{ "cells": [ { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0, "toc": "true" }, "source": [ "

Table of Contents

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" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "# Solving Ordinary Differential Equations" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "## Objectives\n", "In this lecture, we will review the main computational techniques to solve a first order differential problem (initial value problem) of the type:\n", "\n", "\\begin{equation}\n", "\\left\\lbrace\n", "\\begin{array}{lll}\n", "\\displaystyle{\\frac{dy}{dt}} & = & f(y,t) \\\\\n", "y(t_0) & = & y_0\n", "\\end{array}\n", "\\right.\n", "\\end{equation}\n", "\n", "Specifically you will be able to:\n", "\n", "- link the ODE to a physical problem\n", "\n", "- develop different computational methods to solve the initial value problem\n", "\n", "- understand the concept of stability and accuracy of a computation methods\n", "\n", "- describe Euler forward and backward approaches\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "## Introduction\n", "\n", "Ordinary differential equations (ODEs) arise in many physical\n", "situations. For example, there is the second-order Newton’s Second Law of Mechanics $F=ma$ .\n", "\n", "In general, higher-order equations, such as Newton’s force equation, can\n", "be rewritten as a system of first-order equations. So the generic\n", "problem in ODEs is a set of N coupled first-order differential equations\n", "of the form defined above.\n", "\n", "$$\n", " \\frac{d{\\bf y}}{dt} = f({\\bf y},t)\n", "$$\n", "\n", "where ${\\bf y}$ is a vector of variables. As an example, the Newton's law of motion could be written as\n", "\n", "$$\n", "M\\frac{d^2 x}{dt^2} = F(x,t)\n", "$$\n", "\n", "Which is a second order differential equation. But nothing can prevent us from describing the same problem through two first-order coupled ODEs:\n", "\n", "\\begin{equation}\n", "\\left\\lbrace\n", "\\begin{array}{lll}\n", "\\displaystyle{\\frac{dx}{dt}} & = & v_x(x,t) \\\\\n", "\\displaystyle{m\\frac{dv_x}{dt}} & = & F(x,t) \\\\\n", "\\end{array}\n", "\\right.\n", "\\end{equation}\n", "\n", "That means that we can limit our next discussions to the first order problems." ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "## Euler's methods\n", "\n", "In engineering, conceptual models will describe how a certain quantity (mass, energy, position, ...) will evolve: it is about identifying which processes will affect that quantity, and translate them into a mathematical form.\n", "\n", "### From the mass balance to the ODE\n", "#### Source and sink terms\n", "\n", "For the sulfates TMF problem, we had identified 4 contributions to the sulfate concentration.\n", "\n", "1. $M_{\\text{pit}}$ is the mass of sulfates added over $\\Delta t$ from the mill (mg).\n", "2. $M_{\\text{mill}}$ is the mass of sulfates added over $\\Delta t$ from the pit (mg).\n", "3. $M_{\\text{pore}}$ is the mass of sulfates added over $\\Delta t$ from that exchange term with the tailings porewater (mg)\n", "4. $M_{\\text{discharge}}$ is the mass of sulfates removed over $\\Delta t$ from the discharge flux (mg)\n", "\n", "\n", "Therefore, if the mass of sulfates at a certain time $t_0$ is $m(t_0) = m_0$, the mass of sulfates at $t_0 + \\Delta t$ is:\n", "\\begin{equation}\n", "m(t_0 + \\Delta t) = m(t_0) + M_{\\text{pit}} + M_{\\text{mill}} + M_{\\text{pore}} - M_{\\text{dis}}\n", "\\end{equation}\n", "\n", "\n", "And we have identified these terms:\n", "\n", "\\begin{equation}\n", "\\left\\lbrace\n", "\\begin{array}{lll}\n", "M_{\\text{pit}} & = & Q_{\\text{pit}}c_{\\text{pit}} \\Delta t\\\\\n", "M_{\\text{mill}} & = & Q_{\\text{mill}}c_{\\text{mill}} \\Delta t\\\\\n", "M_{\\text{pore}} & = & kA_{\\text{bottom}}(c_{\\text{pore}} - c_{\\text{TMF}}) \\Delta t\\\\\n", "M_{\\text{discharge}} & = & Q_{\\text{dis}}c_{\\text{TMF}} \\Delta t\\\\\n", "\\end{array}\n", "\\right.\n", "\\end{equation}\n", "\n", "So we can rewrite the previous equation as:\n", "\n", "\\begin{equation}\n", "\\begin{array}{llll}\n", "& m(t_0 + \\Delta t) & = & m(t_0) + M_{\\text{pit}} + M_{\\text{mill}} + M_{\\text{pore}} - M_{\\text{dis}} \\\\\n", "\\Longleftrightarrow & m(t_0 + \\Delta t) & = & m(t_0) + \\left(Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + A_{\\text{bottom}} k \\left( c_{\\text{pore}} - c_{\\text{TMF}} \\right) -Q_{\\text{dis}} c_{\\text{TMF}}\\right) \\Delta t \\\\\n", "\\Longleftrightarrow & \\frac{m(t_0 + \\Delta t)-m(t_0)}{\\Delta t} & = & Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + A_{\\text{bottom}} k \\left( c_{\\text{pore}} - c_{\\text{TMF}} \\right) -Q_{\\text{dis}} c_{\\text{TMF}} \\\\\n", "\\end{array}\n", "\\end{equation}\n", "\n", "The link between concentration and mass of sulfates is the volume of water\n", "\\begin{equation}\n", "c_{\\text{TMF}}(t) = \\frac{m(t)}{V(t)}\n", "\\end{equation}\n", "\n", "But, since the volume of water is constant ($V_0$), we have:\n", "\\begin{equation}\n", "c_{\\text{TMF}}(t) = \\frac{m(t)}{V_0}\n", "\\end{equation}\n", "\n", "Therefore, if we divide the latter equation by $V_0$, we have:\n", "\\begin{equation}\n", "\\frac{c_{\\text{TMF}}(t + \\Delta t)-c_{\\text{TMF}}(t)}{\\Delta t} = \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + A_{\\text{bottom}} k \\left( c_{\\text{pore}} - c_{\\text{TMF}} \\right) -Q_{\\text{dis}} c_{\\text{TMF}}}{V_0}\n", "\\end{equation}\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "So, after each time step, we can compute the evolution of $c_{\\text{TMF}}$. How should we choose the timestep?\n", "\n", "\n", "1. It is irrelevant\n", "2. It should be as small as possible\n", "3. The bigger the faster\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "The choice of the timestep matters. Look at the previous equation:\n", "\n", "\\begin{equation}\n", "\\frac{\\color{blue}{c_{\\text{TMF}}}(t + \\Delta t)-c_{\\text{TMF}}(t)}{\\Delta t} = \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + A_{\\text{bottom}} k \\left( c_{\\text{pore}} - {\\color{red}{c_{\\text{TMF}}}} \\right) -Q_{\\text{dis}} \\color{red}{c_{\\text{TMF}}}}{V_0}\n", "\\end{equation}\n", "\n", "To compute the solution $c_{\\text{TMF}}(t+ \\Delta t)$, we use the value of $c_{\\text{TMF}}$. But, at what time? Intrinsically, at every time between $t$ and $ t + \\Delta t$, the value of $c_{\\text{TMF}}$ changes.\n", "\n", "So the timestep should be small!" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "#### ODE\n", "\n", "Going to the limit $\\Delta t \\rightarrow 0$ (i.e. $\\Delta t= dt$, $c(t+\\Delta t)-c(t)= dc$), the equation becomes:\n", "\\begin{equation}\n", "\\frac{dc_{\\text{TMF}}}{dt} = \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + A_{\\text{bottom}} k \\left( c_{\\text{pore}} - c_{\\text{TMF}} \\right) -Q_{\\text{dis}} c_{\\text{TMF}}}{V_0}\n", "\\end{equation}\n", "which is a 1$^{st}$ order linear ODE.