extra_biot_oscillations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ed69d213",
   "metadata": {},
   "source": [
    "# Biot oscillations\n",
    "Supplementary material to _The essence of Biot waves in an oscillator with two degrees of freedom_ by [Dominik Kern](https://orcid.org/0000-0002-1958-2982) and [Thomas Nagel](https://orcid.org/0000-0001-8459-4616), 2024."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b1f94559",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sp\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import fsolve\n",
    "#import tikzplotlib   # time-consuming, enable only when needed to generate plots for the paper"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93ed51d1",
   "metadata": {},
   "source": [
    "Spatial discretization by the mode shape ($k$th mode) of an elastic bar and the assumption that the fluid moves proportional to the solid\n",
    "\\begin{align}\n",
    "u_\\text{f}(t,x) &= U_\\text{f}(t)\\sin\\biggl(\\Bigl(k+ \\frac{1}{2}\\Bigr)\\pi x \\biggr), \\\\\n",
    "u_\\text{s}(t,x) &= U_\\text{s}(t)\\sin\\biggl(\\Bigl(k+ \\frac{1}{2}\\Bigr)\\pi x \\biggr), \n",
    "\\end{align}\n",
    "corresponding to a bar of unit-length with boundary conditions fixed-symmetric for S-waves (one half of a fixed-fixed bar) or fixed-free for P-waves.\n",
    "Temporal discretization (exponential ansatz with $\\hat{U}_\\text{f}$, $\\hat{U}_\\text{s}$, $\\delta\\in\\mathbb{C}$), such that coefficients in characteristic equation remain in real domain ($a_i\\in\\mathbb{R}$)\n",
    "\\begin{align}\n",
    "U_\\text{f}(t) &= \\hat{U}_\\text{f}\\, e^{\\delta t}, \\\\\n",
    "U_\\text{s}(t) &=  \\hat{U}_\\text{s}\\, e^{\\delta t}.\n",
    "\\end{align}\n",
    "\n",
    "Note that the real part of the coefficient $\\mathfrak{R}\\{\\delta\\}$ describes decay and the imaginary part $\\mathfrak{I}\\{\\delta\\}$ oscillation.\n",
    "\n",
    "The original equations are related to domain length $L$ and reference time $T=L\\sqrt{m_\\text{v} \\rho_\\text{s}}$ to make them non-dimensional, note $c_\\mathrm{P}^\\mathrm{el}=\\frac{1}{\\sqrt{m_\\text{v} \\rho_\\text{s}}}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb9700fc",
   "metadata": {},
   "source": [
    "## Parameters\n",
    "The continuum parameters are\n",
    "- $L$ domain length [m],\n",
    "- $\\varrho_\\text{f}$ fluid density [kg m$^{-3}$],\n",
    "- $\\varrho_\\text{s}$ solid density [kg m$^{-3}$],\n",
    "- $m_\\text{v}$ one-dimensional soil compressibility (inverse P-wave modulus) [Pa$^{-1}$],\n",
    "- $G$ shear modulus (bulk) [Pa],\n",
    "- $S_\\text{p}$ storativity [Pa$^{-1}$],\n",
    "- $\\mu$ viscosity [Pa s],\n",
    "- $\\kappa$ permeability [m$^2$],\n",
    "- $\\alpha$ Biot coefficient,\n",
    "- $n$ porosity,\n",
    "- $\\tau$ tortuosity factor."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "597c7d09",
   "metadata": {},
   "source": [
    "Declare variables for dimensional and non-dimensional parameters (symbolic computations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "5c29b39e",
   "metadata": {},
   "outputs": [],
   "source": [
    "L, G, m_v, n, S, alpha, kappa, mu, rho_f, rho_s, tau = sp.symbols('L G m_v n S alpha kappa mu rho_f rho_s tau') \n",
    "I_11, I_12, I_22, M, D_P, D_S, RHO, x, k, g = sp.symbols('I_11 I_12 I_22 M D_P D_S RHO x k g') "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff620c9d",
   "metadata": {},
   "source": [
    "Example of some soil [Verruijt2010: table 5.1], except $\\tau\\ne 0$ to check the influence of tortuosity and bulk compressibility $m_\\text{v}<S_\\text{}p$ to make frequency dependency visible. A Poisson ratio $\\nu=0.2$ is assumed to derive the shear modulus from the bulk modulus (for S-waves)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e1c85085",
   "metadata": {},
   "outputs": [],
   "source": [
    "rho_s_num = 2650\n",
    "rho_f_num = 1000\n",
    "kappa_num = 1e-10   # note, Verruijt lists hydraulic conductivity k=0.001 m/s\n",
    "n_num = 0.4\n",
    "tau_num = 0.1   # different from Verruijt's example (tau=0)\n",
    "alpha_num = 1.0\n",
    "m_v_num = 2e-9   # different from Verruijt's example (mb_v=2e-10)\n",
    "S_num = n_num*5e-10 + (alpha_num-n_num)*0   # S_p = n*C_f + (alpha-n)*C_s\n",
    "mu_num = 1e-3\n",
    "\n",
    "nu_num = 0.2   # Poisson ratio\n",
    "G_num = (1/m_v_num)*(1-2*nu_num)/(2*(1-nu_num))   # shear modulus"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b542af2c",
   "metadata": {},
   "source": [
    "Bar length, plausible values are some meters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "0b30c2a3",
   "metadata": {},
   "outputs": [],
   "source": [
    "L_num = 20    # influences eigenfrequency range, set to center it around f_crit"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0de6e1b6",
   "metadata": {},
   "source": [
    "Weight function integrals $I_{ij} = \\int_0^1 \\tilde{w}_i(x)\\tilde{w}_j(x)\\,\\text{d}x$ as they occur in Galerkin's method (introduce separate values to indicate where different weight functions would enter, although same value here)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "88858454",
   "metadata": {},
   "outputs": [],
   "source": [
    "I_11_k = sp.integrate(sp.sin(x*sp.pi*(1/2+k))**2, (x,0,1))\n",
    "I_12_k = sp.integrate(sp.sin(x*sp.pi*(1/2+k))**2, (x,0,1))\n",
    "I_22_k = sp.integrate(sp.sin(x*sp.pi*(1/2+k))**2, (x,0,1))\n",
    "k0 = 0 # base mode discretization\n",
    "I_11_k0 = float(I_11_k.subs(k, k0))\n",
    "I_12_k0 = float(I_12_k.subs(k, k0))\n",
    "I_22_k0 = float(I_22_k.subs(k, k0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20497933",
   "metadata": {},
   "source": [
    "Lists to substitute numerical values into analytic expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "54f9805d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(L, 20),\n",
       " (n, 0.4),\n",
       " (alpha, 1.0),\n",
       " (G, 187499999.99999994),\n",
       " (m_v, 2e-09),\n",
       " (S, 2.0000000000000003e-10),\n",
       " (rho_f, 1000),\n",
       " (rho_s, 2650),\n",
       " (kappa, 1e-10),\n",
       " (mu, 0.001),\n",
       " (tau, 0.1)]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "continuum_num = [(L, L_num),\n",
    "                 (n, n_num),\n",
    "                 (alpha,   alpha_num),\n",
    "                 (G, G_num),\n",
    "                 (m_v, m_v_num),\n",
    "                 (S, S_num),\n",
    "                 (rho_f, rho_f_num), \n",
    "                 (rho_s, rho_s_num), \n",
    "                 (kappa, kappa_num), \n",
    "                 (mu, mu_num), \n",
    "                 (tau, tau_num)]\n",
    "\n",
    "display(continuum_num)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "846ba26a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(I_11, 0.5),\n",
       " (I_12, 0.5),\n",
       " (I_22, 0.5),\n",
       " (D_S, 113.49230434110538),\n",
       " (D_P, 69.4995588420911),\n",
       " (n, 0.4),\n",
       " (alpha, 1.0),\n",
       " (M, 9.999999999999998),\n",
       " (RHO, 2.65),\n",
       " (tau, 0.1)]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "RHO_num = float((rho_s/rho_f).subs(continuum_num).evalf())\n",
    "M_num =   float((m_v/S).subs(continuum_num).evalf())\n",
    "D_S_num = float((L*n*mu/(kappa*(G*rho_s)**(1/2))).subs(continuum_num).evalf())\n",
    "D_P_num = float((L*n*mu*sp.sqrt(m_v*rho_s)/(kappa*rho_s)).subs(continuum_num).evalf())\n",
    "\n",
    "nondim_num_k0 = [(I_11, I_11_k0),\n",
    "                 (I_12, I_12_k0),\n",
    "                 (I_22, I_22_k0),\n",
    "                 (D_S, D_S_num),\n",
    "                 (D_P, D_P_num),\n",
    "                 (n,   n_num),\n",
    "                 (alpha,   alpha_num),\n",
    "                 (M, M_num),\n",
    "                 (RHO, RHO_num),\n",
    "                 (tau, tau_num)]\n",
    "\n",
    "display(nondim_num_k0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee6f27cb",
   "metadata": {},
   "source": [
    "Non-dimensional Biot's characteristic frequency [Biot1956: equation (7.4)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6dfff2c6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f_crit = 29.312175580925828\n"
     ]
    }
   ],
   "source": [
    "f_characteristic = (mu*n/(2*sp.pi*kappa*rho_f))*L*(m_v*rho_s)**(1/2)\n",
    "f_characteristic_num = float(f_characteristic.subs(continuum_num).evalf())   \n",
    "print(\"f_crit = {}\".format(f_characteristic_num))   # non-dimensional"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f4db004",
   "metadata": {},
   "source": [
    "Wave velocities (elastic S, elastic P, undrained P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "b319a908",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c_el_S = 265.997588299466 m/s\n",
      "c_el_P = 434.372242763069 m/s\n",
      "c_un_S = 306.954565901272 m/s\n",
      "c_un_P = 1662.47378790686 m/s\n"
     ]
    }
   ],
   "source": [
    "c_el_S = (G/rho_s)**(1/2)\n",
    "c_el_S_num = c_el_S.subs(continuum_num).evalf()\n",
    "c_el_P = (m_v*rho_s)**(-1/2)\n",
    "c_el_P_num = c_el_P.subs(continuum_num).evalf()\n",
    "c_un_S = (G/((1-n)*rho_s + n*rho_f))**(1/2)    # drained is same as elastic\n",
    "c_un_S_num = c_un_S.subs(continuum_num).evalf()\n",
    "c_un_P = (((1/m_v) + (alpha**2)/S)/((1-n)*rho_s + n*rho_f))**(1/2)\n",
    "c_un_P_num = c_un_P.subs(continuum_num).evalf()\n",
    "print(\"c_el_S = {} m/s\".format(c_el_S_num))\n",
    "print(\"c_el_P = {} m/s\".format(c_el_P_num))\n",
    "print(\"c_un_S = {} m/s\".format(c_un_S_num))\n",
    "print(\"c_un_P = {} m/s\".format(c_un_P_num))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bfc932e",
   "metadata": {},
   "source": [
    "Variables for MBS models (same variables, but different expressions for S- and P-waves)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "9ef1e133",
   "metadata": {},
   "outputs": [],
   "source": [
    "m_1, m_2, m_Delta, d_Delta, c_1, c_2, c_Delta, delta= sp.symbols('m_1 m_2 m_Delta d_Delta c_1 c_2 c_Delta delta')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de27edaf",
   "metadata": {},
   "source": [
    "## S-waves\n",
    "Use isochronic one-term Galerkin discretization of the poroelastic bar (1D) in non-dimensional formulation, with fixed BC on the left (x=0), symmetry BC on the right (x=1), to illustrate the motion of shear waves, since oscillations are standing waves and simpler to describe while wave velocity, wave length and frequency are still related ($c=\\frac{\\omega}{k}$ in non-dissipative case, otherwise close by).\n",
    "\n",
    "Non-dimensional equations of motion for fluid phase (total displacement $u_\\mathrm{f}$) and solid phase (total displacement $u_\\mathrm{s}$)\n",
    "\\begin{align}\n",
    "\\frac{n}{\\rho}\\ddot{u}_\\text{f} + \\frac{n\\tau}{\\rho}(\\ddot{u}_\\text{f} - \\ddot{u}_\\text{s}) + n D_S (\\dot{u}_\\text{f}-\\dot{u}_\\text{s}) &&=0, \\\\\n",
    "(1-n)\\ddot{u}_\\text{s} - \\frac{n\\tau}{\\rho}(\\ddot{u}_\\text{f} - \\ddot{u}_\\text{s}) - n D_S(\\dot{u}_\\text{f}-\\dot{u}_\\text{s})& - u_\\text{s}'' &=0,\n",
    "\\end{align}\n",
    "with $\\rho = \\frac{\\varrho_\\mathrm{s}}{\\varrho_\\mathrm{f}}$, and $D_\\mathrm{S}=\\frac{n \\mu L}{\\kappa \\sqrt{G \\varrho_\\mathrm{s}}}$, note the relation $D_\\mathrm{S} = \\frac{c_\\mathrm{P}^\\mathrm{el}}{c_\\mathrm{S}^\\mathrm{el}} D_\\mathrm{P}$ with $\\frac{c_\\mathrm{P}^\\mathrm{el}}{c_\\mathrm{S}^\\mathrm{el}} = \\sqrt{\\frac{1}{m_\\mathrm{v} G}}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8afe415c",
   "metadata": {},
   "source": [
    "### Multi-body system matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ecaa8d8",
   "metadata": {},
   "source": [
