doc/tutorial/tutorial.ipynb

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   "source": [
    "# Importieren der Module"
   ]
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   "source": [
    "Um die Pakete verwenden zu können müssen sie zunächst eingeladen werden. Dies erfolgt in Python mit dem import Befehl.\n",
    "    "
   ]
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    "import os                             # operating system functionality\n",
    "import numpy as np                    # arrays and math\n",
    "import matplotlib.pyplot as plt       # plotting\n",
    "import matplotlib\n",
    "from IPython.display import SVG       # showing svg\n",
    "\n",
    "import parabem                            # python panel methode package\n",
    "from parabem.vtk_export import VtkWriter  # export to vtk file format\n",
    "from parabem.utils import check_path      # check path and create directory\n",
    "import parabem.pan2d as pan2d             # 2d panel-methods and solution-elements\n",
    "import parabem.pan3d as pan3d             # 3d panel-methods and solution-elements"
   ]
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   "source": [
    "def paraview(_file=None):\n",
    "    \"\"\"open a vtk-file in paraview.\"\"\"\n",
    "    if not _file:\n",
    "        command = \"paraview\"\n",
    "    else:\n",
    "        command = \"paraview \" + _file\n",
    "    os.system(command)\n",
    "\n",
    "%matplotlib inline\n",
    "matplotlib.rcParams[\"font.size\"] = 10\n",
    "matplotlib.rcParams[\"figure.figsize\"] = [9, 3]\n",
    "%config InlineBackend.figure_format = 'svg'"
   ]
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   "source": [
    "# Basis Objekte\n",
    "Das Basis-Objekt für die Panelmethode kann ein 2d- oder 3d-\"Berechnungsfall\" (**Case2**, **Case3**) sein. Diesem wird die Geometrie bestehend aus mehren  Flächen (**Panel2**, **Panel3**) vorgegeben. Diese sind planare Flächen bzw. im 2D Fall Linien, die aus mehreren Punkten  (**PanelVector2**, **PanelVector3**) gebildet werden, welche Vektoren (**Vector2**, **Vector3**) mit weiteren Parametern sind."
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   "source": [
    "## Vector2 & Vector3\n",
    "**Vector2** und **Vector3** repräsentieren Eigen::Vector2d bzw Eigen::Vector3d in Python. Diese besitzten einige hilfreiche Funktionen. Einige davon werden hier gezeigt:"
   ]
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   "source": [
    "from __future__ import division       # enable python3 division\n",
    "v1 = parabem.Vector2(0, -1)\n",
    "v2 = parabem.PanelVector2(0, 1)\n",
    "v3 = parabem.Vector3(0, 0, 1)\n",
    "v4 = parabem.PanelVector3(1, 0, 1)"
   ]
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   "source": [
    "### Basis Operatoren"
   ]
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     "text": [
      "Vector2d(0, 0)\n",
      "Vector2d(0, 2)\n",
      "Vector2d(0, -2)\n",
      "Vector2d(0, -0.5)\n"
     ]
    }
   ],
   "source": [
    "print(v1 + v2)\n",
    "print(v2 - v1)\n",
    "print(v1 * 2)\n",
    "print(v1 / 2)\n",
    "v1 += v2\n",
    "v1 -= v2\n",
    "v1 *= 2\n",
    "v1 /= 2"
   ]
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  {
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   },
   "source": [
    "### Skalar- / Kreuzprodukt"
   ]
  },
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     "text": [
      "1.0\n",
      "1.0\n",
      "Vector3d(0, 0, 0)\n",
      "Vector3d(0, 1, 0)\n"
     ]
    }
   ],
   "source": [
    "print(v1.dot(v1))\n",
    "print(v3.dot(v4))\n",
    "print(v3.cross(v3))\n",
    "print(v3.cross(v4))"
   ]
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   "source": [
    "### 2d $\\longleftrightarrow$ 3d\n",
    "Es ist möglich einen 2d Vektor in einen 3d Vektor umszuwandeln, sowie umgekehrt. Die Operatoren können aber immer nur auf Vektoren der glchen Dimension angewendet werden."
