06_MEF_Elemento_Triangular_3nodos.ipynb

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    " # Elemento triangular de 3 nodos en elasticidad plana\n",
    "\n",
    "Análisis de Estructuras. Grado en Ingeniería Civil. \n",
    "\n",
    "Autores: \n",
    "\n",
    "+ Alejandro E. Martínez-Castro, Esther Puertas García, Rafael Gallego.\n",
    "+ Departamento de Mecánica de Estructuras e Ingeniería Hidráulica. Universidad de Granada.\n",
    "\n",
    "\n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\"><img alt=\"Licencia de Creative Commons\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Cuadernos MEF: Elemento Triangular de 3 nodos</span> by <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">Alejandro Martínez Castro, Esther Puertas García, Rafael Gallego Sevilla</span> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional License</a>."
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    "## Formulación en un elemento\n",
    "\n",
    "La formulación del MEF para un elemento parte de la expresión del PTV. \n",
    "\n",
    "\\begin{equation}\n",
    "\\int_{\\Omega} \\boldsymbol{\\sigma}^T \\cdot \\boldsymbol{\\varepsilon}^* \\, d\\Omega = \\int_{\\Gamma} \\mathbf{t}^t \\cdot \\mathbf{u}^* \\,d \\Gamma + \\int_{\\Omega} \\mathbf{b}^T \\,\\cdot \\, \\mathbf{u}^* \\,d \\Omega + \\sum_{k=1}^n \\mathbf{F}_k^T \\,\\mathbf{u}_k^*\n",
    "\\end{equation}\n",
    "\n",
    "El siguiente paso consiste en plantear una discretización para el campo $\\mathbf{u}$, mediante funciones de forma, en función de un vector de grados de libertad $\\mathbf{a}$. \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{u} = \\mathbf{N}\\cdot \\mathbf{a}\n",
    "\\end{equation}\n",
    "\n",
    "Esta relación inducirá una relación para el vector $\\boldsymbol{\\varepsilon}$, \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\varepsilon} = \\mathbf{B}\\cdot \\mathbf{a}\n",
    "\\end{equation}"
   ]
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    "Eligiendo alternativamente $\\mathbf{u^*} = N_i(x,y)$ se obtendrá un conjunto de ecuaciones que formará el sistema matricial dentro de un elemento: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K}\\cdot \\mathbf{a} = \\int_{\\Gamma} \\mathbf{N}^T \\cdot \\mathbf{t} \\,d \\Gamma + \\int_{\\Omega} \\mathbf{N}^T \\,\\cdot \\, \\mathbf{b} \\,d \\Omega + \\mathbf{F_n} = \\mathbf{f}_t + \\mathbf{f}_b + \\mathbf{F_n}\n",
    "\\end{equation}\n",
    "\n",
    "Siendo $\\mathbf{F_n}$ el vector de fuerzas nodales, el cual se cancela entre elementos en el ensamblaje. Estas fuerzas nodales tienen dimensiones de fuerza por unidad de longitud (en $z$). \n",
    "\n",
    "Se ha asumido que los desplazamientos son continuos, y que las funciones de forma $N_i(x,y)$ toman valor $1$ en el nodo $i$ y cero en el resto. "
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    "\n",
    "\n",
    "## Discretización y funciones de forma. \n",
    "\n",
    "El elemento más sencillo para elasticidad es el elemento triangular de 3 nodos. \n",
    "\n",
    "Consideremos un elemento de 3 nodos: \n",
    "- Las coordenadas de los 3 vértices son $(x_1, y_1)$, $(x_2,y_2)$, $(x_3,y_3)$. \n",
    "- Los desplazamientos se interpolan a partir de $(u_1,v_1)$, $(u_2,v_2)$, $(u_3,v_3)$. \n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{pmatrix} u \\\\ v \\end{pmatrix} = \n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "N_1 & 0 & N_2 & 0 & N_3 & 0 \\\\\n",
    "0 & N_1 & 0 & N_2 & 0 & N_3 \n",
    "\\end{matrix}\n",
    "\\right] \\cdot \\begin{pmatrix} u_1 \\\\ v_1 \\\\ u_2 \\\\ v_2 \\\\ u_3 \\\\ v_3 \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Esta matriz genera la relación $\\mathbf{u} = \\mathbf{N}\\,\\cdot\\mathbf{a}$.\n",
    "\n",
    "Las funciones $N_1$, $N_2$ y $N_3$ pueden obtenerse de manera análoga a como se hizo en vigas. \n",
    "\n",
    "La base en este caso es la $\\mathcal{B} = \\lbrace 1, x, y \\rbrace$. "
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     "text": [
      "IPython console for SymPy 0.7.6.1 (Python 2.7.11-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://www.sympy.org\n"
     ]
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   "source": [
    "from sympy import init_session\n",
    "init_session()"
   ]
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      "text/latex": [
       "$$\\left[\\begin{matrix}1 & x & y\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[1  x  y]"
      ]
     },
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   ],
   "source": [
    "base = Matrix([[1,x,y]])\n",
    "base"
   ]
  },
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      "text/latex": [
       "$$\\left[\\begin{matrix}1 & x_{1} & y_{1}\\\\1 & x_{2} & y_{2}\\\\1 & x_{3} & y_{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  x₁  y₁⎤\n",
       "⎢         ⎥\n",
       "⎢1  x₂  y₂⎥\n",
       "⎢         ⎥\n",
       "⎣1  x₃  y₃⎦"
      ]
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   "source": [
    "x1,y1,x2,y2,x3,y3 = symbols('x1 y1 x2 y2 x3 y3')\n",
    "matM = Matrix.vstack(base.subs([(x,x1),(y,y1)]),base.subs([(x,x2),(y,y2)]),base.subs([(x,x3),(y,y3)]))\n",
    "matM"
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tI/wgKdczgaVFpKs9qINWH8IE3M1GLj4uSmO3j2BrBOumHD2yBo/d7hPNLjp9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\ntv8NoAy0WJlVhWELd/8/kzCGw8pcwB8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{x_{1}}{- x_{1} + x_{2}} - \\frac{\\left(-1 + \\frac{- x_{1} + x_{3}}{- x_{1} + x_{2}}\\right) \\left(x_{1} - x_{2}\\right) \\left(- \\frac{x_{1} \\left(- y_{1} + y_{2}\\right)}{- x_{1} + x_{2}} + y_{1}\\right)}{\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)} + 1 & - \\frac{x_{1}}{- x_{1} + x_{2}} + \\frac{\\left(- x_{1} + x_{3}\\right) \\left(x_{1} - x_{2}\\right) \\left(- \\frac{x_{1} \\left(- y_{1} + y_{2}\\right)}{- x_{1} + x_{2}} + y_{1}\\right)}{\\left(- x_{1} + x_{2}\\right) \\left(\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)\\right)} & - \\frac{\\left(x_{1} - x_{2}\\right) \\left(- \\frac{x_{1} \\left(- y_{1} + y_{2}\\right)}{- x_{1} + x_{2}} + y_{1}\\right)}{\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)}\\\\- \\frac{\\left(-1 + \\frac{- x_{1} + x_{3}}{- x_{1} + x_{2}}\\right) \\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{2}\\right)}{\\left(- x_{1} + x_{2}\\right) \\left(\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)\\right)} - \\frac{1}{- x_{1} + x_{2}} & \\frac{1}{- x_{1} + x_{2}} + \\frac{\\left(- x_{1} + x_{3}\\right) \\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{2}\\right)}{\\left(- x_{1} + x_{2}\\right)^{2} \\left(\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)\\right)} & - \\frac{\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{2}\\right)}{\\left(- x_{1} + x_{2}\\right) \\left(\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)\\right)}\\\\\\frac{\\left(-1 + \\frac{- x_{1} + x_{3}}{- x_{1} + x_{2}}\\right) \\left(x_{1} - x_{2}\\right)}{\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)} & - \\frac{\\left(- x_{1} + x_{3}\\right) \\left(x_{1} - x_{2}\\right)}{\\left(- x_{1} + x_{2}\\right) \\left(\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)\\right)} & \\frac{x_{1} - x_{2}}{\\left(x_{1} - x_{2}\\right) \\left(- y_{1} + y_{3}\\right) + \\left(x_{1} - x_{3}\\right) \\left(y_{1} - y_{2}\\right)}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡             ⎛     -x₁ + x₃⎞           ⎛  x₁⋅(-y₁ + y₂)     ⎞                \n",
