74qcPZF5h+3hC/PT71a9RRvjbHE8us23+BF8kfzkOwseyzV8VX8G+qDymQxeLbg+GvflY7iosbBR44oVewkZhq9D7Ht4Pof9sMyUc1yZC5zx+nyd+EDYLe4XdTiq4oGDTsO2CC3VY9/uAhw7s3bx3eeuu6+pq+VrQayfr6AzxsZiPdRM5gOjj9t5fCM3x3+mQO098MvOD420a+IMxPtZLuvdstNqR9Qx92WUZzs/7CeQYmkHX5i5SR1u9RRNfygcPWKYdmS+rCbrFD6+5hHAOtlf+PZvjbxDLOoLjeB5hb/wwf1fy/3P13qS+7OM7iH7Cr8NOi24n6JR7MZwzPGtsk/OKgKdBsw4S66JydbIusR4LzfpIrJdi16y/TO/13AmOYlseD154YKfwW/BfGIN9sj347asqsU3Kt2wHxPYg67K9EtutjE+a97IzYlgDGT/yX+O8lS4OEbrO8c2O95ds+Taz9jyHfvrHC5r/+gqXJsp75qfD+iLzdNz5KN2vnCPzFy6tRU0e+kje3134Mv341yihP64WTue2pDv+WKBd/R1TTrN+avAdPGWdhOwUePrAXzZB7ge5i2wQOxA3ED9cP6YkriBeIFYgZuBbyBi6CZrlw3N6fa6/rQK98hia5zR+VUHfzHuWn8/1vTXZRp+V98tGbOV9bnVcf/sE2+k8oXl9hT3AlyK2IMb4dUDiiktXxyN7u7unJX+fKu+L9tVWF6+0Fhr5TfGt9UK//uEANT2nofD//ud1fUOPz5b3R3rVdIbMOyF0rp7pyR9zCzlHtpWfO4auXrTOMT6BfarPxM3VeXmO8QmckzjGJ8BuTY7NfHdMLs3n8HAuy3r+Hd77OL5xXN4Ae2Q7+o1zz7OwBeZxKecJDubxcDxkncxHDq/we1D+PnL5lqr4/H75uvkPaOalYp4KDR1nf0luzJ38FGzGfb/dw/EMNQLintqb9BHmR4zyhaZM4DyzvkdyzEo/y96wLvbXPfEltf0G0bM/lsp+sBf8hpynUWwGnz1Q7HfjlHmsXwsc/97IzV1EN8jIMa/VdM5RD7n8PzKCvk1p7cBfxe/aSMj3oJMHStbweTvx+b9gXV3BeUcIIS5Exefzut1kzgErP6V3Fp6WeQb9NJfroIbyfsylt2mJ71d5P7toGDX/oK68Zx3kvKwm+fWK40EVoZkP7N86CA1/jHVB9+623vdeaTWhU7/+yuka3k7ouCkDnfSH3XmWvpvhfL2tldAVH7/vjH2oQNbVG+c7zUqChf6n9F/U81wNGfNjz3Tng7Ou3hb3XeV7uNc0yQfWbn3WN/D3RmIvE7zPUtyZ72XMKwcjuBYJkm+/uL7Bqdrsb3lfOvVF53yfRKHbjXqrze/hS32vvrpb3Qm+hnqKEC/w97moH8SHGn2GjY8Pui51IOKLpavajME8+BZzYR7Mgf/xLcbje3yL7/A/YgpiBXQCsY/zVtbto5RffZKa4OV6autBqrx7hWozZT7XHLvhu5E7cs3lo6+Cs9XSejEUOtgnZ7yyM5uqbv1E6F9bxNHUzZF+uXjJ6LBVC2tT87oydfNx+JD5H22hlHtTUYtyXrCDa8M4Qj7W5+b39PTTS8UfXqzzPZ8pS7690Oag88OcZkJPKQ2iio9HCN1r2mWW9duufEvrUuHrw9x9zp9ETX/+QOjKu6Np3yPRQlu1ubZqc71vZBR1eniS0LMa9aBf7s9xxzyoL7SpI8BP8BI8RR4VHe1lno8l5DnNtm+mceNGkZvbFHGMTSTkPBfrbKbRp0ZAfp6AgB2CASDvKovZwDLqT8gNeni3sYxmiowgH8gJ+cn4oIM8ZgVyFw/8053gxYQcA/N0Ljsl3xqZIt8w+gCbfeAHhiFPJ9gwxmMdrFd0O5j3H828LyTkI/zw+dYSYvHZMxm892KREeQDOYHOfbSQ6l0+JjKCfCAnvz+RGhO+a2iFDGfjlG9MjBO5Yzzzjtd1ec75vsN5P7nx/TDbyjTm22d8llDmQa74Gdg6fA3GDG99xnmvdJhfdg2p5da5wgfwALzAuQzfwAdjXzi78c84F86Es+EsOAfOA52EPkIvTQ4DnYZOQh+hl8Zn+vVYV//jE7qUmy20qT0R942dIo4be3fndN/jXCamgDY+HOONXWM/F+ucFp4HSq1UjkdpU/9ijGs7p+W9hVNpC6fSpnb2x1DsQ/tjqNDDCvc60S/8ov9v7DWuY2o7iaPO6ZqLfvPnMe6YX6KfIv3Lr/pij+u6JPGK0zb5kMY3qHuSmhZrfLPlTkO6VKFUYxxy2CrqP5wnXJB5bly/rrvXuwyaY8lVnEdyWPw+POuojMHvnHvJ3vA7z6Fr1Yr0qeFlzIsDHD/fck61P83xdQfzb78nPe0S5ys/amBbvXqdZFl6NXCrKuo4r1uIvFvhd2/DrRr4RtMaa/X7g49oYCOPrNug3/Qd05W69fOVJBbrl/LO6VmPdXQ+qLtF3/5PKfaDfAo+VOol/GVfqv2+WniyOm89575XWMZntd8/ywOs4FrkNdbL8xy7dir8HhW/VwM3wLuAgEMa2AjOh3MiJ8Rf8IXzFh7/E+dzlzWwFM5ZNP7310Ey/0T9IRl5Bbq4pbyvPmseGdkNK3ydjIye/OIzMjICjmPkAls1skCOYmSBtQz/IZezZ85wvbxH5AU+gC9+HFJojHHHsu/n3KJWrSPs3/PZFouczmXf8XkXsxwP+yBDNXwZ8lpf0e3dfK5krnc7UPLx9fqxN49xjV6PLhfu5BrgNPSQFsxYrLHv0MPnHDPms9ObnI4NkvWVG7s5VsQ53eutZL4e1MCsrm+I0y/TDg1cq+rHGRpyBX/YTyP+Cc0+AD5R6Cfe2Ma6863QU7stQn4m9IOz57CfK+RcNklo9mmo/40P0G6em4QcE7k4fivniYvVumNmz5pVTuNbrIm1zTxYDzTWxz5AY02s7eZoOeX7wfrYR9nLifRq1Z36t4Gn9ecAKl/bqJ+Zflxv+jOZ/sxcqotCS3T/J2bQL9HZ2pf8vcgLe0Qu2b3e4PL9+P2c4CnYvxmDvHNv0hLWx/0io8PnZrMctstaf3Wfq7E29IoffXDiJtENjnlSD8DHcw0h2NLcZzY7W2gLzz2NbXi1E5qyTmcdfZffN3KOVd7CfJiqR59q71yss4H/9uMzt+L4uVlzzQtMjP3xGmA1yH+Rx+Mvcmfk9/irdw9rTtVDBuk31hTqwmqP0Ad1J+jMSz599Pt3aVu7HvrPEWtlnrGpSg+cuUr2ie/83xPXbrrxgJX6p5BCp0HElzr+dy/sztn8cmf97z/WaGCVP/cbrSff3Yy1OaeL1Dtar2UdWqjyO4GvWXyevZ7i5iW6QsZS+CUFf1MWM53tuZqKCNzJ+jBXA0s3vgjYu/l245R/OIc/xvNt0MAhja9w9bAcH0Pti7pPaPAGNSXorKMz2N90MX4Avgm1HuvhexoPaIzFN24MriQYHGh8h+9BYx2s1y/kbpvRpzbpggufa2D+QfkFbMcf8NnTfavz1jDPOgEn8IyMWMG/vyg41qKu0/SCGVs08OfIiYv070u+8/vJMdqvWwr2B7vz56JCA0OGznDsF70x/hZ4svF76GV0yC3SN+MStdvj2MTffq7Rs9g45RD7+9VCG11yeyLr5C96GXuT9jF/0rTbp4Ctv6HRa/A2XM0+tr1GfyE6eiXzJFyjrzH61Ao+e3ON/sXZM0vYxzxeboN+bJMsbFNshG1FaGNH8KvGFuAfjC24MnJ5AvvNSp2mU7/dYuowsUf4cOCOnz7/rX/+jzTWcP1YXcFVQVu1P/wPsEyRe+eyJ1GnKdfXldf+ZGGwZGGw9EnF+Wr0kjNCT5hTqCIn1pP5LXySLAyTkmueUFee7yy0hWeSVR/Rj/v7qWm7P5ZvE5L7q66nTso+TR0E2qr9ydT4oM/3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       "    var render_items = [{\"docid\":\"3fd2b816-c36a-44ee-aa1a-98ca22b5100b\",\"root_ids\":[\"1031\"],\"roots\":{\"1031\":\"f0b427c2-3cbb-4b02-be6e-a90902d6b2cf\"}}];\n",
       "    root.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",
       "  if (root.Bokeh !== undefined && root.Bokeh.Panel !== undefined && ( root['vtk'] !== undefined) && ( root['vtk'] !== undefined) && ( root['vtk'] !== undefined)) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined && root.Bokeh.Panel !== undefined && (root['vtk'] !== undefined) && (root['vtk'] !== undefined) && (root['vtk'] !== undefined)) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else if (document.readyState == \"complete\") {\n",
       "        attempts++;\n",
       "        if (attempts > 200) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 25, root)\n",
       "  }\n",
       "})(window);</script>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "uh.name = \"Deformation\"\n",
    "\n",
    "    \n",
    "\n",
    "# # write mesh to XDMF file\n",
    "# with io.XDMFFile(msh.comm, \"out/deformation.xdmf\", \"w\") as xdmf:\n",
    "#     xdmf.write_mesh(msh)\n",
    "#     xdmf.write_function(uh)\n",
    "\n",
    "\n",
    "# visualize using pyvista\n",