    "Conversion between multi-body and non-dimensional continuum parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "5fe39e7f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "m_1(k=0) = 0.0754716981132076\n",
      "m_2(k=0) = 0.300000000000000\n",
      "m_Delta(k=0) = 0.00754716981132076\n",
      "d_Delta(k=0) = 22.6984608682211\n",
      "c_2(k=0) = 1.23370055013617\n"
     ]
    }
   ],
   "source": [
    "mbs2nondim_S = [(m_1,     I_11*(n/RHO)), \n",
    "                (m_2,     I_22*(1-n)), \n",
    "                (m_Delta, I_12*n*tau/RHO), \n",
    "                (d_Delta, I_12*n*D_S), \n",
    "                (c_2,     I_22*(sp.pi*(1/2+k))**2)]\n",
    "\n",
    "print(\"m_1(k=0) = {}\".format(m_1.subs(mbs2nondim_S).subs(nondim_num_k0).evalf()))\n",
    "print(\"m_2(k=0) = {}\".format(m_2.subs(mbs2nondim_S).subs(nondim_num_k0).evalf()))\n",
    "print(\"m_Delta(k=0) = {}\".format(m_Delta.subs(mbs2nondim_S).subs(nondim_num_k0).evalf()))\n",
    "print(\"d_Delta(k=0) = {}\".format(d_Delta.subs(mbs2nondim_S).subs(nondim_num_k0).evalf()))\n",
    "print(\"c_2(k=0) = {}\".format(c_2.subs(mbs2nondim_S).subs(nondim_num_k0 + [(k, k0)]).evalf()))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80deb975",
   "metadata": {},
   "source": [
    "mass matrix (symmetric)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "4cee07b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}m_{1} + m_{\\Delta} & - m_{\\Delta}\\\\- m_{\\Delta} & m_{2} + m_{\\Delta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[m_1 + m_Delta,      -m_Delta],\n",
       "[     -m_Delta, m_2 + m_Delta]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mass_S = sp.Matrix(((m_1+m_Delta, -m_Delta), (-m_Delta, m_2+m_Delta)))\n",
    "display(mass_S)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "793915e2",
   "metadata": {},
   "source": [
    "damping matrix (symmetric)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "217df3a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}d_{\\Delta} & - d_{\\Delta}\\\\- d_{\\Delta} & d_{\\Delta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ d_Delta, -d_Delta],\n",
       "[-d_Delta,  d_Delta]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "damping_S = sp.Matrix(((d_Delta, -d_Delta), (-d_Delta, d_Delta)))\n",
    "display(damping_S)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0e86788",
   "metadata": {},
   "source": [
    "stiffness matrix (symmetric)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8de78f68",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & c_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0,   0],\n",
       "[0, c_2]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "stiffness_S = sp.Matrix(((0, 0), (0, c_2)))\n",
    "display(stiffness_S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "efa44a40",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle c_{2} d_{\\Delta} \\delta + c_{2} \\delta^{2} m_{1} + c_{2} \\delta^{2} m_{\\Delta} + d_{\\Delta} \\delta^{3} m_{1} + d_{\\Delta} \\delta^{3} m_{2} + \\delta^{4} m_{1} m_{2} + \\delta^{4} m_{1} m_{\\Delta} + \\delta^{4} m_{2} m_{\\Delta}$"
      ],
      "text/plain": [
       "c_2*d_Delta*delta + c_2*delta**2*m_1 + c_2*delta**2*m_Delta + d_Delta*delta**3*m_1 + d_Delta*delta**3*m_2 + delta**4*m_1*m_2 + delta**4*m_1*m_Delta + delta**4*m_2*m_Delta"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(sp.Determinant(mass_S*delta**2+damping_S*delta+stiffness_S))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "95f578a6",
   "metadata": {},
   "source": [
    "_Swap relation_ [Hagedorn2012], to show that damping matrix cannot be diagonalized (except $c_2=0$ or $d_\\Delta=0$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "47a2e453",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{c_{2} d_{\\Delta} m_{1} m_{\\Delta}}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "c_2*d_Delta*m_1*m_Delta/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{c_{2} d_{\\Delta} \\left(- m_{1} m_{\\Delta} + m_{2} m_{\\Delta} - m_{2} \\left(m_{1} + m_{\\Delta}\\right)\\right)}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "c_2*d_Delta*(-m_1*m_Delta + m_2*m_Delta - m_2*(m_1 + m_Delta))/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{c_{2} d_{\\Delta} m_{1} \\left(m_{1} + m_{\\Delta}\\right)}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "c_2*d_Delta*m_1*(m_1 + m_Delta)/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{c_{2} d_{\\Delta} m_{1} m_{\\Delta}}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "-c_2*d_Delta*m_1*m_Delta/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "inv_mass_S = sp.Matrix.inv(mass_S)\n",
    "zero_or_not = (inv_mass_S*damping_S)*(inv_mass_S*stiffness_S) - (inv_mass_S*stiffness_S)*(inv_mass_S*damping_S)\n",
    "display(sp.simplify(zero_or_not[0,0]))\n",
    "display(sp.simplify(zero_or_not[0,1]))\n",
    "display(sp.simplify(zero_or_not[1,0]))\n",
    "display(sp.simplify(zero_or_not[1,1])) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cea5d110",
   "metadata": {},
   "source": [
    "### Free vibrations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53e2a075",
   "metadata": {},
   "source": [
    "Eigenproblem and characteristic equation $a_4 \\delta^4 + a_3 \\delta^3 + a_2 \\delta^2 + a_1 \\delta + a_0 = 0$ with $a_4,a_3,a_2,a_1\\in\\mathbb{R}^+$ and $a_0=0$ on physical grounds ($0\\le \\tau$,$\\rho_\\mathrm{f}$,$\\rho_\\mathrm{s}$,$G$,$\\kappa$,$\\mu$ and $0\\le n\\le 1$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "6a25e904",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\delta \\left(c_{2} d_{\\Delta} + \\delta^{3} \\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right) + \\delta^{2} \\left(d_{\\Delta} m_{1} + d_{\\Delta} m_{2}\\right) + \\delta \\left(c_{2} m_{1} + c_{2} m_{\\Delta}\\right)\\right)$"
      ],
      "text/plain": [
       "delta*(c_2*d_Delta + delta**3*(m_1*m_2 + m_1*m_Delta + m_2*m_Delta) + delta**2*(d_Delta*m_1 + d_Delta*m_2) + delta*(c_2*m_1 + c_2*m_Delta))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle c_{2} d_{\\Delta} + \\delta^{3} \\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right) + \\delta^{2} \\left(d_{\\Delta} m_{1} + d_{\\Delta} m_{2}\\right) + \\delta \\left(c_{2} m_{1} + c_{2} m_{\\Delta}\\right)$"
      ],
      "text/plain": [
       "c_2*d_Delta + delta**3*(m_1*m_2 + m_1*m_Delta + m_2*m_Delta) + delta**2*(d_Delta*m_1 + d_Delta*m_2) + delta*(c_2*m_1 + c_2*m_Delta)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evp_mbs_S = (delta**2)*mass_S + delta*damping_S + stiffness_S\n",
    "ceq_mbs_S = sp.simplify(sp.Determinant(evp_mbs_S).doit())\n",
    "ceq_mbs_reduced_S = ceq_mbs_S/delta\n",
    "display(ceq_mbs_S.collect(delta))\n",
    "display(ceq_mbs_reduced_S.collect(delta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "d6f5f2b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{n \\left(D_{S} I_{12} I_{22} \\rho^{2} \\left(\\delta^{2} \\left(1 - n\\right) + \\pi^{2} \\left(k + 0.5\\right)^{2}\\right) + I_{11} I_{12} \\delta^{3} n \\tau + \\rho \\delta \\left(D_{S} I_{11} I_{12} \\delta n + \\pi^{2} I_{11} I_{22} \\left(k + 0.5\\right)^{2} + \\pi^{2} I_{12} I_{22} \\tau \\left(k + 0.5\\right)^{2} + \\delta^{2} \\left(- I_{11} I_{22} \\left(n - 1\\right) - I_{12} I_{22} \\tau \\left(n - 1\\right)\\right)\\right)\\right)}{\\rho^{2}}$"
      ],
      "text/plain": [
       "n*(D_S*I_12*I_22*RHO**2*(delta**2*(1 - n) + pi**2*(k + 0.5)**2) + I_11*I_12*delta**3*n*tau + RHO*delta*(D_S*I_11*I_12*delta*n + pi**2*I_11*I_22*(k + 0.5)**2 + pi**2*I_12*I_22*tau*(k + 0.5)**2 + delta**2*(-I_11*I_22*(n - 1) - I_12*I_22*tau*(n - 1))))/RHO**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evp_nondim_S = evp_mbs_S.subs(mbs2nondim_S)\n",
    "ceq_nondim_reduced_S = sp.simplify(ceq_mbs_reduced_S.subs(mbs2nondim_S))\n",
    "#display(evp_continuum_S)\n",
    "display(ceq_nondim_reduced_S.collect(delta))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca84011c",
   "metadata": {},
   "source": [
    "#### Numerical evaluation (example)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "216c7ec7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.0830188679245283 \\delta^{2} + 22.6984608682211 \\delta & - 0.00754716981132076 \\delta^{2} - 22.6984608682211 \\delta\\\\- 0.00754716981132076 \\delta^{2} - 22.6984608682211 \\delta & 0.307547169811321 \\delta^{2} + 22.6984608682211 \\delta + 1.23370055013617\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  0.0830188679245283*delta**2 + 22.6984608682211*delta,                 -0.00754716981132076*delta**2 - 22.6984608682211*delta],\n",
       "[-0.00754716981132076*delta**2 - 22.6984608682211*delta, 0.307547169811321*delta**2 + 22.6984608682211*delta + 1.23370055013617]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.000569597721609114 \\delta^{3} + 6.80953826046632 \\delta^{2} + 0.150943396226415 \\delta \\left(0.165 \\delta^{2} + 11.3492304341105 \\delta + 0.678535302574893\\right) + 28.0031036603687$"
      ],
      "text/plain": [
       "0.000569597721609114*delta**3 + 6.80953826046632*delta**2 + 0.150943396226415*delta*(0.165*delta**2 + 11.3492304341105*delta + 0.678535302574893) + 28.0031036603687"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evp_num_S = evp_nondim_S.subs(nondim_num_k0 + [(k,k0)]).evalf()\n",
    "ceq_num_reduced_S = ceq_nondim_reduced_S.subs(nondim_num_k0 + [(k,k0)]).evalf()\n",
    "display(evp_num_S)\n",
    "display(ceq_num_reduced_S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b0d0087d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-334.54317408329626+0j),\n",
       " (-0.0010979619344772904-1.8126653015384215j),\n",
       " (-0.0010979619344772904+1.8126653015384215j)]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eigenvalue_sp_S = sp.solve(ceq_num_reduced_S, delta)\n",
    "eigenvalue_num_S = [complex(EVSP) for EVSP in eigenvalue_sp_S]\n",
    "display(eigenvalue_num_S)\n",
    "number_of_eigenvalues = len(eigenvalue_num_S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "d54e3b6e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- 0. mode ---\n",
      "eigenvalue = (-334.54317408329626+0j)  \n",
      "eigenvector = [(0.9697847794016882+0j), (-0.24396204959136356+0j)]\n",
      "--- 1. mode ---\n",
      "eigenvalue = (-0.0010979619344772904-1.8126653015384215j)  \n",
      "eigenvector = [(0.7071003661969636+0j), (0.7071003533653427-0.004261736116557033j)]\n",
      "--- 2. mode ---\n",
      "eigenvalue = (-0.0010979619344772904+1.8126653015384215j)  \n",
      "eigenvector = [(0.7071003661969636+0j), (0.7071003533653427+0.004261736116557033j)]\n",
      "t_scale = 0.002973041888700111\n",
      "Tr0 = 2.989151e-03   Tr1 = 9.107784e+02   Ti1 = 3.466269e+00\n"
     ]
    }
   ],
   "source": [
    "eigenvector_num_S = np.zeros((number_of_eigenvalues, 2), dtype=complex)\n",
    "for mode_n, delta_n in enumerate(eigenvalue_num_S):\n",
    "    EVP_S = evp_num_S.subs(delta, delta_n)\n",
    "    ev0r = np.array([1.0, -EVP_S[0,0].evalf()/EVP_S[0,1].evalf()], dtype=complex)   # first row --> eigenvector\n",
    "    ev0r_norm = np.linalg.norm(ev0r) \n",
    "    if ev0r_norm>0:\n",
    "        ev0rn = ev0r/ev0r_norm  # normalize\n",
    "    else:\n",
    "        ev0rn = ev0r\n",
    "    #ev1r = np.array([1.0, -EVP_S[1,0].evalf()/EVP_S[1,1].evalf()], dtype=complex)   # second row, redundant\n",
    "    #ev1rn = ev1r/np.linalg.norm(ev1r)   # normalize  \n",
    "    print(\"--- {}. mode ---\".format(mode_n))\n",
    "    print(\"eigenvalue = {}  \".format(delta_n))\n",
    "    print(\"eigenvector = [{}, {}]\".format(ev0rn[0], ev0rn[1]))\n",
    "    #print(\"eigenvector = [{}, {}] (redundant)\".format(ev1rn[0], ev1rn[1]))\n",
    "    eigenvector_num_S[mode_n,:] = ev0rn\n",
    "    \n",
    "t_scale = 1.0/(np.abs(min(np.imag(eigenvalue_num_S))) + np.abs(min(np.real(eigenvalue_num_S))))\n",