       "⎢             ⎜-1 + ────────⎟⋅(x₁ - x₂)⋅⎜- ───────────── + y₁⎟                \n",
       "⎢     x₁      ⎝     -x₁ + x₂⎠           ⎝     -x₁ + x₂       ⎠              x₁\n",
       "⎢  ──────── - ──────────────────────────────────────────────── + 1     - ─────\n",
       "⎢  -x₁ + x₂      (x₁ - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂)              -x₁ +\n",
       "⎢                                                                             \n",
       "⎢            ⎛     -x₁ + x₃⎞                                                  \n",
       "⎢            ⎜-1 + ────────⎟⋅(x₁ - x₂)⋅(-y₁ + y₂)                             \n",
       "⎢            ⎝     -x₁ + x₂⎠                                    1         1   \n",
       "⎢- ─────────────────────────────────────────────────────── - ────────  ───────\n",
       "⎢  (-x₁ + x₂)⋅((x₁ - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂))   -x₁ + x₂  -x₁ + x\n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                     ⎛     -x₁ + x₃⎞                                         \n",
       "⎢                     ⎜-1 + ────────⎟⋅(x₁ - x₂)                               \n",
       "⎢                     ⎝     -x₁ + x₂⎠                                         \n",
       "⎢             ──────────────────────────────────────────                     ─\n",
       "⎣             (x₁ - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂)                     (\n",
       "\n",
       "                                 ⎛  x₁⋅(-y₁ + y₂)     ⎞                       \n",
       "            (-x₁ + x₃)⋅(x₁ - x₂)⋅⎜- ───────────── + y₁⎟                  -(x₁ \n",
       "                                 ⎝     -x₁ + x₂       ⎠                       \n",
       "─── + ───────────────────────────────────────────────────────        ─────────\n",
       " x₂   (-x₁ + x₂)⋅((x₁ - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂))        (x₁ - x₂)\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                (-x₁ + x₃)⋅(x₁ - x₂)⋅(-y₁ + y₂)                               \n",
       "─ + ────────────────────────────────────────────────────────   ───────────────\n",
       "₂             2                                                (-x₁ + x₂)⋅((x₁\n",
       "    (-x₁ + x₂) ⋅((x₁ - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂))                  \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                -(-x₁ + x₃)⋅(x₁ - x₂)                                         \n",
       "──────────────────────────────────────────────────────               ─────────\n",
       "-x₁ + x₂)⋅((x₁ - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂))               (x₁ - x₂)\n",
       "\n",
       "      ⎛  x₁⋅(-y₁ + y₂)     ⎞            ⎤\n",
       "- x₂)⋅⎜- ───────────── + y₁⎟            ⎥\n",
       "      ⎝     -x₁ + x₂       ⎠            ⎥\n",
       "─────────────────────────────────       ⎥\n",
       "⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂)       ⎥\n",
       "                                        ⎥\n",
       "                                        ⎥\n",
       "                                        ⎥\n",
       "  -(x₁ - x₂)⋅(-y₁ + y₂)                 ⎥\n",
       "────────────────────────────────────────⎥\n",
       " - x₂)⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂))⎥\n",
       "                                        ⎥\n",
       "                                        ⎥\n",
       "                                        ⎥\n",
       "                                        ⎥\n",
       "        x₁ - x₂                         ⎥\n",
       "─────────────────────────────────       ⎥\n",
       "⋅(-y₁ + y₃) + (x₁ - x₃)⋅(y₁ - y₂)       ⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Inv_matM = matM.inv()\n",
    "Inv_matM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "40b38f63-46d3-4d03-be49-7ba1d86ee091"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Esta expresión resulta más sencilla si se obtiene la inversa mediante otro procedimiento. La matriz inversa puede calcularse como la matriz de los adjuntos dividida entre el determinante. \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{M}^{-1} = \\dfrac{\\mathrm{adj} \\mathbf{M}^T}{|\\mathbf{M}|}\n",
    "\\end{equation}\n",
    "\n",
    "Esto puede hacerse en SymPy como sigue: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "nbpresent": {
     "id": "460304fc-805a-47ab-b6ac-1d7324c93cdd"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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SO2EVbCdAtyqKLC4X33Bf+dfIugQWDU/drrOLoXA4m0bMYTvs/FftgLY9Hgl1\nngHcMV/i8CU4X4Dm2TRCDttR64JO2wEWahLSXIjZaL4A5wuwPJ9FSGI77LdWdN+GO2hZ+RH2qIxX\nRBfmXb70Vx5dn2TvG9fS6CMYDuAit/FuyC5FPf16zn9WcypoW+N15t4opNlz7h/v67Al5z6K0uz7\nCLuBca10fQjloDqsRUVEW9At6gBj4TmiuRASmPsybFrFXRjDsuuTaPWNa6XvI+CsqtJfC4ugb67o\nvg3X3bkB9tNbuhly7aML86oWHQWoXeNqGl2EyXDgzKthGYHvyMxoO+IVKR5FNGPP4dG+Dls491GU\nZt9HCPaNq2n0IZQDpViNyoi6oNvWu4KMRgHNkXvH1pdhE+cujGHZ9UnUusbV0ncR5B5sb1sNq7qV\nr7EVsB0AChWPLM1Lb7gfWz7B9XKB8Acvcfs+4ts1rqbRRRBwcgCraTUsI3B+zmg74hWpMwpodjxH\nh/s6bOHcR1E1Bz6Dgq6XfoDvpdqSXJEP3zKdwXYAjMqQbZ7mYoh3GMmw4RztwyjLvo/Q6hpXK9dF\nSOjuHF0Nq5J9a0XbDdd1qHx7T3+Oqvnjw9TAuc8/QE/t8RCFxfDWEy+UMZX3MdV0xkJDRZVRyykb\nAR9YlQNhAtgQlQH6LfCMFsS7iV0uBkx+QnPlCGhWES1Bj5Dc+zrMnFuUgIhBCeYgTcwU+CgNbwyk\nc5xy3sB3wsqfq+Wqyp7yX4AaxME7P+P0nMHWAPhkLKC3eprW3/PzCOI/kOEyFVWWmGrIwxnXKJfz\nBrxP/Lsq6haaS8YWKDCqVu0vzXyHyvdJOm09/ZMeL+/65gEN6I5WpE/lTw23vWhDFC4KWLUt5gv/\n6CCm8j5mVcHY0LAypD+ANPfH1VkMAq2ap3IATskOL9NzEGfNuuEmtCDeAJTGvMzFCk5MWDUJT5PO\naUCCiACCJprcIDPdayn1MFiFMECBEXPKBKQ58FEaAACLSDpyMrOEmYCyckgZp8e4IIybncv/2HBz\nQmOAL6moT5tiWcKQme6NlHocUUMYGFVrFpRrx/uIZOABI2gMldNZDIJnbFYVSjSErZc8QvhDaK+i\nOjF6ZzMZaEcsGcBoklCt2g1XrtD/y7cIoEPl39fUS/b0kYRb0VX2bdaXMCUqRkExaMWkMhXqNPCR\n3IoXAdARVcNVCXLSWRRhorUwFpMH8UeiOJ0yjXB+ptgg3gDMjXk1Fys4MGnVJCJQuMszEysRRNBE\nk3dRUt1rKYmCqBimoNCoLDX3gQ9pEAAsNFxTIyczS5gJhXRSuQMST1ikqjPmkS1oRMsAfElFyQ9p\nq1hMNJQBjZ/npAmDNIdo1z8AAAAF90lEQVQwNGJSu5h3VXQovc4SZqJ5OhR3QABcqrmO+t9SRTW+\nSEg1oJ1iTRONJgnDqv1IAW1itZ2E/Jn16fk0oSll3LISrUjfeGU6NzBF1NSgNM1pi1UnzW9U9VSN\nT2hsaJzXfDNzqCDm7KpDInWTaWNGNRNaUafxKADg7qSq3RsZ4qs0uwBAkOKOSwF30GFFmzRbmmE9\nOOdMc+gTGsGiSFdLAaqcpYaAmWswX9BWEMOCIK6K0HfQvDyGACTfKl4jbq9o3RxYTs8tpyhWoZdn\nCBOzrBVvfEIjRBkqh4rWCJ7xl1W0rhAmBneneO3eyCAnY14r80npNlzXofK33JbxTzZ5XWq8EElQ\n8p2YZCytSOdb2sxxREUovKyaaEUPC57mAx+Zpby3Sn0KyZaG+OgDXFc231QOgGB2OJCeicqLWWvl\nwYQWxhMA3VyRC6Sr4Jgpk/A06wDiIyLEp5J0b6R0aUYwRKERBdXTfODTL2gsHTlxFkmd+MyEQjqp\nxgVhXCUo3kEz2Bjgiyrq0qbaXG2RDGADzoRB1BCGRmitYlJx52PLAWOzripttWSYxSKoFYx1VQEG\nyeH1/OxSrczLFUU8JNSzkEwMIjLFCccrreRTb7hRh8rfT9Nb/qahaUpZJLbucyvS1/xRuD1eoiIU\nPTdohc5/+PUJrSZ4wEfYe6OlYTSQK/3P6V3IZVyPYK3IUzkAaM4Or9JzFGftcs7+m18nNB9fAZRu\nrhQDRCpAWpGEp2n9LcESQQSLP66oRUFUBOPrAZYTaQ58+gUNpLOcOIukHmQCyuQAhRYKgji452db\n0IDWV1c0SntQCute1ldO2h5HmgMYuaRpTz8Vc1dFF6Q3FR0UVFcVSrQAi1ThvqaiVqoioaoRATqp\nLM16w406VL49nz7kp6uqKWXRQN3ZijTfzqDHGRWh4NxI7wDzHHxPeObdzqym8xFOrbGmUalqOHEW\nRTBWds9VDgRqbtawzViZKZ3T4O+v35/yeYw8MpqLl6jT2/tJPpafyB2JkkgK5wNWJhHQpHNFEBFA\nqPDHFVWWmmYE09YDPxoKIdIc+AwK2t4nE2aW83aZKGVyoECjgmgc3V1BPa2vrqgpBQrK08dW1MnA\n9ZU5GxhGbauoirmvokPpmZpI7zIxhVESqNEQlqnCW56bU9RXVKVSCSOpiAmjJmFo1hsuY2RQ3mNe\n5aa9D/k92squsqf6W7mIClEg5QQrJ02fGJcHqul9xKExakfUhgbAyrN/+1QHMJbfHupBjAawGgfn\n+jmjjeLlSlhuBhGhI8GJBWskFZ06gzltINSEoSRCe1IyKoQBCozKUtUc+AwKOg2kM7MIfeAjE7Ny\nfUVHqEyVSNVgBhsAUEKoEQPCqll4mtXE+mLfKdpISVYgwgNpIqgJY8Sy7+MBKErDQpOaR+tOUV1V\njB8UxKwEutvBUkWVO9SopAIUjCqVpdnbcNmhUj4BfktMqqaUrmWltiKF+vP0iApRJsDAyklfdNvu\n+qQJWmOPxkym/M9ZLAI9wHgyHGjk+tMjGDEOB+rnglarU7mAO8SIAWFlEiHNCri8QAQQanwoyUjQ\n0XMumxgVwgAFRswpn6ywon0fmaA1KouB9DpLogiITDf9R8rKgbYBqsaptxkB7HoVZdpQm4kmmk6G\nnpSMGsLAyEmRv5nK+VgeMCqNgXKcxSKkcX6QsVlVsDXLVQ+nkQbWx+dXyKjPi9yRTAgIoyYB5DTP\nvOHmjme8waBi8/Jn2x+1feOZZWHGKK31NaDS+lh0Eb4lGdOogsYvIg7y2UeY3RgpWYG2It7nYuFb\nK4Ctz3jcIoy9xRpzHsI4Y0DT+VgigTGmYYMWxgGHXnILSGIm2ApaQTJmgtZKZOMzHrYIY2+xxpyH\nMK0xYtn6VDy8MWZRBS28iEh0kltAEjPBVvDyyRh8ZwRy2p7lIVOd8oMfm5rgHcOXfEfDjkANSR9p\nnvs4l0aHw05You2M76pB4K7H+YYLcL4EzXNpxBx2oirYToBuWRS563K24XzOl2B5NouYxE5YBdsJ\n0K0KkP/M++z0f7VgVSW0fjrAAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{x_{2} y_{3} - x_{3} y_{2}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)} & \\frac{- x_{1} y_{3} + x_{3} y_{1}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)} & \\frac{x_{1} y_{2} - x_{2} y_{1}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)}\\\\\\frac{y_{2} - y_{3}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)} & \\frac{- y_{1} + y_{3}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)} & \\frac{y_{1} - y_{2}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)}\\\\\\frac{- x_{2} + x_{3}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)} & \\frac{x_{1} - x_{3}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)} & \\frac{- x_{1} + x_{2}}{x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                            x₂⋅y₃ - x₃⋅y₂                                    \n",
       "⎢──────────────────────────────────────────────────────────────────────  ─────\n",
       "⎢x₁⋅y₂ + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)  x₁⋅y₂\n",
       "⎢                                                                             \n",
       "⎢                               y₂ - y₃                                       \n",
       "⎢──────────────────────────────────────────────────────────────────────  ─────\n",
       "⎢x₁⋅y₂ + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)  x₁⋅y₂\n",
       "⎢                                                                             \n",
       "⎢                               -x₂ + x₃                                      \n",
       "⎢──────────────────────────────────────────────────────────────────────  ─────\n",
       "⎣x₁⋅y₂ + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)  x₁⋅y₂\n",
       "\n",
       "                       -x₁⋅y₃ + x₃⋅y₁                                         \n",
       "─────────────────────────────────────────────────────────────────  ───────────\n",
       " + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)  x₁⋅y₂ + x₃⋅\n",
       "                                                                              \n",
       "                          -y₁ + y₃                                            \n",
       "─────────────────────────────────────────────────────────────────  ───────────\n",
       " + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)  x₁⋅y₂ + x₃⋅\n",
       "                                                                              \n",
       "                          x₁ - x₃                                             \n",
       "─────────────────────────────────────────────────────────────────  ───────────\n",
       " + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)  x₁⋅y₂ + x₃⋅\n",
       "\n",
       "                 x₁⋅y₂ - x₂⋅y₁                             ⎤\n",
       "───────────────────────────────────────────────────────────⎥\n",
       "(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)⎥\n",
       "                                                           ⎥\n",
       "                    y₁ - y₂                                ⎥\n",
       "───────────────────────────────────────────────────────────⎥\n",
       "(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)⎥\n",
       "                                                           ⎥\n",
       "                    -x₁ + x₂                               ⎥\n",
       "───────────────────────────────────────────────────────────⎥\n",
       "(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matM.inverse_ADJ()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "93eab622-5673-4585-8f49-2cd39ee188a0"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La función _adjugate_ obtiene la matriz de adjuntos traspuesta. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "nbpresent": {
     "id": "29afc9fe-03a5-4cf5-9f8d-a8b23cdd2d0e"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAABLCAMAAABZYzCHAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nXxSOo/gXjO5zxys1qk8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}x_{2} y_{3} - x_{3} y_{2} & - x_{1} y_{3} + x_{3} y_{1} & x_{1} y_{2} - x_{2} y_{1}\\\\y_{2} - y_{3} & - y_{1} + y_{3} & y_{1} - y_{2}\\\\- x_{2} + x_{3} & x_{1} - x_{3} & - x_{1} + x_{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡x₂⋅y₃ - x₃⋅y₂  -x₁⋅y₃ + x₃⋅y₁  x₁⋅y₂ - x₂⋅y₁⎤\n",
