{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Meet PyMKS\n", "\n", "In this short introduction, we will demonstrate the functionality of PyMKS to compute 2-point statistics in order to objectively quantify microstructures, predict effective properties using homogenization and predict local properties using localization. If you would like more technical details amount any of these methods please see the [theory section](THEORY.html)." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "%load_ext autoreload\n", "%autoreload 2\n", "\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###Quantify Microstructures using 2-Point Statistics\n", "\n", "Lets make two dual phase microstructures with different morphologies." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from pymks.datasets import make_microstructure\n", "\n", "X_1 = make_microstructure(n_samples=1, grain_size=(25, 25))\n", "X_2 = make_microstructure(n_samples=1, grain_size=(15, 95))\n", "\n", "X = np.concatenate((X_1, X_2))\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Throughout PyMKS `X` is used to represent microstructures. Now that we have made the two microstructures, lets take a look at them." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAnEAAAEaCAYAAAB+VgE8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3V1oZGcdx/Ffk+2YnaSt7MValzRkGBmL0yG6W4VUxKbS\n", "i6KlCnZwtyhxbL2yF70p4rYbI9ulvuBNe5mmoV2hjLQWEUQCRXy5sFKxRLwY8rIvoUUKW1mYDNnu\n", "ZrxY5niye86ceTlnzvM88/1AYDInOXPmvDzzn///PM9zS7PZbAoAAABWGUl7AwAAANA9gjgAAAAL\n", "EcQBAABYiCAOAADAQgRxAAAAFiKIAwAAsNCBtDcAAABgmHz44Yd6/vnntb29rVdffVUjI//PqV26\n", "dEkvvPCCrl69qnK5rFKpFLoeMnEAAAADNDExoVOnTqlQKNy07M0339Tx48d18uRJvfHGG23XE5mJ\n", "e++993rfSkP4xzOenJxMcUvic8stt3iPt7e3U9wSu7XOjbTPi0Eez3bvudftOHLkSN/b5X/tTv7G\n", "lPPexPalm/3UyzVg4nHolCvHS2p/Ddt2XG4U9D5dmpvg1ltv1a233hq47OLFi15wNzY2pkajoYMH\n", "Dwb+rdPl1NYBv+uuu1Leknj4L/SLFy+muCXuCGrw/A2FK+eOnyuNPGC71rVIew6/vb0973E2m1W9\n", 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This function computes all of the autocorrelations and cross-correlation(s) for a microstructure. Before we compute the 2-point statistics, we will discretize them using the `PrimitiveBasis` function." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from pymks import PrimitiveBasis\n", "from pymks.stats import correlate\n", "\n", "prim_basis = PrimitiveBasis(n_states=2, domain=[0, 1])\n", "X_ = prim_basis.discretize(X)\n", "X_corr = correlate(X_, periodic_axes=[0, 1])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's take a look at the two autocorrelations and the cross-correlation for these two microstructures." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(101, 101, 3)\n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABDQAAAEsCAYAAAA8U+MrAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXuYFMW5/78zswvLxUUREAkg3uKCJsFECQpefwpGjx6N\n", 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"display_data" } ], "source": [ "from pymks.tools import draw_correlations\n", "\n", "print X_corr[0].shape\n", "\n", "draw_correlations(X_corr[0])\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABDQAAAEsCAYAAAA8U+MrAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXucFMW5//+Zmd1luS24AiIComJcQBOMiATBWxCNBn+o\n", "QAQ0oqASc4zGKN/giRDEC0IiRjlguGi8g4BRjiYYjx4QjIZoNAiC6HFF5H5RcNkbOzO/P2aru7r7\n", "merqnu657D7v14vXDtXd1VXVNc88XfVcIslkMgmGYRiGYRiGYRiGYZgCIprrBjAMwzAMwzAMwzAM\n", "w3iFFzQYhmEYhmEYhmEYhik4eEGDYRiGYRiGYRiGYZiCgxc0GIZhGIZhGIZhGIYpOHhBg2EYhmEY\n", "hmEYhmGYgoMXNBiGYRiGYRiGYRiGKTh4QYNhGIZhGIZhGIZhmIKDFzQYJkO2b9+Ojz/+ONfNYBiG\n", "YcAymWEYJt9gucyESVGuG9BUePfdd/Haa6/hiy++QH19PTp06IAzzjgDw4YNw1FHHZXTtv3Xf/0X\n", "vvrqKzzwwAPa1/z9739HfX09zjvvvIzrauq8/PLLaNu2LXr37p3rpjAM0wjL5OYLy2SGyU9YLjdf\n", 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statistics provide an object way to compare microstructures, and have been shown as an effective input to machine learning methods." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###Predict Homogenized Properties" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section of the intro, we are going to predict the effective stiffness for two phase microstructures using the `MKSHomogenizationModel`, but we could have chosen any other effective material property. \n", "\n", "First we need to make some microstructures and their effective stress values to fit our model. Let's create 200 random instances 3 different types of microstructures, totaling to 600 microstructures." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from pymks.datasets import make_elastic_stress_random\n", "\n", "grain_size = [(47, 6), (4, 49), (14, 14)]\n", "n_samples = [200, 200, 200]\n", "\n", "X_train, y_train = make_elastic_stress_random(n_samples=n_samples, size=(51, 51),\n", " grain_size=grain_size, seed=0)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Once again, `X_train` is our microstructures. Throughout PyMKS `y` is used as either the prpoerty or the field we would like to predict. In this case `y_train` is the effective stress values for `X_train`. Let's look at one of each of the three different types of microstructures." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA6EAAAEdCAYAAAD9zGENAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3V9oZOd9P/6Ptba6Hm3s7l4saSqLHVTU0IlQa7cFpZRG\n", "prkwbXALragbWlSR9Kq5yE3/bWtVZWNCCr1JeucoIkmhKMQ1pVCKwIT+uahLSo1KL4T+2Fo1oYR1\n", "imEktJal70X4zW+PdY5mZmeemTkzrxcIpHNmnnNmzr/nrWfOZx45Pz8/DwAAAOiBsX6vAAAAAKND\n", "CAUAAKBnhFAAAAB6RggFAACgZ4RQAAAAekYIBQAAoGeEUAAAAHrm0WYPeOSRR3qxHiOt6D0+PDzs\n", "8ZrQLUVfvzs5OdnjNem969evx9tvv/3Qz//Od75zYVre+/nUU09dmNbOMZPXZort0+nxXbQvtfr6\n", "29kX89a1TOehFPtJq8tpZ9/Je5/v3r2b+9gU698rne677fjQhz700M/NO+fk6dU5o+z0aXqnn/vk\n", "qG/nTs45Zff9738/Pv/5z8fh4WF87Wtfi7Gx/39M8+23344vfvGLcXp6GouLizE7O1vYjpFQAAAA\n", "mrp27Vq8+OKLMTMzc2Heq6++Gi+88ELcvn07XnnllUvbaToSCgAAAI899lg89thjufPu3r3bCKdX\n", 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We will also specify which correlations we want." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from pymks import MKSHomogenizationModel\n", "\n", "prim_basis = PrimitiveBasis(n_states=2, domain=[0, 1])\n", "homogenize_model = MKSHomogenizationModel(basis=prim_basis,\n", " correlations=[(0, 0), (1, 1), (0, 1)])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's fit our model with the data we created." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [], "source": [ "homogenize_model.fit(X_train, y_train, periodic_axes=[0, 1])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's make some new data to see how good our model is." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "n_samples = [10, 10, 10]\n", "X_test, y_test = make_elastic_stress_random(n_samples=n_samples, size=(51, 51),\n", " grain_size=grain_size, seed=100)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will try and predict the effective stress of our `X_test` microstructures." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": true }, "outputs": [], "source": [ "y_pred = homogenize_model.predict(X_test, periodic_axes=[0, 1])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `MKSHomogenizationModel` generates low dimensional representations of microstructures and regression methods to predict effective properties. Take a look at the low dimensional representations." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAi0AAAEmCAYAAACuxqAsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsnXlAVOX+/18zDKtssruvgIILbqUmZpmllkupmJZambe6\n", "3r7V7d76WWnZduvbbf/anqWmJpapJO5J4pYrogwYibjjsDogArP9/hjnOMOcgWEHfV7/AOecec7n\n", "POcwz/t8ns/n8yhMJpMJgUAgEAgEgmaOsqkNEAgEAoFAIHAGIVoEAoFAIBC0CIRoEQgEAoFA0CIQ\n", "okUgEAgEAkGLQIgWgUAgEAgELQIhWgQCgUAgELQIhGhp4UydOpWFCxc2tRmNRlJSElOnTiUpKamp\n", "TamW+Ph4pk6dilqtbmpTBM2IRYsWMXXqVPLy8praFIGgxaFqagPqk6lTpwKwatWqJrbEedLS0nj9\n", "9ddttrm5ueHl5UVYWBjh4eEMGzaMzp07N42BzRSFQtHUJtQ78fHx/PzzzzbbVCoVgYGB9OrVi/vv\n", "v5/g4OAmsu7G5bXXXiM9Pb3evjcs9/HVV18lKirKbv+N+OwKBI3FDSVaWjLBwcGMGDECAL1ej1ar\n", "JSsri4SEBBISErjtttv429/+hoeHh83nPvzwQ9zd3ZvA4qbhlltuISIiAn9//6Y2pcGIiooiOjoa\n", "gOLiYo4fP8727dvZt28fb731Fm3atGliCwV1Yfr06UycOJHWrVs3tSkCQYtDiJZmQnBwMJMnT7bb\n", 