Newer
Older
Lars Hubatsch
committed
{
"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] > 0.48) & (x[0] < 0.52) & (x[1] > 0.48) & (x[1] < 0.52):\n",
Lars Hubatsch
committed
"# 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-02 # time step\n",
Lars Hubatsch
committed
"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, 96, CellType.Type.quadrilateral)\n",
Lars Hubatsch
committed
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
"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",
Lars Hubatsch
committed
"while (t < T):\n",
" file << (u.split()[0], t)\n",
Lars Hubatsch
committed
" t += dt\n",
" u0.vector()[:] = u.vector()\n",
" solver.solve(problem, u.vector())"
Lars Hubatsch
committed
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
]
},
{
"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"
]
}
],
"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
}