Solving formatting issues

Author: ArasuCandassamy

chapter2/linear-elasticity-v1.ipynb

@@ -13,20 +13,17 @@
     "This leads to the following variational formulation:\n",
     "\n",
     "$$\n",
-    "\n",
     "\\boxed{\n",
     "\\begin{aligned}\n",
     "&\\text{Find } u \\in V \\text{ such that:} \\\\\n",
     "&\\qquad a(u, v) = L(v) \\quad \\forall v \\in V\n",
     "\\end{aligned}\n",
     "}\n",
-    "\n",
     "$$\n",
     "\n",
     "with\n",
     "\n",
     "$$\n",
-    "\n",
     "\\begin{aligned}\n",
     "&a : \n",
     "\\begin{cases}\n",
@@ -39,7 +36,6 @@
     "v \\longmapsto \\int_\\Omega f \\cdot v \\, dx + \\int_{\\partial\\Omega_T} g_T \\cdot v \\, ds\n",
     "\\end{cases}\n",
     "\\end{aligned}\n",
-    "\n",
     "$$\n",
     "\n",
     "\n",

chapter2/linear-elasticity-v2.ipynb

@@ -13,46 +13,39 @@
     "In the case of high values of the Lamé parameter $ \\lambda $, the pressure field $ p $ can be defined as:\n",
     "\n",
     "$$\n",
-    "\n",
     "p := -\\lambda \\nabla \\cdot u \\hspace{1cm} \\text{ in } \\Omega\n",
-    "\n",
     "$$\n",
     "\n",
     "As $u \\in H^1(\\Omega)$, the pressure field $p \\in L^2(\\Omega)$.\n",
     "\n",
     "## The strong formulation of the problem\n",
     "The strong formulation of the problem can be expressed as follows:\n",
-    "Find $ (u, p) \\in V \\times L^2(\\Omega) $ such that\n",
+    "Find $(u, p) \\in V \\times L^2(\\Omega)$ such that\n",
     "\n",
     "$$\n",
-    "\n",
     "\\begin{align}\n",
     "    -\\nabla \\cdot \\sigma(u,p) &= f & \\text{in } & \\Omega \\\\\n",
     "    u &= 0 & \\text{on } & \\partial \\Omega_D \\\\\n",
     "    \\sigma(u,p) \\cdot n &= g_T & \\text{on } & \\partial \\Omega_T \\\\\n",
     "    \\nabla \\cdot u + \\frac{1}{\\lambda} p &= 0 & \\text{in } & \\Omega\n",
     "\\end{align}\n",
-    "\n",
     "$$\n",
     "\n",
     "## The Variational Formulation (Pressure-Displacement Formulation)\n",
-    "To derive the weak formulation of the problem, we multiply the first equation by a test function $ v \\in V $ and integrate by parts over the domain $ \\Omega $ and we add the fourth equation multiplied by a test function $ q \\in L^2(\\Omega) $ integrated over the domain $ \\Omega $:\n",
+    "To derive the weak formulation of the problem, we multiply the first equation by a test function $v \\in V$ and integrate by parts over the domain $\\Omega$ and we add the fourth equation multiplied by a test function $q \\in L^2(\\Omega)$ integrated over the domain $\\Omega$:\n",
     "\n",
     "$$\n",
-    "\n",
     "\\boxed{\n",
     "\\begin{aligned}\n",
     "&\\text{Find } (u, p) \\in V \\times L^2(\\Omega) \\text{ such that:} \\\\\n",
     "&\\qquad \\tilde{a}((u, p), (v, q)) = l(v, q) \\quad \\forall (v, q) \\in V \\times L^2(\\Omega)\n",
     "\\end{aligned}\n",
     "}\n",
-    "\n",
     "$$\n",
     "\n",
     "with\n",
     "\n",
     "$$\n",
-    "\n",
     "\\begin{aligned}\n",
     "&\\tilde{a} : \n",
     "\\begin{cases}\n",
@@ -65,7 +58,6 @@
     "v \\longmapsto \\int_\\Omega f \\cdot v \\, dx + \\int_{\\partial\\Omega_N} t_N \\cdot v \\, ds\n",
     "\\end{cases}\n",
     "\\end{aligned}\n",
-    "\n",
     "$$"
    ]
   },
@@ -469,24 +461,20 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "p_simulated = -lambda_ * compute_divergence(u1, u2, u3, x_vals, y_vals, z_vals)\n",
-    "exact_p = -lambda_ * np.pi * np.sin(np.pi * X) * np.sin(np.pi * Y) * np.cos(np.pi * Z)\n",
-    "error_p = p_simulated - exact_p\n",
-    "\n",
     "fig = plt.figure(figsize=(24, 6 * len(z_plane_list)))\n",
     "for i, z_plane in enumerate(z_plane_list):\n",
     "    list_index = int(z_plane * (h - 1))\n",
     "\n",
     "    # Simulated pressure\n",
     "    ax1 = fig.add_subplot(len(z_plane_list), 3, 3 * i + 1, projection='3d')\n",
-    "    ax1.plot_surface(X[:, :, list_index], Y[:, :, list_index], p_simulated[:, :, list_index], cmap='viridis')\n",
+    "    ax1.plot_surface(X[:, :, list_index], Y[:, :, list_index], p_sim[:, :, list_index], cmap='viridis')\n",
     "    ax1.set_title('Simulated Pressure on z={:.2f} plane'.format(z_plane))\n",
     "    ax1.set_xlabel('X-axis')\n",
     "    ax1.set_ylabel('Y-axis')\n",
-    "    ax1.set_zlabel('p_simulated')\n",
+    "    ax1.set_zlabel('p_sim')\n",
     "    ax1.set_xlim(0, x_lim)\n",
     "    ax1.set_ylim(0, y_lim)\n",
-    "    ax1.set_zlim(p_simulated[:, :, list_index].min(), p_simulated[:, :, list_index].max())\n",
+    "    ax1.set_zlim(p_sim[:, :, list_index].min(), p_sim[:, :, list_index].max())\n",
     "\n",
     "    # Exact pressure\n",
     "    ax2 = fig.add_subplot(len(z_plane_list), 3, 3 * i + 2, projection='3d')\n",
@@ -501,14 +489,14 @@
     "\n",
     "    # Error in pressure\n",
     "    ax3 = fig.add_subplot(len(z_plane_list), 3, 3 * i + 3, projection='3d')\n",
-    "    mappable = ax3.plot_surface(X[:, :, list_index], Y[:, :, list_index], error_p[:, :, list_index], cmap='viridis')\n",
+    "    mappable = ax3.plot_surface(X[:, :, list_index], Y[:, :, list_index], p_sim[:, :, list_index] - exact_p[:, :, list_index], cmap='viridis')\n",
     "    ax3.set_title('Error in Pressure on z={:.2f} plane'.format(z_plane))\n",
     "    ax3.set_xlabel('X-axis')\n",
     "    ax3.set_ylabel('Y-axis')\n",
     "    ax3.set_zlabel('Error in pressure')\n",
     "    ax3.set_xlim(0, x_lim)\n",
     "    ax3.set_ylim(0, y_lim)\n",
-    "    ax3.set_zlim(error_p[:, :, list_index].min(), error_p[:, :, list_index].max())\n",
+    "    ax3.set_zlim((p_sim[:, :, list_index] - exact_p[:, :, list_index]).min(), (p_sim[:, :, list_index] - exact_p[:, :, list_index]).max())\n",
     "    fig.colorbar(mappable, ax=ax3, shrink=0.5, aspect=5)\n",
     "\n",
     "plt.tight_layout()\n",