    "print(\"t_scale = {}\".format(t_scale))\n",
    "t_decay0 = 1/np.abs(np.real(eigenvalue_num_S[0]))\n",
    "t_decay1 = 1/np.abs(np.real(eigenvalue_num_S[1]))\n",
    "t_oscillation1 = 2*np.pi/np.abs(np.imag(eigenvalue_num_S[1]))\n",
    "print(\"Tr0 = {:e}   Tr1 = {:e}   Ti1 = {:e}\".format(t_decay0, t_decay1, t_oscillation1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c8d215d",
   "metadata": {},
   "source": [
    "#### Visualize modes\n",
    "The real root leads to exponential decay, note that state of rest can be for any relative displacement between masses $m_1$ and $m_2$. The pair of complex-conjugated roots leads to damped oscillations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "47054775",
   "metadata": {},
   "outputs": [],
   "source": [
    "t_modeshow = np.linspace(0, 25*t_scale)   # try to show oscillation and decay\n",
    "timesteps_modeshow = len(t_modeshow)\n",
    "u_f = np.zeros((number_of_eigenvalues, timesteps_modeshow))\n",
    "u_s = np.zeros((number_of_eigenvalues, timesteps_modeshow))\n",
    "\n",
    "for mode_n, delta_n in enumerate(eigenvalue_num_S):\n",
    "    u_f[mode_n,:] = np.real(eigenvector_num_S[mode_n,0]*np.exp(delta_n*t_modeshow))\n",
    "    u_s[mode_n,:] = np.real(eigenvector_num_S[mode_n,1]*np.exp(delta_n*t_modeshow))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce97d8ac",
   "metadata": {},
   "source": [
    "Note that for complex-conjugated pairs the real parts of its contributions are identical (imaginary parts cancel on summation)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "5391757a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_modeshow,u_f[0,:],'b',   t_modeshow,u_s[0,:],'k');\n",
    "plt.xlabel(\"$t$\");\n",
    "plt.ylabel(\"$u_\\\\mathrm{f}$,$u_\\\\mathrm{s}$\");\n",
    "#plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ac68354",
   "metadata": {},
   "source": [
    "Solid (black) and fluid (blue) move towards each other and slow down to rest."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "9d637e2d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_modeshow, u_f[1,:],'b',   t_modeshow, u_s[1,:],'k');\n",
    "plt.xlabel(\"$t$\");\n",
    "plt.ylabel(\"$u_\\\\mathrm{f}$,$u_\\\\mathrm{s}$\");\n",
    "#plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17aa6510",
   "metadata": {},
   "source": [
    "Solid (black) and fluid (blue) motion are almost in phase with almost same amplitudes.  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bcd4724",
   "metadata": {},
   "source": [
    "Note, that second root of complex-conjugated pair gives the same plot as first."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "552fe957",
   "metadata": {},
   "source": [
    "### Discriminant\n",
    "There is a discriminant for cubic polynomials indicating the number of real roots. However, for now we claim that there always is the rigid body mode ($\\delta_0=0$), a decaying mode ($\\delta_1\\in\\mathbb{R}$ and $\\delta_1<0$) and an oscillation mode ($\\delta_2, \\delta_3\\in\\mathbb{C}$ and $\\delta_2=\\bar{\\delta}_3$), i.e. S-waves exist always for physically plausible values. \n",
    "For the proof, we refer to this characteristic polynomial (S-waves) as a special case ($c_1=c_\\Delta=0$) of the more general characteristic polynomial for the P-wave model in the next section (same structure, only different values of the coefficients)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "738852e7",
   "metadata": {},
   "source": [
    "### Higher modes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5baacc40",
   "metadata": {},
   "source": [
    "Characteristics of S-waves (takes some seconds)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "e9c5d40d",
   "metadata": {},
   "outputs": [],
   "source": [
    "Nk = 1000\n",
    "d = np.zeros(Nk)   # decay constant\n",
    "c = np.zeros(Nk)   # phase velocity approximation\n",
    "f = np.zeros(Nk)   # frequency\n",
    "AR = np.zeros(Nk)   # amplitude ratio\n",
    "phi = np.zeros(Nk)   # phase angle\n",
    "\n",
    "index_S = 1    # second and third eigenvalue compose wave mode\n",
    "\n",
    "for k_num in range(Nk):\n",
    "    \n",
    "    I_11_num = float(I_11_k.subs(k, k_num))\n",
    "    I_12_num = float(I_12_k.subs(k, k_num))\n",
    "    I_22_num = float(I_22_k.subs(k, k_num))\n",
    "    \n",
    "    # redefine, since I_ij changes with higher modes\n",
    "    nondim_num_k = [(I_11,  I_11_num),\n",
    "                    (I_12,  I_12_num),\n",
    "                    (I_22,  I_22_num),\n",
    "                    (D_S,   D_S_num),\n",
    "                    (D_P,   D_P_num),\n",
    "                    (n,     n_num),\n",
    "                    (alpha, alpha_num),\n",
    "                    (M,     M_num),\n",
    "                    (RHO,   RHO_num),\n",
    "                    (tau,   tau_num)]\n",
    "    \n",
    "    ceq_num_reduced_Sk = ceq_nondim_reduced_S.subs(nondim_num_k + [(k, k_num)]).evalf()\n",
    "    eigenvalue_sp = sp.solve(ceq_num_reduced_Sk, delta)\n",
    "    eigenvalue_num = [complex(EVSP) for EVSP in eigenvalue_sp]\n",
    "    d[k_num] = np.real(eigenvalue_num[index_S])\n",
    "    omega = np.abs(np.imag(eigenvalue_num[index_S]))\n",
    "    c[k_num] = omega/((k_num+1/2)*np.pi)    \n",
    "    f[k_num] = omega/(2*np.pi)\n",
    "    \n",
    "    evp_num_Sk = evp_nondim_S.subs(nondim_num_k + [(k, k_num)]).evalf()\n",
    "    EVP = evp_num_Sk.subs(delta, eigenvalue_num[index_S])\n",
    "    evkr = np.array([1.0, -EVP[0,0].evalf()/EVP[0,1].evalf()], dtype=complex)   # first row --> eigenvector\n",
    "    us_uf = evkr[1]/evkr[0]   # TODO exclude division by zero\n",
    "    AR[k_num] = 1.0/np.absolute(us_uf)   # fluid-to-solid\n",
    "    phi[k_num] = np.angle(us_uf)   #np.abs(np.angle(us_uf))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "33b00d8c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nondimensional c_el_S = 0.612372435695794\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"nondimensional c_el_S = {}\".format(c_el_S_num/c_el_P_num))\n",
    "\n",
    "fig1, ax1 = plt.subplots()\n",
    "ax2 = ax1.twinx()\n",
    "ax1.semilogx(f, c, 'g')\n",
    "ax2.semilogx(f, np.abs(d)/f, 'b')\n",
    "ax1.set_xlabel('frequency');\n",
    "ax1.set_xticks(list(ax1.get_xticks()) + [f_characteristic_num], list(ax1.get_xticklabels()) + ['$f_c$'])   # with extra tick\n",
    "ax1.set_ylabel('phase velocity', color='g');\n",
    "ax2.set_ylabel('decay constant', color='b');\n",
    "#tikzplotlib.save(\"s_wave_eigenvalue.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW')  \n",
    "\n",
    "fig2, ax3 = plt.subplots()\n",
    "ax4 = ax3.twinx()\n",
    "ax3.semilogx(f, AR, 'g')\n",
    "ax4.semilogx(f, phi, 'b')\n",
    "ax3.set_xlabel('frequency');\n",
    "ax3.set_xticks(list(ax3.get_xticks()) + [f_characteristic_num], list(ax3.get_xticklabels()) + ['$f_c$'])   # with extra tick\n",
    "ax3.set_ylabel('amplitude ratio', color='g');\n",
    "ax4.set_ylabel('phase angle [RAD]', color='b');\n",
    "#tikzplotlib.save(\"s_wave_eigenvector.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW') "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72ef048f",
   "metadata": {},
   "source": [
    "Numbers for TikZ-plots (complex amplitudes and time history)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "1c63917b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "318.749559645193\n",
      "0.09413243679788895\n",
      "0.10577309551358847\n",
      "-0.12230081864595586\n"
     ]
    }
   ],
   "source": [
    "k_num = 500   # 0, 50, 500\n",
    "print(f[k_num])\n",
    "print(-d[k_num]/f[k_num])   # delta, with time axis scaled to period\n",
    "print(AR[k_num]*np.cos(phi[k_num])) # u_fluid, real part, with u_solid = 1\n",
    "print(AR[k_num]*np.sin(phi[k_num])) # u_fluid, imaginary part, with u_solid = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1a2ca07",
   "metadata": {},
   "source": [
    "## P-waves\n",
    "Non-dimensional equations of motion for fluid phase (displacement $u_\\text{f}$) and solid phase (displacement $u_\\text{s}$)\n",
    "\\begin{align}\n",
    "\\frac{n}{\\rho}\\ddot{u}_\\mathrm{f} + &\\frac{n\\tau}{\\rho}(\\ddot{u}_\\mathrm{f}- \\ddot{u}_\\mathrm{s}) &+& n D_\\text{P} (\\dot{u}_\\mathrm{f}-\\dot{u}_\\mathrm{s}) &-& nM\\bigl(n u_\\mathrm{f}'' - (n-\\alpha)u_\\mathrm{s}''\\bigr) &=0, \\\\\n",
    "(1-n) \\ddot{u}_\\mathrm{s} -& \\frac{n\\tau}{\\rho} (\\ddot{u}_\\mathrm{f} - \\ddot{u}_\\mathrm{s}) &-& n D_\\text{P} (\\dot{u}_\\mathrm{f}-\\dot{u}_\\mathrm{s}) &-& (n-\\alpha)M \\bigl(-n u_\\mathrm{f}'' + (n-\\alpha)u_\\mathrm{s}''\\bigr) - u_\\mathrm{s}'' &=0,\n",
    "\\end{align}\n",
    "with $\\rho = \\frac{\\varrho_\\text{s}}{\\varrho_\\text{f}}$, $M = \\frac{m_\\text{v}}{S_\\text{p}}$ and $D_\\mathrm{P} = \\frac{n\\mu L\\,\\sqrt{m_\\text{v}}\\, }{\\kappa \\sqrt{\\varrho_\\text{s}}}$, note relation $D_\\mathrm{P} = \\frac{c_\\mathrm{S}^\\mathrm{el}}{c_\\mathrm{P}^\\mathrm{el}}D_S$ with $\\frac{c_\\mathrm{S}^\\mathrm{el}}{c_\\mathrm{P}^\\mathrm{el}} = \\sqrt{m_\\text{v} G}$. \n",
    "\n",
    "**Remarks** about the resulting formulation as multi-body system\n",
    "- negative masses (interaction mass $m_\\Delta$ in off-diagonal entries of mass matrix),\n",
    "- complex eigenvectors (diagonalization of damping matrix with undamped eigenvectors is impossible),\n",
    "- non-dimensional damping $D_\\mathrm{P}$ in P-waves is similar to the ratio $\\frac{L f_\\mathrm{c}}{V_\\mathrm{c}}$ with bar length $L$ times characteristic frequency $f_\\mathrm{c}$ \\[Biot1956: eq.(7.4)\\], which divides low- from high-frequency range, and reference velocity $V_\\mathrm{c}$ in his notation \\[Biot1956: equation (5.4)\\],\n",
    "- Biot refers to the velocity of dilatational waves in the undrained case $V_\\mathrm{c} = \\sqrt{\\frac{1/m_\\text{v} + \\alpha^2/S_\\text{p}}{(1-n)\\varrho_\\text{s} + n \\varrho_\\text{f}}}$ and a characteristic frequency $f_c=\\frac{1}{2\\pi}\\frac{\\mu n}{\\kappa \\varrho_\\text{f}}$ \\[Biot1956: equations (5.4) and (7.4)\\]. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7f27916",
   "metadata": {},
   "source": [
    "### Multi-body system matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16115758",
   "metadata": {},
   "source": [
    "Conversion between multi-body and non-dimensional continuum parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "2e0e6e4e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "m_1(k=0)     = 0.0754716981132076\n",
      "m_2(k=0)     = 0.300000000000000\n",
      "m_Delta(k=0) = 0.00754716981132076\n",
      "d_Delta(k=0) = 13.8999117684182\n",
      "c_1(k=0)     = 4.93480220054468\n",
      "c_2(k=0)     = 8.63590385095319\n",
      "c_Delta(k=0) = -2.96088132032681\n"
     ]
    }
   ],
   "source": [
    "mbs2nondim_P = [(m_1,     I_11*n/RHO), \n",
    "                (m_2,     I_22*(1-n)), \n",
    "                (m_Delta, I_12*n*tau/RHO), \n",
    "                (d_Delta, I_12*n*D_P), \n",
    "                (c_1,     I_11*(((k+1/2)*sp.pi)**2)*n*M*alpha),\n",
    "                (c_2,     I_22*(((k+1/2)*sp.pi)**2)*(1 - (n-alpha)*M*alpha)),\n",
    "                (c_Delta, I_12*(((k+1/2)*sp.pi)**2)*(n-alpha)*M*n)]\n",
    "\n",
    "m_1_num_P =         m_1.subs(mbs2nondim_P).subs(nondim_num_k0).evalf()\n",
    "m_2_num_P =         m_2.subs(mbs2nondim_P).subs(nondim_num_k0).evalf()\n",
    "m_Delta_num_P = m_Delta.subs(mbs2nondim_P).subs(nondim_num_k0).evalf() \n",
    "d_Delta_num_P = d_Delta.subs(mbs2nondim_P).subs(nondim_num_k0).evalf()\n",
    "c_1_num_P =         c_1.subs(mbs2nondim_P).subs(nondim_num_k0 + [(k,k0)]).evalf()\n",
    "c_2_num_P =         c_2.subs(mbs2nondim_P).subs(nondim_num_k0 + [(k,k0)]).evalf()\n",
