fem_ue3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Für dieses Script brauchen wir die Matplotlib und Numpy. Falls diese noch nicht installiert sind, hole dies nach."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "# pip install numpy\n",
    "# pip install matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In diesem Script wollen wir die Verformung eines Netzes in 2 Dimensionen mit homogenen Eigenschaften brechnen. Daher legen wir zunächst die Parameter fest.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "Lx = 1.0   # Länge in x-Richtung\n",
    "Ly = 1.0   # Länge in y-Richtung\n",
    "nx = 10     # Anzahl der Elemente in x-Richtung\n",
    "ny = 10    # Anzahl der Elemente in y-Richtung\n",
    "E = 100.0  # Elastizitätsmodul\n",
    "A = 1.0    # Querschnittsfläche\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Einige Parameter ergeben sich logisch aus den gegenbenen Parametern"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_nodes_x = nx + 1\n",
    "n_nodes_y = ny + 1\n",
    "n_nodes = n_nodes_x * n_nodes_y\n",
    "\n",
    "x_coords = np.linspace(0, Lx, n_nodes_x)\n",
    "y_coords = np.linspace(0, Ly, n_nodes_y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So auf die Koordinaten der Knoten."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "nodes = []\n",
    "for j in range(n_nodes_y):\n",
    "    for i in range(n_nodes_x):\n",
    "        nodes.append((x_coords[i], y_coords[j]))\n",
    "nodes = np.array(nodes)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nun erfolgt die Zuordnung der Elemente.\n",
    "Die Fläche wird hierbei in Quadratische Elemente unterteilt und deren Flächendehnung und Scherungbeanspruchung berechnent. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "elements = []\n",
    "for j in range(ny):\n",
    "    for i in range(nx):\n",
    "        n1 = j * n_nodes_x + i\n",
    "        n2 = n1 + 1\n",
    "        n3 = n1 + n_nodes_x\n",
    "        n4 = n3 + 1\n",
    "        elements.append([n1, n2, n4, n3])\n",
    "elements = np.array(elements)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot der Netzstruktur zur Kontrolle\n",
    "fig, ax = plt.subplots()\n",
    "for element in elements:\n",
    "    x = [nodes[element[i]][0] for i in range(4)] + [nodes[element[0]][0]]\n",
    "    y = [nodes[element[i]][1] for i in range(4)] + [nodes[element[0]][1]]\n",
    "    ax.plot(x, y, 'b')\n",
    "\n",
    "ax.set_aspect('equal')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('2D Finite Elemente Netz')\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}