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     "text": [
      "Vector3d(0, -1, 0)\n",
      "Vector2d(0, 0)\n"
     ]
    }
   ],
   "source": [
    "print(parabem.Vector3(v1))\n",
    "print(parabem.Vector2(v3))"
   ]
  },
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   },
   "source": [
    "### Länge und Normalenrichtung"
   ]
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     "text": [
      "1.0\n",
      "1.0\n",
      "Vector2d(1, 0)\n"
     ]
    }
   ],
   "source": [
    "print(v1.norm())\n",
    "print(abs(v1))\n",
    "print(v1.normal)"
   ]
  },
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   "source": [
    "## Panel2\n",
    "Das **Panel2** besteht aus 2 Punkten der Klasse **PanelVector2**. Einige in der Panelmethode öfter benutzte Eigenschaften werden direkt als Attribute im Panel gespeichert. Dies sind zum Beispiel die Länge **l**, die Ausrichtung **n, t** und der Mittelpunkt **center**."
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       "(2.8284271247461903,\n",
       " Vector2d(-0.707107, -0.707107),\n",
       " Vector2d(-0.707107, 0.707107),\n",
       " Vector2d(2, 3))"
      ]
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   "source": [
    "l = [parabem.PanelVector2(1, 2), parabem.PanelVector2(3, 4)]\n",
    "p = parabem.Panel2(l)\n",
    "p.l, p.t, p.n, p.center"
   ]
  },
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   "source": [
    "## Panel3\n",
    "Das Panel3 besteht aus 3 oder mehr Punkten der Klasse **PanelVector3**. Einige in der Panelmethode öfter benutzte Eigenschaften werden direkt als Attribute im Panel gespeichert. Dies sind zum Beispiel die Fläche **area**, die Ausrichtung **n, m, l** und der Mittelpunkt **center**."
   ]
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      "text/plain": [
       "(2.5495097567963927,\n",
       " Vector3d(-0.588348, 0.196116, 0.784465),\n",
       " Vector3d(-0.57735, 0.57735, -0.57735),\n",
       " Vector3d(-0.566139, -0.792594, -0.226455),\n",
       " Vector3d(1.33333, 1.66667, 0.333333))"
      ]
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   "source": [
    "l = [parabem.PanelVector3(1, 2, 0), \n",
    "     parabem.PanelVector3(3, 4, 1), \n",
    "     parabem.PanelVector3(0, -1, 0)]\n",
    "p = parabem.Panel3(l)\n",
    "p.area, p.n, p.l, p.m, p.center"
   ]
  },
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   "source": [
    "## Einflussfunktionen\n",
    "\n",
    "Die Einflussfunktionen stellen die Kernfunktionen für die Panel Methode dar. Sie sind alle Lösungen der Laplace Gleichung und werden in Potential- und Geschwindigkeitsfunktionen (**v**) unterschieden. Die ersten zwei Argumente für diese Funktionen sind der Zielpunkt (**target**) und das Störungsobjekt (**source**).\n",
    "\n",
    "\n",
    "Die Einflussfunktionen können aus dem Paket pan2d bzw. pan3d geladen werden.\n"
   ]
  },
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       "    </rdf:RDF>\n",
       "  </metadata>\n",
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       "    <path d=\"m 677.58854,141.54559 11.28776,-49.40701 24.42608,0\" id=\"path3447\" inkscape:connector-curvature=\"0\" sodipodi:nodetypes=\"ccc\" style=\"fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.61065191px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#DotL);marker-end:url(#Arrow1Lend)\"/>\n",
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       "    <path d=\"m 615.66007,141.66803 18.25756,-73.955528 24.42607,0\" id=\"path3460\" inkscape:connector-curvature=\"0\" sodipodi:nodetypes=\"ccc\" style=\"fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.61065191px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#marker5410);marker-end:url(#marker4618)\"/>\n",
       "    <text id=\"text3462\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"575.46948\" xml:space=\"preserve\" y=\"24.966869\"><tspan id=\"tspan3464\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"575.46948\" y=\"24.966869\">Bezeichnung des Elements</tspan></text>\n",