    "c_Delta_num_P = c_Delta.subs(mbs2nondim_P).subs(nondim_num_k0 + [(k,k0)]).evalf()\n",
    "# masses\n",
    "print('m_1(k=0)     = ' + str(m_1_num_P))\n",
    "print('m_2(k=0)     = ' + str(m_2_num_P))\n",
    "print('m_Delta(k=0) = ' + str(m_Delta_num_P))\n",
    "# damping\n",
    "print('d_Delta(k=0) = ' + str(d_Delta_num_P))\n",
    "# springs\n",
    "print('c_1(k=0)     = ' + str(c_1_num_P))\n",
    "print('c_2(k=0)     = ' + str(c_2_num_P))\n",
    "print('c_Delta(k=0) = ' + str(c_Delta_num_P))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f97c692",
   "metadata": {},
   "source": [
    "mass matrix (symmetric)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "77f2a02f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}m_{1} + m_{\\Delta} & - m_{\\Delta}\\\\- m_{\\Delta} & m_{2} + m_{\\Delta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[m_1 + m_Delta,      -m_Delta],\n",
       "[     -m_Delta, m_2 + m_Delta]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mass_P = sp.Matrix(((m_1+m_Delta, -m_Delta), (-m_Delta, m_2+m_Delta)))\n",
    "display(mass_P)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65bc46e4",
   "metadata": {},
   "source": [
    "damping matrix (symmetric)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "f65ec0ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}d_{\\Delta} & - d_{\\Delta}\\\\- d_{\\Delta} & d_{\\Delta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ d_Delta, -d_Delta],\n",
       "[-d_Delta,  d_Delta]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "damping_P = sp.Matrix(((d_Delta, -d_Delta), (-d_Delta, d_Delta)))\n",
    "display(damping_P)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da641df5",
   "metadata": {},
   "source": [
    "stiffness matrix (symmetric)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "114e059e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{1} + c_{\\Delta} & - c_{\\Delta}\\\\- c_{\\Delta} & c_{2} + c_{\\Delta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[c_1 + c_Delta,      -c_Delta],\n",
       "[     -c_Delta, c_2 + c_Delta]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "stiffness_P = sp.Matrix(((c_1+c_Delta, -c_Delta), (-c_Delta, c_2+c_Delta)))\n",
    "display(stiffness_P)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a46ba276",
   "metadata": {},
   "source": [
    "check (swap relation) whether diagonalization with eigenvectors of undamped system is possible (answer is no, except degenerate cases)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "198cf63f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d_{\\Delta} \\left(- m_{1} \\left(c_{\\Delta} \\left(m_{2} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{2} + c_{\\Delta}\\right)\\right) + m_{2} \\left(c_{\\Delta} \\left(m_{1} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{1} + c_{\\Delta}\\right)\\right)\\right)}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "d_Delta*(-m_1*(c_Delta*(m_2 + m_Delta) - m_Delta*(c_2 + c_Delta)) + m_2*(c_Delta*(m_1 + m_Delta) - m_Delta*(c_1 + c_Delta)))/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d_{\\Delta} \\left(m_{1} \\left(c_{\\Delta} \\left(m_{2} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{2} + c_{\\Delta}\\right)\\right) - m_{2} \\left(c_{\\Delta} m_{\\Delta} - \\left(c_{1} + c_{\\Delta}\\right) \\left(m_{2} + m_{\\Delta}\\right)\\right) + m_{2} \\left(c_{\\Delta} m_{\\Delta} - \\left(c_{2} + c_{\\Delta}\\right) \\left(m_{1} + m_{\\Delta}\\right)\\right) - m_{2} \\left(c_{\\Delta} \\left(m_{2} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{2} + c_{\\Delta}\\right)\\right)\\right)}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "d_Delta*(m_1*(c_Delta*(m_2 + m_Delta) - m_Delta*(c_2 + c_Delta)) - m_2*(c_Delta*m_Delta - (c_1 + c_Delta)*(m_2 + m_Delta)) + m_2*(c_Delta*m_Delta - (c_2 + c_Delta)*(m_1 + m_Delta)) - m_2*(c_Delta*(m_2 + m_Delta) - m_Delta*(c_2 + c_Delta)))/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d_{\\Delta} \\left(m_{1} \\left(c_{\\Delta} m_{\\Delta} - \\left(c_{1} + c_{\\Delta}\\right) \\left(m_{2} + m_{\\Delta}\\right)\\right) - m_{1} \\left(c_{\\Delta} m_{\\Delta} - \\left(c_{2} + c_{\\Delta}\\right) \\left(m_{1} + m_{\\Delta}\\right)\\right) - m_{1} \\left(c_{\\Delta} \\left(m_{1} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{1} + c_{\\Delta}\\right)\\right) + m_{2} \\left(c_{\\Delta} \\left(m_{1} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{1} + c_{\\Delta}\\right)\\right)\\right)}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "d_Delta*(m_1*(c_Delta*m_Delta - (c_1 + c_Delta)*(m_2 + m_Delta)) - m_1*(c_Delta*m_Delta - (c_2 + c_Delta)*(m_1 + m_Delta)) - m_1*(c_Delta*(m_1 + m_Delta) - m_Delta*(c_1 + c_Delta)) + m_2*(c_Delta*(m_1 + m_Delta) - m_Delta*(c_1 + c_Delta)))/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d_{\\Delta} \\left(m_{1} \\left(c_{\\Delta} \\left(m_{2} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{2} + c_{\\Delta}\\right)\\right) - m_{2} \\left(c_{\\Delta} \\left(m_{1} + m_{\\Delta}\\right) - m_{\\Delta} \\left(c_{1} + c_{\\Delta}\\right)\\right)\\right)}{\\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right)^{2}}$"
      ],
      "text/plain": [
       "d_Delta*(m_1*(c_Delta*(m_2 + m_Delta) - m_Delta*(c_2 + c_Delta)) - m_2*(c_Delta*(m_1 + m_Delta) - m_Delta*(c_1 + c_Delta)))/(m_1*m_2 + m_1*m_Delta + m_2*m_Delta)**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "inv_mass_P = sp.Matrix.inv(mass_P)\n",
    "zero_or_not = (inv_mass_P*damping_P)*(inv_mass_P*stiffness_P) - (inv_mass_P*stiffness_P)*(inv_mass_P*damping_P)\n",
    "display(sp.simplify(zero_or_not[0,0]))\n",
    "display(sp.simplify(zero_or_not[0,1]))\n",
    "display(sp.simplify(zero_or_not[1,0]))\n",
    "display(sp.simplify(zero_or_not[1,1]))        "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0357f83",
   "metadata": {},
   "source": [
    "### Free vibrations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24520e27",
   "metadata": {},
   "source": [
    "Eigenproblem and characteristic equation $a_4 \\delta^4 + a_3 \\delta^3 + a_2 \\delta^2 + a_1 \\delta + a_0 = 0$ with $a_4,a_3,a_2,a_1,a_0\\in\\mathbb{R}^+$ on physical grounds ($0\\le \\tau$,$\\rho_\\mathrm{f}$,$\\rho_\\mathrm{s}$,$m_\\mathrm{v}$,$S_\\mathrm{p}$,$\\kappa$,$\\mu$ and $0\\le \\alpha,n\\le 1$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "7a9797e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle c_{1} c_{2} + c_{1} c_{\\Delta} + c_{2} c_{\\Delta} + \\delta^{4} \\left(m_{1} m_{2} + m_{1} m_{\\Delta} + m_{2} m_{\\Delta}\\right) + \\delta^{3} \\left(d_{\\Delta} m_{1} + d_{\\Delta} m_{2}\\right) + \\delta^{2} \\left(c_{1} m_{2} + c_{1} m_{\\Delta} + c_{2} m_{1} + c_{2} m_{\\Delta} + c_{\\Delta} m_{1} + c_{\\Delta} m_{2}\\right) + \\delta \\left(c_{1} d_{\\Delta} + c_{2} d_{\\Delta}\\right)$"
      ],
      "text/plain": [
       "c_1*c_2 + c_1*c_Delta + c_2*c_Delta + delta**4*(m_1*m_2 + m_1*m_Delta + m_2*m_Delta) + delta**3*(d_Delta*m_1 + d_Delta*m_2) + delta**2*(c_1*m_2 + c_1*m_Delta + c_2*m_1 + c_2*m_Delta + c_Delta*m_1 + c_Delta*m_2) + delta*(c_1*d_Delta + c_2*d_Delta)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evp_mbs_P = (delta**2)*mass_P + delta*damping_P + stiffness_P\n",
    "ceq_mbs_P = sp.simplify(sp.Determinant(evp_mbs_P).doit())\n",
    "display(ceq_mbs_P.collect(delta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "908c9966",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{n \\left(I_{11} I_{12} \\delta^{4} n \\tau + \\rho^{2} \\left(- D_{P} I_{12} I_{22} \\delta^{3} \\left(n - 1\\right) - \\pi^{4} I_{11} I_{12} M^{2} \\alpha n \\left(\\alpha - n\\right) \\left(k + 0.5\\right)^{4} + \\pi^{4} I_{11} I_{22} M \\alpha \\left(k + 0.5\\right)^{4} \\left(M \\alpha \\left(\\alpha - n\\right) + 1\\right) - \\pi^{4} I_{12} I_{22} M \\left(\\alpha - n\\right) \\left(k + 0.5\\right)^{4} \\left(M \\alpha \\left(\\alpha - n\\right) + 1\\right) + \\delta^{2} \\left(- \\pi^{2} I_{11} I_{22} M \\alpha \\left(k + 0.5\\right)^{2} \\left(n - 1\\right) + \\pi^{2} I_{12} I_{22} M \\left(\\alpha - n\\right) \\left(k + 0.5\\right)^{2} \\left(n - 1\\right)\\right) + \\delta \\left(\\pi^{2} D_{P} I_{11} I_{12} M \\alpha n \\left(k + 0.5\\right)^{2} + \\pi^{2} D_{P} I_{12} I_{22} \\left(k + 0.5\\right)^{2} \\left(M \\alpha \\left(\\alpha - n\\right) + 1\\right)\\right)\\right) + \\rho \\delta^{2} \\left(D_{P} I_{11} I_{12} \\delta n + \\pi^{2} I_{11} I_{12} M \\alpha n \\tau \\left(k + 0.5\\right)^{2} - \\pi^{2} I_{11} I_{12} M n \\left(\\alpha - n\\right) \\left(k + 0.5\\right)^{2} + \\pi^{2} I_{11} I_{22} \\left(k + 0.5\\right)^{2} \\left(M \\alpha \\left(\\alpha - n\\right) + 1\\right) + \\pi^{2} I_{12} I_{22} \\tau \\left(k + 0.5\\right)^{2} \\left(M \\alpha \\left(\\alpha - n\\right) + 1\\right) + \\delta^{2} \\left(- I_{11} I_{22} \\left(n - 1\\right) - I_{12} I_{22} \\tau \\left(n - 1\\right)\\right)\\right)\\right)}{\\rho^{2}}$"
      ],
      "text/plain": [
       "n*(I_11*I_12*delta**4*n*tau + RHO**2*(-D_P*I_12*I_22*delta**3*(n - 1) - pi**4*I_11*I_12*M**2*alpha*n*(alpha - n)*(k + 0.5)**4 + pi**4*I_11*I_22*M*alpha*(k + 0.5)**4*(M*alpha*(alpha - n) + 1) - pi**4*I_12*I_22*M*(alpha - n)*(k + 0.5)**4*(M*alpha*(alpha - n) + 1) + delta**2*(-pi**2*I_11*I_22*M*alpha*(k + 0.5)**2*(n - 1) + pi**2*I_12*I_22*M*(alpha - n)*(k + 0.5)**2*(n - 1)) + delta*(pi**2*D_P*I_11*I_12*M*alpha*n*(k + 0.5)**2 + pi**2*D_P*I_12*I_22*(k + 0.5)**2*(M*alpha*(alpha - n) + 1))) + RHO*delta**2*(D_P*I_11*I_12*delta*n + pi**2*I_11*I_12*M*alpha*n*tau*(k + 0.5)**2 - pi**2*I_11*I_12*M*n*(alpha - n)*(k + 0.5)**2 + pi**2*I_11*I_22*(k + 0.5)**2*(M*alpha*(alpha - n) + 1) + pi**2*I_12*I_22*tau*(k + 0.5)**2*(M*alpha*(alpha - n) + 1) + delta**2*(-I_11*I_22*(n - 1) - I_12*I_22*tau*(n - 1))))/RHO**2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evp_nondim_P = evp_mbs_P.subs(mbs2nondim_P)\n",
    "ceq_nondim_P = sp.simplify(ceq_mbs_P.subs(mbs2nondim_P))\n",
    "#display(evp_continuum)\n",
    "display(ceq_nondim_P.collect(delta))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d9be3853",
   "metadata": {},
   "source": [
    "#### Numerical evaluation (example)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "e6b909b3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.0830188679245283 \\delta^{2} + 13.8999117684182 \\delta + 1.97392088021787 & - 0.00754716981132076 \\delta^{2} - 13.8999117684182 \\delta + 2.96088132032681\\\\- 0.00754716981132076 \\delta^{2} - 13.8999117684182 \\delta + 2.96088132032681 & 0.307547169811321 \\delta^{2} + 13.8999117684182 \\delta + 5.67502253062638\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  0.0830188679245283*delta**2 + 13.8999117684182*delta + 1.97392088021787, -0.00754716981132076*delta**2 - 13.8999117684182*delta + 2.96088132032681],\n",
       "[-0.00754716981132076*delta**2 - 13.8999117684182*delta + 2.96088132032681,    0.307547169811321*delta**2 + 13.8999117684182*delta + 5.67502253062638]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.000569597721609114 \\delta^{4} + 4.16997353052547 \\delta^{3} + 0.150943396226415 \\delta^{2} \\left(0.165 \\delta^{2} + 6.94995588420911 \\delta + 3.51604656788808\\right) + 0.592176264065361 \\delta^{2} + 188.63161675096 \\delta + 2.43522727585006$"
      ],
      "text/plain": [
       "0.000569597721609114*delta**4 + 4.16997353052547*delta**3 + 0.150943396226415*delta**2*(0.165*delta**2 + 6.94995588420911*delta + 3.51604656788808) + 0.592176264065361*delta**2 + 188.63161675096*delta + 2.43522727585006"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "evp_num_P = evp_nondim_P.subs(nondim_num_k0 + [(k, k0)]).evalf()\n",
    "ceq_num_P = ceq_nondim_P.subs(nondim_num_k0 + [(k, k0)]).evalf()\n",
    "display(evp_num_P)\n",
    "display(ceq_num_P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "ea9229fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-204.8276454930922+0j),\n",
