0028_sim_kapacita.ipynb

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    "# <font color='navy'> <b> Porovnanie analytického výpočtu a FEM simulácie </b></font>\n",
    "\n",
    "Pre testy a kontrolu nastavenia podmienok simulácie je vhodné porovnanie parametrov simulovaných modelov s parametrami získanými analytickými postupmi.\n",
    "\n",
    "## <font color='teal'> <b> Príklad 4. Kapacita dvoch vodivých gulí</b></font>\n",
    "\n",
    "Úlohou je vypočítať vlastnú a vzájomnú kapacitu dvoch vodivých gulí analytickým postupom a porovnať výsledky s FEM simuláciou.\n",
    "\n",
    "Predpokladajme, že máme dve vodivé gule A, B s rovnakým priemerom $r_1 = r_2$, stredy ktorých sú od seba vzdialené $d$, jedna z gulí sa nachádza v počiatku súradnicovej sústavy. Vzťah pre kapacitu odvodíme metódou zrkadlenia náboja. \n",
    "\n",
    "<img src=\"./img/c1.png\" width=\"600\">"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### <font color='teal'> <b> Teoretické odvodenie </b></font>\n",
    "\n",
    "Pre kapacitu ľubovolného vodivého objektu platí všeobecný vzťah\n",
    "\n",
    "\\begin{equation*}\n",
    "C = \\frac{q} {\\phi}\n",
    "\\end{equation*}\n",
    "\n",
    "ktorý deklaruje kapacitu ako schopnosť telesa zhromažďovať náboj. Ak sa na teleso dostane nejaký náboj $q$, tak jeho potenciál bude mať hodnotu $\\phi$ voči referenčnému nulovému potenciálu v nekonečne, ale na nulovom potenciáli môže byť aj nejaké iné teleso. Potenciál je ekvivalentom práce, ktorú musíme vynaložiť na to, aby sme na teleso prepravili jednotkový náboj z bodu s nulovým potenciálom.\n",
    "\n",
    "V našom príklade budeme preto hľadať *vzájomnú kapacitu* gule A voči uzemnenej guli B a jej *vlastnú kapacitu* voči nulovému potenciálu v nekonečne.\n",
    "\n",
    "### <font color='purple'> <b>  Potenciál na povrchu nabitej gule</b></font>\n",
    "\n",
    "Potenciál na povrchu vodivej nabitej gule je rovnaký, ako potenciál bodového náboja umiestneného v jej strede vo vzdialenosti rovnej polomeru gule. Ak preto do začiatku súradnicovej sústavy umiesnime bodový náboj $q$, vo vzdialenosti $r_1$ bude potenciál daný vzťahom\n",
    "\n",
    "\\begin{equation*}\n",
    "\\phi(r) = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q}{r_1}\n",
    "\\end{equation*}\n",
    "\n",
    "Princíp riešenia úlohy spočíva v tom, že musíme nájsť takú konfiguráciu nábojov (veľkosť, poloha), ktorá vytvoria guloplochu o priemere $r_1$ s potenciálom $\\phi$ so stredom v počiatku súradnicovej sústavy a druhú guloplochu o priemere $r_2$ a stredom vzdialenm o $d$ od počiatku s nulovým poteciálom. Pretože ide o nesymetrické usporiadanie gulí vzhľadom k počiatku súradnicovej sústavy, je zrejmé, že náboje nebudú umiestnené v strede gulí.  "
   ]
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   "metadata": {},
   "source": [
    "### <font color='purple'> <b> Metóda zrkadlenia náboja</b></font>\n",
    "\n",
    "\n",
    "Do priestoru bez gulí do miesta kde by sa nachádzal stred gule A, t.j. v počiatku súradnicovej sústavy umiestnime náboj $q$. Tento okolo seba vytvorí elektrostatické pole, ktorého intenzita vo vzdialenosti $r_1$ bude rovnaká ako na povrchu vodivej guli s rovnakým polomerom nabitej rovnakým nábojom. Veľkosť náboja je určená zadaným potenciálom a priemerom gule.   \n",