       "    <text id=\"text3462-7\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"636.09851\" xml:space=\"preserve\" y=\"48.956791\"><tspan id=\"tspan3464-1\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"636.09851\" y=\"48.956791\">Dimension (2d, 3d)</tspan></text>\n",
       "    <text id=\"text3462-7-4\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"660.08838\" xml:space=\"preserve\" y=\"73.382866\"><tspan id=\"tspan3464-1-0\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"660.08838\" y=\"73.382866\">Ordnung des Elements (0=konstant)</tspan></text>\n",
       "    <text id=\"text3462-7-4-9\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"717.22797\" xml:space=\"preserve\" y=\"97.372765\"><tspan id=\"tspan3464-1-0-7\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"717.22797\" y=\"97.372765\">Integrationsschema, Berechnung</tspan></text>\n",
       "    <text id=\"text3462-7-4-9-1\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"770.44196\" xml:space=\"preserve\" y=\"120.92648\"><tspan id=\"tspan3464-1-0-7-8\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"770.44196\" y=\"120.92648\">Geschwindigkeitsformulierung</tspan></text>\n",
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       "<IPython.core.display.SVG object>"
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SVG(filename='tutorial_files/kernfunktionen_bezeichnung.svg') "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.128074999682\n",
      "-0.0636619772368\n",
      "-0.176208191175\n",
      "Vector2d(0.127324, 0.063662)\n"
     ]
    }
   ],
   "source": [
    "import parabem.pan2d as pan2d\n",
    "\n",
    "target = parabem.Vector2(1, 1)\n",
    "source_point = parabem.PanelVector2(-1, 0)\n",
    "source_point_1 = parabem.PanelVector2(1, 0)\n",
    "source_panel = parabem.Panel2([source_point, source_point_1])\n",
    "\n",
    "print(pan2d.source_2(target, source_point))\n",
    "print(pan2d.doublet_2(target, source_point, parabem.Vector2(1, 0)))\n",
    "print(pan2d.doublet_2_0(target, source_panel))\n",
    "print(pan2d.doublet_2_0_v(target, source_panel))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Case\n",
    "Der **Case** stellt einen \"Berechnungsfall\" dar in dem eine Rand-Integral-Gleichung aufgestellt und das daraus resultierende Gleichungssystem gelöst wird. Die berechneten Werte werden am Panel gesetzt und die Kräfte und Momente werden aufsummiert.\n",
    "\n",
    "Die Berechnungsfälle können aus dem Paket pan2d bzw. pan3d geladen werden."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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       "        <dc:title/>\n",
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       "  </metadata>\n",
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       "    <text id=\"text3336\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"472.31497\" xml:space=\"preserve\" y=\"166.68758\"><tspan id=\"tspan3338\" sodipodi:role=\"line\" x=\"472.31497\" y=\"166.68758\">DirichletDoublet0Source0Case3</tspan></text>\n",
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       "    <path d=\"m 622.44111,140.99073 24.42608,-97.704302 24.42608,0\" id=\"path3443\" inkscape:connector-curvature=\"0\" style=\"fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.61065191px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#marker5420);marker-end:url(#marker4640)\"/>\n",
       "    <path d=\"m 734.01711,141.54559 11.28776,-49.40701 24.42608,0\" id=\"path3447\" inkscape:connector-curvature=\"0\" sodipodi:nodetypes=\"ccc\" style=\"fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.61065191px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#DotL);marker-end:url(#Arrow1Lend)\"/>\n",
       "    <path d=\"m 824.82685,141.47076 6.10652,-24.98094 24.42607,0\" id=\"path3449\" inkscape:connector-curvature=\"0\" sodipodi:nodetypes=\"ccc\" style=\"fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.61065191px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#marker5662);marker-end:url(#marker4702)\"/>\n",
