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_sources/FEM/myFiniteElement.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "5dc5bf2d",
+   "metadata": {},
+   "source": [
+    "# Implementation of Finite Elements\n",
+    "\n",
+    "This lecture shows how to implement our own finite elements in C++,\n",
+    "and how to use them within the NGSolve language. We implement first order and second order triangular finite elements.\n",
+    "\n",
+    "The finite element is implemented in small package on github: \n",
+    "[https://github.com/TUWien-ASC/NGS-myfe](https://github.com/TUWien-ASC/NGS-myfe)\n",
+    "\n",
+    "* basis functions are implemented in myelement.cpp/hpp\n",
+    "* the transformation from reference-element to physical elements in in mydiffop.hpp\n",
+    "* the connectivity is implemented in the `FESpace` in myfespace.cpp/hpp\n",
+    "* the Python-bindings are in mymodule.cpp"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "fb29e374",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Loading libmyfe library\n"
+     ]
+    }
+   ],
+   "source": [
+    "from ngsolve import *\n",
+    "from ngsolve.webgui import Draw\n",
+    "from libmyfe import *\n",
+    "mesh = Mesh(unit_square.GenerateMesh(maxh=0.2, quad_dominated=False))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "74066dcd",
+   "metadata": {},
+   "source": [
+    "We can now create an instance of our own finite element space:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "122e9720",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Constructor of MyFESpace\n",
+      "Flags = secondorder = 1\n",
+      "secondorder\n",
+      "dirichlet = 0: 1\n",
+      "1: 3\n",
+      "2: 4\n",
+      "\n",
+      "\n",
+      "You have chosen second order elements\n",
+      "Update MyFESpace, #vert = 39, #edge = 94\n"
+     ]
+    }
+   ],
+   "source": [
+    "fes = MyFESpace(mesh, secondorder=True, dirichlet=\"left|bottom|top\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "23837763",
+   "metadata": {},
+   "source": [
+    "and use it within NGSolve such as the builtin finite element spaces:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "627a9cfb",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ndof =  133\n"
+     ]
+    }
+   ],
+   "source": [
+    "print (\"ndof = \", fes.ndof)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "ae0deee7",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "caf94b64283e48149e05292ef58c4bc0",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "WebGuiWidget(layout=Layout(height='50vh', width='100%'), value={'gui_settings': {}, 'ngsolve_version': '6.2.24…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "13271b2614e74350ab92e97d0976bfc0",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "WebGuiWidget(layout=Layout(height='50vh', width='100%'), value={'gui_settings': {}, 'ngsolve_version': '6.2.24…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "gfu = GridFunction(fes)\n",
+    "gfu.Set(x*y)\n",
+    "\n",
+    "Draw (gfu)\n",
+    "Draw (grad(gfu)[0], mesh);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b93211e8",
+   "metadata": {},
+   "source": [
+    "and solve the standard problem:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "5402e517",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "d0797c3ccb57429fa5790d8da899ecee",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "WebGuiWidget(layout=Layout(height='50vh', width='100%'), value={'gui_settings': {}, 'ngsolve_version': '6.2.24…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "u,v = fes.TnT()\n",
+    "a = BilinearForm(grad(u)*grad(v)*dx).Assemble()\n",
+    "f = LinearForm(1*v*dx).Assemble()\n",
+    "gfu.vec.data = a.mat.Inverse(fes.FreeDofs())*f.vec\n",
+    "Draw (gfu);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bfc19b37",
+   "metadata": {},
+   "source": [
+    "Draw basis functions:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a735336a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "gfu.vec[:] = 0\n",
+    "gfu.vec[mesh.nv-3] = 1\n",
+    "gfu.vec[fes.ndof-1] = 1\n",
+    "Draw (gfu, order=2);"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8edb5e20",
+   "metadata": {},
+   "source": [
+    "Documentation provided in the DocInfo structure is available in Python - help. Look for 'secondorder'."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "89d8a842",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "help (m.MyFESpace)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ad3d3978",
+   "metadata": {},
+   "source": [
+    "**Exercises:**\n",
+    "\n",
+    "Extend MyFESpace by the following 1st and 2nd order elements:\n",
+    "\n",
+    "- 1D finite elements (ET_SEGM), as needed for boundary conditions\n",
+    "- quadrilateral elements (ET_QUAD), use geom.GenerateMesh(quad_dominated=True)\n",
+    "- tetrahedral elements (ET_TET), test it for 3D domains\n",
+    "\n",
+    "Next, implement 3rd order elements"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7c1edee8",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}