From 889acf40251fbf3602f3714991430b0b91864a1c Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Mon, 13 Feb 2017 16:54:49 -0500 Subject: [PATCH] update HW3 --- HW3/README.md | 6 +- lecture_07/.lecture_07.md.swo | Bin 0 -> 16384 bytes lecture_07/.lecture_07.md.swp | Bin 0 -> 16384 bytes "lecture_08/\020\267Y\001" | 0 .../lecture_08-checkpoint.ipynb | 6 + lecture_08/Auchain_model.gif | Bin 0 -> 17748 bytes lecture_08/au_chain.jpg | Bin 0 -> 234971 bytes lecture_08/goldenratio.png | Bin 0 -> 94208 bytes lecture_08/lecture_08.ipynb | 1941 +++++++++++++++++ lecture_08/lennard_jones.m | 4 + lecture_08/octave-workspace | Bin 0 -> 4894 bytes "lecture_08/\34040\001" | 0 12 files changed, 1954 insertions(+), 3 deletions(-) create mode 100644 lecture_07/.lecture_07.md.swo create mode 100644 lecture_07/.lecture_07.md.swp create mode 100644 "lecture_08/\020\267Y\001" create mode 100644 lecture_08/.ipynb_checkpoints/lecture_08-checkpoint.ipynb create mode 100644 lecture_08/Auchain_model.gif create mode 100644 lecture_08/au_chain.jpg create mode 100644 lecture_08/goldenratio.png create mode 100644 lecture_08/lecture_08.ipynb create mode 100644 lecture_08/lennard_jones.m create mode 100644 lecture_08/octave-workspace create mode 100644 "lecture_08/\34040\001" diff --git a/HW3/README.md b/HW3/README.md index 494e0d5..eb93c41 100644 --- a/HW3/README.md +++ b/HW3/README.md @@ -12,7 +12,7 @@ *Disable the plotting routine for the solvers* d. Use the four solvers `falsepos.m`, `incsearch.m`, `newtraph.m` and `mod_secant.m` - to solve for the angle needed to reach h=1.72 m, with an initial speed of 1.5 m/s. + to solve for the angle needed to reach h=1.72 m, with an initial speed of 15 m/s. e. The `newtraph.m` function needs a derivative, calculate the derivative of your function with respect to theta, `dprojectile_dtheta.m`. This function should @@ -29,7 +29,7 @@ | solver | initial guess(es) | ea | number of iterations| | --- | --- | --- | --- | |falsepos | | | | - |incsearch | | | | + |bisect | | | | |newtraph | | | | |mod_secant | | | | ``` @@ -49,7 +49,7 @@ using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain guesses that do not converge. a. Calculate the first 5 iterations for the Newton-Raphson method with an initial - guess of x_i=2. + guess of x_i=2 for f(x)=x*exp(-x^2). b. 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maximum based on given constraints. Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The Lennard-Jones potential is commonly used to model interatomic bonding. \n", + "\n", + "$E_{LJ}(x)=4\\epsilon \\left(\\left(\\frac{\\sigma}{x}\\right)^{12}-\\left(\\frac{\\sigma}{x}\\right)^{6}\\right)$\n", + "\n", + "Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no force applied to the chain and even for tension, F. This will allow us to calculate the nonlinear spring constant of a 1-D gold chain. \n", + "\n", + "![TEM image of Gold chain](au_chain.jpg)\n", + "\n", + "Computational Tools to Study and Predict the Long-Term Stability of Nanowires.