       " (-0.012910896190741171+0j),\n",
       " (-0.012903346378002107-6.012249185383537j),\n",
       " (-0.012903346378002107+6.012249185383537j)]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eigenvalue_sp_P = sp.solve(ceq_num_P, delta)\n",
    "eigenvalue_num_P = [complex(EVSP) for EVSP in eigenvalue_sp_P]\n",
    "display(eigenvalue_num_P)\n",
    "number_of_eigenvalues = len(eigenvalue_num_P)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb9c199d",
   "metadata": {},
   "source": [
    "Plot characteristic equation, intersections of zero-levels of real- and imaginary parts are roots of characteristic equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "fca9170a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+0sx+ZmbLmtq6hJndZmZbzWy7mX00i+8pIiIXN5mB9bPJ9m3Aa919GMDM3k6YufXuJrWN5Pu0A58B3gwMAL80s2+5+zPN/L4iInJxkwmRfWb2JWAJMBs4AeDu3zGzv2li21K3ANvdfSeAmX0duB0YN0QGB+FvsmiZTK8j84F74JvXwp7mf7t9+8L23nth06bmf7/S+9kb4Pxs0HuvkJYubezzzC8yE8bM5gK/DTwOfAL4HvA0cCPwbndv6vReM/s3wG3u/ofJ/d8FXu3ud9U87w7gjnDv5pthYzObJSJSKtdfD08+aZvcfcNUPu+ilYi7HwO+CGBm7wHuBH4PeBl479Sb2hzufj9wP8DNN2/wxx7LuUEydTt2wJXr4Ytfgt/5naZ/u5/9DG69Fb79bbjttqZ/u/K7+274h3+AEyfybok0aMaMqX/OZJY96XX3/TASKJ+c6DlNsB/oq7i/Mtk3LjPomExHncSlA2AY2n1yHa2XqLMzbNvb9fsyLdougA1n8rOTeExmdtZ3J/GcBy+1IRP4JbDOzNaY2UzgfcC3mvj9RERkkiY7sP4ocIiwdtYA8HngTwlrZy0Gtjerge5+3szuAr5PWADyC+7+dLO+n4iITN5kQuQdwLWMruJ7A/Bj4M+BrcAp4IlmNC7l7t9lchWRiIhkaDID6w5srtj1iJltSAayRaaPrq0uUjgNrZ3l7h+Y7oaIjMh4DS0t2SXSuEmFiJm91sz+k5m9Obn/r8zsc2bW1LPVpcWoNBApnMlWIjuBvwD+tZk9CPwEuBy4y8xe36S2SatSaVBc+keg5Uw2RD4APAXcBzwM/Lq73wa8kzDwLnLp9Aeo2BT+LWmypwW9FfhHd3/FzL7i7ofNbAZwF3C8ec2TlqQ/RiKFMdlK5CSwO1lR9y+SfT8mhMgXm9AuaUWqREQKZ7KVyF8DnyKccPiFZF8ncJW7H21Gw6QFpSGSUSWizGoCvagtZ1Ih4u6bkgH07mT9LIDfB/QbI9NPU3yLSS9kS5r0UmnJxaiOVdx/qiktktal/2JFCqehkw1FmiLj7ixpAv0j0HIUIhIPjYkUm8K/JSlEJB45VSL62zeNlMwtRyEi8dFf9WLSz60lKUQkHvovttjM4MKFvFshGVOISDzSP0Bt2fxaKrOmmSqRlqQQkXhoTKT4lMwtRyEi8dAU32Jra1OItCCFiMRDU3yLLe2G1LhIS1GISDzUnVVsCpGWpBCReKg7q9ja28N2eDjfdkimFCISD83OKraOZCm+8+fzbYdkSiEi8VB3VrF1dobt0FC+7ZBMKUQkHqpEiq2nJ2yP62KnrUQhIvFQJVJsy5aF7eBgvu2QTClEJB6qRIrtyivD9pFH8m2HZEohIvFQJVJc58/De98LV1wBn/60ErqFTPrKhiJNp0qkuO67DzZtGr2/bVsIFCk9VSISj4xDJKVKZBp84xthe/XV4QX92tfybY9kRiEi8VAlUkzHj8Pjj4fb586FCuTJJ/Ntk2RGISLx0JhIMT30UAiP/v5wjsgNN8Avf6mUbhEKEYmHKpFievbZsP3VXw0D7LfeCgMDsGNHvu2STChEJB4aEymmzZvh8sth3rywbtbNN4f9zzyTa7MkGwoRiYcqkWJ67DHYsCH83C5cGJ2V9dxz+bZLMqEQkXioEimeY8dgz54QImYhmefPh5UrNbjeIhQiEg9VIsWzc2fY9vePViIQxkc2bsyvXZKZqEPEzD5uZvvN7Ink4215t0maKKcQUSVyCbZuDdsrrhitRADWr4ddu3RtkRYQdYgkPuXuNyQf3827MdJE6s4qni1bwsWo1q+vDpH+/jDtd9++fNsnTVeEEJFWkf7Xml4hr8nUnTUNtmyBdetg1qzqEEkXY0yn/0ppFSFE7jKzzWb2BTNbMN6TzOwOM9toZhsPHz6cZftkuqg7q3j27oU1a8LtyhfymmvCdvPm7Nskmco9RMzsYTPbUufjduB/AGuBG4BB4L+N93Xc/X533+DuGxYvXpxN42V6qTureHbtCueIpNJkXrgQli4NCzFKqeW+iq+7v2kyzzOzfwQebHJzJE+anVUsR47Ayy/D2rXhfmV3FsDq1WH6r5Ra7pXIRMxsecXddwFb8mqLZCCnMRFVIg1KTyZcvz5sa0Nk7VotfdICog4R4BNm9pSZbQZeD/xp3g2SJlJ3VrGkXVWVIVJp3bpQiZw5k227JFO5d2dNxN1/N+82SIbUnVUsAwNhu3Jl/cevuir8TJ9/Hq67Lrt2SaZir0Sklag7q1j27IHFi2H27NF9td1ZALt3Z9osyZZCROKhECmW2plZtWMiq1aFrQbXS00hIvHQmEix7NgxWm3A2BdyyRLo6hpdX0tKSSEi8dAZ68Vx7lyoMNatG9m15fBS3nDmO7z8crLDLITM9u35tFEyoRCReKg7qzgGB0Pl2Nc3suuN//tOHrnwGzz6aMXz1q8fXaRRSkkhIvHQsifFkS6smMzMGh6GQ6fmVD0EhCVR9u5V2VdiChGJhyqR4kiTIhk8f+ml0Yeqzi/s64OhITh0KLu2SaYUIhIPhUhx7NoVtqtXA9UZUTWOns7e0jTf0lKISDwyDpGUQqQBO3eG2Vc9PQCkC2fP5lT1OHp/f9hq+ZPSUohIPDQ7qzj27Kk6RyStRK63zdU9V/39IaU1Q6u0FCISjzRENLAevxdegBUrRu4ePBi26+350Sm+EM5mX75c3VklphCReAwPhwDJ6K+6QuQSDAxUrZl16BC02QX6bRenT4ex9BF9fbpMbokpRCQew8OZjocoRBp0/DgcPVp1jsjBg7C46ySXeZim9corFc9ftUpLn5SYQkTioRAphjQQ0rWxCNenWjT7FAvsFYDqLq01a8LnpOcBSakoRCQe589DR3ZXJ1CINCid3pteW51QnMyZOcTCpBKpCpH+fjh7dnTpeCkVhYjEQ5VIMdScaAhw7BjM6RwaqUSOHKl4frpIYxo+UioKEYmHQqQYXnghTIBYsmRk1/HjMHfmEJcRKpHKM9hHxk40uF5KChGJh0KkGPbvh2XLqn5WaSWSdmdVVSJpiOzdm2EjJSsKEYmHxkSKYWCgamYWpGMiZ5nPK7S11VQiXV2waJFmaJWUQkTioUqkGPburRoPcU+6szqHaOMCPT2hMqmyZo3GREpKISLxUCUSP/cxIXLyZJi9O7cznGE4Zw6cOFHzeWvXav2sklKISDxUicTvpZfgzJkxM7MA5nWeAXd6euqEyKpVYWBd54qUjkJE4qFKJH4vvBC2y5eP7EpDZE7n2bCdE7q3qvT1hUvqpsv9SmkoRCQeCpH4pSFSsfjiSCUyawjc64dIWrlocL10FCISj4y7s9KeFYXIFKRnnff2juw6ejRsK8dExoSILk5VWgoRiUdOlUhGK8+Xw8BASN2KEKmqRAghMmZ2VhoimqFVOnr7SDwyDpGUKpEp2LMnjIfMmDGy6+TJsO2eeQ4YpxKZOxcuu0yVSAkpRCQeOVUiMgV1TjRMQ6RrRgiRuXN8bCUC4fojWoSxdBQiEo+MQyQdE1F31hQMDlbNzAI4dSps00pk3rywaG/VhalAF6cqKb19JB6anRW/msviQp1KZG64nw64j9DFqUpJISLxUIjE7fTpcKGQOiHS2Qkd7eEFnTsnbOsOrr/ySp10kSJTiEg81J0Vt/37w7bi2uoQQqSri5E0njcv7K+7fhbAzp3Na6NkTm8fiYcqkbil4xl1QqS7e/R+T7eP7K+yenXYakn4UlGISDwUInFLZ1bVmZ3V3c3IC9nVFfaPWT8rDR/N0CoVhYjE49y5qvMPmk0hMkV1ljyBECI9PaP3x61Eli4NP1/N0CoVhYjEQ5VI3AYHQ1pUJgbjVyJjQqStLVQxmqFVKlGEiJm9x8yeNrMLZrah5rG7zWy7mW01s7fk1UbJgAbW41bnHBEYW4l0d41TiUCYoaWz1ksllrfPFuDdwE8rd5rZ1cD7gGuA24DPmll2K/RJts6dUyUSs/37q9bMSp04UV2JzOnxkf1j6AqHpRNFiLj7s+6+tc5DtwNfd/chd98FbAduybZ1kpnz5zUmErOBgTEzs6CiEkleyNmzw/66IdLXB4cOhVPapRSiCJEJ9AKVo3ADyT4po4wH1tWdNQUXLoRKpE6IjFQiiba2cL9uiKxcGdI7PedECi+zvgMzexhYVuehe9z9m9Pw9e8A7gBYVXHpTikQDazH6/Dh8PMZpzuraqx9vEvkwujFqfbuHT35UAots3esu7+pgU/bD1ROSl+Z7Kv39e8H7gfYsGGD1mctIk3xjdeBA2G7rPr/wHPnwkflGevp1Q3rhkh6wqFmaJVG7IX8t4D3mVmnma0B1gG/yLlN0izqzopXnWurQ1hOC5JxkIo0vmglohlapRHF28fM3mVmA8Brge+Y2fcB3P1p4H8BzwDfA/7I3Yfza6k0jXv4q67urDgNDoZtzYmGaYik54YAI91ZYy5MBTBrVggihUhpZH8ZuTrc/QHggXEeuxe4N9sWSebOhWXE1Z0VqXEqkfRaIrNnA8dHX8g5c8IkrLp0capSiaISEckjRNSdNQUDA7BoUagkKoxXidS9RG5KF6cqFb19JA6qROI2zjkiaSVSNbBOGBOpe8Y6hHGRfft0feKSUIhIHM6fD1uNicTpwIG6S55UdWel3OnqmiBEVq8OD7700rQ3U7KnEJE4qDsrboODY6b3Qk13VkUad3ePBswY6fkhWv6kFPT2kTjkGCKqRC5ieBgOHpywEqkdE+nqCiubpAVmlXSar8ZFSkEhInHIaUxEATIJL74YgqRmei+MPyaSLoNStxrRxalKRSEicVCIxCtd52qCEKkdE0lDpO64SDrLS5VIKShEJA45dWdpPGQS0hCps25WGiKVS8FfNETMdHGqEtFbSOKgSiRe6YmGE4RIbXfWuFc3TK1erRApCYWIxCG9voQqkfikS54sWTLmoVOnQnZ0dlbsTJY9gXHWzwJdnKpE9BaSOKSVyMyZmX1LhcgkHTgQxjHqBPzJk6HqMKNud9a403z7+sLy8kNDTWmyZEdvIYmDurPiNc611SGExMj03poz1mGCSkQztEpDISJxOH8+lAUZhghk/u2K6ejRujOzIGT/yFUN29qgvX2kO8sMzpwZ52uuWhWee/BgU5os2TEv4fo1ZnYYqB21WwS8mENzslDWYyvrcUF5j62sxwXlPbbK41rt7oun8smlDJF6zGyju2/Iux3NUNZjK+txQXmPrazHBeU9tks9LnVniYhIwxQiIiLSsFYKkfvzbkATlfXYynpcUN5jK+txQXmP7ZKOq2XGREREZPq1UiUiIiLTTCEiIiINK3WImNlHzMzNbFFy38zs02a23cw2m9lNFc/9oJltSz4+mF+rx2dmf520+wkz+4GZrUj2F/q4AMzsk2b2XNL+B8xsfsVjdyfHttXM3lKx/7Zk33Yz+2guDb8IM3uPmT1tZhfMbEPNY4U9rnqK2m4AM/uCmR0ysy0V+xaa2UPJe+chM1uQ7B/3/RYjM+szs0fM7Jnkd/GPk/3Tc3zuXsoPoA/4PuGkw0XJvrcB/xcw4DXAY8n+hcDOZLsgub0g72Ooc0xzK25/GPhcGY4raetvAh3J7b8D/i65fTXwJNAJrAF2AO3Jxw6gH5iZPOfqvI+jznFdBawHfgxsqNhf6OOqc5yFbHdF+18H3ARsqdj3CeCjye2PVvxO1n2/xfoBLAduSm7PAZ5Pfv+m5fjKXIl8CvhzoHLmwO3AP3nwKDDfzJYDbwEecvcj7v4y8BBwW+Ytvgh3P1Zxt5vRYyv0cQG4+w/cPb2Y6qNAsrgStwNfd/chd98FbAduST62u/tOdz8LfD15blTc/Vl331rnoUIfVx1FbTcA7v5T4EjN7tuBLye3vwy8s2J/vfdblNx90N0fT24fB54Fepmm4ytliJjZ7cB+d3+y5qFeoPJyagPJvvH2R8fM7jWzfcAHgI8luwt/XDU+RPhPCMp3bKmyHVdR2z2Rpe6erIPPAWBpcruwx2pmlwM3Ao8xTcfXMf3NzIaZPQwsq/PQPcB/JHSPFM5Ex+Xu33T3e4B7zOxu4C7gLzNt4CW42LElz7kHOA98Jcu2XYrJHJcUm7u7mRX6fAgz6wH+BfgTdz9mFasuX8rxFTZE3P1N9fab2bWEPuYnkxdpJfC4md0C7CeMlaRWJvv2A79Rs//H097oSRjvuOr4CvBdQohEf1xw8WMzs98Dfgt4oyeds4x/bEywP1NT+JlViv64pmii4ymqg2a23N0Hk+6cQ8n+wh2rmc0gBMhX3P0bye7pOb68B32a/QHsZnRg/e1UDxj9Itm/ENhFGHxekNxemHfb6xzLuorb/x74P2U4rqSttwHPAItr9l9D9QD0TsIgbkdyew2jA7nX5H0cExzfj6keWC/FcVUcTyHbXXMMl1M9sP5JqgeeP5Hcrvt+i/Ujaec/Af+9Zv+0HF/uB5jBC1gZIgZ8hjCL5KmaN/WHCIOb24Hfz7vd4xzLvwBbgM3At4HeMhxX0s7thH7YJ5KPz1U8dk9ybFuBt1bsfxthpskOQtdR7sdR57jeRehTHgIOAt8vw3GNc6yFbHfS9q8Bg8C55Of1B8BlwA+BbcDDJP+ATfR+i/ED+HXCJJzNFe+vt03X8WnZExERaVgpZ2eJiEg2FCIiItIwhYiIiDRMISIiIg1TiIiISMMUIiIi0jCFiMg0MrNZdfa1mdmDZvaUmfVV7J+XLHu/KXnsD5P995nZs2b2+izbLtKIwi57IpK35HounwFmAE8D3wK+nSyQ+QZ3fyl56rXAEne/tuZL/DZw3N1vTr7ebAB3/zMz+wXhRNFHmn8kIo1TJSLSuPcRzkJ/F/DvgF8DVhOWm3lrxfPmM7ouUaXHgVvNbKOZ/RXhrPbUgeTzRKKmEBFp3HeBdwPvAJ4jLBdxGjgL/LLiee3AhcpPNLN5hIsCXUdYn+j1VF9/40LyeSJRU4iINMjdnwP+K+H6DKeB6wnrm33Wqy9EdQNhPaZKdxLW0Trq4WJcP6d6Ofn9wBX1xlhEYqIQEWmQmXUALwH3ArOA+4EfERbvS5/zVeDjwOdrPv1GwjhK5f2n0jvuvoOwqvHe5PIGIlFSiIg0KKkgvuPuZwgrwP6UsBrxlWbWlTzn/cB/AP6k5tNfJgQHZvZ2YC7ws/RBM7uecL3yXnd/CpFIKUREGmRmC4CfmNn/A/7Kw5LYTxCW3f6YmS1MnrqVMNhe6ZPAu8zsScKg/LvdvXLcZAGw293PNfMYRC6VpviKNK6DMN5xTVotuPt+M3sr8Ji7H02edwpYbGaWBA3uvgu4ZYKvvQQ42bSWi0wTXU9EpMnMbCbwVWAd8Fvuvu8iz78PeB1wt7s/lEETRRqmEBERkYZpTERERBqmEBERkYYpREREpGEKERERaZhCREREGqYQERGRhilERESkYQoRERFp2P8H/veFlKJbci8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Xmin = min(np.real(eigenvalue_num_P))\n",
    "Xmax = max(np.real(eigenvalue_num_P))\n",
    "dX = Xmax - Xmin\n",
    "Ymin = min(np.imag(eigenvalue_num_P))\n",
    "Ymax = max(np.imag(eigenvalue_num_P))\n",
    "dY = Ymax - Ymin\n",
    "X = np.linspace(Xmin-dX, Xmax+dX, 500)\n",
    "Y = np.linspace(Ymin-0.5*dY, Ymax+0.5*dY, 500)\n",
    "XX, YY = np.meshgrid(X, Y)\n",
    "\n",
    "ceq_f = sp.lambdify(delta, ceq_num_P, modules='numpy')\n",
    "ZZ = ceq_f(XX+1j*YY)\n",
    "\n",
    "plt.contour(XX, YY, np.real(ZZ), [0], colors='r');\n",
    "plt.contour(XX, YY, np.imag(ZZ), [0], colors='b');\n",
    "plt.xlabel(\"$\\mathfrak{R}\\{\\delta\\}$\");\n",
    "plt.ylabel(\"$\\mathfrak{I}\\{\\delta\\}$\");\n",
    "#plt.show();\n",
    "#tikzplotlib.save(\"characteristic_roots.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW')   # im,re "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "f6f5f0d5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- 0. mode ---\n",
      "eigenvalue = (-204.8276454930922+0j)  \n",
      "eigenvector = [(0.9697323191635941+0j), (-0.2441704920124398+0j)]\n",
      "--- 1. mode ---\n",
      "eigenvalue = (-0.012910896190741171+0j)  \n",
      "eigenvector = [(0.8682438348551128+0j), (-0.4961377260762252+0j)]\n",
      "--- 2. mode ---\n",
      "eigenvalue = (-0.012903346378002107-6.012249185383537j)  \n",
      "eigenvector = [(0.7074140365689353+0j), (0.7065534274602864+0.018644972763910937j)]\n",
      "--- 3. mode ---\n",
      "eigenvalue = (-0.012903346378002107+6.012249185383537j)  \n",
      "eigenvector = [(0.7074140365689353+0j), (0.7065534274602864-0.018644972763910937j)]\n"
     ]
    }
   ],
   "source": [
    "eigenvector_num_P = np.zeros((number_of_eigenvalues, 2), dtype=complex)\n",
    "for mode_n, delta_n in enumerate(eigenvalue_num_P):\n",
    "    EVP_P = evp_num_P.subs(delta, delta_n)\n",
    "    ev0r = np.array([1.0, -EVP_P[0,0].evalf()/EVP_P[0,1].evalf()], dtype=complex)   # first row --> eigenvector\n",
    "    ev0rn = ev0r/np.linalg.norm(ev0r)   # normalize\n",
    "    #ev1r = np.array([1.0, -EVP_P[1,0].evalf()/EVP_P[1,1].evalf()], dtype=complex)   # second row, redundant\n",
    "    #ev1rn = ev1r/np.linalg.norm(ev1r)   # normalize  \n",
    "    print(\"--- {}. mode ---\".format(mode_n))\n",
    "    print(\"eigenvalue = {}  \".format(delta_n))\n",
    "    print(\"eigenvector = [{}, {}]\".format(ev0rn[0], ev0rn[1]))\n",
    "    #print(\"eigenvector = [{}, {}] (redundant)\".format(ev1rn[0], ev1rn[1]))\n",
    "    eigenvector_num_P[mode_n,:] = ev0rn\n",
    "t_scale = 1.0/(np.abs(min(np.imag(eigenvalue_num_P))) + np.abs(min(np.real(eigenvalue_num_P))))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35d1e576",
   "metadata": {},
   "source": [
    "#### Visualize modes\n",
    "Real roots lead to real coefficients, slow and fast decay, two of them together can match initial position and initial velocity of one component (either $u_\\mathrm{f}$ or $u_\\mathrm{s}$). Pair of complex-conjugated roots leads to damped oscillations, which can match initial conditions by amplitude and phase shift."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "ea4f0acb",
   "metadata": {},
   "outputs": [],
   "source": [
    "t_modeshow = np.linspace(0, 20*t_scale)\n",
    "timesteps_modeshow = len(t_modeshow)\n",
    "u_f = np.zeros((number_of_eigenvalues, timesteps_modeshow))\n",
    "u_s = np.zeros((number_of_eigenvalues, timesteps_modeshow))\n",
    "\n",
    "for mode_n, delta_n in enumerate(eigenvalue_num_P):\n",
