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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Extended Standard Cahn-Hilliard Example to Binary Flory Huggins as discussed on 28/11/2019\n",
"# Example can be found at https://bitbucket.org/fenics-project/dolfin/src/master/python/demo/documented/cahn-hilliard/demo_cahn-hilliard.py.rst#rst-header-id1\n",
"# Runs with fenics 2019.01\n",
"# The resulting .pvd file can be opened using default settings in ParaView\n",
"\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"from scipy.optimize import curve_fit\n",
"\n",
"import random\n",
"from dolfin import *\n",
"# Class representing the intial conditions\n",
"class InitialConditions(UserExpression):\n",
" def __init__(self, **kwargs):\n",
" random.seed(2 + MPI.rank(MPI.comm_world))\n",
" super().__init__(**kwargs)\n",
" def eval(self, values, x):\n",
" if x[0] > 0.5:\n",
"# values[0] = 0.63 + 0.02*(0.5 - random.random())\n",
" values[0] = 0.6\n",
" else:\n",
" values[0] = 0.35 \n",
" values[1] = 0.0\n",
"# values[0] = 0.63 + 0.02*(0.5 - random.random())\n",
" values[1] = 0.0\n",
" def value_shape(self):\n",
" return (2,)\n",
"# Class for interfacing with the Newton solver\n",
"class CahnHilliardEquation(NonlinearProblem):\n",
" def __init__(self, a, L):\n",
" NonlinearProblem.__init__(self)\n",
" self.L = L\n",
" self.a = a\n",
" def F(self, b, x):\n",
" assemble(self.L, tensor=b)\n",
" def J(self, A, x):\n",
" assemble(self.a, tensor=A)\n",
"# Model parameters\n",
"lmbda = 1.0e-02 # surface parameter\n",
"dt = 5.0e-03 # time step\n",
"theta = 0.5 # time stepping family, e.g. theta=1 -> backward Euler, theta=0.5 -> Crank-Nicolson\n",
"# Form compiler options\n",
"parameters[\"form_compiler\"][\"optimize\"] = True\n",
"parameters[\"form_compiler\"][\"cpp_optimize\"] = True\n",
"# Create mesh and build function space\n",
"mesh = UnitSquareMesh.create(96, 1, CellType.Type.quadrilateral)\n",
"P1 = FiniteElement(\"Lagrange\", mesh.ufl_cell(), 1)\n",
"ME = FunctionSpace(mesh, P1*P1)\n",
"# Define trial and test functions\n",
"du = TrialFunction(ME)\n",
"q, v = TestFunctions(ME)\n",
"# Define functions\n",
"u = Function(ME) # current solution\n",
"u0 = Function(ME) # solution from previous converged step\n",
"\n",
"# Split mixed functions\n",
"dc, dmu = split(du)\n",
"c, mu = split(u)\n",
"c0, mu0 = split(u0)\n",
"# Create intial conditions and interpolate\n",
"u_init = InitialConditions(degree=1)\n",
"u.interpolate(u_init)\n",
"u0.interpolate(u_init)\n",
"# Compute the chemical potential df/dc\n",
"c = variable(c)\n",
"# f = 100*c**2*(1-c)**2\n",
"# dfdc = diff(f, c)\n",
"X = 2.5\n",
"dfdc = ln(c/(1-c))+(1-2*X*c)\n",
"# mu_(n+theta)\n",
"mu_mid = (1.0-theta)*mu0 + theta*mu\n",
"# Weak statement of the equations\n",
"L0 = c*q*dx - c0*q*dx + dt*c*(1-c)*dot(grad(mu_mid), grad(q))*dx\n",
"L1 = mu*v*dx - (ln(c/(1-c))+(1-2*X*c))*v*dx - lmbda*dot(grad(c), grad(v))*dx\n",
"L = L0 + L1\n",
"# Compute directional derivative about u in the direction of du (Jacobian)\n",
"a = derivative(L, u, du)\n",
"# Create nonlinear problem and Newton solver\n",
"problem = CahnHilliardEquation(a, L)\n",
"solver = NewtonSolver()\n",
"solver.parameters[\"linear_solver\"] = \"lu\"\n",
"solver.parameters[\"convergence_criterion\"] = \"incremental\"\n",
"solver.parameters[\"relative_tolerance\"] = 1e-6\n",
"# Output file\n",
"file = File(\"output.pvd\", \"compressed\")\n",
"\n",
"# Step in time\n",
"t = 0.0\n",
"T = 3000*dt\n",
"while (t < T):\n",
" t += dt\n",
" u0.vector()[:] = u.vector()\n",
" solver.solve(problem, u.vector())\n",
" file << (u.split()[0], t)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def tanh_fit(x, a, b, c, d):\n",
" return np.tanh((x-a)/b)*c+d\n",
"xdata = np.linspace(0, 96, 97)\n",
"popt, pcov = curve_fit(tanh_fit, xdata, u.compute_vertex_values()[0:97])\n",
"plt.plot(xdata, u.compute_vertex_values()[0:97])\n",
"a, b, c, d = popt\n",
"plt.plot(xdata, tanh_fit(xdata, a, b, c, d))# np.tanh((xdata-52)/12.4)*0.36+0.5"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"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": "3.7.3"
}
},
"nbformat": 4,
"nbformat_minor": 4
}