\n", + "By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva \n", + "\n", + "![Model of Gold chain, from molecular dynamics simulation](Auchain_model.gif)\n", + "\n", + "[Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/)\n", + "\n", + "### First, let's find the minimum energy $\\min(E_{LJ}(x))$\n", + "\n", + "## Brute force" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "setdefaults\n", + "epsilon = 0.039; % kcal/mol\n", + "sigma = 2.934; % Angstrom\n", + "x=linspace(2.8,6,200); % bond length in Angstrom\n", + "\n", + "Ex = lennard_jones(x,sigma,epsilon);\n", + "\n", + "[Emin,imin]=min(Ex);\n", + "\n", + "plot(x,Ex,x(imin),Emin,'o')" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 3.2824\n", + "ans = 3.2985\n", + "ans = 3.3146\n" + ] + } + ], + "source": [ + "x(imin-1)\n", + "x(imin)\n", + "x(imin+1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Golden Search Algorithm\n", + "\n", + "We can't just look for a sign change for the problem (unless we can take a derivative) so we need a new approach to determine whether we have a maximum between the two bounds.\n", + "\n", + "Rather than using the midpoint of initial bounds, here the problem is more difficult. We need to compare the values of 4 function evaluations. The golden search uses the golden ratio to determine two interior points. \n", + "\n", + "![golden ratio](goldenratio.png)\n", + "\n", + "Start with bounds of 2.5 and 6 Angstrom. " + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "current_min = -0.019959\r\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% define Au atomic potential\n", + "epsilon = 0.039; % kcal/mol\n", + "sigma = 2.934; % Angstrom\n", + "Au_x= @(x) lennard_jones(x,sigma,epsilon);\n", + "\n", + "% calculate golden ratio\n", + "phi = 1/2+sqrt(5)/2;\n", + "% set initial limits\n", + "x_l=2.8; \n", + "x_u=6; \n", + "\n", + "% Iteration #1\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Iteration #2\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Iteration #3\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_6a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Iteration #3\n", + "d=(phi-1)*(x_u-x_l);\n", + "\n", + "x1=x_l+d; % define point 1\n", + "x2=x_u-d; % define point 2\n", + "\n", + "% evaluate Au_x(x1) and Au_x(x2)\n", + "\n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n", + "hold on;\n", + "\n", + "if f2\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% define Au atomic potential\n", + "epsilon = 0.039; % kcal/mol\n", + "sigma = 2.934; % Angstrom\n", + "Au_x= @(x) lennard_jones(x,sigma,epsilon);\n", + "\n", + "% set initial limits\n", + "x_l=2.8; \n", + "x_u=3.5; \n", + "\n", + "% Iteration #1\n", + "x1=x_l;\n", + "x2=mean([x_l,x_u]);\n", + "x3=x_u;\n", + "\n", + "% evaluate Au_x(x1), Au_x(x2) and Au_x(x3)\n", + " \n", + "f1=Au_x(x1);\n", + "f2=Au_x(x2);\n", + "f3=Au_x(x3);\n", + "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n", + "x_fit = linspace(x1,x3,20);\n", + "y_fit = polyval(p,x_fit);\n", + "\n", + "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n", + "hold on\n", + "if