       "⎢                                            ⎥\n",
       "⎢   y₂ - y₃        -y₁ + y₃        y₁ - y₂   ⎥\n",
       "⎢                                            ⎥\n",
       "⎣  -x₂ + x₃        x₁ - x₃        -x₁ + x₂   ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matM.adjugate()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "ad95e55f-7d67-4ccc-9d65-c1823345463b"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Observe que en la expresión de la inversa aparece un denominador comun"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "nbpresent": {
     "id": "e77df9bf-f721-4b80-a170-7f280f5a6a0c"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAmsAAAAVBAMAAADr1bRPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gvlrfzog0CwEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x_{1} y_{2} + x_{3} \\left(x_{2} y_{2} + y_{1}\\right) + y_{1} + y_{3} \\left(- x_{1} + x_{2}\\right) - \\left(- x_{2} - 1\\right) \\left(- x_{3} y_{2} - y_{1}\\right)$$"
      ],
      "text/plain": [
       "x₁⋅y₂ + x₃⋅(x₂⋅y₂ + y₁) + y₁ + y₃⋅(-x₁ + x₂) - (-x₂ - 1)⋅(-x₃⋅y₂ - y₁)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denominador = matM.adjugate()[0,0]/matM.inverse_ADJ()[0,0]\n",
    "denominador "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "df7e7740-3203-49aa-9004-888cca9bc843"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Simplificando"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "nbpresent": {
     "id": "72869c8b-6fe0-40ad-864c-571b97a06250"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAAQBAMAAAD5QenfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYFmd+VhQTZT+J/AXVrA7ZlZYT8YAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x_{1} y_{2} - x_{1} y_{3} - x_{2} y_{1} + x_{2} y_{3} + x_{3} y_{1} - x_{3} y_{2}$$"
      ],
      "text/plain": [
       "x₁⋅y₂ - x₁⋅y₃ - x₂⋅y₁ + x₂⋅y₃ + x₃⋅y₁ - x₃⋅y₂"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denominador.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "cd2917c1-1cfd-4662-8ee9-3976449ac789"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Que coincide con el determinante de la matriz $\\mathbf{M}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "nbpresent": {
     "id": "36f1abab-0a55-46c6-bd52-327a2a861cd2"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAAQBAMAAAD5QenfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYFmd+VhQTZT+J/AXVrA7ZlZYT8YAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x_{1} y_{2} - x_{1} y_{3} - x_{2} y_{1} + x_{2} y_{3} + x_{3} y_{1} - x_{3} y_{2}$$"
      ],
      "text/plain": [
       "x₁⋅y₂ - x₁⋅y₃ - x₂⋅y₁ + x₂⋅y₃ + x₃⋅y₁ - x₃⋅y₂"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matM.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "6dccd6b8-4002-440e-b711-ef4ac275c1c6"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Si se elige la numeración tal que los nodos 1-2-3 se recorren en sentido contrario a las agujas del reloj, puede demostrarse que el anterior producto coincide con módulo del producto vectorial de los vectores 1-2 y 1-3. \n",
    "\n",
    "En efecto. \n",
    "\n",
    "El vector $\\mathbf{e_1}$, de $(x1,y1)$ a $(x_2,y2)$ es $\\mathbf{e_1} = [x_2-x_1,y_2-y_1]^T$.\n",
    "\n",
    "El vector $\\mathbf{e_2}$, de $(x1,y1)$ a $(x_3,y3)$ es $\\mathbf{e_2} = [x_3-x_1,y_3-y_1]^T$.\n",
    "\n",
    "El producto vectorial de ambos vectores será un vector orientado en la dirección del eje $z$ positivo. La componente $z$ será positivo, porque se ha elegido según la orientación 1-2-3 con recorrido contrario a las agujas del reloj. \n",
    "\n",
    "El módulo del producto vectorial coincidirá por tanto con la componente z. \n",
    "\n",
    "Además, el módulo de este producto vectorial coincide con el área del paralelogramo formado por los dos vectores. Y a su vez, será el doble de área del triángulo formado por los puntos 1-2-3. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "nbpresent": {
     "id": "f41e47fa-2c9c-4147-b3a7-a571750642ed"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOkAAABLCAMAAACiPVA0AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}i & j & k\\\\- x_{1} + x_{2} & - y_{1} + y_{2} & 0\\\\- x_{1} + x_{3} & - y_{1} + y_{3} & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   i         j      k⎤\n",
       "⎢                     ⎥\n",
       "⎢-x₁ + x₂  -y₁ + y₂  0⎥\n",
       "⎢                     ⎥\n",
       "⎣-x₁ + x₃  -y₁ + y₃  0⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i,j,k = symbols('i j k')\n",
    "prodvect = Matrix([[i,j,k], [x2-x1,y2-y1,0],[x3-x1,y3-y1,0]])\n",
    "prodvect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "nbpresent": {
     "id": "efc85cc5-7661-4995-9dc0-997539c9d224"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAAQBAMAAAD5QenfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYFmd+VhQTZT+J/AXVrA7ZlZYT8YAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x_{1} y_{2} - x_{1} y_{3} - x_{2} y_{1} + x_{2} y_{3} + x_{3} y_{1} - x_{3} y_{2}$$"
      ],
      "text/plain": [
       "x₁⋅y₂ - x₁⋅y₃ - x₂⋅y₁ + x₂⋅y₃ + x₃⋅y₁ - x₃⋅y₂"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(prodvect.det()/k).simplify()"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "nbpresent": {
     "id": "229baf68-b84e-4c4e-b4c0-e849d2f987aa"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Por tanto, la inversa puede escribirse a partir del área del triángulo , como: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "nbpresent": {
     "id": "7fc26bf3-2ffd-49e5-a1d5-518d835ec0c2"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAhIAAABOBAMAAACHwOIpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8X/yO6ORD5WYogAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{2 A} \\left(x_{2} y_{3} - x_{3} y_{2}\\right) & \\frac{1}{2 A} \\left(- x_{1} y_{3} + x_{3} y_{1}\\right) & \\frac{1}{2 A} \\left(x_{1} y_{2} - x_{2} y_{1}\\right)\\\\\\frac{y_{2} - y_{3}}{2 A} & \\frac{- y_{1} + y_{3}}{2 A} & \\frac{y_{1} - y_{2}}{2 A}\\\\\\frac{- x_{2} + x_{3}}{2 A} & \\frac{x_{1} - x_{3}}{2 A} & \\frac{- x_{1} + x_{2}}{2 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡x₂⋅y₃ - x₃⋅y₂  -x₁⋅y₃ + x₃⋅y₁  x₁⋅y₂ - x₂⋅y₁⎤\n",
       "⎢─────────────  ──────────────  ─────────────⎥\n",
       "⎢     2⋅A            2⋅A             2⋅A     ⎥\n",
       "⎢                                            ⎥\n",
       "⎢   y₂ - y₃        -y₁ + y₃        y₁ - y₂   ⎥\n",
       "⎢   ───────        ────────        ───────   ⎥\n",
       "⎢     2⋅A            2⋅A             2⋅A     ⎥\n",
       "⎢                                            ⎥\n",
       "⎢  -x₂ + x₃        x₁ - x₃        -x₁ + x₂   ⎥\n",
       "⎢  ────────        ───────        ────────   ⎥\n",