    "# to use it, ipycanvas and ipyvtklink py needs to be installed\n",
    "# Just run in terminal:\n",
    "# ~# pip install ipyvtklink ipycanvas\n",
    "pyvista.start_xvfb()\n",
    "\n",
    "# Create plotter and pyvista grid\n",
    "p = pyvista.Plotter(window_size=[400, 400])\n",
    "topology, cell_types, geometry = plot.create_vtk_mesh(msh)\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=1.0)\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(\"deformation.png\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "fa5a2805-a260-4e9d-8329-d20f0f74c7b5",
   "metadata": {},
   "outputs": [],
   "source": [
    "def macro_strain(i):\n",
    "    \"\"\"returns the macroscopic strain for the 3 elementary load cases\"\"\"\n",
    "    ϵ = np.zeros((6,), dtype = np.float64)\n",
    "    ϵ[i] = 1;\n",
    "    ϵ[3:] /= 2;\n",
    "    return np.array([[ϵ[0], ϵ[5], ϵ[4]],\n",
    "                     [ϵ[5], ϵ[1], ϵ[3]],\n",
    "                     [ϵ[4], ϵ[3], ϵ[2]]]);\n",
    "ε = fem.Constant\n",
    "def stress2Voigt(s):\n",
    "    return ufl.as_vector([s[0,0], \n",
    "                          s[1,1], \n",
    "                          s[2,2],\n",
    "                          s[1,2],\n",
    "                          s[0,2],\n",
    "                          s[0,1]])\n",
    "\n",
    "def strein2Voigt(s):\n",
    "    return ufl.as_vector([s[0,0], \n",
    "                          s[1,1], \n",
    "                          s[2,2],\n",
    "                          2*s[1,2],\n",
    "                          2*s[0,2],\n",
    "                          2*s[0,1]])\n",
    "def indexVoigt(i, j):\n",
    "    if i==j:\n",
    "        return i\n",
    "    elif (i==0):\n",
    "        if (j==1):\n",
    "            return 5;\n",
    "        else:\n",
    "            return 4;\n",
    "    else:\n",
    "        return 3; \n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4290b06c-b5c6-498e-bb1f-626ef99917ac",
   "metadata": {},
   "source": [
    "\n",
    "# Homogenization of elastic properties\n",
    "\n",
    "For homogeinzation procedure we wil use the Voigt notation, according to which the stress and strein tensors are represented as a vectors:\n",
    "\n",
    "$$\n",
    "    \\mathbf{\\varepsilon} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix}\n",
    "            \\varepsilon_{11} \\\\ \n",
    "            \\varepsilon_{22} \\\\ \n",
    "            \\varepsilon_{33} \\\\ \n",
    "            \\varepsilon_{23} \\\\ \n",
    "            \\varepsilon_{13} \\\\ \n",
    "            \\varepsilon_{12} \\\\\n",
    "        \\end{matrix}\n",
