{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from pystencils.session import *\n", "sp.init_printing()\n", "frac = sp.Rational" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Tutorial 06: Phase-field simulation of dentritic solidification\n", "\n", "This is the second tutorial on phase field methods with pystencils. Make sure to read the previous tutorial first. \n", "\n", "In this tutorial we again implement a model described in **Programming Phase-Field Modelling** by S. Bulent Biner.\n", "This time we implement the model from chapter 4.7 that describes dentritic growth. So get ready for some beautiful snowflake pictures.\n", "\n", "We start again by adding all required arrays fields. This time we explicitly store the change of the phase variable φ in time, since the dynamics is calculated using an Allen-Cahn formulation where a term $\\partial_t \\phi$ occurs." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "dh = ps.create_data_handling(domain_size=(300, 300), periodicity=True, \n", " default_target='cpu')\n", "φ_field = dh.add_array('phi', latex_name='φ')\n", "φ_field_tmp = dh.add_array('phi_temp', latex_name='φ_temp')\n", "φ_delta_field = dh.add_array('phidelta', latex_name='φ_D')\n", "t_field = dh.add_array('T')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This model has a lot of parameters that are created here in a symbolic fashion. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle \\frac{{{φ}_{(0,0)}}^{4}}{4} - {{φ}_{(0,0)}}^{3} \\left(\\frac{1}{2} - \\frac{m}{3}\\right) + {{φ}_{(0,0)}}^{2} \\left(\\frac{1}{4} - \\frac{m}{2}\\right) + \\frac{ε^{2} \\left({\\partial_{0} {{φ}_{(0,0)}}}^{2} + {\\partial_{1} {{φ}_{(0,0)}}}^{2}\\right)}{2}$" ], "text/plain": [ " 4 2 ⎛ 2 2⎞\n", "φ_C 3 ⎛1 m⎞ 2 ⎛1 m⎞ ε ⋅⎝D(φ[0,0]) + D(φ[0,0]) ⎠\n", "──── - φ_C ⋅⎜─ - ─⎟ + φ_C ⋅⎜─ - ─⎟ + ────────────────────────────\n", " 4 ⎝2 3⎠ ⎝4 2⎠ 2 " ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ε, m, δ, j, θzero, α, γ, Teq, κ, τ = sp.symbols(\"ε m δ j θ_0 α γ T_eq κ τ\")\n", "εb = sp.Symbol(\"\\\\bar{\\\\epsilon}\")\n", "\n", "φ = φ_field.center\n", "φ_tmp = φ_field_tmp.center\n", "T = t_field.center\n", "\n", "def f(φ, m):\n", " return φ**4 / 4 - (frac(1, 2) - m/3) * φ**3 + (frac(1,4)-m/2)*φ**2\n", "\n", "free_energy_density = ε**2 / 2 * (ps.fd.Diff(φ,0)**2 + ps.fd.Diff(φ,1)**2 ) + f(φ, m)\n", "free_energy_density" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The free energy is again composed of a bulk and interface part." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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