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"output_type": "display_data" } ], "source": [ "from pymks.tools import draw_components\n", "\n", "draw_components([homogenize_model.reduced_fit_data, homogenize_model.reduced_predict_data], \n", " ['Training Data', 'Testing Data'])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's look at a goodness of fit plot for our `MKSHomogenizationModel`." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAZQAAAEpCAYAAAC0kdQLAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4lGXa+P3v9GSSTHqhQyCFBAQEESFApAjiUhWWsoCi\n", "vKvrFv25Pq66gKiw7rO71sX+KEVE6UWQKoFQYiAQwEyCEUggQJj0STLJZNr7xziTDJlACKlwfY7D\n", "40juet23Yc652nlJbDabDUEQBEG4TdKWLoAgCIJwZxABRRAEQWgUIqAIgiAIjUIEFEEQBKFRiIAi\n", "CIIgNAoRUARBEIRGIW/pAghCa/Dss88CsGzZshYuSetz6tQp1q1bR05ODhUVFQwYMIAXX3zxtq+b\n", "kJDARx99xDPPPEN8fPztF1RocSKgCI3uypUr7NmzB61Wi06no7KyEk9PT9q1a0d0dDRDhgwhPDy8\n", 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"display_data" } ], "source": [ "from pymks.tools import draw_goodness_of_fit\n", "\n", "fit_data = np.array([y_train, \n", " homogenize_model.predict(X_train, periodic_axes=[0, 1])])\n", "pred_data = np.array([y_test, y_pred])\n", "\n", "draw_goodness_of_fit(fit_data, pred_data, ['Training Data', 'Testing Data'])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Looks good.\n", "\n", "The `MKSHomogenizationModel` can be used to predict effective properties and processing-structure evolutions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###Predict Local Properties" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this section of the intro, we are going to predict the local strain field in a microstructure using `MKSLocalizationModel`, but we could have predicted another local property.\n", "\n", "First we need some data, so let's make some." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from pymks.datasets import make_elastic_FE_strain_delta\n", "\n", "X_delta, y_delta = make_elastic_FE_strain_delta()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Once again, `X_delta` is our microstructures and `y_delta` is our local strain fields. We need to discretize the microstructure again so we will also use the same basis function." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from pymks import MKSLocalizationModel\n", "\n", "prim_basis = PrimitiveBasis(n_states=2)\n", "localize_model = MKSLocalizationModel(basis=prim_basis)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's use the data to fit our `MKSLocalizationModel`." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": true }, "outputs": [], "source": [ "localize_model.fit(X_delta, y_delta)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have fit our model, we will create a random microstructure and compute its local strain field using finite element analysis. We will then try and reproduce the same strain field with our model." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from pymks.datasets import make_elastic_FE_strain_random\n", "\n", "X_test, y_test = make_elastic_FE_strain_random()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's look at the microstructure and its local strain field." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAoMAAAExCAYAAAAHhKhtAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xt8lOWd9/HvTEiYTA6SAEIhmZxwBENAUNQgghTUivog\n", "IFHqiaVP11qrXXVtZRFCIN1dq60iiKeWZbGeUkSsVXEjntBilS5nowkJJIMogQQ5zAyQkDx/8DAS\n", "cpPkutUh7f15v155vWDu6zvXNYcMP35zH1zNzc3NAgAAgCO5T/UCAAAAcOpQDAIAADgYxSAAAICD\n", "UQwCAAA4GMUgAACAg1EMAgAAOBjFIAAAgINRDH7HZs+erWuvvfZULwMAAMBSl1O9gM7s+CLukUce\n", 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processing-structure evolutions." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }