{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Defining the target Wo type function\n", "Let calculate the derivate expression of the family of functions we want to play with. \n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 97, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "exp(sin(t))*cos(t)\n" ] } ], "source": [ "#Importing all stuff\n", "\n", "from sympy import *\n", "\n", "#////////////// Lets define here the expression of Wo we want to calculate the fold state. /////////////\n", "\n", "\n", "t = Symbol('t') #Defining the symbol t\n", "\n", "Wo = exp(sin(t)) #Plug here your expression. Lets derivate it with sympy in case its get complex. \n", " # IMPORTANT. Plug it in sympy notation so there is no chance to mistake. Later on we will re write it in np. \n", " \n", "dWo = diff(Wo) #Now, let derive the expression and print the result so we can manually :( translate into np. \n", "print(dWo)\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, as it was explained in the theory, the differential equation in which we have to implement this expression is the following: " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "$$\\frac{\\partial z}{\\partial t}= \\sqrt{1-tan(\\gamma )^{2}\\left ( \\frac{\\partial W_{0} }{\\partial t} \\right )^{2}} $$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will implement a function dzdt in python in which we will implement the previous expression in the $\\frac{\\partial z}{\\partial t}$" ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "scrolled": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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We are asking later in the euler integration the slope of the\n", " z(t) function at different x values.\n", " \n", " '''\n", " fold_ang = Gamma/180 * np.pi # Degrees\n", " return np.sqrt(1-(np.tan(fold_ang)**2)*((dWodt(t))**2))\n", "\n", "\n", "\n", "\n", "def euler(n, Gamma, length, steps):\n", " ax.clear()\n", " fold_ang = Gamma/180 * np.pi # Degrees\n", " x0 = 0 #Initial Conditions. Mandatory to solve the ODE\n", " y0 = 0 #Initial Conditions. Mandatory to solve the ODE\n", " xf = 2*np.pi #Limit of the calculation. Any value furder than 2pi will be a repetition. \n", " h = (xf-x0)/n #Tiny steps definition\n", " x = [x0] #Array of the discrete estimated x values\n", " y = [y0] #Array of the discrete estimated y values\n", "\n", " for i in range(n):\n", " y0 = y0 + h * dzdt(x0,y0, Gamma)\n", " x0 = x0 + h\n", " x.append(x0)\n", " y.append(y0)\n", " \n", " z = []\n", " for i in x:\n", " z.append(Wo(t)/np.cos(fold_ang))\n", " \n", " h = []\n", " for i in y:\n", " h.append(0)\n", " print(len(h))\n", " print(len(z))\n", " print(range(n+1))\n", " \n", " mesh_h = length / steps\n", " for stp in range(steps):\n", " for i in range(n+1):\n", " y.append(y[i])\n", " z.append(z[i]-stp*mesh_h*np.cos(fold_ang))\n", " h.append((-stp)*np.sin(fold_ang))\n", " \n", " #Ahora tengo todos los vectores con los mismos valores. \n", " \n", " #y,z,h = np.meshgrid(y, z,h)\n", " ax.scatter(y,z,h ,color = 'black',linewidth = 1, label = '$Wgamma(t)$')\n", " ax.plot(x, Wo(x), color = 'magenta', linewidth = 1 ,label = '$Wo(t)$')\n", " ax.plot_trisurf(y, z, h, vmax =0.1, shade = True, color = 'pink', alpha = 0.5)\n", " ax.set_xlim(0,6.3)\n", " ax.set_ylim(-10,0) \n", " ax.set_zlim(0,-6) \n", "\n", " \n", " \n", " plt.legend()\n", " plt.show()\n", " return x, y, z\n", "\n", "\n", "\n", "length= widgets.FloatSlider(min=0, max=45, value=10, description='Side Length:')\n", "steps = widgets.IntSlider(min=1, max=10, value=5, description='Mesh def:')\n", "n = widgets.IntSlider(min=1, max=600, value=50, description='n:')\n", "Gamma = widgets.FloatSlider(min=0, max=45, value=0, description='Fold it:')\n", " \n", "widgets.interactive(euler, n=n, Gamma=Gamma, length = length, steps = steps)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "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": "3.6.9" } }, "nbformat": 4, "nbformat_minor": 4 }