    "\n",
    "\\begin{equation*}\n",
    "q =  {4 \\pi \\epsilon_0} \\, \\phi(r) \\, {r_1}\n",
    "\\end{equation*}\n",
    "\n",
    "Niekde v blízkosti miesta, kde by sa nachádzal stred gule B umiestnime teraz (neznámy) náboj $q'$ tak, aby tento vytvoril okolo seba guloplochu s nulovým potenciálom priemerom pôvodnej gule B, ktorá by bola ekvivalentom uzemnenej gule B. Intuitívne je zrejmé, že náboj $q'$ bude musieť mať **opačnú** polaritu ako náboj $q$. Hladanými  veličinami sú veľkosť náboja $q'$ a jeho poloha."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color='purple'> <b> Guloplocha s nulovým potenciálom </b></font>\n",
    "\n",
    "<img src=\"./img/c2.png\" width=\"600\">"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Pre potenciál množiny nábojov platí princíp superpozície, takže pre hodnoty potenciálu v bodoch P1 a P2, ktoré sa nachádzajú na hladanej guloploche s nulovým potenciálom, platí\n",
    "\n",
    "\\begin{equation*}\n",
    "\\phi(P1) = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q}{r + d} + \\frac{1}{4 \\pi \\epsilon_0} \\frac{q{'}}{r + d{'}} = 0\n",
    "\\end{equation*}\n",
    "\n",
    "Po úprave dostaneme pre body P1 a  P2 sústavu rovníc, z ktorej vieme určiť hodnotu náboja $q{'}$ a jeho polohu $d{'}$\n",
    "\n",
    "\\begin{align}\n",
    "P1: \\quad &\\, \\frac{q}{d+r} + \\frac{q{'}}{r+d{'}} = 0 \\\\\n",
    "P2: \\quad &\\, \\frac{q}{d-r} + \\frac{q{'}}{r-d{'}} = 0 \\\\\n",
    "\\end{align}\n",
    "\n",
    "Triviálnou úpravou rovníc dostaneme sústavu\n",
    "\\begin{align*}\n",
    "2 q r + 2 q{'} d = 0 \\\\\n",
    "2 q d{'} +  2 q{'} r = 0 \\\\\n",
    "\\end{align*}\n",
    "\n",
    "ktorej riešením je\n",
    "\\begin{equation}\n",
    "q{'} = - \\frac{r}{d} \\, q \\qquad d{'} = - q{'} \\, \\frac{r}{q} = \\frac{r^2}{d}\n",
    "\\end{equation}"
   ]
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   "metadata": {},
   "source": [
    "Je zrejmé, že pole záporného náboja  $q'$ zrejme zdeformovalo ekvipotenciálne hladiny v okolí kladného náboja $q$ a tieto už asi nebudú mať gulová tvar. Túto deformáciu  môžeme kompenzovať novým (kladným) nábojom $q''$, umiestneným niekde v blízkosti stredu gule A, počiatku súradnicovej sústavy. \n",
    "Dôležité je to, že náboj $q''$ slúži pre korekciu tvaru plochy, **potenciál v mieste korigovanej guloplochy zostáva rovnaký** ako na začiatku, pred umiestnením náboja $q'$.\n",
    "\n",
    "Ak teraz odstránime pôvodný náboj $q$, dostaneme pododobnú situáciu ako v predchádzajúcom prípade a budeme požadovať, aby sa v mieste gule A nachádzala guloplocha s nulovým potenciálom.\n",
    "\n",
    "### <font color='purple'> <b> Korekcia náboja </b></font>\n",
    "\n",
    "<img src=\"./img/c3.png\" width=\"600\">"
   ]
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   "source": [
    "Pre potenciál v bodoch P3 a P4 potom obdobne platí\n",
    "\\begin{align*}\n",
    "P3: \\quad &\\, \\frac{q''}{r+d''} + \\frac{q{'}}{x+r} = 0 \\\\\n",
    "P4: \\quad &\\, \\frac{q''}{r-d''} + \\frac{q{'}}{x-r} = 0 \\\\\n",
    "\\end{align*}\n",
    "\n",
    "kde\n",
    "\n",
    "\\begin{equation*}\n",
    "x = d-d' = d - \\frac{r^2}{d} = \\frac{1}{d} (d^2 - r^2) \n",
