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"cells": [
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"source": [
"# Cahn-Hilliard Example\n",
"\n",
"This example demonstrates how to use PyMKS to solve the Cahn-Hilliard equation. The first section provides some background information about the Cahn-Hilliard equation as well as details about calibrating and validating the MKS model. The example demonstrates how to generate sample data, calibrate the influence coefficients and then pick an appropriate number of local states when state space is continuous. The MKS model and a spectral solution of the Cahn-Hilliard equation are compared on a larger test microstructure over multiple time steps."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Cahn-Hilliard Equation\n",
"\n",
"The Cahn-Hilliard equation is used to simulate microstructure evolution during spinodial decomposition and has the following form,\n",
"\n",
"$$ \\dot{\\phi} = \\nabla^2 \\left( \\phi^3 - \\phi \\right) - \\gamma \\nabla^4 \\phi $$\n",
"\n",
"where $\\phi$ is a conserved ordered parameter and $\\sqrt{\\gamma}$ represents the width of the interface. In this example, the Cahn-Hilliard equation is solved using a semi-implicit spectral scheme with periodic boundary conditions, see [Chang and Rutenberg](http://dx.doi.org/10.1103/PhysRevE.72.055701) for more details."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"scrolled": false
},
"outputs": [],
"source": [
"import warnings\n",
"warnings.filterwarnings('ignore')\n",
"import numpy\n",
"import dask.array as da\n",
"import matplotlib.pyplot as plt\n",
"from dask_ml.model_selection import train_test_split, GridSearchCV\n",
"from distributed import Client\n",
"from sklearn.pipeline import Pipeline\n",
"\n",
"from pymks import (\n",
" solve_cahn_hilliard,\n",
" plot_microstructures,\n",
" PrimitiveTransformer,\n",
" LocalizationRegressor,\n",
" ReshapeTransformer,\n",
" coeff_to_real\n",
")"
]
},
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"cell_type": "code",
"execution_count": 2,
"metadata": {
"scrolled": true
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"outputs": [
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"source": [
"#PYTEST_VALIDATE_IGNORE_OUTPUT\n",
"\n",
"%matplotlib inline\n",
"%load_ext autoreload\n",
"%autoreload 2\n",
"\n",
"Client()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Modeling with MKS\n",
"\n",
"In this example the MKS equation will be used to predict microstructure at the next time step using \n",
"\n",
"$$p[s, 1] = \\sum_{r=0}^{S-1} \\alpha[l, r, 1] \\sum_{l=0}^{L-1} m[l, s - r, 0] + ...$$\n",
"\n",
"where $p[s, n + 1]$ is the concentration field at location $s$ and at time $n + 1$, $r$ is the convolution dummy variable and $l$ indicates the local states varable. $\\alpha[l, r, n]$ are the influence coefficients and $m[l, r, 0]$ the microstructure function given to the model. $S$ is the total discretized volume and $L$ is the total number of local states `n_states` choosen to use.\n",
"\n",
"The model will march forward in time by recursively replacing discretizing $p[s, n]$ and substituing it back for $m[l, s - r, n]$.\n",
"\n",
"### Calibration Datasets\n",
"\n",
"Unlike the elastostatic examples, the microstructure (concentration field) for this simulation doesn't have discrete phases. The microstructure is a continuous field that can have a range of values which can change over time, therefore the first order influence coefficients cannot be calibrated with delta microstructures. Instead, a large number of simulations with random initial conditions are used to calibrate the first order influence coefficients using linear regression.\n",
"\n",
"The function `solve_cahn_hilliard` provides an interface to generate calibration datasets for the influence coefficients."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"scrolled": false
},
"outputs": [],
"source": [
"da.random.seed(99)\n",
"\n",
"x_data = (2 * da.random.random((100, 41, 41), chunks=(20, 41, 41)) - 1).persist()\n",
"\n",
"y_data = solve_cahn_hilliard(x_data, delta_t=1e-2).persist()"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"scrolled": true
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"outputs": [
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