       "    <path d=\"m 680.64775,141.66803 18.25756,-73.955528 24.42607,0\" id=\"path3460\" inkscape:connector-curvature=\"0\" sodipodi:nodetypes=\"ccc\" style=\"fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:0.61065191px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-start:url(#marker5410);marker-end:url(#marker4618)\"/>\n",
       "    <text id=\"text3462\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"575.46948\" xml:space=\"preserve\" y=\"24.966869\"><tspan id=\"tspan3464\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"575.46948\" y=\"24.966869\">Einbringung der Randbedingung am umströmten Körper</tspan></text>\n",
       "    <text id=\"text3462-7\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"673.47418\" xml:space=\"preserve\" y=\"48.956791\"><tspan id=\"tspan3464-1\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"673.47418\" y=\"48.956791\">Verwendetes Element</tspan></text>\n",
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       "    <text id=\"text3462-7-4-9\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"773.65649\" xml:space=\"preserve\" y=\"97.372765\"><tspan id=\"tspan3464-1-0-7\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"773.65649\" y=\"97.372765\">weitere Elemente</tspan></text>\n",
       "    <text id=\"text3462-7-4-9-1\" sodipodi:linespacing=\"125%\" style=\"font-style:normal;font-weight:normal;font-size:24.42607689px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1\" x=\"859.72766\" xml:space=\"preserve\" y=\"120.92648\"><tspan id=\"tspan3464-1-0-7-8\" sodipodi:role=\"line\" style=\"font-size:15.26629829px\" x=\"859.72766\" y=\"120.92648\">3d oder 2d Anwendung</tspan></text>\n",
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    "SVG(filename='tutorial_files/Case_bezeichnung.svg') "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Zylinderumströmung mittels Neumannrandbedingung und konstanten Quellenpanels"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "* Erstellen der Geometrie (numpy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
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   "source": [
    "phi = np.linspace(0, 2 * np.pi, 30 + 1)\n",
    "x = np.cos(phi)[:-1]\n",
    "y = np.sin(phi)[:-1]\n",
    "xy = np.transpose(np.array([x, y]))\n",
    "\n",
    "plt.axes().set_aspect(\"equal\", \"datalim\")\n",
    "plt.plot(list(x) + [x[0]], list(y) + [y[0]], marker=\"x\", c=\"black\")\n",
    "plt.grid()\n",
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  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "* Erstellen der **Panels** aus den Geometriedaten (parabem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
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   "source": [
    "points = [parabem.PanelVector2(x, y) for x, y in xy]\n",
    "points += [points[0]]\n",
    "panels = [parabem.Panel2([point, points[i+1]]) \n",
    "          for i, point in enumerate(points[:-1])]"
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  {
   "cell_type": "markdown",
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    "* Erstellen eines **Case**"
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   "execution_count": 15,
   "metadata": {
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   "source": [
    "case = pan2d.NeumannDoublet0Case2(panels)\n",
    "case.v_inf = parabem.Vector2(1, 0)\n",
    "case.run()"
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   "source": [
    "* Visualisierung der berechneten Werte"
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   "execution_count": 16,
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   "source": [
    "print(\"lift: \", case.cl)     # kein Auftrieb, weil kein Nachlauf definiert ist\n",
    "plt.plot([p.cp for p in panels], c=\"g\")\n",
    "plt.ylabel(\"$cp$\")\n",
