Created using Colab

The coupled equation in the time domain.ipynb

@@ -0,0 +1,103 @@
+{
+  "nbformat": 4,
+  "nbformat_minor": 0,
+  "metadata": {
+    "colab": {
+      "private_outputs": true,
+      "provenance": [],
+      "authorship_tag": "ABX9TyN0cY6Dqpal4FkyyfXXQdFk",
+      "include_colab_link": true
+    },
+    "kernelspec": {
+      "name": "python3",
+      "display_name": "Python 3"
+    },
+    "language_info": {
+      "name": "python"
+    }
+  },
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "id": "view-in-github",
+        "colab_type": "text"
+      },
+      "source": [
+        "<a href=\"https://colab.research.google.com/github/GEORMC/Nnumerical_Methods_Course/blob/main/The%20coupled%20equation%20in%20the%20time%20domain.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "execution_count": null,
+      "metadata": {
+        "id": "boju8AYKJmLI"
+      },
+      "outputs": [],
+      "source": [
+        "import numpy as np\n",
+        "\n",
+        "# Define parameters\n",
+        "theta = 0.5  # Time-stepping parameter\n",
+        "delta_t = 0.01  # Time step size\n",
+        "num_steps = 100  # Number of time steps\n",
+        "\n",
+        "# Example matrices (replace with your specific problem matrices)\n",
+        "n_dof = 3  # Degrees of freedom (example size)\n",
+        "K_e = np.array([[10, -2, 0], [-2, 10, -2], [0, -2, 10]])  # Example stiffness matrix\n",
+        "Q = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])  # Coupling matrix\n",
+        "S = np.eye(n_dof)  # Storage matrix\n",
+        "H = 0.1 * np.eye(n_dof)  # Hydraulic conductivity matrix\n",
+        "\n",
+        "# Initial conditions\n",
+        "u_n = np.zeros(n_dof)  # Initial displacement\n",
+        "p_n = np.zeros(n_dof)  # Initial pore pressure\n",
+        "f_u = np.zeros(n_dof)  # External force\n",
+        "f_p = np.zeros(n_dof)  # Fluid source\n",
+        "\n",
+        "# Time-stepping loop\n",
+        "u_history = [u_n]  # Displacement history\n",
+        "p_history = [p_n]  # Pressure history\n",
+        "\n",
+        "# Assemble constant matrices\n",
+        "LHS = np.block([\n",
+        "    [theta * K_e, -theta * Q],\n",
+        "    [Q.T, S + delta_t * theta * H]\n",
+        "])\n",
+        "\n",
+        "for step in range(num_steps):\n",
+        "    # Calculate RHS\n",
+        "    RHS_u = (1 - theta) * K_e @ u_n + f_u\n",
+        "    RHS_p = S @ p_n - (1 - theta) * delta_t * H @ p_n + delta_t * f_p\n",
+        "    RHS = np.hstack([RHS_u, RHS_p])\n",
+        "\n",
+        "    # Solve for next step\n",
+        "    solution = np.linalg.solve(LHS, RHS)\n",
+        "    u_n1, p_n1 = solution[:n_dof], solution[n_dof:]\n",
+        "\n",
+        "    # Update history\n",
+        "    u_history.append(u_n1)\n",
+        "    p_history.append(p_n1)\n",
+        "\n",
+        "    # Update for next iteration\n",
+        "    u_n, p_n = u_n1, p_n1\n",
+        "\n",
+        "# Post-process results\n",
+        "u_history = np.array(u_history)\n",
+        "p_history = np.array(p_history)\n",
+        "\n",
+        "# Example: plot results\n",
+        "import matplotlib.pyplot as plt\n",
+        "\n",
+        "plt.figure(figsize=(12, 6))\n",
+        "plt.plot(u_history, label=\"Displacement (u)\")\n",
+        "plt.plot(p_history, label=\"Pore Pressure (p)\", linestyle=\"--\")\n",
+        "plt.xlabel(\"Time Step\")\n",
+        "plt.ylabel(\"Value\")\n",
+        "plt.legend()\n",
+        "plt.title(\"Coupled System Solution\")\n",
+        "plt.show()\n"
+      ]
+    }
+  ]
+}