       "⎣    2⋅A             2⋅A            2⋅A      ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = symbols('A')\n",
    "Inversa = 1/(2*A) * matM.adjugate()\n",
    "Inversa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "64d50bb8-1555-4071-85fa-9ac830b67214"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La expresión de las funciones de forma es, "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "nbpresent": {
     "id": "07476ba3-3ab7-4a0b-bc8f-11bf388c8816"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "Inv_matM = Inversa\n",
    "N1 = base.dot(Inv_matM.col(0))\n",
    "N2 = base.dot(Inv_matM.col(1))\n",
    "N3 = base.dot(Inv_matM.col(2))\n",
    "N1 = N1.simplify()\n",
    "N2 = N2.simplify()\n",
    "N3 = N3.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "nbpresent": {
     "id": "70c6c67d-396b-4fdb-9b57-495e4fcab4a8"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY4AAAAqBAMAAABW0KEDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/m09pwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{2 A} \\left(x \\left(y_{2} - y_{3}\\right) + x_{2} y_{3} - x_{3} y_{2} - y \\left(x_{2} - x_{3}\\right)\\right)$$"
      ],
      "text/plain": [
       "x⋅(y₂ - y₃) + x₂⋅y₃ - x₃⋅y₂ - y⋅(x₂ - x₃)\n",
       "─────────────────────────────────────────\n",
       "                   2⋅A                   "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "nbpresent": {
     "id": "71a0fe4f-b81a-4cc5-8733-e7aad7a50c77"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ4AAAAqBAMAAABxfiDrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{2 A} \\left(- x \\left(y_{1} - y_{3}\\right) - x_{1} y_{3} + x_{3} y_{1} + y \\left(x_{1} - x_{3}\\right)\\right)$$"
      ],
      "text/plain": [
       "-x⋅(y₁ - y₃) - x₁⋅y₃ + x₃⋅y₁ + y⋅(x₁ - x₃)\n",
       "──────────────────────────────────────────\n",
       "                   2⋅A                    "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "nbpresent": {
     "id": "4b73ca12-7102-4910-84cf-dd285bfaa65d"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY4AAAAqBAMAAABW0KEDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{2 A} \\left(x \\left(y_{1} - y_{2}\\right) + x_{1} y_{2} - x_{2} y_{1} - y \\left(x_{1} - x_{2}\\right)\\right)$$"
      ],
      "text/plain": [
       "x⋅(y₁ - y₂) + x₁⋅y₂ - x₂⋅y₁ - y⋅(x₁ - x₂)\n",
       "─────────────────────────────────────────\n",
       "                   2⋅A                   "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "10df3901-0dac-4161-924f-91596e2eab5e"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Nótese que para que esta expresión pueda aplicarse, interpretándose $A$ como el área del triángulo con símbolo positivo, la orientación del triángulo ha de ser positiva, es decir, los puntos 1-2-3 deben orientarse en sentido antihorario. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "660bee45-7079-4ec5-868c-6d67dbb1ff19"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Expresión del vector de deformaciones. \n",
    "\n",
    "El vector de deformaciones se expresa mediante el siguiente producto: \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\varepsilon} = \\begin{pmatrix} \\varepsilon_{nx} \\\\ \\varepsilon_{ny} \\\\ \\gamma_{xy} \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Y en función de las funciones de forma: \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\varepsilon} = \\left[ \n",
    "\\begin{matrix}\n",
    "\\dfrac{\\partial N_1}{\\partial x} & 0 & \\dfrac{\\partial N_2}{\\partial x} & 0 & \\dfrac{\\partial N_3}{\\partial x} & 0 \\\\\n",
    "0 & \\dfrac{\\partial N_1}{\\partial x} & 0 & \\dfrac{\\partial N_2}{\\partial x} & 0 & \\dfrac{\\partial N_3}{\\partial x} \\\\\n",
    "\\dfrac{\\partial N_1}{\\partial y} & \\dfrac{\\partial N_1}{\\partial x} & \\dfrac{\\partial N_2}{\\partial y} & \\dfrac{\\partial N_2}{\\partial x} & \\dfrac{\\partial N_3}{\\partial y} & \\dfrac{\\partial N_3}{\\partial x}\n",
    "\\end{matrix} \\right] \\cdot\n",
    "\\begin{pmatrix} u_1 \\\\ v_1 \\\\ u_2 \\\\ v_2 \\\\ u_3 \\\\ v_3 \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Relación que puede escribirse como: $\\boldsymbol{\\varepsilon} = \\mathbf{B} \\cdot \\mathbf{a}$.\n",
    "\n",
    "La matriz $\\mathbf{B}$ en este contexto se denomina _matriz de deformación_.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "nbpresent": {
     "id": "8b02ec30-af37-4bb3-9cc7-3c9c79ead397"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAf4AAABNCAMAAABwmPZ4AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n8KQpLp/3qLr6HxbfyxZcVwf+AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{y_{2} - y_{3}}{2 A} & 0 & \\frac{- y_{1} + y_{3}}{2 A} & 0 & \\frac{y_{1} - y_{2}}{2 A} & 0\\\\0 & \\frac{- x_{2} + x_{3}}{2 A} & 0 & \\frac{x_{1} - x_{3}}{2 A} & 0 & \\frac{- x_{1} + x_{2}}{2 A}\\\\\\frac{- x_{2} + x_{3}}{2 A} & \\frac{y_{2} - y_{3}}{2 A} & \\frac{x_{1} - x_{3}}{2 A} & \\frac{- y_{1} + y_{3}}{2 A} & \\frac{- x_{1} + x_{2}}{2 A} & \\frac{y_{1} - y_{2}}{2 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡y₂ - y₃             -y₁ + y₃            y₁ - y₂           ⎤\n",
       "⎢───────      0      ────────     0      ───────      0    ⎥\n",
       "⎢  2⋅A                 2⋅A                 2⋅A             ⎥\n",
       "⎢                                                          ⎥\n",
       "⎢          -x₂ + x₃            x₁ - x₃             -x₁ + x₂⎥\n",
       "⎢   0      ────────     0      ───────      0      ────────⎥\n",
       "⎢            2⋅A                 2⋅A                 2⋅A   ⎥\n",
       "⎢                                                          ⎥\n",
       "⎢-x₂ + x₃  y₂ - y₃   x₁ - x₃   -y₁ + y₃  -x₁ + x₂  y₁ - y₂ ⎥\n",
       "⎢────────  ───────   ───────   ────────  ────────  ─────── ⎥\n",
       "⎣  2⋅A       2⋅A       2⋅A       2⋅A       2⋅A       2⋅A   ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nu = Matrix([[N1, 0, N2, 0, N3, 0]])\n",
    "Nv = Matrix([[0, N1, 0, N2, 0, N3]])\n",
    "B = Matrix.vstack(Nu.diff(x),Nv.diff(y),Nu.diff(y)+Nv.diff(x))\n",
    "B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "1ea26948-7b51-4a5e-8d94-a424bd4ba59b"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Esta matriz puede escribirse en función de seis constantes, $b_1,c_1,b_2,c_2,b_3,c_3$, como \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{B} = \\dfrac{1}{2\\,A}\n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "b_1 & 0 & b_2 & 0 & b_3 & 0 \\\\\n",
    "0 & c_1 & 0 & c_2 & 0 & c_3 \\\\\n",
    "c_1 & b_1 & c_2 & b_2 & c_3 & b_3\n",
    "\\end{matrix}\n",
    "\\right] = \n",
    "\\left[\n",
    "\\begin{matrix}\n",
    "\\mathbf{B_1} & \\mathbf{B_2} & \\mathbf{B_3}\n",
    "\\end{matrix}\n",
    "\\right]\n",
    "\\end{equation}\n",
    "\n",
    "con $\\mathbf{B_i} =\\dfrac{1}{2\\,A}\\, \\left[ \\begin{matrix} b_i & 0 \\\\ 0 & c_i \\\\ c_i & b_i  \\end{matrix} \\right]$\n",
    "\n",
    "siendo,\n",
    "\n",
    "\\begin{align}\n",
    "b_1 & = y_2 - y_3 \\\\\n",
    "b_2 & = y_3 - y_1 \\\\\n",
    "b_3 & = y_1 - y_2 \\\\\n",
    "c_1 & = x_3 - x_2 \\\\\n",
    "c_2 & = x_1 - x_3 \\\\\n",
    "c_3 & = x_2 - x_1\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "nbpresent": {
     "id": "96048ad5-dad1-48aa-af8b-032eb08adea4"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARIAAABOCAMAAADb0mkpAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRM2JIt27Zu9sobNGMgAABbVJREFUeAHtnOGCoygQhHFQd84YNev7v+sBAhvpxlSr\nucjs+WMmMT2V5gsCUtNRanV0t9vq+alPChXv9akU1mKFiFezO7588rdm3YpTn11bfFhIKFXNujZH\nIDHfT4WwFru2+GQ5fM0WSfWUdzebs+3TiTMfliA+EST10KraDicwl1b3tX6mmofoxau+H8G+uEN8\nqr+wbBQrTpGMtVLTTXX9A0xa2fGhHaBoLz4q1Q15cM+v7BDvVANOmqw4RfLolNJ2qMUaafg9bBN6\nKAsvbmLbGbo4d4gb3b5/ppp9zItTJDbVhx1qUSSjm6oaqIlRXDVYL9kjXvWmFyIHL06RmKFkcpIo\nktkh6eYwY20lE8XbG3ShqV3iqoO6bEacIqm073YgknZ2Szszl2yx8K8F8RYcu/eI2zyguT4jTpHE\nhoFI7rO7cKvlV/zrzQetvrfQ5b5DvDVDW+WGt80UzIsZ8SySqp41cinkhDfzsatEaCzJZL0p3ph1\nFTQJZ8SzSDbf9fnFduke2IXz/IfI40+Ir5EsS/zk56vUl7GkeTm8JrLL0wuKr5G8SpB9fXDz0wRN\nwqzA1skPiKdIej3qP2soaNVdu7FMA/NeGeIJkt6sFr5i46bZrGRfH4Nd0D9ej2iFiCdI7MRbhUm9\n1fP0GojB0WNjfCHiCRJ7D3IPfaNvh1M32AoRT5DYThHuVqpajeDNAtKXlpgCxBkkg1+Zm1sXDa2n\ncCDmXvL64hRJ2DCezKBZmw2mU48SxAmSyQ8fZhzRejx5sVGEeIqkM3djnZ1Ptb15X+/KHu4vZYgn\nSKpb0zSjWas1y/SLzcIgq0LEEyQPd+eh1DjbueY+zgO0MsGYFCKeIMHa9rOj/kdCPt/jSMxqHtli\nJO8MnfiEOIuEdXwyTRi6ZVsP3Ew1d0S4EaYW8fcYYUGcGGEsEtbx4ZFoMwy31ZuMMC9u3uINRlgQ\nJ0YYh4R3fHgkzvGxL4Gb1zIjbNn8fZMRtohTI4xDwjs+PJKwkQAj2SX+FiPMG0nECOOQSOykm91k\nsvcAaC/ZJf4OIyxmnhphDJKM48P3kruup9ruTIJIdom/xQjzmVMjjEGSsTd4JPEsiGSP+H9shJ2E\n5CcZYQyST9hJsbMdenBO5hbJr+9/VpmgXpXaY1ZdX/z3N90VEdhJcmfm+uLMhaMEXpXc9rm+OIdE\noV6Vm3mlts/lxVkkqFel9jgzlxdnkawG21dPCnBmMk3IZH4cSQHOTAZJJvPDSEpwZngkucyPIinC\nmWGRZDM/iKQMZ4ZDks/8GJJCnBkGyUbmx5AU4swwSDYyP4bEbIr+vLK3o0hUGLeZT+L4qY+IH0Zi\nd/PfdnxE/DCSuB/9Di4fEeeQSCw2cWXa9cUZJCL/Tlr2JvLvpGVvXpz4d2wHDuIkmiKR+XfCsrdg\nsWH+nbDsLYgT/45FEsRJNEUi8+98ZRpqWnhx0L/z4mD0Umtm/FiskM2L02iKJA5pkA0RK9Og6KfS\nIaSQLYpjbl/InPh3bC8J4iSaIokuGNTIWJkGRasoDvl3URyK/iOe+ncskiieRlMkMv8uVKaBbl8Q\nx/y7II5Fq6x/xyLx4pDbFwSwz31XNOzfOXVhNFzIZsWZsjfaS3wbYf/OxcuiFVzI5sRl0QouZHPi\nNDqLxJP5C3+tkQjtO1m4LFpoJZ4ovkbyF/YJ2uQUidDRlIXLos2+g6SoTha9IZ4gEdaaycJl0UoW\nLoveEk+Q2IlX4GjKwmXRQnf1PPEEidDRlIXLooXu6nniCRI72GR8QToOLWdk4bJoYS7niDNIMr5g\nDoksXBYtrHs7R5wiCVvAYCGbLFwWHfe6sVxOEidIsr4g30tk4bJoJQuXRefFFyRu7bd8WY2skC1v\nI3IEZdHmH/MlRXWyaF48ftVca79gra7t/zlv+IJcG2XhsmhhLqeIu6+aS2ppNnxBDoksXBatZOGy\n6Bfi/wLsnYA0KL9h5QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{1}}{2 A} & 0 & \\frac{b_{2}}{2 A} & 0 & \\frac{b_{3}}{2 A} & 0\\\\0 & \\frac{c_{1}}{2 A} & 0 & \\frac{c_{2}}{2 A} & 0 & \\frac{c_{3}}{2 A}\\\\\\frac{c_{1}}{2 A} & \\frac{b_{1}}{2 A} & \\frac{c_{2}}{2 A} & \\frac{b_{2}}{2 A} & \\frac{c_{3}}{2 A} & \\frac{b_{3}}{2 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ b₁        b₂        b₃     ⎤\n",
       "⎢───   0   ───   0   ───   0 ⎥\n",
       "⎢2⋅A       2⋅A       2⋅A     ⎥\n",
       "⎢                            ⎥\n",
       "⎢      c₁        c₂        c₃⎥\n",
       "⎢ 0   ───   0   ───   0   ───⎥\n",
       "⎢     2⋅A       2⋅A       2⋅A⎥\n",
       "⎢                            ⎥\n",
       "⎢ c₁   b₁   c₂   b₂   c₃   b₃⎥\n",
       "⎢───  ───  ───  ───  ───  ───⎥\n",
       "⎣2⋅A  2⋅A  2⋅A  2⋅A  2⋅A  2⋅A⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b1,b2,b3,c1,c2,c3 = symbols('b1 b2 b3 c1 c2 c3')\n",
    "B = Matrix([[b1,0,b2,0,b3,0],\n",
    "            [0,c1,0,c2,0,c3],\n",
    "            [c1,b1,c2,b2,c3,b3]])\n",
    "B = B / (2*A)\n",
    "B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "40a9f743-579b-4347-a8d8-2b1bf263d2c3"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Tensor de constantes elásticas\n",
    "Para elasticidad isótropa, la matriz $\\mathbf{D}$ sólo depende de tres coeficientes. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "nbpresent": {
     "id": "7535e185-d9e2-4293-b61f-3727ebe924b3"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAABLCAMAAACRBk89AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n/JJB9xlj08BflFI4srxoZNYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}d_{11} & d_{12} & 0\\\\d_{12} & d_{11} & 0\\\\0 & 0 & d_{33}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡d₁₁  d₁₂   0 ⎤\n",
       "⎢             ⎥\n",
       "⎢d₁₂  d₁₁   0 ⎥\n",
       "⎢             ⎥\n",
       "⎣ 0    0   d₃₃⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d11,d12,d33 = symbols('d11 d12 d33')\n",
    "D = Matrix([[d11, d12, 0],[d12,d11,0],[0,0,d33]])\n",
    "D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "b2d6691f-b95f-4823-afa0-55ba8718a126"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Matriz de rigidez\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} = \\int_{\\Omega} \\mathbf{B}^T \\cdot \\mathbf{D} \\cdot \\mathbf{B} \\, d \\Omega \n",
    "\\end{equation}\n",
    "\n",
    "Teniendo en cuenta que el tensor $\\mathbf{D}$ sólo contiene constantes, y que la matriz $\\mathbf{B}$ es constante dentro del elemento, la integral puede verse como la integral en el área del triángulo de una constante. Claramente, esta integral será el producto de la constante por el área del triángulo. Por tanto: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} = \\mathbf{B}^T \\cdot \\mathbf{D} \\cdot \\mathbf{B} \\cdot A\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true,
    "nbpresent": {
     "id": "0763ccaf-e12b-48a8-9843-4a05ab63cc44"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "krigidez = B.T.multiply(D * A).multiply(B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbpresent": {
     "id": "810eac1c-86c2-43fc-8e78-6f40f362cc9f"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "La matriz de rigidez $\\mathbf{K}$ puede expresarse como: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K} = \\left[ \\begin{matrix}\n",
    "\\mathbf{K_{11}} & \\mathbf{K_{12}} & \\mathbf{K_{13}} \\\\\n",
    "\\mathbf{K_{21}} & \\mathbf{K_{22}} & \\mathbf{K_{23}} \\\\\n",
    "\\mathbf{K_{31}} & \\mathbf{K_{32}} & \\mathbf{K_{33}} \\end{matrix} \\right] \n",
    "\\end{equation}\n",
    "\n",
    "Con \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_{ij}} = \\mathbf{B_i}^T \\cdot \\mathbf{D} \\cdot \\mathbf{B_j} \\,A\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true,
    "nbpresent": {