    "plt.xlabel(\"$nr$\")\n",
    "plt.show()"
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  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "nx = 200\n",
    "ny = 200\n",
    "space_x = np.linspace(-2, 2, nx)\n",
    "space_y = np.linspace(-2, 2, ny)\n",
    "grid = [parabem.Vector2(x, y) for y in space_y for x in space_x]\n",
    "\n",
    "velocity = list(map(case.off_body_velocity, grid))\n",
    "pot = list(map(case.off_body_potential, grid))\n",
    "\n",
    "file_name = check_path(\"/tmp/parabem_results/cylinder/field.vtk\")\n",
    "with open(file_name, \"w\") as _file:\n",
    "    writer = VtkWriter()\n",
    "    writer.structed_grid(_file, \"cylinder\", [nx, ny, 1])\n",
    "    writer.points(_file, grid)\n",
    "    writer.data(_file, velocity, name=\"velocity\",\n",
    "                _type=\"VECTORS\", data_type=\"POINT_DATA\")\n",
    "    writer.data(_file, pot, name=\"pot\", \n",
    "                _type=\"SCALARS\", data_type=\"POINT_DATA\")\n",
    "\n",
    "paraview(file_name)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Zylinder Umströmung durch Überlagerung eines Dipols mit einer Parallel-Strömung"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from parabem.pan2d import doublet_2, doublet_2_v, vortex_2, vortex_2_v\n",
    "\n",
    "\n",
    "source = parabem.Vector2(0, 0)                 # center of the circle\n",
    "\n",
    "def cylinder_field(target, circulation=0, r=1, v_inf=parabem.Vector2(1, 0)):\n",
    "    \n",
    "    direction = parabem.Vector2(-1, 0)         # direction of doublet (-v_inf)\n",
    "    mu = v_inf.norm() * 2 * np.pi * r**2   # solve mu * doublet_2_v(t, s) + v_v_inf == 0\n",
    "    return (\n",
    "            # potential influence\n",
    "            mu * doublet_2(target, source, -v_inf) + \n",
    "            v_inf.dot(target) + vortex_2(target, source, direction) * circulation,\n",
    "            \n",
    "            # velocity influence\n",
    "            mu * doublet_2_v(target, source, -v_inf) + \n",
    "            v_inf + vortex_2_v(target, source) * circulation\n",
    "        )\n",
    "def cp(velocity, v_inf=parabem.Vector2(1, 0)):\n",
    "    return 1 - velocity.dot(velocity) / v_inf.dot(v_inf)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Druckverteilung an der Oberfläche"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "slideshow": {
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "phi = np.linspace(0, np.pi * 2, 100)\n",
    "x = list(np.cos(phi) + source.x)\n",
    "y = list(np.sin(phi) + source.y)\n",
    "xy = list(zip(x, y))\n",
    "pot, vel = zip(*[cylinder_field(parabem.Vector2(xi, yi)) for xi, yi in xy])\n",
    "_cp = list(map(cp, vel))\n",
    "vel = [v.norm() for v in vel]\n",
    "\n",
    "plt.axes().set_aspect(\"equal\", \"datalim\")\n",
    "plt.grid()\n",
    "plt.plot(x, y, label=\"surface\", color=\"black\")\n",
    "plt.plot(x, pot, label=\"$\\\\frac{\\phi}{r_K u_{\\infty}}$\", color=\"b\")\n",
    "plt.plot(x, vel, label=\"$\\\\frac{u}{u_{\\infty}}$\", color=\"r\")\n",
    "plt.plot(x, _cp, label=\"$cp$\", color=\"g\")\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.xlim(-2, 3)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Potential- und Geschwindigkeitsverteilung im Raum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "x_grid = np.linspace(-2, 2, 100)\n",
    "y_grid = np.linspace(-2, 2, 100)\n",
    "grid = [parabem.Vector2(x, y) for x in x_grid for y in y_grid]\n",
    "pot, vel = zip(*[cylinder_field(point) for point in grid])\n",
    "writer = VtkWriter()\n",
    "\n",
    "filename = check_path(\"/tmp/parabem_results/cylinder.vtk\")\n",
    "with open(filename, \"w\") as _file:\n",
    "    writer = VtkWriter()\n",
    "    writer.structed_grid(_file, \"z_plane\", [100, 100, 1])\n",
    "    writer.points(_file, grid)\n",
    "    writer.data(_file, pot, name=\"potential\", \n",
    "                _type=\"SCALARS\", data_type=\"POINT_DATA\")\n",