    "    \\right\\}\n",
    "    = \n",
    "    \\left\\{\n",
    "        \\begin{matrix}\n",
    "            \\varepsilon_{x} \\\\ \n",
    "            \\varepsilon_{y} \\\\ \n",
    "            \\varepsilon_{z} \\\\ \n",
    "            \\gamma_{yz}\\\\\n",
    "            \\gamma_{xz}\\\\\n",
    "            \\gamma_{xy}\\\\ \n",
    "        \\end{matrix}\n",
    "    \\right\\};\\hspace{2cm}\n",
    "    \\mathbf{\\sigma} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix}\n",
    "            \\sigma_{11} \\\\ \n",
    "            \\sigma_{22} \\\\ \n",
    "            \\sigma_{33} \\\\ \n",
    "            \\sigma_{23} \\\\ \n",
    "            \\sigma_{13} \\\\ \n",
    "            \\sigma_{12} \\\\\n",
    "        \\end{matrix}\n",
    "    \\right\\}\n",
    "    = \n",
    "    \\left\\{\n",
    "        \\begin{matrix}\n",
    "            \\sigma_{x} \\\\ \n",
    "            \\sigma_{y} \\\\ \n",
    "            \\sigma_{z} \\\\ \n",
    "            \\tau_{yz}\\\\\n",
    "            \\tau_{xz}\\\\\n",
    "            \\tau_{xy}\\\\ \n",
    "        \\end{matrix}\n",
    "    \\right\\};\n",
    "$$\n",
    "Then the stiffness can be determined as\n",
    "$$\n",
    "    \\hat{\\sigma} = \\mathbf{\\hat{C}}\\hat{\\varepsilon},\n",
    "$$\n",
    "where $\\mathbf{\\hat{C}}$ is averaged matrix of elastic properties, $ \\hat{\\varepsilon}~-$  an average strein within the RVEvolume, which is determined by the BC and $\\hat{\\sigma}~-$  an average stress within the RVE's volume, which is determined according to\n",
    "$$\n",
    "    \\hat{\\sigma} = \\frac{1}{V}\\int_\\Omega{\\sigma (x) dx}, \n",
    "$$\n",
    "where V - RVE volume and $\\Omega$ - RVE domain.\n",
    "\n",
    "\n",
    "To determine an avarage elastic properties of the material we need to solve 6 different  boundary problems:\n",
    "$$\n",
    "    \\hat{\\varepsilon}_{1} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix} 1\\\\ 0\\\\ 0\\\\ 0\\\\ 0\\\\ 0\\\\ \\end{matrix}\n",
    "    \\right\\};\\ \n",
    "    \\hat{\\varepsilon}_{2} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix} 0\\\\ 1\\\\ 0\\\\ 0\\\\ 0\\\\ 0\\\\ \\end{matrix}\n",
    "    \\right\\};\\ \n",
    "    \\hat{\\varepsilon}_{3} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix} 0\\\\ 0\\\\ 1\\\\ 0\\\\ 0\\\\ 0\\\\ \\end{matrix}\n",
    "    \\right\\};\\ \n",
    "    \\hat{\\varepsilon}_{4} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix} 0\\\\ 0\\\\ 0\\\\ 1\\\\ 0\\\\ 0\\\\ \\end{matrix}\n",
    "    \\right\\};\\ \n",
