wave3d_reflection.ipynb

{
  "cells": [
    {
      "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": "markdown",
      "metadata": {},
      "source": [
        "# Elastodynamik 3D"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Module laden"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "from IPython.display import display, Math, Latex\n",
        "import sympy as sp\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from ipywidgets import widgets\n",
        "from ipywidgets import interact\n",
        "from sympy.interactive import printing\n",
        "import plot_functions.plot_settings\n",
        "%run plot_functions/wave3d_reflection_interaction.ipynb\n",
        "printing.init_printing(use_latex='mathjax')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "_Last line enables special comment **##:** for LaTeX formatting of output_ "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Reflexion einer P-Welle am freien Rand"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "![wave3d_freeBC.svg](https://github.com/nagelt/soil_dynamics/raw/master/Images/wave3d_freeBC.svg)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Variablen für symbolische Rechnung (sympy) deklarieren"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [],
      "source": [
        "mu, Lambda, alpha, beta, AP_in, AP_re, AS_re, kappa_P, kappa_S, c_P, c_S, x,y,z,t = sp.symbols('mu Lambda alpha beta AP_in AP_re AS_re kappa_P kappa_S c_P c_S x y z t')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Phasen der einfallenden P-Welle, der reflektierten P-Welle und der reflektierten SH-Welle"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [],
      "source": [
        "phase_Pin = sp.I*kappa_P*(x*sp.sin(alpha) + y*sp.cos(alpha) - c_P*t)\n",
        "phase_Pre = sp.I*kappa_P*(x*sp.sin(alpha) - y*sp.cos(alpha) - c_P*t)\n",
        "phase_Sre = sp.I*kappa_S*(x*sp.sin( beta) - y*sp.cos( beta) - c_S*t)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Komplexe Darstellung der Wellenausbreitung aller drei Wellen"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {},
      "outputs": [],
      "source": [
        "eP_in = AP_in * sp.exp(phase_Pin)\n",
        "eP_re = AP_re * sp.exp(phase_Pre)\n",
        "eS_re = AS_re * sp.exp(phase_Sre)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Veschiebungen (x,y) der drei Wellenfelder"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [],
      "source": [
        "uP_in = sp.sin(alpha) * eP_in\n",
        "uP_re = sp.sin(alpha) * eP_re\n",
        "uS_re = sp.cos( beta) * eS_re"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {},
      "outputs": [],
      "source": [
        "vP_in = sp.cos(alpha) * eP_in\n",
        "vP_re =-sp.cos(alpha) * eP_re\n",
        "vS_re = sp.sin( beta) * eS_re"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Spannungen $\\sigma_{xy}$ und $\\sigma_{yy}$, die relevant für die Reflexion sind"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [],
      "source": [
        "sigma_xy_Pin = mu*( sp.diff(uP_in, y) + sp.diff(vP_in, x) )\n",
        "sigma_xy_Pre = mu*( sp.diff(uP_re, y) + sp.diff(vP_re, x) )\n",
        "sigma_xy_Sre = mu*( sp.diff(uS_re, y) + sp.diff(vS_re, x) )"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {},
      "outputs": [],
      "source": [
        "sigma_yy_Pin = (2*mu+Lambda)*sp.diff(vP_in, y) + Lambda*sp.diff(uP_in, x)\n",
        "sigma_yy_Pre = (2*mu+Lambda)*sp.diff(vP_re, y) + Lambda*sp.diff(uP_re, x)\n",
        "sigma_yy_Sre = (2*mu+Lambda)*sp.diff(vS_re, y) + Lambda*sp.diff(uS_re, x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Auswertung der Spannungen am Rand"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {},
      "outputs": [],
      "source": [
        "sigma_xy_BC = sp.simplify( sigma_xy_Pin*sp.exp(-phase_Pin)/sp.I + sigma_xy_Pre*sp.exp(-phase_Pre)/sp.I + sigma_xy_Sre*sp.exp(-phase_Sre)/sp.I)\n",
        "sigma_yy_BC = sp.simplify( sigma_yy_Pin*sp.exp(-phase_Pin)/sp.I + sigma_yy_Pre*sp.exp(-phase_Pre)/sp.I + sigma_yy_Sre*sp.exp(-phase_Sre)/sp.I)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/latex": [