    "\\end{equation*}\n",
    "\n",
    "Riešením sústavy dostaneme\n",
    "\\begin{equation}\n",
    "q{''} = - \\frac{r}{x} q' = \\frac{r^2}{d^2-r^2}q \\qquad d'' = - \\frac{q'' r}{q'} = \\frac{r^2}{d-\\frac{r^2}{d}}\n",
    "\\end{equation}"
   ]
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   "source": [
    "Doplnením náboja $q''$ do sústavu sa ale zase zmenilo pole v okolí náboja $q'$, čo môžeme zase kompenzovať ďaľším nábojom $q'''$ atď. \n",
    "\n",
    "Polohy a veľkosti nábojov dostaneme opakovaním predchádajúceho postupu, je zrejmé, že hodnoty nábojov a ich vzdialeností tvoria nekonečnú postupnosť. Niekoľko členov postupnosti pre veľkosti a polohy nábojov pre pravú a lavú gulu je uvedených v nasledujúcich tabuľkách:\\\\\n",
    "\n",
    "\n",
    "**Lavá gula (A)**\n",
    "\n",
    "\\begin{align*}\n",
    "&\\textrm{Náboj}   \\,\\,\\,\\,&   &\\textrm{Vzdialenosť}  \\\\\n",
    "&q                \\,\\,\\,\\,&   &0\\\\\n",
    "\\\\\n",
    "&\\displaystyle q'' = \\frac{\\displaystyle r^2}{\\displaystyle d^2-r^2} \\, q            \\,\\,\\,\\,& \n",
    "&\\displaystyle d''=  \\frac{\\displaystyle r^2}{\\displaystyle d - \\frac{r^2}{d}} \\\\\n",
    "\\\\\n",
    "&\\displaystyle q'''' = \\frac{\\displaystyle r^4}{\\displaystyle d^4-3d^2r^2+r^4} \\, q  \\,\\,\\,\\,& \n",
    "&\\displaystyle d'''' = \\frac{\\displaystyle r^2}{\\displaystyle d - \\frac{r^2}{\\displaystyle d-\\frac{r^2} {\\displaystyle d-\\frac{r^2}{d}}}}\n",
    "\\end{align*}"
   ]
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   "source": [
    "\n",
    "**Pravá gula (B)**\n",
    "\n",
    "\\begin{align*}\n",
    "&\\textrm{Náboj}   \\,\\,\\,\\,&   &\\textrm{Vzdialenosť}  \\\\\n",
    "&q' = - \\frac{\\displaystyle r}{\\displaystyle d} \\, q  \\,\\,\\,\\,&  &d'= \\frac{\\displaystyle r^2}{\\displaystyle d} \\\\  \n",
    "\\\\\n",
    "&q'''=-\\frac{\\displaystyle r^2}{\\displaystyle d^2-r^2} \\frac{\\displaystyle r}{\\displaystyle d} \\, q \\,\\,\\,\\,&\n",
    "&d'''= \\frac{\\displaystyle r^2}{\\displaystyle d - \\frac{\\displaystyle r^2}{\\displaystyle d-\\frac{\\displaystyle r^2}{\\displaystyle d}}}\n",
    "\\\\\n",
    "&q'''''=-\\frac{\\displaystyle r^4}{\\displaystyle d^4-4d^2r^2+3r^4} \\frac{\\displaystyle r}{\\displaystyle d} q  \\,\\,\\,\\,&\n",
    "&d'''''= \\frac{\\displaystyle r^2}{\\displaystyle d - \\frac{r^2}{\\displaystyle d-\\frac{r^2}{\\displaystyle d-\\frac{r^2}{\\displaystyle d-\\frac{r^2}{d}}}}}\n",
    "\\end{align*}"
   ]
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   "source": [
    "### <font color='purple'> <b> Simulácia ekvipotenciálnych hladín v 2D </b></font>\n",
    "\n",
    "Zjednodušená simulácia potenciálu v okolí bodových nábojov $q$ a $q'$ bola použitá najmä na odhad potrebného rozmeru simulovaného priestoru s ohľadom na okrajové pomienky. Červeno je vyznačená nulová potenciálová hladina.  \n",
    "\n",
    "<img src=\"./img/s2.png\" width=\"600\">\n",
    "\n",
    "<img src=\"./img/s1.png\" width=\"600\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color='teal'> <b> Numerický výpočet potenciálu  </b></font>\n",
    "\n",
    "Numerický vypočet priebehu potenciálu pri postupných iteráciách hodnôt a polôh nábojov."