    "    \\hat{\\varepsilon}_{5} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix} 0\\\\ 0\\\\ 0\\\\ 0\\\\ 1\\\\ 0\\\\ \\end{matrix}\n",
    "    \\right\\};\\ \n",
    "    \\hat{\\varepsilon}_{6} = \n",
    "    \\left\\{\n",
    "        \\begin{matrix} 0\\\\ 0\\\\ 0\\\\ 0\\\\ 0\\\\ 1\\\\ \\end{matrix}\n",
    "    \\right\\};\\\n",
    "$$\n",
    "\n",
    "in each case we obtain we obtain a certain value of Voigt stress vector. Full series of calculation will give us following SLE:\n",
    "\n",
    "$$\n",
    "    \\begin{bmatrix}\n",
    "        \\hat{\\sigma}_{x}^1 & \\hat{\\sigma}_{x}^2 & \\hat{\\sigma}_{x}^3 & \\hat{\\sigma}_{x}^4 & \\hat{\\sigma}_{x}^5 & \\hat{\\sigma}_{x}^6 \\\\\n",
    "        \\hat{\\sigma}_{y}^1 & \\hat{\\sigma}_{y}^2 & \\hat{\\sigma}_{y}^3 & \\hat{\\sigma}_{y}^4 & \\hat{\\sigma}_{y}^5 & \\hat{\\sigma}_{y}^6 \\\\ \n",
    "        \\hat{\\sigma}_{z}^1 & \\hat{\\sigma}_{z}^2 & \\hat{\\sigma}_{z}^3 & \\hat{\\sigma}_{z}^4 & \\hat{\\sigma}_{z}^5 & \\hat{\\sigma}_{z}^6 \\\\ \n",
    "        \\hat{\\tau}_{yz}^1 & \\hat{\\tau}_{yz}^2 & \\hat{\\tau}_{yz}^3 & \\hat{\\tau}_{yz}^4 & \\hat{\\tau}_{yz}^5 & \\hat{\\tau}_{yz}^6 \\\\ \n",
    "        \\hat{\\tau}_{xz}^1 & \\hat{\\tau}_{xz}^2 & \\hat{\\tau}_{xz}^3 & \\hat{\\tau}_{xz}^4 & \\hat{\\tau}_{xz}^5 & \\hat{\\tau}_{xz}^6 \\\\\n",
    "        \\hat{\\tau}_{xy}^1 & \\hat{\\tau}_{xy}^2 & \\hat{\\tau}_{xy}^3 & \\hat{\\tau}_{xy}^4 & \\hat{\\tau}_{xy}^5 & \\hat{\\tau}_{xy}^6 \\\\\n",
    "    \\end{bmatrix}\n",
    "    =\n",
    "    \\mathbf{\\hat{C}}\n",
    "    \\begin{bmatrix}\n",
    "        1 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "        0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    "        0 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    "        0 & 0 & 0 & 1 & 0 & 0 \\\\\n",
    "        0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
    "        0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "since Voigt strein matrix actualy is a unit matrix, then the avareged elastic matrix $\\mathbf{\\hat{C}}$ will be equal to the matrix of calulated stress \n",
    "$\\begin{bmatrix} \\hat{\\sigma}^1 &\\hat{\\sigma}^2 & \\hat{\\sigma}^3 & \\hat{\\sigma}^4 & \\hat{\\sigma}^5 & \\hat{\\sigma}^6 \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "de98ce9b-43f9-4058-bed3-aa6e5fabb343",
   "metadata": {},
   "outputs": [],
   "source": [
    "m_σ = np.zeros((6,6), dtype = np.float64);\n",
    "m_ε = np.zeros((6,6), dtype = np.float64);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "2fa5f824-c102-48da-bfd9-a65de4f8431e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "εxx = 0.9999999999999678; σxx = 28269230769.231506 \n",
      "εyy = -5.389438735344007e-31; σyy = 12115384615.384476 \n",
      "εzz = -1.4620985850350768e-30; σzz = 12115384615.38449 \n",
      "εyz = 3.335020392273665e-31; σyz = 1.2770400808460253e-20 \n",
      "εxz = 1.942128394518956e-17; σxz = 1.5686421648035594e-07 \n",
      "εxy = 2.987459485116179e-17; σxy = 2.4129480456711447e-07 \n",
      "εxx = 7.634951462078143e-18; σxx = 4.1472398789091397e-07 \n",
      "εyy = 1.641629955726085e-17; σyy = 5.565765340438759e-07 \n",
      "εzz = 3.615958089387602e-31; σzz = 2.9139015658044627e-07 \n",
      "εyz = 8.659371309736123e-18; σyz = 6.994107596325321e-08 \n",
      "εxz = 4.747136246213405e-18; σxz = 3.8342254296327384e-08 \n",
      "εxy = 0.9999999999999695; σxy = 8076923076.92304 \n",
      "εxx = -1.2647639152731243e-17; σxx = -2.697534850896955e-07 \n",
      "εyy = 4.16273084928637e-32; σyy = -6.544546800718181e-08 \n",
      "εzz = 7.245790999758572e-18; σzz = 5.1601925065916336e-08 \n",
      "εyz = 1.4322995842840904e-17; σyz = 1.1568573565368427e-07 \n",
      "εxz = 0.9999999999999685; σxz = 8076923076.923052 \n",
      "εxy = -2.47817747490144e-19; σxy = -2.001604883556909e-09 \n",
      "εxx = -1.7454699227972007e-31; σxx = 12115384615.384487 \n",
      "εyy = 0.9999999999999682; σyy = 28269230769.231567 \n",
      "εzz = -3.404840480622218e-31; σzz = 12115384615.384502 \n",
      "εyz = 5.48413078705546e-18; σyz = 4.429490251083316e-08 \n",
      "εxz = 4.503666541448838e-31; σxz = 5.087142110640619e-21 \n",
      "εxy = 1.805100544369803e-17; σxy = 1.457965824298875e-07 \n",
      "εxx = -3.295588526110828e-31; σxx = 5.313738475851858e-08 \n",
      "εyy = 2.3415596107352646e-20; σyy = 5.351563669566252e-08 \n",
      "εzz = 4.3625272728491195e-18; σzz = 1.236089791660778e-07 \n",
      "εyz = 0.9999999999999697; σyz = 8076923076.923036 \n",
      "εxz = 7.318947174879163e-18; σxz = 5.9114573335605045e-08 \n",
      "εxy = -1.1145995024503543e-17; σxy = -9.002534442869561e-08 \n",
      "εxx = -1.5914087033440358e-30; σxx = 12115384615.384508 \n",
      "εyy = -1.336332742905965e-30; σyy = 12115384615.384497 \n",
      "εzz = 0.9999999999999691; σzz = 28269230769.231552 \n",
      "εyz = 2.1245490282911098e-18; σyz = 1.7159819074618846e-08 \n",
      "εxz = -2.2558884267039992e-17; σxz = -1.8220637292609962e-07 \n",
      "εxy = -2.139193088630169e-32; σxy = -7.483504325475705e-21 \n"
     ]
    }
   ],
   "source": [
    "dx = ufl.Measure('dx', domain=msh);\n",
    "volume = fem.assemble_scalar(fem.form(fem.Constant(msh, ScalarType(1.0)) * dx));\n",
    "for i in range(3):\n",