\n", "\n", "#### Asymptotic behavior\n", "\n", "Let us build a bit of physical sense is often a good idea. One should realize that this problem is bound to have a steady solution (a solution which does not evolve in time). This is not always true. But, in this case, at some point, the concentration in the TMF will be so that the sink term will exactly compensate the source terms.\n", "\n", "This happens when the derivative is zero!\n", "\n", "\\begin{equation}\n", "\\begin{array}{lll}\n", "& \\frac{dc_{\\text{TMF}}}{dt} & = & 0 \\\\\n", "\\Longleftrightarrow & 0 & = & \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + A_{\\text{bottom}} k \\left( c_{\\text{pore}} - c_{\\text{TMF}} \\right) -Q_{\\text{dis}} c_{\\text{TMF}}}{V_0} \\\\\n", "\\Longleftrightarrow & c_{\\text{TMF}} & = & \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + kA c_{\\text{pore}} }{Q_{\\text{dis}}+kA} \\equiv c_\\infty\n", "\\end{array}\n", "\\end{equation}\n", "\n", "The asymptotic/final concentration (the steady-state concentration) $c_\\infty$ corresponds to the final solution. \n", "\n", "Can you try to guess which values $c_\\infty$ could be?\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us analyse this by investigating limit cases:\n", "\n", "1. Very fast production from the porewater: $k \\rightarrow \\infty$\n", "\\begin{equation}\n", "\\begin{array}{lll}\n", "c_\\infty & = & \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + kA c_{\\text{pore}} }{Q_{\\text{dis}}+kA} \\\\\n", " & = & \\frac{\\color{blue}{Q_{\\text{pit}} c_{\\text{pit}}} + \\color{blue}{Q_{\\text{mill}} c_{\\text{mill}}} + kA c_{\\text{pore}} }{\\color{blue}{Q_{\\text{dis}}}+kA} \\\\\n", " & \\approx & \\frac{kA c_{\\text{pore}} }{kA} \\\\\n", " & \\approx & c_{\\text{pore}} = 2000 \\text{ mg/L}\n", "\\end{array}\n", "\\end{equation}\n", "\n", "2. Very slow production from the porewater: $k \\rightarrow 0$\n", "\\begin{equation}\n", "\\begin{array}{lll}\n", "c_\\infty & = & \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + kA c_{\\text{pore}} }{Q_{\\text{dis}}+kA} \\\\\n", " & = & \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}} + \\color{blue}{kA c_{\\text{pore}}} }{Q_{\\text{dis}}+\\color{blue}{kA}} \\\\\n", " & \\approx & \\frac{Q_{\\text{pit}} c_{\\text{pit}} + Q_{\\text{mill}} c_{\\text{mill}}}{Q_{\\text{dis}}} \\\\\n", " & \\approx & 256.82 \\text{ mg/L}\n", "\\end{array}\n", "\\end{equation}\n", "\n", "Therefore, the asymptotic concentration should never be outside these two bounds!" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "### Analytical Solution\n", "\n", "The ODE can be rewritten in the following way:\n", "\n", "\\begin{equation}\n", "\\frac{dc_{\\text{TMF}}}{dt} + \\lambda c_{\\text{TMF}} = Q\n", "\\end{equation}\n", "with\n", "\\begin{equation}\n", "\\left\\lbrace\n", "\\begin{array}{lll}\n", "Q & = & \\frac{Q_{\\text{pit}}c_{\\text{pit}} + Q_{\\text{mill}}c_{\\text{mill}}+ kAc_{\\text{pore}}}{V_0} \\\\\n", "\\lambda & = & \\frac{Q_{\\text{dis}}+kA}{V_0}\n", "\\end{array}\n", "\\right.\n", "\\end{equation}\n", "\n", "You can notice that\n", "\\begin{equation}\n", "c_\\infty = \\frac{Q}{\\lambda}\n", "\\end{equation}\n", "\n", "The associated homogeneous problem is\n", "\\begin{equation}\n", "\\begin{array}{ll}\n", "& \\frac{dc_{\\text{TMF}}}{dt} + \\lambda c_{\\text{TMF}} = 0 \\\\\n", "\\Longleftrightarrow & \\frac{dc_{\\text{TMF}}}{dt} = -\\lambda c_{\\text{TMF}} \\\\\n", "\\end{array}\n", "\\end{equation}\n", "\n", "In other words, we are trying to find a function which is proportional to its opposite.\n", "\n", "Any idea?\n" ] }, { "cell_type": "markdown", "metadata": { "lines_to_next_cell": 0 }, "source": [ "The solution of the homogeneous problem $c_H$ is\n", "\\begin{equation}\n", "c_H(t) = A \\mathrm{exp}(-\\lambda t)\n", "\\end{equation}\n", "\n", "The solution to the particular problem should arise from our physical sense. We know that the solution should reach a steady-state, so a particular solution is a constant solution $c_p(t) = c$.\n", "\n", "By injecting that solution into the differential problem, we find:\n", "\n", "\\begin{equation}\n", "c_p(t) = c = \\frac{Q}{\\lambda} = c_\\infty\n", "\\end{equation}\n", "\n", "which corresponds to our physical sense!\n", "\n", "So that the general solution to the ODE is:\n", "\\begin{equation}\n", "c_{\\text{TMF}}(t) = c_\\infty + A \\mathrm{exp}(-\\lambda t)\n", "\\end{equation}\n", "\n", "The last thing to do is to find the value of A so that the initial concentration meets our given problem. Usually, initial means at $t=0$, but you can take any other value of time $t_0$, as long as you are consistent with that value later.\n", "\n", "We know that:\n", "\\begin{equation}\n", "c(0) = c_0 (=93) \\Longleftrightarrow c_\\infty + A = c_0 \\Longleftrightarrow A = c_0 - c_\\infty\n", "\\end{equation}\n", "\n", "So the exact solution of the problem:\n", "\\begin{equation}\n", "c_{\\text{TMF}}(t) = c_\\infty + \\left( c_0 - c_\\infty \\right) \\mathrm{exp}(-\\lambda t)\n", "\\end{equation}\n", "\n", "So, what would happen if you arbitrarly chose $t_0$ to be something else than 0? For example, if you want $t_0$ to be the age of the universe, then, $c_{\\text{TMF}}$ has to match 93 mg/L after 4.343 $\\times$ 10$^{17}$seconds (13.772 billions of years!). That would give you a value A. As long as you stick to that value of A and to that initial $t_0$, the following solution will be the same. But computationnally, the difference between 13 billions of years and 13 billions + 1 year might be lost in rounding erros... So it's probably not the greatest idea!\n", "\n", "The fact that we know the exact solution to a problem allows us to assess the accuracy of a computational approach. Let us take a look at the solution!" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "6.358024691358025e-09\n", "3.2469135802469134e-06\n" ] } ], "source": [ "# Parameters definition\n", "import numpy as np\n", "from matplotlib import pyplot as plt\n", "\n", "c_pit = 50\n", "c_pore = 2000\n", "c_mill = 700\n", "Q_pit = 30\n", "Q_mill = 14\n", "Q_dis = 44\n", "V0 = 8.1e9\n", "c0 = 93 # initial concentration\n", "k = 2.5e-5\n", "Area = 3e5\n", "#\n", "# Eqn 13\n", "#\n", "c_inf = (Q_pit * c_pit + Q_mill * c_mill + Area * k * c_pore) / (Q_dis + Area * k)\n", "A = c0 - c_inf\n", "lam = (Area * k + Q_dis) / V0 # units are here in s^{-1}\n", "#\n", "# Eqn 15\n", "#\n", "Q = (Q_pit * c_pit + Q_mill * c_mill + Area * k * c_pore) / V0\n", "print(lam)\n", "print(Q)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Asymptotic concentration is: 510.6796116504854 mg/L,\n", "exact concentration after 50 years is 510.6611233781182\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This is the routine to plot the exact solution for the concentration at every day\n", "dt = 1 # day\n", "Tf = 50 # years\n", "n = int(1 + 365 * Tf / dt) # int() is to convert thhe result into an integer\n", "c_real = np.zeros(\n", " n, float\n", ") # represents the concentration of sultates (mg/L) at each time\n", "timeY = np.zeros(n, float) # time in years\n", "c_real[0] = c0\n", "c_asymptotic = c_inf * np.ones(\n", " n, float\n", ") # this is an array of size n, whose values are all c_inf\n", "\n", "#\n", "# Eq. 22\n", "#\n", "for i in range(n - 1):\n", " timeY[i + 1] = (i + 1) * dt / 365\n", " c_real[i + 1] = c_inf + (c0 - c_inf) * np.exp(-lam * (i + 1) * dt * 3600 * 24)\n", "\n", "fig, (ax1) = plt.subplots(1, 1, figsize=(8, 4))\n", "ax1.plot(timeY, c_real, label=\"Real Concentration\")\n", "ax1.plot(timeY, c_asymptotic, label=\"Asymptotic value\")\n", "ax1.set(xlabel=\"Time (years)\", ylabel=\"Concentration (mg/L)\")\n", "ax1.legend()\n", "\n", "print(\n", " f\"Asymptotic concentration is: {c_inf} mg/L,\\n\"\n", " f\"exact concentration after {Tf} years is {c_real[n-1]}\"\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Integral form of the ODE\n", "\n", "The general 1$^{st}$ order ODE is written in the form\n", "\n", "\\begin{equation}\n", "\\frac{dy}{dt} = f(y,t) \\Longleftrightarrow dy = f(y,t)dt\n", "\\end{equation}\n", "\n", "If the ODE is not separable (meaning the function $f(y,t)$ cannot be split into function $f(y,t)=Y(y)T(t)$, we cannot apply the variable separation method.\n", "\n", "We can therefore write:\n", "\\begin{equation}\n", "\\int dy = \\int_{t_0}^{t_0+\\Delta t} f(y,t)dt\n", "\\end{equation}\n", "\n", "which leads to\n", "\n", "\\begin{equation}\n", "y(t_0+\\Delta t) = y(t_0) + \\int_{t_0}^{t_0+\\Delta t} f(y,t)dt\n", "\\end{equation}\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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Xc53ff/8dwJ9//qm58KOPPlKnzv3799u0afPJJ59ormBvb79hwwbh508//bRJkyZVf1YKrl271rZtW/WGQh9i48aN6hWETsm2bds0t5o+fXqLFi0yMzPVv87BwUFzhaysrGbNmk2dOlVzJ59++ml120d4eYsXL1YqlZ988on6N1bh2Wef9fb2Vj8cMmSIj49PjVPns88+GzBgQNXr5OTk/K2HkydPVvGeOXjwoM5427dv36pVq0JDQx8Zt0wdk0qdhQsfS+To//5pyHPY5HK0amW0LmF1t2jcuHHHjh0rLn/yyScfuW1SUtK1a9f69++vubB///6NGjU6dOjQq6++ql4oEom0Tnc3btzY2dlZc6Fm9Z3ExMS7d+9aWlpKpVL1Qltb29TUVPXDDh062NraPvJFdunSRTjxc+/evePHj9+6datRo0b37t2requXXnpp9erVMTExH3zwgc4V7O3t+/fvf+jQoXXr1qkXDhgwoLrt06VLlx9//PHNN9/8999//fz8Onfu/MgjOnXqlHDG6/jx49evX79///4jD6fqqQTCObwqtG3btvZzDeLj4wcMGKD1J5s/f36rVq0+//xzhUIxevToTz/99JEvhkyBRIKwMGRmoh7X3GrI53WsrKBUGuk/4xIyoHnz5poLLSwsLC0tL168WEWGXbhwoVWrVlWcMbp27ZrwYddEw5YtWz777LNq5aL6887d3X3WrFlKpdLNzc3CwuKRm3Tq1AmAMPhWxTrp6ema5ye0XpKe7fPaa6/5+voKY3H6HE5GRsYHH3zg6emZlZX17LPP1qZwUUFBwT///OPi4mKEd0tCQoLWL7p9+/ayZcumTp0qfOcIDAxcuHAhP5BNX3g4wsKQlIT6XeaR1+vUR8KnpNZ5b4VCUVRUpPUBqvVB36NHj4SEhCr27OjoKPQnXnjhhcrW0Sc8ABw6dOitt97atWuXp6en1oZKpbKynQgz1nr37l3Fnq9fv96zZ0/Ny4y09qZn+6SlpT355JPPPffcu+++++eff1ZdcTUzM3PIkCFjx4794Ycf1L1V4YeysrKKQb5ly5aPP/64sr39/fffCoVCPYEtPz+/la5+eVRU1N69e/XpNy9evPiJJ56o+FRhYeHJkyfVXxqEXySVSps1a6ZOZVtb27i4uDt37uj/fYIaHqkUQUH1pOYNU6fhGThwYPv27Y8dO+bt7a1eePLkSaVS6SHcG6MSr7766pYtWxITEwcNGqT5vVtzqKpr165RUVF+fn7qhcXFxb///nt1S+asW7euX79+6si5e/euQqEAUFRUtHLlyjlz5qh3rrlVbGxsy5YtNccAtVaQyWQpKSnjx4+vZfs8ePBg9erVa9asycvLGzRo0BdffPHVV19Vsc8ff/wxJydHs0+QnZ0tXOA5f/78RYsWCb0rdQ/sypUrjxz1EkbP4uPjCwsLX3nllYqrDRo0yMrKSp/UqezisBMnTpSUlAjxdu/evc2bN8+YMePixYuaASO8jKysLKaOKUeOWNwgIoepU1tKpbK4uLjiVSa11Lx58++++27cuHEzZszo3r278HV72bJlY8aMUWeD8PGXl5en+Xnk5eW1ZcuWlStX7tixQ/19/8iRI+qv7ZaWlhs3bhw9evTZs2cHDhwoLFy7dq2bm5t6t3l5efq8SIVCoXn1zB9//PHUU089ePDg5s2bmucqIiIi/Pz8hH7Gv//+++23365YsULzLEt2dnZMTIz6rMO8efM6dOiwePFircOsVvvcu3fPy8tLIpE0bdq0bdu227dvd3V1FTo9VRyO+v8ALl++3Lx58wcPHmjm4jPPPKOufqSsctw1LS1NqPFaVFR06NChZcuW6VzNzs6ulhfZpKWltWvXToiTtWvXCqNqRUVFzZo102yuim1IpkMmU9W8KX/6s15/bnIOWw3s2rVr1KhRgwYNsrGxsbW1dXFx8fLyysjIqMNfER0d/dJLLy1duvS77757++2358+fr54P7efn5+joaGNjM3DgwNGjR6vnGSuVyoKCgilTpvj6+v7yyy/ffPONRCLx9fUF8Nxzzx04cEBYJy4ubtiwYXPmzNm+ffvnn39+6NAhpVL533//vfXWWx06dLC2tnZ2dp4+fXrVL+/cuXMDBw709/f/7bff1q1bd+TIkV27dnXp0mXy5Mk3b95UTz9btGjR9OnTf/7559DQ0Oeee+7nn3/W3Ikwh+3rr78OCQnZt2+fr6/vu+++q56OtWrVqqFDh1pbW/fo0eOtt946evSoPu0zcuTITp06tWjRQj1V7+OPP27evHnr1q1Hjhy5b9++yuYfv/POOyNGjDhw4MC2bds2btx4+fJlOzu7yZMnR0dHC+tcuXLFwcFh4cKFEokkLi6uisaJiooaPHjw/v3758+frzm9u87duHHDwcFh7969y5cv/+eff4SFa9assbe3V68jnOvSZyYbOIetwR14drayfXtlQkJ9OHBw5rRpuHHjRkpKikKhqNZWRUVFycnJBQUFwmflpUuXKs7YvnHjxqVLl2r58tLT00+dOnXv3j3hofoHzZnTpaWlKSkpV69erbi5eub0nTt3zpw5I7xgI7RPZWQy2cmTJ4XUFCK8qKio/LUQJf/8848+QZKVlSVcWmRo+fn5586d0/zj/vXXX02bNlU/PHHihJWVVYkeU2mZOg3swIXIWbiwnhw4TH/mtHno0KFDDSZTWVpaqicWC2NQdbVnLQ4ODg4atWxtbGwqrtOoUaM+ffpUvZ82bdq0adPGaO1TGVtbW80pyFrT5AA0adJE85xZFezt7Y3zDrGysurXr5/mkueff97W1jYjI0P401y4cOGNN97g7WtNjUIBb+/6WfOG53Xo8RBOyajPlFS2TtUrUA00a9YsPDz8s88+mzp16p07d3bv3r1582Y2i6lFjocH7OwaXOSAddjIQNavX//mm282bdo0ODhY5zn8mzdvjh49eteuXVlZWa+99tovv/zCRqtDLi4u33//vaWlZffu3ffv39+uXTu2iUnx9QWA7dsb4mu3UBr9EkjVL7ao3q+u7vr0eJWWlqonzpWUlFS8x49SqSwrKxPW0fyZHufHQZ3+K4uNjTXOdbL1jcEPXCJBfDwiI+vb1aB6vn84wkYGoRkhOm8rZ2FhoV5H82ciekTkhIUhLQ0N9kQdR9iIiBqIw4dVNW/0uLi43mJfh4ioIZBK8frryM5GA799H/s6RET1XnKyqgBBw79jLFOHiKh+k8ng7o6EBAwbZgJHw9QhIqrfkSMSISTENCKHqUNEVI8pFBCJEBgIjfLqDV1Dmk2g531fiIhMJHI8PBpizRsTSR2TuUSUl87xwIn0jRwAs2eb2JFxhI2IqP5ZsgRAPSxAYEZ9HSIicyGRICICp0/DFCuFM3WIiOqT0FBVAQITvTkFR9iIiOoNqRTBwUhKMoGrQdnXISKq95EjFpt25LCvQ0RUP8hkqpo3AwaY9oEydYiI6kHkiEQmU/OGqUNEVO8jJzDQHCKHqUNE9FgpFPD2Nr0CBEwdIqJ6GTkeHrCzM5/IYeoQET0+vr4AsH27WR30o1Pn1KlT48aN02df9+/fl0gk/v7+r7/++qxZs7KysvimIiLSTSKBTGaSNW+qVunRnj59esuWLYWFhdHR0S1btnzkjm7fvj1lypSlS5c6ODjcv3/f39+/d+/ev/3229ChQ/nuIiLSjpywMKSlmVvkVNXXeeqpp+bNm7d58+bnn39enx3NnTs3ODjYwcEBgLW19datW5s0aeLj41NWVsY3GBHRQ4cPq2reWFmZ4dFXmjo2Njbt2rXTf0eHDh0aNmxYbm6u8NDKysrZ2fnKlSsXL17ke4yISEUqhb+/yRcgqEnqVFeXLl1KSkqKi4vVS1q1agUgOzubbzMiIlXkiMXYs8dsIwd1WIctPj5eoVA0bdpUvSQpKalp06b9+/fnO42ICDIZPD3NpACBMfo6FhYWmpFz6tSpM2fOTJs2zdbWlm82ImLkQCRCSIiZRw4MdL1OcXGxv7//O++8s0S4HR4RkTmTy1U1b7y92RgWSqWy6jXGjBlz7ty5Cxcu6L/Tzz777MaNG9u3b29S+aRACwuLBQsWVPass7Mz/zZEZAofsqWlA/7v/+6JRFc//NAEDicuLq6ypxYsWPDIQDFI6mzZsuXo0aPfffddo0ZVdaQsLCz0eX2mJzY21sXFhQfOA+eBm8WBCzVvABw5Yvr5qt+neh2PsB06dOjs2bMbN24UIuf8+fPV6iQREZmUoCAAiIxkS9RB6pSUlKivzhEcP3786NGja9eutbCwEJb8+uuvzZo1YysTkTmSSJCSYoY1b6r26LYoKioqKiqquNzV1TUxMTEjI8POzg7ApUuX3n///ZdffjkwMFDIpJycnLNnz86aNYutTETmGDlCAQJGjp6pc+7cOYlEcuvWreTk5NLS0qFDh3bu3Nnf399DGKMEHB0d7969qy7R5uPjk5aWlpaWprmTgQMHVn12h4jIBEmlqsgx46tBq506/fv33717dxVbbtq0SfPhqVOn2JpERDbJyZg2DdnZjByd2BEhIqo7ycmDp01DQgIjh6lDRGRgMhnc3f9Zt44FCKrA01xERHUUOSIR5szJY/FJ9nWIiAxLoVDVvJkyhY3B1CEiMnDkeHggMBDz5