06aa_soil_column_FDM.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false
   },
   "source": [
    "<div style=\"display: flex; align-items: center;\">\n",
    "    <img src=\"https://github.com/nagelt/Teaching_Scripts/raw/9d9e29ecca4b04eaf7397938eacbf116d37ddc93/Images/TUBAF_Logo_blau.png\" width=\"500\" height=\"auto\" height=\"auto\" style=\"margin-right: 100px;\" />\n",
    "    <div>\n",
    "        <p><strong>Prof. Dr. Thomas Nagel</strong></p>\n",
    "        <p>Chair of Soil Mechanics and Foundation Engineering<br>Geotechnical Institute<br>Technische Universität Bergakademie Freiberg.</p>\n",
    "        <p><a href=\"https://tu-freiberg.de/en/soilmechanics\">https://tu-freiberg.de/en/soilmechanics</a></p>\n",
    "    </div>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "#Some plot settings\n",
    "import plot_functions.plot_settings"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 6 - Finite differences: soil column under gravity and top load\n",
    "\n",
    "![soil_column.png](Images/soil_column.png)\n",
    "\n",
    "## Governing differential equation\n",
    "\n",
    "Consider the equilibrium conditions on an infinitesimal element of the soil column:\n",
    "\n",
    "$$\n",
    "    \\uparrow: \\quad F_z - \\frac{\\partial F_z}{\\partial z}\\text{d}z - F_z - \\varrho g A \\text{d}z = 0\n",
    "$$\n",
    "\n",
    "The vertical force is determined by $F_z = \\sigma_{zz}A = E_\\text{s} A \\epsilon_{zz} = -E_\\text{s} A u_{z,z}$. Therefore, the equilibrium conditions read:\n",
    "\n",
    "$$\n",
    "    0 = \\left[ \\frac{\\partial}{\\partial z} \\left(E_\\text{s} A \\frac{\\partial u_z}{\\partial z}\\right) - \\varrho g A \\right]\\text{d}z\n",
    "$$\n",
    "\n",
    "Considering, that the equation has to hold for an arbitrary infinitesimal non-zero $\\text{d}z$, we find the ordinary differential equation\n",
    "\n",
    "$$\n",
    "    0 = \\frac{\\partial}{\\partial z} \\left(E_\\text{s} A \\frac{\\partial u_z}{\\partial z}\\right) - \\varrho g A\n",
    "$$\n",
    "\n",
    "Nota bene: compare this to the differential equation for a rod ('Zugstab'): $(EAu')' + n = 0$. While the cross section of a rod can vary along its length, the cross-sectional area of a soil column is considered a constant, simplifying the equation further:\n",
    "\n",
    "$$\n",
    "    0 = \\frac{\\partial}{\\partial z} \\left(E_\\text{s} \\frac{\\partial u_z}{\\partial z}\\right) - \\varrho g\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's put this to test."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finite difference discretization\n",
    "\n",
    "We assume $E_\\text{s} = \\text{const.}$ and arrive at\n",
    "\n",
    "$$\n",
    "    \\frac{\\partial^2 u_z}{\\partial z^2} = \\frac{\\gamma}{E_\\text{s}}\n",
    "$$\n",
    "\n",
    "In the finite difference method, we introduce a grid made up, in the one-dimensional case, of a series of points. The differentials are the evaluated by finite differences between values at these points:\n",
    "\n",
    "$$\n",
    "    \\frac{\\partial u_z}{\\partial z} \\approx \\frac{\\Delta u_z}{\\Delta z} = \\frac{u_i - u_{i-1}}{z_i - z_{i-1}} = u_i'\n",
    "$$\n",
    "\n",
    "where here, backward differences were chosen (viz. forward and central differences). We also assume a constant $\\Delta z$ here.\n",
    "\n",
    "A second derivative can likewise be approximated (now using forward differences):\n",
    "\n",
    "$$\n",
    "    \\frac{\\partial^2 u_z}{\\partial z^2} \\approx u_i'' = \\frac{u_{i+1}' - u_i'}{\\Delta z}\n",
    "$$\n",
    "\n",
    "Substitution of the first derivatives yields:\n",
    "\n",
    "$$\n",
    "    \\frac{\\partial^2 u_z}{\\partial z^2} \\approx u_i'' = \\frac{u_{i+1} - 2 u_i + u_{i-1}}{\\Delta z^2}\n",
    "$$\n",
    "\n",
    "Our differential equation has now transformed into a finite difference equation evaluated at a series of points $i$ as:\n",
    "\n",
    "$$\n",
    "u_{i+1} - 2 u_i + u_{i-1} = \\Delta z^2 \\frac{\\gamma_i}{E_{\\text{s},i}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We collect all solutions $u_i$ and $z_i$ in a vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "H = 10.\n",
    "number_of_points = 10\n",
    "Delta_z = H/(number_of_points - 1)\n",
    "z = np.linspace(0,H,number_of_points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The right-hand side is identical in every point, unless the stiffness or the specific weight changes with depth."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.00019523, 0.00019523, 0.00019523, 0.00019523, 0.00019523,\n",
       "       0.00019523, 0.00019523, 0.00019523, 0.00019523, 0.00019523])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gamma = 2600. * (1.-0.38) * 9.81 # N/m³\n",
    "Es = 1.e8 #Pa\n",
    "RHS = np.ones(number_of_points) * gamma/Es * Delta_z**2\n",
    "RHS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The left hand side can be expressed via the finite difference matrix $A_{ij}$, so that the equation system finally can be written as $A_{ij} u_{j} = b_i$ for a system with $n=10$ points:\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "    -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