    "    u_f[mode_n,:] = np.real(eigenvector_num_P[mode_n,0]*np.exp(delta_n*t_modeshow))\n",
    "    u_s[mode_n,:] = np.real(eigenvector_num_P[mode_n,1]*np.exp(delta_n*t_modeshow))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e19a5c0",
   "metadata": {},
   "source": [
    "From the solution list above we see that first (0) and second mode (1) correspond to aperiodic decay (real roots), and third and fourth mode to damped oscillations (complex-conjugated roots):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "0d184bf7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_modeshow,u_f[0,:],'b',   t_modeshow,u_s[0,:],'k');\n",
    "plt.plot(t_modeshow,u_f[1,:],'b--', t_modeshow,u_s[1,:],'k--');\n",
    "plt.xlabel(\"$t$\");\n",
    "plt.ylabel(\"$u_f$,$u_s$\");\n",
    "#plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cffaf7f",
   "metadata": {},
   "source": [
    "Solid motion (black) and fluid (blue) of first mode (solid line) and second mode (dashed line)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "c601341a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_modeshow, u_f[2,:],'b',   t_modeshow, u_s[2,:],'k');\n",
    "plt.plot(t_modeshow, u_f[3,:],'b--', t_modeshow, u_s[3,:],'k--');\n",
    "plt.xlabel(\"$t$\");\n",
    "plt.ylabel(\"$u_f$,$u_s$\");\n",
    "#plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ead6946c",
   "metadata": {},
   "source": [
    "Solid motion (black) and fluid (blue) of third mode (solid line) and fourth mode (dashed line)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ae99c1f",
   "metadata": {},
   "source": [
    "Note: \n",
    "- for complex-conjugated roots the curves of two modes are identical (no difference between dashed line and solid line) \n",
    "- the  waves (oscillations in reduced model) where fluid and solid move in counter-phase decay fast than waves with motion in-phase (physically due to higher dissipation). "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7cd6daa",
   "metadata": {},
   "source": [
    "#### Specific initial conditions \n",
    "$U_\\mathrm{f}(0)$, $\\dot{U}_\\mathrm{f}(0)$, $U_\\mathrm{s}(0)$, $\\dot{U}_\\mathrm{s}(0)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "7e29a44d",
   "metadata": {},
   "outputs": [],
   "source": [
    "u_f0 = 2.0\n",
    "ut_f0 = 3.0\n",
    "u_s0 = 0.7\n",
    "ut_s0 = 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "db2adcf8",
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.matrix([[eigenvector_num_P[mode_n, 0] for mode_n in range(number_of_eigenvalues)],\n",
    "               [eigenvector_num_P[mode_n, 1] for mode_n in range(number_of_eigenvalues)],\n",
    "               [eigenvector_num_P[mode_n, 0]*eigenvalue_num_P[mode_n] for mode_n in range(number_of_eigenvalues)],\n",
    "               [eigenvector_num_P[mode_n, 1]*eigenvalue_num_P[mode_n] for mode_n in range(number_of_eigenvalues)]],\n",
    "              dtype=complex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "ed1289bd",
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.matrix([[u_f0], [u_s0], [ut_f0], [ut_s0]], dtype=complex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "413b72e0",
   "metadata": {},
   "outputs": [],
   "source": [
    "amplitude = np.linalg.solve(A, b)   # matrix with one column "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "ef342a8a",
   "metadata": {},
   "outputs": [],
   "source": [
    "u_f_fit = lambda N,T: amplitude[N,0]*eigenvector_num_P[N,0]*np.exp(eigenvalue_num_P[N]*T) \n",
    "u_s_fit = lambda N,T: amplitude[N,0]*eigenvector_num_P[N,1]*np.exp(eigenvalue_num_P[N]*T) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "e0baceec",
   "metadata": {},
   "outputs": [],
   "source": [
    "t_oscillation = np.linspace(0, 30*t_scale, 250)\n",
    "timesteps_oscillation = len(t_oscillation)\n",
    "u_f_oscillation = np.zeros(timesteps_oscillation, dtype=complex)\n",
    "u_s_oscillation = np.zeros(timesteps_oscillation, dtype=complex)\n",
    "\n",
    "for mode_n in range(number_of_eigenvalues):\n",
    "    u_f_oscillation += u_f_fit(mode_n, t_oscillation)\n",
    "    u_s_oscillation += u_s_fit(mode_n, t_oscillation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "0abca9bb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_oscillation,np.real(u_f_oscillation),'b', t_oscillation,np.real(u_s_oscillation),'k');\n",
    "plt.xlabel(\"$t$\");\n",
    "plt.ylabel(\"$u_f$, $u_s$\");\n",
    "#plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14e201c8",
   "metadata": {},
   "source": [
    "### Discriminant\n",
    "Discriminant $D_4$ of a quartic equation $ax^4 + bx^3 + cx^2 + dx + e = 0 $. For real coefficients $a,b,c,d,e \\in \\mathbb{R}$\n",
    "- $D_4 = 0$ at least two roots are equal,\n",
    "- $D_4 < 0$ two real roots and one complex-conjugated pair,\n",
    "- $D_4 > 0$ four real roots or two complex-conjugated pairs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "2a641cc2",
   "metadata": {},
   "outputs": [],
   "source": [
    "quartic_discriminant = lambda a,b,c,d,e: 256 * a**3 * e**3 \\\n",
    "    -192 * a**2 * b * d * e**2 \\\n",
    "    -128 * a**2 * c**2 * e**2 \\\n",
    "    +144 * a**2 * c * d**2 * e \\\n",
    "    -27 * a**2 * d**4 \\\n",
    "    +144 * a * b**2 * c * e**2 \\\n",
    "    -6 * a * b**2 * d**2 * e \\\n",
    "    -80 * a * b * c**2 * d * e \\\n",
    "    +18 * a * b * c * d**3 \\\n",
    "    +16 * a * c**4 *e \\\n",
    "    -4 * a * c**3 * d**2 \\\n",
    "    -27 * b**4 * e**2 \\\n",
    "    +18 * b**3 * c * d * e \\\n",
    "    -4 * b**3 * d**3 \\\n",
    "    -4 * b**2 * c**3 * e \\\n",
    "    + b**2 * c**2 * d**2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93c689c4",
   "metadata": {},
   "source": [
    "#### Dependency on MBS parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "10651986",
   "metadata": {},
   "outputs": [],
   "source": [
    "coefficients_mbs_P = ceq_mbs_P.collect(delta, evaluate=False)\n",
    "a4_mbs_P = coefficients_mbs_P[delta**4] \n",
    "a3_mbs_P = coefficients_mbs_P[delta**3] \n",
    "a2_mbs_P = coefficients_mbs_P[delta**2] \n",
    "a1_mbs_P = coefficients_mbs_P[delta] \n",
    "a0_mbs_P = coefficients_mbs_P[sp.S.One] "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "1d539759",
   "metadata": {},
   "outputs": [],
   "source": [
    "D4_mbs = sp.simplify(quartic_discriminant(a4_mbs_P, a3_mbs_P, a2_mbs_P, a1_mbs_P, a0_mbs_P))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "68dad06b",
   "metadata": {},
   "outputs": [],
   "source": [
    "D4_mbs_num = D4_mbs.subs([(m_1,m_1_num_P), (m_2,m_2_num_P), (m_Delta,m_Delta_num_P), (c_1,c_1_num_P), (c_2,c_2_num_P), (c_Delta, c_Delta_num_P)])\n",
    "D4_mbs_d = sp.lambdify(d_Delta, D4_mbs_num, modules='numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "01772b89",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "d_Delta_range = np.linspace(0.0, 0.0215*float(d_Delta_num_P))\n",
    "D4_mbs_range = D4_mbs_d(d_Delta_range)\n",
    "plt.plot(d_Delta_range, D4_mbs_range, color='black', linewidth=2);\n",
    "plt.plot([0,0.3],[0, 0], color='gray', linewidth=1);\n",
    "#plt.xlim((0,0.3))\n",
    "#plt.ylim((-2,2))\n",
    "plt.xlabel('$d_\\Delta$');\n",
    "plt.ylabel('$D_4$');\n",
    "#plt.show();\n",
    "#tikzplotlib.save(\"d_crit.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW')   "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fbdbd7a3",
   "metadata": {},
   "source": [
    "The value of $d_\\Delta$, when the discriminant $D_4$ changes its sign, corresponds to the change from oscillation ($D_4>0$), i.e. slow P-wave exists, to asymptotic decay ($D_4<0$), i.e. no slow P-wave."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "af6b803a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "critical d_Delta = 0.2353623276385464\n"
     ]
    }
   ],
   "source": [
    "d_Delta_crit_P = fsolve(D4_mbs_d, 0.8)\n",
    "print('critical d_Delta = '+ str(d_Delta_crit_P[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76f2628a",
   "metadata": {},
   "source": [
    "The dissipation seems most relevant, as it introduces the asymmetry to the characteristic polynomial, however the precise value of the critical $d_\\Delta$ also depends on other parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cfd2aed2",
   "metadata": {},
   "source": [
    "### Higher modes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b1625c2",
   "metadata": {},
   "source": [
    "Find wave characteristics for increasing values of eigenfrequencies (higher modes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "e39a713d",
   "metadata": {},
   "outputs": [],
   "source": [
    "Nk = 1000\n",
    "# slow P-wave\n",
    "d1 = np.zeros(Nk)   # decay constant\n",
    "c1 = np.zeros(Nk)   # phase velocity approximation\n",
    "f1 = np.zeros(Nk)   # frequency \n",
    "AR1 = np.zeros(Nk)  # amplitude ratio\n",
    "phi1 = np.zeros(Nk) # phase angle  \n",
    "# fast P-wave\n",
    "d2 = np.zeros(Nk)   # decay constant\n",
    "c2 = np.zeros(Nk)   # phase velocity approximation\n",
    "f2 = np.zeros(Nk)   # frequency\n",
    "AR2 = np.zeros(Nk)  # amplitude ratio\n",
    "phi2 = np.zeros(Nk) # phase angle  \n",
    "\n",
    "index_slow = 0      # first and second eigenvalue compose first mode (slow)\n",
    "index_fast = 2      # third and fourth eigenvalue compose first mode (fast) \n",
    "\n",
    "for k_num in range(Nk):\n",
    "    \n",
    "    I_11_num = float(I_11_k.subs(k, k_num))\n",
    "    I_12_num = float(I_12_k.subs(k, k_num))\n",
    "    I_22_num = float(I_22_k.subs(k, k_num))  \n",
    "\n",
    "    # redefine, since I_ij changes with higher modes\n",
    "    nondim_num_k = [(I_11,  I_11_num),\n",
    "                    (I_12,  I_12_num),\n",
    "                    (I_22,  I_22_num),\n",
    "                    (alpha, alpha_num),  \n",
    "                    (M,     M_num),\n",
    "                    (D_P,   D_P_num),\n",
    "                    (n,     n_num),\n",
    "                    (RHO,   RHO_num),\n",
    "                    (tau,   tau_num)]\n",
    "    \n",
    "    ceq_num_P = ceq_nondim_P.subs(nondim_num_k + [(k, k_num)]).evalf()  \n",
    "    eigenvalue_sp = sp.solve(ceq_num_P, delta)\n",
    "    eigenvalue_num = [complex(EVSP) for EVSP in eigenvalue_sp]\n",
    "    \n",
    "    \n",
    "    d1[k_num] = np.real(eigenvalue_num[index_slow])   \n",
    "    omega = np.abs(np.imag(eigenvalue_num[index_slow]))\n",
    "    c1[k_num] = omega/((k_num+1/2)*np.pi)   \n",
    "    f1[k_num] = omega/(2*np.pi)\n",
    "    \n",
    "    d2[k_num] = np.real(eigenvalue_num[index_fast])   \n",
    "    omega = np.abs(np.imag(eigenvalue_num[index_fast]))\n",
    "    c2[k_num] = omega/((k_num+1/2)*np.pi)   \n",
    "    f2[k_num] = omega/(2*np.pi)\n",
    "    \n",
    "    evp_num_P = evp_nondim_P.subs(nondim_num_k + [(k, k_num)]).evalf()\n",
    "    \n",
    "    EVP = evp_num_P.subs(delta, eigenvalue_num[index_slow])\n",
    "    evkr = np.array([1.0, -EVP[0,0].evalf()/EVP[0,1].evalf()], dtype=complex)   # first row --> eigenvector\n",
    "    us_uf = evkr[1]/evkr[0]   # TODO exclude division by zero\n",
    "    AR1[k_num] = 1.0/np.absolute(us_uf)   # fluid-to-solid\n",
    "    phi1[k_num] = np.angle(us_uf)   # np.abs(np.angle(us_uf))\n",
    "    \n",
    "    EVP = evp_num_P.subs(delta, eigenvalue_num[index_fast])\n",
    "    evkr = np.array([1.0, -EVP[0,0].evalf()/EVP[0,1].evalf()], dtype=complex)   # first row --> eigenvector\n",
    "    us_uf = evkr[1]/evkr[0]   # TODO exclude division by zero\n",
    "    AR2[k_num] = 1.0/np.absolute(us_uf)  # fluid-to-solid\n",
    "    phi2[k_num] = np.angle(us_uf)   # np.abs(np.angle(us_uf))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d78de00c",
   "metadata": {},
   "source": [