    "    writer.data(_file, vel, name=\"velocity\", \n",
    "                _type=\"VECTORS\", data_type=\"POINT_DATA\")\n",
    "paraview(filename)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Joukowsky Profil - konforme Abbildung\n",
    "## Druckverteilung an der Oberfläche eines Joukowsky Profil mittels konformer Abbildung."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "from parabem.airfoil.conformal_mapping import JoukowskyAirfoil\n",
    "airfoil = JoukowskyAirfoil(midpoint=-0.1 + 0.05j)\n",
    "\n",
    "alpha = np.deg2rad(3)\n",
    "\n",
    "vel = airfoil.surface_velocity(alpha, num=70)\n",
    "vel = np.sqrt(vel.imag ** 2 + vel.real ** 2)\n",
    "cp = airfoil.surface_cp(alpha, num=100)\n",
    "coordinates = airfoil.coordinates(100)\n",
    "\n",
    "plt.grid()\n",
    "plt.axes().set_aspect(\"equal\", \"datalim\")\n",
    "plt.plot(coordinates.real, coordinates.imag, \n",
    "         label=\"joukowsky -0.1 + 0.05j\", c=\"black\", marker=\"x\")\n",
    "plt.plot(coordinates.real, cp, label=\"$cp$\", c=\"g\")\n",
    "plt.legend()\n",
    "\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Joukowsky Profil - 2D Panelmethode\n",
    "## Druckverteilung an der Oberfläche"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
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   ],
   "source": [
    "# translate the complex coordinates to (x, y) coordinates\n",
    "coordiantes = list(zip(\n",
    "        airfoil.coordinates(num=70).real, \n",
    "        airfoil.coordinates(num=70).imag))\n",
    "vertices = [parabem.PanelVector2(*v) for v in coordiantes[:-1]]\n",
    "vertices[0].wake_vertex = True\n",
    "panels = [parabem.Panel2([vertices[i], \n",
    "                      vertices[i + 1]]) \n",
    "          for i in range(len(vertices[:-1]))]\n",
    "panels.append(parabem.Panel2([vertices[-1], vertices[0]]))\n",
    "\n",
    "case = pan2d.DirichletDoublet0Source0Case2(panels)\n",
    "case.v_inf = parabem.Vector2(np.cos(alpha), np.sin(alpha))\n",
    "case.run()\n",
    "\n",
    "pan_center_x = [panel.center.x for panel in panels]\n",
    "pan_vel = [panel.velocity.norm() for panel in panels]\n",
    "pan_cp = [panel.cp for panel in panels]\n",
    "\n",
    "plt.grid()\n",
    "plt.axes().set_aspect(\"equal\", \"datalim\")\n",
    "plt.plot(coordinates.real, coordinates.imag, \n",
    "         label=\"joukowsky -0.1 + 0.05j\", c=\"black\", marker=\"x\")\n",
    "plt.plot(pan_center_x, pan_cp, label=\"$cp$\", c=\"g\")\n",
    "plt.legend()\n",
    "\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Potential- und Geschwindigkeitsverteilung im Raum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "nx = 200\n",
    "ny = 200\n",
    "space_x = np.linspace(-3, 3, nx)\n",
    "space_y = np.linspace(-1, 1, ny)\n",
    "grid = [parabem.Vector2(x, y) for y in space_y for x in space_x]\n",
    "\n",
    "velocity = list(map(case.off_body_velocity, grid))\n",
    "pot = list(map(case.off_body_potential, grid))\n",
    "\n",
    "file_name = check_path(\"/tmp/parabem_results/airfoil_2d_linear/field.vtk\")\n",
    "with open(file_name, \"w\") as _file:\n",
    "    writer = VtkWriter()\n",
    "    writer.structed_grid(_file, \"airfoil\", [nx, ny, 1])\n",
    "    writer.points(_file, grid)\n",
    "    writer.data(_file, velocity, name=\"velocity\", \n",
    "                _type=\"VECTORS\", data_type=\"POINT_DATA\")\n",
    "    writer.data(_file, pot, name=\"pot\", _type=\"SCALARS\", \n",
    "                data_type=\"POINT_DATA\")\n",
    "\n",
    "paraview(file_name)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Umströmung einer Kugel\n",
    "## Überlagerung eines Dipols und einer Parallelströmung"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from parabem.pan3d import doublet_3, doublet_3_v\n",
    "\n",
    "\n",
    "def sphere_field(target, r=1, v_inf=parabem.Vector3(1, 0, 0)):\n",
    "    source = parabem.Vector3(0, 0, 0)\n",