    "    1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
    "    0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
    "    0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n",
    "    0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0\\\\\n",
    "    0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0\\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0\\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0\\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1\\\\\n",
    "    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2\\\\\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "    u_0 \\\\ u_1 \\\\ u_2 \\\\ u_3 \\\\ u_4 \\\\ u_5 \\\\ u_6 \\\\ u_7 \\\\ u_8 \\\\ u_9\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\Delta z^2\n",
    "\\begin{pmatrix}\n",
    "    \\frac{\\gamma_0}{E_{\\text{s},0}} \\\\ \\frac{\\gamma_1}{E_{\\text{s},1}} \\\\ \\frac{\\gamma_3}{E_{\\text{s},3}} \\\\ \\frac{\\gamma_4}{E_{\\text{s},4}} \\\\ \\frac{\\gamma_5}{E_{\\text{s},5}} \\\\ \\frac{\\gamma_6}{E_{\\text{s},6}} \\\\ \\frac{\\gamma_7}{E_{\\text{s},7}} \\\\ \\frac{\\gamma_8}{E_{\\text{s},8}} \\\\ \\frac{\\gamma_9}{E_{\\text{s},9}}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "Let's build this matrix now ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0., -2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0., -2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0., -2.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0., -2.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0., -2.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0., -2.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0., -2.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -2.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -2.]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Central_Diag = np.diag([-2.]*number_of_points,0)\n",
    "Central_Diag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "Upper_Diag = np.diag([1.]*(number_of_points-1),1)\n",
    "Lower_Diag = np.diag([1.]*(number_of_points-1),-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 1., -2.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  1., -2.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  1., -2.,  1.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  1., -2.,  1.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  1., -2.,  1.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  1., -2.,  1.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  1., -2.,  1.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1., -2.,  1.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1., -2.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Central_Diag + Upper_Diag + Lower_Diag\n",
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we have everything we need: we have the load vector (right-hand side), the system matrix (vertex connectivity and second derivative), and can solve for the vector of unknown vertex displacements. Let's give it a shot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.linalg.solve(A,RHS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(u*100,z)\n",
    "plt.xlabel('$u_z$ / cm')\n",
    "plt.ylabel('$z$ / m');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That doesn't look plausible. What's missing are the boundary conditions: \n",
    "\n",
    "$$\n",
    "    u (z = 0) = 0 \\qquad u'(z=H) = 0\n",
    "$$\n",
    "\n",
    "They can be integrated easily by manipulating the first and last equations (rows).\n",
    "\n",
    "Let's start with the lower boundary condition, $u(z=0) = u_0 = 0$. Thus, we want the first equation to yield $u_0 = 0$ which can be achieved by manipulating the RHS and System matrix such that the first equation reads\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 & 0 & \\dots & 0 \n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "u_0\n",
    "\\\\\n",
    "\\vdots\n",
    "\\\\\n",
    "u_9\n",
    "\\end{pmatrix}\n",
    "= \n",
    "0\n",
    "$$\n",
    "\n",
    "Nota bene: if a non-zero displacement were given as the boundary condition, the RHS entry would have to be set to that given value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "#lower bc\n",
    "A[0,:] = 0.\n",
    "A[0,0] = 1.\n",
    "RHS[0] = 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we get to the upper boundary condition, $u'(z=H) = 0$. In other words, we want the slope of the $u_z - z$ curve to be zero at the upper end. Because of \n",
    "\n",
    "$$\n",
    "        0 = u'(z=H) \\approx \\frac{u_9 - u_8}{\\Delta z}\n",
    "$$\n",
    "\n",
    "this means that the displacements of the two final vertices should be equal. Thus, we want the final equation to yield $u_8 - u_9 = 0$ which can be achieved by manipulating the RHS and System matrix such that the first equation reads\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "0 & \\dots & 0 & 1 & -1\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "u_0\n",
    "\\\\\n",
    "\\vdots\n",
    "\\\\\n",
    "u_7\n",
    "\\\\\n",
    "u_8\n",
    "\\\\\n",
    "u_9\n",
    "\\end{pmatrix}\n",
    "= \n",
    "0\n",
    "$$\n",
    "\n",
    "Nota bene: if a non-zero displacement were given as the boundary condition, the RHS entry would have to be set to that given value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "#upper bc\n",
