{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Contenido bajo licencia Creative Commons BY 4.0 y código bajo licencia MIT. © Juan Gómez y Nicolás Guarín-Zapata 2020. Este material es parte del curso Modelación Computacional en el programa de Ingeniería Civil de la Universidad EAFIT."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Ensamblaje paso a paso"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Este notebook presenta de manera automática las matrices de rigidez locales para cada elemento y la matriz de rigidez global luego de ensamblar el elemento correspondiente. Se sugiere usar este notebook para verificar el proceso de ensamblaje hecho a mano. Adicionalmente, se sugiere no usarse para sistemas con más de 10 elementos ya que requeriría de matrices muy grandes para resultar práctico.\n",
"\n",
"\n",
"Este notebok está acompañado de un [notebook auxiliar](./06b_ensamblaje.ipynb) que explica cómo construir el sistema de ecuaciones para un sistema de masas y resortes con un ejemplo paso a paso.\n",
"\n",
"**Nota:** Este notebook utiliza los mismos archivos de texto del sistema de resortes del notebook principal y disponibles en la carpeta `files` del repositorio. Para poder continuar con el resto del notebook es necesario que estos archivos esten disponibles en memoria."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from IPython.display import display, Markdown\n",
"import pandas as pd"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def readin():\n",
" nodes = np.loadtxt('files/sprnodes.txt', ndmin=2)\n",
" mats = np.loadtxt('files/sprmater.txt', ndmin=2)\n",
" elements = np.loadtxt('files/spreles.txt', ndmin=2, dtype=np.int)\n",
" loads = np.loadtxt('files/sprloads.txt', ndmin=2)\n",
" return nodes, mats, elements, loads"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def eqcounter(nodes):\n",
" nn = nodes.shape[0]\n",
" IBC = np.zeros((nn, 1), dtype=np.integer)\n",
" neq = 0\n",
" for cont in range(nn):\n",
" IBC[cont] = int(nodes[cont, 2])\n",
" if IBC[cont] == 0:\n",
" IBC[cont] = neq\n",
" neq = neq + 1\n",
" return neq, IBC"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"def DME(nodes, elements):\n",
" nels = elements.shape[0]\n",
" DME_mat = np.zeros((nels, 2), dtype=np.integer)\n",
" neq, IBC = eqcounter(nodes)\n",
" ndof = 2\n",
" nnodes = 2\n",
" for ele in range(nels):\n",
" for node in range(nnodes):\n",
" pos = elements[ele, node + 3]\n",
" DME_mat[ele, node] = IBC[pos]\n",
" return DME_mat, IBC, neq"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"def uelspring(kcof):\n",
" \"\"\"\n",
" Esta rutina calcula la matriz de rigidez para\n",
" un elemento tipo resorte conformado por 2 nodos.\n",
"\n",
" Parámetros\n",
" ----------\n",
" kcof : float\n",
" Coeficiente de rigidez del resorte (>0).\n",
"\n",
" Retorna\n",
" -------\n",
" kloc : ndarray\n",
" Matriz de coeficientes de rigidez local para\n",
" el elemento tipo resorte (2, 2).\n",
"\n",
"\n",
" \"\"\"\n",
" kloc = np.array([\n",
" [kcof, -kcof],\n",
" [-kcof, kcof]])\n",
" return kloc"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"def assem_msg(el, kloc, kglob):\n",
" \"\"\"Muestra la matriz local y global para un elemento dado\"\"\"\n",
" display(Markdown(\"### Elemento {:d}\".format(el)))\n",
" display(Markdown(\"Matriz local\"))\n",
" display(pd.DataFrame(kloc))\n",
" display(Markdown(\"Matriz global luego de ensamblar\"))\n",
" display(pd.DataFrame(kglob))\n",
" return None"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"def assembly(elements, mats, nodes, neq, DME_mat, steps=True):\n",
" IELCON = np.zeros([2], dtype=np.integer)\n",
" KG = np.zeros((neq, neq))\n",
" nels = elements.shape[0]\n",
" nnodes = 2\n",
" ndof = 2\n",
" for el in range(nels):\n",
" elcoor = np.zeros((nnodes))\n",
" im = np.int(elements[el, 2])\n",
" par = mats[im]\n",
" for j in range(nnodes):\n",
" IELCON[j] = elements[el, j+3]\n",
" elcoor[j] = nodes[IELCON[j], 1]\n",
" kloc = uelspring(par[0])\n",
" dme = DME_mat[el, :ndof]\n",
" for row in range(ndof):\n",
" glob_row = dme[row]\n",
" if glob_row != -1:\n",
" for col in range(ndof):\n",
" glob_col = dme[col]\n",
" if glob_col != -1:\n",
" KG[glob_row, glob_col] = KG[glob_row, glob_col] +\\\n",
" kloc[row, col]\n",
" if steps:\n",
" assem_msg(el, kloc, KG)\n",
" return KG"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"def loadasem(loads, IBC, neq, nl):\n",
" \"\"\"\n",
" Ensambla el vector de cargas o de valores conocidos\n",
" en el sistema global de ecuaciones.\n",
"\n",
" Parámetros\n",
" ----------\n",
" loads : ndarray\n",
" Arreglo con las cargas impuestas a las partículas.\n",
" IBC : ndarray (int)\n",
" Arreglo que almacena el identificador de grado de libertad\n",
" a cada partícula.\n",
" neq : int\n",
" Numero de ecuaciones en el sistema.\n",
" nl : int\n",
" Numero de cargas en el sistema.\n",
"\n",
" Retorna\n",
" -------\n",
" rhs_glob : ndarray\n",
" Arreglo con las cargas impuestas a las partículas,\n",
" rhs se refiere a lado derecho (del inglés right-hand-side).\n",
"\n",
" \"\"\"\n",
" rhs_glob = np.zeros((neq))\n",
" for cont in range(nl):\n",
" il = int(loads[cont, 0])\n",
" ilx = IBC[il]\n",
" if ilx != -1:\n",
" rhs_glob[ilx] = loads[cont, 1]\n",
" return rhs_glob"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"nodes, mats, elements, loads = readin()\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"La siguiente celda ensambla el sistema de ecuaciones y muestra cada uno de los pasos."