    "    mu = v_inf.norm() * np.pi * r**3 * 2\n",
    "    return (\n",
    "        mu * doublet_3(target, source, -v_inf) + v_inf.dot(target),\n",
    "        mu * doublet_3_v(target, source, -v_inf) + v_inf\n",
    "        )\n",
    "def cp_(velocity, v_inf=parabem.Vector3(1, 0, 0)):\n",
    "    return 1 - velocity.dot(velocity) / v_inf.dot(v_inf)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Potential- und Geschwindigkeitsverteilung im Raum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
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   "source": [
    "phi = np.linspace(0, np.pi * 2, 300)\n",
    "x = np.cos(phi)\n",
    "y = np.sin(phi)\n",
    "pot, vel = zip(*[sphere_field(\n",
    "            parabem.Vector3(\n",
    "                np.cos(p), \n",
    "                np.sin(p),\n",
    "                0)) for p in phi])\n",
    "cp = list(map(cp_, vel))\n",
    "vel = [v.norm() for v in vel]\n",
    "\n",
    "\n",
    "plt.plot(x, y, label=\"surface\", color=\"black\")\n",
    "plt.plot(x, pot, label=\"$\\\\frac{\\phi}{r_K u_{\\infty}}$\", color=\"b\")\n",
    "plt.plot(x, vel, label=\"$\\\\frac{u}{u_{\\infty}}$\", color=\"r\")\n",
    "plt.plot(x, cp, label=\"$cp$\", color=\"g\")\n",
    "\n",
    "\n",
    "plt.grid()\n",
    "plt.axes().set_aspect(\"equal\", \"datalim\")\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "nx, ny, nz = 30, 30, 30\n",
    "\n",
    "x_grid = np.linspace(-2, 2, nx)\n",
    "y_grid = np.linspace(-2, 2, ny)\n",
    "z_grid = np.linspace(-2, 2, nz)\n",
    "grid = [parabem.Vector3(x, y, z)for z in z_grid\n",
    "                            for y in y_grid \n",
    "                            for x in x_grid]\n",
    "pot, vel = zip(*[sphere_field(point) for point in grid])\n",
    "writer = VtkWriter()\n",
    "\n",
    "filename = check_path(\"/tmp/parabem_results/sphere.vtk\")\n",
    "with open(filename, \"w\") as _file:\n",
    "    writer = VtkWriter()\n",
    "    writer.structed_grid(_file, \"points\", [nx, ny, nz])\n",
    "    writer.points(_file, grid)\n",
    "    writer.data(_file, pot, name=\"potential\", \n",
    "                _type=\"SCALARS\", data_type=\"POINT_DATA\")\n",
    "    writer.data(_file, vel, name=\"velocity\", \n",
    "                _type=\"VECTORS\", data_type=\"POINT_DATA\")\n",
    "\n",
    "paraview(filename)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 3D Panelmethode Kugel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "from parabem.mesh import mesh_object\n",
    "from parabem.vtk_export import CaseToVTK\n",
    "\n",
    "# create panels from mesh\n",
    "mesh = mesh_object.from_OBJ(\"../../examples/mesh/sphere_low_tri.obj\")\n",
    "\n",
    "# create case from panels\n",
    "case = pan3d.DirichletDoublet0Case3(mesh.panels)\n",
    "\n",
    "# set boundary condition far away from the body\n",
    "case.v_inf = parabem.Vector3(1, 0, 0.)\n",
    "\n",
    "# solve for constant potential inside the object\n",
    "case.run()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
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    "center_x, surf_vel, surf_cp, surf_pot = [], [], [], []\n",
    "for panel in case.panels:\n",
    "    center_x.append(panel.center.x)\n",
    "    surf_vel.append(panel.velocity.norm())\n",
    "    surf_cp.append(panel.cp)\n",
    "    surf_pot.append(panel.potential)\n",
    "\n",
    "phi = np.linspace(0, np.pi * 2, 300)\n",
    "x = np.cos(phi)\n",
    "y = np.sin(phi)\n",
    "\n",
    "plt.plot(x, y, label=\"surface\", color=\"black\")\n",
    "plt.scatter(center_x, surf_pot, marker=4, color=\"b\", label=\"$\\\\frac{\\phi}{r_K u_{\\infty}}$\")\n",
    "plt.scatter(center_x, surf_vel, marker=\"+\", color=\"r\", label=\"$\\\\frac{u}{u_{\\infty}}$\")\n",
    "plt.scatter(center_x, surf_cp, marker=\"*\", color=\"g\", label=\"$cp$\")\n",
    "\n",
    "\n",
    "plt.grid()\n",
    "plt.axes().set_aspect(\"equal\", \"datalim\")\n",