              "$\\displaystyle \\sigma_{xy}^\\mathrm{BC}=\\mu \\left(AP_{in} \\kappa_{P} \\sin{\\left(2 \\alpha \\right)} - AP_{re} \\kappa_{P} \\sin{\\left(2 \\alpha \\right)} - AS_{re} \\kappa_{S} \\cos{\\left(2 \\beta \\right)}\\right)$"
            ],
            "text/plain": [
              "<IPython.core.display.Math object>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        },
        {
          "data": {
            "text/latex": [
              "$\\displaystyle \\sigma_{yy}^\\mathrm{BC}=AP_{in} \\kappa_{P} \\left(\\Lambda + 2 \\mu \\cos^{2}{\\left(\\alpha \\right)}\\right) + AP_{re} \\kappa_{P} \\left(\\Lambda + 2 \\mu \\cos^{2}{\\left(\\alpha \\right)}\\right) - AS_{re} \\kappa_{S} \\mu \\sin{\\left(2 \\beta \\right)}$"
            ],
            "text/plain": [
              "<IPython.core.display.Math object>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "display(Math(r'\\sigma_{xy}^\\mathrm{BC}=' + sp.latex(sigma_xy_BC))) \n",
        "display(Math(r'\\sigma_{yy}^\\mathrm{BC}=' + sp.latex(sigma_yy_BC))) "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Gleichungssystem der Randbedingungen (freier Rand) nach den Amplituden der reflektierten Wellen lösen"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "image/png": 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\n",
            "text/latex": [
              "$\\displaystyle \\left\\{ AP_{re} : \\frac{AP_{in} \\left(\\mu \\sin{\\left(2 \\alpha \\right)} \\sin{\\left(2 \\beta \\right)} - \\left(\\Lambda + 2 \\mu \\cos^{2}{\\left(\\alpha \\right)}\\right) \\cos{\\left(2 \\beta \\right)}\\right)}{\\mu \\sin{\\left(2 \\alpha \\right)} \\sin{\\left(2 \\beta \\right)} + \\left(\\Lambda + 2 \\mu \\cos^{2}{\\left(\\alpha \\right)}\\right) \\cos{\\left(2 \\beta \\right)}}, \\  AS_{re} : \\frac{2 AP_{in} \\kappa_{P} \\left(\\Lambda + 2 \\mu \\cos^{2}{\\left(\\alpha \\right)}\\right) \\sin{\\left(2 \\alpha \\right)}}{\\kappa_{S} \\left(\\mu \\sin{\\left(2 \\alpha \\right)} \\sin{\\left(2 \\beta \\right)} + \\left(\\Lambda + 2 \\mu \\cos^{2}{\\left(\\alpha \\right)}\\right) \\cos{\\left(2 \\beta \\right)}\\right)}\\right\\}$"
            ],
            "text/plain": [
              "⎧           ⎛                      ⎛           2   ⎞         ⎞                \n",
              "⎪      APᵢₙ⋅⎝μ⋅sin(2⋅α)⋅sin(2⋅β) - ⎝Λ + 2⋅μ⋅cos (α)⎠⋅cos(2⋅β)⎠                \n",
              "⎨APᵣₑ: ───────────────────────────────────────────────────────, ASᵣₑ: ────────\n",
              "⎪                                ⎛           2   ⎞                        ⎛   \n",
              "⎩          μ⋅sin(2⋅α)⋅sin(2⋅β) + ⎝Λ + 2⋅μ⋅cos (α)⎠⋅cos(2⋅β)           κ_S⋅⎝μ⋅s\n",
              "\n",
              "           ⎛           2   ⎞                  ⎫\n",
              "2⋅APᵢₙ⋅κ_P⋅⎝Λ + 2⋅μ⋅cos (α)⎠⋅sin(2⋅α)         ⎪\n",
              "──────────────────────────────────────────────⎬\n",
              "                   ⎛           2   ⎞         ⎞⎪\n",
              "in(2⋅α)⋅sin(2⋅β) + ⎝Λ + 2⋅μ⋅cos (α)⎠⋅cos(2⋅β)⎠⎭"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "solution = sp.solve( [sigma_xy_BC, sigma_yy_BC], [AP_re, AS_re] )\n",
        "for sol in solution:\n",
        "    solution[sol] = solution[sol].simplify()\n",
        "display(solution)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Grafische Darstellung der Reflexion"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "application/vnd.jupyter.widget-view+json": {
              "model_id": "82d8f1b783ca49aabb8f511763f45d17",
              "version_major": 2,
              "version_minor": 0
            },
            "text/plain": [
              "interactive(children=(IntSlider(value=45, description=r'$\\alpha$ [°]:', max=90), IntSlider(value=25, descripti…"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "interactive_reflection_plot()  "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Frage: In welchem Fall wird nur eine P-Welle reflektiert und in welchem nur eine S-Welle?"
      ]
    }
  ],
  "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": "3.9.18"
    },
    "widgets": {
      "application/vnd.jupyter.widget-state+json": {
        "state": {},
        "version_major": 2,
        "version_minor": 0
      }
    }
  },
  "nbformat": 4,
  "nbformat_minor": 4
}