example.ipynb

{
 "metadata": {
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.4"
  },
  "orig_nbformat": 2,
  "kernelspec": {
   "name": "python3",
   "display_name": "Python 3",
   "language": "python"
  }
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 "nbformat": 4,
 "nbformat_minor": 2,
 "cells": [
  {
   "source": [
    "# Example Usage of the M1EPMA Software\n",
    "\n",
    "- calculation of a forward solution\n",
    "- calculation of k-ratios (standardized by a constant)\n",
    "- calculation of the jacobian of these k-ratios w.r.t. the material parameters"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from physics import Copper, Nickel, Chromium, XRay, Material, Detector, ElectronBeam\n",
    "from scipy.constants import kilo, electron_volt, nano\n",
    "import matplotlib.pyplot as plt\n",
    "from experiment import Experiment, k_ratios, mass_fractions_from_parameters, k_ratios_jacobian\n",
    "import m1model"
   ]
  },
  {
   "source": [
    "Definition of the elements and k-ratios / xrays to be considered in the material."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "elements = [Copper(), Nickel(), Chromium()]\n",
    "n_elements = len(elements)\n",
    "x_rays = [XRay(e, 1.e-12) for e in elements]"
   ]
  },
  {
   "source": [
    "Definition of the energy range. ($\\epsilon_{initial} > \\epsilon_{beam}$, otherwise the beam is not captured)"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "eps_initial_keV = 17.\n",
    "eps_cutoff_keV = 10."
   ]
  },
  {
   "source": [
    "Definition of the material domain and grid (reconstruction  pixel, and finite volume cells), the detector (position) and the electron beam (size, position, energy).\n"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = Material(1, 1, 70, 70, [0., 1000.*nano], [-800.*nano, 0.])\n",
    "detector = Detector(500.*nano, 50.*nano, m)\n",
    "e_beam = ElectronBeam([(30.*nano)**2, (30.*nano)**2], [500. * nano, 0. * nano], 16., 0.1)\n",
    "#plt.plot(np.linspace(0, 1000.*nano), [e_beam.intensity_dist(x, 0.*nano, e_beam.beam_energy_keV) for x in np.linspace(0, 1000.*nano)]) # plots to visualize the beam distribution over the computationl domain\n",
    "#plt.plot(np.linspace(eps_cutoff_keV, eps_initial_keV), [e_beam.intensity_dist(500.*nano, 0.*nano, e) for e in np.linspace(eps_cutoff_keV, eps_initial_keV)])\n"
   ]
  },
  {
   "source": [
    "Definition of the Experiment (a specific electron beam / detector setup)"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "e = Experiment(m, detector, e_beam, elements, x_rays, eps_initial_keV, eps_cutoff_keV, 100)"
   ]
  },
  {
   "source": [
    "Definition of the parameter vector (the shape of the vector is given by the material definition).\n",
    "Mass fractions can be calculated from the parameters."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "params = np.full(e.parameter_dimensions, 0.33)\n",
    "mass_fractions = mass_fractions_from_parameters(m.n_x, m.n_y, params)"
   ],
   "cell_type": "code",
   "metadata": {},
   "execution_count": 6,
   "outputs": []
  },
  {
   "source": [
    "Solving the m1model and displaying a snapshot of it."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "calculating forward solution\n"
     ]
    }
   ],
   "source": [
    "sol = m1model.solve_forward(e, mass_fractions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "energy_step=50\n",
    "plt.imshow(sol['solution'][energy_step][0, :, :].T, origin=\"lower\", extent=(m.dim_x[0], m.dim_x[1], m.dim_y[0], m.dim_y[1]));\n",
    "plt.xlabel(\"x in [m]\");\n",
    "plt.ylabel(\"y in [m]\");\n",
    "plt.title(\"$\\psi^0$ where $\\epsilon = {:.1f}keV$\".format(e.epsilons_keV[energy_step]));"
   ]
  },
  {
   "source": [
    "Calculation of k_ratios (internally solves the m1model again). Note that the k-ratios are not standardized (only a constant factor is used)."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "calculating forward solution\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "DeviceArray([0.36621377, 0.47702644, 0.93333039], dtype=float64)"
      ]
     },
     "metadata": {},
     "execution_count": 9
    }
   ],
   "source": [
    "k = k_ratios(e, params)\n",
    "k"
   ]
  },
  {
   "source": [
    "Calculation of the k-ratios and their jacobian w.r.t. the material parameters. (This might take a while, because jax compiles the derivative functions)\n",
    "The adjoint solution is calculated 3 times, once for each k-ratio (output)."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "solving forward\n",
      "calculating forward solution\n",
      "extraction operator\n",
      "extraction opertator jacobian\n",
      "calculating\n",
      "solving adjoint\n",
      "calculating adjoint solution\n",
      "scalar product jacobian\n",
      "adding to jacobian\n",
      "solving adjoint\n",
      "calculating adjoint solution\n",
      "scalar product jacobian\n",
      "adding to jacobian\n",
      "solving adjoint\n",
      "calculating adjoint solution\n",
      "scalar product jacobian\n",
      "adding to jacobian\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "DeviceArray([[[[ 1.13907627,  0.01088031]]],\n",
       "\n",
       "\n",
       "             [[[ 0.04228116,  1.46389582]]],\n",
       "\n",
       "\n",
       "             [[[-2.70486691, -2.75114424]]]], dtype=float64)"
      ]
     },
     "metadata": {},
     "execution_count": 10
    }
   ],
   "source": [
    "k, jac = k_ratios_jacobian(e, params)\n",
    "jac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "DeviceArray([0.36621377, 0.47702644, 0.93333039], dtype=float64)"
      ]
     },
     "metadata": {},
     "execution_count": 11
    }
   ],
   "source": [
    "k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ]
}