     "id": "e9d2c559-66bc-4bec-87c1-02faa6607885"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "k11 = krigidez[0:2,0:2]\n",
    "k22 = krigidez[2:4,2:4]\n",
    "k33 = krigidez[4:6,4:6]\n",
    "k12 = krigidez[0:2,2:4]\n",
    "k13 = krigidez[0:2,4:6]\n",
    "k23 = krigidez[2:4,4:6]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "nbpresent": {
     "id": "24ebe345-0763-4e1e-b3a2-6fc5b23e0421"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUUAAABACAMAAAC6EJVoAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nug0+pwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{1}^{2} d_{11}}{4 A} + \\frac{c_{1}^{2} d_{33}}{4 A} & \\frac{b_{1} c_{1} d_{12}}{4 A} + \\frac{b_{1} c_{1} d_{33}}{4 A}\\\\\\frac{b_{1} c_{1} d_{12}}{4 A} + \\frac{b_{1} c_{1} d_{33}}{4 A} & \\frac{b_{1}^{2} d_{33}}{4 A} + \\frac{c_{1}^{2} d_{11}}{4 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2         2                             ⎤\n",
       "⎢  b₁ ⋅d₁₁   c₁ ⋅d₃₃    b₁⋅c₁⋅d₁₂   b₁⋅c₁⋅d₃₃⎥\n",
       "⎢  ─────── + ───────    ───────── + ─────────⎥\n",
       "⎢    4⋅A       4⋅A         4⋅A         4⋅A   ⎥\n",
       "⎢                                            ⎥\n",
       "⎢                           2         2      ⎥\n",
       "⎢b₁⋅c₁⋅d₁₂   b₁⋅c₁⋅d₃₃    b₁ ⋅d₃₃   c₁ ⋅d₁₁  ⎥\n",
       "⎢───────── + ─────────    ─────── + ───────  ⎥\n",
       "⎣   4⋅A         4⋅A         4⋅A       4⋅A    ⎦"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "nbpresent": {
     "id": "8a877d82-d4af-4c3d-9cd2-cf5faa631f98"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUUAAABACAMAAAC6EJVoAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACXHtMAAAAEnRSTlMARM1U\nEImZInar3btm7zK9se1I/ktvAAAH40lEQVR4Ae1cbZukKAyku9W2X6b3zv//Yy9gCBAhJI7O3u1z\nflhAoSiKoI3WjnPd4zOOr3u3VqfCISCdPn7j5cvij6vA4OXc5SlcV106BETV049WegX1bu6y3OC4\nCH2DwsMyCBU0lw4B0XT0s3UuXrzFq9js93KNIXqjWJyWT7N+50ICcd9A6XTyOy5/iSq6cVpJDVe6\nLw5t0TsDyEAgtjuV/1OXOypeb2E0wyOt51sMT+s4cxC3GwV6TSukScEe6grQ9gLqqLiEEBym+zBG\nxjE8Y1mbFiAU5NrWRb0+hR2h3gdtLyBZxcsyzxCGT3gOhfviMM0zhmcxrE5hnOc5gbidKKmTPoUd\nod4HbS8gWcX5ObgZb40wiOH5cZ/1YY1LPFvpaZA897y4z1c6SSif9/yAJxUmqUI3t64QqZoisHjz\nPmh7Ackqvmbn3uk+OIGgQY/L+OWXOiacDytP8EtxyJ7rhPK6hNnFhLUSirhCUo0Q61TcF+odUHkB\nySp+wa/I6eHitgMqu/kR6D7XZzYmNIJaxreCI4I4QoEn1gi3W0xqTevncIUQ4BrrsUih7lxYKfF8\nHYvOyqBpGVYXkKyiX71QA7cdgw/613vwz2uDirhU4t4lQ/mMcNLrGxIaUC+DKyQCYqxTkS2YeL6D\nKoM6eQHJKsJt8Q0s4rbjy98WP2/Pp63isEZpIn3126KJQFyGAj9aQj1MUhsxhyuEWK2xnhfzBRPP\nF5Abkk4G7SwgWcXP5JecP8K24/KYb/PUUfEG99LiuE/ze/bxu4I4QvEVIVAxKdqIBVwhETA9FjzJ\nLNTjVGO/OeaGpBNBc9TaApJVpI7zbUc42Y7FLcGIwkAGeG7DswqTWEmR4goBxcKOCmM9FrNQR5Ks\nX9/DlqQM2llAOhWLbQeQ+MzLBOsIk3LgW4J4nYPAUMIvHUxKFKkUVwgCxljHIoV6vO1s+gXsLUkZ\ntLOAVCqW2w5pgFWCawMTSKePcJkBsqKvEmKxcr6mIvbIarMixHt1AalUpL0LdiUm22leq5tAxB7w\nIgNkRVopm/O+eYtktsMKvWwbVxeQSkWkrUuaBHXNf6bWwSRFFcNrXMU/OPLx6o/nV0ho36hoH6qo\n5dMCsnoiSccqa4sIKqqoHlhe8eBpzqGPyx9M0qAi/PyOh/QtSyaoBIkd6VMbsEySetWC6lW8rS9z\nfA/vRfhKIxLUgtA4tBkjsEiS+lSDqlUcRlJxmJawf6HeioxEUA1SICoKVmCJJHWnB1WrOKevgOPw\npGcHdUkZiaAahNCUGSuwRJK61INqVfzcSMXP7F7ZDYM6xcylHad6EA7aKZuBBZLUlQFUq+IIG/Lw\nRgHe2sEbGvquSn1qMoeA1Do6BdgAqlTxfScV3/5V7a7voIeA1EQ8BdgCqlNxgLdXGItwT5ymV4zL\n2pha5w4BqYGfAmwC1al4A+WmJbxZnPxLWHhXWxuOfO4QkFoXpwCbQHUqeu73EH+39dkh/dSpDTSe\nOwQkguXpKcBqULWK79fygC8wi38631/Ls/0kzgfH8oeAMMxQBOBff41/f4+dR8q/denZqlWscf9X\nnYPp/b49MH6oM47sz1Gx+pXKqEb6UGdr+AeomHxKydlndzsl2Xag/AEqkvEj+0q1w+0UZdyDUqqY\npjViblPzNJ8CmvM60h7o8m9das9UqSJNa06S5e3TrLAe2UEzVusH6eJDk6LLDCDL7kNhKn7HfpZx\nKbOngKYuDrIHutIfaPJMMRWTzyCxZDn7NJ8Cmli17IFVX1JqxnOlP5B7ptA6xRthuVTxW/azRg/g\nvg9W0uzyEU65BNeyB9qMfcwfSO4mlcmwVFFrPzOZ2jioazjljLFDMrbsgfAiz2DsAyH8Efcuvph7\npqLRgnotMqWKWvsZTpBuu8BAXcspZ4udNAq/v4dhIJnM7VTzJaVmLIe3nS1K9Eyh6Yc1W4ulilr7\nGc6MbrvAQIOJDTqPjWnWbbGTRtO2B1qMfcwfuPVM6VXEaQWG4Qd8eiyEYjbNcYIqpraNNZCBerMd\nHh41AzXFTsSARdiwB0rGvg1JFz1TOPSNZ4pGnDpOuTIW9fYzg6mNgbq2U84SO2kIlMt2HXCu7kvC\nysLXqxIF6uNQ9bEYp7XuaUv2swid/9CPg9kQZKA069iYZl2KnQgupZxM1ZeEABuSBMxRJJMhNSpj\nEU8XP+BhVvP/NIRVwszULjTtWLzyplz1tBHRbobjiQ2aKppQqIuqisxvxorQFidoewGutQjyyrwM\nDfeZQnEsGzwaYyXTIsmdd5WmtVNVFWsV1eeaBNUIP1DxYJKFilq/GatXjpoTZJW1xRJUKO0C5CTV\nzrs6kULFehX12bp/Ud38ZyqeQlKhIvycj4fkuIt1NtMcLxSpEbRoKxWUuB2SShQi0ldRbT9DzA7B\ntZYVlPh2MlpcmaQWhch0VdTbzxBTJrhWMoMSXzmjxhVJqlGITFdFvf0MMUWCWMcMSnzljBpXJKlG\nITI9FQ32M8QUCa517KDEV8zocSWSehQi01PRYD9DTIU10A5KfMWMHlciqUchMh0VLfYzwuxlTgGF\nTo/B3YMiq2iyn/XEi9dPAQXwY3B3ocgqmuxnUaVeegoodHoM7i4UWUUviNp+1lMvv34K6GFkzeyC\nin4rCi/wq4feflZtXj95Cih0dQyuCSX+tbbB/70x8a+11aX4/6xXIPy1ttv9H1ZgeQE86bCrAAAA\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{2}^{2} d_{11}}{4 A} + \\frac{c_{2}^{2} d_{33}}{4 A} & \\frac{b_{2} c_{2} d_{12}}{4 A} + \\frac{b_{2} c_{2} d_{33}}{4 A}\\\\\\frac{b_{2} c_{2} d_{12}}{4 A} + \\frac{b_{2} c_{2} d_{33}}{4 A} & \\frac{b_{2}^{2} d_{33}}{4 A} + \\frac{c_{2}^{2} d_{11}}{4 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2         2                             ⎤\n",
       "⎢  b₂ ⋅d₁₁   c₂ ⋅d₃₃    b₂⋅c₂⋅d₁₂   b₂⋅c₂⋅d₃₃⎥\n",
       "⎢  ─────── + ───────    ───────── + ─────────⎥\n",
       "⎢    4⋅A       4⋅A         4⋅A         4⋅A   ⎥\n",
       "⎢                                            ⎥\n",
       "⎢                           2         2      ⎥\n",
       "⎢b₂⋅c₂⋅d₁₂   b₂⋅c₂⋅d₃₃    b₂ ⋅d₃₃   c₂ ⋅d₁₁  ⎥\n",
       "⎢───────── + ─────────    ─────── + ───────  ⎥\n",
       "⎣   4⋅A         4⋅A         4⋅A       4⋅A    ⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "nbpresent": {
     "id": "bb39055f-9f29-4ad4-bdf6-8f682799004a"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUUAAABACAMAAAC6EJVoAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{3}^{2} d_{11}}{4 A} + \\frac{c_{3}^{2} d_{33}}{4 A} & \\frac{b_{3} c_{3} d_{12}}{4 A} + \\frac{b_{3} c_{3} d_{33}}{4 A}\\\\\\frac{b_{3} c_{3} d_{12}}{4 A} + \\frac{b_{3} c_{3} d_{33}}{4 A} & \\frac{b_{3}^{2} d_{33}}{4 A} + \\frac{c_{3}^{2} d_{11}}{4 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2         2                             ⎤\n",
       "⎢  b₃ ⋅d₁₁   c₃ ⋅d₃₃    b₃⋅c₃⋅d₁₂   b₃⋅c₃⋅d₃₃⎥\n",
       "⎢  ─────── + ───────    ───────── + ─────────⎥\n",
       "⎢    4⋅A       4⋅A         4⋅A         4⋅A   ⎥\n",
       "⎢                                            ⎥\n",
       "⎢                           2         2      ⎥\n",
       "⎢b₃⋅c₃⋅d₁₂   b₃⋅c₃⋅d₃₃    b₃ ⋅d₃₃   c₃ ⋅d₁₁  ⎥\n",
       "⎢───────── + ─────────    ─────── + ───────  ⎥\n",
       "⎣   4⋅A         4⋅A         4⋅A       4⋅A    ⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k33"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "nbpresent": {