   ]
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy import *\n",
    "from scipy.constants import epsilon_0 \n",
    "import pylab as plt\n",
    "\n",
    "q=4*pi*epsilon_0 * 0.5 #  0.5e-10   # [C] charge\n",
    "r=0.4     # [m] sphere diameter\n",
    "d=2.0     # [m] distance between center of spheres\n",
    "\n",
    "k = r/d; w=r**2/d\n",
    "q0 = q                   # q\n",
    "d0 = 0.0\n",
    "\n",
    "q1 = -k*q                # q'\n",
    "d1 = d-w \n",
    "\n",
    "q2 = w/(d-w)*q           # q''\n",
    "d2 = r**2/(d-w)\n",
    "\n",
    "q3 = -w/(d-2*w)*k*q      # q'''\n",
    "d3 = d-r**2/(d-r**2/(d-w))\n",
    "\n",
    "def phi(q, d, x):\n",
    "    q=q/(4*pi*8.854e-12)\n",
    "    if d==x:\n",
    "        if x<d1+d2/2.:\n",
    "            return 1e6\n",
    "        else:\n",
    "            return -1e6\n",
    "    if d>x:\n",
    "        return q/(d-x)\n",
    "    else:\n",
    "        return -q/(d-x)\n",
    "    \n",
    "x=[]; p0=[]; p1=[]; p2=[]; p3=[]\n",
    "step=0.001\n",
    "for k in arange(d0-1, d3+1.5, step):\n",
    "    x.append(k)\n",
    "    s1=phi(q0, d0, k)\n",
    "    s2=phi(q1, d1, k)\n",
    "    s3=phi(q2, d2, k)\n",
    "    s4=phi(q3, d3, k)\n",
    "    \n",
    "    s=s1;   p0.append(s)\n",
    "    s=s+s2; p1.append(s)\n",
    "    s=s+s3; p2.append(s)\n",
    "    s=s+s4; p3.append(s)\n",
    "\n",
    "plt.plot(x,p0,'b:', label=r\"$q$\")\n",
    "plt.plot(x,p1,'r',  label=r\"$q+q^{'}$\")\n",
    "plt.plot(x,p3,'y', label=r\"$q+q^{'}+q^{''}+q^{'''}$\")\n",
    "plt.plot([-1,3], [0,0], 'r', alpha=0.4)\n",
    "plt.plot([0,0], [-5,5], 'r', alpha=0.4)\n",
    "\n",
    "plt.plot([1.5,2.5], [0,0], 'b', linewidth=3, alpha=0.3)\n",
    "plt.plot([-0.5,0.5], [1, 1], 'g', linewidth=3, alpha= 0.3)\n",
    "\n",
    "plt.text(1.8, -0.35,  '2r', color='blue')\n",
    "plt.text(0.1, 1.25,  '2r', color='green')\n",
    "\n",
    "plt.xlabel(r'$x\\, [m]$')\n",
    "plt.ylabel(r'$Potential \\,\\,\\, U\\,[V]$')\n",
    "plt.ylim(-5.0, 5.0)\n",
    "plt.xlim(-1, 3)\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color='teal'> <b> Výpočet kapacity </b></font>\n",
    "\n",
    "Výsledný celkový náboj lavej gule A je \n",
    "\n",
    "\\begin{equation*}\n",
    "q_{sum} = q + q'' + q'''' \\dots = q \\left( 1 + \\frac{r^2}{d^2-r^2} + \\frac{r^4}{d^4-3d^2r^2+r^4}  + \\dots \\right) \n",
    "\\end{equation*}\n",
    "\n",
    "a jej celková kapacita je potom\n",
    "\n",
    "\\begin{equation*}\n",
    "C_{sum} = \\frac{q}{\\phi} = \\frac{q}{\\frac{\\displaystyle q}{\\displaystyle 4 \\pi \\epsilon_0 r}} = 4 \\pi \\epsilon_0 \\, r  \\left( 1 + \\frac{r^2}{d^2-r^2} + \\frac{r^4}{d^4-3d^2r^2+r^4}  + \\dots \\right) \n",
    "\\end{equation*}\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Celková kapacita pozostáva z dvoch častí, z kapacity voči nulovému potenciálu v nekonečne a kapacite voči druhej guli. Vzájomná kapacita je úmerná indukovanému náboju gule B na guli A, pretože veľkosť náboja gule B poznáme, môžeme náboj na guli A rozdeliť\n",
    "\n",
    "\\begin{align*}\n",
    "C_{sum} &= C_{gnd} + C_{mut} = \\frac{q - q' + q''- q ''' + q'''' + \\dots + q' + q''' - \\dots  }{\\phi} =\n",
    "\\\\\n",
    "&= 4 \\pi \\epsilon_0 \\, r  \\left( 1 -\\frac{r}{d} + \\frac{r^2}{d^2-r^2} -  \\frac{r^2}{d^2-r^2} \\frac{r}{d} + \\frac{r^4}{d^4-3d^2r^2+r^4}  - \\frac{r^4}{d^4-4d^2r^2+3r^4} \\frac{r}{d} + \\dots \\right) +\n",
    "\\\\\n",
