{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Finite Elements [(ipynb)](pset5.ipynb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## [problem 1 (pdf)](pset6.pdf)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1c. Hermite Polynomials" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sympy import *\n", "import numpy as np\n", "from matplotlib import pyplot as plt" ] }, { "cell_type": "code", "execution_count": 393, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": 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u25G4giQWQviAo0fN0Lrc3PqHqCjYtg02brzBQYmJ5k44Z06vxSWy6L77zMKm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\n+7I2ba3dYf1GVn4WV797NfHH4ll35zou73q53SHVSz38/dk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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x, i, h = symbols('x i h')\n", "\n", "expr1 = Piecewise(\n", " (0, (x-i) > h),\n", " (0, (x-i) < -h),\n", " (2*((x-i)/h)**3-3*((x-i)/h)**2+1, (x-i) > 0),\n", " (-2*((x-i)/h)**3-3*((x-i)/h)**2+1, (x-i) <= 0)\n", " )\n", "\n", "expr2 = Piecewise(\n", " (0, (x-i) > h),\n", " (0, (x-i) < -h),\n", " (((x-i)/h)**3-2*((x-i)/h)**2+(x-i)/h, (x-i)>0),\n", " (((x-i)/h)**3+2*((x-i)/h)**2+(x-i)/h, (x-i)<=0)\n", " )\n", "\n", "L=1\n", "h_val = 1\n", "\n", "X = np.linspace(-2,2, 100)\n", "\n", "Y1 = [expr1.subs([(i,0),(h,h_val),(x,val)]) for val in X]\n", "Y2 = [expr2.subs([(i,0),(h,h_val),(x,val)]) for val in X]\n", "Y3 = [expr1.subs([(i,1),(h,h_val),(x,val)]) for val in X]\n", "Y4 = [expr2.subs([(i,1),(h,h_val),(x,val)]) for val in X]\n", "\n", "plt.plot(X, Y1)\n", "plt.plot(X, Y2)\n", "plt.plot(X, Y3)\n", "plt.plot(X, Y4)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Matrix Coefficients" ] }, { "cell_type": "code", "execution_count": 410, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "h = 1\n", "\n", " A = \n", "\n", "[26/35 0 9/70 -13/420]\n", "[0 2/105 13/420 -1/140]\n", "[9/70 13/420 26/35 0]\n", "[-13/420 -1/140 0 2/105]\n", "\n", " B = \n", "\n", "[12/5 1/5 -7/10 0]\n", "[-1/5 4/15 0 -1/20]\n", "[-17/10 -1/5 12/5 1/5]\n", "[1/5 -1/60 -1/5 4/15]\n" ] } ], "source": [ "h_val = 1\n", "\n", "exprs = [expr1.subs([(i,0),(h,h_val)]), expr2.subs([(i,0),(h,h_val)]), expr1.subs([(i,1),(h,h_val)]), expr2.subs([(i,1),(h,h_val)])]\n", "\n", "rows = np.array([[integrate(a*b, (x,-1,2)) for a in exprs] for b in exprs])\n", "\n", "rowsB = np.array([[integrate(diff(a,x)*diff(b,x)-a*diff(b), (x,-1,2)) for a in exprs] for b in exprs])\n", "\n", "\n", "rows[0,1]=rows[1,0]=rows[2,3]=rows[3,2]=0\n", "\n", "print \"h = %d\" % h_val\n", "\n", "print \"\\n A = \\n\"\n", "\n", "for row in rows:\n", " print row\n", " \n", "print \"\\n B = \\n\"\n", "\n", "for row in rowsB:\n", " print row" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Beam Bending" ] }, { "cell_type": "code", "execution_count": 397, "metadata": { "collapsed": false }, "outputs": [], "source": [ "x, i, h = symbols('x i h')\n", "\n", "expr1 = Piecewise(\n", " (0, (x-i) > h),\n", " (0, (x-i) < -h),\n", " (2*((x-i)/h)**3-3*((x-i)/h)**2+1, (x-i) > 0),\n", " (-2*((x-i)/h)**3-3*((x-i)/h)**2+1, (x-i) <= 0)\n", " )\n", "\n", "expr2 = Piecewise(\n", " (0, (x-i) > h),\n", " (0, (x-i) < -h),\n", " (((x-i)/h)**3-2*((x-i)/h)**2+(x-i)/h, (x-i)>0),\n", " (((x-i)/h)**3+2*((x-i)/h)**2+(x-i)/h, (x-i)<=0)\n", " )\n", "\n", "L=1\n", "h_val = 0.1\n", "\n", "exprs2 = [expr1.subs([(i,0),(h, h_val)]), expr2.subs([(i,0),(h, h_val)]), expr1.subs([(i,1),(h, h_val)]), expr2.subs([(i,1),(h, h_val)])]\n", "\n", "rows2 = np.array([[integrate(diff(a,x,x)*diff(b,x,x), (x,-1,2)) for a in exprs2] for b in exprs2])\n", "\n", "dim = int(2*L/h_val)\n", "A = np.zeros((dim,dim))\n", "\n", "A[0,0]=1\n", "A[1,1]=1\n", "\n", "for ind in range(2,dim-2,2):\n", " A[ind:ind+4,ind:ind+4] = rows2\n", " \n" ] }, { "cell_type": "code", "execution_count": 403, "metadata": { "collapsed": false }, "outputs": [], "source": [ "\n", "def final_exp(f):\n", " \n", " B = np.zeros((dim,1))\n", "\n", " for ind in range(2,dim-1,2):\n", " B[ind:ind+2,0] = [integrate(a*f,(x,-1,2)) for a in exprs2[:2]]\n", " \n", " a_vals = np.linalg.inv(A).dot(B) \n", " bases = []\n", "\n", " for ind in np.arange(0,float(L),h_val):\n", " bases.extend([expr1.subs([(i, ind),(h,h_val)]),expr2.subs([(i, ind),(h,h_val)])])\n", "\n", " return sum([bases[ind]*a_vals[ind] for ind in range(20)]),bases\n", " \n" ] }, { "cell_type": "code", "execution_count": 404, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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LZj4M/HVWiHcA/5ZOiP6EV0VWBfw0HX+hWP6UeYiZzQCGp+MIm1kmS9jonEsc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\n3HIhZGZmZrnlQsjMzMxy6/8APgDPfSIn76QAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Xs = np.arange(0,L,h_val/5)\n", "\n", "for f in np.linspace(0,-1,5):\n", " exp,bases = final_exp(f)\n", " Ys = [exp.subs(x,ind) for ind in Xs]\n", " plt.plot(Xs, Ys)\n", "plt.show()\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.12" } }, "nbformat": 4, "nbformat_minor": 1 }