f2x2\n", + " plot(x4,f4,'*',[x1,x2],[f1,f2])\n", + " x1=x2;\n", + " f1=f2;\n", + " else\n", + " plot(x4,f4,'*',[x3,x2],[f3,f2])\n", + " x3=x2;\n", + " f3=f2;\n", + " end\n", + " x2=x4; f2=f4;\n", + "else\n", + " error('no minimum in bracket')\n", + "end\n", + "hold off" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f3 = -0.038878\r\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n", + "x_fit = linspace(x1,x3,20);\n", + "y_fit = polyval(p,x_fit);\n", + "\n", + "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n", + "hold on\n", + "if f2x2\n", + " plot(x4,f4,'*',[x1,x2],[f1,f2])\n", + " x1=x2;\n", + " f1=f2;\n", + " else\n", + " plot(x4,f4,'*',[x3,x2],[f3,f2])\n", + " x3=x2;\n", + " f3=f2;\n", + " end\n", + " x2=x4; f2=f4;\n", + "else\n", + " error('no minimum in bracket')\n", + "end\n", + "hold off\n" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f3 = -0.038878\r\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.02\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.04\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.06\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.08\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", + "\t \n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_5a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n", + "x_fit = linspace(x1,x3,20);\n", + "y_fit = polyval(p,x_fit);\n", + "\n", + "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n", + "hold on\n", + "if f2x2\n", + " plot(x4,f4,'*',[x1,x2],[f1,f2])\n", + " x1=x2;\n", + " f1=f2;\n", + " else\n", + " plot(x4,f4,'*',[x3,x2],[f3,f2])\n", + " x3=x2;\n", + " f3=f2;\n", + " end\n", + " x2=x4; f2=f4;\n", + "else\n", + " error('no minimum in bracket')\n", + "end\n", + "hold off\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Parabolic interpolation does not converge in many scenarios even though it it a bracketing method. Instead, functions like `fminbnd` in Matlab and Octave use a combination of the two (Golden Ratio and Parabolic)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using the solutions to minimization for the nonlinear spring constant\n", + "\n", + "Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain\n", + "\n", + "$E_{total}(\\Delta x) = E_{LJ}(x_{min}+\\Delta x) - F \\cdot \\Delta x$" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "xmin = 3.2933\r\n" + ] + } + ], + "source": [ + "xmin = fminbnd(Au_x,2.8,6)\n", + "\n", + "Etotal = @(dx,F) Au_x(xmin+dx)-F.*dx;\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_08/lennard_jones.m b/lecture_08/lennard_jones.m new file mode 100644 index 0000000..d18a6ad --- /dev/null +++ b/lecture_08/lennard_jones.m @@ -0,0 +1,4 @@ +function E_LJ =lennard_jones(x,sigma,epsilon) + E_LJ = 4*epsilon*((sigma./x).^12-(sigma./x).^6); +end + diff --git a/lecture_08/octave-workspace b/lecture_08/octave-workspace new file mode 100644 index 0000000000000000000000000000000000000000..f505b0a7d58e9e91f9e09b3ff78567b472aeda59 GIT binary patch literal 4894 zcmdT|cUV)|5|5&Ugcec(5s|f^SWvM885=4Jt_@L;>L}ATMu!UlW~~6UY%g z@gyBjXMw<8_^mUvF)u!s0>P(wVbpKEAAw?g8kHx@}{U@#HFr38a zhlNS3yzKo%?&5EQgM-6}f3?JqI^R#}!QIiN{N--afu#}0`6~?-l((!o!tYr7BFO!K zgipT|y5~z?4u5gN;?UuZX?)*NIxm~cWBCR5?