]
},
{
"cell_type": "code",
"execution_count": 10,
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"source": [
"DME_mat, IBC, neq = DME(nodes, elements)\n",
"KG = assembly(elements, mats, nodes, neq, DME_mat)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0.002 0.0025 0.0045]\n"
]
}
],
"source": [
"RHSG = loadasem(loads, IBC, neq, 3)\n",
"UG = np.linalg.solve(KG, RHSG)\n",
"print(UG)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Formato del notebook"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"La siguiente celda cambia el formato del Notebook."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
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"<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
"<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
"\n",
"<style>\n",
"\n",
"/*\n",
"Template for Notebooks for Modelación computacional.\n",
"\n",
"Based on Lorena Barba template available at:\n",
"\n",
" https://github.com/barbagroup/AeroPython/blob/master/styles/custom.css\n",
"*/\n",
"\n",
"/* Fonts */\n",
"@font-face {\n",
"font-family: \"Computer Modern\";\n",
"src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
"}\n",
"\n",
"/* Text */\n",
"div.cell{\n",
"width:800px;\n",
"margin-left:16% !important;\n",
"margin-right:auto;\n",
"}\n",
"h1 {\n",
"font-family: 'Alegreya Sans', sans-serif;\n",
"}\n",
"h2 {\n",
"font-family: 'Fenix', serif;\n",
"}\n",
"h3{\n",
"font-family: 'Fenix', serif;\n",
"margin-top:12px;\n",
"margin-bottom: 3px;\n",
"}\n",
"h4{\n",
"font-family: 'Fenix', serif;\n",
"}\n",
"h5 {\n",
"font-family: 'Alegreya Sans', sans-serif;\n",
"}\t\n",
"div.text_cell_render{\n",
"font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
"line-height: 135%;\n",
"font-size: 120%;\n",
"width:600px;\n",
"margin-left:auto;\n",
"margin-right:auto;\n",
"}\n",
".CodeMirror{\n",
"font-family: \"Source Code Pro\";\n",
"font-size: 90%;\n",
"}\n",
"/* .prompt{\n",
"display: None;\n",
"}*/\n",
".text_cell_render h1 {\n",
"font-weight: 200;\n",
"font-size: 50pt;\n",
"line-height: 100%;\n",
"color:#CD2305;\n",
"margin-bottom: 0.5em;\n",
"margin-top: 0.5em;\n",
"display: block;\n",
"}\t\n",
".text_cell_render h5 {\n",
"font-weight: 300;\n",
"font-size: 16pt;\n",
"color: #CD2305;\n",
"font-style: italic;\n",
"margin-bottom: .5em;\n",
"margin-top: 0.5em;\n",
"display: block;\n",
"}\n",
".warning{\n",
"color: rgb( 240, 20, 20 )\n",
"}\n",
"</style>\n",
"\n",
"<script>\n",
"/* Equations */\n",
"\n",
"MathJax.Hub.Config({\n",
"TeX: {\n",
"extensions: [\"AMSmath.js\"]\n",
"},\n",
"tex2jax: {\n",
"inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
"displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
"},\n",
"displayAlign: 'center', // Change this to 'center' to center equations.\n",
"\"HTML-CSS\": {\n",
"styles: {'.MathJax_Display': {\"margin\": 4}}\n",
"}\n",
"});\n",
"</script>\n",
"\n",
"\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.core.display import HTML\n",
"def css_styling():\n",
" styles = open('./nb_style.css', 'r').read()\n",
" return HTML(styles)\n",
"css_styling()"
]
}
],
"metadata": {
"celltoolbar": "Raw Cell Format",
"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": "3.7.3"
},
"varInspector": {
"cols": {
"lenName": 16,
"lenType": 16,
"lenVar": 40
},
"kernels_config": {
"python": {
"delete_cmd_postfix": "",
"delete_cmd_prefix": "del ",
"library": "var_list.py",
"varRefreshCmd": "print(var_dic_list())"
},
"r": {
"delete_cmd_postfix": ") ",
"delete_cmd_prefix": "rm(",
"library": "var_list.r",
"varRefreshCmd": "cat(var_dic_list()) "
}
},
"types_to_exclude": [
"module",
"function",
"builtin_function_or_method",
"instance",
"_Feature"
],
"window_display": false
}
},
"nbformat": 4,
"nbformat_minor": 1
}