    "A[-1,-1] = -1.\n",
    "RHS[-1] = 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check whether the manipulations were successful:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 1., -2.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  1., -2.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  1., -2.,  1.,  0.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  1., -2.,  1.,  0.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  1., -2.,  1.,  0.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  1., -2.,  1.,  0.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  1., -2.,  1.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1., -2.,  1.],\n",
       "       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1., -1.]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.00019523, 0.00019523, 0.00019523, 0.00019523,\n",
       "       0.00019523, 0.00019523, 0.00019523, 0.00019523, 0.        ])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RHS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's solve the system again with the newly implemented boundary conditions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u = np.linalg.solve(A,RHS)\n",
    "plt.plot(u*100,z)\n",
    "plt.xlabel('$u_z$ / cm')\n",
    "plt.ylabel('$z$ / m');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks plausible. Let's go ahead and perform a basic exploration of the numerical properties of the scheme.\n",
    "\n",
    "We first define a simulation run and then vary the number of grid points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def run_sim(number_of_points):\n",
    "    #discretization\n",
    "    Delta_z = H/(number_of_points - 1)\n",
    "    z = np.linspace(0,H,number_of_points)\n",
    "    #RHS\n",
    "    RHS = np.ones(number_of_points) * gamma/Es * Delta_z**2\n",
    "    #FD Matrix\n",
    "    A = np.diag([-2.]*number_of_points,0) + np.diag([1.]*(number_of_points-1),1) + np.diag([1.]*(number_of_points-1),-1)\n",
    "    #lower bc\n",
    "    A[0,:] = 0.\n",
    "    A[0,0] = 1.\n",
    "    RHS[0] = 0\n",
    "    #upper bc\n",
    "    A[-1,-1] = -1.\n",
    "    RHS[-1] = 0.\n",
    "    #solution\n",
    "    return np.linalg.solve(A,RHS), z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot the numerical result against the analytical solution\n",
    "\n",
    "$$\n",
    "u_z(z) = \\frac{\\varrho gH^2}{2E_\\text{s}} \\left[ \\left(\\frac{z}{H}\\right)^2 - 2 \\frac{z}{H} \\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# analytical solution\n",
    "reference_z = np.linspace(0,H,100)\n",
    "reference_solution = ((reference_z/H)**2 - 2.*reference_z/H) * gamma * H**2 / (2.*Es)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c0577b1cfad341deb768f1389ac1dbac",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=10, description='nodes', min=5), Output()), _dom_classes=('widget-intera…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Interactive playground\n",
    "\n",
    "from ipywidgets import widgets\n",
    "from ipywidgets import interact\n",
    "#Compute reference solution with 100 cells\n",
    "\n",
    "@interact(num_nodes=widgets.IntSlider(min=5, max=100, value=10, step=1, description='nodes'))\n",
    "def plot(num_nodes=10):\n",
    "    fig,ax = plt.subplots()\n",
    "    ax.set_ylabel('$z$ / m')\n",
    "    ax.set_xlabel('$u_z$ / cm')\n",
    "    solution, z = run_sim(num_nodes)\n",
    "    ax.plot(reference_solution*1e2, reference_z, '--', color='k', label='reference solution')\n",
    "    ax.plot(solution*1e2,z,label='solution')\n",
    "    ax.set_ylim([0,10])\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convergence of this simple scheme is rather low. A comparatively simple finite element scheme converges much faster, as we shall see in one of the following exercises.\n",
    "\n",
    "If we plot the convergence rate of the surface settlements with respect to the grid resolution, we observe a linear behaviour:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#HIDDEN\n",
    "fig, ax = plt.subplots()\n",
    "disc_n = np.logspace(1,3,20)\n",
    "analytical_top = reference_solution[-1]\n",
    "for nn in disc_n:\n",
    "    numerical, z = run_sim(int(nn))\n",
    "    numerical_top = numerical[-1]\n",
    "    relerr = np.abs((numerical_top - analytical_top)/analytical_top)\n",
    "    ax.plot(int(nn),relerr,ls='',marker='d',color='blue')\n",
    "ax.plot([1e2,3e2],[5e-3,5e-3/3],ls='-',color='black')\n",
    "ax.text(3.1e2,5e-3/3,'1')\n",
    "ax.plot([1e2,3e2],[5e-3,5e-3/9],ls='-',color='black')\n",
    "ax.text(3.1e2,5e-3/9,'2')\n",
    "ax.set_xscale('log')\n",
    "ax.set_yscale('log')\n",
    "ax.set_xlabel('$n_\\\\mathrm{nodes}$')\n",
    "ax.set_ylabel('$\\\\left| \\\\frac{u_z^\\\\mathrm{numerical} - u_z^\\\\mathrm{analytical}}{u_z^\\\\mathrm{analytical}} \\\\right|_{z=H}$',size=24)\n",
    "fig.tight_layout();"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  },
  "widgets": {
   "application/vnd.jupyter.widget-state+json": {
    "state": {},
    "version_major": 2,
    "version_minor": 0
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}