rExmDpERIaPHACzZ7MxmDpERAYmVNZnAQK9sZmIiGpKIkFEBE6fZuQwdYiIDCw0lDVvaoAjbERE1SeVIjiYNW/Y1yEiMkrkiMWMHPZ1iIgMLyMDYjESEjBgABuDqUNEZEgyGZyckJDAmjdMHSIiw0eOUICAkcPUISIyLIUC3t4sQMDUISIySuR4eKBvX0YOU4eIyPCEmjchIWwJpg4RkYFJJABr3tQZNiIRUZWRExaGtDRGDvs6REQGdviwquaNlRUbg6lDRGRIUin8/VmAoM6xz0hEpCtyhAIEjBz2dYiIDEsmg6cnCxAwdYiIjBI5IhFCQhg5TB0iIgOTy1U1b7y92RhMHSIiQ1IoMGoUa94wdYiIjBI5QgECRg5Th4jI4IKCACAyki3B1CEiMjCJBCkprHljHGxiIjL7yBEKEDBy2NchIjIsqVQVObwalH0dIiKDR45YjPR0Rg77OkREBpaRoap54+DAxmDqEBEZkkwGJyfWvGHqEBEZJXJEIsyZw8hh6hARGZhCATc3BAZiyhQ2BlOHiMjAkePhgdGjWYCAqUNEZJTIATB7NhuDqUNEZGBLlgBgAYLHjq1PRGZAKECQmcnIYV+HiMjAQkNZ84apQ0RkFFIpgoNZ86b+YPITkUlHjljMyGFfh4jI8GQyVc2bAQPYGEwdIiIDR45IxJo3TB0iImNFTmAgI4epQ0RkYAoFvL0RGMgCBEwdIiLDR46HB/r2ZeQwdYiIDE+oeRMSwpZg6hARGZhEArDmTX3Hvw0RmUrkhIUhLY2Rw74OEZGBHT6sqnljZcXGYOoQERmSVAp/fxYgaCjYFSWiBh45QgECRg77OkREhiWTwdOTBQiYOkRERokckQibNzNymDpERAYml6tq3owcycZg6hARGZJCgVGjWPOGqUNEZJTIEQoQMHKYOkREBhcUBACRkWwJpg4RkYFJJEhJYc2bBo1/OSJqOJEjFCBg5LCvQ0RkWFKpKnJ4NSj7OkREBo8csRjp6Ywc9nWIiAwsI0NV88bBgY3B1CEiMiSZDE5OrHnD1CEiMkrkiESYM4eRw9QhIjIwhQJubggMxJQpbAymDhGRgSPHwwMuLixAwNQhIjI8oeZNSAhbgqlDRGRgEgkAFiAwVfyjElE9i5ywMGRmMnLY1yEiMrDwcNa8YeoQERmFVIqgINa8MXn8QkFE9SNyhAIEjBz2dYiIDEsmg6cnCxAwdYiIjBI5IhE2b2bkMHWIiAxMLodIhMBAjBzJxmDqEBEZkkKBUaMQGMgCBEwdIiLDR46HB+zsGDlMHSIiw/P1BYDt29kSTB0iIgOTSCCTseaNeeKfnIiMHjksQMC+DhGRMUilqsjh1aDs6xARGVHlGZ0AACAASURBVDxyxGJkZzNy2NepS1lZWV5eXnl5eWxcInooOZk1b6gu+zp37txZtGhRXl5ecnLyyZMn169fb2Njw/YlIgCQyeDuzpo3VJepY21tPX369M6dO4eFhZ08eZItS0QPI0ckQkgII4fqMnWaNm3atWtXNigRlaNQqGreeHuzMQicw0ZEho0cDw/WvCGmDhEZnEVpKTw8AGD2bLYGMXWIyLB6btgAgAUISMvjfDcsXLiwsqecnZ1NuNFjY2PN893GAzcf3b7/3u7MmbjvvlMmJPAvbkri4uJquwtlXQsNDQVw69atqlczxK9uEGJiYnjgPHATt369sn37hL17+Rc3K3p+qnOEjYjqlFSK4GAkJZU88QQbg+rXCBsRmWDkiMVIT2cBAqoM+zpEVEcyMlQ1bxwc2BhkvNQpKipS/5+IzIVMBicn1ryhR6qzEbaysjIfHx+ZTJaSkmJjY+Ph4dGrVy9nZ+cZM2awlYlMP3JEIsyZw8gh46VOo0aNdu7cyQYlMjsKBdzcEBiIKVPYGPTosGATEFGtIsfDAy4urHlDTB0iMkrkAAgJYWMQU4eIDGzJEoA1b6h6+F4hohqRSBAWhsxMRg6xr0NEBhYejrAwJCUxcoipQ0QGJpUiKAhJSSxAQDXA7ylEVM3IEYsZOcS+DhEZnkymqnkzYAAbg5g6RGTgyBGJcOgQCxAQU4eIjBI5gYEYOZKNQUwdIjIkhQLe3ggMZAECYuoQkeEjx8MDdnaMHGLqEJHh+foCwPbtbAli6hCRgUkkkMlY84bqEN9JRFR55ISFIS2NkUPs6xCRgR0+rKp5Y2XFxiD2dYjIkKRSvP46srNZgID0IZdDJmPqEFHNJCerChAwckgXmQy3biE5GWlpuHgRwk2k3dyYOkRUs08Ud3ckJLAAAWllzPHjSElBdLQqY/r2xQsv4J13sHat6vuJhQVTh4iq+wEjEiEkhJFjtjIy8N9/uHoVBw/i/HkkJQHAe+/B0REvvIDp02FnV9szfUwdIgIAKBSqmjfe3mwMM/mDX7+OlBRcuYKjR1UZ0749XF3x4ot4/3307VsHGcPUIaJKPoE8PFjzxrTJ5bhyBcnJD/sxIhH69XuYMZ07G2OSPFOHiJGjgIcHAEaOiZHJkJqKxEQcPYqYGNy6BTc3DB9u1Ixh6hBRBUFBABAZyZZo6DIykJKCU6ceTi177z28+CImT8aSJXBwqBcvkqlDZN4kEsTG4vRpFiBoiH3U69dx9CiOH0ds7MOzMm+8UW5qWX3D9xmRGQsNVRUgYOQ0BOpJzAcPqkbMRCK4uKhmlz2uETOmDhHpRypFcDCSkng1aP3vzahjRpjE/P779WjEjKlDRPpFjliM9HRGTj3s0KSm4o8/EB+P6GhVb6ZBxwxTh8jsZWSoat6YxseYKXZoXnwRq1ahRw8TLL7K1CEyv+/STk6secMODVOHiIzyaScUIGDksEPD1CEig3/yeXuzAIHRyOU4c8ZMOzRMHSJGjgIeHujbl5Fj6KRJTrb56y9ERCApyXw7NEwdIrMn1LwJCWFLGCLQL1xAXBx+/RXR0Rg82MHPDz/+iD59eB2UDryDNZEZkEgAIDKSn4J1KDkZoaF45RU0bYr33weAVatQUoKQkKQpUzBgABubfR0is42csDBkZvJTsPYyMlQzAnbuhEiE0aOxYAH27ePoGVOHiATh4ax5U0syGU6exO+/Y906tG8PLy+8/z42bmTSMHWISItUiqAg1rypTdLs3g0AXl549VV8+SUbkqlDRFVEjlCAgJ+U+lEocPw4/vjj4fSzN97A9Oks4MDUISJ9vqt7erIAgT7kcsTF4ccfsXMn3Nzw9tv49VcmDVOHiKoVOSIRNm9m5FTdSLt2YeNGVbeGp2qYOkRU06/uQs2bkSPZGFqEC2v27kVYGAAEBmLDBrzwAmdaMHWIqMYfq6NGseZNxSA+cwa7d2PdOohECAhAVBQGDGDDMHWIqJaR4+EBOztGjkAmw5Ej2LoV0dFwc8P06ZyEVi+wNgGRqfD1BYDt2828GYSSAQMHwt4ex49jwQLk5+PIEYwcychhX4eI6opEApnMbGveCJOed+9WXV4TGIgff+QYGlOHiAwXOWZZgEChQFRUuUnPvCKWqUNEBiaVqiLHbD5u1T2bdetUJ2w46ZmpQ0TGihyxGNnZZhI5ycmqec/29pg5E/n5DBumDhEZ8zPYPGreyGQIC1NdZDNnDofRGjbOYSNqsJ/E7u6mXfNGLlfNRhOJACAqCjdvYsoURg5Th4iMHzkiEebMMcnIkctx+DBeeQWtWuHyZWzYgMxMzJvHOWkmgiNsRA2NQgE3NwQGYsoUEzssrTkCvPcp+zpEVA8+mz08MHq0KRUgSE6GRIKmTTFpEl599eFFnYwc9nWIqB5EDoDZs+v5K719G9evt1AqYWFR6TrqOQL29ggIMJ+JeOzrEFFDsWQJgHo+8JSSAicn2Nrigw9esLPDtm3aK1ScI3D2LOcIsK9DRPWNRIKICJw+XZ8j59YtDB+OO3ce9nj8/ADgo48AIDkZS5di505Mm4YNG3j3H6YOEdVboaENouZNcPDDyFH77DM0bYoVK5CdjZAQ1hFg6hBRPSeVIji4QVwbefy4joW3b+Onn9i5IaYOUUOJHLG4oVyO/+STupdv24a2bfm3JICzCYjqtYwMVc2bBnKFpLu7joVOTowcYuoQ1X8yGZycGkrNm+RkeHsjKAjdupVbbmuLLVv4tySmDlH9jxyRCIGB9TxyFAqEh2PgQLi74403kJ+PK1ewYQNefx19+9779FOcPw9HR/456SGe1yGql5/l3t4IDKzPBQgyMrB9O+bPx3vvac8UCAxEYCBiYxNdXFz4xyT2dYjqfeR4eKBv3/oZOQoFpFK88gqcnAAgOxvh4ZycRuzrEDVcQs2bkJD69rrkcmzdiuBg2Ntj6VK4u7NOGrGvQ9TQSSRAvat5I8wUEO47cOwYzp5laU5iX4fINCInLAyZmfXnE10qxaRJrClATB0i0xMeXq9q3mjmjZcXezbE1CEyJVIpgoLqSQEC5g0xdYhMPXKEAgSPO3KEvAEacM20w4fx6qto3JjvqvqLswmIHiuZDJ6ej70AgVSKgQMxaRI2bMDZsw14JvTTT+Pdd3H/Pt9YTB0i0hk5IhE2b35cH/NCZYEOHepj3vzyC5KTH71aURGk0ocPe/XC7NkYOxalpXx7MXWISJNcrqp5M3Lk48qbLl2wfDn27Kl3/ZuiIgQEwNf30eHx7bcYO7bcksGDMWgQgoP5DmPqEJHmp/6oUY+l5k09zxt1R6dJE5w5g9DQR6x5+jSeflp74axZ2LQJ167xfcbUISL8r+aNnZ2RI0edNwcPqvLmxRfraQtFROCPP9C8OebOxY0bVa156RJ699Ze2KoVvLywfDnfakwdIgLg6wsA27c/lrw5dkxVOS0vT1UJob7591906YL+/TFrFu7fx/TpulcrKMDVq7h0Cfn52LwZu3eXe3bMGISHo6SE7zamDpGZk0ggkxmt5o1cjtDQcnnj4KB6KiAAPj4G/NXJyTa//16TDTdtgr8/AMyahaeewu7diIrSXufPPzFjBlauRG4uevfG009DLC63wgsvwNISf/3FNxxTh8jMIycsDOHhRogcuRwSCVq1wtGj2nkD4Oef0bcvevXS3urmTVy4UDcv4M4dyxp86CsUSE1F//4A0KwZvv4aACZPRmFhudVeeglhYRgzBgA++gjDh6NjR+1d9euHs2f5nmPqEJktqVRV88bAV4Oq8+biRaSna+eNYPlyBATo2HbyZCxZUjcvo02bkqysam918CDeeOPhw1degZcX0tKwbJmOlVNTYWODTp1076pPH73mXhNTh8hEI0csNkLkhIejV6+q8gZAYiKaNdPROUAlU8JqmjrF2dnV3mr3bnh5lVuyejWsrbFsGS5f1l758mX07Vvprnr2xJUrfOcxdYjMUHKyEWreJCdj4EAsX65jPK1iAr78so7lBQXIzNQxJazGqVPdvk5mJtq21a5s3bEjFi1CUREmT9aROn36VLq3li1RVsY3H1OHyMw0zc2Fu7tBa97IZPD2hrs7li7F2bNV5Y3g2DEd6+TlITYWSiX+/RdhYTh6tLavqlWr0tzc6m2ydSs+/ljH8ilT8MwzOHIEu3aVW56a+rCvExmpvVWzZnz3MXWIzI1M9tzHHyMkxECRo1BAIoG9PV58EZmZqioHt25h82Zcv/5wtXv34OODvDzVw+PH0bVruf1s2IC5c/Htt2jUCH37YvBg9OtXq1d18ybS0lo9eIAdO7BqFWbOhJ8fRo7EkCGIi9O9VVkZzp7FoEE6nmrcGGFhsLDA9OkPj0IISGE8sLgYSUnaWxUWwtqab8F6p6qJNLm5uWvWrLG0tGzRooWlpeWECROaVDnxpqysbNu2bTdu3CgoKCgtLQ0ICOjZsyebmMyXQgGR6Mbbb3f39jbE7sPDERQEV1fk5z8clTp7FvHx6NgRI0fi7FlYWABAXBzCwx9e55+fr90PEEpNSyQ4fx5vv13uqcxMrF+Prl2RkwN7e9WaFWVkwN8fMhlKS9GkCeztAXQpKcGlS+jWDU8/jXbtVP9V1gU5cgSvvlrpwQ4dioAAfPcd5s7F2rUAVNfitGihSs3339fe5OpVPP8834UNJ3Xy8vJGjBixfv16Z2dnAFu3bn399dcjIyMbNaq0ezRjxowJEyb06dMHQHZ2to+Pz+LFi1944QW2Mplp5Hh4IDDw6ogR3et638nJqg/ZimNlu3cjOBiHDuHcOeTnq77sp6SgfXu0afNwtaIiHbvVHLBS9z/efhsHD6qmHowZg1atVBe5aunWDbt3o00bqD8hYmMvLF/ezs8P3fU7/p07sX69quVycpCVhawsZGc//L8wm+Drr/HRRxg0CJaWWLMG33yDS5fg7KxjJltGBj76iG/EhpM6CxYs6Nq1qxA5APz8/FasWLFp06bx48frXP/QoUPdu3fv879Te/b29qtXrw4KCvq9ZteJEZlA5ACYPRsJCXW4Y5kMn3yCmBhs3qyjauj9+7C1BYDERHTv/nB86eLFcmfdGzfGgwc6dn75Ml56qdySo0dRUPBwttvIkVi0SHfqWFigbVvthe3aIStLr9TJzsaBAzh1CllZuH0bSqX2ClZWaNcOQ4fC3h4pKaqBuAkTMGGC7h2WliIuDhs28L3YQFJHqVSGh4d//vnnmgudnJx++umnylLn+PHjD8q/kXv06PHvv/+yickcCde81GkBAoUCS5Zg/nysX4/t23XvuEUL1USvmJhy1+pfuIDnnnv4sGdPZGToTh2teWKXLpWbUdavH9LScOsW2rfX6zULqaOPyEh07Ah7e4hEsLeHvT3atSv3Q8uW1Wuu+Hg4O+PJJ/lebCCpk5iYmJWVZWNjo7nQxsYmPj7+3r17rVu3rrhJx44dJ02a1KVLl2nTpllYWAA4cODAyMdRwp3oMZNIEBGB06frMHLUp3Cys6uafS38wuJiHDtWrlrzxYv44IOHDwcMQGqq9rY5Obh7t9yUMA8P3Lv38Ow9oOownTtX96nz4Yf48MO6/CN8+y1mzeJ7sT7SfZLm0qVLAFq1aqWVOgqFIj09XecmH330Uffu3T/99FNnZ+dLly4dPHgwOjp66dKlbGIyL6GhCAtDdHRdRY76KpyoKISH63XBz40bKCh4OAntv/+Ql1duhG3QIBw/rqOjA6imhJ09C4UCANq0wfXrD696sbaGhQVycvR98fb2qEF5gtr77Tc4OOCZZ/h2bDipk5eXB8Cq/MVaQtcnT/Obj4bmzZvHxsYOGzYsPj5+4MCB69at++6777T2QGTipFIEB9dVAQL1VTgzZ+LsWQwYoO+GwlD3E0+oHsbHP+ymCN54A2fOaOeBMCWseXOUlGDPHtVJo6efVpV2VrO0hK7BDt169lSllzEpFDh2DIsX8+1YT+n+OlZYWAjA0tJSc2Hjxo0BPNB5FhIA0KxZM5FI9OKLL4aGhh45csTDw+P7779v165dZevHxsaaZ6PzwE2STXLy4GnTTm7eLD9/vpYHXlpq8eOPXbdt6/7RR+k//nitcWNltXZQWoo2bV7cvv3fV17Jysmx/OILkZVV89TUBM1RteeeG7BiRc6bb95ULykrw8iRT7/7blnXrgUvvZQdH18i7MrG5sVduy4OHXpHWM3CYvidOydjYwv1eSVFRbGurjD+n93VVZW1fKs3mNSxtrYGUFr+5rFCL8e6ksuukpOTJ0+evHfv3rZt206cOHHChAm///7722+/fezYMeE0T0VxlV0tBsyfP9+EP3ldXFzMM3JM+cAzMuDqioSEIRWuBq3ugZc/hdMdqMm86+horF7dRyrt06YNXnwRt25B6zWsXQsfn7bLl/fWHAjUmMD2lPqnoCCkp4uEcyTHjsHZGT4+Q/kXN9sDX7hwYW13odQlIiICwM8//6y5cMaMGUK66NxkyJAhMTExmksWLVoE4PDhwzrXr+xXmzytVuKBm4LsbGX79sqEhFoeeFKSUiRSikTKpKRavZyyMmVhoern0lJlt27K0FAdq40bp9yy5dF7Ky1VLlqknDFDuWKFcuZM5Z07/IvzwJW1+VTX3dcZOnSohYVFVvlx37y8PBsbG0dHx4rr5+bmnjt3Tn1xj+DLL7+MiYlJTU31EC5cIDJJMhlEIgQG1qbmjVyOgADExCAkBLWvY+DigkaNEBMDAAcOoHFj3VfYrFmD117D888/ovhNo0b48kv+manO6J5N0LFjx6FDh2pNV7t58+bLL7+sLopTUlKS+7/afi1btmzUqFHFiQZdu3btL9yeicgkKRTw9kZgIObNq/E+Dh9Gr16ws0NmJuqkdM7Nm6prJ2/cwKJF2LlTdzkyKyv8/DNmzsSNG/xD0uNOHQATJ07ct2+f4n8TUHJzc+Pi4qZr3MHc1dW1U6dOMpkMgKWl5bvvvrtmzRqtlJLJZMOHD2crk8lGjocH+vatceTI5fD2hr8/oqKwdm2dXd6zeDFSU7FwIUJCsHdvVbXI2rdHRESd3TyUSB+Vvs19fX2TkpLGjRv31VdfyeXySZMmLVu2TDNCHB0d79692/J/Vwxv2LAhICBg4sSJ3t7ebdq0OX78eEJCwoYNG7QmwhGZDmHoOCSkxl0cf394eSEzs47vZ611V7SqNWum+147RMZOHQArV67MyMiIjIwsLCz85ptvupavjb5p06by791m27dvT01NPX78+IULF0Qi0bhx49i+ZLIkEqCGNW/UZ3GioqpxFQ6R6acOAAcHB4dH3iJKw9NPP/10Xd3/lqg+R05YGNLSahA5huviEJlC6hCRjtwIC0NSEqpZeoNdHCLwXqJE1SOVwt+/BjVvNCeqMXKIfR0i0i9yxGIkJFQrcgoKGnl7s4tDxL4OUbXIZPD0REJCta4GPXwYPj4vsItDxL4OUTUjRyTC5s36R476LM7KlUkffzyETUjEvg6R3gEi1LzR+y6FmmdxevSQswmJ2Nch0o9CgVGj9K95w4lqROzrENUicoQCBPpFDieqEbGvQ1QLQUEAEBnJLg4R+zpEBiaRICVFn5o3ycns4hCxr0NUy8gRChBUGTkKBYKCsHs39uypze11iNjXITJnUqkqcqq8GjQ5GV26QCZDWhojh4h9HaIaR45YjPT0KiJHocCSJQgLw+bN+s+mJiKmDlHF/otQ86byUutyOUaNAoC0tOrW/yQijrARqclkcHevuuaNMHFg+HBERjJyiNjXIapN5IhEmDOnisgJD4ePDw4d4qgaEVOHqDYUCri5ITAQU6ZU9rwwVy07u7q3OCAipg6RVqR4eGD06MoKEMhkcHODvT3v/klUB3hehxg5HgAwe7bO56VSiEQICMCRI4wcIvZ1iGppyRIAlRUgCA1FcDCvACVi6hDVCaEAga6BM6ELlJ1dg3tVE1FVOMJG5io8vLKaNxkZ6NIFffvi9GlGDhH7OkS1J5UiKEhnR+bwYbz+OnbsgLc3m4mIqUNUJ5EjFleMHHWdm/T0KkoTEBFTh0h/Mpmq5k35exLIZKrODevcEBkUz+uQmUWOSFSx5k1yMkQi1rkhYl+HqM4jJzBQK3KEOjdVVl8jIqYOUbUoFPD2RmCgZgEChQK+voiJYZ0bIuPhCBuZR+R4eMDOTjNyZDI8+ywAZGYycoiYOkR1yNcXALZvVy9Q17kJD2edGyKmDpmEY8egUNRw2zNnIJfX0euQSCCTada8kUjg6Yk9eyorME1EBsSveWQQ4eHIyYGTUw03f+opBATg66/Rpk2tIycsDGlpQuSoS32yzg0R+zpkOk6exN69mDq15nuwssLcufD3h1JZi9dx+LCq5o2VFQCZTFXnJjKSkUPE1CFTUVaGSZOwdi0sLLSfWrcOXl767qdPHzz1FLZtq+nrkErh76/u1AgnckJCsHYtT+QQPU7890d1bMcO9OqFjh11PLVpE1xdq7GradMwdCh8fGBpWf3IEQoQ/C9yhEe8IoeIfR0yNT/9hPHjdSy/f7/JuXMYMaIau+rUCf37Y9++ar4CmQyenuqQYeQQMXWoAcvNRX5+pc/ev4+//sLQoTqeSk5urVRCLK7erxOLsWdPNSNHGEobNgz/m66Wnc3IIWLqUAMUGoqDByEW4/LlhwslEsyZo/r5t9/w9NNo0aLcVp98Aj8/bNjQ084Os2bh449x6ZK+v3HYMMTH6/36FApVzRtvb/xv/hqnqxExdahBOnwYffvi/fdRXIy9ex8uDwtDq1aqn8+fx9NPa2+4di22bkWLFqVjx2LrVmzZgt699f2lXbvi1i08eKBf5Hh4CDVvFApIJIiIYOQQ1TucTUD6OnYMixbhv/9w4QJEItXCCxdw69bDOQLZ2XjiCR3b5uXhypVWLi7ay8+cwe+/o1Ur9O4NNzcdGz75JADIZOjWTY/IAYTIEX48fZrT1YjY16EG6//+DwB274a1NdT58eefsLJSFTQTUsfGRse2R4+irAzOzuUWRkdj0SIEBWHyZEilWLZMx4atW6NRI8hkj3pxQUEAEBmpjhyNWgRExNShBsjaGgB27sQ776B5c9XC2FiIxWjaVPWweXPdVXASEtCjh7xt23ILZ85EQIAqG2bORHAw7t/X3rCsDGVlD3+dbhIJUlIQGSkvaiIU+WTkEDF1yBTk5eHkSbz1luqhUom4OGiOm9nbIy9Px4bx8Rg0KFf4WZgCl5uLxETY2z+MK6USR45ob3jvnmq3VUVOWBjCw2V3m/TqheHDWdCTiKlDpuLiRQAPb/2ckgKZrFzqtGuHrCztrYqLceIEBg5Upc6KFQCQmgqlUnXaRtC2rY5tc3LQuDG0OkkPSaXCNDUZ7ITJaxq3MiAipg41cO3aqbo4gmXLYGWF5557uMLgwUhM1N7q+nUUFqJbNzmAvXvx8ssAUFQEAM2aPVyteXMd/aTERAwbhsaNK4kcsRjHjiXfshMu0WHkENV/HImgaujWDTNmYMkSjBmDEydw7BjE4nLDWU5OyMlBVpYqnwTdu+Pll5GQYFtcDBsb1ZwCYdJBbi46dFCtlpenOnWk6cQJjBql66UkJwslB6T/ObD0ABH7OmSyQkIQHIzu3TFrFm7dwpgx5Z5t1QqenuWu5gFgYYHff8fQoXc++OBh9c8+fdC8OXJzH66Wl4fBg8ttqFTi0CG8/XaFFyGTwd0dCQlSDGPkEDF1yDRdvIjnnsORI+jcGf36YdMmWFsLRQDK+fRTbNyovbBxY/ToIdccT7O0xMiRD4sUXLuGtm0fzsAWREeja1f06FEhckQizJkTmjjM0xPp6YwcooaEI2ykrz//RH4+bG0BICMDX32FvXu1i98AePZZDBqE337Da689Yofr12PcOFhbw84O69djx46HM7AFq1Zh9ery2ygUcHNDYKDkzhRWuyFi6pApe/NNXL6M3FysX4/oaOzfj0GDdK8ZEoJRozBoULmzOxV17IgDB5CcjIICbNsm3HqtXOR4eKBfv/KR4+GB0aMlmMfIIWqgOMJG+urSBUuWoGVL/L//h19/rTRyALRujW3bsGIFysoesc/GjfHMM3By0o6ckyehUGDaNO3IUSrxwYUv4uORlsbIIWJfh0xdixZ44QW91uzWDYsX67idqJ769cOQIeUXLVmiVMIDkaU5TVh6gIipQ6QjomqsZcvyjyWSsj0RI+1Pl1owcogaNo6wUb0XGlr6ddiArOgXRzQ5coSRQ8TUITIcqbR0UbAISe9OtmPpASITwO+NVK8jB2Kxm21S4Hy7KVPYHERMHSLDkckgFouRsHzfAF4HSmQyOMJG9TRySvqKxEhYnjCMkUPE1CEybOSU9hetKZzIyCEyPRxho3pGoSj2fG/tg4mdvp3PyCFiX4fIwJHz8muHTtgVfD6/Yl1RImLqENUl5Qe+//yD8zN/4CRpIlPFETaqL8oWSM5EZUcH/fblAr4tiZg6RAaOnLtLw+YNS9v3Jd+TRKaMI2xUDxw+fHdp2PihSfuirFjwhoh9HSJDkkrz3/14/NDkXX/YMXKImDpEho0ciMWTe8YzcoiYOkQGJpPlvzba+8mEzceGMXKIzATP69Bji5yi3qKZTUI2XxzGu4ISMXWI6lJBQUFGRkZxcbHqsUJR5ChaVzJxQaoPI4eIqUNUZ+7evfv+++/b2Nj4+flZW1t//vnnhfn5d508ltwJfO3ofEYOkbnhaDoZUFlZ2VtvvZWQkCA8LC4uXrly5Y3vd/jJ+rknzBswgC1ExL4OUd3Zu3evOnLUwmU3b25dwcqeREwdojp2/PhxncsLS06xcYiYOkR17Mknn9S5vG3btmwcIqYOUR3z8PBo3Lix1kIbG5vhw4ezcYiYOkR1bODAgQsXLtRc0qRJk2+//daOc9eIzBXnsJFhzZw5JyJiyI0bW1u3/sfJyWnatGmDBw9msxAxdYjqnkIBDw+0bet+4oR7G8mfxAAAGBBJREFUQkKsi4sL24TIzHGEjQxoyRIAiIwEy6wREfs6ZFgSCSIicPo0I4eImDpk+MgJC0NSEiOHiMrhCBvVvfBwVeRwqhoRMXXIsKRSBAUxcohIt6qGP3Jzc9esWWNpadmiRQtLS8sJEyY00WO4ZPfu3SkpKaWlpQCmTZvGKzPMLXLEYqSnM3KIqJp9nby8vBEjRri6us6ePXv69OktW7Z8/fXXy8rKqthXQUGBj4/PhQsX5s6du2jRIkdHx3HjxrGJzYdMBrEYCQlwcGBjEFE1U2fBggVdu3Z1dnYWHvr5+V27dm3Tpk1V7GvixIkymWz+/PlCEZSFCxeePXuWTWwmFAqIRNixAywmTUTVTh2lUhkeHu7q6qq50MnJ6aeffqpsR4mJiT/88MPixYvVS3766aeff/6ZTWwmPDzg5QVvb326RDK5XM4WM9H+rkyhULAdqNqpk5iYmJWVZWNjo7nQxsYmPj7+3r17OjcJDg5u3br1kCFD1EsGDx6s+ZBMmESC7GyEhOjTJVKIRCK2mKk6cuSIr68v24GqnTqXLl0C0KpVK63UUSgU6enpOvtGf/75Z/fu3TMzM7/88svly5fPmTPn8uXLbF9zIJUiLAzR0XpdmhMUFBQSEmJlZcV2M0ne3t4ymezw4cNsCqqM7s+JvLw8AFofDULXR3hKS25u7t27d/v167dnz55FixZZWFhkZmaKxeKtW7e+9NJLbGUTJpPB0xN79jx60lpBQcErr7wyfPhwb32G4ajBCg8Pd3Nzu3LlSv/+/dkapG/qFBYWArC0tNRcKMwRePDgQcX18/PzAfz999/h4eEWFhYAunTp4u7uPn78+PPnzzdr1kznb9Gqga9JPYvBJMXGxprGgZSWWowf/6yX182Skv8eeUxSqTQ6Ojo6Onr+/Pn8h2fypk6dunfvXpN5q5vtv/GK4uLiarsLpS6bN28GcPDgQc2FCxYsAJCQkFBx/aKiokaNGvXo0UNz4Zo1awD8+eefOn9FZb/a5MXExJjMsbz3nvK996px4IcOHWrfvn1+fj7/4iYsISGhffv26enp5nbgZvsXr+6nuu7zOk888YQwJFJx2K1169YV17e0tGzfvr3W7YpbtGgB4Pz58/zeZ5JCQ3H+PLZvr8YmI0eO3Lx5c0BAAFvPVMnlck9Pz6SkJAdetEWV0J06Q4cOtbCwyMrK0kodGxsbR0dHnZt07dpVazqsUJ7A1taWrWx6pFIEB+s7g0AreABkZGSwDU1SSEhISEgIK5JQFXR/ZnTs2HHo0KFa09Vu3rz58ssvq4vilJSUyOVyoVcEYOzYsZ9//nlxcbH6bFBWVpaFhcVzzz3HVjYx+s8g0Ck8PJxtaKrmzZvHRqCa9HUATJw4cd++feoLvnJzc+Pi4qZPn65ewdXVtVOnTjKZTHg4fvz4Nm3aREVFqVeIior6+OOPe/XqxVY2JQoF3NwQGMgaBERUd30dAL6+vklJSePGjfvqq6/kcvmkSZOWLVs2fPhw9QqOjo53795t2bKl8LBFixb79++fPHlycXGxg4PDpk2bunXrtnr1ajaxiQkKgr09+I2WiOo4dQCsXLkyIyMjMjKysLDwm2++6dq1q+azFWuyvfDCC0eOHImNjT19+vT48eOfeeYZtq+JCQ/H7t3IzGRLEJEBUgeAg4NDteaitGnT5p133mGzmqTkZPj4IDubtwcloprjXd1ILzIZ3N2RkMAb5xARU4cMTKGAtzdnEBARU4eMIigIAGcQEFEd4Ag9PYIwgyAtjS1BREwdMjBhBkF6OnhrAiKqExxho0rJ5XB3x6FDYEktImLqkGEpFBg1CoGBGDmSjUFETB0ysCVLAGD2bLYEETF1yMAOH0ZYGPbt4wWhNZGamurq6lpcXMymIGLq0KNlZOD113HsGGcQ1NDChQtjY2O3bNnCpiBi6tAjyOVwcuIMgpq7ePFifn5+o0aNlixZUnV3Jy8vz9HRMTExkY1GTB0yU8IMAi8vziCouRUrVmzYsOG9997LzMysursTHR2dmpqqVVSXiKlDZkSYQRASwpaooUuXLtnY2HTq1Gnu3LmP7O7Ex8f369evbdu2bDdi6pA5kkoRFobwcM4gqFVH5/PPPwfQu3dvobuzdevWylb+66+/RowYwUYjpg6ZI5kMYjGiolhSulYdHWtr606dOgkP1d2dkpISzdViYmL8/PzGjBmTmJh4+fJlPz+/FStWsPXIfPBrLUGhgEiEHTswYAAbo+a++uqrhQsXqh/27t177Nix4eHhkZGRr7zyinq5q6urq6vr/v37IyIiwsPDbW1t2XTEvg6ZFw8PeHnB25stUXOpqanCGR3NhUJ3Z8eOHRXP7sTHx/fv35+RQ0wdMjsSCcAZBHXR0fnss8+0Fjo6Oo4dOzY7O3vbtm0VU8fV1ZXtRkwdMi/JyZxBUDcdHWtr644dO1Z8au7cuRYWFsHBwZpndwoKCv755x8XFxc2HTF1yIwoFHB3x+bNnEFgkI6Ourvz0ksvXbt2TXMy299//61QKNQT2PLz89mGxNQh0xcUBFdXXhBqwI6OwNfXV2syW3x8/IABA4QrdeLj448dO8ZmJPPBgRUzlZzMO4TWjcWLFxcUFEydOrWyFa5fv25ra3v16tVt27YFBAQASEtL69u3L4CioqJDhw4tW7aMzUhMHTJl6rE11vespZs3b+7cuVOhUOiz8qpVq4TU+eCDD2bNmnXgwIHTp0/PnDmTzUhMHTJxHFurKx06dNC6CLSi2NhYrYkDr7zyysCBAx88ePDmm2+yDYmpQyZOKuXY2uNnb2/PRiDzxNkE5kUuh6cnoqI4tkZETB0yvIAAeHmx8g0RPTYcYTMjhw8jJgaZmWwJImJfhwxMLoe/P6KiWIaAiJg6ZHgcWyOi+oDfe80Cx9aIiH0dMhKOrRERU4eMZ9QoBAZybI2ImDpkeOHhyM7G7NlsCSKqFzjmYspkMvj4ID2dY2tExL4OGZ63N9avh4MDW4KImDpkYMLY2sSJbAkiqkc48mKaOLZGROzrkPFwbI2ImDpkJBxbI6J6i+MvpiYjAz4+yM7m2BoRsa9DBqZQ4O23sWMH7OzYGETE1CED++Yb2NvD25stQUT1FEdhTEdGBqZORXY2W4KI2NchA+PYGhExdch4OLZGRA0CR9hMAcfWiIh9HTIShQJOThxbIyKmDhlFUBBcXTm2RkQNA0fYGrbkZOzejbQ0tgQRsa9DBqZQwN0dmzfDyoqNQURMHTIwYWxt5Ei2BBE1GBxha6g4tkZE7OuQkXBsjYiYOmQ8HFsjogaKI2wNz+HDHFsjIvZ1yCjkcvj7IyqKY2tExNQhwwsJgasrBgxgSxBRg8QRtoYkIwPz5yM/ny1BROzrkOHNno316zm2RkTs65DhSaWIicH27WwJImJfhwxMoYCnJ/bsQRN+TyAipg4Z2u7d6N8fw4axJYioYeM35wZALoePD9LT2RJExL4OGd7s2Zg2DQ4ObAkiYl+HDCwjg5UIiIh9HTKWt99GSAhnSxMRU4cM7/BhAPDyYksQkYngCFv9pVDA35+zpYmIfR0yim++gasrZ0sTEfs6ZHgyGaZORXY2W4KI2Nchw/vkEyxcCDs7tgQRsa9DBpaczJJrRMS+DhnL++8jJISTCIiIqUOGFx4OAN7ebAkiMkH8Ol2/KBQICsKxY2wJImJfhwxvyRK4urLkGhGxr0OGxxtUExH7OmQ8vEE1EbGvQ0bCG1QTEfs6ZCQKBSZNwubNnC1NREwdMrzdu2Fvj5Ej2RJExNSpKaVSWfUKCxcuNMMWj4uL01oi3KB640azO3Cz/YvzwHng5qyqAZ3c3Nw1a9ZYWlq2aNHC0tJywoQJTfQeAIqJifnmm2927drFJn4k3qCaiJg6yMvLGzFixPr1652dnQFs3br19ddfj4yMbNTo0d2jBw8eBAQE9OnTh+37SBkZWLeOs6WJyFxUGiELFizo2rWrEDkA/Pz8rl27tmnTJn12unr1agsLCzauPgICsGMHZ0sTkXmnjlKpDA8Pd3V11Vzo5OT0008/PXKPx48f79Gjh62tLRv3kQ4fRnY2b1BN9P/bO/+oJqs3gN8xwGxjLBBwEOAx8QABS5owZtmh5UxYRmknWRIexUPYwTSqU7asb1CHfviLg5NCPJXI8QcTIkWKHQ13xAV15nD8mM0YAgYDBTbm3M/3+8d7vu9529iw0zeKy/38weE+772w57nv7vPe9z73uYg573WUSuXw8DCDwSALGQyGXC6fmJjw8uesVqtUKs3OzkaWnRb8gGqJBEVLIxCIOe91NBoNAIBOp7t4Hbvd3tvb6+XPHThwYPv27cis9wI6oBqBQMxBpn7MNhgMAADaH1cb8KkPfmlKVCrVggULHnzwQWTWaTGZQGEh8OrBEQgEAkawqdi/fz8A4Ny5c2Th3r173YUEdrv9jTfecDqdeJHL5QqFQswzc97wNQDsRrcfAoGA3qG4MPVcJyAgAADgcDjcJ0D4JXckEklBQcG9h645HI7i4mJPV99///250Uf/QbcpAoGYTWOW59399zhuT+11mEwmAMBsNrt7ncDAQPf6165dM5vNwcHBRKyB3W632WwTExNUKtVlfQjHx8dnzrgWBAKBgIS/Pm5P7XW4XC6FQhkeHnbxOgwGIzY21r2+Xq/v7+8Xi8WEpKurKzQ0VCwWL1q0qKioCHUVAoFAIAAAFE9LLDwej8fjff7554REKBT6+/ufPn0aL9psNpPJhM+K3GGxWBwO57vvvkMmRiAQCASBx9wEr7zySn19vd1ux4vj4+MtLS07d+4kKqSnp0dERIyMjLi3dTqdJpPJS7QbAoFAIJDX+QMvv/xyVlZWXl7eyMiITqcTiUSlpaWPP/44USE2NnbRokX333+/S8P8/PwVK1YAADo6Ovh8fmlpKbIyAoFAIHAo3oOYdTrdxYsX7969+/TTT0dFRf1f/mVPT09VVdXChQvtdntcXNzatWshs+kPP/zQ3NwcHh5+584doVDIZrO9VG5oaDhz5kxmZmZ0dLTBYNBoNEqlUiKRQGYTq9UqFoszMzOJzH6QMTw8XFhYePjwYZeMHu689NJLHA4nJSWFRqP19fW1trby+XyBQACNKYxG4759+/r6+oaGhhITE3fu3BkWFgZTX2u12oqKioGBAYvFsnLlym3bts2bN89T5cHBwcLCwqysrLi4OAzDfvvtN5lMVlJSsnDhQmgMIpPJmpubfX19h4aGuFxubm6uv7+/twbYzNLe3p6QkHDz5k0Mw5xOZ25ubnFxMQYRFRUVGRkZNpsNw7Dx8XEOhyOTybzXJ3dHQkJCZ2cnTAYpKSnJy8vDny2OHz+OwcWtW7d27NixefPm5cuXAwCGhoambUIebvz9/cViMbHLDQJGR0c3bNjQ29uLYZjBYHjhhRcCAwMvX74MjYJyuXz79u0GgwHDMK1WGx8fHxsbe/v2bU/1e3p6yF/wBQsWSKVSmL4Chw4dqq+vx3+/fft2VFTUU089ZbVavW3WnOGPuGzZss8++4woDg4O+vv7QzPO3rx5k0ajtbe3E5Ljx49HRkbiTsiT19m9e3dxcfFHH33U2NjocDggG5dv3LhhNpvVajWUXsdqtfb19TkcjvLy8nv0OgKBQCKRvPPOO5WVlTqdDjKDFBQUXL9+nShOTk4GBwcvXrwYmhs7IyODrMvly5cBALm5uV68zosvvrhnz5733nuvpqZmYmICpu52Op1BQUEcDoeQFBYW4u97/vQu0b+Jrq4upVJZSTo1Mzw8PCIiora2dvduGDbq19XVUSgUDodDSNLS0vr7+1tbW1euXOmplUAgWAFvOrbIyEiI31D7+fn92TfPoaGhBQUFsBrk7NmzdXV13d3deHQrjUZ74oknTp8+3dPTEx8fP9u16+3tbWxsxE8awyVcLjcoKKixsdFLq8TExNdffx3WHo+IiCDnE8DzqI2Pj3tp4jOTn6+pqQn8L58bAYPBOHPmDBwd0NTU5K4dAAAaBRGIaR8ybDab1WolJPgmcb1eD4F2dDqdyWSaTCYX4djYmM1mm4PdTaFQrly58ssvvxCSS5cu0el078u3MzrX8ZTKuqOjA44+0Gg0LtrhCYSuXbvmpZXNZvv6668NBsPdu3e5XC45UBABJWq1+sKFCw6Hg0qlbtq0yVOWqdmIXC632+1+fn6EpKOjw8/PLyEhAQLtQkJChoeHydqNjY0NDAyw2Wyy0J1z585dv37dYrGwWKzs7GyYDr0kny4tl8sVCsWRI0dCQ0P/LV7HUypro9GIYRgEPWEwGFgsFllCpVJpNJr3rUvl5eUHDx4MCwszm80CgeCZZ55566230NAMKyqVSq1W46+/GxoaUlNTv//+e2jeQ1IoFPL4+/PPP1+5cqWoqAiaYx5dorOqqqqcTueHH37opcmpU6fWrFmzZs0aAEBRUdHJkydra2t94TpWa+/evV1dXT/++OOFCxemXy+YyaWnrKwsAIDZbCYL8egmF+EshclkpqamuggZDIa7kECn0+HxMDiNjY0+Pj4qlQqyRWZYowkI7j2aoKOjg1xMSUlZv349lDaxWCxJSUnPPfecxWKBUkGtVkun0z/99FMvde7cuaPVaomi0WikUqnl5eXwWWNycvLYsWNRUVE1NTXea87ouo6nVNa+vr733XcfBA4/ICDARTv8PvPyCiU6Opp8NTk52el0fvnll2hOACuJiYnkYnJycm1t7ejoKHya7tq16+GHHz558uQ0uzdmJ3a7PScnp7S09M033/RSbf78+Q899BBRpNPpMTExhw4dgs8gNBpNJBJlZ2eLRKKGhgZvL+Vm8mN5SmU9ZR7r2QiTyXTRbnJyEsMwTwqaTKbq6mpyvAe+LNTZ2YlGZyhRKBQtLS1kCaw9fuTIkfHx8erqal9IT2jPz8/Py8t79dVXvVeTSqW//vqrS49rNBoi2Rhk4PudS0pK/i1eh8fjAQDcU1mnpqbCYXEej+euHQDAk4LffPNNTk5OWVkZIcGDf4KDg9EADSXPPvusQCAgxztB2eNnz55VqVSVlZX4UnNnZ2d3dzdMCorFYj6fv3nzZrxYV1fn8pIDp7u7e/369fn5+WSh1WoNDAyEwxkfPXo0LCzs6NGjhCQoKAgAoNVq/y1eJzMzc968eb2kc5sdDoder1+9ejUc9+Lzzz9/69Yto9FISH7//XcAAFlBcr7U8PDwdevW5ebmEhJ8J3NaWhoaoOHAZrOR57JsNvuLL74gr7f39PQwmcy4uDhoVP7pp59aW1sPHDhAxAd9++23XnLGzDoqKirYbLZIJCJ7WSqViv8+MTFBBI6zWKxly5aRj0Gz2Wy9vb3QfMGVSqVeryfP1Pv7+/H73EurGfW3AQEBIpGotrZWKBTikvPnz8+fP3/jxo1w9EF6enpMTIxUKt20aRPxELRq1aqkpCTiEenjjz9uaGjALbBq1SqZTEaOX6qurl66dOm2bdsgG3wtFgvxE0o8KZienq5UKnU6XUhICABg3bp15BOq+vv7L168KJFIiDFrtqPRaDZu3Mjn8/GdsDabbXR0VKVSvf3223AoeOrUqYMHDz722GPnz58HAJjN5pGREaL7xsbGoqOjY2Nj29raAABMJvPJJ58kb4+tr6+3Wq2ffPIJHNbIysrS6XRkt9rU1EShUHbt2uWt2QzHORgMBj6fX1ZWNjk52draunz5crlcDlMgx9WrV5OTk5ubm00m07Fjx7hc7uDgIHH18OHDERERbW1thKS9vX3r1q0ymaytrW3Hjh1paWnkiBcI+OCDDzIyMpYsWcJgMCIjI1evXr1lyxZotHM4HBs2bODz+SwWi8FgxMfHr127ds+ePUSFLVu2xMfH48t7eP133313//79KpXqxIkTiYmJ5MoQ8Oijj7oPMmw2Gw7tRkdHp5y0vfbaa3gFs9mclJSUk5NDNNHr9Vu3bj1x4oRKpSorK1u6dKn3bDGzjqqqqvz8fKlUqlar9+3bFxIS8tVXX3lvMk3O6b8DDMNUKpVCoWAwGEKhcNocvbPxsVehUKhUqiVLlggEgmlf4BqNxpaWloGBgUceeSQlJYW86woBJVevXlUoFMHBwTweD6bcw4gpcTgcly5dUqvVMTExaWlpLhvJIQA/fe3GjRuLFy/m8XgPPPCA9/r/gNdBIBAIxJwFPVYjEAgEAnkdBAKBQCCvg0AgEAgE8joIBAKBQF4HgUAgEAjkdRAIBALxT/FfSB/qBwbXQiQAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import Image\n", "\n", "Image(filename=\"figures/figinit.png\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Euler's approach\n", "\n", "The two Euler's methods are a way to compute this integral term. They both work on the approximation that the function $f(y,t)$ is constant over the timestep [$t$;$t+\\Delta t$].