Created using Colab

1DElement.ipynb

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+{
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+  "metadata": {
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+      "authorship_tag": "ABX9TyNYWx+I2cFrt4q7k9DFe8h8",
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+    {
+      "cell_type": "markdown",
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+      "source": [
+        "<a href=\"https://colab.research.google.com/github/GEORMC/Nnumerical_Methods_Course/blob/main/1DElement.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": "EVc6sM8n_6WC"
+      },
+      "outputs": [],
+      "source": [
+        "import numpy as np\n",
+        "import matplotlib.pyplot as plt\n",
+        "\n",
+        "# Define the function to assemble the global stiffness matrix\n",
+        "def assemble_stiffness_matrix(nodal_coords, nodal_connectivity, material_props):\n",
+        "    num_nodes = len(nodal_coords)\n",
+        "    K_global = np.zeros((num_nodes, num_nodes))\n",
+        "\n",
+        "    for element in nodal_connectivity:\n",
+        "        node1, node2 = element\n",
+        "        x1, x2 = nodal_coords[node1], nodal_coords[node2]\n",
+        "        length = x2 - x1\n",
+        "        k = material_props['E'] * material_props['A'] / length\n",
+        "\n",
+        "        # Element stiffness matrix for 1D linear element\n",
+        "        K_local = (k / length) * np.array([[1, -1], [-1, 1]])\n",
+        "\n",
+        "        # Assembly into the global stiffness matrix\n",
+        "        K_global[node1:node2+1, node1:node2+1] += K_local\n",
+        "\n",
+        "    return K_global\n",
+        "\n",
+        "# Apply boundary conditions (Dirichlet)\n",
+        "def apply_boundary_conditions(K, F, bc):\n",
+        "    for node, value in bc.items():\n",
+        "        # Modify stiffness matrix and force vector to enforce boundary condition\n",
+        "        K[node, :] = 0\n",
+        "        K[:, node] = 0\n",
+        "        K[node, node] = 1\n",
+        "        F[node] = value\n",
+        "    return K, F\n",
+        "\n",
+        "# Define function to solve FEM problem\n",
+        "def fem_1d(nodal_coords, nodal_connectivity, material_props, boundary_conditions, load):\n",
+        "    num_nodes = len(nodal_coords)\n",
+        "\n",
+        "    # Step 1: Assemble global stiffness matrix\n",
+        "    K_global = assemble_stiffness_matrix(nodal_coords, nodal_connectivity, material_props)\n",
+        "\n",
+        "    # Step 2: Assemble global force vector\n",
+        "    F_global = np.zeros(num_nodes)\n",
+        "    for node, value in load.items():\n",
+        "        F_global[node] = value\n",
+        "\n",
+        "    # Step 3: Apply boundary conditions\n",
+        "    K_global, F_global = apply_boundary_conditions(K_global, F_global, boundary_conditions)\n",
+        "\n",
+        "    # Step 4: Solve the system of equations\n",
+        "    displacements = np.linalg.solve(K_global, F_global)\n",
+        "\n",
+        "    return displacements\n",
+        "\n",
+        "# Example Input Data\n",
+        "nodal_coords = np.array([0.0, 0.5, 1.0])  # Coordinates of nodes\n",
+        "nodal_connectivity = np.array([[0, 1], [1, 2]])  # Elements defined by node numbers\n",
+        "material_props = {'E': 210e9, 'A': 1e-4}  # Material properties: E (Young's modulus) and A (Cross-sectional area)\n",
+        "boundary_conditions = {0: 0.0}  # Boundary condition: node 0 is fixed (Dirichlet condition)\n",
+        "load = {2: 1000.0}  # Load applied at node 2\n",
+        "\n",
+        "# Solve the FEM problem\n",
+        "displacements = fem_1d(nodal_coords, nodal_connectivity, material_props, boundary_conditions, load)\n",
+        "\n",
+        "# Output displacements\n",
+        "print(\"Nodal Displacements:\", displacements)\n",
+        "\n",
+        "# Plot the displacements\n",
+        "plt.plot(nodal_coords, displacements, '-o')\n",
+        "plt.xlabel('Position (m)')\n",
+        "plt.ylabel('Displacement (m)')\n",
+        "plt.title('1D FEM Nodal Displacements')\n",
+        "plt.grid(True)\n",
+        "plt.show()\n"
+      ]
+    }
+  ]
+}