    "    for j in range(i,3):\n",
    "\n",
    "        ub_ = fem.Function(V);\n",
    "        full_bc = lambda x: KUBC(x, i,j, unit_disp);\n",
    "        ub_.interpolate(full_bc);\n",
    "        nonbottom_dofs = fem.locate_dofs_topological(V,\n",
    "                                                 facets_tags.dim,\n",
    "                                                 marked_facets);\n",
    "        bc_ = fem.dirichletbc(ub_, nonbottom_dofs);\n",
    "        problem = fem.petsc.LinearProblem(a, L, bcs=[bc_], petsc_options={\"ksp_type\": \"preonly\", \"pc_type\": \"lu\"})\n",
    "\n",
    "        uh = problem.solve()\n",
    "        for (k, case) in enumerate([\"xx\", \"yy\", \"zz\", \"yz\", \"xz\", \"xy\"]):\n",
    "            ϵ_i = fem.assemble_scalar(fem.form(strein2Voigt(epsilon(uh))[k]*dx)) / volume;\n",
    "            σ_i = fem.assemble_scalar(fem.form(stress2Voigt(sigma(uh))[k]*dx)) / volume;\n",
    "            m_σ[indexVoigt(i, j), k] = σ_i;\n",
    "            m_ε[indexVoigt(i, j), k] = ϵ_i;\n",
    "            print(\"ε{} = {}; σ{} = {} \".format(case, ϵ_i, case, σ_i));\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "20a75e27-bd41-4c49-9ddf-80eddfb4bbcd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.83e+10,  1.21e+10,  1.21e+10,  1.28e-20,  1.57e-07,  2.41e-07],\n",
       "       [ 1.21e+10,  2.83e+10,  1.21e+10,  4.43e-08,  5.09e-21,  1.46e-07],\n",
       "       [ 1.21e+10,  1.21e+10,  2.83e+10,  1.72e-08, -1.82e-07, -7.48e-21],\n",
       "       [ 5.31e-08,  5.35e-08,  1.24e-07,  8.08e+09,  5.91e-08, -9.00e-08],\n",
       "       [-2.70e-07, -6.54e-08,  5.16e-08,  1.16e-07,  8.08e+09, -2.00e-09],\n",
       "       [ 4.15e-07,  5.57e-07,  2.91e-07,  6.99e-08,  3.83e-08,  8.08e+09]])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.set_printoptions(precision=2);\n",
    "m_σ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "73649cfd-5b61-4209-8ee2-14fe9d42047d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00e+00, -5.39e-31, -1.46e-30,  3.34e-31,  1.94e-17,  2.99e-17],\n",
       "       [-1.75e-31,  1.00e+00, -3.40e-31,  5.48e-18,  4.50e-31,  1.81e-17],\n",
       "       [-1.59e-30, -1.34e-30,  1.00e+00,  2.12e-18, -2.26e-17, -2.14e-32],\n",
       "       [-3.30e-31,  2.34e-20,  4.36e-18,  1.00e+00,  7.32e-18, -1.11e-17],\n",
       "       [-1.26e-17,  4.16e-32,  7.25e-18,  1.43e-17,  1.00e+00, -2.48e-19],\n",
       "       [ 7.63e-18,  1.64e-17,  3.62e-31,  8.66e-18,  4.75e-18,  1.00e+00]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m_ε"
   ]
  }
 ],
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