    "Slow P-waves may exist only above a threshold frequency (depending on parameters, particularly dissipation). Maybe the actual threshold frequency is lower, but cannot be resolved more accurately by an integer number of modes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "728f1e67",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Slow P-waves exist from 31. mode on corresponding to a frequency of f = 3.7815127464432794\n"
     ]
    }
   ],
   "source": [
    "nonzero_f = f1.nonzero();    # only when there is an frequency (wave exists)\n",
    "index_first_nonzero_f = nonzero_f[0][0]\n",
    "print(\"Slow P-waves exist from {}. mode on corresponding to a frequency of f = {}\".format(index_first_nonzero_f+1, f1[index_first_nonzero_f]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c8384a8",
   "metadata": {},
   "source": [
    "Characteristics of slow P-waves"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "fa62ebee",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig3, ax5 = plt.subplots();\n",
    "ax6 = ax5.twinx();\n",
    "ax5.semilogx(f1[nonzero_f], c1[nonzero_f], 'g');\n",
    "ax6.semilogx(f1[nonzero_f], np.abs(d1[nonzero_f])/f1[nonzero_f], 'b');   # frequency may be zero so decay instead of damping ratio\n",
    "ax5.set_xlabel('frequency');\n",
    "#ax5.set_ylim([0, 1])\n",
    "ax5.set_xticks(list(ax5.get_xticks()) + [f_characteristic_num], list(ax5.get_xticklabels()) + ['$f_c$'])   # with extra tick\n",
    "ax5.set_ylabel('phase velocity', color='g');\n",
    "ax6.set_ylabel('decay constant', color='b');   \n",
    "#ax6.set_ylim([0, 10])\n",
    "#tikzplotlib.save(\"p_wave_slow_eigenvalue.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW')   \n",
    "\n",
    "fig4, ax7 = plt.subplots();\n",
    "ax8 = ax7.twinx();\n",
    "ax7.semilogx(f1[nonzero_f], AR1[nonzero_f], 'g');\n",
    "ax8.semilogx(f1[nonzero_f], phi1[nonzero_f], 'b');\n",
    "ax7.set_xlabel('frequency');\n",
    "#ax7.set_ylim([0,2.5])\n",
    "ax7.set_xticks(list(ax7.get_xticks()) + [f_characteristic_num], list(ax7.get_xticklabels()) + ['$f_c$'])   # with extra tick\n",
    "ax7.set_ylabel('amplitude ratio', color='g');\n",
    "ax8.set_ylabel('phase angle [RAD]', color='b');\n",
    "#ax8.set_ylim([-1.2,0])\n",
    "#tikzplotlib.save(\"p_wave_slow_eigenvector.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW') "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d0c5b13",
   "metadata": {},
   "source": [
    "Characteristics of fast P-waves"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "c987bd6b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "nondimensional c_un_P = 3.82730207927597\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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IrPRWk3+glOO3icgSEVkkIjNEpEPYsQe961aKyIWxitGSlqnxguiEEbKzbi7U2UntLWcHE4AxHhFJBUbiVpTvAAwLT0qeV1X1dFXtDPwNeNK7tgNwDdARuAh4xrtf1FnSMjXat9/Chg3BJa1P134MLeaxZWWbYAIw5ohMYJWqrlHVw8A43Ory3yuxJmJ9wBsaz1BgnKoeUtW1uEnQY9K1yZKWqdFC7VlBJa2PVn1Ei1PyWL4sjQMHgonBGI/fleTvEJHVuJLWnZFcGw2WtEyNlpsLKSmu63m87Tiwg9yNuZzZqw5FRW4qKWNiKE1E5oW9hlfmJqo6UlVPAO4HHopuiBWzpGVqtNxc6NgR6teP/7Mnr5lMsRZz3UXtgR9OnmtMDBSqavew16gSxyNdSX4ccEklr600S1qmxlINthPGx6s+5ug6RzOwa2datrSkZQKXC7QXkXYiko7rWDEh/AQRaR+2ORj42vs8AbhGRGqLSDugPRCT72ibxsnUWOvWwfbtwSWtKeumcE7bc0hLSaNHjyPta8YEQVULRWQE8DGQCoxR1WUi8ggwT1UnACNEZABQAOwEbvSuXSYibwDLgULgDlUtikWclrRMjRVkJ4z1u9azdtdaftvrt9/H8M47sGMHHHNM/OMxBkBVJwGTSux7OOzzb8q59lHg0bKOi3C3jxD2qfJ8eSdY9aCpsXJz3fx/p58e/2dPXTcVgP5t+wNHEue8efGPxZg4uQ9oAGSU87qnoptYScvUWLm5rtdgenr8nz1l3RQa121Mx2YdgSNTSOXmwgUXxD8eY+JgrCqPlHeCCBV2ibKSlqmRiorcRLlBtWdNXTeV/m37kyLuR/Doo+Gkk6xdy1RfqvxPNM6xpGVqpJUrYe/eYJLWul3rWL97/fdVgyHWGcNUdyKcLMLfRZjovZ4Q4eRI7mFJy9RIQXbCmLJ2CsCPklZmJmzcCPkxGd1iTLBE6A1MBfYAo4B/A/uAKSL08nsfa9MyNVJuLmRkwMkR/Y0XHVPXT6VJvSZ0aPrDuUhDCTQ3F1rEZAIcYwL1MDBMlalh+94V4XMgCzdRb4V8l7QkWxpItjTwfb7IGBHZIiJLyzjeUETeF5EvRWSZiNzs997GVFVuLnTr5qZwiidV/VF7VkjnzpCWZlWEpto6oUTCAkCVL4Dj/d6kwh9ZyZbTJVsWAsuA5ZIt8yVbTvNx75dwU9SX5Q5guaqeAfQH/u6NwjYmpg4fhkWLAhqftXs93+z+hrNb/3gpkrp14bTTLGmZamtPOcf2+b2Jn+rB54G7NUunAEi29MfVR/Yp7yJVnSYibcs7BcgQt/JdA2AHbiS1MTG1eLFLXEEkrZnfzATgzNZnlno8MxPeeMNNMWVrQppqppUI/yxlvxDBjPB+klb9UMIC0CydKtkSjelFn8bNV7URN6jsalUtLu1Ebzbi4QDpQQyqMdVKkJ0wZm6YSYP0Bpx+bOkjmnv0gFGjYNUqaN++1FOMSVb3lXPM97B6P0lrjWTL/wPGetvXA2v8PqAcFwKLgHOBE4BPRWR6iUXGAPBmIx4FUL9+fS153JhI5OZCkybQJoB1F2dumEmvlr1ISyn9Ry+8M4YlLVOdqPJyaftFqAP81O99/DRD/xxoCoz3Xk29fVV1MzBenVXAWuCUKNzXmHKFZnaPd/Xbd4e+Y8nmJfRt1bfMczp2dG1b1q5lqjMRUkUYJMJYYD1wtd9rKyxpaZbu5MjqlNH0DXAeMF1EjgVOJjolOGPKtG8fLF8Ol10W/2fPyZuDouUmrbQ06NrVkpapnkToB1wLDMItXdIXaKfKfr/3KDNpSbY8pVl6l2TL+7hOEz+gWTqk/ODkNVyvwCYikofrh18LQFWfA/4IvCQiS3ANcfer6ja/gRtTGQsWQHFxcJ0wUiSFXi3LH0fZowc8/zwUFrokZkx1IEIerrDyLHCvKntEWBtJwoLyS1qhNqwnKhOgqg6r4PhGwKYGNXEVdCeMTsd2IqN2Rrnn9egBTz0Fy5bBGWfEJzZj4uAt3ErHVwNFIrxHKQWiipTZpqVZOt/72Fmz9IvwF9A58niNCV5uLrRuDcceG9/nFhYXMidvTrlVgyHhnTGMqS5UuQtoB/wdVwu3EmgqwlUi+J64wk9HjBtL2XeT3wcYk0hCnTDibfHmxewr2OcraZ14opv13ZKWqW5UUVWmqDIcl8CuBYYC6/zeo7w2rWHeDdtJtkwIO5SBGwhsTFLZtg1Wr4bhw+P/7FkbZgHQp1W5Y/IB16uxRw/IyYl1VMb8kIhcBPwvkAqMVtXHShy/G/gFbiKIrcDPVXW9d6wIWOKd+o1q+f0eVCkA3hfhK+CQ3xjLa9OaBWwCmuCKcyF7gMV+H2BMogglgZ494//smRtm0iKjBa0btvZ1fo8e8Ne/woEDrgu8MbEmIqnASOB8IA/IFZEJqro87LSFQHdV3S8itwN/40h39QOq2rns+9MJ10eiOfCu96yngZ78MMeUq8ykpVm6Htd/vrffmxmTyObOdRPkdusW/2fPyZtD71a9EZ+Dw3r0cAtVLloEve0n0MRHJrBKVdcAiMg4XNXd90lL9cjsSMAc3GQTfv0b13NwNm5e2kXAy8B1qhz0e5MKO9RKtvQC/gWcCqTjio37NEuPiiBYYwI3d66bkLaB7ybf6Niybwvrdq3jjh53+L4mM9O95+Za0jJRkyYi4dMljfJmGwppAWwI287DlYLKcgvwYdh2He/+hcBjqvpuifNrq/KS93mlCL/xs1JxSX5GgTwNXAO8CXQHfgacFOmDjAmSqqsevOKK+D87N9/1qMhsken7mubN3cvatUwUFapq92jcSESux+WDfmG726hqvogcD3wuIktUdXXY8ToidMGNywU4FL6tygI/z/Y1dFGzdJVkS6pmaRHwordUyYN+rjUmEXz9NezcGUx7Vk5+DimSQtfjukZ0XY8e1oPQxFU+0Cpsu6W37wdEZADwe6Cfqn7fgUJV8733NSIyFegChCetb4Eny9hW3Dy0FfKTtPZLtqQDiyRb/obrnBHnpfOMqZq5c917IElrYw6nNTuNBumR1Uv26AHvvQe7drku8MbEWC7QXkTa4ZLVNbge5N8TkS645aouUtUtYfsbAftV9ZCINMFNz/S38GtV6R+NIP0knxu880bgFupqBVwejYcbEy9z57q2rFNPje9zVZWc/Bwym/uvGgwJtWvNn1/+ecZEg6oW4n7PfwysAN5Q1WUi8oiIhLqvP45b//BNEVkk8v1wqFOBeSLyJTAF16YV3usQESqsavBzTrklLcmWVODPmqXXAQeB7IpuaEwimjvXlVxSU+P73NU7V7PjwI6I2rNCunutDzk5cN55UQ7MmFKo6iRgUol9D4d9HlDGdbOA0heJO+JFEfpzpE2rNC/gqhXLVG5Jy2vDauNVDxqTlA4ehC+/PFJyiaecfNeTojJJq1EjNzuGtWuZaqIhML+CV0FFN/G1CCQw05sVY19op2bpk2VfYkziWLQICgqC64RRN60uHZt1rNT1PXrA9OlRDsqYAKjSNhr38dOmtRr4wDs3I+xlTFIItBNGfg7dmncrc6XiimRmQl4ebNoU5cCMSVJ+FoG0diyT1ObOhZYt3bineCooKmDBpgURDSouKXzG9yHlzuRmTM1gXddNtTd3bjClrMWbF3Oo6BA9W1b+4V26uM4j1q5ljGNJy1RrW7fCmjXBVQ1C5TphhNSr56aespkxTHUhgohwvQgPe9utRfD9Q2JJy1RrQc7snrMxh6b1mtKmYZsq3adXL1daLC6OUmDGBOsZ3ETsodXt9+BmfPelwqQl2XKSZMtnki1Lve1Oki0PVSZSY+Jt7lxXvRbEzO45+Tlktsj0PbN7Wfr0gd27YcWKKAVmTLB6qnIHbuwvquzETcbui5+S1r9x8wwWAGiWLsZN72FMwgvN7F6/fnyf+92h71ixdUWVqgZDQrO8z5pV5VsZkwgKREjFzTeICE0B3/UIfpJWPc3SkjXqhf7jMyYYxcWuejCIqsH5G+ejaFSS1oknQpMmlrRMtfFP4B2gmQiPAjOAP/u92M/gkW2SLScQyorZcgVu0lxjEtrKlW6y2SCS1tx8NzgsGklLxFURzp5d5VsZEzhVXhFhPnAebkqnS1TxXfntJ2ndAYwCTpFsyQfWEtlqlcYEYto093722fF/dk5+DicecyLH1D0mKvfr3RsmTIBt21ypy5hkI0L4D8MW4LXwY6rs8HMfP4OL1wADJFvqAymapXsiDdaYIEyfDj/5CZxwQvyfnZOfQ7+2/So+0ac+fdz7nDlw8cVRu60x8TQfV2MX3jMptK3A8X5uUmbSkmy5u4z97kkVzD0oImOAi4EtqnpaGef0B54CagHbVDV6P+Wmxps+Hc46y1WvxVP+d/nk78mv1HIkZeneHdLSXLuWJS2TjFRpF437lFfSCs0veDLQAwitm/JTwM9Qx5eAp4H/lHZQRI7G9de/SFW/EZFmPu5pjC/r18M338C998b/2bkb3fQV0WjPCqlXDzp3tnYtk/zKWDNrN7BeteJOfmUmrdCcg5It04CuoWpByZY/ABMrurGqThORtuWcci0wXlW/8c7fUs65xkQkNDN6UO1ZaSlpdP5J56jet08fGD3azVhfq1ZUb21MPD0DdAUW46oGTweWAg1FuF2VT8q72E+X92OBw2Hbh719VXUS0EhEporIfBH5WRTuaQzgklbDhm6MVrzNzZ/LGceeQd1adaN63z59YP9+WLw4qrc15nsicpGIrBSRVSLyQCnH7xaR5SKyWEQ+E5E2YcduFJGvvdeN5TxmI9BFle6qdAM645bAOh/4W0Ux+uk9+B8gR7LlHW/7EuBlH9f5eXY3XLfHusBsEZmjql+VPFFEhgPDAdLTbT1KU7Hp06Fv3/ivVFysxeTm53J9p+h3sA0NMp49O5gZPkz1JiKpuOmUzgfygFwRmaCqy8NOWwh0V9X9InI7LslcLSLHAFlAd1ynivnetTtLedRJqiwLbaiyXIRTVFnjp/25wpKWZumjwM3ATu91s2ap74Fg5cgDPlbVfaq6DZgGnFFqDKqjVLW7qnZPS6vcukSm5ti61U15dNZZ8X/2ym0r2XN4T1Tbs0JatYIWLWDmzKjf2hiATGCVqq5R1cPAOGBo+AmqOkVV93ubc4CW3ucLgU9VdYeXqD4FLirjOctEeFaEft7rGWC5CLXxsXKxn7kHWwPbcCOY3wG2e/uq6j3gTBFJE5F6QE/wP8DMmLLMmOHeg0ha0ZjZvSwirvQ4fTqoRv32xrQANoRt53n7ynIL8GElrr0JWAXc5b3WePsKgHMqCtJPsWUi3mwYuGq8dsBKoNz1w0XkNaA/0ERE8nBFx1oAqvqcqq4QkY9wjXHFwGhVXeojHmPKNX061KnjuonHW05+DhnpGZzc+OSY3L9fP3jjDVi7Fo73NarFmO+lici8sO1RqjqqMjcSketxVYERD1NS5QDwd+9V0t6KrvczuPj08G3Jlq7AryoOTIf5OOdx4PGKzjMmEtOnu6mbateO/7NzNubQvXl3UlNi05gW6g05bZolLROxQlUt70+5fKBV2HZLb98PiMgA4PdAP1U9FHZt/xLXTi3tISL0Bf4AtCEsB6n6G1wc8XpamqULcFV5xiScvXth4cJgqgYPFh7ky2+/pGeL2P14dOgAjRvDF1/E7BGm5soF2otIOxFJx63mMSH8BBHpAjwPDCkxTOlj4AIRaSQijYALvH2leQF4EjgTNwY49PKlwpJWiZkxUnD96zf6fYAx8TR7NhQVBZO0Fn27iILigpi0Z4WkpLh/W2heRWOiRVULRWQELtmkAmNUdZmIPALMU9UJuJqxBsCb3jpx36jqEFXdISJ/xCU+gEdUtay5BHerft8WFjE/bVoZYZ8LcW1cb1f2gcbE0rRprpt7qHt4PIU6YfRo4fuPxkrp1w/efRfy8qBlywpPN8Y3VZ0ETCqx7+GwzwPKuXYMMMbHY6aI8DgwHghVL6LKAj8x+klayzVL3wzfIdlyJfBmGecbE5jp06FLF8jIqPjcaJuTN4cWGS1oeVRsM0l4u9a118b0UcbEQqj+PLx9TYFz/Vzsp03rQZ/7jAnUoUNupeIgqgbBzYTRs2Xsm3vPOAOOOsratUxyUuWcUl6+EhaUP8v7QGAQ0EKy5Z9hh47CVi42CWj+fDh4MJiktXXfVtbsXMMvu/0y5s9KTYUzz7R2LZO8RBiMGzZVJ7RPlUf8XFteSWsjMA84iFsHJfSagBv9bExCCf0SP/PM+D871J4Vy56D4fr1g//+FzZvjsvjjIkaEZ4DrgZ+jZsw90pc93dfypvl/UvgS8mWVzRLrWRlEt706XDqqdC0afyfPTd/LimSQrfm8ZkUMNSuNX06XHFFXB5pTLT0UaWTCItVyRbh7+C/N2F51YNvaJZeBSyUbPnRpDGapZ0qF68x0VdU5Obku/rqYJ4/J28Opzc7nQbpDeLyvG7d3BpbX3xhScsknQPe+34RmgPbgeP8Xlxe78HfeO+2TqpJeEuXwu7dwbRnFWsxOfk5XN0xfhmzVi03D+GUKXF7pDHR8oEIR+PGfC3A9Rz8t9+Ly6se3OS9r69igMbE3OTJ7r1fxDOhVd1X279i96Hdcek5GG7AALj/fti0CY7z/XeqMcFS5Y/ex7dF+ACoo8puv9eXVz24hyMT5YJrMNPQu2bpUZWI15iYmDjRLfjYqlXF50bb3Ly5QPw6YYScf75LWpMnww03xPXRxkSFKocIG2DsR3klrQCGZxoTud27XYeEe+4J5vlz8+eSkZ7BKU1OietzzzgDmjSxpGVqFl8rKnozu5+JK2nN0CxdGNOojInA5MlQWAiDBwfz/Dl5c8hskRmzmd3LkpIC553n/v2qbr0tY6o7P4tAPgy8DDQGmgAvSbY8FOvAjPFr4kQ4+uhg5hvcX7CfxZsXx71qMGTAANi40a3UbEwyEEFEuF6Eh73t1iL4nmXazzRO1wE9NMv9B/QCrDLCJITiYpg0CS68ENJ81RtE14JNCyjSorh3wgg5/3z3HuqIYkwSeAboDYTWXNwDjPR7sZ+ktZGwqTaA2pSyMJgxQViwwM0KEVTVYFCdMELatIETT4RPPw3k8cZURk9V7sDNtoQqO4F0vxf7+dt0N7BMsuVTXJvW+UBOaD5CzdI7Iw7ZmCiZONG15Vx0UTDPn5s/lzYN23Bsg2ODCQBX2ho7FgoK3PgtYxJcgQipeL3TRWgKFPu92E/Sesd7hUyNJDpjYmnSJOjZM5ipm8B1wujTqk8wD/cMGADPPutmuA9i3kVjIvRPXE5pJsKjwBWA734SFSYtzdKXKx+bMbGzZQvk5sIjvuaGjr5Nezax4bsNgVUNhpx7rpv5/cMPLWmZxKfKKyLMB87Djfu9RBXfXYn89B68WLJloWTLDsmW7yRb9ki2fFeFmI2Jig8/dF29Bw0K5vlz8732rIA6YYQcfbRLVhMnBhqGqQZE5CIRWSkiq0TkgVKOny0iC0SkUESuKHGsSEQWea8JZT+DE4C1qowElgLne9M6+eKnI8ZTwI1AY83SozRLM2w2DJMIJk500xd16RLM8+fmzSUtJY0uPwkogDCDB8OXX0JeXtCRmGQlIqm4XnwDgQ7AMBHpUOK0b4CbgFdLucUBVe3svYaU86i3gSIRTgSeB1qVcb9S+UlaG4ClmqU/mundmKAUFMDHH7tSVlCDaufkz+GMY8+gbq26wQQQJtR7ctKkYOMwSS0TWKWqa1T1MDAOGBp+gqquU9XFRNBxohTFqhQClwFPq3IfUZrlPeR/gEmSLV8QNkeUZumTkUZqTLTMnAnffRdcV/fDRYeZkzeH4V2HBxNACaeeCm3butLn8MQIySSeNBGZF7Y9SlVHhW23wBVSQvKASOq+63j3LwQeU9V3yzivQIRhwM+An3r7fPd79VPSehTYjxurlRH2KpeIjBGRLSKytILzepRWP2pMeSZNct27BwwI5vnzN87nYOFBzmoTwFoopRBxCXzyZDh4MOhoTIIqVNXuYa9RFV8SkTaq2h24FnhKRE4o47ybcYOLH1VlrQjtgLF+H+KnpNVcs/Q0vzcM8xLwNPCfsk7w6lD/CnxSifubGmziRLd6b0ZA0zpP/2Y6AGe1ToykBS5pjRwJU6cGN27NJLV8XPtSSEsimEhCVfO99zUiMhXoAqz+8XksB+4M216LywO++ClpTZJsucDvDY8EotOAHRWc9mtco9yWSO9vaq5162D58uCqBsElrfbHtA90UHFJ/ftD3brWi9BUWi7QXkTaiUg6cA1QZi/AcCLSSERqe5+bAH2B5aWfS3sR3hJhuQhrQi+/QfpJWrcDH0m2HIhml3cRaQFcCjxb1XuZmiX0SzmopFWsxcz8ZmZClbLAJazzzoMPPnBDAYyJhKoWAiOAj4EVwBuqukxEHhGRIfB9c04ecCXwvIgs8y4/FZgnIl8CU3BtWqUmLeBF3O/9QuAcXG3c//mN08/g4lhVwDwF3K+qxVJB9y8RGQ4MB0hP9z1FlammJk508+2ddFIwz1+2ZRk7D+5MmPascEOHuqT15ZfQuXPQ0Zhko6qTgEkl9j0c9jkXV21Y8rpZwOk+H1NXlc9EEFXWA3/wBhs/XNGF4H89rUZAe8ImztUsneYzwLJ0B8Z5CasJMEhECkvrceI1GI4CqF+/vv0NWYPt3w9TpsAvfxlcDInYnhUydKj72rz9tiUtk7AOiZACfC3CCFy7WQO/F/uZEeMXwDRckTHbe/9DpUINo6rtVLWtqrYF3gJ+VU4XSWMAl7AOHgxuFgxwSeu4BsdxfKPjgwuiDE2bQr9+LmkZk6B+A9TDdcbohlvq6ka/F/tp0/oN0ANYr1l6Dq5HyK6KLhKR14DZwMkikicit4jIbSJym9/gjClp4kSoX9/9Yg6CqjJ9/XTOanMWFVVrB+Xyy92ikLYwpElEquSqsleVPFVuVuUyVeb4vd5P0jqoWXoQQLKltmbpf4GTKw5Mh6nqcapaS1VbquoLqvqcqj5Xyrk3qepbfoM2NZOqS1oDBkDt2sHEsG7XOvL35Cdk1WDIpZe6dyttmUQkwkki/FuET0T4PPTye72fNq08yZajgXeBTyVbdgLrKxeuMZW3bBl88w085HsRg+hL5PaskObNoU8fl7SC/FoZU4Y3geeAfwNFkV7sp/eg93cbf5BsmQI0BD6K9EHGVNWrr0JKSsDjs9ZPp2HthpzWrDLj7ePn8svhnntgzRo4PvGa3kzNVqha+aFOokk2oKN+/fq6b9++oMMwcXboELRqBX37wjvvVHx+rJzy9CmccMwJTLw2sUfwrlsH7drBY4/B/fcHHY1JBCKyX1XrB/d8jvE+3ombUOIdwuez1QonowD8tWkZE7i33oKtW+FXvwouhq37trJy+8qErhoMadsWeveGV14JOhJjvjcfmIfrKXgfMMvbF9rvi69xWsYE7ZlnoH17N+NDUKaumwpAvzYBdV2M0HXXwYgRsHgxdOoUdDSmplOlXTTuYyUtk/AWLYJZs1wpKyXA79jJayaTkZ5BjxY9ggsiAlddBWlpVtoyiUWEOiLcLcJ4Ed4W4S6RIxNXVMSSlkl4zzzj5tW70ffww9iYvHYy57Q7h7SU5KigaNoULrzQdWAprsqSfcZE13+AjsC/cCuBdCSCpUksaZmEtmuXKylcey00ahRcHGt3rmXNzjUMaBfQAl6VdP31kJcH06o66Zox0XOaKreoMsV73YpLXL5Y0jIJ7eWX3XyDQXbAAPhs7WcADDg+uZLWkCHQoAH8n+85tI2JuQUi9AptiNCTCDpiWNIyCUvVVQ326gVduwYby+Q1k2me0ZxTmpwSbCARqlfPjdl6802wkSImQXQDZomwToR1uOn+eoiwRITFFV2cHJXzpkb6/HP46iv4T5lrX8dHsRbz2drPGNR+UMLON1ien//clVjffBNuuinoaIyhSutqW9IyCWvkSGjSBK68Mtg4Fn27iG37t3FeuwD721fBWWfBySfDqFGWtEzwvDW0Ks2qB01CysuD996DW26BOr47w8bGxK8mIggXnVilPxADIwK33gqzZ8PSpUFHYxKZiFwkIitFZJWIPFDK8bNFZIGIFIrIFSWO3SgiX3uvmPX1taRlEtKoUa5NK8jFHkMmfj2RHi160Kx+s6BDqbQbb4T0dPj3v4OOxCQqEUkFRgIDgQ7AMBHpUOK0b4CbgFdLXHsMkAX0BDKBLBGJSX9fS1om4Rw+7JLW4MFu/rwgbd23lZz8HAa3D3CW3iho0sQtWTJ2LBw4EHQ0JkFlAqtUdY2qHgbGAUPDT1DVdaq6GCg58u9C4FNV3aGqO4FPqWLbVVksaZmE8847sHlz8N3cAT5c9SGKJn3SAhg+HHbuhNdfDzoSk6BaABvCtvO8fbG+NiKWtEzCeeYZt5zGhRcGHYmrGvxJg5/Q5bguQYdSZeecAx07wlNPuapXU+Okici8sNfwoAOqDEtaJqEsWeJmb7j99mDnGQQoKCrg41UfM+jEQaRI8v+oiMBdd8GXX8IXXwQdjQlAoap2D3uNKnE8H2gVtt3S2+dHVa6NSPL/JJpq5dlnoXZtuPnmoCOBmRtmsvvQbga1HxR0KFFz3XWufesf/wg6EpOAcoH2ItJORNKBa4AJPq/9GLhARBp5HTAu8PZFnSUtkzC++851FLjmGmjcOOho4O3lb1MnrQ4XnHBB0KFETd26cNtt8P77sGpV0NGYRKKqhcAIXLJZAbyhqstE5BERGQIgIj1EJA+4EnheRJZ51+4A/ohLfLnAI96+qLOVi03CGDnSrf+UkwM9Al79o1iLaflkS3q17MX4q8cHG0yUbdoEbdq4jhlPPx10NCZegl65OFqspGUSQmiewe7dg09YADO/mcmmvZu4skPA03HEwHHHwQ03wAsvwLffBh2NMZGxpGUSwrRpsHx5YnRzB3hz+ZvUTq3NxSddHHQoMfHgg2483BNPBB2JMZGxpGUSwsiRbr2sa64JOhJXNfj2ircZ2H4gGbUzgg4nJk48EYYNcx1ftm0LOhpj/LOkZQK3caMbUPzzn7uOAkGbtWEWG/dsrJZVg+F+/3s3O4b1JDTJJGZJS0TGiMgWESl1ik4RuU5EFovIEhGZJSJnxCoWk9hGj4bCQterLRG8sewNaqfW5qcn/TToUGLq1FPhiivgX/+y0pZJHrEsab1E+XNPrQX6qerpuK6SJQe6mRpg61ZXRXXRRa7KKmiHCg/x6pJX+enJP622VYPh/vAHtzjkn/4UdCTG+BOzpKWq04Ay++mr6ixvYkWAObgR1KYGOXgQhg6FXbvgj38MOhpnwsoJbD+wnV90+UXQocRFhw6uWvaZZ2DNmqCjMaZiidKmdQvwYVkHRWR4aL6swsLCOIZlYqW42C1IOHu2G1DcvXvQETmjF46m1VGtGHD8gKBDiZvsbEhLg4ceCjoSYyoWeNISkXNwSev+ss5R1VGh+bLS0myx5erg4YfdbON//atrV0kE63et59PVn/LzLj8nNSU16HDipnlz+O1v4bXXYP78oKMxpnyBJi0R6QSMBoaq6vYgYzHx8+KL8OijbjXd++4LOpojXlz0IgA3d06AiQ/j7H/+B5o1czOSFJdcKcmYBBJY0hKR1sB44AZV/SqoOEx8ff65mz7o/PPd2CyRoCNyioqLeHHRi5x/wvm0ObpN0OHEXcOG8PjjMGeO+6PCmEQVs7kHReQ1oD/QBNiMW4q5FoCqPicio4HLgfXeJYWqWmHLhs09mLxWrIDevaFlS5g50/2iTBSTvp7E4FcH8/oVr3NVx6uCDicQqtCvn5uZZOXKxJi02ERPdZl70CbMNXGxZQv06gX797u/5tu2DTqiH+r/Un9W71zN6jtXk56aHnQ4gVmyBLp0cZ1kRo8OOhoTTdUlaQXeEcNUfwcOuK7t334LEyYkXsKavWE2X6z/gnt631OjExbA6afD3Xe7yXQ/+SToaIz5MStpmZgqLnbzCb71Frz9Nlx6adAR/diQ14Ywa8Ms1t+1nvrpSf+HaJUdPAhdu8KePa7kdfTRQUdkosFKWsb48Pvfw5tvukb+RExYS7cs5f2v3ufOnndawvLUqQMvv+zW3brrrqCjMeaHLGmZmBk9Gh57DH75S1fllIgem/EYDdIbMCJzRNChJJQePeB3v3PJ6623go7GmCMsaZmYmDwZbr8dLrzQrY6bKF3bw63ZuYZxS8fxy26/5Ji6xwQdTsJ56CHXeebnP4evvw46GhMPInKRiKwUkVUi8kApx2uLyOve8bki0tbb31ZEDojIIu/1XKxitKRlom7ZMrj8cjeL+BtvuCmCEtF9n95Hemo6v+3126BDSUjp6W7Wklq13KwlBw4EHZGJJRFJBUYCA4EOwDAR6VDitFuAnap6IvAP4K9hx1aramfvFbM1GyxpmajavBkGD4Z69eCDD+Coo4KOqHQfrfqI8SvG89DZD9HiqBZBh5OwWreG//s/WLzYrSqdZP22TGQygVWqukZVDwPjgKElzhkKvOx9fgs4TyS+9SiWtEzU7N8PQ4a4MVnvv+9+4SWig4UH+fWHv+akxidxT+97gg4n4Q0cCFlZ8NJLbq5Ik7TSQhOPe6/hJY63ADaEbed5+0o9R1ULgd1AaBh6OxFZKCJfiMhZMYgfgAStuDHJprgYfvYzyM2F8eMTZ9b20jw+83FW7VjFJ9d/Qu202kGHkxSysuCrr+DBB+GEE+DK6r2oc3Xla9ahStoEtFbV7SLSDXhXRDqq6nfRfpCVtExUPPigG4f197/DJZcEHU3Z1u5cy59n/JkrO1zJ+SecH3Q4SUMExoyBvn3dHyfTpwcdkYmBfKBV2HZLb1+p54hIGtAQ2K6qh0KTnqvqfGA1cFIsgrSkZaps1Cj4299cm0cij+sp1mLumHQHqZLKkxc+GXQ4SadOHXjnHWjTxrVbzp0bdEQmynKB9iLSTkTSgWuACSXOmQDc6H2+AvhcVVVEmnodORCR44H2QEyWFbWkZarkk09csho4EP73fxOza3tI9tRsPlz1IX857y+0PMoWyq6Mpk3hs8/c+4UXwoIFQUdkosVroxoBfAysAN5Q1WUi8oiIDPFOewFoLCKrgLuBULf4s4HFIrII10HjNlUtc+X6qrBpnEylLV0KffpAu3auuihRewoCvLnsTa566ypu6nwTY4aMIc4dnqqd9evh7LPdVE8ffOC+D0xiqy7TOFnSMpXy7bfQsycUFLhqolatKr4mKAs3LaTvmL50Oa4Ln//sc+t8ESXr1sEFF0Benps1Y9CgoCMy5akuScuqB03E9u+Hn/4Utm1zf2UncsLavHczQ8cNpXG9xoy/arwlrChq2xZmzHCDyIcMcR01jIk1S1omIkVFcP31MH8+jBvnZgNPVLsP7ubS1y9l2/5tTLhmAsc2ODbokKqdZs1gyhQ491y45RYYMcKVvo2JFUtaJiL33+96kP3jH660lahWbltJz9E9yd2Yy9hLx9LluC5Bh1RtHXUUTJoE994LI0fCeefBxo1BR2WqK0taxrdnn3XjsEaMgDvvDDqasn349Yf0HN2T7Qe289nPPuPyDpcHHVK1l5bmlp955RWYNw86dXKDzI2JNktaxpePPoJf/9qNz/nHPxKza7uq8reZf2Pwq4Np16gd826dx9ltzg46rBrl2mth4ULX3nX55W6G+J07g47KVCfWe9BUqLAQOnSA+vVd1/YGDYKO6Md2HtjJHZPu4LWlr3FVx6sYM2SMLeoYoIICeOQR+POfoXFjeOIJuOGGxPxjp6aoLr0HLWkZX9avh9RUaJlgY3I37dnEP+b8g2fnPcu+w/t49NxHeeDMB2wcVoJYtMitqzZnDpx1lktemZlBR1UzWdIKiCUtA7B6x2oen/U4Ly56kcLiQq7ueDUPnPkAnY7tFHRopoTiYtcd/sEH3TCJSy6BP/0JOnYMOrKaxZJWQCxp1WyLNy/msRmP8fqy10lLSePmzjdzX5/7OOGYE4IOzVRgzx7XHvrEE7B3L1x6qetx2Lt30JHVDJa0AmJJq+bYX7CfBZsWMDdvLjkbc5ibN5f1u9eTkZ7B7d1v565ed3FcxnFBh2kitG0bPPmk6426a5dLWnfe6UpgdeoEHV31ZUmrohuLjAEuBrao6mmlHBfgf4FBwH7gJlWtcPpNS1rVU1FxESu2rSAnP+f7JLVk8xKKtAiAtke3pWeLnvRp1YcbOt1Ao7qNAo7YVNXevfDii670tXYtNGoE110HN98MXbpYp41os6RV0Y1Fzgb2Av8pI2kNAn6NS1o9gf9V1Z4V3deSVmIqLC5kz6E97Dm8p8z3vYf3HtkXtn/3wd0s27qMvYf3AnB0naPJbJFJzxY9yWyRSWaLTJrVbxbwv9DESnExfP65a/caPx4OHXILTV52mXtlZkKKDc6pMktafm4u0hb4oIyk9TwwVVVf87ZXAv1VdVN597SkFYz/9/n/Y+X2lWUmpUNFh3zdJy0ljYz0DBqkNyCjdgYZ6Rlk1M7glManuETVsicnHnMiKWK/pWqinTvd5Lvjx7slUAoK4Ljj3NI3f/mLmzbKVE51SVppAT67BbAhbDvP2/ejpCUiw4HhAOnp6XEJzvzQwm8Xsnrn6u+TTJN6TY4kHW9facmo5Hvt1NrWHd2UqVEjuPVW99q1CyZOhPfec4PbR44MOjqTCIIsaX0APKaqM7ztz4D7VXVeefe0kpYxNY+qtXFVVXUpaQVZB5MPhC9q0dLbZ4wxP2AJy4QEmbQmAD8Tpxewu6L2LGOMMbEjIheJyEoRWSUiD5RyvLaIvO4dn+vVpoWOPejtXykiF8Yqxpi1aYnIa0B/oImI5AFZQC0AVX0OmITrObgK1+X95ljFYowxpnwikgqMBM7H9THIFZEJqro87LRbgJ2qeqKIXAP8FbhaRDoA1wAdgebAZBE5SdUbsxJFMUtaqjqsguMK3BGr5xtjjIlIJrBKVdcAiMg4YCgQnrSGAn/wPr8FPO2NuR0KjFPVQ8BaEVnl3W92tIO0fsXGGFMzpInIvLDX8BLHy+rRXeo5qloI7AYa+7w2KoLs8m6MMSZ+ClW1e9BBVJWVtIwxxoC/Ht3fnyMiaUBDYLvPa6PCkpYxxhiAXKC9iLQTkXRcx4oJJc6ZANzofb4C+NzrnzABuMbrXdgOaA/kxCLIpKse3L9/v4rIgUpengqU15ulrOOl7Q/fV/J4aLu0c9KAwoiiLj82P+eUFV9p2xV9Tsb4w/fFKv40QEs5p6LvnZLbQcVf2e/9ktvlxV/Z2MuLr6Lj1T3+SH731C0vQFUtFJERwMfeNWNUdZmIPALMU9UJwAvAWK+jxQ5cYsM77w1cp41C4I5Y9BwMBZpUL++LV9lrR1XmeGn7w/eVPB7aLu2cysZfUeyRxF/edkWfkzH+EvtiEj8wr6LvkwSPv1Lf+5HEn4g/u8kefzx+9yTaq6ZVD75fyeOl7X+/nOPv+zgnUn6u9xt/edt+PldGkPFXNXa/96jo+6SsfYkQf2W/90tuW/z+4/BzPBF+9ySUpFsEUkTmaRL3gLH4gxWr+OP1dUnmr38yxw4Wf6JIxpLWqKADqCKLP1ixij9eX5dk/vonc+xg8SeEpCtpGWOMqbmSsaRljDGmhrKkZYwxJmlY0jLGGJM0qlXSEpFLROTf3novFwQdT6RE5HgReUFE3go6Fj9EpL6IvOx9za8LOp5IRePrLSJ/F5EvReRf0YzN57OT/fv9VBF5TkTeEpHbg46nMryfgXkicnHQsURKRPqLyHTv/0H/oOPxK2GSloiMEZEtIrK0xP5yFyULp6rvquqtwG3A1bGMt6Qoxb9GVW+JbaTli/DfcRnwlvc1HxL3YEsRSfxV/XqLyAlAX1U9Q1V/XcXQQ/eMJP7Avt/LEmH8K1T1NuAqoG8Q8ZZUiZ/j+4E34htl2SKMX4G9QB3crOzJIejRzWGjts8GugJLw/alAquB44F04EugA3A68EGJV7Ow6/4OdE3i+N9Kkv8PDwKdvXNeDfp7KNL4q/L1Bk7GLcWwCVgI1A8w/rh/v0crftwfOx8C1wYde6Tx4xZLvAa4Cbg46NgrEX+Kd/xY4JWgY/f7Spi5B1V1moQt3ewpdVEyVf0L8KPiuLcY2WPAh6q6IMYh/0A04k8Ekfw7cH+dtQQWkSCl9gjjX04lqepKEXkZWKeqoyt7n1Lu6zt+EVlBQN/vZYn0669uPrsJIjIReDWuwZYiwvgbAPVxCeCAiExS1eJ4xltShL+HQt//O4Ha8YuyahLiF005Il1Y7NfAAOAKEbktloH5FFH8ItJYRJ4DuojIg7EOLgJl/TvGA5eLyLMk9lQxpcYfha/36bi/WmOtrK9/on2/l6Wsr39/EfmniDwPTAomNF9KjV9Vf6+qd+GS7b+DTljlKOvrf5n3tR8LPB1IZJWQMCWtaFDVfwL/DDqOylLV7bj2iaSgqvuAm4OOo7Ki8PXuCCwVkVTgcVwbwXrv+zDmqsH3+1RgasBhVJmqvhR0DJWhquNxf3gmlURPWnFbWCxGkj3+kGT/d0Q9fhHJAApU9YC45RzeU9UvqnLPctjXP1gWfwJJ9OpBP4uSJbJkjz8k2f8dsYj/NCDUQ6sbMLOK9yuPff2DZfEnkqB7goT1cHkN1xOrAFfneou3fxDwFa73y++DjrO6xl9d/h1BxI9rlH8BeAI4JtniT/avv8VffeL387IJc40xxiSNRK8eNMYYY75nScsYY0zSsKRljDEmaVjSMsYYkzQsaRljjEkalrSMMcYkDUtapkYRkTtFZIWIvBJ0LMaYyNk4LVOjiMh/gQGqmhe2L01VCwMMyxjjk5W0TI3hzeh+PPChiOwWkbEiMhMYKyJNReRtEcn1Xn29axqLyCciskxERovIehFpIiJtwxfaE5F7ReQP3ucTROQjEZkvbmXYU7z9L3mzms8SkTUickXY9feLyBJxqyA/5t1jQdjx9uHbxtRUiT5hrjFRo6q3ichFwDnACOCnwJnqJr19FfiHqs4QkdbAx8CpQBYwQ1UfEZHBgJ+VjkcBt6nq1yLSE3gGONc7dhxwJnAKbv63t0RkIG4qqJ6qul9EjlHVHV5i7ayqi3Cz6b8Yna+EMcnLkpapySao6gHv8wCgg1tHFICjRKQBbiXYywBUdaKI7Czvht41fYA3w+4VvsDeu+rWXVouIseGPftFVd3vPWeHt380cLOI3A1cjVvMz5gazZKWqcn2hX1OAXqp6sHwE8IST0mF/LB6vU7YfXapaucyrjsUfvsK4nsbV9L7HJivbv0vY2o0a9MyxvkEtxIwACLS2fs4DbjW2zcQaOTt3ww089q8agMXA6jqd8BaEbnSu0ZE5IwKnv0prkRVz7vmGO9eB3HVlM9iVYPGAJa0jAm5E+guIotFZDlHVjTOBs4WkWW4asJvAFS1AHgEyMElnf+G3es64BYR+RJYhmuvKpOqfoRr35onIouAe8MOvwIU45KqMTWedXk3JgIisg7orqrb4vS8e4GGqvr/4vE8YxKdtWkZk6BE5B3gBI70PDSmxrOSljHGmKRhbVrGGGOShiUtY4wxScOSljHGmKRhScsYY0zSsKRljDEmaVjSMsYYkzT+P0wrslslpJbpAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"nondimensional c_un_P = {}\".format(c_un_P_num/c_el_P_num))\n",
    "\n",
    "fig5, ax9 = plt.subplots();\n",
    "ax10 = ax9.twinx();\n",
    "ax9.semilogx(f2, c2, 'g');\n",
    "ax10.semilogx(f2, np.abs(d2)/f2, 'b');\n",
    "#ax9.set_ylim([0,2])\n",
    "ax9.set_xlabel('frequency');\n",
    "ax9.set_xticks(list(ax9.get_xticks()) + [f_characteristic_num], list(ax9.get_xticklabels()) + ['$f_c$'])   # with extra tick\n",
    "ax9.set_ylabel('phase velocity', color='g');\n",
    "ax10.set_ylabel('decay constant', color='b');\n",
    "#ax10.set_ylim([0,1.2e-5])\n",
    "#tikzplotlib.save(\"p_wave_fast_eigenvalue.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW')   \n",
    "\n",
    "fig6, ax11 = plt.subplots();\n",
    "ax12 = ax11.twinx();\n",
    "#ax11.set_ylim([0,1])\n",
    "ax11.semilogx(f2, AR2, 'g');\n",
    "ax12.semilogx(f2, phi2, 'b');\n",
    "ax11.set_xlabel('frequency');\n",
    "#ax12.set_ylim([-1.0,0])\n",
    "ax11.set_xticks(list(ax11.get_xticks()) + [f_characteristic_num], list(ax11.get_xticklabels()) + ['$f_c$'])   # with extra tick\n",
    "ax11.set_ylabel('amplitude ratio', color='g');\n",
    "ax12.set_ylabel('phase angle [RAD]', color='b');\n",
    "#tikzplotlib.save(\"p_wave_fast_eigenvector.tex\", axis_height = '\\\\figH', axis_width = '\\\\figW') "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79c44a23",
   "metadata": {},
   "source": [
    "Numbers for TikZ-plots (complex amplitudes and time history)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "11a55570",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30\n",
      "--\n",
      "3.7815127464432794\n",
      "23.951735298792855\n",
      "-0.5497164147632475\n",
      "-0.011182524789751964\n",
      "--\n",
      "61.80646848927272\n",
      "0.19187941811069834\n",
      "0.5013781513666287\n",
      "-0.1592354367022788\n"
     ]
    }
   ],
   "source": [
    "k_num = index_first_nonzero_f   # first mode with slow P-wave\n",
    "print(k_num)\n",
    "print('--')\n",
    "print(f1[k_num])\n",
    "print(-d1[k_num]/f1[k_num])   # delta, with time axis scaled to period\n",
    "print((1/AR1[k_num])*np.cos(-phi1[k_num])) # u_solid, real part, with u_fluid = 1\n",
    "print((1/AR1[k_num])*np.sin(-phi1[k_num])) # u_solid, imaginary part, with u_fluid = 1\n",
    "print('--')\n",
    "print(f2[k_num])\n",
    "print(-d2[k_num]/f2[k_num])   # delta, with time axis scaled to period\n",
    "print(1/AR2[k_num]*np.cos(-phi2[k_num])) # u_solid, real part, with u_fluid = 1\n",
    "print(1/AR2[k_num]*np.sin(-phi2[k_num])) # u_solid, imaginary part, with u_fluid = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "924d9256",
   "metadata": {},
   "source": [
    "## References\n",
    "- \\[Biot1956\\]   M.A. Biot; _Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid -- I. Low-frequency Range_; Journal of the Acoustical Society of America 28(2); 1956.\n",
    "- \\[Verruijt2010\\]   A. Verruijt; _An Introduction to Soil Dynamics_; Springer Dordrecht Heidelberg London New York; 2010.\n",
    "- \\[Hagedorn2012\\] P. Hagedorn, D. Hochlehnert; _Technische Schwingungslehre: Schwingungen linearer diskreter mechanischer Systeme_; Europa-Lehrmittel, 2021."
   ]
  }
 ],
 "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": "3.10.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}