    "plt.xlabel(\"$x$\", fontsize=15)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
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   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
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   },
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   "source": [
    "lin = np.linspace(-0.5, 0.5, 5)\n",
    "grid = [[-2, k, j] for j in lin for k in lin]\n",
    "\n",
    "file_name = \"/tmp/parabem_results/sphere_case\"\n",
    "vtk_writer = CaseToVTK(case, file_name)\n",
    "vtk_writer.write_panels(data_type=\"cell\")\n",
    "vtk_writer.write_field([-2, 2, 20], [-2, 2, 20], [-2, 2, 20])\n",
    "\n",
    "paraview(file_name + \"/panels.vtk\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Umströmung eines Gleitschirms\n",
    "## Erstellen des \"Case\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
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     "slide_type": "subslide"
    }
   },
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   "source": [
    "from openglider.jsonify import load\n",
    "from openglider.utils.distribution import Distribution\n",
    "from openglider.glider.in_out.export_3d import parabem_Panels\n",
    "\n",
    "from parabem.utils import v_inf_deg_range3\n",
    "\n",
    "\n",
    "# load glider\n",
    "file_name = \"../../examples/openglider/glider/referenz_schirm_berg.json\"\n",
    "with open(file_name) as _file:\n",
    "    parGlider = load(_file)[\"data\"]\n",
    "    parGlider.shape.set_const_cell_dist()\n",
    "    glider = parGlider.get_glider_3d()\n",
    "\n",
    "# create the panels and get the trailing edge\n",
    "_, panels, trailing_edge = parabem_Panels(\n",
    "    glider,\n",
    "    midribs=0,\n",
    "    profile_numpoints=50,\n",
    "    distribution=Distribution.nose_cos_distribution(0.2),\n",
    "    num_average=0,\n",
    "    symmetric=True)\n",
    "\n",
    "# setup the case with panels and trailing edge\n",
    "case = pan3d.DirichletDoublet0Source0Case3(panels, trailing_edge)\n",
    "\n",
    "# set the boundarycondition far away from the wing\n",
    "case.v_inf = parabem.Vector3(*parGlider.v_inf)\n",
    "\n",
    "# create the wake (length, number of wake panels per column)\n",
    "case.create_wake(length=10000, count=10)\n",
    "\n",
    "# set reference values\n",
    "case.mom_ref_point = parabem.Vector3(1.25, 0, -5)\n",
    "case.A_ref = parGlider.shape.area\n",
    "\n",
    "# set farfield-factor\n",
    "case.farfield = 5\n",
    "\n",
    "# chooce between \"on_body\" or \"trefftz\"\n",
    "case.drag_calc = \"on_body\"\n",
    "\n",
    "# if \"trefftz\" is used, this point represents the position of the trefftz-plane\n",
    "case.trefftz_cut_pos = case.v_inf * 100\n",
    "\n",
    "# run the case with fixed wake for different aoa (v_inf)\n",
    "polars = case.polars(v_inf_deg_range3(case.v_inf, -15, 15, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Polaren"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc63fdb48d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cL, cD, cP = [], [], []\n",
    "alpha = []\n",
    "for i in polars.values:\n",
    "    alpha.append(np.rad2deg(i.alpha))\n",
    "    cL.append(i.cL)\n",
    "    cD.append(i.cD)\n",
    "    cP.append(i.cP)\n",
    "    \n",
    "plt.ylabel(\"$\\\\alpha$\", fontsize=15)\n",
    "plt.plot(cL, alpha)\n",
    "plt.plot(cD, alpha)\n",
    "plt.plot(cP, alpha)\n",
    "plt.legend([\"$c_A$\", \"$c_{Wi}$\", \"$c_N$\"])\n",
    "\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Visualisierung"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "file_name = \"/tmp/parabem_results/vtk_glider_case\"\n",
    "vtk_writer = CaseToVTK(case, file_name)\n",
    "vtk_writer.write_panels(data_type=\"cell\")\n",
    "vtk_writer.write_wake_panels()\n",
    "vtk_writer.write_body_stream(panels, 100)\n",
    "paraview(file_name + \"/panels.vtk\")"
   ]
  }
 ],
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