+;g{h4LlY4^A!GY0bYP9Bnd*>4A*~ z3SQZaa1_1XFnpxPUc4B+Yi5gqS{JM$!19=I!6exLx~ zR}Z!(Vgb&$b1knVya1_@`ZW% zm^{pQho~bDxgRc^o8_E`g{67xzcd`g!{(`n`>YP)!9FuLj|aK<(8pqOh(#{GdzN@% zV_gnr@seXlFVDe2+p8*{s=-@T->qpL~ipR7U+^Gd=to1_8{%_NMRvv7J}aw1M?u3=gABw*HE_HV5l6R?)5 zwJGOPJeC-$y>Ximj}^?Xosb%bW#u;cJh?c`?4a+yvr`WV;?^As*PYe?ZYc8 zOv4-+W3Vf3`8MIU82r6|+SKU{d-3O%YULFBy{MItWf0dIjScVnUowQzI2V1ytUsdg zw0n10^xi1cPsmHNQ;))3>(&IN6z{=|iKjSf{5|NQKUaM3(QZt7R{Nw{up8UW_4YT@ zcH@&?10(GtkyvirxZBe-65SU}*>JTr0*ySYF0o@HaNg_iK`Vm@q!{&@e!_*QIpL_zae0$=FAQsMmTa=08-{jz zsR?!kq4-$PbS_*a6n9>)vOX2M6MblnTN=7U(DJVEU7u?RhBr??($lm9D}0uU&ROq3 z-HNe^3+sb1a^$8|PwQYTthIDAY6`-qgN9}~EkGDaNVgW*6=Ui&6RT#C7os zVmz1~NzKm)L=EK+c>|LJu~_AdekKNBXQAZ?7s~*=bYtVKO`ZPu@j$hk!#;n^KKzar zJkcMw#q^Cyt?|Rkz;qjls~=iTmt4r9`=MdcW4FMQzUcHa=*2`!Upz`ttC4;aVXh62 z`Km~S_h+1MeY8P@F4-wZ#KT3H)Fo=iQXiCy9v3cg@WDXc`2rrt2bFZ|-*avXv1^ai zdw-}9U%v87ik~gSz>Kp?-`0EMnUdz!32VGjZdO`m_jLijEUNllo-e?&TR3KT!3$MY z+YOZFdtrK;u}$N7Pwe-uY@2NCiC4})o%iah2ky&wDF4aK17F?fv)cS z9iN^YuKlXZ9lNW{S#Ni_V~ziq1TM!NL%2&Q8;jkLeRtZJoJDTvRprs#^xPFs7*HeB z16(m8P04>A%M~qz3c@*ME?E0G)lp%+3%0nl1t0q4jNJtxn!~c3(bc!8fep?WP`x(m zajO$9`WFR2N@38z1 zA>^L2;&W{<9RIYZ5+i z+@JtY*w@q7`71!>VBO|L=?ZYT{hpRVg#yGqdfce}Q~`XeUPzSt6kyA-X6LUOG+310 zrq{86270^RcRbrl169wDCu&1z@Y%B8ry`dIFBef}7go}sdW!s)v_={n<0)N?q|hNn zrU)H^-tiXMFQ$Xt)6pY-cBI3KYd_sE*-eLOCnoKidYBH}%<1M@HFOvt)orE2Yr(PbA4lZtRlFM+$@nER|JbL8|RceiZHw0i4ai{ zv}(3@c&jRbT&eJcje!!hX9xJ0uT_Fyqe2L%mEfFDMDOHOCCIbhdqJaI38I!K>{NK9 z1cI4q(ECXVHjK`a`<<%{Mhf{g59ca_?uWxM*Q}I*-f(PnS&%Yx)t0IjWGTbb)1*a| z0c8eJhC+hNYd{&|{ceu3)>Z*O+g}>Y_$si~>`}pdI~7AafAlc)fpZ2QP- zD#V^|OFp)!KtofHuw{fQT)*{Y*8TCSaQy7I&sCc-Ea`Mqj}I z^VTuPcUdt&W@a}A7~UW6Eev8n5#c?pF$@@WS%0Q&76TH>PNY6AX29^G8K2BAG9Vzg z-17P@1`vy#GkU=Q8GLUUaBrva4*dZJER{Kq8dSI~G11gk1AUvCyyR(WaB%&~VKlxP zFjwAo2w$THdra0;_u8sKAEBkUPz}f^_&r<=nzWy`ZA(^z^=uL`HMl_|0at?sgLcZ* zSJmLe#}>jf)Ihi0N-EFbC}SDBFm)lOmIB&se(F( z36G1#8eyhPFw6fV__HMwD#`G1Wx|ZuVJ5!?GU0IeZo(CqpdLUS_ArA9aUS~|Rvl%6 zY}@lp@FiATP{)MdSFxf?pEF^LEFhS0XO0H3Boh`*$st^x1?6MMC?{&MK#wO8$WLWK zj?#Dn8WyO^0AvB_b)GgX_)MBxz=G|~29{gHSn%=*=}0X2S+*bxs;{08zH*8M^9YpZ zU17no5;Q*bD+{!xi}NNnvmhnoIwh}%1@Z}`2e2V1^47I@78~A)Eyj-=#|AqA>FI2E z=(wTn!%{Y^Afm@(BO5MSkP*#>X-i)!Z3$q5WIoBs2JZBhUsfDoL(Djr8C6HvAaj>< zY!J!V(Q1C~_eg zIiO{&Ey!u&K+@WRUs&B7kXt^EFdY{HjE)dT3>Vr7F9_1*g3ZK|jomZ3@GEhexh>(s zQs!y(ruAGnN1YSA$%zY72F}&r_2WWb&q8C-Tw z;)LQt)8)Ex*-yA2J4V~MP)ml#7cLmd9GV9wiRch(@IW_avr^Y29;B0QwSWh7S>W?v zr#oq29>|U-4<0y?<_O`zQ^FdT;&`x%|NiW(Tpm;!k_z)+CaFdx4+_Tw)@R=3fd;4F zNUf0vvas*ufh;Le)Ma0>;-Tqzmh7Q@Cl5`}7HdzsJGy`W1{M!Z&sjDREeA4xkcXz{ zKWCGd6k=N!cR!ghvJCn6?8+tx)IUX8H5tb-OljaWb`|%X6%9hvpHS;OXppAr*jE@z zgX8j@?UBhexIPg4!tM}}Nj}QoHLjq+r;ZfVtfPT)i*{-63mQy#UMzXoNduE#=cJVm zr-Nnf)jbIuItb7r&_j<7(Py8#tyoBhyb_l!<1Oe=aj0k2aC;qZ>Hh!L@yAt-Y=EL9yNLO(7Y!s#LH`pbordoP zRg*?bsNq@(+GQRs&)6eDu-+utaaMv=4sVaWXptbL#LerHvJ}G?+%D5Jlwy@4FQnH} ziupP#Z>|fFVw2b*Yf1{yvK>2j7fTU7n$>PaDeCYv$CutC_;%hKxu=1kPZVr_(k4Ya l&WCkHoy54QE%x>h>s~B7d7WsheBb@VB*w$NRkg@c?O#Po{_p?* literal 0 HcmV?d00001 diff --git "a/lecture_08/\34040\001" "b/lecture_08/\34040\001" new file mode 100644 index 0000000..e69de29

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