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   "source": [
    "# Example: One-dimensional heat flow "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example is from the CALFEM manual (exs2.py).\n",
    "\n",
    "**Purpose:**\n",
    "\n",
    "Analysis of one-dimensional heat flow.\n",
    "\n",
    "**Description:**\n",
    "\n",
    "Consider a wall built up of concrete and thermal insulation. The outdoor\n",
    "temperature is −17 ◦C and the temperature inside is 20 ◦C. At the inside of\n",
    "the thermal insulation there is a heat source yielding $10 ~W/m^2$.\n",
    "\n",
    "![](../images/exs2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The wall is subdivided into five elements and the one-dimensional spring\n",
    "(analogy) element `spring1e` is used. Equivalent spring stiffnesses are\n",
    "$k_i = λ A/L$ for thermal conductivity and $k_i = A/R$ for thermal\n",
    "surface resistance. Corresponding spring stiffnesses per $m^2$ of the wall\n",
    "are:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "k_1 &= 1/0.04 = 25.0 ~W/K  \\\\\n",
    "k_2 &= 1.7/0.070 = 24.3 ~W/K  \\\\\n",
    "k_3 &= 0.040/0.100 = 0.4 ~W/K  \\\\\n",
    "k_4 &= 1.7/0.100 = 17.0 ~W/K  \\\\\n",
    "k_5 &= 1/0.13 = 7.7 ~W/K \n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A global system matrix K and a heat flow vector f are defined. The heat source\n",
    "inside the wall is considered by setting $f_4 = 10$. The element matrices\n",
    "`Ke` are computed using `spring1e`, and the function `assem` assembles the\n",
    "global stiffness matrix.\n",
    "\n",
    "The system of equations is solved using `solveq` with considerations to the\n",
    "boundary conditions in `bc` and `bcVal`. The prescribed temperatures are \n",
    "$T_1 = −17 ~^{\\circ}C$ and $T_2 = 20~^{\\circ}C$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Necessary modules are first imported."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import calfem.core as cfc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, the element topology is defined"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "Edof = np.array([\n",
    "    [1,2],\n",
    "    [2,3],\n",
    "    [3,4],\n",
    "    [4,5],\n",
    "    [5,6]\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create stiffness matrix K and load vector f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = np.mat(np.zeros((6,6)))\n",
    "f = np.mat(np.zeros((6,1)))\n",
    "f[3] = 10.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define element properties (ep) and create element matrices for the different material layers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "ep1 = 25.0\n",
    "ep2 = 24.3\n",
    "ep3 = 0.4\n",
    "ep4 = 17.0\n",
    "ep5 = 7.7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Element stiffness matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ke1 = cfc.spring1e(ep1)\n",
    "Ke2 = cfc.spring1e(ep2)\n",
    "Ke3 = cfc.spring1e(ep3)\n",
    "Ke4 = cfc.spring1e(ep4)\n",
    "Ke5 = cfc.spring1e(ep5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assemble all element matrices into the global stiffness matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Stiffness matrix K:\n",
      "[[ 25.  -25.    0.    0.    0.    0. ]\n",
      " [-25.   49.3 -24.3   0.    0.    0. ]\n",
      " [  0.  -24.3  24.7  -0.4   0.    0. ]\n",
      " [  0.    0.   -0.4  17.4 -17.    0. ]\n",
      " [  0.    0.    0.  -17.   24.7  -7.7]\n",
      " [  0.    0.    0.    0.   -7.7   7.7]]\n"
     ]
    }
   ],
   "source": [
    "cfc.assem(Edof[0,:], K, Ke1)\n",
    "cfc.assem(Edof[1,:], K, Ke2) \n",
    "cfc.assem(Edof[2,:], K, Ke3)\n",
    "cfc.assem(Edof[3,:], K, Ke4)\n",
    "cfc.assem(Edof[4,:], K, Ke5)\n",
    "\n",
    "print(\"Stiffness matrix K:\")\n",
    "print(K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the boundary conditions and solve the system of equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Displacements a:\n",
      "[[-17.        ]\n",
      " [-16.43842455]\n",
      " [-15.86067203]\n",
      " [ 19.23779344]\n",
      " [ 19.47540439]\n",
      " [ 20.        ]]\n",
      "Reaction forces r:\n",
      "[[-1.40393862e+01]\n",
      " [ 0.00000000e+00]\n",
      " [ 0.00000000e+00]\n",
      " [ 0.00000000e+00]\n",
      " [ 5.68434189e-14]\n",
      " [ 4.03938619e+00]]\n"
     ]
    }
   ],
   "source": [
    "bc = np.array([1,6])\n",
    "bc_val = np.array([-17.0, 20.0])\n",
    "a,r = cfc.solveq(K, f, bc, bc_val)\n",
    "\n",
    "print(\"Displacements a:\")\n",
    "print(a)\n",
    "\n",
    "print(\"Reaction forces r:\")\n",
    "print(r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
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