\n", "\n", "#### Forward Euler\n", "\n", "The forward Euler approach consists in the following approximation:\n", "\n", "\\begin{equation}\n", "f(y,t) \\approx f(y_0,t_0) \\text{ between [$t_0$;$t_0+\\Delta t$]}\n", "\\end{equation}\n", "\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Image(filename=\"figures/figExplicitEuler.png\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This approximation allows us to write:\n", "\n", "\\begin{equation}\n", "\\begin{array}{lll}\n", "\\int_{t_0}^{t_0 + \\Delta t} f(y,t)dt & \\approx & f(y_0,t_0) \\int_{t_0}^{t_0 + \\Delta t} dt \\\\\n", "& \\approx & f(y,t_0) \\Delta t\n", "\\end{array}\n", "\\end{equation}\n", "\n", "The forward Euler method takes the solution at time $t_n$ and advances\n", "it to time $t_{n+1}$ using the value of the derivative $f(y_n,t_n)$ at\n", "time $t_n$\n", "\n", "$$y(t_0+\\Delta t) = y(t_0) + \\Delta t f(y(t_0),t_0)$$\n", "\n", "This is also referred to as the explicit approach, as the new value of function $y$ can be computed immediately (i.e. explicitly) from the previous expression." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Backward Euler\n", "\n", "The backward Euler method is very similar in principle but its application leads to very different results (forward approach is almost never used for stability issues, see below). The forward Euler approach consists in the following approximation:\n", "\n", "\\begin{equation}\n", "f(y,t) \\approx f(y,t_0+\\Delta t) \\text{ between [$t_0$;$t_0+\\Delta t$]}\n", "\\end{equation}\n", "\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Image(filename=\"figures/figImplicitEuler.png\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This approximation allows us to write:\n", "\n", "\\begin{equation}\n", "\\begin{array}{lll}\n", "\\int_{t_0}^{t_0 + \\Delta t} f(y,t)dt & \\approx & f(y,t_0+\\Delta t) \\int_{t_0}^{t_0 + \\Delta t} dt \\\\\n", "& \\approx & f(y,t_0+\\Delta t) \\Delta t\n", "\\end{array}\n", "\\end{equation}\n", "\n", "Back in the initial ODE, this yields:\n", "\\begin{equation}\n", "y(t_0+\\Delta t) = y(t_0) + f(y,t_0+\\Delta t) \\Delta t\n", "\\end{equation}\n", "\n", "Which is an equation which has to be solved. Depending on the form of the function $f$, this might be a nonlinear equation to solve. We will see later examples of methods to solve these problems (Newton method, for example)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Examples of application\n", "\n", "We have shown that the associated ODE to the sulate problem could be written as:\n", "\n", "\\begin{equation}\n", "\\frac{dc_{\\text{TMF}}}{dt} + \\lambda c_{\\text{TMF}} = Q\n", "\\end{equation}\n", "with\n", "\\begin{equation}\n", "\\left\\lbrace\n", "\\begin{array}{lllll}\n", "Q & = & \\frac{Q_{\\text{pit}}c_{\\text{pit}} + Q_{\\text{mill}}c_{\\text{mill}}+ kAc_{\\text{pore}}}{V_0} & \\approx & 3.247 \\times 10^{-6} mg . L^{-1} . s^{-1} \\\\\n", "\\lambda & = & \\frac{Q_{\\text{dis}}+kA}{V_0} & \\approx & 6.358 \\times 10^{-9} s^{-1}\n", "\\end{array}\n", "\\right.\n", "\\end{equation}\n", "\n", "The forward Euler's method yields:\n", "\\begin{equation}\n", "c(t_0 + \\Delta t) = c(t_0) + \\left( Q - \\lambda c(t_0) \\right) \\Delta t\n", "\\end{equation}\n", "\n", "While the backward Euler's method yields:\n", "\\begin{equation}\n", "\\begin{array}{llll}\n", "& c(t_0 + \\Delta t) & = & c(t_0) + \\Delta t (Q - \\lambda c(t_0+\\Delta t)) \\\\\n", "\\Longleftrightarrow & c(t_0 + \\Delta t) \\left( 1 + \\lambda \\Delta t \\right) & = & c(t_0) + Q \\Delta t \\\\\n", "\\Longleftrightarrow & c(t_0 + \\Delta t) & = & \\displaystyle{\\frac{c(t_0) + Q \\Delta t}{1 + \\lambda \\Delta t}} \\\\\n", "\\end{array}\n", "\\end{equation}\n", "\n", "So, let us compare these solutions to the analytical solution. We will therefore compute the two Euler approaches at each timestep and assess the error of the solution.\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "lines_to_next_cell": 2 }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This is the routine to plot the exact solution for the concentration at every day\n", "dt = 1 # day\n", "Tf = 50 # years\n", "n = int(1 + 365 * Tf / dt) # int() is to convert thhe result into an integer\n", "c_real = np.zeros(\n", " n, float\n", ") # represents the concentration of sultates (mg/L) at each time\n", "timeY = np.zeros(n, float) # time in years\n", "c_real[0] = c0\n", "c_asymptotic = c_inf * np.ones(\n", " n, float\n", ") # this is an array of size n full, whose values are all c_inf\n", "Forward_Euler = np.zeros(n, float)\n", "Backward_Euler = np.zeros(n, float)\n", "err_Backward = np.zeros(n, float)\n", "err_Forward = np.zeros(n, float)\n", "Forward_Euler[0] = c0\n", "Backward_Euler[0] = c0\n", "\n", "\n", "for i in range(n - 1):\n", " timeY[i + 1] = (i + 1) * dt / 365\n", " c_real[i + 1] = c_inf + (c0 - c_inf) * np.exp(-lam * (i + 1) * dt * 3600 * 24)\n", " Forward_Euler[i + 1] = 0 #ADAPT THIS\n", " Backward_Euler[i + 1] = # ADAPT THIS\n", " err_Backward[i + 1] = (c_real[i + 1] - Backward_Euler[i + 1]) / c_real[i + 1]\n", " err_Forward[i + 1] = (Forward_Euler[i + 1] - c_real[i + 1]) / c_real[i + 1]\n", "\n", "fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 8))\n", "ax1.plot(timeY, c_real, label=\"Real Concentration\")\n", "ax1.plot(timeY, Forward_Euler, label=\"Concentration - Forward Euler\")\n", "ax1.plot(timeY, Backward_Euler, label=\"Concentration - Backward Euler\")\n", "ax1.set(xlabel=\"Time (years)\", ylabel=\"Concentration (mg/L)\")\n", "ax1.legend()\n", "\n", "ax2.plot(timeY, 100 * err_Backward, label=\"Error - backward Euler\")\n", "ax2.plot(timeY, 100 * err_Forward, label=\"Error - forward Euler\")\n", "ax2.set(xlabel=\"Time (years)\", ylabel=\"Error (%)\")\n", "ax2.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can see that the two approaches are quite close to each other and to the analytical solution. This means that the approximation is good.