     "id": "adf011ac-4da7-4377-9f0c-2c87b134a71c"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAA1CAMAAAADIxJjAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n3SUOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{1} b_{2} d_{11}}{4 A} + \\frac{c_{1} c_{2} d_{33}}{4 A} & \\frac{b_{1} c_{2} d_{12}}{4 A} + \\frac{b_{2} c_{1} d_{33}}{4 A}\\\\\\frac{b_{1} c_{2} d_{33}}{4 A} + \\frac{b_{2} c_{1} d_{12}}{4 A} & \\frac{b_{1} b_{2} d_{33}}{4 A} + \\frac{c_{1} c_{2} d_{11}}{4 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⋅b₂⋅d₁₁   c₁⋅c₂⋅d₃₃  b₁⋅c₂⋅d₁₂   b₂⋅c₁⋅d₃₃⎤\n",
       "⎢───────── + ─────────  ───────── + ─────────⎥\n",
       "⎢   4⋅A         4⋅A        4⋅A         4⋅A   ⎥\n",
       "⎢                                            ⎥\n",
       "⎢b₁⋅c₂⋅d₃₃   b₂⋅c₁⋅d₁₂  b₁⋅b₂⋅d₃₃   c₁⋅c₂⋅d₁₁⎥\n",
       "⎢───────── + ─────────  ───────── + ─────────⎥\n",
       "⎣   4⋅A         4⋅A        4⋅A         4⋅A   ⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "nbpresent": {
     "id": "e0ec15a7-f48b-46f8-b59d-78c9cc427ab4"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAA1CAMAAAADIxJjAAAAM1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{1} b_{3} d_{11}}{4 A} + \\frac{c_{1} c_{3} d_{33}}{4 A} & \\frac{b_{1} c_{3} d_{12}}{4 A} + \\frac{b_{3} c_{1} d_{33}}{4 A}\\\\\\frac{b_{1} c_{3} d_{33}}{4 A} + \\frac{b_{3} c_{1} d_{12}}{4 A} & \\frac{b_{1} b_{3} d_{33}}{4 A} + \\frac{c_{1} c_{3} d_{11}}{4 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⋅b₃⋅d₁₁   c₁⋅c₃⋅d₃₃  b₁⋅c₃⋅d₁₂   b₃⋅c₁⋅d₃₃⎤\n",
       "⎢───────── + ─────────  ───────── + ─────────⎥\n",
       "⎢   4⋅A         4⋅A        4⋅A         4⋅A   ⎥\n",
       "⎢                                            ⎥\n",
       "⎢b₁⋅c₃⋅d₃₃   b₃⋅c₁⋅d₁₂  b₁⋅b₃⋅d₃₃   c₁⋅c₃⋅d₁₁⎥\n",
       "⎢───────── + ─────────  ───────── + ─────────⎥\n",
       "⎣   4⋅A         4⋅A        4⋅A         4⋅A   ⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "nbpresent": {
     "id": "f62cb181-033c-42de-a5ee-b95925e1a582"
    },
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{b_{2} b_{3} d_{11}}{4 A} + \\frac{c_{2} c_{3} d_{33}}{4 A} & \\frac{b_{2} c_{3} d_{12}}{4 A} + \\frac{b_{3} c_{2} d_{33}}{4 A}\\\\\\frac{b_{2} c_{3} d_{33}}{4 A} + \\frac{b_{3} c_{2} d_{12}}{4 A} & \\frac{b_{2} b_{3} d_{33}}{4 A} + \\frac{c_{2} c_{3} d_{11}}{4 A}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₂⋅b₃⋅d₁₁   c₂⋅c₃⋅d₃₃  b₂⋅c₃⋅d₁₂   b₃⋅c₂⋅d₃₃⎤\n",
       "⎢───────── + ─────────  ───────── + ─────────⎥\n",
       "⎢   4⋅A         4⋅A        4⋅A         4⋅A   ⎥\n",
       "⎢                                            ⎥\n",
       "⎢b₂⋅c₃⋅d₃₃   b₃⋅c₂⋅d₁₂  b₂⋅b₃⋅d₃₃   c₂⋅c₃⋅d₁₁⎥\n",
       "⎢───────── + ─────────  ───────── + ─────────⎥\n",
       "⎣   4⋅A         4⋅A        4⋅A         4⋅A   ⎦"
      ]
     },
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    "k23"
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    "Nótese que la matriz $\\mathbf{K}$ es simétrica, lo cual no implica que se de la relación $\\mathbf{K_{ij}} = \\mathbf{K_{ji}}$, pues esta relación es entre matrices, y por tanto, la relación correcta es: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{K_{ij}} = \\mathbf{K_{ji}}^T\n",
    "\\end{equation}"
   ]
  },
  {
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    "## Vector de fuerzas debido a tensiones en el contorno\n",
    "\n",
    "En la expresión del PTV las tensiones en el contorno $\\Gamma$, denominadas $\\mathbf{t}$ intervienen mediante la integral: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_t = \\int_{\\Gamma} \\mathbf{N}^T \\cdot \\mathbf{t} \\,d \\Gamma\n",
    "\\end{equation}\n",
    "\n",
    "Las fuerzas nodales se obtienen, por tanto, por integración en el elemento. \n",
    "\n",
    "Tras el ensamblaje, este vector sólo produce componentes no nulas cuando uno de los lados coincide con un lado cargado del sólido. "
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Si el lado cargado es el 1-2, con un vector $\\mathbf{t} = [t_x,t_y]^T$ constante, queda: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_t = \\dfrac{L_{12}}{2} \\cdot \\begin{pmatrix} t_x \\\\ t_y \\\\ t_x \\\\ t_y \\\\0 \\\\ 0 \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "donde $L_{12}$ es la longitud del lado 1-2. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
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   },
   "source": [
    "De forma análoga, si el lado cargado es el 1-3, se obtendrá: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_t = \\dfrac{L_{13}}{2} \\cdot \\begin{pmatrix} t_x \\\\ t_y \\\\ 0 \\\\ 0 \\\\t_x \\\\ t_y \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Y si el lado cargado es el 2-3 se obtendrá: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_t = \\dfrac{L_{23}}{2} \\cdot \\begin{pmatrix} 0 \\\\ 0 \\\\ t_x \\\\ t_y  \\\\t_x \\\\ t_y \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Puede demostrarse que, para cualquier distribución de tensiones, el elemento lineal reparte estas cargas a los dos nodos mediante la reducción del sistema de vectores deslizantes original a dos fuerzas en los nodos. Este reparto puede denominarse _isostático_ y expresa simplemente la idea básica de que las dos fuerzas equivalentes son tales que generan la misma resultante y el mismo momento que el sistema original. Bajo este punto de vista, el cálculo de las fuerzas nodales equivalentes es muy sencillo, y no requiere integración para su evaluación. "
   ]
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   "metadata": {
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   },
   "source": [
    "## Vector de fuerzas debido a fuerzas volumétricas. \n",
    "\n",
    "En la expresión del PTV interviene el vector de fuerzas interiores, volumétricas, $\\mathbf{b}$, generando fuerzas nodales equivalentes. Para un elemento: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_b = \\int_{\\Omega} \\mathbf{N}^T \\cdot \\mathbf{b} \\,d \\Gamma\n",
    "\\end{equation}\n",
    "\n",
    "La integración genera fuerzas nodales en los tres nodos: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_b = \\dfrac{A}{3} \\,\\begin{pmatrix} b_x \\\\b_y \\\\b_x \\\\b_y \\\\ b_x \\\\ b_y \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Si la fuerza por unidad de volumen corresponde al peso propio, entonces $b_x = 0$ y $b_y = - \\rho \\,g$, siendo $\\rho$ la densidad volumétrica del material, y $g$ la constante gravitatoria. \n",
    "\n",
    "Por tanto, para el caso de fuerzas debidas a peso propio, se tomará: \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{f}_b = \\dfrac{A}{3} \\,\\begin{pmatrix} 0 \\\\ -\\rho \\,g \\\\ 0 \\\\- \\rho \\,g  \\\\ 0 \\\\ -\\rho \\,g \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "\n"
   ]
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