    "&+ 4 \\pi \\epsilon_0 \\, r  \\left(\\frac{r}{d} +  \\frac{r^2}{d^2-r^2} \\frac{r}{d} + \\frac{r^4}{d^4-4d^2r^2+3r^4} \\frac{r}{d}  + \\dots \\right) \n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Na základe vyššie uvedených vzťahov môžeme spočítať hodnotu kapacity (pre prvé tri členy radov) s hodnotami\n",
    "\n",
    "  - $r = 0.5 \\,[m]$ - polomer gulí\n",
    "  - $d = 2.0\\, [m]$ - vzdialenosť stredov gulí  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "C12 = 14.972907573290463 [pF]\n",
      "C10 = 46.42671205015176 [pF]\n"
     ]
    }
   ],
   "source": [
    "eps = 8.8542e-12 # F/m\n",
    "\n",
    "def C12(a,d):\n",
    "    # vzajomna kapacita\n",
    "    c = 4*pi*eps*(a**2)/d*(1 + a**2/(d**2 - 2*(a**2)) + \\\n",
    "        (a**4)/(d**4 - 4*(d**2)*(a**2) + 3*(a**4)) )\n",
    "    return c\n",
    "        \n",
    "def C10(a,d):\n",
    "    # vlastna kapacita\n",
    "    c = 4*pi*eps*a*(1 - a/d + a**2/(d**2 - (a**2)) + \\\n",
    "    (a**3)/(d**3 - 2*d*(a**2)))\n",
    "    return c\n",
    "\n",
    "print('C12 =', C12(.5, 2.) * 1e12, '[pF]')\n",
    "print('C10 =', C10(.5, 2.) * 1e12, '[pF]')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <font color='teal'> <b> FEM Simulácia </b></font>\n",
    "\n",
    "Pri simulácii príkladu pomocou FEM postupujeme štandardne, vytvoríme model (FreeCAD, gmsh ...) a vygenerujeme mriežku.\n",
    "\n",
    "<img src=\"./img/s6.png\" width=\"600\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "V povelovom súbore simulácie definujeme výpočet kapacitnej matice a v okrajových podmienkach identicikáciu telies, ktorých kapacitu budeme počitať. Pretože na výpočet kapacity je použitý iný algoritmus ako na výpočet polí, tokov a pod. je vhodné mať pre každý typ simulácie vlastný povelový súbor.  \n",
    "\n",
    "    Solver 1\n",
    "      Equation = Electrostatics\n",
    "      Calculate Capacitance Matrix = True              ! vypocet kapacitnej matice\n",
    "      Calculate Electric Flux = True\n",
    "      Variable = Potential\n",
    "      Capacitance Matrix Filename = cap_matrix.dat     ! ulozenie vysledkov\n",
    "      Procedure = \"StatElecSolve\" \"StatElecSolver\"\n",
    "      Calculate Electric Field = True\n",
    "      Calculate Electric Energy = True\n",
    "      ...\n",
    "    End\n",
    "    \n",
    "    Boundary Condition 1\n",
    "      Target Boundaries(1) = 4 \n",
    "      Name = \"1v\"\n",
    "      Capacitance Body = 2                              ! doplnenie ID telies pre vypocet kapacity\n",
    "      Potential = 1\n",
    "    End\n",
    "\n",
    "    Boundary Condition 2\n",
    "      Target Boundaries(1) = 3 \n",
    "      Name = \"-1v\"\n",
    "      Capacitance Body = 3\n",
    "      Potential = -1\n",
    "    End"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pre výpočet bola použitá kapacitná matica z solveru pre elektrostatické úlohy, simulačná oblasť bola gula s priemerom 5m. Výpočítané hodnôt z 3D simulácie sú\n",
    "\n",
    "\\begin{equation*}\n",
    "C_{gnd} = 44.70 \\,\\, pF \\qquad C_{mut} = 14.90 \\,\\, pF\n",
    "\\end{equation*}\n",
    "\n",
    "Pre porovnanie - analytické hodnoty\n",
    "\n",
    "\\begin{equation*}\n",
    "C_{gnd} = 46.42 \\,\\, pF \\qquad C_{mut} = 14.97 \\,\\, pF\n",
    "\\end{equation*}\n",
    "\n",
    "Rozdiel v hodnotách je zrejme spôsobený v nedostatočnej hustote siete a okrajovými efektami pri malej simulačnej oblasti. Simulačný model môžeme samozrejme využiť aj na štandardnú simuláciu a zobrazenie vlastností elektrostatického poľa.\n",
    "\n",
    "<img src=\"./img/s5.png\" width=\"600\">\n",
    "\n",
    "--------------"
   ]
  }
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