\n", "When you calculated the sulfates concentration after 1 day, you saw that it had increased from 93 to 93.23 mg/L, hence a relative increase of 0.2%. This yields that the approximation that $c_{\\text{TMF}}$ is constant over a day is pretty good, hence a relatively small error!\n", "\n", "So, the bigger the timestep, the less accurate is the approximation. Let us check that!\n", "We will set a timestep of 5 years and see the effect on the result and the error." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "dt = 5 * 365 # day\n", "Tf = 50 # years\n", "n = int(1 + 365 * Tf / dt) # int() is to convert thhe result into an integer\n", "\n", "timeY = np.linspace(0, Tf, n) # this is a way to create an array of size n,\n", "# which starts at 0, ends at Tf and every elements\n", "# are evenly spaced\n", "c_real = np.zeros(\n", " n, float\n", ") # represents the concentration of sultates (mg/L) at each time\n", "# timeY = np.zeros(n, float) # time in years\n", "c_real[0] = c0\n", "c_asymptotic = c_inf * np.ones(\n", " n, float\n", ") # this is an array of size n full, whose values are all c_inf\n", "Forward_Euler = np.zeros(n, float)\n", "Backward_Euler = np.zeros(n, float)\n", "err_Backward = np.zeros(n, float)\n", "err_Forward = np.zeros(n, float)\n", "Forward_Euler[0] = c0\n", "Backward_Euler[0] = c0\n", "\n", "\n", "for i in range(n - 1):\n", " # timeY[i+1] = (i+1)*dt/365\n", " c_real[i + 1] = c_inf + (c0 - c_inf) * np.exp(-lam * timeY[i + 1] * 365 * 3600 * 24)\n", " Forward_Euler[i + 1] = 0\n", " Backward_Euler[i + 1] = 0\n", " err_Backward[i + 1] = 0\n", " err_Forward[i + 1] = 0\n", "\n", "fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 8))\n", "ax1.plot(timeY, c_real, label=\"Real Concentration\")\n", "ax1.plot(timeY, Forward_Euler, label=\"Concentration - Forward Euler\")\n", "ax1.plot(timeY, Backward_Euler, label=\"Concentration - Backward Euler\")\n", "ax1.set(xlabel=\"Time (years)\", ylabel=\"Concentration (mg/L)\")\n", "ax1.legend()\n", "\n", "ax2.plot(timeY, 100 * err_Backward, label=\"Error - backward Euler\")\n", "ax2.plot(timeY, 100 * err_Forward, label=\"Error - forkward Euler\")\n", "ax2.set(xlabel=\"Time (years)\", ylabel=\"Error (%)\")\n", "ax2.legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This defines the accuracy of the algorithm. The accuracy decreases when the timestep increases!\n", "\n", "Basically, Euler's method states that here, for the first 5 years, the source term from the tailings porewater is at its maximum, and the discharge is at its minimum for the forward Euler's approach. We therefore have a very important overestimation of the source and underestimation of the sink, which yields an important overestimation of the error.\n", "\n", "Now, do the same with an even bigger timestep (25 years and see what is happening over a longer timescale, like 250 years)." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "### YOUR CODE HERE" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This defines the concept of stability.\n", "\n", "We can show that Euler's backward method is unconditionnally stable (for every timestep), while Euler's forward (explicit) method has a restrictive timestep condition for stability.\n", "\n", "That is one of the reason why Euler's backward (implicit) method is usually preferred than the Euler's forward (explicit approach). However, this yields\n", "1. In most case (depending on $f(y,t)$), there is an equation to solve (often numerically) to find the value of $y(t+\\Delta t)$\n", "2. Stability does not mean accuracy\n", "\n", "These methods are referred to as 1$^{st}$ order methods. The order of the method is linked to the Taylor approximation\n", "\n", "\\begin{equation}\n", "y(t+\\Delta t) = y(t) + (\\Delta t) y^\\prime(t) + \\frac{1}{2}(\\Delta t)^2 y^{\\prime\\prime}(t) +\\cdots\n", "\\end{equation}\n", "\n", "So, as the function f is the derivative of $y$, the Euler's approach omits all the term but the 1$^{st}$ order: it is called a first order method (also we use $y = y_0 + \\Delta t f$). We compute the evolution of the solution based on the first order approximation of $y$ (we assume a constant value for its derivative $f$).\n", "\n", "The truncation error corresponds to the omission of all the higher order terms:\n", "\n", "\\begin{equation}\n", "\\text{Truncation error } = \\frac{1}{2}(\\Delta t)^2 y^{\\prime\\prime}(t) +\\cdots\n", "\\end{equation}\n", "\n", "So, the intrinsic error we make with Euler's methods is proportional to $(\\Delta t)^2$. In words, we say that forward Euler is first order accurate with errors of second order. It is therefore clear that reducing the timestep reduces the errors.\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Runge Kutta methods\n", "\n", "In a lot of applications, Euler's implicit method is used. But in some cases, higher order and more accurate methods need to be used. They basically compute the integral\n", "\n", "$$\n", "y(t_0+\\Delta t) = y(t_0) + \\int_{t_0}^{t_0+\\Delta t} f(t,y) dt\n", "$$\n", "\n", "with more accuracy (with a higher-order). These are commonly known as the Runge-Kutta methods (Euler's methods are a 1st order Runge-Kutta method). The idea of the Runge-Kutta schemes is to take advantage of derivative information at the times between $t_n$ and $t_{n+1}$ to increase the order of accuracy.\n", "\n", "For example, in the midpoint method, the derivative at the initial time\n", "is used to approximate the derivative at the midpoint of the interval,\n", "$f(y_n+\\frac{1}{2}hf(y_n,t_n), t_n+\\frac{1}{2}h)$. The derivative at the\n", "midpoint is then used to advance the solution to the next step. The\n", "method can be written in two *stages* $k_i$,\n", "\n", "\n", "\\begin{equation}\n", " \\begin{array}{lll}\n", " k_1 & = & \\Delta t f(y(t),t)\\\\\n", " k_2 & =& \\Delta t \\, f\\left( y(t)+\\frac{1}{2}k_1, t+\\frac{1}{2}\\Delta t \\right)\\ \\\\\\\n", " y(t+\\Delta t) & =& y(t) + k_2\n", " \\end{array}\n", "\\end{equation}\n", "\n", "The midpoint method is known as a 2-stage Runge-Kutta formula.\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Image(filename=\"figures/figMidpoint.png\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In general, an *explicit* 2-stage Runge-Kutta method can be written as,\n", "\n", "\\begin{equation}\n", " \\begin{array}{l}\n", " k_1 = h f(y(t),t)\\\\\n", " k_2 = \\Delta t f \\left( y(t)+b_{21}k_1, t+a_2h \\right)\\ \\ \\ \\\\\n", " y(t+\\Delta t) = y(t) + c_1k_1 +c_2k_2\n", " \\end{array}\n", "\\end{equation}\n", "\n", "It is said explicit because the new solution only depends on the previous solution. The different coefficient can be determined by comparison with the 2$^{nd}$ order Taylor approximation." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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