{ "cells": [ { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import pyvista\npyvista.OFF_SCREEN = True\npyvista.set_jupyter_backend('static')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n# BEAM2 reference geometry \u2014 Hermite cubic shape functions\n\nThe 2-node 3D beam element (Euler-Bernoulli slender-beam limit)\nmaps to the natural-coordinate segment $s \\in [-1, +1]$.\nTranslations and torsion use linear shape functions; transverse\ndisplacement and slope use **Hermite cubics** so $C^1$\ncontinuity holds across element boundaries.\n\nLinear shape functions (axial + torsion):\n\n\\begin{align}N_1^L(s) = \\tfrac{1 - s}{2}, \\qquad\n N_2^L(s) = \\tfrac{1 + s}{2}.\\end{align}\n\nHermite cubic shape functions (transverse displacement and\nslope, mapped to the physical length $L$):\n\n\\begin{align}H_1(\\xi) = 2\\xi^{3} - 3\\xi^{2} + 1, \\quad\n H_2(\\xi) = L\\,(\\xi^{3} - 2\\xi^{2} + \\xi),\\end{align}\n\n\\begin{align}H_3(\\xi) = -2\\xi^{3} + 3\\xi^{2}, \\quad\n H_4(\\xi) = L\\,(\\xi^{3} - \\xi^{2}),\\end{align}\n\nwith $\\xi = (s + 1)/2 \\in [0, 1]$. ``H_1`` and ``H_3``\ninterpolate the two nodal displacements; ``H_2`` and ``H_4``\ninterpolate the two nodal slopes.\n\n## References\n* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002)\n *Concepts and Applications of Finite Element Analysis*, 4th\n ed., Wiley, \u00a72.4\u2013\u00a72.6, Table 16.3-1.\n* Zienkiewicz, O. C., Taylor, R. L. (2013) *The Finite Element\n Method: Its Basis and Fundamentals*, 7th ed., \u00a72.5.1\n eqs. (2.26)\u2013(2.27).\n* Przemieniecki, J. S. (1968) *Theory of Matrix Structural\n Analysis*, McGraw-Hill, \u00a75.\n\nImplementation: :class:`femorph_solver.elements.beam2.Beam2`.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from __future__ import annotations\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pyvista as pv" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hermite cubic basis on [0, 1]\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "L = 1.0 # element length\nxi = np.linspace(0.0, 1.0, 200)\nH1 = 2.0 * xi**3 - 3.0 * xi**2 + 1.0\nH2 = L * (xi**3 - 2.0 * xi**2 + xi)\nH3 = -2.0 * xi**3 + 3.0 * xi**2\nH4 = L * (xi**3 - xi**2)\n\nfig, ax = plt.subplots(1, 1, figsize=(6.4, 4.0))\nax.plot(xi, H1, label=\"$H_1$ \u2014 node-1 displacement\", color=\"#1f77b4\", lw=2)\nax.plot(xi, H2, label=\"$H_2$ \u2014 node-1 slope ($\\\\theta_1$)\", color=\"#ff7f0e\", lw=2)\nax.plot(xi, H3, label=\"$H_3$ \u2014 node-2 displacement\", color=\"#2ca02c\", lw=2)\nax.plot(xi, H4, label=\"$H_4$ \u2014 node-2 slope ($\\\\theta_2$)\", color=\"#d62728\", lw=2)\nax.axhline(0.0, color=\"black\", lw=0.5)\nax.axhline(1.0, color=\"grey\", lw=0.5, ls=\":\")\nax.scatter([0.0, 1.0], [0.0, 0.0], color=\"black\", zorder=5)\nax.set_xlabel(r\"$\\xi = (s + 1) / 2$\")\nax.set_ylabel(\"$H_i(\\\\xi)$\")\nax.set_title(\"BEAM2 Hermite cubic shape functions ($L = 1$)\")\nax.legend(loc=\"upper center\", ncol=2, fontsize=9, framealpha=0.95)\nax.set_xlim(0.0, 1.0)\nax.set_ylim(-0.2, 1.05)\nax.grid(True, ls=\":\", alpha=0.5)\nfig.tight_layout()\nfig.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sanity \u2014 boundary conditions of the Hermite basis\n\nThe four Hermite cubics satisfy a Kronecker-delta-like property\nat the two endpoints $\\xi = 0$ (node 1) and\n$\\xi = 1$ (node 2):\n\n+-------+--------+--------+--------+--------+\n| | H1 | H2 | H3 | H4 |\n+-------+--------+--------+--------+--------+\n| \u03be=0 | 1 | 0 | 0 | 0 |\n| H' | 0 | L | 0 | 0 |\n| \u03be=1 | 0 | 0 | 1 | 0 |\n| H' | 0 | 0 | 0 | L |\n+-------+--------+--------+--------+--------+\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "vals = np.array([[H1[0], H2[0], H3[0], H4[0]], [H1[-1], H2[-1], H3[-1], H4[-1]]])\nnp.testing.assert_allclose(vals[0], [1, 0, 0, 0], atol=1e-12)\nnp.testing.assert_allclose(vals[1], [0, 0, 1, 0], atol=1e-12)\nprint(\"OK \u2014 Hermite cubics interpolate displacements and slopes at the nodes.\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2-node beam reference (s \u2208 [-1, +1])\n\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [ "cells = np.array([2, 0, 1])\ncell_types = np.array([pv.CellType.LINE], dtype=np.uint8)\nbeam = pv.UnstructuredGrid(cells, cell_types, np.array([[-1.0, 0.0, 0.0], [+1.0, 0.0, 0.0]]))\n\n# 2-point Gauss-Legendre on [-1, +1].\ng = 1.0 / np.sqrt(3.0)\ngauss = np.array([[-g, 0.0, 0.0], [+g, 0.0, 0.0]])\n\nplotter = pv.Plotter(off_screen=True, window_size=(640, 240))\nplotter.add_mesh(beam, color=\"black\", line_width=4)\nplotter.add_points(\n np.array([[-1.0, 0.0, 0.0], [+1.0, 0.0, 0.0]]),\n render_points_as_spheres=True,\n point_size=18,\n color=\"black\",\n label=\"end nodes (2)\",\n)\nplotter.add_points(\n gauss,\n render_points_as_spheres=True,\n point_size=14,\n color=\"#d62728\",\n label=\"2-pt Gauss-Legendre\",\n)\nplotter.view_xy()\nplotter.camera.zoom(1.6)\nplotter.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.12.3" } }, "nbformat": 4, "nbformat_minor": 0 }